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At least 73 records · Page 4

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NASA NTRS1984

An iwatsubo-based solution for labyrinth seals - comparison with experimental results

The basic equations are derived for compressible flow in a labyrinth seal. The flow is assumed to be completely turbulent in the circumferential direction where the friction factor is determined by the Blasius relation. Linearized zeroth and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equation. The first-order equations are solved by a separation of variables solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are compared to published test results.

Childs, D. W.↗

NASA NTRS1987

A Comparison of Experimental and Theoretical Results for Labyrinth Gas Seals

The basic equations are derived for a two control volume model for compressible flow in a labyrinth seal. The flow is assumed to be completely turbulent and isoenergetic. The wall friction factors are determined using the Blasius formula. Jet flow theory is used for the calculation of the recirculation velocity in the cavity. Linearized zeroth and first order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth order pressure distribution is found by satisfying the leakage equation. The circumferential velocity distribution is determined by satisfying the momentum equations. The first order equations are solved by a separation of variable solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are compared to experimental test results.

Scharrer, Joseph Kirk↗

NASA NTRS1986

Absorption and scattering of light by nonspherical particles

Using the example of the polarization of scattered light, it is shown that the scattering matrices for identical, randomly ordered particles and for spherical particles are unequal. The spherical assumptions of Mie theory are therefore inconsistent with the random shapes and sizes of atmospheric particulates. The implications for corrections made to extinction measurements of forward scattering light are discussed. Several analytical methods are examined as potential bases for developing more accurate models, including Rayleigh theory, Fraunhoffer Diffraction theory, anomalous diffraction theory, Rayleigh-Gans theory, the separation of variables technique, the Purcell-Pennypacker method, the T-matrix method, and finite difference calculations.

Bohren, C. F.↗

NASA NTRS1987

Theory versus experiment for the rotordynamic coefficients of labyrinth gas seals. I - A two control volume model

The basic equations are derived for a two-control-volume model for compressible flow in a labyrinth seal. The recirculation velocity in the cavity is incorporated into the model for the first time. The flow is assumed to be completely turbulent and isoenergetic. The wall friction factors are determined using the Blasius formula. Jet flow theory is used for the calculation of the recirculation velocity in the cavity. Linearized zeroth- and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equations. The first-order equations are solved by a separation of variable solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients.

Scharrer, J. K.↗

NASA NTRS1989

Rotordynamic coefficients for stepped labyrinth gas seals

The basic equations are derived for compressible flow in a stepped labyrinth gas seal. The flow is assumed to be completely turbulent in the circumferential direction where the friction factor is determined by the Blasius relation. Linearized zeroth and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equations. The first order equations are solved by a separation of variables solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are presented in the form of a parametric study, since there are no known experimental data for the rotordynamic coefficients of stepped labyrinth gas seals. The parametric study investigates the relative rotordynamic stability of convergent, straight and divergent stepped labyrinth gas seals. The results show that, generally, the divergent seal is more stable, rotordynamically, than the straight or convergent seals. The results also show that the teeth-on-stator seals are not always more stable, rotordynamically, then the teeth-on-rotor seals as was shown by experiment by Childs and Scharrer (1986b) for a 15 tooth seal.

Scharrer, Joseph K.↗

NASA NTRS1989

A normal-mode approach to Jovian atmospheric dynamic

A nonlinear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics is proposed, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the nonlinear case is then derived. It is shown that, for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the barotropic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal-mode model with one baroclinic mode is dynamically equivalent to a one-layer model with solid lower topography. It is also shown that, for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode is applicable to Jupiter. The accuracy of the normal-mode model for Jupiter is tested using the following simple problems: (1) forced, vertically propagating Rossby waves, using two and three baroclinic modes, and (2) baroclinic instability, using two baroclinic modes. It is found that the normal-mode model provides qualitatively correct results, even with only a very limited number of vertical degrees of freedom.

