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Dromions of flexural-gravity waves
- Mohammad-Reza Alam
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- 19 February 2013, pp. 1-13
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Here we show that weakly nonlinear flexural-gravity wave packets, such as those propagating on the surface of ice-covered waters, admit three-dimensional fully localized solutions that travel with a constant speed without dispersion or dissipation. These solutions, that are formed at the intersection of line-soliton mean-flow tracks, have exponentially decaying tails in all directions and are called dromions in contrast to lumps that decay only algebraically. We derive, by asymptotic expansion and assuming multiple scales for spatial and temporal variations, the three-dimensional weakly nonlinear governing equations that describe the coupled motion of the wavepacket envelope and the underlying mean current. We show that in the limit of long waves and strong flexural rigidity these equations reduce to a system of nonlinear elliptic–hyperbolic partial differential equations similar to the Davey–Stewartson I (DSI) equation, but with major differences in the coefficients. Specifically, and contrary to DSI equations, the elliptic and hyperbolic operators in the flexural-gravity equations are not canonical resulting in complications in analytical considerations. Furthermore, standard computational techniques encounter difficulties in obtaining the dromion solution to these equations owing to the presence of a spatial hyperbolic operator whose solution does not decay at infinity. Here, we present a direct (iterative) numerical scheme that uses pseudo-spectral expansion and pseudo-time integration to find the dromion solution to the flexural-gravity wave equation. Details of this direct simulation technique are discussed and properties of the solution are elaborated through an illustrative case study. Dromions may play an important role in transporting energy over the ice cover in the Arctic, resulting in the ice breaking far away from the ice edge, and also posing danger to icebreaker ships. In fact we found that, contrary to DSI dromions that only exist in water depths of less than 5 mm, flexural-gravity dromions exist for a broad range of ice thicknesses and water depths including values that may be realized in polar oceans.
Optimal Taylor–Couette flow: direct numerical simulations
- Rodolfo Ostilla, Richard J. A. M. Stevens, Siegfried Grossmann, Roberto Verzicco, Detlef Lohse
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- 19 February 2013, pp. 14-46
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We numerically simulate turbulent Taylor–Couette flow for independently rotating inner and outer cylinders, focusing on the analogy with turbulent Rayleigh–Bénard flow. Reynolds numbers of $R{e}_{i} = 8\times 1{0}^{3} $ and $R{e}_{o} = \pm 4\times 1{0}^{3} $ of the inner and outer cylinders, respectively, are reached, corresponding to Taylor numbers $Ta$ up to $1{0}^{8} $. Effective scaling laws for the torque and other system responses are found. Recent experiments with the Twente Turbulent Taylor–Couette (${T}^{3} C$) setup and with a similar facility in Maryland at very high Reynolds numbers have revealed an optimum transport at a certain non-zero rotation rate ratio $a= - {\omega }_{o} / {\omega }_{i} $ of about ${a}_{\mathit{opt}} = 0. 33$. For large enough $Ta$ in the numerically accessible range we also find such an optimum transport at non-zero counter-rotation. The position of this maximum is found to shift with the driving, reaching a maximum of ${a}_{\mathit{opt}} = 0. 15$ for $Ta= 2. 5\times 1{0}^{7} $. An explanation for this shift is elucidated, consistent with the experimental result that ${a}_{\mathit{opt}} $ becomes approximately independent of the driving strength for large enough Reynolds numbers. We furthermore numerically calculate the angular velocity profiles and visualize the different flow structures for the various regimes. By writing the equations in a frame co-rotating with the outer cylinder a link is found between the local angular velocity profiles and the global transport quantities.
Inertial waves in a differentially rotating spherical shell
- C. Baruteau, M. Rieutord
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- 19 February 2013, pp. 47-81
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We investigate the properties of small-amplitude inertial waves propagating in a differentially rotating incompressible fluid contained in a spherical shell. For cylindrical and shellular rotation profiles and in the inviscid limit, inertial waves obey a second-order partial differential equation of mixed type. Two kinds of inertial modes therefore exist, depending on whether the hyperbolic domain where characteristics propagate covers the whole shell or not. The occurrence of these two kinds of inertial modes is examined, and we show that the range of frequencies at which inertial waves may propagate is broader than with solid-body rotation. Using high-resolution calculations based on a spectral method, we show that, as with solid-body rotation, singular modes with thin shear layers following short-period attractors still exist with differential rotation. They exist even in the case of a full sphere. In the limit of vanishing viscosities, the width of the shear layers seems to weakly depend on the global background shear, showing a scaling in ${E}^{1/ 3} $ with the Ekman number $E$, as in the solid-body rotation case. There also exist modes with thin detached layers of width scaling with ${E}^{1/ 2} $ as Ekman boundary layers. The behaviour of inertial waves with a corotation resonance within the shell is also considered. For cylindrical rotation, waves get dramatically absorbed at corotation. In contrast, for shellular rotation, waves may cross a critical layer without visible absorption, and such modes can be unstable for small enough Ekman numbers.
