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Energy dissipation rates of ensemble eddy viscosity models of turbulence: the periodic box

William Layton, Nanda Nechingal Raghunathan

Abstract

Classical eddy viscosity models of turbulence add an eddy viscosity term based on the Kolmogorov-Prandtl parameterization by a turbulent length scale $l$ and a turbulent kinetic energy $k^{\prime }$. Approximations of the unknowns $l,k^{\prime }$ are typically constructed by solving multi-parameter systems of nonlinear convection-diffusion-reaction equations. Often these over-diffuse so additional fixes are added. Alternately, one can solve an ensemble of NSE's with perturbed data and simply compute directly $k^{\prime }$(without modeling). The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? We prove herein that for turbulence in a periodic box it does not.

Energy dissipation rates of ensemble eddy viscosity models of turbulence: the periodic box

Abstract

Classical eddy viscosity models of turbulence add an eddy viscosity term based on the Kolmogorov-Prandtl parameterization by a turbulent length scale and a turbulent kinetic energy . Approximations of the unknowns are typically constructed by solving multi-parameter systems of nonlinear convection-diffusion-reaction equations. Often these over-diffuse so additional fixes are added. Alternately, one can solve an ensemble of NSE's with perturbed data and simply compute directly (without modeling). The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? We prove herein that for turbulence in a periodic box it does not.

Paper Structure

This paper contains 9 sections, 7 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Energy Dissipation and Turbulence Phenomenology \newlabelEnergy Dissipation and Turbulence Phenomenology
  3. Related work
  4. Notation and preliminaries
  5. The energy dissipation rate
  6. Estimation of EEV energy dissipation rates
  7. Proof of the theorem
  8. Model Existence
  9. Conclusions

Key Result

Proposition 2.1

Consider the model (eq:EEVmodel) with boundary conditions (eq:PeriodicBCs) and $l(x,t)=|u^{\prime }|_{e}\tau$. For a weak solution satisfying (EQEnergyIneq) the following are uniformly bounded in $T$

Theorems & Definitions (11)

  • Proposition 2.1: Uniform Bounds
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • ...and 1 more