质点积分守恒重映方法

针对大变形流体动力学数值计算中经常需要应用的网格重构与物理量重映技术,提出了一种逻辑简单的质点积分守恒重映方法。将旧网格细分为众多有体积的质点,并将旧网格的物理量分配到各个质点,新网格各守恒量的积分直接由落在新网格内的所有质点的物理量累加。建立了收敛速度极快的计算格式,采用的控制体很好地解决了速度的重映计算问题。分析了此守恒重映方法的收敛性与守恒性,研究了积分控制体对速度计算的影响。     

拉格朗日高阶中心型守恒气体动力学格式

提出了拉格朗日高阶中心型守恒气体动力学格式。用产生于当前时刻子网格密度和当前时刻网格声速的子网格压力构造了子网格力,用加权本质无震荡方法构造的高阶子网格力构造了高阶空间通量,借助时间中点通量的泰勒展开完成了高阶时间通量离散,利用动量守恒条件使得格点速度以与网格面的数值通量相容的方式计算。编制了拉格朗日高阶中心型守恒气体动力学格式,对Saltzman活塞问题进行了数值模拟,数值结果表明,拉格朗日高阶中心型守恒气体动力学格式的有效性和精确性.     

基于非结构网格求解二维浅水方程的高精度有限体积方法

采用HLL格式,在三角形非结构网格下采用有限体积离散,建立了求解二维浅水方程的高精度的数值模型.本文采用多维重构和多维限制器的方法来获得高精度的空间格式以及防止非物理振荡的产生,时间离散采用三阶Runge-Kutta法以获得高阶的时间精度.基于三角形网格,底坡源项采用简单的斜底模型离散,为保证计算格式的和谐性,对经典的HLL格式计算的数值通量中的静水压力项进行了修正.算例证明本文提出的方法的和谐性并具有高精度的间断捕捉能力和稳定性.     

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.     

基于GPU加速的三阶有限体积格式管道瞬变流求解模型

为高效和高精度求解长距离输水系统瞬变流变化过程,应用三阶ENO有限体积格式求解一维管道非恒定流方程组,基于Lax-Friedrichs通量裂分法重构界面通量,上下游界面采用虚拟网格技术并结合交叉管网边界条件建立了一套高效和高精度求解管道瞬变流水锤波的数值模型。引入GPU加速技术,实现对大型输水系统的高效计算。通过特征线法、一阶及二阶Godunov有限体积格式对模型进行验证,结果表明,三阶ENO格式在极低的Courant数时也能保持较好的间断捕捉性能且无非物理振荡。同时,对Courant数的高度不敏感性,使得模型划分网格时具有高度的灵活性并能显著提高计算速度。应用GPU加速技术,发现模型在较多网格数时有明显的加速效果,且加速效果随网格数增多而显著。本文模型可为长距离输水系统非恒定瞬变过程的高效精准快速模拟预测提供理论支撑。     

非结构混合网格高超声速绕流与磁场干扰数值模拟

对均匀磁场干扰下的二维钝头体无粘高超声速流场进行了基于非结构混合网格的数值模拟.受磁流体力学方程组高度非线性的影响及考虑到数值模拟格式的精度,目前在此类流场的数值模拟中大多使用结构网格及有限差分方法,因而在三维复杂外形及复杂流场方面的研究受到限制.本文主要探索使用非结构网格(含混合网格)技术时的数值模拟方法.控制方程为耦合了Maxwell方程及无粘流体力学方程的磁流体力学方程组,数值离散格式采用Jameson有限体积格心格式,5步Runge-Kutta显式时间推进.计算模型为二维钝头体,初始磁场均匀分布.对不同磁感应强度影响下的高超声速流场进行了数值模拟,并与有限的资料进行了对比,得到了较符合的结果.     

