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Curvilinear coordinate systems were used extensively to solve partial differential equations on arbitrary regions. An analysis of truncation error in the computation of derivatives revealed why numerical results may be erroneous. A more accurate method of computing derivatives is presented.
The computer program AFTBDY generates a body fitted curvilinear coordinate system for a wedge curved after body. This wedge curved after body is being used in an experimental program. The coordinate system generated by AFTBDY is used to solve 3D compressible N.S. equations. The coordinate system in the physical plane is a cartesian x,y,z system, whereas, in the transformed plane a rectangular xi, eta, zeta system is used. The coordinate system generated is such that in the transformed plane coordinate spacing in the xi, eta, zeta direction is constant and equal to unity. The physical plane coordinate lines in the different regions are clustered heavily or sparsely depending on the regions where physical quantities to be solved for by the N.S. equations have high or low gradients. The coordinate distribution in the physical plane is such that x stays constant in eta and zeta direction, whereas, z stays constant in xi and eta direction. The desired distribution in x and z is input to the program. Consequently, only the y-coordinate is solved for by the program AFTBDY.
The use of curvilinear coordinates is sometimes indicated by the inherent geometry of a fluid dynamics problem, but this introduces fictitious forces into the momentum equations that spoil the strict conservative form. If one is willing to work in three dimensions, these fictitious forces can be eliminated by solving for rectangular (Cartesian) momentum components on a curvilinear mesh. A thoroughly geometric approach to fluid dynamics on spacetime demonstrates this transparently, while also giving insight into a greater unity of the relativistic and nonrelativistic cases than is usually appreciated.
The computer program COORDC generates a body fitted curvilinear coordinate system for corner geometry with or without corner fillets. It is assumed that at any given xi, x remains constant; consequently the only variation is in y and z. It is also assumed that for all xi's in the physical plane the coordinate system in y-z plane is similar. This enables solution of coordinate system for one particular xi = 1 (x for xi = 1 is arbitrarily chosen to be 0.0) and the solution for all other xi plane can be easily specified once the coordinates in the physical plane on the line 1 or = to xi or = to IMAX, eta = 1, zeta = 1 are specified.
General curvilinear coordinate systems are considered along with the error induced by coordinate systems, basic differential models for coordinate generation, elliptic grid generation, conformal grid generation, algebraic grid generation, orthogonal grid generation, patched coordinate systems, and solid mechanics applications of boundary fitted coordinate systems. Attention is given to coordinate system control and adaptive meshes, the application of body conforming curvilinear grids for finite difference solution of external flow, the use of solution adaptive grids in solving partial differential equations, adaptive gridding for finite difference solutions to heat and mass transfer problems, and the application of curvilinear coordinate generation techniques to the computation of internal flows. Other topics explored are related to the solution of nonlinear water wave problems using boundary-fitted coordinate systems, the numerical modeling of estuarine hydrodynamics on a boundary-fitted coordinate system, and conformal grid generation for multielement airfoils.
Hellmann-Feynman theorem in curvilinear coordinates for exact and for floating variational wave functions
The basic governing equations for the second-order three-dimensional hypersonic thermal and chemical nonequilibrium boundary layer are derived by means of an order-of-magnitude analysis. A two-temperature concept is implemented into the system of boundary-layer equations by simplifying the rather complicated general three-temperature thermal gas model. The equations are written in a surface-oriented non-orthogonal curvilinear coordinate system, where two curvilinear coordinates are non-orthogonial and a third coordinate is normal to the surface. The equations are described with minimum use of tensor expressions arising from the coordinate transformation, to avoid unnecessary confusion for readers. The set of equations obtained will be suitable for the development of a three-dimensional nonequilibrium boundary-layer code. Such a code could be used to determine economically the aerodynamic/aerothermodynamic loads to the surfaces of hypersonic vehicles with general configurations. In addition, the basic equations for three-dimensional stagnation flow, of which solution is required as an initial value for space-marching integration of the boundary-layer equations, are given along with the boundary conditions, the boundary-layer parameters, and the inner-outer layer matching procedure. Expressions for the chemical reaction rates and the thermodynamic and transport properties in the thermal nonequilibrium environment are explicitly given.
