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An implicit method for the solution of transonic flows modelled by the time‐dependent Euler equations is presented. The method is characterized by a robust linearization for first‐ and second‐order versions of Roe's flux‐difference splitting scheme, an implicit treatment of the boundary conditions and the implementation of an adaptive grid strategy for global efficiency. The performance of the method is investigated for the GAMM test circular‐arc bump configuration and for the RAE 2822 aerofoil.

Partially‐decoupled upwind‐based total‐variation‐diminishing (TVD) finite‐difference schemes for the solution of the conservation laws governing two‐dimensional non‐equilibrium vibrationally relaxing and chemically reacting flows of thermally‐perfect gaseous mixtures are presented. In these methods, a novel partially‐decoupled flux‐difference splitting approach is adopted. The fluid conservation laws and species concentration and vibrational energy equations are decoupled by means of a frozen flow approximation. The resulting partially‐decoupled gas‐dynamic and thermodynamic subsystems are then solved alternately in a lagged manner within a time marching procedure, thereby providing explicit coupling between the two equation sets. Both time‐split semi‐implicit and factored implicit flux‐limited TVD upwind schemes are described. The semi‐implicit formulation is more appropriate for unsteady applications whereas the factored implicit form is useful for obtaining steady‐state solutions. Extensions of Roe's approximate Riemann solvers, giving the eigenvalues and eigenvectors of the fully coupled systems, are used to evaluate the numerical flux functions. Additional modifications to the Riemann solutions are also described which ensure that the approximate solutions are not aphysical. The proposed partially‐decoupled methods are shown to have several computational advantages over chemistry‐split and fully coupled techniques. Furthermore, numerical results for single, complex, and double Mach reflection flows, as well as corner‐expansion and blunt‐body flows, using a five‐species four‐temperature model for air demonstrate the capabilities of the methods.

A flux‐difference splitting based on the polynomial character of the flux vectors is applied to steady Euler equations, discretized with a vertex‐centred finite volume method. In first order accurate form, a discrete set of equations is obtained which is both conservative and positive. Due to the positivity, the set of equations can be solved by collective relaxation methods in multigrid form. A full multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. Second order accuracy is obtained by the Chakravarthy‐Osher flux‐extrapolation technique, using the Roe‐Chakravarthy minmod limiter. In second order form, direct relaxation of the discrete equations is no longer possible due to the loss of positivity. A defect‐correction is used in order to solve the second order system.

This paper aims to develop an efficient numerical method for simulating multicomponent flows by solving the system of conservative equations closed by a general two parameters equation of state.

A finite difference method for solving the two‐dimensional Euler or Navier‐Stokes equations for multicomponent flows in a general curvilinear coordinate system is developed. The system of conservative equations (mass, momentum and energy) is closed with a general two parameters equation of state (ρe=(p+γp_∞)/(γ−1)), which, associated to a γ‐formulation, allows easy computation of multicomponent flows. In order to enforce the stability of the numerical scheme, the Roe's flux‐difference splitting is adopted for the numerical treatment of the inviscid fluxes. The method is adapted to treat also unsteady flows by implementing an explicit Euler scheme.

The method was applied to compute various configurations of flows, ranging from incompressible to compressible fluid, including cases of single component flows or multicomponent ones. Computations show that the use of primitive variables instead of conservative ones, especially at low Mach numbers, improves the iteration process when the resolution is performed with a relaxation procedure such as Gauss‐Seidel method. Simulations of compressible flows with a strong shock show the ability of the present method to capture shocks correctly even with the use of primitive variables. To complete numerical tests, flows involving two fluids with the presence of interactions between a shock and a discontinuity surface have been treated successfully. Also, a case of cavitating flow has been considered in this work.

The present method permits the simulation of a large variety of multicomponent complexes flows with an efficient numerical taking advantage of Roe's flux‐difference splitting in curvilinear coordinate system.

A computational fluid dynamics code for the calculation of laminar hypersonic multi‐species gas flows in chemical non‐equilibrium in axisymmetric or two‐dimensional configuration on shared and distributed memory parallel computers is presented and validated. The code is designed to work efficiently in combination with an automatic domain decompositioning method developed to facilitate efficient parallel computations of various flow problems.

The baseline implicit numerical method developed is the lower‐upper symmetric Gauss‐Seidel scheme, which is combined with a sub‐iteration scheme to achieve time‐accuracy up to third‐order. The spatial discretisation is based on Roe's flux‐difference splitting and various non‐linear flux limiters maintaining total‐variation diminishing properties and up to third‐order spatial accuracy in continuous regions of flow. The domain subdivision procedure is designed to work for single‐ and multi‐block domains without being constrained by the block boundaries, and an arbitrary number of processors used for the computation.

The code developed reproduces accurately various types of flows, e.g. flow over a flat plate, diffusive mixing and oscillating shock induced combustion around a projectile fired into premixed gas, and demonstrates close to linear scalability within limits of load imbalance.

The cases considered are axisymmetric or two‐dimensional, and assume laminar flow. An extension to three‐dimensional turbulent flows is left for future work.

Results of a parallel computation, utilising a newly developed automatic domain subdivision procedure, for oscillating shock‐induced combustion around a projectile and various other cases are presented. The influence of entropy correction in Roe's flux‐difference splitting algorithm on diffusive mixing of multi‐species flows was examined.

