Constitutive relations for stress and heat flux in a shock wave
Mathematical analysis of shock wave structure incorporating Maxwell moment methods and Boltzmann equation for stress and heat flux constitutive relations
Baganoff, D.↗
SEARCH · Search NASA
Results for “MOMENT EQUATION”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Buildup of laser oscillations from quantum noise
Laser oscillation buildup from quantum noise, deriving equations of motion for moments of photon distribution and time dependence
Lamb, W. E., Jr.↗
Analysis of the dynamics of a nutating body
The equations for the displacement, velocity, and acceleration of a point in a nutating body are developed. These are used to derive equations for the inertial moment developed by a nutating body of arbitrary shape. Calculations made for a previously designed nutating plate transmission indicate that that device is severely speed limited because of the very high magnitude inertial moment.
Anderson, W. J.↗
An unsteady wake model for a hingeless rotor
A simple nonsteady wake model derived from the unsteady moment of the momentum equation for zero advance ratio is correlated with cyclic pitch frequency response tests conducted with a small hingeless rotor model. Two and three or more bladed rotor analyses are presented.
Crews, S. T.↗
Estimation of material properties of a nonlinearly elastic bar
A method of determining certain characteristic flexural rigidities and elastic properties of nonlinearly elastic materials is presented. An estimation method utilizing perturbation methods and a least squares fitting technique is used to solve the nonlinear differential equation derived from the moment curvature relation, subject to boundary values representing deflections of the bar at discrete points. Deflection data from numerical simulations of a nonlinearly elastic, prismatic bar are used to demonstrate the estimation method. Numerical experiments relating the accuracy of the identification to the number and accuracy of the boundary values are presented. Conclusions based on the numerical experiments are included.
Baker, B. E.↗
Measurement of the inertial properties of the Helios F-1 spacecraft
A gravity pendulum method of measuring lateral moments of inertia of large structures with an error of less than 1% is outlined. The method is based on the fact that in a physical pendulum with a knife-edge support the distance from the axis of rotation to the system center of gravity determines the minimal period of oscillation and is equal to the system centroidal radius of gyration. The method is applied to results of a test procedure in which the Helios F-1 spacecraft was placed in a roll fixture with crossed flexure pivots as elastic constraints and system oscillation measurements were made with each of a set of added moment-of-inertia increments. Equations of motion are derived with allowance for the effect of the finite pivot radius and an error analysis is carried out to find the criterion for maximum accuracy in determining the square of the centroidal radius of gyration. The test procedure allows all measurements to be made with the specimen in upright position.
Gayman, W. H.↗
On the accelerations of the moon and sun, the constant of gravitation, and the origin of mountains
On the assumption of a constant value for the gravitational constant G, the lunar and solar tidal couples and the rate of change of angular velocity of the earth are recalculated on the basis of the improved values for the apparent accelerations of the moon and sun arrived at by Muller and Stephenson (1977). These new values are then used to derive general dynamical equations with a changing moment of inertia and changing G connecting the apparent accelerations to the lunar and solar couples. The effect of a changing G is considered, and a value of (dG/dt)/G = -3 times 10 to the -11th power per yr can be reconciled only weakly with the improved accelerations, especially for the case of linear friction. For quadratic friction the situation is more favorable. The new accelerations, however, are best reconciled with a nonchanging G, and certainly cannot be reconciled with a value of (dG/dt)/G = -6 times 10 to the -11th power per yr.
Lyttleton, R. A.↗
Electron temperature anisotropy in the polar wind
The steady state flow of a fully ionized H(+)-O(+)-electron plasma along geomagnetic field lines in the high-latitude topside ionosphere is investigated theoretically, with emphasis on the electron temperature anisotropy and heat flow in the polar wind. The 13-moment system of transport equations developed by Schunk (1975, 1977), which contains a continuity, momentum, internal energy, stress tensor and heat flow equation for each species, is employed to represent the electrons, with a simplified set of transport equations used for the ions, and the transport equations are solved at altitudes from 1500 to 12,000 km for both subsonic and supersonic H(+) outflows. For subsonic H(+) outflows, the electron gas is found to remain collision dominated to high altitudes. For supersonic H(+) outflows, the electron gas is also found to be collision dominated at altitudes below 2500 km, with an anisotropy in electron temperature distribution developing such that the temperature of the perpendicular electrons is greater than that of the parallel electrons at higher altitudes. In addition, the magnitude of downward electron heat flux at 1500 km is shown to have a dramatic effect on the individual parallel and perpendicular electron temperatures.
