Arguments for and against Engesser's buckling formulas

Summary To account for shear in buckling pioblems, one uses either the approach of Engesser or the socalled modified approach, proposed by Haringx in connection with helical springs. Both are based on onedimensional reasoning. While most authors seem to prefer the second method, Nänni [4] has shown by means of the three-dimensional theory of elasticity that, for bars, the first one is superior. The present paper is based on a more fundamental one-dimensional approach and confirms Nänni's and hence Engesser's results. It also shows that, together with shear, shortening of the bar must be taken into account, and it improves Engesser's formulas accordingly. It finally shows why Haringx' approach is correct for springs.

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Argumente für und gegen Engessers Knickformeln

Übersicht Um bei Knickproblemen den Schub zu berücksichtigen, verwendet man entweder das Engessersche oder das sogenannte modifizierte Verfahren, das von Haringx zur Behandlung von Schraubenfedern vorgeschlagen worden ist. Während die meisten Autoren die zweite Methode vorzuziehen scheinen, hat Nänni [4] mittels der dreidimensionalen Elastizitätstheorie gezeigt, daß bei Stäben die erste überlegen ist. Die vorliegende Arbeit stützt sich auf ein fundamentaleres eindimensionales Verfahren und bestätigt Nännis und damit Engessers Ergebnisse. Sie zeigt auch, daß mit dem Schub zusammen die Verkürzung des Stabes zu berücksichtigen ist, und sie verbessert Engessers Formeln dementsprechend. Schließlich zeigt sie, weshalb das Verfahren von Haringx bei Federn korrekt ist.

     

Application of ‘Operational Quadrature Methods’ in Time Domain Boundary Element Methods

The usual time domain Boundary Element Method (BEM) contains fundamentalsolutions which are convoluted with time-dependent boundary data andintegrated over the boundary surface. Here, a new approach for theevaluation of the convolution integrals, the so-called OperationalQuadrature Methods developed by Lubich, is presented. In thisformulation, the convolution integral is numerically approximated by aquadrature formula whose weights are determined using the Laplacetransform of the fundamental solution and a linear multisep method. Tostudy the behaviour of the method, the numerical convolution of afundamental solution with a unit step function is compared with theanalytical result. Then, a time domain Boundary Element formulationapplying the Operational Quadrature Methods is derived. For thisformulation only the fundamental solutions in Laplace domain arenecessary. The properties of the new formulation are studied with anumerical example.     

Real-Space Renormalization Estimates for Two-Phase Flow in Porous Media

We present a spatial renormalization group algorithm to handle immiscibletwo-phase flow in heterogeneous porous media. We call this algorithmFRACTAM-R, where FRACTAM is an acronym for Fast Renormalization Algorithmfor Correlated Transport in Anisotropic Media, and the R stands for relativepermeability. Originally, FRACTAM was an approximate iterative process thatreplaces the L × L lattice of grid blocks, representing the reservoir,by a (L/2) × (L/2) one. In fact, FRACTAM replaces the original L× L lattice by a hierarchical (fractal) lattice, in such a way thatfinding the solution of the two-phase flow equations becomes trivial. Thistriviality translates in practice into computer efficiency. For N=L ×L grid blocks we find that the computer time necessary to calculatefractional flow F(t) and pressure P(t) as a function of time scales as N^1.7 for FRACTAM-R. This should be contrasted with thecomputational time of a conventional grid simulator N^2.3. The solution we find in this way is an accurateapproximation to the direct solution of the original problem.     

Effect of an inhibitor on high-speed turbulent flames and the transition to detonation

The influence of an inhibitor (CF₃Br or Halon 1301) on the propagation of high-speed turbulent flames, quasi-detonations and the transition to detonation has been investigated for methane-air, propane-air and acetylene-air mixtures. The experiments are carried out in a 13 m tube (15 cm diameter) filled with regularly spaced orifice plates (blockage ratio of 0.39) to ensure rapid flame acceleration. In all cases, the addition of the inhibitor reduces the turbulent flame velocity and extinguishes the flame with sufficient inhibitor concentration (2.7% and 7.5% for methane-air and propane-air, respectively). For acetylene-air mixtures, the quasi-detonation speed is progressively reduced with increasing inhibitor concentration and eventually causes the failure of the quasi-detonation and transition back to a fast turbulent flame. The inhibitor also narrows the propagation limits in all cases. To elucidate the inhibition mechanism, detailed modelling of both the turbulent flame structure as well as the chemical kinetics are required.     

