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NASA NTRS1972

Calculation of turbulent boundary layers with heat transfer and pressure gradient utilizing a compressibility transformation. Part 3: Computer program manual

A computer program is described capable of determining the properties of a compressible turbulent boundary layer with pressure gradient and heat transfer. The program treats the two-dimensional problem assuming perfect gas and Crocco integral energy solution. A compressibility transformation is applied to the equation for the conservation of mass and momentum, which relates this flow to a low speed constant property flow with simultaneous mass transfer and pressure gradient. The resulting system of describing equations consists of eight ordinary differential equations which are solved numerically. For Part 1, see N72-12226; for Part 2, see N72-15264.

Schneider, J.↗

NASA NTRS1976

The oblique impingement of an axisymmetric jet

The mechanics of the oblique impingement of an axisymmetric jet on a plane surface are examined in detail. The stagnation point is discussed. A schematic drawing of the problem and coordinate system used to describe the flow field are given. The kinematic features of the flow above the plate are examined in the context of the conservation of mass, the vorticity of the jet, and the vorticity introduced by the jetplate interaction. The dynamic features of the flow are examined in terms of the surface pressure distribution and the cause-effect relationships which exist between the pressure and velocity/vorticity distributions. Flow calculations performed are given. The investigation is relevant to the flow resulting from the interaction of the propulsion jet with the main airfoil (STOL aircraft), and is appropriate to an over- or under- wing configuration.

Foss, J. F.↗

NASA NTRS1975

Three-dimensional finite element analysis for high velocity impact

A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model.

Chan, S. T. K.↗

NASA NTRS1977

Equations and closure methods

Basic differential equations governing compressible turbulent boundary layer flow are reviewed, including conservation of mass and energy, momentum equations derived from Navier-Stokes equations, and equations of state. Closure procedures were broken down into: (1) simple or zeroth-order methods, (2) first-order or mean field closure methods, and (3) second-order or mean turbulence field methods.

Source record↗

NASA NTRS1980

The equatorial rotation rate of the supergranulation cells

The equatorial rotation rate of the supergranulation cells has been observed to be 14.72 plus or minus 0.07 deg/day. Velocity patterns observed at different times are cross-correlated to derive the rotation rate. The observed rate is 3% faster than recent observations of the surface rotation rate by Doppler shifts. The difference between the cell rate and surface rate is consistent with a model of the supergranular convection in which angular momentum per unit mass is conserved in the radial flow (Foukal, 1977).

Duvall, T. L., Jr.↗

NASA NTRS1980

Molecular clouds and galactic spiral structure

Two large-scale 2.6 mm CO surveys of the galactic plane, one in the first quadrant (l = 12 to 60 deg, b = -1 to +1 deg), the other in the second (l = 105 to 139 deg, b = -3 to +3 deg), have provided evidence that, contrary to previous findings, molecular clouds constitute a highly specific tracer of spiral structure. Molecular counterparts of five of the classical 21-cm spiral arms have been identified: the Perseus arm, the local arm (including Lindblad's local expanding ring), the Sagittarius arm, the Scutum arm, and the 4-kpc arm. The region between the local arm and the Perseus arm is apparently devoid of molecular clouds, and the interarm regions of the inner Galaxy appear largely so. CO spiral structure implies that the mean lifetime of molecular clouds cannot be greater than 100 million years, the time required for interstellar matter to cross a spiral arm. Conservation of mass then sets a limit on the fraction of the interstellar medium in the form of molecular clouds: it cannot exceed one-half at any distance from the galactic center in the range 4-12 kpc.

Cohen, R. S.↗

NASA NTRS1981

Implicit boundary conditions for the solution of the parabolized Navier-Stokes equations for supersonic flows

A fully implicit set of boundary conditions is developed for the solution of the parabolized Navier-Stokes equations for supersonic flow in two dimensions. Shock fitting is employed at the shock and the body has no-slip and specified temperature conditions. A specified heat transfer condition at the wall can be handled in a similar manner. In addition, the shock location is advanced in space in a fully implicit manner by utilizing the Rankine-Hugoniot conditions along with global conservation of mass.

