HLLC scheme for the preconditioned pseudo-compressibility Navier–Stokes equations for incompressible viscous flows

A new HLLC (Harten-Lax-van leer contact) approximate Riemann solver with the preconditioning technique based on the pseudo-compressibility formulation for numerical simulation of the incompressible viscous flows has been proposed, which follows the HLLC Riemann solver (Harten, Lax and van Leer solver with contact resolution modified by Toro) for the compressible flow system. In the authors' previous work, the preconditioned Roe's Riemann solver is applied to the finite difference discretisation of the inviscid flux for incompressible flows. Although the Roe's Riemann solver is found to be an accurate and robust scheme in various numerical computations, the HLLC Riemann solver is more suitable for the pseudo-compressible Navier--Stokes equations, in which the inviscid flux vector is a non-homogeneous function of degree one of the flow field vector, and however the Roe's solver is restricted to the homogeneous systems. Numerical investigations have been performed in order to demonstrate the efficiency and accuracy of the present procedure in both two- and three-dimensional cases. The present results are found to be in good agreement with the exact solutions, existing numerical results and experimental data.     

A robust high‐resolution finite volume scheme for the simulation of long waves over complex domains

The propagation, runup and rundown of long surface waves are numerically investigated, initially in one dimension, using a well‐balanced high‐resolution finite volume scheme. A conservative form of the nonlinear shallow water equations with source terms is solved numerically using a high‐resolution Godunov‐type explicit scheme coupled with Roe's approximate Riemann solver. The scheme is also extended to handle two‐dimensional complex domains. The numerical difficulties related to the presence of the topography source terms in the model equations along with the appearance of the wet/dry fronts are properly treated and extended. The resulting numerical model accurately describes breaking waves as bores or hydraulic jumps and conserves volume across flow discontinuities. Numerical results show very good agreement with previously presented analytical or asymptotic solutions as well as with experimental benchmark data. Copyright © 2007 John Wiley & Sons, Ltd.     

Developing a hybrid flux function suitable for hypersonic flow simulation with high‐order methods

In this paper, we develop a new hybrid Euler flux function based on Roe's flux difference scheme, which is free from shock instability and still preserves the accuracy and efficiency of Roe's flux scheme. For computational cost, only 5% extra CPU time is required compared with Roe's FDS. In hypersonic flow simulation with high‐order methods, the hybrid flux function would automatically switch to the Rusanov flux function near shock waves to improve the robustness, and in smooth regions, Roe's FDS would be recovered so that the advantages of high‐order methods can be maintained. Multidimensional dissipation is introduced to eliminate the adverse effects caused by flux function switching and further enhance the robustness of shock‐capturing, especially when the shock waves are not aligned with grids. A series of tests shows that this new hybrid flux function with a high‐order weighted compact nonlinear scheme is not only robust for shock‐capturing but also accurate for hypersonic heat transfer prediction. Copyright © 2015 John Wiley & Sons, Ltd.     

Improved shock-capturing of Jameson's scheme for the Euler equations

It is known that Jameson's scheme is a pseudo-second-order-accurate scheme for solving discrete conservation laws. The scheme contains a non-linear artificial dissipative flux which is designed to capture shocks. In this paper, it is shown that the, shock-capturing of Jameson's scheme for the Euler equations can be improved by replacing the Lax-Friedrichs' type of dissipative flux with Roe's dissipative flux. This replacement is at a moderate expense of the calculation time.     

A FINITE VOLUME SOLVER FOR THE SHALLOW WATER EQUATIONS IN VARYING BED ENVIRONMENTS

Abstract

A numerical scheme for solving the shallow-water equations is presented. An analogy is made between flows governed by shallow-water equations and the Euler system of equations used in gas dynamics. An emphasis is placed on the difference presented by the bathymetry in hydraulic systems. The discretization of the governing equations is based on Roe's flux difference-splitting solver, initially developed for solving inviscid compressible flows. The spatial discretization is handled within a finite-volume context by using triangles or quadrilaterals as the basic control-volume cells. This approach enables an easy and flexible treatment of general geometries. A development of the boundary conditions tailored for the current scheme is given. Fundamental validation tests 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 numerical simulation of boundary layer effects in a shock tube

A numerical scheme is used to investigate boundary layer effects in a shock tube. The method consists of a mixture of Roe's approximate Riemann solver and central differences for the convective fluxes and central differences for the viscous fluxes and is implicit in one space dimension. Comparisons are made with experimental data and with solutions obtained via boundary layer equations. Examination of the calculated flow field explains the observed behaviour and highlights the approximate nature of boundary layer solutions.     

