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NASA NTRS2018

High-resolution Wave Propagation Method for Stratified Flows

The implementation of the multidimensional f-waves Riemann solver for the time-dependent, three-dimensional, nonhydrostatic, meso- and microscale atmospheric flows is described in detail. The Riemann solver employs flux-based wave decomposition (f-waves) for the calculation of Godunov fluxes in which the flux differences are written directly as the linear combination of the right eigenvectors of the hyperbolic system. The scheme incorporates the source term due to gravity without introducing discretization errors which is an important property in the context of atmospheric flows. The resulting flow solver is conservative, accurate, stable, and well-balanced. The implementation of the solver is evaluated using benchmark test cases for atmospheric dynamics.

Riemann problem↗

NASA NTRS1984

A spectral element method for fluid dynamics - Laminar flow in a channel expansion

A spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible Navier-Stokes equations. In the spectral element discretization, the computational domain is broken into a series of elements, and the velocity in each element is represented as a high-order Lagrangian interpolant through Chebyshev collocation points. The hyperbolic piece of the governing equations is then treated with an explicit collocation scheme, while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle. The implementation of the technique is demonstrated on a one-dimensional inflow-outflow advection-diffusion equation, and the method is then applied to laminar two-dimensional (separated) flow in a channel expansion. Comparisons are made with experiment and previous numerical work.

Patera, A. T.↗

NASA NTRS2002

The Overgrid Interface for Computational Simulations on Overset Grids

Computational simulations using overset grids typically involve multiple steps and a variety of software modules. A graphical interface called OVERGRID has been specially designed for such purposes. Data required and created by the different steps include geometry, grids, domain connectivity information and flow solver input parameters. The interface provides a unified environment for the visualization, processing, generation and diagnosis of such data. General modules are available for the manipulation of structured grids and unstructured surface triangulations. Modules more specific for the overset approach include surface curve generators, hyperbolic and algebraic surface grid generators, a hyperbolic volume grid generator, Cartesian box grid generators, and domain connectivity: pre-processing tools. An interface provides automatic selection and viewing of flow solver boundary conditions, and various other flow solver inputs. For problems involving multiple components in relative motion, a module is available to build the component/grid relationships and to prescribe and animate the dynamics of the different components.

Chan, William M.↗

NASA NTRS1973

A refined analysis of composite laminates

The purpose of this paper is to develop a sufficiently accurate analysis, which is much simpler than exact three-dimensional analysis, for statics and dynamics of composite laminates. The governing differential equations and boundary conditions are derived by following a variational approach. The displacements are assumed piecewise linear across the thickness and the effects of transverse shear deformations and rotary inertia are included. A procedure for obtaining the general solution of the above governing differential equations in the form of hyperbolic-trigonometric series is given. The accuracy of the present theory is assessed by obtaining results for free vibrations and flexure of simply supported rectangular laminates and comparing them with results from exact three-dimensional analysis.

Srinivas, S.↗

NASA NTRS1995

Explicit and implicit compact high-resolution shock-capturing methods for multidimensional Euler equations 1: Formulation

Two classes of explicit compact high-resolution shock-capturing methods for the multidimensional compressible Euler equations for fluid dynamics are constructed. Some of these schemes can be fourth-order accurate away from discontinuities. For the semi-discrete case their shock-capturing properties are of the total variation diminishing (TVD), total variation bounded (TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1-D scalar hyperbolic conservation laws and are positive schemes in more than one dimension. These fourth-order schemes require the same grid stencil as their second-order non-compact cousins. One class does not require the standard matrix inversion or a special numerical boundary condition treatment associated with typical compact schemes. Due to the construction, these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on one-dimensional extensions. However, one class has a more desirable high-resolution shock-capturing property and a smaller operation count in 3-D than the other class. The extension of these schemes to coupled nonlinear systems can be accomplished using the Roe approximate Riemann solver, the generalized Steger and Warming flux-vector splitting or the van Leer type flux-vector splitting. Modification to existing high-resolution second- or third-order non-compact shock-capturing computer codes is minimal. High-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax-Friedrichs numerical flux without the use of Riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock-capturing counterparts. The simplest extension to viscous flows can be achieved by using the standard fourth-order compact or non-compact formula for the viscous terms.

Yee, H. C.↗

NASA NTRS1989

Full potential unsteady computations including aeroelastic effects

A unified formulation is presented based on the full potential framework coupled with an appropriate structural model to compute steady and unsteady flows over rigid and flexible configurations across the Mach number range. The unsteady form of the full potential equation in conservation form is solved using an implicit scheme maintaining time accuracy through internal Newton iterations. A flux biasing procedure based on the unsteady sonic reference conditions is implemented to compute hyperbolic regions with moving sonic and shock surfaces. The wake behind a trailing edge is modeled using a mathematical cut across which the pressure is satisfied to be continuous by solving an appropriate vorticity convection equation. An aeroelastic model based on the generalized modal deflection approach interacts with the nonlinear aerodynamics and includes both static as well as dynamic structural analyses capability. Results are presented for rigid and flexible configurations at different Mach numbers ranging from subsonic to supersonic conditions. The dynamic response of a flexible wing below and above its flutter point is demonstrated.

