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.     

Modified Multidimensional Dissipation Scheme on Unstructured Meshes for High-speed Compressible Flow Analysis

A Roe's flux-difference splitting scheme, combining with the entropy fix method according to Van Leer et al., and the H-correction entropy fix method by Pandolfi and D'Ambrosio, is proposed. The presented scheme eliminates unphysical flow behaviors such as a spurious bump of the carbuncle phenomenon that occurs on the bow shock from flow over a blunt body, and the expansion shock generated from flow over a forward facing step. The proposed scheme is further extended to obtain high-order spatial and temporal solution accuracy. The scheme is, in addition, combined with an adaptive meshing technique that generates unstructured triangular meshes to resemble the flow phenomena for reducing computational effort. The entire procedure is evaluated by solving several benchmarks as well as complex steady-state and transient high-speed compressible flow problems.     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A weighted least square scheme for compressible flows

The article describes the development of a high order finite volume method for the solution of transonic flow problems. The method is based on a reconstruction procedure similar to the weighted essentially non-oscillatory scheme (WENO). The analysis of accuracy and stability of the method is carried out for the case of smooth data and for simple discontinuity. The computational results demonstrate the performance of the WLSQR method for the solution of several flow problems in 2D and 3D using both structured and unstructured meshes.     

A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.     

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.     

Adaptive delaunay triangulation with multidimensional dissipation scheme for high-speed compressible flow analysis

IntroductionHigh-speed compressible flows normally involve many complex flow phenomena,suchas shock waves,flow expansions,and shock-shock interactions[1].Effects of thesephenomena are critical in the design of high-speed structures.These flows are charact…     

A two‐dimensional finite volume morphodynamic model on unstructured triangular grids

We discuss the application of a finite volume method to morphodynamic models on unstructured triangular meshes. The model is based on coupling the shallow water equations for the hydrodynamics with a sediment transport equation for the morphodynamics. The finite volume method is formulated for the quasi‐steady approach and the coupled approach. In the first approach, the steady hydrodynamic state is calculated first and the corresponding water velocity is used in the sediment transport equation to be solved subsequently. The second approach solves the coupled hydrodynamics and sediment transport system within the same time step. The gradient fluxes are discretized using a modified Roe's scheme incorporating the sign of the Jacobian matrix in the morphodynamic system. A well‐balanced discretization is used for the treatment of source terms. We also describe an adaptive procedure in the finite volume method by monitoring the bed–load in the computational domain during its transport process. The method uses unstructured meshes, incorporates upwinded numerical fluxes and slope limiters to provide sharp resolution of steep bed gradients that may form in the approximate solution. Numerical results are shown for a test problem in the evolution of an initially hump‐shaped bed in a squared channel. For the considered morphodynamical regimes, the obtained results point out that the coupled approach performs better than the quasi‐steady approach only when the bed–load rapidly interacts with the hydrodynamics. Copyright © 2009 John Wiley & Sons, Ltd.     

Computation of Compressible Flows using a Two-equation Turbulence Model on Unstructured Grids

This paper presents a numerical method for solving compressible turbulent flows using a k - l turbulence model on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multi-stage Runge-Kutta time stepping scheme, while the turbulence equations are advanced using a multi-stage point-implicit scheme. The positivity of turbulence variables is achieved using a simple change of dependent variables. The developed method is used to compute a variety of turbulent flow problems. The results obtained are in good agreement with theoretical and experimental data, indicating that the present method provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.     

A finite volume Navier-Stokes algorithm for adaptive grids

A compact, finite volume, time-marching scheme for the two-dimensional Navier-Stokes equations of viscous fluid flow is presented. The scheme is designed for unstructured (locally refined) quadrilateral meshes. An earlier inviscid equation (Euler) scheme is employed for the convective terms and the emphasis is on treatment of the viscous terms. An essential feature of the algorithm is that all necessary operations are restricted to within each cell, which is very important when dealing with unstructured grids. Numerical issues which have to be addressed when developing a Navier-Stokes scheme are investigated. These issues are not limited to the particular Navier-Stokes scheme developed in the present work but are general problems. Specifically, the extent of the numerical molecule, which is related to the compactness of the scheme and to its suitability for unstructured grids, is examined. An approach which considers suppression of odd-even mode decoupling of the solution when designing a scheme is presented. In addition, accuracy issues related to grid stretching as well as boundary layer solution contamination due to artificial dissipation are addressed. Although the above issues are investigated with respect to the specific scheme presented, the conclusions are valid for an entire class of finite volume algorithms. The Navier-Stokes solver is validated through test cases which involve comparisons with analytical, numerical and experimental results. The solver is coupled to an adaptive algorithm for high-Reynolds-number aerofoil flow computations.     

