New scaling for compressible wall turbulence

Classical Mach-number (M) scaling in compressible wall turbulence was suggested by van Driest (Van Driest E R. Turbulent boundary layers in compressible fluids. J Aerodynamics Science, 1951, 18(3): 145-160) and Huang et al. (Huang P G, Coleman G N, Bradshaw P. Compressible turbulent channel flows: DNS results and modeling. J Fluid Mech, 1995, 305: 185-218). Using a concept of velocity-vorticity correlation structure (VVCS), defined by high correlation regions in a field of two-point cross-correlation coefficient between a velocity and a vorticity component, we have discovered a limiting VVCS as the closest streamwise vortex structure to the wall, which provides a concrete Morkovin scaling summarizing all compressibility effects. Specifically, when the height and mean velocity of the limiting VVCS are used as the units for the length scale and the velocity, all geometrical measures in the spanwise and normal directions, as well as the mean velocity and fluctuation (r.m.s) profiles become M-independent. The results are validated by direct numerical simulations (DNS) of compressible channel flows with M up to 3. Furthermore, a quantitative model is found for the M-scaling in terms of the wall density, which is also validated by the DNS data. These findings yield a geometrical interpretation of the semi-local transformation (Huang et al., 1995), and a conclusion that the location and the thermodynamic properties associated with the limiting VVCS determine the M-effects on supersonic wall-bounded flows.