Reynolds Stress Equation Research Articles

Predicting Buoyant Shear Flows Using Anisotropic Dissipation Rate Models

This paper examines the modeling of two-dimensional homogeneous stratified turbulent shear flows using the Reynolds-stress and Reynolds-heat-flux equations. Several closure models have been investigated; the emphasis is placed on assessing the effect of modeling the dissipation rate tensor in the Reynolds-stress equation. Three different approaches are considered; one is an isotropic approach while the other two are anisotropic approaches. The isotropic approach is based on Kolmogorov's hypothesis and a dissipation rate equation modified to account for vortex stretching. One of the anisotropic approaches is based on an algebraic representation of the dissipation rate tensor, while another relies on solving a modeled transport equation for this tensor. In addition, within the former anisotropic approach, two different algebraic representations are examined; one is a function of the Reynolds-stress anisotropy tensor, and the other is a function of the mean velocity gradients. The performance of these closure models is evaluated against experimental and direct numerical simulation data of pure shear flows, pure buoyant flows and buoyant shear flows. Calculations have been carried out over a range of Richardson numbers (Ri) and two different Prandtl numbers (Pr); thus the effect of Pr on the development of counter-gradient heat flux in a stratified shear flow can be assessed. At low Ri, the isotropic model performs well in the predictions of stratified shear flows; however, its performance deteriorates as Ri increases. At high Ri, the transport equation model for the dissipation rate tensor gives the best result. Furthermore, the results also lend credence to the algebraic dissipation rate model based on the Reynolds stress anisotropy tensor. Finally, it is found that Pr has an effect on the development of counter-gradient heat flux. The calculations show that, under the action of shear, counter-gradient heat flux does not occur even at Ri = 1 in an air flow.

Prediction of Forced Gas Flows in Circular Tubes at High Heat Fluxes Accompanied by Laminarization.

The application of high heat fluxes to turbulent gas flows causes significant variation of the gas properties due to the large temperature increase, invalidating the use of design relations such as the popular Dittus-Boelter correlation and of conventional turbulence models. To develop turbulence models for such flows, the predictive capabilities of a Reynolds stress equation model (RSM) with turbulence heat flux equations and of a model were examined. We have employed the turbulence stress-thermal energy gradient production, in the pressure-temperature fluctuation gradient correlation, for the turbulence heat flux equations, in order to improve the accuracy of predictions for heat transfer accompanied by laminarization. The model has already been validated for gas flows in annular tubes; in this study its applicability to a circular tube was examined. Validations were performed by comparison of predictions with constant fluid properties at fully-established conditions and to experimental measurements extending to laminarization conditions. The turbulence models both predict laminar and turbulent results and the approximate transition Reynolds number for the fully-established flow at low heat fluxes. The results predicted with both models also agreed well with data on laminarizing flows. Thus, the models presented here are capable of predicting the flow reasonably from low to high heat fluxes.

Turbulence Attenuation by Small Particles in Simple Shear Flows

This paper is concerned with the influence of a dispersed phase on carrier flow turbulence. The carrier flow is assumed to be a simple homogeneous shear, in which the fluid Reynolds stress tensor is independent of spatial location, but in which there is a linear mean shear across the flow. In the shear flow, the Reynolds stress is nondiagonal, and we find extra dissipation terms in the Reynolds stress equations compared with those arising in isotropic turbulence. The source terms are used to develop a simple model to predict changes in turbulence levels in particle-laden shear flows. It is shown that the general effect of particles is to attenuate turbulence. The resulting expression is not explicitly dependent on the shear. The theory is used to predict turbulence attenuation in near-homogeneous particle-laden flows in pipes and channels and is compared with experimental data. [S0098-2202(00)00401-6]

