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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

Two inviscid computational simulations of separated flow about airfoils

Two inviscid computational simulations of separated flow about airfoils are described. The basic computational method is the line relaxation finite-difference method. Viscous separation is approximated with inviscid free-streamline separation. The point of separation is specified, and the pressure in the separation region is calculated. In the first simulation, the empiricism of constant pressure in the separation region is employed. This empiricism is easier to implement with the present method than with singularity methods. In the second simulation, acoustic theory is used to determine the pressure in the separation region. The results of both simulations are compared with experiment.

Barnwell, R. W.↗

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 NTRS1975

Analysis and testing of two-dimensional slot nozzle ejectors with variable area mixing sections

Finite difference computer techniques have been used to calculate the detailed performance of air-to-air two-dimensional ejectors with symmetric variable area mixing sections and coaxial converging primary nozzles. The analysis of the primary nozzle assumed correct expansion of the flow and is suitable for subsonic and slightly supersonic velocity levels. The variation of the mixing section channel walls is assumed to be gradual so that the static pressure can be assumed uniform on planes perpendicular to the axis. A test program was run to provide two-dimensional ejector test data for verification of the computer analysis. A primary converging nozzle with a discharge geometry of 3.18 mm x 203 mm was supplied with 0.340 kg/sec of air at about 2.43 bar and 356 K. This nozzle was combined with two mixing section geometries with throat sizes of 31.8 mm x 203 mm and 47.6 mm x 203 mm and was tested at a total of 11 operating points.

Gilbert, G. B.↗

NASA NTRS1976

Investigation of slot cooling at high subsonic speeds

The paper discusses the results of an experimental and numerical investigation of tangential slot injection film cooling with zero pressure gradients in subsonic boundary layers at freestream Mach numbers of 0.4, 0.6, and 0.8. The results are compared with the predictions obtained from a finite-difference boundary-layer program developed by NASA for slot injection into turbulent supersonic boundary layers. The two sets of results are found to compare favorably. The numerical results point to the existence of a unique relation between isothermal effectiveness and adiabatic effectiveness, thereby confirming the existence of similarity conditions for temperature and velocity profiles.

Ferri, A.↗

NASA NTRS1976

Aeroacoustics: Fan noise and control; Duct acoustics; Rotor noise

The recent progress in duct acoustic research is reflected by fourteen papers which cover essentially the areas of linear duct acoustics, the characteristics of absorbent materials, and nonlinear duct acoustics. With respect to propagation in ducts, emphasis is placed on computational methods aimed at attaining effective economical modeling of propagation in variable-geometry and in hard-wall or soft-wall ducts. Computational methods developed to reduce computer processing and storage requirements are outlined, along with a wave envelope technique for reducing processing and storage needs. The latter technique is applied to the optimum segmented acoustic duct linear design. A finite difference technique is used in combination with conformal mapping to compute the optimum multisection duct linear design.

Schwartz, I. R.↗

NASA NTRS1976

A note on the leap-frog scheme in two and three dimensions

The paper considers the leap-frog finite-difference method (Kreiss and Oliger, 1973) for systems of partial differential equations of the form du/dt = dF/dx + dG/dy + dH/dz, where d denotes partial derivative, u is a q-component vector and a function of x, y, z, and t, and the vectors F, G, and H are functions of u only. The original leap-frog algorithm is shown to admit a modification that improves on the stability conditions for two and three dimensions by factors of 2 and 2.8, respectively, thereby permitting larger time steps. The scheme for three dimensions is considered optimal in the sense that it combines simple averaging and large time steps.

Abarbanel, S.↗

NASA NTRS1976

High-latitude truncation errors of box-type primitive equation models

The 'box-type' finite-difference method includes a weighted average of the pressure gradient with weights proportional to the surface of the grid walls. It is shown that this averaging introduces first-order truncation errors near the poles. An example is shown in which the relative error is of zero order and the scheme produces large distortions in the solution at high latitudes.

