On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998     

Turbulence Modeling in Noninertial Frames of Reference

The effect of an arbitrary change of frame on the structure of turbulence models is examined from a theoretical standpoint. It is proven, as a rigorous consequence of the Navier-Stokes equations, that turbulence models must be form invariant under arbitrary translational accelerations of the reference frame and should only be affected by rotations through the intrinsic mean vorticity. A direct application of this invariance property along with the Taylor-Proudman theorem, material frame-indifference in the limit of two-dimensional turbulence, and Rapid Distortion Theory is shown to yield powerful constraints on the allowable form of turbulence models. Most of the commonly used turbulence models are demonstrated to be in violation of these constraints and consequently are inconsistent with the Navier-Stokes equations in noninertial frames. Alternative models with improved noninertial properties are developed and some simple applications to rotating turbulent flows are considered.This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18107 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665, U.S.A. Partial support was also provided by the Office of Naval Research under Contract No. N00014-85-K-0238.     

Modeling of Three-Dimensional Turbulent Jet and Boundary-Layer Flows

New anisotropic algebraic constitutive relations for the Reynolds stress tensor are formulated. These relations make it possible to model correctly three-dimensional turbulent flows which cannot be described on the basis of traditional modern semi-empirical models of turbulence. Along with the well-known nonlinear Saffman term, these relations contain new nonlinear terms which take the wall effect into account. Several two- and three-dimensional turbulent flows are calculated nu\-merically using the averaged Navier-Stokes equations. The calculation results are compared with known experimental data.     

非线性湍流模式研究及进展

现代湍流模式研究已经超出了经典的Ｂｏｕｓｓｉｎｅｓｑ涡粘性概念和线性的雷诺应力输运范畴,湍流运动过程中的非线性本质已成为模式研究人员所关心的中心问题。其目的在于使湍流模式能更加真实地再现湍流运动的复杂性,提高模式的适用范围,使复杂湍流能够得到合理的模拟,非线性湍流模式在解决复杂湍流运动的计算中已经取得可喜进展,正逐步应用于工程湍流的计算。同时,工程中的湍流问题计算也已走出了简单剪切流动类型及传统的ｋ－ε（及其它形式的）二方程模式框架,二阶矩封闭模式在先进的工程计算中已被用来解决诸如可压缩的空气动力学、发动机气缸及三维复杂几何场内等具有重要应用背景的流动问题,并逐步进入计算流体力学商业软件包。      

Stress and heat flux in non-inertial reference frames

We compute the stress tensor and the heat flux in a non-inertial reference frame, using the BGK model to any order in the product between the microscopic collision time and the angular velocity of the reference frame, a product that is assumed to be of order one. We find that both the stress and the heat flux change both in magnitude and in orientation with respect to its inertial counterparts, and that the non-inertiality always lowers their absolute value. Received: October 7, 1996 / Accepted: October 7, 1996     

基于基因表达式编程的数据驱动湍流建模

计算流体动力学是湍流研究的重要手段, 其中雷诺平均模拟在航空航天等实际工程中得到了广泛应用. 雷诺平均模拟的结果很大程度上依赖于湍流模型的预测精度, 而实际工程应用中常用的模型往往精度有限. 近年来, 数据驱动的湍流建模方法得到越来越多的关注. 本文介绍了基于基因表达式编程 (gene-expression programming, GEP) 方法的湍流建模相关进展. 本文首先讨论基因表达式编程应用于湍流建模的具体方法, 包括基本算法、显式代数应力模型和湍流传热两种建模框架、模型测试方法以及损失函数设置等. 在此基础上, 基因表达式编程方法被应用于涡轮叶栅尾流混合、竖直平板间自然对流、三维横向流中的射流等问题. 结果表明, GEP可以有效提升常用模型对于尾流混合损失、壁面热通量等关键参数的预测精度. 基因表达式编程方法可以显式给出模型方程, 因此模型具有可解释性强等特点. 基于双向耦合方法得到的模型还被证明具有较好的后验测试精度和鲁棒性. 基因表达式编程方法还被初步应用于大涡模拟亚格子应力和边界层转捩等问题的建模, 在不同湍流建模领域表现出很大的潜力.      

