A two‐scale low‐Reynolds number turbulence model

In this study, a two‐scale low‐Reynolds number turbulence model is proposed. The Kolmogorov turbulence time scale, based on fluid kinematic viscosity and the dissipation rate of turbulent kinetic energy (ν, ε), is adopted to address the viscous effects and the rapid increasing of dissipation rate in the near‐wall region. As a wall is approached, the turbulence time scale transits smoothly from a turbulent kinetic energy based (k, ε) scale to a (ν, ε) scale. The damping functions of the low‐Reynolds number models can thus be simplified and the near‐wall turbulence characteristics, such as the ε distribution, are correctly reproduced. The proposed two‐scale low‐Reynolds number turbulence model is first examined in detail by predicting a two‐dimensional channel flow, and then it is applied to predict a backward‐facing step flow. Numerical results are compared with the direct numerical simulation (DNS) budgets, experimental data and the model results of Chien, and Lam and Bremhorst respectively. It is proved that the proposed two‐scale model indeed improves the predictions of the turbulent flows considered. Copyright © 2000 John Wiley & Sons, Ltd.     

A k–$${\varepsilon}$$ turbulence closure model of an isothermal dry granular dense matter

The turbulent flow characteristics of an isothermal dry granular dense matter with incompressible grains are investigated by the proposed first-order k–\({\varepsilon}\) turbulence closure model. Reynolds-filter process is applied to obtain the balance equations of the mean fields with two kinematic equations describing the time evolutions of the turbulent kinetic energy and dissipation. The first and second laws of thermodynamics are used to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated by a quasi-linear theory. The established closure model is applied to analyses of a gravity-driven stationary flow down an inclined moving plane. While the mean velocity decreases monotonically from its value on the moving plane toward the free surface, the mean porosity increases exponentially; the turbulent kinetic energy and dissipation evolve, respectively, from their minimum and maximum values on the plane toward their maximum and minimum values on the free surface. The evaluated mean velocity and porosity correspond to the experimental outcomes, while the turbulent dissipation distribution demonstrates a similarity to that of Newtonian fluids in turbulent shear flows. When compared to the zero-order model, the turbulent eddy evolution tends to enhance the transfer of the turbulent kinetic energy and plane shearing across the flow layer, resulting in more intensive turbulent fluctuation in the upper part of the flow. Solid boundary as energy source and sink of the turbulent kinetic energy becomes more apparent in the established first-order model.     

Statistics and Modelling of Turbulent Kinetic Energy Transport in Different Regimes of Premixed Combustion

The statistical behaviour of turbulent kinetic energy transport in turbulent premixed flames is analysed using data from three-dimensional Direct Numerical Simulation (DNS) of freely propagating turbulent premixed flames under decaying turbulence. For flames within the corrugated flamelets regime, it is observed that turbulent kinetic energy is generated within the flame brush. By contrast, for flames within the thin reaction zones regime it has been found that the turbulent kinetic energy decays monotonically through the flame brush. Similar trends are observed also for the dissipation rate of turbulent kinetic energy. Within the corrugated flamelets regime, it is demonstrated that the effects of the mean pressure gradient and pressure dilatation within the flame are sufficient to overcome the effects of viscous dissipation and are responsible for the observed augmentation of turbulent kinetic energy in the flame brush. In the thin reaction zones regime, the effects of the mean pressure gradient and pressure dilatation terms are relatively much weaker than those of viscous dissipation, resulting in a monotonic decay of turbulent kinetic energy across the flame brush. The modelling of the various unclosed terms of the turbulent kinetic energy transport equation has been analysed in detail. The predictions of existing models are compared with corresponding quantities extracted from DNS data. Based on this a-priori DNS assessment, either appropriate models are identified or new models are proposed where necessary. It is shown that the turbulent flux of turbulent kinetic energy exhibits counter-gradient (gradient) transport wherever the turbulent scalar flux is counter-gradient (gradient) in nature. A new model has been proposed for the turbulent flux of turbulent kinetic energy, and is found to capture the qualitative and quantitative behaviour obtained from DNS data for both the corrugated flamelets and thin reaction zones regimes without the need to adjust any of the model constants.     

