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NASA NTRS1988

Universality classes for deterministic surface growth

A scaling theory for the generalized deterministic Kardar-Parisi-Zhang (1986) equation with beta greater than 1, is developed to study the growth of a surface through deterministic local rules. A one-dimensional surface model corresponding to beta = 1 is presented and solved exactly. The model can be studied as a limiting case of ballistic deposition, or as the deterministic limit of the Eden (1961) model. The scaling exponents, the correlation functions, and the skewness of the surface are determined. The results are compared with those of Burgers' (1974) equation for the case of beta = 2.

Krug, J.↗

NASA NTRS1988

Static and dynamic properties of incommensurate smectic-A(IC) liquid crystals

The elasticity, topological defects, and hydrodynamics of the incommensurate smectic A(IC) phase liquid crystals are studied. The phase is characterized by two colinear mass density waves of incommensurate spatial frequency. The elastic free energy is formulated in terms of a displacement field and a phason field. It is found that the topological defects of the system are dislocations with a nonzero phason field and phason field components. A two-dimensional Burgers lattice for these dislocations is introduced. It is shown that the hydrodynamic modes of the phase include first- and second-sound modes whose direction-dependent velocities are identical to those in ordinary smectics.

Lubensky, T. C.↗

NASA NTRS1988

Optical matrix-vector processing for computational fluid dynamics

An optical processor to solve partial differential equations for computational fluid dynamics applications is considered. This application is new and original for optical processors. The algorithms that are used are optical realizations of the Newton-Raphson method for nonlinear equations and a new optical LU direct decomposition and Gauss-Seidel iterative solution to the resultant linear algebraic equations. These algorithms are used to solve Burger's equation (a specific form of the momentum equation in fluid dynamics). The nonlinear equations provide 1-D velocity data at each time step. Simulation results of optical processing with these algorithms on computational fluid dynamics data is included.

Perlee, Caroline J.↗

NASA NTRS1989

A package for unstructured grid generation and finite element flow solvers

A set of programs for unstructured mesh construction, fluid flow calculation and flow field visualization in two and three dimensions is described. The grid generators are based on the advancing front concept. The flow solvers use the finite element methods with Flux Corrected Transport techniques to solve several sets of equations including the Euler and Burger's equations. Several examples show the flexibility and accuracy of the methods.

Gumbert, Clyde↗

NASA NTRS1990

Triangle based TVD schemes for hyperbolic conservation laws

A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.

Durlofsky, Louis J.↗

NASA NTRS1990

One dimensional heavy ion beam transport: Energy independent model

Attempts are made to model the transport problem for heavy ion beams in various targets, employing the current level of understanding of the physics of high-charge and energy (HZE) particle interaction with matter are made. An energy independent transport model, with the most simplified assumptions and proper parameters is presented. The first and essential assumption in this case (energy independent transport) is the high energy characterization of the incident beam. The energy independent equation is solved and application is made to high energy neon (NE-20) and iron (FE-56) beams in water. The numerical solutions is given and compared to a numerical solution to determine the accuracy of the model. The lower limit energy for neon and iron to be high energy beams is calculated due to Barkas and Burger theory by LBLFRG computer program. The calculated values in the density range of interest (50 g/sq cm) of water are: 833.43 MeV/nuc for neon and 1597.68 MeV/nuc for iron. The analytical solutions of the energy independent transport equation gives the flux of different collision terms. The fluxes of individual collision terms are given and the total fluxes are shown in graphs relative to different thicknesses of water. The values for fluxes are calculated by the ANASTP computer code.

Farhat, Hamidullah↗

NASA NTRS1990

Development of comprehensive numerical schemes for predicting evaporating gas-droplets flow processes of a liquid-fueled combustor

An existing Computational Fluid Dynamics code for simulating complex turbulent flows inside a liquid rocket combustion chamber was validated and further developed. The Advanced Rocket Injector/Combustor Code (ARICC) is simplified and validated against benchmark flow situations for laminar and turbulent flows. The numerical method used in ARICC Code is re-examined for incompressible flow calculations. For turbulent flows, both the subgrid and the two equation k-epsilon turbulence models are studied. Cases tested include idealized Burger's equation in complex geometries and boundaries, a laminar pipe flow, a high Reynolds number turbulent flow, and a confined coaxial jet with recirculations. The accuracy of the algorithm is examined by comparing the numerical results with the analytical solutions as well as experimented data with different grid sizes.

