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Longitudinal propulsion coupling system. Volume 2 - Digital computer solution to equations of motion
Computer program for solving equations of motion governing pogo oscillations
Multigrid, Fractional-Step Computation Of Flow
Speed of computer code solving three-dimensional Navier-Stokes equations of flow of incompressible fluid by fractional-step method increased significantly by use of multigrid procedures. In method, equations solved on general nonorthogonal curvilinear coordinate grid, using volume fluxes. At each time step, computations performed in two fractional steps. In first step, equations of conservation of momentum solved by use of gradient of pressure from previous time step via explicit approximate-factorization method, yielding approximate flow field that does not satisfy equation of conservation of mass. In second step, discrete Poisson-like equation with Neumann-type boundary conditions, formed by combining equations of conservation of momentum and mass, solved iteratively.
A Fourier collocation time domain method for numerically solving Maxwell's equations
A new method for solving Maxwell's equations in the time domain for arbitrary values of permittivity, conductivity, and permeability is presented. Spatial derivatives are found by a Fourier transform method and time integration is performed using a second order, semi-implicit procedure. Electric and magnetic fields are collocated on the same grid points, rather than on interleaved points, as in the Finite Difference Time Domain (FDTD) method. Numerical results are presented for the propagation of a 2-D Transverse Electromagnetic (TEM) mode out of a parallel plate waveguide and into a dielectric and conducting medium.
Analysis of opposed jet hydrogen-air counter flow diffusion flame
A computational simulation of the opposed-jet diffusion flame is performed to study its structure and extinction limits. The present analysis concentrates on the nitrogen-diluted hydrogen-air diffusion flame, which provides the basic information for many vehicle designs such as the aerospace plane for which hydrogen is a candidate as the fuel. The computer program uses the time-marching technique to solve the energy and species equations coupled with the momentum equation solved by the collocation method. The procedure is implemented in two stages. In the first stage, a one-step forward overal chemical reaction is chosen with the gas phase chemical reaction rate determined by comparison with experimental data. In the second stage, a complete chemical reaction mechanism is introduced with detailed thermodynamic and transport property calculations. Comparison between experimental extinction data and theoretical predictions is discussed. The effects of thermal diffusion as well as Lewis number and Prandtl number variations on the diffusion flame are also presented.
A New Look into the Effect of Large Drops on Radiative Transfer Process
Recent studies indicate that a cloudy atmosphere absorbs more solar radiation than any current 1D or 3D radiation model can predict. The excess absorption is not large, perhaps 10-15 W/sq m or less, but any such systematic bias is of concern since radiative transfer models are assumed to be sufficiently accurate for remote sensing applications and climate modeling. The most natural explanation would be that models do not capture real 3D cloud structure and, as a consequence, their photon path lengths are too short. However, extensive calculations, using increasingly realistic 3D cloud structures, failed to produce photon paths long enough to explain the excess absorption. Other possible explanations have also been unsuccessful so, at this point, conventional models seem to offer no solution to this puzzle. The weakest link in conventional models is the way a size distribution of cloud particles is mathematically handled. Basically, real particles are replaced with a single average particle. This "ensemble assumption" assumes that all particle sizes are well represented in any given elementary volume. But the concentration of larger particles can be so low that this assumption is significantly violated. We show how a different mathematical route, using the concept of a cumulative distribution, avoids the ensemble assumption. The cumulative distribution has jumps, or steps, corresponding to the rarer sizes. These jumps result in an additional term, a kind of Green's function, in the solution of the radiative transfer equation. Solving the cloud radiative transfer equation with the measured particle distributions, described in a cumulative rather than an ensemble fashion, may lead to increased cloud absorption of the magnitude observed.
Parallel Finite Element Domain Decomposition for Structural/Acoustic Analysis
A domain decomposition (DD) formulation for solving sparse linear systems of equations resulting from finite element analysis is presented. The formulation incorporates mixed direct and iterative equation solving strategics and other novel algorithmic ideas that are optimized to take advantage of sparsity and exploit modern computer architecture, such as memory and parallel computing. The most time consuming part of the formulation is identified and the critical roles of direct sparse and iterative solvers within the framework of the formulation are discussed. Experiments on several computer platforms using several complex test matrices are conducted using software based on the formulation. Small-scale structural examples are used to validate thc steps in the formulation and large-scale (l,000,000+ unknowns) duct acoustic examples are used to evaluate the ORIGIN 2000 processors, and a duster of 6 PCs (running under the Windows environment). Statistics show that the formulation is efficient in both sequential and parallel computing environmental and that the formulation is significantly faster and consumes less memory than that based on one of the best available commercialized parallel sparse solvers.