A novel multidimensional solution reconstruction and edge‐based limiting procedure for unstructured cell‐centered finite volumes with application to shallow water dynamics

On unstructured meshes, the cell‐centered finite volume (CCFV) formulation, where the finite control volumes are the mesh elements themselves, is probably the most used formulation for numerically solving the two‐dimensional nonlinear shallow water equations and hyperbolic conservation laws in general. Within this CCFV framework, second‐order spatial accuracy is achieved with a Monotone Upstream‐centered Schemes for Conservation Laws‐type (MUSCL) linear reconstruction technique, where a novel edge‐based multidimensional limiting procedure is derived for the control of the total variation of the reconstructed field. To this end, a relatively simple, but very effective modification to a reconstruction procedure for CCFV schemes, is introduced, which takes into account geometrical characteristics of computational triangular meshes. The proposed strategy is shown not to suffer from loss of accuracy on grids with poor connectivity. We apply this reconstruction in the development of a second‐order well‐balanced Godunov‐type scheme for the simulation of unsteady two‐dimensional flows over arbitrary topography with wetting and drying on triangular meshes. Although the proposed limited reconstruction is independent from the Riemann solver used, the well‐known approximate Riemann solver of Roe is utilized to compute the numerical fluxes, whereas the Green–Gauss divergence formulation for gradient computations is implemented. Two different stencils for the Green–Gauss gradient computations are implemented and critically tested, in conjunction with the proposed limiting strategy, on various grid types, for smooth and nonsmooth flow conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

The composite finite volume method on unstructured meshes for the two‐dimensional shallow water equations

A composite finite volume method (FVM) is developed on unstructured triangular meshes and tested for the two‐dimensional free‐surface flow equations. The methodology is based on the theory of the remainder effect of finite difference schemes and the property that the numerical dissipation and dispersion of the schemes are compensated by each other in a composite scheme. The composite FVM is formed by global composition of several Lax–Wendroff‐type steps followed by a diffusive Lax–Friedrich‐type step, which filters out the oscillations around shocks typical for the Lax–Wendroff scheme. To test the efficiency and reliability of the present method, five typical problems of discontinuous solutions of two‐dimensional shallow water are solved. The numerical results show that the proposed method, which needs no use of a limiter function, is easy to implement, is accurate, robust and is highly stable. Copyright © 2001 John Wiley & Sons, Ltd.     

NEAR-WAKE FLOW CALCULATIONS BY AN IMPLICIT 2D NAVIER-STOKES ROE RIEMANN SOLVER

Abstract

A flux formulation using a projected 2D Roe Riemann solver on unstructured grids (R2D Solver) is introduced for solving the Navier-Stokes equations and is applied to calculations of axisymmetric laminar near-wake flows behind a spherical-conical body. The numerical framework was first developed by P. L. Roe et al, in the late eighties. They looked for unsteady solutions to Euler's equations using a rather simple but exact three state linearization on triangular grids and decomposing the solution using some effective wave models. Our approach differs from their techniques by constructing a second order accurate and conservative flux functions under the well-known classical finite volume formulation. However, our Riemann Solver is obtained by a suitable linearization procedure upon all three prescribed nodal values given on each triangle. Our numerical method is applied to a Mach 4.3 flow problem for refined unstructured triangular grid behind the body. Numerical results indicate that our technique is stable, accurate and converges successfully to a stationary solution as the cell size is reduced from the coarse lo the finest grid.     

基于非结构网格的TTGC有限元格式的实现及在超声速流动中的应用

基于非结构网格系统,实现了时空三阶精度的TTGC有限元格式,并在三阶TTGC格式上发展了基于人工粘性的激波捕捉技术。在非结构网格下,采用这种方法对若干典型的超声速流动问题(SOD激波管、马赫数为3的前台阶流动以及马赫数为8的高超声速圆柱流动)进行了验证计算。结果表明,TTGC格式分辨率高,在粗糙网格下能够准确的模拟超声速流场中的激波、接触间断等复杂流动现象,并且能有效的控制间断附近的数值色散现象。与传统的有限体积方法相比,本文实现的TTGC有限元格式在模拟超声速流动问题方面具有格式精度高、数值耗散小等优点。     

Assessment of different reconstruction techniques for implementing the NVSF schemes on unstructured meshes

