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Longer time accuracy for the Ladyzhenskya model with the EMAC formulation

Rihui Lan, Jorge Reyes

Abstract

In this paper, we incorporate the EMAC formulation into the Ladyzhenskaya model (LM), a large eddy simulation (LES) of incompressible flows. The EMAC formulation, which conserves energy, linear momentum, and angular momentum even with weak enforcement of incompressibility, has been shown to provide tangible benefits over the popular skew-symmetric for direct numerical simulation and regularized models of the Navier Stokes equations (NSE). The combination of EMAC with the LM addresses the known over-dissipation issues associated with the classical Smagorinsky model (SM). We develop a finite element discretization for the EMAC-LM system and analyze its stability and derive numerical error estimates, showing improved long-time behavior compared to the standard LM approach, particularly due to EMAC's favorable Gronwall constant independent of the Reynolds number. Benchmark simulations demonstrate that the EMAC-LM model yields more accurate flow structures, especially at high Reynolds numbers.

Longer time accuracy for the Ladyzhenskya model with the EMAC formulation

Abstract

In this paper, we incorporate the EMAC formulation into the Ladyzhenskaya model (LM), a large eddy simulation (LES) of incompressible flows. The EMAC formulation, which conserves energy, linear momentum, and angular momentum even with weak enforcement of incompressibility, has been shown to provide tangible benefits over the popular skew-symmetric for direct numerical simulation and regularized models of the Navier Stokes equations (NSE). The combination of EMAC with the LM addresses the known over-dissipation issues associated with the classical Smagorinsky model (SM). We develop a finite element discretization for the EMAC-LM system and analyze its stability and derive numerical error estimates, showing improved long-time behavior compared to the standard LM approach, particularly due to EMAC's favorable Gronwall constant independent of the Reynolds number. Benchmark simulations demonstrate that the EMAC-LM model yields more accurate flow structures, especially at high Reynolds numbers.
Paper Structure (8 sections, 14 theorems, 78 equations, 3 figures, 1 table)

This paper contains 8 sections, 14 theorems, 78 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Numerical Scheme and Analysis
  4. Convergence Analysis
  5. Numerical Experiments
  6. Taylor Green Vortex Problem
  7. The channel flow past a forward-backward facing step
  8. Conclusions

Key Result

Lemma 1

For any fixed $r \in [2,\infty]$, there exists a constant $C >0$ depending only on $\Omega$ such that

Figures (3)

  • Figure 1: Plots of magnitude of the simulated velocity at times $t$ = 10, 20, 30 and 40 generated by LM-EMAC scheme for the channel flow past a forward-backward facing step problem with the viscosity $\mathrm{Re}=10^4$.
  • Figure 2: Plots of magnitude of the simulated velocity at times $t$ = 10, 20, 30 and 40 generated by LM-SKEW for the channel flow past a forward-backward facing step problem with the viscosity $\mathrm{Re} = 10^4$.
  • Figure 3: Evolutions of the simulated kinetic energy (left), momentum (middle) and angular momentum (right) of the numerical solution generated by LM-EMAC and LM-SKEW schemes for the step problem with the viscosity $\mathrm{Re} = 10^4$. The time step size $\Delta t=0.01$.

Theorems & Definitions (18)

  • Lemma 1: Stability of Stokes Projection GiraultNochettoScott
  • Lemma 2: Error estimate of Stokes Projection de2018analysis
  • Lemma 3: Inverse inequalities BrennerScott
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6: Strong monotonicity minty1962monotonelions1969quelques
  • Lemma 7: Lipschitz Continuity du1991analysisReyesGSM
  • Lemma 8
  • Lemma 9: Discrete Gronwall Lemma heywood1990finite
  • ...and 8 more