Instability

Related publications (100)

Spatio-temporal stability of the Kármán vortex street and the effect of confinement

Cristobal Manuel Arratia Martinez, François Gallaire, Saviz Mowlavi

The instability of the Kármán vortex street is revisited under a spatio-temporal perspective that allows the taking into account of the advection of the vortices by the external flow. We analyse a simplified point vortex model and show through numerical simulations of its linear impulse response that the system becomes convectively unstable above a certain critical advection velocity. This critical velocity decreases as the aspect ratio approaches its specific value for temporal stability, and increases with the confinement induced by lateral walls. In the limiting unconfined case, direct application of the Briggs–Bers criterion to the dispersion relation gives results in excellent agreement with the numerical simulations. Finally, a direct numerical simulation of the Re=100 flow past a confined cylinder is performed, and the actual advection velocity of the resulting vortex street is found to be much larger than the critical advection velocity for convective instability given by our model. The Kármán vortex street is therefore strongly convectively unstable.

Cambridge University Press2016

Non-linear modelling of saturated internal and external MHD instabilities in tokamaks

Andreas Kleiner

Macroscopic instabilities and their unfavourable effects on plasma confinement pose a central challenge for the development of reactor relevant tokamak scenarios. Some promising operation scenarios feature extended regions of low magnetic shear. These are in the core region for hybrid scenarios, and in the pedestal-edge region for e.g. the ELM-free quiescent H-mode (QH-mode). This thesis presents a non-linear study of macroscopic magnetohydrodynamic (MHD) instabilities with a focus on plasmas with regions of low magnetic shear. In the first part, we investigate the triggering of fast growing resistive modes in hybrid tokamak plasmas, where infernal modes can couple to neoclassical tearing modes (NTMs). Numerical simulations with the non-linear resistive initial value code XTOR-2F with and without inclusion of bootstrap current effects allow us to determine the evolution of MHD modes from the linear to the late non-linear phase. An analytical model is developed to describe the vanishing of mode coupling in the early non-linear regime. In this context, we extend the tearing stability parameter

\Delta'

from the linear to the non-linear phase and calculate the individual contributions to the growth of the resistive mode. This allows for an identification of the triggering mechanism in the initial phase and the dominant terms in the non-linear phase. A comparison with the numerical results shows that infernal mode coupling can destabilise otherwise stable NTMs and thus seed

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magnetic islands. The helically perturbed bootstrap current is found to further destabilise the magnetic islands in the non-linear phase. In the second part of the thesis, 3D free-boundary equilibrium computations are employed to describe saturated external kink-type modes. The approach is first demonstrated to capture the salient features of non-linearly saturated external kink modes in standard baseline tokamak scenarios and is then applied to QH-mode plasmas. A method to conveniently extract the saturated displacement amplitude from edge-corrugated VMEC equilibria in terms of straight field line Fourier modes is presented. For standard current-driven modes the amplitude is compared with a non-linear analytical model, for which a numerical solver is implemented. In QH-mode plasmas, dangerous edge localised modes (ELMs) are replaced by benign edge harmonic oscillations (EHOs). The latter are commonly assumed to be connected to saturated external kink states. In our study of QH-mode plasmas, we consider two different driving mechanisms for external kink type-modes separately. We find that standard current-driven external kinks are linearly unstable, and non-linearly stable in a wide parameter range, especially where

q_{\text{edge}} \lesssim m/n

. But, where standard current-driven kinks are linearly stable we find that coupling of pressure-driven infernal modes can cause instability, and their upper sideband drives edge corrugations that appear to have external kink features. Both types of modes are identified with the VMEC equilibrium code, and the spectra are compared favourably with those of linear numerical approaches and analytic methods. EHOs could be connected to both type of modes.

