The approximate similarity between higher-order and second-order stress–velocity cross-spectra and its significance for the convection velocity of Reynolds stress fluctuations

Stress–velocity cross-spectra provide critical insights into the wall turbulence dynamics, where second-order cross-spectra have been used to characterise the amplitude modulation of large-scale motions on smaller scales. Here, we investigate the higher-order stress–velocity cross-spectra. Through theoretical analysis, we derive an exact relationship demonstrating that the difference in convection velocity between streamwise Reynolds normal stress fluctuations ( $r$ ) and streamwise velocity fluctuations ( $u$ ) – termed the $r{-}u$ convection velocity difference – is governed jointly by the second- and fourth-order cross-spectra. A new ‘coherence similarity’ (CS) model is proposed, which reveals an approximate similarity between higher-order and second-order cross-spectra. As a result, the $r{-}u$ convection velocity difference can be explained in terms of second-order cross-spectral properties. Numerical validation confirms that the CS model predicts higher-order cross-spectra and the convection velocity difference accurately. Furthermore, the contours of stress–velocity cross-spectra undergo a structural transition from single-lobe to triple-lobe patterns with increasing wall distance, suggesting the presence of complex space–time coupling between $r$ and $u$ .