Browsing by Author "Toro, E. F."
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Item Open Access An adaptive gridding technique for conservation laws on complex domains(Cranfield University, 1997-05) Boden, E. P.; Toro, E. F.Obtaining accurate solutions to flows that involve discontinuous features still re- mains one of the most difficult tasks in computational fluid dynamics today. Some discontinuous features, such as shear waves and material interfaces, are quite deli- cate, yet they have a profound effect on the rest of the flow field. The accuracy of the numerical scheme and the quality of the grid discretisation of the flow domain, are both critical when computing multi-dimensional discontinuous solutions. Here, the second order WAF scheme is used in conjuction with an adaptive grid algorithm, which is able to automatically modify the grid in regions of discontinuous features and solid boundaries. The grid algorithm is a combination of two successful ap- proaches, namely Chimera and Cartesian grid Adaptive Mesh Refinement (AMR). The Chimera approach is able to accurately represent non-Cartesian boundaries, whilst the AMR approach yields significant savings in memory storage and cPu time. The combined algorithm has been thoroughly validated for convection test problems in gas dynamics. The computed solutions compare well with other numerical and experimental results. These tests have also been used to assess the efficiency of the grid adaption algorithms. Finally, the approach is applied to axi-symmetric, two- dimensional, two-phase, reactive flows in the context of internal ballistics problems. Again, the computed results are compared with other numerical and experimental results.Item Open Access CFD modelling of solid propellant ignition(Cranfield University, 1996-07) Lowe, C.; Clarke, J. F.; Toro, E. F.Solid propellant is the highly energetic fuel burnt in the combustion chamber of ballistic weapons. It is manufactured, for this purpose, in either granular or stick form. Internal ballistics describes the behavior within the combustion chamber throughout the ballistic cycle upto projectile exit from the muzzle of the gun barrel. Over the last twenty years this has been achieved by modelling the process using two-phase flow equations. The solid granules or sticks constitute the first phase, which can be assumed to be incompressible over typical pressure ranges within the chamber. The gas-phase is composed of both the original ambient gas contained around the propellant and additional gas produced by the propellant gasifying on heating. Equations can be derived that describe the conservation of mass, momentum and energy in terms of average flow variables. The equations are a highly non-linear system of partial-differential- equations. High-speed flow features are observed in internal ballistics and ordinary fini te- difference methods are unsuitable numerical methods due to inaccurate prediction of discontinuous flow features. Modern shock-capturing methods are employed, which solve the system of equations in conservation form, with the ability to capture shocks and contact discontinuities. However, although the numerical solutions compare well with experiment over the bulk of the combustion chamber, the ignition models used in internal ballistics are unreliable. These are based on either gas or solid-surface temperature achieving some empirically measured 'ignition temperature' after which the propellant burns according to an empirical pressure dependent burning law. Observations indicate that this is not an adequate representation of ignition. Time differences between first solid gasification and ignition imply two distinct processes occurring. ]Further, ignition occurring in gas-only regions indicates that ignition is controlled by a gas-phase reaction. This thesis develops simple ideas to describe possible mechanisms for these physical observations. The aim is to provide an improved model of the ignition of solid propellant. A two stage reaction process is described involving endothermic gasification of the solid, to produce a source of reactant gas, followed by a very exothermic gas-phase ignition reaction. Firstly the gas-phase ignition is considered. A very simple reaction is suggested which is assumed to control the combustion of reactant gas, produced by solid gasification. Ignition is, by definition, the initiation of this exothermic reaction. Chemical kinetics are included in the gas-phase flow equations to explore the evolution of the reactant gas that is subject to changes in temperature and pressure. By assuming spatial uniformity, analytical solutions of the problem are deduced. The physical interpretation of the solution is discussed, in particular, the relationship between temperature, reactant concentration and ignition is explored. Numerical methods are required to solve the one-dimensional flow equations. Development of suitable CFD methods provides a method of solution. Finite-volume schemes, based on the original work by Godunov, are used to solve the conservation form of the equations. A simple test problem is considered whereby reactant gas is injected into a cylindrical combustion chamber. By examining the resulting flow histories, valuable information is gathered about the complicated coupling of chemistry and flow. Chemistry is included into a system of two-phase flow equations. By using standard averaging methods along with an equation for gas-phase species, equations are derived that describe the rate of change of average flo%v variables for both gas and particle phases. Numerical schemes are developed and some of the difficulties involved in two-phase flow systems, that are not an