Browsing by Author "Roe, P. L."
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Item Open Access An adaptive grid algorithm for computational shock hydrodynamics(Cranfield University, 1991-01) Quirk, J. J.; Roe, P. L.During the development of computational methods that solve time dependent shock hydrodynamic problems, two underlying strategies have emerged that enable flow features to be resolved clearly. One, employ a numerical scheme of inherently high resolution, usually a second-order Godunov-type method. Two, locally refine the computational mesh in regions of interest. It has been demonstrated by Berger & Collela that a combination of both strategies is necessary if a solution of very high resolution is sought. The present study combines Roe's flux-difference splitting scheme with an adaptive mesh refinement algorithm developed from the ideas of Berger. The result being a general purpose scheme that can fully resolve complicated flows but which requires only modest computing power. The material in this thesis reflects three broad aims. First, to explain the methodology and intricacies of our scheme. Compared to non-adaptive methods our scheme is undeniably complicated, for it contains many elements which must be carefully co-ordinated. Second, to vindicate this complexity. To this end, computational results are presented which are comparable in resolution to Schlieren photographs, yet the calculations were performed on a small desktop workstation. Third, to give sufficient details of our implementation so as to allay the apprehensions of any person who might wish to code up the scheme.Item Open Access Efficient upwind algorithms for solution of the Euler and Navier-stokes equations(Cranfield University, 1995-10) McNeil, C. Y.; Roe, P. L.; Qin, N.An efficient three-dimensionasl tructured solver for the Euler and Navier-Stokese quations is developed based on a finite volume upwind algorithm using Roe fluxes. Multigrid and optimal smoothing multi-stage time stepping accelerate convergence. The accuracy of the new solver is demonstrated for inviscid flows in the range 0.675 :5M :5 25. A comparative grid convergence study for transonic turbulent flow about a wing is conducted with the present solver and a scalar dissipation central difference industrial design solver. The upwind solver demonstrates faster grid convergence than the central scheme, producing more consistent estimates of lift, drag and boundary layer parameters. In transonic viscous computations, the upwind scheme with convergence acceleration is over 20 times more efficient than without it. The ability of the upwind solver to compute viscous flows of comparable accuracy to scalar dissipation central schemes on grids of one-quarter the density make it a more accurate, cost effective alternative. In addition, an original convergencea cceleration method termed shock acceleration is proposed. The method is designed to reduce the errors caused by the shock wave singularity M -+ 1, based on a localized treatment of discontinuities. Acceleration models are formulated for an inhomogeneous PDE in one variable. Results for the Roe and Engquist-Osher schemes demonstrate an order of magnitude improvement in the rate of convergence. One of the acceleration models is extended to the quasi one-dimensiona Euler equations for duct flow. Results for this case d monstrate a marked increase in convergence with negligible loss in accuracy when the acceleration procedure is applied after the shock has settled in its final cell. Typically, the method saves up to 60% in computational expense. Significantly, the performance gain is entirely at the expense of the error modes associated with discrete shock structure. In view of the success achieved, further development of the method is proposed.Item Open Access FAR - field boundaries and their numerical treatment(1989-07) Karni, S.; Roe, P. L.Many computational problems of theoretical and practical interest are not naturally bounded by physical boundaries. Aerodynamic examples include flow calculations past aerofoils or past wing-body configurations, semi-bounded channel flows etc. Other examples include simulations of Turbomachinery flows, problems in Underwater Acoustics etc. To obtain a numerical solution, the problem has first to be converted to a finite region, by introducing an artificial boundary at some finite distance. Boundary conditions must be specified at the artificial boundary for well-posedness of the truncated problem. They should simulate an open boundary across which the fluid flows and should ideally allow outgoing waves to pass through without generating reflections. Indeed, reflections at the boundary not only degrade the accuracy of transient solutions but also inhibit convergence to steady-state. In many problems of practical interest, perfect absorption cannot be achieved. Instead one aims at minimizing the amount of reflected energy using asymptotic expansions based on various asymptotic arguments. The more accurate the boundary statements, the closer the artificial boundaries can be located to the regions of aerodynamic interest, thereby reducing the computational domain and costs. We present a thorough numerical study of the efficiency of several widely used boundary conditions in absorbing outgoing waves. We identify the key parameters upon which the level of absorption at the boundaries depends and expose the limitations of some of the existing recipes. We show that substantial reflections may occur even under conditions which are considerably milder than those encountered in practical calculations. We then introduce an unconventional approach to the treatment of artificial boundaries. It is proposed that in the far field the governing equations are modified in a boundary-layer like manner. Two closely related far field modifications are derived and analysed: (a) Slowing down the outgoing waves and (b) Attenuating the outgoing waves. Under the first modification the outgoing waves are prevented from reaching the boundary hence from reflecting. Under the second, the outgoing waves are attenuated to practically zero strength before reaching the boundary. Both modifications do not alter the propagation of the incoming waves to allow the launching of correct information from the boundary into the interior. Analytic conditions are derived to ensure that no reflections are generated due to the change of coefficients in the governing equations. Reflection analysis is also performed on the discrete level. Well-posedness of the modified systems is established as well as stability of the resulting interface problem. The modifications are extended to two space dimensions and are applied to a variety of one and multidimensional test problems. Results indicate that the proposed far field modifications are attractive in genuinely time-dependent calculations. Preliminary steady state calculations with the unsteady 2D Euler equations show significantly improved convergence properties.Item Open Access Topics in Numerical Computation of Compressible Flow(Cranfield University, 1990-07) Lin, Hong-Chia; Roe, P. L.This thesis aims to assist the development of a multiblock implicit Navier-Stokes code for hypersonic flow applications. There are mainly three topics, which concern the understanding of basic Riemann solvers, the implementing of implicit zonal method, and grid adaption for viscous flow. Three problems of Riemann solvers are investigated. The post-shock oscillation problem of slowly moving shocks is examined, especially for Roe's Riemann solver, and possible cures are suggested for both first and second order schemes. The carbuncle phenomenon associated with blunt body calculation is cured by a formula based on pressure gradient, which will not degrade the solutions for viscous calculations too much. The grid-dependent characteristic of current upwind schemes is also demonstrated. Several issues associated with implicit zonal methods are discussed. The effects of having different mesh sizes in different zones when shock present are examined with first order explicit scheme and such effects are shown to be unwanted therefore big mesh size change should be avoided. Several implicit schemes are tested for hypersonic flow. The conservative DDADI scheme is found to be the most robust one. A simple and robust implicit zonal method is demonstrated. A proper treatment of the diagonal Jacobian and choosing the updating method are found to be crucial. The final topic concerns the calculation and grid adaption of viscous flow. We study the linear advection-diffusion equation thoroughly. The results are unfortunately not applicable to Navier-Stokes equations directly. Nevertheless a suggestion on the mesh size control for viscous flow is made and demonstrated. An attempt to construct a cell-vertex TVD scheme is described in the appendix.