diff --git "a/4NFAT4oBgHgl3EQfExwU/content/tmp_files/load_file.txt" "b/4NFAT4oBgHgl3EQfExwU/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/4NFAT4oBgHgl3EQfExwU/content/tmp_files/load_file.txt" @@ -0,0 +1,3600 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf,len=3599 +page_content='Under consideration for publication in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1 Adjoint-based variational optimal mixed models for large-eddy simulation of turbulence Zelong Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yunpeng Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiaoning Wang and Jianchun Wang† 1Department of Mechanics and Aerospace Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Southern University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' People’s Republic of China 2Guangdong–Hong Kong–Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Southern University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' People’s Republic of China (Received xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' revised xx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' accepted xx) An adjoint-based variational optimal mixed model (VOMM) is proposed for subgrid-scale (SGS) closure in large-eddy simulation (LES) of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The stabilized adjoint LES equations are formulated by introducing a minimal regularization to address the numerical instabilities of the long-term gradient evaluations in chaotic turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model parameters are optimized by minimizing the discrepancy of energy dissipation spectra between LES calculations and a priori knowledge of direct numerical simulation (DNS) using the gradient- based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori performance of the VOMM model is comprehensively examined in LES of three turbulent flows, including the forced homogeneous isotropic turbulence, decaying homogenous isotropic turbulence, and temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model outperforms the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM) and approximate deconvolution model (ADM) in predictions of various turbulence statistics, including the velocity spectrum, structure functions, statistics of velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy, dissipation rate, momentum thickness and Reynolds stress, as well as the instantaneous vortex structures at different grid resolutions and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the VOMM model only takes up 30% time of the DMM model for all flow scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results demonstrate that the proposed VOMM model improves the numerical stability of LES and has high a posteriori accuracy and computational efficiency by incorporating the a priori information of turbulence statistics, highlighting that the VOMM model has a great potential to develop advanced SGS models in the LES of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Key words: subgrid-scale model, variational optimal models, adjoint-based optimization, large- eddy simulation, incompressible turbulence 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Introduction Large-eddy simulation (LES) has become an effective tool for the investigation of turbulent flows, and has been widely applied to many industrial problems including the aeroacoustics, combustions, meteorological physics, interfacial mixing, etc (Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dominant large-scale motions of turbulence are directly resolved by the LES, leaving the effects of residual subgrid scales (SGS) on the resolved large scales modeled by the SGS models (Lesieur & Metais 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Meneveau & Katz 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, direct numerical simulation (DNS) of turbulence requires a sufficiently high mesh resolution to fully resolve all flow scales down † Email address for correspondence: wangjc@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='08423v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='flu-dyn] 20 Jan 2023 2 to the size of the Kolmogorov eddies, whose computational cost is prohibitively expensive at a high Reynolds number (Pope 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore, LES is much more computationally efficient than the DNS by significantly reducing the degrees of freedom of turbulence, meanwhile accurately reconstructing large-scale flow structures (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Durbin 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The modeling of the unclosed SGS stress is crucial for the accuracy of predictions in LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' SGS models can be generally categorized into functional models, structural models and mixed models (Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garnier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The functional SGS models utilize the explicit dissipative terms to correctly reconstruct the forward kinetic energy cascade from large scales to small scales (Rozema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Abkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Smagorinsky model is one of the most popular functional SGS models and is favored for its substantial numerical stability and excellent robustness of LES calculations (Smagorinsky 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the functional SGS models generally exhibit excessive dissipation and fail to predict the sophisticated small-scale flow structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the structural SGS models recover the unclosed SGS stress with high a priori accuracy by exactly truncating the Taylor series expansions or the assumption of scale similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These structural models include the approximate deconvolution method (Stolz & Adams 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001), scale-similarity model (Bardina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994), velocity gradient model (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1979), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The structural SGS models can accurately capture the spatial distribution of SGS energy flux and backscatter of the kinetic energy, but suffer from the numerical instability without sufficient SGS dissipation in the a posteriori studies of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The mixed models consist of the structural models and functional eddy-viscosity models to balance the numerical stability and accuracy of LES and compensate their inherent model deficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Clark model combines the velocity gradient model with the Smagorinsky eddy viscosity (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Erlebacher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (1992) proposed a mixed model which consists of the scale-similarity model and the dissipative Smagorinsky term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the early stage, the SGS model parameters were either theoretically derived from the isotropic turbulent flows (Lilly 1967) or estimated by the a priori analysis of DNS and experimental observations (Deardorff 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1979), yielding poor predictions in the a posteriori LES (Lesieur & Metais 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Meneveau & Katz 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A pioneering dynamical procedure with the Germano identity was developed to determine the Smagorinsky coefficient adaptively by the least-squares algorithm (Germano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Subsequently, the dynamic versions of mixed models were successively proposed, including the one-parameter (Zang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1992) and two-parameter dynamical mixed models (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2008), the dynamic Clark model (Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994) and dynamic ADM model (Habisreutinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2007), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The coefficients of a general multi- parameter dynamic mixed model (DMM) can be conveniently determined by the Germano- identity-based dynamic approach (Sagaut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, extensive previous studies have shown that these DMM models are excessively dissipative in the transitional regions, but underestimate the SGS dissipation in situations of coarse mesh resolutions and grid anisotropy (Meneveau & Katz 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the dissipative Smagorinsky part in the DMM models is usually dominant over the structural part, leading to little advantage in the high a priori accuracy of structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The basis tensors of the DMM model, comprising the functional eddy-viscosity and the accurate structural part, give a complete representation of the SGS stress and SGS energy flux (SGS dissipation), which is essential for the SGS modeling of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2022) preliminarily explored a scale-similarity dynamic procedure (SSD) with a dynamic nonlinear algebraic model, yielding more accurate predictions of various turbulence statistics and instantaneous vortex structures for both a priori and a posteriori analyses of LES than the Germano-identity-based dynamic (GID) approach in the homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the SSD procedure still suffers from the numerical instability at coarse-grid-resolution cases, where the spatial discretization error dominates the SGS modeling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It might be challenging to develop a general dynamic framework for the model coefficient determination at various grid resolutions applicable to different types of turbulence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results 3 demonstrate that the adjustment of SGS model parameters can effectively improve the accuracy of SGS modeling and enhance the predictions of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, additional artificial viscous or penalized regularization terms have been also in- troduced to enhance the a posteriori stability of structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A secondary filtering regularization technique was proposed by Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2001) and Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2004) to maintain the numerical stability of ADM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Vollant et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2016) efficiently regularized the velocity gradient model by dynamically clipping the SGS backscatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A spectral-vanishing- viscosity method (Tadmor 1989) was proposed to effectively suppress the Gibbs oscillations at high wavenumbers (Cerutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2000) and has been successfully applied to the prediction of turbulent channel flows (Karamanos & Karniadakis 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2020a) used a hyperviscosity term to address the stability issue of the spatial-artificial-neural-network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The effective hyperviscosity term was further applied to other data-driven SGS models (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2021b) developed a small-scale eddy-viscosity model to enhance the a posteriori stability of dynamic iterative approximate deconvolution models, without affecting the accurate predictions of large-scale flow structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A kinetic-energy-flux constrained SGS model proposed by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2022) regularizes the DSM model by the correct kinetic energy flux approximated by the tensor-diffusivity model and accurately predicts the transition to turbulence of a compressible flat-plate boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It is noteworthy that additional numerical parameters would be introduced for most regularization techniques, which are sensitive to the grid resolution of LES, requiring multiple tedious testings for different turbulence scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To our knowledge, there might not be a unified adaptive regularization framework proposed for the stability of structural SGS models that can be universally applied to various types of turbulence with different grid resolutions of LES calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dependence of SGS model parameters on grid resolutions of LES might be effectively addressed by incorporating the a priori knowledge of DNS or experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In recent years, many data-driven closure approaches (Tracey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Maulik & San 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Park & Choi 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022) have been extensively developed to improve the modeling of unclosed terms in turbulence, as more high-fidelity DNS or experimental data become available (Kutz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Duraisamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2016b) proposed a representative tensor-basis- neural-network (TBNN) model with the multiplicative layer that predicts coefficients of the basis tensors for the modeled Reynolds stress by taking velocity invariants as input to preserve Galilean invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The TBNN architecture can accurately reconstruct the anisotropy of Reynolds stress and predict the flow separation better than the baseline linear or nonlinear eddy-viscosity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2020c) further developed the artificial-neural-network-based nonlinear algebraic models yielding better predictions of LES statistics than classical dynamic SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gene- expression-programming technique was proposed to acquire the explicit mathematical expression of the unclosed SGS stress modeled by basis functions for LES using an evolutionary algorithm (Schoepplein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The multi-agent reinforcement-learning framework was developed to discover Smagorinsky model coefficients using the control policy rewarded by the statistical discrepancy of energy spectrum (Novati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Kurz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2023), and further applied to modeling the near-wall dynamics (Bae & Koumoutsakos 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Although the machine-learning-based closure models can improve the a priori accuracy of turbulence models fairly well, they have been reported to suffer from the ill-conditioned issues in the a posteriori studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The small a priori errors of the modeled Reynolds stress can be significantly amplified and then propagated into the mean velocity field in the a posteriori testings (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Gamahara & Hattori (2017) established an artificial-neural-network framework for the SGS closures of turbulent channel flows, which accurately predicts the unclosed SGS stress in a priori studies, but shows no obvious advantages over the Smagorinsky model in the reconstruction of the mean velocity profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The recurrent neural network was employed to 4 learn the coarse-grained discretization errors of LES and expected to construct the perfect LES formulation (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these perfect SGS closure terms also encounter serious a posteriori instability issues, even though the a priori predictions show high correlations with the exact unclosed terms (Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results indicate that most current data-driven closure approaches can acquire sufficiently high a priori accuracy after being trained by the high-fidelity DNS or experimental data, but still lack indispensable extrapolation capabilities and are difficult to be applied to the a posteriori testings of out-of-sampling flow scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The data-assimilation techniques can effectively remedy the deficiencies of insufficient a posteriori accuracy of closure models by iteratively evaluating and minimizing the discrep- ancies between coarse-grained a posteriori calculations and benchmark high-fidelity DNS or experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The data-assimilation approaches can be generally classified into three categories: ensemble-based statistical methods (Colburn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022), adjoint-based variational approaches (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Delport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Badreddine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2014) and their mixed variants (Mons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ensemble-based statistical techniques use ensemble statistics to approximately measure the model uncertainty and continuously correct the measurement errors of observations by the classical Kalman-filtering strategies or nudging methods (Clark Di Leoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These statistical assimilation methods allow the convenient inference of flow states and statistics, without any detailed information of dynamical systems, facilitating their wide application in complex practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the state estimations of these ensemble-based approaches frequently evaluate the matrix multiplication and inverse operations, resulting in the massive computation expense and large memory usage for the high degree-of-freedom turbulence problems at a high Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the adjoint- based variational techniques employ the optimal control strategy to efficiently optimize the model parameters or state variables by minimizing the discrepancies between the benchmark observations and a posteriori predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Singh & Duraisamy (2016) proposed a field-inversion procedure to infer model discrepancies in the source terms of Reynolds-averaged Navier–Stokes (RANS) transport equations using Bayesian posterior estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2018) simplified the field-inversion strategy and employed the continuous adjoint formulation to optimize a spatially varying turbulence production term in the Spalart–Allmaras model of RANS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In comparison with the extensive studies of data-assimilation-based RANS models (Kato & Obayashi 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Kato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xiao & Cinnella 2019), investigations on SGS models of LES assimilated with high-fidelity simulation data are still preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A spatially-varying parameter in a local uncertainty model and initial conditions were optimized based on experimental observations of the cylindrical wake flow using the discrete adjoint algorithm (Chandramouli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Mons et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2021) developed a non-intrusive ensemble-variational approach (EnVar) to enhance the predictions of the mean flow and Reynolds stresses by adjusting the wall-normal distribution of the Smagorinsky coefficient or injecting an artificial steady force in the LES momentum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS force modeled by the artificial neural network was optimized by the point-to-point errors of the filtered velocity field using the discrete adjoint method for the decaying isotropic turbulence and plane jet flows (Sirignano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' MacArt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these discrete adjoint or ensemble-based variational methods require massive matrix operations with significant memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In this paper, a variational optimal mixed model (VOMM) is proposed to reconstruct the unclosed SGS stress by assimilating the turbulence statistics of high-fidelity filtered DNS data using the continuous adjoint approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The main difference from the previous work is that we derive adjoint LES equations with the general SGS model and conduct the energy budget analysis of adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The continuous adjoint algorithm can enhance the physical understanding of the adjoint-based sensitivities and provide flexibility in selecting the discretization scheme for the adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The quadratic terms of shear strain rate in adjoint LES equations turn out to be responsible for the exponential temporal growth of the adjoint-based gradients, giving rise to the 5 numerical divergence in a long time horizon for the chaotic turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Hence, the stabilized adjoint LES equations are correspondingly formulated to enhance the numerical stability of the adjoint LES calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To the extent of the authors’ knowledge, few previous studies have given detailed derivations of the adjoint LES equations with general SGS mixed models and formulated the stabilized version for long-term gradient evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the selected cost functional is essential for the convergence and performance of adjoint-based gradient optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to the previous studies, turbulence statistical discrepancies rather than the chaotic point-to-point prediction errors are adopted to quantify the multiscale statistical behaviours of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a priori information about statistics of turbulence acquired from experimental data or DNS results, including energy spectra, structure functions, and probability density functions of physical quantities, can be used to determine or correct SGS model parameters to improve the a posteriori accuracy of LES greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Turbulent statistical assimilation can effectively alleviate the impact of chaotic field observations on the performance of data assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, the a posteriori performance of VOMM model is comprehensively investigated and compared to classical SGS models at multiple grid resolutions in different turbulence scenarios, including the forced and decaying homogeneous isotropic turbulence, as well as the temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2 describes the governing equations of the large-eddy simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The conventional subgrid-scale models, including DSM, DMM and ADM models, are briefly introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4, we first derive the adjoint LES equations with a general form of mixed SGS models, then conduct the energy budget analysis of adjoint equations, and correspondingly propose the stabilized adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Afterwards, the adjoint-based variational optimal mixed model is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5 further investigates the a posteriori performance of the VOMM model in comparison to the classical SGS models for three turbulent flow scenarios, including the forced homogeneous isotropic turbulence, decaying homogeneous isotropic turbulence, and temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Conclusions are finally drawn in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Governing equations of the large-eddy simulation The three dimensional incompressible turbulence is governed by the Navier-Stokes equations (Pope 2000), namely 𝜕𝑢𝑖 𝜕𝑥𝑖 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) 𝜕𝑢𝑖 𝜕𝑡 + 𝜕 �𝑢𝑖𝑢 𝑗 � 𝜕𝑥 𝑗 = − 𝜕𝑝 𝜕𝑥𝑖 + 𝜈 𝜕2𝑢𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + F𝑖, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) where 𝑢𝑖 is the 𝑖-th component of velocity, 𝑝 denotes the pressure divided by the constant density, 𝜈 is the kinematic viscosity, and F𝑖 represents the large-scale forcing on the fluid momentum in the 𝑖-th coordinate direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The summation convection for the repeated indices is adopted by default for simplicity in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, the dimensionless governing parameter for the incompressible turbulence, namely, the Taylor microscale Reynolds number 𝑅𝑒𝜆 is defined as (Pope 2000) 𝑅𝑒𝜆 = 𝑢rms𝜆 √ 3𝜈 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) where 𝑢rms = √︁ ⟨𝑢𝑖𝑢𝑖⟩ represents the root-mean-square (rms) value of the velocity magnitude, and ⟨·⟩ represents a spatial average along the homogeneous direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', average over the entire domain for the isotropic turbulence and the horizontal average for the temporally evolving mixing 6 layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝜆 = 𝑢rms√︁ 5𝜈/𝜀 is the Taylor microscale, where 𝜀 = 2𝜈 � 𝑆𝑖 𝑗𝑆𝑖 𝑗 � represents the average dissipation rate and 𝑆𝑖 𝑗 = 1 2 �𝜕𝑢𝑖/𝜕𝑥 𝑗 + 𝜕𝑢 𝑗/𝜕𝑥𝑖 � denotes the strain-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To obtain the governing equations of the large-eddy simulation, a spatial filtering operation, ¯𝑓 (x) = ∫ Ω 𝑓 (x′) 𝐺 �x − x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� 𝑑x′ is applied to the Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, an overbar denotes the spatial filtering, Ω is the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐺 and ¯Δ are the filter kernel and filter width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The governing equations for the LES can be correspondingly derived as (Sagaut 2006) 𝜕 ¯𝑢𝑖 𝜕𝑥𝑖 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) 𝜕 ¯𝑢𝑖 𝜕𝑡 + 𝜕 � ¯𝑢𝑖 ¯𝑢 𝑗 � 𝜕𝑥 𝑗 = − 𝜕 ¯𝑝 𝜕𝑥𝑖 − 𝜕𝜏𝑖 𝑗 𝜕𝑥 𝑗 + 𝜈 𝜕2 ¯𝑢𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + ¯F𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) Here, the unclosed SGS stress tensor 𝜏𝑖 𝑗 = 𝑢𝑖𝑢 𝑗 − ¯𝑢𝑖 ¯𝑢 𝑗 cannot be directly calculated using the resolved variables ¯𝑢𝑖, and additional SGS stress modeling is required to make the LES equations solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Conventional subgrid-scale models for LES The SGS models aim to establish the approximate constitutive equation for SGS unclosed terms using the known resolved variables, and reconstruct the nonlinear interactions between the resolved large scales and unsolved small scales as accurately as possible (Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Johnson 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The explicit SGS models consist of the functional and structural models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The functional modeling adopts the eddy-viscosity forms to mimic the forward kinetic energy transfer from the resolved large scales to the residual small scales, while the structural models can accurately recover the unclosed SGS stress by the hypothesis of scale similarity or using the truncated series expansions with high a priori accuracy (Sagaut 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' One of the most widely-used functional models is the Smagorinsky model (Smagorinsky 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1967), expressed as 𝜏𝐴 𝑖 𝑗 = 𝜏𝑖 𝑗 − 𝛿𝑖 𝑗 3 𝜏𝑘𝑘 = −2𝐶2 𝑆 ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) where 𝛿𝑖 𝑗 denotes the Kronecker delta operator, ¯𝑆𝑖 𝑗 = 1 2 �𝜕 ¯𝑢𝑖/𝜕𝑥 𝑗 + 𝜕 ¯𝑢 𝑗/𝜕𝑥𝑖 � is the filtered strain-rate tensor and | ¯𝑆| = (2 ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗)1/2 represents the characteristic filtered strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The superscript “A” represents the trace-free anisotropic part of the arbitrary variables, namely, (•) 𝐴 𝑖 𝑗 = (•)𝑖 𝑗 − (•)𝑘𝑘𝛿𝑖 𝑗/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The isotropic SGS stress 𝜏𝑘𝑘 is absorbed into the pressure term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶2 𝑆 is the Smagorinsky coefficient and can be determined empirically or by a theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The most common approach is based on the least-squares dynamic procedure using the Germano identity, giving rise to the dynamic Smagorinsky model (DSM), whose coefficient is given by (Germano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Lilly 1992) 𝐶2 𝑆 = ⟨𝐿 𝐴 𝑖 𝑗M𝑖 𝑗⟩ ⟨M𝑘𝑙M𝑘𝑙⟩ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) where the Leonard stress 𝐿𝑖 𝑗 = � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗, 𝐿 𝐴 𝑖 𝑗 = 𝐿𝑖 𝑗 − 1 3𝛿𝑖 𝑗𝐿𝑘𝑘 and M𝑖 𝑗 = ˜𝛼𝑖 𝑗 − 𝛽𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, a tilde stands for the test filtering operation at the double-filtering scale ˜Δ = 2 ¯Δ, the variables 𝛼𝑖 𝑗 = 2 ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗 and 𝛽𝑖 𝑗 = 2 ˜Δ2| ˜¯𝑆| ˜¯𝑆𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The scale-similarity model 𝜏𝑖 𝑗 = � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗 is a typical structural model and can correctly reconstruct the SGS stress with high a priori accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these structural models often exhibit insufficient dissipation and numerical instability in the a posteriori testings of LES due to the underestimation of the forward kinetic energy cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dynamic mixed model (DMM) combines the scale-similarity model with the dissipative 7 Smagorinsky term, and is given by (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2008) 𝜏𝑖 𝑗 = 𝐶1 ¯Δ2 �� ¯𝑆 �� ¯𝑆𝑖 𝑗 + 𝐶2 � � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) Similar to the DSM model, model coefficients of the DMM model 𝐶1 and 𝐶2 are dynamically determined by the least-squares algorithm using the Germano identity, expressed respectively as (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020) 𝐶1 = � 𝑁2 𝑖 𝑗 � � 𝐿𝑖 𝑗 𝑀𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 � � 𝐿𝑖 𝑗𝑁𝑖 𝑗 � � 𝑁2 𝑖 𝑗 � � 𝑀2 𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) 𝐶2 = � 𝑀2 𝑖 𝑗 � � 𝐿𝑖 𝑗𝑁𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 � � 𝐿𝑖 𝑗 𝑀𝑖 𝑗 � � 𝑁2 𝑖 𝑗 � � 𝑀2 𝑖 𝑗 � − � 𝑀𝑖 𝑗𝑁𝑖 𝑗 �2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) where 𝑀𝑖 𝑗 = 𝐻1,𝑖 𝑗 − ˜ℎ1,𝑖 𝑗, and 𝑁𝑖 𝑗 = 𝐻2,𝑖 𝑗 − ˜ℎ2,𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, ℎ1,𝑖 𝑗 = −2 ¯Δ2 �� ¯𝑆 �� ¯𝑆𝑖 𝑗, ℎ2,𝑖 𝑗 = � ¯𝑢𝑖 ¯𝑢 𝑗 − ˜¯𝑢𝑖 ˜¯𝑢 𝑗, 𝐻1,𝑖 𝑗 = −2 ˜Δ2 ��� ˜¯𝑆 ��� ˜¯𝑆𝑖 𝑗, and 𝐻2,𝑖 𝑗 = � ˜¯𝑢𝑖 ˜¯𝑢 𝑗 − ˆ˜¯𝑢𝑖 ˆ˜¯𝑢 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The hat stands for the test filtering at scale ˆΔ = 4 ¯Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The unfiltered variables can be accurately recovered by the resolved filtered field using the iterative approximate deconvolution procedure, namely (Stolz & Adams 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001) 𝑢∗ 𝑖 = 𝐴𝐷 𝑁 ( ¯𝑢𝑖) = 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ ¯𝑢𝑖, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) where the asterisk represents the approximately unfiltered variables, 𝐴𝐷 𝑁 is the abbreviation of the 𝑁-th order approximate deconvolution, 𝐼 is the identity, and the symbol “⊗” stands for the spatial convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For any two functions 𝑓 and 𝑔, 𝑓 ⊗ 𝑔 = ∫ +∞ −∞ 𝑓 (x′) 𝑔 (x − x′) 𝑑x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The unclosed SGS stress then can be recovered with the scale-similarity form by the approximate deconvolution method (ADM), given by (Bardina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1980) 𝜏𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗 − ¯𝑢∗ 𝑖 ¯𝑢∗ 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) The number of iterations for the ADM model is recommended to be 𝑁 =3 ∼ 5 (Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The accuracy of the ADM model becomes higher, while the numerical stability drops, as the number of iterations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Hence, 𝑁 = 5 is selected in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to maintain the numerical stability of the a posteriori testings of LES [𝜕 ¯𝑢𝑖/𝜕𝑡 = ¯𝑅𝑖 ( ¯𝑢𝑖, 𝑡)], Stolz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2001) and Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2004) introduced a secondary filtering relaxation term [𝜕 ¯𝑢𝑖/𝜕𝑡 = ¯𝑅𝑖 ( ¯𝑢𝑖, 𝑡)+ ¯𝑆𝑖 ( ¯𝑢𝑖)], yielding ¯𝑆𝑖 ( ¯𝑢𝑖) = −𝜒 � 𝐼 − 𝐺 ⊗ 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 � ⊗ ¯𝑢𝑖, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) where 𝜒 is the empirical regularization coefficient, which is approximately insensitive to the LES results in previous studies, and we choose 𝜒 = 0 and 1 for comparisons in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Adjoint-based variational optimal mixed models (VOMM) The mixed model is composed of the structural parts and the dissipative functional terms, and its general form can be written as (Sagaut et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2000) 𝜏𝑖 𝑗 �𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� = 𝑁 ∑︁ 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ�, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) 8 where 𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� represents the 𝑛-th basis stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) denotes the corre- sponding model coefficient and 𝑁 is the number of basis stress tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The model coefficients are generally respectively determined by the multivariate least-squares algorithm proposed by Germano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (1991) and Lilly (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Many previous studies have shown that the dynamic mixed models give rise to an excessive dissipation of energy in the transitional regions and dissipation underestimation if the filter scales are sufficiently large, especially in situations of grid anisotropy (Meneveau & Katz 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Moser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In recent years, data-driven based high-accuracy SGS models are successively proposed (Kutz 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Duraisamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2019a) proposed an artificial-neural-network-based mixed model which accurately recovers the unclosed SGS terms by estimating mixed model coefficients with local flow characteristics as inputs of the machine-learning strategy, yielding better predictions of LES statistics than the classical dynamic mixed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The input features of the data-driven closure models are crucial for the accuracy of SGS models (Gamahara & Hattori 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Park & Choi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Incorporating the accurate structural parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', filtered velocity gradients at the neighboring stencil turn out to improve the performance of data-driven SGS models effectively (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019b, 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Moreover, the spatial flow structures at scales between ��Δ/2 and 2 ¯Δ are found to be essential for the SGS modeling of LES at the filter scale ¯Δ (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The strategy of the blind deconvolution with the artificial neural network was proposed to recover the unknown original unfiltered variables from the known filtered quantities with high accuracy (Maulik & San 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Maulik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A deconvolutional-artificial-neural-network (DANN) framework was further proposed to accurate reconstruct the SGS unclosed terms both in a priori and a posteriori analyses of isotropic turbulence (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020, 2021a), and successfully applied to the chemically reacting compressible turbulence (Teng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It was demonstrated that the DANN models embed the properties of symmetry and realizability conditions, which preserve the physical reliability of the DANN framework (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to enhance the interpretability of black-box machine-learning SGS models, a semi-explicit ANN-based spatial gradient model and constant-coefficient spatial gradient models are successively proposed by the elaborate Taylor expansions of velocity gradients in the neighboring stencil locations (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The machine-learning-based SGS models trained by high-fidelity simulation data can be regarded as the structural models with high a priori accuracy, requiring additional indispensable dissipation to account for the spatial discretization effect and ensure the numerical stability in the a posteriori studies of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition to the machine-learning-assisted SGS models, some a priori information about statistics of turbulence acquired from experimental data or DNS results like energy spectra, structure functions, and probability density functions of physical quantities can be used to determine or correct the model coefficients of SGS models to improve the model accuracy greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These a priori knowledge of turbulent statistical quantities can be dynamically assimilated into the closure models via the data-assimilation based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Among these data-assimilation techniques, adjoint-based variational methods adopt the optimal control strategy to efficiently calculate all the gradients of cost functionals for the model coefficients by solving the forward governing equations and the backward adjoint equations (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Then, the model coefficients of SGS models are iteratively updated using the gradient-based optimization algorithm until the optimal values are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The cost functionals measure the discrepancies of statistical quantities in turbulence between the LES results and measurements from the experimental or DNS data, which can greatly alleviate the impact of chaotic field observations on the performance of data assimilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In this work, we resort to the state-of-art adjoint-based data-assimilation approaches to establish a general optimal SGS framework to determine model parameters adaptively for various grid resolutions of LES in different turbulence scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Adjoint LES equations and gradient evaluations with the mixed model We optimize the model coefficients of the SGS closure model to minimize the statistical discrepancies between the LES calculations and the reference values acquired from the experi- mental or DNS data, which can be defined as the minimal optimization problem constrained by the governing equations (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The constrained optimization problem for the turbulent closure modeling is expressed as min 𝐶𝑛 J � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , 𝜙 � ¯𝑢ref 𝑖 �� , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑅0 ( ¯𝑢𝑖) = 𝜕 ¯𝑢𝑖 𝜕𝑥𝑖 = 0, 𝑅𝑖 ( ¯𝑢𝑖, ¯𝑝) = 𝜕 ¯𝑢𝑖 𝜕𝑡 + 𝜕( ¯𝑢𝑖 ¯𝑢𝑗) 𝜕𝑥𝑗 + 𝜕 ¯𝑝 𝜕𝑥𝑖 − 𝜈 𝜕2 ¯𝑢𝑖 𝜕𝑥𝑗𝜕𝑥𝑗 − F 𝑖 + 𝜕𝜏𝑖 𝑗 𝜕𝑥𝑗 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) where J � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , 𝜙 � ¯𝑢ref 𝑖 �� = 𝑇∫ 0 ∫ Ω 𝐽 � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛, x, 𝑡) , 𝜙 � ¯𝑢ref 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' x, 𝑡�� 𝑑x𝑑𝑡 denotes the total cost functions, 𝐽 � 𝜙 ( ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛, x, 𝑡) , 𝜙 � ¯𝑢ref 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' x, 𝑡�� is the discrepancy of statistical quantities 𝜙 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' kinetic energy spectra, structure functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=') between the LES results ¯𝑢𝑖 and reference values ¯𝑢ref 𝑖 (experimental or DNS data) at a certain state (𝐶𝑛, x, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) denotes model coefficients of the SGS mixed