Kalnay-Rivas, E.↗

NASA NTRS1977

Vector sweep

Program is numerical procedure for calculating radiation/geometrical configuration factors and/or space-environmental heat-flux histories. System can accomodate complex systems of diverse geometric shapes residing in unlimited variety of mutual shadow conditions. Basic analytical device is finite-difference employment of unit-sphere method of calculating configuration factors.

Moutrie, C. L.↗

NASA NTRS1976

Analysis and testing of two-dimensional vented Coanda ejectors with asymmetric variable area mixing sections

The analysis of asymmetric, curved (Coanda) ejector flow has been completed using a finite difference technique and a quasi-orthogonal streamline coordinate system. The boundary layer type jet mixing analysis accounts for the effect of streamline curvature in pressure gradients normal to the streamlines and on eddy viscosities. The analysis assured perfect gases, free of pressure discontinuities and flow separation and treated three compound flows of supersonic and subsonic streams. Flow parameters and ejector performance were measured in a vented Coanda flow geometry for the verification of the computer analysis. A primary converging nozzle with a discharge geometry of 0.003175 m x 0.2032 m was supplied with 0.283 cu m/sec of air at about 241.3 KPa absolute stagnation pressure and 82 C stagnation temperature. One mixing section geometry was used with a 0.127 m constant radius Coanda surface. Eight tests were run at spacing between the Coanda surface and primary nozzle 0.01915 m and 0.318 m and at three angles of Coanda turning: 22.5 deg, 45.0 deg, and 75.0 deg. The wall static pressures, the loci of maximum stagnation pressures, and the stagnation pressure profiles agree well between analytical and experimental results.

Maroti, L. A.↗

NASA NTRS1976

Large deflections of shallow conical membrane

Large deflections of a shallow elastic conical membrane fixed at the outer edge and loaded by either uniform or hydrostatic pressure are investigated. The overning equations were solved by the method of matched asymptotic expansions and by a finite difference method. Agreement between the two methods was excellent for the small values of the perturbation parameter.

Chang, W. H.↗

NASA NTRS1976

Numerical and approximate solution of the high Reynolds number small separation problem

Several possible methods of solving the small separation problem at high Reynolds number are investigated. In addition to using analytical methods, there are several numerical approaches which are used. High Reynolds number laminar two dimensional problems are used for simplicity. A brief discussion is given of the finite difference methods since these methods are discussed in detail. Most of the emphasis is placed on developing an approximate integral method. As a model problem the supersonic compression ramp problem is chosen since several numerical solutions along with experimental data are available. The techniques discussed are modified and applied to other similar type wall geometries.

Davis, R. T.↗

NASA NTRS1976

The relative merits of several numerical techniques for solving the compressible Navier-Stokes equations

Four explicit finite difference techniques designed to solve the time-dependent, compressible Navier Stokes equations are compared. These techniques are: (1) MacCormack, (2) modified Du Fort-Frankel, (3) modified hopscotch, and (4) Brailovskaya. The comparison was made numerically by solving the quasi-one dimensional Navier Stokes equations for the flow in a converging-diverging nozzle. Solutions with and without standing normal shock waves were computed for unit Reynolds numbers (based on total conditions) ranging from 45374 to 2269. The results indicate that all four techniques are comparable in accuracy; however, the modified hopscotch scheme is two to three times faster than the Brailovskaya and MacCormack schemes and three to six times faster than the modified Du Fort-Frankel scheme.

Holst, T. L.↗

NASA NTRS1976

Calculation of transonic flows using an extended integral equation method

An extended integral equation method for transonic flows is developed. In the extended integral equation method velocities in the flow field are calculated in addition to values on the aerofoil surface, in contrast with the less accurate 'standard' integral equation method in which only surface velocities are calculated. The results obtained for aerofoils in subcritical flow and in supercritical flow when shock waves are present compare satisfactorily with the results of recent finite difference methods.

