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Separation of variables for linear partial differential equations of second order in two independent variables
NASA’s Urban Air Mobility (UAM) Sub-Project is engaged in research to support the introduction of air taxis into the National Airspace System. Such operations will require a range of communication, navigation, and surveillance systems. Air vehicles for UAM are under development and will initially have human pilots. Separation from other aircraft, obstacles, and weather may be a pilot responsibility or provided by an operator’s ground-based systems. Eventually, air taxis may be flown from the ground or fly autonomously. The variable separation concept adds separation buffers to air taxis depending upon their situation or location.
Two schemes of closing turbulent moment equations are proposed both of which make double correlation equations separated into single-point equations. The first is based on neglected triple correlation, leading to an equation differing from small perturbed gasdynamic equations where the separation constant appears as the frequency. Grid-produced turbulence is described in this light as time-independent, cylindrically-isotropic turbulence. Application to wall turbulence guided by a new asymptotic method for the Orr-Sommerfeld equation reveals a neutrally stable mode of essentially three dimensional nature. The second closure scheme is based on an assumption of identity of the separated variables through which triple and quadruple correlations are formed. The resulting equation adds, to its equivalent of the first scheme, an integral of nonlinear convolution in the frequency describing a role due to triple correlation of direct energy-cascading.
A separation of variables solution has been obtained for transient radiative cooling of an absorbing-scattering plane layer. The solution applies after an initial transient period required for adjustment of the temperature and scattering source function distributions. The layer emittance, equal to the instantaneous heat loss divided by the fourth power of the instantaneous mean temperature, becomes constant. This emittance is a function of only the optical thickness of the layer and the scattering albedo; its behavior as a function of these quantities is considerably different than for a layer at constant temperature.
Partial separation of variables applied to two p-electron calculations - spin orbit, angular momentum, energy transfer, atomic excitation
Three terms, ''Waterman's T-matrix method'', ''extended boundary condition method (EBCM)'', and ''null field method'', have been interchangeable in the literature to indicate a method based on surface integral equations to calculate the T-matrix. Unlike the previous method, the invariant imbedding method (IIM) calculates the T-matrix by the use of a volume integral equation. In addition, the standard separation of variables method (SOV) can be applied to compute the T-matrix of a sphere centered at the origin of the coordinate system and having a maximal radius such that the sphere remains inscribed within a nonspherical particle. This study explores the feasibility of a numerical combination of the IIM and the SOV, hereafter referred to as the IIMþSOV method, for computing the single-scattering properties of nonspherical dielectric particles, which are, in general, inhomogeneous. The IIMþSOV method is shown to be capable of solving light-scattering problems for large nonspherical particles where the standard EBCM fails to converge. The IIMþSOV method is flexible and applicable to inhomogeneous particles and aggregated nonspherical particles (overlapped circumscribed spheres) representing a challenge to the standard superposition T-matrix method. The IIMþSOV computational program, developed in this study, is validated against EBCM simulated spheroid and cylinder cases with excellent numerical agreement (up to four decimal places). In addition, solutions for cylinders with large aspect ratios, inhomogeneous particles, and two-particle systems are compared with results from discrete dipole approximation (DDA) computations, and comparisons with the improved geometric-optics method (IGOM) are found to be quite encouraging.
For a time-dependent, n-dimensional, special diagonal Hamilton-Jacobi equation a necessary and sufficient condition for the separation of variables to yield a complete integral of the form was established by specifying the admissible forms in terms of arbitrary functions. A complete integral was then expressed in terms of these arbitrary functions and also the n irreducible constants. As an application of the results obtained for the two-dimensional Hamilton-Jacobi equation, analysis was made for a comparatively wide class of dynamical problems involving a particle moving in Euclidean three-dimensional space under the action of external forces but constrained on a moving surface. All the possible cases in which this equation had a complete integral of the form were obtained and these are tubulated for reference.
Wave function of multivalent atom from linear determinants formulated from laplace theorem
Shutter varies gas throughput of porous plug. Variable area exposed on porous plug allows to pass varying rates of vapor flow while blocking flow of liquid helium II from cryogenic bath. Applications in refining operations, industrial chemistry, and steam-powered equipment.
Radial, diaphragm separated injection method for variable thrust liquid rocket engines with or without mixture ratio variation
A numerical technique to solve the Euler equations for steady, one dimensional flows is presented. The technique is essentially implicit, but is structured as a sequence of explicit solutions for each Riemann variable separately. Each solution is obtained by integrating in the direction prescribed by the propagation of the Riemann variables. The technique is second-order accurate. It requires very few steps for convergence, and each step requires a minimal number of operations. Therefore, it is three orders of magnitude more efficient than a standard time-dependent technique. The technique works very well for transonic flows and provides shock fitting with errors as small as 0.001. Results are presented for subsonic problems. Errors are evaluated by comparison with exact solutions.
