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We prove a cell entropy inequality for a class of high order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one dimensional scalar convex case.
The problem of how a boundary layer responds to the motion of a convexed vortex on a porous wall was investigated. The wall velocity is approximately given by Darcy's law. The vorticity-stream function approach was adopted for solving Navier-Stokes equations of two dimensional incompressible viscous flows. The weighted-mean scheme was used for constructing finite difference approximations of spatial derivatives. Several test problems were solved and numerical results demonstrate clearly the accuracy, stability, and efficiency of the scheme. The weighted mean scheme then can be applied to the vortical flow problem.
The electromagnetic (EM) scattering from a planar termination located inside relatively arbitrarily shaped open-ended waveguide cavities with smoothly curved interior walls is analyzed using a Gaussian Beam (GB) expansion of the incident plane wave fields in the open end. The cavities under consideration may contain perfectly-conducting interior walls with or without a thin layer of material coating, or the walls may be characterized by an impedance boundary condition. In the present approach, the GB's are tracked only to the termination of the waveguide cavity via beam reflections from interior waveguide cavity walls. The Gaussian beams are tracked approximately only along their beam axes; this approximation which remains valid for relatively well focussed beams assumes that an incident GB gives rise to a reflected GB with parameters related to the incident beam and the radius of curvature of the wall. It is found that this approximation breaks down for GB's which come close to grazing a convex surface and when the width of the incident beam is comparable to the radius of curvature of the surface. The expansion of the fields at the open end depend on the incidence angle only through the expansion coefficients, so the GB's need to be tracked through the waveguide cavity only once for a wide range of incidence angles. At the termination, the sum of all the GB's are integrated using a result developed from a generalized reciprocity principle, to give the fields scattered from the interior of the cavity. The rim edge at the open end of the cavity is assumed to be sharp and the external scattering from the rim is added separately using Geometrical Theory of Diffraction. The results based on the present approach are compared with solutions based on the hybrid asymptotic modal method. The agreement is found to be very good for cavities made up of planar surfaces, and also for cavities with curved surfaces which are not too long with respect to their width.
The requirements of symmetry and convexity in the application of algorithms for the minimization or maximization of a function are discussed. It is argued that if a function of a single variable is convex and symmetric in a neighborhood of an extremum, the extremum may be approximated to the precision that increases by at least a power of two per functional evaluation. The procedure may be used to drive a complex optimization procedure in the multivariate area estimation problem encountered in remote sensing.
Iterative solution methods for minimizing convex, differentiable function in Euclidean space
A computer model capable of analyzing the flow field in the transition liner of small gas turbine engines is developed. A FORTRAN code has been assembled from existing codes and physical submodels and used to predict the flow in several test geometries which contain characteristics similar to transition liners, and for which experimental data was available. Comparisons between the predictions and measurements indicate that the code produces qualitative results but that the turbulence models, both K-E and algebraic Reynolds Stress, underestimate the cross-stream diffusion. The code has also been used to perform a numerical experiment to examine the effect of a variety of parameters on the mixing process in transition liners. Comparisons illustrate that geometries with significant curvature show a drift of the jet trajectory toward the convex wall and weaker wake region vortices and decreased penetration for jets located on the convex wall of the liner, when compared to jets located on concave walls. Also shown were the approximate equivalency of angled slots and round holes and a technique by which jet mixing correlations developed for rectangular channels can be used for can geometries.
In order to determine the conditions of engine operation causing the most severe thermal stresses in the hot parts of a turbojet engine, a J47-25 engine was instrumented with thermocouples and operated to obtain engine material temperatures under steady-state and transient conditions. Temperatures measured during rated take-off conditions of nozzle guide vanes downstream of a single combustor differed on the order of 400 degrees F depending on the relation of the blades position to the highest temperature zone of the burner. Under the same operation conditions, measured midspan temperatures in a nozzle guide vane in the highest temperature zone of a combustor wake ranged from approximately 1670 degrees F at leading and trailing edges to 1340 degrees F at midchord on the convex side of the blade. The maximum measured nozzle-guide-vane temperature of 1920degrees at the trailing edge occurred during a rapid acceleration from idle to rated take-off speed following which the tail-pipe gas temperature exceeded maximum allowable temperature by 125 degrees F.
