Conditional Moment Closure for Large Eddy Simulations

A conditional moment closure (CMC) based combustion model for large-eddy simulations (LES) of turbulent reacting flow is proposed and evaluated. Transport equations for the conditionally filtered species are derived that are consistent with the LES formulation and closures are suggested for the modelling of the conditional velocity, conditional scalar dissipation and the fluctuations around the conditional mean. A conventional β-probability density distribution of the scalar is used together with dynamic modelling for the sub-grid fluxes. The model is validated by comparison of simulations with measurements of a piloted, turbulent methane-air jet diffusion flame.     

饱和两相介质动力固结问题时域求解的精细时程积分方法

针对u-p形式的饱和两相介质波动方程,采用精细时程积分方法计算固相位移u,采用向后差分算法求解流体压力p,建立了饱和两相介质动力固结问题时域求解的精细时程积分方法。针对标准算例,对该方法的计算精度进行了校核。开展了该方法相关算法特性的研究,对采用不同数值积分方法计算非齐次波动方程特解项计算精度的差异进行了对比研究,并对采用不同积分点数目的高斯积分法计算特解项条件下计算精度的差异进行了对比研究。研究结果表明,(1)该方法具有良好的计算精度。(2)计算非齐次波动方程特解项的数值积分方法中,梯形积分法的计算精度最差,高斯积分法、辛普生积分法和科茨积分法都具有较好的计算精度。(3)增加高斯积分点数目对于提高计算精度的作用并不显著。     

Development of Gas-Particle Euler-Euler LES Approach: A Priori Analysis of Particle Sub-Grid Models in Homogeneous Isotropic Turbulence

A new large eddy simulation (LES) approach for particle-laden turbulent flows in the framework of the Eulerian formalism for inertial particle statistical modelling is developed. Local instantaneous Eulerian equations for the particle cloud are first written using the mesoscopic Eulerian formalism (MEF) proposed by Février et al. (J Fluid Mech 533:1–46, 2005), which accounts for the contribution of an uncorrelated velocity component for inertial particles with relaxation time larger than the Kolmogorov time scale. Second, particle LES equations are obtained by volume filtering the mesoscopic Eulerian ones. In such an approach, the particulate flow at larger scales than the filter width is recovered while sub-grid effects need to be modelled. Particle eddy-viscosity, scale similarity and mixed sub-grid stress (SGS) models derived from fluid compressible turbulence SGS models are presented. Evaluation of such models is performed using three sets of particle Lagrangian results computed from discrete particle simulation (DPS) coupled with fluid direct numerical simulation (DNS) of homogeneous isotropic decaying turbulence. The two phase flow regime corresponds to the dilute one where two-way coupling and inter-particle collisions are not considered. The different particle Stokes number (based on Kolmogorov time scale) are initially equal to 1, 2.2 and 5.1. The mesoscopic field properties are analysed in detail by considering the particle velocity probability function (PDF), correlated velocity power spectra and random uncorrelated velocity moments. The mesoscopic fields measured from DPS+DNS are then filtered to obtain large scale fields. A priori evaluation of particle sub-grid stress models gives comparable agreement than for fluid compressible turbulence models. It has been found that the standard Smagorinsky eddy-viscosity model exhibits the smaller correlation coefficients, the scale similarity model shows very good correlation coefficient but strongly underestimates the sub-grid dissipation and the mixed model is on the whole superior to pure eddy-viscosity model.     

“Perfect” modeling framework for dynamic SGS model testing in large eddy simulation

A perfect modeling framework for the systematic study of the effect of filter shape on the resolved scales of motion in large eddy simulation is developed. The effects of the explicit and implicit filtering approaches in large eddy simulation are considered. A simple model for smooth filtering is proposed and the related effects are analyzed. The proposed approach provides an effective research tool for assessing the behavior of sub-grid scale models in a dynamic fashion. The performances of various classical models are examined by using the perfect modeling formalism for simulating the large and/or the small residual scales effect. Numerical experiments are performed for decaying isotropic turbulence. The consistency of the sub-grid scale models with the effective composite filter employed in real simulations is discussed. The necessity of using mixed models when solving doubly-filtered Navier–Stokes equations is verified. It is found that time evolution of large scale velocity field is more sensitive to sub-grid large scale models like Bardina model, while the grid-filtered sub-filter scale model is necessary to ensure the proper energy dissipation.     