绕壁面小障碍物的湍流边界层研究

采用RNGK－ε模型，对统平板壁面上的条柱、方柱等多种不同形状二维小障碍物的固壁剪切湍流进行数值研究，获得了小障碍物下游回流区及其发展区的时均速度场、回流分离点长度和湍流边界层厚度，计算结果与实验比较吻合良好．结果表明，小障碍物在壁面上的设置，明显地改变了平板壁面边界层的湍流形态和结构．     

AN ANALYTICAL DESIGN METHODOLOGY FOR AN ANNULAR COMBUSTOR

SUMMARY

This paper describes a computational procedure for the optimization of the performance parameters of a simulated annular combustor. This method has been applied to analyze the influence of the performance parameters and geometries on the annular combustor characteristics and provide a good understanding of combustor internal flow fields, and therefore it can be used for guiding the combustor design process. The approach is based on the solution of governing nonlinear, elliptic partial differential equations for 3-D axisymmetric recirculating turbulent reacting swirling flows and the modelling of turbulence, combustion, thermal radiation and pollutant formation. The turbulence effects are introduced via the modified two-equation κ-ε model. Turbulent combustion is modelled using the κ-ε-g model and a two-step turbulent combustion model is employed for the excess emission of carbon monoxide CO. For the evaluation of the NO pollutant formation rate, the NO pollutant formation model, which takes into account the influence of turbulence, presented here. The radiative heat transfer is handled by the heat flux model. The predictions of the combustor character-istics and performance parameters are made using the present approach.

Predictions of velocity, length of the recirculation zone, combustion efficiency and wall temperature are compared with measurements. Agreement between the predictions and experimental data is very satisfactory.     

Presumed Mapping Functions for Eulerian Modelling of Turbulent Mixing

The mapping closure of Chen et al. [Phys. Rev. Lett., 63, 1989] is a transported probability density function (PDF) method that has proven very efficient for modelling of turbulent mixing in homogeneous turbulence. By utilizing a Gaussian reference field, the solution to the mapping function (in homogeneous turbulence) can be found analytically for a range of initial conditions common for turbulent combustion applications, e.g. for binary or trinary mixing. The purpose of this paper is to investigate the possibility of making this solution a presumed mapping function (PMF) for inhomogeneous flows. The PMF in turn will imply a presumed mixture fraction PDF that can be used for a wide range of models in turbulent combustion, e.g. flamelet models, the conditional moment closure (CMC) or large eddy simulations. The true novelty of the paper, though, is in the derivation of highly efficient, closed algebraic expressions for several existing models of conditional statistics, e.g. for the conditional scalar dissipation/diffusion rate or the conditional mean velocity. The closed form expressions nearly eliminates the overhead computational cost that usually is associated with nonlinear models for conditional statistics. In this respect it is argued that the PMF is particularly well suited for CMC that relies heavily on manipulations of the PDF for consistency. The accuracy of the PMF approach is shown with comparison to DNS of a single scalar mixing layer to be better than for the β-PDF. Not only in the shape of the PDF itself, but also for all conditional statistics models computed from the PDF.     

A hybrid k-ε turbulence model of recirculating flow

This investigation deals with the modification of streamline curvature effects in the k-ε turbulence model for the case of recirculating flows. Based upon an idea that the modification of curvature effects in C₂ should not be made in regions where the streamline curvature is small, a hybrid k-ε model extended from the modification originally proposed by Srinivasan and Mongia is developed. A satisfactory agreement of model predictions with experimental data reveals that the hybrid k-ε model can perform better simulation of recirculating turbulent flows.     