A numerical solution of flow and heat transfer characteristics is obtained by the finite analytic method for a two dimensional laminar channel flow over a two-dimensional square cavity. The finite analytic method utilizes the local analytic solution in a small element of the problem region to form the algebraic equation relating an interior nodal value with its surrounding nodal values. Stable and rapidly converged solutions were obtained for Reynolds numbers ranging to 1000 and Prandtl number to 10. Streamfunction, vorticity and temperature profiles are solved. Local and mean Nusselt number are given. It is found that the separation streamlines between the cavity and channel flow are concave into the cavity at low Reynolds number and convex at high Reynolds number (Re greater than 100) and for square cavity the mean Nusselt number may be approximately correlated with Peclet number as Nu(m) = 0.365 Pe exp 0.2.
The theory developed by Hahne (1992) for the diffraction of time-harmonic electromagnetic waves from fixed obstacles is briefly summarized and extended. Applications of the theory are considered which comprise, first, a spherical harmonic expansion of the so-called radiation impedance operator in the theory, for a spherical surface, and second, a reconsideration of familiar short-wavelength approximation from the new standpoint, including a derivation of the so-called physical optics method on the basis of quasi-planar approximation to the radiation impedance operator, augmented by the method of stationary phase. The latter includes a rederivation of the geometrical optics approximation for the complete Green's function for the electromagnetic field in the presence of a smooth- and a convex-surfaced perfectly electrically conductive obstacle.
The Lax-Friedrichs scheme, approximating the scalar, genuinely nonlinear conservation law u sub t + f sub x (u) = 0 where f(u) is, say, strictly convex double dot f dot a sub asterisk 0 is studied. The divided differences of the numerical solution at time t do not exceed 2 (t dot a sub asterisk) to the -1. This one-sided Lipschitz boundedness is in complete agreement with the corresponding estimate one has in the differential case; in particular, it is independent of the initial amplitude in sharp contrast to liner problems. It guarantees the entropy compactness of the scheme in this case, as well as providing a quantitive insight into the large-time behavior of the numerical computation.
This paper investigates a convex optimization based method that can rapidly generate the fuel optimal asteroid powered descent trajectory. The ultimate goal is to autonomously design the optimal powered descent trajectory on-board the spacecraft immediately prior to the descent burn. Compared to a planetary powered landing problem, the major difficulty is the complex gravity field near the surface of an asteroid that cannot be approximated by a constant gravity field. This paper uses relaxation techniques and a successive solution process that seeks the solution to the original nonlinear, nonconvex problem through the solutions to a sequence of convex optimal control problems.
In the analytical study of ice accretions that form on aerodynamic surfaces (airfoils, engine inlets, etc.) it is often necessary to be able to calculate convective heat transfer rates. In order to do this, local convective heat transfer coefficients for the ice accretion shapes must be known. In the past, coefficients obtained for circular cylinders were used as an approximation to the actual coefficients since no better information existed. The purpose of this experimental study was to provide local convective heat transfer coefficients for four shapes that represent ice accretions. The shapes were tested with smooth and rough surfaces. The experimental method chosen was the thin-skin heat rate technique. Using this method local Nusselt numbers were determined for the ice shapes. In general it was found that the convective heat transfer was higher in regions where the model's surfaces were convex and lower in regions where the model's surfaces were concave. The effect of roughness was to increase the heat transfer in the high heat transfer regions by approximately 100% while little change was apparent in the low heat transfer regions.
A systematic procedure for constructing semidiscrete, second order accurate, variation diminishing, five point band width, approximations to scalar conservation laws, is presented. These schemes are constructed to also satisfy a single discrete entropy inequality. Thus, in the convex flux case, convergence is proven to be the unique physically correct solution. For hyperbolic systems of conservation laws, this construction is used formally to extend the first author's first order accurate scheme, and show (under some minor technical hypotheses) that limit solutions satisfy an entropy inequality. Results concerning discrete shocks, a maximum principle, and maximal order of accuracy are obtained. Numerical applications are also presented.