The evolution of miscible gravity currents in horizontal porous layers
- M. L. Szulczewski, R. Juanes
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- 19 February 2013, pp. 82-96
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Gravity currents of miscible fluids in porous media are important to understand because they occur in important engineering projects, such as enhanced oil recovery and geologic CO2 sequestration. These flows are often modelled based on two simplifying assumptions: vertical velocities are negligible compared with horizontal velocities, and diffusion is negligible compared with advection. In many cases, however, these assumptions limit the validity of the models to a finite, intermediate time interval during the flow, making prediction of the flow at early and late times difficult. Here, we consider the effects of vertical flow and diffusion to develop a set of models for the entire evolution of a miscible gravity current. To gain physical insight, we study a simple system: lock exchange of equal-viscosity fluids in a horizontal, vertically confined layer of permeable rock. We show that the flow exhibits five regimes: (i) an early diffusion regime, in which the fluids diffuse across the initially sharp fluid–fluid interface; (ii) an S-slumping regime, in which the fluid–fluid interface tilts in an S-shape; (iii) a straight-line slumping regime, in which the fluid–fluid interface tilts as a straight line; (iv) a Taylor-slumping regime, in which Taylor dispersion at the aquifer scale enhances mixing between the fluids and causes the flow to continuously decelerate; and (v) a late diffusion regime, in which the flow becomes so slow that mass transfer again occurs dominantly though diffusion.
Vortex-induced vibration of a neutrally buoyant tethered sphere
- H. Lee, K. Hourigan, M. C. Thompson
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- 19 February 2013, pp. 97-128
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A combined numerical and experimental study examining vortex-induced vibration (VIV) of a neutrally buoyant tethered sphere has been undertaken. The study covered the Reynolds-number range $50\leq \mathit{Re}\lesssim 12\hspace{0.167em} 000$, with the numerical ($50\leq \mathit{Re}\leq 800$) and experimental ($370\leqslant \mathit{Re}\lesssim 12\hspace{0.167em} 000$) ranges overlapping. Neutral buoyancy was chosen to eliminate one parameter, i.e. the influence of gravity, on the VIV behaviour, although, of course, the effect of added mass remains. The tether length was also chosen to be sufficiently long so that, to a good approximation, the sphere was constrained to move within a plane. Seven broad but relatively distinct sphere oscillation and wake states could be distinguished. For regime I, the wake is steady and axisymmetric, and it undergoes transition to a steady two-tailed wake in regime II at $\mathit{Re}= 210$. Those regimes are directly analogous to those of a fixed sphere. Once the sphere begins to vibrate at $\mathit{Re}\simeq 270$ in regime III, the wake behaviour is distinct from the fixed-sphere wake. Initially the vibration frequency of the sphere is half the shedding frequency in the wake, with the latter consistent with the fixed-sphere wake frequency. The sphere vibration is not purely periodic but modulated over several base periods. However, at slightly higher Reynolds numbers ($\mathit{Re}\simeq 280$), planar symmetry is broken, and the vibration shifts to the planar normal (or azimuthal) direction, and becomes completely azimuthal at the start of regime IV at $\mathit{Re}= 300$. In comparison, for a fixed sphere, planar symmetry is broken at a much higher Reynolds number of $\mathit{Re}\simeq 375$. Interestingly, planar symmetry returns to the wake for $\mathit{Re}\gt 330$, in regime V, for which the oscillations are again radial, and is maintained until $\mathit{Re}= 450$ or higher. At the same time, the characteristic vortex loops in the wake become symmetrical, i.e. two-sided. For $\mathit{Re}\gt 500$, in regime VI, the trajectory of the sphere becomes irregular, possibly chaotic. That state is maintained over the remaining Reynolds-number range simulated numerically ($\mathit{Re}\leq 800$). Experiments overlapping this Reynolds-number range confirm the amplitude radial oscillations in regime V and the chaotic wandering for regime VI. At still higher Reynolds numbers of $\mathit{Re}\gt 3000$, in regime VII, the trajectories evolve to quasi-circular orbits about the neutral point, with the orbital radius increasing as the Reynolds number is increased. At $\mathit{Re}= 12\hspace{0.167em} 000$, the orbital diameter reaches approximately one sphere diameter. Of interest, this transition sequence is distinct from that for a vertically tethered heavy sphere, which undergoes transition to quasi-circular orbits beyond $\mathit{Re}= 500$.
