Wall shear stress, pressure, and heat flux are of significant importance in engineering applications. In this two-part study, we investigate the compressibility effects on wall shear stress, pressure, and heat flux fluctuations in compressible wall-bounded turbulence by exploiting direct numerical simulation databases. In Paper I, we primarily deal with the one-point statistics, whereas in this second part, we report the effects of compressibility on the frequency spectra, wavenumber-frequency spectra of these flow quantities, and the two-point cross-correlations between them. It is found that the scaling laws of the spectra at low and high frequencies are retained as those of incompressible flows, whereas the spectra intensities at mid frequencies increase with the enhancement of compressibility effects, which is identified to be related to the ever-predominating traveling wave packets. These wave packets are convected downstream at the same velocity of [Formula: see text] as that of pressure fluctuations, higher than that of the streaky structures [Formula: see text] ( Ub the bulk velocity), and enhance the space and time cross correlation between wall shear stress, pressure, and heat flux fluctuations. By extracting the envelopes of the traveling wave packets and inspecting the time and space correlations between the envelopes and the streaky structures, we found that the emergence of traveling wave packets comes later than the streaky structures, both in time and space. Based on these observations, we provide a depiction of the physical processes regarding the formation and evolution of the traveling wave packets.

In this paper, a direct numerical simulation is performed to study the wake structures of a micro-ramp vortex generator in the hypersonic flow. The whole evolution process of vortical structures of the wake flow has been carefully analyzed. First, the scaling laws for the profiles of mean velocities and fluctuations have been explored. The results show that the scaling laws based on the supersonic experiments can be directly applied in the hypersonic wake flow with a few discrepancies near the wall. Then, the detailed structures of arc-like vortices are examined, and the legs of vortices are found not being connected with each other. No full vortex ring is detected in the present simulation. At the downstream of the wake flow, the intensity of arc-like vortices has been augmented when passing the separation shock. The results reveal that a vortex train composed of arc-like vortices and shear layer is formed. With the help of the proper orthogonal decomposition, the impact of arc-like vortices on the heat flux has been revealed. The alternative positive and negative strips in the streamwise direction suggest the impinging process of arc-like vortices. The frequencies of the corresponding modes further prove that the strips are the footprint of arc-like vortices.

No abstract

The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$ , with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

This two-part study investigates the effects of Mach number and wall temperature on the statistics of wall shear stress, pressure, and heat flux fluctuations in compressible wall-bounded turbulence. In the first part, we focus on their one-point statistics, including the root mean square (r.m.s.), skewness factor (third-order moment), flatness factor (fourth-order moment), and their correlations. By exploiting the direct numerical simulation databases, we found that the r.m.s. of the streamwise wall shear stress and pressure, the skewness factor of all the flow quantities considered, and the flatness factor of streamwise wall shear stress monotonically vary with the friction Mach number ([Formula: see text]), while for the rest, the wall heat flux and global temperature parameters should be taken into account as well for a monotonic trend of variation. The correlation coefficients between wall shear stress, pressure, and heat flux fluctuations increase with the Mach number [Formula: see text], suggesting the underlying interactions between dynamic and thermodynamic processes. The distributions of spectra and probability density functions indicate that the increased correlation is induced by the highly intermittent traveling wave packets among the streaky structures, as reflected by the “double-peak” feature of the spectra that gradually emerges with the increasing compressibility effects. The probability density distribution also manifests the alteration of the occurrence of extreme events caused by these structures. By accordingly decomposing the fluctuations with cutoff filtering, it is found that the root mean squares of streamwise wall shear stress and heat flux fluctuations related to the streaky structures are Mach number-independent, while those related to the traveling wave packets monotonically increase with the friction Mach number.