Adaptive ANOVA decomposition of stochastic incompressible and compressible flows

Realistic representation of stochastic inputs associated with various sources of uncertainty in the simulation of fluid flows leads to high dimensional representations that are computationally prohibitive. We investigate the use of adaptive ANOVA decomposition as an effective dimension–reduction technique in modeling steady incompressible and compressible flows with nominal dimension of random space up to 100. We present three different adaptivity criteria and compare the adaptive ANOVA method against sparse grid, Monte Carlo and quasi-Monte Carlo methods to evaluate its relative efficiency and accuracy. For the incompressible flow problem, the effect of random temperature boundary conditions (modeled as high-dimensional stochastic processes) on the Nusselt number is investigated for different values of correlation length. For the compressible flow, the effects of random geometric perturbations (simulating random roughness) on the scattering of a strong shock wave is investigated both analytically and numerically. A probabilistic collocation method is combined with adaptive ANOVA to obtain both incompressible and compressible flow solutions. We demonstrate that for both cases even draconian truncations of the ANOVA expansion lead to accurate solutions with a speed-up factor of three orders of magnitude compared to Monte Carlo and at least one order of magnitude compared to sparse grids for comparable accuracy.     

Development of Lattice Boltzmann Flux Solver for Simulation of Incompressible Flows

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.     

Lattice Boltzmann computational fluid dynamics in three dimensions

The recent development of the lattice gas method and its extension to the lattice Boltzmann method have provided new computational schemes for fluid dynamics. Both methods are fully paralleled and can easily model many different physical problems, including flows with complicated boundary conditions. In this paper, basic principles of a lattice Boltzmann computational method are described and applied to several three-dimensional benchmark problems. In most previous lattice gas and lattice Boltzmann methods, a face-centered-hyper-cubic lattice in four-dimensional space was used to obtain an isotropic stress tensor. To conserve computer memory, we develop a model which requires 14 moving directions instead of the usual 24 directions. Lattice Boltzmann models, describing two-phase fluid flows and magnetohydrodynamics, can be developed based on this simpler 14-directional lattice. Comparisons between three-dimensional spectral code results and results using our method are given for simple periodic geometries. An important property of the lattice Boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency, while all other methods, including the spectral method, are unable to model complicated geometries efficiently.     

一个新的用于不可压磁流体动力学的格子波尔兹曼模型

绝大多数现有的格子波尔兹曼磁流体动力学模型其实是用可压缩方法来模拟不可压磁流体。而这些可压缩效应在数值模拟中往往会带来意想不到的误差。在这篇文章中，我们提出了一个全新的可用于的不可压格子波尔兹曼磁流体动力学模型，并且进行了哈特曼流的数值模拟。模拟结果与哈特曼流的解析解非常吻合。这个方法需要一个假设条件来消除误差。我们做了大量的数值试验，并且与Dellar教授的模型进行了详细的分析与比较。     

A improved incompressible lattice Boltzmann model for time-independent flows

It is well known that the lattice Boltzmann equation method (LBE) can model the incompressible Navier-Stokes (NS) equations in the limit where density goes to a constant. In a LBE simulation, however, the density cannot be constant because pressure is equal to density times the square of sound speed, hence a compressibility error seems inevitable for the LBE to model incompressible flows. This work uses a modified equilibrium distribution and a modified velocity to construct an LBE which models time-independent (steady) incompressible flows with significantly reduced compressibility error. Computational results in 2D cavity flow and in a 2D flow with an exact solution are reported.     

Lattice Boltzmann Simulation of Free-Surface Temperature Dispersion in Shallow Water Flows

We develop a lattice Boltzmann method for modeling free-surface temperature dispersion in the shallow water flows. The governing equations are derived from the incompressible Navier-Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The thermal effects are incorporated in the momentum equation by using a Boussinesq approximation. The dispersion of free-surface temperature is modelled by an advection-diffusion equation. Two distribution functions are used in the lattice Boltzmann method to recover the flow and temperature variables using the same lattice structure. Neither upwind discretization procedures nor Riemann problem solvers are needed in discretizing the shallow water equations. In addition, the source terms are straightforwardly included in the model without relying on well-balanced techniques to treat flux gradients and source terms. We validate the model for a class of problems with known analytical solutions and we also present numerical results for sea-surface temperature distribution in the Strait of Gibraltar.     

