多介质可压缩流体动力学界面捕捉方法

研究多介质流界面捕捉方法的主要目的是消除多介质流体在界面处压力、速度可能出现的非物理振荡现象 ,并通过流体动力学方程和界面捕捉方程的耦合 ,将多介质流体动力学计算形式上转化为单介质流体计算 ,从而可以采用对计算单介质有效的高精度计算方法来处理多介质流动问题。推广了Shyue界面捕捉和其等效方程的推导方法 ,给出的结果可以适用于具有状态方程 p =( ,e,a1 , ,an) +( ,e,b1 , ,bn)e的介质 ,并通过了数值试验验证。     

可压缩多介质流动的间断有限元方法

采用间断有限元方法、LS方法和通量装配技术相结合,建立了一种计算可压缩多介质流动的有效 方法。计算中以光滑Heavside函数构造流体比热比和重新初始化方程中的符号距离函数,并采用通量装配 技术抑制界面附近的非物理振荡。为解决可压缩多介质流动提供一种新的手段。     

多孔介质——自由流界面应力跳跃条件下流动特性解析解

对非对称多孔介质--自由流复合通道内多孔介质内部及多孔介质与自由流体界面处复杂质量、动量输运特性进行研究. 在多孔介质区采用Brinkman-extended Darcy模型并结合速度连续,剪切应力跳跃的界面条件对此复合通道内流体的传递现象进行求解,提出了考虑界面应力跳跃时非对称复合通道各区域流体运动速度及摩擦系数的解析式,分析了界面应力跳跃系数,达西数及无量纲多孔层偏心厚度对流体速度及摩擦系数的影响. 结果表明:改变界面性质可在一定条件下明显控制各区域流体速度分布;在达西数、多孔层偏心厚度一定情况下,界面应力系数的增大会使界面流速减小,而使流体摩擦系数增大,特别是界面应力系数小于0的情况下变化更明显,此时若不考虑界面应力系数则会造成较大误差. 当界面应力系数及多孔层偏心厚度均为较小负数值时,改变多孔层偏心厚度对界面速度的影响要大于改变界面应力系数的情况;而当界面应力系数及多孔层偏心厚度为较大正数值时,情况则相反. 较大达西数下,界面应力系数及多孔层偏心厚度对流体摩擦系数的影响均较大,继续减小达西数至一定程度时,界面应力系数对流体摩擦系数的影响可忽略不计而认为只与多孔层偏心厚度相关,且对较大多孔层偏心厚度更敏感.      

基于格子Boltzmann方法的粘弹性流体圆柱绕流分析

将光滑界面法引入到格子Boltzmann方法中分析粘弹性流体绕流问题,分别采用单松弛模型和对流扩散模型求解运动方程和Oldroyd-B本构方程,针对圆形和椭圆内部边界条件,给出连续界面插值函数,在此基础上,运用光滑界面法将内部边界转换为作用力项施加到演化方程中。首先分析圆柱绕流问题,给出不同材料参数情况下的流场分布和阻力系数计算结果,比较发现与宏观数值模拟结果相吻合。将模型拓展到绕椭圆流动中,分析椭圆形状和材料参数对粘弹性流体绕柱流的影响,发现随着椭圆长轴与短轴比值的增加和维森伯格数的增加,阻力系数逐渐下降,并且长短轴比对迭代收敛有较大影响。     

激波作用下三维涡的形成及其界面捕捉计算

针对三维多介质可压缩流体,给出了可压缩多介质流体三维高精度数值计算方法,以及界面捕捉方程和带重新初始化的三维LevelSet方法,对初始压力间断和密度间断条件形成激波、接触间断以及稀疏波的三维复杂流场相互作用情况进行数值计算,给出流场中涡的形成过程和界面位置。并对计算方法进行理论验证。     

非结构动网格在多介质流体数值模拟中的应用

采用非结构动网格方法对含多介质的流场进行数值模拟.采用改进的弹簧方法来处理由于边界运动而产生的网格变形.采用基于格心的有限体积方法求解守恒型的ALE(Arbitrary Lagrangiall-Eulerian)方程,控制面通量的计算采用HLLC(Hartem,Lax,van Leer,Contact)方法,采用几何构造的方法使空间达到二阶精度,时间离散采用四阶Runge-Kutta方法.物质界面的处理采用虚拟流体方法.本文对含动边界的激波管、水下爆炸等流场进行数值模拟,取得较好的结果,不同时刻界面的位置和整个扩张过程被准确模拟.     