Progress in Favré–Reynolds stress closures for compressible flows

A closure for the compressible portion of the pressure-strain covariance is developed. It is shown that, within the context of a pressure-strain closure assumption linear in the Reynolds stresses, an expression for the pressure-dilatation can be used to construct a representation for the pressure-strain. Additional closures for the unclosed terms in the Favré–Reynolds stress equations involving the mean acceleration are also constructed. The closures accommodate compressibility corrections depending on the magnitude of the turbulent Mach number, the mean density gradient, the mean pressure gradient, the mean dilatation, and, of course, the mean velocity gradients. The effects of the compressibility corrections on the Favré–Reynolds stresses are consistent with current DNS results. Using the compressible pressure-strain and mean acceleration closures in the Favré–Reynolds stress equations an algebraic closure for the Favré–Reynolds stresses is constructed. Noteworthy is the fact that, in the absence of mean velocity gradients, the mean density gradient produces Favré–Reynolds stresses in accelerating mean flows. Computations of the mixing layer using the compressible closures developed are described. Favré–Reynolds stress closure and two-equation algebraic models are compared to laboratory data for the mixing layer. Experimental data from diverse laboratories for the Favré–Reynolds stresses appears inconsistent and, as a consequence, comparison of the Reynolds stress predictions to the data is not conclusive. Reductions of the kinetic energy and the spread rate are consistent with the sizable decreases seen in these classes of flows.

AN ANALYSIS OF PHASE DISTRIBUTION AND TURBULENCE IN DISPERSED PARTICLE/LIQUID FLOWS

An analysis of dilute, turbulent particle/liquid two-phase flow is presented. The three-dimensional conservation equations that govern turbulent motion in solid/fluid flows are derived using ensemble averaging, and the unknown terms in these equations are constituted to achieve closure. These closure terms include the shear stress due to interparticle collisions, the corresponding terms in the Reynolds stress equation, the force and dissipation due to particle-wall collisions, and the interfacial work due to particle/turbulent eddy interaction. The resultant two-fluid model was then evaluated using a computational fluid dynamic (CFD) solver and the predictions were compared with experimental data. Good agreement was observed for a variety of flow conditions.

Application of a second-moment closure model to simulate the turbulent dispersion from an elevated line source

In this study, a second-moment closure model and an algebraic stress and flux model are used to simulate the turbulent dispersion from an elevated line source located in a turbulent boundary layer. In the modeling of the Reynolds stress equation, several models for the triple velocity correlation are examined by comparing their predicting results with the direct numerical simulation data. Predicted mean and fluctuating correlation quantities of velocity and temperature by the second-moment closure model and the algebraic model are compared with the experimental data. From the comparisons, it is found that the second-moment closure is capable of reproducing the experimental results satisfactorily. However, predictions of the algebraic stress and flux model are quite poor.

Measurement and modelling of homogeneous axisymmetric turbulence

A new method for determining the slow and rapid pressure-strain rate terms directly from wind-tunnel experiments has been developed with the aid of a newly developed theoretical description of the kinematics of homogeneous axisymmetric turbulence. Both the straining and the return-to-isotropy process of homogeneous axisymmetric turbulence are studied with the aim of improving Reynolds stress closures. Direct experimental determination of the different terms in the transport equation for the Reynolds stress tensor plays a major role in the validation and development of turbulence models. For the first time it is shown that the pressure{strain correlation can be determined with good accuracy without balancing it out from the Reynolds stress transport equation (and without measuring the pressure). Instead it is determined through evaluation of integrals containing second- and third-order two-point velocity correlations. All the terms in the Reynolds stress equations are measured directly and balance is achieved.

Turbulence in Compressible Mixing Layers

The direct numerical simulation (DNS) of two-dimensional compressible turbulent mixing layers is reported in this paper for convective Mach numbers Mc = 0.5, 0.8 and 1.0. All scales of flow are resolved with a 2562 grid, although results are also obtained for 642, 962 and 1282 grids for the purpose of determining the effective accuracy and grid-independence of our calculations. The effect of Mach number is also reported for all the Reynolds stress tensor components and for the “shear” components of the anisotropy tensor, the dissipation tensor, pressure-strain, and the triple correlation tensor. The short-time behaviors of some of these quantities are similar to those reported by Sarkar (1995) for homogeneous shear flow, in spite of the differences in the problem type and initial and boundary conditions. The relative magnitudes and signs of the unclosed terms in the Reynolds stress equations provide information on those that have to be retained for turbulence modeling as well as the sense of their contribution.

