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At least 343 records · Page 19

Optimal trajectories based on linear equations

The Principal results of a recent theory of fuel optimal space trajectories for linear differential equations are presented. Both impulsive and bounded-thrust problems are treated. A new form of the Lawden Primer vector is found that is identical for both problems. For this reason, starting iteratives from the solution of the impulsive problem are highly effective in the solution of the two-point boundary-value problem associated with bounded thrust. These results were applied to the problem of fuel optimal maneuvers of a spacecraft near a satellite in circular orbit using the Clohessy-Wiltshire equations. For this case two-point boundary-value problems were solved using a microcomputer, and optimal trajectory shapes displayed. The results of this theory can also be applied if the satellite is in an arbitrary Keplerian orbit through the use of the Tschauner-Hempel equations. A new form of the solution of these equations has been found that is identical for elliptical, parabolic, and hyperbolic orbits except in the way that a certain integral is evaluated. For elliptical orbits this integral is evaluated through the use of the eccentric anomaly. An analogous evaluation is performed for hyperbolic orbits.

Carter, Thomas E.↗

COMOC 2: Two-dimensional aerodynamics sequence, computer program user's guide

The COMOC finite element fluid mechanics computer program system is applicable to diverse problem classes. The two dimensional aerodynamics sequence was established for solution of the potential and/or viscous and turbulent flowfields associated with subsonic flight of elementary two dimensional isolated airfoils. The sequence is constituted of three specific flowfield options in COMOC for two dimensional flows. These include the potential flow option, the boundary layer option, and the parabolic Navier-Stokes option. By sequencing through these options, it is possible to computationally construct a weak-interaction model of the aerodynamic flowfield. This report is the user's guide to operation of COMOC for the aerodynamics sequence.

Manhardt, P. D.↗

Numerical investigation of internal high-speed viscous flows using a parabolic technique

A feasibility study has been conducted to assess the applicability of an existing parabolic analysis (ADD-Axisymmetric Diffuser Duct), developed previously for subsonic viscous internal flows, to mixed supersonic/subsonic flows with heat addition simulating a SCRAMJET combustor. A study was conducted with the ADD code modified to include additional convection effects in the normal momentum equation when supersonic expansion and compression waves are present. A set of test problems with weak shock and expansion waves have been analyzed with this modified ADD method and stable and accurate solutions were demonstrated provided the streamwise step size was maintained at levels larger than the boundary layer displacement thickness. Calculations made with further reductions in step size encountered departure solutions consistent with strong interaction theory. Calculations were also performed for a flow field with a flame front in which a specific heat release was imposed to simulate a SCRAMJET combustor. In this case the flame front generated relatively thick shear layers which aggravated the departure solution problem. Qualitatively correct results were obtained for these cases using a marching technique with the convective terms in the normal momentum equation suppressed. It is concluded from the present study that for the class of problems where strong viscous/inviscid interactions are present a global iteration procedure is required.

Anderson, O. L.↗

Superconvergence analysis for nonlinear parabolic equation with EQ{sub 1}{sup rot} nonconforming finite element

EQ{sub 1}{sup rot} nonconforming finite element method (FEM) applied to a class of nonlinear parabolic equation is discussed. First, by use of two typical characters of this element (one is that the associated FE interpolation operator is identical to its traditional Ritz projection operator; the other is that the consistency error is of order O(h{sup 2}), one order higher than the interpolation error, when the exact solution of the problems belongs to H{sup 3}(Ω)), the supercloseness of order O(h{sup 2}) in broken H{sup 1}-norm for semi-discrete scheme is obtained without the boundedness of numerical solution in L{sup ∞}-norm through splitting the numerical solution into several parts. Moreover, we get the desired result with requirement of u,u{sub t}∈H{sup 3}(Ω) only. Secondly, a linearized Crank–Nicolson fully discrete scheme is proposed and the superclose property of order O(h{sup 2}+τ{sup 2}) is derived by constructing a suitable auxiliary problem. Finally, a numerical example is carried out to confirm our theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

97 MATHEMATICS AND COMPUTING↗

Transformation of two and three-dimensional regions by elliptic systems

The research during this period continued to expand the class of numerical algorithms that can be accurately and efficiently implemented on overlapping grids. Whereas previous calculations have been used to solve elliptic equations and to find the steady-state solution of parabolic equations, the present work is aimed towards developing time-accurate solution techniques for parabolic and hyperbolic equations. The primary difficulty here is in the correct treatment of the interior boundary nodes that must be updated at each iteration. The implementation of explicit methods is straightforward. However, the common practice of lagging these values when using an implicit methods leads to inconsistencies in the difference equation. One way to avoid this problem is to alternately calculate with an implicit and an explicit method on each subgrid. With this procedure, the explicit method generates boundary values at the next time level which are then used by the implicit step. It can be shown that when a backward implicit method is combined with a forward explicit method, the composite method is second order accurate and unconditionally stable for linear problems. A second area in which progress can be reported is in the distribution of grid points on curves and surfaces.

