Steady Buoyant Navier-Stokes for Differentially Heated Cavity

Aim of this tutorial: learn how the indirect reconstruction algorithm (two step approach) works, combining the methods discussed in the previous tutorial. This tutorial is of particular interest for multi-physics problems with partial observations: it is legitimate to investigate the possibility that a field carries information about another field: for instance, the temperature and the velocity in a buoyancy-driven problem. Moreover, this tutorial will show how to import snapshots from OpenFOAM simulations exploiting the fluidfoam module.

IR-scheme

This two step method is based on the following steps Introini et al. (2023): at first, the characteristic parameter is estimation by solving an optimisation problem whose input are the measures of the observable field; then, the reconstruction of the unobservable field is performed by using POD with Interpolation.

The Differentially Heated Cavity is taken from the ROSE-ROM4FOAM tutorial. It’s 2D Cavity with velocity imposed at the left and right boundary, along with [Sah18].

cavity

The governing equations are the Navier-Stokes with energy equation under the Boussinesq approximation

\[\begin{split}\left\{ \begin{aligned} \nabla \cdot \mathbf{u}&=0 \quad &\text{ in } \Omega\\ (\mathbf{u} \cdot \nabla)\mathbf{u} - \nu \Delta \mathbf{u}+ \nabla p - \mathbf{g}\cdot\beta(T-T_\infty) &=0 & \text{ in } \Omega \\ \mathbf{u} \cdot \nabla T - \alpha \Delta T&= 0\quad& \text{ in } \Omega \end{aligned} \right.\end{split}\]

given \(\Omega\) as the domain.