Theory of pressure-strain-rate correlation for Reynolds-stress turbulence closures. Part 1. Off-diagonal element

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Abstract
A theoretical calculation is made of (an off-diagonal element of) the pressure–strain-rate term ρ−10〈p[∇u + (∇u)T]〉 for a simple turbulent shear flow at high Reynolds number. This calculation is described as follows. (1) An expression for the pressure–strain-rate term is analytically derived in terms of measurable quantities (velocity spectra) - this derivation makes use of a cumulant discard. (2) It is proved that, to the lowest order in the spectral anisotropy, the (nonlinear part of) the pressure-strain-rate term is linearly proportional to the Reynolds stress. (3) A formula is derived for the constant of this proportionality (the Rotta constant) in terms of arbitrary velocity spectra. (4) This formula is used to analytically calculate Rotta's constant, Cxz, for a class of models of velocity spectra (the variation of Rotta's constant caused by variations in the spectral shapes is examined). (5) It is found that Cxz is surprisingly insensitive to the large-wavelength part of the spectrum. This insensitivity suggests that Cxz should not vary much from one turbulence application to another provided that the Reynolds number is very large. However, it is also shown that Cxz is unexpectedly sensitive to the short-wavelength part of the spectrum, and varies with Reynolds number when the latter is less than about 30.The calculation is based on a straightforward solution of the Navier–Stokes equation to obtain formal expressions for u and p. These expressions are then used to write the pressure–strain-rate in terms of a two-time fourth-order velocity correlation. The latter correlation is evaluated by a standard cumulant discard. Simplifying assumptions of the calculation are that average quantities vary little in space and time, and that the mean flow are unidirectional. These simplifications are made in order to emphasize the method of calculation and its details.

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