Mastin, E. W.↗

Multiscale turbulence effects in underexpanded supersonic jets

A modified version of the multiscale turbulence model of Hanjalic has been applied to the problem of underexpanded supersonic jets. In particular, the shock-cell decay resulting from shock-mixing layer interactions has been studied for both mildly interacting and strongly resonant jet conditions. A version of the Hanjalic model which accounts for nonequilibrium energy transfer between two spectral scales of turbulence was incorporated into an existing shock-capturing, parabolized Navier-Stokes computational model in order to perform numerical experiments. Results are presented for nominal initial jet Mach numbers of 2.0, 1.4, and 1.0 and are compared to experiments and to predictions made using single-scale models. The results show significant effects of multi-spectral turbulent energy transfer on the predicted shock-cell decay particularly for the lower jet Mach numbers.

Abdol-Hamid, Khaled S.↗

A fourth-order accurate finite volume method for ideal MHD via upwind constrained transport

Here we present a fourth-order accurate finite volume method for the solution of ideal magnetohydrodynamics (MHD). The numerical method combines high-order quadrature rules in the solution of semi-discrete formulations of hyperbolic conservation laws with the upwind constrained transport (UCT) framework to ensure that the divergence-free constraint of the magnetic field is satisfied. A novel implementation of UCT that uses the piecewise parabolic method (PPM) for the reconstruction of magnetic fields at cell corners in 2D is introduced. The resulting scheme can be expressed as the extension of the second-order accurate constrained transport (CT) Godunov-type scheme that is currently used in the Athena astrophysics code. After validating the base algorithm on a series of hydrodynamics test problems, we present the results of multidimensional MHD test problems which demonstrate formal fourth-order convergence for smooth problems, robustness for discontinuous problems, and improved accuracy relative to the second-order scheme.

97 MATHEMATICS AND COMPUTING↗

The Shenandoah concentrator

A 7 meter diameter, parabolic dish solar collector was designed and developed for first application at Shenandoah, Georgia. Key features and requirements for the collector are outlined. Performance test results for collector testing at Sandia Laboratories in Albuquerque are summarized. The key features, requirements and performance of the solar collector subassemblies/subsystems are discussed: mount and drives, reflector, receiver, and collector control unit. Problems experienced during collector testing in Albuquerque are identified and solutions described.

Poche, A. J.↗

A Sounding Rocket Mission Concept to Acquire High-Resolution Radiometric Spectra Spanning the 9 nm - 31 nm Wavelength Range

