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     

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.     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

Reynolds Stress Budgets in Couette and Boundary Layer Flows

Reynolds stress budgets for both Couette and boundary layer flows are evaluated and presented. Data are taken from direct numerical simulations of rotating and non-rotating plane turbulent Couette flow and turbulent boundary layer with and without adverse pressure gradient. Comparison of the total shear stress for the two types of flows suggests that the Couette case may be regarded as the high Reynolds number limit for the boundary layer flow close to the wall. The limit values of turbulence statistics close to the wall for the boundary layer for increasing Reynolds number approach the corresponding Couette flow values. The direction of rotation is chosen so that it has a stabilizing effect, whereas the adverse pressure gradient is destabilizing. The pressure-strain rate tensor in the Couette flow case is presented for a split into slow, rapid and Stokes terms. Most of the influence from rotation is located to the region close to the wall, and both the slow and rapid parts are affected. The anisotropy for the boundary layer decreases for higher Reynolds number, reflecting the larger separation of scales, and becomes close to that for Couette flow. The adverse pressure gradient has a strong weakening effect on the anisotropy. All of the data presented here are available on the web [36]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Predicting Noninertial Effects with Linear and Nonlinear Eddy-Viscosity, and Algebraic Stress Models

Three types of turbulence models which account for rotational effects in noninertial frames of reference are evaluated for the case of incompressible, fully developed rotating turbulent channel flow. The different types of models are a Coroiolis-modified eddy-viscosity model, a realizable nonlinear eddy-viscosity model, and an algebraic stress model which accounts for dissipation rate anisotropies. A direct numerical simulation of a rotating channel flow is used for the validation of the turbulence models. This simulation differs from previous studies in that significantly higher rotation numbers are investigated. Flows at these higher rotation numbers are characterized by a relaminarization on the cyclonic or suction side of the channel, and a linear velocity profile on the anticyclonic or pressure side of the channel. The predictive performance of the three types of models are examined in detail, and formulation deficiencies are identified which cause poor predictive performance for some of the models. Criteria are identified which allow for accurate prediction of such flows by algebraic stress models and their corresponding Reynolds stress formulations.     

Anholonomic configuration spaces and metric tensors in finite elastoplasticity

Deformation mappings are considered that correspond to the motions of lattice defects, elastic stretch and rotation of the lattice, and initial defect distributions. Intermediate (i.e., relaxed) configuration spaces associated with these deformation maps are identified and then classified from the differential-geometric point of view. A fundamental issue is the proper selection of coordinate systems and metric tensors in these configurations when such configurations are classified as anholonomic. The particular choice of a global, external Cartesian coordinate system and corresponding covariant identity tensor as a metric on an intermediate configuration space is shown to be a constitutive assumption often made regardless of the existence of geometrically necessary crystal defects associated with the anholonomicity (i.e., the non-Euclidean nature) of the space. Since the metric tensor on the anholonomic configuration emerges necessarily in the definitions of scalar products, certain transpose maps, tensorial symmetry operations, and Jacobian invariants, its selection should not be trivialized. Several alternative (i.e., non-Euclidean) representations proposed in the literature for the metric tensor on anholonomic spaces are critically examined.     

Some results for approximate strain and rotation tensor formulations in geometrically non-linear Reissner-Mindlin plate theory

Finite element deflection and stress results are presented for four flat plate configurations and are computed using kinematically approximate (rotation tensor, strain tensor or both) non-linear Reissner-Mindlin plate models. The finite element model is based on a mixed variational principle and has both displacement and force field variables. High order interpolation of the field variables is possible through p-type discretization. Results for some of the higher order approximate models are given for what appears to be the first time. It is found that for the class of example problems examined, exact strain tensor but approximate rotation tensor theories can significantly improve the solution over approximate strain tensor models such as the von Kármán and moderate rotation models when moderate deflections/rotations are present. However, for each of the problems examined (with the exception of a postbuckling problem) the von Kármán and moderate rotation model results compared favorably with the higher order models for deflection magnitudes which could be reasonably expected in typical aeroelastic configurations.     