一类平衡大气边界层边界条件研究

基于标准k-ε湍流模型,首先利用湍流粘度方程和剪切应力在整个边界层内恒定的假设,推导出一类耗散率表达式,并根据常用的湍动能入口剖面方程以及平均风速剖面方程,计算获得相应的耗散率方程;然后在输运方程中添加自定义源项,通过已经确定的平均速度方程、湍动能方程、耗散率方程计算得到相应输运方程的自定义源项表达式,并进行空风洞数值模拟,从而得到了一类满足平衡大气边界层的来流边界条件．通过将这种边界条件与由湍流平衡条件得到的边界条件进行比较,表明本方法获得的边界条件更适用．并且,本方法无需考虑修正壁面函数和修正湍流模型常数,因而计算更为简单,可为平衡大气边界层的研究提供一种新的思路．     

FINITE ELEMENT SOLUTIONS OF LAMINAR AND TURBULENT MIXED CONVECTION IN A DRIVEN CAVITY

This paper presents laminar and turbulent mixed convection solutions of a driven cavity flow using the finite element method. For the laminar flow, distributions of velocity and temperature with and without the effect of buoyancy force are presented and compared. For the turbulent flow, governing partial differential equations of the thermal turbulence two-equati on model and kinetic turbulence two-equation model are used. Corresponding results such as kinetic eddy diffusivity, kinetic eddy energy, thermal eddy energy and their dissipations are presented.     

Turbulence in multiphase flow

We establish in this paper the foundations of a two-field turbulent flow model that includes two turbulent fields. The case of dispersed particles in an incompressible carrier fluid is treated here, but the very presence of these two fields allows for the generalization of the model to the instability-induced turbulent mixing of two materials. This model describes both cases of turbulent mass diffusion and small drag regime, “wave-like” interpenetration of the two components. It also includes the damping of the turbulence due to the presence of the particles. In addition, a theoretical derivation of the drag-induced decay of the large-scale turbulence kinetic energy is proposed as another mechanism specific to turbulent multiphase flow.     

On the interfacial roughness scale in turbulent stratified two-phase flow: 3D lattice Boltzmann numerical simulations with forced turbulence and surfactant

Numerical 3D simulations of turbulent, stratified two-phase shear flow with a surfactant laden interface were used to test and develop a phenomenological interfacial roughness scale model where the energy required to deform the interface (buoyancy, interfacial tension, and viscous work) is proportional to the turbulent kinetic energy adjacent to the interface.The turbulence was forced in the upper and lower liquids in the simulations, to emulate the interfacial dynamics without requiring (prohibitively) large simulation domains and Reynolds numbers. The addition of surfactant lead to an increased roughness scale (for the same turbulent kinetic energy) due to the introduction of interfacial dilatational elasticity that suppressed horizontal motion parallel to the interface, and enhanced the vertical motion.The phenomenological roughness scale model was not fully developed for dilatational elasticity in this work, but we proposed a source term that represents surfactant induced pressure fluctuations near the interface. This source term should be developed further to account for the relation between surfactant density fluctuations and turbulence adjacent to the interface. We foresee that the roughness scale model can be used as a basis for more general interfacial closure relations in Reynolds averaged turbulence models, where also mobile surfactant is accounted for.     

可压缩各向同性衰减湍流直接数值模拟研究

采用五阶有限差分WENO格式直接模拟了高初始湍流Mach数的可压缩均匀各向同性湍流,主要分析了湍流的统计特性 和压缩性的影响,包括能谱特征、激波串、耗散率、标度律等. 研究表明,湍动能主要来自于速度场螺旋分量的贡献;各向同性湍流的小尺度脉动对压缩性更为敏感,并且压缩性的增强加快了湍流大 尺度脉动向小尺度脉动的湍动能输运;随着湍流Mach数的升高,胀量(压缩)耗散率所占比率也显著增长. 标度律分析表明,强可压缩湍流的横向速度结构函数仍然具有扩展自相似性;当阶数较高(p ≥ 5)时,纵向速度结构函数的扩展自相似性则不再成立. 对于压缩性较弱的湍流,与不可压缩湍流一致,横向湍流脉动的间歇性要强于纵向湍流脉动;而对于强可压缩湍流,纵向湍流脉动的 间歇性要强于横向湍流脉动.     

一种基于湍动能方程的转捩判定方法简

转捩现象是阻碍阻力高精度求解的主要问题之一. Menter 和Langtry 所提出的γ-θ转捩模型通过引入涡量雷诺数和间歇因子输运方程来驱动转捩,但是其中很多经验公式的理论立足点有待商榷. 驱使层流转变到湍流依赖的仍然是平均速度的一阶和二阶相关量,它们组合构成了湍动能方程的耗散尺度. 在湍动能方程中做合适的耗散平衡后,仅仅依靠湍动能方程可以有效地捕捉转捩现象. 采用自然转捩和旁路转捩测试算例进行了验证,结果证明该方法与试验值匹配较好,具有一定的工程实用价值.     

