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A solution of the Navier-Stokes equations is given for an incompressible stagnation point flow whose magnitude oscillates in time about a constant, non-zero, value (an unsteady Hiemenz flow). Analytic approximations to the solution in the low and high frequency limits are given and compared with the results of numerical integrations. The application of these results to one aspect of the boundary layer receptivity problem is also discussed.
Numerical integrations using a potential enstrophy-conserving scheme are presented for the flow within a mixed layer over hilly terrain using the hydrostatic shallow-water equations with a quadratic drag law. The mesoscale area treated is 150 km on a side; cyclic lateral boundary conditions are used. It is found that for the idealized conditions treated (no surface heating, no entrainment and no pressure adjustments aloft), the topography quickly induces a steady state flow pattern by means of surface friction. Unsteadiness does not occur unless a surface-friction Reynolds number is greater than approximately 100. Effects of varying the Rossby number, Froude number and terrain-height parameter are examined.
Rayleigh's integral formula is evaluated numerically for planar radiators of any shape, with any specified velocity in the source plane using the fast Fourier transfrom algorithm. The major advantage of this technique is its speed of computation - over 400 times faster than a straightforward two-dimensional numerical integration. The technique is developed for computation of the radiated pressure in the nearfield of the source and can be easily extended to provide, with little computation time, the vector intensity in the nearfield. Computations with the FFT of the nearfield pressure of baffled rectangular plates with clamped and free boundaries are compared with the 'exact' solution to illuminate any errors. The bias errors, introduced by the FFT, are investigated and a technique is developed to significantly reduce them.
Analytical results are obtained by numerical integration of a cylindrically symmetric integral representation of the external solutions of the Helmholtz equation. The accuracy of this method is checked by comparisons of computed results with 'exact' solutions generated by the point source method. In all cases, the average error for the amplitude and phase of the points calculated in the field is found to be less than ten percent. Theoretical studies which model experiments run for the NASA Langely Bellmouth inlet configuration are presented and comparisons are made with the experimental results. In all cases very good agreement is obtained between the experimental and theoretically calculated values.
This paper involves a numerical simulation of the final stages of transition to turbulence in plane channel flow at a Reynolds number of 7500. Three-dimensional, incompressible Navier-Stokes equations are numerically integrated to obtain the time-evolution of two- and three-dimensional finite-amplitude disturbances. Computations are performed on the CYBER-203 vector processor for a 32 x 33 x 32 grid. Solutions indicate the existence of structures similar to those observed in the laboratory and which are characteristic of various stages of transition that lead to final breakdown. Details of the resulting flow field after breakdown indicate the evolution of streak-like formations found in turbulent flows. Although the flow field does approach a steady-state (turbulent channel flow), implementation of subgrid-scale terms are necesary to obtain proper turbulent statistics.
Array-fed reflector antennas are used extensively in today's contour and multiple beam satellite antennas. To determine the directivity of these antennas theoretically, the total radiated power of the array feed must be accurately computed. In this paper, a closed-form expression for the radiated power is obtained for arrays with the well-known (cos theta q-type element patterns. The formulation is general, and takes into account polarization, nonsymmetric E- and H-plane element patterns, nonuniform element spacings and arbitrary complex excitation coefficients. Selective numerical data are presented to demonstrate the usefulness of this result. Comparisons are made with the available results obtained using direct numerical integration techniques, and with other available data based on less general formulations. Excellent agreement is observed for all cases. In particular, data are presented on the directivity of seven-element cluster feeds used in multiple beam designs, and an array feed for producing a contour beam covering the United States Eastern Time Zone (ETZ).
The paths of dust particles in the Jovian ring are investigated using a numerical integration program, including the acceleration due to gravity and the Lorentz and drag accelerations arising from the motions of the charged dust through the Jovian plasma. It is determined that the orbit of a 2.5 micron radius spherical dust particle with a density of 2 g/cu cm -10V will become significantly perturbed. The ring will tend to warp northwards near 130-160 deg longitude, with the maximum excursion of the Jupiter ring grains equalling about 0.1 deg (consistent with a distance of 220 km above the equatorial plane). It is found that either the particles are larger or the voltages on them less than what has been determined by previous investigators, while the plasma near the ring may be considerably cooler than was estimated. Calculations show that particles of 0.3 micron with -10 V potentials are spread from 1.68-1.98 of the radius of Jupiter and inclined up to 7 deg out of the equatorial plane. The paths of these particles do not follow Keplerian orbits, and the particle positions are not symmetric about the equatorial plane. Particles of 0.4 micron radius have less asymmetric orbits than 0.3 micron particles, while particles less than 0.2 micron are perturbed into Jupiter cloudtops within a few tens of hours.
