BGK方法在非结构网格上的应用

采用旋转局部坐标的方法,发展了一种针对非结构网格的BGK计算方法.该方法属于有限体积法,大致分为两个步骤:①空间离散:②通量计算及时间推进.在第①步中,采用基于最小二乘法的高阶ENO格式来获得宏观物理量的高阶导数;在第②步中,采用旋转坐标轴的方法来计算非结构网格单元各边的通量.并得出了后台阶绕流(Backward Facing Step)及翼型绕流(Flow Over an Airfoil)两个算例的计算结果.     

三维非结构网格自动生成

采用各向异性平面非结构网格生成技术对曲面的参数平面进行三角剖分,从而得到曲面的非结构网格,作为三维非结构网格的边界网格.应用推进面法生成网格内点,增量法将生成的内点逐点插入现有网格进行网格细化,得到三维计算域的Delaunay非结构网格.讨论了非结构网格质量优化方法.给出几个算例说明方法的应用.     

多重网格法在非结构网格中的应用

将多重网格法运用于非结构网格.网格是通过聚合法得到的,网格之间是相互关联的.方程的求解采用Jamson的有限体积法.给出了二维、三维情况的数值算例.     

三维非匹配网格上的扩散方程数值求解

将子网格剖分的支撑算子方法,拓展应用于三维非匹配网格上的扩散方程求解.算例表明该方法在正交非匹配网格上能够精确获得线性解;在一般非匹配网格上可以达到二阶精度;在求解曲面网格和节点不共面网格时,精度比平面近似的方法要高,也可以达到2阶精度,同时也适合求解含有物质界面的混合介质网格.     

自适应间断有限元方法求解三维欧拉方程

将龙格库塔间断有限元方法(RDDG)与自适应方法相结合,求解三维欧拉方程.区域剖分采用非结构四面体网格,依据数值解的变化采用自适应技术对网格进行局部加密或粗化,减少总体网格数目,提高计算效率.给出四种自适应策略并分析不同自适应策略的优缺点.数值算例表明方法的有效性.     

一种求解Euler方程的BGK型非结构化自适应算法

本文构造了一种BGK型二阶非结构化网格自适应算法,用于求解Euler方程.若干一维、二维标准算例表明,本文所构造的算法在模拟复杂流动时具有很高的流场分辨率.     

复杂流道内N-S方程的非交错网格数值求解

本文试图用非交错网格求解二维曲线坐际系下的N-S方程,文中对压力修正方程进行了改造,导出了曲线速度分量的修正表达式,作为计算方法的检验,本文针对90°弯曲管道和波瓣喷管混合器三维流动进行了两个算例分析,结果表明该方法是稳定收敛的。     

三维非结构粘性网格生成方法

描述了一套适合粘性流动计算的三维非结构网格自动生成方法.在物面附近的粘性作用区域,采用推进层方法生成各向异性的"扁平"四面体网格,并通过一定的网格伸长控制参数,实现整个流场区域网格高度的平滑过渡.当粘性网格的推进高度达到预定要求时,推进层方法自动停止,转而采用阵面推进方法生成常规意义的尽量接近正四面体的各向同性网格.同时给出了利用该方法生成的M6机翼非结构粘性网格来求解机翼粘性绕流的简单算例.     

流场自适应叉树网格数值模拟三维复杂超声速流场

建立了基于分区结构网格的三维贴体叉树形网格的数据结构,并阐述了在此基础上进行自适应分裂/合并判别方法.为节省网格量和保证流场结构捕捉质量,提出对自适应程度进行区域性控制,以及对流场结构进行"保护"性预加密的优化方式.通过应用该网格对三维复杂超声速流场算例的计算,证明该方法对网格加密控制方便,对流场结构分辨率高.     

多点择优推进阵面法生成曲面三角形网格

在现有曲面非结构网格生成法的基础上,提出了一种新的曲面网格生成法——多点择优推进阵面法.它可在曲面上直接进行三角形网格划分,克服了映射法的网格变形问题,并且可以在网格生成结束后,对曲面网格直接进行Laplace格点松弛光顺.该方法使用简单,不受曲面块类型的限制,且网格质量高,可以为三维非结构网格生成提供高质量的初始阵面,并给出了若干个算例.     

