基于WENO重构的真正多维黎曼求解器

具有良好守恒性与网格适应性的有限体积格式在流体力学的数值计算中占有重要地位。其中，求解数值流通量是实施有限体积法的关键步骤。一维情形下，通过求解局部黎曼问题来获得数值流通量的相关理论已经比较成熟。但是在计算多维问题时，传统的维度分裂方法仅考虑沿界面法向传播的信息，这不仅影响格式的精度，还可能会造成数值不稳定性从而诱发非物理现象。本文基于对流-压力通量分裂方法来构造真正多维的黎曼求解器，通过求解网格顶点处的多维黎曼问题来实现格式的多维特性。采用五阶WENO重构方法来获得空间的高阶精度，时间离散采用三阶TVD龙格-库塔格式。一系列数值实验的结果表明，真正多维的黎曼求解器不仅具有更高的分辨率还能有效克服多维强激波模拟中的数值不稳定性。     

An extension of Osher's Riemann solver for chemical and vibrational non-equilibrium gas flows

In this paper we study an extension of Osher's Riemann solver to mixtures of perfect gases whose equation of state is of the form encountered in hypersonic applications. As classically, one needs to compute the Riemann invariants of the system to evaluate Osher's numerical flux. For the case of interest here it is impossible in general to derive simple enough expressions which can lead to an efficient calculation of fluxes. The key point here is the definition of approximate Riemann invariants to alleviate this difficulty. Some of the properties of this new numerical flux are discussed. We give 1D and 2D applications to illustrate the robustness and capability of this new solver. We show by numerical examples that the main properties of Osher's solver are preserved; in particular, no entropy fix is needed even for hypersonic applications.     

间断Galerkin有限元和有限体积混合计算方法研究

通过局部坐标变换而建立的非正交单元间断Galerkin(DG)有限元计算方法计算精度高， 但计算量大、内存需求大；而非结构网格有限体积方法虽然准确计算热流的问题目 前还没有完全解决，但其具有计算速度快和内存需求小的优点. 该研究是将有 限元和有限体积方法的优点结合，发展有限元和有限体积的混合方法. 在物面 附近黏性占主导作用的区域内采用有限元方法进行计算，在远离物面的区域采用快速的有限 体积方法进行计算，在有限元和有限体积方法结合处要保证通量守恒. 通过算例说明有 限元和有限体积混合方法既能保证黏性区域的热流计算精度和流场结构的分辨率，又能 降低内存需求和提高计算效率，使有限元方法应用于复杂外形(实际工程问题)的计 算成为可能.     

Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes

A 2D, depth-integrated, free surface flow solver for the shallow water equations is developed and tested. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov-type second-order upwind finite volume formulation, whereby the inviscid fluxes of the system of equations are obtained using Roe's flux function. The eigensystem of the 2D shallow water equations is derived and is used for the construction of Roe's matrix on an unstructured mesh. The viscous terms of the shallow water equations are computed using a finite volume formulation which is second-order-accurate. Verification of the solution technique for the inviscid form of the governing equations as well as for the full system of equations is carried out by comparing the model output with documented published results and very good agreement is obtained. A numerical experiment is also conducted in order to evaluate the performance of the solution technique as applied to linear convection problems. The presented results show that the solution technique is robust. © 1997 John Wiley & Sons, Ltd.     

Contributions to numerical developments in shock waves attenuation in porous filters

The paper deals with the numerical method of the compressible gas flow through a porous filter emphasizing the treatment of the interface between a pure gaseous phase and a solid phase. An incident shock wave is initiated in the gaseous phase interacting with a porous filter inducing a transmitted and a reflected wave. To take into account the discontinuity jump in the porosity between the gaseous phase and the porous filter, an approximate Riemann solver is used to compute homogeneous non-conservative Euler equations in porous media using ideal gas state law. The discretization of this problem is based on a finite volume method where the fluxes are evaluated by a “volumes finis Roe” (VFRoe) scheme. A stationary solution is determined with a continuous variable porosity in order to test the numerical scheme. Numerical results are compared with the two-phase shock tube experiments and simulations of a shock wave attenuation and gas filtration in porous filters are presented.      

