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NASA NTRS1980

A simple numerical orthogonal coordinate generator for fluid dynamic applications

An application of a simple numerical technique which allows for the rapid construction of orthogonal coordinate systems about two dimensional and axisymmetric bodies is presented. The technique is based on a predictor corrector numerical method. It can be used to generate orthogonal meshes which have unequally spaced points in two directions. These orthogonal meshes in their transformed computational plane are, however, equally spaced so that the differencing for the metric coefficients and the fluid dynamic equation terms can be easily determined using equally spaced central finite differences. Solutions to the Navier-Stokes equations for flow over blunt bodies with reverse curvature are presented. The coupling of the time dependent fluid dynamic equations and the coordinate generator worked well with no undesirable effects noted.

Graves, R. A., Jr.↗

NASA NTRS1980

Numerical techniques for solving nonlinear instability problems in smokeless tactical solid rocket motors

The selection of a satisfactory numerical method for calculating the propagation of steep fronted shock life waveforms in a solid rocket motor combustion chamber is discussed. A number of different numerical schemes were evaluated by comparing the results obtained for three problems: the shock tube problems; the linear wave equation, and nonlinear wave propagation in a closed tube. The most promising method--a combination of the Lax-Wendroff, Hybrid and Artificial Compression techniques, was incorporated into an existing nonlinear instability program. The capability of the modified program to treat steep fronted wave instabilities in low smoke tactical motors was verified by solving a number of motor test cases with disturbance amplitudes as high as 80% of the mean pressure.

Baum, J. D.↗

NASA NTRS1981

Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics

A comprehensive overview of the state of the art of well-posedness and stability analysis of difference approximations for initial boundary value problems of the hyperbolic type is presented. The applicability of recent theoretical development to practical calculations for nonlinear gas dynamics is examined. The one dimensional inviscid gas dynamics equations in conservation law form are selected for numerical experiments. The class of implicit schemes developed from linear multistep methods in ordinary differential equations is chosen and the use of linear extrapolation as an explicit or implicit boundary scheme is emphasized. Specification of boundary data in the primitive variables and computation in terms of the conservative variables in the interior is discussed. Some numerical examples for the quasi-one-dimensional nozzle are given.

Yee, H. C.↗

NASA NTRS1981

Numerical simulation of steady supersonic flow

A noniterative, implicit, space-marching, finite-difference algorithm was developed for the steady thin-layer Navier-Stokes equations in conservation-law form. The numerical algorithm is applicable to steady supersonic viscous flow over bodies of arbitrary shape. In addition, the same code can be used to compute supersonic inviscid flow or three-dimensional boundary layers. Computed results from two-dimensional and three-dimensional versions of the numerical algorithm are in good agreement with those obtained from more costly time-marching techniques.

Schiff, L. B.↗

NASA NTRS1981

Global function approach in structural analysis: Basic approach, numerical results

The structural response to a given environment is described by the differential equations of motion of deformable bodies. Analytic solutions of such problems for a reasonably large class of structural configurations are not within the realm of the possible. Consequently, the mathematical problem is recast into a numerical problem for solution on the computer. New technology in the space and energy fields led to a growing demand for accurate analysis which at times cannot be met due to the limits set by available budgets for computer time. In response to this need for more efficient numerical analysis, the possibilities of reducing the number of freedoms in the system through a revival of the global function approach were explored.

Almroth, B. O.↗

NASA NTRS1980

Numerical prediction of vortex cores of the leading and trailing edges of delta wings

The purpose of the present paper is to predict the roll-up of the vortex sheets emanating from the leading- and trailing-edges of delta wings with emphasis on the interaction of vortex cores beyond the trailing edge. The motivation behind the present work is the recent experimental data published by Hummel. The Nonlinear Discrete-Vortex method (NDV-method) is modified and extended to predict the leading- and trailing-vortex cores beyond the trailing edge. The present model alleviates the problems previously encountered in predicting satisfactory pressure distributions. This is accomplished by lumping the free-vortex lines during the iteration procedure. The leading- and trailing-edge cores and their feeding sheets are obtained as parts of the solution. The numerical results show that the NDV-method is successful in confirming the formation of a trailing-edge core with opposite circulation and opposite roll-up to those of the leading-edge core. This work is a breakthrough in the high angle of attack aerodynamics and moreover, it is the first numerical prediction done on this problem

Kandil, O. A.↗

NASA NTRS1980

Numerical lifting line theory applied to drooped leading-edge wings below and above stall