Achterberg, Richard K.↗

NASA NTRS1989

Some aspects of transient cooling of a radiating rectangular medium

The emission from a gray radiating medium is analyzed for transient cooling in surroundings at a low temperature. The medium is rectangular with no variations in the direction normal to the cross section. The integral equation for the transient temperature distribution is solved numerically using a two-dimensional Gaussian integration subroutine. The emissive ability for a rectangle at uniform temperature is compared with that for transient cooling where the temperature distribution of the region has reached a fully developed shape, as shown by a separation of variables solution. The two solutions provide the upper and lower bounds for the emittance of a rectangle during transient cooling. The emittances for various aspect ratios are presented as a function of the mean length of the rectangle and are compared with results for a plane layer and a cylinder.

Siegel, Robert↗

NASA NTRS1991

Transfer function modeling of damping mechanisms in viscoelastic plates

This work formulates a method for the modeling of material damping characteristics in plates. The Sophie German equation of classical plate theory is modified to incorporate hysteresis effects represented by complex stiffness using the transfer function approach proposed by Golla and Hughes, (1985). However, this procedure is not limited to this representation. The governing characteristic equation is decoupled through separation of variables, yielding a solution similar to that of undamped classical plate theory, allowing solution of the steady state as well as the transient response problem.

Slater, J. C.↗

NASA NTRS1992

Nonlinear Maneuver Autopilot

Trajectory-control laws based on singular-perturbation theory and nonlinear dynamical modeling. Nonlinear maneuver autopilot commands flight-test trajectories of F-15 airplane. Underlying theory of controller enables separation of variables processed in fast and slow control loops, reducing amount of computation required.

Menon, P. K. A.↗

NASA NTRS1992

A variational approach to three-dimensional elasticity solutions of laminated composite plates

The displacements in a laminated composite are represented as products of two sets of unknown functions, one of which is only a function of the thickness coordinate and the other is a function of the in-plane coordinates (i.e., separation of variables approach), and the minimization of the total potential energy is reduced to a sequence of iterative linear problems. Analytical solutions are developed for cross-ply and angle-ply laminated composite rectangular plates. The solution for simply-supported cross-ply plates under sinusoidal transverse load reduces to that of Pagano. Numerical results for stresses and is placements for antisymmetric angle-ply laminates are presented. The three-dimensional elasticity solutions developed are important because they can be used to study the behavior of composite laminates, in addition to serving as reference for approximate solutions by numerical methods and two-dimensional theories.

Savoia, M.↗

NASA NTRS1994

Modal element method for potential flow in non-uniform ducts: Combining closed form analysis with CFD

An analytical procedure is presented, called the modal element method, that combines numerical grid based algorithms with eigenfunction expansions developed by separation of variables. A modal element method is presented for solving potential flow in a channel with two-dimensional cylindrical like obstacles. The infinite computational region is divided into three subdomains; the bounded finite element domain, which is characterized by the cylindrical obstacle and the surrounding unbounded uniform channel entrance and exit domains. The velocity potential is represented approximately in the grid based domain by a finite element solution and is represented analytically by an eigenfunction expansion in the uniform semi-infinite entrance and exit domains. The calculated flow fields are in excellent agreement with exact analytical solutions. By eliminating the grid surrounding the obstacle, the modal element method reduces the numerical grid size, employs a more precise far field boundary condition, as well as giving theoretical insight to the interaction of the obstacle with the mean flow. Although the analysis focuses on a specific geometry, the formulation is general and can be applied to a variety of problems as seen by a comparison to companion theories in aeroacoustics and electromagnetics.

Baumeister, Kenneth J.↗

NASA NTRS1993

Impactor control of central peak and peak-ring formation

The relation between the depth and diameter of excavation for impacts typically is assumed to be proportional. Such an assumption is consistent with the constant aspect ratio (diameter:depth) observed for simple craters found in a wide range of planetary settings and crater-scaling laws derived from laboratory experiments. Although complex craters exhibit evidence for floor uplift and rim collapse of a transient profile, they are typically thought to resemble initially smaller, simple craters. At large scales, however, early-time processes consume a greater fraction of crater growth and the assumption of late-time equivalence of energy release as a point source becomes inappropriate. The authors propose instead that crater diameter, depth, and impactor penetration represent separable dependent variables that underscore the fundamental difference between impact and point-source explosion excavation processes. An important consequence of this perspective is that central pits, peaks, and rings may represent contrasting target responses to impactor penetration and could provide an important indicator of impactor dimensions.

