An attempt has been made to find a new mixing-length model which will account for the effects of compressibility on the turbulence. Through an analysis of the Reynolds-stress formula and following the ideas of Prandtl’s original mixing-length hypothesis, the effort has essentially been directed to classes of compressible flow problems where there exist large changes of density and Mach number. The new mixing-length model which has been found here is a generalization to include gradients of Mach number, pressure and density; and the local parameters values, such as speed of sound, density, Mach number, ratio of specific heats and a new parameter S which is introduced in the new turbulent transport formula. The new parameter S acts like an effective turbulent Schmidt Number for mixtures of gases or a turbulent Prandtl Number for a homogeneous gas.

The new turbulent model and Prandtl’s original model are both applied to six test cases including: tangential slot injection problems, the very complex cases of shock-wave/turbulent shear-layer and boundary-layer interactions, and high Mach number turbulent boundary layer flows over a flat plate. The effective numerical method of flux-splitting with Roe’s scheme has been adopted in the present work. The results are all compared with experimental data. The predictions with the new mixing-length model are generally in good agreement with the measured data. The predictions with Prandtl’s original model have some discrepancies, especially for heated injection and higher Mach number turbulent boundary layers. The inability of Prandtl's model to predict the spreading rate of the free shear layer in slot injection into supersonic flows and higher Mach number turbulent boundary layer flows is due to the occurrence of large density changes which are not accounted for in the original Prandtl model. The new mixing-length model obtained here is the first to explicitly include these effects, and the results obtained show that these effects are important.