Study on the numerical schemes for hypersonic flow simulation

Hypersonic flow is full of complex physical and chemical processes, hence its investigation needs careful analysis of existing schemes and choosing a suitable scheme or designing a brand new scheme. The present study deals with two numerical schemes Harten, Lax, and van Leer with Contact (HLLC) and advection upstream splitting method (AUSM) to effectively simulate hypersonic flow fields, and accurately predict shock waves with minimal diffusion. In present computations, hypersonic flows have been modeled as a system of hyperbolic equations with one additional equation for non-equilibrium energy and relaxing source terms. Real gas effects, which appear typically in hypersonic flows, have been simulated through energy relaxation method. HLLC and AUSM methods are modified to incorporate the conservation laws for non-equilibrium energy. Numerical implementation have shown that non-equilibrium energy convect with mass, and hence has no bearing on the basic numerical scheme. The numerical simulation carried out shows good comparison with experimental data available in literature. Both numerical schemes have shown identical results at equilibrium. Present study has demonstrated that real gas effects in hypersonic flows can be modeled through energy relaxation method along with either AUSM or HLLC numerical scheme.     

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.     

Solving the Savage–Hutter equations for granular avalanche flows with a second‐order Godunov type method on GPU

In this article, we apply Davis's second‐order predictor‐corrector Godunov type method to numerical solution of the Savage–Hutter equations for modeling granular avalanche flows. The method uses monotone upstream‐centered schemes for conservation laws (MUSCL) reconstruction for conservative variables and Harten–Lax–van Leer contact (HLLC) scheme for numerical fluxes. Static resistance conditions and stopping criteria are incorporated into the algorithm. The computation is implemented on graphics processing unit (GPU) by using compute unified device architecture programming model. A practice of allocating memory for two‐dimensional array in GPU is given and computational efficiency of two‐dimensional memory allocation is compared with one‐dimensional memory allocation. The effectiveness of the present simulation model is verified through several typical numerical examples. Numerical tests show that significant speedups of the GPU program over the CPU serial version can be obtained, and Davis's method in conjunction with MUSCL and HLLC schemes is accurate and robust for simulating granular avalanche flows with shock waves. As an application example, a case with a teardrop‐shaped hydraulic jump in Johnson and Gray's granular jet experiment is reproduced by using specific friction coefficients given in the literature. Copyright © 2014 John Wiley & Sons, Ltd.     

A finite volume shock‐capturing solver of the fully coupled shallow water‐sediment equations

This paper describes a numerical solver of well‐balanced, 2D depth‐averaged shallow water‐sediment equations. The equations permit variable horizontal fluid density and are designed to model water‐sediment flow over a mobile bed. A Godunov‐type, Harten–Lax–van Leer contact (HLLC) finite volume scheme is used to solve the fully coupled system of hyperbolic conservation laws that describe flow hydrodynamics, suspended sediment transport, bedload transport and bed morphological change. Dependent variables are specially selected to handle the presence of the variable density property in the mathematical formulation. The model is verified against analytical and semi‐analytical solutions for bedload transport and suspended sediment transport, respectively. The well‐balanced property of the equations is verified for a variable‐density dam break flow over discontinuous bathymetry. Simulations of an idealised dam‐break flow over an erodible bed are in excellent agreement with previously published results, validating the ability of the model to capture the complex interaction between rapidly varying flow and an erodible bed and validating the eigenstructure of the system of variable‐density governing equations. Flow hydrodynamics and final bed topography of a laboratory‐based 2D partial dam breach over a mobile bed are satisfactorily reproduced by the numerical model. Comparison of the final bed topographies, computed for two distinct sediment transport methods, highlights the sensitivity of shallow water‐sediment models to the choice of closure relationships. Copyright © 2016 John Wiley & Sons, Ltd.     

Three-dimensional reduced Navier–Stokes solutions for subsonic separated and non-separated flows,using a global pressure relaxation procedure

The reduced Navier–Stokes and thin layer approximations to the Navier–Stokes equations are used to obtain solutions for viscous subsonic three-dimensional flows. A spatial marching method is combined with a direct sparse matrix solver to obtain successive solutions in a global relaxation process. Results have been obtained for flow fields with and without regions of flow reversal.     

