Maximum Entropy Method for Solving the Turbulent Channel Flow Problem

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 (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 Re = 400 and 1000.

Asymmetrical Order in Wall-Bounded Turbulent Flows

“…These are evidently new and unique concepts, nonetheless they lead to a complete and symmetric set of transport equations (Eqs. 4-6) for u'v', u' 2 and v' 2 , and agree quite well with DNS results as shown above and in previous works [9,10,16].…”

“…It is the dissipation (du' 2 /dy+) or the gradient profiles that scale with the Reynolds number in an asymmetrical but ordered manner. This bears some implications toward the internal organization processes in turbulence as predicated by the global constraints and transport dynamics, and also renders possible computations of the full flow structure starting from a reference profile [9,10].…”