The rapid expansion of a supersonic turbulent flow: role of bulk dilatation

The rapid expansion of a turbulent boundary layer in supersonic flow is studied analytically and experimentally. Emphasis is placed on the effect of bulk dilatation on turbulent fluctuations. The hypotheses made in the analysis are similar to those in the rapid distortion theory and are used to simplify second-order closures. By assuming that the fluctuating velocity is solenoidal an extension of classical subsonic models is proposed. A new variable is defined, which takes into account the mean density variations, and behaves like the Reynolds stress tensor in subsonic flows with weak inhomogeneities and a weak dissipation rate. The results of the analysis are compared with turbulence measurements performed in a supersonic boundary layer subjected to an expansion fan. The proposed approximations describe correctly the evolution of turbulence intensities: bulk dilatation contributes predominantly to the Reynolds stress evolution. The boundary layer is ‘relaminarized’ by the expansion. Downstream of the latter, the layer returns to equilibrium. Measurements show that the turbulence decays slowly in the outer layer and increases rapidly in the inner layer.