# [PhD Thesis Presentation] -Dongrong Zhang- Spectral Theory of Turbulent Flows

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[Thesis Abstract]

In this doctoral thesis we formulate a spectral theory of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows and use the spectral theory along with dimensional analysis and similarity methods to shed some new theoretical light on several outstanding problems in the interpretation of the extensive empirical data available on such flows. The spectral theory applies to turbulent flows of both constant-density and stratified fluids, and is predicated on the derivation, which we accomplish via a control-volume analysis, of a “spectral link” between the MVPs and the eddy velocity distribution (or, alternatively, the spectrum of turbulent kinetic energy) of the phenomenological theory of turbulence. This spectral link is bijective in that it relates each successive layer of a MVP to, and only to, the corresponding range of the eddy velocity distribution (the buffer layer to the dissipative range, the log layer to the inertial range, and the wake to the energetic range), with the implication that it is possible to infer the internal structure of a turbulent state, as one would by parsing through the successive ranges of the eddy velocity distribution, by parsing instead through the successive layers of the attendant MVP. We use the spectral theory to argue that a number of well-known, and as yet unresolved, disparities in the classical experimental data on friction factor in turbulent plane Couette flows stem from the existence of multiple states of turbulence that differ only at the largest length scales in the flow, corresponding to the energetic range, and are therefore consistent with small-scale universality. We also apply the spectral theory to thermally-stratified plane Couette flows of a type relevant to the atmosphere and conclude that the spectral link remains bijective in the presence of thermal stratification, but the layers of the MVPs are altered by buoyancy. For the intermediate region, which is the only to have been previously theoretically studied, the spectral theory makes predictions in accord with the classical scaling of the Monin-Obukhov similarity theory (MOST); for the other regions (that which is near the wall and that which is far from the wall and extends beyond the intermediate region), the predictions of the spectral theory are in accord with a new set of scaling laws, which we are able to derive on the basis of dimensional analysis and similarity assumptions. We hope that our findings will foster the formulation of better spectral models which might help opening up new prospects of gaining further theoretical insight into turbulence, “the most important unsolved problem of classical physics”

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