Scale separation for implicit large eddy simulation

With implicit large eddy simulation (ILES) the truncation error of the discretization scheme acts as subgrid-scale (SGS) model for the computation of turbulent flows. Although ILES is comparably simple, numerically robust and easy to implement, a considerable challenge is the design of numerical discretization schemes resulting in a physically consistent SGS model. In this work, we consider the implicit SGS modeling capacity of the adaptive central-upwind weighted-essentially-non-oscillatory scheme (WENO-CU6) [X.Y. Hu, Q. Wang, N.A. Adams, An adaptive central-upwind weighted essentially non-oscillatory scheme, J. Comput. Phys. 229 (2010) 8952–8965] by incorporating a physically-motivated scale-separation formulation. Scale separation is accomplished by a simple modification of the WENO weights. The resulting modified scheme maintains the shock-capturing capabilities of the original WENO-CU6 scheme while it is also able to reproduce the Kolmogorov range of the kinetic-energy spectrum for turbulence at the limit of infinite Reynolds number independently of grid resolution. For isentropic compressible turbulence the pseudo-sound regime of the dilatational kinetic-energy spectrum and the non-Gaussian probability-density function of the longitudinal velocity derivative are reproduced.     

A fully discrete,kinetic energy consistent finite-volume scheme for compressible flows

A robust, implicit, low-dissipation method suitable for LES/DNS of compressible turbulent flows is discussed. The scheme is designed such that the discrete flux of kinetic energy and its rate of change are consistent with those predicted by the momentum and continuity equations. The resulting spatial fluxes are similar to those derived using the so-called skew-symmetric formulation of the convective terms. Enforcing consistency for the time derivative results in a novel density weighted Crank–Nicolson type scheme. The method is stable without the addition of any explicit dissipation terms at very high Reynolds numbers for flows without shocks. Shock capturing is achieved by switching on a dissipative flux term which tends to zero in smooth regions of the flow. Numerical examples include a one-dimensional shock tube problem, the Taylor–Green problem, simulations of isotropic turbulence, hypersonic flow over a double-cone geometry, and compressible turbulent channel flow.     

Suitability of artificial bulk viscosity for large-eddy simulation of turbulent flows with shocks

The artificial bulk viscosity method to numerically capture shocks is investigated for large-eddy simulation (LES). Different variations of this method are tested on a turbulent flow over a cylinder at Reynolds number of 10,000 and free-stream Mach number of 0.85. The artificial bulk viscosity model by Cook and Cabot, which is parameterized by the strain rate magnitude, is found to provide unnecessary bulk viscosity in turbulent regions away from shocks. While developed turbulent structures are found unaffected, this extra bulk viscosity is shown to significantly damp the sound field. An alternative formulation of the model which is parameterized by the rate of dilatation is proposed. This formulation is shown to avoid the unnecessary bulk viscosity and enhance the sound-prediction capability of the model. It was found that standard LES combined with artificial bulk viscosity is a promising approach for simulation of turbulent flows with shocks. The formulation of the model on curvilinear coordinates is presented in the appendix.     

On the use of the discontinuous Galerkin method for numerical simulation of two-dimensional compressible turbulence with shocks

In this paper, the discontinuous Galerkin (DG) method combined with localized artificial diffusivity is investigated in the context of numerical simulation of broadband compressible turbulent flows with shocks for under-resolved cases. Firstly, the spectral property of the DG method is analyzed using the approximate dispersion relation (ADR) method and compared with typical finite difference methods, which reveals quantitatively that significantly less grid points can be used with DG for comparable numerical error. Then several typical test cases relevant to problems of compressible turbulence are simulated, including one-dimensional shock/entropy wave interaction, two-dimensional decaying isotropic turbulence, and two-dimensional temporal mixing layers. Numerical results indicate that higher numerical accuracy can be achieved on the same number of degrees of freedom with DG than high order finite difference schemes. Furthermore, shocks are also well captured using the localized artificial diffusivity method. The results in this work can provide useful guidance for further applications of DG to direct and large eddy simulation of compressible turbulent flows.     