Shankar, Vijaya↗

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 NTRS1989

Two high-velocity encounters of elliptical galaxies

This paper describes results obtained on a simulation of two high-velocity encounters of NGC 4782/4783 and NGC 2672/2673 binary elliptical galaxies which differ substantially in mass ratio (about 1 for the first pair, and about 10 for the second). CCD images and velocities obtained from digital spectra were used to constrain simulations of the galaxy collisions. The binary orbital elements, the orientation of the orbit in the sky, the time since pericenter, and the dynamical mass of the pair were derived. Results suggested that the dumb-bell galaxy NGC 4782/4783 is not a supermassive galaxy, as was claimed earlier on the basis of the high relative velocity and high central dispersion, but has a moderate mass to luminosity ratio M/L(B) of about 10. It was concluded that its trajectory changed from hyperbolic to elliptical as a result of energy lost during the collision. It was found that the NGC 2672/2673 also has a moderate M/L(B) of about 7.

Balcells, Marc↗

NASA NTRS1979

Physical and dynamical evolution of long-period comets

The source of long-period comets was investigated using a Monte Carlo simulation of comet evolution under the influence of a combination of physical and dynamical processes. The perturbation of cometary orbits by major planets and by non-gravitational forces was modeled, as was physical loss of comets due to random disruption (splitting) and planetary collision; a model was also derived for loss of all volatiles. The importance of each of these processes was examined. The primary end states found for long-period comets were: ejection on hyperbolic orbit, 65.2%; random disruption, 27.6%; and formation of silicate crusts, 7.1%. The basic correctness of the Ort hypothesis was confirmed.

Weissman, P. R.↗

NASA NTRS1983

Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations

The application of a new implicit unconditionally stable high resolution total variation diminishing (TVD) scheme to steady state calculations. It is a member of a one parameter family of explicit and implicit second order accurate schemes developed by Harten for the computation of weak solutions of hyperbolic conservation laws. This scheme is guaranteed not to generate spurious oscillations for a nonlinear scalar equation and a constant coefficient system. Numerical experiments show that this scheme not only has a rapid convergence rate, but also generates a highly resolved approximation to the steady state solution. A detailed implementation of the implicit scheme for the one and two dimensional compressible inviscid equations of gas dynamics is presented. Some numerical computations of one and two dimensional fluid flows containing shocks demonstrate the efficiency and accuracy of this new scheme.

Yee, H. C.↗

NASA NTRS1983

Implicit total variation diminishing (TVD) schemes for steady-state calculations

Yee, H. C.↗

NASA NTRS1988

Subharmonic resonance in a mixing layer

The subharmonic resonance phenomenon in a spatially evolving mixing layer is studied using direct simulations of the 2-D Navier-Stokes equations. The computational domain extends to + or - infinity in the cross-stream direction with U(1) = 1.25 and U(2) = 0.25 imposed at + infinity and - infinity respectively. The domain is finite in the streamwise direction with inflow and outflow boundary conditions imposed at x/delta sub omega = 0 and 100, respectively. A hyperbolic-tangent mean velocity profile is assumed at the inlet and the Reynolds number based on the inlet vorticity thickness and velocity difference is Re = 600. It is observed that the phase angle between the fundamental and its subharmonic plays a key role in the spatial development of these modes. Contour plots of vorticity show that varying the phase will have a dramatic effect on the dynamics of the vortices. Pairing or shredding is observed depending on the phase. Fourier decomposition of the time traces show that the fundamental grows, saturates and decays with the downstream distance. The subharmonic has a similar behavior. However, the level at which the modes will saturate is affected by the phase. At 0 deg phase, it was found that as the fundamental saturates, the growth rate of the subharmonic is enhanced. At 90 deg phase, it was found that as the fundamental saturates, the growth rate of the subharmonic is inhibited. In the later case, the growth rate of the subharmonic recovers after saturation of the fundamental. These results are in qualitative agreement with experimental data.

Mansour, N. N.↗

NASA NTRS1988

Finite elements: Theory and application

Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.

