Hierarchical random additive model for wall-bounded flows at high Reynolds numbers

Abstract

High Reynolds number wall-bounded flows involve dynamically important scales that cover many orders of magnitude in time and space. Fluid motions that dominate momentum transport also cover wide ranges of scales and all of these scales must be accounted for when modeling turbulence near walls. In this review, we summarize recent developments as related to the so-called hierarchical random additive process (HRAP), which models the near-wall momentum-carrying parts of turbulence by exploiting the self-similarity and the tree-like hierarchical organization of attached eddies. The HRAP model follows Townsend (1980 The Structure of Turbulent Shear Flow) and models boundary layer flows as collections of space-filling, self-similar, wall-attached eddies. However, instead of invoking specifically-shaped typical eddy, HRAP isolates the basic scaling properties of the phenomena by parameterizing contributions of near-wall eddies to generic flow quantities using random addends, a superposition of which leads to the real observable (fluid velocity, wall-shear stress, etc). This compact and relatively simple representation proves to be quite useful, leading to scaling predictions of many turbulence statistics. So far, the HRAP model has been used to provide estimates for scalings of single- and two-point velocity central moments, single- and two-point moment generating functions of velocity fluctuations, and variance of wall-shear stress fluctuations as a function of Reynolds number.