A new mixed subgrid-scale model for large eddy simulation of turbulent drag-reducing flows of viscoelastic fluids

A mixed subgrid-scale(SGS) model based on coherent structures and temporal approximate deconvolution(MCT) is proposed for turbulent drag-reducing flows of viscoelastic fluids. The main idea of the MCT SGS model is to perform spatial filtering for the momentum equation and temporal filtering for the conformation tensor transport equation of turbulent flow of viscoelastic fluid, respectively. The MCT model is suitable for large eddy simulation(LES) of turbulent dragreducing flows of viscoelastic fluids in engineering applications since the model parameters can be easily obtained. The LES of forced homogeneous isotropic turbulence(FHIT) with polymer additives and turbulent channel flow with surfactant additives based on MCT SGS model shows excellent agreements with direct numerical simulation(DNS) results. Compared with the LES results using the temporal approximate deconvolution model(TADM) for FHIT with polymer additives, this mixed SGS model MCT behaves better, regarding the enhancement of calculating parameters such as the Reynolds number.For scientific and engineering research, turbulent flows at high Reynolds numbers are expected, so the MCT model can be a more suitable model for the LES of turbulent drag-reducing flows of viscoelastic fluid with polymer or surfactant additives.     

Dynamic subgrid scale modeling of turbulent flows using lattice-Boltzmann method

In this paper, we discuss the incorporation of dynamic subgrid scale (SGS) models in the lattice-Boltzmann method (LBM) for large-eddy simulation (LES) of turbulent flows. The use of a dynamic procedure, which involves sampling or test-filtering of super-grid turbulence dynamics and subsequent use of scale-invariance for two levels, circumvents the need for empiricism in determining the magnitude of the model coefficient of the SGS models. We employ the multiple relaxation times (MRT) formulation of LBM with a forcing term, which has improved physical fidelity and numerical stability achieved by proper separation of relaxation time scales of hydrodynamic and non-hydrodynamic modes, for simulation of the grid-filtered dynamics of large-eddies. The dynamic procedure is illustrated for use with the common Smagorinsky eddy-viscosity SGS model, and incorporated in the LBM kinetic approach through effective relaxation time scales. The strain rate tensor in the SGS model is locally computed by means of non-equilibrium moments of the MRT-LBM. We also discuss proper sampling techniques or test-filters that facilitate implementation of dynamic models in the LBM. For accommodating variable resolutions, we employ conservative, locally refined grids in this framework. As examples, we consider the canonical anisotropic and inhomogeneous turbulent flow problem, i.e. fully-developed turbulent channel flow at two different shear Reynolds numbers Re_∗ of 180 and 395. The approach is able to automatically and self-consistently compute the values of the Smagorinsky coefficient, C_S. In particular, the computed value in the outer or bulk flow region, where turbulence is generally more isotropic, is about 0.155 (or the model coefficient ) which is in good agreement with prior data. It is also shown that the model coefficient becomes smaller and approaches towards zero near walls, reflecting the dampening of turbulent length scales near walls. The computed turbulence statistics at these Reynolds numbers are also in good agreement with prior data. The paper also discusses a procedure for incorporation of more general scale-similarity based SGS stress models.     

混合通量差分格式的比较

系统研究了几种混合通量差分格式的构造方法和耗散模型,分别对低速平板绕流、二维跨音速喷管流动和高超音速钝头体无粘绕流进行了数值模拟,结合先进的EASM湍流模型对格式的粘性分辨率和激波稳定性进行了细致的比较分析.结果表明混合通量差分格式兼顾了FDS和FVS格式的优点,具有较高的间断分辨率和数值稳定性.     

Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves

Flows in which shock waves and turbulence are present and interact dynamically occur in a wide range of applications, including inertial confinement fusion, supernovae explosion, and scramjet propulsion. Accurate simulations of such problems are challenging because of the contradictory requirements of numerical methods used to simulate turbulence, which must minimize any numerical dissipation that would otherwise overwhelm the small scales, and shock-capturing schemes, which introduce numerical dissipation to stabilize the solution. The objective of the present work is to evaluate the performance of several numerical methods capable of simultaneously handling turbulence and shock waves. A comprehensive range of high-resolution methods (WENO, hybrid WENO/central difference, artificial diffusivity, adaptive characteristic-based filter, and shock fitting) and suite of test cases (Taylor–Green vortex, Shu–Osher problem, shock-vorticity/entropy wave interaction, Noh problem, compressible isotropic turbulence) relevant to problems with shocks and turbulence are considered. The results indicate that the WENO methods provide sharp shock profiles, but overwhelm the physical dissipation. The hybrid method is minimally dissipative and leads to sharp shocks and well-resolved broadband turbulence, but relies on an appropriate shock sensor. Artificial diffusivity methods in which the artificial bulk viscosity is based on the magnitude of the strain-rate tensor resolve vortical structures well but damp dilatational modes in compressible turbulence; dilatation-based artificial bulk viscosity methods significantly improve this behavior. For well-defined shocks, the shock fitting approach yields good results.     

