Singular perturbation methods for slow–fast dynamics

Recently, geometric singular perturbation theory has been extended considerably while at the same time producing many new applications. We will review a number of aspects relevant to non-linear dynamics to apply this to periodic solutions within slow manifolds and to review a number of non-hyperbolic cases. The results are illustrated by examples.     

Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables

In this paper, the nonlinear dynamics of a two-neuron Hopfield network with slow and fast variables is investigated. By means of the geometric singular perturbation theory, the condition that ensures the existence of the relaxation oscillation is obtained, the period of the relaxation oscillation is determined analytically, and the shape of attractive basin of every stable equilibrium is figured out. By using the method of stability switches, the delay effect on the characteristics of the relaxation oscillation and the attractive basins is studied. Case studies are given to demonstrate the validity of theoretical analysis.     

Memory decay rates of viscoelastic solids: not too slow, but not too fast either

Fading memory is a distinguishing characteristic of viscoelastic solids. Its assessment is often achieved by measuring the stress due to harmonic strain histories at different frequencies: from the experimental point of view, the storage and loss moduli are, hence, introduced. On the other side, the mathematical modeling of viscoelastic materials is usually based on the consideration of a kernel function whose decay rate is sufficiently fast. For several different solid materials, we have collated experimental evidence showing an high sensitivity to frequency variations of both the storage and loss moduli. By contrast, we prove that the commonly employed viscoelastic kernels (Prony series, continuous kernel, etc.) cannot reproduce this experimental behavior, as the resulting frequency sensitivity of the storage modulus is always zero when assessed at low frequency. This leads to identification problems of the material parameters which are strongly ill conditioned. However, we identify the specific kernel property which is responsible for this misbehavior: the long-term material memory must not decrease too fast. Some viscoelastic kernels, showing the correct memory??s rate of decay, are introduced and their improved ability to match the experimental data analyzed.     

Computer-aided Verification of Fatigue-machine Performance

Experimental Techniques -     

Extremal paths of plastic work and deformation

Optimal paths of deformation between homogeneous states of finite strain are identified on the basis that the work expended should be an absolute minimum. The materials considered are classically rigid/plastic, isotropic and nonhardening, with any convex yield surface. A related minimum principle is derived for a class of functionals of the strain history alone, without reference to work or to material properties. Plane isochoric deformations are subsequently treated in particular detail, supported by simple parametric representations of the competing paths and associated kinematic data.     

Computer-aided heuristic analysis of fluid models

A new kind of symbolic program to aid the heuristic simplification of fluid models is presented. The program, AOM, employs order of magnitude analysis and method of dominant balance to generate simplified models. It has two novel features: (1) it uses heuristic techniques to decide what equations to solve and what algebra to do, and (2) it explains its deduction steps. The basic operation of AOM consists of five steps: (1) assign order of magnitude estimates to terms in the equations, (2) find maximal terms of each equation, i.e., terms that are not dominated by any other terms in the same equation, (3) consider all possible n-term dominant balance assumptions, (4) propagate the effects of the balance assumptions, and (5) remove partial models based on inconsistent balance assumptions. AOM also exploits constraints among equations and submodels to simplify complicated fluid models such as the triple-deck equations. Three annotated examples are presented to explain the operations of AOM. The implications for the development of computer-aided analysis programs for fluid dynamics and education are discussed.This research was funded in part by NSF NYI Award ECS-9357773 and CCR-9109567.     

Velocity fidelity of flow tracer particles

Recent developments concerning the unsteady dynamic forces on a spherical particle at finite Reynolds number are reviewed for solid particles and clean micro-bubble. A particle frequency :response function and an energy transfer function are derived for a solid particle or a contaminated micro-bubble in gas or liquid flow. A simple, unified method for estimating the cut-off frequency, or cut-off size, of a solid particle or a contaminated bubble is developed. Particle motion in isotropic turbulence is examined. Responses of the tracer particle to integral length scale structure, to turbulence energy, and to Taylor micro-scale structure are discussed in terms of the particle turbulence diffusivity, the particle turbulence intensity, and the ensemble average of the second invariant of fluid turbulence deformation tensor evaluated on the particle trajectory.The author is grateful to R. J. Adrian and C. Kent for their encouragement and support in writing this paper. This work is supported by the Engineering Research Center (ERC) for Particle Science and Technology at the University of Florida, the National Science Foundation (EEC-9402989), and industrial partners of ERC.     

Computer-aided calibration and measurements with a quadruple hotwire probe

A method for calibration and measurement with a four-wire probe is described. For each of the wires a three dimensional calibration field is determined, thus no assumption like King's law or the cosine law need to be made. The velocity vector can then be detected in a fairly large angular range (± 40°) with a numerical search algorithm. First measurements in a free jet and a confined, strongly swirling flow are presented.A version of this paper was presented at the 11th Symposium on Turbulence, University of Missouri-Rolla, Oct. 17–19, 1988     

Computer-aided analysis of flow past a surface-mounted obstacle

The incompressible, laminar, isothermal flow of a Newtonian fluid at steady state past a surface-mounted obstacle (flow over a step) is studied in a two-dimensional numerical experiment using the Galerkin finite element method. The dimensionless Navier–Stokes equations are solved in the whole range of the laminar flow regime. The numerical predictions are compared with available experimental data. The emphasis in the discussion of the results is on the presentation of the streamlines for various Reynolds numbers, the pressure distribution over and downstream of the step, the shear stress distribution along the surface of the step and the length of the recirculation region as a function of the Reynolds number. This analysis may be used in numerous applications from agricultural to civil, mechanical and chemical engineering. © 1997 John Wiley & Sons, Ltd.     