The mechanical integrity of hard, erosion-resistant coatings were tested using the Structural Dynamics Laboratory at the NASA Glenn Research Center. Under the guidance of Structural Mechanics and Dynamics Branch personnel, fixturing and test procedures were developed at Glenn to simulate engine vibratory conditions on coated polymer-matrix- composite bypass vanes using a slip table in the Structural Dynamics Laboratory. Results from the high-frequency mechanical bench testing, along with concurrent erosion testing of coupons and vanes, provided sufficient confidence to engine-endurance test similarly coated vane segments. The knowledge gained from this program will be applied to the development of oxidation- and erosion-resistant coatings for polymer matrix composite blades and vanes in future advanced turbine engines. Fan bypass vanes from the AE3007 (Rolls Royce America, Indianapolis, IN) gas turbine engine were coated by Engelhard (Windsor, CT) with compliant bond coatings and hard ceramic coatings. The coatings were developed collaboratively by Glenn and Allison Advanced Development Corporation (AADC)/Rolls Royce America through research sponsored by the High-Temperature Engine Materials Technology Project (HITEMP) and the Higher Operating Temperature Propulsion Components (HOTPC) project. High-cycle fatigue was performed through high-frequency vibratory testing on a shaker table. Vane resonant frequency modes were surveyed from 50 to 3000 Hz at input loads from 1g to 55g on both uncoated production vanes and vanes with the erosion-resistant coating. Vanes were instrumented with both lightweight accelerometers and strain gauges to establish resonance, mode shape, and strain amplitudes. Two high-frequency dwell conditions were chosen to excite two strain levels: one approaching the vane's maximum allowable design strain and another near the expected maximum strain during engine operation. Six specimens were tested per dwell condition. Pretest and posttest inspections were performed optically at up to 60 magnification and using a fluorescent-dye penetrant. Accumulation of 10 million cycles at a strain amplitude of two to three times that expected in the engine (approximately 670 Hz and 20g) led to the development of multiple cracks in the coating that were only detectable using fluorescent-dye penetrant inspection. Cracks were prevalent on the trailing edge and on the convex side of the midsection. No cracking or spalling was evident using standard optical inspection at up to 60 magnification. Further inspection may reveal whether these fine cracks penetrated the coating or were strictly on the surface. The dwell condition that simulated actual engine conditions produced no obvious surface flaws even after up to 80 million cycles had been accumulated at strain amplitudes produced at approximately 1500 Hz and 45g.
The on-surface radiation-condition method and the WKBJ method for approximating solutions to differential equations are used to derive an analytic formula for the surface currents on a two-dimensional perfectly conducting convex target. The currents are induced by an incident TE-polarized plane wave. The case of a circular cylinder is used to demonstrate the usefulness of the combined methods. It is shown that a two-term expansion yields good results for the surface currents and excellent results for the ensuing bistatic radar cross section.
An incompressible three-dimensional laminar boundary-layer flow over a swept wing is used as a model to study both the wall-curvature and streamline-curvature effects on the stationary crossflow instability. The basic state is obtained by solving the full Navier-Stokes (N-S) equations numerically. The linear disturbance equations are cast on a fixed, body-intrinsic, curvilinear coordinate system. Those nonparallel terms which contribute mainly to the streamline-curvature effect are retained in the formulation of the disturbance equations and approximated by their local finite difference values. The resulting eigenvalue problem is solved by a Chebyshev collocation method. The present results indicate that the convex wall curvature has a stabilizing effect, whereas the streamline curvature has a destabilizing effect. A validation of these effects with an N-S solution for the linear disturbance flow is provided.
An afocal monolithic optical element formed of a shallow cylinder of optical material (glass, polymer, etc.) with fast aspheric surfaces, nominally confocal paraboloids, configured on the front and back surfaces. The front surface is substantially planar, and this lends itself to deposition of multi-layer stacks of thin dielectric and metal films to create a filter for rejecting out-of-band light. However, an aspheric section (for example, a paraboloid) can either be ground into a small area of this surface (for a Cassegrain-type telescope) or attached to the planar surface (for a Gregorian-type telescope). This aspheric section of the surface is then silvered to create the telescope's secondary mirror. The rear surface of the cylinder is figured into a steep, convex asphere (again, a paraboloid in the examples), and also made reflective to form the telescope's primary mirror. A small section of the rear surface (approximately the size of the secondary obscuration, depending on the required field of the telescope) is ground flat to provide an unpowered surface through which the collimated light beam can exit the optical element. This portion of the rear surface is made to transmit the light concentrated by the reflective surfaces, and can support the deposition of a spectral filter.
Two nozzle flowfields are evaluated numerically using a Low Reynolds k-epsilon turbulence model to examine possible boundary layer relaminarization. Flow enters the scramjet-like nozzles at a Mach Number of approximately 4.5 and is expanded around smooth shoulders to an exit Mach number of about 6.5. The strong streamwise acceleration and large favorable pressure gradients act with the convex curvature to form small regions of laminar-like boundary layer behavior along the expansion surface. Two relaminarization criteria are evaluated and they both predict relaminarization to be occurring, with the trends accurately modeled by the Low-Reynolds Number turbulence model that is used. The Baldwin-Lomax turbulence model cannot model the near-wall changes that take place in the rapidly accelerating boundary layer, but it proves to be useful in providing the acceleration parameters used in predicting the potential for relaminarization.