model 𝜏𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 , and 𝑡 ∈ [0,𝑇] is the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, “s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='t.” stands for the abbreviation of “subject to”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑅0 and 𝑅𝑖 (𝑖 = 1, 2, 3) represent the LES continuity equation and momentum equations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Lagrangian functional L is introduced to take the dynamics of LES variables ¯v = [ ¯𝑝, ¯𝑢1, ¯𝑢2, ¯𝑢3]𝑇 into account and convert the constrained optimization (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) into the un- constrained optimization problem, namely (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2006) min 𝐶𝑛 L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , where L = J � 𝜙 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , 𝜙 � ¯vref�� − 3 ∑︁ 𝑘=0 𝑇 ∫ 0 ∫ Ω 𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) · ¯𝑣† 𝑘𝑑x𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) Here, ¯v† = � ¯𝑝†, ¯𝑢† 1, ¯𝑢† 2, ¯𝑢† 3 �𝑇 are the adjoint LES variables of ¯v, where ¯𝑝† and ¯𝑢† 𝑖 are the adjoint pressure and adjoint velocity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the sake of brevity, the inner product of time and space is defined by ⟨f, g⟩x,𝑡 = 𝑇∫ 0 ∫ Ω f (x, 𝑡) · g (x, 𝑡) 𝑑x𝑑𝑡, where f (x, 𝑡) and g (x, 𝑡) denote the arbitrary physical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Lagrangian functional L can be simplified as L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛)− 3� 𝑘=0 � 𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , ¯v†� x,𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivity of the Lagrangian functional L can be derived by 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = 𝛿J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) − 3 ∑︁ 𝑘=0 � 𝑅𝑘 (𝛿¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) , ¯v†� x,𝑡 − 3 ∑︁ 𝑘=0 � 𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝛿𝐶𝑛) , ¯v†� x,𝑡, = 𝛿J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕¯v 𝛿¯v, ¯v† � x,𝑡 − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕𝐶𝑛 𝛿𝐶𝑛, ¯v† � x,𝑡 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) where 𝜕𝑅𝑘/𝜕¯v and 𝜕𝑅𝑘/𝜕𝐶𝑛 are the tangent operators of the governing equations 𝑅𝑘 (𝑘 = 0, 1, 2, 3) for the variables ¯v and parameters 𝐶𝑛 with the perturbation field 𝛿¯v = ¯v (𝐶𝑛 + 𝛿𝐶𝑛) − ¯v (𝐶𝑛) , 𝑛 ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 is the sensitivity of the cost functional J and calculated as the Gâteaux-Fréchet derivative (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001) 10 of J at 𝐶𝑛 in the direction 𝛿𝐶𝑛, namely 𝛿J (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝛿𝐶𝑛) = lim 𝜀→0 𝑑 𝑑𝜀 J (¯v (𝐶𝑛 + 𝜀𝛿𝐶𝑛)) = � 𝜕𝐽 𝜕¯v, 𝛿¯v � x,𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) The adjoint identity (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001) can be obtained via the integral by part, given by � R (¯v) , ¯v†� x,𝑡 = � ¯v, R† � ¯v†�� x,𝑡 + � ¯F, ¯v†� 𝑡 �� Γ + �¯v, ¯v†� x ��𝑇 0 = � ¯v, R† � ¯v†�� x,𝑡 + 𝐵𝑇, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) where the partial differential equations R (¯v) = 𝜕¯v/𝜕𝑡 + 𝜕 ¯F/𝜕x = 0 with the associated adjoint operator R† �¯v†�, ¯F denotes the fluxes and Γ is the boundary of the domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝐵𝑇 = � ¯F, ¯v†� 𝑡 �� Γ + �¯v, ¯v†� x ��𝑇 0 represents the boundary and temporal integral terms, which determines the boundary and terminal conditions of the adjoint equations to give 𝐵𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ⟨f, g⟩𝑡 = 𝑇∫ 0 f (x, 𝑡) · g (x, 𝑡) 𝑑𝑡 and ⟨f, g⟩x = ∫ Ω f (x, 𝑡) · g (x, 𝑡) 𝑑x denote the temporal and spatial inner products, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 can be expressed with the adjoint identity, namely (Bewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Delport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2009, 2011) � 𝜕𝑅𝑘 𝜕¯v · 𝛿¯v, ¯v† � x,𝑡 = � 𝛿¯v, � 𝜕𝑅𝑘 𝜕¯v �† ¯v† � x,𝑡 + 𝐵𝑇, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) where (𝜕𝑅𝑘/𝜕¯v)† is the adjoint operator of the LES tangent Jacobian tensor 𝜕𝑅𝑘/𝜕¯v, (𝑘 = 0, 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Substitute the Fréchet derivative 𝛿J (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) and the adjoint identity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) into the sensitivity of the Lagrangian functional L (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4), and we get 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = � 𝜕𝐽 𝜕¯v − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 𝜕¯v �† ¯v†, 𝛿¯v � x,𝑡 − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕𝐶𝑛 𝛿𝐶𝑛, ¯v† � x,𝑡 − 𝐵𝑇, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) To avoid calculating the perturbation field 𝛿¯v in the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8, the inner product should be equal to 0 and the corresponding adjoint LES equations can be derived by 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 𝜕¯v �† ¯v† − 𝜕𝐽 𝜕¯v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='9) Substitute the specific forms of the LES equations 𝑅𝑘 (𝑘 = 0, 1, 2, 3) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2), and the adjoint LES equations can be written as 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑖 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10) 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + 𝜕 ¯𝑝† 𝜕𝑥𝑖 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕𝜏† 𝑖 𝑗 𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11) where 𝜏† 𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛),† 𝑖 𝑗 denotes the adjoint SGS mixed model and 𝑇 (𝑛),† 𝑖 𝑗 is the 𝑛-th adjoint basis stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed derivation of the adjoint LES equations can refer to the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terminal conditions of the adjoint LES equations is determined by the last term of adjoint identity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6), namely �¯v†, 𝛿¯v � x ��𝑇 0 = �¯v† (𝑇) , 𝛿¯v (𝑇) � x − �¯v† (0) , 𝛿¯v (0) � x = �¯v† (𝑇) , 𝛿¯v (𝑇) � x, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='12) where 𝛿¯v (0) = 0, since the unperturbed initial LES field is exactly given by the filtered DNS (fDNS) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terminal conditions ¯v† (𝑇) = � ¯𝑢† 𝑖 (𝑇) , ¯𝑝† (𝑇) �𝑇 = 0 make the temporal integral 11 terms �� 𝛿¯v, ¯v†� x �𝑇 0 equal to zero and the calculation of the terminal perturbation 𝛿¯v (����) is obviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terminal conditions ( ¯𝑢† 𝑖 (𝑇) = 0, ¯𝑝† (𝑇) = 0) and boundary conditions of the adjoint LES equations are identified by setting 𝐵𝑇 = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivity of the Lagrangian functional L can be further expressed as 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) = − 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝜕𝐶𝑛 𝛿𝐶𝑛, ¯v† � x,𝑡 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='13) where 𝜕𝑅0/𝜕𝐶𝑛 = 0, and 𝜕𝑅𝑖/𝜕𝐶𝑛 = 𝜕 𝜕𝐶𝑛 � 𝜕𝜏𝑖 𝑗 𝜕𝑥𝑗 � = 𝜕𝑇 (𝑛) 𝑖 𝑗 /𝜕𝑥 𝑗 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) denotes the 𝑛-th SGS basis force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) temporally advances forward in the time horizon 𝑡 ∈ [0,𝑇] and the adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11) are integrated backward with zero terminal conditions, the gradients of Lagrangian functional for the SGS model coefficients can be calculated efficiently by 𝜕L 𝜕𝐶𝑛 = 𝛿L (¯v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛) 𝛿𝐶𝑛 = − � 𝜕𝑇 (𝑛) 𝑖 𝑗 𝜕𝑥 𝑗 , ¯𝑢† 𝑖 � x,𝑡 , (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14) The adjoint-based gradient evaluations are independent of the parameter perturbations 𝛿𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁), which are very efficient compared to the finite difference algorithm and forward sensitivity analysis with at least 𝑁 parameter perturbations and 𝑁 + 1 LES equation calculations for each optimization iteration (Chandramouli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sirignano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' MacArt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Energy budget analysis of the adjoint LES equations Before proceeding to the introduction of the variational optimal mixed models, it is essential to analyze the energy budget of the adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint LES kinetic energy ( ¯E† = ¯𝑢† 𝑖 ¯𝑢† 𝑖 /2) equation is derived through multiplying the adjoint velocity ¯𝑢† 𝑖 on both sides of the adjoint LES momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11), namely 𝜕 ¯E† 𝜕𝑡 + 𝜕 ¯P𝑗 𝜕𝑥 𝑗 = ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 + ¯𝐷† − ¯Π† − ¯𝐽†, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='15) where ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 denotes the adjoint energy production term due to the shear strain rate ¯𝑆𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, ¯P𝑗 is the adjoint spatial transport flux, ¯𝐷 is the adjoint viscous dissipation term, ¯Π† is the adjoint variable of the SGS energy flux ¯Π = −𝜏𝑖 𝑗 ¯𝑆𝑖 𝑗 and ¯𝐽† is the energy injected from the discrepancy between LES results and reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These terms are respectively defined by ¯P𝑗 = ¯E† ¯𝑢 𝑗 + � ¯𝑝† + ¯𝑢𝑖 ¯𝑢† 𝑖 � ¯𝑢† 𝑗 + � 𝜈 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜏† 𝑖 𝑗 � ¯𝑢† 𝑖 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='16) ¯𝐷 = 𝜈 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='17) ¯Π† = −𝜏† 𝑖 𝑗 ¯𝑆† 𝑖 𝑗, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='18) ¯𝐽† = ¯𝑢† 𝑖 𝜕𝐽 𝜕 ¯𝑢𝑖 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='19) 12 where ¯𝑆† 𝑖 𝑗 = � 𝜕 ¯𝑢† 𝑖 /𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗/𝜕𝑥𝑖 � /2 represents the adjoint strain-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The backward evolution of the adjoint volume-averaged kinetic energy can be written as − 𝜕 � ¯E†� 𝜕𝑡 = − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � − � ¯𝐷†� + � ¯Π†� + � ¯𝐽†� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='20) where � ¯𝐷†� is pure dissipation term that drains out the adjoint energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯Π†� denotes the adjoint SGS energy transport term which represents the forward adjoint energy transfer from large scales to unsolved residual scales if � ¯Π†� > 0, otherwise stands for the adjoint SGS energy backscatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The accurate reconstruction of � ¯Π†� is crucial for the SGS modeling of LES and gradient evaluations with respect to the SGS model coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯𝐽†� is the loss-induced adjoint energy injection term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯𝐷†� is the viscous dissipation which enhances the numerical stability of the adjoint LES field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' � ¯𝐽†� is the adjoint energy production due to the discrepancy between LES evaluation and reference data, which dominates the accuracy of the sensitivity calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The large-scale strain-rate tensor ¯𝑆𝑖 𝑗 can be decomposed into its principal components using the eigendecomposition approach, such that (Wang & Gao 2013) ¯𝑆𝑖 𝑗=𝜆1𝑞(1) 𝑖 𝑞(1) 𝑗 + 𝜆2𝑞(2) 𝑖 𝑞(2) 𝑗 + 𝜆3𝑞(3) 𝑖 𝑞(3) 𝑗 = 3 ∑︁ 𝑘=1 𝜆𝑘𝑞(𝑘) 𝑖 𝑞(𝑘) 𝑗 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='21) where 𝜆1, 𝜆2 and 𝜆3 are the eigenvalues of the shear strain rate, with 𝑞(1) 𝑖 , 𝑞(2) 𝑖 and 𝑞(3) 𝑖 being the associated eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here,𝜆1+𝜆2+𝜆3 = 0 for the trace-free strain rate ¯𝑆𝑖 𝑗 in the incompressible turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Hence, the quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='20 is further expressed as − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � = − 3 ∑︁ 𝑘=1 � 𝜆𝑘 � 𝑞(𝑘) 𝑖 ¯𝑢† 𝑖 � � 𝑞(𝑘) 𝑗 ¯𝑢† 𝑗 �� = − 3 ∑︁ 𝑘=1 � 𝜆𝑘 � 𝑞(𝑘) 𝑖 ¯𝑢† 𝑖 �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='22) The sign of the eigenvalues 𝜆𝑘, (𝑘 = 1, 2, 3) determines the contribution of the adjoint energy from the quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � is productive or dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The quadratic terms with negative eigenvalues of the shear strain rate produce the positive adjoint energy production, while those with positive eigenvalues drain out the adjoint energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In previous studies of chaotic adjoint methods, the adjoint-based gradients are found to grow exponentially with time and finally numerically diverge in a long time horizon for the chaotic flows (Wang & Gao 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Ashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garai & Murman 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The terms � ¯𝐷†� , � ¯Π†� and � ¯𝐽†� in the volume-averaged adjoint energy equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='20) are less likely to cause the exponential growth of the adjoint energy, since the adjoint energy term � ¯E†� does not appears explicitly in these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It can be further shown that the quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � plays the dominant role in the exponential growth of the adjoint variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We apply the Cauchy-Schwarz inequality to the inner product terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='22, such that (Talnikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2017) � 𝑞(𝑘) 𝑖 ¯𝑢† 𝑖 �2 ⩽ � 𝑞(𝑘) 𝑖 𝑞(𝑘) 𝑖 � � ¯𝑢† 𝑖 ¯𝑢† 𝑖 � = 2 ���q(𝑘)��� E † (𝑘 = 1, 2, 3) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='23) where “∥·∥” denotes the L2 norm of the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the quadratic terms with negative eigenvalues (adjoint energy production), the evolution of the adjoint energy can be approximated using the leading principal vectors as − 𝜕 � E †� 𝜕𝑡 ≈ 2|𝜆|∞∥q∥∞ � E †� , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='24) 13 where |𝜆|∞ = max Ω {−𝜆1, −𝜆2, −𝜆3} denotes the magnitude of the leading negative eigenvalue in the entire domain Ω and ∥q∥∞ represents the corresponding eigenvector magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint energy is then calculated by the backward time interval, namely � E †� (𝑡) ≈ � E †� (𝑇) exp [2|𝜆|∞∥q∥∞ (𝑇 − 𝑡)] , 𝑡 ∈ [0,𝑇] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='25) The quadratic term − � ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � with negative eigenvalues makes the adjoint energy grow exponentially over time and numerically unstable if it cannot be suppressed by the adjoint dissipation in a long time horizon 𝑡 ∈ [0,𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to stabilize the adjoint equations during every iteration, an additional symmetric tensor ¯𝑆𝑎 𝑖 𝑗 (Ashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garai & Murman 2021) is introduced to maintain the numerical stability of the adjoint momentum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11), and the stabilized adjoint momentum equations are then expressed as 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 + 𝜕 ¯𝑝† 𝜕𝑥𝑖 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕𝜏† 𝑖 𝑗 𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) Consequently, the stabilized adjoint kinetic energy equation is written by 𝜕E † 𝜕𝑡 + 𝜕P 𝑗 𝜕𝑥 𝑗 = ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 + ¯𝐷† − ¯Π† − ¯𝐽†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='27) Here, the quadratic term ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 < 0 � − ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 > 0 � is responsible for the exponential growth of the adjoint energy, and the minimal artificial symmetric tensor is added to keep the adjoint variables numerically stable in advancing backward of the adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The artificial symmetric tensor ¯𝑆𝑎 𝑖 𝑗 can be optimized by the suboptimal minimization problem (Ashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Garai & Murman 2021), such that min ¯𝑆𝑎 𝑖 𝑗 1 2 ¯𝑆𝑎 𝑖 𝑗 ¯𝑆𝑎 𝑖 𝑗, 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='28) We use the sequential quadratic programming (SQP) approach (Boggs & Tolle 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Chung & Freund 2022) to efficiently solve the suboptimal problem, and the augmented Lagrangian functional L is applied to the constrained minimization problem, namely L = 1 2 ¯𝑆𝑎 𝑖 𝑗 ¯𝑆𝑎 𝑖 𝑗 + 𝜆 � ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='29) where 𝜆 is the Lagrangian multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Karush–Kuhn–Tucker (KKT) optimal conditions (Kuhn & Tucker 1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Blonigan & Wang 2018) are obtained by taking the derivatives of the cost functional with respect to the augmented optimal variables ( ¯𝑆𝑎 𝑖 𝑗 and 𝜆), derived by 𝜕L 𝜕 ¯𝑆𝑎 𝑖 𝑗 = ¯𝑆𝑎 𝑖 𝑗 − 𝜆 � ¯𝑢† 𝑖 ¯𝑢† 𝑗 � = 0 ⇒ ¯𝑆𝑎 𝑖 𝑗 = 𝜆 � ¯𝑢† 𝑖 ¯𝑢† 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30) 𝜕L 𝜕𝜆 = ¯𝑢† 𝑖 � ¯𝑆𝑖 𝑗 − ¯𝑆𝑎 𝑖 𝑗 � ¯𝑢† 𝑗 = 0 ⇒ ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 = ¯𝑢† 𝑖 ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='31) By multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30 by ¯𝑢† 𝑖 from the left and right by ¯𝑢† 𝑗 , and then substituting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='31, the Lagrangian multiplier 𝜆 is calculated by 𝜆 = ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 � ¯𝑢† 𝑘 ¯𝑢† 𝑘 �2 = ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 4E †2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Initial SGS parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='SGS parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='SGS model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Forward LES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='LES statistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Loss function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop criterion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Adjoint SGS model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='L-BFGS gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Gradient Calculations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Reference statistics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='No ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Start ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Yes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Initial velocity field ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='L-BFGS gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Terminal condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Loss sensitivity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Backward adjoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='LES equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Backward adjoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='LES equations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Stop criterion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 1: Schematic diagram of the adjoint-based variational optimal mixed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The minimal artificial symmetric tensor ¯𝑆𝑎 𝑖 𝑗 can be obtained by substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='32 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30, yielding ¯𝑆𝑎 𝑖 𝑗 = ��� ��� ¯𝑢† 𝑚 ¯𝑆𝑚𝑛 ¯����† 𝑛 4E †2 ¯𝑢† 𝑖 ¯𝑢† 𝑗, if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 < 0, 0 , if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='33) The artificial momentum term ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 is thus additionally calculated in the stabilized adjoint momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26), namely ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 = �� �� ¯𝑢† 𝑚 ¯𝑆𝑚𝑛 ¯𝑢† 𝑛 2E † ¯𝑢† 𝑖 , if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 < 0, 0 , if ¯𝑢† 𝑖 ¯𝑆𝑖 𝑗 ¯𝑢† 𝑗 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='34) The minimal stabilization term ¯𝑆𝑎 𝑖 𝑗 ¯𝑢† 𝑗 can efficiently maintain the numerical stability of LES adjoint variables in the long-term chaotic turbulent calculations as much as possible, without deteriorating the correct evaluations of the adjoint-based gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Adjoint-based