Combined finite volume–finite element method for shallow water equations

This paper is concerned with solving the viscous and inviscid shallow water equations. The numerical method is based on second-order finite volume–finite element (FV–FE) discretization: the convective inviscid terms of the shallow water equations are computed by a finite volume method, while the diffusive viscous terms are computed with a finite element method. The method is implemented on unstructured meshes. The inviscid fluxes are evaluated with the approximate Riemann solver coupled with a second-order upwind reconstruction. Herein, the Roe and the Osher approximate Riemann solvers are used respectively and a comparison between them is made. Appropriate limiters are used to suppress spurious oscillations and the performance of three different limiters is assessed. Moreover, the second-order conforming piecewise linear finite elements are used. The second-order TVD Runge–Kutta method is applied to the time integration. Verification of the method for the full viscous system and the inviscid equations is carried out. By solving an advection–diffusion problem, the performance assessment for the FV–FE method, the full finite volume method, and the discontinuous Galerkin method is presented.     

A residual-based scheme for computing compressible flows on unstructured grids

A second-order residual-based compact scheme, initially developed for computing flows on Cartesian and curvilinear grids, is extended to general unstructured grids in a finite-volume framework. The scheme is applied to the computation of several inviscid and viscous compressible flows governed by the Euler and Navier-Stokes equations. Its efficiency and accuracy properties are compared with those of conventional second-order upwind schemes based on variable reconstruction.     

An implicit MacCormack scheme for unsteady flow calculations

This paper describes the implicit MacCormack scheme [1] in finite volume formulation. Unsteady flows with moving boundaries are considered using arbitrary Lagrangian–Eulerian approach.The scheme is unconditionally stable and does not require solution of large systems of linear equations. Moreover, the upgrade from explicit MacCormack scheme to implicit one is very simple and straightforward.Several computational results for 2D and 3D flows over profiles and wings are presented for the case of inviscid and viscous flows.     

On some approaches to solve CFD problems

This paper presents two efficient methods for spatial flows calculations. In order to simulate of incompressible viscous flows, a second-order accurate scheme with an incomplete LU decomposed implicit operator is developed. The scheme is based on the method of artificial compressibility and Roe flux-difference splitting technique for the convective terms. The numerical algorithm can be used to compute both steady-state and time-dependent flow problems. The second method is developed for modeling of stationary compressible inviscid flows. This numerical algorithm is based on a simple flux-difference splitting into physical processes method and combines a multi level grid technology with a convergence acceleration procedure for internal iterations. The capabilities of the methods are illustrated by computations of steady-state flow in a rotary pump, unsteady flow over a circular cylinder and stationary subsonic flow over an ellipsoid.     

A SVD-GFD scheme for computing 3D incompressible viscous fluid flows

A generalized finite difference (GFD) scheme for the simulation of three-dimensional (3D) incompressible viscous fluid flows in primitive variables is described in this paper. Numerical discretization is carried out on a hybrid Cartesian cum meshfree grid, with derivative approximation on non-Cartesian grids being carried out by a singular value decomposition (SVD) based GFD procedure. The Navier-Stokes equations are integrated by a time-splitting pressure correction scheme with second-order Crank-Nicolson and second-order discretization of time and spatial derivatives respectively. Axisymmetric and asymmetric 3D flows past a sphere with Reynolds numbers of up to 300 are simulated and compared with the results of Johnson and Patel [Johnson TA, Patel VC. Flow past a sphere up to a Reynolds number of 300. J Fluid Mech 1999;378:19-70] and others. Flows past toroidal rings are also simulated to illustrate the ability of the scheme to deal with more complex body geometry. The current method can also deal with flow past 3D bodies with sharp edges and corners, which is shown by a simple 3D case.     

