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NASA NTRS1984

A numerical transport scheme which avoids negative mixing ratios

A new scheme for numerically integrating the transport equation has been developed. This square root method avoids the problem of negative mixing ratios by using the square root of the concentration instead of the concentration itself as an advective time step. Conservation of total mass at every time step is achieved by extending the concept of quadratic conservation to the numerical time integration method. A few time schemes that fulfill the requirement of step-by-step quadratic conservation are described, the simplest of which is the modified Lax-Wendroff method.

Schneider, H.-R.↗

NASA NTRS1984

Numerical formulation for a higher order plane finite dynamic element

The paper describes the development of an eight-node plane rectangular finite dynamic element and presents detailed descriptions of the associated numerical formulation involving the higher order dynamic correction terms pertaining to the related stiffness and inertia matrices. Numerical test results of free vibration analyses are presented in detail for the newly developed eight-node element and also the corresponding four-node element in order to make a clear comparison of the relative efficiencies of the corresponding finite element and dynamic element procedures. Such results indicate a superior pattern of solution convergence of the presently developed dynamic element.

Gupta, K. K.↗

NASA NTRS1984

Numerical analysis of some problems related to the mechanics of pneumatic tires: Finite deformation/rolling contact of a viscoelastic cylinder and finite deformation of cord-reinforced rubber composites

The formulation and numerical analysis of several problems related to the behavior of pneumatic tires are considered. These problems include the general rolling contact problem of a rubber-like viscoelastic cylinder undergoing finite deformations and the finite deformation of cord-reinforced rubber composites. New finite element models are developed for these problems. Numerical results obtained for several representative cases are presented.

Oden, J. T.↗

NASA NTRS1984

Numerical analysis of flow and heat transfer in the VAFB LOX storage Dewar tank

The present report describes numerical simulation of three-dimensional transient distributions of velocity and temperature of liquid oxygen (LOX) in the LOX Dewar tank of Vendenberg Air Force Base (VAFB). The present analyses cover the replenish time period only. Four test cases have been considered. For all four cases, the input boundary conditions are comprised of LOX facility heat loads, drain flow rates, recirculation flow rates and dewar heating. All the quantities are prescribed as functions of time. The first two test cases considered sensitivity of results to the computational grid. In Case 3, system heat load was changed, while in Case 4, a lower LOX level was specified. Cases 1 and 2 showed that the temperatures were not sensitive to the grid refinement. This provided a basic check on the numerical model. Cases 3 and 4 showed that the thermal boundary layer motion near the tank surface becomes more significant at the late time, e.g., 5 1/2 hours from replenish start. Comparison between results of Cases 3 and 4 showed, as expected, that the smaller initial LOX volume given in Case 4, results in higher temperature level. All calculated velocity and temperature distributions were found to be plausible.

Tam, L. T.↗

NASA NTRS1984

Direct numerical simulation of curved turbulent channel flow

Low Reynolds number, mildly curved, turbulent channel flow has been simulated numerically without subgrid scale models. A new spectral numerical method developed for this problem was used, and the computations were performed with 2 million degrees of freedom. A variety of statistical and structural information has been extracted from the computed flow fields. These include mean velocity, turbulence stresses, velocity skewness, and flatness factors, space time correlations and spectra, all the terms in the Reynolds stress balance equations, and contour and vector plots of instantaneous velocity fields. The effects of curvature on this flow were determined by comparing the concave and convex sides of the channel. The observed effects are consistent with experimental observations for mild curvature. The most significant difference in the turbulence statistics between the concave and convex sides was in the Reynolds shear stress. This was accompanied by significant differences in the terms of the Reynolds shear stress balance equations. In addition, it was found that stationary Taylor-Gortler vortices were present and that they had a significant effect on the flow by contributing to the mean Reynolds shear stress, and by affecting the underlying turbulence.

Moser, R. D.↗

NASA NTRS1984

A numerical investigation of planetesimal collision trajectories with a Moon accumulating in Earth orbit

In the scenario of lunar origin in which the Moon is assumed to have accreted most of its mass while in orbit about the Earth, ismals on the accrea knowledge of the relative impact rates of heliocentric planetting Earth and Moon is essential for any attempt to establish dynamical constraints on lunar origin. Numerical integrations of the regularized equations of motion for four bodies (Sun, Earth, Moon, planetismal) were done. A planetismal impact trajectory was calculated by assuming that the planetismal has hit the surface of the Moon at an assumed location, traveling in an assumed direction, and with an assumed impact speed. Next, the equations of motion were numerically integrated backward in time in order to determine from where the planetismal has come. In this way those volumes in heliocentric orbital element space which contribute trajectories that directly impact the Moon.

