Turbulent shear-layer scaling in the limit of infinite Reynolds number derived from the unsteady equations of motion

By considering the time-mean motion, an asymptotic description of turbulent shear flows in the limit of high Reynolds numbers, conveniently carried out by adopting the method of matched asymptotic expansions, is inherently based on (a few) assumptions regarding the essential physical properties of the flow in the respective layer considered. Although this procedure has substantiated some classical results, as, for instance, the logarithmic law of the wall, it is, nevertheless, not completely satisfactory from a rational point of view: for this reason, we first demonstrate that a rigorous application of the matching principle alone is not capable of deducing the aforementioned wall law in the case of boundary layer flows by taking into account the widely accepted presumption of the so-called ‘inviscid’ nature of the fully turbulent main portion of the shear layer. Therefore, to some extent, the usual approaches turn out to be incomplete, for (i) the motivation for the assumptions commonly employed seems not stringent and, unfortunately, (ii) are, strictly speaking, insufficient for developing a complete asymptotic theory. To be more precise, assumptions that are usually expressed by dimensional considerations can not adequately enter the matching conditions derived from the non-dimensional form of the governing equations, which, however, is required for an asymptotic analysis. Consequently, the time-mean scaling of the asymptotic structure of the flow is seen to result from an asymptotic analysis of the Navier–Stokes equations by, as the only fundamental assumption, considering an asymptotically slender turbulent shear layer. Amongst others, it is demonstrated briefly how a combination of multiple-scales techniques and matched asymptotic expansions leads to the internal boundary layer splitting. Also, it is outlined how the latter also reflects the time-mean characteristics of the flow, which have already been adopted successfully in former studies, e.g. by using rather conventional mixing length arguments.