Toward a Turbulence Constitutive Relation for Geophysical Flows

Rapidly rotating turbulent flows are frequently in approximate geostrophic balance. Single-point turbulence closures, in general, are not consistent with a geostrophic balance. This article addresses and resolves the possibility of a constitutive relation for single-point second-order closures for classes of rotating and stratified flows relevant to geophysics. Physical situations in which a geostrophic balance is attained are described. Closely related issues of frame-indifference, horizontal divergence, and the Taylor–Proudman theorem are discussed. It is shown that, in the absence of vortex stretching along the axis of rotation, turbulence is frame-indifferent. Unfortunately, no turbulence closures are consistent with this frame-indifference that is frequently an important feature of rotating or quasi-geostrophic flows. A derivation and discussion of the geostrophic constraint which ensures that the modeled second-moment equations are frame-invariant, in the appropriate limit, is given. It is shown that rotating, stratified, and shallow water flows are situations in which such a constitutive relation procedure is useful. A nonlinear nonconstant coefficient representation for the rapid-pressure strain covariance appearing in the Reynolds stress and heat flux equations, consistent with the geostrophic balance, is described. The rapid-pressure strain closure features coefficients that are not constants determined by numerical optimization but are functions of the state of turbulence as parametrized by the Reynolds stresses and the turbulent heat fluxes as is required by tensor representation theory. These issues are relevant to baroclinic and barotropic atmospheric and oceanic flows. The planetary boundary layers in which there is a transition, with height or depth, from a thermally or shear driven turbulence to a geostrophic turbulence is a classic geophysical example to which the considerations in this article are relevant.

Some developments in computational modeling of turbulent flows

In this paper, some recent developments of two turbulence closure schemes at ICOMP, NASA Lewis will be discussed. One is the Reynolds-stress algebraic equation model and the other is the Reynolds-stress transport equation model. Various model constraints required by the rapid distortion theory, the invariant theory and the realizability principle, etc. will be described in the model development. The models discussed are for high-turbulent Reynolds number flows, so that the near-wall turbulence and the low-Reynolds-number turbulence are not discussed here.

Fully explicit and self-consistent algebraic Reynolds stress model

A fully-explicit, self-consistent algebraic expression for the Reynolds stress, which is the exact solution to the Reynolds stress transport equation in the `weak equilibrium'' limit for two-dimensional mean flows for all linear and some quasi-linear pressure-strain models, is derived. Current explicit algebraic Reynolds stress models derived by employing the `weak equilibrium'' assumption treat the production-to-dissipation (P varepsilon) ratio implicitly, resulting in an effective viscosity that can be singular away from the equilibrium limit. In the present paper, the set of simultaneous algebraic Reynolds stress equations are solved in the full non-linear form and the eddy viscosity is found to be non-singular. Preliminary tests indicate that the model performs adequately, even for three dimensional mean flow cases. Due to the explicit and non-singular nature of the effective viscosity, this model should mitigate many of the difficulties encountered in computing complex turbulent flows with the algebraic Reynolds stress models.

A Numerical Study of a Bluff-body Stabilized Diffusion Flame. Part 1. Influence of Turbulence Modeling and Boundary Conditions

Abstract This is the first part of a paper on numerical prediction of a bluff-body stabilized turbulent diffusion flame of syngas and air. This part considers the influence of turbulence modeling and boundary conditions on the predictions. Part 2 investigates the effect of the turbulence-chemistry interaction model and the effect of finite-rate chemistry. Results based on the “standard” k-s model and a Reynolds-stress-equation (RSE) model are compared. Measurements are taken from the literature. The RSE model predicts results in better agreement with the measurements than the k-e model. The two models predict significantly different composition and temperature levels in the recirculation bubble created by the bluff body. The specification of the turbulence level in the fuel-jet has a substantial influence on the axial decay of mixture fraction. Grid-resolution studies show that a relatively coarse grid is capable of representing the present flow with sufficient accuracy to evaluate the various sub-models.

