An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part I Non-isothermal flow

Tensor representation theory is used to derive an explicit algebraic model that consists of an explicit algebraic stress model (EASM) and an explicit algebraic heat flux model (EAHFM) for two-dimensional (2-D) incompressible non-isothermal turbulent flows. The representation methodology used for the heat flux vector is adapted from that used for the polynomial representation of the Reynolds stress anisotropy tensor. Since the methodology is based on the formation of invariants from either vector or tensor basis sets, it is possible to derive explicit polynomial vector expansions for the heat flux vector. The resulting EAHFM is necessarily coupled with the turbulent velocity field through an EASM for the Reynolds stress anisotropy. An EASM has previously been derived by Jongen and Gatski [10]. Therefore, it is used in conjunction with the derived EAHFM to form the explicit algebraic model for incompressible 2-D flows. This explicit algebraic model is analyzed and compared with previous formulations including its ability to approximate the commonly accepted value for the turbulent Prandtl number. The effect of pressure-scrambling vector model calibration on predictive performance is also assessed. Finally, the explicit algebraic model is validated against a 2-D homogeneous shear flow with a variety of thermal gradients. Dedicated to the memory of the late Professor Charles G. Speziale of Boston University     

Accounting for Buoyancy Effects in the Explicit Algebraic Stress Model: Homogeneous Turbulent Shear Flows

Most explicit algebraic stress models are formulated for turbulent shear flows without accounting for external body force effects, such as the buoyant force. These models yield fairly good predictions of the turbulence field generated by mean shear. As for thermal turbulence generated by the buoyant force, the models fail to give satisfactory results. The reason is that the models do not explicitly account for buoyancy effects, which interact with the mean shear to enhance or suppress turbulent mixing. Since applicable, coupled differential equations have been developed describing these thermal turbulent fields, it is possible to develop corresponding explicit algebraic stress models using tensor representation theory. While the procedure to be followed has been employed previously, unique challenges arise in extending the procedure for developing the algebraic representations to turbulent buoyant flows. In this paper the development of an explicit algebraic stress model (EASM) is confined to the homogeneous buoyant shear flow case to illustrate the methodology needed to develop the proper polynomial representations. The derivation is based on the implicit formulation of the Reynolds stress anisotropy at buoyant equilibrium. A five-term representation is found to be necessary to account properly for the effect of the thermal field. Thus derived, external buoyancy effects are represented in the scalar coefficients of the basis tensors, and structural buoyancy effects are accounted for in additional terms in the stress anisotropy tensor. These terms will not vanish even in the absence of mean shear. The performance of the new EASM, together with a two-equation (2-Eq) model, the non-buoyant EASM of Gatski and Speziale (1993) and a full second-order model, is assessed against direct numerical simulations of homogeneous, buoyant shear flows at two different Richardson numbers representing weak and strong buoyancy effects. The calculations show that this five-term representation yields better results than the 2-Eq model and the EASM of Gatski and Speziale where buoyancy effects are not explicitly accounted for. Received 5 March 2001 and accepted 15 January 2002     

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. This revised version was published online in July 2006 with corrections to the Cover Date.     

Binary diffusion and heat transfer in laminar mixed convection channel flows with uniform wall heat flux: extremely thin film thickness

A detailed numerical study has been performed to investigate the combined heat and mass transfer in laminar mixed convection channel flows with uniform wall heat flux. In an initial effort the liquid film on the channel wall is assumed to be extremely thin in thickness. Major dimensionless groups governing the present problem areGr _T,Gr _Mx,Pr,Sc, φ andRe. Results are specifically presented for an air-water system under various conditions. The effects of wall heating flux, the Reynolds number and the relative humidity of the moist air in the ambient on the momentum, heat and mass transfer in the flow are investigated in great detail.     