Solution of the hyperbolic mild‐slope equation using the finite volume method

A finite volume solver for the 2D depth‐integrated harmonic hyperbolic formulation of the mild‐slope equation for wave propagation is presented and discussed. The solver is implemented on unstructured triangular meshes and the solution methodology is based upon a Godunov‐type second‐order finite volume scheme, whereby the numerical fluxes are computed using Roe's flux function. The eigensystem of the mild‐slope equations is derived and used for the construction of Roe's matrix. A formulation that updates the unknown variables in time implicitly is presented, which produces a more accurate and reliable scheme than hitherto available. Boundary conditions for different types of boundaries are also derived. The agreement of the computed results with analytical results for a range of wave propagation/transformation problems is very good, and the model is found to be virtually paraxiality‐free. Copyright © 2003 John Wiley & Sons, Ltd.     

A Godunov‐type fractional semi‐implicit method based on staggered grid for dam‐break modeling

A semi‐implicit finite volume model based upon staggered grid is presented for solving shallow water equation. The model employs a time‐splitting scheme that uses a predictor–corrector method for the advection term. The fluxes are calculated based on a Riemann solver in the prediction step and a downwind scheme in the correction step. A simple TVD scheme is employed for shock capturing purposes in which the Minmond limiter is used for flux functions. As a consequence of using staggered grid, an ADI method is adopted for solving the discretized equations for 2‐D problems. Several 1‐D and 2‐D flows have been modeled with satisfactory results when compared with analytical and experimental test cases. The model is also capable of simulating supercritical as well as subcritical flow. Copyright © 2010 John Wiley & Sons, Ltd.     

Toward a reduction of mesh imprinting

This paper is the initial investigation into a new Lagrangian cell‐centered hydrodynamic scheme that is motivated by the desire for an algorithm that resists mesh imprinting and has reduced complexity. Key attributes of the new approach include multidimensional construction, the use of flux‐corrected transport (FCT) to achieve second order accuracy, automatic determination of the mesh motion through vertex fluxes, and vorticity control. Toward this end, vorticity preserving Lax–Wendroff (VPLW) type schemes for the two‐dimensional acoustic equations were analyzed and then implemented in a FCT algorithm. Here, mesh imprinting takes the form of anisotropic dispersion relationships. If the stencil for the LW methods is limited to nine points, four free parameters exist. Two parameters were fixed by insisting that no spurious vorticity be created. Dispersion analysis was used to understand how the remaining two parameters could be chosen to increase isotropy. This led to new VPLW schemes that suffer less mesh imprinting than the rotated Richtmyer method. A multidimensional, vorticity preserving FCT implementation was then sought using the most promising VPLW scheme to address the problem of spurious extrema. A well‐behaved first order scheme and a new flux limiter were devised in the process. The flux limiter is unique in that it acts on temporal changes and does not place a priori bounds on the solution. Numerical results have demonstrated that the vorticity preserving FCT scheme has comparable performance to an unsplit MUSCL‐H algorithm at high Courant numbers but with reduced mesh imprinting and superior symmetry preservation. Copyright © 2014 John Wiley & Sons, Ltd.     

An improved axisymmetric lattice Boltzmann flux solver for axisymmetric isothermal/thermal flows

In this work, an improved axisymmetric lattice Boltzmann flux solver (LBFS) is proposed for simulation of axisymmetric isothermal and thermal flows. This solver globally resolves the axisymmetric Navier-Stokes (N-S) equations through the finite volume strategy and locally reconstructs numerical fluxes with solutions to the lattice Boltzmann equation. Compared with previous axisymmetric LBFS, some novel strategies are adopted in this work to simplify the formulations and improve the accuracy. First, the macroscopic equations are reformulated to reduce the number of source terms and remove spatial derivatives involved in the source terms. Second, the local reconstruction of numerical fluxes utilizes relationships given by the Chapman-Enskog analysis and combines the radial coordinate (r) with the local solution to the standard LB equation. By adopting these two modifications, the present axisymmetric LBFS avoids the fractional-step formulation and the finite-difference approximation adopted in the previous solver, which reduces the complexity of implementation. Moreover, an alternative way of predicting intermediate pressure is proposed, which could effectively fix the inaccurate resolution of the pressure field in previous axisymmetric LBFS. Further extensions are made to enrich the applicability of the present solver in thermal axisymmetric flows. Validations on various benchmark tests are carried out for comprehensive evaluation of the robustness and flexibility of the proposed solver.     

