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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 NTRS1984

A dynamical basis for the parameterization of organized deep convection in large-scale numerical models

A hierarchy of steady, nonlinear, semianalytic models of different types of convection were produced. These provide a theoretical framework for determining cloud outflow fluxes of both dynamic and thermodynamic quantities, which can be used to formulate dynamical transports in parameterization schemes. This was achieved by exploiting certain Lagrangian conservation properties of steady flow, from which an equation for the vertical displacement of particles can be obtained and the outflow entropy, energy and momentum fluxes and the infow/outflow mass fluxes can be determined from solution to the equation. These fluxes are determined in terms of grid scale parameters such as convective available potential energy (CAPE), cloud layer shear, and horizontal pressure gradients. Five main types of system models are identified, respectively representing archtypes of convection in zero shear, large shear, midlatitude squall lines, tropical squall lines and cellular convection. The downdraught is an important aspect in the first four of these and the cloud scale transport of momentum is very distinctive.

Moncrieff, M. W.↗

NASA NTRS1984

A two-point difference scheme for computing steady-state solutions to the conservative one-dimensional Euler equations

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 for the flux derivatives with dissipation added at supersonic points via the retarded density concept. Application of the method to 1-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. For 1-dimensional Euler calculations, it is shown that the scheme offers two advantages over the more widely-used three-point schemes. The first is in regard to application of boundary conditions, and the second relates to the fact that the two-point algorithm is well-conditioned for large time steps.

Wornom, S. F.↗

NASA NTRS1986

Preconditioned methods for solving the incompressible and low speed compressible equations

Acceleration methods are presented for solving the steady state incompressible equations. These systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state. The compressible equations in conservation form with slow flow are also considered. Two arbitrary functions, alpha and beta, are introduced in the general preconditioning. An analysis of this system is presented and an optimal value for beta is determined given a constant, alpha. It is further shown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed. Several generalizations to the compressible equations are presented which generalize previous results.

Turkel, E.↗

NASA NTRS1987

Preconditioned methods for solving the incompressible and low speed compressible equations

The introduction of artificial time derivatives which permit faster convergence to the steady state is used to precondition steady state incompressible equation-solving acceleration methods. An optimal value is obtained for beta, given constant alpha, where alpha and beta are arbitrary functions in the general preconditioning stage. A symmetric hyperbolic system is formed by the resulting incompressible equations. Compressible equations in conservation form with slow flow are also considered.

Turkel, Eli↗

NASA NTRS1988

Prediction of unsteady loads on maneuvering delta wings using time-accurate Euler schemes

Three-dimensional steady and unsteady vortex-dominated flows around sharp-edged delta wings are considered in this paper. The problem is formulated by using the unsteady conservative Euler equations for the flow relative motion with respect to a moving frame of reference. An implicit approximately-factored finite volume scheme is used to solve the resulting equations on a three-dimensional computational grid which is generated by using a modified Joukowski transformation in cross-flow planes at the grid chord stations. The scheme is applied to a delta wing undergoing pitching oscillation around a large angle of attack. The initial conditions correspond to a steady flow around a delta wing of aspect ratio of one, freestream Mach number of 0.3 and mean angle of attack of 20.5. The steady flow results are compared with those of an explicit computational scheme and the experimental data, and they are in good agreement.

Kandil, Osama A.↗

NASA NTRS1988

Baroclinic instability in vertically discrete systems

The Charney-Phillips (1953) standard vertical grid for quasi-geostrophic models is compared with the Lorenz (1960) vertical grid (which departs from the standard grid by carrying horizontal velocity and potential temperature at same levels) in the framework of the quasi-geostrophic potential vorticity equation and baroclinic instability. It is demonstrated that the Charney-Phillips grid makes it possible to maintain important dynamical constraints on quasi-geostrophic flow, such as the conservation of quasi-geostrophic potential vorticity through horizontal advection and resulting integral constraints. The Lorenz grid, on the other hand, is not straightforward even to define quasi-geostrophic potential vorticity. Moreover, due to an extra degree of freedom in potential temperature, the Lorenz grid can falsely satisfy the necessary condition for baroclinic instability near the lower and the upper boundaries.