Computing Flows Using Chimera and Unstructured Grids
DRAGONFLOW is a computer program that solves the Navier-Stokes equations of flows in complexly shaped three-dimensional regions discretized by use of a direct replacement of arbitrary grid overlapping by nonstructured (DRAGON) grid. A DRAGON grid (see figure) is a combination of a chimera grid (a composite of structured subgrids) and a collection of unstructured subgrids. DRAGONFLOW incorporates modified versions of two prior Navier-Stokes-equation-solving programs: OVERFLOW, which is designed to solve on chimera grids; and USM3D, which is used to solve on unstructured grids. A master module controls the invocation of individual modules in the libraries. At each time step of a simulated flow, DRAGONFLOW is invoked on the chimera portion of the DRAGON grid in alternation with USM3D, which is invoked on the unstructured subgrids of the DRAGON grid. The USM3D and OVERFLOW modules then immediately exchange their solutions and other data. As a result, USM3D and OVERFLOW are coupled seamlessly.
Prediction and measurement of turbulent aerodynamic trailing edge flows
A viscous-inviscid interaction algorithm is developed for prediction of two-dimensional mean and fluctuating velocity distributions in the wake immediately downstream of an airfoil trailing edge. A composite pressure field is defined, and a Poisson equation solved for transverse pressure variations. A parabolized form of the time-averaged steady Navier-Stokes equations are solved in conjunction with a viscous-augmented two-dimensional inviscid potential flow analysis. A tensor constitutive equation is employed to predict Reynolds stress distributions from solutions of a turbulence kinetic energy two equation closure model. Numerical predictions compared favorably with detailed experimental data for mean and fluctuating velocities, and Reynolds shear stress distributions, in the trailing edge region of a NACA 63-012 airfoil.
A fully vectorized numerical solution of the incompressible Navier-Stokes equations
A vectorizable algorithm is presented for the implicit finite difference solution of the incompressible Navier-Stokes equations in general curvilinear coordinates. The unsteady Reynolds averaged Navier-Stokes equations solved are in two dimension and non-conservative primitive variable form. A two-layer algebraic eddy viscosity turbulence model is used to incorporate the effects of turbulence. Two momentum equations and a Poisson pressure equation, which is obtained by taking the divergence of the momentum equations and satisfying the continuity equation, are solved simultaneously at each time step. An elliptic grid generation approach is used to generate a boundary conforming coordinate system about an airfoil. The governing equations are expressed in terms of the curvilinear coordinates and are solved on a uniform rectangular computational domain. A checkerboard SOR, which can effectively utilize the computer architectural concept of vector processing, is used for iterative solution of the governing equations.
A numerical method for solving the Vlasov equation
A numerical procedure is derived for the solution of the Vlasov-Poisson system of equations in two phase-space variables. Derivatives with respect to the phase-space variables are approximated by a weighted sum of the values of the distribution function at property chosen neighboring points. The resulting set of ordinary differential equations is then solved by using an appropriate time intergration scheme. The accuracy of the proposed method is tested with some simple model problems. The results for the free streaming case, linear Landau damping, and nonlinear Landau damping are investigated and compared with those of the splitting scheme. The proposed method is found to be very accurate and efficient.
Propagation of plane waves in a variable area duct carrying a compressible subsonic flow
A complete numerical solution is obtained for plane wave propagation in a variable-area duct which carries a mean compressible flow. The equation solved is a matrix formulation of the equations first derived by Tsien in 1952. Sample calculations are presented for four particular flow configurations: (1) a duct-nozzle-duct combination with the acoustic wave traveling with the flow, (2) a duct-nozzle-duct combination with the acoustic wave traveling against the flow, (3) an inlet-nozzle-duct configuration with the acoustic wave traveling against the flow, and (4) a duct-nozzle-exit configuration with the acoustic wave traveling with the flow. For each of these cases, the variation of energy flux with frequency is shown for particular geometry and flow condition.
