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NASA NTRS1978

Navier-Stokes calculations for laminar and turbulent hypersonic flow over indented nosetips

A time-accurate finite-difference Navier-Stokes code has been used to calculate the viscous flow over a severely indented blunt body in a supersonic stream. An algebraic turbulence model is used and the results are compared with experimental data from wind-tunnel tests. Qualitative agreement is obtained for the surface pressure distribution and flow-field structure, including the separated bubble in the indented region. However, uncertainties still exist in the heating calculations, which are attributed to the turbulence model. For both laminar and turbulent calculations, the flow exhibits a fundamental unsteady character at a frequency of about 50 kHz.

Rakich, J. V.↗

NASA NTRS1978

Forced oscillations in quadratically damped systems

Bayliss (1975) has studied the question whether in the case of linear differential equations the relationship between the stability of the homogeneous equations and the existence of almost periodic solutions to the inhomogeneous equation is preserved by finite difference approximations. In the current investigation analogous properties are considered for the case in which the damping is quadratic rather than linear. The properties of the considered equation for arbitrary forcing terms are examined and the validity is proved of a theorem concerning the characteristics of the unique solution. By using the Lipschitz continuity of the mapping and the contracting mapping principle, almost periodic solutions can be found for perturbations of the considered equation. Attention is also given to the Lipschitz continuity of the solution operator and the results of numerical tests which have been conducted to test the discussed theory.

Bayliss, A.↗

NASA NTRS1978

Shear flow beneath oceanic plates - Local nonsimilarity boundary layers for olivine rheology

The principle of local similarity, which has been used to model the two-dimensional boundary layers in the oceanic upper mantle, permits calculation of the temperature, velocity, and stress fields with essentially analytic techniques. Finite difference numerical methods are hard pressed to resolve the detail required by the large variation of viscosity between the lithosphere and the asthenosphere. In this paper the local similarity approximation has been justified by quantitatively evaluating the effect of nonsimilarity due to viscous heating, nonlinear temperature- and pressure-dependent rheology, buoyancy, adiabatic cooling, etc. Nonsimilar effects produce only small modifications of the locally similar boundary layers; important geophysical observables such as surface heat flux and ocean floor topography are given to better than 10 percent by the locally similar solution. A posteriori evaluations of the terms neglected in the boundary layer simplification of the complete equations have been conducted on the locally similar temperature and velocity profiles close to the spreading ridge. The boundary layer models are valid to depths of 100 km at 3 m.y. and 10 km at 0.3 m.y.

Yuen, D. A.↗

NASA NTRS1978

Extrapolation methods for dynamic partial differential equations

Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. Computational results are presented confirming the analytic discussions.

Turkel, E.↗

NASA NTRS1978

Predictive capabilities of series solutions for laminar free convection boundary layer heat transfer

Various types of series solutions for predicting laminar, free-convection boundary-layer heat transfer over both isothermal and nonisothermal boundaries are reviewed. The methods include finite difference, Merk series, Blasius series, and Goertler series. Comparative results are presented for heat transfer over an isothermal, horizontal, elliptical cylinder in both slender and blunt configurations.

Lin, F. N.↗

NASA NTRS1978

Numerical solutions and laser-Doppler measurements of spin-up

The spin-up flow in a cylinder of homogeneous fluid has been examined both experimentally and numerically. A series of laser-Doppler measurements was made of the zonal flow over a range of Ekman numbers and Rossby numbers at various locations in the interior of the flow. These measurements exceed previous ones in accuracy. The weak inertial modes excited by the impulsive start are detectable. The numerical simulations used the primitive equations in axisymmetric form and employed finite-difference techniques on both constant and variable grids. The number of grid points necessary to resolve the Ekman layers was determined. A thorough comparison of the simulations and the experimental measurements is made which includes the details of the amplitude and frequency of the inertial modes. Agreement to within the experimental tolerance is achieved. Analytical results for conditions identical to those in the experiments are not available but some similar linear and nonlinear theories are also compared with the experiments.

Warn-Varnas, 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 NTRS1977

Navier-Stokes solution for the laminar and turbulent supersonic flow near the trailing edge of a flat plate

The present paper deals with problems of two-dimensional laminar and turbulent flow of a viscous incompressible fluid near the trailing edge of a thin flat plate. The complete system of Navier-Stokes equations is solved by means of MacCormack's finite-difference scheme. In the laminar case, the results are compared with the triple deck solution of Daniels (1973). The comparisons indicate that the asymptotic triple deck theory for supersonic trailing-edge flow is accurate within five percent for Reynolds numbers greater than 1000. In the turbulent case, the Prandtl-Van Driest-Clausner algebraic eddy viscosity model is used, showing that the region of upstream influence is roughly of the order of the boundary layer thickness.

Hussaini, M. Y.↗

NASA NTRS1977

Impact-induced energy partitioning, melting, and vaporization on terrestrial planets

The reported investigation was conducted in connection with the desirability to obtain information regarding the response of a solidified planetary surface to the primary impact of a meteoroid whose dimensions are small compared to that of the planet. The investigation is an extension of studies conducted by Ahrens and O'Keefe (1977) concerned with a calculation via finite-difference techniques of the entire flow field, the energy partitioning, and the quantity of melt and vapor produced upon impact of an iron object on a gabbroic anorthosite half-space at 15 km/sec. The assumptions used in these calculations are summarized. Calculations based on an equation of state for the low- and high-pressure mineral assemblage of gabbroic anorthosite, initial density 2.94 g/cu cm are considered. An algorithm is formulated for the description of the phase transformation upon shock loading. It is shown that the relative volume of melt and the equivalent volume of shock-induced vapor is proportional to the meteoroid kinetic energy (at high speeds) using the similarity scaling of projectile and target properties as suggested by Dienes and Walsh (1970).

