Nonlinear RDT theory of near-wall turbulence

A WKB method was recently used to extend rapid distortion theory (RDT) to initially inhomogeneous turbulence strained by irrotational mean flows [S.V. Nazarenko, N. Kevlahan, B. Dubrulle, J. Fluid Mech. 390 (1999) 325]. This theory takes into account the feedback of turbulence on the mean flow, and it was used by Nazarenko et al. to explain the effect of strain reduction caused by turbulence observed by Andreotti et al. [B. Andreotti, S. Douady,Y. Couder, in: O. Boratav, A. Eden, A. Erzan (Eds.), Turbulence Modeling and Vortex Dynamics, Proceedings of a Workshop held at Istanbul, Turkey, 2–6 September 1996, pp. 92–108]. In this paper, we develop a similar WKB RDT approach for shear flows. We restrict ourselves to problems where the turbulence is small-scale with respect to the mean flow length-scale and turbulence vorticity is weak compared to the mean shear. We show that the celebrated log-law of the wall exists as an exact analytical solution in our model if the initial turbulence vorticity (debris of the near-wall vortices penetrating into the outer regions) is statistically homogeneous in space and shortly correlated in time. We demonstrate that the main contribution to the shear stress comes from very small turbulent scales which are close to the viscous cut-off and which are elongated in the stream-wise direction (streaks). We also find that anisotropy of the initial turbulent vorticity changes the scaling of the shear stress, but leaves the log-law essentially unchanged.