Multimodal analysis of weakly nonlinear sloshing in a spherical tank
- Odd M. Faltinsen, Alexander N. Timokha
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- 19 February 2013, pp. 129-164
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Sloshing in a spherical tank due to horizontal excitation is studied by using the nonlinear multimodal method which involves the analytically approximate sloshing modes by Faltinsen & Timokha (J. Fluid Mech., vol. 703, 2012, pp. 391–401). General fully and weakly nonlinear modal equations are derived but an emphasis is on the Moiseev–Narimanov asymptotic modal system which implies that the forcing frequency is close to the lowest natural sloshing frequency and there are no secondary resonances in the forcing frequency range leading to a nonlinear resonant amplification of double and triple harmonics in higher modes. The Moiseev–Narimanov modal system is used to construct an asymptotic time-periodic solution and, thereby, classify the corresponding steady-state wave regimes appearing as stable and unstable planar waves and swirling. The results on the stability boundaries are compared with experiments by Sumner & Stofan (1963, Tech. Rep. TN D-1991, NASA Technical Note) and Sumner (1966, Tech. Rep. TN D-3210, NASA). A good agreement is established for $0. 2\leq h\lesssim 1$. Discrepancy for higher liquid depths $1\lesssim h\lt 2$ are explained by secondary resonance.
An alternative view of generalized Lagrangian mean theory
- Rick Salmon
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- 19 February 2013, pp. 165-182
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If the variables describing wave–mean flow interactions are chosen to include a set of fluid–particle labels corresponding to the mean flow, then the generalized Lagrangian mean (GLM) theory takes the form of an ordinary classical field theory. Its only truly distinctive features then arise from the distinctive feature of fluid dynamics as a field theory, namely, the particle-relabelling symmetry property, which corresponds by Noether’s theorem to the many vorticity conservation laws of fluid mechanics. The key feature of the formulation is that all the dependent variables depend on a common set of space–time coordinates. This feature permits an easy and transparent derivation of the GLM equations by use of the energy–momentum tensor formalism. The particle-relabelling symmetry property leads to the GLM potential vorticity law in which pseudo-momentum is the only wave activity term present. Thus the particle-relabelling symmetry explains the prominent importance of pseudo-momentum in GLM theory.
Sensitivity analysis of a time-delayed thermo-acoustic system via an adjoint-based approach
- Luca Magri, Matthew P. Juniper
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- 19 February 2013, pp. 183-202
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We apply adjoint-based sensitivity analysis to a time-delayed thermo-acoustic system: a Rijke tube containing a hot wire. We calculate how the growth rate and frequency of small oscillations about a base state are affected either by a generic passive control element in the system (the structural sensitivity analysis) or by a generic change to its base state (the base-state sensitivity analysis). We illustrate the structural sensitivity by calculating the effect of a second hot wire with a small heat-release parameter. In a single calculation, this shows how the second hot wire changes the growth rate and frequency of the small oscillations, as a function of its position in the tube. We then examine the components of the structural sensitivity in order to determine the passive control mechanism that has the strongest influence on the growth rate. We find that a force applied to the acoustic momentum equation in the opposite direction to the instantaneous velocity is the most stabilizing feedback mechanism. We also find that its effect is maximized when it is placed at the downstream end of the tube. This feedback mechanism could be supplied, for example, by an adiabatic mesh. We illustrate the base-state sensitivity by calculating the effects of small variations in the damping factor, the heat-release time-delay coefficient, the heat-release parameter, and the hot-wire location. The successful application of sensitivity analysis to thermo-acoustics opens up new possibilities for the passive control of thermo-acoustic oscillations by providing gradient information that can be combined with constrained optimization algorithms in order to reduce linear growth rates.