The motion of an ion in a space-charge wave on a strongly magnetized electron beam is investigated. The motions of the ion perpendicular and parallel to the beam direction are coupled by a nonlinear term in the ion Hamiltonian that is proportional to the wave amplitude, and this coupling causes the motion of the ion to deviate significantly from that of a linear harmonic oscillator in certain resonant regions of phase space. A sequence of canonical transformations is employed to investigate the motion of the ion in these regions. It is determined that wave amplitudes that are too small to trap beam electrons are too small to cause these resonances to overlap. When this overlap does not occur, the motion is found not to be discernibly ergodic in any three-dimensional subspace of the energy hypersurface because there exists a third constant of the motion in addition to the total energy and angular momentum. The numerically integrated ion trajectories are studied using surface-of-section techniques in order to verify these findings. It is found that the third constant of the motion constrains an ion initially trapped in a potential well of the wave to remain trapped in this well. It is concluded that ergodic behavior poses no threat at attempts at collective ion acceleration in space-charge waves on an electron beam.
A simulation program for verifying and checking the performance of the Galileo Spacecraft's Attitude and Articulation Control Subsystem's (AACS) flight software is discussed. The program, which is called Functional Simulator (FUNSIM), provides a simple method of interfacing user-supplied mathematical models coded in FORTRAN which describes spacecraft dynamics, sensors, and actuators; this is done with the AACS flight software, coded in HAL/S (High-level Advanced Language/Shuttle). It is thus able to simulate the AACS flight software accurately to the HAL/S statement level in the environment of a mainframe computer system. FUNSIM also has a command and data subsystem (CDS) simulator. It is noted that the input/output data and timing are simulated with the same precision as the flight microprocessor. FUNSIM uses a variable stepsize numerical integration algorithm complete with individual error bound control on the state variable to solve the equations of motion. The program has been designed to provide both line printer and matrix dot plotting of the variables requested in the run section and to provide error diagnostics.
Four deflected thrust nozzle concepts, designed to operate at the low pressure ratio typical of high bypass-ratio turbofan engines for medium speed (subsonic) V/STOL aircraft, were studied. Maps of overall performance characteristics and exit velocity distributions are used to highlight similarities and differences between the four concepts. Analytically determined secondary flows at the exit of a 90 deg circular pipe bend are compared with the experimental results from the more complex three dimensional geometries. The relative impact of total-pressure losses and secondary flows on nozzle thrust coefficient is addressed by numerical integration of exit velocity measurements. Previously announced in STAR as N83-25657
A procedure for analyzing the vibrations of rotating turbomachinery blades has been previously developed. This procedure is based upon shallow shell theory, and utilizes the Ritz method to determine frequencies and mode shapes. However, it has been limited heretofore to blades of uniform thickness, uniform curvature, and/or twist and rectangular planform. The present work shows how the procedure may be generalized to eliminate the aforementioned restrictions. Nonrectangular planforms are dealt with by a suitable coordinate transformation. This, as well as variable thickness, curvature and twist, require using numerical integration. The procedure is demonstrated on four examples of cantilevered blades for which theoretical and experimental data have been previously published: (1) flat plate with spanwise taper, (2) flat plate with chordwise taper, (3) twisted plate with chordwise taper, and (4) cylindrical shell with chordwise taper.
The steady isothermal solar wind equations are shown to admit, under certain circumstances, mutliple transonic solutions when, for example, momentum deposition gives rise to multiplee critical points in the flow. These multiple solutions consist of a continuous solution and solutions which involve shock transitions between critical solutions. The ambiguity arising from the multiplicity of the solutions can be resolved by following the time evolution of a wind profile with one critical point. Results of the numerical integration of the time-dependent equations with momentum addition show that each of these multiple solutions is physically accessible and depends on the rate of change of momentum deposition. These results suggest that standing shocks are likely to be present in the inner solar wind flow.
