Graetz Problem Dataset

Dataset Description

This dataset contains thermal simulations of the Graetz problem with varying geometric and flow parameters.

Dataset Summary

The Graetz dataset provides numerical simulations of heat transfer in a channel with developing thermal boundary layer. The problem is characterized by varying channel geometry (omega2_length) and Péclet number, making it suitable for parametric reduced-order modeling and thermal-fluid analysis.

Dataset Structure

Data Instances

The dataset consists of three configurations:

• geometry: Mesh information (nodes and connectivity) - varies with geometric parameter
• snapshots: Temperature field solutions
• parameters: Geometric and flow parameters for each simulation

Data Fields

Geometry Configuration

• node_coordinates_x: Sequence of x-coordinates of mesh nodes (float64)
• node_coordinates_y: Sequence of y-coordinates of mesh nodes (float64)
• connectivity: Sequence of element connectivity (triangular elements, int32)

Snapshots Configuration

• temperature: Temperature field at each node (float64)

Parameters Configuration

• omega2_length: Geometric parameter defining the channel configuration (float64)
• peclet: Péclet number characterizing the heat transfer regime (float64)

Data Splits

• default: Contains all simulations with varying parameters

Dataset Creation

Source Data

The dataset was generated using finite element simulations of the convection-diffusion equation representing the Graetz problem. The mesh geometry varies with the omega2_length parameter, making this a parametrized geometry problem.

Preprocessing

Each simulation has its own mesh corresponding to the geometric parameter value. Solutions are stored as 1D arrays corresponding to the nodal values on each respective mesh.

Usage

from datasets import load_dataset
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.tri as mtri

# Load geometry
ds_geom = load_dataset("SISSAmathLab/graetz", name="geometry")

# Load snapshots
ds_data = load_dataset("SISSAmathLab/graetz", name="snapshots")

# Load parameters
ds_params = load_dataset("SISSAmathLab/graetz", name="parameters")

# Visualize temperature distribution for simulation 16
idx = 16
pts_x = np.asarray(ds_geom['default']['node_coordinates_x'][idx]).flatten()
pts_y = np.asarray(ds_geom['default']['node_coordinates_y'][idx]).flatten()
connectivity = ds_geom['default']['connectivity'][idx]
temperature = ds_data['default']['temperature'][idx]

omega2_length = ds_params['default']['omega2_length'][idx]
peclet = ds_params['default']['peclet'][idx]

triang = mtri.Triangulation(pts_x, pts_y, connectivity)
plt.tripcolor(triang, temperature, cmap='coolwarm')
plt.colorbar(label='Temperature')
plt.title(f'Graetz Problem (ω₂={omega2_length:.2f}, Pe={peclet:.1f})')
plt.xlabel('x')
plt.ylabel('y')
plt.axis('equal')
plt.show()

Contact

For questions or issues, please contact SISSA mathLab.