On the nature of multitude scalings in decaying isotropic turbulence

We report multitude scaling laws for isotropic fully developed decaying turbulence through group theoretic method employing on the spectral equations both for modelling and without any modelling of nonlinear energy transfer. For modelling, besides the existence of classical power law scalings, an exponential decay of turbulent energy in time is obtained subject to exponentially decaying integral length scale at infinite Reynolds number limit. For the transfer without modelling, at finite Reynolds number, in addition to general power law decay of turbulence intensity with integral length scale growing as a square root of time, an exponential decay of energy in time is explored when integral length scale remains constant. Both the power and exponential decaying laws of energy agree to the theoretical results of George (1992), George and Wang (2009) and experimental results of fractal grid generated turbulence by Hurst and Vassilicos (2007). At infinite Reynolds number limit, a general power law scaling is obtained from which all classical scaling laws are recovered. Further, in this limit, turbulence exhibits a general exponential decaying law of energy with exponential decaying integral length scale depending on two scaling group parameters. The role of symmetry group parameters on turbulence dynamics is discussed in this study.     

A Novel Framework for Integration of Random-Walk Particle-Tracking Simulation in Subsurface Multi-phase Flow Modeling

Coarse-scale models are generally preferred in the numerical simulation of multi-phase flow due to computational constraints. However, capturing the effects of fine-scale heterogeneity on flow and isolating the impacts of numerical (artificial) dispersion, which increases with scale, are not trivial. In this paper, a particle-tracking method is devised and integrated in a scale-up workflow to estimate the conditional probability distributions of multi-phase flow functions, which can be considered as inputs in coarse-scale simulations with existing commercial packages. First, a novel particle-tracking method is developed to solve the saturation transport equation. The transport calculation is coupled with a velocity update, following the implicit pressure, explicit saturation framework, to solve the governing equations of two-phase immiscible flow. Each phase particle is advanced in a deterministic convection step according to the phase velocity, as well as in a stochastic dispersion step based on the random Brownian motion. A kernel-based formulation is proposed for computation of fluid saturation in accordance with the phase particle distribution. A novel aspect is that this method employs the kernel approach to construct saturation from phase particle distribution, which is an important improvement to the conventional box method that necessitates a large number of particles per grid cell for consistent saturation interpolation. The model is validated against various analytical solutions. Finally, the validated model is integrated in a statistical scale-up procedure to calibrate effective, or “pseudo,” multi-phase flow functions (e.g., relative permeability functions). The proposed scale-up framework does not impose any length scale requirement regarding the distribution of sub-grid heterogeneities.     

非平稳随机激励下结构体系动力可靠度时域解法

将结构动力方程写成状态方程形式,采用精细积分法对其进行数值求解,导出了非平稳激励下结构随机响应的时域显式表达式,该过程的计算量仅相当于两次确定性时程分析的计算量. 基于该显式表达式,结合首次超越失效准则,提出了非平稳随机激励下结构体系动力可靠度的数值模拟算法. 与功率谱方法相比,该方法无需同时在时频域内进行大量数值积分,也无需引入关于响应过程跨越界限次数概率分布, 以及各失效模式相关性等方面的假定. 通过数值算例, 对比了该方法与泊松过程法、马尔可夫过程法、传统蒙特卡罗法的计算精度和计算效率,结果显示该方法具有理想的精度和相当高的效率.      

考虑惯性效应的渗流方程及相关阻力标度律

本文基于体积平均法推导得到了多孔介质中考虑惯性效应时的局部线化宏观流动方程,由此可以递推得到较大雷诺数Re 时Navier-Stokes 方程的解,从而避免了直接求解Navier-Stokes 方程所带来的计算成本高和计算稳定性差的问题．针对正方形周期排列模型的算例表明,平均速度方向与宏观压力梯度方向并不总是一致,一般情况下,随着Re 增大,二者差异也会增大．当固定平均速度方向v ? 时,压差阻力在较大的Re 范围内存在标度律,标度指数约为3．该标度指数与弱惯性区域标度指数一致,但弱惯性区域Re 范围仅为0相似文献   

Large-eddy simulation of circular cylinder flow at subcritical Reynolds number: Turbulent wake and sound radiation

The flows past a circular cylinder at Reynolds number 3900 are simulated using large-eddy simulation(LES) and the far-field sound is calculated from the LES results. A low dissipation energy-conserving finite volume scheme is used to discretize the incompressible Navier–Stokes equations. The dynamic global coefficient version of the Vreman's subgrid scale(SGS) model is used to compute the sub-grid stresses. Curle's integral of Lighthill's acoustic analogy is used to extract the sound radiated from the cylinder. The profiles of mean velocity and turbulent fluctuations obtained are consistent with the previous experimental and computational results. The sound radiation at far field exhibits the characteristic of a dipole and directivity. The sound spectra display the-5/3 power law. It is shown that Vreman's SGS model in company with dynamic procedure is suitable for LES of turbulence generated noise.     