Dwoyer, D. L.↗

NASA NTRS2005

Baseline Computational Fluid Dynamics Methodology for Longitudinal-Mode Liquid-Propellant Rocket Combustion Instability

A computational method for the analysis of longitudinal-mode liquid rocket combustion instability has been developed based on the unsteady, quasi-one-dimensional Euler equations where the combustion process source terms were introduced through the incorporation of a two-zone, linearized representation: (1) A two-parameter collapsed combustion zone at the injector face, and (2) a two-parameter distributed combustion zone based on a Lagrangian treatment of the propellant spray. The unsteady Euler equations in inhomogeneous form retain full hyperbolicity and are integrated implicitly in time using second-order, high-resolution, characteristic-based, flux-differencing spatial discretization with Roe-averaging of the Jacobian matrix. This method was initially validated against an analytical solution for nonreacting, isentropic duct acoustics with specified admittances at the inflow and outflow boundaries. For small amplitude perturbations, numerical predictions for the amplification coefficient and oscillation period were found to compare favorably with predictions from linearized small-disturbance theory as long as the grid exceeded a critical density (100 nodes/wavelength). The numerical methodology was then exercised on a generic combustor configuration using both collapsed and distributed combustion zone models with a short nozzle admittance approximation for the outflow boundary. In these cases, the response parameters were varied to determine stability limits defining resonant coupling onset.

Litchford, R. J.↗

NASA NTRS1987

Analytical investigation of the dynamics of tethered constellations in Earth orbit (phase 2)

Simulation of two short distance crawling maneuvers of the elevator both with and without environmental perturbations acting upon the system is discussed. These simulation runs were performed in order to provide results useful for the interpretation of the data from the tests, on the ground, of a scaled down engineering model of the elevator. In these simulation runs the elevator crawls along the tether in accordance to the developed mirror image motion control law (MIMCL). Results from the simulation of the 4 km long maneuver run were compared to those obtained by adopting the modified hyperbolic tangent control law (MHTCL). A preprocessor was developed for setting up the initial conditions of a tethered system with L platforms, M longitudinal dampers, and N lumped masses (platforms plus tether beads). A short test run of the 4-platform system with 3 longitudinal dampers and 10 lumper mass without any perturbation acting upon the system is illustrated. In support of the Tether Applications Working Group, SAO is preparing a catalog of tether simulations, has prepared a set of simulator test cases, obtained results from SKYHOOK, and solicited results from simulators at other institutions, and prepared a paper on a specific analytic solution.

Lorenzini, Enrico C.↗

NASA NTRS2005

A Robust Absorbing Boundary Condition for Compressible Flows

An absorbing non-reflecting boundary condition (NRBC) for practical computations in fluid dynamics and aeroacoustics is presented with theoretical proof. This paper is a continuation and improvement of a previous paper by the author. The absorbing NRBC technique is based on a first principle of non reflecting, which contains the essential physics that a plane wave solution of the Euler equations remains intact across the boundary. The technique is theoretically shown to work for a large class of finite volume approaches. When combined with the hyperbolic conservation laws, the NRBC is simple, robust and truly multi-dimensional; no additional implementation is needed except the prescribed physical boundary conditions. Several numerical examples in multi-dimensional spaces using two different finite volume schemes are illustrated to demonstrate its robustness in practical computations. Limitations and remedies of the technique are also discussed.

Loh, Ching Y.↗

NASA NTRS1983

Dynamical evolution of the Oort cometary cloud

The dynamical evolution of comets in the Oort cloud under the influence of stellar perturbations has been modeled using Monte Carlo techniques. It is shown that the cloud has been depleted over the history of the solar system. Comets are lost from the cloud by direct ejection due to close stellar encounters, diffusion of aphelia to distances beyond the sun's sphere of influence, or diffusion of perihelia into the planetary region where Jupiter and Saturn perturbations either eject them on hyperbolic trajectories or capture them to short-period orbits. The population of the cloud is estimated to be 1.0 - 1.5 x 10 to the 12th comets and the total mass is on the order of 1.9 earth masses. In addition to random passing stars, less frequent encounters with giant molecular clouds may play a significant role in randomizing the orbits of comets in the cloud and reducing the effective radius of the sun's sphere of influence.

Weissman, P. R.↗

NASA NTRS1989

Advances in fluid dynamics

Various papers on advances in fluid dynamics are presented. Individual topics addressed include: nonunique solutions of the Euler equations; numerical solution of the biharmonic equation using a spectral multigrid method; finite-volume shape and placement for efficiency, accuracy, and physics; NND schemes and their application to numerical simulation of complex plume flow problems; symbolic analysis and computational algorithm construction; and velocity and mass flux distribution measurements of spherical glass beads in air flow in a 90-deg vertical-to-horizontal bend. Also discussed are: hyperbolic solution adaptive mesh generation; progress and future directions in CFD; numerical simulation of shock focusing; microbubble formation and splitting in a turbulent boundary layer for turbulence reduction; numerical study of spherical Taylor-Couette flow, trajectories of vortex lines beneath a free surface or above a plane; and a mathematical model for response of the coronary circulation to high sustained gravitational force fields.

Ballhaus, W. F., Jr.↗