Spectral link of the generalized Townsend-Perry constants in turbulent boundary layers

Abstract

We propose a first minimal theory for boundary layer turbulence that captures very well the profile of the mean-square velocity fluctuations in the streamwise direction and give a quantitative prediction of the Townsend-Perry constants. Our theory is based on connecting all moments of velocity fluctuations as a function of the distance to the wall with the turbulent energy spectrum. A similar spectral theory was proposed in G. Gioia and P. Chakraborty [Phys. Rev. Lett. 96, 044502 (2006)PRLTAO0031-900710.1103/PhysRevLett.96.044502] to explain the friction factor and the von Kármán law in G. Gioia, N. Guttenberg, N. Goldenfeld, and P. Chakraborty [Phys. Rev. Lett. 105, 184501 (2010)PRLTAO0031-900710.1103/PhysRevLett.105.184501]. We generalized it by including fluctuations in the wall-shear stress and the streamwise velocity. The theoretical predictions for the mean velocity and mean-square fluctuations reproduce the shape of the velocity profiles in the buffer and inertial layer obtained from wind tunnel experiments.