Arakawa, Akio↗

NASA NTRS1998

Asymptotic Spreading Rate of Initially Compressible Jets: Experiment and Analysis

Experimental results for the spreading and centerline velocity decay rates for round, compressible jets, from a convergent and a convergent-divergent nozzle, are presented. The spreading rate is determined from the variation of streamwise mass flux obtained from Pitot probe surveys. Results for the far asymptotic region show that both spreading and centerline velocity decay rates, when nondimensionalized by parameters at the nozzle exit, decrease with increasing "jet Mach number" M(sub J). Dimensional analysis with the assumption of momentum conservation, together with compressible flow calculations for the conditions at the nozzle exit, predict this Mach number dependence well. The analysis also demonstrates that an increase in the "potential core length" of the jet occurring with increasing M(sub J), a commonly observed trend, is largely accounted for simply by the variations in the density and static pressure at the nozzle exit. The effect of decreasing mixing efficiency with increasing compressibility is inferred to contribute only partially to the latter trend.

Zaman, K. B. M. Q.↗

NASA NTRS1998

Asymptotic Spreading Rate of Initially Compressible Jets-Experiment and Analysis

Experimental results for the spreading and centerline velocity decay rates for round, compressible jets, from a convergent and a convergent-divergent nozzle, are presented. The spreading rate is determined from the variation of streamwise mass flux obtained from Pitot probe surveys. Results for the far asymptotic region show that both spreading and centerline velocity decay rates, when nondimensionalized by parameters at the nozzle exit, decrease with increasing "jet Mach number" M(sub j). Dimensional analysis with the assumption of momentum conservation, together with compressible flow calculations for the conditions at the nozzle exit, predict this Mach number dependence well. The analysis also demonstrates that an increase in the "potential core length" of the jet occurring with increasing M(sub j), a commonly observed trend, is largely accounted for simply by the variations in the density and static pressure at the nozzle exit. The effect of decreasing mixing efficiency with increasing compressibility is inferred to contribute only partially to the latter trend.

Zaman, K. B. M. Q.↗

NASA NTRS1984

Numerical simulation of the interaction of a vortex with stationary airfoil in transonic flow

A perturbation form of an implicit conservative, noniterative numerical algorithm for the two-dimensional thin layer Navier-Stokes and Euler equations is used to compute the interaction flow-field of a vortex with stationary airfoil. A Lamb-like analytical vortex having a finite core is chosen to interact with a thick (NACA 0012) and a thin (NACA 64A006) airfoil independently in transonic flow. Two different configurations of vortex interaction are studied, viz., (1) when the vortex is fixed at one location in the flowfield, and (2) when the vortex is convecting past the airfoil at freestream velocity. Parallel computations of this interacting flowfield are also done using a version of the Transonic Small Disturbance Code (ATRAN2). A special treatment of the leading edge region for thin airfoils is included in this code. With this, the three methods gave qualitatively similar results for the weaker interactions considered in this study. However, the strongest interactions considered proved to be beyond the capabilities of the small disturbance code. The results also show a far greater influence of the vortex on the airfoil flowfield when the vortex is stationary than when it is convecting with the flow.

Srinivasan, G. R.↗

NASA NTRS1995

General Fluid System Simulation Program to Model Secondary Flows in Turbomachinery

The complexity and variety of turbomachinery flow circuits created a need for a general fluid system simulation program for test data anomaly resolution as well as design review. The objective of the paper is to present a computer program that has been developed to support Marshall Space Flight Center's turbomachinery internal flow analysis efforts. The computer program solves for the mass. energy and species conservation equation at each node and flow rate equation at each branch of the network by a novel numerical procedure which is a combination of both Newton-Ralphson and successive substitution method and uses a thermodynamic property program for computing real gas properties. A generalized, robust, modular, and 'user-friendly' computer program has been developed to model internal flow rates, pressures, temperatures, concentrations of gas mixtures and axial thrusts. The program can be used for any network for compressible and incompressible flows, choked flow, change of phase and gaseous mixturecs. The code has been validated by comparing the predictions with Space Shuttle Main Engine test data.

