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An approximation framework is presented for computation (in finite dimensional spaces) of Riccati operators that can be guaranteed to converge to the Riccati operator in feedback controls for abstract evolution systems in a Hilbert space. It is indicated how these results may be used in the linear optimal regulator problem for a large class of parabolic systems.
The case of a parabolic reflector and a point feed is considered, taking into account the question regarding the location in which the feed should be placed for an achievement of maximum directivity. Based on the tracing of geometrical rays, the obvious answer is obtained that the feed should be placed at the focal point. In the present paper, it is shown that this answer is not always correct. There are situations in which the maximum directivity is achieved when the feed is axially displaced toward the reflector or away from it. This 'focal shift' phenomenon is a result of three competing factors which affect the directivity of a reflector. The factors are related to phase synchronism over the reflector aperture, aperture illumination efficiency, and spillover loss. For achieving the maximum directivity, it is necessary to find the best compromise among the three factors.
A parabolic combustion code called CHARNAL was run for a variety of measured and estimated inlet boundary conditions. A parametric study was carried out to determine what compromises in accuracy would occur if estimated, rather than experimentally measured, inlet conditions were used and what values of the dissipation length scale would give the best results. The conclusions were that a dissipation length scale of approximately 0.03 gave the best results regardless of the type of inlet conditions used. Both subsonic and supersonic flow were examined and the effect of finite-rate versus equilibrium-chemistry calculations was briefly considered. No significant compromises in accuracy were found when estimated inlet boundary conditions were used.
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 separating 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.
A feasibility study has been conducted to assess the applicability of an existing parabolic analysis (ADD-Axisymmetric Diffuser Duct), developed previously for subsonic viscous internal flows, to mixed supersonic/subsonic flows with heat addition simulating a SCRAMJET combustor. A study was conducted with the ADD code modified to include additional convection effects in the normal momentum equation when supersonic expansion and compression waves are present. A set of test problems with weak shock and expansion waves have been analyzed with this modified ADD method and stable and accurate solutions were demonstrated provided the streamwise step size was maintained at levels larger than the boundary layer displacement thickness. Calculations made with further reductions in step size encountered departure solutions consistent with strong interaction theory. Calculations were also performed for a flow field with a flame front in which a specific heat release was imposed to simulate a SCRAMJET combustor. In this case the flame front generated relatively thick shear layers which aggravated the departure solution problem. Qualitatively correct results were obtained for these cases using a marching technique with the convective terms in the normal momentum equation suppressed. It is concluded from the present study that for the class of problems where strong viscous/inviscid interactions are present a global iteration procedure is required.
A stable global-iteration procedure is applied to a fully implicit noniterative parabolized Navier-Stokes algorithm to improve the solutions of the single-sweep method as well as to resolve small separation bubbles due to shock-wave/boundary-layer interactions. A modified form of the FLARE approximation is employed in the separated regions and some requirements for stable iterations are discussed. The results of the global-iteration scheme for two test cases of supersonic internal flows are compared against the results of the single-sweep method as well as the results of the full Navier-Stokes solver.
Downstream marching iterative schemes for the solution of the Parabolized or Thin Layer (PNS or TL) Navier-Stokes equations are described. Modifications of the primitive equation global relaxation sweep procedure result in efficient second-order marching schemes. These schemes take full account of the reduced order of the approximate equations as they behave like the SLOR for a single elliptic equation. The improved smoothing properties permit the introduction of Multi-Grid acceleration. The proposed algorithm is essentially Reynolds number independent and therefore can be applied to the solution of the subsonic Euler equations. The convergence rates are similar to those obtained by the Multi-Grid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Extensions to three-dimensional and compressible subsonic flows are discussed. Numerical results are presented.
The development history of the data acquisition and data analysis software is discussed. The software development occurred between 1978 and 1984 in support of solar energy module testing at the Jet Propulsion Laboratory's Parabolic Dish Test Site, located within Edwards Test Station. The development went through incremental stages, starting with a simple single-user BASIC set of programs, and progressing to the relative complex multi-user FORTRAN system that was used until the termination of the project. Additional software in support of testing is discussed including software in support of a meteorological subsystem and the Test Bed Concentrator Control Console interface. Conclusions and recommendations for further development are discussed.
The parabolized Navier-Stokes (PNS) equations are used to calculate the flow-field characteristics about the hypersonic research aircraft X-24C. A comparison of the results obtained using elliptic, hyperbolic and algebraic grid generators is presented. The outer bow shock is treated as a sharp discontinuity, and the discontinuities within the shock layer are captured. Surface pressures and heat-transfer results at angles of attack of 6 deg and 20 deg, obtained using the three grid generators, are compared. The PNS equations are marched downstream over the body in both Cartesian and cylindrical base coordinate systems, and the results are compared. A robust marching procedure is demonstrated by successfully using large marching-step sizes with the implicit shock fitting procedure. A correlation is found between the marching-step size, Reynolds number and the angle of attack at fixed values of smoothing and stability coefficients for the marching scheme.