Sur une méthode de pénalisation tensorielle pour la résolution des équations de Navier–Stokes

An original reformulation of the viscous stress tensor is proposed for the motion equations dedicated to an incompressible fluid. Four different tensors appear in this decomposition, associated with viscosities of compression, elongation, shearing and rotation. This new model allows us to build a numerical solver of the Navier–Stokes equations based on a technique of tensorial penalization which is generalized with all the stresses acting on a flow. The processing of incompressibility and the rotation of a rigid body in a flow are described thanks to the model. Several numerical applications are proposed to illustrate the abilities of the new penalization method.     

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     

General invariant representations of the constitutive equations for isotropic nonlinearly elastic materials

This paper develops general invariant representations of the constitutive equations for isotropic nonlinearly elastic materials. Different sets of mutually orthogonal unit tensor bases are constructed from the strain argument tensor by using the representation theorem and corresponding irreducible invariants are defined. Their relations and geometrical interpretations are established in three dimensional principal space. It is shown that the constitutive law linking the stress and strain tensors is revealed to be a simple relationship between two vectors in the principal space. Relative to two different sets of the basis tensors, the constitutive equations are transformed according to the transformation rule of vectors. When a potential function is assumed to exist, the vector associated with the stress tensor is expressed in terms of its gradient with respect to the vector associated with the strain tensor. The Hill’s stability condition is shown to be that the scalar product of the increment of those two vectors must be positive. When potential function exists, it becomes to be that the 3 × 3 constitutive matrix derived from its second order derivative with respect to the vector associated with the strain must be positive definite. By decomposing the second order symmetric tensor space into the direct sum of a coaxial tensor subspace and another one orthogonal to it, the closed form representations for the fourth order tangent operator and its inversion are derived in an extremely simple way.     

Flow of a quasi-Newtonian fluid through a planar contraction

Flows through abrupt contractions are dominated by the rapid extension experienced in passing through the contraction. Thus, it is useful to employ a fluid model which considers the extensional viscosity explicitly in its constitutive equation. In this paper, the quasi-Newtonian fluid model, which admits shear thinning and extension thickening of the viscosity depending on the local type of flow as proposed by Schunk and Scriven [P. Schunk, L. Scriven, J, Rheol 34 (1990) 1085], is applied to the numerical simulation of the flow of a dilute polyacrylamide solution through a planar 4 : 1 contraction. In this theory the extra stress tensor does not only depend on the rate of strain tensor but also on the relative rate of rotation of the fluid. The material function – the viscosity function – is allowed to depend on the invariants of these two kinematic tensors yielding a local distinction between extensional, shear or rotation dominated flow. The governing equations are discretized using a finite volume method. Different model parameters are varied and the simulation results are compared with the generalized Newtonian fluid and experimental data.     

Calculation of strain histories in Protean coordinate systems

Integral equations for the relative deformation gradient tensors are solved to give analytical expressions which involve velocities and velocity gradients along streamlines. For some Protean coordinate systems, metric tensors are presented, and deformation gradients and strain histories are calculated. The results are tested for two types of flow: rotational shearing flow and extensional flow. They are found to give the existing exact relations for the Finger strain tensor.     

强压缩湍流雷诺应力模型中压力应变快速项的探索

对压力应变快速项的五个模型作了压缩性修正,即在模型中引入了由于平均流可压而导入的不为零的平均速度散度,并把五个模型计算所得的雷诺应力各向异性张量分量、平均湍能及压力应变快速项的值与快速畸变理论的计算结果作了比较。结果表明,包含湍流应变中效应的线性模型可达到四阶非线性模型的精度。     