Turbulence modulation in jet diffusion flames: Modeling and experiments

Turbulence in a diffusion flame is modulated by thermal expansion, buoyancy effects and lift-off. Aside from simple shear-generated sources density-velocity correlations, π′u ″ i represent additional source terms in the Reynolds-stress equations that distort the shear-generated turbulence anisotropy. A modeled transport equation and several zero-order models of π′u ″ i are analyzed and their effect on flame predictions with a Favre averaged second-order moment closure for velocity and scalar transport is investigated. The chemical reaction is described by chemical equilibrium and laminar flamelet modeling. The latter is shown to have limitations in application due to differential diffusion. Two attached, vertical H 2/N 2-air flames with the same Reynolds numbers but different Froude numbers are investigated numerically and experimentally. The desired data base for an overall comparison is provided by comprehensive 3D-LDV, coherent anti-Stokes Raman spectroscopy and spontaneous Raman spectroscopy measurements. The calculations yield correct results in all measured profiles of velocity, temperature and species concentrations. It is shown that only one zero-order model and the transport equation of π′u ″ i are adequate. The neglect of those terms will falsify the prediction of decay rates, fluctuations and flame shapes. The magnitude of errors depends on the local Froude number which decreases downstream. The increase in the influence of buoyancy leads to smaller decay rates of axial velocity and to enhanced scalar mixing. Furthermore, turbulence intensities are reduced, and scalar fluctuations and anisotropy are enlarged. The experimentally observed visible flame length shortening with decreasing, density weighted Froude number is reproduced by the presented model.

Three-dimensional shock/boundary-layer interaction using Reynolds stress equation turbulence model

The three-dimensional crossing shock wave/turbulent-boundary-layer interaction caused by an asymmetric 7 x 11-deg double fin with incoming Mach number 3.95 was simulated using the Reynolds-averaged Navier-Stokes equations with a full Reynolds stress equation turbulence model. An implicit approximate factorization method is used for the temporal integration. Roe's scheme is used for evaluation of the convective terms of the mean flow and Reynolds stress equations with a third-order MUSCL-type differencing. The computed surface pressure is in good agreement with the experiment. The computed heat transfer coefficient shows a modest improvement compared with previous results obtained using the k-c model with Chien's low-Reynolds-number correction. Both computations of the heat transfer display significant deviations from experiment. Tests of the grid refinement, different upstream boundary-layer profiles, and different isothermal wall temperature also are presented.

Ak-ɛ-PDF two-phase turbulence model for simulating sudden-expansion particle-laden flows

Ak-ɛ-PDF model based on statistical theory for turbulent gas-particle flows is proposed, and a numerical procedure combining the finite difference and finite fluctuating-velocity-group methods is used. The obtained statistically averaged equations have the same form as those obtained by using the Reynolds averaging. Using thek-ɛ-PDF model (PDF particle turbulence model combined with thek-ɛ gas turbulence model), many terms, such as the diffusion term in particle Reynolds Stress equations, can be accurately calculated for verifying the second-moment-closure model. Thek-ɛ-PDF model is used to simulate sudden-expansion particle-laden flow. comparison of the predictions using bothk-ɛ-PDF and thek-ɛ-kp models with experimental results shows that thek-ɛ-PDF model give more reasonable non-isotropic features of particle turbulence.

Numerical Analysis of Three-Dimensional Turbulent Sructure in a Periodicaly-Ribbed Channel.

This paper presents a numerical analysis to investigate three-dimensional turbulent structure and fluid flow behavior in a rectangular channel flow with streamwise-periodic ribs mounted on facing walls. In the calculation, an algebraic Reynolds stress model together with boundary-fitted coordinate system is applied to a periodically ribbed channel in order to solve the accelerated, separated and recirculating flows. The calculated results are compared with the experimental results available to examine the validity of the present method. As a result of this calculation, it was found that the present method predicted well the distributions of mean flow velocity and reattachment length. Moreover, the calculated results of turbulent energy and shear stress are in good agreement with the experimental data. Although it seems that the convection term of the Reynolds stress equation has an effect on the distributions of Reynolds stresses in this periodically ribbed channel flow, an algebraic Reynolds stress model, in which convection and diffusion terms are transformed from the differential form to algebraic form, predicts the experimental results of Reynolds stresses with little discrepancy. As regards the secondary flow, calculated flow patterns differ with the experimental results. In calculation, large variations of secondary flow direction are predicted along the streamwise direction whereas the experimental results show an unchanged flow pattern in both regions with and without recirculating flow.

