Abstract
We show that the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law’—can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also show that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.
Highlights
The mean-velocity profile (MVP) of a wall-bounded turbulent flow is the function u(y) that gives the mean velocity of the flow, u, at any given distance to the wall, y [1,2,3]
Known as the law of the wall, the defect law and the log law, the classic laws underly the customary division of MVPs into layers, which in turn informs the very way in which MVPs
That formula for τt may be combined with the equation of momentum balance, τt + ρνu = τw(1 − y/δ) [2,3] (where ρνu is the viscous shear stress and τw(1 − y/δ) is the total shear stress), to obtain an expression for u that links the MVPs to the spectrum of turbulent energy: yu 1 − yy−1 + cvy, (4.1)
Summary
The mean-velocity profile (MVP) of a wall-bounded turbulent flow is the function u(y) that gives the mean (time-averaged) velocity of the flow, u, at any given distance to the wall, y [1,2,3]. The classic laws have been shown to be consistent with models of wall-bounded turbulent flows, notably the attached-eddy hypothesis [4,5]. E(k), which may be readily measured experimentally, represents the way in which turbulent energy is allotted among eddies of different sizes in a flow. As it turns out, a derivation of the classic laws can be completed, starting from the spectral analogues and the spectral link, if E(k) satisfies a sufficient condition that spells out the general scope of viscous and finite-domain effects in the energetics of wall turbulence. Our task will be to identify, and put to the empirical test, that sufficient condition
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