Nixon, D.↗

NASA NTRS1976

Equations of state and impact-induced shock-wave attenuation on the moon

Current equation of state formulations, used for finite difference cratering flow calculations, are cast into a framework permitting comparison of peak pressures attained upon impact of a sphere, with a half-space, along the impact symmetry axis, to one dimensional impedance match solutions. On the basis of this formulation and application of thermochemical data, the regimes of melting and vaporization are examined. For the purpose of identifying material which will, upon isentropic release from the impact-induced shock state, result in a solid just brought to its melting point, i.e., incipiently melted (IM); completely melted (CM); just brought to its boiling point, i.e., incipiently vaporized (IV); and completely vaporized (CV) state, the pressures at which the critical isentropes intersect the Hugoniots of iron and gabbroic anorthosite (GA) are examined in detail. The latter rock type is assumed to be representative of the lunar highlands. The Hugoniot pressures, for which IM, CM, IV, and CV will occur upon isentropic expansion, are calculated to be 2.2, 2.6, 4.2, and 16.8 Mbar, respectively.

Ahrens, T. J.↗

NASA NTRS1976

Diffusion analysis for two-phase metal-matrix composite

Diffusion controlled filament matrix interaction in a metal matrix composite, where the filaments and matrix comprise a two phase binary alloy system, was mathematically modeled. The problem of a diffusion controlled, two phase moving interface by means of a one dimensional, variable grid, finite difference technique was analyzed. Concentration dependent diffusion coefficients and equilibrium solubility limits were used, and the change in filament diameter and compositional changes in the matrix were calculated as a function of exposure time at elevated temperatures. With the tungsten nickel (W-Ni) system as a model composite system, unidirectional composites containing from 0.06 to 0.44 initial filament volume fraction were modeled. Compositional changes in the matrix were calculated by superposition of the contributions from neighboring filaments. Alternate methods for determining compositional changes between first and second nearest neighbor filaments were also considered. The results show the relative importance of filament volume fraction, filament diameter, exposure temperature, and exposure time as they affect the rate and extent of filament matrix interaction.

Tenney, D. R.↗

NASA NTRS1976

Experiments with the fourth order GISS general circulation model

The new GISS GCM with 4th order conservative horizontal finite difference is briefly described. A simple forecast performed with real initial data indicates a significant increase in skill with respect to standard 2nd order GISS GCM. The improvement is limited by other sources of errors, especially poor initial data and vertical truncation. It is estimated that with a 4th order - 200 km grid model, errors introduced by horizontal truncation will be negligible for about a week for most synoptic scales (wavelengths larger than 1000 km).

Kalnay-Rivas, E.↗

NASA NTRS1977

Numerical solutions of Navier-Stokes equations for the structure of a trailing vortex

The structure and decay of a trailing vortex were analyzed during the numerical solutions of the full Navier-Stokes equations. Unsteady forms of the governing equations were recast in terms of circulation, vorticity, and stream function as dependent variables, and a second upwind finite difference scheme was used to integrate them with prescribed initial and boundary conditions. The boundary conditions at the outer edge and at the outflow section of the trailing vortex were considered. Different models of the flow were postulated, and solutions were obtained describing the development of the flow as integration proceeds in time. A parametric study was undertaken with a view to understanding the various phenomena that may possibly occur in the trailing vortex. Using the Hoffman and Joubert law of circulation at the inflow section, the results of this investigation were compared with experimental data for a Convair 990 wind model and a rectangular wing. With an exponentially decaying law of circulation at the inflow section and an adverse pressure gradient at the outer edge of the trailing vortex, solutions depict vortex bursting through the sudden expansion of the core and/or through the stagnation and consequent reversal of the flow on the axis. It was found that this bursting takes place at lower values of the swirl ratio as the Reynolds number increases.

Jain, A. C.↗