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The MAC discretization of fluid flow is analyzed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard H(sup 1) and L(sup 2) norms are given for the velocity and pressure, respectively.
A static (wind-off) test was conducted in the Static Test Facility of the 16-ft transonic tunnel to determine the performance and turning effectiveness of post-exit yaw vanes installed on two-dimensional convergent-divergent nozzles. One nozzle design that was previously tested was used as a baseline, simulating dry power and afterburning power nozzles at both 0 and 20 degree pitch vectoring conditions. Vanes were installed on these four nozzle configurations to study the effects of vane deflection angle, longitudinal and lateral location, size, and camber. All vanes were hinged at the nozzle sidewall exit, and in addition, some were also hinged at the vane quarter chord (double-hinged). The vane concepts tested generally produced yaw thrust vectoring angles much less than the geometric vane angles, for (up to 8 percent) resultant thrust losses. When the nozzles were pitch vectored, yawing effectiveness decreased as the vanes were moved downstream. Thrust penalties and yawing effectiveness both decreased rapidly as the vanes were moved outboard (laterally). Vane length and height changes increased yawing effectiveness and thrust ratio losses, while using vane camber, and double-hinged vanes increased resultant yaw angles by 50 to 100 percent.
A method for treating nonideal gas flows through converging-diverging nozzles is described. The method incorporates the Redlich-Kwong equation of state. The Runge-Kutta method is used to obtain a solution. Numerical results were obtained for methane gas. Typical plots of pressure, temperature, and area ratios as functions of Mach number are given. From the plots, it can be seen that there exists a range of reservoir conditions that require the gas to be treated as nonideal if an accurate solution is to be obtained.
A three-dimensional code (PAB3D) has been under development to simulate underexpanded and overexpanded supersonic jet exhaust effects on a nonaxisymmetric nozzle/afterbody. The code is a multiblock/multizone technique solving the simplified Navier-Stokes equations. This method was applied to the solution of two nozzle configurations to verify the recent implementation of a nozzle performance calculation package. Static pressure distributions, discharge coefficient and thrust ratio quantities were calculated for on-design and off-design operating conditions for rectangular convergent-divergent nozzles with different throat designs. Approaches involved in computing the flows for these nozzles are presented. Computation of performance parameters were typically within 1 percent of experimental on-design and off-design data.
Improved techniques and well-optimized basis sets are presented for application of the outgoing wave variational principle to calculate converged quantum mechanical reaction probabilities. They are illustrated with calculations for the reactions D + H2 yields HD + H with total angular momentum J = 3 and F + H2 yields HF + H with J = 0 and 3. The optimization involves the choice of distortion potential, the grid for calculating half-integrated Green's functions, the placement, width, and number of primitive distributed Gaussians, and the computationally most efficient partition between dynamically adapted and primitive basis functions. Benchmark calculations with 224-1064 channels are presented.
The application of formal methods to the analysis of computing systems promises to provide higher and higher levels of assurance as the sophistication of our tools and techniques increases. Improvements in tools and techniques come about as we pit the current state of the art against new and challenging problems. A promising area for the application of formal methods is in real-time and distributed computing. Some of the algorithms in this area are both subtle and important. In response to this challenge and as part of an ongoing attempt to verify an implementation of the Interactive Convergence Clock Synchronization Algorithm (ICCSA), we decided to undertake a proof of the correctness of the algorithm using the Boyer-Moore theorem prover. This paper describes our approach to proving the ICCSA using the Boyer-Moore prover.
The paper reports converged quantum total integral reactive cross sections for the reaction F + H2 yielding HF + H, for initial rotational states j sub i = 0 and 1, using a time-dependent method. The results are compared to classical results and to the experimental results of Neumark et al. (1985). Strong quantum effects are found in the threshold region for both initial states (i.e., in the dependence of the reaction on initial state for low energies). The classical results agree better with experiment than do the quantum results; this appears to be due to errors in the potential used.
The paper summarizes the information obtained on the South Pacific convergence zone (SPCZ) since it was first depicted in the surface analyses by Bergeron (1930). Particular attention is given to the average structure and characteristics of the SPCZ, the current theories regarding its origin and maintenance, and its relation to global-scale circulations. Some of the findings related to the quasi-periodic behavior of the SPCZ are presented.
An experimental investigation of multiaxis thrust vectoring nozzles with spherical convergent flaps (SCF) and a convertible (in-flight deployable) centerline plug were tested in the NASA-Langley 16-Foot Transonic Tunnel Static Test Facility. Parameters tested during the static test included plug length, external shroud length, shroud internal angle, and yaw vector angle. Results indicated that the SCF convertible-plug nozzle flow is highly three dimensional and characterized by internal flow separation and shock effects. The resultant thrust and discharge coefficient levels were lower than previous wedge nozzle results. In addition, increasing shroud internal angle increased resultant thrust ratio but decreased discharge coefficient. Increasing shroud length increased resultant thrust ratio performance for NPR's greater than 6.0 but for NPR's less than 6.0, resultant thrust ratio decreased as shroud length increased. Increasing plug length caused an increase in resultant thrust ratio and an increase in discharge coefficient. Increasing geometric yaw vector angle had no effect on resultant thrust ratio and little effect on discharge coefficient.
