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NASA NTRS1976

Analysis of the radiation patterns of reflector antennas

The development and application of a numerical technique for rapid calculation of the far-field radiation patterns of a reflector antenna from either a measured or computed feed pattern are reported. The reflector is defined by the intersection of a cone with any surface of revolution or an offset sector of any surface of revolution. The feed is assumed to be linearly polarized and can have an arbitrary location. Both the copolarized and the cross-polarized reflector radiation patterns are computed. Calculations using the technique compare closely with measured radiation patterns of a waveguide-fed offset parabolic reflector. Unique features of this technique are freedom from restrictive feed assumptions and the numerical methods used in preparing the aperture-plane electric-field data for integration.

Kauffman, J. F.↗

NASA NTRS1976

Numerical computation of transonic flows with shock waves

Some recent developments in numerical methods for calculating solutions to the transonic potential flow equation are reviewed, including (1) the construction of the stable coordinate independent difference schemes; (2) the use of conservation form to insure proper shock jump conditions; (3) analysis of the relaxation method by the time dependent analogy; (4) accelerated iterative schemes.

Jameson, A.↗

NASA NTRS1976

Heat Transfer and Fluid Mechanics Institute, Meeting, 25th, University of California, Davis, Calif., June 21-23, 1976, Proceedings

Major research advances in heat transfer and fluid dynamics are outlined, with particular reference to relevant energy problems. Of significant importance are such topics as synthetic fuels in combustion, turbulence models, combustion modeling, numerical methods for interacting boundary layers, and light-scattering diagnostics for gases. The discussion covers thermal convection, two-phase flow and boiling heat transfer, turbulent flows, combustion, and aerospace heat transfer problems. Other areas discussed include compressible flows, fluid mechanics and drag, and heat exchangers. Featured topics comprise heat and salt transfer in double-diffusive systems, limits of boiling heat transfer in a liquid-filled enclosure, investigation of buoyancy-induced flow stratification in a cylindrical plenum, and digital algorithms for dynamic analysis of a heat exchanger. Individual items are announced in this issue.

Mckillop, A. A.↗

NASA NTRS1976

Gradient optimization and nonlinear control

The book represents an introduction to computation in control by an iterative, gradient, numerical method, where linearity is not assumed. The general language and approach used are those of elementary functional analysis. The particular gradient method that is emphasized and used is conjugate gradient descent, a well known method exhibiting quadratic convergence while requiring very little more computation than simple steepest descent. Constraints are not dealt with directly, but rather the approach is to introduce them as penalty terms in the criterion. General conjugate gradient descent methods are developed and applied to problems in control.

Hasdorff, L.↗

NASA NTRS1976

A numerical solution of the flow field over a transonic airfoil including strong-shock-induced flow separation

An approximate numerical method has been developed for computing the entire transonic flow field over an airfoil, including the boundary layer, the shock-wave/boundary-layer interaction, and the shock-induced-separation. The inviscid flow, the turbulent boundary layer in the weak interaction zone, and the strong interaction elliptic region are solved separately. The inviscid flow is represented by the solution of the transonic small-disturbance equation, while the boundary layer is solved by the generalized Galerkin method, and the strong interaction region (including the wake) is computed from the time-averaged Navier-Stokes equations. A second-order interaction is computed iteratively using the inviscid pressure distribution as a boundary condition for the boundary layer and Navier-Stokes codes, and the viscous displacement thickness to modify the airfoil geometry for the inviscid computation. The computed pressure distributions and shock-wave profiles are compared with experimental data. Agreement is very good except that the pressure recovery on the trailing edge is greater than the experimental data.

Seginer, A.↗

NASA NTRS1976

Numerical solution of periodic transonic flow through a fan stage

A numerical method of solution of the inviscid, compressible, two-dimensional unsteady flow on a blade-to-blade stream surface through a stage (rotor and stator), or a single blade row, of an axial flow compressor or fan is described. A cyclic procedure has been developed for representation of adjacent blade-to-blade passages, which asymptotically achieves the correct phase between all passages of a stage. A shock-capturing finite-difference method is employed in the interior of the passage, and a method-of-characteristics technique is used at the boundaries. The blade slipstreams form two of the passage boundaries, and are treated as moving contact surfaces capable of supporting jumps in entropy and tangential velocity. The Kutta condition is imposed by requiring the slipstreams to originate at the trailing edges, which are assumed to be sharp. Results are presented for several transonic fan rotors, and compared with available experimental data consisting of holographic observations of shock structure and pressure contour maps. A subcritical stator solution is also compared with results from a relaxation method. Finally, a periodic solution for a stage consisting of 44 rotor blades and 46 stator blades is discussed.

Erdos, J. I.↗

NASA NTRS1976

Numerical solution of nonlinear partial differential equations of mixed type

A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

Jameson, A.↗

NASA NTRS1976

A two-dimensional finite difference solution for the transient thermal behavior of tubular solar collector

A numerical procedure was established using the finite-difference technique in the determination of the time-varying temperature distribution of a tubular solar collector under changing solar radiancy and ambient temperature. Three types of spatial discretization processes were considered and compared for their accuracy of computations and for selection of the shortest computer time and cost. The stability criteria of this technique was analyzed in detail to give the critical time increment to ensure stable computations. The results of the numerical analysis were in good agreement with the analytical solution previously reported. The numerical method proved to be a powerful tool in the investigation of the collector sensitivity to two different flow patterns and several flow control mechanisms.

