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NASA NTRS1981

Test problems for inviscid transonic flow

The paper discusses some results obtained in the solving of test problems for inviscid transonic flow with shock waves using the TRANDES program. The method used employs the full inviscid perturbation-potential flow equation in a Cartesian grid system that is stretched to infinity. The equation is represented by a nonconservative system of finite-difference equations that includes at supersonic points a rotated difference scheme and is solved by column relaxation. Except for the NACA 0012 case with the freestream Mach number equal to 0.95, all test problems were solved straightforwardly and appeared to be converged or close to convergence. The only difficulty was some sensitivity to grid placement, which is typical of the Cartesian formulation

Carlson, L. A.↗

NASA NTRS1982

Three-dimensional inviscid analysis of the scramjet inlet flow field

A computer code has been developed to analyze the inviscid flow field in a supersonic combustion ramjet (scramjet) inlet. The code uses the three-dimensional Euler equations in full conservation form to describe the inlet flow. An algebraic numerical coordinate transformation is used to generate a set of boundary-fitted curvilinear coordinates. The governing equations are solved by a time-asymptotic, unsplit, two-step, finite-difference method. This method is highly efficient on the vector processing computers for which the current code is written. Detailed results are presented for two scramjet inlet configurations over a range of Mach numbers. The calculated results are compared with the available experimental and theoretical results.

Kumar, A.↗

NASA NTRS1982

Computation of blast wave-obstacle interactions

Numerical simulations of the interaction of a planar blast wave with various obstacles are presented. These obstacles are either ground structures or vehicles flying in the atmosphere. For a structure on the ground, the blast wave encounter is side-on, while for the flying vehicles the encounter is either head-on or oblique. Second-order accurate, finite-difference, and shock-capturing procedures are employed to solve the two-dimensional, axisymmetric, and three-dimensional unsteady Euler equations. Results are presented for the flow field consisting of blast wave striking obstacles that are at rest, moving subsonically and moving supersonically. Comparison of the numerical results with experimental data for a configuration at rest substantiates the validity of this approach and its potential as a flow analysis tool.

Champney, J. M.↗

NASA NTRS1982

Computations of transonic flow over an oscillating airfoil with shock-induced separation

A numerical study is presented on unsteady transonic flow over an airfoil in pitching oscillation at a moderately high incidence. The study was focused on the effects of unsteady shock-wave/boundary-layer interactions and subsequent flow separations. An implicit finite-difference scheme was used to solve the full time-dependent compressible Navier-Stokes equations in conservation-law form. The algorithm adopted for this study is the Beam-Warming spatially factored scheme. It is, however, constructed in a body-fitted curvilinear coordinate system for efficient unsteady flow computations. The computational results are compared with experimental data measured in the Ames 11- by 11-Foot Transonic Wind Tunnel.

Chyu, W. J.↗

NASA NTRS1982

Influence of land-surface evapotranspiration on the earth's climate

Land-surface evapotranspiration is shown to strongly influence global fields of rainfall, temperature and motion by calculations using a numerical model of the atmosphere, confirming the general belief in the importance of evapotranspiration-producing surface vegetation for the earth's climate. The current version of the Goddard Laboratory atmospheric general circulation model is used in the present experiment, in which conservation equations for mass, momentum, moisture and energy are expressed in finite-difference form for a spherical grid to calculate (1) surface pressure field evolution, and (2) the wind, temperature, and water vapor fields at nine levels between the surface and a 20 km height.

Shukla, J.↗

NASA NTRS1982

A numerical study of the pitch-angle scattering of cosmic rays

The results are presented of a careful study of finite-difference solutions to the problem of cosmic-ray transport, including pitch-angle scattering. In contrast to some recent studies, the diffusion approximation is confirmed in cases where the scattering mean free path is small compared with other length scales. The reasons for the discrepancy with the conclusions of Gombosi and Owens are discussed.

