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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

High resolution finite volume methods on arbitrary grids via wave propagation

A generalization of Godunov's method for systems of conservation laws has been developed and analyzed that can be applied with arbitrary time steps on arbitrary grids in one space dimension. Stability for arbitrary time steps is achieved by allowing waves to propagate through more than one mesh cell in a time step. The method is extended here to second order accuracy and to a finite volume method in two space dimensions. This latter method is based on solving one dimensional normal and tangential Rieman problems at cell interfaces and again propagating waves through one or more mesh cells. By avoiding the usual time step restriction of explicit methods, it is possible to use reasonable time steps on irregular grids where the minimum cell area is much smaller than the average cell. Boundary conditions for the Euler equations are discussed and special attention is given to the case of a Cartesian grid cut by an irregular boundary. In this case small grid cells arise only near the boundary, and it is desirable to use a time step appropriate for the regular interior cells. Numerical results in two dimensions show that this can be achieved.

Leveque, Randall J.↗
NASA NTRS1995

Advanced in Visualization of 3D Time-Dependent CFD Solutions

Numerical simulations of complex 3D time-dependent (unsteady) flows are becoming increasingly feasible because of the progress in computing systems. Unfortunately, many existing flow visualization systems were developed for time-independent (steady) solutions and do not adequately depict solutions from unsteady flow simulations. Furthermore, most systems only handle one time step of the solutions individually and do not consider the time-dependent nature of the solutions. For example, instantaneous streamlines are computed by tracking the particles using one time step of the solution. However, for streaklines and timelines, particles need to be tracked through all time steps. Streaklines can reveal quite different information about the flow than those revealed by instantaneous streamlines. Comparisons of instantaneous streamlines with dynamic streaklines are shown. For a complex 3D flow simulation, it is common to generate a grid system with several millions of grid points and to have tens of thousands of time steps. The disk requirement for storing the flow data can easily be tens of gigabytes. Visualizing solutions of this magnitude is a challenging problem with today's computer hardware technology. Even interactive visualization of one time step of the flow data can be a problem for some existing flow visualization systems because of the size of the grid. Current approaches for visualizing complex 3D time-dependent CFD solutions are described. The flow visualization system developed at NASA Ames Research Center to compute time-dependent particle traces from unsteady CFD solutions is described. The system computes particle traces (streaklines) by integrating through the time steps. This system has been used by several NASA scientists to visualize their CFD time-dependent solutions. The flow visualization capabilities of this system are described, and visualization results are shown.

Lane, David A.↗
NASA NTRS1994

A Navier-Strokes Chimera Code on the Connection Machine CM-5: Design and Performance

We have implemented a three-dimensional compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the 'chimera' approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. A parallel machine like the CM-5 is well-suited for finite-difference methods on structured grids. The regular pattern of connections of a structured mesh maps well onto the architecture of the machine. So the first design choice, finite differences on a structured mesh, is natural. We use centered differences in space, with added artificial dissipation terms. When numerically solving the Navier-Stokes equations, there are liable to be some mesh cells near a solid body that are small in at least one direction. This mesh cell geometry can impose a very severe CFL (Courant-Friedrichs-Lewy) condition on the time step for explicit time-stepping methods. Thus, though explicit time-stepping is well-suited to the architecture of the machine, we have adopted implicit time-stepping. We have further taken the approximate factorization approach. This creates the need to solve large banded linear systems and creates the first possible barrier to an efficient algorithm. To overcome this first possible barrier we have considered two options. The first is just to solve the banded linear systems with data spread over the whole machine, using whatever fast method is available. This option is adequate for solving scalar tridiagonal systems, but for scalar pentadiagonal or block tridiagonal systems it is somewhat slower than desired. The second option is to 'transpose' the flow and geometry variables as part of the time-stepping process: Start with x-lines of data in-processor. Form explicit terms in x, then transpose so y-lines of data are in-processor. Form explicit terms in y, then transpose so z-lines are in processor. Form explicit terms in z, then solve linear systems in the z-direction. Transpose to the y-direction, then solve linear systems in the y-direction. Finally transpose to the x direction and solve linear systems in the x-direction. This strategy avoids inter-processor communication when differencing and solving linear systems, but requires a large amount of communication when doing the transposes. The transpose method is more efficient than the non-transpose strategy when dealing with scalar pentadiagonal or block tridiagonal systems. For handling geometrically complex problems the chimera strategy was adopted. For multiple zone cases we compute on each zone sequentially (using the whole parallel machine), then send the chimera interpolation data to a distributed data structure (array) laid out over the whole machine. This information transfer implies an irregular communication pattern, and is the second possible barrier to an efficient algorithm. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran. We make use of the Connection Machine Scientific Software Library (CMSSL) for the linear solver and array transpose operations.

