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.

The propagation speed Vc of the streamwise velocity fluctuations u' in turbulent channel flows is calculated using direct numerical simulation (DNS) data at four Mach numbers (M=0, 0.8, 2.0, and 3.0). The profiles of Vc are shown to display remarkable similarity at different M. Quantitative models are developed based on a statistical structure called Velocity-Vorticity Correlation Structure (VVCS), defined as the vorticity region most correlated to velocity fluctuations at a fixed location. Good agreement with DNS-measured propagation velocities is obtained throughout the channel and for all M. The result confirms earlier speculation that the near-wall propagation is due to an advection by coherent vortex structures, and validates the concept of the VVCS.

A novel notion of turbulent structure-the local cascade structure-is introduced to study the convection phenomenon in a turbulent channel flow. A space-time cross-correlation method is used to calculate the convection velocity. It is found that there are two characteristic convection speeds near the wall, one associated with small-scale streaks of a lower speed and another with streamwise vortices and hairpin vortices of a higher speed. The new concept of turbulent structure is powerful to illustrate the dominant role of coherent structures in the near-wall convection, and to reveal also the nature of the convection-the propagation of patterns of velocity fluctuations-which is scale-dependent.