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A simple, robust, and accurate HLLC-type Riemann solver for the compressible Euler equations at various Mach numbers is built. To cure shock instability of the HLLC solver at strong shocks, a pressure-control technique, which plays a role in limiting the propagation of erroneous pressure perturbation, is proposed. With an all Mach correction method for the compressible Euler system, the proposed method is further extended to compute flow problems at low Mach numbers. The proposed all Mach HLLC-type scheme has been implemented and used to compute a variety of flow problems ranging from hypersonic compressible to low Mach incompressible flow regimes. Various numerical results demonstrate that the obtained all Mach HLLC-type scheme is both accurate and stable for all speed ranges.     

A Free-Lagrange method for unsteady compressible flow: simulation of a confined cylindrical blast wave

A Free-Lagrange numerical procedure for the simulation of two-dimensional inviscid compressible flow is described in detail. The unsteady Euler equations are solved on an unstructured Lagrangian grid based on a density-weighted Voronoi mesh. The flow solver is of the Godunov type, utilising either the HLLE (2 wave) approximate Riemann solver or the more recent HLLC (3 wave) variant, each adapted to the Lagrangian frame. Within each mesh cell, conserved properties are treated as piece-wise linear, and a slope limiter of the MUSCL type is used to give non-oscillatory behaviour with nominal second order accuracy in space. The solver is first order accurate in time. Modifications to the slope limiter to minimise grid and coordinate dependent effects are described. The performances of the HLLE and HLLC solvers are compared for two test problems; a one-dimensional shock tube and a two-dimensional blast wave confined within a rigid cylinder. The blast wave is initiated by impulsive heating of a gas column whose centreline is parallel to, and one half of the cylinder radius from, the axis of the cylinder. For the shock tube problem, both solvers predict shock and expansion waves in good agreement with theory. For the HLLE solver, contact resolution is poor, especially in the blast wave problem. The HLLC solver achieves near-exact contact capture in both problems. Received May 25, 1995 / Accepted September 11, 1995     

Thermodynamic analysis and numerical resolution of a turbulent - fully ionized plasma flow model

This paper is devoted to the modeling and numerical resolution of a non-dissipative compressible turbulent plasma flow model involving three temperatures (turbulence, ions and electrons). The first step is to derive such a model. To do this, an analysis of the Reynolds averaged Euler equations (the k-model) is carried out. It is shown that thermodynamic requirements enable the derivation of an equation of state for turbulent variables. This equation of state is of the same type as those of an ideal gas. In this context, the various thermodynamic variables of turbulence can be obtained (energy, pressure, temperature etc.). This hyperbolic conservative model has exactly the same structure as the two temperatures plasma model of Zeldovich. Thanks to the clear structure of these two models, the turbulent plasma model is derived and involves three temperatures. The second step is to derive an accurate numerical scheme for its solution. A linearized Riemann solver and a positive HLLC type solver are derived and embedded into a conventional Godunov scheme. It is shown that this method requires important corrections to preserve contact discontinuities and temperatures monotonicity. The corrections are based upon a non-conservative formulation of the turbulence and electrons energy equations, while total energy conservation is preserved. The modified method behaves correctly with contacts and shocks.Received: 11 February 2002, Revised: 19 June 2003, Accepted: 7 August 2003, Published online: 14 October 2003 Correspondence to: R. Saurel     

Simulation of dam‐ and dyke‐break hydrodynamics on dynamically adaptive quadtree grids

Flooding due to the failure of a dam or dyke has potentially disastrous consequences. This paper presents a Godunov‐type finite volume solver of the shallow water equations based on dynamically adaptive quadtree grids. The Harten, Lax and van Leer approximate Riemann solver with the Contact wave restored (HLLC) scheme is used to evaluate interface fluxes in both wet‐ and dry‐bed applications. The numerical model is validated against results from alternative numerical models for idealized circular and rectangular dam breaks. Close agreement is achieved with experimental measurements from the CADAM dam break test and data from a laboratory dyke break undertaken at Delft University of Technology. Copyright © 2004 John Wiley Sons, Ltd.     

