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At least 487 records · Page 27

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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

Timing formulas for dissection algorithms on vector computers

The use of the finite element and finite difference methods often leads to the problem of solving large, sparse, positive definite systems of linear equations. MACSYMA plays a major role in the generation of formulas representing the time required for execution of the dissection algorithms. The use of MACSYMA in the generation of those formulas is described.

Poole, W. G., Jr.↗

NASA NTRS1977

Liquid jet impingement normal to a disk in zero gravity

The free surface shapes of circular liquid jets impinging normal to sharp-edged disks in zero gravity are determined. Zero gravity drop tower experiments yielded three distinct flow patterns that were classified in terms of the relative effects of surface tension and inertial forces. An order of magnitude analysis was conducted that indicated regions where viscous forces were not significant in the computation of free surface shapes. The free surface analysis was simplified by transforming the governing potential flow equations and boundary conditions into the inverse plane, where the stream function and velocity potential became the coordinates. The resulting nonlinear equations were solved by standard finite difference methods, and comparisons were made with the experimental data for the inertia dominated regime.

Labus, T. L.↗

NASA NTRS1977

Study of effects of injector geometry on fuel-air mixing and combustion

An implicit finite-difference method has been developed for computing the flow in the near field of a fuel injector as part of a broader study of the effects of fuel injector geometry on fuel-air mixing and combustion. Detailed numerical results have been obtained for cases of laminar and turbulent flow without base injection, corresponding to the supersonic base flow problem. These numerical results indicated that the method is stable and convergent, and that significant savings in computer time can be achieved, compared with explicit methods.

Bangert, L. H.↗

NASA NTRS1977

Numerical solution of the viscous hypersonic flow past blunted cones at angle of attack

Hypersonic viscous flow over spherically blunted cones of large half angle is computed at small angles of attack in the plane of symmetry of the flow field. Time-dependent viscous shock-layer equations in body-oriented coordinates are used to describe the flow field. The shock wave is treated as a discontinuity, across which the Rankine-Hugoniot relations are used to compute the flow conditions behind the shock. A time-marching second-order finite-difference method is used to solve the equations for a perfect gas. The local CFL (Courant-Friedrich-Lewy) time increment is used to advance the solution in time at each grid point. A fourth-order damping is used to damp the oscillations in the flow quantities. The numerical results of the present analysis for quantities such as shock standoff distance, surface-pressure distribution, and heating rates compare well with the existing theoretical and experimental results.

Kumar, A.↗

NASA NTRS1977

Heat transfer in cooled guide vanes

A numerical study to determine the temperature distribution in the guide vanes of a radial inflow turbine is presented. A computer program has been developed to calculate the temperature distribution when the vanes are cooled internally using a combination of impingement and film cooling techniques. The study is based on the use of the finite difference method in a two dimensional heat conduction problem. The results are then compared to determine the best cooling configuration for a certain coolant to primary mass flow ratio.

Tabakoff, W.↗

NASA NTRS1977

Numerical solution of axisymmetric boattail flow fields with plume simulators

Turbulent separating flows over axisymmetric afterbody-boattail configurations with solid sting plume simulators are computed with a time-dependent finite-difference method to solve the compressible Navier-Stokes equations. The Reynolds stress terms are replaced with a two-layer eddy viscosity model including a relaxation formula to model the nonequilibrium effects of the separated flow. The mesh is alined with the boattail body through an analytic transformation which accommodates a wide variety of boattail geometries. Numerical results for a series of boattail geometries over a wide range of Reynolds number (140,000 to 140 million) are presented and, when possible, compared with experimental data or independent numerical results.

Holst, T. L.↗

NASA NTRS1977

An investigation of temperature distribution in cooled guide vanes

A numerical study to determine the temperature distribution in the guide vane blades of a radial inflow turbine is presented. A computer program was developed which permits the temperature distribution to be calculated when the blade is cooled internally using a combination of impingement and film cooling techniques. The study is based on the use of the finite difference method in a two dimensional heat conduction problem. The results are then compared to determine the best cooling configuration for a certain coolant to primary mass flow ratio.

Kotwal, R.↗

NASA NTRS1977

The separated turbulent boundary layer over a wavy wall

A study and application of the fourth order spline collocation procedure, numerical solution of boundary layer like differential equations, is presented. A simple inversion algorithm for the simultaneous solution of the resulting difference equations is given. Particular attention is focused on the boundary condition representation for the spline second derivative approximations. Solutions using the spline procedure, as well as the three point finite difference method, are presented for several model problems in order to assess and improve the spline numerical scheme. Application of the resulting algorithm to the incompressible laminar self similar boundary layer equations is presented.

Polak, A.↗

NASA NTRS1978

A computer program for calculating laminar and turbulent boundary layers for two-dimensional time-dependent flows

A computer program is described which provides solutions of two dimensional equations appropriate to laminar and turbulent boundary layers for boundary conditions with an external flow which fluctuates in magnitude. The program is based on the numerical solution of the governing boundary layer equations by an efficient two point finite difference method. An eddy viscosity formulation was used to model the Reynolds shear stress term. The main features of the method are briefly described and instructions for the computer program with a listing are provided. Sample calculations to demonstrate its usage and capabilities for laminar and turbulent unsteady boundary layers with an external flow which fluctuated in magnitude are presented.