variational optimal mixed models (VOMM) In this research, we select the mixed model comprised of the Smagorinsky dissipative term (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) and approximate deconvolution model (ADM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) in the scale-similarity form, expressed as (Sagaut 2006) 𝜏𝑖 𝑗 = 𝐶1 � ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗 � + 𝐶2 � 𝑢∗ 𝑖 𝑢∗ 𝑗 − 𝑢∗ 𝑖 𝑢∗ 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) where 𝑢∗ 𝑖 denotes the approximate unfiltered velocity recovered by the iterative van Cittert procedure (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In previous studies (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020), we have conducted error analyses to validate that deconvolutional-type SGS models with scale-similarity form (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) perform better than those with theconventionaldirect-modelingform(𝜏𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗− ¯𝑢𝑖 ¯𝑢 𝑗),satisfyingtheproperties of symmetry and realizability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The model coefficients 𝐶1 and 𝐶2 are optimally identified by minimizing the discrepancy between statistical quantities calculated by the LES results and those measured by the filtered DNS (fDNS) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The selected statistics should be able to sufficiently quantify the multiscale transport behaviours of turbulence, meanwhile facilitating the practical measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS stress 𝜏𝑖 𝑗 and SGS force 𝜕𝜏𝑖 𝑗/𝜕𝑥 𝑗 are intermediate variables, 15 𝑅𝑒𝜆 𝐸𝑘 𝑘max𝜂 𝜂/ℎDNS 𝐿𝐼 /𝜂 𝜆/𝜂 𝑢rms 𝜔rms 𝜀 252 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='01 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='30 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='73 Table 1: One-point statistics for the DNS of forced homogeneous isotropic turbulence with grid resolution of 10243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and their statistics are relatively difficult to be obtained through the actual observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the statistics of velocity are more convenient to measure and the velocity spectrum clearly quantifies the turbulent kinetic energy distributions at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS modeling is especially concerned with the accurate reconstruction of small scales near the filter width, therefore we select the dissipation spectrum as the optimization statistical quantities 𝜙 ( ¯𝑢𝑖) to increase the weights of small scales, namely (Pope 2000) 𝜙 ( ¯𝑢𝑖, 𝑘, 𝑡) = 𝐷 (𝑘, 𝑡) = ∫ k 𝜈𝑘2 ¯𝑣∗ 𝑖 (k, 𝑡) ¯𝑣𝑖 (k, 𝑡) 𝛿 (|k| − 𝑘) 𝑑k, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='36) where 𝛿 (·) denotes the Dirac delta function and the star symbol represents complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑘 and k stand for the wavenumber magnitude and wavenumber vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, ¯𝑣 𝑗 (𝜅, 𝑡) = F � ¯𝑢 𝑗 (x, 𝑡) � = � k ¯𝑢 𝑗 (x, 𝑡) 𝑒−𝑖k·x is the 𝑗-th velocity component in Fourier space, where F {·} represents the 3D Fourier transform, and 𝑖 is the imaginary unit with 𝑖2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The optimization problem constrained by the governing equations for the SGS parameters 𝐶1 and 𝐶2 is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2, where the cost functional for the dissipation spectrum 𝐷 (𝑘, 𝑡) is given by J � 𝜙, 𝜙fDNS� = 𝑇 ∫ 0 𝑘max ∑︁ 𝑘=1 𝐽 � 𝐷 (𝑘, 𝑡) , 𝐷fDNS (𝑘, 𝑡) � 𝑑𝑡, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='37) where 𝑘max = 𝑁LES/3 is the effective maximum wavenumber, 𝑁LES is the number of LES grids, and the discrepancy function 𝐽 � 𝐷 (𝑘, 𝑡) , 𝐷fDNS (𝑘, 𝑡) � = � 𝐷 (𝑘, 𝑡) − 𝐷fDNS (𝑘, 𝑡) �2 takes the L2 norm of the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gradients of the loss function with respect to the model coefficients 𝐶1 and 𝐶2 are evaluated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14, where the adjoint variables ¯𝑢† 𝑖 are calculated by backward advancing the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) with zero terminal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivity term 𝜕𝐽/𝜕 ¯𝑢𝑖 is calculated by the chain rule, namely 𝜕𝐽 𝜕 ¯𝑢𝑖 = 𝜕𝐽 𝜕𝐷 𝜕𝐷 𝜕 ¯𝑢𝑖 = 2 � 𝐷 − 𝐷fDNS� F−1 � 2𝜈𝑘2 ¯𝑣𝑖 (k, 𝑡) 𝛿 (|k| − 𝑘) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='38) where F−1 {·} denotes the 3D inverse Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the stabilized adjoint momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26), the adjoint SGS stress is given by 𝜏† 𝑖 𝑗 = 𝐶1𝑇 (1),† 𝑖 𝑗 + 𝐶2𝑇 (2),† 𝑖 𝑗 , where the associated adjoint basis stress tensors 𝑇 (1),† 𝑖 𝑗 and 𝑇 (2),† 𝑖 𝑗 are expressed in detail as 𝑇 (1),† 𝑖 𝑗 = − ¯Δ2 � �� ¯𝑆 �� ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 �� ¯𝑆 �� ¯𝑆𝑖 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='39) 𝑇 (2),† 𝑖 𝑗 = 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='40) 16 (𝑎) �� � �� � �� � �� � � �� �� �� �� �� �� �� �� �� � �� � ���� � ���� � � � ��� DNS ��� ���� � � � �� (𝑏) �� � �� � �� � �� � � �� �� �� �� �� �� �� �� �� �� �� �� ���� � ���� � � � ��� DNS ��� ���� � � � �� Figure 2: Velocity and dissipation spectra of DNS and filtered DNS in forced homogeneous isotropic turbulence with grid resolution of 10243: (𝑎) velocity spectra, and (𝑏) dissipation spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Diamond represent the cutoff wavenumber 𝑘𝑐=16 ( ¯Δ = 32ℎDNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' where 𝑁 = 5 denotes the number of iterations for the AD procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed derivation of the adjoint SGS stress tensors for the VOMM model can refer to the Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To our knowledge, few previous works have studied the mixed SGS models and given the detailed derivations of the adjoint SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the gradients of the cost functional for the model coefficients are obtained by successively solving the forward LES equations and backward stabilized adjoint LES equations, a gradient- based iterative optimization procedure can be established, namely (Liu & Nocedal 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Badreddine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2014) 𝐶 (𝑘+1) 𝑛 = 𝐶 (𝑘) 𝑛 + 𝛾(���)𝑑 (𝑘) 𝑛 , (𝑛 = 1, 2, · · · , 𝑁) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) where 𝐶 (𝑘) 𝑛 is the 𝑛-th model coefficient during the 𝑘-th gradient-based optimal iteration, 𝑑 (𝑘) 𝑛 denotes the updated direction of the 𝑛-th model coefficient and 𝛾(𝑘) represents the step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We use a popular quasi-Newton method named limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) algorithm to update the directions 𝑑 (𝑘) 𝑛 (Liu & Nocedal 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The step size 𝛾(𝑘) is calculated by the backtracking-Armijo line search method in the L-BFGS algorithm (Armijo 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In summary, the diagram of the VOMM model is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1, and the calculation steps are listed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (1) We first select the pure structural ADM model without the dissipative Smagorinsky term as the initial SGS model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) with model coefficients 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (2) The LES transient statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the dissipation spectrum shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='36) is then evaluated by forward calculating the LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) initialized by the filtered DNS velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The statistical discrepancy (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='37) between the LES statistics and the a priori measurable benchmark data (fDNS data) is measured to evaluate the performance of the SGS model with current parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (3) Afterwards, the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) are integrated back- ward with zero terminal conditions, driven by the loss sensitivity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='38) and corresponding adjoint SGS model (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='39 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of augmented functional with respect to the model coefficients (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='14) are sequentially evaluated using the adjoint variables and the SGS basis forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (4) The L-BFGS gradient-based optimization algorithm (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) is adopted to iteratively update the SGS model parameters by repeating the above calculations until the stopping criteria are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 17 � �� �� �� �� �� �� �� �� �� ��� ���������� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � J�J0 � � � ��� DNS ��� � �� � � �� � ��� � �� � � �� � ��� � �� � � ��� � Figure 3: The evolution of the normalized cost function in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' FGR LES Resolution 𝐶 (0) 1 𝐶 (0) 2 𝐶opt 1 𝐶opt 2 1 323 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0529 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='229 2 643 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0101 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='027 4 1283 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0030 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='000 Table 2: The initial and optimal parameters of the VOMM model for LES computations with the filter width ¯Δ = 32ℎDNS in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The stop criteria for the VOMM model for the optimization iterations are summarized as follows: (a) the number of iterations reaches the maximum number of iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) the ratio of the current loss to the initial loss is smaller than a given error threshold 𝜖0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝜖0 = 1%) , namely, J (𝑘)/J (0) ⩽ 𝜖0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) the difference of model coefficients between two successive iterations is negligible, namely, ���𝐶 (𝑘+1) 𝑛 − 𝐶 (𝑘) 𝑛 ��� / ���𝐶 (0) 𝑛 ��� ⩽ 𝜖0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Eventually, the optimal parameters of the VOMM model are automatically obtained after reaching the given stopping optimization criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A posteriori studies of the VOMM models In order to examine the performance of the proposed VOMM model, the a posteriori evaluations are respectively carried out for the forced, decaying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The results of the filtered direct numerical simulation (DNS) are the benchmark for the performance evaluations of the large-eddy simulation (LES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We first introduce the detailed settings of DNS for these three turbulent problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The DNS data are then explicitly filtered by the commonly-used Gaussian filter, which is expressed 18 Model(FGR=1,𝑁 = 323) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='066 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='584 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='273 Model(FGR=2,𝑁 = 643) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='870 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='418 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='594 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='285 Model(FGR=4,𝑁 = 1283) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='512 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='517 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='588 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='240 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='645 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='321 Table 3: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations with the filter width ¯Δ = 32ℎDNS in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' as (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sagaut 2006) 𝐺 �r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� = � 6 𝜋 ¯Δ2 �1/2 exp � −6r2 ¯Δ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) The filter scale ¯Δ = 32ℎDNS is selected for both the forced and decaying homogeneous isotropic turbulence, while ¯Δ = 8ℎDNS for the temporally evolving turbulent mixing layer, where ℎDNS denotes the grid spacing of DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Three conventional SGS models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', the dynamic Smagorinsky model (DSM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1), the dynamic mixed model (DMM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) and the approximate deconvolution model with standard secondary filtering regularization (ADM, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) are adopted to compare against the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The consistent instantaneous snapshots of the filtered DNS data are used to initialize the LES calculations for different SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Both the turbulent statistics and transient contours are evaluated and compared with different SGS models for the a posteriori testings of the three canonical turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Forced homogeneous isotropic turbulence We perform the direct numerical simulation of forced incompressible isotropic turbulence using the uniform grid resolution 𝑁 = 10243 in a cubic box of (2𝜋)3 with periodic boundary conditions (ℎDNS = 2𝜋/1024) (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020a,c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pseudo-spectral method is used for the spatial discretization of the governing equations (Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Peyret 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The nonlinear advection terms are fully dealiased by the two-thirds dealiasing rule (Canuto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A second-order two-step Adams-Bashforth explicit scheme is used for time integration (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The kinematic viscosity is chosen as 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001, and large-scale forcing is applied to the two lowest wavenumber shells to maintain the turbulence in statistical equilibrium, giving rise to the Taylor Reynolds number Re𝜆 ≈ 250 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed one-point statistics of DNS data for the forced isotropic turbulence are summarized in Table 1 (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝑘max = 2𝜋 3ℎDNS denotes the largest effective wavenumber after the fully dealiasing, and 𝜔rms = √︁ ⟨𝜔𝑖𝜔𝑖⟩ represents the root-mean-square value of the vorticity magnitude, where 𝜔 = ∇ × u stands for the vorticity which is the curl of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Kolmogorov length ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='scale 𝜂 and the integral length scale 𝐿𝐼 stand for the smallest resolved scale and the largest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 4: Velocity spectra for different SGS models in the a posteriori analysis of forced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS: (a) log-log for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) semi-log for FGR=1, 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) log-log for FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) semi-log for FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (e) log-log for FGR=4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 1283;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (f) semi-log for FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' characteristic scale of turbulence, and are defined respectively by 𝜂 = � 𝜈3 𝜀 �1/4 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) 𝐿𝐼 = 3𝜋 2(𝑢rms)2 ∫ +∞ 0 𝐸 (𝑘) 𝑘 𝑑𝑘, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) where 𝜀 is the spatial average dissipation rate of kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The total turbulent kinetic energy 𝐸𝑘 = ⟨𝑢𝑖𝑢𝑖⟩ /2 = ∫ +∞ 0 𝐸 (𝑘) 𝑑𝑘, and 𝐸 (𝑘) represents the velocity spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The resolution parameters 𝑘max𝜂 ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1 and 𝜂/ℎDNS ⩾ 1 indicate that the grid resolution is sufficient to capture ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 5: Second-order structure functions of the filtered velocity for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the smallest turbulent eddy scales and ensure the convergence of turbulent kinetic energy at all scales (Ishihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2007, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to alleviate the impact of initial conditions, the forced homogeneous isotropic turbulence is run for a long period after the flow gradually reaches a statistically steady state (more than 50 large-eddy turnover times 𝜏 = 𝐿𝐼 /𝑢rms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We select data of the last ten large-eddy turnover times as a benchmark for LES comparisons (total forty flow-field snapshots of DNS data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In this paper, the Gaussian filter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) is used as the explicit filter to calculate the filtered physical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Theselected filter width ¯Δ = 32ℎDNS and the correspondingcutoff wavenumber is 𝑘𝑐 = 𝜋/ ¯Δ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity and dissipation spectra of the DNS and filtered DNS at ¯Δ = 32ℎDNS are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The filtered velocity spectrum nearly overlaps with the DNS data in a Kolmogorov scaling law of 𝑘−5/3 at the low wavenumber region, while it drops significantly at the region larger than the truncated wavenumber 𝑘𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Overall 12% of the turbulent kinetic energy is filtered out in the residual velocity field at the filter scale ¯Δ = 32ℎDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the filtered dissipation spectrum gradually grows with the power of law scaling 𝑘1/3 at the low-wavenumber inertial region, and drops sharply where the cutoff wavenumber exceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The small scales near the truncated wavenumbers are essential for the reconstruction of the filtered dissipation spectrum and also very important for the residual SGS modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, these small scales account for a very small proportion of the turbulent kinetic energy, almost several orders of magnitude smaller than the large scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Thus, the dissipation spectrum instead of the kinetic energy spectrum is chosen as the optimization objective function of the proposed VOMM model in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori testings of LES are essential to validate the practical performance of the SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES calculations use the same kinematic viscosity (𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001) with the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The filter width is fixed to ¯Δ = 32ℎDNS and the impact of the spatial discretization errors on the SGS models is investigated by changing the grid resolution of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Three different filter-to-grid ratios FGR= ¯Δ/ℎLES=1, 2 and 4 are chosen to study the influence of spatial discretization on the SGS modeling, and the corresponding grid points of LES are 𝑁 = 323, 643 and 1283, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The proposed VOMM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) is compared against the classical SGS models, including the dynamic Smagorinsky model (DSM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1), the dynamic mixed model (DMM, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) and the standard approximate deconvolution model with secondary filtering regularization (ADM, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The relaxation factors of ADM model 𝜒=0 and 1 are chosen for comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ratios of the time steps for LES and DNS are Δ𝑡LES/Δ𝑡DNS = {10, 10, 5} for different grids (FGR=1, 2 and 4 with 𝑁 = 323, 643 and 1283).