The equilibrium states and the stability analysis of Reynolds stress equations for homogeneous turbulent shear flows

An analysis is performed to examine the equilibrium states and the stability of modeled Reynolds stress equations for homogeneous turbulent shear flows. The system of the governing equations consists of four coupled ordinary differential equations for b11, b22, b12, and ε/Sk. Depending on the model constants adopted, the solution of this system may converge, diverge, oscillate, or the equilibrium solution may not exist. It is shown that the rapid part of the pressure–strain correlation is most responsible for such different behaviors. Constraints for the model constants in various Reynolds stress models are obtained by stability conditions for the equilibrium states as well as by their physically realizable bounds. It is observed that most models for the pressure–strain correlation that are linear in the anisotropy tensor are stable and produce reasonable equilibrium solutions. Whereas quadratic or cubic models often oscillate and converge onto unrealistic states, especially under high shear conditions.

A rapid-pressure covariance representation consistent with the Taylor—Proudman theorem materially frame indifferent in the two-dimensional limit

A nonlinear variable-coefficient representation for the rapid-pressure covariance appearing in the Reynolds stress and heat-flux equations, consistent with the Taylor–Proudman theorem, is presented. The representation ensures that the modelled second-order equations are frame indifferent with respect to rotation in a number of different flows for which such an invariance is required. The model coefficients are functions of the state of the turbulence; they are valid for all states of a mechanical turbulence, attaining their limiting values only when the limit state is achieved. This is accomplished by a special ansatz that is used to obtain – analytically – the coefficients valid away from the realizability limit. Unlike other rapid-pressure representations in which extreme states are used to set model constants, here the coefficients are variable functions asymptotically consisted with – not fixed by – the limit states of the turbulence field. The mathematical principles invoked do not specify all the coefficients in the model; undetermined coefficients appear as free parameters which are used to ensure that the representation is asymptotically consistent with an experimentally determined equilibrium state of homogeneous sheared turbulence. This is done by ensuring that the modelled evolution equations have the same fixed points as those obtained from numerical and laboratory experiments for the homogeneous shear. Results of computations of homogeneous shear, with rotation and with curvature, are shown. Results are better, in a wide class of planar flows for which the model has not been calibrated, than those of other nonlinear models.

Reynolds stress model assessment using round jet experimental data

Experimental data of Hussein et al. (1993) for the turbulent round jet are used to evaluate individual components of Reynolds stress turbulence models. Models for terms in the Reynolds stress equations are reviewed, with particular emphasis on linear and nonlinear pressure-strain models. Improved coefficients for the Choi and Lumley return-to-isotropy expressions have been developed by the authors. These coefficients are valid for a wider range of flows than the currently used coefficients. Pressure-strain and transport model components are compared to the experimental data for the jet, and agreement is very good, indicating that the models are reasonably correct. Predictions using the linear models are generally as good as those obtained using nonlinear models, indicating that nonlinear models may not be necessary for engineering accuracy for this flow.

A general non-orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second-moment turbulence-transport closure, Part 1: Computational implementation

A computational procedure has been developed for predicting separated turbulent flows in complex two-dimensional and three-dimensional geometries. The procedure is based on the fully conservative, structured finite volume framework within which the volumes are non-orthogonal and collocated such that all flow variables are stored at one and the same set of nodes. To ease the task of discretization and to enhance the conservative property of the scheme, a Cartesian or datum-line-adapted decomposition of the velocity field has been used. The solution algorithm is iterative in nature, approaching the steady solution with the aid of a pressure-correction scheme. Convection is approximated with a range of schemes, among them higher-order upstream-weighted approximations and a TVD-type MUSCL form, the last applied principally to the transport equations governing turbulence properties. Effects of turbulence are represented either by two-equation eddy-viscosity models or by a full Reynolds-stress-transport closure. The former category includes both high- and low-Reynolds-number variants in two- as well as three-dimensional conditions. To achieve a stable implementation of the Reynolds-stress equations, a special interpolation practice, analogous to that of Rhie and Chow for momentum [1], has been introduced within the general framework. The procedure has been formulated so as to apply to both incompressible and compressible flows. The latter may contain shocks and highly supersonic regions. To achieve this range of applicability, the retarded-density concept has been combined with the basic pressure-based algorithm to capture shock waves. A ‘Full Approximation Multigrid’ scheme for convergence acceleration has been incorporated and applied in conjunction with all turbulence models including second-moment closure. The present first part of a twin paper focuses on numerical and turbulence-modelling issues. In Part 2, computational results are presented for six representative applications out of fifteen recently predicted with the algorithm within an extensive validation exercise.