On the turbulent mixing of compressible free shear layers

For over a quarter of a century it has been recognized that turbulent shear flows are dominated by large-scale structures. Yet the majority of models for turbulent mixing fail to include the properties of the structures either explicitly or implicitly. The results obtained using these models may appear to be satisfactory, when compared with experimental observations, but in general these models require the inclusion of empirical constants, which render the predictions only as good as the empirical database used in the determination of such constants. Existing models of turbulence also fail to provide, apart from its stochastic properties, a description of the time-dependent properties of a turbulent shear flow and its development. In this paper we introduce a model for the large-scale structures in a turbulent shear layer. Although, with certain reservations, the model is applicable to most turbulent shear flows, we restrict ourselves here to the consideration of turbulent mixing in a two-stream compressible shear layer. Two models are developed for this case that describe the influence of the large-scale motions on the turbulent mixing process. The first model simulates the average behaviour by calculating the development of the part of the turbulence spectrum related to the large-scale structures in the flow. The second model simulates the passage of a single train of large-scale structures, thereby modelling a significant part of the time-dependent features of the turbulent flow. In both these treatments the large-scale structures are described by a superposition of instability waves. The local properties of these waves are determined from linear, inviscid, stability analysis. The streamwise development of the mean flow, which includes the amplitude distribution of these instability waves, is determined from an energy integral analysis. The models contain no empirical constants. Predictions are made for the effects of freestream velocity and density ratio as well as freestream Mach number on the growth of the mixing layer. The predictions agree very well with experimental observations. Calculations are also made for the time-dependent motion of the turbulent shear layer in the form of streaklines that agree qualitatively with observation. For some other turbulent shear flows the dominant structure of the large eddies can be obtained similarly using linear stability analysis and a partial justification for this procedure is given in the Appendix. In wall-bounded flows a preliminary analysis indicates that a linear, viscous, stability analysis must be extended to second order to derive the most unstable waves and their subsequent development. The extension of the present model to such cases and the inclusion of the effects of chemical reactions in the models are also discussed.