Solution of viscous transonic flow over wings

Knowledge of the viscous flow about wings is very important in 3-D wing design. In transonic flow about a typical supercritical wing, the viscous effect results in a sizable reduction of the lift-to-drag ratio. The Reynolds number dependence of the flow is not clearly defined, and no known similitude exists that can be used to scale the experimental data for a particular design. Recent advances in computer technology and numerical technique have relieved the difficulty of obtaining a theoretical solution somewhat, but the lack of a proper reliable method of treating the turbulence in a time-averaged Navier-Stokes solution remains the major stumbling block.For this paper, a “zonal” approach has been used for a viscid-inviscid interaction analysis to yield an iterative solution for the viscous flow about wings in the transonic flow regime. The chord Reynolds number considered was of the order of 10⁶ and above so that the flow was predominantly turbulent. The inviscid flow field was obtained by solving the 3-D potential flow equation. A parabolic coordinate mapping was used in the computation, in conjunction with a finite volume formulation. A new approximate factorization scheme has been developed for the iterative solution of the inviscid flow. A special far field asymptotic boundary condition that improves the accuracy and convergence of the method was derived. For the 3-D boundary layer calculation, the integral method of Myring-Smith-Stock was extensively modified to make it suitable for the interaction calculation. The effect of wing thickness was taken into account and the 3-D viscous wake was computed. The interaction calculation was formulated with a set of coupling conditions that includes the source flux distribution due to the surface boundary layer on the wing, the flux jump distribution due to the viscous wake, and the effect of the viscous wake curvature. The transpiration boundary conditions have been used for the inviscid flow in the coupled calculation. In addition, a method was devised so that the results of an analysis of the trailing edge strong interaction solution for a 2-D viscous airfoil could be adapted for the normal pressure correction near the trailing edge. The theory has been applied to supercritical wing geometries of practical interest. The converged viscous flow results compare favorably with experimental pressure data.     

Cartesian cut cell approach for simulating incompressible flows with rigid bodies of arbitrary shape

In this paper, a Cartesian grid method with cut cell approach has been developed to simulate two dimensional unsteady viscous incompressible flows with rigid bodies of arbitrary shape. A collocated finite volume method with nominally second-order accurate schemes in space is used for discretization. A pressure-free projection method is used to solve the equations governing incompressible flows. For fixed-body problems, the Adams-Bashforth scheme is employed for the advection terms and the Crank-Nicholson scheme for the diffusion terms. For moving-body problems, the fully implicit scheme is employed for both terms. The present cut cell approach with cell merging process ensures global mass/momentum conservation and avoid exceptionally small size of control volume which causes impractical time step size. The cell merging process not only keeps the shape resolution as good as before merging, but also makes both the location of cut face center and the construction of interpolation stencil easy and systematic, hence enables the straightforward extension to three dimensional space in the future. Various test examples, including a moving-body problem, were computed and validated against previous simulations or experiments to prove the accuracy and effectiveness of the present method. The observed order of accuracy in the spatial discretization is superlinear.     

A finite element method for diffusion dominated unsteady viscous flows

A general conforming finite element scheme for computing viscous flows is presented which is of second-order accuracy in space and time. Viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm. A method for solving the algebraic equations at each time step is given. The method is demonstrated on two test problems, one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step.     

Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations

We propose a new third-order multidimensional upwind algorithm for the solution of the flow equations on tetrahedral cells unstructured grids. This algorithm has a compact stencil (cell-based computations) and uses a finite element idea when computing the residual over the cell to achieve its third-order (spatial) accuracy. The construction of the new scheme is presented. The asymptotic accuracy for steady or unsteady, inviscid or viscous flow situations is proved using numerical experiments. The new high-order discretization proves to have excellent parallel scalability. Our studies show the advantages of the new compact third-order scheme when compared with the classical second-order multidimensional upwind schemes.     

A finite volume method to solve the 3D Navier–Stokes equations on unstructured collocated meshes

A new method to solve the Navier–Stokes equations for incompressible viscous flows and the transport of a scalar quantity is proposed. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. The governing equations are discretized using a collocated, cell-centered arrangement of velocity and pressure. The solution variables are stored at the cell-circumcenters. Theoretical results and numerical properties of the scheme are provided. Predictions of lid-driven cavity flow, flows past a cylinder and heat transport in a cylinder are performed to validate the method.     