A numerical iterative solution to the classical Prandtl lifting-line theory, suitably modified for poststall behavior, is used to study the aerodynamic characteristics of straight rectangular finite wings with and without leading-edge droop. This study is prompted by the use of such leading-edge modifications to inhibit stall/spins in light general aviation aircraft. The results indicate that lifting-line solutions at high angle of attack can be obtained that agree with experimental data to within 20%, and much closer for many cases. Therefore, such solutions give reasonable preliminary engineering results for both drooped and undrooped wings in the poststall region. However, as predicted by von Karman, the lifting-line solutions are not unique when sectional negative lift slopes are encountered. In addition, the present numerical results always yield symmetrical lift distributions along the span, in contrast to the asymmetrical solutions observed by Schairer in the late 1930's. Finally, a series of parametric tests at low angle of attack indicate that the effect of drooped leading edges on aircraft cruise performance is minimal.

Anderson, J. D., Jr.↗

NASA NTRS1981

A reattaching free shear layer in compressible turbulent flow - A comparison of numerical and experimental results

An investigation of a two-dimensional, free turbulent shear layer reattaching on an inclined surface at Mach 2.92 and at a high Reynolds number is described. The test geometry is specifically designed to isolate the reattachment process of a high-speed separated flow. A numerical solution of the time-dependent, Reynolds-averaged, Navier-Stokes equations for the entire flow field, employing a two-equation eddy viscosity turbulence model, is presented. Detailed comparisons of prediction and experiment are made in the free shear layer, at reattachment, and in the developing boundary layer downstream. These comparisons include mean surface quantities as well as mean and fluctuating flowfield quantities. Although the overall features of this complex flow field are predicted, there are several deficiencies in the numerical solution, particularly in the region downstream of reattachment. Modifications of the turbulence model to correct these deficiencies are discussed.

Horstman, C. C.↗

NASA NTRS1980

Numerical techniques in linear duct acoustics - A status report

A review is presented covering both finite difference and finite element analysis of small amplitude (linear) sound propagation in straight and variable area ducts with flow, as might be found in a typical turbojet engine duct, muffer, or industrial ventilation system. Both 'steady' state and transient theories are discussed. Emphasis is placed on the advantages and limitations associated with the various numerical techniques. Examples of practical problems are given for which the numerical techniques have been applied.

Baumeister, K. J.↗

NASA NTRS1981

A split explicit reformulation of the regional numerical weather prediction model of the Japan Meteorological Agency

The split explicit integration scheme for numerical weather prediction models is employed in a version of the regional numerical weather prediction model of the Japan Meteorological Agency. The finite-difference scheme of the model is designed in the manner proposed by Okamura (1975). The horizontal advection terms in the governing equations are integrated with a time step limited by the wind speed while the terms which describe inertial-gravity oscillations are integrated in a succession of shorter time steps. The physical processes included within the model are precipitation, small-scale convection, surface exchanges of sensible and latent heat, and radiative heating and cooling. An example of a surface pressure forecast over Europe is shown for initial data observed at 0000 GMT 29 December 1979. Quantitative precipitation forecasts over Europe and North America for the 24 h period beginning at 0000 GMT 30 December 1979 are also shown. It is concluded that the model is capable of realistically depicting the evolution of synoptic-scale systems.

Duffy, D. G.↗

NASA NTRS1981

Numerical simulation of axisymmetric turbulent jet flow

The transport equations for the turbulent Reynolds stresses and energy dissipation rate in conjunction with the governing equations for the vorticity and stream function are numerically solved for in the case of an axisymmetric jet into stagnant surroundings. The time dependence of the equations is retained allowing for any transient results to be interpreted in terms of conditionally averaged flow quantities. The general form of the governing differential equations is presented as well as the numerical procedure used in the solution. Comments concerning the inherent limitations in solving the time-dependent set of equations are made and the versatility of the approach examined. Comparisons of the various components of the stationary Reynolds stress tensor are made with experimental results.

Gatski, T. B.↗

NASA NTRS1981

A numerical solution for two-dimensional Fredholm integral equations of the second kind with kernels of the logarithmic potential form

Two dimensional Fredholm integral equations with logarithmic potential kernels are numerically solved. The explicit consequence of these solutions to their true solutions is demonstrated. The results are based on a previous work in which numerical solutions were obtained for Fredholm integral equations of the second kind with continuous kernels.