Stationary plume induced by carbon dioxide dissolution
- F. Nadal, P. Meunier, B. Pouligny, E. Laurichesse
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- 19 February 2013, pp. 203-229
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In this paper, laminar convection flows induced by carbon dioxide absorption are addressed from experimental, numerical and theoretical points of view. A vertical glass tube (of centimetre scale) filled with distilled water is subjected to a sudden increase in the partial pressure of carbon dioxide. As a result of the diffusion of the gas into the unsaturated solution, a thin layer of fluid located underneath the surface becomes heavier. This initial density gradient first destabilizes to form a plume, which goes downwards through the entire cell. After a first transient pulsating regime (periodic succession of such Rayleigh–Bénard plumes), a stationary flow settles in the tube, which is maintained by the constant supply of gas at the surface. At late stages, this stationary regime is followed by an aperiodic regime, which lasts until the complete saturation of the solution (thermodynamic equilibrium). The present study only focuses on the stationary regime, whose characteristics appear to be almost independent of the Bond number and the aspect ratio but strongly dependent on the chemical Rayleigh number. Three decades of Rayleigh numbers are explored using particle image velocimetry measurements, which allows for a precise determination of the scaling exponents for the vertical velocity amplitude and the plume width. The assumption that gravity and a constant pressure gradient balance the viscous effects enables us to derive an analytic expression for the stationary vertical velocity on the axis, which scales as ${\mathit{Ra}}^{2/ 3} \mathop{(\ln \mathit{Ra})}\nolimits ^{1/ 3} $. As a consequence, the width of the plume scales as ${\mathit{Ra}}^{- 1/ 6} \mathop{(\ln \mathit{Ra})}\nolimits ^{- 1/ 3} $ and the mass Nusselt number as $\mathop{(\mathit{Ra}/ \ln \mathit{Ra})}\nolimits ^{1/ 3} $. These scalings are in excellent agreement with the experimental and numerical results. The multiplicative constants of these scalings can also be calculated and show a fairly good agreement if a rigid boundary condition (no-slip) is assumed at the free surface.
Explicit filtering to obtain grid-spacing-independent and discretization-order-independent large-eddy simulation of two-phase volumetrically dilute flow with evaporation
- Senthilkumaran Radhakrishnan, Josette Bellan
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- 19 February 2013, pp. 230-267
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Predictions from conventional large-eddy simulation (LES) are known to be grid-spacing and spatial-discretization-order dependent. In a previous article (Radhakrishnan & Bellan, J. Fluid Mech., vol. 697, 2012a, pp. 399–435), we reformulated LES for compressible single-phase flow by explicitly filtering the nonlinear terms in the governing equations so as to render the solution grid-spacing and discretization-order independent. Having shown in Radhakrishnan & Bellan (2012a) that the reformulated LES, which we call EFLES, yields grid-spacing-independent and discretization-order-independent solutions for compressible single-phase flow, we explore here the potential of EFLES for evaporating two-phase flow where the small scales have an additional origin compared to single-phase flow. Thus, we created a database through direct numerical simulation (DNS) that when filtered serves as a template for comparisons with both conventional LES and EFLES. Both conventional LES and EFLES are conducted with two gas-phase SGS models; the drop-field SGS model is the same in all these simulations. For EFLES, we also compared simulations performed with the same SGS model for the gas phase but two different drop-field SGS models. Moreover, to elucidate the influence of explicit filtering versus gas-phase SGS modelling, EFLES with two drop-field SGS models but no gas-phase SGS models were conducted. The results from all these simulations were compared to those from DNS and from the filtered DNS (FDNS). Similar to the single-phase flow findings, the conventional LES method yields solutions which are both grid-spacing and spatial-discretization-order dependent. The EFLES solutions are found to be grid-spacing independent for sufficiently large filter-width to grid-spacing ratio, although for the highest discretization order this ratio is larger in the two-phase flow compared to the single-phase flow. For a sufficiently fine grid, the results are also discretization-order independent. The absence of a gas-phase SGS model leads to build-up of energy near the filter cut-off indicating that while explicit filtering removes energy above the filter width, it does not provide the correct dissipation at the scales smaller than this width. A wider viewpoint leads to the conclusion that although the minimum filter-width to grid-spacing ratio necessary to obtain the unique grid-independent solution might be different for various discretization-order schemes, the grid-independent solution thus obtained is also discretization-order independent.