Computer-aided calibration of X-probes using a look-up table

A simple method for the computer-aided calibration of an X-probe is described. This method requires the X-probe to be pitched in the free-stream at several velocities. From the corresponding output voltages, a calibration look-up table can be generated. The technique requires fewer assumptions than traditional methods based on King's law.     

Motion of tracer particles in supersonic flows

 Factors that may act on particle motion in high-speed flow are investigated. The classical expressions of drag coefficient C _D for a sphere are reviewed. Then, a drag expression is proposed, extending Cunningham’s method to higher velocities and Knudsen numbers. This law, valid from continuum to free molecule conditions, for Re≲200 and M≲1 (where Re and M are, respectively, the Reynolds and Mach numbers based on relative velocity), is used to compare calculated and experimental values of the drag coefficient, as well as the particle velocities across an oblique shock wave. Calculated results are found to be in agreement with experiments. Received: 3 June 1997/Accepted: 16 August 1998     

Classification of compatibility paths of SDOF mechanisms

The compatibility paths of mechanisms with a single degree-of-freedom typically form sets of curves in the global representation space. We classify the different cases of compatibility by introducing an energy function. The result obtained also depends on which element of the mechanism is regarded as driven. The different singularity types are demonstrated by examples (split-vanish point, limit point, asymmetric bifurcation, infinitely degenerate bifurcation, hilltop point, compatibility surface).     

Interpretation of internal tracer experiments and local sojourn time distributions

Tracer experiments are a valuable tool in the study of transport phenomena. Most of the theoretical work about tracers has been concerned either with systems having clearly defined inlet and outlet or with recirculating flows. In this article the methods are generalized to more complex situations and an attempt is made to provide a uniform theoretical treatment for local tracer experiments performed at several sites inside the flow systems. Local purging rate and mixing rate are defined and their measurements and applications are discussed. Several sojourn time distribution are discussed, especially that of local remaining life which might be useful in the study of local processes in large flow systems. The results should be of interest in a wide variety of areas in which tracer experiments are used, such as chemical reactor design, physiology, hydrology and the study of dispersion processes in the atmosphere and in oceans.     

Computer-aided fringe-pattern analyzer—A case of photoelastic fringe

A real-time “Computer-Aided FRinge-pattern AN-alyzer” (CAFRAN) system for the analysis of photoelastic fringe is developed. The pattern-analyzing process which utilizes image analysis or pattern-recognition technique and discretized-computation process in terms of finite-element method play an important role in this system. The former makes it possible to readout the fringe position and the fringe order of both isochromatic and isoclinic informations, and the latter analyzes the sum of the principal stresses which constitutes the Laplace field. Both results are combined to give the distributions of stress components over the domain under consideration. As far as the results of classical problems solved by this technique are concerned, the CAFRAN is promising in reading the fractional fringe order and encouraging the use of this technique to such problems as stress concentration.     

Distortion of LDA fringe pattern by tracer particles

For precise flow velocity measurements laser Doppler anemometry (LDA) is wide-spread in use in the laboratories of industry and universitarian research institutions. The LDA method has the advantage of being not intrusive and able to discriminate between forward and reverse velocities. So far, laser Doppler anemometry is believed to be one of the most accurate flow measuring techniques. However, recent investigations have shown that the period lengths of LDA signal bursts are not constant within an individual burst. This can induce an additional scatter in the signal frequency and in the determination of the flow velocity. Until now, the reason for the period variations has not been investigated in detail although the problem was observed before. This paper describes experimental investigations which show that the particle passage through the laser beams shortly before the point of superposition, i.e. the LDA measuring volume, yields a distorted LDA fringe pattern. Thus, the signal period length from an individual particle, passing the center of the measuring volume at the same time, varies according to the distortion of the fringe spacing.     

Design of conducting paths based on topology optimization

In this paper, a new method based on topology optimization is developed to solve the volume-to-point heat conduction problem with distributed heat sources, which can construct the conducting paths by distributing high conductive materials. The closer natural tree of conducting paths is obtained by the present method when the heat source is uniform. The better heat transfer performance of the conducting paths for the volume-to-point (VP) problem can be obtained than those by the existing methods. The difference among the existing methods is discussed.     

Computer-aided spillway design using the boundary element method and non-linear programming

The general context of this paper is to support the design of spillways by a direct mathematical approach instead of trial-and-error methods. First, a two-dimensional model is formulated to determine the free surface and the discharge for a stationary, incompressible, homogeneous, non-viscous and irrotational flow over a fixed spillway. The flow satisfies the Laplace equation and the Bernoulli equation (potential flow). An important feature of the model is that it can be extended to design the spillway structure when the spillway is not fixed but the pressure on the spillway is described by a cavitation criterion. Next, the continuous model is discretized by the boundary element method (BEM). We use a non-linear programming algorithm to calculate the pressures and the shape of the spillway. A computer-aided design package is developed on a PC using the equations describing the free surface, the BEM and standard optimization techniques. The input and output of the model are realized using graphical routines. Finally, we discuss the convergence and the computation time of the algorithms.