Schultz, Peter H.↗

NASA NTRS1994

Transfer function modeling of damping mechanisms in distributed parameter models

This work formulates a method for the modeling of material damping characteristics in distributed parameter models which may be easily applied to models such as rod, plate, and beam equations. The general linear boundary value vibration equation is modified to incorporate hysteresis effects represented by complex stiffness using the transfer function approach proposed by Golla and Hughes. The governing characteristic equations are decoupled through separation of variables yielding solutions similar to those of undamped classical theory, allowing solution of the steady state as well as transient response. Example problems and solutions are provided demonstrating the similarity of the solutions to those of the classical theories and transient responses of nonviscous systems.

Slater, J. C.↗

NASA NTRS1993

Exact element modal analysis of beam/oscillator systems

Exact modal analysis of a beam/many-oscillator system is presented. Each oscillator consists of a discrete mass having rotational inertia, and two locally-dependent springs: one being rotational and the other, translational. A constant longitudinal force is applied to the mass, and its effect is coupled with the dynamic modes. The beam's distributed mass and other material properties, and its geometric properties, do not vary within spans bounded by the oscillators, but may vary from span to span. The classical separation-of-variables technique is used to obtain a dosedform description of the dynamic matrix, and the modal frequencies are determined as the zeros of this determinant.

Broome, Taft H., Jr.↗

NASA NTRS1995

Analysis of gas absorption to a thin liquid film in the presence of a zero-order chemical reaction

The paper presents a detailed theoretical analysis of the process of gas absorption to a thin liquid film adjacent to a horizontal rotating disk. The film is formed by the impingement of a controlled liquid jet at the center of the disk and subsequent radial spreading of liquid along the disk. The chemical reaction between the gas and the liquid film can be expressed as a zero-order homogeneous reaction. The process was modeled by establishing equations for the conservation of mass, momentum, and species concentration and solving them analytically. A scaling analysis was used to determine dominant transport processes. Appropriate boundary conditions were used to solve these equations to develop expressions for the local concentration of gas across the thickness of the film and distributions of film height, bulk concentration, and Sherwood number along the radius of the disk. The partial differential equation for species concentration was solved using the separation of variables technique along with the Duhamel's theorem and the final analytical solution was expressed using confluent hypergeometric functions. Tables for eigenvalues and eigenfunctions are presented for a number of reaction rate constants. A parametric study was performed using Reynolds number, Ekman number, and dimensionless reaction rate as parameters. At all radial locations, Sherwood number increased with Reynolds number (flow rate) as well as Ekman number (rate of rotation). The enhancement of mass transfer due to chemical reaction was found to be small when compared to the case of no reaction (pure absorption), but the enhancement factor was very significant when compared to pure absorption in a stagnant liquid film. The zero-order reaction processes considered in the present investigation included the absorption of oxygen in aqueous alkaline solutions of sodiumdithionite and rhodium complex catalyzed carbonylation of methanol. Present analytical results were compared to previous theoretical results for limiting conditions, and were found to have very good agreement.

