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NASA NTRS1986

Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, A.↗

NASA NTRS1987

Lewis inverse design code (LINDES): Users manual

The method of complex characteristics and hodograph transformation for the design of shockless airfoils was introduced by Bauer, Garabedian, and Korn and has been extended by the author to design subcritical and supercritical cascades with high solidities and large inlet angles. This new capability was achieved by introducing a new conformal mapping of the hodograph domain onto an ellipse and expanding the solution in terms of Chebyshev polynomials. A new computer code, the NASA Lewis inverse design code, was developed based on this idea. This new design code is an efficient method for the design of airfoils in cascade. In particular, the design of subcritical cascades of airfoils is a very fast, robust, and versatile process. The inverse design code can be made to interact with a turbulent boundary layer calculation to obtain airfoils with no separated flows at the design condition. This report is intended to serve as a users manual for this design code. Material previously reported by the author is included here for completeness and quick access to the user. The manual contains a description of the method followed by a discussion of the design procedure and examples. The input parameters necessary to run the code are then described and their default values given. Output listings corresponding to six different blade shapes designed with the code are given, as well as the necessary input data to reproduce the computer runs. The examples have been chosen to show that a wide range of applications can be covered with the code, ranging from supercritical propeller sections to wind tunnel turning vanes that can operate with a large inlet flow angle range.

Sanz, Jose M.↗

NASA NTRS1987

Shuttle computational grid generation

The well known Karman-Trefftz conformal transformation, consisting of repeated applications of the same basic formula, were found to be quite successful to body, wing, and wing-body cross sections. This grid generation technique is extended to cross sections of more complex forms, and also more automatic. Computer programs were written for the selection of hinge points on cross section with angular shapes, the Karman-Trefftz tranformation of arbitrary shapes, and the special transform of hinge point on the imaginary axis. A feasibility study is performed for the future application of conformal mapping grid generation to complex three dimensional configurations. Examples such as Orbiter vehicle section and a few others were used.

Ing, Chang↗

NASA NTRS1987

Numerical simulation of viscous flows over transonic aircraft configurations

Numerical simulation of compressible viscous flow fields is performed for a transonic transport configuration. A single structured grid system is constructed using analytical transformations such as conformal mapping, shearing/twisting/rotating/clustering/stretching transformations. The Reynolds-averaged, thin-layer Navier-Stokes equations are solved on a supercomputer, FACOM VP-400, using the LU-ADI factorization method.

Takanashi, Susumu↗

NASA NTRS1988

Vortex methods for separated flows

The numerical solution of the Euler or Navier-Stokes equations by Lagrangian vortex methods is discussed. The mathematical background is presented in an elementary fashion and includes the relationship with traditional point-vortex studies, the convergence to smooth solutions of the Euler equations, and the essential differences between two- and three-dimensional cases. The difficulties in extending the method to viscous or compressible flows are explained. The overlap with the excellent review articles available is kept to a minimum and more emphasis is placed on the area of expertise, namely two-dimensional flows around bluff bodies. When solid walls are present, complete mathematical models are not available and a more heuristic attitude must be adopted. The imposition of inviscid and viscous boundary conditions without conformal mappings or image vortices and the creation of vorticity along solid walls are examined in detail. Methods for boundary-layer treatment and the question of the Kutta condition are discussed. Practical aspects and tips helpful in creating a method that really works are explained. The topics include the robustness of the method and the assessment of accuracy, vortex-core profiles, timemarching schemes, numerical dissipation, and efficient programming. Calculations of flows past streamlined or bluff bodies are used as examples when appropriate.

Spalart, Philippe R.↗

NASA NTRS1988

An improved version of NCOREL: A computer program for 3-D nonlinear supersonic potential flow computations

A computer code called NCOREL (for Nonconical Relaxation) has been developed to solve for supersonic full potential flows over complex geometries. The method first solves for the conical at the apex and then marches downstream in a spherical coordinate system. Implicit relaxation techniques are used to numerically solve the full potential equation at each subsequent crossflow plane. Many improvements have been made to the original code including more reliable numerics for computing wing-body flows with multiple embedded shocks, inlet flow through simulation, wake model and entropy corrections. Line relaxation or approximate factorization schemes are optionally available. Improved internal grid generation using analytic conformal mappings, supported by a simple geometric Harris wave drag input that was originally developed for panel methods and internal geometry package are some of the new features.