The differential formulations and computational techniques currently used for the incompressible Navier-Stokes (NS) and parabolic Navier-Stokes (PNS) equations are reviewed. In particular, attention is given to problems associated with the choice of difference equations, the method of solution and the choice of algorithm, the coupling of dependent variables and discretized equations, the application of boundary conditions, and grid generation. A new composite velocity NS and PNS formulation in (u,v,p) variables is presented, and the applicability of a 'forward' difference global pressure iteration for the (u,v,p) PNS system is demonstrated.
Asymptotic solutions of order k to the nth are developed for the reduced wave equation. Here k is a dimensionless wave number and n is the arbitrary order of the approximation. These approximations are an extension of geometric acoustics theory, and provide corrections to that theory in the form of multiplicative functions which satisfy parabolic partial differential equations. These corrections account for the diffraction effects caused by variation of the field normal to the ray path and the interaction of these transverse variations with the variation of the field along the ray. The theory is applied to the example of radiation from a piston, and it is demonstrated that the higher order approximations are more accurate for decreasing values of k.
Episodes of emesis unaccompanied by the usual prodomal signs of motion sickness have been reported by astronauts in the Space Shuttle program. Such reports have raised the issue whether space motion sickness has different characteristics from terrestrial motion sickness. Evidence is presented here from parabolic flight experiments that sudden vomiting can occur in response to a provocative vestibular stimulus even when no premonitory symptoms are being experienced. Accordingly, in chronic exposure conditions, the absence of prominent signs or symptoms of motion sickness does not necessarily mean an absence of sensitization.
A new upwind algorithm based on Roe's scheme has been developed to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method does not require the addition of user specified smoothing terms for the capture of discontinuities such as shock waves. Thus, the method is easy to use and can be applied without modification to a wide variety of supersonic flowfields. The advantages and disadvantages of this adaptation are discussed in relation to those of the conventional Beam-Warming scheme in terms of accuracy, stability, computer time and storage, and programming effort. The new algorithm has been validated by applying it to three laminar test cases including flat plate boundary-layer flow, hypersonic flow past a 15 deg compression corner, and hypersonic flow into a converging inlet. The computed results compare well with experiment and show a dramatic improvement in the resolution of flowfield details when compared with the results obtained using the conventional Beam-Warming algorithm.
An asymptotic, large wave number approximation for the equations governing the propagation of acoustic disturbances through a stratified moving medium is developed. The theory is an extension of the geometric acoustics approximation and provides corrections to that approximation in the form of multiplicative functions which satisfy parabolic differential equations of second order. By properly accounting for variations in the acoustic field in directions normal to the rays both caustic surfaces and the secularity of the geometric theory may be avoided.
Downstream marching iterative schemes for the solution of the Parabolized or Thin Layer (PNS or TL) Navier-Stokes equations are described. Modifications of the primitive equation global relaxation sweep procedure result in efficient second-order marching schemes. These schemes take full account of the reduced order of the approximate equations as they behave like the SLOR for a single elliptic equation. The improved smoothing properties permit the introduction of Multi-Grid acceleration. The proposed algorithm is essentially Reynolds number independent and therefore can be applied to the solution of the subsonic Euler equations. The convergence rates are similar to those obtained by the Multi-Grid solution of a single elliptic equation; the storage is also comparable as only the pressure has to be stored on all levels. Extensions to three-dimensional and compressible subsonic flows are discussed. Numerical results are presented.
A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems. Unlike any classical counterpart, this class offers arbitrarily high, optimal order convergence. In support of this claim, error estimates are proved, and computational results are presented. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.
Compound parabolic concentrators are nonimaging cone-shaped optics with useful angular transmission characteristics. Two cones used throat-to-throat accept radiant flux within one well-defined acceptance angle and redistribute it into another. If the entrance cone is fed with Lambertian flux, the exit cone produces a beam whose half-angle is the exit cone's acceptance angle and whose cross section shows uniform irradiance from near the exit mouth to infinity. (The pair is a beam angle transformer). The design of one pair of cones is discussed, also an experiment to map the irradiance of the emergent beam, and a raytracing program which models the cones fed by Lambertian flux. Experimental results compare favorably with raytrace results.
The present, residuals' L(2)-norms analysis of the rate of convergence to steady state for parabolic ADI solvers allows the prediction of the number of iterations required for convergence, as a function of the Courant number alpha. A modification of current ADI codes is presented which significantly improves the convergence rate and is insensitive to the Courant number over a large range of alpha. This corrected algorithm is tested for the cases of Dirichlet problems for uniform grids of many mesh sizes, mixed Dirichlet-Neumann problems, and problems defined on stretched grids and/or problems with variable coefficients.