When studying Solar Extreme Ultraviolet (EUV) emissions, both single-wavelength, two- dimensional (2D) spectroheliograms and multi-wavelength, one-dimensional (1D) line spectra are important, especially for a thorough understanding of the complex processes in the solar magnetized plasma from the base of the chromosphere through the corona. 2D image data are required for a detailed study of spatial structures, whereas radiometric (i.e., spectral) data provide information on relevant atomic excitation/ionization state densities (and thus temperature). Using both imaging and radiometric techniques, several satellite missions presently study solar dynamics in the EUV, including the Solar Dynamics Observatory (SDO), Hinode, and the Solar-Terrestrial Relations Observatory (STEREO). The EUV wavelengths of interest typically span 9 nm to 31 nm, with the shorter wavelengths being associated with the hottest features (e.g., intense flares and bright points) and the longer wavelengths associated with cooler features (e.g., coronal holes and filaments). Because the optical components of satellite instruments degrade over time, it is not uncommon to conduct sounding rocket underflights for calibration purposes. The authors have designed a radiometric sounding rocket payload that could serve as both a calibration underflight for and a complementary scientific mission to the upcoming Solar Ultraviolet Imager (SUVI) mission aboard the GOES-R satellite (scheduled for a 2015 launch). The challenge to provide quality radiometric line spectra over the 9-31 nm range covered by SUVI was driven by the multilayer coatings required to make the optical components, including mirrors and gratings, reflective over the entire range. Typically, these multilayers provide useful EUV reflectances over bandwidths of a few nm. Our solution to this problem was to employ a three-telescope system in which the optical components were coated with multilayers that spanned three wavelength ranges to cover the three pairs of SUVI bands. The complete system was designed to fit within the Black Brandt-IX 22.-diameter payload skin envelope. The basic optical path is that of a simple parabolic telescope in which EUV light is focused onto a slit and shutter assembly and imaged onto a normal-incidence diffraction grating, which then disperses the light onto a 2048 2048 CCD sensor. The CCD thus records 1D spatial information along one axis and spectral information along the other. The slit spans 40 arc-minutes in length, thus covering a solar diameter out to +/- 1.3 solar radii. Our operations concept includes imaging at three distinct positions: the north-south meridian, the northeast-southwest diagonal, and real-time pointing at an active region. Six 10-second images will be obtained at each position. Fine pointing is provided by the SPARCS-VII attitude control system typically employed on Black Brandt solar missions. Both before and after launch, all three telescopes will be calibrated with the EUV line emission source and monochromater system at NASA's Stray Light Facility at Marshall Spaceflight Center. Details of the payload design, operations concept, and data application will be presented.

Krause, L. Habash↗

Principal normal curvature of surfaces

Certain principal normal curvatures of differential geometry were developed for use in curvature matrices associated with the asymptotic solution of electromagnetic diffraction problems. The effort is directed toward microwave antenna simulations and high speed digital computer analysis of radiometric instruments used to obtain soil moisture, sea state, salinity and temperature data. It is shown that the methods used to develop the principal normal curvatures for paraboloid, hyperboloid, ellipsoid, sphere, and cone can be applied to other radiometer geometries such as the parabolic torus, even though the surface parameterizations are different. It is concluded that deployable offset geometries, distorted by rotational forces and solar loads may be analyzed by similar means given a suitable surface description.

Schmidt, R. F.↗

Improving the convergence rate to steady state of parabolic ADI methods

The present, residuals' L(2)-norms analysis of the rate of convergence to steady state for parabolic ADI solvers allows the prediction of the number of iterations required for convergence, as a function of the Courant number alpha. A modification of current ADI codes is presented which significantly improves the convergence rate and is insensitive to the Courant number over a large range of alpha. This corrected algorithm is tested for the cases of Dirichlet problems for uniform grids of many mesh sizes, mixed Dirichlet-Neumann problems, and problems defined on stretched grids and/or problems with variable coefficients.

Abarbanel, Saul S.↗

A Sub-Hertz, Low-Frequency Vibration Isolation Platform

One of the major technical problems deep-space optical communication (DSOC) systems need to solve is the isolation of the optical terminal from vibrations produced by the spacecraft navigational control system and by the moving parts of onboard instruments. Even under these vibration perturbations, the DSOC transceivers (telescopes) need to be pointed l000 fs of times more accurately than an RF communication system (parabolic antennas). Mechanical resonators have been extensively used to provide vibration isolation for groundbased, airborne, and spaceborne payloads. The effectiveness of these isolation systems is determined mainly by the ability of designing a mechanical oscillator with the lowest possible resonant frequency. The Low-Frequency Vibration Isolation Platform (LFVIP), developed during this effort, aims to reduce the resonant frequency of the mechanical oscillators into the sub-Hertz region in order to maximize the passive isolation afforded by the 40 dB/decade roll-off response of the resonator. The LFVIP also provides tip/tilt functionality for acquisition and tracking of a beacon signal. An active control system is used for platform positioning and for dampening of the mechanical oscillator. The basic idea in the design of the isolation platform is to use a passive isolation strut with an approximately equal to 100-mHz resonance frequency. This will extend the isolation range to lower frequencies. The harmonic oscillator is a second-order lowpass filter for mechanical disturbances. The resonance quality depends on the dissipation mechanisms, which are mainly hysteretic because of the low resonant frequency and the absence of any viscous medium. The LFVIP system is configured using the well-established Stewart Platform, which consists of a top platform connected to a base with six extensible struts (see figure). The struts are attached to the base and to the platform via universal joints, which permit the extension and contraction of the struts. The struts ends are connected in pairs to the base and to the platform, forming an octahedron. The six struts provide the vibration isolation due to the properties of mechanical oscillators that behave as second-order lowpass filters for frequencies above the resonance. At high frequency, the ideal second-order low-pass filter response is spoiled by the distributed mass and the internal modes of membrane and of the platform with its payload.

