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This paper introduces a novel algorithm for solving the two‐dimensional Euler and Navier‐Stokes compressible equations using a one‐step effective flux vector‐splitting implicit method. The new approach makes a contribution by deriving a simple and yet effective implicit scheme which has the features of an exact factorization and avoids the solving of block‐diagonal system of equations. This results in a significant improvement in computational efficiency as compared to the standard Beam‐Warming and Steger implicit factored schemes. The current work has advantageous characteristics in the creation of higher order numerical implicit terms. The scheme is stable if we could select the correct values of the scalars (λ±ξ and λ±η) for the respective split flux‐vectors (F± and G±) along the ξ− and η−directions. A simple solving procedure is suggested with the discussion of the implicit boundary conditions, stability analysis, time‐step length and convergence criteria. This method is spatially second‐order accurate, fully conservative and implemented with general co‐ordinate transformations for treating complex geometries. Also, the scheme shows a good convergence rate and acceptable accuracy in capturing the shock waves. Results calculated from the program developed include transonic flows through convergence‐divergence nozzle and turbine cascade. Comparisons with other well‐documented experimental data are presented and their agreements are very promising. The extension of the algorithm to 3D simulation is straightforward and under way.

Deals with the formulation and application of temporal and spatial spectral element approximations for the solution of convection‐diffusion problems. Proposes a new high‐order splitting space‐time spectral element method which exploits space‐time discontinuous Galerkin for the first hyperbolic substep and space continuous‐time discontinuous Galerkin for the second parabolic substep. Analyses this method and presents its characteristics in terms of accuracy and stability. Also investigates a subcycling technique, in which several hyperbolic substeps are taken for each parabolic substep; a technique which enables fast, cost‐effective time integration with little loss of accuracy. Demonstrates, by a numerical comparison with other coupled and splitting space‐time spectral element methods, that the proposed method exhibits significant improvements in accuracy, stability and computational efficiency, which suggests that this method is a potential alternative to existing schemes. Describes several areas for future research.

The flux vector splitting (FVS) schemes are known for their higher resistance to shock instabilities and carbuncle phenomena in high-speed flow computations, which are generally accompanied by relatively large numerical diffusion. However, it is desirable to control the numerical diffusion of FVS schemes inside the boundary layer for improved accuracy in viscous flow computations. This study aims to develop a new methodology for controlling the numerical diffusion of FVS schemes for viscous flow computations with the help of a recently developed boundary layer sensor.

The governing equations are solved using a cell-centered finite volume approach and Euler time integration. The gradients in the viscous fluxes are evaluated by applying the Green’s theorem. For the inviscid fluxes, a new approach is introduced, where the original upwind formulation of an FVS scheme is first cast into an equivalent central discretization along with a numerical diffusion term. Subsequently, the numerical diffusion is scaled down by using a novel scaling function that operates based on a boundary layer sensor. The effectiveness of the approach is demonstrated by applying the same on van Leer’s FVS and AUSM schemes. The resulting schemes are named as Diffusion-Regulated van Leer’s FVS-Viscous (DRvLFV) and Diffusion-Regulated AUSM-Viscous (DRAUSMV) schemes.

The numerical tests show that the DRvLFV scheme shows significant improvement over its parent scheme in resolving the skin friction and wall heat flux profiles. The DRAUSMV scheme is also found marginally more accurate than its parent scheme. However, stability requirements limit the scaling down of only the numerical diffusion term corresponding to the acoustic part of the AUSM scheme.

To the best of the authors’ knowledge, this is the first successful attempt to regulate the numerical diffusion of FVS schemes inside boundary layers by applying a novel scaling function to their artificial viscosity forms. The new methodology can reduce the erroneous smearing of boundary layers by FVS schemes in high-speed flow applications.

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.

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.

To present conversion of the advection upwind splitting method (AUSM+) from the conventional density‐based and coupled formulation to the pressure‐based and segregated formulation.

The spatial discretization is done by a finite volume method. A collocated grid cell‐center formulation is used. The pressure‐correction procedure is set up in the usual way for a compressible flow problem. The conventional Rhie‐Chow interpolation methodology for the determination of the transporting velocity, and the conventional central interpolation for the pressure at the control volume faces, are replaced by AUSM+ definitions.

The AUSM+ flux definitions are spontaneously well suited for use in a collocated pressure‐correction formulation. The formulation does not require extensions to these flux definitions. As a consequence, the results of a density‐based fully coupled method, are identical to the results of a pressure‐based segregated formulation. The advantage of the pressure‐correction method with respect to the density‐based method, is the higher efficiency for low Mach number applications. The advantage of the AUSM+ flux definition for the transporting velocity with respect to the conventional Rhie‐Chow interpolation, is the improved accuracy in high Mach number flows. As a consequence, the combination of AUSM+ with a pressure‐correction method leads to an algorithm with improved performance for flows at all Mach numbers.

A new methodology, with obvious advantages, is composed by the combination of ingredients from an existing spatial discretization method (AUSM+) and an existing time stepping method (pressure‐correction).

A class of flux‐splitting explicit second‐order finite difference schemes is set up. An ‘optimal’ scheme is defined for 1‐D flows and applied to 2‐D flows with CFL being able to reach 2. The results obtained show that this ‘optimal’ scheme is well adapted to the unsteady flows.

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.

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.

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.