Turbulence-Model Predictions for Turbulent Boundary Layers

A set of turbulence model equations, originally postulated by Saffman, forms the basis of this three-part study of steady turbulent-boundary-layer structure above a flat plate. In one part of the study the turbulence equations are integrated through the viscous sublayer by means of time-marching numerical integration techniques and the constant in the law of the wall is predicted as a function of wall roughness. In a second part, the Van Driest compressible law of the wall is deduced by classical mathematical methods. Then, using the Van Driest law as a wall boundary condition, the model equations are integrated through the entire boundary layer, again by time-marching methods, for a freestream Mach number of 2.96. NCREASINGLY, turbulent-flow studies have concentrated on phenomenologi cal-model equations describing turbulent motion. The most promising approach seeks rate equations which describe evolution of the Reynolds stresses; development of equations of this type has progressed for both low- and highspeed flows.1"5 This paper focuses upon one such set of turbulence-mo del equations originally devised by Saffman 2 and subsequently modified by Wilcox and Alber3 to account for the effects of compressibility; the model equations are based on the hypothesis that transfer of momentum and heat by turbulence can be described by an eddy viscosity which is a function of two turbulence densities, viz, an energy density, e, and a pseudovorticity density, Q. Previous work 2'3 has concentrated mostly on formulation of the nonlinear diffusion equations satisfied by the turbulence densities; included in the present study is some model development which is directed primarily to the boundary conditions imposed at or near a solid boundary. As a result, uncertainties regarding boundary conditions have been eliminated. A concurrent study has been made of the predictions of the model equations for boundary layers on a flat plate under both compressible and incompressible flow conditions. Past studies have left at least two areas of boundary-condition uncertainty. On the one hand, Saffman2 indicated that the most natural boundary conditions for the mean velocity and the turbulence densities follow from matching to the law of the wall. Matching is valid for the incompressible Saffman model because law-of-the-wall behavior near a solid boundary is contained within the equations; however, Wilcox and Alber3 did not generalize the matching concept for compressible flows. On the other hand, for some applications, integration through the sublayer might be necessary (because of, e.g., numerical reasons or sublayer-structure uncertainties). Thus, the question of boundary conditions for the turbulence densities appropriate to a solid boundary was considered briefly by Saffman and in a little more detail by Wilcox and Alber. Although the boundary condition on e is straightforward (it vanishes at solid boundaries), both studies fall short of specifying a boundary condition on the vorticity density.