Maximum Entropy Method for Solving the Turbulent Channel Flow Problem.

T.-W Lee

doi:10.3390/e21070675

Abstract

There are two components in this work that allow for solutions of the turbulent channel flow problem: One is the Galilean-transformed Navier-Stokes equation which gives a theoretical expression for the Reynolds stress (u′v′); and the second the maximum entropy principle which provides the spatial distribution of turbulent kinetic energy. The first concept transforms the momentum balance for a control volume moving at the local mean velocity, breaking the momentum exchange down to its basic components, u′v′, u′2, pressure and viscous forces. The Reynolds stress gradient budget confirms this alternative interpretation of the turbulence momentum balance, as validated with DNS data. The second concept of maximum entropy principle states that turbulent kinetic energy in fully-developed flows will distribute itself until the maximum entropy is attained while conforming to the physical constraints. By equating the maximum entropy state with maximum allowable (viscous) dissipation at a given Reynolds number, along with other constraints, we arrive at function forms (inner and outer) for the turbulent kinetic energy. This allows us to compute the Reynolds stress, then integrate it to obtain the velocity profiles in channel flows. The results agree well with direct numerical simulation (DNS) data at Reτ = 400 and 1000.

Highlights

Analytical solutions to turbulence problems have become a rarified genre, in part due to rapid advances in numerics that can solve many problems of fundamental and practical significance
We have taken an alternate route for solving turbulence problems with some modest success, in deriving the turbulence energy spectra from the maximum entropy principle [1] and in determining the Reynolds stress from the first principles [2,3,4]
Turbulence can be considered as a large ensemble of energetic eddies which achieves dissipative equilibrium state due to its rapid mixing properties, so that it is an opportune phenomenon to apply the maximum entropy principle

Summary

Introduction

Analytical solutions to turbulence problems have become a rarified genre, in part due to rapid advances in numerics that can solve many problems of fundamental and practical significance. The above treatment of turbulence momentum to obtain directly the Reynolds stress is a unique development, and similar to the use of the maximum entropy principle below provides a viable physics-based approach to solving one of the most difficult problems in (fluid) physics. U’ profile obtained from Equation u0 v0 -transport, pressure and the viscous terms combined. Equation (2) is an expression that relates the off-diagonal Reynolds stress term,02u’v’, with other turbulence variables, but for closure we still need the diagonal components, u02 and v . For channel (2)(see is an expression that relates the off-diagonal stressu’term term, with other

Maximum Entropy Principle and Turbulence

For Reaτrapid

Findings

Conclusions