Barnett, M.↗

NASA NTRS1981

On the formation of protostellar disks

An analysis is presented of the hydrodynamic aspects of the growth of protostellar disks from the accretion of a rotating gas cloud. Streamline equations for the flow onto the disk are derived, including the special case where the disk is very thin. For the radial structure of the disk, detailed equations of conservation of mass, momentum, angular momentum, and energy are given. Exact solutions are found for the case of no internal viscosity. For the more general case where there is viscous stress, hypotheses are formulated with the objective of discovering the main qualitative characteristics of a turbulent system and to estimate the magnitude of the turbulent stress. Solutions are also found for boundary conditions, and the large and small viscosity limits. The relation of energy balance to disk thermal structure is detailed. Finally, trends in the collapse of the dust cloud are discussed in the light of the solutions given.

Cassen, P.↗

NASA NTRS1982

Application of a finite element algorithm for the solution of steady transonic Euler equations

A finite element algorithm for the solution of two-dimensional, steady Euler equations is presented which, through a Clebsch-type transformation for the velocity vector, solves the conservation of mass equation with one primary variable and two additional equations for the convection of two new variables. The accuracy and efficiency of this scheme is discussed, and the second-order accuracy attained in the analysis of the convection of the vorticity is demonstrated together with the efficient treatment of rotational and irrotational flow subregions. A sample problem is used to show the accuracy of the numerical scheme and its convergence characteristics.

Akay, H. U.↗

NASA NTRS1982

A numerical experiment on the equilibrium and stability of a rotating galactic bar

A self-consistent, three-dimensional numerical experiment is performed on an N-body system whose initial state is a realization of a certain theoretical model of a rotating triaxial galaxy. The model is a stellar-dynamical counterpart of a uniformly rotating polytrope of index equal to 0.5. The aim of the experiment is to study the equilibrium of the system and, in particular, to test its stability. The experimental system behaves in the mean like a realization of the theoretical model for at least seven crossing times. The principal departure of the system from equilibrium is an oscillation which is identified as a radial pulsation. There is no indication in its behavior that the system is unstable with respect to anu mode with an e-folding time shorter than or of the order of two crossing times. Certain changes that occur in the state of the system are interpreted, with the aid of the theoretical model, as secular changes which result from a slight failure of our numerical methods to conserve the mass, energy, and angular momentum of the system; these effects are small enough that they do not vitiate the experiment on a dynamical time scale.

Miller, R. H.↗

NASA NTRS1983

Analysis of aerothermal loads on spherical dome protuberances

Hypersonic flow over spherical dome protuberances was investigated to determine increased pressure and heating loads to the surface. The configuration was mathematically modeled in a time-dependent three-dimensional analysis of the conservation of mass, momentum (Navier-Stokes), and energy equations. A boundary mapping technique was used to obtain a rectangular parallelepiped computational domain, a MacCormack explicit time-split predictor-corrector finite difference algorithm was used to obtain solutions. Results show local pressures and heating rates for domes one-half, one, and two boundary layer thicknesses high were increased by factors on the order of 1.4, 2, and 6, respectively. Flow over the lower dome was everywhere attached while flow over the intermediate dome had small windward and leeside separations. The higher dome had an unsteady windward separation region and a large leeside separation region. Trailing vortices form on all domes with intensity increasing with dome height. Discussion of applying the results to a thermally bowed thermal protection system are presented.

Olsen, G. C.↗

NASA NTRS1982

Computation techniques and computer programs to analyze Stirling cycle engines using characteristic dynamic energy equations

The basic equations that are used to describe the physical phenomena in a Stirling cycle engine are the general energy equations and equations for the conservation of mass and conversion of momentum. These equations, together with the equation of state, an analytical expression for the gas velocity, and an equation for mesh temperature are used in this computer study of Stirling cycle characteristics. The partial differential equations describing the physical phenomena that occurs in a Stirling cycle engine are of the hyperbolic type. The hyperbolic equations have real characteristic lines. By utilizing appropriate points along these curved lines the partial differential equations can be reduced to ordinary differential equations. These equations are solved numerically using a fourth-fifth order Runge-Kutta integration technique.

Larson, V. H.↗

NASA NTRS1983

Analysis of aerothermal loads on spherical dome protuberances

Hypersonic flow over spherical dome protuberances was investigated to determine increased pressure and heating loads to the surface. The configuration was mathematically modeled in a time-dependant three-dimensional analysis of the conservation of mass, momentum (Navier-Stokes), and energy equations. A boundary mapping techique was used to obtain a rectangular parallelepiped computational domain, and a MacCormack explicit time-split predictor-corrector finite difference algorithm was used to obtain solutions. Results show local pressures and heating rates for domes one-half, one, and two boundary layer thicknesses high were increased by factors on the order of 1.4, 2, and 6, respectively. However, because lee-side pressure and thermal loads were reduced the two lower height domes did not experience any net increase in total loads. The total loads on the higher dome were increased by twenty-five percent. Flow over the lower dome was everywhere attached while flow over the intermediate dome had small windward and leeside separations. The higher dome had an unsteady windward separation region and a large leeside separation region. Trailing vortices form on all domes with intensity increasing with dome height. Discussions of applying the results to a thermally bowed thermal protection system are presented.

