Variational multiscale turbulence modelling in a high order spectral element method

In the variational multiscale (VMS) approach to large eddy simulation (LES), the governing equations are projected onto an a priori scale partitioning of the solution space. This gives an alternative framework for designing and analyzing turbulence models. We describe the implementation of the VMS LES methodology in a high order spectral element method with a nodal basis, and discuss the properties of the proposed scale partitioning. The spectral element code is first validated by doing a direct numerical simulation of fully developed plane channel flow. The performance of the turbulence model is then assessed by several coarse grid simulations of channel flow at different Reynolds numbers.     

Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

A new finite element method for the efficient discretization of elliptic homogenization problems is proposed. These problems, characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed. Recently, the finite element heterogeneous multiscale method (FE-HMM) has been proposed for such problems, based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh. Departing from the approach used in the FE-HMM, we show that interpolation techniques based on the reduced basis methodology (an offline-online strategy) allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions. This new method, called the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is significantly more efficient than the FE-HMM for high order macroscopic discretizations and for three-dimensional problems, when the repeated computation of micro problems over the whole computational domain is expensive. A priori error estimates of the RB-FE-HMM are derived. Numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method.     

A hierarchical fracture model for the iterative multiscale finite volume method

An iterative multiscale finite volume (i-MSFV) method is devised for the simulation of multiphase flow in fractured porous media in the context of a hierarchical fracture modeling framework. Motivated by the small pressure change inside highly conductive fractures, the fully coupled system is split into smaller systems, which are then sequentially solved. This splitting technique results in only one additional degree of freedom for each connected fracture network appearing in the matrix system. It can be interpreted as an agglomeration of highly connected cells; similar as in algebraic multigrid methods. For the solution of the resulting algebraic system, an i-MSFV method is introduced. In addition to the local basis and correction functions, which were previously developed in this framework, local fracture functions are introduced to accurately capture the fractures at the coarse scale. In this multiscale approach there exists one fracture function per network and local domain, and in the coarse scale problem there appears only one additional degree of freedom per connected fracture network. Numerical results are presented for validation and verification of this new iterative multiscale approach for fractured porous media, and to investigate its computational efficiency. Finally, it is demonstrated that the new method is an effective multiscale approach for simulations of realistic multiphase flows in fractured heterogeneous porous media.     

肋片对后台阶湍流流动影响的数值模拟

采用二阶全展开ETG有限元方法作为大涡模拟空间离散格式,计算了Reynolds数为47625条件下的后台阶湍流流动,结果与相关实验资料符合良好,在此基础上分析了附加肋片高度和肋距对后台阶湍流流动的影响.计算结果表明,不同肋高和肋距对台阶下游流动具有较大的影响,相应条件下台阶下游的涡系及其时变过程都发生了很大的变化,计算给出了台阶回流段长度随肋高和随肋距变化的曲线,并指出,在台阶下游附加肋片可以作为后台阶湍流流动一种简单有效的被动控制方式.     

Large eddy simulation of turbulent premixed flames using level-set G-equation

Level-set G-equation and stationary flamelet chemistry are used in large eddy simulation of a propane/air premixed turbulent flame stabilized by a bluff body. The aim was to study the interaction between the flame front and turbulent eddies, and in particular to examine the effect of sub-grid scale (SGS) eddies on the wrinkling of the flame surface. The results indicated that the two types of turbulence eddies—the resolved large scale eddies and the unresolved SGS eddies—have different effects on the flame. The fluctuation of the flame surface, which is responsible for the broadening of the time averaged mean flame brush by turbulence, depends on the large resolved turbulence eddies. Time averaged mean flow velocity, temperature, and major species concentrations mainly depend on the large scale resolved eddies. The unresolved SGS eddies contribute to the wrinkling at the SGS level and play an important role in the enhancement of the propagation speed of the resolved flame front. In addition, the spatially filtered intermediate species, such as radicals, and the spatially filtered reaction rates strongly depend on the small SGS eddies. The asymptotic behavior of flame wrinkling by the SGS eddies, with respect to the decrease in filter size and grid size, is investigated further using a simplified level-set equation in a model shear flow. It is shown that to minimize the influence of the SGS eddies, fine grid and filter size may have to be used.     

A variational multiscale method for turbulent flow simulation with adaptive large scale space

In turbulent flows it is only feasible to simulate large flow structures. Variational multiscale (VMS) methods define these flow structures by projections onto appropriate function spaces. This paper presents a finite element VMS method which chooses the large scale projection space adaptively. The adaption controls the influence of an eddy viscosity model and it is based on the size of the so-called resolved small scales. The adaptive procedure is described in detail. Numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented. It is shown that the method selects the large scale space in a reasonable way and that appropriately chosen parameters improve the results compared to the basic method, which uses the same local large scale space in the whole domain and for all times.     

