Extended Mixing-Length Applications to Compressible Turbulent Boundary Layers
NUMERICAL calculations based on the compressible boundary-layer equations and an integral form of the kinetic-energy-of-turbulence (IKET) equation are presented for a variety of conditions. The addition of the IKET equation permits the streamwise computation of an additional dependent variable normally taken as an empirical constant in conventional mixing-length formulations. This socalled extended mixing-length hypothesis is not new, having been developed originally by McDonald and Camarata1 and applied by Chan. 2 Examples given include relaminarization, adverse and favorable pressure gradients, acoustic-energyinduced transition, and surface roughness. The extended mixing-length hypothesis is shown to be considerably more flexible than conventional mixing-length turbulence models. Contents Theoretical Method The extended mixing-length hypothesis represents a turbulence modeling approach that lies between the conventional mixing-length formulations (zero-equation model of turbulence) and the one-equation turbulence models. The IKET equation was obtained by integrating across the boundary layer an equation for the transport of turbulent kinetic energy. The resulting IKET equation contains a source term representing the absorption of incident acoustic energy, the driving force for the transition process. A two-layer model of the turbulent boundary layer was adopted following the classical inner-outer region approach in which separate functional relationships are prescribed in each region, with continuity of the functions between each region. The innerregion damping expressions were taken from Wolfshtein3 and are similar in form to the well-known expression of Van Driest. In this analysis, the ratio of the outer-region length scale to the boundary-layer thickness (A/6) was the additional parameter calculated by streamwise solution of the IKET equation. To improve the fidelity of the IKET analysis, the values of some turbulence model constants were altered to reflect roughness and pressure gradient effects. The alterations can be expressed consistently in terms of K, the von Karman constant, and A^ft the effective value of the van Driest damping constant in the inner region. Pressure gradient effects on A^ and K were assessed using equilibrium expressions in conjunction with a lag analysis.