Okeefe, J. D.↗

NASA NTRS1978

An implicit algorithm for the conservative transonic full potential equation using an arbitrary mesh

A new, implicit approximate factorization (AF) algorithm designed to solve the conservative full-potential equation for the transonic flow past arbitrary airfoils has been developed. The new algorithm uses an upwind bias of the density coefficient to provide stability in supersonic regions. This allows the simple two- and three-banded matrix form of the AF scheme to be retained over the entire flow field, even in regions of supersonic flow. A numerical transformation is used to establish an arbitrary body-fitted finite-difference mesh. Airfoil pressure distributions have been computed and are in good agreement with independent results.

Holst, T. L.↗

NASA NTRS1978

Non-linear effects on sound propagation in a variable area duct carrying high subsonic Mach number flows

The general one-dimensional non-linear equation for the acoustic velocity potential in a variable area duct carrying high subsonic Mach number flows is presented and solved numerically using an implicit finite difference scheme. For the linearized equation with no flow, the present scheme is compared with the exact solution and the fourth-order Runge-Kutta method with excellent agreement for dimensionless time periods. Non-linear solutions are more sensitive to Mach numbers and exciting amplitudes and less sensitive to exciting frequencies. In general, non-linear effects can be safely neglected for low Mach number flows but must be accounted for when high Mach number flows are encountered.

Lumsdaine, E.↗

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

Theoretical and experimental study of the drag of multielement airfoils

The viscous/potential flow past single-element and multielement airfoils is studied theoretically and experimentally. A computerized analysis, based on iteratively coupled potential-flow and boundary-layer analysis, is used to predict the flow field of the airfoil. The method yields detailed characteristics of conventional laminar and turbulent boundary layers, turbulent wakes, and confluent boundary layers. The viscous flows are analyzed with a method that uses finite-difference solutions of the boundary-layer equations. Reynolds stress in the boundary layers and wakes is simulated with eddy viscosity models for the various flow zones. The viscous calculations are carried into the wake of the airfoil where the drag is found from the defect in the wake momentum.

Olson, L. E.↗

NASA NTRS1975

Shock effects from a large impact on the moon

The paper calculates the shock and shear deformation-induced internal energy distribution associated with a major basin-forming hypervelocity meteorite impact on the moon. The Hageman and Walsh formulation of the axisymmetric two-dimensional conservation equations in finite difference form is used, and the flow field induced upon impact of an iron meteorite traveling at 15 km/sec with a gabbroic anorthosite lunar crust is calculated for sequential time steps over a grid of hoop-shaped zones fixed in space. If an energy of 5 times 10 to the 32nd power ergs for the projectile energy is required to excavate a basin of Imbrium size, than a flow field of approximately 210 km in radius approximately 19 seconds after impact is indicated. The meteorite residue, the melt, and the ejecta are discussed, and the results are compared with the Gault and Heitowit (1963) formulation.

Okeefe, J. D.↗

NASA NTRS1978

Application of the method of integral relations to boundary layer flows over blunt bodies

Calculations of boundary layer flows past blunt bodies at angles of incidence are presented. Using the method of integral relations together with the method of lines, the full three-dimensional boundary layer equations are reduced to a system of first-order ordinary differential equations. The streamwise shear stress function and the cross-flow velocity component are represented as suitable functions of the streamwise velocity component. The role of the zone of dependence is automatically satisfied by the choice of differencing in the method of lines. Solutions correct to the second order are obtained in the positive shear region for flow over an ellipsoid at 30-deg incidence. The results are compared with corresponding finite difference solutions.

Modarress, D.↗

NASA NTRS1980

Transonic flow past swept wings

FLO-22 computer program aids in numerical analysis of transonic potential flow past lifting, swept wing. FLO-22 uses relaxation method to solve finite-difference approximation of full-potential equation for transonic flow past configuration consisting of wing or arbitrary planform and dihedral extending from symmetry plane or wall. Comparisons of FLO-22 calculated results with experimental data for both conventional and super-critical transport wings show good agreement.

Caughey, D. A.↗

NASA NTRS1978

Navier-Stokes calculations with a coupled strongly implicit method. Part 2: Spline solutions

A coupled strongly implicit method is combined with a deferred-corrector spline solver for the vorticity-stream function form of the Navier-Stokes equation. Solutions for cavity, channel and cylinder flows are obtained with the fourth-order spline 4 procedure. The strongly coupled spline corrector method converges as rapidly as the finite difference calculations and also allows for arbitrary large time increments for the Reynolds numbers considered. In some cases fourth-order smoothing or filtering is required in order to suppress high frequency oscillations.

Rubin, S. G.↗

NASA NTRS1978

Operational manual for two-dimensional transonic code TSFOIL

This code solves the two-dimensional, transonic, small-disturbance equations for flow past lifting airfoils in both free air and various wind-tunnel environments by using a variant of the finite-difference method. A description of the theoretical and numerical basis of the code is provided, together with complete operating instructions and sample cases for the general user. In addition, a programmer's manual is also presented to assist the user interested in modifying the code. Included in the programmer's manual are a dictionary of subroutine variables in common and a detailed description of each subroutine.

Stahara, S. S.↗