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Among the filtered DNS data of the ten large- eddy turnover periods, the data of the first two large-eddy turnover times are used for the adjoint optimization of the VOMM model (only the dissipation spectrum is used, stored once every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1𝜏, twenty sets in total), and the remaining data of the last eight large-eddy turnover times are used for the a posteriori accuracy validation of the LES models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 6: Fourth-order structure functions of the filtered velocity for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' At the adjoint-based optimization stage of the VOMM model, the calculations of the adjoint equations are consistent with the primary LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We adopt the same pseudo-spectral numerical scheme to spatially discrete the stabilized adjoint momentum equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A second-order two-step Adams-Bashforth explicit scheme is applied for the time backward integration with zero terminal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Since the large-scale forcing is assumed to be nearly independent of the filtered velocity, the large-scale forcing term does not appear in the adjoint momentum equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' During the adjoint optimization stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1) of the VOMM model, the pure structural ADM model without the dissipative Smagorinsky term is selected as the initial SGS model with model coefficients 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The LES forward evolution is initialized by the filtered DNS velocity field and the dissipation spectrum is calculated when the filtered DNS data are available (every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1𝜏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The statistical discrepancy of the dissipation spectrum between the LES and fDNS data is evaluated and recorded as the cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of the cost functional with respect to the model coefficients are calculated through backward integrating the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26) with zero terminal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS model coefficients are then iteratively updated by the gradient-based L-BFGS optimization algorithm (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) until reaching the stopping criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 3 shows the evolution of the cost function normalized by the initial discrepancy during the adjoint-based optimization in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions (prediction errors of dissipation spectra between LES and fDNS data) for all three different filter-to-grid ratio cases (FGR=1,2 and 4) gradually converge and become stationary within less than twenty iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The error is significantly reduced by nearly an order of magnitude for the cases of FGR=1 and 2 within about ten iterations, and is drastically reduced to 20% of the initial state at FGR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results indicate that the adjoint-based L-BFGS gradient optimization is very efficient and effectively obtains the optimal model coefficients within several iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The optimal parameters of the VOMM model are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The magnitude of the eddy- viscosity coefficient (( ���𝐶opt 1 ���) ) dramatically reduces from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0529 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='003 with the increasing of FGR and LES resolutions, while the coefficient of the ADM part (𝐶opt 2 ) gradually approaches unity, which is identical to the theoretical value derived from the Taylor series expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the optimal model coefficients are obtained, we further examine the a posteriori performance of the VOMM model using the filtered DNS data of the last eight large-eddy turnover periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Table 3 gives the average computational cost for the SGS stress modeling at the same filter width ¯Δ = 32ℎDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For all three different grid resolutions, the computation time of the VOMM model is only about 30% of that of the DMM model, without significantly increasing the computational cost in comparison to the ADM models (𝜒 = 0 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='The velocity spectra of different SGS models with the filter scale ¯Δ = 32ℎDNS in comparison to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 7: Sixth-order structure functions of the filtered velocity for LES in the a posteriori ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='analysis of forced homogeneous isotropic turbulence with the same filter scale ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' those of the DNS and filtered DNS (fDNS) data are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity spectrum of DNS data exhibits a sufficiently long inertial range with a typical 𝑘−5/3 scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The spectrum of fDNS almost overlaps with that of DNS at the low-wavenumber region, but is obviously lower than that of DNS near the truncated wavenumber since the small-scale kinetic energy at high wavenumbers is filtered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES only solves the large-scale variables with the filtered Navier-Stokes equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5), leaving the effect of residual small scales to be approximately reconstructed by the SGS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore the statistics of an ideal LES would overlap with that of the fDNS data as closely as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' When the grid resolution of LES is sufficiently coarse and the grid spacing of LES is equal to the filter scale (FGR=1, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figs 4a and 4b), the spatial discretization error is significant and deteriorates the accuracy of the SGS stress modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES calculations with traditional SGS models are very difficult to obtain accurate predictions of the turbulent kinetic energy cascade at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity spectra predicted by the ADM models with 𝜒 = 0 and 1 exhibit numerical unstable, and kinetic energy at high wavenumbers is obviously overestimated due to the insufficient dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM and DMM models also have dramatic overestimations at high-wavenumber regions, with predictions even larger than that of the DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, VOMM model predicts the velocity spectra most accurately among these SGS models whose results nearly coincide with that of fDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the cases of fine grid resolutions (FGR=2 and 4), the pure ADM model (𝜒 = 0) is still numerically unstable since the pure structural model itself cannot produce sufficient SGS dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM model with the standard secondary-filtering regularization (𝜒 = 1) exhibits excessively dissipative, and the small-scale kinetic energy at high wavenumbers is extremely exhausted and much lower than that of fDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The predictions of DSM and DMM models illustrate the obviously tilted distribution, where kinetic energy at low wavenumbers is accumulated, while that near the truncated wavenumber is diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dynamic least-square procedure for both DSM and DMM models would overestimate the eddy-viscosity coefficient for the cases of fine grid resolutions (FGR=2 and 4), and small-scale flow structures near the truncated wavenumbers are exhausted by the excessive dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent kinetic energy is transferred from large scales to small scales through the forward energy cascade process of the nonlinear advection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The lack of the sufficient flow structures near the cutoff wavenumber leads to the energy accumulation in the intermediate wavenumber region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model is superior to the other SGS models and can accurately predict the velocity spectra at all different grid resolutions of LES, with the predictions very close to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to further examine the reconstruction of multiscale properties of turbulence by the SGS models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' we calculate the longitudinal structure functions of the filtered velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' namely ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' �� �� � � � �� �� �� �� �� �� �� �� �� �� � �� � ��� � � �� � � � � � ��� DNS ��� � � ���� ��� ��� ����� � �� ����� � �� ���� (𝑑) �� �� �� �� �� � � � � � � � � � � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' �� �� � � � �� �� �� �� �� �� �� �� �� �� � �� � ��� � � �� � � � � � ��� DNS ��� � � ���� ��� ��� ����� � �� ����� � �� ���� Figure 8: PDFs of the normalized velocity increments 𝛿r ¯𝑢/ ¯𝑢rms for LES at grid resolution of 323 in the a posteriori analysis of forced homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS: (a) r = ¯Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) r = 2 ¯Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) r = 3 ¯Δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) r = 4 ¯Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2018, 2019a) ¯𝑆𝑛(𝑟) = ����� 𝛿𝑟 ¯𝑢 ¯𝑢rms ���� 𝑛� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4) where 𝑛 represents the order of structure function and 𝛿𝑟 ¯𝑢 = [¯u (x + r) − ¯u (x)] · ˆr denotes the longitudinal velocity increment at the separation r with the unit distance vector ˆr = r/|r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figures 5, 6 and 7 respectively compare the second-order, fourth-order and sixth-order structure functions of the filtered velocity for different SGS models with the filtered DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For all three grid resolutions of LES (FGR=1, 2 and 4), all SGS models predict the lower-order structure functions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5) much better than the higher-order structure functions (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 6 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, the predictions of structure functions are improved greatly with the increasing of the grid resolution, and those of all SGS models almost coincide with each other at large separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM models (both 𝜒 = 0 and 1) give the worst predictions and obviously overestimate the structure function at small distances r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM and DMM models also predict the structure functions greater than the fDNS data at small separations but underestimate the structure functions at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model can accurately reconstruct the structure functions with different orders at both small and large separations, almost overlapping with those of the filtered DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We then evaluate the probability density functions (PDFs) of the filtered velocity increments to measure the spatial correlations of turbulence, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 8, where the velocity increments 𝛿𝑟 ¯𝑢/ ¯𝑢rms are normalized by the root-mean-square value of velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The cases of fine grid resolutions (FGR=2 and 4) are very similar to that of FGR=1 and not shown in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The PDFs of the velocity increments exhibit approximately symmetrical distribution, relatively 24 (a) fDNS (b) DMM (c) ADM (𝜒=0) (d) ADM (𝜒=1) (e) DSM (f) VOMM Figure 9: Contours of the normalized vorticity ¯𝜔/ ¯𝜔rms fDNS at an arbitrary 𝑥1-𝑥2 plane at 𝑡/𝜏 ≈ 4 for LES at a grid resolution of 643 (FGR=2) in forced homogeneous isotropic turbulence with the filter width ¯Δ = 32ℎDNS: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' rms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2πrms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2πrms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2π25 � �� �� �� �� �� �� �� �� �� ��� ���������� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � J�J0 � � � ��� DNS ��� � �� � � �� � ��� � �� � � �� � ��� � �� � � ��� � Figure 10: The evolution of the normalized cost function in decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' FGR LES Resolution 𝐶 (0) 1 𝐶 (0) 2 𝐶opt 1 𝐶opt 2 1 323 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0398 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='150 2 643 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0094 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='326 4 1283 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0020 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='101 Table 4: The initial and optimal parameters of the VOMM model for LES computations with the filter width ¯Δ = 32ℎDNS in decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' concentrated at small distances while gradually becoming wider as the distance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The PDFs predicted by the ADM, DSM and DMM models are significantly wider than those of the fDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In comparison with these traditional SGS models, the VOMM model gives the most accurate prediction of the velocity increments for different distances, which are in reasonable agreement with the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We finally examine the reconstruction of instantaneous spatial flow structures by plotting the contours of the normalized vorticity magnitude as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The vorticity contours are consistently extracted on an arbitrary 𝑥1-𝑥2 plane for the isotropic turbulence at the same time with approximately four large-eddy turnover periods ( 𝑡/𝜏 ≈ 4) at a grid resolution of 643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' It is noteworthy that the exact point-to-point correlations are difficult to achieve under the long- term forecasting of LES due to the chaotic nature of the turbulence and extreme sensitivity to perturbations (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022c, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model overpredicts some unrealistic small-scale structures, which are obviously different from the band-like or strip-like spatial structures of the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The DSM, DMM and ADM (𝜒 = 1) models only predict the large-scale vorticity structures and some small scales are excessively dissipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these traditional SGS models, the VOMM model predicts the vortex structures very similar to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 26 Model(FGR=1,𝑁 = 323) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='070 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='590 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='269 Model(FGR=2,𝑁 = 643) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='026 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='589 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='553 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='317 Model(FGR=4,𝑁 = 1283) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='026 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='287 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='521 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='531 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='393 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='586 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='330 Table 5: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations with the filter width ¯Δ = 32ℎDNS in decaying homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 11: Temporal evolutions of the turbulent kinetic energy 𝐸𝑘 for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of decaying homogeneous isotropic turbulence with the same filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='scale ¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Decaying homogeneous isotropic turbulence In order to investigate the impact of turbulent unsteady evolution on SGS stress modeling, the numerical simulation of decaying homogeneous isotropic turbulence in a cubic box of (2𝜋)3 with periodic boundary conditions is investigated in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The numerical simulation method is consistent with the forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We spatially discretize the governing equations using the pseudo-spectral method with the two-thirds dealiasing rule at a uniform grid resolution of 𝑁 = 10243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The temporal discretization scheme adopts the second- order two-step Adams-Bashforth explicit method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The statistically steady isotropic turbulence data of the forced isotropic turbulence (detailed statistics see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1) is used as the initial field for DNS decaying turbulence without the large-scale forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The kinematic viscosity is set to 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001 and the initial Taylor Reynolds number is Re𝜆 ≈ 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We calculate the DNS data of decaying turbulence for about six large-eddy turnover times (𝜏 = 𝐿𝐼 /𝑢rms), the first two of which are used for the adjoint-based optimization to determine the model coefficients of VOMM model (only the dissipation spectrum is used, stored once every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1𝜏, twenty sets in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori studies of LES adopt the consistent kinematic viscosity (𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='001) with the DNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Gaussian filter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) is selected as the explicit filter with the given filter width ¯Δ = 32ℎDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Similar to the forced isotropic turbulence, three different filter-to-grid ratios FGR= ¯Δ/ℎLES=1,2 and 4 are chosen to investigate the impact of the spatial discretization on the SGS stress modeling with the corresponding grid resolutions of LES 𝑁 = 323, 643 and 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(𝑎) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='Figure 12: Temporal evolutions of the average dissipation rate ¯𝜀 for LES in the a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='posteriori analysis of decaying homogeneous isotropic turbulence with the same filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='scale ¯Δ = 32ℎDNS: (a) FGR=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' adjoint-based optimization of the VOMM model (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1) is first performed to determine the optimal model coefficients using the dissipation spectra as the cost functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model without the Smagorinsky part is used as the initial SGS model with parameters 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of the cost functional for the model coefficients are evaluated by successively forward solving the LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) and backward integrating the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gradient-based L-BFGS optimization algorithm (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) is used for iteratively updating the SGS model parameters until reaching the stopping criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The evolution of the cost function normalized by the initial loss during the adjoint-based optimization for the decaying isotropic turbulence is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions for all three cases of different grid resolutions (FGR=1,2 and 4) drop rapidly at the beginning and gradually reach a plateau within approximately twenty iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The prediction errors of the optimization objective are considerably reduced to 10% of the initial state for both FGR=1 and 2, and substantially decreased to about 20% of the original value at FGR=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradient optimization can quickly obtain the optimal model parameters within a limited number of iterations (less than 100 optimization iterations, namely, 200 LES evaluations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Table 4 gives the optimal parameters of the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The magnitude of the dissipative Smagorinsky coefficient ( ���𝐶opt 1 ���) significantly drops from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0398 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='002 as the LES resolution increases, which is slightly lower than that in forced homogeneous isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the coefficient of the structural part (𝐶opt 2 ) is asymptotically close to unity as the grid spacing of LES becomes smaller, similar to the results of forced isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori performance of the VOMM model is further validated after determining the optimal SGS model coefficients by the adjoint-based gradient optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We compare the proposed VOMM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='35) with the classical SGS models including the DSM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1), DMM model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3) and the ADM model regularized by the standard secondary- filtering technique (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The time steps of LES are given as Δ𝑡LES/Δ𝑡DNS = {10, 10, 5} for different grid resolutions (FGR=1, 2 and 4 with 𝑁 = 323, 643 and 1283).