Turbulence Modeling Effects on the Prediction of Equilibrium States of Buoyant Shear Flows

The effects of turbulence modeling on the prediction of equilibrium states of turbulent buoyant shear flows were investigated. The velocity field models used include a two-equation closure, a Reynolds-stress closure assuming two different pressure-strain models and three different dissipation rate tensor models. As for the thermal field closure models, two different pressure-scrambling models and nine different temperature variance dissipation rate ɛ_τ) equations were considered. The emphasis of this paper is focused on the effects of the ɛ_τ-equation, of the dissipation rate models, of the pressure-strain models and of the pressure-scrambling models on the prediction of the approach to equilibrium turbulence. Equilibrium turbulence is defined by the time rate of change of the scaled Reynolds stress anisotropic tensor and heat flux vector becoming zero. These conditions lead to the equilibrium state parameters, given by /ɛ, _τ/ɛ_τ, , Sk/ɛ and G/ɛ, becoming constant. Here, and _τ are the production of turbulent kinetic energy k and temperature variance , respectively, ɛ and ɛ_τ are their respective dissipation rates, R is the mixed time scale ratio, G is the buoyant production of k and S is the mean shear gradient. Calculations show that the ɛ_τ-equation has a significant effect on the prediction of the approach to equilibrium turbulence. For a particular ɛ_τ-equation, all velocity closure models considered give an equilibrium state if anisotropic dissipation is accounted for in one form or another in the dissipation rate tensor or in the ɛ-equation. It is further found that the models considered for the pressure-strain tensor and the pressure-scrambling vector have little or no effect on the prediction of the approach to equilibrium turbulence. Received 21 April 2000 and accepted 21 February 2001     

Turbulence modelling of problem aerospace flows

Unsteady Reynolds averaged Navier–Stokes (URANS) and detached eddy simulation (DES) related approaches are considered for high angle of attack NACA0012 airfoil, wing–flap, generic tilt‐rotor airfoil and double‐delta geometry flows. These are all found to be problem flows for URANS models. For DES fifth‐order upwinding is found too dissipative and the use of, for high speed flows, instability prone centred differencing essential. An existing hybrid ILES–RANS modelling approach, intended for flexible geometry, relatively high numerical dissipation codes is tested along with differential wall distance algorithms. The former gives promising results. The standard turbulence modelling approaches are found to give perhaps a surprising results variation. Results suggest that for the problem flows, the explicit algebraic stress and Menter shear stress transport (SST) URANS models are more accurate than the economical Spalart–Allmaras (SA). However, the explicit algebraic stress model (EASM) in its k–ε form is impractically expensive to converge. Here, SA predictions lack a rotation correction term and this is likely to improve these results. Copyright © 2005 John Wiley & Sons, Ltd.     

Applications of URANS on predicting unsteady turbulent separated flows

Accurate prediction of unsteady separated turbulent flows remains one of the toughest tasks and a practi cal challenge for turbulence modeling. In this paper, a 2D flow past a circular cylinder at Reynolds number 3,900 is numerically investigated by using the technique of unsteady RANS （URANS）. Some typical linear and nonlinear eddy viscosity turbulence models （LEVM and NLEVM） and a quadratic explicit algebraic stress model （EASM） are evaluated. Numerical results have shown that a high-performance cubic NLEVM, such as CLS, are superior to the others in simulating turbulent separated flows with unsteady vortex shedding.     

RAS one-equation turbulence model with non-singular eddy-viscosity coefficient

A simplified consistency formulation for P_k/ε (production to dissipation ratio) is devised to obtain a non-singular C_μ (coefficient of eddy-viscosity) in the explicit algebraic Reynolds stress model of Gatski and Speziale. The coefficient C_μ depends non-linearly on both rotational/irrotational strains and is used in the framework of an improved RAS (Rahman–Agarwal–Siikonen) one-equation turbulence model to calculate a few well-documented turbulent flows, yielding predictions in good agreement with the direct numerical simulation and experimental data. The strain-dependent C_μ assists the RAS model in constructing the coefficients and functions such as to benefit complex flows with non-equilibrium turbulence. Comparisons with the Spalart–Allmaras one-equation model and the shear stress transport k-ω model demonstrate that C_μ improves the response of RAS model to non-equilibrium effects.     