用改进的ghost fluid方法模拟强激波与界面的相互作用

为了解决原来的ghost fluid方法在计算强激波和界面相互作用时界面附近出现的速度和压力振荡问题,对原来的ghost fluid方法进行了改进,通过在界面处构造Riemann问题并求出界面的压力和速度,ghost fluid流体的压力和速度分别用界面的压力和速度代替,ghost流体的密度通过熵常数外推得到。改进的ghost fluid保持了原来的ghost fluid的简单性,对一维强激波与气-气、气-液界面的相互作用问题以及射流问题进行了数值计算,得到了分辨率较高的计算结果。     

一种追踪多介质流体界面运动的NND数值模拟方法

把界面捕捉等效方程、Level-Set方程和欧拉方程组耦合,在Stiffened状态方程下,采用高分辨率NND格式求解流体力学方程组并用Level-Set函数捕捉界面的位置。对二维情况下激波和气泡相互作用的问题进行数值模拟,并与波传算法的模拟结果进行比较。计算结果表明该方法能有效的抑制间断附近的非物理振荡,有很强的捕捉界面的能力。     

MOF界面重构方法在超高速运动中的应用

应用MFPPM(Multi-Fluid Parabolic Piecewise Method)方法进行超高速运动数值模拟时,由于其使用的PPM(Parabolic Piecewise Method)方法通常采用几个过渡网格来描述间断面,使得界面分辨率不高。为了更精确地对界面进行描述,将MOF(Moment-of-Fluid)界面重构方法与具有自主知识产权的MFPPM程序相结合,并首次将MOF方法运用到超高速运动中,对瞬时起爆的TNT炸药内聚压缩不同形状(三角形、正方形、正五边形、正六边形)的气腔进行了数值模拟;分别对四种构型在t=3?s和t=6.5?s时的VOF和MOF结果进行了对比。结果表明:MOF方法不仅能有效地捕捉到超高速运动中的构型几何特征,且在界面处不需要多个网格进行过渡,提高了界面重构的精度和分辨率。本文研究为界面不稳定性等问题中的复杂界面重构提供了一种新的方法。     

折线裂纹和两相材料界面的相互作用

利用螺位错基本解建立了和界面相交的折线裂纹的Cauchy型积分方程.根据奇异积分方程理论,得出了确定折线裂纹和界面交点处的奇性应力指数的特征方程,以及交点处各角形域内的奇性应力.利用所得的交点处的奇性应力定义了折线裂纹和界面交点处的应力强度因子.对所得积分方程进行数值求解,可得裂纹端点以及裂纹和界面交点处的应力强度因子.     

Assessment of shock capturing schemes for discontinuous Galerkin method

This paper carries out systematical investigations on the performance of several typical shock-capturing schemes for the discontinuous Galerkin （DG） method, including the total variation bounded （TVB） limiter and three artificial diffusivity schemes （the basis function-based （BF） scheme, the face residual-based （FR） scheme, and the element residual-based （ER） scheme）. Shock-dominated flows （the Sod problem, the Shu- Osher problem, the double Mach reflection problem, and the transonic NACA0012 flow） are considered, addressing the issues of accuracy, non-oscillatory property, dependence on user-specified constants, resolution of discontinuities, and capability for steady solutions. Numerical results indicate that the TVB limiter is more efficient and robust, while the artificial diffusivity schemes are able to preserve small-scale flow structures better. In high order cases, the artificial diffusivity schemes have demonstrated superior performance over the TVB limiter.     

Adaptive Runge–Kutta discontinuous Galerkin method for complex geometry problems on Cartesian grid

A Cartesian grid method using immersed boundary technique to simulate the impact of body in fluid has become an important research topic in computational fluid dynamics because of its simplification, automation of grid generation, and accuracy of results. In the frame of Cartesian grid, one often uses finite volume method with second order accuracy or finite difference method. In this paper, an h‐adaptive Runge–Kutta discontinuous Galerkin (RKDG) method on Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is developed. A ghost cell immersed boundary treatment with the modification of normal velocity is presented. The method is validated versus well documented test problems involving both steady and unsteady compressible flows through complex bodies over a wide range of Mach numbers. The numerical results show that the present boundary treatment to some extent reduces the error of entropy and demonstrate the efficiency, robustness, and versatility of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

A high-order Runge-Kutta discontinuous Galerkin method with a subcell limiter on adaptive unstructured grids for two-dimensional compressible inviscid flows

A robust, adaptive unstructured mesh refinement strategy for high-order Runge-Kutta discontinuous Galerkin method is proposed. The present work mainly focuses on accurate capturing of sharp gradient flow features like strong shocks in the simulations of two-dimensional inviscid compressible flows. A posteriori finite volume subcell limiter is employed in the shock-affected cells to control numerical spurious oscillations. An efficient cell-by-cell adaptive mesh refinement is implemented to increase the resolution of our simulations. This strategy enables to capture strong shocks without much numerical dissipation. A wide range of challenging test cases is considered to demonstrate the efficiency of the present adaptive numerical strategy for solving inviscid compressible flow problems having strong shocks.     