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The average computational costs for the SGS stress modeling with different grid resolutions using different SGS models at the same filter scale ¯Δ = 32ℎDNS are summarized in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The computation time of the VOMM model only accounts for approximately 30% of the time of DMM model and slightly increases in computational cost compared to the ADM models with 𝜒 = 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figures 11 and 12 respectively compare the temporal evolutions of the turbulent kinetic energy and the resolved dissipation rate ( ¯𝜀 = 2𝜈 � ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗 � ) of different SGS models with the filtered DNS (fDNS) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent kinetic energy gradually decays from the initial statistically steady state over time, since there are no additional forcing driving the dissipative turbulent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' All the classical SGS models (DSM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DMM 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='DNS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� � ��� ��� ���� ��� ��� ����� � �� ����� � �� ���� ( 𝑓 ) �� � �� � �� � �� �� �� �� �� �� �� �� �� �� �� � �� � � � ��� � � � � � ��� DNS ��� � �� !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� � ��� ��� ���� ��� ��� ����� � �� ����� � �� ���� Figure 13: Velocity spectra for different SGS models in the a posteriori analysis of decaying homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS at 𝑡/𝜏 ≈ 2 and 4: (a) FGR=1, 𝑁 = 323 at 𝑡/𝜏 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=1, 𝑁 = 323 at 𝑡/𝜏 ≈ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) FGR=2, 𝑁 = 643 at 𝑡/𝜏 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) FGR=2, 𝑁 = 643 at 𝑡/𝜏 ≈ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (e) FGR=4, 𝑁 = 1283 at 𝑡/𝜏 ≈ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (f) FGR=4, 𝑁 = 1283 at 𝑡/𝜏 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' throughout the time, which differs significantly from the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model gives reasonable predictions of the turbulent kinetic energy, which is the closest to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The average dissipation rate displays a decline trend with time, similar to that of the turbulent kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, all conventional SGS models wrongly predict the non- monotonic tendency of the average dissipation rate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the case of sufficiently coarse grid resolution of LES (FGR=1 with 𝑁 = 323), DSM, DMM and ADM models overpredict the dissipation rate with an erroneous temporal evolution that first increases and then decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' When the grid resolution of LES becomes fine (FGR=2 and 4 with 𝑁 = 643 and 1283), DSM and DMM models obviously underestimate the dissipative rate at the early stage of decaying turbulence (𝑡/𝜏 ⩽ 3), then DMM model gradually becomes closer to the fDNS data while DSM model 29 (𝑎) � ��� � ��� � ��� � � �� � � #"$ ��� � ��� ��� ��� ��� ��� ��� ��� � � �� � � � � � ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� � �� %�� � ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� � ��� � ��� � � �� � � #"$ ��� � ��� ��� ��� ��� ��� ��� ��� � � �� � � � � � ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� � �� %�� � ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑐) � ��� � ��� � ��� � � �� � � #"$ ��� � ��� ��� ��� ��� ��� ��� ��� � � ��� � � � � � ��!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� � �� %�� � ��� ��� ��� ��� ����� � �� ����� � �� ���� Figure 14: PDFs of the normalized vorticity ¯𝜔/ ¯𝜔rms fDNS for LES in the a posteriori analysis of decaying homogeneous isotropic turbulence with the same filter scale ¯Δ = 32ℎDNS at 𝑡/𝜏 ≈ 4: (a) FGR=1, 𝑁 = 323;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 643;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and (c) FGR=4, 𝑁 = 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' overestimates the dissipation rate with the decaying of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model ( 𝜒 = 0 ) always gives the overestimations of the dissipation rate for all three different grid resolutions of LES, even though the pure ADM model can accurately predict the turbulent kinetic energy at a sufficiently high grid resolution (FGR=4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results demonstrate that the pure structural ADM model without any dissipative terms might not accurately predict all physical quantities of LES (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', the average dissipation rate), even if the grid resolution is high enough compared to the filter scale (FGR=4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM model with standard secondary-filtering regularization (𝜒 = 1) provides excessive dissipation similar to the DSM model with mispredictions of first underestimating and then overestimating the average dissipation rate over time at FGR=2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In comparison to these classical SGS models, the VOMM model accurately predicts the temporal evolutions of average dissipation rate for all three different grid resolutions, which agrees fairly well with the benchmark filtered DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The transient velocity spectra of different SGS models at the filter width ¯Δ = 32ℎDNS with two different time instants 𝑡/𝜏 ≈ 2 and 4 are further illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The velocity spectra exhibit an overall decrease, and the kinetic energy at all wavenumbers declines with the decaying of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' All the classical SGS models (DSM, DMM and ADM models) overpredict the kinetic energy at high wavenumbers for the coarse grid-resolution case (FGR=1 with 𝑁 = 323 ) and the excessive kinetic energy stacked at small scales leads to the numerical instability of LES, which gradually intensifies with the evolution of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The conventional SGS models provide insufficient model dissipation to balance the discretization errors and the small-scale kinetic energy cannot be effectively dissipated in time at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the fine grid-resolution cases (FGR=2 and 4 with 𝑁 = 643 and 1283), the dissipation of the traditional SGS models (DSM, DMM models, and ADM model with 𝜒 = 1) is too strong to diminish most small-scale flow structures near the truncated wavenumber, which hinders the normal transmission of turbulent kinetic energy cascades from large scales to small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore, the kinetic energy of classical SGS models accumulates in the region of intermediate wavenumbers, leading to the overestimations of the turbulent kinetic energy with time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 11) at FGR=2 and 4 with 𝑁 = 643 and 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' LES using the pure ADM model with 𝜒 = 0 is always numerically unstable and lacks necessary SGS dissipation to drain out the small-scale kinetic energy for all different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these classical SGS models, the VOMM model can accurately reconstruct the kinetic energy cascade with the predictions that nearly coincide with those of fDNS at all three different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, we compare the PDFs of the normalized vorticity magnitude at the dimensionless time 𝑡/𝜏 ≈ 4 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The vorticity is normalized by the root-mean-square values of the vorticity calculated by the fDNS data for comparisons of different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM models with 𝜒 = 0 gives the worst prediction of the vorticity with erroneous peaks 30 (a) fDNS (b) DMM (c) ADM (𝜒=0) (d) ADM (𝜒=1) (e) DSM (f) VOMM Figure 15: Contours of the normalized vorticity ¯𝜔/ ¯𝜔rms fDNS at an arbitrary 𝑥1-𝑥2 plane at 𝑡/𝜏 ≈ 4 for LES at a grid resolution of 643 (FGR=2) in decaying homogeneous isotropic turbulence with the filter width ¯Δ = 32ℎDNS: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2πrms 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 C1/2π31 (𝑎) (𝑏) π Figure 16: Diagram of the temporally evolving mixing layer with the mean velocity profile: (a) schematic of the mixing layer, (b) mean streamwise velocity profile ⟨𝑢1⟩ along the normal (𝑥2) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' of PDFs significantly different from the fDNS data for all three grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The secondary filtering technique (𝜒 = 1) of the ADM model cannot improve the prediction of vorticity very well, whose estimations are still obviously different from the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM and DMM models underestimate the PDF of vorticity and have wrong predictions of the PDF peak at the coarse grid-resolution case (FGR=1 with 𝑁 = 323 ), while greatly improving the predictions of PDFs with the increasing of the grid resolution (FGR=2 and 4 with 𝑁 = 643 and 1283).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model outperforms these classical SGS models at all three different grid resolutions, which gives a reasonably good prediction for both the locations and the peaks of the PDFs of the vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The reconstruction of transient spatial vorticity structures are finally demonstrated by the contours of the normalized vorticity magnitude shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The instantaneous snapshots are selected on an arbitrary 𝑥1-𝑥2 slice at the consistent time instant 𝑡/𝜏 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model predicts the excessive stochastic small-scale structures, which significantly differ from the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The other SGS models can predict the large-scale vorticity structures quite well, but the VOMM model reconstruct the spatial vortex structures very similar to the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model can accurately recover more flow structures and the temporal evolution of the vortex with suitable SGS dissipation and accurate structural modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Temporally evolving turbulent mixing layer The turbulent mixing layer is one of the cardinal flows in the fluid-mechanics community, which is widely applied to the investigation of turbulent combustion, chemical reaction mixing process, and fundamental studies of flow instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent mixing layer involves the unsteady shear process of vortex shedding and transition from laminar to turbulent flows, which are remarkably suitable for investigating the impact of non-uniform turbulent shear and mixing on the SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The temporally evolving turbulent mixing layer characterized by the Kelvin–Helmholtz instability induced by the initial velocity difference is considered in this paper (Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The free-shear mixing layer is governed by the same Navier-Stokes equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2) without the forcing term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 16 illustrates the diagram of the flow configuration for the temporally evolving turbulent mixing layer with the initial hyperbolic tangent streamwise velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The numerical simulation of mixing layer is performed in a cuboid domain with lengths 𝐿1×𝐿2×𝐿3 = 8𝜋×8𝜋×4𝜋 at the uniform grid resolution of 𝑁1 × 𝑁2 × 𝑁3 = 512 × 512 × 256 where 𝑥1 ∈ [−𝐿1/2, 𝐿1/2], 𝑥2 ∈ [−𝐿2/2, 𝐿2/2] and 𝑥3 ∈ [−𝐿3/2, 𝐿3/2] denote the streamwise, transverse and spanwise directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To enable a periodic configuration in the normal direction, the initial 32 𝑁1 × 𝑁2 × 𝑁3 𝐿1 × 𝐿2 × 𝐿3 𝜈∞ 𝑅𝑒𝜃 𝛿0 𝜃 Δ𝑈 Δ𝑑/ℎDNS ℎDNS Δ𝑡DNS 512 × 512 × 256 8𝜋 × 8𝜋 × 4𝜋 5 × 10−4 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='08 2 8 𝜋/64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='002 Table 6: Numerical parameters for the DNS of the temporally evolving mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' mean streamwise velocity (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 16b) is given by (Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a) ⟨𝑢1⟩ = Δ𝑈 2 � tanh � 𝑥2 2𝛿0 𝜃 � − tanh � 𝑥2 + 𝐿2/2 2𝛿0 𝜃 � − tanh � 𝑥2 − 𝐿2/2 2𝛿0 𝜃 �� , for − 𝐿2 2 ⩽ 𝑥2 ⩽ 𝐿2 2 , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5) where Δ𝑈 = 2 is the velocity difference between two equal and opposite free streams across the shear layer, 𝛿0 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='08 denotes the initial momentum thickness, and ⟨·⟩ stands for a spatial average over all the homogeneous directions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑥1 and 𝑥3 directions for the mixing layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The initial mean transverse and spanwise velocities are both set to zero, namely, ⟨𝑢2⟩ = ⟨𝑢3⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Since the initial mean velocity field is periodic in all three directions, the triply periodic boundary conditions are adopted and the pseudo-spectral method with the two-thirds dealiasing rule is used for the spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' An explicit two-step Adam-Bashforth scheme is selected as the time-advancing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to effectively suppress the influence of the top and bottom boundaries on the central mixing layer, two numerical diffusion buffer zones are applied near the vertical edges of domain (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The thickness of the buffer layer is set to 15𝛿0 𝜃 in the paper, which is sufficiently large and has a negligible effect on the calculations of mixing layer (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The digital filter method is used to generate the spatially-correlated initial perturbation imposed on the mean velocities with the digital filter width Δ𝑑 = ¯Δ = 8ℎDNS consistent to the filter scale of LES (Klein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The initial Reynolds stress distribution (𝑅𝑖 𝑗 = � 𝑢′ 𝑖𝑢′ 𝑗 � where 𝑢′ 𝑖 = 𝑢𝑖 − ⟨𝑢𝑖⟩ represents the fluctuated velocity) of the digital filter method is assumed as a vertical distribution of 𝑅𝑖 𝑗 = 𝐴 � 1 − ⟨𝑢1⟩2� 𝐼𝑖 𝑗 with the identity 𝐼𝑖 𝑗 and peak amplitude 𝐴 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='025Δ𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The kinematic viscosity of shear layer is set to 𝜈∞ = 5 × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The momentum thickness quantifies the range of turbulence region in the mixing layer, which is defined by (Rogers & Moser 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019) 𝛿𝜃 = 𝐿2/4 ∫ −𝐿2/4 � 1 4 − � ⟨ ¯𝑢1⟩ Δ𝑈 �2� 𝑑𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6) Correspondingly, the Reynolds number based on the momentum thickness 𝑅𝑒𝜃 is expressed as 𝑅𝑒𝜃 = Δ𝑈𝛿𝜃 𝜈∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='7) Here, the initial momentum thickness Reynolds number is 𝑅𝑒0 𝜃 = 320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The detailed numerical parameters of DNS for the temporally evolving mixing layer is summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We calculate the DNS of the mixing layer for total of eight hundred time units (𝑡/𝜏𝜃 = 800) normalized by 𝜏𝜃 = 𝛿0 𝜃/Δ𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to reduce the impact of initial random disturbances on the temporal development of the shear layer, six numerical experiments with different random initializations are performed, one of which is adopted for the parameter optimization of the VOMM model, while the remaining five are used to evaluate the ensemble-averaged physical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori studies of LES are conducted using the explicit Gaussian filter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='1) with 33 � �� �� �� �� �� �� �� �� �� ��� ���������� � ��� ��� ��� ��� ��� ��� ��� ��� ��� � J�J0 � � � �� DNS ��� � �� � � �� � � �� ��� � �� � � ��� � � �� Figure 17: The evolution of the normalized cost function in temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' FGR LES Resolution 𝐶 (0) 1 𝐶 (0) 2 𝐶opt 1 𝐶opt 2 1 642 × 32 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0637 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='188 2 1282 × 64 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0126 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='000 Table 7: The initial and optimal parameters of the VOMM model for LES computations with the filter width ¯Δ = 8ℎDNS in temporally evolving mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the given filter scale ¯Δ = 8ℎDNS and initialized by the same instantaneous velocity field of the filtered DNS at 𝑡/𝜏𝜃 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Two different filter-to-grid ratios FGR= ¯Δ/ℎLES=1 and 2 are selected to study the influence of the spatial resolution or discretization error on the SGS stress modeling with the corresponding grid resolutions of LES: 𝑁 = 642 × 32 and 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The results from the previous two turbulence problems (forcing and decaying homogenous isotropic turbulence) indicate that the statistics of turbulence are very close and similar when the grid resolution is sufficiently fine (FGR=2 and 4) and the discretization error is considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the statistics of LES with a relatively coarse grid resolution (FGR=1) are distinctly