Redundant variables in celestial mechanics
Three degree of freedom perturbed two body problem, applying theory of redundant variables to Lagrangian equations of motion
Ananda, M.↗
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Spherical balloon response to three-dimensional time-dependent flows
The concept of the Lagrangian displacement of a balloon is introduced. It is shown that the general balloon response problem is extremely complicated because the wind-forcing functions in the balloon equations of motion are functions of the wind velocity vector and its Eulerian first derivatives evaluated at the location of the balloon. The linear perturbation equations for a spherical balloon are derived by perturbing the components of velocity of the balloon about a terminal velocity state which is in equilibrium with a space-time invariant mean horizontal flow. The atmospheric flow is also perturbed such that the resulting equations can be used to analyze the responses of spherical balloons to three-dimensional time-dependent flows. The wind field is represented in terms of a four-fold Fourier integral that involves three orthogonal wave numbers and a frequency, while the balloon components of velocity are represented as Fourier integrals involving a frequency which, in turn, is a function of the wind field wave numbers and frequency and the unperturbed flow components of velocity.
Fichtl, G. H.↗
Spatially homogeneous rotating world models.
The mathematical problem encountered when looking for the simplest expanding and rotating model of the universe without the compactness condition for the space sections is formulated. The Lagrangian function is derived for four different rotating universes simultaneously. These models correspond in a certain sense to Godel's (1950) ?symmetric case.'
Ozsvath, I.↗
A systematic exposition of the conservation equations for blast waves.
In order to provide a rational background for the analysis of experimental observations of blast wave phenomena, the conservation equations governing their nonsteady flow field are formulated in a general manner, without the usual restrictions imposed by an equation of state, and with proper account taken, by means of source terms, of other effects which, besides the inertial terms that conventionally dominate these equations, can affect the flow. Taking advantage of the fact that a blast wave can be generally considered as a spatially one-dimensional flow field whose nonsteady behavior can be regarded, consequently, as a function of just two independent variables, two generalized blast wave coordinates are introduced, one associated with the front of the blast wave and the other with its flow field. The conservation equations are accordingly transformed into this coordinate system, acquiring thereby a comprehensive character, in that they refer then to any frame of reference, being applicable, in particular, to problems involving either space or time profiles of the gas-dynamic parameters in the Eulerian system, or time profiles in the Lagrangian system.
Oppenheim, A. K.↗
Comparison of some approximations for isotropic turbulence.
Study of several related turbulence approximations with regard to dynamical properties and agreement of numerical predictions with laboratory and computer experiments. The approximations considered include the direct-interaction equations (Kraichnan, 1964), Herring's (1966) self-consistent-field theory, a generalization of Edwards' (1964) theory, the abridged Lagrangian-history, direct-interaction approximation (Kraichnan, 1966), the test-field model (Kraichnan, 1971), and an approximation, not previously described, in which one velocity field passively suffers convection by another. Most of the cited approximations are representable by stochastic model equations for the velocity amplitude. Explicit constructions are given for the stochastic models, in a form that can be approximated on a digital computer. These constructions are used to discuss the physical and mathematical differences between the model dynamics and actual Navier-Stokes dynamics.-
Herring, J. R.↗
CNO abundances and hydrodynamic models of the nova outburst.
We have used a fully implicit, Lagrangian, hydrodynamic computer code incorporating a nuclear reaction network to follow thermonuclear runaways in the hydrogen-rich envelopes of white dwarfs in order to produce a nova outburst. Because of the short time-scales and the high nuclear burning rates produced in our models, the nuclear reactions are far out of equilibrium and the beta-plus unstable nuclei become the most abundant nuclei in the envelope except for hydrogen and helium. Our models have ejected 1.00017 solar mass with kinetic energies of 8 times 10 to the 44-th power ergs, a value that agrees quite closely with the observed values for novae.
Starrfield, S.↗
A new theory of gravity
A new relativistic theory of gravity is presented. This theory agrees with all experiments to date. It is a metric theory, it is Lagrangian-based, and it possesses a preferred frame with conformally-flat space slices. With an appropriate choice of certain adjustable functions and parameters, this theory possesses precisely the same post-Newtonian limit as general relativity.
Ni, W.↗
The EOLE experiment: Early results and current objectives
The EOLE experiment with 480 constant level balloons released in the Southern Hemisphere is described. Each balloon floating freely at approximately the 200 mb level, is a precise tracer of the horizontal motion of air masses, the accuracy of which is limited only by the laminated structure of the stratospheric flow, within an RMS uncertainty of 1.5 m/sec. The balloons were found after 2 months to distribute at random over the whole hemisphere outside the tropics, irrespective of their original launching site. Early results of Eulerian and Lagrangian averages of the EOLE wind data are given for describing the mean 200 mb zonal and meridional circulations. The effect of the small scale eddies of two-dimensional turbulence has been studied with respect to the relative eddy diffusion of pairs of balloons and the relative dispersion of triangular clusters. New estimates of the RMS divergence of the 200 mb flow are given, together with their scale dependence which was found to be a logarithmic law.
