Co-spectrum and mean velocity in turbulent boundary layers

Abstract:

Among the most significant phenomenological theories of turbulence, two are commonly singled out: the von Karman-Prandtl logarithmic law for the mean velocity profile and the Kolmogorov hypothesis for the local structure of the turbulent velocity. These two developments have often been regarded as separate given that the log-law is an outcome of restrictive boundary effects on eddy sizes responsible for mixing, while the local structure of turbulence is associated with locally homogeneous and isotropic turbulence far from any boundary. Furthermore, these two theories have stirred significant debate with some stating that the von Karman-Prandtl law should be abandoned and replaced by a power-law when viscosity is small but finite. Here, connections are explored between spectral descriptions of turbulence and the mean velocity profile in the equilibrium layer of wall-bounded flows using a modeled budget for the co-spectral density. Employing a standard model for the wall normal velocity variance and a Rotta-like return-to-isotropy closure for the pressure-strain effects, the co-spectrum describing the momentum flux is derived. The approach establishes a relation between the von Karman, the one-dimensional Kolmogorov and the Rotta constants. Depending on the choices made about small-scale intermittency corrections and how it is introduced into the energy cascade, the logarithmic mean velocity profile or a power-law profile with an exponent that depends on the intermittency correction are derived thereby offering a new perspective on this long standing debate.