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NASA NTRS1983

A static model of chromospheric heating in solar flares

The response of the solar chromosphere to flare processes, namely nonthermal electrons, thermal conduction, and coronal pressure, is modeled. Finite difference methods employing linearization and iteration are used in obtaining simultaneous solutions to the equations of steady-state energy balance, hydrostatic equilibrium, radiative transfer, and atomic statistical equilibrium. The atmospheric response is assumed to be confined to one dimension by a strong vertical magnetic field. A solution is obtained to the radiative transfer equation for the most important optically thick transitions of hydrogen, magnesium, and calcium. The theoretical atmospheres discussed here are seen as elucidating the role of various physical processes in establishing the structure of flare chromospheres. At low coronal pressures, conduction is found to be more important than nonthermal electrons in establishing the position of the transition region. Only thermal conduction can adequately account for the chromospheric evaporation in compact flares. Of the mechanisms considered, only nonthermal electrons bring about significant heating below the flare transition region.

Ricchiazzi, P. J.↗

NASA NTRS1983

An improved finite-difference analysis of uncoupled vibrations of tapered cantilever beams

An improved finite difference procedure for determining the natural frequencies and mode shapes of tapered cantilever beams undergoing uncoupled vibrations is presented. Boundary conditions are derived in the form of simple recursive relations involving the second order central differences. Results obtained by using the conventional first order central differences and the present second order central differences are compared, and it is observed that the present second order scheme is more efficient than the conventional approach. An important advantage offered by the present approach is that the results converge to exact values rapidly, and thus the extrapolation of the results is not necessary. Consequently, the basic handicap with the classical finite difference method of solution that requires the Richardson's extrapolation procedure is eliminated. Furthermore, for the cases considered herein, the present approach produces consistent lower bound solutions.

Subrahmanyam, K. B.↗

NASA NTRS1984

Surface grid generation for wing-fuselage bodies

In the application of finite-difference methods to obtain numerical solutions of viscous compressible fluid flow about wing-fuselage bodies, it is advantageous to transform the governing equations to an idealized boundary-fitted coordinate system. The advantages are reduced computational complexity and added accuracy in the application of boundary conditions. The solution process requires that a grid be superimposed on the physical solution domain which corresponds to a uniform grid on a rectangular computational domain (uniform rectangular parallel-epiped). Grid generation is the determination of a one to one relationship between grid points in the physical domain and grid points in the computational domain. A technique for computing wing-fuselage surface grids using the Harris geometry and software for smooth-surface representation is described. Grid spacing control concepts which govern the relationship between the wing-fuselage surface and the computational grid are also presented.

Smith, R. E.↗

NASA NTRS1982

COSAL: A black-box compressible stability analysis code for transition prediction in three-dimensional boundary layers

A fast computer code COSAL for transition prediction in three dimensional boundary layers using compressible stability analysis is described. The compressible stability eigenvalue problem is solved using a finite difference method, and the code is a black box in the sense that no guess of the eigenvalue is required from the user. Several optimization procedures were incorporated into COSAL to calculate integrated growth rates (N factor) for transition correlation for swept and tapered laminar flow control wings using the well known e to the Nth power method. A user's guide to the program is provided.

Malik, M. R.↗

NASA NTRS1984

Large deflections of circular isotropic membranes subjected to arbitrary axisymmetric loading

Circular membranes with fixed peripheral edges, subjected to arbitrary axisymmetric loading are analyzed. A single governing differential equation in terms of radial stress is used. This nonlinear governing equation is solved using the finite difference method in conjunction with Newton-Raphson method. Three loading cases, namely (1) uniformly loaded membrane, (2) a membrane with uniform load over an inner portion, and (3) a membrane with ring load, are analyzed. Calculated central displacement and the central and edge radial stresses for uniformly loaded membrane, agree extremely well with the classical solution.

Kelkar, A.↗

NASA NTRS1984

Computation of viscous flow in planar and axisymmetric ducts by an implicit marching procedure

A streamwise marching procedure, approximately 200 times faster than a full Navier-Stokes procedure with comparable accuracy, is presented for solving problems of compressible viscous subsonic flow. Results are presented and compared with experimental data for the cases of developing turbulent flow in a circular pipe; turbulent flow in a two-dimensional S-duct; and turbulent flow in a typical subsonic diffuser. Prior to each main marching step, a preliminary marching step is taken in which the integral continuity equation and an uncoupled form of the streamwise momentum equation are solved simultaneously to obtain the viscous pressure correction. During the main marching step the equations for continuity, streamwise momentum, cross-flow momentum, and energy are solved simultaneously as a coupled system using an implicit finite-difference method, with the viscous pressure correction treated as a source term. The analysis may be used for flows with both favorable and adverse pressure gradients and to predict the location of flow separation.

