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A splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.
The concept of splitting is used extensively to simulate complex three dimensional flows on modern computer architectures. Used in all aspects, from initial grid generation to the determination of the final converged solution, splitting is used to enhance code vectorization, to permit solution driven grid adaption and grid enrichment, to permit the use of concurrent processing, and to enhance data flow through hierarchal memory systems. Three examples are used to illustrate these concepts to complex three dimensional flow fields: (1) interactive flow over a bump; (2) supersonic flow past a blunt based conical afterbody at incidence to a free stream and containing a centered propulsive jet; and (3) supersonic flow past a sharp leading edge delta wing at incidence to the free stream.
Two time-level, five-point explicit and implicit upwind difference schemes based on the characteristic flux difference splitting concept have been developed for the two-dimensional Euler equations. The method is conservative, second-order accurate in time and space, and general coordinate systems are used to treat complex geometries. Nonlinear flux limiters are employed to yield oscillation free sharp shock profiles. Upstream interpolation is used to yield a class of higher-order upwind schemes which closely mimic the locally one-dimensional method of characteristics (with fixed time intervals) through operator splitting. Numerical results have been obtained for a plane shock reflection and for flow over a circular arc in a channel. Characteristics of upwind TVD schemes, as applied to two-dimensional flows with embedded shocks are discussed.
The structure, torsional potentials, vibrational spectra, and harmonic force fields for s-cis and s-trans isomers of methacryloyl fluoride are examined to understand the conformational properties of the molecules and their relationship to macroscopic polymer properties. The structure is found to be in good agreement with experiment. It is shown by calculations that the energy difference between the cis and the transisomers is less than 1 kcal/mol at both the split valence and the split valence polarized levels, with the trans form favored. Analysis of the torsional potentials indicates that a rigid rotor model provides a reasonable description of the motion of the COF group in the molecule. The torsional barrier to interconvert the s-trans to the s-cis form is found to be 7.0 kcal/mol. A fit of the data to a three-term Fourier series shows that it is possible to reproduce the experimentally derived barrier, even though a direct determination indicates that the barrier is higher.
Computational aspects of modern high-resolution upwind finite-difference schemes for hyperbolic systems of conservation laws are examined. An operational unification is demonstrated for constructing a wide class of flux-difference-split and flux-split schemes based on the design principles underlying total variation diminishing (TVD) schemes. Consideration is also given to TVD scheme design by preprocessing, the extension of preprocessing and postprocessing approaches to general control volumes, the removal of expansion shocks and 'glitches', relaxation methods for implicit TVD schemes, and a new family of high-accuracy TVD schemes.
The results of an experimental assessment of some advanced Stirling engine component concepts are presented. High performance piston rings, reciprocating oil scrapers and heat pipes with getters and with mechanical couplings were tested. The tests yielded the following results: (1) Bonded, split, pumping piston rings, in preliminary testing, proved a promising concept, exhibiting low leakage and friction losses. Solid piston rings proved impractical in view of their sensitivity to the operating temperature; (2) A babbit oil scraper in a compliant housing performed well in atmospheric endurance testing. In pressurized tests the scraper did not perform well as a containment seal. The latter tests suggest modifications which may adapt Ti successfully to that application; and (3) Heat pipe endurance tests indicated the adequacy of simple, inexpensive fabrication and filling procedures. Getters were provided to increase the tolerance of the heat pipes to the presence of air and commercially available couplings were demonstrated to be suitable for heat pipe application. In addition to the above tests, the program also included a design effort for a split shaft applicable to a swashplate driven engine with a pressurized crank-case. The design is aimed, and does accomplish, an increase in component life to more than 10,000 hours.
The asymmetric nature of the fluid film stiffness and damping properties in rotors supported on fluid film bearings causes a forward or a backward whirl depending on the bearing parameters and the speed of the rotor. A rotor was designed to exhibit backward synchronous whirl. The rotor-bearing system exhibited split criticals, and a backward whirl was observed between the split criticals. The orbital diagrams show the whirl pattern.
Observations are reported which confirm Paczynski's 1986 speculation that the previously known (Arp and Hazard, 1980) pair of 19th magnitude quasars 1146+111B,C are actually two images of a single object produced by a gravitational lens. The image split is 157 arcsec, more than 20 times greater than any previously reported, thus indicating an exceptionally massive lensing object. The data supporting the lens hypothesis are remarkably similar, high signal-to-noise, moderate resolution spectra of the two components. Both spectra show strong Mg II 2798-angstrom emission at z = 1.012 plus or minus 0.001 with indistinguishable redshifts (delta v = 126 plus or minus 309 km/s), widths (FWHM = 64 plus or minus 4 angstroms) and detailed profile shapes. Both spectra also show broad troughs at 6180 angstroms and several weaker continuum features. Neither object exhibits the O II 3727-A forbidden line which is frequently strong in Mg II emission objects. If the foreground galaxy clustering apparent in a deep R band charge-coupled device image proves insufficient to explain the large image splitting, other possibilities such as massive dark objects (e.g. a black hole of about 10 to the 15th solar masses) or a cosmic string may be indicated.
Boron/aluminum specimens were made with crack-like slits in the center and with various proportions of 0 and + or - 45 deg plies. They were fatigue loaded and then fractured to determine their residual strengths. The fatigue loads were generally in the range of 60 to 80 percent of the static tensile strength of the specimen as determined from a previous study, and the stress ratio was .05. For virtually all of the specimens the fatigue loading was continued for 100,000 cycles. The specimens were radiographed after the fatigue loading to determine the nature of the fatigue damage. A few specimens were sectioned and examined in a scanning electron microscope after being radiographed in order to verify the interpretation of the radiographs and also to get a better insight into the nature of the fatigue damage. The results indicate that the fatiguing does not significantly affect the strength of the specimens tested. The results of the radiography and of the scanning electron microscopy indicate that the 45 deg plies suffer extensive damage in the form of split and broken fibers and matrix cracking in the vicinity of the ends of the split. By contrast, the only significant damage to the 0 deg plies was a single 0 deg matric crack growing from the ends of the slit and between the 0 deg fibers.