Revisit of dilation-based shock capturing for discontinuous Galerkin methods

The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement.     

适用于间断Galerkin方法的限制器研究

开发了一种适用于高精度间断Galerkin方法的斜率（多项式系数）限制器。与现有的斜率限制器不同,该限制器实施过程不考虑网格单元类型（三角形或四边形）,通过全微分方法构造新的多项式系数,因此,该限制器能够适用于各种类型网格——结构化网格、具有单一单元的非结构化网格和具有混合单元的非结构化网格。由于该限制器能够方便地应用于具有混合单元的非结构化网格,因此,本文使用的程序能够方便地求解具有复杂几何结构的流动问题。本文利用一些典型算例对其性能进行了验证,表明该限制器适用于不同类型的网格单元,能够在光滑解区保证高的精度,并能够在阊断区抑帛3非物理振荡。     

A limiter-based well-balanced discontinuous Galerkin method for shallow-water flows with wetting and drying: Triangular grids

A novel wetting and drying treatment for second-order Runge-Kutta discontinuous Galerkin methods solving the nonlinear shallow-water equations is proposed. It is developed for general conforming two-dimensional triangular meshes and utilizes a slope limiting strategy to accurately model inundation. The method features a nondestructive limiter, which concurrently meets the requirements for linear stability and wetting and drying. It further combines existing approaches for positivity preservation and well balancing with an innovative velocity-based limiting of the momentum. This limiting controls spurious velocities in the vicinity of the wet/dry interface. It leads to a computationally stable and robust scheme, even on unstructured grids, and allows for large time steps in combination with explicit time integrators. The scheme comprises only one free parameter, to which it is not sensitive in terms of stability. A number of numerical test cases, ranging from analytical tests to near-realistic laboratory benchmarks, demonstrate the performance of the method for inundation applications. In particular, superlinear convergence, mass conservation, well balancedness, and stability are verified.     

High-order strand grid methods for shock turbulence interaction

ABSTRACT

In this work, we examine the flux correction method for three-dimensional transonic turbulent flows on strand grids. Building upon previous work, we treat flux derivatives along strands with high-order summation-by-parts operators and penalty-based boundary conditions. A finite-volume like limiting strategy is implemented in the flux correction algorithm in order to sharply capture shocks. To achieve turbulence closure in the Reynolds-Averaged Navier–Stokes equations, a robust version of the Spalart–Allmaras turbulence model is employed that accommodates negative values of the turbulence working variable. Validation studies are considered which demonstrate the flux correction method achieves a high degree of accuracy for turbulent shock interaction flows.     

An improvement of classical slope limiters for high‐order discontinuous Galerkin method

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of a second‐order Runge–Kutta discontinuous Galerkin scheme for the shallow water equations with source terms

The present work addresses the numerical prediction of discontinuous shallow water flows by the application of a second‐order Runge–Kutta discontinuous Galerkin scheme (RKDG2). The unsteady flow of water in a one‐dimensional approach is described by the Saint Venant's model which incorporates source terms in practical applications. Therefore, the RKDG2 scheme is reformulated with a simple way to integrate source terms. Further, an adequate boundary conditions handling, by the theory of characteristics, was overviewed to be adapted to the external points of the mesh, as well as to some points of local invalidity of the Saint Venant's model. To validate the proposed technique, steady and transient test problems (all having a reference solution) were considered and computed by means of the overall method. The results were illustrated jointly with the reference solution and the results carried out by a traditional second‐order finite volume (FV2) scheme implemented with the same techniques as the RKDG2. The proposed method has proven its practical consideration when solving discontinuous shallow water flow involving: non‐prismatic channels, various cross‐sections, smoothly varying bed topography and internal boundary conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

High-order hybridisable discontinuous Galerkin method for the gas kinetic equation

ABSTRACT

The high-order hybridisable discontinuous Galerkin (HDG) method is used to find steady-state solution of gas kinetic equations on two-dimensional geometry. The velocity distribution function and its traces are approximated in piecewise polynomial space on triangular mesh and mesh skeleton, respectively. By employing a numerical flux derived from the upwind scheme and imposing its continuity on mesh skeleton, the global system for unknown traces is obtained with fewer coupled degrees of freedom, compared to the original DG method. The solutions of model equation for the Poiseuille flow through square channel show the higher order solver is faster than the lower order one. Moreover, the HDG scheme is more efficient than the original DG method when the degree of approximating polynomial is larger than 2. Finally, the developed scheme is extended to solve the Boltzmann equation with full collision operator, which can produce accurate results for shear-driven and thermally induced flows.