A three-dimensional numerical solver based on finite-difference solution of three-dimensional elastodynamic equations in generalized curvilinear coordinates has been developed and used to generate data such as radial and tangential stresses over various gear component geometries under rotation. The geometries considered are an annulus, a thin annular disk, and a thin solid disk. The solution is based on first principles and does not involve lumped parameter or distributed parameter systems approach. The elastodynamic equations in the velocity-stress formulation that are considered here have been used in the solution of problems of geophysics where non-rotating Cartesian grids are considered. For arbitrary geometries, these equations along with the appropriate boundary conditions have been cast in generalized curvilinear coordinates in the present study.
A method for automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multi-connected region containing any number of arbitrarily shaped bodies is presented. 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. Numerical solutions for the lifting and nonlifting potential flow about Joukowski and Karman-Trefftz airfoils using this coordinate system generation show excellent comparison with the analytic solutions. The application to fields with multiple bodies is illustrated by a potential flow solution for multiple airfoils.
Equations of anisotropic shell elasticity theory in curvilinear coordinates
A method of automatic body-fitted curvilinear coordinate generation is described and used to construct a finite-difference solution of the full incompressible time-dependent Navier-Stokes equations for the unsteady laminar viscous flow arbitrary two-dimensional airfoils or any other two-dimensional body. A method of controlling the spacing of the coordinate lines encircling the body is developed in order to treat high Reynolds number flows, since the coordinate lines must concentrate near the surface to a greater degree as the Reynolds number increases. Multiple airfoils and submerged hydrofoils are treated as illustrative examples. The solution shows good agreement with the Blasius boundary layer solution for the flow past a semi-infinite flat plate.
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.
The MEIT equations are derived in orthogonal streamline curvilinear coordinates defined by the inviscid velocity at the wall and the surface-normal direction. The resulting formulations is (1) is not restricted to flat-plate or axisymmetric geometries, (2) accommodates vortical inviscid flows (e.g., induced by curved shocks) and wall mass blowing, and (3) allows for an arbitrary definition of the edge state. The streamline ODEs are also extended to a surface PDE framework.
Application of digital computers for deriving motion equations of point mass in curvilinear coordinate systems by use of tensor calculus
Various finite difference techniques used to solve Laplace's equation are compared. Curvilinear coordinate systems are used on two dimensional regions with irregular boundaries, specifically, regions around circles and airfoils. Truncation errors are analyzed for three different finite difference methods. The false boundary method and two point and three point extrapolation schemes, used when having the Neumann boundary condition are considered and the effects of spacing and nonorthogonality in the coordinate systems are studied.
A noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves was developed. The basic generating equation is the Gaussian equation for an Euclidean space which is solved analytically. The method is applied in many cases and these test results demonstrate that the proposed method can be readily applied to a wide variety of problems. The method can also be used for simply connected regions only by obtaining the solution of the linear equation under the changed boundary conditions.
Description of a new method of writing the conservation equations of gasdynamics in curvilinear coordinates which eliminates undifferentiated terms. It is thus possible to readily apply difference schemes derived for Cartesian coordinates which conserve mass, momentum, and energy in the total flow field. The method is derived for orthogonal coordinates, and then extended to cover the most general class of coordinate transformations, using general tensor analysis. Several special features of the equations are discussed.
Progress in the direct-inverse wing design method in curvilinear coordinates has been made. A spanwise oscillation problem and proposed remedies are discussed. Test cases are presented which reveal the approximate limits on the wing's aspect ratio and leading edge wing sweep angle for a successful design, and which show the significance of spanwise grid skewness, grid refinement, viscous interaction, the initial airfoil section and Mach number-pressure distribution compatibility on the final design. Furthermore, preliminary results are shown which indicate that it is feasible to successfully design a region of the wing which begins aft of the leading edge and terminates prior to the trailing edge.