Schunk, R. W.↗
Proton temperature anisotropy in the polar wind
The steady state flow of a fully ionized H(+)-O(+) electron plasma along geomagnetic field lines in the high-latitude topside ionosphere is studied. The theoretical formulation is based on a 13-moment system of transport equations, and allows for different species temperatures parallel and perpendicular to the geomagnetic field and nonclassical heat flows. For subsonic and supersonic flows, an appreciable H(+) temperature anisotropy occurs at all altitudes above 1500 km, and tends to be regulated at high altitudes. The direction of the temperature anisotropy is related to the direction of the H(+) heat flow to some extent; for supersonic flow an upward flow of heat from the lower ionosphere is required, while for subsonic flow solutions can be obtained with a downward H(+) heat flow. For subsonic flow, H(+)-H(+) collisions have an important effect on the H(+) stress and heat flow balance at all altitudes between 1500 and 12,000 km.
Schunk, R. W.↗
Thermohydrodynamic analysis for laminar lubricating films
A Galerkin-type analysis to include thermal effects in laminar lubricating films was performed. The lubricant properties were assumed constant except for a temperature-dependent Newtonian viscosity. The cross-film temperature profile is established by collocation at the film boundaries and two interior Lobatto points. The interior temperatures are determined by requiring that the zeroth and first moment of the energy equation be satisfied across the film. The fluidity is forced to conform to a third--degree polynomial appropriate to the Lobatto-point temperatures. Preliminary indications are that the use of just two such sampling points enables satisfactory prediction of bearing performance even in the presence of substantial viscosity variation.
Elrod, H. G.↗
Krylov model reduction algorithm for undamped structural dynamics systems
Krylov vectors furnish an efficient basis for eigenvalue analysis and model reduction of structural dynamics systems. The reduced-order model obtained by the present Krylov model-reduction algorithm for an undamped structural-dynamics system is found to match low-frequency moments. The transformed system equation in Krylov coordinates reflects the structure of a tandem system.
Craig, Roy R., Jr.↗
Anisotropic ion heating and parallel O(+) acceleration in regions of rapid E x B convection
A numerical solution to the 20-moment set of transport equations has been found in order to study subauroral ionospheric outflows during periods of enhanced perpendicular ion drifts. The numerical model solves the time-dependent O(+) density, momentum, and both the parallel and perpendicular energy and heat flow equations in the 200-6000 km altitude range. Assuming perpendicular drifts of 3 km/s relative to the neutral atmosphere, we have found that anisotropic heating of O(+) (a result of ion-neutral collisions) leads to a temperature anisotropy, with perpendicular temperatures exceeding 8000 K and parallel temperatures greater than 5000 K (near 200 km altitude). Above approximately 2000 km, transport processes dominate the effects of collisions and wavelike oscillations in O(+) velocity, temperature and heat flux were noted.
Korosmezey, A.↗
Equations and charts for the rapid estimation of hinge-moment and effectiveness parameters for trailing-edge controls having leading and trailing edges swept ahead of the Mach lines
Explore the source record for details and available documents.
Goin, Kenneth L↗
Propeller blade stresses caused by periodic displacement of the propeller shaft
The present report deals with different vibration stresses of the propeller and their removal by an elastic coupling of propeller and engine. A method is described for protecting the propeller from unstable oscillations and herewith from the thus excited alternating gyroscopic moments. The respective vibration equations are set down and the amount of elasticity required is deduced.
