Euler Solution Using Cartesian Grid with Least Squares Technique

subdomains exist. They include methods using unstructured meshes, body-fitted curvilinear meshes, and Cartesian meshes. This paper discusses an approach that uses “gridless” or “meshless” methods to address the boundary or interface while standard structured grid methods are used everywhere else. The present method uses the Cartesian grid to specify and distribute the computational points on the boundary surface but not to define the geometrical properties. Euler fluxes for the neighbors of cut cells are computed using the gridless method involving a local least-squares curve fit to a “cloud” of grid points. The boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values in a similar least-squares fashion. No halo points are needed. The overall scheme is robust, stable and converges well for a range of Mach numbers tested. Solutions from the proposed approach are computed for the NACA 0012 and RAE 2822 airfoils and the results are compared with those obtained by a standard Euler solver using body-fitted grids. For grids with equal resolution the method is less accurate for capturing shocks but an improvement in resolution of 60% gives a sharper shock front. The approach offers a number of advantages and its extension to three dimensions is straightforward. Unstructured meshes are typically constructed from triangles in two-dimensions or tetrahedral cells in threedimensions (e.g., Refs. 1-4). The main advantage of traditional unstructured grids is the ease of grid generation about complex configurations since the cells may be oriented in any arbitrary way to conform to the geometry. However, the computational time and cost for unstructured mesh computations are generally higher than the structured mesh approach. The unstructured mesh method tends to be inefficient when applied to large scale three-dimensional problems. In addition, mesh quality is also not easily controlled which will impact convergence and accuracy. Body-fitted hexahedral grids, which belong to the structured grid family, have been widely used. The main advantage is the ease in implementing boundary conditions due to the body-aligned nature of the mesh. However, a major drawback for this is the difficulty of mesh generation for complex geometry. Highly distorted or skewed cells may occur in some regions of the domain that adversely influence the computation. Moreover, often it is not possible to use a single grid for very complex geometries. More sophisticated methods such as multiblock, patched grids, Chimera or hybridtype grids are proposed and used in varying degrees of successes (e.g., Refs. 5-7). These methods require transfer of information between the different meshes. Introduction A continuing obstacle of Computational Fluid Dynamics (CFD) for configurations with complex geometry is the problem of mesh generation. Several alternatives for dividing the flow field into discrete Regardless whether an unstructured or a structured method is used, the grid generation involves both surface preparation and griding before the volume grids can begin. In recent years there is a renewed interest in the conceptually simple approach of Cartesian grids. Both structured and unstructured methods of solution can be applied to grids created by this means. It possesses the advantages associated with structured grids and some of its attractions include ease of grid ––––––––––––––––––––––– 1. Associate Scientist, email: [email protected] 2. Principal Research Scientist, email: [email protected] Member AIAA. 3. Associate Professor, email: [email protected] Senior Member AIAA. Copyright  2003 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc with permission 1 American Institute of Aeronautics and Astronautics generation, lower computational storage requirements, and significantly less operational count per cell compared to body-fitted schemes. The convergence properties of the solver are comparatively better since there are no problems related to skewness or distortion of cells. Furthermore, Cartesian grids offer considerable ease in implementing higher order schemes. However, the main challenges in using a Cartesian method are the problems in dealing with the arbitrary boundaries, as the grids are not body-aligned. The cells of a Cartesian mesh near the body can extend through surfaces of solid components. Accurate means of representations for boundary conditions in cells that intersect surfaces are essential for successful Cartesian schemes. An important feature of the Cartesian approach is the elimination of the need to create a case-specific bodyfitted surface mesh which is human labor intensive, and replaces it by a more computer centric solution involving a technique to characterize the intersections between hexahedral mesh cells and body surfaces. The flow grid can be generated independently and the cells of the Cartesian mesh that intersect the surface can be determined without having first to mesh the surface. There is no explicit relation between the surface description and the local flow grid cell structure. The task of surface description, which is unavoidable, is to resolve the geometry, and no longer include tedious surface grid generation process. To ensure adequate representation without the use of excessive grid points, adaptive mesh refinement of the Cartesian grid can be used. A Cartesian based algorithm must first identify cells that intersect solid surfaces and remove or flag cells that are completely enclosed within the solid body. These would not be considered as part of the flow computational domain. Solution using the Cartesian method usually consists of a standard method for the regular cells and a special treatment for the boundary cells. In the literature solvers involving either unstructured or structured grid techniques are reported. Different methods have been proposed in the literature to resolve the boundary conditions. Broadly the methods proposed involve either cut cells for a finitevolume treatment or grid points for a finite-difference construction. A cut-cells method ensures conservation and the same flux computation algorithm is applied. However, the task of computing the volume and fluxes for all the irregularly shaped cut cells entails a considerable increase in complexity. More crucially, the method can at times lead to the creation of very small cells at the boundary which poses problems of numerical stability. To alleviate this problem, rule based cell merging techniques were used to combine cut cells that are too small with neighbouring cells such as in the work of Clarke et al. and Ye et al.. While this approach ensures strict conservation properties, cell formation and merging is not straightforward in threedimensions. Also it is difficult to ensure that the local geometric properties are in full consistent with the original shape particularly when coarse grids are used as in the implementation of multigrid. The introduction of the finite-difference approach on the other hand avoids all the problems associated with cut cells. For instance, in the work of Epstein et al., cut cells were handled by an extrapolation procedure in which halo points are used at the boundary. Special treatment is needed to ensure that halo points are available at regions with thin surfaces (e.g., a trailing edge) where multiple values are defined at a particular point . Unlike the cut cells method, this method is not conservative. Another way to improve accuracy is to use higher order representation for the boundary as in the work of Forrer and Jeltsch to minimize the lack of conservation. To achieve an accurate solution for the boundary, Cartesian methods require a mesh refinement strategy to resolve the geometry. Wu offers an anisotropic refinement strategy that significantly reduces the number of computational cells as compared to traditional isotropic refinement strategy. Other mesh refinement strategies for Cartesian methods can be found in the literature (e.g., Refs. 18-21). In this paper, a different Cartesian method is described for solving the Euler equations. It uses “gridless” or “meshless” methods to address the boundary or interface while standard structured grid methods are used everywhere else. This is different from Batina’s approach and other workers who have proposed a completely gridless technique where the governing equations are solved by a local least-squares curve fit using clouds of grid points. A similar approach to the present work was reported by Kirshman and Liu, who used a finite difference scheme with Van Leer flux splitting technique. In the present work, a finite-volume formulation with central differencing is used with artificial dissipation. Also a different approach is employed in discretizing the surface. The Cartesian grid is used to define and distribute the computational points on the boundary surface. These surface points spacing which are more or less comparable to the Cartesian grid points are better distributed to reduce the possibility of ill-conditioned matrices in the gridless computation with cloud points. Since accuracy is an important issue at the boundary, geometrical properties are defined using the original data of the geometry and not the surface computational points, which may be too coarse. A simplified overall implementation of boundary 2 American Institute of Aeronautics and Astronautics condition is adopted. Euler fluxes for the neighbors of cut cells are computed using a ‘gridless’ method and the resulting derivatives are analytically determined for the flow equations. Boundary conditions implemented on the surface points are automatically satisfied in the process of evaluating the surface values by a similar least-squares method. In the next section, the numerical implementation of the Cartesian method for the two-dimensional Euler Equations will be presented. Determination of the gridless cloud points are described followed by the spatial discretization for the flow field and boundary nodes. The discretized Euler equations are solved using the modified four-stage Runge-Kutta scheme developed by Jameson et al.