This paper describes two accurate and efficient numerical methods for computing unsteady viscous flows. The first one solves the incompressible Navier‐Stokes equations in their vorticity‐velocity formulation, using a staggered‐grid finite‐volume spatial discretization to provide second‐order accuracy on arbitrary grids, and combines effectively an alternating direction implicit scheme for the vorticity transport equation and a multigrid line‐Gauss‐Seidel relaxation for the velocity equations. The second method solves the compressible Reynolds‐averaged Navier‐Stokes equations in strong conservation form, with a k−ω turbulence closure model. The equations are discretized in time using an implicit three‐time‐level scheme, combined with a dual time stepping approach, so that the residual at every physical time step is annihilated using an efficient multigrid Runge‐Kutta iteration with variable time stepping and implicit residual smoothing. The space discretization uses a Roe’s flux difference splitting for the convective terms and standard central differences for the diffusive ones. A turbulent unsteady cascade flow is used to demonstrate the accuracy and efficiency of the method. The authors are currently working towards extending the two approaches described in this paper to three space dimensions.

A higher-order implicit shock-capturing scheme is presented for the Euler equations based on time linearization of the implicit flux vector rather than the residual vector.

The flux vector is linearized through a truncated Taylor-series expansion whose leading-order implicit term is an inner product of the flux Jacobian and the vector of differences between the current and previous time step values of conserved variables. The implicit conserved-variable difference vector is evaluated at cell faces by using the reconstructed states at the left and right sides of a cell face and projecting the difference between the left and right states onto the right eigenvectors. Flux linearization also facilitates the construction of implicit schemes with higher-order spatial accuracy (up to third order in the present study). To enhance the diagonal dominance of the coefficient matrix and thereby increase the implicitness of the scheme, wave strengths at cell faces are expressed as the inner product of the inverse of the right eigenvector matrix and the difference in the right and left reconstructed states at a cell face.

The accuracy of the implicit algorithm at Courant–Friedrichs–Lewy (CFL) numbers greater than unity is demonstrated for a number of test cases comprising one-dimensional (1-D) Sod’s shock tube, quasi 1-D steady flow through a converging-diverging nozzle, and two-dimensional (2-D) supersonic flow over a compression corner and an expansion corner.

The algorithm has the advantage that it does not entail spatial derivatives of flux Jacobian so that the implicit flux can be readily evaluated using Roe’s approximate Jacobian. As a result, this approach readily facilitates the construction of implicit schemes with high-order spatial accuracy such as Roe-MUSCL.

A novel finite-volume-based higher-order implicit shock-capturing scheme was developed that uses time linearization of fluxes at cell interfaces.

The purpose of this paper is to present a new hybrid Euler flux fonction for use in a finite-volume Euler/Navier-Stokes code and adapted to compressible flow problems.

The proposed scheme, called AUFSRR can be devised by combining the AUFS solver and the Roe solver, based on a rotated Riemann solver approach (Sun and Takayama, 2003; Ren, 2003). The upwind direction is determined by the velocity-difference vector and idea is to apply the AUFS solver in the direction normal to shocks to suppress carbuncle and the Roe solver across shear layers to avoid an excessive amount of dissipation. The resulting flux functions can be implemented in a very simple manner, in the form of the Roe solver with modified wave speeds, so that converting an existing AUFS flux code into the new fluxes is an extremely simple task.

The proposed flux functions require about 18 per cent more CPU time than the Roe flux. Accuracy, efficiency and other essential features of AUFSRR scheme are evaluated by analyzing shock propagation behaviours for both the steady and unsteady compressible flows. This is demonstrated by several test cases (1D and 2D) with standard finite-volume Euler code, by comparing results with existing methods.

The hybrid Euler flux function is used in a finite-volume Euler/Navier-Stokes code and adapted to compressible flow problems.

The AUFSRR scheme is devised by combining the AUFS solver and the Roe solver, based on a rotated Riemann solver approach.

The steady compressible Navier—Stokes equations coupled to the k—ε turbulence equations are discretized within a vertex‐centered finite volume formulation. The convective fluxes are obtained by the polynomial flux‐difference splitting upwind method. The first order accurate part results directly from the splitting. The second order part is obtained by the flux‐extrapolation technique using the minmod limiter. The diffusive fluxes are discretized in the central way and are split into a normal and a tangential contribution. The first order accurate part of the convective fluxes together with the normal contribution of the diffusive fluxes form a positive system which allows solution by classical relaxation methods. The source terms in the low‐Reynolds k‐ε equations are grouped into positive and negative terms. The linearized negative source terms are added to the positive system to increase the diagonal dominance. The resulting positive system forms the left hand side of the equations. The remaining terms are put in the right hand side. A multigrid method based on successive relaxation, full weighting, bilinear interpolation and W‐cycle is used. The multigrid method itself acts on the left hand side of the equations. The right hand side is updated in a defect correction cycle.

The purpose of this paper is to present a novel numerical method to solve incompressible flows with natural and mixed convections using pseudo‐compressibility formulation.

The present method is derived using the framework of Harten Lax and van Leer with contact (HLLC) method of Toro, Spruce and Spears, that was originally developed for compressible gas dynamics equations. This work generalizes the algorithm described in the previous paper to the case where heat transfer is involved. Here, the solution of the Riemann problem is approximated by a three‐wave system.

A few test cases involving incompressible laminar flows inside 2D square cavity for various Rayleigh and Reynolds numbers are considered for validating the present method. The computed results from the present method are found to be quite promising.

Although pseudo‐compressibility formulation has been found to have superior performance and has the potential to have numerical treatments similar to compressible flow equations, only two numerical methods have been applied so far; namely Jameson method and Roes flux difference splitting method. A new sophisticated numerical method, following the framework of HLLC method, is derived and implemented for solving pseudo‐compressibility‐based incompressible flow equations with heat transfer.