Variational Multiscale Evolve and Filter Strategies for Convection-Dominated Flows

TL;DR

This paper addresses instability and overdiffusion in convection-dominated Navier–Stokes simulations stabilized by evolve-filter (EF) methods. By embedding EF into a variational multiscale (VMS) framework, the authors develop two correction strategies, VMS-EFFC and VMS-EPFC, that separate large and small scales in the evolved velocity, filter the small scales, and recover accurate flow features, including vortex shedding, on under-resolved meshes. They extend these VMS-based concepts to reduced-order modeling, yielding VMS-EFFC-ROM and VMS-EPFC-ROM that outperform Galerkin ROMs and standard EF-ROM in both qualitative and quantitative metrics for a flow past a cylinder at $Re=1000$. The results demonstrate improved accuracy in velocity norms and flow behavior with manageable computational overhead, highlighting the potential of VMS-based EF corrections for reliable full-order and reduced-order simulations of convection-dominated flows.

Abstract

The evolve-filter (EF) model is a filter-based numerical stabilization for under-resolved convection-dominated flows. EF is a simple, modular, and effective strategy for both full-order models (FOMs) and reduced-order models (ROMs). It is well-known, however, that when the filter radius is too large, EF can be overdiffusive and yield inaccurate results. To alleviate this, EF is usually supplemented with a relaxation step. The relaxation parameter, however, is very sensitive with respect to the model parameters. In this paper, we propose a novel strategy to alleviate the EF overdiffusivity for a large filter radius. Specifically, we leverage the variational multiscale (VMS) framework to separate the large resolved scales from the small resolved scales in the evolved velocity, and we use the filtered small scales to correct the large scales. Furthermore, in the new VMS-EF strategy, we use two different ways to decompose the evolved velocity: the VMS Evolve-Filter-Filter-Correct (VMS-EFFC) and the VMS Evolve-Postprocess-Filter-Correct (VMS-EPFC) algorithms. The new VMS-based algorithms yield significantly more accurate results than the standard EF in both the FOM and the ROM simulations of a flow past a cylinder at Reynolds number Re = 1000.

Figures (36)

• Figure 1: Spatial domain $\Omega$: schematic representation. $\Gamma_D = \Gamma_{\text{in}} \cup \Gamma_\text{w}$. Homogeneous Dirichlet conditions are applied on the solid cyan boundary. The inlet boundary $\Gamma_{\text{in}}$ is represented by a dotted magenta line. The "free flow" boundary $\Gamma_N$ is depicted by a dashed black line.
• Figure 2: Ganeral setting. Coarse mesh (M1) and fine mesh (M2), left and right plots, respectively.
• Figure 3: General setting. Velocity profiles for NSE-no-filtered simulation on the coarse mesh (M1) at $t=1$ and $t=4$, left and right plots, respectively.
• Figure 4: Experiment 1. Velocity profiles at $t=1$: DNS (top left), EF (top right) for $\delta=1.59 \cdot 10^{-3}$, VMS-EFFC (bottom left) for $\delta_1=\delta_2=1.59 \cdot 10^{-3}$, and VMS-EPFC (bottom right) for $\delta=1.59 \cdot 10^{-3}$.
• Figure 5: Experiment 1. Velocity profiles at $t=4$: DNS (top left), EF (top right) for $\delta=1.59 \cdot 10^{-3}$, VMS-EFFC (bottom left) for $\delta_1=\delta_2=1.59 \cdot 10^{-3}$, and VMS-EPFC (bottom right) for $\delta=1.59 \cdot 10^{-3}$.

Theorems & Definitions (12)

• Remark 1: grad-div stabilization
• Remark 2: Boundary conditions
• Remark 3: The choice of $\delta_1$ and $\delta_2$
• Remark 4: Relation to Approximate Deconvolution
• Remark 5: Choosing the reduced filter radii
• Remark 6: More on the relative error
• Remark 7: Choosing the grad-div stabilization parameter $\gamma_P$
• Remark 8: Relaxation
• Remark 9: Robustness of the VMS-EFFC approch
• Remark 10: On the role of $\overline{r}_u$