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At least 1,225 records · Page 68

Solution of 3-dimensional time-dependent viscous flows. Part 1: Investigation of candidate algoriths

There is considerable interest in developing a numerical scheme for solving the time dependent viscous compressible three dimensional flow to aid in the design of helicopter rotors. Numerical algorithms are examined to determine their overall suitability for the efficient and routine solution of an appropriate system of partial differential equations. It is concluded that a consistently split time linearized block implicit scheme using either quintic B spline collocation or the generalized operator compact implicit approach to generate a fourth order accurate algorithm is particularly well suited for use on the present problem. High cell Reynolds number behavior leads to favoring the generalized operator compact implicit approach over the quintic B spline collocation method.

Weinberg, B. C.↗

Contributions on the Subject of Longitudinal Movements of Aircraft in Wind Shears

The effect of downburst-type wind shears on the longitudinal dynamic behavior of an unguided aircraft is simulated numerically on the basis of published meteorological data and the flight characteristics of an A300-B passenger jet. The nonlinear differential equations of the aircraft motion are linearized by conventional methods, and the wind effects are introduced via the linear derivatives of the wind components referred to the wind gradients to obtain simplified technical models of the longitudinal response to all possible types of constant-gradient wind shears during the first 20-60 sec. Graphs, maps, and diagrams are provided, and a number of accidents presumed to have involved wind shears are analyzed in detail.

Krauspe, P.↗

Exact finite difference schemes for the non-linear unidirectional wave equation

Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.

Mickens, R. E.↗

On the structure of parallelism in a highly concurrent PDE solver

A parallel multigrid algorithm for solving elliptic partial differential equations is developed and evaluated. A V-cycle multigrid method is altered to increase the degree of parallelism. A numerical analysis of the resulting concurrent-iteration multigrid algorithm is performed; its architectural implications are considered; highly parallel systems without shared memory are examined (including mesh-connected arrays, mesh-shuffle-connected systems, permutation networks, and direct VLSI embeddings); and the results of numerical experiments are presented in tables and graphs.

Gannon, D.↗

Nonlinear programming extensions to rational function approximations of unsteady aerodynamics

This paper deals with approximating unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft. Two methods of formulating these approximations are extended to include both the same flexibility in constraining them and the same methodology in optimizing nonlinear parameters as another currently used 'extended least-squares' method. Optimal selection of 'nonlinear' parameters is made in each of the three methods by use of the same nonlinear (nongradient) optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is of lower order than that required when no optimization of the nonlinear terms is performed. The free 'linear' parameters are determined using least-squares matrix techniques on a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from the different approaches are described, and results are presented which show comparative evaluations from application of each of the extended methods to a numerical example. The results obtained for the example problem show a significant (up to 63 percent) reduction in the number of differential equations used to represent the unsteady aerodynamic forces in linear time-invariant equations of motion as compared to a conventional method in which nonlinear terms are not optimized.

Tiffany, Sherwood H.↗

A direct algorithm for solution of incompressible three-dimensional unsteady Navier-Stokes equations

A direct, implicit, numerical solution algorithm, with second-order accuracy in space and time, is constructed for the three-dimensional unsteady incompressible Navier-Stokes equations formulated in terms of velocity and vorticity, using generalized orthogonal coordinates to achieve the accurate solution of complex viscous flow configurations. A numerically stable, efficient, direct inversion procedure is developed for the computationally intensive divergence-curl elliptic velocity problem. This overdetermined partial differential operator is first formulated as a uniquely determined, nonsingular matrix-vector problem; this aspect of the procedure is a unique feature of the present analysis. The three-dimensional vorticity-transport equation is solved by a modified factorization technique which completely eliminates the need for any block-matrix inversions and only scalar tridiagonal matrices need to be inverted. The method is applied to the test problem of the three-dimensional flow within a shear-driven cubical box. Coherent streamwise vortex structures are observed within the steady-state flow at Re = 100.

Osswald, G. A.↗

Tropopause adjustment to tropical cyclones as inferred from satellite ozone observations

Data from the Total Ozone Mapping Spectrometer on the Nimbus-7 are used to map the climatological total ozone within and around tropical cyclones. A tropical cyclone numerical model is used to interpret total ozone patterns. The total ozone maps are used to diagnose storm intensity and to differentiate between intesifying and weakening tropical cyclones. The tropical cyclone intensity versus total ozone anomaly distribution is examined and examples of mapping the total ozone of tropical cyclones in the period 1979-1982 are presented.