Jespersen, Dennis C.↗
NASA NTRS2010

Adaptive Numerical Algorithms in Space Weather Modeling

Space weather describes the various processes in the Sun-Earth system that present danger to human health and technology. The goal of space weather forecasting is to provide an opportunity to mitigate these negative effects. Physics-based space weather modeling is characterized by disparate temporal and spatial scales as well as by different physics in different domains. A multi-physics system can be modeled by a software framework comprising of several components. Each component corresponds to a physics domain, and each component is represented by one or more numerical models. The publicly available Space Weather Modeling Framework (SWMF) can execute and couple together several components distributed over a parallel machine in a flexible and efficient manner. The framework also allows resolving disparate spatial and temporal scales with independent spatial and temporal discretizations in the various models. Several of the computationally most expensive domains of the framework are modeled by the Block-Adaptive Tree Solar wind Roe Upwind Scheme (BATS-R-US) code that can solve various forms of the magnetohydrodynamics (MHD) equations, including Hall, semi-relativistic, multi-species and multi-fluid MHD, anisotropic pressure, radiative transport and heat conduction. Modeling disparate scales within BATS-R-US is achieved by a block-adaptive mesh both in Cartesian and generalized coordinates. Most recently we have created a new core for BATS-R-US: the Block-Adaptive Tree Library (BATL) that provides a general toolkit for creating, load balancing and message passing in a 1, 2 or 3 dimensional block-adaptive grid. We describe the algorithms of BATL and demonstrate its efficiency and scaling properties for various problems. BATS-R-US uses several time-integration schemes to address multiple time-scales: explicit time stepping with fixed or local time steps, partially steady-state evolution, point-implicit, semi-implicit, explicit/implicit, and fully implicit numerical schemes. Depending on the application, we find that different time stepping methods are optimal. Several of the time integration schemes exploit the block-based granularity of the grid structure. The framework and the adaptive algorithms enable physics based space weather modeling and even forecasting.

Toth, Gabor↗
NASA NTRS1991

Numerical algorithm comparison for the accurate and efficient computation of high-incidence vortical flow

Computations from two Navier-Stokes codes, NSS and F3D, are presented for a tangent-ogive-cylinder body at high angle of attack. Features of this steady flow include a pair of primary vortices on the leeward side of the body as well as secondary vortices. The topological and physical plausibility of this vortical structure is discussed. The accuracy of these codes are assessed by comparison of the numerical solutions with experimental data. The effects of turbulence model, numerical dissipation, and grid refinement are presented. The overall efficiency of these codes are also assessed by examining their convergence rates, computational time per time step, and maximum allowable time step for time-accurate computations. Overall, the numerical results from both codes compared equally well with experimental data, however, the NSS code was found to be significantly more efficient than the F3D code.

Chaderjian, Neal M.↗
NASA NTRS1991

Three-dimensional blade vortex interactions

A three-dimensional time dependent Navier-Stokes analysis was applied to the rotor blade vortex interaction problem. The numerical procedure is an iterative implicit procedure using three point central differences to represent spatial derivatives. A series of calculations were made to determine the time steps, pseudo-time steps, iterations, artificial dissipation level, etc. required to maintain a nondissipative vortex. Results show the chosen method to have excellent non-dissipative properties provided the correct parameters are chosen. This study was used to set parameters for both two- and three-dimensional blade vortex interaction studies. The case considered was the interaction between a vortex and a NACA0012 airfoil. The results showed the detailed physics during the interaction including the pressure pulse propagating from the blade. The simulated flow physics was qualitatively similar to that experimentally observed. The BVI phenomena is the result of the buildup and violent collapse of the shock waves and local supersonic pockets on the blade surfaces. The resulting pressure pulse build-up appears to be centered at the blade leading edge.