Multigrid has been applied to an existing three-dimensional compressible Euler solver to accelerate the convergence of the implicit symmetric relaxation scheme. This lower-upper symmetric Gauss-Seidel implicit scheme is shown to be an effective multigrid driver in three-dimensions. A grid refinement study is performed including the effects of large cell aspect ratio meshes. Performance figures of the present multigrid code on Cray computers including the new C90 are presented. A reduction of three orders of of magnitude in the residual for a three-dimensional transonic inviscid flow using 920K grid points is obtained in less than 4 minutes on a Cray C90.
An experimental investigation was made at Mach number 1.90 of the performance of a series of perforated convergent-divergent supersonic diffusers operating with initial boundary layer, which was induced and controlled by lengths of cylindrical inlets affixed to the diffusers. Supercritical mass-flow and peak total-pressure recoveries were decreased slightly by use of the longest inlets (4 inlet diameters in length). Combinations of cylindrical inlets, perforated diffusers, and subsonic diffuser were evaluated as simulated wind tunnels having second throats. Comparisons with noncontracted configurations of similar scale indicated conservatively computed power reductions of 25 percent.
An investigation was conducted in the 8-by-6 foot supersonic wind tunnel on the effect of exhaust-gas temperatures on the external and internal characteristics of a convergent-divergent nozzle having an area expansion ratio of 1.83. Data were obtained over a pressure-ratio range from 1 to 20 at free-stream Mach numbers of 1.6 and 2.0 for exhaust temperatures of 860 degrees, 1650 degrees, and 2000 degrees R. Results of this investigation indicated that generally both the internal and external performance characteristics were only slightly affected by a large change in jet temperature. The small differences in performance which did occur were predicted satisfactorily from theoretical considerations.
Internal performance data on a short exhaust nozzle designed by the method of characteristics were obtained over a range of pressure ratios from 1.5 to 22. The peak thrust coefficient was not affected by a shortened divergent section, but it occurred at lower pressure ratios due to reduction in expansion ratio. This nozzle contour based on characteristics solution gave higher thrust coefficients than a conical convergent-divergent nozzle of equivalent length. Abrupt-inlet sections permitted a reduction in nozzle length without a thrust-coefficient reduction.
Numerous experiments were conducted to determine factors which affect the internal performance of convergent-plug exhaust nozzles. The results of these experiments, which include effects of such things as plug shape, nozzle inlet Mach number, and outer-shell characteristics, provide a basis for optimum desifn from the standpoint of weight and size. The results also show performance penalties which can result when the nozzle is too small.
An investigation was conducted in the static test facility of the Langley 16-Foot Transonic Tunnel to evaluate the internal performance of a nonaxisymmetric convergent divergent nozzle designed to have simultaneous pitch and yaw thrust vectoring capability. This concept utilized divergent flap deflection for thrust vectoring in the pitch plane and flow-turning deflectors installed within the divergent flaps for yaw thrust vectoring. Modifications consisting of reducing the sidewall length and deflecting the sidewall outboard were investigated as means to increase yaw-vectoring performance. This investigation studied the effects of multiaxis (pitch and yaw) thrust vectoring on nozzle internal performance characteristics. All tests were conducted with no external flow, and nozzle pressure ratio was varied from 2.0 to approximately 13.0. The results indicate that this nozzle concept can successfully generate multiaxis thrust vectoring. Deflection of the divergent flaps produced resultant pitch vector angles that, although dependent on nozzle pressure ratio, were nearly equal to the geometric pitch vector angle. Losses in resultant thrust due to pitch vectoring were small or negligible. The yaw deflectors produced resultant yaw vector angles up to 21 degrees that were controllable by varying yaw deflector rotation. However, yaw deflector rotation resulted in significant losses in thrust ratios and, in some cases, nozzle discharge coefficient. Either of the sidewall modifications generally reduced these losses and increased maximum resultant yaw vector angle. During multiaxis (simultaneous pitch and yaw) thrust vectoring, little or no cross coupling between the thrust vectoring processes was observed.
In this paper, we present an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is defined in terms of an associated symmetric problem and includes point and line, Jacobi, and Gauss-Seidel iterations. We also study smoothers based entirely on the original operator. One is based on the normal form, that is, the product of the operator and its transpose. Other smoothers studied include point and line, Jacobi, and Gauss-Seidel. We show that the uniform estimates for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not depending on the number of multigrid levels).
Our Sensitivity-Based Minimal Residual (SBMR) method which is based on our earlier Distributed Minimal Residual (DMR) method allows each component of the solution vector in a system of equations to have its own convergence speed. Our global SBMR method was found to consistently outperform the DMR method while requiring considerably less computer memory. Recently, we have developed and tested a new Line SBMR or LSBMR method and a Time-Step-Scaling (TSS) method that are even more robust and computationally efficient than our global SBMR method, especially on highly clustered computational grids in laminar and turbulent flow computations.
The theory of non-oscillatory scalar schemes is developed in this paper in terms of the local extremum diminishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multi-dimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for one-dimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multi-dimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.