Anisotropic damage–plasticity model for concrete

A material model for concrete is proposed here within the framework of a thermodynamically consistent elasto-plasticity–damage theory. Two anisotropic damage tensors and two damage criteria are adopted to describe the distinctive degradation of the mechanical properties of concrete under tensile and compressive loadings. The total stress tensor is decomposed into tensile and compressive components in order to accommodate the need for the above mentioned damage tensors. The plasticity yield criterion presented in this work accounts for the spectral decomposition of the stress tensor and allows multiple hardening rules to be used. This plastic yield criterion is used simultaneously with the damage criteria to simulate the physical behavior of concrete. Non-associative flow rule for the plastic strains is used to account for the dilatancy of concrete as a frictional material. The thermodynamic Helmholtz free energy concept is used to consistently derive dissipation potentials for damage and plasticity and to allow evolution laws for different hardening parameters. The evolution of the two damage tensors is accounted for through the use of fracture-energy-based continuum damage mechanics. An expression is derived for the damage–elasto-plastic tangent operator. The theoretical framework of the model is described here while the implementation of this model will be discussed in a subsequent paper.     

Canonical representations and degree of freedom formulae of orthogonal tensors in n-dimensional Euclidean space

In this paper, with the help of the eigenvalue properties of orthogonal tensors in n-dimensional Euclidean space and the representations of the orthogonal tensors in 2-dimensional space, the canonical representations of orthogonal tensors in n-dimensional Euclidean space are easily gotten. The paper also gives all the constraint relationships among the principal invariants of arbitrarily given orthogonal tensor by use of Cayley-Hamilton theorem; these results make it possible to solve all the eigenvalues of any orthogonal tensor based on a quite reduced equation of m-th order, where m is the integer part of n/2. Finally, the formulae of the degree of freedom of orthogonal tensors are given.     

有限转动张量的保角参量及在多柔体系统中的应用

采用保角转动参数描述了多体系统中的大转动张量．该方法消除了传统的欧拉参数描述所必需的约束方程，并且适于大变形部件的建模需要．利用以上结果建立了含大变形梁状部件的多体系统的力学模型．     

A new hybrid turbulence modelling strategy for industrial CFD

This paper presents a new strategy for turbulence model employment with emphasis on the model's applicability for industrial computational fluid dynamics (CFD). In the hybrid modelling strategy proposed here, the Reynolds stress and mean rate of strain tensors are coupled via Boussinesq's formula as in the standard k–εmodel. However, the turbulent kinetic energy is calculated as the sum of the normal Reynolds‐stress components, representing the solutions of the appropriate transport equations. The equations governing the Reynolds‐stress tensor and dissipation rate have been solved in the framework of a ‘background’ second‐moment closure model. Furthermore, the structure parameter C‐µ has been re‐calculated from a newly proposed functional dependency rather than kept constant. This new definition of C‐µ has been assessed by using direct numerical simulation (DNS) results of several generic flow configurations featuring different phenomena such as separation, reattachment and rotation. Comparisons show a large departure of C‐µ from the commonly used value of 0.09. The model proposed is computationally validated in a number of well‐proven fluid flow benchmarks, e.g. backward‐facing step, 180° turn‐around duct, rotating pipe, impinging jet and three‐dimensional (3D) Ahmed body. The obtained results confirm that the present hybrid model delivers a robust solution procedure while preserving most of the physical advantages of the Reynolds‐stress model over simple k–εmodels. A low Reynolds number version of the hybrid model is also proposed and discussed. Copyright © 2003 John Wiley & Sons, Ltd.     

Second-moment closure modelling of turbulence in a non-inertial frame

This article investigates the performance of second-moment closures in a rotating reference frame and presents a new closure for the rapid pressure-strain rate correlation based on the recently developed materially frame indifference principle. It is observed here that the existing second-moment closures with appropriate near-wall treatment can adequately predict flows in a rotating channel and in an axially rotating pipe for moderate rotation rate. Analysis of the newly proposed model indicates that it is capable of reflecting flow features under strong rotation.     

On 2D Viscoelasticity with Small Strain

An exact two-dimensional rotation–strain model describing the motion of Hookean incompressible viscoelastic materials is constructed by the polar decomposition of the deformation tensor. The global existence of classical solutions is proved under smallness assumptions only on the size of the initial strain tensor. The proof of global existence utilizes the weak dissipative mechanism of motion, which is revealed by passing the partial dissipation to the whole system.