The diophantine problem Y² - X³ = A in a polynomial ring

Let C[z] be the ring of polynomials in z with complex coefficients; we consider the equation Y² — X³ = A, with A[is an element of]C[z] given, and seek solutions of this with X, Y[is an element of]C[z] i.e. we treat the equation as a "polynomial diophantine" problem. We show that when A is of degree 5 or 6 and has no multiple roots, then there are exactly 240 solutions (X, Y) to the problem with deg X ≤ 2 and deg Y ≤ 3

Nozzle fabrication technique

A block of electrically conductive material which is to be formed into a body with internal and/or external surfaces that approximate hyperboloids of one sheet is placed so that its axis is set at a predetermined skew angle with relation to a traveling EDM electrode wire. The electrode wire is then moved into cutting proximity of the body wire. Thereafter, by revolving the body about its own axis, the external and/or internal surfaces of the body will be cut into an approximate hyperbolic surface of revolution depending upon whether the body is positioned with the cutting wire outside of the body or in a previously formed longitudinal passage in the body. As an alternative technique, elongated channels can also be cut into the wall of the body by successively orienting the body to a selected number of angular positions, with the electrode wire being either outside of the body or in a previously formed passage in the body. At each of these angular positions, the electrode wire is moved orthogonally with respect to the axis of the wire, while both the body axis skew angle and the rotational position about that axis is controlled by cutting a channel or groove in the body to relieve stresses in the body material or to convey a coolant fluid

Pressure wave propagation studies for oscillating cascades

The unsteady flow field around an oscillating cascade of flat plates is studied using a time marching Euler code. Exact solutions based on linear theory serve as model problems to study pressure wave propagation in the numerical solution. The importance of using proper unsteady boundary conditions, grid resolution, and time step is demonstrated. Results show that an approximate non-reflecting boundary condition based on linear theory does a good job of minimizing reflections from the inflow and outflow boundaries and allows the placement of the boundaries to be closer than cases using reflective boundary conditions. Stretching the boundary to dampen the unsteady waves is another way to minimize reflections. Grid clustering near the plates does a better job of capturing the unsteady flow field than cases using uniform grids as long as the CFL number is less than one for a sufficient portion of the grid. Results for various stagger angles and oscillation frequencies show good agreement with linear theory as long as the grid is properly resolved

Numerical simulations of unsteady, viscous, transonic flow over isolated and cascaded airfoils using a deforming grid

A compressible, unsteady, full Navier-Stokes, finite difference code was developed for modeling transonic flow through two-dimensional, oscillating cascades. The procedure introduces a deforming grid technique to capture the motion of the airfoils. Results using a deforming grid are presented for both isolated and cascaded airfoils. The load histories and unsteady pressure distributions are predicted for the NASA 64A010 isolated airfoil and compared with existing experimental data. Results show that the deforming grid technique can be used to successfully predict the unsteady flow properties around an oscillating airfoil. The deforming grid technique was extended for modeling unsteady flow in a cascade. The use of a deforming grid simplifies the specification of boundary conditions. Unsteady flow solutions similar to the isolated airfoil predictions are found for a NACA 0012 cascade with zero interblade phase angle and zero stagger. Experimental data for these cases are not available for code validation, but computational results are presented to show sample predictions from the code. Applications of the code to typical turbomachinery flow conditions will be presented in future work

Unsteady-flow-field predictions for oscillating cascades

The unsteady flow field around an oscillating cascade of flat plates with zero stagger was studied by using a time marching Euler code. This case had an exact solution based on linear theory and served as a model problem for studying pressure wave propagation in the numerical solution. The importance of using proper unsteady boundary conditions, grid resolution, and time step size was shown for a moderate reduced frequency. Results show that an approximate nonreflecting boundary condition based on linear theory does a good job of minimizing reflections from the inflow and outflow boundaries and allows the placement of the boundaries to be closer to the airfoils than when reflective boundaries are used. Stretching the boundary to dampen the unsteady waves is another way to minimize reflections. Grid clustering near the plates captures the unsteady flow field better than when uniform grids are used as long as the 'Courant Friedrichs Levy' (CFL) number is less than 1 for a sufficient portion of the grid. Finally, a solution based on an optimization of grid, CFL number, and boundary conditions shows good agreement with linear theory