Realization of contact resolving approximate Riemann solvers for strong shock and expansion flows

The Harten–Lax–van Leer contact (HLLC) and Roe schemes are good approximate Riemann solvers that have the ability to resolve shock, contact, and rarefaction waves. However, they can produce spurious solutions, called shock instabilities, in the vicinity of strong shock. In strong expansion flows, the Roe scheme can admit nonphysical solutions such as expansion shock, and it sometimes fails. We carefully examined both schemes and propose simple methods to prevent such problems. High‐order accuracy is achieved using the weighted average flux (WAF) and MUSCL‐Hancock schemes. Using the WAF scheme, the HLLC and Roe schemes can be expressed in similar form. The HLLC and Roe schemes are tested against Quirk's test problems, and shock instability appears in both schemes. To remedy shock instability, we propose a control method of flux difference across the contact and shear waves. To catch shock waves, an appropriate pressure sensing function is defined. Using the proposed method, shock instabilities are successfully controlled. For the Roe scheme, a modified Harten–Hyman entropy fix method using Harten–Lax–van Leer‐type switching is suggested. A suitable criterion for switching is established, and the modified Roe scheme works successfully with the suggested method. Copyright © 2009 John Wiley & Sons, Ltd.     

A PRESSURE CORRECTION METHOD FOR UNSTRUCTURED MESHES WITH ARBITRARY CONTROL VOLUMES

A pressure correction procedure for general unstructured meshes is presented. It is a cell-centred, collocated finite volume method and the pressure–velocity coupling is treated using SIMPLEC. The cells can have an arbitrary number of grid points (cell vertices). In the present study the number of faces on the cells varies between three and six. The discretized equations are solved using either a symmetric Gauss–Seidel solver or a conjugate gradient solver with a preconditioner. The method is applied to three two-dimensional test cases in which the flow is incompressible and laminar. The extension to three dimensions as well as to turbulent flow using transport models is straightforward. It can also be extended to handle compressible flow.     

ITERATIVE SOLUTION OF INCOMPRESSIBLE NAVIER–STOKES EQUATIONS ON THE MEIKO COMPUTING SURFACE

The numerical discretization of the equations governing fluid flow results in coupled, quasi-linear and non-symmetric systems. Various approaches exist for resolving the non-linearity and couplings. During each non-linear iteration, nominally linear systems are solved for each of the flow variables. Line relaxation techniques are traditionally employed for solving these systems. However, they could be very expensive for realistic applications and present serious synchronization problems in a distributed memory parallel environment. In this paper the discrete linear systems are solved using the generalized conjugate gradient method of Concus and Golub. The performance of this algorithm is compared with the line Gauss–Seidel algorithm for laminar recirculatory flow in uni- and multiprocessor environments. The uniprocessor performances of these algorithms are also compared with that of a popular iterative solver for non-symmetric systems (the GMRES algorithm).     

Upwind discretization of the steady Navier–Stokes equations

A discretization method is presented for the full, steady, compressible Navier–Stokes equations. The method makes use of quadrilateral finite volumes and consists of an upwind discretization of the convective part and a central discretization of the diffusive part. In the present paper the emphasis lies on the discretization of the convective part. The solution method applied solves the steady equations directly by means of a non-linear relaxation method accelerated by multigrid. The solution method requires the discretization to be continuously differentiable. For two upwind schemes which satisfy this requirement (Osher's and van Leer's scheme), results of a quantitative error analysis are presented. Osher's scheme appears to be increasingly more accurate than van Leer's scheme with increasing Reynolds number. A suitable higher-order accurate discretization of the convection terms is derived. On the basis of this higher-order scheme, to preserve monotonicity, a new limiter is constructed. Numerical results are presented for a subsonic flat plate flow and a supersonic flat plate flow with oblique shock wave–boundary layer interaction. The results obtained agree with the predictions made. Useful properties of the discretization method are that it allows an easy check of false diffusion and that it needs no tuning of parameters.     

Implicit application of non‐reflective boundary conditions for Navier–Stokes equations in generalized coordinates

The non‐reflective boundary conditions (NRBC) for Navier–Stokes equations originally suggested by Poinsot and Lele (J. Comput. Phys. 1992; 101 :104–129) in Cartesian coordinates are extended to generalized coordinates. The characteristic form Navier–Stokes equations in conservative variables are given. In this characteristic‐based method, the NRBC is implicitly coupled with the Navier–Stokes flow solver and are solved simultaneously with the flow solver. The calculations are conducted for a subsonic vortex propagating flow and the steady and unsteady transonic inlet‐diffuser flows. The results indicate that the present method is accurate and robust, and the NRBC are essential for unsteady flow calculations. Copyright © 2005 John Wiley & Sons, Ltd.     