Majumdar, Alok K.↗

NASA NTRS1979

Boundary conditions for implicit solutions to the compressible Navier-Stokes equations in finite computational domains

The admissible boundary conditions for the unsteady Navier-Stokes equations are insufficient to determine completely the flow variables at grid points on the boundaries of a finite computational domain. A general method is presented in which the boundary conditions are supplemented by conservatively differenced forms of the flow equations. For implicit algorithms, the method provides a fully implicit coupling between boundary and interior points. Implicit boundary-point computation schemes are given for most types of flow boundary encountered in internal and external flow problems. Convergence to steady state is generally faster than with locally explicit boundary-point schemes.

Thomas, P. D.↗

NASA NTRS1985

Conserving Purge Gases

Two-step flow control adjusts to dimensional changes in purged equipment. Two pressure actuated valves provide two-step control. When turbopump starts, helium flows through both valves. As pump operates its parts expand, clearance decrease and back pressure builds up on valves. At preset back pressure value, one valve closes, reducing helium flow.

Kan, K.↗

NASA NTRS1988

A study of numerical methods for hyperbolic conservation laws with stiff source terms

The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a simple advection equation with a parameter-dependent source term was studied. Two approaches to incorporate the source term were utilized: MacCormack type predictor-corrector methods with flux limiters, and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Various comparisons over a wide range of parameter values were made. In the stiff case where the solution contains discontinuities, incorrect numerical propagation speeds are observed with all of the methods considered. This phenomenon is studied and explained.

Leveque, R. J.↗

NASA NTRS2013

The Influence of Unsteadiness on the Analysis of Pressure Gain Combustion Devices

Pressure gain combustion (PGC) has been the object of scientific study for over a century due to its promise of improved thermodynamic efficiency. In many recent application concepts PGC is utilized as a component in an otherwise continuous, normally steady flow system, such as a gas turbine or ram jet engine. However, PGC is inherently unsteady. Failure to account for the effects of this periodic unsteadiness can lead to misunderstanding and errors in performance calculations. This paper seeks to provide some clarity by presenting a consistent method of thermodynamic cycle analysis for a device utilizing PGC technology. The incorporation of the unsteady PGC process into the conservation equations for a continuous flow device is presented. Most importantly, the appropriate method for computing the conservation of momentum is presented. It will be shown that proper, consistent analysis of cyclic conservation principles produces representative performance predictions.

Paxson, Daniel E.↗

NASA NTRS1987

Energetics of oscillating lifting surfaces using integral conservation laws

The energetics of oscillating flexible lifting surfaces in two and three dimensions is calculated by the use of integral conservation laws in inviscid incompressible flow for general and harmonic transverse oscillations. Total thrust is calculated from the momentum theorem and energy loss rate due to vortex shedding in the wake from the principle of conservation of mechanical energy. Total power required to maintain the oscillations and hydrodynamic efficiency are also determined. In two dimensions, the results are obtained in closed form. In three dimensions, the distribution of vorticity on the lifting surface is also required as input to the calculations. Thus, unsteady lifting-surface theory must be used as well. The analysis is applicable to oscillating lifting surfaces of arbitrary planform, aspect ratio, and reduced frequency and does not require calculation of the leading-edge thrust.

Ahmadi, Ali R.↗

NASA NTRS1974

A new formulation of the conservation equations of fluid dynamics

The computation of time-dependent flows has inspired a new, higher-dimensional formulation of the conservation equations of fluid dynamics in which time is treated as a fourth coordinate. The formulation is derived for a constant-density flow, and then extended to a variable-density flow by introducing a fifth, fictitious coordinate. This new coordinate can also act as a source coordinate, so that external source terms can be included. The analysis is carried out for both incompressible, stratified flow, and compressible equilibrium flow. The results are then extended to non-equilibrium and magnetohydrodynamic flows. Several applications of the new formulation to the computation of time-dependent flows are discussed.

Vinokur, M.↗

NASA NTRS1991

Evaluation of Rotordynamic Coefficients of Look-through Labyrinths by Means of a Three Volume Bulk Flow Model

To describe the compressible, turbulent flow in a labyrinth seal, a three volume bulk flow model is presented. The conservation equations for mass, momentum, and energy are established in every control volume. A perturbation analysis is performed, yielding zeroth order equations for centric rotor position and first order equations describing the flow field for small rotor motions around the seal center. The equations are integrated numerically. From perturbation pressure, the forces on the shaft and the dynamic coefficients are calculated.

Nordmann, R.↗