ALBERT

Description: The ability to treat arbitrary boundary shapes is one of the most desirable characteristics of a method for generating grids, including those about airfoils. In a grid used for computing aerodynamic flow over an airfoil, or any other body shape, the surface of the body is usually treated as an inner boundary and often cannot be easily represented as an analytic function. The GRAPE computer program was developed to incorporate a method for generating two-dimensional finite-difference grids about airfoils and other shapes by the use of the Poisson differential equation. GRAPE can be used with any boundary shape, even one specified by tabulated points and including a limited number of sharp corners. The GRAPE program has been developed to be numerically stable and computationally fast. GRAPE can provide the aerodynamic analyst with an efficient and consistent means of grid generation. The GRAPE procedure generates a grid between an inner and an outer boundary by utilizing an iterative procedure to solve the Poisson differential equation subject to geometrical restraints. In this method, the inhomogeneous terms of the equation are automatically chosen such that two important effects are imposed on the grid. The first effect is control of the spacing between mesh points along mesh lines intersecting the boundaries. The second effect is control of the angles with which mesh lines intersect the boundaries. Along with the iterative solution to Poisson's equation, a technique of coarse-fine sequencing is employed to accelerate numerical convergence. GRAPE program control cards and input data are entered via the NAMELIST feature. Each variable has a default value such that user supplied data is kept to a minimum. Basic input data consists of the boundary specification, mesh point spacings on the boundaries, and mesh line angles at the boundaries. Output consists of a dataset containing the grid data and, if requested, a plot of the generated mesh. The GRAPE program is written in FORTRAN IV for batch execution and has been implemented on a CDC 6000 series computer with a central memory requirement of approximately 135K (octal) of 60 bit words. For plotted output the commercially available DISSPLA graphics software package is required. The GRAPE program was developed in 1980.

Description: The ability to treat arbitrary boundary shapes is one of the most desirable characteristics of a method for generating grids. 3DGRAPE is designed to make computational grids in or about almost any shape. These grids are generated by the solution of Poisson's differential equations in three dimensions. The program automatically finds its own values for inhomogeneous terms which give near-orthogonality and controlled grid cell height at boundaries. Grids generated by 3DGRAPE have been applied to both viscous and inviscid aerodynamic problems, and to problems in other fluid-dynamic areas. 3DGRAPE uses zones to solve the problem of warping one cube into the physical domain in real-world computational fluid dynamics problems. In a zonal approach, a physical domain is divided into regions, each of which maps into its own computational cube. It is believed that even the most complicated physical region can be divided into zones, and since it is possible to warp a cube into each zone, a grid generator which is oriented to zones and allows communication across zonal boundaries (where appropriate) solves the problem of topological complexity. 3DGRAPE expects to read in already-distributed x,y,z coordinates on the bodies of interest, coordinates which will remain fixed during the entire grid-generation process. The 3DGRAPE code makes no attempt to fit given body shapes and redistribute points thereon. Body-fitting is a formidable problem in itself. The user must either be working with some simple analytical body shape, upon which a simple analytical distribution can be easily effected, or must have available some sophisticated stand-alone body-fitting software. 3DGRAPE does not require the user to supply the block-to-block boundaries nor the shapes of the distribution of points. 3DGRAPE will typically supply those block-to-block boundaries simply as surfaces in the elliptic grid. Thus at block-to-block boundaries the following conditions are obtained: (1) grids lines will match up as they approach the block-to-block boundary from either side, (2) grid lines will cross the boundary with no slope discontinuity, (3) the spacing of points along the line piercing the boundary will be continuous, (4) the shape of the boundary will be consistent with the surrounding grid, and (5) the distribution of points on the boundary will be reasonable in view of the surrounding grid. 3DGRAPE offers a powerful building-block approach to complex 3-D grid generation, but is a low-level tool. Users may build each face of each block as they wish, from a wide variety of resources. 3DGRAPE uses point-successive-over-relaxation (point-SOR) to solve the Poisson equations. This method is slow, although it does vectorize nicely. Any number of sophisticated graphics programs may be used on the stored output file of 3DGRAPE though it lacks interactive graphics. Versatility was a prominent consideration in developing the code. The block structure allows a great latitude in the problems it can treat. As the acronym implies, this program should be able to handle just about any physical region into which a computational cube or cubes can be warped. 3DGRAPE was written in FORTRAN 77 and should be machine independent. It was originally developed on a Cray under COS and tested on a MicroVAX 3200 under VMS 5.1.