Morel, P.↗
Turbulent transport measurements with a laser Doppler velocimeter
The power spectrum of phototube current from a laser Doppler velocimeter operating in the heterodyne mode has been computed. The spectrum is obtained in terms of the space time correlation function of the fluid. The spectral width and shape predicted by the theory are in agreement with experiment. For normal operating parameters the time average spectrum contains information only for times shorter than the Lagrangian integral time scale of the turbulence. To examine the long time behavior, one must use either extremely small scattering angles, much longer wavelength radiation or a different mode of signal analysis, e.g., FM detection.
Edwards, R. V.↗
Turbulent transport measurements with a laser Doppler velocimeter.
The power spectrum of phototube current from a laser Doppler velocimeter operating in the heterodyne mode has been computed. The spectral width and shape predicted by the theory are in agreement with experiment. For normal operating parameters the time-average spectrum contains information only for times shorter than the Lagrangian-integral time scale of the turbulence. To examine the long-time behavior, one must use either extremely small scattering angles, much-longer-wavelength radiation, or a different mode of signal analysis, e.g., FM detection.
Edwards, R. V.↗
Static geometric and material nonlinear analysis.
This paper presents a survey of four aspects of nonlinear analysis - basic formulation, plasticity relations, computational procedures, and methods of solution. This survey is for the most part limited to static geometric and material nonlinear analysis under the assumption of small strains. The two formulations covered are the total Lagrangian and the incremental moving coordinate formulations. A comparison of the two formulations is presented and an attempt is made to clarify the underlying basis of each formulation. The survey of computational procedures reveals that most researchers are now using some form of numerical integration to evaluate nonlinear contributions.
Stricklin, J. A.↗
The Eole experiment - Early results and current objectives.
The Eole experiment with 480 constant level balloons released in the Southern Hemisphere is described. Each balloon, floating freely at approximately the 200-mb level, is a precise tracer of the horizontal motion of air masses, the accuracy of which is limited only by the laminated structure of the stratospheric flow, within an rms uncertainty of 1.5 m/sec. The balloons were found after 2 months to distribute at random over the whole hemisphere outside the tropics, irrespective of their original launching site. Early results of Eulerian and Lagrangian averages of the Eole wind data are given for describing the mean 200-mb zonal and meridional circulations.
Morel, P.↗
A new theory of gravity.
A new relativistic theory of gravity is presented. This theory agrees with all experiments to date. It is a metric theory; it is Lagrangian-based; and it possesses a preferred frame with conformally flat space slices. With an appropriate choice of certain adjustable functions and parameters and of the cosmological model, this theory possesses precisely the same post-Newtonian limit as general relativity.
Ni, W.-T.↗
Scale dependence of the 200-mb divergence inferred from EOLE data.
The EOLE experiment with 480 constant-volume balloons distributed over the Southern Hemisphere approximately at the 200-mb level, has provided a unique, highly accurate set of tracer trajectories in the general westerly circulation. The trajectories of neighboring balloons are analyzed to estimate the horizontal divergence from the Lagrangian derivative of the area of one cluster. The variance of the divergence estimates results from two almost comparable effects: the true divergence of the horizontal flow and eddy diffusion due to small-scale, two-dimensional turbulence. Taking this into account, the rms divergence is found to be of the order of 0.00001 per sec and decreases logarithmically with cluster size. This scale dependence is shown to be consistent with the quasi-geostrophic turbulence model of the general circulation in midlatitudes.
Morel, P.↗
A note on relative motion in the general three-body problem.
It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem, such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10% faster than with barycentric or heliocentric coordinates.
Broucke, R.↗
Numerical calculations of velocity and pressure distribution around oscillating airfoils
An analytical procedure based on the Navier-Stokes equations was developed for analyzing and representing properties of unsteady viscous flow around oscillating obstacles. A variational formulation of the vorticity transport equation was discretized in finite element form and integrated numerically. At each time step of the numerical integration, the velocity field around the obstacle was determined for the instantaneous vorticity distribution from the finite element solution of Poisson's equation. The time-dependent boundary conditions around the oscillating obstacle were introduced as external constraints, using the Lagrangian Multiplier Technique, at each time step of the numerical integration. The procedure was then applied for determining pressures around obstacles oscillating in unsteady flow. The obtained results for a cylinder and an airfoil were illustrated in the form of streamlines and vorticity and pressure distributions.
Bratanow, T.↗
The three dimensional motion and stability of a rotating space station-cable - Counterweight configuration
The three dimensional equations of motion for a cable connected space station - counterweight system are developed using a Lagrangian formulation. The system model employed allows for cable and end body damping and restoring effects. The equations are then linearized about the equilibrium motion and nondimensionalized. To first degree, the out-of-plane equations uncouple from the in-plane equations. Therefore, the characteristic polynomials for the in-plane and out-of-plane equations are developed and treated separately. From the general in-plane characteristic equation, necessary conditions for stability are obtained. The Routh-Hurwitz necessary and sufficient conditions for stability are derived for the general out-of-plane characteristic equation. Special cases of the in-plane and out-of-plane equations (such as identical end masses, and when the cable is attached to the centers of mass of the two end bodies) are then examined for stability criteria.
Bainum, P. M.↗
A general form of the co-moving tensorial derivative
A general expression for the co-moving derivative of a tensor is derived. A variable describing the coordinate velocity field is introduced. Time dependency of the metric elements is expressed in terms of this velocity field. The resulting description of motion is one of which the Eulerian and Lagrangian viewpoints are special cases. This general description is useful in problems involving moving boundaries or discontinuities.
Huckins, E. K., III↗