Wall shear stress and wall heat flux in a supersonic turbulent boundary layer

Abstract

We report the characteristics of wall shear stress (WSS) and wall heat flux (WHF) from direct numerical simulation (DNS) of a spatially developing zero-pressure-gradient supersonic turbulent boundary layer at a free-stream Mach number M∞ = 2.25 and a Reynolds number Reτ = 769 with a cold-wall thermal condition (a ratio of wall temperature to recovery temperature Tw/Tr = 0.75). A comparative analysis is performed on statistical data, including fluctuation intensity, probability density function, frequency spectra, and space–time correlation. The root mean square fluctuations of the WHF exhibit a logarithmic dependence on Reτ similar to that for the WSS, the main difference being a larger constant. Unlike the WSS, the probability density function of the WHF does not follow a lognormal distribution. The results suggest that the WHF contains more energy in the higher frequencies and propagates downstream faster than the WSS. A detailed conditional analysis comparing the flow structures responsible for extreme positive and negative fluctuation events of the WSS and WHF is performed for the first time, to the best of our knowledge. The conditioned results for the WSS exhibit closer structural similarities with the incompressible DNS analysis documented by Pan and Kwon [“Extremely high wall-shear stress events in a turbulent boundary layer,” J. Phys.: Conf. Ser. 1001, 012004 (2018)] and Guerrero et al. [“Extreme wall shear stress events in turbulent pipe flows: Spatial characteristics of coherent motions,” J. Fluid Mech. 904, A18 (2020)]. Importantly, the conditionally averaged flow fields of the WHF exhibit a different mechanism, where the extreme positive and negative events are generated by a characteristic two-layer structure of temperature fluctuations under the action of a strong Q4 event or a pair of strong oblique vortices. Nevertheless, we use the bi-dimensional empirical decomposition method to split the fluctuating velocity and temperature structures into four different modes with specific spanwise length scales, and we quantify their influence on the mean WSS and WHF generation. It is shown that the mean WSS is mainly related to small-scale structures in the near-wall region, whereas the mean WHF is associated with the combined action of near-wall small-scale structures and large-scale structures in the logarithmic and outer regions.