Nonlinear RDT theory of near-wall turbulence

A WKB method was recently used to extend rapid distortion theory (RDT) to i nitially 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 ct al. to explain the effect of strain reduction c aused 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 S eptember 1996, pp. 92-108]. In this paper, we develop a similar WKB RDT app roach for shear flows. We restrict ourselves to problems where the turbulen ce 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 penetrat ing into the outer regions) is statistically homogeneous in space and short ly correlated in time. We demonstrate that the main contribution to the she ar stress comes from very small turbulent scales which are close to the vis cous cut-off and which are elongated in the stream-wise direction (streaks) . We also find that anisotropy of the initial turbulent vorticity changes t he scaling of the shear stress, but leaves the log-law essentially unchange d. (C) 2000 Elsevier Science B.V. All rights reserved.