Davoudzadeh, Farhad↗
NASA NTRS1990

Two- and three-dimensional blade vortex interactions

A three-dimensional time dependent Navier-Stokes analysis was applied to the rotor blade vortex interaction (BVI) problem. The numerical procedure is an iterative implicit procedure using three point central differences to represent spatial derivatives. A series of calculations were made to determine the time steps, pseudo-time steps, iterations, artificial dissipation level, etc. required to maintain a nondissipative vortex. Results show the chosen method to have excellent non-dissipative properties provided the correct parameters are chosen. This study was used to set parameters for both two- and three-dimensional blade vortex interaction studies. The two-dimensional study considered the interaction between a vortex and a NACA0012 airfoil. The results showed the detailed physics during the interaction including the pressure pulse propagating from the blade. The simulated flow physics was qualitatively similar to that experimentally observed. The 2-D BVI phenomena is the result of the buildup and violent collapse of the shock waves and local supersonic pockets on the blade surfaces. The resulting pressure pulse build-up appears to be centered at the blade leading edge. The three-dimensional interaction study considered the case of a vortex at 20 deg incidence to the blade leading edge. Although the qualitative results were similar to that of the two-dimensional interaction, details clearly showed the three-dimensional nature of the interaction process.

Davoudzadeh, F.↗
NASA NTRS1993

Extending substructure based iterative solvers to multiple load and repeated analyses

Direct solvers currently dominate commercial finite element structural software, but do not scale well in the fine granularity regime targeted by emerging parallel processors. Substructure based iterative solvers--often called also domain decomposition algorithms--lend themselves better to parallel processing, but must overcome several obstacles before earning their place in general purpose structural analysis programs. One such obstacle is the solution of systems with many or repeated right hand sides. Such systems arise, for example, in multiple load static analyses and in implicit linear dynamics computations. Direct solvers are well-suited for these problems because after the system matrix has been factored, the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions. On the other hand, iterative solvers in general are ill-suited for these problems because they often must restart from scratch for every different right hand side. In this paper, we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides. Basically, we formulate the overall problem as a series of minimization problems over K-orthogonal and supplementary subspaces, and tailor the preconditioned conjugate gradient algorithm to solve them efficiently. The resulting solution method is scalable, whereas direct factorization schemes and forward and backward substitution algorithms are not. We illustrate the proposed methodology with the solution of static and dynamic structural problems, and highlight its potential to outperform forward and backward substitutions on parallel computers. As an example, we show that for a linear structural dynamics problem with 11640 degrees of freedom, every time-step beyond time-step 15 is solved in a single iteration and consumes 1.0 second on a 32 processor iPSC-860 system; for the same problem and the same parallel processor, a pair of forward/backward substitutions at each step consumes 15.0 seconds.

Farhat, Charbel↗
NASA NTRS1995

Building a better leapfrog

In stellar dynamical computer simulations, as well as other types of simulations using particles, time step size is often held constant in order to guarantee a high degree of energy conservation. In many applications, allowing the time step size to change in time can offer a great saving in computational cost, but variable-size time steps usually imply a substantial degradation in energy conservation. We present a meta-algorithm' for choosing time steps in such a way as to guarantee time symmetry in any integration scheme, thus allowing vastly improved energy conservation for orbital calculations with variable time steps. We apply the algorithm to the familiar leapfrog scheme, and generalize to higher order integration schemes, showing how the stability properties of the fixed-step leapfrog scheme can be extended to higher order, variable-step integrators such as the Hermite method. We illustrate the remarkable properties of these time-symmetric integrators for the case of a highly eccentric elliptical Kepler orbit and discuss applications to more complex problems.

Hut, Piet↗
NASA NTRS1989

Design of optimally smoothing multi-stage schemes for the Euler equations

In this paper, a method is developed for designing multi-stage schemes that give optimal damping of high-frequencies for a given spatial-differencing operator. The objective of the method is to design schemes that combine well with multi-grid acceleration. The schemes are tested on a nonlinear scalar equation, and compared to Runge-Kutta schemes with the maximum stable time-step. The optimally smoothing schemes perform better than the Runge-Kutta schemes, even on a single grid. The analysis is extended to the Euler equations in one space-dimension by use of 'characteristic time-stepping', which preconditions the equations, removing stiffness due to variations among characteristic speeds. Convergence rates independent of the number of cells in the finest grid are achieved for transonic flow with and without a shock. Characteristic time-stepping is shown to be preferable to local time-stepping, although use of the optimally damping schemes appears to enhance the performance of local time-stepping. The extension of the analysis to the two-dimensional Euler equations is hampered by the lack of a model for characteristic time-stepping in two dimensions. Some results for local time-stepping are presented.