湍流模型在复杂流场数值模拟中的应用

采用4种湍流模型:代数Baldwin Lomax(B-L)模型、半方程Johnson King(J-K)模型的两个版本(J-K90A和J K92)以及两方程k-g模型,分别数值模拟了导弹超音速流动、NASATND-712标模和民机翼身组合体(两区C-O网格)跨音速流动.采用中心有限体积和多步Runge-Kutta方法数值积分三维可压缩雷诺平均Navier-Stokes(N-S)方程组.k-g湍流模型方程的求解采用类似于N-S方程组的方法进行.所有湍流模型均能很好地模拟附体及小分离流动;对于大攻角、分离剧烈的导弹流动,k-g和J-K92模型与实验吻合更好;B-L模型在模拟民机跨音速流动时,它所捕捉的激波位置较其余3种模型靠后.利用多块网格模拟民机翼身组合体流场时,k-g模型的模拟能力强于其余3种模型.     

不同亚格子模型在亚声速槽道流大涡模拟中的应用对比

湍流边界层流动是一种广泛存在于飞行器内部和外部的流动现象，是基础理论和模型验证的重要研究对象.能够捕捉大部分流动细节且计算量适中的大涡模拟（large-eddy simulation，LES）方法在湍流数值模拟中得到了越来越广泛的应用.文章基于格心有限差分方法，使用4阶紧致中心格式离散N-S方程无黏项，分别应用5种不同的亚格子（subgrid-scale，SGS）模型，即隐式，SM（Smagorinsky model），DSM（dynamic Smagorinsky model），WALE（wall-adapting local eddy-viscosity model）和CSM（coherent structures model），对Re = 3 000，Ma = 0.5的等温壁面槽道流动进行了大涡模拟研究.与实验值和直接数值模拟（direct numerical simulation，DNS）结果对比后发现，流场平均温度、平均密度等热力学量以及平均流向速度对亚格子模型不敏感，不适宜作为判断模型优劣的判据.亚格子模型在壁面附近的耗散越大，壁面摩擦速度以及阻力系数就越小.对于与速度相关的脉动量来说，不同模型得到的结果在壁面和脉动峰值附近误差比较大，中心线附近较小；显式模型结果在流向速度峰值处均高于参考值，而在展向和壁面法向速度脉动峰值处则均偏低.考虑显式的4种模型在壁面附近的涡黏系数分布，DSM和CSM曲线满足涡黏系数与无量纲壁面距离3次方成正比的分布规律，SM曲线斜率偏小而WALE曲线斜率偏大.      

Explicit algebraic Reynolds stress model for compressible flow turbulence

Algebraic Reynolds stress model (ARSM) is often employed in practical turbulent flow simulations. Most of previous works on ARSM have been carried out for incompressible flows. In the present paper, a new ARSM model is suggested for compressible flows. The model adopts a compressibility factor function involving the turbulent Mach number and the gradient Mach number. Compared to incompressible flow, explicit solution for ARSM for compressible flow can hardly be obtained due to dilatation terms. We propose approximate representations for these dilatation-related terms to obtain an explicit procedure for compressible flow turbulence. The model is applied to compressible mixing layer, supersonic flat-plate boundary and planar supersonic wake flow. It is found that the model works very well yielding results that are in good agreement with the DNS and the experimental data.     

