Finite element and sensitivity analysis of thermally induced flow instabilities

This paper presents a finite element algorithm for the simulation of thermo‐hydrodynamic instabilities causing manufacturing defects in injection molding of plastic and metal powder. Mold‐filling parameters determine the flow pattern during filling, which in turn influences the quality of the final part. Insufficiently, well‐controlled operating conditions may generate inhomogeneities, empty spaces or unusable parts. An understanding of the flow behavior will enable manufacturers to reduce or even eliminate defects and improve their competitiveness. This work presents a rigorous study using numerical simulation and sensitivity analysis. The problem is modeled by the Navier–Stokes equations, the energy equation and a generalized Newtonian viscosity model. The solution algorithm is applied to a simple flow in a symmetrical gate geometry. This problem exhibits both symmetrical and non‐symmetrical solutions depending on the values taken by flow parameters. Under particular combinations of operating conditions, the flow was stable and symmetric, while some other combinations leading to large thermally induced viscosity gradients produce unstable and asymmetric flow. Based on the numerical results, a stability chart of the flow was established, identifying the boundaries between regions of stable and unstable flow in terms of the Graetz number (ratio of thermal conduction time to the convection time scale) and B, a dimensionless ratio indicating the sensitivity of viscosity to temperature changes. Sensitivities with respect to flow parameters are then computed using the continuous sensitivity equations method. We demonstrate that sensitivities are able to detect the transition between the stable and unstable flow regimes and correctly indicate how parameters should change in order to increase the stability of the flow. Copyright © 2009 John Wiley & Sons, Ltd.     

Airfoil design with external flow suction

A method is proposed for the numerical-analytic design of a high-lift airfoil with slit air suction from the external flow within the framework of ideal fluid theory. The suction slit is represented by a channel with constant wall velocity. Examples of the design of wing profiles with non-detached flows having lift coefficientsC _y =2.68 and 4 and maximum relative velocities over the profile [_max|/=2 and 2.2 are given. A scheme for an aircraft of the flying wing type with internally located engines and pay load is briefly discussed.Kazan', Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 23–28, November–December, 1996.     

Airfoil cascade in unsteady eddy flow

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 85–91, July–August, 1994.     

An experimental investigation of thermally induced flow instabilities in a convective boiling upflow system

The two-phase flow instabilities in a single channel, forced convection, open loop, up-flow system have been investigated experimentally using R-11 as the working fluid. The effects of mass flow rate, heat input, inlet liquid temperature and upstream compressible volume on two-phase flow instabilities have been investigated. Two heater surfaces were tested at five different heat inputs with constant inlet temperature, and four different inlet temperatures with constant heat input. For each case, the mass flow rate was varied over a wide range covering the entire scope for boiling two-phase flows. Experiments were also conducted at different compressible volumes, with constant heat input, inlet fluid temperature, and average mass flow rate. The oscillations of inlet mass flow rate, heater inlet pressure, and heater wall temperature were recorded. The effects of mass flow rate, inlet liquid temperature and heat input on the amplitudes and periods of inlet pressure and thermal oscillations are presented in tabular and graphical forms.

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Experimentelle Untersuchung der thermisch bedingten Strömungsinstabilitäten bei konvektiven Sieden in einer Aufwärtsströmung

Zusammenfassung Mit R-11 als Arbeitsfluid werden Zweiphasen-Strömungsinstabilitäten im Einzelkanal bei erzwungener Aufwärtsströmung ohne Rückführung experimentell untersucht, und zwar besonders im Hinblick auf den Einfluß der Massenstromdichte, der Wärmezufuhr, der Eintrittstemperatur und des in Strömungsrichtung zunehmenden kompressiblen Volumenanteils. Zwei Heizflächen wurden einmal mit fünf verschiedenen Heizflächenbelastungen und konstanter Eintrittstemperatur getestet und dann bei vier verschiedenen Wärmeeinträgen und konstanter Eintrittstemperatur. In jedem der Fälle erfolgte die Variation der Massenstromdichte in einem so weiten Bereich, daß das gesamte Feld der Zweiphasen-Siedeströmungen durchfahren werden konnte. Es wurden auch Experimente mit verschiedenen kompressiblen Volumenanteilen bei konstanter Wärmezufuhr, Fluideintrittstemperatur und mittlerer Massenstromdichte durchgeführt. Die Oszillationen der Massenstromdichte und des Druckes am Eintritt in die Heizstrecke sowie deren Wandtemperatur wurden aufgezeichnet. Die Enflüsse der Massenstromdichte, der Fluideintrittstemperatur und der Wärmezufuhr auf Amplituden und Perioden der Druck- und Temperaturoszillationen sind tabellarisch und in Diagrammform dargestellt.

