ALBERT

Description: A theoretical calculation is made of (the diagonal elements of) pressure-strain-rate calculation Po 1〈p[Vu + (Vu)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. It is found that Rotta's expression for [formula omitted] 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. © 1982, Cambridge University Press. All rights reserved.

Description: A theoretical calculation is made of the decay of turbulence energy in the presence of coherent internal gravity waves of various intensities. The wave-turbulence interaction considered is energy production by wave shear. The production term (stress) is calculated by a second-order closure, with temperature fluctuations accounted for by buoyancy subrange theory. This formalism applies to large or small turbulence Froude number, both extremes of which are often encountered in experimental turbulence decay. The theoretical turbulence decay is shown to be a universal function of the wave shear (strain rate) and wave frequency provided that the energy is expressed in terms of the buoyancy wavenumber &B, and time is expressed in terms of N, the Brunt—Vaisala frequency. With the amplitude of wave shear characterized by a gradient Richardson number Ri0, the turbulence decay is found to undergo a sudden transition from rapid decay to a much slower oscillating decay when Ri0is less than about 0.4. The transition time occurs at about t « 2nN-1. If Ri0exceeds 0.8 the rate of decay exceeds that of a neutral fluid. A transition in turbulence decay was observed in experiments by Dickey & Mellor (1980). It might explain the continued presence of turbulence in dynamically stable regions of oceans or atmosphere. The theory is compared in much detail with the Dickey & Mellor experiment. A briefer comparison is also made with other experiments, and with previous calculations of turbulence maintenance by steady mean shear. A simple explanation is proposed of why a transition is observed in a vertical grid experiment but not in horizontal grid experiments. © 1984, Archives Européenes de Sociology. All rights reserved.

Description: Although models of the pressure-strain term explain many features of nearly uniform homogeneous shear flows, a discrepancy remains (Leslie 1980). It is suggested that the discrepancy is caused by use of an empirical expression for the fluctuation part of the pressure-strain term, the part usually denoted by ϕij, 1. The discrepancy is eliminated by replacement of the empirical ϕij1a recent theoretical expression. Relatedly, the Launder, Reece & Rodi (1975) model for the mean-field part ϕij2isshown to be a good approximation for both a strongly and weakly sheared flow. This model of ϕij2whencombined with the theoretical ϕij1is found to provide an explanation for experiments of both Champagne, Harris & Corrsin (1970) and Harris, Graham & Corrsin (1977). Full correction requires that deviations from local isotropy be accounted for. Special emphasis is given to a theoretical demonstration that the pressure-strain term does not cause a retun to isotropy but, rather, it resists large anisotropy-a weaker effect. © 1985, Cambridge University Press. All rights reserved.

Description: The influence of buoyancy on the pressure-strain term is calculated approximately by an analytical theory. It is shown that the buoyancy contribution to (ϕijϕji)1the fluctuation part of the pressure-strain term, is approximately equal to the buoyancy contribution which comes from the mean-field part of the pressure-strain term, provided that the mean buoyancy does not vary rapidly in space or time. The latter, but not the former, buoyancy contribution was previously obtained by Launder (1975) and by Zeman & Lumley (1976). Both contributions are shown to be accounted for by use of a single numerical coefficient CθThe value of Cθpredicted from purely theoretical considerations is 0.7, and a value determined from an experiment is 0.9. The theoretical method has some generality and can be applied to higher than second-order correlations of velocity and temperature fluctuations. © 1986, Cambridge University Press. All rights reserved.

Description: A calculation is made of the turbulent transport terms (third moments) that occur in the Reynolds stress equation for buoyant and/or sheared fluids. This calculation is based on neglect of a two-time fourth-order cumulant – a weaker approximation than neglect of the usual single-time fourth-order cumulant. The previously used eddy-damping assumption for single point moments is avoided. This assumption is then examined critically. Comparison is afterward made between the turbulent transport terms derived here and those derived previously by the eddy-damping method, and between the respective derivations. Also the dissipation of third moments is calculated. The calculation is formally limited to mean quantities which vary but slowly in space and time, and to small anisotropy.

Description: A theoretical calculation is made of (an off-diagonal element of) the pressure-strain-rate term Po-1⟨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. © 1981, Cambridge University Press. All rights reserved.

Description: A theoretical expression for the slow part (the nonlinear fluctuation part) of the pressure—strain rate is compared with simulations of anisotropic homogeneous flows. The purpose is to determine the quantitative accuracy of the theory and to test its qualitative predictions that the generalized Rotta coefficient, a non-dimensionalized ratio of slow term to kinetic energy anisotropy, varies with direction and can be negative (this is counter to isotropy return). Comparisons are made between theoretical and simulated values of the slow term and of the generalized Rotta coefficients. Also compared to simulations is an extension of the theory to account for non-stationary turbulence fields. The implication of the comparison for two-point closure theories and for Reynolds stress modelling is pointed out. © 1989, Cambridge University Press. All rights reserved.