Multiscaling in the Presence of Indeterminacy: Wall-Induced Turbulence

Abstract

This paper provides a multiscale analytical study of steady incompressible turbulent flow through a channel of either Couette or pressure-driven Poiseuille type. Mathematically, the paper’s two most novel features are that (1) the analysis begins with an underdetermined singular perturbation problem, namely the Reynolds averaged mean momentum balance equation, and (2) it leads to the existence of an infinite number of length scales. (These two features are probably linked, but the linkage will not be pursued.) The paper develops a credible assumption of a mathematical nature which, when added to the initial underdetermined problem, results in a knowledge of almost the complete layer (scaling) structure of the mean velocity and Reynolds stress profiles. This structure in turn provides a lot of other important information about those profiles. The possibility of almost-logarithmic sections of the mean velocity profile is given special attention. The sense in which the length scales are asymptotically proportional to the distance from the wall is determined. Most traditional theoretical analyses of these wall-bounded flows are based ultimately on either the classical overlap hypothesis, mixing length concepts, or similarity arguments. The present paper avoids those approaches and their attendant assumptions. Empirical data are also not used, except that the Reynolds stress takes on positive values. Instead, reasonable criteria are proposed for recognizing scaling layers in the flow, and they are then used to determine the scaling structure and much more information.