Gabrielsen, R. E.↗

NASA NTRS1981

Optimum employment of satellite indirect soundings as numerical model input

The characteristics of satellite-derived temperature soundings that would significantly affect their use as input for numerical weather prediction models were examined. Independent evaluations of satellite soundings were emphasized to better define error characteristics. Results of a Nimbus-6 sounding study reveal an underestimation of the strength of synoptic scale troughs and ridges, and associated gradients in isobaric height and temperature fields. The most significant errors occurred near the Earth's surface and the tropopause. Soundings from the TIROS-N and NOAA-6 satellites were also evaluated. Results again showed an underestimation of upper level trough amplitudes leading to weaker thermal gradient depictions in satellite-only fields. These errors show a definite correlation to the synoptic flow patterns. In a satellite-only analysis used to initialize a numerical model forecast, it was found that these synoptically correlated errors were retained in the forecast sequence.

Horn, L. H.↗

NASA NTRS1982

Numerical simulation of one-dimensional heat transfer in composite bodies with phase change

A numerical simulation was developed to investigate the one dimensional heat transfer occurring in a system composed of a layered aircraft blade having an ice deposit on its surface. The finite difference representation of the heat conduction equations was done using the Crank-Nicolson implicit finite difference formulation. The simulation considers uniform or time dependent heat sources, from heaters which can be either point sources or of finite thickness. For the ice water phase change, a numerical method which approximates the latent heat effect by a large heat capacity over a small temperature interval was applied. The simulation describes the temperature profiles within the various layers of the de-icer pad, as well as the movement of the ice water interface. The simulation could also be used to predict the one dimensional temperature profiles in any composite slab having different boundary conditions.

Dewitt, K. J.↗

NASA NTRS1982

A numerical study of drop-on-demand ink jets

Ongoing work related to development and utilization of a numerical model for treating the fluid dynamics of ink jets is discussed. The model embodies the complete nonlinear, time dependent, axi-symmetric equations in finite difference form. The jet nozzle geometry with no-slip boundary conditions and the existence of a contact circle are included. The contact circle is allowed some freedom of movement, but wetting of exterior surfaces is not addressed. The principal objective in current numerical experiments is to determine what pressure history, in conjunction with surface forces, will lead to clean drop formation.

Fromm, J.↗

NASA NTRS1982

A numerical method for integrating the kinetic equations of droplet spectra evolution by condensation/evaporation and by coalescence/breakup processes

An extension of the method of moments is developed for the numerical integration of the kinetic equations of droplet spectra evolution by condensation/evaporation and by coalescence/breakup processes. The number density function n sub k (x,t) in each separate droplet packet between droplet mass grid points (x sub k, x sub k+1) is represented by an expansion in orthogonal polynomials with a given weighting function. In this way droplet number concentrations, liquid water contents and other moments in each droplet packet are conserved and the problem of solving the kinetic equations is replaced by one of solving a set of coupled differential equations for the number density function moments. The method is tested against analytic solutions of the corresponding kinetic equations. Numerical results are obtained for different coalescence/breakup and condensation/evaporation kernels and for different initial droplet spectra. Also droplet mass grid intervals, weighting functions, and time steps are varied.

Emukashvily, I. M.↗

NASA NTRS1981

Numerical prediction of the monsoon depression of 5-7 July 1979

A well defined monsoon depression was used for two assimilation and forecast experiments: (1) using conventional surface and upper air data, (2) using these data plus Monex data. The data sets were assimilated and used with a general circulation model to make numerical predictions. The model, the analysis and assimilation procedure, the differences in the analyses due to different data inputs, and the differences in the numerical predictions are described. The MONEX data have a positive impact, although the differences after 24 hr are not significant. The MONEX assimilation does not agree with manual analysis location of depression center. The 2.5 x 3 deg horizontal resolution of the prediction model is too coarse. The assimilation of geopotential height data derived from satellite soundings generated gravity waves with amplitudes similar to the meteorologically significant features investigated.

Shukla, J.↗

NASA NTRS1981

Applications of estimation theory to numerical weather prediction

Numerical weather prediction (NWP) is an initial value problem for a system of nonlinear partial differential equations in which the initial values are known only incompletely and inaccurately. Data at initial time can be supplemented, however, by observations of the system distributed over a time interval preceding it. Estimation theory was successful in approaching such problems for models governed by systems of ordinary differential equations and of linear PDEs. Estimation-theoretic methods for NWP are developed. A model exhibiting many features of large scale atmospheric flow important in NWP is the one governed by the shallow fluid equations. The estimation problem for a linearized formulation of these equations is studied. A finite difference version of the equations is used as a forecast model to simulate the numerical models used in NWP.

Cohn, S.↗