Kota, J.↗

NASA NTRS1982

A numerical solution of the supersonic flow over a rearward facing step with transverse non-reacting hydrogen injection

The present investigation represents an application of computational fluid dynamics to a problem associated with the flow in the combustor region of a supersonic combustion ramjet engine (scramjet). The governing equations are considered, taking into account the Navier-Stokes equations, a molecular viscosity calculation, the molecular thermal conductivity, molecular diffusion, and a turbulence model. The employed numerical solution is patterned after the explicit, time-dependent, unsplit, predictor-corrector, finite-difference method given by MacCormack (1969). The calculation is concerned with the supersonic flow over a rearward-facing step with transverse H2 injection at conditions germane to the combustor region of a scramjet engine. The H2 jet acts as an effective body which essentially shields the primary flow from the rearward-facing step, thus substantially changing the wave pattern in the primary flow.

Berman, H. A.↗

NASA NTRS1982

A conservative finite difference algorithm for the unsteady transonic potential equation in generalized coordinates

An implicit, approximate-factorization, finite-difference algorithm has been developed for the computation of unsteady, inviscid transonic flows in two and three dimensions. The computer program solves the full-potential equation in generalized coordinates in conservation-law form in order to properly capture shock-wave position and speed. A body-fitted coordinate system is employed for the simple and accurate treatment of boundary conditions on the body surface. The time-accurate algorithm is modified to a conventional ADI relaxation scheme for steady-state computations. Results from two- and three-dimensional steady and two-dimensional unsteady calculations are compared with existing methods.

Bridgeman, J. O.↗

NASA NTRS1982

Application of two-point difference schemes to the conservative Euler equations for one-dimensional flows

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 with dissipation added at supersonic points via the retarded density concept. Application of the method to the one-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. It is shown that the scheme offers certain advantages over the more widely-used three-point schemes, especially in regard to application of boundary conditions.

Wornom, S. F.↗

NASA NTRS1982

Incompressible Navier-Stokes and parabolized Navier-Stokes solution procedures and computational techniques

Recent developments with finite-difference techniques are emphasized. The quotation marks reflect the fact that any finite discretization procedure can be included in this category. Many so-called finite element collocation and galerkin methods can be reproduced by appropriate forms of the differential equations and discretization formulas. Many of the difficulties encountered in early Navier-Stokes calculations were inherent not only in the choice of the different equations (accuracy), but also in the method of solution or choice of algorithm (convergence and stability, in the manner in which the dependent variables or discretized equations are related (coupling), in the manner that boundary conditions are applied, in the manner that the coordinate mesh is specified (grid generation), and finally, in recognizing that for many high Reynolds number flows not all contributions to the Navier-Stokes equations are necessarily of equal importance (parabolization, preferred direction, pressure interaction, asymptotic and mathematical character). It is these elements that are reviewed. Several Navier-Stokes and parabolized Navier-Stokes formulations are also presented.

Rubin, S. G.↗

NASA NTRS1981

Use of interactive graphics to analyze QUICK-geometry

The QUICK InterActive Graphics Analysis (QUIAGA) program and its advantages for displaying aircraft QUICK-geometry to aid in detection and analysis of errors are described. The QUICK-geometry system generates a completely analytical aircraft geometry description for use by finite-difference flow codes. The QUIAGA program was developed to exercise the QUICK-geometry subroutines to examine the analytic definition of a configuration by plotting cross sections and body lines on a graphics terminal. A number of options are available, including multiple cross-section views, hidden-line removal, and display of control point locations. Use of these options for the detection and analysis of errors in the QUICK-geometry definition can be of great assistance in speedily arriving at a correct analytical geometry description for flow-field computation. The QUIAGA program has been used in developing a QUICK-geometry model of the NASA Space Shuttle Orbiter, and examples from this experience are given to show some of the program's features. Details of program usage and an example session are given in the appendixes.

Townsend, J. C.↗

NASA NTRS1983

Modeling of Fluctuating Mass Flux in Variable Density Flows

The approach solves for both Reynolds and Favre averaged quantities and calculates the scalar pdf. Turbulent models used to close the governing equations are formulated to account for complex mixing and variable density effects. In addition, turbulent mass diffusivities are not assumed to be in constant proportion to turbulent momentum diffusivities. The governing equations are solved by a combination of finite-difference technique and Monte-Carlo simulation. Some preliminary results on simple variable density shear flows are presented. The differences between these results and those obtained using conventional models are discussed.