A rotating reference frame‐based lattice Boltzmann flux solver for simulation of turbomachinery flows

In this paper, the newly developed lattice Boltzmann flux solver (LBFS) is developed into a version in the rotating frame of reference for simulation of turbomachinery flows. LBFS is a finite volume solver for the solution of macroscopic governing differential equations. Unlike conventional upwind or Godunov‐type flux solvers which are constructed by considering the mathematical properties of Euler equations, it evaluates numerical fluxes at the cell interface by reconstructing local solution of lattice Boltzmann equation (LBE). In other words, the numerical fluxes are physically determined rather than by some mathematical approximation. The LBE is herein expressed in a relative frame of reference in order to correctly recover the macroscopic equations, which is also the basis of LBFS. To solve the LBE, an appropriate lattice Boltzmann model needs to be established in advance. This includes both the determinations of the discrete velocity model and its associated equilibrium distribution functions. Particularly, a simple and effective D1Q4 model is adopted, and the equilibrium distribution functions could be efficiently obtained by using the direct method. The present LBFS is validated by several inviscid and viscous test cases. The numerical results demonstrate that it could be well applied to typical and complex turbomachinery flows with favorable accuracy. It is also shown that LBFS has a delicate dissipation mechanism and is thus free of some artificial fixes, which are often needed in conventional schemes. Copyright © 2016 John Wiley & Sons, Ltd.     

A height–flux method for simulating free surface flows and interfaces

A new technique for the numerical simulation of the free surface flows is developed. This technique is based on the finite element method with penalty formulation, and a flux method for surface advection. The advection part which is completely independent of the momentum solver is based on subdividing the fluid domain into small subvolumes along one of the co-ordinate axis. The subvolumes are then used to find the height function which will later describe the free surface. The free surface of the fluid in each subvolume is approximated by a line segment and its slope is calculated using the volume of the fluid in the two neighbouring subvolumes. Later, the unidirectional volume flux from one subvolume to its neighbouring one is calculated using the conservation laws, and the new surface line segments are reconstructed. This technique, referred to as the Height–Flux Method (HFM) is implemented to simulate the temporal instability of a capillary jet. The results of the numerical simulation well predict the experimental data. It is also shown that the HFM is computationally more efficient than the techniques which use a kinematic boundary condition for the surface advection.     

On high‐resolution schemes for solving unsteady compressible two‐phase dilute viscous flows

A high‐resolution numerical scheme based on the MUSCL–Hancock approach is developed to solve unsteady compressible two‐phase dilute viscous flow. Numerical considerations for the development of the scheme are provided. Several solvers for the Godunov fluxes are tested and the results lead to the choice of an exact Riemann solver adapted for both gaseous and dispersed phases. The accuracy of the scheme is proven step by step through specific test cases. These simulations are for one‐phase viscous flows over a flat plate in subsonic and supersonic regimes, unsteady flows in a low‐pressure shock tube, two‐phase dilute viscous flows over a flat plate and, finally, two‐phase unsteady viscous flows in a shock tube. The results are compared with well‐established analytical and numerical solutions and very good agreement is achieved. Copyright © 1999 John Wiley & Sons, Ltd.     

An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

Shallow flow simulation on dynamically adaptive cut cell quadtree grids

A computationally efficient, high‐resolution numerical model of shallow flow hydrodynamics is described, based on dynamically adaptive quadtree grids. The numerical model solves the two‐dimensional non‐linear shallow water equations by means of an explicit second‐order MUSCL‐Hancock Godunov‐type finite volume scheme. Interface fluxes are evaluated using an HLLC approximate Riemann solver. Cartesian cut cells are used to improve the fit to curved boundaries. A ghost‐cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The numerical model is validated through simulations of reflection of a surge wave at a wall, a low Froude number potential flow past a circular cylinder, and the shock‐like interaction between a bore and a circular cylinder. The computational efficiency is shown to be greatly improved compared with solutions on a uniform structured grid implemented with cut cells. Copyright © 2006 John Wiley & Sons, Ltd.     