基于结构网格的高超声速流动大规模并行数值模拟研究

本文采用MPI消息传递模式自主开发出适用于高超声速流动数值模拟的并行计算软件,该软件以三维Navier-Stokes方程为基本控制方程来求解层流问题,应用基于结构网格的有限体积法对计算域进行离散,采用AUSMPW+格式求解对流通量,利用MUSCL插值方法获得高阶精度,时间格式上采用LU-SGS方法进行时间迭代以加快求解定常流动的收敛过程。在高性能计算机上针对不同高超声速流动进行大规模并行计算的结果表明,所开发的CFD并行计算软件具有较高的并行计算效率,为高超声速飞行器气动力/热的准确预测提供了高效工具。     

A computerized analysis of supersonic nonuniform flows over sharp and spherically blunted cones at angle of attack

A system of computer programs has been developed to predict supersonic inviscid and viscous nonuniform flow fields over sharp and spherically blunted cones at angle of attack. For blunt cones the flow fields considered were axisymmetric wake flows positioned such that the flow in the subsonic nose region remained axisymmetric. For sharp cones, both axisymmetric wake flows and two-dimensional shear flows were considered. The programs used in solving inviscid flow fields incorporate a modified inverse method for solving subsonic flow regions and modified axiymmetric and three-dimensional method of characteristics procedures for solving the supersonic flow regions. Body properties predicted by the inviscid solutions were used as edge data for solution of the corresponding laminar boundary-layers over the bodies. The viscous flow solutions were obtained using axisymmetric and full three-dimensional boundary-layer programs. Typical results from inviscid calculations have shown the development of strong adverse pressure gradients over both sharp and blunt cones in wake flows. In addition a thin entropy layer was found near the surface of both bodies; however, the normal pressure gradient was found to be negligible for the nonuniform flows considered. For the sharp cone in shear flow, property variation along the body was found to be almost linear. In all cases the aerodynamic coefficients were found to be significantly affected by the free-stream nonuniformity. Typical viscous flow field results have shown that relative to uniform flow values the skin friction and heat transfer increase along the windward streamline of both blunt and sharp cones in the nonuniform flows considered. Decreasing the width of the wake in wake flow increases the heat transfer and skin friction.     

Vertex-centroid finite volume scheme on tetrahedral grids for conservation laws

Vertex-centroid schemes are cell-centered finite volume schemes for conservation laws which make use of both centroid and vertex values to construct high-resolution schemes. The vertex values must be obtained through a consistent averaging (interpolation) procedure while the centroid values are updated by the finite volume scheme. A modified interpolation scheme is proposed which is better than existing schemes in giving positive weights in the interpolation formula. A simplified reconstruction scheme is also proposed which is also more efficient and leads to more robust schemes for discontinuous problems. For scalar conservation laws, we develop limited versions of the schemes which are stable in maximum norm by constructing suitable limiters. The schemes are applied to compressible flows governed by the Euler equations of inviscid gas dynamics.     

A computational scheme in generalized coordinates for viscous incompressible flows

This paper presents a computational scheme suitable for analyzing viscous incompressible flows in generalized curvilinear coordinate system. The scheme is based on finite volume algorithm with an overlapping staggered grid. The pseudo-diffusive terms arising from the coordinate transformation are treated as source terms. The system of nonlinear algebraic equations is solved by a semi-implicit procedure based upon line-relaxation and a generalization of Patankar's pressure correction algorithm. Examples of the application of the algorithm to flow in convergent channels, developing flow in a U-bend, and flow past backward facing step, are given. In addition, the case of flow past backward facing step is analyzed in detail, and the computed flowfields are found to be in close agreement with previous experimental and numerical results for expansion ratio (defined as the ratio of step height to channel height) of 0.5. The results are summarized in the form of a correlation relating the primary separation length, Reynolds number and expansion ratio.     