Finite difference lattice Boltzmann method with arbitrary specific heat ratio applicable to supersonic flow simulations

The current finite difference lattice Boltzmann method (FDLBM) gives a fixed specific heat ratio because internal energy is limited to the translational freedom of the space. Yan et al. and Kataoka et al. clarified the conditions for deriving models with arbitrary specific heat ratio and proposed Euler models. However, these model applications to numerical simulations showed the weakness in the numerical stability. In this paper, a two-dimensional FDLBM Navier Stokes model and a three-dimensional FDLBM Euler model, which allow arbitrary values to be set for the specific heat ratio, were proposed. These models stably performed numerical simulations from subsonic to supersonic ranges.     

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.     

A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows

A momentum exchange-based immersed boundary-lattice Boltzmann method is presented in this Letter for simulating incompressible viscous flows. This method combines the good features of the lattice Boltzmann method (LBM) and the immersed boundary method (IBM) by using two unrelated computational meshes, an Eulerian mesh for the flow domain and a Lagrangian mesh for the solid boundaries in the flow. In this method, the non-slip boundary condition is enforced by introducing a forcing term into the lattice Boltzmann equation (LBE). Unlike the conventional IBM using the penalty method with a user-defined parameter or the direct forcing scheme based on the Navier–Stokes (NS) equations, the forcing term is simply calculated by the momentum exchange of the boundary particle density distribution functions, which are interpolated by the Lagrangian polynomials from the underlying Eulerian mesh. Numerical examples show that the present method can provide very accurate numerical results.     

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.     

基于总能形式的耦合的双分布函数热晶格玻尔兹曼数值方法

双分布函数热晶格玻尔兹曼数值方法在微尺度热流动系统中得到广泛的应用. 本文基于晶格玻尔兹曼平衡分布函数低阶Hermite展开式, 创新性地提出了包含黏性热耗散和压缩功的耦合的双分布函数热晶格玻尔兹曼数值方法, 将能量场内温度的变化以动量源的形式引入晶格波尔兹曼动量演化方程, 实现了能量场与动量场之间的耦合. 研究了考虑黏性热耗散和压缩功的和不考虑的两种热自然对流模型, 重点分析了不同瑞利数和普朗特数下流场内的流动情况以及温度、速度和平均努赛尔数的变化趋势. 本文实验结果与文献结果一致, 验证了本文数值方法的可行性和准确性. 研究结果表明: 随着瑞利数和普朗特数的增大, 方腔内对流传热作用逐渐增强, 边界处形成明显的边界层; 考虑黏性热耗散和压缩功的模型对流作用相对增强, 黏性热耗散和压缩功对自然对流的影响在微尺度流动过程中不能忽略.     

激波的格子玻尔兹曼方法模拟

格子玻尔兹曼方法已被广泛应用于模拟流体,但主要应用于模拟不可压流体。应用二维正方格阵上的3速9点模型模拟激波的传播。考察了激波关系式、激波的宽度和结构。模拟结果与由Chapman-Enskog展开导出的流体方程的解相符。这说明格子玻尔兹曼方法能够模拟可压缩流体。     

Lattice Boltzmann Study of Mixed Convection in aCubic Cavity

The problem of the mixed convection in a cubic cavity is studied with lattice Boltzmann method. A multiple-relaxation-time lattice Boltzmann model for incompressible flow in the cubic cavity and another thermal lattice Boltzmann model for solving energy/temperature equation are proposed. The present models are first validated through a comparison with some available results, and then, we present a detailed parameter study on the mixed
convection in the cubic cavity. The numerical results show that the flow and temperature patterns change greatly with variations of the Reynolds and Richardson numbers.     

Lattice Bhatnagar-Gross-Krook Simulations in 2-D Incompressible Magnetohydrodynamics

Lattice Boltzmann Method is recently developed within numerical schemes for simulating a variety of physical systems. In this paper a new lattice Bhatnagar-Gross-Krook (LBGK) model for two-dimensional incompressible magnetohydrodynamics (IMHD) is presented. The model is an extension of a hydrodynamics lattice BGK model with 9 velocities on a square lattice, resulting in a model with 17 velocities. Most of the existing LBGK models for MHD can be viewed as compressible schemes to simulate incompressible flows. The compressible effect might lead to some undesirable errors in numerical simulations. In our model the compressible effect has been overcome successfully. The model is then applied to the Hartmann flow, giving reasonable results.     