Cox, L. P.↗

NASA NTRS1984

Numerical Modeling of the Atmosphere

Numerical models and numerical analyses of observations were used to improve the understanding of the physical processes important in global weather and climate. Results from a study of the effect of baroclinic waves on mid-latitude vertical temperature structure show that the waves' vertical eddy heat flux tend to eliminate the potential vorticity gradient near the steering level by causing the static stability to decrease rapidly with height in the lower troposphere above the boundary layer. An empirical study of the relationship between eddy heat fluxes and the meridional temperature gradient has been completed. The results indicate that the feedback in the flux-gradient system is comparable to dissipation on all time scales between the synoptic and the seasonal. The definition of the Eliassen-Palm flux and the Eliassen-Palm and non-acceleration theorems has been generalized to include eddy forcing of condensation. Calculations based on the generalized diagnostics showed that the annual mean eddy forcing of the zonal mean zonal wind is two and one half times stronger when the condensation effects are included.

Stone, P. H.↗

NASA NTRS1984

Numerical Prediction of the President's Day Cyclone

Numerical-diagnostic studies of major anomalous weather events are being conducted in order to investigate the relevant physical processes associated with the events and the role of satellite observing systems in the analysis and prediction of these phenomena. One component of this study is concerned with the numerical prediction of intense coastal and oceanic cyclogenesis. The specific objectives of this research are: (1) to assess the accuracy of Goddard Laboratory for Atmospheric Sciences model predictions of cyclogenesis, (2) to determine the importance of large scale dynamical processes and diabatic heating to the prediction, and (3) to evaluate the sensitivity of the model predictions to the initial conditions. It was shown that diabatic heating resulting from oceanic fluxes significantly contributed to the generation of low level cyclonic vorticity and the intensification and slow rate movement of an upper level ridge over the western Atlantic. As an upper level short-wave trough approached this ridge, diabatic heating associated with the release of latent heat intensified, and the gradient of vorticity, vorticity advection, upper level divergence, and upward vertical motion in advance of the trough were greatly increased, providing strong large-scale forcing for the surface cyclogenesis.

Atlas, R.↗

NASA NTRS1984

Comparison of uniform perturbation solutions and numerical solutions for some potential flows past slender bodies

Approximate solutions for potential flow past an axisymmetric slender body and past a thin airfoil are calculated using a uniform perturbation method and then compared with either the exact analytical solution or the solution obtained using a purely numerical method. The perturbation method is based upon a representation of the disturbance flow as the superposition of singularities distributed entirely within the body, while the numerical (panel) method is based upon a distribution of singularities on the surface of the body. It is found that the perturbation method provides very good results for small values of the slenderness ratio and for small angles of attack. Moreover, for comparable accuracy, the perturbation method is simpler to implement, requires less computer memory, and generally uses less computation time than the panel method. In particular, the uniform perturbation method yields good resolution near the regions of the leading and trailing edges where other methods fail or require special attention.

Wong, T. C.↗

NASA NTRS1984

High altitude chemically reacting gas particle mixtures. Volume 1: A theoretical analysis and development of the numerical solution

The overall contractual effort and the theory and numerical solution for the Reacting and Multi-Phase (RAMP2) computer code are described. The code can be used to model the dominant phenomena which affect the prediction of liquid and solid rocket nozzle and orbital plume flow fields. Fundamental equations for steady flow of reacting gas-particle mixtures, method of characteristics, mesh point construction, and numerical integration of the conservation equations are considered herein.

Smith, S. D.↗

NASA NTRS1985

Numerical studies of interacting vortices

To get a basic understanding of the physics of flowfields modeled by vortex filaments with finite vortical cores, systematic numerical studies of the interactions of two dimensional vortices and pairs of coaxial axisymmetric circular vortex rings were made. Finite difference solutions of the unsteady incompressible Navier-Stokes equations were carried out using vorticity and stream function as primary variables. Special emphasis was placed on the formulation of appropriate boundary conditions necessary for the calculations in a finite computational domain. Numerical results illustrate the interaction of vortex filaments, demonstrate when and how they merge with each other, and establish the region of validity for an asymptotic analysis.

