samples/texts/1513018/page_5.md

• Using academic examples, we prove the accuracy of the proposed numerical procedure especially convergence curves towards a 3d non-stationary analytical solution with wet-dry interfaces have been obtained (see paragraph 6.2.1).

Most of the numerical models in the literature for environmental stratified flows use finite difference or finite element schemes solving the free surface Navier-Stokes equations. We refer in particular to [30, 37] and references therein for a partial review of these methods. Since the layer-averaged model has the form of a conservation law with source terms, we single out a finite volume scheme. Moreover, the kinetic interpretation of the continuous model leads to a kinetic solver endowed with strong stability properties (well-balancing, domain invariant, discrete entropy [10]). The viscous terms are discretized using a finite element approach. Considering various analytical solutions we emphasize the accuracy of the discrete model and we also show the applicability of the model to real geophysical situations. The numerical method is implemented in Freshkiss3d [49] and other various academic tests are documented on the web site.

The outline of the paper is as follows. In Section 2, we recall the incompressible and hydrostatic Navier-Stokes equations and the associated boundary conditions. The layer-averaged system obtained by a vertical discretization of the hydrostatic model is described in Section 3. The kinetic interpretation of the model is given in Section 4 allowing to derive a numerical scheme presented in Section 5. Numerical validations and application to real test cases are shown in Section 6.

2 The hydrostatic Navier-Stokes system

We consider the three-dimensional hydrostatic Navier-Stokes system [38] describing a free surface gravitational flow moving over a bottom topography $z_b(x, y)$. For free surface flows, the hydrostatic assumption consists in neglecting the vertical acceleration, see [22, 35, 41, 23] for justifications of such hydrostatic models.

The incompressible and hydrostatic Navier-Stokes system consists in the model

$$\begin{array}{cccc}& \mathrm{\nabla }\cdot \mathbf{U}=0,& & \text{(1)}\end{array}\backslash nabla\; \backslash cdot\; \backslash mathbf\{U\}\; =\; 0,\; \backslash tag\{1\}$$

$$\begin{array}{cccc}& \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+{\mathrm{\nabla }}_{x,y}(\mathbf{u}\otimes \mathbf{u})+\frac{\mathrm{\partial }\mathbf{u}w}{\mathrm{\partial }z}=\frac{1}{{\rho }_0}{\mathrm{\nabla }}_{x,y}\sigma +\frac{\mu }{{\rho }_0}\frac{{\mathrm{\partial }}^2\mathbf{u}}{\mathrm{\partial }{z}^2},& & \text{(2)}\end{array}\backslash frac\{\backslash partial\; \backslash mathbf\{u\}\}\{\backslash partial\; t\}\; +\; \backslash nabla\_\{x,y\}\; (\backslash mathbf\{u\}\; \backslash otimes\; \backslash mathbf\{u\})\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{u\}w\}\{\backslash partial\; z\}\; =\; \backslash frac\{1\}\{\backslash rho\_0\}\; \backslash nabla\_\{x,y\}\; \backslash sigma\; +\; \backslash frac\{\backslash mu\}\{\backslash rho\_0\}\; \backslash frac\{\backslash partial^2\; \backslash mathbf\{u\}\}\{\backslash partial\; z^2\},\; \backslash tag\{2\}$$

$$\begin{array}{cccc}& \frac{\mathrm{\partial }p}{\mathrm{\partial }z}=-{\rho }_{0}g,& & \text{(3)}\end{array}\backslash frac\{\backslash partial\; p\}\{\backslash partial\; z\}\; =\; -\backslash rho\_0\; g,\; \backslash tag\{3\}$$

where $\mathbf{U}(t,x,y,z) = (u,v,w)^T$ is the velocity, $\mathbf{u}(t,x,y,z) = (u,v)^T$ is the horizontal velocity, $\sigma = -pI_d + \mu\nabla_{x,y}\mathbf{u} = -pI_d + \Sigma$ is the total stress tensor, $p$ is the fluid pressure, $g$ represents the gravity acceleration and $\rho_0$ is the fluid density. The quantity $\nabla$ denotes $\nabla = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z})^T$, $\nabla_{x,y}$ corresponds to the projection of $\nabla$ on the horizontal plane i.e. $\nabla_{x,y} = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y})^T$. We assume a Newtonian fluid, $\mu$ is the viscosity coefficient and we will make use of $\nu = \mu/\rho_0$.

We consider a free surface flow (see Fig. 1-(a)), therefore we assume

$$z_b(x,y)\le z\le \eta (t,x,y):=h(t,x,y)+z_b(x,y),z\_b(x,\; y)\; \backslash leq\; z\; \backslash leq\; \backslash eta(t,\; x,\; y)\; :=\; h(t,\; x,\; y)\; +\; z\_b(x,\; y),$$

with $z_b(x, y)$ the bottom elevation and $h(t, x, y)$ the water depth. Due to the hydrostatic assumption in Eq. (3), the pressure gradient in Eq. (2) reduces to $\rho_0 g \nabla_{x,y} \eta$.