Olsen, G. C.↗

NASA NTRS1983

Formation of standing shocks in stellar winds and related astrophysical flows

Stellar winds and other analogous astrophysical flows can be described, to lowest order, by the familiar one dimensional hydrodynamic equations which, being nonlinear, admit in some instances discontinuous as well as continuous transonic solutions for identical inner boundary conditions. The characteristics of the time dependent differential equations of motion are described to show how a perturbation changes profile in time and, under well defined conditions, develops into a stationary shock discontinuity. The formation of standing shocks in wind type astrophysical flows depends on the fulfillment of appropriate necessary conditions, which are determined by the conservation of mass, momentum and energy across the discontinuity, and certain sufficient conditions, which are determined by the flow's history.

Tsinganos, K.↗

NASA NTRS1984

Simplified combustion noise theory yielding a prediction of fluctuating pressure level

The first order equations for the conservation of mass and momentum in differential form are combined for an ideal gas to yield a single second order partial differential equation in one dimension and time. Small perturbation analysis is applied. A Fourier transformation is performed that results in a second order, constant coefficient, nonhomogeneous equation. The driving function is taken to be the source of combustion noise. A simplified model describing the energy addition via the combustion process gives the required source information for substitution in the driving function. This enables the particular integral solution of the nonhomogeneous equation to be found. This solution multiplied by the acoustic pressure efficiency predicts the acoustic pressure spectrum measured in turbine engine combustors. The prediction was compared with the overall sound pressure levels measured in a CF6-50 turbofan engine combustor and found to be in excellent agreement.

Huff, R. G.↗

NASA NTRS1984

Aerothermal loads analysis for high speed flow over a quilted surface configuration

Attention is given to hypersonic laminar flow over a quilted surface configuration that simulates an array of Space Shuttle Thermal Protection System panels bowed in a spherical shape as a result of thermal gradient through the panel thickness. Pressure and heating loads to the surface are determined. The flow field over the configuration was mathematically modeled by means of time-dependent, three-dimensional conservation of mass, momentum, and energy equations. A boundary mapping technique was then used to obtain a rectangular, parallel piped computational domain, and an explicit MacCormack (1972) explicit time-split predictor corrector finite difference algorithm was used to obtain steady state solutions. Total integrated heating loads vary linearly with bowed height when this value does not exceed the local boundary layer thickness.

Olsen, G. C.↗

NASA NTRS1983

Streamwise computation of three-dimensional parabolic flows

A new approach to calculate three-dimensional parabolic flows is presented. The flow field is computed by calculating velocity along a set of streamlines. The dependent variables commonly used in the computation of three-dimensional flows are the three velocity components. In contrast, the dependent variables in the present approach are the streamwise velocity and the two coordinates, in the cross-stream plane, of the chosen streamlines. The streamwise velocity is calculated from the finite difference equations obtained by applying Euler's momentum theorem to streamtubes constructed around the chosen streamlines; the streamline coordinates are calculated from the conservation of mass. Results of the calculations, based on the present approach, are compared with the experimental data for flow through rectangular ducts; the agreement is satisfactory.

Greywall, M. S.↗

NASA NTRS1984

Numerical study of two-dimensional non-plane MHD wave propagation in a supersonic, superalfvenic magnetohydrodynamic flow

The features of a 2.5-dimensional time-dependent MHD numerical code used to simulate the propagation of finite amplitude MHD waves through an inhomogeneous, supersonic superalfvenic medium are described. Basic equations for conservation of mass, momentum, and free energy in a unit volume plasma gas and for magnetic induction are defined. Initial conditions are functions of the radial coordinates and disturbances are introduced at the lower boundary. A set of finite difference equations based on a Lax-Wendroff scheme is used for the simulation. The model is applied to analyzing a solar flare shock wave in steady-state and global transient conditions while propagating at 1 AU heliolongitude.

Han, S. M.↗