A physically based correlation for the effects of power law rheology and inclination on slug bubble rise velocity

A study has been made of the motion of long bubbles in inclined pipes containing viscous Newtonian and non-Newtonian liquids. A semi-theoretical expression for the rise velocity of air bubbles in water is derived on the hypothesis that the dominant factor is the momentum exchange of the bubble underflow, i.e. the bubble nose shape. The correlation calls on empirical inputs from established literature on bubble rise speeds at high Reynolds number. The effects of increasing Newtonian viscosity are analysed with reference to the momentum exchange and it is shown how viscosity reduces the inclination dependence of the bubble Froude number. Results from an experimental survey in seven different non-Newtonian liquids in three different diameter pipes are presented. These data are correlated so as to decouple the effects of surface tension and viscosity. An empirical relation is proposed for the effective shear rate in the fluid travelling around the bubble nose. Our correlation is compared to literature data from a broad range of Reynolds numbers with excellent agreement except at shallow angles.     

Deep field noise in holographic particle image velocimetry (HPIV): numerical and experimental particle image field modelling

 Holographic recording overcomes the limits in 2-D particle image velocimetry (PIV) to cover a 3-D flow field volume. Interrogation by focusing on single planes in a reconstructed particle field is disturbed by noise from out-of-focus particles. A numerical simulation models image reconstruction and shows how validation rates depend on aperture and volume depth. An experimental model environment of scattering particles in moveable plastic slices gives support to the numerical results. Simulations and tests are carried out for interrogation by autocorrelation and crosscorrelation techniques and furnish guidelines for system design. Received: 27 December 1996 / Accepted: 14 August 1997     

Discontinuous fold bifurcations in mechanical systems

Summary  This paper treats discontinuous fold bifurcations of periodic solutions of discontinuous systems. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. A Floquet multiplier of a discontinuous system can jump through the unit circle, causing a discontinuous bifurcation. Numerical examples are treated, which show discontinuous fold bifurcations. A discontinuous fold bifurcation can connect stable branches to branches with infinitely unstable solutions. Received 20 September 2000; accepted for publication 26 June 2001     

Dynamics of hybrid shafts

In this paper the dynamic performance and cross-section deformation of shafts made of metals (steel and aluminum), composites (CFRP and GFRP) and hybrids of metals and composites have been studied. A layered finite degenerated shell element with transverse shear deformation and dynamic behavior is employed. Results obtained show that improvements in dynamic performance and reduction of cross-section deformation of hybrid shafts over metallic and composite shafts are possible.     

Some aspects of linear and nonlinear viscoelastic behaviour of polymer melts in shear

In linear viscoelastic investigations the frequency dependence of the phase shift between stress and strain appears to be very characteristic of the molecular structure of the material. This function is also a good approximation of the slope of the double logarithmic plot of the absolute value of the shear modulusG _d vs. the angular frequency. The product (G _d /) sin 2 comes very close to the relaxation spectrumH(), with = 1/, in all physical states of the material.The experimentally observed separability of time and strain effects in nonlinear viscoelasticity of highly viscous isotropic polymer fluids imposes restraints to the form of the constitutive equation. A single integral superposition equation of the Boltzmann type containing the product of a time function and a nonlinear strain function gives good results in describing experimental data in shear as well as in elongation. The molecular structure affects both functions in a different way. A universal definition of the nonlinear tensorial strain measure has not yet been developed. There are some indications that a definition on the basis of the principal stretch ratios may be fruitful.Invited paper, presented at the First Conference of European Rheologists at Graz (Austria), April 14–16, 1982.     

Secondary cells and separation in developing laminar curved-pipe flows

Laminar flows through 180° curved bends of circular cross section are investigated numerically. For small curvature ratio, , defined as pipe radius over mean bend radius, the governing equations could be parabolized. The equations are solved for an range of from 0.04 to 0.143, a Dean number (De) range of from 277.5 to 1360, and for a uniform flow, a potential vortex, and a parabolic flow inlet condition. In all these studies a zero cross-stream flow at the inlet is assumed. A detailed study of the effects of , De, and inlet condition on the secondary flow pattern is carried out. Within the range of parameters investigated, up to three secondary cells are found in the cross-stream half-plane of a curved pipe. They are the Dean-type secondary cell, a secondary separation cell near the inner bend (closest to the center of curvature of the bend), and a third cell near the pipe center. The number of secondary cells in the cross-stream half-plane is greatly influenced by the inlet flow, and to a much lesser extent by and De. For example, only the Dean cell is found in a curved-pipe flow where and De are small and the inlet flow is either uniform or a potential vortex. When the inlet condition of the same case is changed to a parabolic flow, a three-cell structure results. Furthermore, as De increases to 1180, incipient axial flow separation begins at around 23° downstream of the curved-pipe entrance. The formation and extent of the separation and third cells are investigated together with their dependence on the parameters studied. This investigation further shows that, within the range of parameters examined, there is no secondary cell occurring near the outer bend, contrary to some earlier findings on fully developed curved-pipe flows.This work was supported by the Office of Naval Research under Grant No. N0014-81-K-0428 and by DTRC, Annapolis, Maryland, under Contract No. N00167-86-K-0075. Also, support in the form of an IPA awarded to RMCS during his sabbatical leave at DTRC, Annapolis, Maryland, in the spring of 1990 is gratefully acknowledged.