Convergence acceleration of segregated algorithms using dynamic tuning additive correction multigrid strategy

A convergence acceleration method based on an additive correction multigrid–SIMPLEC (ACM‐S) algorithm with dynamic tuning of the relaxation factors is presented. In the ACM‐S method, the coarse grid velocity correction components obtained from the mass conservation (velocity potential) correction equation are included into the fine grid momentum equations before the coarse grid momentum correction equations are formed using the additive correction methodology. Therefore, the coupling between the momentum and mass conservation equations is obtained on the coarse grid, while maintaining the segregated structure of the single grid algorithm. This allows the use of the same solver (smoother) on the coarse grid. For turbulent flows with heat transfer, additional scalar equations are solved outside of the momentum–mass conservation equations loop. The convergence of the additional scalar equations is accelerated using a dynamic tuning of the relaxation factors. Both a relative error (RE) scheme and a local Reynolds/Peclet (ER/P) relaxation scheme methods are used. These methodologies are tested for laminar isothermal flows and turbulent flows with heat transfer over geometrically complex two‐ and three‐dimensional configurations. Savings up to 57% in CPU time are obtained for complex geometric domains representative of practical engineering problems. Copyright © 1999 John Wiley & Sons, Ltd.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

An efficient numerical method for reactive flow with general equation of states

This paper presents an efficient method to simulate the reactive flow for general equation of states with the compressible fluid model coupled with reactive rate equation. The important aspect is to deal with mixture of different phases in one cell, which will inevitably happen in the Eulerian method for reactive flow. Physical variables such as the pressure,velocity, and speed of sound in each cell need to be reconstructed for the Harten‐Lax‐Leer‐Contact (HLLC) Riemann solver, which will result in nonlinear algebra equations, and these reconstructed variables are used to obtain the flux. Numerical examples of stable and unstable detonations with different equation of states demonstrate the accuracy and efficiency of this method. Copyright © 2016 John Wiley & Sons, Ltd.     

A general approximate‐state Riemann solver for hyperbolic systems of conservation laws with source terms

An approximate‐state Riemann solver for the solution of hyperbolic systems of conservation laws with source terms is proposed. The formulation is developed under the assumption that the solution is made of rarefaction waves. The solution is determined using the Riemann invariants expressed as functions of the components of the flux vector. This allows the flux vector to be computed directly at the interfaces between the computational cells. The contribution of the source term is taken into account in the governing equations for the Riemann invariants. An application to the water hammer equations and the shallow water equations shows that an appropriate expression of the pressure force at the interface allows the balance with the source terms to be preserved, thus ensuring consistency with the equations to be solved as well as a correct computation of steady‐state flow configurations. Owing to the particular structure of the variable and flux vectors, the expressions of the fluxes are shown to coincide partly with those given by the HLL/HLLC solver. Computational examples show that the approximate‐state solver yields more accurate solutions than the HLL solver in the presence of discontinuous solutions and arbitrary geometries. Copyright © 2006 John Wiley & Sons, Ltd.     

High‐order numerical simulations of flow‐induced noise

In this paper, the flow/acoustics splitting method for predicting flow‐generated noise is further developed by introducing high‐order finite difference schemes. The splitting method consists of dividing the acoustic problem into a viscous incompressible flow part and an inviscid acoustic part. The incompressible flow equations are solved by a second‐order finite volume code EllipSys2D/3D. The acoustic field is obtained by solving a set of acoustic perturbation equations forced by flow quantities. The incompressible pressure and velocity form the input to the acoustic equations. The present work is an extension of our acoustics solver, with the introduction of high‐order schemes for spatial discretization and a Runge–Kutta scheme for time integration. To achieve low dissipation and dispersion errors, either Dispersion‐Relation‐Preserving (DRP) schemes or optimized compact finite difference schemes are used for the spatial discretizations. Applications and validations of the new acoustics solver are presented for benchmark aeroacoustic problems and for flow over an NACA 0012 airfoil. Copyright © 2010 John Wiley & Sons, Ltd.