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NASA NTRS1981

Oblique-incidence secondary emission from charged dielectronics

Experimental measurements and computer simulation of secondary electron emission coefficients for FEP-Teflon, for normal and oblique incidence in the presence of a normal electric field are reported. Knowledge of the electrostatic environment surrounding the specimen calculation of particle trajectories are considered. A simulation using a conformal mapping, a Green's integral, and a trajectory generator provides the necessary mathematical support for the measurements which have been made with normal fields of 1.5 and 2.7 kV/mm. When incidence is normal and energy exceeds the critical energy, the coefficient is given by V.58 and for oblique incidence this expression may be divided by the cosine of the angle. The parameter V sub 0 is a function of normal field. Measurements for values of V sub f are presented.

Robinson, J. W.↗

NASA NTRS1982

An analytical procedure for computing smooth transitions between two specified cross sections with applications to blended wing body configuration

An analytical procedure is described for designing smooth transition surfaces for blended wing-body configurations. Starting from two specified cross section shapes, the procedure generates a gradual transition from one cross section shape to the other as an analytic blend of the two shapes. The method utilizes a conformal mapping, with subsequent translation and scaling, to transform the specified and shapes to curves that can be combined more smoothly. A sample calculation is applied to a blended wing-body missile type configuration with a top mounted inlet.

Barger, R. L.↗

NASA NTRS1982

Calculation of three-dimensional, inviscid supersonic, steady flows

A numerical technique is described for the calculation of three dimensional, inviscid, supersonic, steady flows over wing-body configurations. A high degree of accuracy without increasing the number of computational nodes is obtained by means of a powerful conformal mapping technique. Results are presented for some simple body configurations and for a more complex arrow wing airframe. The numerical results show good agreement with experimental measurements.

Moretti, G.↗

NASA NTRS1982

Fast generation of three-dimensional computational boundary-conforming periodic grids of C-type

A fast computer program, GRID3C, was developed to generate multilevel three dimensional, C type, periodic, boundary conforming grids for the calculation of realistic turbomachinery and propeller flow fields. The technique is based on two analytic functions that conformally map a cascade of semi-infinite slits to a cascade of doubly infinite strips on different Riemann sheets. Up to four consecutively refined three dimensional grids are automatically generated and permanently stored on four different computer tapes. Grid nonorthogonality is introduced by a separate coordinate shearing and stretching performed in each of three coordinate directions. The grids are easily clustered closer to the blade surface, the trailing and leading edges and the hub or shroud regions by changing appropriate input parameters. Hub and duct (or outer free boundary) have different axisymmetric shapes. A vortex sheet of arbitrary thickness emanating smoothly from the blade trailing edge is generated automatically by GRID3C. Blade cross sectional shape, chord length, twist angle, sweep angle, and dihedral angle can vary in an arbitrary smooth fashion in the spanwise direction.

Dulikravich, D. S.↗

NASA NTRS1981

Heat transfer in cooled porous region with curved boundary

Heat transfer characteristics are analyzed for a cooled two-dimensional porous medium having a curved boundary. A general analytical procedure is given in combination with a numerical conformal mapping method used to transform the porous region into an upper half plane. To illustrate the method, results are evaluated for a cosine shaped boundary subjected to uniform external heating. The results show the effects of coolant starvation in the thick regions of the medium, and the extent that internal heat conduction causes the heated surface to have a more uniform temperature.

Siegel, R.↗

NASA NTRS1981

Green's function for cracks of equal length from a circular hole in an infinite sheet

Stress-intensity factors are obtained for twin collinear cracks emanating from a hole in an infinite plate. Point forces that tend to make the crack surfaces separate and slide are applied to one of the cracks. A conformal mapping and the Muskelishvili formulation involving series representation of the Kolosov stress functions are used to derive the solutions. The variation of stress-intensity factor with crack length and with point of application of the loads is presented for both crack tips.

Shivakumar, V.↗

NASA NTRS1982

Cauchy integral method for two-dimensional solidification interface shapes

A method is developed to determine the shape of steady state solidification interfaces formed when liquid above its freezing point circulates over a cold surface. The solidification interface, which is at uniform temperature, will form in a shape such that the non-uniform energy convected to it is locally balanced by conduction into the solid. The interface shape is of interest relative to the crystal structure formed during solidification; regulating the crystal structure has application in casting naturally strengthened metallic composites. The results also pertain to phase-change energy storage devices, where the solidified configuration and overall heat transfer are needed. The analysis uses a conformal mapping technique to relate the desired interface coordinates to the components of the temperature gradient at the interface. These components are unknown because the interface shape is unknown. A Cauchy integral formulation provides a second relation involving the components, and a simultaneous solution yields the interface shape.

Siegel, R.↗

NASA NTRS1984

Radial-Cascade Analysis

Conformal mapping transforms radial cascade to axial cascade. Report describes analysis of pressure distributions on radial diffuser geometries within Space Shuttle main and preburner pumps. Analysis uses modified version of Douglas-Neuman (D-N) procedure for two-dimensional axial cascades.

