Maximum Entropy Method for Solving the Turbulent Channel Flow Problem

TL;DR

This paper introduces a novel approach combining a Galilean-transformed Navier-Stokes equation and the maximum entropy principle to analytically solve the turbulent channel flow problem, producing results consistent with DNS data.

Contribution

It presents a new theoretical framework using maximum entropy and Galilean transformation to derive velocity profiles in turbulent channel flows.

Findings

Results agree well with DNS data at Re=400 and 1000.

Provides analytical expressions for Reynolds stress and velocity profiles.

Validates the maximum entropy approach for turbulence modeling.

Abstract

There are two components in this work that allow solutions of the turbulent channel problem: one is the Galilean-transformed Navier-Stokes equation which gives a theoretical expression for the Reynolds stress; 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. 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 Re = 400 and 1000.