# On explicit algebraic stress models for complex turbulent flows

01 Nov 1992-Vol. 93, pp 15481

TL;DR: Explicit algebraic stress models that are valid for three-dimensional turbulent flows in noninertial frames are systematically derived from a hierarchy of second-order closure models.

Abstract: Explicit algebraic stress models that are valid for three-dimensional turbulent flows in noninertial frames are systematically derived from a hierarchy of second-order closure models. This represents a generalization of the model derived by Pope who based his analysis on the Launder, Reece, and Rodi model restricted to two-dimensional turbulent flows in an inertial frame. The relationship between the new models and traditional algebraic stress models -- as well as anistropic eddy visosity models -- is theoretically established. The need for regularization is demonstrated in an effort to explain why traditional algebraic stress models have failed in complex flows. It is also shown that these explicit algebraic stress models can shed new light on what second-order closure models predict for the equilibrium states of homogeneous turbulent flows and can serve as a useful alternative in practical computations.

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TL;DR: This paper presents a method of using deep neural networks to learn a model for the Reynolds stress anisotropy tensor from high-fidelity simulation data and proposes a novel neural network architecture which uses a multiplicative layer with an invariant tensor basis to embed Galilean invariance into the predicted anisotropic tensor.

Abstract: There exists significant demand for improved Reynolds-averaged Navier–Stokes (RANS) turbulence models that are informed by and can represent a richer set of turbulence physics. This paper presents a method of using deep neural networks to learn a model for the Reynolds stress anisotropy tensor from high-fidelity simulation data. A novel neural network architecture is proposed which uses a multiplicative layer with an invariant tensor basis to embed Galilean invariance into the predicted anisotropy tensor. It is demonstrated that this neural network architecture provides improved prediction accuracy compared with a generic neural network architecture that does not embed this invariance property. The Reynolds stress anisotropy predictions of this invariant neural network are propagated through to the velocity field for two test cases. For both test cases, significant improvement versus baseline RANS linear eddy viscosity and nonlinear eddy viscosity models is demonstrated.

1,159 citations

### Cites background from "On explicit algebraic stress models..."

...These nonlinear models have not gained widespread usage because they do not give consistent performance improvement over the LEVM and often lead to worsened convergence properties (Gatski & Speziale 1993)....

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TL;DR: In this article, an explicit algebraic Reynolds stress turbulence model (EARSM) is presented for both incompressible and compressible three-dimensional wall-bounded turbulent flows, which represents a solution of implicit ARSM equations, where the production to dissipation ratio is obtained as a solution to a nonlinear algebraic relation.

Abstract: 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.

753 citations

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Ansys

^{1}TL;DR: The article gives an overview of the Scale-Adaptive Simulation (SAS) method and the motivation for the formulation of the SAS method is given and a detailed explanation of the underlying ideas is presented.

Abstract: The article gives an overview of the Scale-Adaptive Simulation (SAS) method developed by the authors during the last years. The motivation for the formulation of the SAS method is given and a detailed explanation of the underlying ideas is presented. The derivation of the high-Reynolds number form of the equations as well as the calibration of the constants is provided. The concept of SAS is explained using several generic examples and test cases. In a companion article, the model is applied to more complex industrial-type applications.

724 citations

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TL;DR: The Spalart-Allmaras (SA) one-equation turbulence model as mentioned in this paper was developed for aerodynamic flow simulations and was shown to be quite competitive with advanced nonlinear and Reynolds-stress models and to be much more accurate than the original SA model.

Abstract: Aunie edapproachtosystem-rotationandstreamline-curvatureeffectsintheframeworkofsimpleeddy-viscosity turbulence models is exercised in a range of rotating and curved channel e ows. The Spalart ‐Allmaras (SA) oneequation turbulence model (Spalart, P. R., and Allmaras, S. R., “ A One-Equation Turbulence Model for Aerodynamic Flows,” AIAAPaper 92-0439, 1992 )modie ed in thismanner is shown to bequitecompetitivewith advanced nonlinear and Reynolds-stress models and to be much more accurate than the original SA model and other eddyviscosity models that are widely used for industrial e ow computations. The new term adds about 20% to the computing cost, but does not degrade convergence.

510 citations

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TL;DR: In this article, a new approach to time-dependent Reynolds-averaged Navier-Stokes (RANS) computations and very large-eddy simulations (VLES) is presented in which subgrid scale models are proposed that allow a direct numerical simulation (DNS) to go continuously to a RANS computation in the coarse mesh/infinite Reynolds number limit.

