A Class of Functional Equations of Neutral Type
Class of functional integral equations in space of continuous functions
INTEGRAL EQUATION↗
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A Continuous Square Root Information Filter-Smoother with Discrete Data Update
Data filters may be separated into two categories being continuous filters or discrete filters. Continuous data filters are described by a matrix differential equation or Ricatti equation and discrete filters are obtained by solving the continuous equations over some finite time interval.
filter↗
The equations describing the cosmic-ray gas in the interplanetary region.
Continuity and momentum equations for cosmic ray gas particles in interplanetary region
Gleeson, L. J.↗
Approximate factorization for incompressible flow
For computational solution of the incompressible Navier-Stokes equations, the approximate factorization (AF) algorithm is used to solve the vectorized momentum equation in delta form based on the pressure calculated in the previous time step. The newly calculated velocities are substituted into the pressure equation (obtained from a linear combination of the continuity and momentum equation), which is then solved by means of line SOR. Computational results are presented for the NACA 66 sub 3 018 airfoil at Reynolds numbers of 1000 and 40,000 and attack angles of 0 and 6 degrees. Comparison with wind tunnel data for Re = 40,000 indicates good qualitative agreement between measured and calculated pressure distributions. Quantitative agreement is only fair, however, with the calculations somewhat displaced from the measurements. Furthermore, the computed velocity profiles are unrealistically thick around the airfoil, due to the excessive amount of artificial viscosity needed for stability. Based on the performance of the algorithm with regard to stability, it is concluded that AF/SOR is suitable for calculations at Reynolds numbers less than 10,000. Speedwise, the method is faster than point SOR by at least a factor of two.
Bernard, R. S.↗
Compressibility effects on rotor forces in the leakage path between a shrouded pump impeller and its housing
A modified approach to Childs' previous work on fluid-structure interaction forces in the leakage path between an impeller shroud and its housing is presented in this paper. Three governing equations consisting of continuity, path-momentum, and circumferential-momentum equations were developed to describe the leakage path inside a pump impeller. Radial displacement perturbations were used to solve for radial and circumferential force coefficients. In addition, impeller-discharge pressure disturbances were used to obtain pressure oscillation responses due to precessing impeller pressure wave pattern. Childs' model was modified from an incompressible model to a compressible barotropic-fluid model (the density of the working fluid is a function of the pressure and a constant temperature only). Results obtained from this model yielded interaction forces for radial and circumferential force coefficients. Radial and circumferential forces define reaction forces within the impeller leakage path. An acoustic model for the same leakage path was also developed. The convective, Coriolis, and centrifugal acceleration terms are removed from the compressible model to obtain the acoustics model. A solution due to impeller discharge pressure disturbances model was also developed for the compressible and acoustics models. The results from these modifications are used to determine what effects additional perturbation terms in the compressible model have on the acoustic model. The results show that the additional fluid mechanics terms in the compressible model cause resonances (peaks) in the force coefficient response curves. However, these peaks only occurred at high values of inlet circumferential velocity ratios greater than 0.7. The peak pressure oscillation was shown to occur at the wearing ring seal. Introduction of impeller discharge disturbances with n = 11 diametral nodes showed that maximum peak pressure oscillations occurred at nondimensional precession frequencies of f = 6.4 and f = 7.8 for this particular pump. Bolleter's results suggest that for peak pressure oscillations to occur at the wearing ring seal, the nondimensional excitation frequency should be on the order of f = 2.182 for n = 11. The resonances found in this research do not match the excitation frequencies predicted by Bolleter. At the predicted peak excitation frequencies given by Bolleter, the compressible model shows an attenuation of the pressure oscillations at the seal exit. The compressibility of the fluid does not have a significant influence on the model at low values of nondimensional excitation frequency. At high values of nondimensional frequency, the effects of compressibility become more significant. For the acoustic analysis, the convective, Coriolis, and centrifugal acceleration terms do affect the results to a limited extent for precession excitation and to a large extent for a pressure excitation when the fluid operates at relatively high Mach numbers.
Cao, Nhai The↗
Stability criteria for n-th order, homogeneous linear differential equations
Stability criteria for n-th order homogeneous linear differential equations for real continuous functions obtained by Liapunov method
LIAPUNOV FUNCTION↗
Radiation-Controlled, Opposed-Flow Flame Spread in a Microgravity Environment
The effects of surface and gas-phase radiation on the rate and the structure of laminar flame spread over thin fuels are investigated using a flame-spread model which consists of the continuity, momentum, species, and energy equations in the gas and the continuity and energy equations in the solid. Numerical calculations, complemented by scaling arguments, show that, at high velocities of the oxidizer flow, radiation effects are unimportant; the spread rate decreases with increasing opposing velocity due to finite-rate gas-phase kinetics. However, radiation becomes progressively important when the opposing velocity is below a certain value: the flame cools, shrinks in size, and its spread rate falls sharply with decreasing opposing velocity.