Lansing, F. L.↗

NASA NTRS1976

Macroscopic aspects of interfacial reactions

The extent of interdiffusion and formation of new phases is determined by the constitution diagram of the alloy system, the interdiffusion coefficients of the phases present, and the thermal conditions (temperature and time) associated with the bonding process and/or subsequent use of the bonded structure. In many instance, the kinetics of interdiffusion and phase formation can be predicted from known parameters using numerical methods and computer techniques. Predictions are compared with experimentally determined parameters for a variety of metallurgical alloy systems.

Heckel, R. W.↗

NASA NTRS1976

Numerical solution of supersonic three-dimensional free-mixing flows using the parabolic-elliptic Navier-Stokes equations

A numerical method is presented for solving the parabolic-elliptic Navier-Stokes equations. The solution procedure is applied to three-dimensional supersonic laminar jet flow issuing parallel with a supersonic free stream. A coordinate transformation is introduced which maps the boundaries at infinity into a finite computational domain in order to eliminate difficulties associated with the imposition of free-stream boundary conditions. Results are presented for an approximate circular jet, a square jet, varying aspect ratio rectangular jets, and interacting square jets. The solution behavior varies from axisymmetric to nearly two-dimensional in character. For cases where comparisons of the present results with those obtained from shear layer calculations could be made, agreement was good.

Hirsh, R. S.↗

NASA NTRS1976

Singularity computations

An approach is described for singularity computations based on a numerical method for elastoplastic flow to delineate radial and angular distribution of field quantities and measure the intensity of the singularity. The method is applicable to problems in solid mechanics and lends itself to certain types of heat flow and fluid motion studies. Its use is not limited to linear, elastic, small strain, or two-dimensional situations.

Swedlow, J. L.↗

NASA NTRS1977

Separating and turbulent boundary layer calculations using polynomial interpretation

Higher order numerical methods derived from polynomial spline interpolation or Hermitian differencing are applied to a separating laminar boundary layer, i.e., the Howarth problem, and the turbulent flat plate boundary layer flow. Preliminary results are presented. It is found that accuracy equal to that of conventional second order accurate finite difference methods is achieved with many fewer mesh points and with reduced computer storage and time requirements.

Rubin, S. G.↗

NASA NTRS1977

Numerical calculation of the transonic flow past a swept wing

A numerical method is presented for analyzing the transonic potential flow past a lifting, swept wing. A finite difference approximation to the full potential equation is solved in a coordinate system which is nearly conformally mapped from the physical space in planes parallel to the symmetry plane, and reduces the wing surface to a portion of one boundary of the computational grid. A coordinate invariant, rotated difference scheme is used, and the difference equations are solved by relaxation. The method is capable of treating wings of arbitrary planform and dihedral, although approximations in treating the tips and vortex sheet make its accuracy suspect for wings of small aspect ratio. Comparisons of calculated results with experimental data are shown for examples of both conventional and supercritical transport wings. Agreement is good for both types, but it was found necessary to account for the displacement effect of the boundary layer for the supercritical wing, presumably because of its greater sensitivity to changes in effective geometry.

Jameson, A.↗

NASA NTRS1977

Surface heat flux determination: An analytical and experimental study using a single embedded thermocouple

A numerical method by which data from a single embedded thermocouple can be used to predict the transient thermal environment for both high- and low-conductivity materials is described. The results of an investigation performed to verify the method clearly demonstrate that accurate, transient, surface heating conditions can be obtained from a thermocouple l.016 centimeters from the heating surface in a low-conductivity material. Space shuttle orbiter thermal protection system materials having temperature- and pressure-dependent properties, and typical orbiter entry heating conditions were used to verify the accuracy of the analytical procedure. Analytically generated, as well as experimental, data were used to compare predicted and measured surface temperatures.

Williams, S. D.↗

NASA NTRS1976

Solution of the comoving-frame equation of transfer in spherically symmetric flows. IV - Frequency-dependent source functions for scattering by atoms and electrons

A numerical method is presented of solving the radiative transfer equation in the comoving frame of a spherically symmetric expanding atmosphere in which both the line and the electron-scattering source function can depend on frequency (i.e., when there is partial frequency redistribution in the scattering process). This method is used to assess the adequacy of various assumptions regarding frequency redistribution in the comoving frame and to discuss the effects of electron scattering more accurately than previously possible. The methods developed here can be used in realistic model atmospheres to account for the (major) effects of electron scattering upon emergent flux profiles.

Mihalas, D.↗

NASA NTRS1977

Effects of inviscid parallel shear flows on steady and unsteady aerodynamics and flutter

A general theory of planar disturbances in inviscid parallel shear flows, analogous to thin-wing theory in potential flows, has been developed. Integral relations between surface pressure and deformation are obtained which are similar to, and can be solved by the same numerical methods as, those of potential flow. Computed results are shown which illustrate the effects of a model turbulent boundary layer on various lifting and nonlifting surfaces, including an elastic panel in low supersonic flow and an airfoil control surface in subsonic flow.

Williams, M. H.↗

NASA NTRS1976

Turbulence and numerical wind tunnels

The feasibility and possible significance of aerodynamic simulations are considered. Despite certain limitations, there is reason to believe that many useful and important flows containing turbulence can be computed with an accuracy acceptable for use in aircraft designs. Attention is given to the equations for unsteady fluid flow, the direct calculation of unsteady viscous flows, the direct calculation of nonlinear equations, the computation of flows about bodies using turbulence modeling, future developments in numerical methods, and future developments in computers.

Lomax, H.↗

NASA NTRS1977

The resolvent of singular integral equations

The investigation reported is concerned with the construction of the resolvent for any given kernel function. In problems with ill-behaved inhomogeneous terms as, for instance, in the aerodynamic problem of flow over a flapped airfoil, direct numerical methods become very difficult. A description is presented of a solution method by resolvent which can be employed in such problems.

Williams, M. H.↗