Abstract: Reynolds stress models and traditional large-eddy simulations are reexamined with a view toward developing a combined methodology for the computation of complex turbulent flows. More specifically, an entirely new approach to time-dependent Reynolds-averaged Navier-Stokes (RANS) computations and very large-eddy simulations (VLES) is presented in which subgrid scale models are proposed that allow a direct numerical simulation (DNS) to go continuously to a RANS computation in the coarse mesh/infinite Reynolds number limit. In between these two limits, we have a large eddy simulation (LES) or VLES, depending on the level of resolution. The Reynolds stress model that is ultimately recovered in the coarse mesh/infinite Reynolds number limit has built in nonequilibrium features that make it suitable for time-dependent RANS. The fundamental technical issues associated with this new approach, which has the capability of bridging the gap between DNS, LES and RANS, are discussed in detail. Illustrative calculations are presented along with a discussion of the future implications of these results for the simulation of the turbulent flows of technological importance.

460 citations

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TL;DR: In this paper, the authors present a review of the applicability and applicability of numerical predictions of turbulent flow, and advocate that computational economy, range of applicability, and physical realism are best served by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy k and its dissipation rate ϵ, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour.

Abstract: The paper reviews the problem of making numerical predictions of turbulent flow. It advocates that computational economy, range of applicability and physical realism are best served at present by turbulence models in which the magnitudes of two turbulence quantities, the turbulence kinetic energy k and its dissipation rate ϵ, are calculated from transport equations solved simultaneously with those governing the mean flow behaviour. The width of applicability of the model is demonstrated by reference to numerical computations of nine substantially different kinds of turbulent flow.

11,866 citations

### "On explicit algebraic stress models..." refers methods in this paper

...experimental data of Johnston et al. (1972). The computations were done using law of the wall boundary conditions....

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TL;DR: In this article, the authors developed 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.

Abstract: 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.

3,855 citations

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TL;DR: In this article, a two-equation model and Reynolds stress transport model are developed for turbulent shear flows and tested for homogeneous shear flow and flow over a backward facing step.

Abstract: Turbulence models are developed by supplementing the renormalization group (RNG) approach of Yakhot and Orszag [J. Sci. Comput. 1, 3 (1986)] with scale expansions for the Reynolds stress and production of dissipation terms. The additional expansion parameter (η≡SK/■) is the ratio of the turbulent to mean strain time scale. While low‐order expansions appear to provide an adequate description for the Reynolds stress, no finite truncation of the expansion for the production of dissipation term in powers of η suffices−terms of all orders must be retained. Based on these ideas, a new two‐equation model and Reynolds stress transport model are developed for turbulent shear flows. The models are tested for homogeneous shear flow and flow over a backward facing step. Comparisons between the model predictions and experimental data are excellent.

2,347 citations

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TL;DR: In this paper, the authors examined the modeling of the pressure-strain correlation of turbulent flows from a basic theoretical standpoint with a view toward developing improved second-order closure models and proved that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor.

Abstract: The modeling of the pressure-strain correlation of turbulence is examined from a basic theoretical standpoint with a view toward developing improved second-order closure models. Invariance considerations along with elementary dynamical systems theory are used in the analysis of the standard hierarchy of closure models. In these commonly used models, the pressure-strain correlation is assumed to be a linear function of the mean velocity gradients with coefficients that depend algebraically on the anisotropy tensor. It is proven that for plane homogeneous turbulent flows the equilibrium structure of this hierarchy of models is encapsulated by a relatively simple model which is only quadratically nonlinear in the anisotropy tensor. This new quadratic model - the SSG model - is shown to outperform the Launder, Reece, and Rodi model (as well as more recent models that have a considerably more complex nonlinear structure) in a variety of homogeneous turbulent flows. Some deficiencies still remain for the description of rotating turbulent shear flows that are intrinsic to this general hierarchy of models and, hence, cannot be overcome by the mere introduction of more complex nonlinearities. It is thus argued that the recent trend of adding substantially more complex nonlinear terms containing the anisotropy tensor may be of questionable value in the modeling of the pressure-strain correlation. Possible alternative approaches are discussed briefly.

1,556 citations

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TL;DR: In this article, a simple way to model the pressure-containing correlations which appear in the transport equations for Reynolds stress and heat flux was proposed, which accounts for gravitational effects and the modification of the fluctuating pressure field by the presence of a wall.

Abstract: Proposals are made for modelling the pressure-containing correlations which appear in the transport equations for Reynolds stress and heat flux in a simple way which accounts for gravitational effects and the modification of the fluctuating pressure field by the presence of a wall. The predicted changes in structure are shown to agree with Young's (1975) measurements in a free stratified shear flow and with the Kansas data on the atmospheric surface layer.

1,482 citations