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NASA NTRS1986

Transformation of two and three-dimensional regions by elliptic systems

Grid smoothing and orthogonalization procedures were developed and implemented in the construction of two and three dimensional grids. The procedures are based on the variational methods of grid generation. The two-dimensional examples were computed using the MSU IRIS Graphics Workstation. It was demonstrated that the elliptic grid generation equations, with arbitrary forcing functions, can be solved, in their variational formulation, using a gradient method. Since gradient methods have a global convergence property, the divergence problems often encountered when using SOR iterative methods can be avoided. It is not to be concluded, however, that SOR methods should be abandoned, since gradient methods tend to converge very slowly. In fact, slow convergence was the major problem encountered in the three-dimensional grids. Further progress was made on the continuing effort to develop conservative interpolation formulas for overlapping grids.

Mastin, C. Wayne↗

NASA NTRS1991

Effect of nose shape on three-dimensional stagnation region streamlines and heating rates

A new method for calculating the three-dimensional inviscid surface streamlines and streamline metrics using Cartesian coordinates and time as the independent variable of integration has been developed. The technique calculates the streamline from a specified point on the body to a point near the stagnation point by using a prescribed pressure distribution in the Euler equations. The differential equations, which are singular at the stagnation point, are of the two point boundary value problem type. Laminar heating rates are calculated using the axisymmetric analog concept for three-dimensional boundary layers and approximate solutions to the axisymmetric boundary layer equations. Results for elliptic conic forebody geometries show that location of the point of maximum heating depends on the type of conic in the plane of symmetry and the angle of attack, and that this location is in general different from the stagnation point. The new method was found to give smooth predictions of heat transfer in the nose region where previous methods gave oscillatory results.

Hassan, Basil↗

NASA NTRS1994

Canonical-variables multigrid method for steady-state Euler equations

A novel approach for the solution of inviscid flow problems for subsonic compressible flows is described. The approach is based on canonical forms of the equations, in which subsystems governed by hyperbolic operators are separated from those governed by elliptic ones. The discretizations used as well as the iterative techniques for the different subsystems are inherently different. Hyperbolic parts, which describe, in general, propagation phenomena, are discretized using upwind schemes and are solved by marching techniques. Elliptic parts, which are directionally unbiased, are discretized using h-elliptic central discretizations, and are solved by pointwise relaxations together with coarse grid acceleration. The resulting discretization schemes introduce artificial viscosity only for the hyperbolic parts of the system; thus a smaller total artificial viscosity is used, while the multigrid solvers used are much more efficient. Solutions of the subsonic compressible Euler equations are achieved at the same efficiency as the full potential equation.

Taasan, Shlomo↗

NASA NTRS1960

The Numerical Calculation of Flow Past Conical Bodies Supporting Elliptic Conical Shock Waves at Finite Angles of Incidence

The inverse method, with the shock wave prescribed to be an elliptic cone at a finite angle of incidence, is applied to calculate numerically the supersonic perfect-gas flow past conical bodies not having axial symmetry. Two formulations of the problem are employed, one using a pair of stream functions and the other involving entropy and components of velocity. A number of solutions are presented, illustrating the numerical methods employed, and showing the effects of moderate variation of the initial parameters.

Briggs, Benjamin R.↗

NASA NTRS1974

Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.

Born, G. H.↗

NASA NTRS1988

Body force circulation and the Antarctic Ozone minimum

A dynamical mechanism, which is relevant to the ozone-depletion problem and which involves a 'body force' circulation discussed by Eliassen (1951), is considered. The body force problem and its application to 'generic' upper-tropospheric wave-driving are discussed theoretically. Following a review of the interhemispheric differences in the winter and early-spring circulation, the elliptic equation governing the residual mean meridional streamfunction is derived, and its analytic-solution properties in negligible shear are discussed. Finally, a numerical solution is derived using the observed August winds under the assumption of radiative equilibrium, and the effect of diabatic heating is discussed.

Dunkerton, Timothy J.↗

NASA NTRS1976

Theoretical correction for the lift of elliptic wings

In the wing section theory the magnitude of the circulation, and hence of the lift, is determined by the velocity that would be induced near the trailing edge of the section in a non-lifting potential flow. In three dimensional flow the problem is complicated by the presence of the wake and no simple basic solution has been found. Treatment of the problem of a wing of finite span is reported on the basis of the two dimensional theory, corrected for the effect of the wake.