Nonlinear disturbance growth during sedimentation in dilute fibre suspensions
- Feng Zhang, Anders A. Dahlkild, Fredrik Lundell
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- 19 February 2013, pp. 268-294
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Disturbances in a dilute fibre suspension are studied with an Eulerian approach. Based on a linear stability analysis, it is shown that inertia and hydrodynamic diffusion damp perturbations at long wavelengths and short wavelengths, respectively, leading to a wavenumber selection. For small but finite Reynolds number of the fluid bulk motion, the most unstable wavenumber is a finite value, which increases with Reynolds number. Furthermore, the diffusion narrows the range of unstable wavenumbers. Numerical simulations of the full nonlinear evolution in time of a normal-mode perturbation show that the induced flow may either die out or saturate on a finite amplitude. The character of this long-time behaviour is dictated by the wavenumber and the presence or absence, as well as nature, of the translational and rotational diffusivities.
Capillary-wave scattering from an infinitesimal barrier and dissipation at dynamic contact lines
- Likun Zhang, David B. Thiessen
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- 19 February 2013, pp. 295-313
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The interaction of pure capillary waves with boundaries that constrain the contact line are of interest for problems involving liquids contained by minimal solid contact for applications in low gravity and at small scales in normal gravity. Time-harmonic capillary waves on a liquid cylinder axially incident on and scattered by an infinitesimal concentric barrier are considered theoretically in the inviscid limit. The barrier is taken to be infinitesimally small in the sense that its immersed depth is of the order of the amplitude of contact-line motion. Edge conditions on the barrier that are investigated include a pinned contact line and a moving contact line by an effective-slip model, assuming that contact-line velocity is proportional to the deviation of the contact angle from equilibrium multiplied by a slip coefficient. The incident waves are taken to be those with wavelengths short enough to be stable on the liquid cylinder. Scattering and dissipation by the contact line are determined as a function of wavenumber and slip coefficient. Zero transmission is approached in the long-wave limit. The short-wave limit agrees with established results for the scattering of planar gravity–capillary waves on a deep liquid by a surface-piercing vertical barrier in the limit of zero barrier depth and zero gravity. We find that contact-line dissipation at the barrier is a maximum for incident waves whose phase speed is of the order of the slip coefficient, which is interpreted as an effect of impedance matching. Transmission past an infinitesimal barrier is found to be low over all parameter space, illustrating the importance of contact-line constraints.
Recurrent solutions of the Alber equation initialized by Joint North Sea Wave Project spectra
- A. Ribal, A. V. Babanin, I. Young, A. Toffoli, M. Stiassnie
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- 19 February 2013, pp. 314-344
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Linear instability of two-dimensional wave fields and its concurrent evolution in time is here investigated by means of the Alber equation for narrow-banded random surface waves in deep water subject to inhomogeneous disturbances. The probability of freak waves in the context of these simulations is also discussed. The instability is first studied for the symmetric Lorentz spectrum, and continued for the realistic asymmetric Joint North Sea Wave Project (JONSWAP) spectrum of ocean waves with variable directional spreading and steepness. It is found that instability depends on the directional spreading and parameters $\alpha $ and $\gamma $ of the JONSWAP spectrum, where $\alpha $ and $\gamma $ are the energy scale and the peak enhancement factor, respectively. Both influence the mean steepness of waves with such a spectrum, although in different ways. Specifically, if the instability stops as a result of the directional spreading, increase of the steepness by increasing $\alpha $ or $\gamma $ can reactivate it. A criterion for the instability is suggested as a dimensionless ‘width parameter’, $\Pi $. For the unstable conditions, long-time evolution is simulated by integrating the Alber equation numerically. Recurrent evolution is obtained, which is a stochastic counterpart of the Fermi–Pasta–Ulam recurrence obtained for the cubic Schrödinger equation. This recurrence enables us to study the probability of freak waves, and the results are compared to the values given by the Rayleigh distribution. Moreover, it is found that stability–instability transition, the most unstable mode, recurrence duration and freak wave probability depend solely on the dimensionless ‘width parameter’, $\Pi $.