Study on anisotropic buoyant turbulence model

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Fully explicit and self-consistent algebraic Reynolds stress model

A fully explicit, self-consistent algebraic expression (for 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/) ratio as a constant, resulting in an effective viscosity that can be singular away from the equilibrium limit. In this paper the set of simultaneous algebraic Reynolds stress equations in the weak-equilibrium limit are solved in the full nonlinear form and the eddy viscosity is found to be nonsingular. Preliminary tests indicate that the model performs adequately, even for three-dimensional mean-flow cases. Due to the explicit and nonsingular 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.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480.     

Buoyancy effects on turbulent swirling flows

A three-parameter model of turbulence applicable to free boundary layers has been developed and applied for the prediction of axisymmetric turbulent swirling flows in uniform and stagnant surroundings under the action of buoyancy forces. The turbulent momentum and heat fluxes appearing in the time-averaged equations for the mean motion have been determined from algebraic expressions, derived by neglecting the convection and diffusion terms in the differential transport equations for these quantities, which relate the turbulent fluxes to the kinetic energy of turbulence, k, the dissipation length scale of turbulence, L, and the temperature covariance, T ′². Differential transport equations have been used to determine these latter quantities. The governing equations have been solved using fully implicit finite difference schemes. The turbulence model is capable of reproducing the gross features of pure jet flows, buoyant flows and swirling flows for weak and moderate swirl. The behaviour of a turbulent buoyant swirling jet has been found to depend solely on exit swirl and Froude numbers. The predicted results indicate that the incorporation of buoyancy can cause significant changes in the behaviour of a swirling jet, particularly when the buoyancy strength is high. The jet exhibits similarity behaviour in the initial region for weak swirl and weak buoyancy strengths only, and the asymptotic case of a swirling jet under the action of buoyancy forces is a pure plume in the far field. The predicted results have been found to be in satisfactory agreement with the available experimental data and in good qualitative agreement with other predicted results.     

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. Received 14 October 1996 and accepted 9 June 1997     

Investigation of transitional and turbulent heat and momentum transport in a rotating cavity

The paper gives the results of the DNS/LES which was performed to investigate the transitional and turbulent non-isothermal flows within a rotor/stator cavity. Computations were performed for the cavity of aspect ratio L = 2–35, Rm = 1.8 and for rotational Reynolds numbers up to 290000. The main purpose of the investigations was to analyze the influence of aspect ratio and Reynolds number on the flow structure and heat transfer. The numerical solution is based on a pseudo-spectral Chebyshev–Fourier–Galerkin collocation approximation. The time scheme is semi-implicit second-order accurate, which combines an implicit treatment of the diffusive terms and an explicit Adams–Bashforth extrapolation for the non-linear convective terms. In the paper we analyze distributions of the Reynolds stress tensor components, the turbulent heat flux tensor components, Nusselt number distributions and the turbulent Prandtl number and other structural parameters, which can be useful for modeling purposes. Selected results are compared with the experimental data obtained for single heated rotating disk by Elkins and Eaton (2000).     

Conjugate heat transfer study of incompressible turbulent offset jet flows

In the present case, the conjugate heat transfer involving a turbulent plane offset jet is considered. The bottom wall of the solid block is maintained at an isothermal temperature higher than the jet inlet temperature. The parameters considered are the offset ratio (OR), the conductivity ratio (K), the solid slab thickness (S) and the Prandtl number (Pr). The Reynolds number considered is 15,000 because the flow becomes fully turbulent and then it becomes independent of the Reynolds number. The ranges of parameters considered are: OR = 3, 7 and 11, K = 1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds number two-equation model (k–ε) has been used for turbulence modeling. Results for the solid–fluid interface temperature, local Nusselt number, local heat flux, average Nusselt number and average heat transfer have been presented and discussed.     