Ortiz, Gerardo, G.↗

Massive separation and dynamic stall on a cusped trailing-edge airfoil

The cross-over from a predominantly attached two-dimensional flow to the bluff body form of separation is modeled via the interacting boundary layer approximation. The initial breakdown of the predominantly attached flow on a cusped trailing edge airfoil is examined using the Hilbert integral form of the unsteady interacting boundary layer equations. In addition, an interacting boundary layer technique is developed for calculating bluff body separation. This new model eliminates the severe scaling problems associated with bluff body separation through the use of a realistic inviscid eddy model, based on the infinite eddy Kirchhoff free-streamline description of separation. Brief consideration is given to the cross-over from bluff body separation to a predominantly attached flow, the extension to finite eddies and cascade flows, and the possible coupling with full, or parabolized, Navier-Stokes calculations.

Rothmayer, A. P.↗

Three dimensional PNS solutions of hypersonic internal flows with equilibrium chemistry

An implicit procedure for solving parabolized Navier-Stokes equations under the assumption of a general equation of state for a gas in chemical equilibrium is given. A general and consistent approach for the evaluation of Jacobian matrices in the implicit operator avoids the use of unnecessary auxiliary quantities and approximations, and leads to a simple expression. Applications to two- and three-dimensional flow problems show efficiency in computer time and economy in storage.

Liou, May-Fun↗

Simulations of Boundary-Layer Transition

For incompressible benchmark flows, we have demonstrated the capability of the parabolized stability equations (PSE) to simulate the transition process in excellent agreement with microscopic experiments and direct Navier-Stokes simulations at modest computational cost. Encouraged by these results, we have developed the PSE methodology of three-dimensional boundary-layers in general curvilinear coordinates for the range from low to hypersonic speeds, and for both linear and nonlinear problems. For given initial and boundary conditions, the approach permits simulations from receptivity through linear and secondary instabilities into the late stages of transition where significant changes in skin friction and heat transfer coefficients occur. We have performed transition simulations for a variety of two- and three-dimensional similarity solutions and for realistic flows over swept wings at subsonic and supersonic speeds, the pressure ans suction side of turbine blades at low and medium turbulence levels, and over a blunt cone at Mach number Ma = 8. We present selected results for different transition mechanisms with emphasis on the late stage of transition and the evolution of wall-shear stress and heat transfer.

Herbert, Thorwald↗

Calculation of three-dimensional, viscous flow through turbomachinery blade passages by parabolic marching

The three-dimensional compressible Navier-Stokes equations are formulated in a rotating coordinate system, so as to include centrifugal and Coriolis forces. The equations are parabolized by using a previously calculated inviscid static pressure field. The thin layer Navier-Stokes approximation, which neglects streamwise diffusion, is used. A body-fitted coordinate system is used. The streamwise momentum equation is uncoupled from the cross-stream momentum equation by using contravariant momentum components, and then using the contravariant velocity components as primary unknowns. To reduce problems with small separated regions, the Reyhner and Flugge-Lotz approximation is used. The energy equation is included to allow for calculation of heat transfer. The flow may be laminar, or a simple eddy-viscosity turbulence may be used. A number of curved ducts and an axial stator were analyzed, including cases for which experimental data are available.

Katsanis, T.↗

Calculation of three-dimensional, viscous flow through turbomachinery blade passages by parabolic marching

The three-dimensional compressible Navier-Stokes equations are formulated in a rotating coordinate system, so as to include centrifugal and Coriolis forces. The equations are parabolized by using a previously calculated inviscid static pressure field. The thin layer Navier-Stokes approximation, which neglects streamwise diffusion, is used. A body-fitted coordinate system is used. The streamwise momentum equation is uncoupled from the cross-stream momentum equation by using contravariant momentum components, and then using the contravariant velocity components as primary unknowns. To reduce problems with small separating regions, the Reyhner and Flugge-Lotz approximation is used. The energy equation is included to allow for calculation of heat transfer. The flow may be laminar, or a simple eddy-viscosity turbulence may be used. A number of curved ducts and an axial stator were analyzed, including cases for which experimental data are available.

Katsanis, T.↗