The shape stability of a lipid vesicle in a uniaxial extensional flow
- Hong Zhao, Eric S. G. Shaqfeh
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- 19 February 2013, pp. 345-361
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The dynamics of a lipid vesicle in a uniaxial extensional flow are investigated by using a spectral boundary integral equation method. The vesicle at its stationary state assumes an axisymmetric shape of mirror symmetry, with its surface velocity vanishing everywhere. When the reduced volume of the vesicle is less than 0.75, there exists a critical capillary number, beyond which the stationary shape is unstable. The most unstable mode breaks the mirror symmetry of the shape so that the vesicle deforms into a dumbbell shape with two unequally sized ends. This is followed by the formation of a thin tube bridging the two dumbbell ends, whose length increases with time. The numerical results are in qualitative agreement with experimental observations.
A model of bubble-induced turbulence based on large-scale wake interactions
- Guillaume Riboux, Dominique Legendre, Frédéric Risso
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- 19 February 2013, pp. 362-387
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Navier–Stokes simulations of the agitation generated by a homogeneous swarm of high-Reynolds-number rising bubbles are performed. The bubbles are modelled by fixed momentum sources of finite size randomly distributed in a uniform flow. The mesh grid is regular with a spacing close to the bubble size. This allows us to simulate a swarm of a few thousand bubbles in a computational domain of a hundred bubble diameters, which corresponds to a gas volume fraction $\alpha $ from 0.6 % to 4 %. The small-scale disturbances close to the bubbles are not resolved but the wakes are correctly described from a distance of a few diameters. This simple model reproduces well all the statistical properties of the vertical velocity fluctuations measured in previous experiments: scaling as ${\alpha }^{0. 4} $, self-similar probability density functions and power spectral density including a subrange evolving as the power $- 3$ of the wavenumber $k$. It can therefore be concluded that bubble-induced agitation mainly results from wake interactions. Considering the flow in a frame that is fixed relative to the bubbles, the combined use of both time and spatial averaging makes it possible to distinguish two contributions to the liquid fluctuations. The first is the spatial fluctuations that are the consequence of the bubble mean wakes. The second corresponds to the temporal fluctuations that are the result of the development of a flow instability. Note that the latter is not due to the destabilization of individual bubble wakes, since a computation with a single bubble leads to a steady flow. It is a collective instability of the randomly distributed bubble wakes. The spectrum of the time fluctuations shows a peak around a frequency ${f}_{cwi} $, which is independent of $\alpha $. From the present results it is possible to determine the origin of the overall properties of the total fluctuations observed in the experiments. The scaling of the velocity fluctuation as ${\alpha }^{0. 4} $ is a combination of the scalings of the spatial and temporal fluctuations, which are different from each other. As the time fluctuations are symmetric in the vertical direction, the asymmetry of the probability density function of the vertical velocity comes from that of the spatial fluctuations. Both contributions exhibit a ${k}^{- 3} $ spectral behaviour around the same range of wavenumbers, which explains why it is observed regardless of the nature of the dominant contribution.
Falling styles of disks
- Franck Auguste, Jacques Magnaudet, David Fabre
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- 19 February 2013, pp. 388-405
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We numerically investigate the dynamics of thin disks falling under gravity in a viscous fluid medium at rest at infinity. Varying independently the density and thickness of the disk reveals the influence of the disk aspect ratio which, contrary to previous belief, is found to be highly significant as it may completely change the route to non-vertical paths as well as the boundaries between the various path regimes. The transition from the straight vertical path to the planar fluttering regime is found to exhibit complex dynamics: a bistable behaviour of the system is detected within some parameter range and several intermediate regimes are observed in which, although the wake is unstable, the path barely deviates from vertical. By varying independently the body-to-fluid inertia ratio and the relative magnitude of inertial and viscous effects over a significant range, we set up a comprehensive map of the corresponding styles of path followed by an infinitely thin disk. We observe the four types of planar regimes already reported in experiments but also identify two additional fully three-dimensional regimes in which the body experiences a slow horizontal precession superimposed onto zigzagging or tumbling motions.
Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow
- G. Dergham, D. Sipp, J.-Ch. Robinet
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- 19 February 2013, pp. 406-430
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Methods for investigating and approximating the linear dynamics of amplifier flows are examined in this paper. The procedures are derived for incompressible flow over a two-dimensional backward-facing step. First, the singular value decomposition of the resolvent is performed over a frequency range in order to identify the optimal and suboptimal harmonic forcing and responses of the flow. These forcing/responses are shown to be organized into two categories: the first accounting for the Orr and Kelvin–Helmholtz instabilities in the shear layer and the second for the advection and diffusion of perturbations in the free stream. Next, we investigate the dynamics of the flow when excited by a white in space and time noise. We compute the predominant patterns of the random flow which optimally account for the sustained variance, the empirical orthogonal functions (EOFs), as well as the predominant forcing structures which optimally contribute to the sustained variance, the stochastic optimals (SOs). The leading EOFs and SOs are expressed as a linear combination of the suboptimal forcing and responses of the flow and are related to particular instability mechanisms and/or frequency intervals. Finally, we use the leading EOFs, SOs and balanced modes (obtained from balanced truncation) to build low-order models of the flow dynamics. These models are shown to accurately recover the time propagator and resolvent of the original dynamical system. In other words, such models capture the entire flow response from any forcing and may be used in the design of efficient closed-loop controllers for amplifier flows.
Steady base states for non-Newtonian granular hydrodynamics
- Francisco Vega Reyes, Andrés Santos, Vicente Garzó
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- 19 February 2013, pp. 431-464
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We study in this work steady laminar flows in a low-density granular gas modelled as a system of identical smooth hard spheres that collide inelastically. The system is excited by shear and temperature sources at the boundaries, which consist of two infinite parallel walls. Thus, the geometry of the system is the same that yields the planar Fourier and Couette flows in standard gases. We show that it is possible to describe the steady granular flows in this system, even at large inelasticities, by means of a (non-Newtonian) hydrodynamic approach. All five types of Couette–Fourier granular flows are systematically described, identifying the different types of hydrodynamic profiles. Excellent agreement is found between our classification of flows and simulation results. Also, we obtain the corresponding nonlinear transport coefficients by following three independent and complementary methods: (i) an analytical solution obtained from Grad’s 13-moment method applied to the inelastic Boltzmann equation; (ii) a numerical solution of the inelastic Boltzmann equation obtained by means of the direct simulation Monte Carlo method; and (iii) event-driven molecular dynamics simulations. We find that, while Grad’s theory does not describe quantitatively well all transport coefficients, the three procedures yield the same general classification of planar Couette–Fourier flows for the granular gas.
Orientation of non-spherical particles in an axisymmetric random flow
- Dario Vincenzi
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- 19 February 2013, pp. 465-487
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The dynamics of non-spherical rigid particles immersed in an axisymmetric random flow is studied analytically. The motion of the particles is described by Jeffery’s equation; the random flow is Gaussian and has short correlation time. The stationary probability density function of orientations is calculated exactly. Four regimes are identified depending on the statistical anisotropy of the flow and on the geometrical shape of the particle. If $\boldsymbol{\lambda} $ is the axis of symmetry of the flow, the four regimes are: rotation about $\boldsymbol{\lambda} $, tumbling motion between $\boldsymbol{\lambda} $ and $- \boldsymbol{\lambda} $, combination of rotation and tumbling, and preferential alignment with a direction oblique to $\boldsymbol{\lambda} $.
Modelling circulation, impulse and kinetic energy of starting jets with non-zero radial velocity
- Michael Krieg, Kamran Mohseni
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- 19 February 2013, pp. 488-526
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The evolution of starting jet circulation, impulse and kinetic energy are derived in terms of kinematics at the entrance boundary of a semi-infinite axisymmetric domain. This analysis is not limited to the case of parallel jet flows; and the effect of non-zero radial velocity is specifically identified. The pressure distribution along the entrance boundary is also derived as it is required for kinetic energy modelling. This is done without reliance on an approximated potential function (i.e. translating flat plate), making it a powerful analytical tool for any axisymmetric jet flow. The pressure model indicates that a non-zero radial velocity is required for any ‘over-pressure’ at the nozzle exit. Jet flows are created from multiple nozzle configurations to validate this model. The jet is illuminated in cross-section, and velocity and vorticity fields are determined using digital particle image velocimetry (DPIV) techniques and circulation, impulse and kinetic energy of the jet are calculated from the DPIV data. A non-zero radial velocity at the entrance boundary has a drastic effect on the final jet. Experimental data showed that a specific configuration resulting in a jet with a converging radial velocity, with a magnitude close to 40 % of the axial velocity at its maximum, attains a final circulation which is 90–100 % larger than a parallel starting jet with identical volume flux and nozzle diameter, depending on the stroke ratio. The converging jet also attains a final impulse which is 70–75 % larger than the equivalent parallel jet and a final kinetic energy 105–135 % larger.