different from those of LES with satisfactory grid resolutions (FGR=2 and 4), since the spatial discretization error of FGR=1 is considerably significant and dominates the SGS modelling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Therefore, the a posteriori testings of LES at both FGR=1 and 2 are essential for performance evaluations of the SGS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dissipation spectrum of the filtered DNS is consistently used as the objective function to optimize the model parameters of the VOMM model during the period (assess every 𝑡/𝜏𝜃 = 10 with total thirty-six groups at 50 ⩽ 𝑡/𝜏𝜃 ⩽ 400) of the adjoint-based optimization (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model without the dissipative term is adopted as the initial SGS model with coefficients 𝐶 (0) 1 = 0 and 𝐶 (0) 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We calculate the adjoint-based gradients of the cost functional for the model parameters by backward integrating the stabilized adjoint LES equations (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='10 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS model coefficients are iteratively updated by the L-BFGS optimization method (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='41) until the stopping criterion is ultimately satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 17 gives the optimization 34 Model(FGR=1,𝑁 = 642 × 32) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='646 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='362 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='590 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='330 Model(FGR=2,𝑁 = 1282 × 64) DSM DMM ADM(𝜒=0) ADM(𝜒=1) VOMM t(CPU·s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='756 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='370 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='465 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='460 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='908 t/tDMM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='590 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='230 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='300 Table 8: The average computational cost of SGS stress modeling 𝜏𝑖 𝑗 for LES computations with the filter width ¯Δ = 8ℎDNS in temporally evolving turbulent mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' process of the cost function during the adjoint-based optimization for the temporally evolving mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions for both FGR=1 and 2 drop dramatically and reach a steady plateau within less than ten iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The cost function of FGR=1 shows a more distinct reduction with approximately 8% of the initial level than that of FGR=2 decreasing to the 10% of original value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The optimal parameters of VOMM model are quickly obtained by the effective gradient- based optimization within a limited number of iterations (around 10 optimization evaluations, namely, 20 LES calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Table 7 summarizes the optimal parameters of the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The parameter magnitude of the dissipative Smagorinsky term ( ���𝐶opt 1 ���) obviously decreases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0637 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='0126 when the FGR increases from 1 to 2, while the ADM coefficient (𝐶opt 2 ) generally tends towards unity, similar to the cases of isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We then examine the a posteriori performance of the proposed VOMM model once the SGS model coefficients are determined by the adjoint-based gradient optimization strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to demonstrate the generality of the optimal model parameters that are insensitive to the initial perturbations, ensemble-averaged quantities are evaluated by five numerical experiments with different initial random disturbances from the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The time steps of LES are set as Δ𝑡LES/Δ𝑡DNS = {10, 5} to guarantee the consistent CFL number for different grid resolutions (FGR=1 and 2 with 𝑁 = 642 × 32 and 1282 × 64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model is compared with the conventional SGS models (DSM, DMM and ADM models), and the average modeling costs for different SGS models are listed in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model evaluates efficiently with about 30% computational cost of the DMM model which is similar to those of the ADM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Figure 18 illustrates the temporal evolutions of the momentum thickness 𝛿𝜃 in LES calculations of different SGS models compared to the benchmark fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' At the case of coarse grid resolution (FGR=1 with 𝑁 = 642 × 32), all conventional SGS models underpredict the momentum thickness at the early stage of shear layer development (𝑡/𝜏𝜃 ⩽ 300) but give obvious overestimations in the linear growth region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the fine-grid-resolution case (FGR=2 with 𝑁 = 1282 × 64), DMM and ADM (𝜒=1) models can capture the growth rate of momentum thickness well at the beginning of temporal development, but still overpredict the thickness with the developing of shear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The prediction of the pure ADM model with 𝜒 = 0 is irregular and nonlinear all the time without an apparent linear self-similar region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The DSM model at different grid resolutions gives the clearly tilted temporal evolutions, where the momentum thickness is underestimated at the beginning of transition region and overpredicted in the region of linear growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the predictions of the VOMM model always coincide well with those of fDNS, and they accurately capture the temporal growth rate in the linear region at both grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, the evolutions of the turbulent kinetic energy in the streamwise and spanwise 35 (𝑎) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ��� ��� ��� ��� ��� � � ��� � �� � � �� � ��� � � � �" DNS !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ��� ��� ��� ��� ��� � � ��� � �� � � ��� � ��� � � � �" DNS !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ��� ��� ����� � �� ����� � �� ���� Figure 18: Temporal evolutions of the momentum thickness 𝛿𝜃 for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (𝑎) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� ���� � "� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� ���� � "� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� Figure 19: Temporal evolutions of the streamwise turbulent kinetic energy 𝐸𝑘1 for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' directions are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 19 and 20, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The comparisons of transverse turbulent kinetic energy for different SGS models are very similar to those in the spanwise direction, not shown in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The turbulent kinetic energy of DNS in different directions gradually increases with the developing of the shear layer, since the initial perturbated velocity field is approximately laminar and steadily transitions to turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The temporal development of the streamwise kinetic energy can be approximately regarded as a linear growth with time, which is distinctly different from that of spanwise kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' All classical SGS models predict both streamwise and spanwise kinetic energy much larger than the benchmark fDNS results at both grid resolutions of LES, except that the pure ADM model gives underestimations of kinetic energy in the fine-grid-resolution case (FGR=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these traditional models, the VOMM model accurately predicts the kinetic energy at different grid resolutions in both streamwise and spanwise directions, and is the closest to the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The profiles of the resolved Reynolds shear stress component ¯𝑅12 = � ¯𝑢′ 1 ¯𝑢′ 2 � at time instants 𝑡/𝜏𝜃 ≈ 500 and 800 are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 21, which is the dominant Reynolds stress term due to the intense mixing along the streamwise and normal directions (Vreman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sharan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The normal distribution of the Reynolds stress is a second-order statistic of turbulence which has high requirements for the accuracy of SGS modeling of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The ADM models underpredict the Reynolds stress, while DSM and DMM models give obvious overestimations at 36 (𝑎) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� � "� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) � ��� ��� ��� ��� ��� ��� ��� ��� #�� � � ����� ���� ����� � "� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS ��� ��� ��� ����� � �� ����� � �� ���� Figure 20: Temporal evolutions of the spanwise turbulent kinetic energy 𝐸𝑘3 for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS: (a) FGR=1, 𝑁 = 642 × 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=2, 𝑁 = 1282 × 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (𝑎) ���� ���� ���� � ��� ��� ��� x 2 /4π ����� � ���� ���� ���� ���� ���� ���� � � � �� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � "�� � � ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) ���� ���� ���� � ��� ��� ��� x 2 /4π ����� � ���� ���� ���� ���� ���� ���� � � � �� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � "�� � � ��� ��� ��� ��� ����� � �� ����� � �� ���� Figure 21: The transient profile of the resolved Reynolds shear stress ¯𝑅12 = � ¯𝑢′ 1 ¯𝑢′ 2 � along the cross-stream direction for LES in the a posteriori analysis of temporally evolving turbulent mixing layer with filter scale ¯Δ = 8ℎDNS at grid resolution of 𝑁 = 1282 × 64: (a) 𝑡/𝜏𝜃 ≈ 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) 𝑡/𝜏𝜃 ≈ 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Compared to these classical SGS models, the VOMM model gives the prediction closest to the fDNS results, and accurately recovers the transient profiles of Reynolds stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We further compare the velocity spectra of different SGS models with the DNS and filtered DNS data at time instants 𝑡/𝜏𝜃 ≈ 500 and 800, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The spectra of DNS at 𝑡/𝜏𝜃 ≈ 500 and 800 are very similar since the instantaneous velocity fields at different moments are both at the self-similar stage of mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the coarse grid-resolution case at FGR=1 with 𝑁 = 642 × 32, the conventional SGS models (DSM, DMM and ADM models) always give the overestimations of the small-scale kinetic energy at high wavenumbers, and the excess kinetic energy accumulates at small scales and exacerbates the numerical instability of LES over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS dissipation provided by these conventional SGS models is insufficient to stabilize the numerical perturbations induced by the spatial discretization errors, which cannot effectively drain out the small-scale kinetic energy in time at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' For the case of fine grid resolution at FGR=2 with 𝑁 = 1282×64, the pure ADM model is still numerically unstable, whose prediction distinctly deviates from the fDNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' And the velocity spectra predicted by the other conventional SGS models (DSM, DMM and ADM with 𝜒=0) diminish at high-wavenumber regions and accumulate in the region of intermediate wavenumbers, since these traditional SGS models are too dissipative at the fine grid-resolution case to recover the effect of small-scale flow structures near the cutoff wavenumber, giving rise to the blockage of the kinetic energy cascade from large scales to small scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the kinetic energy cascade can be correctly constructed with high accuracy by 37 (𝑎) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑏) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � �� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑐) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� (𝑑) �� � � � �� � �� � �� �� " �� �� �� �� �� �� �� �� �� �� �� �� ��"� ��� � �� � � ��� � � ��� � � � �!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DNS � #�� � � ��� ��� ��� ��� ��� ����� � �� ����� � �� ���� Figure 22: Velocity spectra for different SGS models in the a posteriori analysis of temporally evolving turbulent mixing layer with the same filter scale ¯Δ = 8ℎDNS at 𝑡/𝜏𝜃 ≈ 500 and 800: (a) FGR=1, 𝑁 = 642 × 32 at 𝑡/𝜏𝜃 ≈ 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (b) FGR=1, 𝑁 = 642 × 32 at 𝑡/𝜏𝜃 ≈ 800;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (c) FGR=2, 𝑁 = 1282 × 64 at 𝑡/𝜏𝜃 ≈ 500;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (d) FGR=2, 𝑁 = 1282 × 64 at 𝑡/𝜏𝜃 ≈ 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the VOMM model, and the predictions are always in reasonable agreement with those of fDNS at different grid resolutions and time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The reconstruction of vortex structures is finally compared with different SGS models by displaying the iso-surface of the Q-criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Q-criterion is a useful visualization tool for observing vortex structures in turbulent flows, and is the second invariant of velocity gradient tensor, namely (Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Dubief & Delcayre 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019) 𝑄 = 1 2 � ¯Ω𝑖 𝑗 ¯Ω𝑖 𝑗 − ¯𝑆𝑖 𝑗 ¯𝑆𝑖 𝑗 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8) where ¯Ω𝑖 𝑗 = 1 2 �𝜕 ¯𝑢𝑖/𝜕𝑥 𝑗 − 𝜕 ¯𝑢 𝑗/𝜕𝑥𝑖 � represents the rotation-rate tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The instantaneous iso- surface of Q at 𝑡/𝜏𝜃 ≈ 500 is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 23 during the self-similar stage of the mixing layer for Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 colored by the streamwise velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The Q iso-surface of fDNS contains a large number of elaborate vortex structures near the middle 𝑥1-𝑥3 plane of the shear layer, including the rib-like vortices, hairpin vortices and complex helical vortices, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' DSM, DMM and ADM (𝜒=1) models exhibit an excessive dissipation that only large-scale rib-like vortex structures remain, while the pure ADM model with 𝜒=0 suffers from numerical instability of LES and overpredicts many nonphysical small-scale structures caused by numerical noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model can accurately reconstruct much more vortex structures, highlighting its advantage in improving the accuracy of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 38 (a) fDNS (b) DMM (c) ADM (𝜒=0) (d) ADM (𝜒=1) (e) DSM (f) VOMM Figure 23: The iso-surface of the Q-criterion at 𝑄=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 colored by the streamwise velocity at 𝑡/𝜏𝜃 ≈ 500 in the a posteriori analysis of temporally evolving turbulent mixing layer with filter scale ¯Δ = 8ℎDNS at grid resolution of 𝑁 = 1282 × 64: (a) fDNS, (b) DMM, (c) ADM(𝜒=0), (d) ADM(𝜒=1), (e) DMM, and (f) VOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' u1 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='5 14元 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='539 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Conclusion In this work, an adjoint-based variational optimal mixed model (VOMM) is developed for the large-eddy simulation of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' We first derive the original adjoint LES equations with the general SGS model, and then carry out the energy budget analysis of adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These detailed derivations demonstrate that the quadratic term with negative eigenvalues of the shear strain rate is responsible for the exponential temporal growth of the adjoint-based gradients, giving rise to the numerical divergence in a long time horizon for the chaotic turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' This issue might greatly limits the application of the adjoint-based variational methods and optimal control strategy in turbulence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' An additional stabilization term is introduced to maintain the numerical stability of the adjoint LES equations and is efficiently calculated by the sequential quadratic programming (SQP) approach, without degrading the accuracy of gradient evaluations for the SGS model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Subsequently, the stabilized adjoint LES equations are correspondingly formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The approximate deconvolution model (ADM) in the scale-similarity form and the dissipative Smagorinsky term are selected as the basis tensors of the proposed VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The parameters of the VOMM model are optimally identified by minimizing the statistical discrepancies between dissipation spectra of the LES and those of the benchmark filtered DNS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The adjoint-based gradients of cost functional for model coefficients are efficiently evaluated by successively forward solving the LES equations and backward integrating the stabilized adjoint LES equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The gradient-based L-BFGS optimization algorithm is adopted for iteratively updating the VOMM model parameters until the optimal values are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Three turbulent flow scenarios including the forced homogeneous isotropic turbulence, de- caying homogeneous isotropic turbulence and temporally evolving turbulent mixing layer are investigated to examine the a posteriori performance of the VOMM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure structural ADM model without the dissipative Smagorinsky term is selected as the initial SGS model for the parameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The loss functions of the dissipation spectra can dramatically converge and reach the optimal state of only about 10% of the initial value within less than twenty iterations (about forty LES evaluations) during the adjoint-based gradient optimization at different grid resolutions for these three types of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results indicate that the adjoint-based gradient optimization is an effective tool to obtain the optimal parameters of VOMM model within only a few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Meanwhile, the computational efficiency of the proposed method is independent of the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Once the optimal SGS model coefficients are determined by the adjoint-based gradient optimization, the a posteriori accuracy of the VOMM model is further tested in comparison with the classical SGS models, including the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), the pure ADM model and ADM model with the standard secondary- filtering regularization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The various statistics of turbulence and the instantaneous flow structures are comprehensively compared for LES calculations of different SGS models with the benchmark filtered DNS data at different grid resolutions of three turbulent flow scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the cases of forced and decaying homogeneous isotropic turbulence, the filter scale is fixed to ¯Δ = 32ℎDNS and the impact of the spatial discretization errors on the SGS modeling is studied by changing the grid resolution of LES with three different filter-to-grid ratios FGR=1, 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The a posteriori performance of the proposed VOMM model is systematically evaluated by comparison