Description: This computer program is designed to calculate the flow fields in two-dimensional and three-dimensional axisymmetric supersonic inlets. The method of characteristics is used to compute arrays of points in the flow field. At each point the total pressure, local Mach number, local flow angle, and static pressure are calculated. This program can be used to design and analyze supersonic inlets by determining the surface compression rates and throat flow properties. The program employs the method of characteristics for a perfect gas. The basic equation used in the program is the compatibility equation which relates the change in stream angle to the change in entropy and the change in velocity. In order to facilitate the computation, the flow field behind the bow shock wave is broken into regions bounded by shock waves. In each region successive rays are computed from a surface to a shock wave until the shock wave intersects a surface or falls outside the cowl lip. As soon as the intersection occurs a new region is started and the previous region continued only in the area in which it is needed, thus eliminating unnecessary calculations. The maximum number of regions possible in the program is ten, which allows for the simultaneous calculations of up to nine shock waves. Input to this program consists of surface contours, free-stream Mach number, and various calculation control parameters. Output consists of printed and/or plotted results. For plotted results an SC-4020 or similar plotting device is required. This program is written in FORTRAN IV to be executed in the batch mode and has been implemented on a CDC 7600 with a central memory requirement of approximately 27k (octal) of 60 bit words.

Description: Elliptic partial differential equations are used to generate a smooth grid that permits a one-to-one mapping in such a way that mesh lines of the same family do not cross. Problems that arise due to lack of clustering at crucial points or intersections of mesh lines at highly acute angles, are examined and various forcing or source terms are used (to correct the problems) that are either compatible with the maximum principle or are so locally controlled that mesh lines do not intersect. Attention is given to various schematics of unclustered grids and grid detail about (highly cambered) airfoils.

Description: An algorithm for generating computational grids about arbitrary three-dimensional bodies is developed. The elliptic partial differential equation (PDE) approach developed by Steger and Sorenson and used in the NASA computer program GRAPE is extended from two to three dimensions. Forcing functions which are found automatically by the algorithm give the user the ability to control mesh cell size and skewness at boundary surfaces. This algorithm, as is typical of PDE grid generators, gives smooth grid lines and spacing in the interior of the grid. The method is applied to a rectilinear wind-tunnel case and to two body shapes in spherical coordinates.

Description: The inviscid, internal, and external axial corner flows generated by two intersecting wedges traveling supersonically are obtained by use of a second-order shock-capturing, finite-difference approach. The governing equations are solved iteratively in conical coordinates to yield the complicated wave structure of the internal corner and the simple peripheral shock of the external corner. The numerical results for the internal flows compare favorably with existing experimental data.

Description: A method for generating boundary-fitted, curvilinear, two dimensional grids by the use of the Poisson equations is presented. Grids of C-type and O-type were made about airfoils and other shapes, with circular, rectangular, cascade-type, and other outer boundary shapes. Both viscous and inviscid spacings were used. In all cases, two important types of grid control can be exercised at both inner and outer boundaries. First is arbitrary control of the distances between the boundaries and the adjacent lines of the same coordinate family, i.e., stand-off distances. Second is arbitrary control of the angles with which lines of the opposite coordinate family intersect the boundaries. Thus, both grid cell size (or aspect ratio) and grid cell skewness are controlled at boundaries. Reasonable cell size and shape are ensured even in cases wherein extreme boundary shapes would tend to cause skewness or poorly controlled grid spacing. An inherent feature of the Poisson equations is that lines in the interior of the grid smoothly connect the boundary points (the grid mapping functions are second order differentiable).

Description: The hyperbolic scheme is used to efficiently generate smoothly varying grids with good step size control near the body. Although only two dimensional applications are presented, the basic concepts are shown to extend to three dimensions.

Description: Transonic flow fields about a low-aspect-ratio advanced technology wing have been computed using a viscous/inviscid zonal approach. The flow field near the wing where viscous effects are important was solved using the 'Reynolds-Averaged Navier-Stokes Equations' in 'thin-layer' form. The Euler equations were used to determine the flow field in regions away from the wing where viscous effects are insignificant. A zonal grid using an H-H topology was generated around the wing by first solving a set of Poisson's equations for the global grid. This grid was then subdivided into separate zones of viscous or inviscid flow as suggested by the flow physics. A series of flow cases were computed and compared with corresponding sets of experimental data. All cases showed good agreement with experiment in terms of the pressure field. Also, a good correlation between computed separated surface flow and experimental oil flow was obtained.