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The stability of three dimensional rotating disk flow and the effects of Coriolis forces and streamline curvature were investigated. It was shown that this analysis gives better growth rates than Orr-Sommerfeld equation. Results support the numerical prediction that the number of stationary vortices varies directly with the Reynolds number.
An analytical technique was developed to predict the behavior of a rotor system subjected to sudden unbalance. The technique is implemented in the Turbine Engine Transient Rotor Analysis (TETRA) computer program using the component element method. The analysis was particularly aimed toward blade-loss phenomena in gas turbine engines. A dual-rotor, casing, and pylon structure can be modeled by the computer program. Blade tip rubs, Coriolis forces, and mechanical clearances are included. The analytical system was verified by modeling and simulating actual test conditions for a rig test as well as a full-engine, blade-release demonstration.
The vertical structure of the large scale galactic shock along spiral arms is studied in order to understand the extended features or the wings of spiral arms observed by Kepner (1970) and recently checked by Soukup (1979). Numerical calculations of two-dimensional gas flow were performed for two models of the gas. The results show that the shock front, standing perfectly perpendicular to the galactic plane, extends impressively above the scale height of the interstellar gas. In the extreme case of isothermal gas, the shock solution exists at any distance from the plane. In a realistic model in which the kinetic temperature of the gas is stratified, increasing with height, the shock can extend up to 700 pc. For both models, although the velocity component perpendicular to the galactic plane is small, the solutions corresponding to layered one-dimensional flow contain no shocks at such heights. In order to study the geometry of the shock front and the flow pattern near it, a simplified model is adopted, in which the Coriolis force is neglected. However, estimates show that including it will enhance the shock strength. The compression at the shock front makes the gas observable at high latitudes.
The stability of three-dimensional rotating disk flow is investigated, including the effects of Coriolis forces and streamline curvature. The numerical results show that the critical Reynolds number for establishment of stationary vortex flow is 287. These vortices spiral outward at an angle of about 11.2 deg, and transition to turbulence occurs when their total amplification is about e to the 11th. New experimental results are also reported on the spatial growth rates of the stationary vortices. It is shown that the analysis gives growth rates that compare much better with the experimental results than do results obtained using the Orr-Sommerfeld equation. The experimental results tend to support the numerical prediction that the number of stationary vortices varies directly with the Reynolds number. The calculations also indicate the existence of weakly unstable propagating (type II) modes at low Reynolds numbers (Critical Reynolds Number being approximately equal to 49).
Near the equator where the Coriolis force vanishes, the momentum balance for the axially symmetric circulation is established between horizontal and vertical diffusion, which, a priori, does not impose constraints on the direction or magnitude of the zonal winds. Solar radiation absorbed at low latitudes is a major force in driving large scale motions with air rising near the equator and falling at higher latitudes. In the upper leg of the meridional cell, angular momentum is redistributed so that the atmosphere tends to subrotate (or corotate) at low latitudes and superrotate at high latitudes. In the lower leg, however, the process is reversed and produces a tendency for the equatorial region to superrotate. The outcome depends on the energy budget which is closely coupled to the momentum budget through the thermal wind equation; a pressure (temperature) maximum is required to sustain equatorial superrotation. Such a condition arises in regions which are convectively unstable and the temperature lapse rate is superadiabatic. It should arise in the tropospheres of Jupiter and Saturn; planetary energy from the interior is carried to higher altitudes where radiation to space becomes important. Upward equatorial motions in the direct and indirect circulations (Ferrel-Thomson type) imposed by insolation can then trap dynamic energy for equatorial heating which can sustain the superrotation of the equatorial region.