Performance of RNG Turbulence Modelling for Forced Convective Heat Transfer in Ducts

This investigation concerns numerical calculation of turbulent forced convective heat transfer and fluid flow in straight ducts using the RNG (Re-Normalized Group) turbulence method.

A computational method has been developed to predict the turbulent Reynolds stresses and turbulent heat fluxes in ducts with different turbulence models. The turbulent Reynolds stresses and other turbulent flow quantities are predicted with the RNG κ?ε model and the RNG non-linear κ-ε model of Speziale. The turbulent heat fluxes are modeled by the simple eddy diffusivity (SED) concept, GGDH and WET methods. Two wall functions are used, one for the velocity field and one for the temperature field. All the models arc implemented for an arbitrary three dimensional duct.

Fully developed condition is achieved by imposing cyclic boundary conditions in the main flow direction. The numerical approach is based on the finite volume technique with a non-staggered grid arrangement. The pressure-velocity coupling is handled by using the SIMPLEC-algorithm. The convective terms are treated by the QUICK, scheme while the diffusive terms are handled by the central-difference scheme. The hybrid scheme is used for solving the κ and ε equations.

The overall comparison between the models is presented in terms of friction factor and Nusselt number. The secondary flow generation is also of major concern.     

Extended Mac-Millan's equation

The principle of classical dynamics and Appell-Chetayev ’s assumption are extended to non-inertial frame, from which extended Mac-Millan ’s equation is derived for non-holonomic system in non-inertial system.     

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。     

A dynamic two-equation Sub Grid Scale model

In this paper we demonstrate that the transport equation of the generalised subgrid scale (SGS) turbulent stress tensor is form-invariant but not frame-indifferent under Euclidean transformations of the frame. A new closure equation between the generalized SGS turbulent stress tensor and the resolved kinematic quantities is proposed. The closure equation at the basis of the proposed model (Two-Equation Model, TEM): a) respects the principle of the turbulence frame indifference [1]; b) takes into account both the anisotropy of the turbulence velocity scales and turbulence length scales; c) removes any balance assumption between the production and dissipation of SGS turbulent kinetic energy; d) assumes scale similarity in the definition of the second-order tensor representing the turbulent velocity scales. In the proposed model: a) the closure coefficient C which appears in the constitutive equation is uniquely determined without using Germanos dynamic procedure [2]; b) the generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy (which is calculated by means of its balance equation) and the modified Leonard tensor; c) the viscous dissipation of the generalized SGS turbulent kinetic energy is calculated by solving the balance equation. The proposed model is tested for a turbulent channel flow at Reynolds numbers (based on friction velocity and channel half-width) ranging from 180 to 2340.Received: 11 February 2004, Accepted: 20 August 2004, Published online: 22 February 2005PACS: 02.60.Cb, 47.27.Eq, 47.11. + j Correspondence to: F. Gallerano     

Correcting Anisotropic Gradient Transport of k

The gradient transport model for k is extended to classes of turbulent flows for which the gradient transport hypothesis is relevant but the anisotropy of the Reynolds stress, to which the eddy diffusivity is proportional, is large and variable. In highly anisotropic turbulence the standard isotropic model used in engineering practice is fundamentally wrong and the conventional anisotropic approximation inadequate. The work is motivated by the important observations that the eddy diffusivity coefficient for a standard gradient transport model for various transported quantities is a factor of 3–10 times larger in highly anisotropic turbulence than that used in standard engineering models. While the conventional anisotropic eddy diffusivity approximation appears adequate for material conserved scalars it is inadequate for k. The problem is solved by addressing the anisotropy of the turbulent transport of k at the level of the underlying third order tensor. It is shown that, unlike the traditional transport models for k, the orientation of the anisotropy with respect to the direction of the gradient plays a crucial role not accounted for in conventional models used in engineering calculations. The new anisotropic eddy diffusivity tensor is quadratic in the anisotropy (the traditional model is linear in the anisotropy). It is shown that the new more rigorous anisotropic eddy diffusivity varies 300% more than the standard model comparing the isotropic limit to the value for the two-dimensional limit. The two-dimensional limit is important in strongly stably stratified flows, in pressure gradient or shock driven flows and in rotating flows. Using the simple shear and the homogeneous non-equilibrium Rayleigh Taylor turbulence the new anisotropic diffusivity tensor is validated in inhomogeneous Rayleigh Taylor turbulence at early and late times.     

