A quadtree-based adaptively-refined Cartesian-grid algorithm for solution of the Euler equations.

Abstract

A method for solution of the steady two-dimensional Euler equations is presented. The scheme is designed to overcome the difficulties associated with geometric complexity and the existence of disparate length scales in the computed flow-fields. An adaptively-refined Cartesian grid defined by a tree-based data structure is used. Connectivity information is obtained from the data tree via the parent/children relationships of the cells. Initial grid generation is enhanced by geometry-based cell adaptation. The solution is converged to a steady state using a linear reconstruction and an approximate Riemann solver. Multi-stage time stepping and multigrid convergence acceleration are used to advance the solution in time. Solution adaptation is achieved through the use of solution-based gradient information. This enables the grid resolution to match more closely the local length scales of the flow. The initial grid is generated with a minimum of user input for any complex configuration. The user need only specify a set of points defining the bodies, the base grid resolution, and cell size thresholds for the geometry-based adaptation. With proper thresholds, the grid is automatically adapted to the curvature of a body, providing the resolution required to resolve that body adequately. The grid is then improved through the use of solution-based adaptation. The difficulties associated with the small cut cells created by the arbitrary way that the Cartesian grid cuts through the body are overcome by using local time-stepping, coupled with a linear reconstruction method designed specifically for unstructured grids. The solutions obtained show the second-order global accuracy of the scheme. Results are presented for airfoils at subsonic, transonic, and supersonic speeds. The results compare favorably with benchmark solutions on structured grids with substantially more cells. Also included are a channel flow, several axisymetric jet flows, and several multi-element airfoil flows. In all cases, the small cut cells generated by the intersection of the body with the Cartesian grid have no adverse effect on the smoothness of the solution. The broad range of results presented demonstrates the geometric flexibility of this approach, as well as the accuracy and efficiency attainable by solution-based adaptation.