Bhattacharjee, Subrata↗
Power Operations of the Mars Exploration Rovers
The rovers of the National Aeronautics and Space Administration’s (NASA) Mars Exploration Rovers (MER) project, Spirit (MER-A) and Opportunity (MER-B), safely landed on the surface of Mars three weeks apart during January 2004. Spirit and Opportunity were built and operated by the Jet Propulsion Laboratory (JPL), which is managed by the California Institute of Technology (Caltech) for NASA. Spirit landed at Gusev Crater, 14.8 degrees south of the equator, and operated continuously on the surface of Mars from January 4, 2004 until last contact from Spirit on March 22, 2010. Opportunity landed at Meridiani Planum, a location 2.5 degrees south of the Martian equator. Opportunity operated continuously on the Martian surface from January 25, 2004 until the last received transmission from Opportunity on June 10, 2018. The goal of the MER project was to determine if Mars ever had a habitable environment, in particular, if it ever had water. During their missions, both Spirit and Opportunity found evidence that liquid water once flowed on the surface of Mars. Both Spirit and Opportunity used a 1.33 m2 triple-junction solar array as their power sources. Based on observations of the original Mars rover, the Sojourner rover of the Mars Pathfinder mission that landed on Mars on July 4, 1997, the expectation was that the Martian dust would rapidly accumulate on the solar arrays of Spirit and Opportunity, and that the rovers would not have enough energy to continue operations after 90 Martian days (sols). Instead, due in part to lower dust accumulation rates than expected and numerous dust cleaning events, the Spirit and Opportunity rovers continued to operate on the Martian surface for over 2000 sols (MER-A) and 5000 sols (MER-B), respectively. During this time, the rovers experienced multiple Martian winters and several dust storms. Because the sources of solar array energy loss were known, the solar array energy output offered a method to scientifically estimate the loading and aeolian removal of dust from the solar arrays each sol. The MER Power operations team called this value that they calculated the solar array Dust Factor (DF). Dust Factor was defined as the fraction of sunlight that penetrates the accumulated dust on the surface of the solar array. A Dust Factor of 1.0 would indicate that the solar array was perfectly clean. A Dust Factor of 0.6 would indicate that only 60% of the available sunlight was able to penetrate the accumulated dust on the solar arrays. The MER Power subsystem operations team used the Multi-Mission Power Analysis Tool (MMPAT) to perform these Dust Factor calculations. The MMPAT software tool modeled the behavior of the solar arrays and the batteries as they interacted with the spacecraft power loads over the mission timeline. MMPAT also had knowledge (through telemetry and user inputs) of telemetered Power subsystem voltages and currents, atmospheric opacity (Tau), rover surface location, rover attitude, the planetary tilt and distance of Mars from the sun based on day of year, the instantaneous elevation of the sun based on time of day, temperatures (internal and external), terrain masking, and shadowing (due to the camera mast and antennas). Once all of the known sources of array energy loss are accounted, the remaining difference between the expected array energy and the actual array energy determines the solar array Dust Factor. The assumptions made while determining the Dust Factor are 1) that the single measured atmospheric opacity value (Tau) is constant over the course of the entire sol, 2) there is no measurable solar cell degradation, 3) there are no shorted strings and therefore, 4) all unexpected solar array energy losses are due to accumulated dust on the solar array. Although it cannot provide an absolute measure of dust loading, the determination of solar array Dust Factor provides a useful way of tracking dust accumulation and aeolian dust removal trends on Martian spacecraft. Any spacecraft on the Martian surface is vulnerable to dust, especially solar-powered spacecraft. The Spirit and Opportunity rovers were operational on the Martian surface for much longer than expected due in part to aeolian removal of dust from their solar arrays. The first few dust removal events were a pleasant surprise to the MER operations teams; however, over time a pattern began to arise. In over three Mars Years on the surface, the Power operations team tracked the solar array Dust Factor at Gusev Crater (location of MER-A) and observed that there were several significant dust removal and deposition events in Mars Year (MY) 27, even in the absence of a large dust storm. In MY 28 at Gusev Crater, the large atmospheric opacity (Tau) increase lagged significant dust removal events. Overall at Gusev Crater, there was a pattern of steady dust accumulation on the solar arrays, with a small number of significant dust cleaning events. At Meridiani Planum, where Opportunity rover operated for over seven Mars Years (late MY 26 to mid MY 34), a clear and consistent pattern of dust movement emerged. Meridiani Planum had a predictable, seasonally dependent pattern of gradual and continuous dust accumulation and removal. In summary, this paper explains the reasons for the development of the solar array Dust Factor and how it was used in mission operations. In particular, this paper describes the MMPAT software package, how it models array energy, including the important assumptions, model inputs and sources of error. And finally, this paper will show how the calculated solar array Dust Factor was used at Gusev Crater and Meridiani Planum to predict dust accumulation rates and weather patterns, and the importance of this generated data set for current and future solar- powered missions to Mars, such as the InSight lander and the planned Mars Sample Return rover.