issue in single-phase flow, are presented. An internal ballistics application is considered as a test case and the solution discussed. The other important reaction involved in the combustion cycle, solid gasification, is explored. The model is based on detailed description of interphase mass and energy transfer at the solid-gas interface. This involves the solution of the heat conduction equation with a moving boundary that divides the solid and gas regions. Similar numerical schemes are constructed to solve the equations. Finally, this model is coupled with the equations of gas-phase reaction. This describes the complete cycle whereby increases in gas temperature cause the solid to increase in temperature and gasify. Subsequent gas-phase combustion of the reactant gases produces heat-transfer between the solid and gas and continues to accelerate gasification. Eventually this results in selfsustained combustion of the solid propellant.Item Open Access A class of upwind methods for conservation laws(1994-06) Billett, S. J.; Toro, E. F.Various new methods for the solution of hyperbolic systems of conservation laws in one, two and three space dimensions are developed. All are explicit, conservative timemarching methods that are second order accurate in space and time in regions of smooth flow and make use of local Riemann problems at intercell boundaries. In one space dimension, the Weighted Average Flux (w af ) approach of Toro is extended to generate a scheme that is stable with timesteps twice as large as those allowed by the stability conditions of the original scheme. A Riemann problem based extension of the Warming-Beam scheme is considered. Total Variation Diminishing (t v d ) conditions are enforced for both schemes. Numerical results for the Euler Equations of Gas Dynamics are presented. In two and three space dimensions, finite volume versions of the waf scheme on Cartesian grids are derived for the linear advection equation. Two two dimensional schemes are found that are second order accurate in space and time. One of these is extended for the solution of nonlinear systems of hyperbolic conservation laws in two separate ways. The resulting schemes are tested on the Shallow Water equations. The equivalent three dimensional schemes are also discussed. The two dimensional schemes are then extended for use on structured, body-fitted grids of quadrilaterals and one of these extensions is used to demonstrate the phenomena of Mach reflection of shallow water bores.Item Open Access The dry-bed problem in shallow-water flows(Cranfield Institute of Technology; College of Aeronautics, 1990) Toro, E. F.Item Open Access Fully discrete arbitrary-order schemes for a model hyperbolic conservation law(1993) Shi, Jian; Toro, E. F.We investigate the fully discrete methodology and establish a formula from which two-level explicit fully discrete arbitrary-order (both in space and time) conservative numerical schemes for a model hyperbolic conservation law can be derived. To illustrate this approach fully discrete second, third and fourth order numerical schemes are presented.Item Open Access Fully discrete high resolution schemes for systems of conservation laws(Cranfield University, 1994-09) Shi, Jian; Toro, E. F.Effective and robust high resolution schemes are of vital importance for simulation of viscous and inviscid flows. Since second-order high resolution schemes in practice are inadquate for many applications, large efforts have been put towards developing higher- order accurate schemes in the past. Although some progress has been made, the efforts were frustrated by the lack of effective and robust new schemes. Therefore this thesis is aimed at challenging this difficult but very important issue. Some new theories and methodologies were established during this research, which covers the linear stability analysis for high-order numerical schemes; the fully discrete techniques for model equations; the formulation of conservative high-order schemes and the high-order Total Variation Diminishing (TVD) schemes. According to these theories arbitrary-order high resolution schemes can be developed. To illustrate the methodologies second-, third-, fourth-, and 20th-order schemes are presented. These high resolution schemes were tested and validated by solving some popular test problems for one and two dimensional Euler and incompressible Navier-Stokes equations. The efficiency and robustness are the features of these high-order schemes.Item Open Access Fully discrete high-order TVD schemes for a scalar hyperbolic conservation law(1993) Shi, Jian; Toro, E. F.In this paper we investigate fully discrete high-order TVD schemes for a scalar hyper- bolic conservation law using flux limiters . Formulae which define Courant number dependent TVD regions for second and third-order TVD schemes are established. A semi-empirical TVD procedure for an m-th order scheme (m ≥ 4) are proposed and tested.Item Open Access A linearised Reimann solver for Gudonov-type methods(1991) Toro, E. F.Very simple linearisations for the solution to the Riemann problem for the time-dependent and for the steady supersonic Euler equations are presented. When used locally in conjunction with Godunov-type methods, computing savings by a factor of about four, relative to the use of exact Riemann solvers, can be achieved. For severe flow regimes however, the linearisation looses accuracy and robustness. We then propose the use of a Riemann-solver adaptation procedure. This retains the accuracy and robustness of the exact Riemann solver and the computational efficiency of the cheap linearised Riemann solver. Also, reliable and simple switching criteria are presented. Numerical