Liu, G. C.↗

NASA NTRS1985

Numerical approach for the aerodynamic analysis if airfoils with laminar separation

A numerical method for simultaneously and efficiently coupling an external subsonic potential flow and an interior viscous flow such that the two flows match at an interfacing boundary is discussed. Both a panel method and a simple point compressible vortex model are used for the outer potential field. The interior flow solvers which were used are the Navier-Stokes and Euler codes of T. J. Coakley and the Euler code of A. Verhoff. In order to test compatibility, the panel method is coupled to the less expensive Euler codes since the coupling procedure is identical with the Navier-Stokes code. The results show significant efficiency improvements can be obtained over the uncoupled approach. Results also indicate the outer potential flow is best represented by the simple point compressible vortex model. The panel method couples smoothly to Coakley's implicit code but is numerically incompatible as coupled with the explicit Euler code. An improved Navier-Stokes code is under initial development which extends the Euler code to include the necessary viscous terms. Results are shown for all infinite length channel with one wavy periodic wall with and without laminar separation.

Halt, D. W.↗

NASA NTRS1985

Numerical simulation of boundary layer excitation by surface heating/cooling

The concept of active control of growing disturbances in an unstable compressible flow by using time periodic, localized surface heating is studied numerically. The simulations are calculated by a fourth-order accurate solution of the compressible, laminar Navier-Stokes equations. Fourth-order accuracy is particularly important for this problem because the solution must be computed over many wavelengths. The numerical results demonstrate the growth of an initially small fluctuation into the nonlinear regime where a local breakdown into smaller scale disturbances can be observed. It is shown that periodic surface heating over a small strip can reduce the level of the fluctuation provided that the phase of the heating current is properly chosen.

Bayliss, A.↗

NASA NTRS1985

Numerical study of porous airfoils in transonic flow

A numerical study was made to examine the effect of a porous surface on the aerodynamic performance of a transonic airfoil. The pressure jump across the normal shock wave on the upper surface of the airfoil was reduced by making the surface below the shock porous. The weakened shock is preceded by an oblique shock at the upstream end of the porous surface where air is blown out of the cavity. The lambda shock structure shown in the numerical result qualitatively agrees with that observed in the wind tunnel. According to the present analysis, the porous airfoil has a smaller drag and a higher lift than the solid airfoil.

Chen, C. L.↗

NASA NTRS1985

The Construction of Curves and Surfaces Using Numerical Optimization Techniques

Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions.

Ferguson, D. R.↗

NASA NTRS1985

Numerical Estimation of the Curvature of Biological Surfaces

Many biological systems may profitably be studied as surface phenomena. A model consisting of isotropic growth of a curved surface from a flat sheet is assumed. With such a model, the Gaussian curvature of the final surface determines whether growth rate of the surface is subharmonic or superharmonic. These properties correspond to notions of convexity and concavity, and thus to local excess growth and local deficiency of growth. In biological models where the major factors controlling surface growth are intrinsic to the surface, researchers thus gained from geometrical study information on the differential growth undergone by the surface. These ideas were applied to an analysis of the folding of the cerebral cortex, a geometrically rather complex surface growth. A numerical surface curvature technique based on an approximation to the Dupin indicatrix of the surface was developed. A metric for comparing curvature estimates is introduced, and considerable numerical testing indicated the reliability of this technique.

Todd, P. H.↗

NASA NTRS1985

On the numerical computation of nonlinear force-free magnetic fields

An algorithm has been developed to extrapolate nonlinear force-free magnetic fields from a source surface, given the proper boundary conditions. The results of this work; describing the mathematical formalism that was developed, the numerical techniques employed, and the stability criteria developed for these numerical schemes are presented. An analytical solution is used for a test case; the results show that the computational accuracy for the case of a nonlinear force-free magnetic field was on the order of a few percent ( 5%).

Wu, S. T.↗

NASA NTRS1985

Utilization of satellite data and regional scale numerical models in short range weather forecasting

Overwhelming evidence was developed in a number of studies of satellite data impact on numerical weather prediction that it is unrealistic to expect satellite temperature soundings to improve detailed regional numerical weather prediction. It is likely that satellite data over the United States would substantially impact mesoscale dynamical predictions if the effort were made to develop a composite moisture analysis system. The horizontal variability of moisture, most clearly depicited in images from satellite water vapor channels, would not be determined from conventional rawinsondes even if that network were increased by a doubling of both the number of sites and the time frequency.

Kreitzberg, C. W.↗