Chin, Keith B.↗
Application of viscous-inviscid interaction methods to transonic turbulent flows
Two different viscous-inviscid interaction schemes were developed for the analysis of steady, turbulent, transonic, separated flows over axisymmetric bodies. The viscous and inviscid solutions are coupled through the displacement concept using a transpiration velocity approach. In the semi-inverse interaction scheme, the viscous and inviscid equations are solved in an explicitly separate manner and the displacement thickness distribution is iteratively updated by a simple coupling algorithm. In the simultaneous interaction method, local solutions of viscous and inviscid equations are treated simultaneously, and the displacement thickness is treated as an unknown and is obtained as a part of the solution through a global iteration procedure. The inviscid flow region is described by a direct finite-difference solution of a velocity potential equation in conservative form. The potential equation is solved on a numerically generated mesh by an approximate factorization (AF2) scheme in the semi-inverse interaction method and by a successive line overrelaxation (SLOR) scheme in the simultaneous interaction method. The boundary-layer equations are used for the viscous flow region. The continuity and momentum equations are solved inversely in a coupled manner using a fully implicit finite-difference scheme.
Lee, D.↗
On the continuous spectra of the compressible boundary layer stability equations
The branch cuts in the complex frequency plane (omega) that correspond to the continuous spectrum of the two-dimensional compressible boundary layer stability equations are computed by looking for solutions that are pure oscillatory in the free-stream. In the complex omega-plane there are three compressible and one incompressible branch cuts. The compressible branch cuts are given as solutions of third order polynomials. Computations are made for a range of Reynolds and Mach numbers, and also for several streamwise wavenumbers.
Ashpis, David E.↗
Three-dimensional fully spectral numerical method for mantle convection with depth-dependent properties
A semi-implicit fully spectral collocation method for the simulation of three-dimensional mantle convection with depth-dependent thermo-dynamic and transport properties is presented. The variable property Navier-Stokes equation expressed in terms of the primitive variable velocity and pressure is solved with the mass continuity and temperature equations. The periodic horizontal boundary conditions allow a Fourier expansion for the two horizontal directions. The stress-free, impermeable isothermal boundary conditions along with the depth dependent coefficients are handled with a Chebyshev expansion in the vertical direction. In the limit of an infinite Prandtl number appropriate to mantle convection, the inertial terms in the momentum equation are unimportant. In this case an explicit solution of a Poisson equation for pressure can be avoided; instead a fourth-order equation for vertical velocity can be solved. Simultaneous imposition of both impermeable and continuity boundary conditions during the vertical velocity evaluation is discussed. The pressure distributions on the top and bottom bounding planes were determined by means of an influence matrix technique. The numerical method employed here avoids time-splitting errors and enforces velocity boundary conditions and continuity over the entire domain, including the boundaries, to machine accuracy. Strongly time-dependent three-dimensional solutions up to a surface Rayleigh number of 1 x 10(exp 7) have been obtained. Strong upwellings, pulsating chaotically, are formed by the collective merging of cylindrical plumes.
Balachandar, S.↗
On the append and continue features in NASTRAN
The APPEND feature is described which is applicable in the case of real eigenvalue analysis. This feature permitted the addition of new eigenvalues and eigenvectors to those already computed in a previously checkpointed run without re-executing the entire problem. The next feature was the CONTINUE feature which was applicable in the case of transient analysis of coupled equations. This feature enabled the integration of coupled equations to be continued beyond the last output time for which the solution was obtained in a previously checkpointed run (without re-executing the entire problem). These two features are illustrated by suitable examples.