Three new far‐upwind reconstruction techniques, New‐Technique 1, 2, and 3, are proposed in this paper, which localize the normalized variable and space formulation (NVSF) schemes and facilitate the implementation of standard bounded high‐resolution differencing schemes on arbitrary unstructured meshes. By theoretical analysis, it is concluded that the three new techniques overcome two inherent drawbacks of the original technique found in the literature. Eleven classic high‐resolution NVSF schemes developed in the past decades are selected to evaluate performances of the three new techniques relative to the original technique. Under the circumstances of arbitrary unstructured meshes, stretched meshes, and uniform triangular meshes, for each NVSF scheme, the accuracies and convergence properties, when implementing the four aforementioned far‐upwind reconstruction techniques respectively, are assessed by the pure convection of several scalar profiles. The numerical results clearly show that New‐Technique‐2 leads to a better performance in terms of overall accuracy and convergence behavior for the 11 NVSF schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient implicit unstructured finite volume solver for generalised Newtonian fluids

An implicit finite volume solver is developed for the steady-state solution of generalised Newtonian fluids on unstructured meshes in 2D. The pseudo-compressibility technique is employed to couple the continuity and momentum equations by transforming the governing equations into a hyperbolic system. A second-order accurate spatial discretisation is provided by performing a least-squares gradient reconstruction within each control volume of unstructured meshes. A central flux function is used for the convective terms and a solution jump term is added to the averaged component for the viscous terms. Global implicit time-stepping using successive evolution–relaxation is utilised to accelerate the convergence to steady-state solutions. The performance of our flow solver is examined for power-law and Carreau–Yasuda non-Newtonian fluids in different geometries. The effects of model parameters and Reynolds number are studied on the convergence rate and flow features. Our results verify second-order accuracy of the discretisation and also fast and efficient convergence to the steady-state solution for a wide range of flow variables.     

复杂无粘流场数值模拟的矩形/三角形混合网格技术

建立了一套模拟复杂无粘流场的矩形／三角形混合网格技术，其中三角形仅限于物面附近，发挥非结构网格的几何灵活性，以少量的网格模拟复杂外型；同时在以外的区域采用矩形结构网格，发挥矩形网格计算简单快速的优势，有效地克服全非结构网格计算方法需要较大内存量和较长ＣＰＵ时间的不足．混合网格系统由修正的四分树方法生成．将ＮＮＤ有限差分与ＮＮＤ有限体积格式有机地融合于混合网格计算，消除了全矩形网格模拟曲边界的台阶效应，同时保证了网格间的通量守恒．数值实验表明本方法在模拟复杂无粘流场方面的灵活性和高效性．     

A staggered control volume scheme for unstructured triangular grids

The purpose of this work is to introduce and validate a new staggered control volume method for the simulation of 2D/axisymmetric incompressible flows. The present study introduces a numerical procedure for solving the Navier–Stokes equations using the primitive variable formulation. The proposed method is an extension of the staggered grid methodology to unstructured triangular meshes for a control volume approach which features ease of handling of irregularly shaped domains. Two alternative elements are studied: transported scalars are stored either at the sides of an element or at its vertices, while the pressure is always stored at the centre of an element. Two interpolation functions were investigated for the integration of the momentum equations: a skewed mass-weighted upwind function and a flow-oriented exponential shape function. The momentum equations are solved over the covolume of a side or of a vertex and the pressure–velocity coupling makes use of a localized linear reconstruction of the discontinuous pressure field surrounding an element in order to obtain the pressure gradient terms. The pressure equation is obtained through a discretization of the continuity equation which uses the triangular element itself as the control volume. The method is applied to the simulation of the following test cases: backward-facing step flow, flow over a two-dimensional obstacle and flow in a pipe with sudden contraction of cross-sectional area. All numerical investigations are compared with experimental data from the literature. A grid convergence and error analysis study is also carried out for flow in a driven cavity. Results compared favourably with experimental data and so the new control volume scheme is deemed well suited for the prediction of incompressible flows in complex geometries. © 1997 John Wiley & Sons, Ltd.     

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.     

A preconditioned dual time-stepping procedure coupled with matrix-free LU-SGS scheme for unsteady low speed viscous flows with moving objects

In this paper an effective method is developed to solve unsteady low speed viscous flow problems with moving objects by using the governing equations of compressible fluids. The method is based on a dual time-stepping scheme, combined with low Mach number preconditioning and an implicit matrix-free Lower-Upper Symmetric Gauss-Seidel iteration on unstructured dynamic meshes. Because preconditioning modifies the governing equations, that induces the change of system's eigenvalues and eigenvectors, characteristic boundary conditions are also modified to suit the preconditioned characteristic system. Several test cases are simulated, including an in-line oscillating cylinder in a fluid at rest, flow over a flapping NACA0014 airfoil and low speed flow past a flapping-wing micro-air vehicle. Compared with experimental results whenever possible, the computed results indicate that this algorithm shows satisfactory improvement of solution efficiency and accuracy for low speed flow problems.