High-Order Numerical Algorithms for Steady and Unsteady Simulation of Viscous Compressible Flow

Stanford University, Stanford CA

Investigators

Abstract

The goal of this project if to produce an efficient high-order methodology for 3D, high-speed unsteady viscous fluid flow in complex domains. No existing solver achieves all these goals, despite the fact that significant progress in relevant research areas has been made during the last few years. To achieve this goal, two major objectives may be identified. Firstly, the development of the high-order Spectral Difference (SD) Method for unstructured grids will be addressed. More specifically, the scheme will be extended to three dimensions, and formulated for mixed-element grids. The latter is necessary for efficient resolution of flows at high Reynolds numbers. Furthermore, upon analyzing and validating viable viscous discretization techniques, the scheme will be extended by the principal investigator and his colleagues to the Navier-Stokes equations on mixed grids. The second major objective is to improve the efficiency and robustness of the high-order methodology. Numerical schemes of third or higher spatial order are not yet efficient enough for many problems of practical engineering interest, in particular high-speed viscous flow. To keep the number of degrees of freedom comparable to that of a second order method, the number of grid cells must be reduced as the order is increased. To resolve discontinuities sharply without oscillations h-p adaptively will be introduced. For both steady-state and time- dependent problems a fast solution technique is necessary to solve the arising systems of nonlinear equations. In order to meet this goal, an h-p multigrid strategy will be combined with efficient time stepping and relaxation schemes for both steady and unsteady flows. Computational simulation has revolutionized the engineering design process and is by now the single most important tool in a wide range of high technology industries. That in turn has had a huge impact in many real life applications of great importance to society and science, from aerospace to health to the environment. The proposed research is targeted at advancing the state of the art of numerical algorithms which will be needed for the U.S. to remain at the forefront of computational simulation for both industrial and scientific applications. In the Aerospace Industry simulation methods have been largely responsible for doubling the fuel efficiency of long range commercial aircraft since the advent of jet transport. The need for further improvement is crucial for both the economics and environmental impact of air transport, a jumbo jet releases its take-off weight (around 400 tons) in carbon dioxide emissions during a single flight. The European Union has recognized the importance of advancing simulation techniques with a variety of European wide and national initiatives (such as the German Mega Flow and Mega Design programs.) Simulations and optimized redesign of the 747 wing can save tons" of fuel, travel time and reduce emissions. Computer simulation is equally crucial to advances in numerous other applications. For example, simulations of blood flow through a beating heart and its valves, helps doctors understand and repair the malfunctioning heart; simulation of air traffic control patterns can move that industry faster toward safer skies, reduce the occurrence of mid-air collisions and the time to reroute flights around weather systems; modeling of air flow around computer chips and over the surface of a fast spinning disk drive improve function and reliability. Today's ability to model huge global weather system can change the fate of nations struggling with tsunami's, floods, the devastation of hurricanes; Astrophysics and many studies of importance to National Security all rely on this new aspect of science and mathematics. At the heart of all these is the development of more advanced numerical algorithms which provide the cornerstone of computational simulation. The proposed research intends to extend these to more efficient high-order 3D methods, While the immediate application of the research is to fluid flow problems related to aerospace, an improved fluid flow simulation capability is immediately transferable to other industrial and scientific applications, including those mentioned above as well as many others such as the automotive and marine industries, wind energy, and a wide range of environmental issues. Even the flow patterns in urban areas around buildings can be used to evaluate how germ or chemical particles would be spread by winds circulating through the 'canyons' of clustered tall buildings. Further advances in all aspects of computational simulation and its underlying numerical methology is essential in order to maintain the Unites States' technological leadership and competitiveness in the rapidly evolving global economy. The results of these developments and techniques will play an increasingly pivotal role and have major implications environmentally, scientifically economically and socially as the results filter down into the many applications using them today, as well as future applications that will need to evaluate complex nonlinear partial differential equations for a variety of physics.