Spectral methods for problems in complex geometries

Boundary-fitted coordinate systems for numerical solution of partial differential equations—A review

The generation of computational grids suitable for obtaining accurate numerical solutions to the three-dimensional Navier-Stokes equations is the subject of intensive research. For a wide class of nozzle configurations, a three-dimensional grid can be constructed by a sequence of two-dimensional grids in successive cross-sectional planes. The present paper is concerned with numerical generation of two-dimensional grids. An effective method of interior grid control is presented based on a modified elliptic system containing free parameters. For a simply connected region, the free parameters are computed from the Dirichlet boundary values. The resulting interior grid point distribution is controlled entirely by a priori selection of the grid point distribution along the boundaries of the section.

Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations

TOMCAT - A code for numerical generation of boundary-fitted curvilinear coordinate systems on fields containing any number of arbitrary two-dimensional bodies

A fully three-dimensional compressible inverse design method for the design of radial and mixed flow turbomachines is described. In this method the distribution of the circumferentially averaged swirl velocity rV-theta-BAR on the meridional geometry of the impeller is prescribed and the corresponding blade shape is computed iteratively. Two approaches are presented for solving the compressible flow problem. In the approximate approach the pitchwise variation in density is neglected and as a result the algorithm is simple and efficient. In the exact approach the velocities and density are computed throughout the three-dimensional flow field by employing a fast fourier transform in the tangential direction. The results of the approximate and exact approach are compared for the case of a high-speed (subsonic) radial-inflow turbine and it is shown that the difference between the blade shapes computed by the two methods is well within the manufacturing tolerances. The method was validated by calculating the flow through a designed high-speed radial-inflow turbine by using a three-dimensional inviscid Euler solver. Very good correlation was obtained between the specified and computed rV-theta-BAR-distributions.

A compressible three‐dimensional design method for radial and mixed flow turbomachinery blades

A method is presented for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected two-dimensional region containing any number of arbitrarily shaped bodies. No restrictions are placed on the shape of the boundaries, which may even be time-dependent, and the approach is not restricted in principle to two dimensions. With this procedure the numerical solution of a partial differential system may be done on a fixed rectangular field with a square mesh with no interpolation required regardless of the shape of the physical boundaries, regardless of the spacing of the curvilinear coordinate lines in the physical field, and regardless of the movement of the coordinate system in the physical plane. A number of examples of coordinate systems and application thereof to the solution of partial differential equations are given. The FORTRAN computer program and instructions for use are included.

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Boundary-fitted curvilinear coordinate systems for solution of partial differential equations on fields containing any number of arbitrary two-dimensional bodies

F. C. Thames

Papers

Boundary-fitted curvilinear coordinate systems for solution of partial differential equations on fields containing any number of arbitrary two-dimensional bodies