Meng, S. Y.↗

NASA NTRS1982

Determination of boundary shape of cooled porous region

The shape of a cooled porous wall section is found that will provide a uniform surface temperature, as dictated by material limitations, when the surface is subjected to spatially nonuniform heating. In the analysis, local temperatures and pressures in the porous material are expressed in terms of a potential function. From the imposed thermal conditions, this potential function is governed by the dual constraints of both its value and its normal derivative being specified along the heated surface. The unknown shape of this surface is obtained by meeting these dual conditions. The analytical method uses a generalized conformal mapping procedure that includes a curved boundary. The coolant flow can be compressible or incompressible, and its viscosity can depend on temperature.

Snyder, A.↗

NASA NTRS1983

A viscous-inviscid interactive procedure for rotational flow in cascades of two-dimensional airfoils of arbitrary shape

A viscous-inviscid interactive calculation procedure is developed for application to flow in cascades of two-dimensional airfoils. This procedure has essentially three components. First, a numerical solution of the Euler equations which can accommodate an arbitrarily specified cascade geometry is carried out on a nonorthogonal curvileanear grid mesh that is fitted to the geometry of the cascade. A method of grid generation has been used which relies in part on a succession of conformal mappings. Second, a viscous solution for use in boundary layer and wake regions has been programmed. Finally, an interactive scheme which takes the form of a source-sink distribution along the blade surface and wake centerline is employed. Results have been obtained with this procedure for several cascade flow situations, and some comparisons with experiment are presented.

Johnston, W.↗

NASA NTRS1982

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

A comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given. Some general mathematical framework and error analysis common to such coordinate systems is also included. The general categories of generating systems are those based on conformal mapping, orthogonal systems, nearly orthogonal systems, systems produced as the solution of elliptic and hyperbolic partial differential equations, and systems generated algebraically by interpolation among the boundaries. Also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation. Dynamically adaptive coordinate systems, coupled with the physical solution, and time-dependent systems that follow moving boundaries are treated. References reporting experience using such coordinate systems are reviewed as well as those covering the system development.

Thompson, J. F.↗

NASA NTRS1983

Design of shock-free compressor cascades including viscous boundary layer effects

A computer code that generates shock-free transonic compressor cascade shapes while taking into account viscosity effects is developed. The mathematical model for the inviscid flow field is the full potential equation. The Kutta-Joukowski condition is satisfied by varying the free stream angle at downstream infinity. A boundary fitted computational grid of C-type is generated using a sequence of conformal mapping and nonorthogonal coordinate stretching and shearing transformations. The inviscid calculation is performed sequentially on up to four consecutively refined grids thereby accelerating the convergence of the solution process. The full potential equation is solved using a finite area technique and rotated, type-dependent finite differencing. Artificial viscosity of the first order is added in a fully conservative form. Shock-free cascade airfoil shapes are obtained using the fictitious gas concept of Sobieczky and the method of characteristics in the rheograph plane. Viscosity effects are incorporated via a boundary layer displacement thickness. The integral boundary layer code is based on Rotta's turbulence model and assumes transition region of zero length.

Dulikravich, G. S.↗

NASA NTRS1983

A viscous-inviscid interactive procedure for rotational flow in cascades of two dimensional airfoils of arbitrary shape

A viscous-inviscid interactive calculation procedure is developed for application to flow in cascades of two-dimensional airfoils. This procedure has essentially three components. First, a numerical solution of the Eulers equations which can accommodate an arbitrarily specified cascade geometry of the cascade. A method of grid generation has been used which relics in part on a succession of conformal mappings. Second, a viscous solution for use in boundary layers and wake regions was programmed. Finally, an interactive scheme which takes the form of a source-sink distribution along the blade surface and wake centerline is employed. Results were obtained with this procedure for several cascade flow situations, and some comparisons with experiment are presented.

Johnston, W. A.↗

NASA NTRS1986

Analytical determination of propeller performance degradation due to ice accretion

A computer code has been developed which is capable of computing propeller performance for clean, glaze, or rime iced propeller configurations, thereby providing a mechanism for determining the degree of performance degradation which results from a given icing encounter. The inviscid, incompressible flow field at each specified propeller radial location is first computed using the Theodorsen transformation method of conformal mapping. A droplet trajectory computation then calculates droplet impingement points and airfoil collection efficiency for each radial location, at which point several user-selectable empirical correlations are available for determining the aerodynamic penalities which arise due to the ice accretion. Propeller performance is finally computed using strip analysis for either the clean or iced propeller. In the iced mode, the differential thrust and torque coefficient equations are modified by the drag and lift coefficient increments due to ice to obtain the appropriate iced values. Comparison with available experimental propeller icing data shows good agreement in several cases. The code's capability to properly predict iced thrust coefficient, power coefficient, and propeller efficiency is shown to be dependent on the choice of empirical correlation employed as well as proper specification of radial icing extent.

Miller, T. L.↗

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↗