A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension

We describe a cell-centered Godunov scheme for Lagrangian gas dynamics on general unstructured meshes in arbitrary dimension. The construction of the scheme is based upon the definition of some geometric vectors which are defined on a moving mesh. The finite volume solver is node based and compatible with the mesh displacement. We also discuss boundary conditions. Numerical results on basic 3D tests problems show the efficiency of this approach. We also consider a quasi-incompressible test problem for which our nodal solver gives very good results if compared with other Godunov solvers. We briefly discuss the compatibility with ALE and/or AMR techniques at the end of this work. We detail the coefficients of the isoparametric element in the appendix.     

基于非结构四边形网格的WENO格式

基于非结构四边形网格发展求解双曲守恒律的三阶加权基本无振荡（WENO）格式.针对任意非结构四边形网格选取重构模板，并给出基于线性多项式的三阶线性重构.但对于一般的非结构四边形网格，会出现非常大的线性权和负权，使得非线性重构的WENO格式对光滑问题也不稳定.本文给出一个处理非常大的线性权的优化重构方法，对优化后得到的负线性权采用分裂方法进行处理.对于非线性权，提出一种考虑局部网格和物理量间断的新光滑度量因子.采用优化重构方法和新的非线性权，当前的三阶WENO格式在质量很差的网格上也具有很好的稳定性.理论的三阶精度在数值精度测试算例中得到验证，同时一范数和无穷范数的误差绝对值不依赖于网格质量；具有强间断的数值结果证明了当前格式的有效性.     

Remapping-free ALE-type kinetic method for flow computations

Based on the integral form of the fluid dynamic equations, a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed. Different from the earlier unified moving mesh gas-kinetic method [C.Q. Jin, K. Xu, An unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175], the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2D case. Since a purely Lagrangian method is always associated with mesh entangling, in order to avoid computational collapsing in multidimensional flow simulation, the mesh velocity is constructed by considering both fluid velocity (Lagrangian methodology) and diffusive velocity (Regenerating Eulerian mesh function). Therefore, we obtain a generalized Arbitrary-Lagrangian–Eulerian (ALE) method by properly designing a mesh velocity instead of re-generating a new mesh after distortion. As a result, the remapping step to interpolate flow variables from old mesh to new mesh is avoided. The current method provides a general framework, which can be considered as a remapping-free ALE-type method. Since there is great freedom in choosing mesh velocity, in order to improve the accuracy and robustness of the method, the adaptive moving mesh method [H.Z. Tang, T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515] can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients.     

Efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes

This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements.     

Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method

The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769–795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air–water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier–Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model’s capability in predicting the interaction dynamics of rising bubbles.     

Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics

This work presents a multi-dimensional cell-centered unstructured finite volume scheme for the solution of multimaterial compressible fluid flows written in the Lagrangian formalism. This formulation is considered in the Arbitrary-Lagrangian–Eulerian (ALE) framework with the constraint that the mesh velocity and the fluid velocity coincide. The link between the vertex velocity and the fluid motion is obtained by a formulation of the momentum conservation on a class of multi-scale encased volumes around mesh vertices. The vertex velocity is derived with a nodal Riemann solver constructed in such a way that the mesh motion and the face fluxes are compatible. Finally, the resulting scheme conserves both momentum and total energy and, it satisfies a semi-discrete entropy inequality. The numerical results obtained for some classical 2D and 3D hydrodynamic test cases show the robustness and the accuracy of the proposed algorithm.     

Gas-kinetic scheme with discontinuous derivative for low speed flow computation

The present paper concerns the improvement of the gas-kinetic scheme (GKS) for low speed flow computation. In the modified GKS scheme, the flow distributions with discontinuous derivatives are used as an initial condition at the cell interface for the flux evaluation. This discontinuity is determined by considering both the flow characteristic and grid’s resolution. Compared with GKS method with a continuous slope for the flow variables at a cell interface, the new scheme is more robust and accurate. In the under resolved flow computation, the new scheme presents much less numerical oscillation. The extension of the current scheme to unstructured mesh is straightforward. To validate the method, both computations of 2D lid-driven cavity flow and 3D flow past a sphere are performed. The numerical results validate the current method.     

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.