Meyer, J↗
Comparison of hydrodynamic and semi-kinetic treatments for plasma flow along closed field lines
Hydrodynamic and semi-kinetic treatments of plasma flow along closed geomagnetic field lines are compared. The hydrodynamic treatment is based on a simplified 16-moment set of transport equations as the equations for the heat flows are not solved; the heat flows are treated heuristically. The semi-kinetic treatment is based on a particle code. The comparison deals with the distributions of the plasma density, flow velocity, and parallel and perpendicular temperatures as obtained from the two treatments during the various stages of the flow. In the kinetic treatment, the appropriate boundary condition is the prescription of the velocity distribution functions for the particles entering the flux tubes at the ionospheric boundaries; those particles leaving the system are determined by the processes occurring in the flux tube. The prescribed distributions are half-Maxwellian with temperature T(sub 0) and density n(sub 0). In the hydrodynamic model, the prescribed boundary conditions are on density (n(sub 0)), flow velocity (V(sub 0)) and temperature (T(sub 0). It was found that results from the hydrodynamic treatment critically depend on V(sub 0); for early stages of the flow this treatment yields results in good agreement with those from the kinetic treatment, when V(sub 0) = square root of (kT(sub 0)/2 (pi)m), which is the average velocity of particles moving in a given direction for a Maxwellian distribution. During this early stage, the flows developing form the conjugate ionospheres show some distinct transitions. For the first hour or so, the flows are highly supersonic and penetrate deep into the opposite hemispheres, and both hydrodynamics and kinetic treatments yield almost similar features. It is found that during this period heatflow effects are negligibly small. When a flow penetrates deep into the opposite hemisphere, the kinetic treatment predicts reflection and setting up of counterstreaming. In contrast, the hydrodynamic treatment yields a shock in the flow. The reasons for this difference in the two treatments is discussed, showing that in view of the relatively warm ions, the coupling of ion beams and the consequent shock formation in the offequatorial region are not likely due to the enhancements in the beam temperatures. The counterstreaming in the kinetic treatment and the shock in the hydrodynamic treatment first advance upward to the equator and then downward to the ionospheric boundary from where the flow originated. The transit time for this advancement is found to be about 1 hour for the respective models. After 2 hours or so, both models predict that the flows from the ionospheric boundaries are generally subsonic with respect to the local ion-sound speed. At late stages of the flow, when a substantial fraction of ions entering the flux tube begin to return back in the kinetic treatment, the hydrodynamic treatment with the boundary condition V(sub 0) = square root of (kT(sub 0)/2(pi)m) yields an over-refilling, and the choice of V(sub 0) becomes uncertain.
Singh, Nagendra↗
Development of an Innovative Algorithm for Aerodynamics-Structure Interaction Using Lattice Boltzmann Method
The lattice Boltzmann equation (LBE) is a kinetic formulation which offers an alternative computational method capable of solving fluid dynamics for various systems. Major advantages of the method are owing to the fact that the solution for the particle distribution functions is explicit, easy to implement, and the algorithm is natural to parallelize. In this final report, we summarize the works accomplished in the past three years. Since most works have been published, the technical details can be found in the literature. Brief summary will be provided in this report. In this project, a second-order accurate treatment of boundary condition in the LBE method is developed for a curved boundary and tested successfully in various 2-D and 3-D configurations. To evaluate the aerodynamic force on a body in the context of LBE method, several force evaluation schemes have been investigated. A simple momentum exchange method is shown to give reliable and accurate values for the force on a body in both 2-D and 3-D cases. Various 3-D LBE models have been assessed in terms of efficiency, accuracy, and robustness. In general, accurate 3-D results can be obtained using LBE methods. The 3-D 19-bit model is found to be the best one among the 15-bit, 19-bit, and 27-bit LBE models. To achieve desired grid resolution and to accommodate the far field boundary conditions in aerodynamics computations, a multi-block LBE method is developed by dividing the flow field into various blocks each having constant lattice spacing. Substantial contribution to the LBE method is also made through the development of a new, generalized lattice Boltzmann equation constructed in the moment space in order to improve the computational stability, detailed theoretical analysis on the stability, dispersion, and dissipation characteristics of the LBE method, and computational studies of high Reynolds number flows with singular gradients. Finally, a finite difference-based lattice Boltzmann method is developed for inviscid compressible flows.
Mei, Ren-Wei↗
Coupling Between Waveguide-Fed Slot Arrays
Coupling between two waveguide-fed planar slot arrays has been investigated using full-wave analysis. The analysis employs the method-of-moments solution to the pertinent coupled integral equations for the aperture electric field of all slots. In order to compute coupling between two arrays, the input port of the first array is excited with a TE(sub 10) mode wave while the second one is match-terminated. After solving the moment method matrix equations, the aperture fields of all slots are obtained and thereby the TE(sub 10) mode wave received at the input port of the second array is determined. Coupling between two arrays is the ratio of the wave amplitude arriving in the second array port to the incident wave amplitude at the first array port. The coupling mechanism has been studied as a function of spacing between arrays in different directions, e.g. the electric field plane, the magnetic field plane, and the diagonal plane. Computed coupling values are presented for different array geometries. This work is novel since it provides a good understanding of coupling between waveguide-fed slot arrays as a function of spacing and orientation for different aperture distributions and array architectures. This serves as a useful tool for antenna design engineers and system engineers.
Rengarajan, Sembiam↗
CFIE MM solution for TE and TM incidence on a 2-D conducting body with dielectric filled cavity
The problem of determining the scattering cross section of an arbitrarily shaped two-dimensional conducting body with an arbitrarily shaped dielectric-filled cavity is considered. The problem is solved using a method-of-moments (MM) solution for the combined field integral equations (CFIE). The particular form of the method of moments solution used has here uses a minimum number of expansion coefficients. Results are given for TE and TM incident waves.
Goggans, Paul M.↗