Model order reduction (POD) for flows

The physical problem.

Starting point are the incompressible Navier-Stokes equation for a velocity \(u\) and pressure \(p\) solving \begin{align} \partial_t u - \operatorname{Re}^{-1} \Delta u + \operatorname{div}( u \otimes u ) - \nabla p & = f, \\ \operatorname{div}( u ) & = 0, \end{align} The problematic case with the incompressible Navier-Stokes equations considered here is the high Reynolds number situation which forbids the resolution of all relevant scales that may appear in a flow. In this project a Large Eddy Simulation (LES) shall be implemented, where only a certain amount of scales can be resolved, but the effect of smaller scales on the resolvable scales is modeled properly, here using a Variational Multiscale (VMS) approach.

Model order reduction

We assume that we are given a working (turbulent) unsteady Navier-Stokes solver. The aim is to extract the "dominant" structures in the flow. The reduction considered here is only "passive" in the sense that we only want to store and detect these structures, they are not yet used to carry out reduced order simulations.

Peter Vogel room

As a toy example we consider the so-called Peter-Vogel room, starting in 2D for reasonable run times. After a few time steps a new state is obtained. A POD algorithm shall be implemented in NGSolve which compresses these states to the dominant modes, stores and lateron allows to visualize them.

Tasks

• Familiarize with the \(H(\operatorname{div})\)-conforming HDG discretization, e.g. the version implemented in NGSolve model templates.
• Implement a POD class that may be called in every time steps, but stores (dynamically) only the \(N\) most dominant modes in the velocity. E.g. consider the "RAPOD" presented here(poster) and here(slides) and here(paper). Note that the \(L^2\) scalar product (rather than the \(\ell^2\) scalar product in \(\mathbb{R}^n\) should be involved to pick the right modes.
• Implement a similar POD class to pick the highest velocity and pressure modes.
• Setup the 2D Peter Vogel configuration and carry out simulations for different Reynolds numbers. What do you observe (are the modes changing, are more modes necessary to represent the flow now, ...?)
• Use the 3D Peter Vogel room together with the VMS model (when finished) and extract the main flow "modes".

Further related references:

[1] M. Oulghelou, C. Allery. A Riemannian Barycentric Interpolation : Application to the Parametric Unsteady Navier-Stokes Reduced Order Model, arXiv:2009.11231.