A numerical simulation of the final stages of transition to turbulence in plane channel flow is reported. Three dimensional, incompressible Navier-Stokes equations are numerically integrated to obtain the time-evolution of two and three dimensional finite amplitude disturbances. Computations are performed on the CYBER-203 vector processor for a 32x51x32 grid. Results are presented for no-slip boundary conditions at the solid walls as well as for periodic suction-blowing to simulate active control of transition by mass transfer. Solutions indicate that the method is capable of simulating the complex character of vorticity dynamics during the various stages of transition and final breakdown. In particular, evidence points to the formation of a lambda-shape vortex and the subsequent system of horseshoe vortices inclined to the main flow direction as the main elements of transition. Calculations involving suction-blowing indicate that interference with a wave of suitable phase and amplitude reduces the disturbance growth rates.
Results obtained from a numerical simulation of the final stages of transition to turbulence in plane channel flow are described. Three dimensional, incompressible Navier-Stokes equations are numerically integrated to obtain the time evolution of two and three dimensional finite amplitude disturbances. Computations are performed on CYBER-203 vector processor for a 32x51x32 grid. Results are presented for no-slip boundary conditions at the solid walls as well as for periodic suction blowing to simulate active control of transition by mass transfer. Solutions indicate that the method is capable of simulating the complex character of vorticity dynamics during the various stages of transition and final breakdown. In particular, evidence points to the formation of a lambda-shape vortex and the subsequent system of horseshoe vortices inclined to the main flow direction as the main elements of transition. Calculations involving periodic suction-blowing indicate that interference with a wave of suitable phase and amplitude reduces the disturbance growth rates.
To identify the fastest algorithm currently available for the numerical integration of chemical kinetic rate equations, several algorithms were examined. Findings to date are summarized. The algorithms examined include two general-purpose codes EPISODE and LSODE and three special-purpose (for chemical kinetic calculations) codes CHEMEQ, CRK1D, and GCKP84. In addition, an explicit Runge-Kutta-Merson differential equation solver (IMSL Routine DASCRU) is used to illustrate the problems associated with integrating chemical kinetic rate equations by a classical method. Algorithms were applied to two test problems drawn from combustion kinetics. These problems included all three combustion regimes: induction, heat release and equilibration. Variations of the temperature and species mole fraction are given with time for test problems 1 and 2, respectively. Both test problems were integrated over a time interval of 1 ms in order to obtain near-equilibration of all species and temperature. Of the codes examined in this study, only CREK1D and GCDP84 were written explicitly for integrating exothermic, non-isothermal combustion rate equations. These therefore have built-in procedures for calculating the temperature.
The equations of motion are developed for a perfectly flexible, inelastic tether with a satellite at its extremity. The tether is attached to a space vehicle in orbit. The tether is allowed to possess electrical conductivity. A numerical solution algorithm to provide the motion of the tether and satellite system is presented. The resulting differential equations can be solved by various existing standard numerical integration computer programs. The resulting differential equations allow the introduction of approximations that can lead to analytical, approximate general solutions. The differential equations allow more dynamical insight of the motion.
A general chemical kinetics code is described for complex, homogeneous ideal gas reactions in any chemical system. The main features of the GCKP84 code are flexibility, convenience, and speed of computation for many different reaction conditions. The code, which replaces the GCKP code published previously, solves numerically the differential equations for complex reaction in a batch system or one dimensional inviscid flow. It also solves numerically the nonlinear algebraic equations describing the well stirred reactor. A new state of the art numerical integration method is used for greatly increased speed in handling systems of stiff differential equations. The theory and the computer program, including details of input preparation and a guide to using the code are given.
The results of a study to determine the accuracy of the three dimensional solid elements available in NASTRAN for predicting displacements is presented. Of particular interest in the study is determining how to effectively use solid elements in analyzing thick optical mirrors, as might exist in a large telescope. Surface deformations due to thermal and gravity loading can be significant contributors to the determination of the overall optical quality of a telescope. The study investigates most of the solid elements currently available in either COSMIC or MSC NASTRAN. Error bounds as a function of mesh refinement and element aspect ratios are addressed. It is shown that the MSC solid elements are, in general, more accurate than their COSMIC NASTRAN counterparts due to the specialized numerical integration used. In addition, the MSC elements appear to be more economical to use on the DEC VAX 11/780 computer.