Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the energy dissipation rate of ensemble eddy viscosity models of turbulence: Shear flows

William Layton

Abstract

Classical eddy viscosity models add a viscosity term with turbulent viscosity coefficient developed beginning with the Kolmogorov-Prandtl parameterization. Approximations of unknown accuracy of the unknown mixing lengths and turbulent kinetic energy are typically constructed by solving associated systems of nonlinear convection-diffusion-reaction equations with nonlinear boundary conditions. Often these over-diffuse so additional fixes are added such as wall laws or using different approximations in different regions (which must also be specified). Alternately, one can solve an ensemble of NSE's with perturbed data, compute the ensemble mean and fluctuation and simply compute directly the turbulent viscosity parameterization. This idea is recent, seems to be of lower complexity and greater accuracy and produces parameterizations with the correct near wall asymptotic behavior. The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? This question is addressed herein.

On the energy dissipation rate of ensemble eddy viscosity models of turbulence: Shear flows

Abstract

Classical eddy viscosity models add a viscosity term with turbulent viscosity coefficient developed beginning with the Kolmogorov-Prandtl parameterization. Approximations of unknown accuracy of the unknown mixing lengths and turbulent kinetic energy are typically constructed by solving associated systems of nonlinear convection-diffusion-reaction equations with nonlinear boundary conditions. Often these over-diffuse so additional fixes are added such as wall laws or using different approximations in different regions (which must also be specified). Alternately, one can solve an ensemble of NSE's with perturbed data, compute the ensemble mean and fluctuation and simply compute directly the turbulent viscosity parameterization. This idea is recent, seems to be of lower complexity and greater accuracy and produces parameterizations with the correct near wall asymptotic behavior. The question then arises: Does this ensemble eddy viscosity approach over-diffuse solutions? This question is addressed herein.
Paper Structure (6 sections, 11 theorems, 71 equations)

This paper contains 6 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The energy inequality
  3. Related work
  4. Notation and preliminaries
  5. Shear Flows
  6. Conclusions

Key Result

Theorem 1

Any weak solution of the eddy viscosity model (eq:EnsembleNSE) satisfying the energy inequality (eq:EnIneq) has its model energy dissipation bounded as

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Definition 1
  • Proposition 1: Uniform Bounds
  • Definition 2
  • Theorem 4
  • Remark 1
  • proof
  • ...and 10 more