Non-universal $k_1^{-1} $ laws in pressure-gradient-driven turbulent wall-bounded flows

The scaling laws for the spectra of streamwise velocity fluctuations in turbulent wall-bounded flows are reformulated with inclusion of the streamwise pressure gradient. These laws indicate that the presence of pressure gradient naturally leads to the so-called incomplete similarity of spectra irrespective of the mean flow acceleration. Interestingly however, the corresponding spectral overlap arguments still lead to the inverse-power variation of the power spectral density of streamwise velocity fluctuations i.e. the $k_1^{-1}$ law. These are, however, the non-universal $k_1^{-1} $ laws arising out of incomplete similarity. Experimental evidence in the literature on pipe and channel flows clearly supports this. Striking experimental evidence is presented in favour of the non-universal $k_1^{-1} $ laws for an accelerating turbulent boundary layer flow. It is observed that the prerequisite condition of ``high Reynolds number'' for having substantial spectral overlap in experiments appears to be remarkably relaxed in the presence of streamwise mean flow acceleration.