Rajagopalan, S.↗

NASA NTRS1994

On the prediction of velocity fields from redshift space galaxy samples

We present a new method for recovering the underlying velocity field from an observed distribution of galaxies in redshift space. The method is based on a kinematic Zel'dovich relation between the velocity and density fields in redshift space. This relation is expressed in a differential equation slightly modified from the usual Poisson equation and which depends nontrivially on Beta identically equal to Omega(exp 0.6)/b. The linear equation can be readily solved by standard techniques of separation of variables by means of spherical harmonics. One can also include a term describing the 'rocket effect' discussed by Kaiser (1987). From this redshift space information alone, one can generate a prediction of the peculiar velocity field for each harmonic (l, m) as a function of distance. We note that for the quadrupole and higher order moments, the equation is a boundary value problem with solutions dependent on both the interior and exterior mass distribution. However, for a shell at distance r, the dipole, as well as the monopole, of the velocity field in the Local Group frame is fully determined by the interior mass distribution. This implies that the shear of the measured velocity field, when fitted to a dipole distortion, should be aligned and consistent with the gravity field inferred from the well determined local galaxy distribution. As a preliminary application we compute the velocity dipole of distant shells as predicted from the 1.2 Jy IRAS survey compared to the measured velocity dipole on shells, as inferred from a recent POTENT analysis. The coherence between the two fields is good, yielding a best estimate of Beta = 0.6 +/- 0.2.

Nusser, Adi↗

NASA NTRS2005

Quantifying Exact Motions Along Lineaments on Europa

Evaluating the precise motions along lineaments on the surface of Jupiter's icy moon, Europa, is a valuable tool for interpreting the development and history of lineaments of various morphologies. Such morphologies include strike-slip faults, dilational bands, ridges, and convergence zones. However, the exact mode of origin and kinematic behavior of these various lineaments are not obvious based on morphology alone. In fact, the apparent motions implied by displaced crosscut features can provide misleading indications of true motions along lineaments. Identifying the precise motions (combinations of sliding and opening/closing) is critical to the accurate characterization and interpretation of each of these lineament types. Lineaments of interest (i.e., those having displaced relatively older features in some manner) are identified on Galileo spacecraft images and measurements are made of the total offset, the separation, and relative orientations of crosscut features with respect to the lineament of interest. Specifically, by using these measured quantities and a series of trigonometric equations, the precise motions (i.e., dilation, convergence, strike-slip, or a combination of strike-slip and dilation or convergence) can be determined. These measurements are, however, limited by the resolution of the available images. This study focuses on motion analysis techniques for Europan lineaments and the precise characterization of fault-orthogonal and/or strike-slip motion along lineaments of varying morphologies. We highlight potential pitfalls of cursory analyses of motion indicators. For example, lineaments with obvious lateral offsets have typically been identified simply as strike-slip faults. This assumption may actually be incorrect, as fault-orthogonal motions may contribute to apparent lateral displacements (offsets or separations). Also, variability in the amount of fault motion along the trace length should theoretically be identifiable using the outlined technique. Strike-slip faults on Europa have conventionally been presumed to have a constant slip magnitude along their lengths; however, we posit that variable distributions of slip may be common, as is typical along terrestrial strike-slip faults.

Vetter, J. C.↗

NASA NTRS2004

Relative Sensor with 4Pi Coverage for Formation Flying Missions

The Terrestrial Planet Finder (TPF) pre-project, an element of NASA s Origins program, is currently developing two architectures for a mission to search for earth-like planets around nearby stars. One of the architectures being developed is the Formation Flying Interferometer (FFI). The FFI is envisioned to consist of up to seven spacecraft (as many as six "collectors" with IR telescopes, and a "combiner") flying in precise formation within +/-1 cm of pre-determined trajectories for synchronized observations. The spacecraft-to-spacecraft separations are variable between 20 m and 100 m or more during observations to support various configurations of the interferometer in the planet-finding mode. The challenges involved with TPF autonomous operations, ranging from formation acquisition and formation maneuvering to high precision formation control during science observations, are unprecedented. In this paper we discuss the development of the formation acquisition sensor, which uses novel modulation and duplexing schemes to enable fast signal acquisition, multiple-spacecraft operation, and mitigation of inherent jamming conditions, while providing precise formation sensing and integrated radar capability. This approach performs delay synthesis and carrier cycle ambiguity resolution to improve range measurement, and uses differential carrier cycle ambiguity resolution to make precise bearing angle measurements without calibration maneuvers.

Tien, Jeffrey Y.↗