Nozzle fabrication technique

This invention relates to techniques for fabricating hour glass throat or convergent divergent nozzle shapes, and more particularly to new and improved techniques for forming rocket nozzles from electrically conductive material and forming cooling channels in the wall thereof. The concept of positioning a block of electrically conductive material so that its axis is set at a predetermined skew angle with relation to a travelling electron discharge machine electrode and thereafter revolving the body about its own axis to generate a hyperbolic surface of revolution, either internal or external is novel. The method will generate a rocket nozzle which may be provided with cooling channels using the same control and positioning system. The configuration of the cooling channels so produced are unique and novel. Also the method is adaptable to nonmetallic material using analogous cutting tools, such as, water jet, laser, abrasive wire and hot wire

From white elephant to Nobel Prize: Dennis Gabor’s wavefront reconstruction

Dennis Gabor devised a new concept for optical imaging in 1947 that went by a variety of names over the following decade: holoscopy, wavefront reconstruction, interference microscopy, diffraction microscopy and Gaboroscopy. A well-connected and creative research engineer, Gabor worked actively to publicize and exploit his concept, but the scheme failed to capture the interest of many researchers. Gabor’s theory was repeatedly deemed unintuitive and baffling; the technique was appraised by his contemporaries to be of dubious practicality and, at best, constrained to a narrow branch of science. By the late 1950s, Gabor’s subject had been assessed by its handful of practitioners to be a white elephant. Nevertheless, the concept was later rehabilitated by the research of Emmett Leith and Juris Upatnieks at the University of Michigan, and Yury Denisyuk at the Vavilov Institute in Leningrad. What had been judged a failure was recast as a success: evaluations of Gabor’s work were transformed during the 1960s, when it was represented as the foundation on which to construct the new and distinctly different subject of holography, a re-evaluation that gained the Nobel Prize for Physics for Gabor alone in 1971. This paper focuses on the difficulties experienced in constructing a meaningful subject, a practical application and a viable technical community from Gabor’s ideas during the decade 1947-1957

Numerical analysis of flow through oscillating cascade sections

The design of turbomachinery blades requires the prevention of flutter for all operating conditions. However, flow field predictions used for aeroelastic analysis are not well understood for all flow regimes. The present research focuses on numerical solutions of the Euler and Navier-Stokes equations using an ADI procedure to model two-dimensional, transonic flow through oscillating cascades. The model prescribes harmonic pitching motions for the blade sections for both zero and nonzero interblade phase angles. The code introduces the use of a deforming grid technique for convenient specification of the periodic boundary conditions. Approximate nonreflecting boundary conditions were coded for the inlet and exit boundary conditions. Sample unsteady solutions were performed for an oscillating cascade and compared to experimental data. Also, test cases were run for a flat plate cascade to compare with the unsteady, small-perturbation, subsonic analyis. The predictions for oscillating cascades with nonzero interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. The zero degree interblade phase angle cases, which were near a resonant condition, differ from the experiment and theory. Studies on reflecting versus nonreflecting inlet and exit boundary conditions show that the treatment of the boundary can have a significant effect on the first harmonic, unsteady pressure distributions for certain flow conditions

OL-AC Phillips Laboratory MPD thruster research program

The topics are presented in viewgraph form and include the following: facility construction; quadruple langmuir probe measurements; hollow/porous anode magnetoplasmadynamic (MPD) thruster; the measurement of the ionization fraction inside of the MPD thruster; and the experimental investigation of the effects of microturbulence on MPD thruster performance