An energy spectrum is preliminarily characterized by its mean and standard deviation. In this study, we derive exact expressions for the means and bandwidths of space-time energy spectra at fixed frequencies. The mean wave numbers are used to determine the phase velocities that bridge from temporal spectra to space-time spectra. The bandwidths are used to measure the well-known spectral broadening. We show that phase velocities alone are insufficient to determine the bandwidths of energy spectra. As a result, the cross-spectral method predicts narrower bandwidths of energy spectra. Therefore, in addition to phase velocities, amplitudes are used to rescale the space-time energy spectra, leading to the correct bandwidths. Existing data from direct numerical simulations of turbulent channel flows validate the rescaling approach.

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In this paper, we consider shock reflection problems, occurring in supersonic and hypersonic intake flow under off-design conditions, in which the incident shock wave is disturbed by the lip-generated upstream expansion wave and the reflected shock wave intersects with a downstream cowl-turning deflected shock wave. The expansion wave and deflected shock wave are here generated with the same magnitude of flow deflection angle or turning angle. With the help of shock interaction theory and numerical simulation, the influence of the turning angle of the lip and cowl on the flow structure and the critical conditions for transition between regular reflection and Mach reflection are analysed. It is found that the dual-solution domain is significantly altered by the interference between the expansion wave and shock waves. The flow structure in the condition of Mach reflection is then analysed with a model updated from a previous study. It is shown that the Mach stem height is an increasing function of the turning angle, while the horizontal position of the Mach stem is shifted in the downstream direction for small turning angle and in the upstream direction for large turning angle.

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