Siclari, Michael J.↗

NASA NTRS1988

Characteristics of broadside coupled cylindrical striplines

This letter presents a method for calculating the even and odd mode impedances of broadside coupled cylindrical stiplines using the conformal mapping technique. Closed form expressions are presented for calculating the even and odd mode characteristic impedances. The data for even and odd mode impedances are presented. The analysis can also be extended to warped broadside coupled striplines.

Reddy, C. Jagadeswara↗

NASA NTRS1988

The stress field in an edge cracked plate in antiplane deformation

The antiplane strain problem of an edge cracked elastic plate subjected to a surface displacement load is solved, and the surface line loads are obtained using conformal mapping techniques. Results are presented for the yield, stress, strain and displacement distributions, and stress intensity factors in the plate. A superposition technique and Green's functions are used to determine the strain field on the upper plate surface due to arbitrarily applied tensile stress on the lower plate surface.

Li, V. C.↗

NASA NTRS1988

Quadrature methods for periodic singular and weakly singular Fredholm integral equations

High-accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are subsequently used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Throughout the development the periodic nature of the problem plays a crucial role. Such periodic equations are used in the solution of planar elliptic boundary value problems such as those that arise in elasticity, potential theory, conformal mapping, and free surface flows. The use of the quadrature methods is demonstrated with numerical examples.

Sidi, Avram↗

NASA NTRS1988

Chemical nonequilibrium and viscous flow computation for conic aerobrake bodies

Three-dimensional analyses are presented for the viscous, reactive flow over a complete entry-body configuration with a wide-angle conic surface. The predictive method uses a split approach that solves iteratively the Navier-Stokes and the continuity equations of chemical species. The finite-difference formulation and the computational grid are adapted to the bow shock and the conformally mapped body such that the velocity components are in the computational spherical-polar space. Combinations of several conic forebody and afterbody configurations have been studied using wind-tunnel, Space Shuttle, and aerobraking orbital transfer vehicle (AOTV) entry conditions. The effects of the borebody bluntness and of finite-rate chemical reactions on the shock layer, the wall catalycity on the boundary layer, the shear-layer impingement on the afterbody, and the base-flow environment are discussed.

Li, C. P.↗

NASA NTRS1989

Instability of jets of arbitrary geometry

This paper describes a calculation technique for determining the stability of jets of arbitrary cross section. In particular, elliptic and rectangular jets are considered. The numerical procedure involves both a conformal transformation between the computational domain and the physical plane and a solution of the transformed stability equation in the computational domain. Modern, efficient, conformal mappings are used for both simply and doubly connected domains. The numerical solution is based on a finite difference/pseudospectral discretization of the stability equation. The technique is verified by comparison with previous calculations for circular and elliptic jets. Calculations are performed for the stability of elliptic and rectangular jets of aspect ratio 2. Growth rates, phase velocities, and pressure eigenfunctions are calculated.

Baty, Roy S.↗

NASA NTRS1989

Aerodynamic stability and heating analyses for the aeroassist flight experiment vehicle

Since ground based flow simulations are presently unable to model flight conditions expected for AOTVs (Aeroassist Orbital Transfer Vehicle) and other hypersonic space vehicles, computer codes are being developed to provide design parameters necessary for structure, guidance, and control aspects. Over the past four years, VRFLO (Viscous Reactive Flow) has been written to model finite-rate chemistry and viscous effects for a variety of aerobrake bodies. VRFLO includes a number of unique features that are summarized as follows: (1) Grid generation is an integral part of the code for several aerobrake configurations which includes the wake flow region; (2) The formulation is valid for three air chemical models; (3) An ADI central difference technique is used to solve the Navier-Stokes and species continuity equations in split groups; and (4) Grid density and numerical damping are minimized by shock-fitting and conformal mapping of body points.

Mcgary, J.↗

NASA NTRS1990

Reynolds stress closure in jet flows using wave models

A collection of papers is presented. The outline of this report is as follows. Chapter three contains a description of a weakly nonlinear turbulence model that was developed. An essential part of the application of such a closure scheme to general geometry jets is the solution of the local hydrodynamic stability equation for a given jet cross-section. Chapter four describes the conformal mapping schemes used to map such geometries onto a simple computational domain. Chapter five describes a solution of a stability problem for circular, elliptic, and rectangular geometries. In chapter six linear models for the shock shell structure in non-circular jets is given. The appendices contain reprints of papers also published during this study including the following topics: (1) instability of elliptic jets; (2) a technique for predicting the shock cell structure in non-circular jets using a vortex sheet model; and (3) the resonant interaction between twin supersonic jets.