Towne, C. E.↗

NASA NTRS1984

Predicted electrothermal deicing of aircraft blades

A finite difference method is presented for the transient two-dimensional simulation of an electrothermal de-icer pad of an aircraft wing or blade. The irregular geometry of the composite ice laden blade is handled by use of a body fitted coordinate transformation. By this approach the various blade layers are mapped into a set of stacked rectangular strips in which the numerical solution takes place. Several heat conduction examples are presented in order to demonstrate the accuracy of the numerical procedure. Ice melting time predictions are made and compared to earlier predictions where possible. Finally, a new graphical presentation of thermal results is shown.

Keith, T. G., Jr.↗

NASA NTRS1984

A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations

An implicit finite-difference method is presented for obtaining steady-state solutions to the time-dependent, conservative Euler equations for flows containing shocks. The method uses a two-point central-difference scheme for the flux derivatives with dissipation added at supersonic points via the retarded density concept. Application of the method to 1-dimensional nozzle flow equations for various combinations of subsonic and supersonic boundary conditions show the method to be very efficient. Residuals are typically reduced to machine zero in approximately 35 time steps for 50 mesh points. For 1-dimensional Euler calculations, it is shown that the scheme offers two advantages over the more widely-used three-point schemes. The first is in regard to application of boundary conditions, and the second relates to the fact that the two-point algorithm is well-conditioned for large time steps.

Wornom, S. F.↗

NASA NTRS1984

Flow simulations for nacelle-propeller configurations using Euler equations

Euler codes for both axisymmetric and general three-dimensional nacelle-propeller flow analysis have been developed. Surface-fitted grids are generated either by a finite difference method or by an algebraic method. The propeller is represented by an actuator disk along a computational plane where proper boundary conditions are assigned to simulate the propeller power loading. Computed results for a NASA SR3 propeller as well as a NASA turboprop configuration are compared with test data. Good agreement has been achieved through the present simulation method.

Yu, N. J.↗

NASA NTRS1984

Development and applications of algorithms for calculating the transonic flow about harmonically oscillating wings

A finite difference method to solve the unsteady transonic flow about harmonically oscillating wings was investigated. The procedure is based on separating the velocity potential into steady and unsteady parts and linearizing the resulting unsteady differential equation for small disturbances. The differential equation for the unsteady velocity potential is linear with spatially varying coefficients and with the time variable eliminated by assuming harmonic motion. An alternating direction implicit procedure was investigated, and a pilot program was developed for both two and three dimensional wings. This program provides a relatively efficient relaxation solution without previously encountered solution instability problems. Pressure distributions for two rectangular wings are calculated. Conjugate gradient techniques were developed for the asymmetric, indefinite problem. The conjugate gradient procedure is evaluated for applications to the unsteady transonic problem. Different equations for the alternating direction procedure are derived using a coordinate transformation for swept and tapered wing planforms. Pressure distributions for swept, untaped wings of vanishing thickness are correlated with linear results for sweep angles up to 45 degrees.

Ehlers, F. E.↗

NASA NTRS1984

Shock capturing finite difference algorithms for supersonic flow past fighter and missile type configurations

The construction of a reliable, shock capturing finite difference method to solve the Euler equations for inviscid, supersonic flow past fighter and missile type configurations is highly desirable. The numerical method must have a firm theoretical foundation and must be robust and efficient. It should be able to treat subsonic pockets in a predominantly supersonic flow. The method must also be easily applicable to the complex topologies of the aerodynamic configuration under consideration. The ongoing approach to this task is described and for steady supersonic flows is presented. This scheme is the basic numerical method. Results of work obtained during previous years are presented.

Osher, S.↗

NASA NTRS1984

A preliminary study of numerical simulation of thermosolutal convection of interest to crystal growth

Calculations were performed with computer models using three types of finite difference methods of thermosolutal convection: horizontal heating of a container filled with a stably stratified solution, finger convection in a container, and finger convection in a horizontally infinite channel. The importance of including thermosolutal convection in models of crystal growth is emphasized, and the difficulties in doing so are demonstrated. It is pointed out that these difficulties, due primarily to the fine structure of the convection, may be partly overcome by the use of fine grids and implicit time stepping methods.