Variational multiscale element-free Galerkin method for 2D Burgers’ equation

A new numerical method, which is based on the coupling between variational multiscale method and meshfree methods, is developed for 2D Burgers’ equation with various values of Re. The proposed method takes full advantage of meshfree methods, therefore, no mesh generation and mesh recreation are involved. Meanwhile, compared with the variational multiscale finite element method, a strong assumption, that is, the fine scale vanishes identically over the element boundaries although non-zero within the elements, is not needed. Subsequently two problems which have an available analytical solution of their own are solved to analyze the convergence behaviour of the proposed method. Finally a 2D Burgers’ equation having large Re is solved and the results have also been compared with the ones computed by two other methods. The numerical results show that the proposed method can indeed obtain accurate numerical results for 2D Burgers’ equation having large Re, which does not refer to the choice of a proper stabilization parameter.     

Aperture-averaging effects for weak to strong scintillations in turbulent atmosphere

Under the approximations of (1) the received irradiance fluctuations of an optical wave caused by small scale turbulent eddies are multiplicatively modulated by the fluctuations caused by large scale turbulent eddies;(2) the scintillations caused by small- and large-scale eddies, respectively, are statistically independent; (3)the Rytov method for optical scintillation collected by the finite-diameter receiving aperture is valid for light wave propagation under weak to saturation fluctuation regime, we develop the applicable apertureaveraging analytic formulas in the week-to-strong-fluctuation for the scintillations of plane and spherical waves, which include the outer- and inner-scale rules of turbulence.     

各向同性紊流能量衰减的大涡模拟

在多重网格驱动的,高效且得到充分验证的有限体积方法的基础上发展了可压缩流大涡模拟的方法.空间离散采用Jameson的中心格式附加二阶和四阶耗散的方法,时间推进则采用了双时间步长的方法.亚格子剪切应力张量和亚格子热通量用Smagorinsky模型进行模拟.通过对各向同性紊流能量衰减的模拟来验证本方法的准确性和高效性,模拟得到的能量谱和紊流动能衰减历程与过滤后的CBC实验数据吻合良好.     

Irradiance scintillation index for Gaussian beam based on the generalized anisotropic von Karman spectrum

Experiments and theoretical investigations have shown that the atmosphere turbulence exhibits both anisotropic and non-Kolmogorov properties. In this paper, based on the anisotropic generalized von Karman spectrum and the Rytov approximation theory, new expression for the irradiance scintillation index of optical waves is derived for Gaussian beam propagating through weak anisotropic non-Kolmogorov turbulence. Compared with previously published results, it considers simultaneously the asymmetry property of turbulence cells or eddies in the orthogonal xy-plane, the general spectral power law in the range 3–4 instead of constant value of 11/3 for the Kolmogorov turbulence, and the finite turbulence inner and outer scales. Two anisotropic factors are introduced to parameterize the anisotropy of turbulence cells or eddies in horizontal and vertical directions. In the special cases of these two anisotropic factors equaling one and the finite turbulence inner and outer scales equaling separately zero and infinite, the derived expression can reduce correctly to the previously published results. Calculations are performed to analyze the derived results.     

多孔复合介质周期结构热传导和质扩散问题的多尺度数值方法

本文针对一类复杂的多孔复合介质的热传导和质扩散问题,给出具体的多尺度渐近展开公式,并在此基础上设计了有限元算法格式,它是宏观和细观相结合的数值方法。理论分析和数值实验均表明：多尺度数值方法对求解多孔复合介质周期结构的热传导和质扩散问题是可行的和有效的。     

A Eulerian PDF scheme for LES of nonpremixed turbulent combustion with second-order accurate mixture fraction

Monte Carlo simulations of joint probability density function (PDF) approaches have been developed in the past largely with Reynolds averaged Navier Stokes (RANS) applications. Current interests are in the extension of PDF approaches to large eddy simulation (LES). As LES resolves accurately the large scales of turbulence in time, the Monte Carlo simulation and the flow field need to be tightly coupled. A tight coupling can be achieved if the consistency between the scalar field solution obtained via finite-volume (FV) methods and that from the stochastic solution of the PDF is ensured. For nonpremixed turbulent flames with two distinct streams, the local reactive mixture is described by the mixture fraction. A Eulerian Monte Carlo method is developed to achieve a second-order accuracy in the instantaneous filtered mixture fraction that is consistent with the corresponding FV. The performances of the proposed scheme are extensively evaluated using a one-dimensional model. Then, the scheme is applied to two cases with LES. The first one is a non-reacting mixing flow of two different fluids. The second case is the Sandia piloted turbulent flame D with a steady state flamelet model. Both results confirm the consistency of the proposed method to the level of filtered mixture fraction.     