基于高阶和高效格式的悬停旋翼可压缩无粘绕流的计算

发展了一种基于高精度和高效格式计算悬停旋翼复杂绕流的隐式有限体积方法。控制方程为Euler方程,其中对流项通量的左右状态量采用五阶加权基本无振荡(WENO)格式重构,时间推进应用隐式LU-SGS算法,为进一步加速收敛,采用三层V循环多重网格松弛方法。考虑到多重网格方法的思想以及五阶WENO格式涉及6个网格单元,建议仅在最细网格上使用WENO格式。计算结果表明本文方法能有效捕捉激波,对尾迹也有较高分辨率,基于隐式LU-SGS算法的多重网格方法能有效提高计算效率。     

GMRES方法在含动边界流场中的应用

以基于格心的有限体积法为基础，空间二阶精度，采用4阶Runge—Kutta，GMRES隐式方法求解基于ALE形式的Euler方程，网格单元边界处守恒量通量的计算采用了Hanel方法，对NACA0012翼型绕流及运动圆球绕流等问题进行数值模拟，取得了较好的结果．GMRES方法克服了以往隐式方法大量耗费内存的弱点，达到了计算耗时短和占用内存少的统一．     

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

谱体积方法单元分割与问题单元标记方法的改进研究

谱体积方法是一种本质上解决网格依赖性的高精度CFD计算方法,本文研究了二维Euler方程的谱体积方法,提出一种基于切比雪夫多项式的单元分割方法,建立了基于WENO的变量限制器方法,并发展了结合谱体积和控制体的问题单元标记方法.采用15°超声速压缩拐角和NACA0012跨声速流动两个典型算例进行验证,结果表明,该分区方法具有更好的计算精度,标记方法可有效识别不连续区域,在较少的网格下即可获得与密网格传统有限体积法相当的计算精度.     

A 3D finite element ALE method using an approximate Riemann solution

Arbitrary Lagrangian–Eulerian finite volume methods that solve a multidimensional Riemann‐like problem at the cell center in a staggered grid hydrodynamic (SGH) arrangement have been proposed. This research proposes a new 3D finite element arbitrary Lagrangian–Eulerian SGH method that incorporates a multidimensional Riemann‐like problem. Two different Riemann jump relations are investigated. A new limiting method that greatly improves the accuracy of the SGH method on isentropic flows is investigated. A remap method that improves upon a well‐known mesh relaxation and remapping technique in order to ensure total energy conservation during the remap is also presented. Numerical details and test problem results are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

A 3D staggered Lagrangian scheme for ideal magnetohydrodynamics on unstructured meshes

In this paper, we propose a 3D staggered Lagrangian scheme for the ideal magnetohydrodynamics (MHD) on unstructured meshes. All the thermal variables and the magnetic induction are defined in the cell centers while the fluid velocity is located at the nodes. The meshes are compatibly discretized to ensure the geometric conservation laws in Lagrangian computation by the classical subcell method, then the momentum equation is discretized using the subcell forces and the specific internal energy equation is obtained by the total energy conservation. Invoking the Galilean invariance, magnetic flux conservation, and the thermodynamic consistency, the expressions of subcell force as well as the cell-centered velocity are derived. Besides, the magnetic divergence-free constraint is fulfilled by a projection method after each time step. Various numerical tests are presented to assert the robustness and accuracy of our scheme.     

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow （Power-Law model）. The collocated （i.e. non-staggered） arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.     

A new modification of the immersed‐boundary method for simulating flows with complex moving boundaries

In this paper, a new immersed‐boundary method for simulating flows over complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. The present method is based on a finite‐difference approach on a staggered mesh together with a fractional‐step method. It must be noted that the immersed boundary is generally not coincident with the position of the solution variables on the grid, therefore, an appropriate strategy is needed to construct a relationship between the curved boundary and the grid points nearby. Furthermore, a momentum forcing is added on the body boundaries and also inside the body to satisfy the no‐slip boundary condition. The immersed boundary is represented by a series of interfacial markers, and the markers are also used as Lagrangian forcing points. A linear interpolation is then used to scale the Lagrangian forcing from the interfacial markers to the corresponding grid points nearby. This treatment of the immersed‐boundary is used to simulate several problems, which have been validated with previous experimental results in the open literature, verifying the accuracy of the present method. Copyright © 2006 John Wiley & Sons, Ltd.     