Miller, T. L.↗

NASA NTRS1985

Use of a hyperbolic grid generation scheme in simulating supersonic viscous flow about three-dimensional winged configuration

The present paper describes a numerical mesh generation technique to be used with an implicit finite difference method for simulating visous supersonic flow about low-aspect-ratio wing body configurations using a single grid strategy. The computational domain is segmented into multiple regions, with borders located in supersonic areas to avoid the otherwise costly interfacing procedure between adjacent segments. The numerical procedure is applied to calculate the turbulent flow around the shuttle orbiter and a canard projectile at supersonic free stream Mach number.

Rizk, Y. M.↗

NASA NTRS1984

Airfoil trailing edge flow measurements and comparison with theory, incorporating open wind tunnel corrections

Trailing edge data for boundary layer-near wake thickness parameters are given for airfoils and flat plates. Reynolds number effects are examined as a function of model size, velocity and boundary layer tripping. These data expand that presented previously by the authors particularly for airfoil non-zero angles of attack. Comparisons are made here with boundary layer calculations using potential flow modeling and a well documented two-dimensional finite-difference method for laminar and turbulent boundary layers. Open wind tunnel corrections to angle of attack and camber are developed and are incorporated in the potential flow modeling to assure correct comparisons for non-zero angles of attack. It was found that although the open tunnel flow turbulence affected boundary layer transition for the higher velocities the theory successfully 'brackets' the data. Comparisons demonstrate the degree of accuracy one might expect for the prediction of boundary layer thickness parameters when given only geometry and nominal flow conditions as input to boundary layer codes.

Brooks, T. F.↗

NASA NTRS1985

A study of the effects of numerical dissipation on the calculation of supersonic separated flows

An extensive investigation of the effect of numerical dissipation on the calculation of supersonic, separated flow over a rearward-facing step is carried out. The complete two-dimensional Navier-Stokes equations are solved by means of MacCormack's standard explicit, unsplit, time-dependent, finite difference method. A fourth-order numerical dissipation term is added explicitly. The magnitude of this term is progressively varied, and its consequences on the flowfield calculations are identified and studied. For a cold-wall, heat transfer case, numerical dissipation had a major effect on the results, particularly in the separated region. However, rather dramatically for an adiabatic wall case, numerical dissipation had virtually no effect on the results. The role of grid size on both the influence of numerical dissipation, and on the overall accuracy of the separated flow solutions is discussed.

Kuruvila, G.↗

NASA NTRS1985

The numerical simulation of steady transonic rotational flow using a dual potential formulation

A finite-difference method is presented that simulates steady transonic rotational flow of an inviscid fluid by representing the velocity field as the sum of scalar and vector potentials. This dual potential velocity decomposition extends the validity of the scalar (full) velocity potential to include vorticity. The inclusion of a vector potential also permits an alternate treatment of lift that does not require a circulation wake cut. This is accomplished by specifying the vector potential as a constant on the airfoil surface in order to satisfy a Kutta condition. The governing equations are solved as iteratively decoupled scalar equations using approximate factorization techniques, and the overall efficiency approaches that of the full potential equation. The governing equations are able to convect entropy and vorticity throughout the flow field and are equivalent to the Euler equations in continuous flow domains, however at shocks the Rankine-Hugoniot entropy jump must be supplied. An entropy correction method is presented and verified with transonic airfoil solutions of the Euler equations.

Chaderjian, N. M.↗

NASA NTRS1985

Full two-dimensional transient solutions of electrothermal aircraft blade deicing

Two finite difference methods are presented for the analysis of transient, two-dimensional responses of an electrothermal de-icer pad of an aircraft wing or blade with attached variable ice layer thickness. Both models employ a Crank-Nicholson iterative scheme, and use an enthalpy formulation to handle the phase change in the ice layer. The first technique makes use of a 'staircase' approach, fitting the irregular ice boundary with square computational cells. The second technique uses a body fitted coordinate transform, and maps the exact shape of the irregular boundary into a rectangular body, with uniformally square computational cells. The numerical solution takes place in the transformed plane. Initial results accounting for variable ice layer thickness are presented. Details of planned de-icing tests at NASA-Lewis, which will provide empirical verification for the above two methods, are also presented.

Masiulaniec, K. C.↗

NASA NTRS1985

A fully implicit scheme for the barotropic primitive equations

An efficient implicit finite-difference method is developed and tested for a global barotropic model. The scheme requires, at each time step, the solution of only one-dimensional block-tridiagonal linear systems. This additional computation is offset by the use of a time step chosen independently of the mesh spacing. The method is second-order accurate in time and fourth-order accurate in space. Present experience indicates that this implicit method is practical for numerical simulation on fine meshes.

Cohn, S. E.↗