Evaluation a priori d'un modèle non-linéaire de turbulenceA priori evaluation and improvement of a non-linear model for turbulent flows

The aim of this work is a priori evaluation and improvement of a non-linear model for turbulent flows using the results from direct numerical simulation of Navier–Stokes equations. The algebraic explicit non-linear model recently proposed by Rumsey C.L. et al. [1] is studied. The data base used here comes from a direct numerical simulation of a turbulent flow through a square duct. For this flow, this study shows that the hypothesis of equilibrium state for the anisotropic tensor is correct. The analysis is made using the maps of the second and third invariants of the Reynolds stress tensor. The approach used permits to conclude that the model using a wall function improves the numerical prediction of the anisotropy. To cite this article: O. El Yahyaoui et al., C. R. Mecanique 330 (2002) 27–34     

Comparisons and applications of numerical simulation methods for predicting aerodynamic heating around complex configurations

Numerical simulation methods of aerodynamic heating were compared by considering the inuence of numerical schemes and turbulence models,and attempting to investigate the applicability of numerical simulation methods on predicting heat flux in engineering applications. For some typical cases provided with detailed experimental data,four spatial schemes and four turbulence models were adopted to calculate surface heat flux. By analyzing and comparing,some inuencing regularities of numerical schemes and turbulence models on calculating heat flux had been acquired. It is clear that AUSM+-up scheme with rapid compressibilitymodified high Reynolds number k-ω model should be appropriate for calculating heat flux. The numerical methods selected as preference above were applied to calculate the heat flux of a 3-D complex geometry in high speed turbulent flows. The results indicated that numerical simulation can capture the complex flow phenomena and reveal the mechanism of aerodynamic heating. Especially,the numerical result of the heat flux at the stagnation point of the wedge was well in agreement with the prediction of Kemp-Riddel formula,and the surface heat flux distribution was consistent with experiment results,which implied that numerical simulation can be introduced to predict heat flux in engineering applications.     

Some Reynolds number effects on two- and three-dimensional turbulent boundary layers

Experimental data for a two-dimensional (2-D) turbulent boundary layer (TBL) flow and a three-dimensional (3-D) pressure-driven TBL flow outside of a wing/body junction were obtained for an approach Reynolds number based on momentum thickness of Re _θ =23,200. The wing shape had a 3:2 elliptical nose, NACA 0020 profiled tail, and was mounted on a flat wall. Some Reynolds number effects are examined using fine spatial resolution (Δy ⁺=1.8) three-velocity-component laser-Doppler velocimeter measurements of mean velocities and Reynolds stresses at nine stations for Re _θ =23,200 and previously reported data for a much thinner boundary layer at Re _θ =5,940 for the same wing shape. In the 3-D boundary layers, while the stress profiles vary considerably along the flow due to deceleration, acceleration, and skewing, profiles of the parameter correlate well and over available Reynolds numbers. The measured static pressure variations on the flat wall are similar for the two Reynolds numbers, so the vorticity flux and the measured mean velocities scaled on wall variables agree closely near the wall. The stresses vary similarly for both cases, but with higher values in the outer region of the higher Re _θ case. The outer layer turbulence in the thicker high Reynolds number case behaves similarly to a rapid distortion of the flow, since stream-wise vortical effects from the wall have not diffused completely through the boundary layer at all measurement stations. Received: 9 June 2000/Accepted: 26 January 2001     

Buoyancy modelling with incompressible SPH for laminar and turbulent flows

This work aims to model buoyant, laminar or turbulent flows, using a two‐dimensional incompressible smoothed particle hydrodynamics model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work [1], we extend the unified semi‐analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds‐averaged Navier–Stokes approach to treat turbulent flows. The k ? ? turbulence model is used, where buoyancy is modelled through an additional term in the k ? ? equations like in mesh‐based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or to impose heat flux (Neumann) wall boundary conditions in incompressible smoothed particle hydrodynamics. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock‐exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a finite volume approach using an open‐source industrial code. Copyright © 2015 John Wiley & Sons, Ltd.     

Explicit algebraic stress modelling of homogeneous and inhomogeneous flows

Capability of the explicit algebraic stress models to predict homogeneous and inhomogeneous shear flows are examined. The importance of the explicit solution of the production to dissipation ratio is first highlighted by examining the algebraic stress models performance at purely irrotational strain conditions. Turbulent recirculating flows within sudden expanding pipes are further simulated with explicit algebraic stress model and anisotropic eddy viscosity model. Both models predict better stress–strain interactions, showing reasonable shear layer developments. The anisotropic stress field are also accurately predicted by the models, though the anisotropic eddy viscosity model of Craft et al. returns marginally better results. Copyright © 2005 John Wiley & Sons, Ltd.