Pamidi, P. R.↗
A note on the plasma moment equations in a dipole field
Thermal plasma motion in magnetic dipole field defined by direction in coordinate system of continuity and momentum equations
Grebowsky, J. M.↗
A space-marching method for incompressible Navier-Stokes equations
This paper deals with the development of a space-marching method for incompressible flows. The method solves the continuity and momentum equations as a coupled system at each streamwise station. The character of the system of equations has been changed from elliptic to hyperbolic/parabolic in order to enable the equations to be marched in space. The present method has many advantages compared to the existing parabolic or space-marching methods for incompressible flow: (1) it avoids the solution of Poisson equations, (2) it conserves the mass flow with no additional computation, (3) it does not require the specification of an assumed pressure field when used in the prediction of duct flows. The present method can capture strong secondary velocities and strong transverse pressure gradients. Predictions of the flow through straight and curved ducts are in good agreement with analytical and experimental results.
Pouagare, M.↗
Unsteady transonic flow in cascades
There is a need for methods to predict the unsteady air loads associated with flutter of turbomachinery blading at transonic speeds. The results of such an analysis in which the steady relative flow approaching a cascade of thin airfoils is assumed to be transonic, irrotational, and isentropic is presented. The blades in the cascade are allowed to undergo a small amplitude harmonic oscillation which generates a small unsteady flow superimposed on the existing steady flow. The blades are assumed to oscillate with a prescribed motion of constant amplitude and interblade phase angle. The equations of motion are obtained by linearizing about a uniform flow the inviscid nonheat conducting continuity and momentum equations. The resulting equations are solved by employing the Weiner Hopf technique. The solution yields the unsteady aerodynamic forces acting on the cascade at Mach number equal to 1. Making use of an unsteady transonic similarity law, these results are compared with the results obtained from linear unsteady subsonic and supersonic cascade theories. A parametric study is conducted to find the effects of reduced frequency, solidity, stagger angle, and position of pitching axis on the flutter.
Surampudi, S. P.↗
Unsteady transonic flow in cascades
There is a need for methods to predict the unsteady air loads associated with flutter of turbomachinery blading at transonic speeds. The results of such an analysis in which the steady relative flow approaching a cascade of thin airfoils is assumed to be transonic, irrotational, and isentropic is presented. The blades in the cascade are allowed to undergo a small amplitude harmonic oscillation which generates a small unsteady flow superimposed on the existing steady flow. The blades are assumed to oscillate with a prescribed motion of constant amplitude and interblade phase angle. The equations of motion are obtained by linearizing about a uniform flow the inviscid nonheat conducting continuity and momentum equations. The resulting equations are solved by employing the Weiner Hopf technique. The solution yields the unsteady aerodynamic forces acting on the cascade at Mach number equal to 1. Making use of an unsteady transonic similarity law, these results are compared with the results obtained from linear unsteady subsonic and supersonic cascade theories. A parametric study is conducted to find the effects of reduced frequency, solidity, stagger angle, and position of pitching axis on the flutter.
Surampudi, S. P.↗
The development of flux-split algorithms for flows with non-equilibrium thermodynamics and chemical reactions
A finite-volume method for the numerical computation of flows with nonequilibrium thermodynamics and chemistry is presented. A thermodynamic model is described which simplifies the coupling between the chemistry and thermodynamics and also results in the retention of the homogeneity property of the Euler equations (including all the species continuity and vibrational energy conservation equations). Flux-splitting procedures are developed for the fully coupled equations involving fluid dynamics, chemical production and thermodynamic relaxation processes. New forms of flux-vector split and flux-difference split algorithms are embodied in a fully coupled, implicit, large-block structure, including all the species conservation and energy production equations. Several numerical examples are presented, including high-temperature shock tube and nozzle flows. The methodology is compared to other existing techniques, including spectral and central-differenced procedures, and favorable comparisons are shown regarding accuracy, shock-capturing and convergence rates.
Grossman, B.↗
High-voltage electron collection by a spherical satellite
The numerical solution to the cold electron flow equations is presented for the region surrounding an ionospheric satellite at high positive potential. The momentum, continuity, and Poisson equations are solved simultaneously for satellite potentials of 1000, 10,000, 30,000, and 100,000 kT(kT = 0.17 eV). The electron velocity vectors in the sheath region are presented. The electron current density to the satellite surface is also reported. Electrons impact the satellite surface with average velocity vectors having angles of incidence (relative to the surface normal) that increase from the satellite's magnetic pole to equator. A toroidal flow develops in the vicinity of the magnetic equator and expands away from the satellite surface as the surface potential is increased. The current density to the satellite surface increases with increasing satellite potential and decreases with increasing polar angle.
Sheldon, J. W.↗