An improved dynamic subgrid-scale model and its application to large eddy simulation of rotating channel flows

Rotating turbulence occurs extensively in nature and engineering circumstances. Meanwhile, understanding physical mechanisms of the rotating turbulence is important to the fundamental research of turbulence. The turbulent flow in rotating frames undergoes two kinds of Coriolis force effects. First, a secondary flow is induced in the case that there is a mean vorticity component perpendicular to the rotating axis. Second, there are augmenting or suppressing effects on the turbulence if there i…     

A Reynolds stress model for turbulent flows of viscoelastic fluids

A second-order closure is developed for predicting turbulent flows of viscoelastic fluids described by a modified generalised Newtonian fluid model incorporating a nonlinear viscosity that depends on a strain-hardening Trouton ratio as a means to handle some of the effects of viscoelasticity upon turbulent flows. Its performance is assessed by comparing its predictions for fully developed turbulent pipe flow with experimental data for four different dilute polymeric solutions and also with two sets of direct numerical simulation data for fluids theoretically described by the finitely extensible nonlinear elastic – Peterlin model. The model is based on a Newtonian Reynolds stress closure to predict Newtonian fluid flows, which incorporates low Reynolds number damping functions to properly deal with wall effects and to provide the capability to handle fluid viscoelasticity more effectively. This new turbulence model was able to capture well the drag reduction of various viscoelastic fluids over a wide range of Reynolds numbers and performed better than previously developed models for the same type of constitutive equation, even if the streamwise and wall-normal turbulence intensities were underpredicted.     

尺度自适应模拟和大涡模拟的关联性分析

采用理论分析和数值模拟相结合的方法,系统研究了尺度自适应模拟（scale-adaptive simulation,SAS）和大涡模拟（large-eddy simulation,LES）的关联性问题.在理论分析方面,对比分析了系综平均和滤波的定义、Spalart-Allmaras（SA）湍流模型和动态亚格子（subgrid-scale,SGS）模型关于湍流黏性系数的求解方式.理论分析结果表明,系综平均等价于盒式直接滤波,SAS和LES的控制方程在数学形式上具有一致性;SAS存在过多的湍流耗散,主要来自于SA输运方程中的扩散项.在数值模拟方面,选取来流Mach数0.55,Reynolds数2×105的圆柱可压缩绕流为分析算例.计算结果表明,SAS和LES预测的大尺度平均流场信息几乎一致,SAS预测的湍流脉动信息略低于LES.SAS在圆柱近尾迹区的湍流耗散过大,而在稍远的尾迹区几乎能够完全等效于LES.      

An adaptive implicit–explicit scheme for the DNS and LES of compressible flows on unstructured grids

An adaptive implicit–explicit scheme for Direct Numerical Simulation (DNS) and Large-Eddy Simulation (LES) of compressible turbulent flows on unstructured grids is developed. The method uses a node-based finite-volume discretization with Summation-by-Parts (SBP) property, which, in conjunction with Simultaneous Approximation Terms (SAT) for imposing boundary conditions, leads to a linearly stable semi-discrete scheme. The solution is marched in time using an Implicit–Explicit Runge–Kutta (IMEX-RK) time-advancement scheme. A novel adaptive algorithm for splitting the system into implicit and explicit sets is developed. The method is validated using several canonical laminar and turbulent flows. Load balance for the new scheme is achieved by a dual-constraint, domain decomposition algorithm. The scalability and computational efficiency of the method is investigated, and memory savings compared with a fully implicit method is demonstrated. A notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed.     

Constrained large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions

Constrained large-eddy simulation (CLES) method has been recently developed by Chen and his colleagues for simulating attached and detached wall-bounded turbulent flows. In CLES, the whole domain is simulated using large-eddy simulation (LES) while a Reynolds stress constraint is enforced on the subgrid-scale (SGS) stress model for near wall regions. In this paper, CLES is used to simulate the separated flow in a channel with streamwise-periodic constrictions at Re = 10,595. The results of CLES are compared with those of Reynolds-averaged Navier-Stokes (RANS) method, LES, detached eddy simulation (DES) and previous LES results by Breuer et al. and Ziefle et al. Although a coarse grid is used, our results from the present LES, DES and CLES do not show large deviations from the reference results using much finer grid resolution. The comparison also shows that CLES performs the best among different turbulence models tested, demonstrating that the CLES provides an excellent alternative model for separated flows. Furthermore, the cross-comparisons among different CLES implementations have been carried out. Our simulation results are in favor of using the constraint from algebraic RANS model or solving the RANS model equations in the whole domain with a length scale modification according to the idea from DES.     