to the conventional SGS models (DSM, DMM and ADM models) in terms of the velocity spectra, structure functions with different orders, PDFs of the velocity increments and vorticity, temporal evolutions of the turbulent kinetic energy and average dissipation rate, as well as the instantaneous vorticity contours at different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The pure ADM model always exhibits numerical instability due to the insufficient sufficient SGS dissipation for all grid- resolution cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The dynamic models and standard regularized ADM model underpredict the 40 model dissipation in the case of coarse grid resolution (FGR=1), with the excess kinetic energy accumulated at small scales leading to the numerical instability of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The SGS dissipation imposed by these classical SGS models is insufficient to suppress the numerical perturbations dominated by the spatial discretization, and it cannot effectively drain out the small-scale kinetic energy in time at FGR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' However, the traditional SGS models are too dissipative that most small- scale flow structures near the truncated wavenumber are diminished, giving rise to the blockage of the kinetic energy cascade from large scales to small scales at situations of satisfactory grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model can correctly reconstruct the kinetic energy cascade and the evolution of dissipation rate with high accuracy, which is essential for the isotropic turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the VOMM model accurately predicts various flow statistics and transient spatial flow structures, which are always in reasonable agreement with the benchmark filtered DNS results at different grid resolutions and times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In the context of the temporally evolving turbulent mixing layer, the unsteady evolution of the shear layer from the initial perturbed velocity field gradually transitions to fully developed turbulence is challenging for the SGS modeling of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The VOMM model can accurately reconstruct the temporal evolutions of characteristic physical quantities of the mixing layer, including the momentum thickness, turbulent kinetic energy in different directions and transient velocity spectra at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The corresponding predictions of VOMM are closest to the filtered DNS results and superior to these conventional SGS models (DSM, DMM and ADM models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The profiles of Reynolds shear stress at the self-similar stage of the shear layer are critical for the development of mixing layer, and all conventional SGS models are not able to accurately predict the vertical distributions with significant deviations from the benchmark fDNS result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In contrast, the VOMM model predicts the Reynolds stress fairly well at different time instants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Besides, it can be clearly observed from the iso-surface of Q-criterion that the VOMM model accurately recovers the diverse spatial vortex structures very similar to the benchmark fDNS data in comparison to the classical SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Furthermore, for the cases of three turbulent flow scenarios with different grid resolutions, the computational cost of the proposed VOMM model is only about 30% the time of the DMM model, which is very efficient and competitive compared to the classical SGS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' These results suggest that the proposed VOMM model has high a posteriori accuracy and computational efficiency by assimilating the a priori knowledge of turbulence statistics, and can be a promising tool to develop advanced SGS models in the LES of turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Eventually, it is noteworthy that fine-tuning a small number of model parameters of some traditional SGS models can significantly improve the a posteriori accuracy of LES using the proposed adjoint-based optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In addition, the predictions of LES in complex turbulent flows using the VOMM model might be dramatically accurate as the number of model coefficients increases, while the computational cost of the adjoint-based approach hardly varies with to the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Although the high-fidelity turbulence statistics is provided by DNS data in the current study, the experimental measurements can also be assimilated using the same optimization procedure to increase the accuracy of LES modeling for a particular type of complex turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In future work, we would further apply the VOMM model with the existing optimal parameters to more complex turbulent flows and generalize to turbulence with different filter scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China (NSFC Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 91952104, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 92052301, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 12172161, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 12161141017), by the National Numerical Windtunnel Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' NNW2019ZT1-A04), by the NSFC Basic Science Center Program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 11988102), by the Shenzhen Science and Technology Program (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' KQTD20180411143441009), by Key Special Project for Introduced Talents Team of Southern Marine Science and Engineering Guangdong Laboratory (Guangzhou) (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 41 GML2019ZD0103), and by Department of Science and Technology of Guangdong Province (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 2019B21203001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' This work was also supported by Center for Computational Science and Engineering of Southern University of Science and Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Declaration of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Derivation of the adjoint large-eddy simulation equations The large-eddy simulation (LES) equations are expressed as (Pope 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Sagaut 2006) 𝑅0 ( ¯𝑢𝑖) = 𝜕 ¯𝑢𝑖 𝜕𝑥𝑖 = 0, (A 1) 𝑅𝑖 ( ¯𝑢𝑖, ¯𝑝) = 𝜕 ¯𝑢𝑖 𝜕𝑡 + 𝜕 � ¯𝑢𝑖 ¯𝑢 𝑗 � 𝜕𝑥 𝑗 + 𝜕 ¯𝑝 𝜕𝑥𝑖 − 𝜈 𝜕2 ¯𝑢𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 − F 𝑖 + 𝜕𝜏𝑖 𝑗 𝜕𝑥 𝑗 = 0, (A 2) where an overbar denotes the filtered variables with filter scale ¯Δ, ¯𝑢𝑖 and ¯𝑝 denote the filtered velocity and pressure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝜈 is the kinematic viscosity, and ¯F𝑖 represents the large-scale forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The unclosed SGS stress 𝜏𝑖 𝑗 = 𝑢𝑖𝑢 𝑗 − ¯𝑢𝑖 ¯𝑢 𝑗 is modeled by the 𝑁-parameter mixed model 𝜏𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� with the basis stress tensors 𝑇 (𝑛) 𝑖 𝑗 and model coefficients 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The sensitivities of the governing equations for the LES variables ¯v = [ ¯𝑝, ¯𝑢1, ¯𝑢2, ¯𝑢3]𝑇 are given by 𝛿𝑅𝑘 = 𝜕𝑅𝑘 𝜕¯v · 𝛿¯v = � 𝜕𝛿 ¯𝑢𝑖 𝜕𝑥𝑖 𝜕𝛿 ¯𝑢𝑖 𝜕𝑡 + 𝜕( ¯𝑢𝑗 𝛿 ¯𝑢𝑖) 𝜕𝑥𝑗 + 𝜕( ¯𝑢𝑖 𝛿 ¯𝑢𝑗) 𝜕𝑥𝑗 + 𝜕𝛿 ¯𝑝 𝜕𝑥𝑖 − 𝜈 𝜕2 𝛿 ¯𝑢𝑖 𝜕𝑥𝑗𝜕𝑥𝑗 + 𝜕𝛿𝜏𝑖 𝑗 𝜕𝑥𝑗 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (A 3) The adjoint LES equations are derived by the adjoint identity acting on the adjoint variables ¯v† = � ¯𝑝†, ¯𝑢† 1, ¯𝑢† 2, ¯𝑢† 3 �𝑇 , namely � 𝜕𝑅𝑘 𝜕¯v · 𝛿¯v, ¯v† � x,𝑡 = � 𝛿¯v, � 𝜕𝑅𝑘 𝜕¯v �† ¯v† � x,𝑡 + 𝐵𝑇, (A 4) where 𝐵𝑇 denotes the boundary and temporal integral terms, and 𝐵𝑇 = 0 can identify the boundary and terminal conditions of the adjoint equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The corresponding adjoint LES equations can be expressed as 3 ∑︁ 𝑘=0 � 𝜕𝑅𝑘 𝜕¯v �† ¯v† − 𝜕𝐽 𝜕¯v = 0, (A 5) where 𝜕𝐽/𝜕¯v = � 0, 𝜕𝐽 𝜕 ¯𝑢1 , 𝜕𝐽 𝜕 ¯𝑢2 , 𝜕𝐽 𝜕 ¯𝑢3 �𝑇 denotesthesensitivityofthecostfunctional 𝐽 � ¯𝑢𝑖, ¯𝑢ref 𝑖 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐶𝑛, x, 𝑡� which quantifies the discrepancy between ¯𝑢𝑖 and the reference data ¯𝑢ref 𝑖 in the LES calculations under the given parameters 𝐶𝑛 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁) at a certain space-time state (x, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, the terms (𝜕𝑅𝑘/𝜕¯v)† · ¯v† (𝑘 = 0, 1, 2, 3) are derived by multiplying the perturbation LES equations (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A 3) with the adjoint LES variables ¯v†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and then integrating by parts to rearrange all of the 42 differential operators without 𝛿¯v onto the adjoint variables ¯v† ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' yielding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ¯𝑝† + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕( ¯𝑢𝑗 𝛿 ¯𝑢𝑖) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕( ¯𝑢𝑖 𝛿 ¯𝑢𝑗) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ 𝜕𝛿 ¯𝑝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 − 𝜈 𝜕2 𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗𝜕𝑥𝑗 + 𝜕𝛿𝜏𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑝 − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑡 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢 𝑗 + 𝜈 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕2 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗𝜕𝑥𝑗 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝜏𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='− ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕2𝜏𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢𝑖𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑖+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='terminal condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ 𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ¯𝑢 𝑗 + 𝜈 𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝜏𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑖 − 𝜈 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ 𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿 ¯𝑝 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑝† + ¯𝑢 𝑗 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='boundary condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (A 6) The adjoint LES equations are written in detail as 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑖 = 0, (A 7) 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢† 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕 𝜕𝑥 𝑗 � ¯𝑢† 𝑘 𝜕𝜏𝑗𝑘 𝜕 ¯𝑢𝑖 � − ¯𝑢† 𝑘 𝜕2𝜏𝑗𝑘 𝜕 ¯𝑢𝑖𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (A 8) It is worth noting that the adjoint SGS term ¯𝑢† 𝑘 𝜕2𝜏𝑗𝑘 𝜕 ¯𝑢𝑖𝜕𝑥𝑗 can lead to the non-conservation of the adjoint momentum and deteriorate the evaluation of the adjoint-based gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' To our knowledge, few previous studies have addressed this critical issues that make the LES adjoint field prone to numerical instability and eventual divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' In order to maintain the momentum conservation in the adjoint equations, we remove ¯𝑢† 𝑘 𝜕2𝜏𝑗𝑘 𝜕 ¯𝑢𝑖𝜕𝑥𝑗 from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' A 8, and the conservative adjoint LES equations are obtained as 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑖 = 0, (A 9) 𝜕 ¯𝑢† 𝑖 𝜕𝑡 + � 𝜕 ¯𝑢�� 𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 � ¯𝑢 𝑗 + 𝜈 𝜕2 ¯𝑢† 𝑖 𝜕𝑥 𝑗𝜕𝑥 𝑗 + 𝜕𝜏† 𝑖 𝑗 𝜕𝑥 𝑗 + 𝜕𝐽 𝜕 ¯𝑢𝑖 = 0, (A 10) where 𝜏† 𝑖 𝑗 = ¯𝑢† 𝑘 𝜕𝜏𝑗𝑘 𝜕 ¯𝑢𝑖 is the adjoint SGS stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' If the unclosed SGS terms is modeled by the 𝑁-parameter mixed model 𝜏𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛) 𝑖 𝑗 � ¯𝑢𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ¯Δ� with the basis stress tensors 𝑇 (𝑛) 𝑖 𝑗 and model coefficients 𝐶𝑛, the adjoint SGS stresses are correspondingly represented as 𝜏† 𝑖 𝑗 = 𝑁� 𝑛=1 𝐶𝑛𝑇 (𝑛),† 𝑖 𝑗 with the associated adjoint basis stress tensors 𝑇 (𝑛),† 𝑖 𝑗 (𝑛 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=', 𝑁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Derivation of the adjoint SGS stress for the VOMM model The present variational optimal mixed model (VOMM) combines the approximate deconvolu- tion model (ADM) in the scale-similarity form with the dissipative Smagorinsky part,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' expressed as 𝜏𝑖 𝑗 = 𝐶1𝑇 (1) 𝑖 𝑗 + 𝐶2𝑇 (2) 𝑖 𝑗 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' with 𝑇 (1) 𝑖 𝑗 = ¯Δ2| ¯𝑆| ¯𝑆𝑖 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑇 (2) 𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗 − 𝑢∗ 𝑖 𝑢∗ 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 1) where 𝑢∗ 𝑖 = 𝑁� 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ ¯𝑢𝑖 stands for the 𝑖-th approximate unfiltered velocity component recovered by the iterative van Cittert procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝑁 is the number of iterations for the AD procedure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 𝐼 is the identity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and the symbol “⊗” is the spatial convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Here, 𝐶1 and 𝐶2 are SGS 43 model coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The variation of the first basis SGS tensor 𝑇 (1) 𝑖 𝑗 with respect to the velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' is derived by 𝛿𝑇 (1) 𝑖 𝑗 = ¯Δ2 � | ¯𝑆|𝛿 ¯𝑆𝑖 𝑗 + �𝛿| ¯𝑆|� ¯𝑆𝑖 𝑗 � = ¯Δ2 � | ¯𝑆| 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑢𝑘 + 𝜕| ¯𝑆| 𝜕 ¯𝑢𝑘 ¯𝑆𝑖 𝑗 � 𝛿 ¯𝑢𝑘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 2) where the derivatives of the shear strain-rate tensor and characteristic strain rate for the velocity are further written as 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑢𝑘 = 1 2 𝜕 𝜕 ¯𝑢𝑘 � 𝜕 ¯𝑢𝑖 𝜕𝑥 𝑗 + 𝜕 ¯𝑢 𝑗 𝜕𝑥𝑖 � = 1 2 � 𝜕𝛿𝑖𝑘 𝜕𝑥 𝑗 + 𝜕𝛿 𝑗𝑘 𝜕𝑥𝑖 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 3) and 𝜕| ¯𝑆| 𝜕 ¯𝑢𝑘 = 𝜕| ¯𝑆| 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑆𝑖 𝑗 𝜕 ¯𝑢𝑘 = ¯𝑆𝑖 𝑗 | ¯𝑆| � 𝜕𝛿𝑖𝑘 𝜕𝑥 𝑗 + 𝜕𝛿 𝑗𝑘 𝜕𝑥𝑖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='(B 4) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='The inner product between the variation of the first basis SGS force and the adjoint velocity is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='derived by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 = − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='= − ¯Δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿𝑖𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 + 𝜕𝛿 𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝛿𝑚𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑛 + 𝜕𝛿𝑛𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 ¯𝑆𝑚𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ¯𝑆𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑘 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='= − ¯Δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� 2 ¯𝑆𝑗𝑘 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='| ¯𝑆| ¯𝑆𝑚𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑛 + 𝜕 ¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝛿 ¯𝑢𝑘 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝜕𝑥𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='¯𝑢† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝛿𝑇 (1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='𝑖 𝑗 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 5) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' the adjoint strain-rate tensor ¯𝑆† 𝑖 𝑗 = � 𝜕 ¯𝑢† 𝑖 /𝜕𝑥 𝑗 + 𝜕 ¯𝑢† 𝑗/𝜕𝑥𝑖 � /2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' and the inner product term can be further expressed as ¯𝑢† 𝑖 𝜕𝛿𝑇 (1) 𝑖 𝑗 𝜕𝑥 𝑗 = � 𝜕 𝜕𝑥 𝑗 � − ¯Δ2 � | ¯𝑆| ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 | ¯𝑆| ¯𝑆𝑖 𝑗 ��� 𝛿 ¯𝑢𝑖 + 𝜕 𝜕𝑥 𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (1) 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 6) Thus, the adjoint basis stress tensor 𝑇 (1),† 𝑖 𝑗 is given by 𝑇 (1),† 𝑖 𝑗 = − ¯Δ2 � | ¯𝑆| ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 | ¯𝑆| ¯𝑆𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 7) The common filter function 𝐺 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' top-hat, Gaussian and spectral filters) is symmetric spatial filter, and is self-adjoint, namely (Vreman 2004) ⟨𝐺 ⊗ 𝑓 , 𝑔⟩x = ⟨ 𝑓 , 𝐺 ⊗ 𝑔⟩x, (B 8) where 𝑓 (x) and 𝑔 (x) are arbitrary variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The 𝐺𝑛 filter with spatially filtering 𝑛 times (𝐺𝑛 = 𝐺 ⊗ 𝐺 ⊗ · · · ⊗ 𝐺) also satisfies the self-adjoint property proved by the mathematical induction method, expressed as ⟨𝐺𝑛 ⊗ 𝑓 , 𝑔⟩x = � 𝐺 ⊗ 𝐺𝑛−1 ⊗ 𝑓 , 𝑔 � x = � 𝐺𝑛−1 ⊗ 𝑓 , 𝐺 ⊗ 𝑔 � x = · · · = ⟨ 𝑓 , 𝐺𝑛 ⊗ 𝑔⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 9) The (𝐼 − 𝐺) filter is also a symmetric filter, and the approximate deconvolution procedure 𝐻 = 𝑁� 𝑛=1 (𝐼 − 𝐺)𝑛−1 is thus the self-adjoint filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' The second basis SGS tensor 𝑇 (2) 𝑖 𝑗 can be described using the AD abbreviated notation, namely 𝑇 (2) 𝑖 𝑗 = 𝑢∗ 𝑖 𝑢∗ 𝑗 − 𝑢∗ 𝑖 𝑢∗ 𝑗 = 𝐺 ⊗ � (𝐻 ⊗ ¯𝑢𝑖) �𝐻 ⊗ ¯𝑢 𝑗 �� − [𝐺 ⊗ (𝐻 ⊗ ¯𝑢𝑖)] � 𝐺 ⊗ �𝐻 ⊗ ¯𝑢 𝑗 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 10) 44 The variation of the second basis SGS tensor 𝑇 (2) 𝑖 𝑗 with respect to the velocity, expressed as 𝛿𝑇 (2) 𝑖 𝑗 = 𝐺 ⊗ � (𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ 𝑗 � +𝐺 ⊗ � 𝑢∗ 𝑖 �𝐻 ⊗ 𝛿 ¯𝑢 𝑗 �� −[𝐺 ⊗ (𝐻 ⊗ 𝛿 ¯𝑢𝑖)] 𝑢∗ 𝑗 −𝑢∗ 𝑖 � 𝐺 ⊗ �𝐻 ⊗ 𝛿 ¯𝑢 𝑗 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 11) The inner product between the variation of the second basis SGS force and the adjoint velocity is given by 𝜕𝛿𝑇 (2) 𝑖 𝑗 𝜕𝑥 𝑗 ¯𝑢† 𝑖 = − 𝜕 ¯𝑢† 𝑖 𝜕𝑥𝑗 𝛿𝑇 (2) 𝑖 𝑗 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � = −2 ¯𝑆† 𝑖 𝑗 � 𝐺 ⊗ � (𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ 𝑗 �� + 2 ¯𝑆† 𝑖 𝑗 [𝐺 ⊗ (𝐻 ⊗ 𝛿 ¯𝑢𝑖)] 𝑢∗ 𝑗 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 12) The inner product term can be further simplified by the self-adjoint property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' such that 𝜕𝛿𝑇 (2) 𝑖 𝑗 𝜕𝑥𝑗 ¯𝑢† 𝑖 = −2 � 𝐺 ⊗ ¯𝑆† 𝑖 𝑗 � � (𝐻 ⊗ 𝛿 ¯𝑢𝑖) 𝑢∗ 𝑗 � + 2 � 𝐺 ⊗ � ¯𝑆† 𝑖 𝑗𝑢∗ 𝑗 �� (𝐻 ⊗ 𝛿 ¯𝑢𝑖) + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � = 𝐻 ⊗ � −2 ¯𝑆† 𝑖 𝑗𝑢∗ 𝑗 + 2 ¯𝑆† 𝑖 𝑗𝑢∗ 𝑗 � 𝛿 ¯𝑢𝑖 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � = � 𝜕 𝜕𝑥𝑗 � 𝐻 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 �� + 𝐻 ⊗ � 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 𝑢∗ 𝑗 − 𝜕 ¯𝑢† 𝑗 𝜕𝑥𝑖 𝑢∗ 𝑗 �� 𝛿 ¯𝑢𝑖 + 𝜕 𝜕𝑥𝑗 � ¯𝑢† 𝑖 𝛿𝑇 (2) 𝑖 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 13) It is quite notable that the second adjoint SGS term makes the non-conservation of the adjoint momentum, therefore we discard the second adjoint SGS term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Thus, the second adjoint basis stress tensor 𝑇 (2),† 𝑖 𝑗 can be written as 𝑇 (2),† 𝑖 𝑗 = 𝐻 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � = 𝑁 ∑︁ 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' (B 14) In summary, the adjoint SGS stress of the proposed VOMM model is represented by 𝜏† 𝑖 𝑗 = 𝐶1𝑇 (1),† 𝑖 𝑗 + 𝐶2𝑇 (2),† 𝑖 𝑗 , (B 15) where the adjoint basis stress tensors are 𝑇 (1),† 𝑖 𝑗 = − ¯Δ2 � | ¯𝑆| ¯𝑆† 𝑖 𝑗 + 2 ¯𝑆𝑘𝑙 ¯𝑆† 𝑘𝑙 | ¯𝑆| ¯𝑆𝑖 𝑗 � and 𝑇 (2),† 𝑖 𝑗 = 𝑁� 𝑛=1 (𝐼 − 𝐺)𝑛−1 ⊗ � ¯𝑢† 𝑖 𝑢∗ 𝑗 − ¯𝑢† 𝑖 𝑢∗ 𝑗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fluids A: Fluid Dynamics 4 (1), 127–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhan, Jie-min, Li, Yu-tian, Wai, Wing-hong Onyx & Hu, Wen-qing 2019 Comparison between the q criterion and rortex in the application of an in-stream structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fluids 31 (12), 121701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhang, Xin-Lei, Xiao, Heng, Luo, Xiaodong & He, Guowei 2022 Ensemble Kalman method for learning turbulence models from indirect observation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' 949, A26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Zhou, Zhideng, He, Guowei, Wang, Shizhao & Jin, Guodong 2019 Subgrid-scale model for large-eddy simulation of isotropic turbulent flows using an artificial neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'} +page_content=' Fluids 195, 104319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFAT4oBgHgl3EQfExwU/content/2301.08423v1.pdf'}