稳态燃烧室进气过程的大涡模拟

为精确描述燃烧室进气过程中的流场动态特性,本文应用大涡模拟模型对燃烧室稳态进气过程进行三维瞬态数值模拟研究,着重考察了三种不同亚网格模型(SGS)的性能,这三种SGS模型是:代数Smagorinsky模型、动态Smagorinsky模型和单方程动态动能输运模型(LDKEM)。计算结果表明LEDKEM模型和动态Smagorinsky模型均能较好地反映流场的瞬变性和随机性,前者性能最佳;而代数Smagorinsky模型精度较差,且不能模拟流场的微结构。     

Local–global multiscale model reduction for flows in high-contrast heterogeneous media

In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coarse grid. Local multiscale methods considered in this paper use special eigenvalue problems in a local domain to systematically identify important features of the solution. In particular, our local approaches are capable of homogenizing localized features and representing them with one basis function per coarse node that are used in constructing a weight function for the local eigenvalue problem. Global model reduction based on balanced truncation methods is used to identify important global coarse-scale modes. This provides a substantial CPU savings as Lyapunov equations are solved for the coarse system. Typical local multiscale methods are designed to find an approximation of the solution for any given coarse-level inputs. In many practical applications, a goal is to find a reduced basis when the input space belongs to a smaller dimensional subspace of coarse-level inputs. The proposed approaches provide efficient model reduction tools in this direction. Our numerical results show that, only with a careful choice of the number of degrees of freedom for local multiscale spaces and global modes, one can achieve a balanced and optimal result.     

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.     

Subgrid combustion modeling of 3-D premixed flames in the thin-reaction-zone regime

Large eddy simulation (LES) is used to investigate three-dimensional (3D) lean premixed turbulent methane–air flames in the thin-reaction-zone regime. In this regime, the Kolmogorov scale is smaller than the preheat zone thickness, but larger than the reaction zone thickness. Past numerical studies of similar flames were primarily direct numerical simulation either in two-dimensions or using the artificially thickened flame approach in 3D. For an LES the effect of small (unresolved) scales on the scalar field must be, modeled accurately to capture the correct flame structure. A subgrid combustion model based on the linear-eddy-mixing (LEM) model is used within an LES framework (called LEM–LES hereafter) to capture the 3D flame-structure of the highly stretched premixed flames. A finite-rate, one-step methane–air chemistry with a non-unity Lewis number formulation is used in this study. The simulated flame structure resembles flames experimentally studied in the thin-reaction-zone regime. Even though the preheat zone is broadened by the penetration of small eddies, the chemical reaction zone remains thin and localized. This feature is captured properly in the current LEM–LES approach. The flame structure and other statistics such as the flame area evolution, curvature, and strain-rate statistics computed using the LEM–LES are also in good agreement with the past DNS studies.     

An Inf-Sup Stabilized Finite Element Method by Multiscale Functions for the Stokes Equations

In the paper, an inf-sup stabilized finite element method by multiscale functions for the Stokes equations is discussed. The key idea is to use a Petrov-Galerkin approach based on the enrichment of the standard polynomial space for the velocity component with multiscale functions. The inf-sup condition for $P_1$-$P_0$ triangular element (or $Q_1$-$P_0$ quadrilateral element) is established. And the optimal error estimates of the stabilized finite element method for the Stokes equations are obtained.     

Density fluctuation spectrum in whistler turbulence

We develop a nonlinear two-dimensional fluid model of whistler turbulence that includes effect of electron fluid density perturbations. The latter is coupled nonlinearly with wave magnetic field. This coupling leads essentially to finite compressibility effects in whistler turbulence model. We find from our simulations that despite strong compressibility effects, the density fluctuations follow the evolution of the wave magnetic field fluctuations. In a characteristic regime where large scale whistlers are predominant, the coupled density fluctuations are found to follow a Kolmogorov-like phenomenology in the inertial range turbulence. Consequently, the turbulent energy is dominated by the large scale (compared to electron inertial length) eddies and it follows a Kolmogorov-like k−7/3 spectrum, where k is a characteristic wavenumber.     

A new phase-screen method for electromagnetic wave propagation in turbulent flows using large-eddy simulation

Use of large-eddy simulation (LES) data in electromagnetic wave propagation modeling is not very common because of the high computational cost involved. A new phase-screen method is proposed to model radio wave propagation, in the atmospheric turbulence, using the resolved scale refractivity obtained from LES. The proposed method offers the same level of accuracy, as the one already existing in the literature, at much cheaper cost.