Stabilized POD Reduced Order Models for convection-dominated incompressible flows

TL;DR

This work tackles the challenge of accurate reduced-order modeling for convection-dominated incompressible flows by comparing two stabilization strategies within a POD-Galerkin framework. It combines offline snapshot-based POD with a pressure-Poisson constraint (PPE) and introduces either a global constant artificial viscosity or a coefficient-dependent viscosity to stabilize the reduced system. Across FDA nozzle-like benchmarks and Reynolds numbers from $2000$ to $6500$, both stabilization approaches significantly improve time-averaged predictions relative to non-stabilized POD-Galerkin ROMs, achieving errors on the order of $10^{-2}$ in velocity and $10^{-1}$ in pressure for many cases, while preserving mass conservation within about $1\%$. The study highlights the trade-offs: stabilization yields steady ROMs at high $Re$ and still struggles to recover unsteady dynamics, motivating future elliptic-filter based enhancements for time-dependent fidelity in convection-dominated regimes.

Abstract

We present a comparative computational study of two stabilized Reduced Order Models (ROMs) for the simulation of convection-dominated incompressible flow (Reynolds number of the order of a few thousands). Representative solutions in the parameter space, which includes either time only or time and Reynolds number, are computed with a Finite Volume method and used to generate a reduced basis via Proper Orthogonal Decomposition (POD). Galerkin projection of the Navier-Stokes equations onto the reduced space is used to compute the ROM solution. To ensure computational efficiency, the number of POD modes is truncated and ROM solution accuracy is recovered through two stabilization methods: i) adding a global constant artificial viscosity to the reduced dimensional model, and ii) adding a different value of artificial viscosity for the different POD modes. We test the stabilized ROMs for fluid flow in an idealized medical device consisting of a conical convergent, a narrow throat, and a sudden expansion. Both stabilization methods significantly improve the ROM solution accuracy over a standard (non-stabilized) POD-Galerkin model.

Figures (30)

• Figure 1: FDA benchmark scheme.
• Figure 2: Comparison between experimental data (dashed lines) and the FOM solution (solid line) for the time-averaged axial pressure (left) and velocity (right) for different Reynolds numbers.
• Figure 3: (a) Cumulative energy of the eigenvalues \ref{['eq:energy']} and (b) mean relative error \ref{['err']} for pressure and velocity.
• Figure 4: Comparison of FOM and ROM time-averaged axial velocity and pressure using 9 modes for both variables and no stabilization technique.
• Figure 5: Qualitative comparison of the ROM time-averaged pressure without stabilization with the corresponding FOM solution.