Two‐dimensional dam break flow simulation

Numerical modelling of shallow water flow in two dimensions is presented in this work with the results obtained in dam break tests. Free surface flow in channels can be described mathematically by the shallow‐water system of equations. These equations have been discretized using an approach based on unstructured Delaunay triangles and applied to the simulation of two‐dimensional dam break flows. A cell centred finite volume method based on Roe's approximate Riemann solver across the edges of the cells is presented and the results are compared for first‐ and second‐order accuracy. Special treatment of the friction term has been adopted and will be described. The scheme is capable of handling complex flow domains as shown in the simulation corresponding to the test cases proposed, i.e. that of a dam break wave propagating into a 45° bend channel (UCL) and in a channel with a constriction (LNEC‐IST). Comparisons of experimental and numerical results are shown. Copyright © 2000 John Wiley & Sons, Ltd.     

GASEOUS FLOW ORIGINATED BY A GUNSHOT,I.

SUMMARY

The numerical simulation of the gaseous flow originated by a gunshot both during the projectile motion within the muzzle and out of it is considered. The numerical solution of the Euler equations is obtained via a second order accurate ENO scheme. A special algorithm of cells reorganization is developed for computing the projectile motion out of the muzzle. It reveals an interesting property of the ENO schemes and Roe's approximate solution of automatically capturing discontinuities by means of the choice of the corresponding grid velocity. The algorithm developed describes sufficiently well all gasdynamics phenomena which accompany the gunshot.     

Roe linearization for the Euler equations augmented by the convective terms from the k–ω turbulence model

This paper describes the method of calculation of the eigenvalues, eigenvectors of the Jacobian matrix of the Euler equations augmented by the convective part of k–ω turbulence model. The equations are 3D and values are expressed in terms of cell normals of a finite volume. The expressions for wave strengths are also found, which are also necessary to calculate the inter‐cell fluxes for Roe's scheme. Copyright © 2007 John Wiley & Sons, Ltd.     

Robustness versus accuracy in shock‐wave computations

Despite constant progress in the development of upwind schemes, some failings still remain. Quirk recently reported (Quirk JJ. A contribution to the great Riemann solver debate. International Journal for Numerical Methods in Fluids 1994; 18 : 555–574) that approximate Riemann solvers, which share the exact capture of contact discontinuities, generally suffer from such failings. One of these is the odd–even decoupling that occurs along planar shocks aligned with the mesh. First, a few results on some failings are given, namely the carbuncle phenomenon and the kinked Mach stem. Then, following Quirk's analysis of Roe's scheme, general criteria are derived to predict the odd–even decoupling. This analysis is applied to Roe's scheme (Roe PL, Approximate Riemann solvers, parameters vectors, and difference schemes, Journal of Computational Physics 1981; 43 : 357–372), the Equilibrium Flux Method (Pullin DI, Direct simulation methods for compressible inviscid ideal gas flow, Journal of Computational Physics 1980; 34 : 231–244), the Equilibrium Interface Method (Macrossan MN, Oliver. RI, A kinetic theory solution method for the Navier–Stokes equations, International Journal for Numerical Methods in Fluids 1993; 17 : 177–193) and the AUSM scheme (Liou MS, Steffen CJ, A new flux splitting scheme, Journal of Computational Physics 1993; 107 : 23–39). Strict stability is shown to be desirable to avoid most of these flaws. Finally, the link between marginal stability and accuracy on shear waves is established. Copyright © 2000 John Wiley & Sons, Ltd.     