Une alternative aux hypothèses de Boussinesq des gaz chauds : L'approximation polytropiqueAn alternative to Boussinesq hypothesis for hot gases,the polytropic approximation

A new approximation is proposed for the balance equations of heated gas flows, taking into account completely the couplings between dynamic and thermal fields, as well as compressibility, stating in a polytropic formulation the relation between the quantities of corresponding state. The polytropic exponent is the new variable that characterises the transfers; it substitutes the temperature for the pressure and the density that are physical quantities difficult to grasp. The method opens interesting prospects for the study of heated gases, for experiments, or for modelling and numerical simulation. To cite this article: C. Rey, S. Benjeddou, C. R. Mecanique 332 (2004).     

燃烧室两相流场亚网格燃烧模型的研究

在三维任意曲线坐标系下采用不同的亚网格燃烧模型对环形燃烧室火焰筒气液两相湍流瞬态反应流进行大涡模拟．计算中所采用的数学度模型有：k方程亚网格尺度模型估算亚网格湍流黏性;热通量辐射模型估算辐射换热,分别采用亚网格EBU燃烧模型（E-A model）、亚网格二阶矩输运方程模型（SOM）和亚网格二阶矩代数模型（SOM-A）估算化学反应速率．并在非交错网格系统下气相采用SIMPLE算法和混合差分格式求解,液相采用Lagrange处理,并用PSIC算法对其进行求解．通过实验结果和计算结果的比较,表明在三维任意曲线坐标系下对燃烧室火焰简两相湍流油雾燃烧流场进行大涡模拟,3种不同的亚网格燃烧模型都能真实反映两相湍流化学反应流流动及实际燃烧过程,而采用亚网格二阶矩输运方程模型稍优于其他两种亚网格燃烧模型．     

Dynamic Response of a Multilayered Poroelastic Half-Space to Harmonic Surface Tractions

The propagator matrix method is developed to study the dynamic response of a multilayered poroelastic half-space to time-harmonic surface tractions. In a cylindrical coordinate system, a method of displacement potentials is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to those equations are obtained. After that, the propagator matrix method and the vector surface harmonics are employed to derive the solutions for a multilayered poroelastic half-space subjected to surface tractions. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. Finally, a high-order adaptive integration method with continued fraction expansions for accelerating the convergence of the truncated integral is adopted to numerically evaluate the integral solutions expressed in terms of semi-infinite Hankel-type integrals with respect to horizontal wavenumber. Furthermore, to validate the present approach, the response of a uniform poroelastic half-space is examined using the formulation proposed in this article. It is shown that the numerical results computed with this approach agree well with those computed with the analytical solution of a uniform half-space.     

Large-Eddy Simulation of the Flow Over a Circular Cylinder at Reynolds Number 3900 Using the OpenFOAM Toolbox

The flow over a circular cylinder at Reynolds number 3900 and Mach number 0.2 was predicted numerically using the technique of large-eddy simulation. The computations were carried out with an O-type curvilinear grid of size of 300 × 300 × 64. The numerical simulations were performed using a second-order finite-volume method with central-difference schemes for the approximation of convective terms. A conventional Smagorinsky and a dynamic k-equation eddy viscosity sub-grid scale models were applied. The integration time interval for data sampling was extended up to 150 vortex shedding periods for the purpose of obtaining a fully converged mean flow field. The present numerical results were found to be in good agreement with existing experimental data and previously obtained large-eddy simulation results. This gives an indication on the adequacy and accuracy of the selected large-eddy simulation technique implemented in the OpenFOAM toolbox.     

Singular loadings in elasticity problems and singular solutions of the corresponding integral equations

The method of singular integral equations is an efficient method for the formulation and numerical solution of plane and antiplane, static and dynamic, isotropic and anisotropic elasticity problems. Here we consider three cases of singular loadings of the elastic medium: by a force, by a moment and by a loading distribution with a simple pole. These loadings cause corresponding singularities in the right-hand side function and in the unknown function of the integral equation. A method for the numerical solution of the singular integral equation under the above singular loadings is proposed and the validity of this equation at the singular points is investigated.     