Chen, C. P.↗

NASA NTRS1990

Comparison of several finite-difference methods

The present numerical study establishes that, with the MacCormack and 'two-four' methods, the solution accuracy obtainable in problems involving wave propagation, shock-wave and contact discontinuities, and viscous effects, will be strongly dependent on Courant number. The application of flux correction to the MacCormack and two-four methods is noted to significantly attenuate dispersion errors; the ensuing solutions capture the discontinuities in the shock-tube problem with improved accuracy and resolution, and are free of dispersion errors for the viscous Burgers' equation.

Biringen, S.↗

NASA NTRS1990

Calculation of shocked one-dimensional flows on abruptly changing grids by mathematical programming

Cell-centered finite differences on cells of an arbitrarily spaced grid are presently used to discretize the steady-state inviscid and nearly inviscid Burgers' equations for steady-state shocked flow in a quasi-one-dimensional nozzle. The everdetermined system of nonlinear algebraic equations thus obtained is solved by a procedure which minimizes a weighted sum of the residuals' absolute values. Numerical solutions for both viscous and inviscid problems are accurate and nonoscillatory, on grids whose abrupt mesh lengths refinements are as great as a factor of 10,000.

Lavery, John E.↗

NASA NTRS1990

The structure and dynamics of bubble-type vortex breakdown

A unique discrete form of the Navier-Stokes equations for unsteady, three-dimensional, incompressible flow has been used to study vortex breakdown numerically. A Burgers-type vortex was introduced along the central axis of the computational domain, and allowed to evolve in space and time. By varying the strength of the vortex and the free stream axial velocity distribution, using a previously developed Rossby number criterion as a guide, the location and size of the vortex breakdown region was controlled. While the boundaries of the vortex breakdown bubble appear to be nominally symmetric, the internal flow field is not. Consequently, the mechanisms for mixing and entrainment required to sustain the bubble region are different from those suggested by earlier axisymmetric models. Results presented in this study, for a Reynolds number of 200, are in good qualitative agreement with higher Reynolds number experimental observations, and a variety of plots have been presented to help illuminate the fluid physics.

Spall, R. E.↗

NASA NTRS1989

Direct simulation of compressible turbulence

Several direct simulations of 3-D homogeneous, compressible turbulence are presented with emphasis on the differences with incompressible turbulent simulations. A fully spectral collocation algorithm, periodic in all directions coupled with a 3rd order Runge-Kutta time discretization scheme is sufficient to produce well-resolved flows at Taylor Reynolds numbers below 40 on grids of 128x128x128. A Helmholtz decomposition of velocity is useful to differentiate between the purely compressible effects and those effects solely due to vorticity production. In the context of homogeneous flows, this decomposition in unique. Time-dependent energy and dissipation spectra of the compressible and solenoidal velocity components indicate the presence of localized small scale structures. These structures are strongly a function of the initial conditions. Researchers concentrate on a regime characterized by very small fluctuating Mach numbers Ma (on the order of 0.03) and density and temperature fluctuations much greater than sq Ma. This leads to a state in which more than 70 percent of the kinetic energy is contained in the so-called compressible component of the velocity. Furthermore, these conditions lead to the formation of curved weak shocks (or shocklets) which travel at approximately the sound speed across the physical domain. Various terms in the vorticity and divergence of velocity production equations are plotted versus time to gain some understanding of how small scales are actually formed. Possible links with Burger turbulence are examined. To visualize better the dynamics of the flow, new graphic visualization techniques have been developed. The 3-D structure of the shocks are visualized with the help of volume rendering algorithms developed in-house. A combination of stereographic projection and animation greatly increase the number of visual cues necessary to properly interpret the complex flow.