An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.
A mathematical model based on the classical shear-lag assumptions is used to study the residual strength and fracture behavior of composite laminates with symmetrically placed buffer strips. The laminate is loaded by a uniform remote longitudinal tensile strain and has initial damage in the form of a transverse crack in the parent laminate between buffer strips. The crack growth behavior as a function of material properties, number of buffer-strip plies, spacing, width of buffer strips, longitudinal matrix splitting, and debonding at the interface is studied. Buffer-strip laminates are shown to arrest fracture and increase the residual strengths significantly over those of one material laminates, with S-glass being a more effective buffer strip material than Kevlar in increasing the damage tolerance of graphite/epoxy panels. For a typical graphite/epoxy laminate with S-glass buffer-strips, the residual strength is about 2.4 times the residual strength of an all graphite/epoxy panel with the same crack length. Approximately 50% of this increase is due to the S-glass/epoxy buffer-strips, 40% due to longitudinal splitting of the buffer strip interface and 10% due to bonding.
A general survey of grid generators is presented with a concern for understanding why grids are necessary, how they are applied, and how they are generated. After an examination of the need for meshes, the overall applications setting is established with a categorization of the various connectivity patterns. This is split between structured grids and unstructured meshes. Altogether, the categorization establishes the foundation upon which grid generation techniques are developed. The two primary categories are algebraic techniques and partial differential equation techniques. These are each split into basic parts, and accordingly are individually examined in some detail. In the process, the interrelations between the various parts are accented. From the established background in the primary techniques, consideration is shifted to the topic of interactive grid generation and then to adaptive meshes. The setting for adaptivity is established with a suitable means to monitor severe solution behavior. Adaptive grids are considered first and are followed by adaptive triangular meshes. Then the consideration shifts to the temporal coupling between grid generators and PDE-solvers. To conclude, a reflection upon the discussion, herein, is given.
Global oscillations of the sun (r-modes) with very long periods of about 1 month are reviewed and studied. Such modes would be trapped in an acoustic cavity formed either by most of the convective envelope or by most of the radiative interior. A turning point frequency giving cavity boundaries is defined and the run of eigenvalues for angular harmonics l less than or equal to 3 are plotted for a conventional solar convection zone. The r-modes show equipartition of oscillatory energy among shells which each contain one antinode in the radial dimension. Toroidal motion is dominant to at least the 14th radial harmonic mode. Viscosity from convective turbulence is strong and would damp any mode in just a few solar rotations if it were the only significant nonadiabatic effect. 'Radial fine splitting' which lifts the degeneracy in n is very small (20 nHz or less) for all n less than or equal to 14 trapped in the envelope. But if splitting could be detected, there would be a valuable new constraint on solar convection theories.
A splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations.
A new algorithm has been developed for the Euler equations that uses flux vector splitting in combination with the concept of rotating the coordinate system to the local streamwise direction. Flux vector biasing is applied along the local streamwise direction and central differencing is used transverse to the flow direction. The flux vector biasing is switched from upwind for supersonic flow to downwind-biased for subsonic flow. This switching is based on the Mach number; hence the proper domain of dependence is used in the supersonic regions and the switching occurs across shock waves. The theoretical basis and the development of the formulas for flux vector splitting are presented. Then several one-dimensional calculations are presented of steady and unsteady transonic flows, which demonstrate the stability and accuracy of the algorithm. Finally results are shown for unsteady transonic flow over an airfoil. The pressure coefficient plots show sharp transonic shock profiles, and the Mach contour plots show smoothly varying contours.
An attempt is made to investigate local time linearization techniques for implicit flux-difference splitting and flux-vector splitting schemes in the simplest settings (i.e., first-order spatial schemes and one-dimensional Euler flows). It is noted that first-order spatial schemes provide the simplest examples of schemes which are collective extensions of scalar TVD schemes. Simple analytical results concerning the local linearizations are highlighted and subsequently verified using a numerical fixed-point analysis on selected problems. It is noted that while primary emphasis is on asymptotic behavior, many of the results have implications for time-accurate calculations as well.
Numerical flux formulas for the convection terms in the Euler or Navier-Stokes equations are analyzed with regard to their accuracy in representing steady nonlinear and linear waves (shocks and entropy/shear waves, respectively). Numerical results are obtained for a one-dimensional conical Navier-Stokes flow including both a shock and a boundary layer. Analysis and experiments indicate that for an accurate representation of both layers the flux formula must include information about all different waves by which neighboring cells interact, as in Roe's flux-difference splitting. In comparison, Van Leer's flux-vector splitting, which ignores the linear waves, badly diffuses the boundary layer. The results of MacCormack's scheme, if properly tuned, are significantly better. The use of a sufficiently detailed flux formula appears to reduce the number of cells required to resolve a boundary layer by a factor 1/2 to 1/4 and thus pays off.
The present work uses the PIC method with operator splitting to hydroelastoplastic multimaterial flow problems in 2D Eulerian coordinates. The operator splitting procedures are carried out in both Lagrangian and remap phases with alternating direction for each time advancement to maintain the accuracy of one-dimensional scheme. A detail description of the numerical approximations is given with a sample calculation of a tungsten rod penetrating an aluminum target.