Van Leer, Bram↗
NASA NTRS1992

Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations

The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility.

Elmiligui, Alaa↗
NASA NTRS2010

Reactive Collision Avoidance Algorithm

The reactive collision avoidance (RCA) algorithm allows a spacecraft to find a fuel-optimal trajectory for avoiding an arbitrary number of colliding spacecraft in real time while accounting for acceleration limits. In addition to spacecraft, the technology can be used for vehicles that can accelerate in any direction, such as helicopters and submersibles. In contrast to existing, passive algorithms that simultaneously design trajectories for a cluster of vehicles working to achieve a common goal, RCA is implemented onboard spacecraft only when an imminent collision is detected, and then plans a collision avoidance maneuver for only that host vehicle, thus preventing a collision in an off-nominal situation for which passive algorithms cannot. An example scenario for such a situation might be when a spacecraft in the cluster is approaching another one, but enters safe mode and begins to drift. Functionally, the RCA detects colliding spacecraft, plans an evasion trajectory by solving the Evasion Trajectory Problem (ETP), and then recovers after the collision is avoided. A direct optimization approach was used to develop the algorithm so it can run in real time. In this innovation, a parameterized class of avoidance trajectories is specified, and then the optimal trajectory is found by searching over the parameters. The class of trajectories is selected as bang-off-bang as motivated by optimal control theory. That is, an avoiding spacecraft first applies full acceleration in a constant direction, then coasts, and finally applies full acceleration to stop. The parameter optimization problem can be solved offline and stored as a look-up table of values. Using a look-up table allows the algorithm to run in real time. Given a colliding spacecraft, the properties of the collision geometry serve as indices of the look-up table that gives the optimal trajectory. For multiple colliding spacecraft, the set of trajectories that avoid all spacecraft is rapidly searched on-line. The optimal avoidance trajectory is implemented as a receding-horizon model predictive control law. Therefore, at each time step, the optimal avoidance trajectory is found and the first time step of its acceleration is applied. At the next time step of the control computer, the problem is re-solved and the new first time step is again applied. This continual updating allows the RCA algorithm to adapt to a colliding spacecraft that is making erratic course changes.