Rodgers, Edward B.↗

Numerical algorithms for steady and unsteady incompressible Navier-Stokes equations

The numerical analysis of the incompressible Navier-Stokes equations are becoming important tools in the understanding of some fluid flow problems which are encountered in research as well as in industry. With the advent of the supercomputers, more realistic problems can be studied with a wider choice of numerical algorithms. An alternative formulation is presented for viscous incompressible flows. The incompressible Navier-Stokes equations are cast in a velocity/vorticity formulation. This formulation consists of solving the Poisson equations for the velocity components and the vorticity transport equation. Two numerical algorithms for the steady two-dimensional laminar flows are presented. The first method is based on the actual partial differential equations. This uses a finite-difference approximation of the governing equations on a staggered grid. The second method uses a finite element discretization with the vorticity transport equation approximated using a Galerkin approximation and the Poisson equations are obtained using a least squares method. The equations are solved efficiently using Newton's method and a banded direct matrix solver (LINPACK). The method is extended to steady three-dimensional laminar flows and applied to a cubic driven cavity using finite difference schemes and a staggered grid arrangement on a Cartesian mesh. The equations are solved iteratively using a plane zebra relaxation scheme. Currently, a two-dimensional, unsteady algorithm is being developed using a generalized coordinate system. The equations are discretized using a finite-volume approach. This work will then be extended to three-dimensional flows.

Hafez, Mohammed↗

Simulation of the collapse and dissipation of Langmuir wave packets

Particle-in-cell (PIC) simulations and Zakharov's partial differential equations (PDEs) are used to investigate the collapse of isolated Langmuir wave packets in two dimensions. Collapse thresholds are determined numerically, and the roles of enhanced Langmuir wave damping and nonlinearities not included in the standard Zakharov equations are discussed. The Langmuir wave and ion dynamics in PIC simulations are compared with the predictions of PDE simulations incorporating enhanced Langmuir damping. Electron heating and coherent acceleration in the PIC simulations are discussed and compared with predictions of the transit-time theory.

Newman, D. L.↗

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology.

Barth, Timothy J.↗

Using adaptive grid in modeling rocket nozzle flow

The mechanical behavior of a rocket motor internal flow field results in a system of nonlinear partial differential equations which cannot be solved analytically. However, this system of equations called the Navier-Stokes equations can be solved numerically. The accuracy and the convergence of the solution of the system of equations will depend largely on how precisely the sharp gradients in the domain of interest can be resolved. With the advances in computer technology, more sophisticated algorithms are available to improve the accuracy and convergence of the solutions. An adaptive grid generation is one of the schemes which can be incorporated into the algorithm to enhance the capability of numerical modeling. It is equivalent to putting intelligence into the algorithm to optimize the use of computer memory. With this scheme, the finite difference domain of the flow field called the grid does neither have to be very fine nor strategically placed at the location of sharp gradients. The grid is self adapting as the solution evolves. This scheme significantly improves the methodology of solving flow problems in rocket nozzles by taking the refinement part of grid generation out of the hands of computational fluid dynamics (CFD) specialists and place it into the computer algorithm itself.

Chow, Alan S.↗

The rollup of a vortex layer near a wall

The behavior of an inviscid vortex layer of non-zero thickness near a wall is studied, both through direct numerical simulation of the two-dimensional vorticity equation at high Reynolds numbers, and using an approximate ordinary nonlinear integro-differential equation which is satisfied in the limit of a thin layer under long-wavelength perturbations. For appropriate initial conditions the layer rolls up and breaks into compact vortices which move along the wall at constant speed. Because of the effect of the wall, they correspond to equilibrium counter-rotating vortex dipoles. This breakup can be related to the disintegration of the initial conditions of the approximate nonlinear dispersive equation into solitary waves. The study is motivated by the formation of longitudinal vortices from vortex sheets in the wall region of a turbulent channel.

Jimenez, Javier↗

Effects of non-unity Lewis numbers in diffusion flames

The purpose of this work is to carry out direct numerical simulations of diffusion controlled combustion with non-unity Lewis numbers for the reactants and products, thus accounting for the differential diffusion effects of the temperature and concentration fields. We use a formulation based on combining the conservation equations in a way to eliminate the reaction terms similar to the method used by Burke and Schumann (1928) for unity Lewis numbers. We present calculations for an axisymmetric fuel jet and for a planar, time evolving mixing layer, leaving out the effects of thermal expansion and variations of the transport coefficients due to the heat release. Our results show that the front of the flame shifts toward the fuel or oxygen sides owing to the effect of the differential diffusion and that the location of maximum temperature may not coincide with the flame. The dependence of the distribution of the reaction products on their Lewis number has been investigated.