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Dedicated to Prof. Dr.-Ing. F. Mayinger's 60th birthday     

The flow field induced by the torsional oscillations of a spherical cell containing a fluid drop

The flow field induced by the torsional oscillations of a spherical cell containing a fluid drop is examined. It has been found that in addition to the oscillating motion of each fluid particle there exist three standing vortices in every quadrant between the drop and the container. The steady streaming into the drop can be directed either clockwise or counterclockwise, depending on the values of the parameter of the fluids inside and outside the drop. Typical flow fields are shown graphically.     

Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping

The dynamics of a system of coupled oscillators possessing strongly nonlinear stiffness and damping is examined. The system consists of a linear oscillator coupled to a strongly nonlinear, light attachment, where the nonlinear terms of the system are realized due to geometric effects. We show that the effects of nonlinear damping are far from being purely parasitic and introduce new dynamics when compared to the corresponding systems with linear damping. The dynamics is analyzed by performing a slow/fast decomposition leading to slow flows, which in turn are used to study transient instability caused by a bifurcation to 1:3 resonance capture. In addition, a new dynamical phenomenon of continuous resonance scattering is observed that is both persistent and prevalent for the case of the nonlinearly damped system: For certain moderate excitations, the transient dynamics “tracks” a manifold of impulsive orbits, in effect transitioning between multiple resonance captures over definitive frequency and energy ranges. Eventual bifurcation to 1:3 resonance capture generates the dynamic instability, which is manifested as a sudden burst of the response of the light attachment. Such instabilities that result in strong energy transfer indicate potential for various applications of nonlinear damping such as in vibration suppression and energy harvesting.     

Coherent neural oscillations induced by weak synaptic noise

Nonlinear Dynamics - We analyze the effect of synaptic noise on the dynamics of the FitzHugh–Nagumo (FHN) neuron model. In our deterministic parameter regime, a limit cycle solution cannot...     

Primary instabilities of liquid film flow sheared by turbulent gas stream

Evolution of excited waves on a viscous liquid film has been investigated experimentally for the annular gas–liquid flow in a vertical tube. For the first time the dispersion relations are obtained experimentally for linear waves on liquid film surface in the presence of turbulent gas flow. Both cocurrent and countercurrent flow regimes are investigated. As an example of comparison with theory, the experimental data are compared to the results of calculations based on the Benjamin quasi-laminar model for turbulent gas flow. The calculation results are found to be in good agreement with experiments for moderate values of film Reynolds number.     

Experimental evidence of flutter and divergence instabilities induced by dry friction

Flutter and divergence instabilities have been advocated to be possible in elastic structures with Coulomb friction, but no direct experimental evidence has ever been provided. Moreover, the same types of instability can be induced by tangential follower forces, but these are commonly thought to be of extremely difficult, if not impossible, practical realization. Therefore, a clear experimental basis for flutter and divergence induced by friction or follower-loading is still lacking. This is provided for the first time in the present article, showing how a follower force of tangential type can be realized via Coulomb friction and how this, in full agreement with the theory, can induce a blowing-up vibrational motion of increasing amplitude (flutter) or an exponentially growing motion (divergence). In addition, our results show the limits of a treatment based on the linearized equations, so that nonlinearities yield the initial blowing-up vibration of flutter to reach eventually a steady state. The presented results give full evidence to potential problems in the design of mechanical systems subject to friction, open a new perspective in the realization of follower-loading systems and of innovative structures exhibiting ‘unusual’ dynamical behaviors.     

Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity

This paper aims at offering an insight into the dynamical behaviors of incommensurate fractional-order singularly perturbed van der Pol oscillators subjected to constant forcing, especially when the forcing is close to Andronov–Hopf bifurcation points. These bifurcation points are predicted thanks to the theorem on stability of incommensurate fractional-order systems, as functions of the forcing and fractional derivative orders. When the forcing is chosen near Andronov–Hopf bifurcation, the dynamics of fractional-order systems show a static-looking transient regime whose length increases exponentially with the closeness to the bifurcation point. This peculiar phenomenon is not common in numerical simulation of dynamical systems. We show that this quasi-static transient behavior is due to the combine action of the slow passage effect at folded saddle-node singularity and fractional derivation memory effect on the slow flow around this singularity; this forces the system to remain for a long time in the vicinity of its equilibrium point, though unstable. The system frees oneself from this quasi-static transient state by spiraling before entering relaxation oscillation. Such a situation results in mixed mode oscillations in the oscillatory regime. One obtains mixed mode oscillations from a very simple system: A two-variable system subjected to constant forcing.     