Numerical investigation of the flow and heat behaviours of an impinging jet

The flow and temperature fields of a turbulent impinging jet are rather complex. In order to accurately describe the flow and heat-transfer process, two important factors that must be taken into account are the turbulence model and the wall function. Several turbulence models, including κ–? turbulence models, κ–ω turbulence models, low-Re turbulence models, the κ–κ_l–ω turbulence model, the Transition SST turbulence model, the V2F turbulence model and the RSM turbulence model, are examined and compared to experimental data. Furthermore, for the near wall region, various wall functions are presented for comparison and they include the standard wall function, the scale wall function, the non-equilibrium wall function and the enhanced wall function. The distribution features of velocity, turbulent kinetic energy and Nusselt number are determined in order to provide a reliable reference for the multiphase impinging jet in the future.     

A turbulent time scale based k–ε model for probability density function modeling of turbulence/chemistry interactions: Application to HCCI combustion

Homogenous Charge Compression Ignition (HCCI) engine technology is known as an alternative to reduce NO_x and particulate matter (PM) emissions. As shown by several experimental studies published in the literature, the ideally homogeneous mixture charge becomes stratified in composition and temperature, and turbulent mixing is found to play an important role in controlling the combustion progress. In a previous study, an IEM model (Interaction by Exchange with the Mean) has been used to describe the micromixing in a stochastic reactor model that simulates the HCCI process. The IEM model is a deterministic model, based on the principle that the scalar value approaches the mean value over the entire volume with a characteristic mixing time. In this previous model, the turbulent time scale was treated as a fixed parameter. The present study focuses on the development of a micro-mixing time model, in order to take into account the physical phenomena it stands for. For that purpose, a (k–ε) model is used to express this micro-mixing time model. The turbulence model used here is based on zero dimensional energy cascade applied during the compression and the expansion cycle; mean kinetic energy is converted to turbulent kinetic energy. Turbulent kinetic energy is converted to heat through viscous dissipation. Besides, in this study a relation to calculate the initial heterogeneities amplitude is proposed. The comparison of simulation results against experimental data shows overall satisfactory agreement at variable turbulent time scale.     

一种基于湍动能方程的转捩判定方法

转捩现象是阻碍阻力高精度求解的主要问题之一. Menter 和Langtry 所提出的γ-θ转捩模型通过引入涡量雷诺数和间歇因子输运方程来驱动转捩,但是其中很多经验公式的理论立足点有待商榷. 驱使层流转变到湍流依赖的仍然是平均速度的一阶和二阶相关量,它们组合构成了湍动能方程的耗散尺度. 在湍动能方程中做合适的耗散平衡后,仅仅依靠湍动能方程可以有效地捕捉转捩现象. 采用自然转捩和旁路转捩测试算例进行了验证,结果证明该方法与试验值匹配较好,具有一定的工程实用价值.      

可压缩均匀剪切湍流的二阶矩模拟

基于对可压缩湍流中脉动压力场和脉动速度场特征的理论分析以及ＤＮＳ结果,建立了可均匀剪切湍流中压力－变形率关联的压缩性修正模式,应用这个模式,加上Ｓａｒｋａｒ等建立的脉动体胀率项（ｄｉｌａｔａｔｉｏｎａｌ ｔｅｒｍｓ）的模式,预测可压缩均匀剪切湍流随时间的发展,所得雷诺应力各是性张量的平衡值与Ｂｌａｉｓｄｅｌｌ等的ＤＮＳ数据非常一致。这个模式准确地预测出均匀剪切湍流中压缩性导致的雷诺应力结构的“流向     

Applications of a higher‐order bounded numerical scheme to turbulent flows

This paper applies the higher‐order bounded numerical scheme Weighted Average Coefficients Ensuring Boundedness (WACEB) to simulate two‐ and three‐dimensional turbulent flows. In the scheme, a weighted average formulation is used for interpolating the variables at cell faces and the weighted average coefficients are determined from a normalized variable formulation and total variation diminishing (TVD) constraints to ensure the boundedness of the solution. The scheme is applied to two turbulent flow problems: (1) two‐dimensional turbulent flow around a blunt plate; and (2) three‐dimensional turbulent flow inside a mildly curved U‐bend. In the present study, turbulence is evaluated by using a low‐Reynolds number version of the k–ω model. For the flow simulation, the QUICK scheme is applied to the momentum equations while either the WACEB scheme (Method 1) or the UPWIND scheme (Method 2) is used for the turbulence equations. The present study shows that the WACEB scheme has at least second‐order accuracy while ensuring boundedness of the solutions. The present numerical study for a pure convection problem shows that the ‘TVD’ slope ranges from 2 to 4. For the turbulent recirculating flow, two different mixed procedures (Method 1 and Method 2) produce a substantial difference for the mean velocities as well as for the turbulence kinetic energy. Method 1 predicts better results than Method 2 does, comparing the analytical solution and the experimental data. For the turbulent flow inside the mildly curved U‐bend, although the predictions of velocity distributions with two procedures are very close, a noticeable difference of turbulence kinetic energy is exhibited. It is noticed that the discrepancy exists between numerical results and the experimental data. The reason is the limit of the two‐equation turbulence model to such complex turbulent flows with extra strain‐rates. Copyright © 2001 John Wiley & Sons, Ltd.     