So, R. M. C.↗

NASA NTRS1981

Mesh generation by conformal and quasiconformal mappings

It is pointed out that many recent advances in the finite-difference solution of elliptic equations have been limited to regions whose boundary contours coincide with coordinate lines of the Cartesian coordinate system. The reason for this is related to the fact that in the case of an arbitrary curvilinear coordinate system the original equation becomes much more complex. However, there is no added complexity if an orthogonal coordinate system is generated from a conformal mapping. In the present investigation, a finite difference method developed for the construction of conformal mappings has been generalized to construct quasi-conformal mappings. It is expected that the use of more sophisticated numerical algorithms could lead to improvements in both speed and accuracy. Quasi-conformal mappings have applications not only in the solution of elliptic equations but also in other areas such as orthogonal mesh generation on surfaces and the solution of certain fluid flow problems.

Mastin, C. W.↗

NASA NTRS1982

Entrainment and detrainment in a simple cumulus cloud model

A cumulus cloud model, analogous to the mixed-layer models of the planetary boundary layer and the upper ocean, is developed using a single, unitary entrainment process in which the motion of the cloud boundary relative to the mean flow is permitted, produced, and controlled by turbulent processes. An alternate theory to the mixing-length theory of Asai and Kasahara (1967) is proposed which completely removes the strong scale-dependence of the Asai-Kasahara model. The model reintroduces scale-dependence by introducing including the pe5turbation pressure term of the equation of vertical motion. It is shown that the model predicts deeper clouds than the Asai-Kasahara model for a given sounding, due to the entrainment assumption and the effects of the perturbation pressure. Lateral entrainment dominates cloud-top entrainment, although finite-difference errors increase the cloud-top entrainment rate from zero to a positive value in actual situations. The fractional entrainment rate for updrafts is determined to vary only slightly with height and to decrease only slowly as the cloud radius increases, while the fractional detrainment rate for updrafts increases with height.

Randall, D. A.↗

NASA NTRS1982

A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables

The application of solution methods for compact finite-difference schemes to a vorticity-velocity form of the two-dimensional unsteady Navier-Stokes equations is described. An account is also given of numerical experiments for stagnation point and driven cavity flows. One experiment treats the flow impinging on a flat plate, and the numerical results can be compared with the analytical steady state solution (Batchelor, 1967). The other deals with the driven cavity problem, the primary purpose being to describe the time evolution of this classical problem (Donovan, 1970).

Gatski, T. B.↗

NASA NTRS1982

Numerical boundary condition procedures and multigrid methods; Proceedings of the Symposium, NASA Ames Research Center, Moffett Field, CA, October 19-22, 1981

Papers presented in this volume provide an overview of recent work on numerical boundary condition procedures and multigrid methods. The topics discussed include implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows; far field boundary conditions for compressible flows; and influence of boundary approximations and conditions on finite-difference solutions. Papers are also presented on fully implicit shock tracking and on the stability of two-dimensional hyperbolic initial boundary value problems for explicit and implicit schemes.

Source record↗

NASA NTRS1982

Idealized dynamic grid computation of physical systems

The construction and utility of an idealized computational space for finite-difference computation of physical systems are considered. All computations are performed in tau space, which at any instant represents the dynamic grid of a subject physical system appropriately stretched into a uniform, orthogonal grid. A strain field model of physical system is described, including the sample computation of strain density fields. Dynamic grid generation using invariant-mapping is addressed, and the tau computational space is considered, including a grid dynamism constraint, a grid orthogonality constraint, and a grid smoothness constraint. The detailed tau computational space method is presented and illustrated with two sample problems.

Anyiwo, J. C.↗

NASA NTRS1981

Viscous-inviscid interaction method including wake effects for three-dimensional wing-body configurations

A viscous-inviscid interaction method has been developed by using a three-dimensional integral boundary-layer method which produces results in good agreement with a finite-difference method in a fraction of the computer time. The integral method is stable and robust and incorporates a model for computation in a small region of streamwise separation. A locally two-dimensional wake model, accounting for thickness and curvature effects, is also included in the interaction procedure. Computation time spent in converging an interacted result is, many times, only slightly greater than that required to converge an inviscid calculation. Results are shown from the interaction method, run at experimental angle of attack, Reynolds number, and Mach number, on a wing-body test case for which viscous effects are large. Agreement with experiment is good; in particular, the present wake model improves prediction of the spanwise lift distribution and lower surface cove pressure.

Streett, C. L.↗