Lattice Bhatnagar-Gross-Krook Simulations in 2-D Incompressible Magnetohydrodynamics

Lattice Boltzmann Method is recently developed within numerical schemes for simulating a variety of physical systems. In this paper a new lattice.Bhatnagar-Gross-Krook （LBGK） model for two-dimensional incompressible magnetohydrodynamics （IMHD） is presented. The model is an extension of a hydrodynamics lattice BGK model with 9 velocities on a square lattice, resulting in a model with 17 velocities. Most of the existing LBGK models for MHD can be viewed as compressible schemes to simulate incompressible flows. The compressible effect might lead to some undesirable errors in numerical simulations. In our model the compressible effect has been overcome successfully. The model is then applied to the Hartmann flow, giving reasonable results.     

格子Boltzmann亚格子模型的研究

为了将格子Boltzmann法应用于大雷诺数流动的模拟,本文将Smagorinsky亚格子模型和LBGK模型相结合,并对该亚格子LBM模型进行了研究。利用该亚格子LBM模型,对二维顶盖驱动流进行了模拟,得到了若干大雷诺数下流线图和方腔中心线上无量纲速度分布。计算结果与基准解进行比较,两者相互吻合。     

A lattice Boltzmann method for nonlinear disturbances around an arbitrary base flow

In this paper we address the problem of the time evolution of a perturbation around a steady base flow with the use of the lattice Boltzmann method (LBM). This approach, named base flow lattice Boltzmann method, is of great interest in particular for aeroacoustic fields where the acoustic perturbation, on the one hand, is almost exclusively influenced by the large scale average structures of the underlying flow, and on the other hand, has a low effect on the large structures. The method is implemented for weakly compressible flows and the results of the base flow lattice Boltzmann are compared with the standard single relaxation time LBM. The boundary conditions for the base flow lattice Boltzmann method are discussed, as well as the implementation of outflow conditions for acoustic waves.     

An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows

The recently proposed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM) [14] is improved in this work to simulate three-dimensional incompressible viscous flows. In the conventional IB-LBM, the restoring force is pre-calculated, and the non-slip boundary condition is not enforced as compared to body-fitted solvers. As a result, there is a flow penetration to the solid boundary. This drawback was removed by the new version of IB-LBM [14], in which the restoring force is considered as unknown and is determined in such a way that the non-slip boundary condition is enforced. Since Eulerian points are also defined inside the solid boundary, the computational domain is usually regular and the Cartesian mesh is used. On the other hand, to well capture the boundary layer and in the meantime, to save the computational effort, we often use non-uniform mesh in IB-LBM applications. In our previous two-dimensional simulations [14], the Taylor series expansion and least squares-based lattice Boltzmann method (TLLBM) was used on the non-uniform Cartesian mesh to get the flow field. The final expression of TLLBM is an algebraic formulation with some weighting coefficients. These coefficients could be computed in advance and stored for the following computations. However, this way may become impractical for 3D cases as the memory requirement often exceeds the machine capacity. The other way is to calculate the coefficients at every time step. As a result, extra time is consumed significantly. To overcome this drawback, in this study, we propose a more efficient approach to solve lattice Boltzmann equation on the non-uniform Cartesian mesh. As compared to TLLBM, the proposed approach needs much less computational time and virtual storage. Its good accuracy and efficiency are well demonstrated by its application to simulate the 3D lid-driven cubic cavity flow. To valid the combination of proposed approach with the new version of IBM [14] for 3D flows with curved boundaries, the flows over a sphere and torus are simulated. The obtained numerical results compare very well with available data in the literature.     

一个三维多松弛时间全速域格子Boltzmann模型

构建一个既适用于低速不可压流体又适用于高速可压缩流体的三维自由参数多松弛时间格子Boltzmann模型.模型中,根据SO（3）群的不可约表述基函数构造转化矩阵,根据恢复可压Navier-Stokes方程的需要选取非守恒矩平衡值.通过von Neumann稳定性分析模型参数对数值稳定性的影响,并给出建议选择范围.模型经过基准问题的验证,模拟结果与解析解及其它数值结果符合较好.