Application of vorticity confinement to the prediction of flows over complex bodies - A survey of recent results

Abstract

Many flows of interest are characterized by large regions of concentrated vortical structures that persist and can convect over long distances. Flows of this nature include those associated with aircraft (and in particular rotorcraft), but also include flows associated with ships, automobiles, bridges, and buildings. Conventional CFD methods tend to dissipate vortical structures, degrading the overall accuracy of the computed flow. This dissipation can be reduced through the use of fine grids, but at the expense of greatly increased computational demands. The confinement method prevents the dissipation of vortical structures on coarse grids. As a result, flows around very complex configurations can be computed on coarse grids at low cost. The method has been incorporated into a flow solver (called HELIX) that is based on Cartesian grid methods; a user supplies only a triangulated surface description of a configuration. As a result, setup time is minimal, even for extremely complex configurations. In this paper, the confinement approach implemented in HELIX is used to calculate flowfields around configurations in the low-speed aerodynamic regime. An extension of the confinement method, called surface confinement, is demonstrated as a model for separating turbulent boundary layers. Senior Research Scientist 0 Research Scientist ^Professor Copyright © 2001 by Flow Analysis, Inc. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Introduction This paper describes recent results that demonstrate the accuracy, efficiency, and simplicity of a relatively new basic CFD technique, The method is applied in this paper to aerodynamic problems characterized by geometric complexity and lowspeed (incompressible) flow fields. As a result of these complexities, application of conventional CFD methods is costly and highly demanding of computational resources. In addition to the need for an efficient and simple solution technique for complex configurations, there is the realization that CFD methods must use modeling rather than solve exact equations for realistic Reynolds numbers, since most vortical regions such as boundary layers and wakes are turbulent. This is true even for conventional CFD solutions for flow over simple configurations. This realization leads to the possibility of developing an entirely new CFD methodology where very different numerical models are used for turbulent vortical regions. Most models involve first hypothesizing a turbulence model based on a partial differential equation and then trying to accurately discretize and resolve it in the thin regions. Resolving the small scales of the flow generally require costly solution strategies involving body-fitted grids, often with extensive refinement near the surface. Once it is realized that the partial differential equations are only approximate models for vortical regions, we are led to the idea of modeling the internal structure of the vortical regions directly on the grid using nonlinear difference equations, rather than using finite difference equations that American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s) Sponsoring Organization. attempt to approximately resolve the model differential equations. This approach allows treatment of vortical structures as near-singular objects spread over only a few grid cells on an essentially uniform Cartesian computational grid. These ideas are implemented in the methodology termed vorticity confinement. For several decades, starting with Lax (Ref. 1) and others, these ideas have also been implemented for the treatment of shocks. However, in those cases characteristics slope inward toward the shock, making the implementation much easier. One of the first confinement-type schemes for contact discontinuites was developed by Harten (Ref. 2), but was specialized to one-dimensional compressible flow. In this paper a short description of confinement will first be given. Then, a sequence of computations are presented for successively more complex bodies. Vorticity Confinement Vorticity confinement is the basic technique behind the method presented in this paper. It is a method to conserve and concentrate on a regular grid; the vortical structures can represent boundary layers over solid surfaces, or separated convected over arbitrarily long distances, confinement can be implemented in a flow solver, for both incompressible or compressible flow, by adding a term to the discretized momentum conservation equations (Ref. 3). For general unsteady incompressible flows, the governing equations with the confinement term are the continuity equation and the momentum equations, with an added term: where q is the velocity vector, p is pressure, p is density, and // is a diffusion coefficient. For the last confinement term £?, 8 is a numerical coefficient that, together with //, controls the size of the convecting vortical regions or vortical boundary layers. The diffusion coefficient is specified explicitly only if required by stability considerations, e.g., a central-difference convective algorithm will generally require // to be specified explicitly; whereas an upwind scheme will not. HELIX has both convective options available. There are many possible forms for the confinement term. The simplest one is