Asymptotic Expansion, Symmetry Groups, and Invariant Solutions of Laminar and Turbulent Wall-Bounded Flows

Abstract

Investigating analytic solutions of wall-bounded flows such as the Blasius or the Falkner-Skan solution for laminar flows and the logarithmic law of the wall for turbulent flows it is demonstrated that symmetry groups and particularly scaling groups are essential for their derivation. Both sub-models for laminar and turbulent wall-bounded flows, i.e. Prandtl's boundary layer equations and the multi-point correlation equations, are derived from the Navier-Stokes equations. The latter have significantly different symmetry properties compared to the former sub-models. Essential to both laminar and turbulent sub-models is that they admit two scaling groups while the Navier-Stokes equations only admit one scaling group. It is important for the understanding of both sub-models that their admitted two scaling groups have essentially different physical and mathematical properties. The scaling groups of boundary layer theory admit independent scaling in streamwise and wall-normal direction. Reynolds number is a fundamental parameter in the equations. In contrast the multi-point correlation equations for turbulent flows admit essentially the same groups as the inviscid equations of fluid motion. Hence to leader order no Reynolds number dependence is contained in the solutions for turbulent flows.