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New method for analyzing free wake of hovering-helicopter rotor produces more reliable results and requires less computer time. Copes with wake instabilities, both physical and numerical, that afflict usual time-stepping analysis methods. Tip-vortex position, in coordinates moving with rotor blade, updated in relaxation procedure. Wake-airflow solution reached when updates converge to steady wake shape.
In a major technical breakthrough, a computer model for Bridgman-Stockbarger crystal growth was developed. The model includes melt convection, solute effects, thermal conduction in the ampule, melt, and crystal, and the determination of the curved moving crystal-melt interface. The key to the numerical method is the use of a nonuniform computational mesh which moves with the interface, so that the interface is a mesh surface. In addition, implicit methods are used for advection and diffusion of heat, concentration, and vorticity, for interface movement, and for internal gracity waves. This allows large time-steps without loss of stability or accuracy. Numerical results are presented for the interface shape, temperature distribution, and concentration distribution, in steady-state crystl growth. Solutions are presented for two test cases using water, with two different salts in solution. The two diffusivities differ by a factor of ten, and the concentrations differ by a factor of twenty.
A nonlinear structural dynamics program with an element library that exploits parallel processing is under development. The aim is to exploit scheduling-allocation so that parallel processing and vectorization can effectively be treated in a general purpose program. As a byproduct an automatic scheme for assigning time steps was devised. A rudimentary form of the program is complete and has been tested; it shows substantial advantage can be taken of parallelism. In addition, a stability proof for the subcycling algorithm has been developed.
A finite difference solution code for the two dimensional Navier-Stokes equations was combined with a moving-grid system. The thin layer Navier-Stokes equations with a turbulence model are solved in a time-accurate manner in order to study the unsteady aerodynamics around airfoils undergoing small amplitude pitching or heaving motions in the transonic regime. The accuracy of the solutions obtained by the use of the present moving-grid technqiue is investigated. The effects of the minimum grid size and the integrating time-step size on the solutions are also checked. Some of the solutions obtained by the present method are compared with experimental results. It is demonstrated that the unsteady aerodynamics around oscillating airfoils can be predicted fairly well by the present code for cases in which the dynamic angle of attack or displacement is small.
A transonic unsteady aerodynamic and aeroelasticity code has been developed for application to realistic aircraft configurations. The new code is called CAP-TSD which is an acronym for Computational Aeroelasticity Program - Transonic Small Disturbance. The CAP-TSD code uses a time-accurate approximate factorization (AF) algorithm for solution of the unsteady transonic small-disturbance equation. The AF algorithm is very efficient for solution of steady and unsteady transonic flow problems. It can provide accurate solutions in only several hundred time steps yielding a significant computational cost savings when compared to alternative methods. The new code can treat complete aircraft geometries with multiple lifting surfaces and bodies including canard, wing, tail, control surfaces, launchers, pylons, fuselage, stores, and nacelles. Applications are presented for a series of five configurations of increasing complexity to demonstrate the wide range of geometrical applicability of CAP-TSD. These results are in good agreement with available experimental steady and unsteady pressure data. Calculations for the General Dynamics one-ninth scale F-16C aircraft model are presented to demonstrate application to a realistic configuration. Unsteady results for the entire F-16C aircraft undergoing a rigid pitching motion illustrated the capability required to perform transonic unsteady aerodynamic and aeroelastic analyses for such configurations.
The effects of quasi-linear interactions on thick-target electron beams in the solar corona are investigated. Coulomb collisions produce regions of positive gradient in electron distributions which are initially monotonic decreasing functions of energy. In the resulting two-stream instability, energy and momentum are transferred from electrons to Langmuir waves and the region of positive slope in the electron distribution is replaced by a plateau. In the corona, the timescale for this quasi-linear relaxation is very short compared to the collision time. It is therefore possible to model the effects of quasi-linear relaxation by replacing any region of positive slop in the distribution by a plateau at each time step, in such a way as to conserve particle number. The X-ray bremsstrahlung and collisional heating rate produced by a relaxed beam are evaluated. Although the analysis is strictly steady state, it is relevant to the theoretical interpretation of hard X-ray bursts with durations of the order of a few seconds (i.e., the majority of such bursts).
An artificial dissipation model, including boundary treatment, that is employed in many central difference schemes for solving the Euler and Navier-Stokes equations is discussed. Modifications of this model such as the eigenvalue scaling suggested by upwind differencing are examined. Multistage time stepping schemes with and without a multigrid method are used to investigate the effects of changes in the dissipation model on accuracy and convergence. Improved accuracy for inviscid and viscous airfoil flow is obtained with the modified eigenvalue scaling. Slower convergence rates are experienced with the multigrid method using such scaling. The rate of convergence is improved by applying a dissipation scaling function that depends on mesh cell aspect ratio.
A method was developed which calculates two-dimensional, transonic, viscous flow in ducts. The finite-volume, time-marching formulation is used to obtain steady flow solutions of the Reynolds-averaged form of the Navier-Stokes equations. The entire calculation is performed in the physical domain. Control volumes are chosen so that smoothing of flow properties, typically required for stability, is not required. Different time steps are used in the different governing equations. A new pressure interpolation scheme is introduced which improves the shock capturing ability of the method. A multi-volume method for pressure changes in the boundary layer allows calculations which use very long and thin control volumes (length/height - 1000). The method is compared with two test cases. Essentially incompressible turbulent boundary layer flow in an adverse pressure gradient is calculated and the computed distributions of mean velocity and shear are in good agreement with the measurements. Transonic viscous flow in a converging diverging nozzle is calculated; the Mach number upstream of the shock is approximately 1.25. The agreement between the calculated and measured shock strength and total pressure losses is good.
Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps solves exactly the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to pseudo acoustic waves and a non-physical stationary wave. The implications of these findings for the design of high-resolution computing schemes are discussed.
A procedure is presented for applying a radiative boundary condition to the lateral boundary of a limited domain model. The model considered, the Terminal Area Simulation System (TASS), has nonhydrostatic compressible and unsteady governing equations. The primary applications to date have been studies of downbursts, wind shear, thunderstorms and mass fires. A time-splitting integration is used with TASS, with the acoustic terms being integrated separately with a smaller time step. A first-order closure approximation is employed to ensure subgrid turbulence closure. TASS applications for two-dimensional axisymmetric and three-dimensional simulations are described. The radiative boundary conditions were applied in the two-dimensional case to the temperature, pressure and non-normal velocity components. Solutions in both cases avoided runaway circulations and experienced minimal domain-wide mass loss. Also, distorsion was minimized at the boundaries.
A fast-slow factored scheme is presented for use with shallow-water primitive equation numerical weather prediction models. The technique was developed to reduce the rotational mode errors which arise when the fast and slow terms of the governing differential equations are treated simultaneously. The method factors out the fast and slow terms along the coordinate directions by means of a modified Crank-Nicolson scheme. A finite-difference spatial discretization is carried out in the zonal and meridional directions to reduce the factorization error to near-zero, and that time steps of 60-90 min can be used to obtain acceptably accurate results, even in the presence of fine spatial structures in the flow.
The drift of trailing vortices in a cross wind near to the ground is modeled by an unsteady, two-dimensional, rotational flowfield with a concentration of large vorticity in spots having finite total strength. The problem is analyzed by a combination of the method of matched asymptotic analyses for the decay of the vortical spots and the Euler solution for the unsteady rotational flow. A primary feature of the numerical scheme is that the grid size and time step depend only the length and velocity scales of the background flow and is independent of the effective core size of a vortical spot which can be much smaller than the grid size. Numerical results are presented to demonstrated the strong interaction between the trajectjories of the vortical spots and the redistribution of vorticity in the background flowfield.
A Lagrangian nodal analysis method for Stirling engines (SEs) is described, validated, and applied to a conventional SE and an isothermalized SE (with fins in the hot and cold spaces). The analysis employs a constant-mass gas node (which moves with respect to the solid nodes during each time step) instead of the fixed gas nodes of Eulerian analysis. The isothermalized SE is found to have efficiency only slightly greater than that of a conventional SE.
The present adaptive FEM technique for convection-dominated problems is based on a Petrov-Galerkin scheme for spatial approximation, whose typical time-step employs test functions chosen to yield an approximate solution coinciding with the exact solutions at the finite element grid nodes. The derivation of truly local a posteriori error estimates is made possible by this procedure, which is also shown to be a very effective solver by the numerical examples presented.
A unified central-difference finite-volume Euler and Navier-Stokes solver with four-stage Runge-Kutta time stepping is presented. The computer code developed for this purpose is capable of solving the standard set and nonstandard sets (zero-total-pressure loss) of Euler equations and the thin-layer and full Navier-Stokes equations. Applications are presented for conical supersonic flows with weak shocks using the standard and nonstandard sets of Euler equations, and the thin-layer and full Navier-Stokes equations for sharp and round-edged delta wings. Applications are also presented for three-dimensional transonic and subsonic flows using the standard set of Euler equations for sharp-edged delta wings. The computational results of the different sets of equations are compared with each other and with the experimental results and conclusions on the validity of these sets to these applications, are presented.
The interaction between vortex rings of finite strength and an axisymmetric rotational background flow is studied by a singular perturbation method, because it is difficult to use a finite-difference method to analyze the viscous decay in the small core of a vortex ring. The analysis is carried out by combining a composite solution of a vortex ring and an unsteady Euler solution for the background rotational flow. Using the method of averaging, a numerical scheme is developed to obtain an Euler solution in which the grid and time-step sizes depend solely on the length and velocity scales of the background flow. Numerical results are presented to illustrate the interaction between the trajectories and decay rates of the vortex rings and the background rotational flow.
A modification of the unsteady finite volume three-dimensional Euler scheme of Denton (1982) is developed to provide more accurate steady state solutions at enhanced convergence rates, and it is assessed for application to detailed axial blade row flow predictions by correlating predictions with data for a wide range of flow fields. Radial and pitchwise smoothing are minimized by the elimination of under-relaxing while time marching, by the addition of an internal transient averaging scheme, and by the use of the maximum stable local time step for each grid element. Results show that the flow field is accurately predicted in regions of primarily axial pressure gradients, though the predictions do not correlate well with the data in regions of strong nonaxial pressure gradients. Though the scheme is second order accurate in the axial direction, it is first order accurate in the other dimensions, smearing the flow field details.
An LU implicit multigrid scheme is developed for the calculation of three-dimensional transonic flow through rotating cascades. This numerical method solves the unsteady Euler equations of gas dynamics in a finite-volume form. The implicit scheme makes it possible to take a much larger time step than is normally permitted in most explicit schemes, while the multigrid method is incorporated to accelerate the convergence rate for steady state calculations. Using this method, computational storage requirements are comparable to those of explicit schemes, while operation counts are considerably less than those found in the more widely-used ADI schemes.