Scharf, Daniel↗
NASA NTRSDate not supplied

Integrated Bosch Process System Models for In-Situ Oxygen and Carbon Production

In-Situ Resource Utilization (ISRU) technology is a vital component to NASA’s mission of a sustainable presence on the Moon and Mars. Local resources can be leveraged to reduce resupply frequency and mass. Elements of the Bosch process, combined with the carbothermal reduction process, can produce oxygen on the lunar surface with minimal consumables. The Bosch process can also produce oxygen on the Martian surface by using the CO 2 -rich environment. Between both systems, adsorption pump, solar thermal energy, carbon formation reactor, and water recovery subsystems are modeled and integrated to create a functional model in MATLAB software. The model is used to simulate performance of the system and reduce mass, power, and volume requirements. This integrated system model provides a tool to scale ISRU technologies for oxygen and carbon production. The MATLAB model is created by developing a system of independent subsystem models that are solved for their quasi-steady state values which can be integrated with respect to time to determine the change in current states. A flexible time stepping method is used to ensure a high level of accuracy during periods of rapid change while still making use of a simple explicit integration method. The flexible time step is calculated for each independent subsystem and the minimum value from those is used as the overall time step. A flexible time step is calculated by dividing a resolution value, or the maximum change per time step, by the variables current rate of change. The maximum value from all points in space is used for subsystem models that contain multiple values. The process is done for every variable that is being monitored in each subsystem and the global minimum is used as that iteration’s timestep. Several assumptions used in the MATLAB model for fluid flow dynamics, such as 1-D gas flow through the sorption pump, are supported by modeling in Ansys Fluent software. The Lunar oxygen production system is outlined in Fig. 1. The carbothermal reduction subsystem uses solar energy to heat a mixture of lunar regolith and carbon powder to produce carbon monoxide. To begin, the carbon monoxide feeds to the modified Bosch subsystem along with hydrogen gas. The reactants then enter the carbon formation reactor where water and carbon powder are produced. Solar thermal energy is used to add energy to the reactor, but waste heat from the carbothermal process is another potential heat source. The water is collected and electrolyzed to produce hydrogen which reenters the Bosch subsystem, and the oxygen is stored for downstream use. The carbon powder is collected and feeds back into the carbothermal subsystem. The Martian oxygen production system uses the full Bosch process and is outlined in Fig 2. A CO 2 adsorption pump thermally cycles to scrub and pressurize CO 2 from the environment. Along with an initial supply of hydrogen, the reactants enter the Reverse Water Gas Shift Reactor (RWGSR) which produces carbon monoxide and water. Carbon monoxide and unreacted hydrogen enter the carbon formation reactor to produce water and carbon powder. The water is collected from both reactors and electrolyzed to reintroduce hydrogen and store oxygen for propellant production or life support. Carbon is removed from the carbon formation reactor and stored. The adsorption pump utilizes rapid cycle temperature swings within a stack of zeolite coated surfaces. The subsystem model solves 1-D quasi-steady conservation laws of the quasi-steady form, shown in Eq. 1, for the gas stream and heat exchange liquid to predict performance parameters such as breakthrough capacity and optimum cycle time. The source term S is used to capture interactions between the fluid flows and the sorbent. A quasi-steady-state scheme is used where no time derivatives appear in the governing equations, except for those in the source terms. This results in an autonomous system, where ∂F/∂x = ƒ(F). The fluxes F are provided at the inlet, and an explicit method is used to solve for the spatial distribution of F. The heat and mass flows to the sorbent are then extracted from the source terms. These flows are numerically integrated to produce a 1-D solution for the system’s state as a function of both time and space. The body of the adsorption pump is separated into two semi-independent models: the heat exchanger fluid flow and gas flow through the zeolite coated surfaces. Both models are solved using the above-described method to find a 1-D solution as a function of space and interact only once a timestep is taken. The interaction point is the sorbent through which all heat transfer between the two models must occur. Sorbent mass adsorption is calculated using the Lagergren model, shown in Eq. 2, where the transfer coefficient, λ D , is found by solving a system of nondimensionalized equations derived by using the heat and mass transfer analogy for transport phenomena. Using Grade 544 Type 13X zeolite as the sorbent material, the equilibrium concentration, θ eq , is calculated using the k-site Langmuir isotherm and fit parameters. Additionally, the enthalpy of adsorption used in the model is computed by interpolation of available data [1]. The subsystem model was validated using the Rapid Cycle Temperature Swing Adsorption (RC-TSA) pump. The solar thermal energy subsystem focuses on a solar concentrator concept with a heat exchanger to heat the reactants before entering the carbon formation reactor. The subsystem model assumes a fixed solar flux and reflector efficiency to calculate the reactant temperature given the incoming temperature, pressure, and exchanger geometry. The receiver is a custom manufactured series of copper blocks with serpentine channels to increase its surface area and the residence time of the reactants to heat up to 550 °C. The subsystem model was validated using a heat exchanger developed at NASA Glenn Research Center (GRC). The solar thermal energy subsystem focuses on a solar concentrator concept with a heat exchanger to heat the reactants before entering the carbon formation reactor. The subsystem model assumes a fixed solar flux and reflector efficiency to calculate the reactant temperature given the incoming temperature, pressure, and exchanger geometry. The receiver is a custom manufactured series of copper blocks with serpentine channels to increase its surface area and the residence time of the reactants to heat up to 550 °C. The subsystem model was validated using a heat exchanger developed at NASA Glenn Research Center (GRC).