The invention, a static phase separator (SPS), uses airflow and capillary wetting characteristics to passively separate a two-phase (liquid and air) flow. The device accommodates highly variable liquid wetting characteristics. The resultant design allows for a range of wetting properties from about 0 to over 90 advancing contact angle, with frequent complete separation of liquid from gas observed when using appropriately scaled test conditions. Additionally, the design accommodates a range of air-to-liquid flow-rate ratios from only liquid flow to over 200:1 air-to-liquid flow rate. The SPS uses a helix input section with an ice-cream-cone-shaped constant area cross section (see figure). The wedge portion of the cross section is on the outer edge of the helix, and collects the liquid via centripetal acceleration. The helix then passes into an increasing cross-sectional area vane region. The liquid in the helix wedge is directed into the top of capillary wedges in the liquid containment section. The transition from diffuser to containment section includes a 90 change in capillary pumping direction, while maintaining inertial direction. This serves to impinge the liquid into the two off-center symmetrical vanes by the airflow. Rather than the airflow serving to shear liquid away from the capillary vanes, the design allows for further penetration of the liquid into the vanes by the air shear. This is also assisted by locating the air exit ports downstream of the liquid drain port. Additionally, any droplets not contained in the capillary vanes are re-entrained downstream by a third opposing capillary vane, which directs liquid back toward the liquid drain port. Finally, the dual air exit ports serve to slow the airflow down, and to reduce the likelihood of shear. The ports are stove-piped into the cavity to form an unfriendly capillary surface for a wetting fluid to carryover. The liquid drain port is located at the start of the containment region, allowing for draining the bulk fluid in a continuous circuit. The functional operation of the SPS involves introducing liquid flow (from a human body, a syringe, or other source) to the two-phase inlet while an air fan pulls on the air exit lines. The fan is operated until the liquid is fully introduced. The system is drained by negative pressure on the liquid drain lines when the SPS containment system is full.
Urban Air Mobility (UAM) has captured the imagination of the public and the aviation industry for someday soon moving people and goods through and around metropolitan areas using Unmanned Aircraft Systems (UAS) that are electrically powered, environmentally friendly, and autonomously operated. Significant investment and rapid development of vehicles for this activity is taking place, with package delivery services already beginning in some areas. However, the ability to manage thousands of these vehicles safely in a congested urban area presents a challenge unprecedented in air traffic management. Initial studies of this problem led by NASA under the UAS Traffic Management (UTM) initiative have primarily focused on geo-fencing and centralized reservation of airspace for individual flights. This paper proposes an extension of UTM using a de-centralized approach employing airborne surveillance, self-separation, and a minimized “design separation” approach to permit the optimization and safety of each flight in very high traffic densities and close proximities. The concept employs Airborne Trajectory Management (ABTM) principles and a novel new concept for variable separation criteria to manage the angular velocity of a passing vehicle, thus eliminating the "startle factor" and perceived hazard of very close operations. ABTM also accomplishes most of the services required for safe planning and execution of normal flights and recovery from abnormal or emergency operations while accommodating conventional piloted flights using the current air traffic control paradigm. The environment for UAM operations is described along with the proposed means for autonomous, tactical separation of the vehicles. Sample geometries of traffic conflicts and resolutions are shown and the airspace definitions, rules for flight within them, and additions and exemptions to the rules for these flights are listed and explained.
The performance of air traffic controllers using a simulated microwave landing system to control the landing of STOL aircraft is investigated. In a series of two experiments, controllers were asked to achieve a 60-sec separation between targets at the missed approach point when only two targets were on the simulated radar scope, and in the presence of 25 targets, with up to 10 on the screen at any one time. In both experiments, the presence of a ground wind is found to degrade separation performance, and an increased work load results in even greater separation variability, as the controllers did not have time to convert aircraft distances into times. In addition, curved courses are found to have an effect on performance. It is thus recommended that in order to create a practical air traffic control system with separation standards based on time, as in the MLS, controller work loads should be reduced and aircraft positions should be displayed with respect to time rather than distance.
Mathematical models which quantify the relation of wheat yield to selected weather-related variables are presented. Other sources of variation (amount of applied nitrogen, improved varieties, cultural practices) have been incorporated in the models to explain yield variation both singly and in combination with weather-related variables. Separate models were developed for fall-planted (winter) and spring-planted (spring) wheats. Meteorological variation is observed, basically, by daily measurements of minimum and maximum temperatures, precipitation, and tabled values of solar radiation at the edge of the atmosphere and daylength. Two different soil moisture budgets are suggested to compute simulated values of evapotranspiration; one uses the above-mentioned inputs, the other uses the measured temperatures and precipitation but replaces the tabled values (solar radiation and daylength) by measured solar radiation and satellite-derived multispectral scanner data to estimate leaf area index. Weather-related variables are defined by phenological stages, rather than calendar periods, to make the models more universally applicable.
Generalized von Zeipel treatment of lunar and artificial satellite theories, generating single canonical transformation by variable separation technique
A two-dimensional detailed study of the behavior of long waves in curved ducts and in junctions between straight and curved ducts will be given. The mathematical treatment of the problem utilizes the method of separation variables. Solutions and expressions for principal mode of the wave are obtained by using the linearized equation of motion solved for its characteristic values. The unavoidable approximations in the numerical solutions of the cylindrical functions are due to use of series expansion of Bessel functions and from restrictions necessary to solve infinite matrices.
A variational technique is used to model sound transmission through a nonuniform duct segment consisting of an axial variation in wall admittance or cross sectional area. The method involves the Ritz minimization of functionals which have the governing equations as stationary conditions. The variational method is verified by application to segments of variables-separable geometry for which eigenfunction expansion techniques offer an alternative solution procedure, and by comparison with the results of stepped duct approximations to the nonuniformity. Quantitative data are presented which indicate the boundary condition matching to be a suitable measure of the accuracy of the transmitted field.