The Reynolds stress in turbulence from a Lagrangian perspective

Abstract

We present a unique method for solving for the Reynolds stress in turbulent canonical flows, based on the momentum balance for a control volume moving at the local mean velocity. A differential transform converts this momentum balance to a solvable form. Validations with experimental and computational data in simple geometries show quite good results. An alternate Lagrangian analytical method is offered, leading to a potential closure method for the Reynolds stress in terms of computable turbulence parameters.

Maximum Entropy Method for Solving the Turbulent Channel Flow Problem

“…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.…”

“…The method is based on the first principles, Lagrangian momentum balance and the maximum entropy principle, and involves no ad-hoc modeling common in turbulence models. In that regard, the current approach has no similar precedents other than those referenced above [ 1 , 2 , 3 , 4 ]. The starting point is the Galilean-transformed Navier-Stokes equations [ 2 ].…”

“…In that regard, the current approach has no similar precedents other than those referenced above [ 1 , 2 , 3 , 4 ]. The starting point is the Galilean-transformed Navier-Stokes equations [ 2 ]. To illustrate, for simple boundary layer flows, we have: …”

“…Also, d / dx has been replaced with C 1 Ud / d y to account for the displacement effect. This concept of converting d / dx was derived from consideration of control volume moving at the mean velocity in boundary layer flows with displacement effects [ 2 , 3 , 4 ], but it also works for channel flows as well (see Appendix A ). Lagrangian methods have been used in analysis of turbulence data [ 6 , 7 ], and also in pdf-modeling [ 8 ].…”

Origin of the Turbulence Structure in Wall-Bounded Flows, and Implications toward Computability

“…Klewicski et al [13] echo this hierarchical concept, through a mathematical analysis based on a hypothetical "test function". Starting from 2016, an alternative Lagrangian formalism for the Reynolds stress has been derived and presented in a series of works [14][15][16][17][18]. In this new perspective, if an observer moves at the local mean velocity then the turbulence fluctuation and attendant transport can be mostly decoupled from the mean velocities, resulting in compact expressions for the Reynolds stress tensor components [17].…”