A hybrid scheme based on finite element/volume methods for two immiscible fluid flows

We have successfully extended our implicit hybrid finite element/volume (FE/FV) solver to flows involving two immiscible fluids. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix‐free implicit cell‐centered FV method. The pressure Poisson equation is solved by the node‐based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. This updating strategy can be rigorously proven to be able to eliminate the unphysical pressure boundary layer and is crucial for the correct temporal convergence rate. Our current staggered‐mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centers and the auxiliary variable at vertices. The fluid interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix‐free FV method, as the one used for momentum equations, is used to solve the advection equation. We will focus on the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm that enforces the conservation of the mass for each fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

Combined swept region and intersection‐based single‐material remapping method

A typical arbitrary Lagrangian–Eulerian algorithm consists of a Lagrangian step, where the computational mesh moves with the fluid flow; a rezoning step, where the computational mesh is smoothed and repaired in case it gets too distorted; and a remapping step, where all fluid quantities are conservatively interpolated on this new mesh. In single‐material simulations, the remapping process can be represented in a flux form, with fluxes approximated by integrating a reconstructed function over intersections of neighboring computational cells on the original and rezoned computational mesh. This algorithm is complex and computationally demanding – Therefore, a simpler approach that utilizes regions swept by the cell edges during rezoning is often used in practice. However, it has been observed that such simplification can lead to distortion of the solution symmetry, especially when the mesh movement is not bound by such symmetry. For this reason, we propose an algorithm combining both approaches in a similar way that was proposed for multi‐material remapping (two‐step hybrid remap). Intersections and exact integration are employed only in certain parts of the computational mesh, marked by a switching function – Various different criteria are presented in this paper. The swept‐based method is used elsewhere in areas that are not marked. This way, our algorithm can retain the beneficial symmetry‐preserving capabilities of intersection‐based remapping while keeping the overall computational cost moderate.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

A fully discrete ALE method over untwisted time–space control volumes

In this paper, a fully discrete high‐resolution arbitrary Lagrangian–Eulerian (ALE) method is developed over untwisted time–space control volumes. In the framework of the finite volume method, 2D Euler equations are discretized over untwisted moving control volumes, and the resulting numerical flux is computed using the generalized Riemann problem solver. Then, the fluid flows between meshes at two successive time steps can be updated without a remapping process in the classic ALE method. This remapping‐free ALE method directly couples the mesh motion into a physical variable update to reflect the temporal evolution in the whole process. An untwisted moving mesh is generated in terms of the vorticity‐free part of the fluid velocity according to the Helmholtz theorem. Some typical numerical tests show the competitive performance of the current method. Copyright © 2016 John Wiley & Sons, Ltd.     

Calculation of two-dimensional shear-driven cavity flows at high reynolds numbers

The time-dependent Navier–Stokes equations are numerically integrated for two-dimensional incompressible viscous flow in a shear-driven square cavity. Using a time-splitting method and finite differences on a staggered mesh, the momentum and pressure equations are directly solved by a tensor product method where one finite difference direction is diagonalized by eigenvalue decomposition. The effects of increasing Reynolds number are studied and the developing boundary layer is captured by using a finely clustered mesh. At Re = 30000 the flow is in a continuously developing unsteady regime. Power spectrum plots indicate that the unsteady flow oscillates with one fundamental frequency and exhibits some characteristics of transition between laminar and turbulent states.     

NND格式在非结构网格中的推广

在张涵信提出的无波动、无自由参数的差分格式（ＮＮＤ格式）的基础上，构造了适用于非结构网格的二阶精度ＮＮＤ有限体积格式，解决了现有非结构网格方法中为抑制激波附近的波动而必须引入含自由参数的人工粘性项的困难，并采用网格自适应技术以提高效率．通过对二维平板激波反射和前台阶在管道内的流动问题的计算，表明本方法可有效地用于Ｅｕｌｅｒ方程的求解．