The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate E. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows. Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112075001814

Progress in the development of a Reynolds-stress turbulence closure

Some new developments of explicit algebraic Reynolds stress turbulence models (EARSM) are presented. The new developments include a new near-wall treatment ensuring realizability for the individual stress components, a formulation for compressible flows, and a suggestion for a possible approximation of diffusion terms in the anisotropy transport equation. Recent developments in this area are assessed and collected into a model for both incompressible and compressible three-dimensional wall-bounded turbulent flows. This model represents a solution of the implicit ARSM equations, where the production to dissipation ratio is obtained as a solution to a nonlinear algebraic relation. Three-dimensionality is fully accounted for in the mean flow description of the stress anisotropy. The resulting EARSM has been found to be well suited to integration to the wall and all individual Reynolds stresses can be well predicted by introducing wall damping functions derived from the van Driest damping function. The platform for the model consists of the transport equations for the kinetic energy and an auxiliary quantity. The proposed model can be used with any such platform, and examples are shown for two different choices of the auxiliary quantity.

An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows

Second-moment closure: present… and future?

Recent advances on the numerical modelling of turbulent flows

Second-moment turbulence models focus directly on the transport equations for the Reynolds stresses rather than supposing the stress and strain fields to be directly linked via an eddy viscosity. This elaboration enables the effects of complex strains and force fields on the turbulence structure to be better captured. The paper summarizes the principal modelling strategies adopted for the unknown processes in these equations and presents the forms that have been found most useful in engineering calculations. Methods adopted for overcoming significant problems of numerical instability and lack of convergence compared with eddy-viscosity-based schemes are also presented. Applications involving momentum and heat transfer in complex flows are drawn from the advanced technology sectors of the power generation and aircraft industries.

Second-moment closure and its use in modelling turbulent industrial flows

A quasi‐isotropic locally homogeneous model for the two‐point double‐velocity correlation tensor is developed by the application of a plausible set of symmetry restrictions to this tensor. The two‐point correlation is related to the single‐point correlation by three independent functions, and reduces to the nonisotropic single‐point correlation and the isotropic two‐point correlation in the limiting cases. It is shown that the model implies that by the use of three independent microscales (and macroscales) it is possible to describe a matrix of 27 elements of microscales (and macroscales) which describe the nonisotropic field. The model is also used to calculate the dissipation rate of the components of the Reynolds stresses. Finally, a formula which expresses the redistribution of the velocity fluctuations due to pressure fluctuations is developed.

Two‐point correlation model and the redistribution of Reynolds stresses

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Two‐point correlation model and the redistribution of Reynolds stresses