Cebeci, T.↗

NASA NTRS1978

Numerical study of transonic flow over oscillating airfoils using the full potential equation

The behavior of unsteady aerodynamic loadings on airfoils oscillating in transonic flow has been investigated numerically with particular attention given to supercritical airfoil sections. A previously developed finite difference method, which is based on the full potential equation and which uses a quasi-conservative scheme for proper capture of a shock wave motion, was employed for the present study. The unsteady aerodynamic pressure and load distributions on several different airfoil sections are presented with particular emphasis on the effects of free-stream Mach number, reduced frequency, and mean angle of attack. These parameters are demonstrated to have a significant effect on the behavior of the unsteady aerodynamic loadings. Comparisons of the present calculations with the exact inviscid solution and with the experimental results are also presented.

Isogai, K.↗

NASA NTRS1977

An implicit-iterative solution of the heat conduction equation with a radiation boundary condition

For the problem of predicting one-dimensional heat transfer between conducting and radiating mediums by an implicit finite difference method, four different formulations were used to approximate the surface radiation boundary condition while retaining an implicit formulation for the interior temperature nodes. These formulations are an explicit boundary condition, a linearized boundary condition, an iterative boundary condition, and a semi-iterative boundary method. The results of these methods in predicting surface temperature on the space shuttle orbiter thermal protection system model under a variety of heating rates were compared. The iterative technique caused the surface temperature to be bounded at each step. While the linearized and explicit methods were generally more efficient, the iterative and semi-iterative techniques provided a realistic surface temperature response without requiring step size control techniques.

Williams, S. D.↗

NASA NTRS1977

Comparison of numerical and experimental 'conical' flow fields in supersonic corners with compression and/or expansion

The flow field produced by the intersection of two plane solid surfaces in a supersonic stream is a complex interference flow. These flows can be fully compressive, fully expansive, or of mixed compression-expansion nature. This paper presents a comparison of the experimentally obtained flow-field structure in an axial corner with that predicted numerically by using a shock-capturing finite-difference method. The effect of sweep and surface deflection are evaluated, and the general influence of each is presented for the three classes of corner flow. The results show that the numerical method is a valuable aid in understanding the flow structure for simple configurations. In addition, confidence in the numerical method is gained for use in solving more general three-dimensional configurations where the flow is nonconical and several wave interaction may be presented.

Anderson, D. A.↗

NASA NTRS1978

Turbulent viscous shock layer solutions for Jovian entry at small angles of attack

The equations governing the laminar and turbulent flows of reacting gas mixtures in chemical equilibrium over axially symmetric blunt bodies at small angles of attack are developed and presented in the unsteady conservative form. Solutions are obtained in the planes of symmetry of the flow field for the typical conditions encountered by a probe entering the Jovian atmosphere at small angles of attack. The eddy-viscosity is approximated by a two-layer model. The shock wave is treated as a discontinuity across which the shock relations are used to compute the flow conditions behind the shock. A time-asymptotic finite-difference method is used to solve the equations. The zero angle of attack results for a hyperboloid are compared with the existing results but neither experimental nor computational results are available for comparison with the present results at angle of attack.

Jumar, A.↗

NASA NTRS1978

An investigation of heat pipe meniscus heat transfer

The use of grooved evaporator surfaces in heat pipes has increased in popularity in the past few years primarily due to the reproducibility achievable with grooved walls and the relatively low costs of the threading or extrusion processes involved in their production. The present study combines both analyses and experiments on square groove geometries, with special emphasis on overcoming the limitations of earlier analyses with finite-difference methods and groove-fillet hydrodynamic simplifications. The groove fillet, which has in previous analyses been assumed constant in radius of curvature, is permitted to change in thickness and curvature consistent with hydrodynamics and heat loss from the groove. A model is developed for accurate determination of the effect of constriction resistance on groove performance. The grooved-surface tests to be conducted are briefly described which will provide data under closely controlled operation to allow comparison and verification of the analyses.

Saaski, E. W.↗

NASA NTRS1978

Numerical solution of two-dimensional turbulent blunt body flows with an impinging shock

An implicit finite-difference method has been developed to compute two-dimensional, turbulent, blunt body flows with an impinging shock wave. The full time-averaged Navier-Stokes equations are solved with algebraic eddy viscosity and turbulent Prandtl number models employed for shear stress and heat flux. The irregular-shaped bow shock is treated as a discontinuity across which the Rankine-Hugoniot equations are applied. A Type III turbulent shock interference flow field has been computed and the numerical results compare favorably with existing experimental data. In addition, comparisons are made between the present implicit code and a previous explicit code.

Tannehill, J. C.↗

NASA NTRS1978

The Incorporation of Truncated Fourier Series into Finite Difference Approximations of Structural Stability Equations

A new trigonometric approach to the finite difference calculus was applied to the problem of beam buckling as represented by virtual work and equilibrium equations. The trigonometric functions were varied by adjusting a wavelength parameter in the approximating Fourier series. Values of the critical force obtained from the modified approach for beams with a variety of boundary conditions were compared to results using the conventional finite difference method. The trigonometric approach produced significantly more accurate approximations for the critical force than the conventional approach for a relatively wide range in values of the wavelength parameter; and the optimizing value of the wavelength parameter corresponded to the half-wavelength of the buckled mode shape. It was found from a modal analysis that the most accurate solutions are obtained when the approximating function closely represents the actual displacement function and matches the actual boundary conditions.

Hannah, S. R.↗