A recently developed explicit algebraic Reynolds stress model (EARSM) by Grigoriev et al. ["A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation," Phys. Fluids 25(10), 105112 (2013)] and the related differential Reynolds stress model (DRSM) are used to investigate the influence of homogeneous shear and compression on the evolution of turbulence in the limit of rapid distortion theory (RDT). The DRSM predictions of the turbulence kinetic energy evolution are in reasonable agreement with RDT while the evolution of diagonal components of anisotropy correctly captures the essential features, which is not the case for standard compressible extensions of DRSMs. The EARSM is shown to give a realizable anisotropy tensor and a correct trend of the growth of turbulence kinetic energy K, which saturates at a power law growth versus compression ratio, as well as retaining a normalized strain in the RDT regime. In contrast, an eddy-viscosity model results in a rapid exponential growth of K and excludes both realizability and high magnitude of the strain rate. We illustrate the importance of using a proper algebraic treatment of EARSM in systems with high values of dilatation and vorticity but low shear. A homogeneously compressed and rotating gas cloud with cylindrical symmetry, related to astrophysical flows and swirling supercritical flows, was investigated too. We also outline the extension of DRSM and EARSM to include the effect of non-homogeneous density coupled with "local mean acceleration" which can be important for, e.g., stratified flows or flows with heat release. A fixed-point analysis of direct numerical simulation data of combustion in a wall-jet flow demonstrates that our model gives quantitatively correct predictions of both streamwise and cross-stream components of turbulent density flux as well as their influence on the anisotropies. In summary, we believe that our approach, based on a proper formulation of the rapid pressure-strain correlation and accounting for the coupling with turbulent density flux, can be an important element in CFD tools for compressible flows.

A direct solution to a tensorial equation which constitutes a basis for explicit algebraic Reynolds stress models is derived. We consider equations linear and quasilinear in the strain tensor and show how the independent tensor groups emerge. Solution of an extended model with a linearly coupled active scalar, governed by a linear in anisotropy tensor equation, is also outlined.

An explicit algebraic model (EARSM) for variable denstiy turbulent flow developed by Grigoriev et al. [Phys. Fluids (2015)] is revisited here. We apply it to a quasi one-dimensional nozzle flow, a wall-jet flow with combustion and large density variation and a supercritical flow of carbon dioxide with heat transfer and buoyancy. It is confirmed that the coupling between strong mean density gradient due to high speed, heat release or thermodynamic variations and the 'local mean acceleration' of the flow produces strong turbulent density and heat fluxes, which strongly affect the turbulence. The possible calibration branches are identified and analyzed. We show that a simple and unified calibration of the model gives good predictions for all cases considered. Therefore, the model is a reliable tool for the computation of compressible flows with large density variation.

The explicit algebraic Reynolds stress model of Grigoriev et al. ["A realizable explicit algebraic Reynolds stress model for compressible turbulent flow with significant mean dilatation," Phys. Fluids 25, 105112 (2013)] is extended to account for the turbulent density flux in variable density flows. The influence of the mean dilatation and the variation of mean density on the rapid pressure-strain correlation are properly accounted for introducing terms balancing a so-called "baroclinic" production in the Reynolds stress tensor equation. Applying the weak-equilibrium assumption leads to a self-consistent formulation of the model. The model together with a K - ω model is applied to a quasi-one-dimensional plane nozzle flow transcending from subsonic to supersonic regimes. The model remains realizable with constraints put on the model parameters. When density fluxes are taken into account, the model is less likely to become unrealizable. The density variance coupled with a "local mean acceleration" also can influence the model acting to increase anisotropy. The general trends of the behaviour of the anisotropy and production components under the variation of model parameters are assessed. We show how the explicit model can be applied to two- and three-dimensional mean flows without previous knowledge of a tensor basis to obtain the general solution. Approaches are proposed in order to achieve an approximate solution to the consistency equation in cases when analytic solution is missing. In summary, the proposed model has the potential to significantly improve simulations of variable-density flows.

An explicit algebraic and differential Reynolds stress models (EARSM and DRSM) are used to investigate the influence of homogeneous shear and compression on the behaviour of turbulence in the limit of rapid distortion theory (RDT). EARSM is shown to give realizable results and to preserve RDT regime, unlike the eddy-viscosity model (EVM). The DRSM version of our model is in reasonable agreement with RDT theory. 

The explicit algebraic Reynolds stress model of Wallin and Johansson [J. Fluid Mech. 403, 89 (2000)] is extended to compressible and variable-density turbulent flows. This is achieved by correctly taking into account the influence of the mean dilatation on the rapid pressure-strain correlation. The resulting model is formally identical to the original model in the limit of constant density. For two-dimensional mean flows the model is analyzed and the physical root of the resulting quartic equation is identified. Using a fixed-point analysis of homogeneously sheared and strained compressible flows, we show that the new model is realizable, unlike the previous model. Application of the model together with a K - omega model to quasi one-dimensional plane nozzle flow, transcending from subsonic to supersonic regime, also demonstrates realizability. Negative "dilatational" production of turbulence kinetic energy competes with positive "incompressible" production, eventually making the total production negative during the spatial evolution of the nozzle flow. Finally, an approach to include the baroclinic effect into the dissipation equation is proposed and an algebraic model for density-velocity correlations is outlined to estimate the corrections associated with density fluctuations. All in all, the new model can become a significant tool for CFD (computational fluid dynamics) of compressible flows.