A three-dimensional hydrodynamic model in curvilinear co-ordinates with collocated grid

A three-dimensional hydrodynamic model has been developed for turbulent flows with free surface. In the horizontal x–y-plane, a boundary-fitted curvilinear co-ordinate system is adopted, while in the vertical direction, a σ-co-ordinate transformation is used to represent the free surface and bed topography or lower boundary. Using the finite volume method, the convection terms are discretized using Roe's second-order-accurate scheme. The governing equations are solved in a collocated grid system by a fractional three-step implicit algorithm that has been developed to handle the velocity–pressure–depth coupling problem of free surface incompressible fluid flows. The present study is the extension of previous work to three-dimensional turbulent flows. The model has been applied to three test cases. Comparison with available data shows that the model developed is successful, and is valuable to engineering application. © 1998 John Wiley & Sons, Ltd.     

A numerical model for the flooding and drying of irregular domains

A numerical technique for the modelling of shallow water flow in one and two dimensions is presented in this work along with the results obtained in different applications involving unsteady flows in complex geometries. A cell‐centred finite volume method based on Roe's approximate Riemann solver across the edges of both structured and unstructured cells is presented. The discretization of the bed slope source terms is done following an upwind approach. In some applications a problem arises when the flow propagates over adverse dry bed slopes, so a special procedure has been introduced to model the advancing front. It is shown that this modification reproduces exactly steady state of still water in configurations with strong variations in bed slope and contour. The applications presented are mainly related with unsteady flow problems. The scheme is capable of handling complex flow domains as will be shown in the simulations corresponding to the test cases that are going to be presented. Comparisons of experimental and numerical results are shown for some of the tests. Copyright © 2002 John Wiley & Sons, Ltd.     

Aeroheating CFD analysis of missile slots

This paper develops a hypersonic aerothermal simulation method for missile slot flow. The finite volume method of structure grid solver is developed for solving Euler and Navier-Stokes equations. The solver includes Park's two temperature model and the air multi-species reaction model. The second-order accuracy TVD numerical method was deduced to compute the hypersonic aeroheating which improves the computational efficiency. Computational results are given to show the high accuracy comparing to the existing experimental data.     

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.     

A lattice Boltzmann method for simulation of multi-species shock accelerated flows

A numerical scheme for simulating multi-species shock accelerated flows using lattice Boltzmann approach has been proposed. It uses the moment conservation approach of Yang, Shu, and Wu and extends it to multi-species fluid problems. The multi-species method of Wang et al. has been modified by use of a predictor–corrector approach. This has helped in preventing the pressure oscillations while handling multi-species. Simulation of 2D shock cylinder interaction with this solver has shown good agreement with the experimental data and could capture material discontinuity and unsteady shocks. The simulation of a single mode Richtmyer–Meshkov instability showed that the solver is able to capture the development of spike and bubble as per the experimental findings of Aure and Jacobs. The dissipation in the proposed scheme was further reduced by the use of fifth-order weighted essentially non-oscillatory (WENO). Validated with multiple problems, this method has been found to capture shock instability with good accuracy with a check on pressure oscillations.     

Flux difference splitting for the Euler equations in one spatial co-ordinate with area variation

An approximate (linearized) Riemann solver is presented for the solution of the Euler equations of gas dynamics in one spatial co-ordinate. This includes cylindrically and spherically symmetric geometries and also applies to narrow ducts with area variation. The method is Roe's flux difference splitting with a technique for dealing with source terms. The results of two problems, a spherically divergent infinite shock and a converging cylindrical shock, are presented. The numerical results compare favourably with those of Noh's recent survey and also with those of Ben-Artzi and Falcovitz using a more complicated Riemann solver.     

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.     

Approximation of shallow water equations by Roe's Riemann solver

The inviscid shallow water equations provide a genuinely hyperbolic system and all the numerical tools that have been developed for a system of conservation laws can be applied to them. However, this system of equations presents some peculiarities that can be exploited when developing a numerical method based on Roe's Riemann solver and enhanced by a slope limiting of MUSCL type. In the present paper a TVD version of the Lax-Wendroff scheme is used and its performance is shown in 1D and 2D computations. Then two specific difficulties that arise in this context are investigated. The former is the capability of this class of schemes to handle geometric source terms that arise to model the bottom variation. The latter analysis pertains to situations in which strict hyperbolicity is lost, i.e. when two eigenvalues collapse into one.     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.