Derivation and analysis of near wall modelsfor channel and recirculating flows

Abstract

The problem of predicting features of turbulent flows occurs in many applications such as geophysical flows, turbulent mixing, pollution dispersal, and even in the design of artificial hearts. One promising approach is large eddy simulation (LES), which seeks to predict local spatial averages u of the fluid's velocity u. In some applications, the LES equations are solved over moderate time intervals and the core difficulty is associated with modeling near wall turbulence in complex geometries. Thus, one important problem in LES is to find appropriate boundary conditions for the flow averages which depend on the behavior of the unknown flow near the wall. Inspired by early works of Navier and Maxwell, we develop such boundary conditions of the form u ¯ ⋅ n = 0 and β ( δ , Re ⁡ , | u ¯ ⋅ τ | ) u ¯ ⋅ τ + 2 Re ⁡ − 1 n ⋅ D ( u ¯ ) ⋅ τ = 0 on the wall. We derive effective friction coefficients β appropriate for both channel flows and recirculating flows and study their asymptotic behavior as the averaging radius S → 0 and as the Reynolds number Re → oc. In the first limit, no-slip conditions are recovered. In the second, free-slip conditions are recovered. Our goal herein is not to develop new theories of turbulent boundary layers but rather to use existing boundary layer theories to improve numerical boundary conditions for flow averages.