Theory of the pressure–strain rate. Part 2. Diagonal elements

Abstract

A theoretical calculation is made of (the diagonal elements of) pressure-strain-rate calculation ρ0−1〈p[∇u + (∇u)T]〉 for a simple turbulent shear flow. This calculation parallels a previous calculation of the off-diagonal element. The calculation is described as follows. (1) Beginning with the Navier-Stokes equation, an expression for the (diagonal) pressure-strain-rate term is derived analytically in terms of measurable quantities (velocity spectra) - this derivation makes use of a cumulant discard. (2) It is proved that, to lowest order in the spectral anisotropy, the diagonal pressure-strain-rate term is linearly proportional to the diagonal Reynolds-stress elements. (3) A formula is derived for the proportionality constants (Rotta constants) in terms of arbitrary spectra. (4) This formula is used to calculate theoretically the numerical value of Rotta's constant Cii for models of velocity spectra (the variation of Cii with variations of spectral shapes and of Reynolds number are also determined). (5) Deficiencies and limitations of Rotta's model are identified and discussed.It is found that Rotta's expression for 2ρ0−1〈p∂ui/∂i〉 is only valid for special spectra. Surprisingly large deviations of Rotta's expression from theory are found for a more complex spectra thought to be typical of simple shear flow. In addition, it is found that Cxz is intrinsically and quantitatively different from Cii because the latter depends importantly on the large-wavenumber part of the spectrum (the inertial subrange) whereas the former does not. The numerical ratio Czz/Cxz is calculated theoretically and shown to be about 2 for the zero-moment model. It is concluded that a linear term in the stress anisotropy as proposed by Rotta does not always exist. The deviation of Rotta's model from theory is understood by distinguishing between the spectral anisotropy and the stress anisotropy.For the zero-moment spectral model, where the Rotta relation is valid, it is found that Cii varies significantly with large Reynolds number but is rather insensitive to the large-wavelength part of the spectrum.