An Efficient Correction Method to Obtain a Formally Third-Order Accurate Flow Solver for Node-Centered Unstructured Grids

A new method of obtaining third-order accuracy on unstructured grid flow solvers is presented. The method involves a simple correction to a traditional linear Galerkin scheme on tetrahedra and can be conveniently added to existing second-order accurate node-centered flow solvers. The correction involves gradients of the flux computed with a quadratic least squares approximation. However, once the gradients are computed, no second derivative information or high-order quadrature is necessary to achieve third-order accuracy. The scheme is analyzed both analytically using truncation error, and numerically using solution error for an exact solution to the Euler equations. Two demonstration cases for steady, inviscid flow reveal increased accuracy and excellent shock capturing with no loss in steady-state convergence rate. Computational timing results are presented which show the additional expense from the correction is modest compared to the increase in accuracy.     

Simulation of supercritical flow in crossroads: Confrontation of a 2D and 3D numerical approaches to experimental results

This work deals with the modeling of a flood in an urban environment. Among the various types of urban flood events, it was decided to study specifically the severe surface flooding events, which take place in highly urbanized areas. This work concerns particularly the numerical resolution of the two-dimensional Saint Venant equations for the study of the propagation of flood through the crossroads in the city. A discontinuous finite-element space discretization with a second-order Runge-Kutta time discretization is used to solve the two-dimensional Saint Venant equations. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally more efficient than comparable second-order finite volume methods. The scheme is applied to several supercritical flows in crossroads, which are investigated by Mignot. The experimental results obtained by the author are used to verify the accuracy and the robustness of the method. The results obtained are compared to those obtained by a second-order finite volume method (Rubar20 (2D)) and by FLUENT (3D). A very good agreement between the numerical solution obtained by the Runge-Kutta discontinuous Galerkin (RKDG) method and the experimental measured data were found. The method is then able to simulate the flow patterns observed experimentally and able to predict well the water depths, the discharge distribution in the downstream branches of the crossroad and the location of the hydraulic jumps and other flow characteristics more than the other methods.     

High resolution finite volume computations on unstructured grids using solution dependent weighted least squares gradients

An improved high resolution finite volume method based on linear and quadratic variable reconstructions using solution dependent weighted least squares (SDWLS) gradients has been presented here. An extended stencil consisting of vertex-based neighbours of a cell is used in the higher order reconstructions for inviscid flux computations. A QR algorithm with Householder transformation is used to solve the weighted least squares problem. In case of Navier–Stokes equations, viscous fluxes are discretized in a central differencing manner based on the Coirier’s diamond path. A few inviscid and viscous test cases are solved in order to demonstrate the efficacy of the present method. Progressive improvements in solution accuracy are observed with the increase in the order of variable reconstructions. In most cases, results of quadratic reconstruction show significant improvements over that of linear reconstruction.     

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier-Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier-Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29] and [12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.     

Applications of stencil-adaptive finite difference method to incompressible viscous flows with curved boundary

This paper investigates the applicability of the stencil-adaptive finite difference method for the simulation of two-dimensional unsteady incompressible viscous flows with curved boundary. The adaptive stencil refinement algorithm has been proven to be able to continuously adapt the stencil resolution according to the gradient of flow parameter of interest [Ding H, Shu C. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J Comput Phys 2006;214:397-420], which facilitates the saving of the computational efforts. On the other hand, the capability of the domain-free discretization technique in dealing with the curved boundary provides a great flexibility for the finite difference scheme on the Cartesian grid. Here, we show that their combination makes it possible to simulate the unsteady incompressible flow with curved boundary on a dynamically changed grid. The methods are validated by simulating steady and unsteady incompressible viscous flows over a stationary circular cylinder.     

An evaluation of cell type finite difference methods for solving viscous flow problems

Cell (integral) type equations are examined and compared with grid point techniques for 1-D compressible flow problems. An anlysis of Burger's equation for weak wave propagation is made in order to establish the nature of the cell scheme for both constant cell properties and for linear variation of cell properties. The results of the analysis yield insight into the bahavior of the Godunov cell method for inviscid flows as well as the Lax-Wendroff technique. This understanding is employed to extend Godunov's method to viscous flow calculations. Tests of the method for the complete 1-D compressible vicous flow equations are conducted.