results for one, two and three-dimensional test problems suggest that the resulting numerical methods are competitive for practical applications, in terms of robustness, accuracy and computational efficiency.Item Open Access A linearised Reimann solver for the time-dependent Euler equations of gas dynamics(1991) Toro, E. F.The time-dependent Euler equations of Gas Dynamics are a set of non-linear hyperbolic conservation laws that admit discontinuous solutions (e.g. shocks). In this paper we are concerned with Riemann-problem based numerical methods for solving the general initial-value problem for these equations. We present an approximate, linearised Riemann solver for the time-dependent Euler equations. The solution is direct and involves few and simple arithmetic operations. The Riemann solver is then used, locally, in conjunction with the WAF numerical method to solve the time-dependent Euler equations in one and two space dimensions with general initial data. For flows with shocks waves of moderate strength the computed results are very accurate. For severe flow regimes we advocate the use of the present linearised Riemann solver in combination with the exact Riemann solver in an adaptive fashion. Numerical experiments demonstrate that such an approach can be very successful. One and two-dimensional test problems show that the linearised Riemann solver is used in over 99% of the flow field producing net computing savings by a factor of about 2. A reliable and simple switching criterion is also presented. Results show that the adaptive approach effectively provides the resolution and robustness of the exact Riemann solver at the computing cost of the simple linearised Riemann solver. The relevance of the present methods concerns the numerical solution of multi-dimensional problems accurately and economically.Item Open Access Multi-dimensional WAF-type schemes for model conservation laws(1993) Toro, E. F.We explore how the Weighted Average Flux (WAF) approach can be used to generate first and second order accurate finite volume schemes in one, two and three space dimensions. The derived schemes have multidimensional upwinding aspects and good stability properties. We construct oscillation-free methods that are second order accurate in smooth flow by making use of one-dimensional limiter functions. Some numerical results are presented.Item Open Access A numerical investigation of the artificial compressibility method for the solution of the Navier-Stokes equations(1992) Elsworth, D. T.; Toro, E. F.The Artificial Compressibility approach is an important numerical method for solving the incompressible Navier-Stokes Equations. The application of high resolution numerical methods to the equations of the artificial compressibility approach is a relatively new phenomenon and deserves further investigation. In this paper we examine the performance of five Riemann solvers: an exact Riemann solver, and four approximate solvers. The application of reflective boundary conditions is investigated, as well as the way in which the artificial compressibility coefficient is chosen.Item Open Access Riemann problems and the WAF method for two-dimensional shallow water equations(Cranfield Institute of Technology; College of Aeronautics, 1990) Toro, E. F.Item Open Access Riemann solvers for solving the incompressible Navier-Stokes equations using the artificial compressibility method(1992) Elsworth, D. T.; Toro, E. F.The solution to the Incompressible Navier-Stokes equations still represents a significant numerical challenge. The reason for this is that there is a lack of coupling between velocity and pressure. This means that the equations themselves provide no way of explicitly updating the pressure field as the velocity field is advanced. The artificial compressibility approach, devised by A. J. Chorin (see Chorin 1967), represents one way of overcoming this difficulty. It is arrived at by altering the incompressible equations in such a way as to result in a system of equations in which the left hand side is hyperbolic. We wish to take advantage of the hyperbolic nature of these equations and use Riemann-problem- based-numerical-methods (or RP methods).Item Open Access Some aspects of shock capturing methods for gas dynamics(1991) Toro, E. F.Three topics on modern shock capturing methods for the time-dependent Euler equations of Gas Dynamics are addressed. First we present the Weighted Average Flux Method (or WAF), one of several Riemann-problem based shock capturing methods. Then we deal with the Riemann problem. We present an efficient exact Riemann solver, a robust non-iterative Riemann solver based on the behaviour of the exact solver, and an improved version of the Harten-Lax-van Leer Riemann solver. Also, a very simple linearised Riemann solver is presented together with a Riemann solver adaptation procedure. We also present a Riemann-solver adaptation procedure that has proved successful. Applications of the WAF method with the various Riemann solvers are presented.Item Open Access TVD regions for the weighted average flux (WAF) method as applied to a model hyperbolic conservation law(Cranfield Institute of Technology; College of Aeronautics, 1989) Toro, E. F.Item Open Access Viscous flux limiters(1991) Toro, E. F.We present Numerical Viscosity Functions, or NVFs, for use with Riemann-problem based shock-capturing methods as applied to viscous flows. In particular, viscous flux limiters are derived. The analysis pertain to a linear convection-diffusion model equation. Our NVFs combine the physical viscosity, the role of which is maximised, with numerical viscosity, whose role is minimised, to capture TVD solutions to viscous flows