Morris, Philip J.↗

NASA NTRS1991

Transformation of two and three-dimensional regions by elliptic systems

A reliable linear system is presented for grid generation in 2-D and 3-D. The method is robust in the sense that convergence is guaranteed but is not as reliable as other nonlinear elliptic methods in generating nonfolding grids. The construction of nonfolding grids depends on having reasonable approximations of cell aspect ratios and an appropriate distribution of grid points on the boundary of the region. Some guidelines are included on approximating the aspect ratios, but little help is offered on setting up the boundary grid other than to say that in 2-D the boundary correspondence should be close to that generated by a conformal mapping. It is assumed that the functions which control the grid distribution depend only on the computational variables and not on the physical variables. Whether this is actually the case depends on how the grid is constructed. In a dynamic adaptive procedure where the grid is constructed in the process of solving a fluid flow problem, the grid is usually updated at fixed iteration counts using the current value of the control function. Since the control function is not being updated during the iteration of the grid equations, the grid construction is a linear procedure. However, in the case of a static adaptive procedure where a trial solution is computed and used to construct an adaptive grid, the control functions may be recomputed at every step of the grid iteration.

Mastin, C. Wayne↗

NASA NTRS1991

Airfoil optimization with efficient gradient calculations

The viscous airfoil design analysis code XFOIL was extended to allow optimization using conformal mapping coefficients as design variables. The optimization technique used was the Steepest Descent method applied to a Penalty Function. The gradients of the aerodynamic variables with respect to the design variables were cheaply calculated as by-products of XFOIL's integral boundary layer Newton solver. The speed of the optimization process further increased by updating the Newton system boundary layer variables after each optimization step using the available gradient information. Two examples are presented.

Sorensen, Thomas↗

NASA NTRS1992

Two-fence concept for efficient trapping of vortices on airfoils

Previous work on the use of a vortex trapped above a wing in order to produce high lift at low angles of attack is extended here. It is first postulated that the optimum way to trap a vortex is to design the airfoil section and wing so that the flow along the vortex core is minimized. It is then shown that a vertical fence both in front of and behind the separation bubble generated by the trapped vortex is an effective way to reduce the mass flow removal and its associated drag to a negligible amount. In order to show that vertical surfaces upstream and downstream of the vortex separation bubble have an opposite effect on the source requirements for vortex trapping, conformal mapping methods are used to obtain the solutions for a variety of simple two-dimensional, inviscid, incompressible flow configurations. Trapped-vortex flowfield solutions for the flow over flat plate and Clark-Y airfoils are then used to demonstrate that the heights of the fences can be tailored to make the required mass withdrawal (and therefore, the drag due to trapping) to be vanishingly small.

Rossow, Vernon J.↗

NASA NTRS1991

Krylov Subspace Methods for Complex Non-Hermitian Linear Systems

We consider Krylov subspace methods for the solution of large sparse linear systems Ax = b with complex non-Hermitian coefficient matrices. Such linear systems arise in important applications, such as inverse scattering, numerical solution of time-dependent Schrodinger equations, underwater acoustics, eddy current computations, numerical computations in quantum chromodynamics, and numerical conformal mapping. Typically, the resulting coefficient matrices A exhibit special structures, such as complex symmetry, or they are shifted Hermitian matrices. In this paper, we first describe a Krylov subspace approach with iterates defined by a quasi-minimal residual property, the QMR method, for solving general complex non-Hermitian linear systems. Then, we study special Krylov subspace methods designed for the two families of complex symmetric respectively shifted Hermitian linear systems. We also include some results concerning the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.

Freund, Roland W.↗

NASA NTRS1992

A composite structured/unstructured-mesh Euler method for complex airfoil shapes

A general two-dimensional Euler zonal method has been developed for computing flows about complex airfoil geometries such as multielement and iced airfoils. The method utilizes a composite structured and unstructured grid generated using conformal mapping and Delaunay triangulation, respectively. The finite-volume Euler method is then modified to couple solutions in the zones with structured and unstructured grids. Solutions about an iced airfoil and a multielement airfoil are given as examples of applications of the scheme.

Hefazi, H.↗