Scaling of coherent structures in compressible wall-bounded turbulence

Semi-local scales have been widely used in compressible wall-bounded turbulence, but it is still unclear whether they are applicable to the scaling of coherent structures, especially under conditions of high Mach number and cold wall temperature. By scrutinizing the direct numerical simulation dataset at different Mach numbers and wall temperatures, this paper demonstrates that the coherent structures normalized by semi-local scales are universal in size. In addition to this, we find that the ratios of Kolmogorov scales to semi-local scales are independent of Mach number and wall temperature. Thus, Kolmogorov scales can achieve the same scaling effect as the semi-local scales. The velocity spectra are also compared to verify the current scaling method quantitatively. A method to determine the threshold for the vortex identification criterion is proposed, allowing the same threshold for different cases to obtain vortices of similar size. The scaling of other statistics including turbulent kinetic energy, streamwise Reynolds normal stress, and root mean square of fluctuating vorticity is also investigated. A new velocity scale is proposed based on the total-stress-based transformation for mean streamwise velocity, which can collapse the profiles of these statistics more accurately than the semi-local velocity scale. The present paper demonstrates that through appropriate normalization, the structures and statistics of compressible turbulence become universal, reaffirming the validity of Morkovin's hypothesis even for the present high Mach number and cold wall cases.