Jones, R. T.↗

NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

Chang, S. C.↗

NASA NTRS1975

Application of local linearization and the transonic equivalence rule to the flow about slender analytic bodies at Mach numbers near 1.0

The local linearization method for axisymmetric flow is combined with the transonic equivalence rule to calculate pressure distribution on slender bodies at free-stream Mach numbers from .8 to 1.2. This is an approximate solution to the transonic flow problem which yields results applicable during the preliminary design stages of a configuration development. The method can be used to determine the aerodynamic loads on parabolic arc bodies having either circular or elliptical cross sections. It is particularly useful in predicting pressure distributions and normal force distributions along the body at small angles of attack. The equations discussed may be extended to include wing-body combinations.

Tyson, R. W.↗

NASA NTRS1975

Accurate integration of geostationary orbits with Burdet's focal elements

The theory of Burdet's focal elements (1969) is outlined. The differential equations are presented, and the initial value problem is described together with the transformation to rectangular coordinates and classical elements. The focal elements are well defined for zero eccentricity and inclination and can be adopted for the computation of elliptic, parabolic, and hyperbolic motion. For the numerical integration of near-geostationary orbits, a comparison of the efficiency is made between focal elements, Kustaanheimo-Stiefel (1971) theory, and rectangular coordinates. For this class of orbits, a higher accuracy has been obtained by integrating elements than integrating rectangular coordinates.

Flury, W.↗

NASA NTRS1983

A least squares finite element scheme for transonic flow around harmonically oscillating airfoils

The present investigation shows that a finite element scheme with a weighted least squares variational principle is applicable to the problem of transonic flow around a harmonically oscillating airfoil. For the flat plate case, numerical results compare favorably with the exact solution. The obtained numerical results for the transonic problem, for which an exact solution is not known, have the characteristics of known experimental results. It is demonstrated that the performance of the employed numerical method is independent of equation type (elliptic or hyperbolic) and frequency. The weighted least squares principle allows the appropriate modeling of singularities, which such a modeling of singularities is not possible with normal least squares.

Cox, C. L.↗

NASA NTRS1987

Reynolds stress closure in jet flows using wave models

Numerical methods were developed that will form the computational part of the turbulence closure scheme. A wave model was developed for the two-dimensional shear layer. This configuration is being used as a test case for the closure schemes. Various numerical schemes were examined to give efficient solutions of the Rayleigh equation for this geometry. These include both spectral and finite difference methods. Secondly, numerical methods are under development to solve the non-separable Rayleigh equation. This solution is required for the closure scheme in more complex geometries. A model problem was used to assist in the algorithm development. Two-dimensional spectral methods and a hybrid spectral/finite difference technique were developed. An analytic solution of the Rayleigh equation for a basic elliptic flow was obtained. This will be used to verify the stability codes developed for arbitrary geometries. Other numerical methods for solving the Rayleigh equation based on the boundary element technique were also examined. These solutions are forming the basis of a model for the shock structure in jets of arbitrary geometry.

Morris, P. J.↗

NASA NTRS1994

An efficient three-dimensional Poisson solver for SIMD high-performance-computing architectures

We present an algorithm that solves the three-dimensional Poisson equation on a cylindrical grid. The technique uses a finite-difference scheme with operator splitting. This splitting maps the banded structure of the operator matrix into a two-dimensional set of tridiagonal matrices, which are then solved in parallel. Our algorithm couples FFT techniques with the well-known ADI (Alternating Direction Implicit) method for solving Elliptic PDE's, and the implementation is extremely well suited for a massively parallel environment like the SIMD architecture of the MasPar MP-1. Due to the highly recursive nature of our problem, we believe that our method is highly efficient, as it avoids excessive interprocessor communication.

Cohl, H.↗

NASA NTRS1999

A Gas-Kinetic Method for Hyperbolic-Elliptic Equations and Its Application in Two-Phase Fluid Flow

A gas-kinetic method for the hyperbolic-elliptic equations is presented in this paper. In the mixed type system, the co-existence and the phase transition between liquid and gas are described by the van der Waals-type equation of state (EOS). Due to the unstable mechanism for a fluid in the elliptic region, interface between the liquid and gas can be kept sharp through the condensation and evaporation process to remove the "averaged" numerical fluid away from the elliptic region, and the interface thickness depends on the numerical diffusion and stiffness of the phase change. A few examples are presented in this paper for both phase transition and multifluid interface problems.