Linan, A.↗

Radial alignment simulation of solar wind streams observed by Pioneers 10 and 11 in 1974

A particularly favorable radial lineup between spacecraft in Earth orbit and Pioneers 10 and 11 (near the ecliptic plane at approximately 5.5 AU and approximately 4.5 AU, respectively) occurred in mid-1974, when the solar corona was in a stable and well-defined warped-dipole configuration. The radial alignment study reported here differs from previous applications of the technique in two respects: (1) It is the first time a two-dimensional (2-D) MHD model has been tested over such a lengthy propagation interval; the 2-D capability is crucial for treatment of the nonradial shearing motions occurring across the stream interface. (2) The three-dimensional (3-D) structure observed in the white light corona at that time is related to the systematic patterns of nonradial flow deflections appearing at the Pioneer corotating interaction region (CIR) fronts. Comparison of predicted and observed flows for pairs of streams in two successive rotations reveals that when the parent coronal hole projects far across the solar equator in a predominantly north-south orientation (i.e., nearest the 2-D idealization), the mapping is accurate down to details of the flow structures. But where the spacecraft tracks along a latitudinal boundary of a hole or the associated stream front is inclined at a shallow angle to the equator, the numerical projections deviate systematically from the observations. Among the sources of error are 3-D dynamical interactions neglected in the model, differential rotation effects, and slow temporal evolution of the coronal structures. A characteristic pattern of north-south and east-west deflections indicative of the 3-D geometry appears across the CIRs, but these patterns evidently reflect primarily the local, as opposed to global, orientation of the 3-D stream fronts. Such patterns appear common in CIRs observed by Pioneer during this period. These findings thus hold special relevance for the analysis of Ulysses observations, since the present coronal configuration is similar to that of 1974.

Pizzo, V. J.↗

De-Aliasing Through Over-Integration Applied to the Flux Reconstruction and Discontinuous Galerkin Methods

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

High-Order↗

Revisiting matrix-based inversion of scanning mobility particle sizer (SMPS) and humidified tandem differential mobility analyzer (HTDMA) data

Abstract. Tikhonov regularization is a tool for reducing noise amplification during data inversion. This work introduces RegularizationTools.jl, a general-purpose software package for applying Tikhonov regularization to data. The package implements well-established numerical algorithms and is suitable for systems of up to ∼ 1000 equations. Included is an abstraction to systematically categorize specific inversion configurations and their associated hyperparameters. A generic interface translates arbitrary linear forward models defined by a computer function into the corresponding design matrix. This obviates the need to explicitly write out and discretize the Fredholm integral equation, thus facilitating fast prototyping of new regularization schemes associated with measurement techniques. Example applications include the inversion involving data from scanning mobility particle sizers (SMPSs) and humidified tandem differential mobility analyzers (HTDMAs). Inversion of SMPS size distributions reported in this work builds upon the freely available software DifferentialMobilityAnalyzers.jl. The speed of inversion is improved by a factor of ∼ 200, now requiring between 2 and 5 ms per SMPS scan when using 120 size bins. Previously reported occasional failure to converge to a valid solution is reduced by switching from the L-curve method to generalized cross-validation as the metric to search for the optimal regularization parameter. Higher-order inversions resulting in smooth, denoised reconstructions of size distributions are now included in DifferentialMobilityAnalyzers.jl. This work also demonstrates that an SMPS-style matrix-based inversion can be applied to find the growth factor frequency distribution from raw HTDMA data while also accounting for multiply charged particles. The outcome of the aerosol-related inversion methods is showcased by inverting multi-week SMPS and HTDMA datasets from ground-based observations, including SMPS data obtained at Bodega Marine Laboratory during the CalWater 2/ACAPEX campaign and co-located SMPS and HTDMA data collected at the US Department of Energy observatory located at the Southern Great Plains site in Oklahoma, USA. Results show that the proposed approaches are suitable for unsupervised, nonparametric inversion of large-scale datasets as well as inversion in real time during data acquisition on low-cost reduced-instruction-set architectures used in single-board computers. The included software implementation of Tikhonov regularization is freely available, general, and domain-independent and thus can be applied to many other inverse problems arising in atmospheric measurement techniques and beyond.

54 ENVIRONMENTAL SCIENCES↗

Nonlinear modeling and performance analysis of cracked beam microgyroscopes

Cracks constitute a common structural defect in microelectromechanical systems that may arise during manufacturing, mechanical fatigue, or shock loading. Areas weakened by cracks increase the flexibility of the microstructure. Furthermore, depending on the crack severity, the increased flexibility can cause dramatic changes in the static and dynamic behaviors of electrically actuated MEMS. In this study, a numerical investigation of the depth and location of multiple edge cracks on the performance of beam microgyroscopes is conducted. Applying the Griffith strain energy release theorem for an edge crack, the vibrational characteristics of microgyroscopes with multiple cracks including the static deflection, natural frequencies, and mode shapes are studied analytically. Then, the differential quadrature method is employed to model the nonlinear dynamic behaviors of the cracked gyroscope system. Cracks initiated along the driving or sensing directions of the microgyroscope are found to have differing effects on its nonlinear dynamic response. This numerical study reveals that the onset of severe single cracks or a series of multiple cracks can significantly affect the sensitivity of the microgyroscope to base rotations and therefore leads to a degradation in the performance of the damaged MEMS sensor. This study also shows that a potential design approach for broadband microgyroscope systems manufactured with cracks can be proposed.

42 ENGINEERING↗