Quasisteady quasihomogeneous description of the scale interactions in near-wall turbulence

Abstract

By introducing a notion of an ideal large-scale filter, a formal statement is given of the hypothesis of the quasi-steady quasi-homogeneous nature of the interaction between the large and small scales in the near-wall part of turbulent flows. This made the derivations easier and more rigorous.A method is proposed to find the optimal large-scale filter by multi-objective optimization, with the first objective being a large correlation between large-scale fluctuations near the wall and in the layer at a certain finite distance from the wall, and the second objective being a small correlation between the small scales in the same layers. The filter was demonstrated to give good results. Within the quasi-steady quasi-homogeneous theory expansions for various quantities were found with respect to the amplitude of the large-scale fluctuations. Including the higher-order terms improved the agreement with numerical data. Interestingly, it turns out that the quasisteady quasi-homogeneous theory implies a dependence of the mean profile log-law constants on the Reynolds number. The main overall result of the present work is the demonstration of the relevance of the quasi-steady quasi-homogeneous theory for near-wall turbulent flows.

A large-scale filter for applications of QSQH theory of scale interactions in near-wall turbulence

“…The comparisons are shown in figure 2. The agreement for α is very good, while γ 1 discrepancy is of the order of 10%, which is also the degree of discrepancy reported in Zhang and Chernyshenko (2016) for other comparisons. While, obviously, at least a part of this 10% can be due to the truncation of the Taylor expansion, the QSQH theory itself is approximate only, as we will discuss in section 3.…”

“…For example, in (4) the amplitude modulation contribution to α is proportional to U(y) and the scale modulation contribution is proportional to y U y d d , which means that their ratio tends to unity as the wall is approached. For 1 g the scale-modulation mechanism is responsible for the section with negative slope of γ 1 (y), where this mechanism overcomes the contribution of the amplitude modulation (see figure 11 and the related discussion in Zhang and Chernyshenko (2016)). Therefore, any considerations based on the QSQH theory should always take into account the scale modulation element of the theory.…”

“…For details of such mathematical derivations the reader is referred to Zhang and Chernyshenko (2016).…”

“…The properties of this filter also determine the results of analytic derivations and qualitative analysis on the basis of the QSQH theory, and how easy the analytic derivations and qualitative analysis will be. In Zhang and Chernyshenko (2016) these two questions were separated. The properties of the filter sufficient for analytic derivations and qualitative analysis were specified as an essential component of the QSQH theory, without specifying the filter exactly.…”

Multi-scale interactions in a compressible boundary layer

“…That study demonstrated that the turbulence statistics of the near-wall small scales remained universal, in the above sense, only up to y + ≈ 30-50, depending on the turbulence correlation in question. Contemporaneously, Chernyshenko et al [42], Zhang & Chernyshenko [43] developed a theoretical "Quasi-Steady -Quasi Homo- . geneous" (QSQH) framework that rests on the quasi-steady hypothesis, assumes the universality of the small-scale motion when scaled by the large-scale shearstress footprints, and configured to predict the response of near-wall turbulence to the Reynolds-number dependent large-scale outer motions.…”