Xu, Kun↗

NASA NTRS1986

Transformation of two and three-dimensional regions by elliptic systems

Efforts in transferring computational work from the LRC computer to the IRIS Graphics Workstation at MSU are reported and the computation of a conservative solution of a simple hyperbolic equation on an overlapping grid is discussed. Several conclusions concerning computations on overlapping grids are apparent. Problems only occur when there is a major difference in grid spacing on the individual component grids. In the case of hyperbolic equations, it is necessary that both interpolation and extrapolation be applied at the grid boundaries. When interpolated values are used at outflow boundary points, excessive oscillations in the numerical solution may be the result. The same conclusions would be valid for more complicated systems of hyperbolic equations such as the Euler equations for inviscid flow. Some of the solution values would be extrapolated at the overlap boundary, the exact number depending on the number of characteristics pointing out of the overlap region. It is also possible that similar boundary conditions may be needed for some parabolic equations such as high Reynolds number viscous flow equations. Efforts were also expended on the development of three-dimensional conservative interpolation procedures. Finally, the investigation of grid smoothing procedures were initiated during this reporting period. It was decided that the first grid smoothing algorithms will be based on the concepts of variational grid generation.

Mastin, C. W.↗

NASA NTRS1980

Targeting with fixed propellant load

The inertial upper stage (IUS) for the space shuttle operation employs solid rocket stages with fixed, propellant loadings. This means that, if for a given mission the satellite weight is less than the maximum, the IUS will deliver higher delta V's than required. One way to waste the excess capability in order to achieve the desired orbit would be to execute a nonoptimal transfer which would require higher than maximum delta V's. An algorithm is presented which defines takeoff points on the parking orbit and the injection points on the orbit for which transfer orbits require a fixed delta V sub 1 and a fixed delta V sub 2 (defined by the satellite weight). It is assumed that both the parking and the target orbit are elliptical. This allows the use of the same algorithm for guidance. The transfer orbit achieved by the erroneous delta V sub 1 is regarded as a new parking orbit and the new transfer problem is solved by assuming a delta, delta V sub 1. The maximum value of delta, delta V sub 1 is the delta V sub 1 variation and its minimum value is the one which still yields a solution by the algorithm. The delta V sub errors are regarded as orbit injection errors and compensated in the usual way.

Gedeon, G. S.↗

NASA NTRS1979

Automatic mesh-point clustering near a boundary in grid generation with elliptic partial differential equations

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.

Steger, J. L.↗

NASA NTRS1994

SolSTUS: Solar Source Thermal Upper Stage

This paper was written by members of the Utah State University (USU) Space Systems Design class, fall quarter 1993. The class is funded by NASA and administered by the University Space Research Association (USRA). The focus of the class is to give students some experience in design of space systems and as a source of original ideas for NASA. This paper is a summary of the work done by members of the Space Systems Design class during the opening phase of the course. The class was divided into groups to work on different areas of the Solar Thermal Rocket (STR) booster in order to produce a design reference mission that would identify the key design issues. The design reference mission focused upon a small satellite mission to Mars. There are several critical components in a Solar Thermal Rocket. STR's produce a very low thrust, but have a high specific impulse, meaning that they take longer to reach the desired orbit, but use a lot less fuel in doing it. The complexity of the rocket is discussed in this paper. Some of the more critical design problems discussed are: (1) the structural and optical complexity of collecting and focusing sunlight onto a specific point, (2) long term storage of fuel (liquid hydrogen), (3) attitude control while thrusting in an elliptical orbit and orienting the mirrors to collect sunlight, and (4) power and communications for the rocket and it's internal systems. The design reference mission discussed here is a very general mission to Mars. A first order trajectory design has been done and a possible basic science payload for Mars has been suggested. This paper summarizes the design reference mission (DRM) formulated by the USU students during fall quarter and identifies major design challenges that will confront the design team during the next two quarters here at USU.

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