A two-dimensional numerical model is developed to study the momentum coupling between the ionosphere and neutral atmosphere in the vicinity of discrete high-latitude features, such as convection channels and plasma density troughs. Based on generalized magnetohydrodynamic equations the model takes account of global pressure gradients, viscous dissipation, ion drag, the Coriolis force, and electrodynamic drifts. Among the findings of an initial steady state investigation are the following: (1) in convection channels, significant shears and rotations of the thermospheric flow can occur below 200 km if a minimum in the electron density profile is present between the E and F regions; (2) in convection channels, the thermospheric wind decreases with height in the F region owing to the effects of horizontal viscosity; and (3) at low altitudes, the boundaries of convection channels may produce Ekman spirals.
Some aspects of adiabatic drift theory are investigated in the regime where the E x B drift velocity is comparable with or larger than the gyro velocity. Particles undergo five drifts in addition to the E x B gradient, and line curvature drifts have three more terms in the parallel equation of motion. The case of the rapidly rotating rigid magnetic field configuration is found to be an exception due to the corotating particle with a nearly periodic motion. The guiding center drift velocity and parallel differential equation of motion are obtained, and the drift velocity in the rotating frame is found to consist of the expected field gradient and line curvature drift, plus centrifugal and coriolis force induced drifts. It is shown that the second invariant is conserved in the rotating frame by the four drifts, so that a particle slowly drifts around on its drift shell and returns to its original field line. Thus, there is no long term energy change, and any energy change is periodic on the bounce and drift time scale.
The considered investigation takes into account wind measurements conducted with the aid of 39 sounding rocket chemical release payloads. Attention is given to the geomagnetic dependence, winds and ion convection patterns, subauroral dayside measurements, evening auroral belt measurements, evening polar cap measurements, postmidnight polar cap measurements, midnight and early morning auroral belt measurements, late evening auroral belt measurements, and anomalous winds. The prevailing theme in presenting and interpreting the wind measurements between 160 and 320 km has been that the gross characteristics of the high-latitude wind system can be readily understood in terms of the pattern distribution of electric fields when the inertial properties of the neutral atmosphere and Coriolis forces are taken into account.
A set of three-dimensional flow-field data for the region around a cylinder impulsively spun-up from rest was derived with a numerical model based on the Navier-Stokes equations. Laser-Doppler anemometer data in the azimuthal direction was employed to test the model predictions, and data was developed for a flowfield with Ekman numbers from 9.18/1,000,000 to 9.18/10,000. The contributions of inviscid and viscous terms were determined as functions of radius and time. It was found that immediately after start-up viscous diffusion is the dominant factor, which is replaced by nonlinear radial advection. The Coriolis force dominates in the later stages of spin-up. The inward radial flow is a maximum near the front, where the vertical velocity is small, but features strong radial gradients, as it does at the edge of the Ekman layer.
The theory of steady transonic winds from condensed bodies is extended to general, two-dimensional, axisymmetric systems. A stream function is used to reduce the gasdynamics equations to a single, second-order differential equation plus an algebraic equation for the density. The approach extends Parker's (1958) quasi-one-dimensional theory, which uses Bernoulli's theorem, to a complete two-dimensional calculation which includes vorticity and rotation. The conceptual basis of the stream function approach is described in detail, and a numerical method for solving the resulting equations is presented. The applications illustrate solutions for stellar (spherical source surface) and galactic (oblate spheroidal source surface) models. Among other things, it is found that for rapidly rotating stellar models the Coriolis force dominates the centrifugal terms and the streamlines bend toward the rotation axis as a consequence of the axisymmetry. For both the rotating and nonrotating galaxy models, the streamlines bend toward the equator.
In order to analyze the dynamic behavior of the first stage compressor/fan of the 'E3' turbofan engine, a classical structural dynamics approach is employed to couple the motions of a flexible bladed disk to a rotating flexible shaft. The analysis accounts for flexible disk displacements which are transverse to the plane of rotation, and radial as well as tangential, and also accounts for rigid disk translations along, and rotations about, axes normal to the undeformed shaft axes. In the case of a wide range of E3 engine shaft flexibilities and speeds, some of the one-diametral node frequencies are shown to be affected by shaft degrees of freedom whose stiffness values are in general range of design practice. Coriolis forces are also found to significantly affect natural frequencies where strong coupling between certain modes is present.