On occurrence of mixed-torus bursting oscillations induced by non-smoothness

Nonlinear Dynamics - The paper devotes to the slow–fast behaviors of a higher-dimensional non-smooth system with the coupling of two scales. Some novel bursting attractors and interesting...     

Small Reynolds number instabilities in two-layer Couette flow

Instabilities in two-layer Couette flow are investigated from a small Reynolds number expansion of the eigenvalue problem governing linear stability. The wave velocity and growth rate are given explicitly, and previous results for long waves and short waves are retrieved as special cases. In addition to the inertial instability due to viscous stratification, the flow may be subject to the Rayleigh–Taylor instability. As a result of the competition of these two instabilities, inertia may completely stabilise a gravity-unstable flow above some finite critical Froude number, or conversely, for a gravity-stable flow, inertia may give rise to finite wavenumber instability above some finite critical Weber number. General conditions for these phenomena are given, as well as exact or approximate values of the critical numbers. The validity domain of the many asymptotic expansions is then investigated from comparison with the numerical solution. It appears that the small-Re expansion gives good results beyond Re = 1, with an error less that 1%. For Reynolds numbers of a few hundred, we show from the eigenfunctions and the energy equation that the nature of the instability changes: instability still arises from the interfacial mode (there is no mode crossing), but this mode takes all the features of a shear mode. The other modes correspond to the stable eigenmodes of the single-layer Couette flow, which are recovered when one fluid is rigidified by increasing its viscosity or surface tension.     

Longitudinal instabilities in exit flow from a slit

Summary Longitudinal instabilities in exit flow from a slit have been analysed. A theory which shows them to be of surface tension origin has been proposed. It predicts an increase in critical wave length with increasing surface tension, decreasing viscosity, and increasing slit opening. This effect is shown experimentally to be observable in the flow of Newtonian fluids besides the observations in viscoelastic fluids reported previously.The theory predicts that these longitudinal instabilities will not be observed if the Ohnesorge number based on half slit widtha is less than 0.945. Also, the observed disturbances will have wave lengths corresponding to wave numbers less than 1.91/a.

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Zusammenfassung Es werden longitudinale Instabilitäten beim Ausströmen aus einem Parallelspalt analysiert. Dabei wird eine Theorie in Ansatz gebracht, welche diese auf die Oberflächenspannung zurückführt. Sie sagt ein Größerwerden der kritischen Wellenlänge mit wachsender Oberflächenspannung, abnehmender Viskosität und zunehmender Spaltweite voraus. Es wird gezeigt, daß dieser Effekt nicht nur, wie früher schon berichtet, in viskoelastischen, sondern auch in newtonschen Flüssigkeiten beobachtet werden kann.Die Theorie sagt weiter voraus, daß derartige longitudinale Instabilitäten nicht beobachtet werden können, solange die mit der halben Spaltweitea gebildete Ohnesorge-Zahl kleiner als 0.945 ausfällt. Schließlich folgt noch, daß die beobachteten Störungen Wellenlängen besitzen, die kleiner als 1,91/a sind.

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With 7 figures     

Flow instabilities in capillary flow of concentrated suspensions

The development of flow instabilities during the capillary flow of two concentrated suspensions filled with 76.5 and 65.6% by volume solids was investigated. The flow instabilities manifested themselves by the development of concentration gradients as a result of the filtering of the binder, superimposed on the bulk motion of the suspension. The effects of apparent shear rate, capillary diameter and the surface roughness of the particles were investigated. The use of the comparison of the filtration rate with the bulk velocity of the suspension during flow is shown to be promising for the prediction of the apparent shear rate at which filtration-based flow instabilities occur.     

A study of particle motion induced by two-dimensional liquid oscillations

The motion of single particles in liquids undergoing two-dimensional oscillations has been studied both experimentally and theoretically. By imposing combined vertical and horizontal oscillations on a liquid, particle mean motion can be controlled so that individual particles either rise against gravity or fall. The one dimensional equation of particle motion was modified to suit this situation and numerical solutions were used to predict the occurrence of the rising motion although agreement with experiment was only qualitative. A dimensional analysis approach combined with experiments allowed prediction of particle behaviour within the range of variables of this study. Extension of the single particle studies to mixtures of quartz and galena in various flowing liquids allowed prediction of the conditions under which separation and upgrading of minerals could occur and this was verified experimentally.     