Mathematical modelling of a human external respiratory system
A closed system of algebraic and common differential equations solved by computer is investigated. It includes equations which describe the activity pattern of the respiratory center, the phrenic nerve, the thrust produced by the diaphragm as a function of the lung volume and discharge frequency of the phrenic nerve, as well as certain relations of the lung stretch receptors and chemoreceptors on various lung and blood characteristics, equations for lung biomechanics, pulmonary blood flow, alveolar gas exchange and capillary blood composition equations to determine various air and blood flow and gas exchange parameters, and various gas mixing and arterial and venous blood composition equations, to determine other blood, air and gas mixing characteristics. Data are presented by means of graphs and tables, and some advantages of this model over others are demonstrated by test results.
Comet Gas and Dust Dynamics Modeling
This software models the gas and dust dynamics of comet coma (the head region of a comet) in order to support the Microwave Instrument for Rosetta Orbiter (MIRO) project. MIRO will study the evolution of the comet 67P/Churyumov-Gerasimenko's coma system. The instrument will measure surface temperature, gas-production rates and relative abundances, and velocity and excitation temperatures of each species along with their spatial temporal variability. This software will use these measurements to improve the understanding of coma dynamics. The modeling tool solves the equation of motion of a dust particle, the energy balance equation of the dust particle, the continuity equation for the dust and gas flow, and the dust and gas mixture energy equation. By solving these equations numerically, the software calculates the temperature and velocity of gas and dust as a function of time for a given initial gas and dust production rate, and a dust characteristic parameter that measures the ability of a dust particle to adjust its velocity to the local gas velocity. The software is written in a modular manner, thereby allowing the addition of more dynamics equations as needed. All of the numerical algorithms are added in-house and no third-party libraries are used.
Advanced methods for the solution of differential equations
This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.
A decade of computer simulations for space shuttle aerodynamics
Ten years ago computer simulations of the flow field around the Space Shuttle Orbiter were limited to inviscid calculations for the windward side of the forebody and viscous calculations for selected 2-D problems. Advances in computer hardware and numerical methods during the past ten years made it possible to calculate viscous flow over the complete orbiter configuration at angle of attack. The equations solved are the Reynolds-averaged, Navier-Stokes equation, simplified by either the thin-layer or parabolized approximation. An algebraic eddy viscosity model is used for turbulent flow. The free stream is assumed to be a perfect gas for wind tunnel conditions and a real gas in thermodynamic equilibrium for flight conditions. Four examples of recent computer simulations are presented. Flow field results include oil flow patterns on the surface and Mach number contours, isobars, and cross-flow velocity vectors in the shock layer.
A decade of computer simulations for Space Shuttle aerodynamics
Ten years ago computer simulations of the flow field around the Space Shuttle Orbiter were limited to inviscid calculations for the windward side of the forebody and viscous calculations for selected 2-D problems. Advances in computer hardware and numerical methods during the past ten years made it possible to calculate viscous flow over the complete orbiter configuration at angle of attack. The equations solved are the Reynolds-averaged, Navier-Stokes equation, simplified by either the thin-layer or parabolized approximation. An algebraic eddy viscosity model is used for turbulent flow. The free stream is assumed to be a perfect gas for wind tunnel conditions and a real gas in thermodynamic equilibrium for flight conditions. Four examples of recent computer simulations are presented. Flow field results include oil flow patterns on the surface and Mach number contours, isobars, and cross-flow velocity vectors in the shock layer.
Flow and Turbulence Modeling and Computation of Shock Buffet Onset for Conventional and Supercritical Airfoils
Flow and turbulence models applied to the problem of shock buffet onset are studied. The accuracy of the interactive boundary layer and the thin-layer Navier-Stokes equations solved with recent upwind techniques using similar transport field equation turbulence models is assessed for standard steady test cases, including conditions having significant shock separation. The two methods are found to compare well in the shock buffet onset region of a supercritical airfoil that involves strong trailing-edge separation. A computational analysis using the interactive-boundary layer has revealed a Reynolds scaling effect in the shock buffet onset of the supercritical airfoil, which compares well with experiment. The methods are next applied to a conventional airfoil. Steady shock-separated computations of the conventional airfoil with the two methods compare well with experiment. Although the interactive boundary layer computations in the shock buffet region compare well with experiment for the conventional airfoil, the thin-layer Navier-Stokes computations do not. These findings are discussed in connection with possible mechanisms important in the onset of shock buffet and the constraints imposed by current numerical modeling techniques.