Nonlinear Development of an Instability Wave in a Turbulent Wake

Abstract

Recent experimental observations appear to indicate that turbulent mean flow velocity profiles with points of inflection are dynamically unstable with respect to traveling wavy disturbances. As an illustration, the description of the nonlinear development of an instability wave in the turbulent wake behind a thin body is presented on the basis of integrals of the equations of mean flow momentum, kinetic energy, and the time-averaged disturbance kinetic energy, with the turbulent dissipation integrals relegated to phenomenological considerations. The over-all physical mechanisms leading to the streamwise distribution of the mean flow decay and disturbance energy amplification and decay are explained through the disturbance energy production integral and the mean flow and disturbance turbulent-dissipation integrals. The growth of the disturbance wave is limited by (1) the rapidity of turbulent mean flow decay which renders the production integral less spectacular and (2) the prominant role of the turbulent dissipation integral which draws energy away from the disturbance wave to the turbulence. These limitations and the rapidity with which events take place in the streamwise direction essentially distinguishes the instability wave in a turbulent mean flow with inflectional velocity profile from that in a corresponding laminar flow.