Particle subgrid scale modelling in large-eddy simulations of particle-laden turbulence

This study is concerned with particle subgrid scale (SGS) modelling in large-eddy simulations (LESs) of particle-laden turbulence. Although many particle-laden LES studies have neglected the effect of the SGS on the particles, several particle SGS models have been proposed in the literature. In this research, the approximate deconvolution method (ADM) and the stochastic models of Fukagata et al. (Dynamics of Brownian particles in a turbulent channel flow, Heat Mass Transf. 40 (2004), 715–726) Shotorban and Mashayek (A stochastic model for particle motion in large-eddy simulation, J. Turbul. 7 (2006), 1–13) and Berrouk et al. (Stochastic modelling of inertial particle dispersion by subgrid motion for LES of high Reynolds number pipe flow, J. Turbul. 8 (2007), pp. 1–20) are analysed. The particle SGS models are assessed using both a priori and a posteriori simulations of inertial particles in a periodic box of decaying, homogeneous and isotropic turbulence with an initial Reynolds number of Re_λ = 74. The model results are compared with particle statistics from a direct numerical simulation (DNS). Particles with a large range of Stokes numbers are tested using various filter sizes and stochastic model constant values. Simulations with and without gravity are performed to evaluate the ability of the models to account for the crossing trajectory and continuity effects. The results show that ADM improves results but is only capable of recovering a portion of the SGS turbulent kinetic energy. Conversely, the stochastic models are able to recover sufficient SGS energy, but show a large range of results dependent on the Stokes number and filter size. The stochastic models generally perform best at small Stokes numbers, but are unable to predict preferential concentration.     

Validity of Taylor's dissipation-viscosity independence postulate in variable-viscosity turbulent fluid mixtures

G. I. Taylor's postulate [Proc. R. Soc. A 151, 421 (1935)] that dissipation is independent of viscosity at high Reynolds numbers is the foundation of many single-fluid turbulence theories and closure models. The validity of this key postulate in an important class of flows, turbulent mixtures, is not yet clearly established. We devise a simple numerical experiment of decaying turbulence in a mixture of two fluids of vastly different viscosities to examine dissipation scaling. Initially, the two fluids are segregated, and dissipation is directly proportional to viscosity. As turbulence evolves and fluids mix, the velocity gradients rapidly adapt to the viscosity field, and within one-half eddy turnover time, dissipation-viscosity independence is established. Viscosity-weighted velocity-gradient skewness is shown to be constant, leading to the validity of Taylor's postulate in turbulent mixtures.     

Evaluating the modulated gradient model in large eddy simulation of channel flow with OpenFOAM

Recently, a new family of subgrid-scale (SGS) models, termed as gradient-based models, has been introduced to calculate the SGS stresses in large eddy simulation (LES). In the present work, the modulated gradient model (MGM) was implemented in the OpenFOAM package, and the pimpleFoam solver was improved to be adopted with non-eddy viscosity models. The MGM is a new, nonlinear model that uses the local equilibrium hypothesis to assess the SGS kinetic energy and the velocity gradient tensor to calculate the relative weight of the different components of the SGS stress tensor. To evaluate the accuracy of the MGM along with the modified pimpleFoam solver, a turbulent channel flow was simulated at the three different frictional Reynolds numbers of 180, 395 and 590. Furthermore, the results were compared with direct numerical simulation data, as well as the numerical results obtained by the established SGS models such as the dynamic Smagorinsky model (DSM). A suitable accuracy for the first- and second-order turbulence parameters was reported. Moreover, it was demonstrated that MGM is computationally efficient compared to the DSM in treating channel flow.     

不同亚格子模式在后台阶湍流流动大涡模拟中的应用

本文用大涡模拟方法研究了湍流后台阶流场中的大涡演变过程,并在此基础上研究了目前大涡模拟中比较常用的六种亚格子模式。在相同的流动几何参数以及计算条件下,给出了不同亚格子模式下湍流流动瞬时压力场以及流场中瞬时粘性大小的分布,尤其给出了流场瞬时演变的大尺度涡结构。将六种亚格子模式从回流区长度、计算时间、计算结果的准确性和流场稳定性等不同角度进行了比较。为选取合适的亚格子模式深入研究湍流耗散机理奠定了基础。