Turbulent cascade modeling of single and bubbly two-phase turbulent flows

The analysis of turbulent two-phase flows requires closure models in order to perform reliable computational multiphase fluid dynamics (CMFD) analyses. A spectral turbulence cascade-transport model, which tracks the evolution of the turbulent kinetic energy from large to small liquid eddies, has been developed for the analysis of the homogeneous decay of isotropic single and bubbly two-phase turbulence. This model has been validated for the decay of homogeneous, isotropic single and two-phase bubbly flow turbulence for data having a 5 mm mean bubble diameter. The Reynolds number of the data based on bubble diameter and relative velocity is approximately 1400.     

A nonlinear kp-εp particle two-scale turbulence model and its application

A particle nonlinear two-scale turbulence model is proposed for simulating the anisotropic turbulent two-phase flow. The particle kinetic energy equation for two-scale fluctuation, particle energy transfer rate equation for large-scale fluctuation, and particle turbulent kinetic energy dissipation rate equation for small-scale fluctuation are derived and closed. This model is used to simulate gas–particle flows in a sudden-expansion chamber. The simulation is compared with the experiment and with those obtained by using another two kinds of tow-phase turbulence model, such as the single-scale two-phase turbulence model and the particle two-scale second-order moment (USM) two-phase turbulence model. It is shown that the present model gives simulation in much better agreement with the experiment than the single-scale two-phase turbulence model does and is almost as good as the particle two-scale USM turbulence model. The project supported by China Postdoctoral Science Foundation (2004036239).     

Turbulent boundary-layer analysis using finite elements

The use of finite element methods for turbulent boundary-layer flow is relatively recent and of limited extent.¹ In the present study, we extend the group variable approach of Fletcher and Fleer^2,3 to treat turbulent boundary layer flows with heat transfer using a two-equation turbulence model. The main concepts in the formulations include a Dorodnitsyn-type transformation which uses a velocity component as the transverse variable, a ‘variational’ formulation for the transformed equations using special test functions and development of a two-equation turbulence model in terms of the turbulent kinetic energy and turbulence dissipation rate as additional field variables. Several numerical test cases have been examined comparing the results with finite difference calculations and comparing the two-equation turbulence model with an algebraic turbulence model.     

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     

Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion

A numerical study based on the Eulerian–Lagrangian formulation is performed for dispersed phase motion in a turbulent flow. The effect of spatial filtering, commonly employed in large-eddy simulations, and the role of the subgrid scale turbulence on the statistics of heavy particles, including preferential concentration, are studied through a priori analysis of DNS of particle-laden forced isotropic turbulence. In simulations where the subgrid scale kinetic energy attains 30–35% of the total we observe the impact of residual fluid motions on particles of a smaller inertia. It is shown that neglecting the influence of subgrid scale fluctuations has a significant effect on the preferential concentration of those particles. A stochastic Langevin model is proposed to reconstruct the residual (or subgrid scale) fluid velocity along particle trajectories. The computation results for a selection of particle inertia parameters are performed to appraise the model through comparisons of particle turbulent kinetic energy and the statistics of preferential concentrations.     

基于雷诺应力模型的平衡大气边界层模拟

基于雷诺应力湍流模型（简称RSM模型）,研究了平衡大气边界层风场数值模拟问题．假设流体不可压,且不计雷诺应力输运方程中的对流项、浮力产生项、系统旋转产生项和扩散项,在准各向同性的条件下,推导出RSM模型湍动能k的表达式是标准k-ε模型k常数表达式的0.893倍．考虑k沿高度变化的修正,根据在标准k-ε模型中满足水平均匀性的湍流来流边界条件,提出在RSM模型中产生平衡大气边界层的湍流来流边界条件．基于空风洞的数值模拟结果表明,与工程上常用的湍流来流边界条件相比,基于本文提出的湍流来流边界条件得到的风场水平均匀性更优,且在整个流域内,得到的雷诺应力剖面更合适．从而验证了该湍流来流边界条件的适用性．