Zang, T. A.↗

NASA NTRS1989

Assessment of numerical methods for the solution of fluid dynamics equations for nonlinear resonance systems

The capability of accurate nonlinear flow analysis of resonance systems is essential in many problems, including combustion instability. Classical numerical schemes are either too diffusive or too dispersive especially for transient problems. In the last few years, significant progress has been made in the numerical methods for flows with shocks. The objective was to assess advanced shock capturing schemes on transient flows. Several numerical schemes were tested including TVD, MUSCL, ENO, FCT, and Riemann Solver Godunov type schemes. A systematic assessment was performed on scalar transport, Burgers' and gas dynamic problems. Several shock capturing schemes are compared on fast transient resonant pipe flow problems. A system of 1-D nonlinear hyperbolic gas dynamics equations is solved to predict propagation of finite amplitude waves, the wave steepening, formation, propagation, and reflection of shocks for several hundred wave cycles. It is shown that high accuracy schemes can be used for direct, exact nonlinear analysis of combustion instability problems, preserving high harmonic energy content for long periods of time.

Przekwas, A. J.↗

NASA NTRS1991

Uniform high order spectral methods for one and two dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗

NASA NTRS1991

Alfven shock trains

The Cohen-Kulsrud-Burgers equation (CKB) is used to consider the nonlinear evolution of resistive, quasi-parallel Alfven waves subject to a long-wavelength, plane-polarized, monochromatic instability. The instability saturates by nonlinear steepening, which proceeds until the periodic waveform develops an interior scale length comparable to the dissipation length; a fast or an intermediate shock then forms. The result is a periodic train of Alfven shocks of one or the other type. For propagation strictly parallel to the magnetic field, there will be two shocks per instability wavelength. Numerical integration of the time-dependent CKB equation shows that an initial, small-amplitude growing wave asymptotes to a stable, periodic stationary wave whose analytic solution specifies how the type of shock embedded in the shock train, and the amplitude and speed of the shock train, depend on the strength and phase of the instability. Waveforms observed upstream of the earth's bowshock and cometary shocks resemble those calculated here.

Malkov, M. A.↗

NASA NTRS1991

A Runge-Kutta discontinuous finite element method for high speed flows

A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.

Bey, Kim S.↗

NASA NTRS1989

Optimal moving grids for time-dependent partial differential equations

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of partial differential equation solutions in the least squares norm.

Wathen, A. J.↗

NASA NTRS1992

Essentially nonoscillatory postprocessing filtering methods

High order accurate centered flux approximations used in the computation of numerical solutions to nonlinear partial differential equations produce large oscillations in regions of sharp transitions. Here, we present a new class of filtering methods denoted by Essentially Nonoscillatory Least Squares (ENOLS), which constructs an upgraded filtered solution that is close to the physically correct weak solution of the original evolution equation. Our method relies on the evaluation of a least squares polynomial approximation to oscillatory data using a set of points which is determined via the ENO network. Numerical results are given in one and two space dimensions for both scalar and systems of hyperbolic conservation laws. Computational running time, efficiency, and robustness of method are illustrated in various examples such as Riemann initial data for both Burgers' and Euler's equations of gas dynamics. In all standard cases, the filtered solution appears to converge numerically to the correct solution of the original problem. Some interesting results based on nonstandard central difference schemes, which exactly preserve entropy, and have been recently shown generally not to be weakly convergent to a solution of the conservation law, are also obtained using our filters.

Lafon, F.↗

NASA NTRS1992

Adaptive mesh strategies for the spectral element method

An adaptive spectral method was developed for the efficient solution of time dependent partial differential equations. Adaptive mesh strategies that include resolution refinement and coarsening by three different methods are illustrated on solutions to the 1-D viscous Burger equation and the 2-D Navier-Stokes equations for driven flow in a cavity. Sharp gradients, singularities, and regions of poor resolution are resolved optimally as they develop in time using error estimators which indicate the choice of refinement to be used. The adaptive formulation presents significant increases in efficiency, flexibility, and general capabilities for high order spectral methods.

Mavriplis, Catherine↗