The results of a dynamic/thermodynamic numerical model of Arctic sea ice are compared with satellite images from the Nimbus 5 electrically scanning microwave radiometer. The model combines aspects of two previous sea ice models those of Parkinson and Washington and Ling, Rasmussen, and Campbell. A solid/fluid model basically follows the formulation of the Parkinson and Washington model with the addition of the constitutive equation and equation of state from the Ling model. The Parkinson and Washington model simulates the seasonal cycle of sea ice thicknesses and concentrations with a horizontal resolution of roughly 200 km and a timestep of 8 hours. The thermodynamics are calculated through energy balances at the interfaces between ice and air, water and ice, and water and air. The ice dynamics are calculated through a momentum equation balancing air stress, water stress, dynamic topography, and Coriolis force, with an adjustment for internal ice resistance.
The analytical simulation of deployment dynamics of these two axis concepts as well as the evolution of practical designs for the add on deployable inertia boom units is described. With the boom free to swing back in response to Coriolis forces as well as outwards in response to centrifugal forces, the kinematics of motion are complex but admit the possibility of absorbing deployment energy in frictional or other damping devices about the radial axis, where large amplitude motions can occur and where the design envelope allows more available volume. An acceptable range is defined for frictional damping for any given spin rate. Inadequate damping allows boom motions which strike the spacecraft; excessive damping causes the boom to swing out and latch with damaging violence. The acceptable range is a design parameter and must accommodate spin rate tolerance and also the tolerance and repeatability of the damping mechanisms.
Report presents two general methods for reducing mathematical complexities of state equations of robot-arm dynamics. Robot arms contain both rotary and linear joints. Both methods start with homogenous coordinates and Lagrangian formulation of mechanics briefly summarized in report. First method uses matrix-analysis techniques; second, vector-analysis techniques. Vector-analysis method includes new differential-vector representation of centripetal and Coriolis forces.
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.
Magneto-convection and gravity waves are numerically simulated with a nonlinear, three-dimensional, time-dependent model of a stratified, rotating, spherical fluid shell heated from below. A Solar-like reference state is specified while global velocity, magnetic field, and thermodynamic perturbations are computed from the anelastic magnetohydrodynamic equations. Convective overshooting from the upper (superadiabatic) part of the shell excites gravity waves in the lower (subadiabatic) part. Due to differential rotation and Coriolis forces, convective cell patterns propagate eastward with a latitudinally dependent phase velocity. The structure of the excited wave motions in the stable region is more time-dependent than of the convective motions above. The magnetic field tends to be concentrated over giant-cell downdrafts in the convective zone but is affected very little by the wave motion in the stable region.
Evidence from three-dimensional numerical modelling is presented that only cometary orbits with a limited range in inclination with respect to the galactic plane are formally stable for the length of time required to cause periodic extinction events. The calculations were done using Cowell's method employing a fourth-order Runge-Kutta integration scheme in an inertial reference frame in orbit about the Galaxy. Tidal perturbations in the radial direction due to the Galaxy and the Coriolis forces are included. The vertical component of the gravitational field of the galactic disk is superimposed on these forces. The results indicate that orbits for Nemesis that are inclined at more than 30 deg to the galactic plane are not allowed and suggests that the search for Nemesis should be concentrated toward the plane of the Galaxy. Perturbations by passing stars or molecular clouds may make even the low-inclination orbits unstable.
Rotating turbulent channel flow has been simulated numerically using the large-eddy-simulation technique. With a three-dimensional, time-dependent, numerical simulation of turbulence, it was possible to obtain detailed flow-field information which had not been available previously from experiments. This information was used to study the structure and statistical properties of the flow with externally imposed rotation. It was found that in agreement with experimental observations, the Coriolis force introduced by the system rotation could stabilize/destabilize turbulence. The large-scale motion observed in the experiment was also found to exist in the computed flow field.