An efficient and accurate algebraic interface capturing method for unstructured grids in 2 and 3 dimensions: The THINC method with quadratic surface representation

We present in this paper an efficient and accurate volume of fluid (VOF) type scheme to compute moving interfaces on unstructured grids with arbitrary quadrilateral mesh elements in 2D and hexahedral elements in 3D. Being an extension of the multi‐dimensional tangent of hyperbola interface capturing (THINC) reconstruction proposed by the authors in Cartesian grid, an algebraic VOF scheme is devised for arbitrary quadrilateral and hexahedral elements. The interface is cell‐wisely approximated by a quadratic surface, which substantially improves the numerical accuracy. The same as the other THINC type schemes, the present method does not require the explicit geometric representation of the interface when computing numerical fluxes and thus is very computationally efficient and straightforward in implementation. The proposed scheme has been verified by benchmark tests, which reveal that this scheme is able to produce high‐quality numerical solutions of moving interfaces in unstructured grids and thus a practical method for interfacial multi‐phase flow simulations. Copyright © 2014 John Wiley & Sons, Ltd.     

Roe‐type Riemann solver for gas–liquid flows using drift‐flux model with an approximate form of the Jacobian matrix

This work presents an approximate Riemann solver to the transient isothermal drift ‐ flux model. The set of equations constitutes a non‐linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix A are expressed through exact analytical expressions. It is also proposed a simplified form of A considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of A . The proposed solver is validated against three benchmark solutions and two experimental pipe flow data. Copyright © 2015 John Wiley & Sons, Ltd.     

A hybrid kinetic WENO scheme for inviscid and viscous flows

In this paper, we propose a high‐order finite volume hybrid kinetic Weighted Essentially Non‐Oscillatory (WENO) scheme for inviscid and viscous flows. Based on the WENO reconstruction technique, a hybrid kinetic numerical flux is introduced for the present method, which includes the mechanisms of both the free transfer and the collision of gas molecules. The collisionless free transfer part of the hybrid numerical flux is constructed from the conventional kinetic flux vector splitting treatment, and the collision contribution is considered by constructing an equilibrium gas state and calculating the corresponding numerical flux at the cell interface. The total variation diminishing Runge–Kutta methods are used for the temporal integration. The high‐order accuracy and good shock‐capturing capability of the proposed hybrid kinetic WENO scheme are validated by many numerical examples in one‐dimensional and two‐dimensional cases. Copyright © 2015 John Wiley & Sons, Ltd.     

A bore-capturing finite volume method for open-channel flows

A high-resolution finite volume hydrodynamic solver is presented for open-channel flows based on the 2D shallow water equations. This Godunov-type upwind scheme uses an efficient Harten–Lax–van Leer (HLL) approximate Riemann solver capable of capturing bore waves and simulating supercritical flows. Second-order accuracy is achieved by means of MUSCL reconstruction in conjunction with a Hancock two-stage scheme for the time integration. By using a finite volume approach, the computational grid can be irregular which allows for easy boundary fitting. The method can be applied directly to model 1D flows in an open channel with a rectangular cross-section without the need to modify the scheme. Such a modification is normally required for solving the 1D St Venant equations to take account of the variation of channel width. The numerical scheme and results of three test problems are presented in this paper. © 1998 John Wiley & Sons, Ltd.     

An efficient upwind/relaxation algorithm for the Euler and Navier–Stokes equations

An efficient Euler and full Navier–Stokes solver based on a flux splitting scheme is presented. The original Van Leer flux vector splitting form is extended to arbitrary body-fitted co-ordinates in the physical domain so that it can be used with a finite volume scheme. The block matrix is inverted by Gauss–Seidel iteration. It is verified that the often used reflection boundary condition will produce incorrect flux crossing the wall and cause too large numerical dissipation if flux vector splitting is used. To remove such errors, an appropriate treatment of wall boundary conditions is suggested. Inviscid and viscous steady transonic internal flows are analysed, including the case of shock-induced boundary layer separation.     

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

In the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM⁺ ? up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM⁺ ? up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.