Nonlinear turbulence models for predicting strong curvature effects

Prediction of the characteristics of turbulent flows with strong streamline curvature, such as flows in turbomachines, curved channel flows, flows around airfoils and buildings, is of great importance in engineering applications and poses a very practical challenge for turbulence modeling. In this paper, we analyze qualitatively the curvature effects on the structure of turbulence and conduct numerical simulations of a turbulent Uduct flow with a number of turbulence models in order to assess their overall performance. The models evaluated in this work are some typical linear eddy viscosity turbulence models, nonlinear eddy viscosity turbulence models （NLEVM） （quadratic and cubic）, a quadratic explicit algebraic stress model （EASM） and a Reynolds stress model （RSM） developed based on the second-moment closure. Our numerical results show that a cubic NLEVM that performs considerably well in other benchmark turbulent flows, such as the Craft, Launder and Suga model and the Huang and Ma model, is able to capture the major features of the highly curved turbulent U-duct flow, including the damping of turbulence near the convex wall, the enhancement of turbulence near the concave wall, and the subsequent turbulent flow separation. The predictions of the cubic models are quite close to that of the RSM, in relatively good agreement with the experimental data, which suggests that these models may be employed to simulate the turbulent curved flows in engineering applications.     

Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies

The unsteady turbulent flow around bodies at high Reynolds number is predicted by an anisotropic eddy-viscosity model in the context of the Organised Eddy Simulation (OES). A tensorial eddy-viscosity concept is developed to reinforce turbulent stress anisotropy, that is a crucial characteristic of non-equilibrium turbulence in the near-region. The theoretical aspects of the modelling are investigated by means of a phase-averaged PIV in the flow around a circular cylinder at Reynolds number 1.4×10⁵. A pronounced stress–strain misalignment is quantified in the near-wake region of the detached flow, that is well captured by a tensorial eddy-viscosity concept. This is achieved by modelling the turbulence stress anisotropy tensor by its projection onto the principal matrices of the strain-rate tensor. Additional transport equations for the projection coefficients are derived from a second-order moment closure scheme. The modification of the turbulence length scale yielded by OES is used in the Detached Eddy Simulation hybrid approach. The detached turbulent flows around a NACA0012 airfoil (2-D) and a circular cylinder (3-D) are studied at Reynolds numbers 10⁵ and 1.4×10⁵, respectively. The results compared to experimental ones emphasise the predictive capabilities of the OES approach concerning the flow physics capture for turbulent unsteady flows around bodies at high Reynolds numbers.     

A qualitative assessment of the role of a viscosity depending on the third invariant of the rate-of-deformation tensor upon turbulent non-Newtonian flow

The numerical simulation of some non-Newtonian effects in wall and wall-free turbulent flows, such as drag reduction in pipe flows or the decrease in transverse normal Reynolds stresses, has been attempted in the past with a limited degree of success on the basis of modified wall functions applied to traditional turbulence models (k–ε), rather than through more realistic rheological constitutive equations. In this work, it is qualitatively shown that if the viscosity function of a generalised Newtonian fluid is assumed to depend on the third invariant of the rate of deformation tensor, there is an increase of the viscous diffusion terms, but especially, of the dissipation of turbulence kinetic energy by a factor equal to the Trouton ratio of the fluid, divided by the Trouton ratio of the solvent, thus indicating a possible way to improve rheological–turbulence modelling.