Large eddy simulation of turbulent incompressible flows by a three‐level finite element method

The variational multiscale method provides a methodical framework for large eddy simulation of turbulent flows. In this work, a particular implementation in the form of a three‐level finite element method separating large resolved, small resolved, and unresolved scales is proposed. Residual‐free bubbles are used for the numerical approximation of the small‐scale momentum equation. A stabilizing term is added, in order to take into account the effect of the small‐scale continuity equation. This implementation guarantees the stability of the method without further provisions and offers substantial computational savings on the small‐scale level. Furthermore, it is accounted for the unresolved scales by a specific dynamic modelling procedure. The method is tested for two different turbulent flow situations. Copyright © 2005 John Wiley & Sons, Ltd.     

A transient solution method for the finite element incompressible Navier-Stokes equations

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a direct time integration method which has unique features in its formulation as well as in its evaluation of the contribution of external functions. Particular processes regarding the continuity conditions and the boundary conditions lead to a set of non-linear recurrence equations which represent evolution of the velocities and the pressures under the incompressibility constraint. An iteration process as to the non-linear convective terms is performed until the convergence is achieved in every integration step. Excessively artificial techniques are not introduced into the present solution procedure. Numerical examples with vortex shedding behind a rectangular cylinder are presented to illustrate the features of the proposed method. The calculated results are compared with experimental data and visualized flow fields in literature.     

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations(BIE)and solved with the newly developed boundary point method(BPM).The model is closely derived from the concept of the equivalent inclusion of Eshelby tensors.Eigenstrains are iteratively determined for each short.fiber embedded in the matrix with various properties via the Eshelby tensors,which can be readily obtained beforehand either through analytical or numerical means.As unknown variables appear only on the boundary of the solution domain,the solution scale of the inhomogeneity problem with the model is greatly reduced.This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM.The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element(RVE),showing the validity and the effectiveness of the proposed computational modal and the solution procedure.     

Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations

A computational model is proposed for short-fiber reinforced materials with the eigenstrain formulation of the boundary integral equations （BIE） and solved with the newly developed boundary point method （BPM）. The model is closely derived from the concept of the equivalent inclusion Of Eshelby tensors. Eigenstrains are iteratively determined for each short-fiber embedded in the matrix with various properties via the Eshelby tensors, which can be readily obtained beforehand either through analytical or numerical means. As unknown variables appear only on the boundary of the solution domain, the solution scale of the inhomogeneity problem with the model is greatly reduced. This feature is considered significant because such a traditionally time-consuming problem with inhomogeneity can be solved most cost-effectively compared with existing numerical models of the FEM or the BEM. The numerical examples are presented to compute the overall elastic properties for various short-fiber reinforced composites over a representative volume element （RVE）, showing the validity and the effectiveness of the proposed computational modal and the solution procedure.     

A boundary integral equation formulation for the Reissner's plates resting on two-parameter foundation

In this paper a boundary integral equation formulation for the Reissner's plates resting on a two-parameter foundation is established. With the aid of the Hormander Operator method, the equations of the corresponding fundamental solutions are converted into a sixth order partial differential equation with a scale function as an unknown. In order to reduce the equation further, two auxiliary functions are introduced. They satisfy a second and a fourth order equation respectively. The expressions of the auxiliary functions can be derived easily. The fundamental solutions of the Reissnei's plates on the two-parameter foundation arc expressed by a linear combination of the auxiliary functions and their derivatives. The boundary integral equations are formulated by the use of the weighted residual procedure. The fundamental solutions obtained are taken as the kernel functions of the boundary integral equations. A few examples are studied. The numerical results show high accuacy and efficiency of the present formulation.This work was supported by the National Natural Science Foundation of China.     

A new velocity–vorticity boundary integral formulation for Navier–Stokes equations

In the present work, an indirect boundary integral method for the numerical solution of Navier–Stokes equations formulated in velocity–vorticity dependent variables is proposed. This wholly integral approach, based on Helmholtz's decomposition, deals directly with the vorticity field and gives emphasis to the establishment of appropriate boundary conditions for the vorticity transport equation. The coupling between the vorticity and the vortical velocity fields is expressed by an iterative procedure. The present analysis shows the usefulness of an integral formulation not only in providing a potentially more efficient computational tool, but also in giving a better understanding to the physics of the phenomenon. Copyright © 2000 John Wiley & Sons, Ltd.