Stretching flow instabilities at the exits of extrusion dies

As liquid leaves an extrusion die, the surface layers are rapidly stretched. Stretching flows may become unstable in two ways: by breaking, or in a ductile manner producing an uneven “necked” sample which, in continuous extrusion and drawing, is sometimes called “draw resonance”. There is a quantitative correlation between the extrusion defect known as “sharkskin” and the cohesive failure of polymer melts. By extruding under closely defined conditions, it is possible to introduce a transitory “structure” into the surface layer of the extrudate greatly enhancing its cohesive strength and eliminating this defect. A similar quantitative correlation is established between the uneven coating thickness sometimes obtained during coextrusion of a high viscosity melt on the surface of a low viscosity melt and the tensile drawing instability known as “draw resonance”. Simple criteria are established to avoid this problem in practical flow engineering.     

Inlet instabilities in the capillary flow of polyethylene melts

Inlet instabilities in the capillary flow of polyethylene melts were studied in this work. Extrudate distortions in branched polyethylenes, produced by unstable upstream flow, were found to be accompanied by pressure oscillations that do not have their origin in the slip phenomenon, but on polymer compressibility. The absence of slip was clearly evidenced in the experiments, and the differences between pressure oscillations occurring in linear and branched polymers are shown.Pressure oscillations in the capillary flow of branched polyethylenes were found to be made up of at least two components of different frequency and amplitude. These two components were identified with different bulk defects appearing in the extrudates. Information about the dynamics of vortices upstream of the contraction and extrudate distortions is obtained from the analysis of pressure oscillations.The influence of capillary entrance angle on flow curves was also investigated. From the results, it is concluded that the extensional component of the flow in the contraction is the main factor responsible for the slope change usually found in the log-log flow curves of both linear and branched polyethylenes.     

Transient boiling flow instabilities in a multi-channel upflow system

In this study, the transient boiling flow instabilities in a four parallel channel upflow system with and without cross-connections have been experimentally investigated using Freon-11 as the test fluid. Several series of transient experiments have been run using a step increase in power input with a constant flow control valve setting and a step decrease in flow rate with a constant heat input. Two major modes of oscillations, namely density-wave type and pressure-drop type oscillations, have been observed. Total system pressure drop has been plotted as a function of mass flow rate for various heat inputs and flow control valve settings, and the stable and unstable regions for transient density-wave type and pressure-drop type oscillations have been determined and the boundaries of sustained instabilities have also been indicated. Experiments have been conducted with equal and unequal heat inputs and the results have been compared with each other. A comparison of the transient instabilities observed in four parallel channel system with and without cross connections has also been made.     

Two-phase flow instabilities in a silicon microchannels heat sink

Two-phase flow instabilities are highly undesirable in microchannels-based heat sinks as they can lead to temperature oscillations with high amplitudes, premature critical heat flux and mechanical vibrations. This work is an experimental study of boiling instabilities in a microchannel silicon heat sink with 40 parallel rectangular microchannels, having a length of 15 mm and a hydraulic diameter of 194 μm. A series of experiments have been carried out to investigate pressure and temperature oscillations during the flow boiling instabilities under uniform heating, using water as a cooling liquid. Thin nickel film thermometers, integrated on the back side of a heat sink with microchannels, were used in order to obtain a better insight related to temperature fluctuations caused by two-phase flow instabilities. Flow regime maps are presented for two inlet water temperatures, showing stable and unstable flow regimes. It was observed that boiling leads to asymmetrical flow distribution within microchannels that result in high temperature non-uniformity and the simultaneously existence of different flow regimes along the transverse direction. Two types of two-phase flow instabilities with appreciable pressure and temperature fluctuations were observed, that depended on the heat to mass flux ratio and inlet water temperature. These were high amplitude/low frequency and low amplitude/high frequency instabilities. High speed camera imaging, performed simultaneously with pressure and temperature measurements, showed that inlet/outlet pressure and the temperature fluctuations existed due to alternation between liquid/two-phase/vapour flows. It was also determined that the inlet water subcooling condition affects the magnitudes of the temperature oscillations in two-phase flow instabilities and flow distribution within the microchannels.     

Transient mixed convection flow instabilities in a vertical pipe

The present paper investigates a numerical study of flow instabilities in transient mixed convection in a vertical pipe. At the entrance of the pipe, the flow is suddenly submitted to a temperature step. The convection heat transfer on the outer surface of the pipe is taken into account. The governing equations are solved using a finite difference explicit scheme. The numerical results show that the time development of streamlines and isotherms is strongly dependent on the inlet temperature steps. For positive temperature steps, the unsteady vortex is significant in the vicinity of the wall and the reversal flow appears below the wave instability. In the case of negative temperature steps and especially for the low Reynolds number, the reversal flow appears on top of the wave instability. During the transient, the apparition of the vortical structures along the wall leads to the wall boundary layer instability. This phenomenon is due to the transient mixed convection flows. The temperature step effects on the heat transfer of the flow are presented in our paper.