In situ Resource Utilization↗
NASA NTRS2006

QUICR-learning for Multi-Agent Coordination

Coordinating multiple agents that need to perform a sequence of actions to maximize a system level reward requires solving two distinct credit assignment problems. First, credit must be assigned for an action taken at time step t that results in a reward at time step t > t. Second, credit must be assigned for the contribution of agent i to the overall system performance. The first credit assignment problem is typically addressed with temporal difference methods such as Q-learning. The second credit assignment problem is typically addressed by creating custom reward functions. To address both credit assignment problems simultaneously, we propose the "Q Updates with Immediate Counterfactual Rewards-learning" (QUICR-learning) designed to improve both the convergence properties and performance of Q-learning in large multi-agent problems. QUICR-learning is based on previous work on single-time-step counterfactual rewards described by the collectives framework. Results on a traffic congestion problem shows that QUICR-learning is significantly better than a Q-learner using collectives-based (single-time-step counterfactual) rewards. In addition QUICR-learning provides significant gains over conventional and local Q-learning. Additional results on a multi-agent grid-world problem show that the improvements due to QUICR-learning are not domain specific and can provide up to a ten fold increase in performance over existing methods.

Agogino, Adrian K.↗
NASA NTRS2004

Algorithm for Training a Recurrent Multilayer Perceptron

An improved algorithm has been devised for training a recurrent multilayer perceptron (RMLP) for optimal performance in predicting the behavior of a complex, dynamic, and noisy system multiple time steps into the future. [An RMLP is a computational neural network with self-feedback and cross-talk (both delayed by one time step) among neurons in hidden layers]. Like other neural-network-training algorithms, this algorithm adjusts network biases and synaptic-connection weights according to a gradient-descent rule. The distinguishing feature of this algorithm is a combination of global feedback (the use of predictions as well as the current output value in computing the gradient at each time step) and recursiveness. The recursive aspect of the algorithm lies in the inclusion of the gradient of predictions at each time step with respect to the predictions at the preceding time step; this recursion enables the RMLP to learn the dynamics. It has been conjectured that carrying the recursion to even earlier time steps would enable the RMLP to represent a noisier, more complex system.

Parlos, Alexander G.↗
NASA NTRS1988

Computation of steady and unsteady vortex-dominated flows with shock waves

The unsteady Euler equations have been derived in the conservation form for the flow relative motion with respect to a rotating frame of reference. The resulting equations are solved by using a central-difference finite-volume scheme with four-state Runge-Kutta time stepping. For steady flow applications local time stepping is used, and for unsteady applications the minimum global time stepping is used. A three-dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wings. The capability of the three-dimensional program has been demonstrated for a rigid sharp-edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow.

Kandil, Osama A.↗
NASA NTRS1987

Computation of steady and unsteady vortex dominated flows

The unsteady Euler equations have been written in the conservation form for the relative motion in a rotating frame of reference. The resulting equations are solved by using a central-difference finite-volume scheme with four-stage Runge-Kutta time stepping. For steady flow problems, local time stepping is used and for unsteady flow problems, the minimum global time stepping is used. A three-dimensional fully vectorized computer program has been developed and applied to steady and unsteady maneuvering delta wing. The capability of the three-dimensional program has been demonstrated for a rigid sharp-edged delta wing undergoing uniform rolling in a conical flow and rolling oscillations in a locally conical flow.

Kandil, Osama A.↗
NASA NTRS1984

Finite-difference fluid dynamics computer mathematical models for the design and interpretation of experiments for space flight

Numerical methods are used to design a spherical baroclinic flow model experiment of the large scale atmosphere flow for Spacelab. The dielectric simulation of radial gravity is only dominant in a low gravity environment. Computer codes are developed to study the processes at work in crystal growing systems which are also candidates for space flight. Crystalline materials rarely achieve their potential properties because of imperfections and component concentration variations. Thermosolutal convection in the liquid melt can be the cause of these imperfections. Such convection is suppressed in a low gravity environment. Two and three dimensional finite difference codes are being used for this work. Nonuniform meshes and implicit iterative methods are used. The iterative method for steady solutions is based on time stepping but has the options of different time steps for velocity and temperature and of a time step varying smoothly with position according to specified powers of the mesh spacings. This allows for more rapid convergence. The code being developed for the crystal growth studies allows for growth of the crystal as the solid-liquid interface. The moving interface is followed using finite differences; shape variations are permitted. For convenience in applying finite differences in the solid and liquid, a time dependent coordinate transformation is used to make this interface a coordinate surface.

Roberts, G. O.↗