EPFL2019

, , ,

We study the flow past a permeable sphere modeled using homogenization theory. The flow through the porous medium is described by the Darcy law, in which the permeability quantifies the resistance for the fluid to pass through the microstructure. A slip condition on the tangential velocity at the interface between the fluid and porous region is employed to account for the viscous effects in the proximity of the interface. The steady and axisymmetric flow is first characterized under the assumption of a homogenous and isotropic porous medium. In a certain range of permeability, the recirculation region penetrates inside the sphere, resulting in a strong modification of the linear stability properties of the flow and in a decrease in the critical Reynolds numbers for the flow instability. However, for very large permeabilities, a critical permeability value is identified, beyond which the steady and axisymmetric flow remains always linearly stable. The hypothesis of a homogenous porous medium is then relaxed, and the effect of polynomial distributions of permeability inside the body is studied. Interestingly, some macroscopic flow properties do not significantly vary with the permeability distributions, provided that their average is maintained constant. The analysis is concluded by outlining a simplified procedure to retrieve the full-scale structure corresponding to a considered distribution of permeability.

AIP Publishing2021

From coating flow patterns to porous body wake dynamics via multiscale models

Pier Giuseppe Ledda

Multiscale phenomena are involved in countless problems in fluid mechanics. Coating flows are known to exhibit a broad variety of patterns, such as wine tears in a glass and dripping of fresh paint applied on a wall. Coating flows are typically modeled under the assumption that the thickness of the fluid layer is much smaller than the characteristic length of the free-surface deformations, i.e. there is a separation of scales between the microscopic variations of the velocity and pressure field along the thin layer and the macroscopic modulations of the free-surface. A different multiscale phenomenon of undeniable interest in the fluid dynamics community is the flow around and through porous objects. Dandelion seeds are transported by the wind thanks to a hairy structure, called pappus, known to present larger values of the aerodynamic drag and a more stable wake compared to an impervious disk in the same flow conditions.This thesis investigates the pattern formation of several coating flows and the wake dynamics past diverse permeable bodies via multiscale models. We initially consider the flow of a thin viscous film underneath an inclined planar substrate. We show the emergence of free-surface structures modulated along the direction transversal to the main flow, called rivulets. These rivulets result from a pure equilibrium between hydrostatic gravity and surface tension effects, and may destabilize with the formation of traveling drops. We determine via a linear stability analysis the critical values of the inclination angle and film thickness beyond which rivulets destabilize. We numerically study the linear and non-linear response with respect to a harmonic forcing in the inlet flow rate, determining the diverse lenses' patterns emerging on a steady rivulet. The dripping problem is deepened by considering a single drop deposited on a very thin film. Very slight inclinations with respect to the horizontal, of the order of degrees, lead to the formation of a rivulet in the wake of a shrinking drop. Subsequently, we investigate the role of these instabilities in karst draperies formation, by coupling the hydrodynamic model with the deposition of calcium carbonate on the substrate. We implement an algorithm which retrieves the asymptotic properties of the two-dimensional linear impulse response from numerical simulations. The analysis shows the predominance of streamwise structures, reminiscent of draperies, growing on the substrate. The role of modifications of the substrate is then investigated in the cases of dewetting of very thin polymer films, in the context of production of optical metasurfaces, and in the case of three-dimensional spreading of a mass of fluid. The last part of the thesis is devoted to the modifications of wake flows instabilities past bluff bodies when composed of a permeable microstructure, with a focus on the case of a porous sphere and a cylindrical circular membrane. We develop an inverse procedure to optimize and retrieve the microstructure based on flow objectives. The analysis is concluded by studying the path instability of a freely-falling permeable disk. A complex series of bifurcations occurs but, as the ratio between voids and solid structure increases, all wake and path instabilities are damped.

EPFL2022

The influence of shear layer thickness on the stability of confined two-dimensional wakes

Luca Biancofiore, François Gallaire

The influence of confinement onto the inviscid and incompressible linear stability of the family of wakes introduced by Monkewitz [Phys. Fluids 31, 999 (1988)] is examined. The nondimensional parameters of the model, the velocity ratio Λ, defined as ratio of the velocity gap to the mean velocity, the profile shape parameter, which controls the shear layer thickness δw, and the confinement parameter, are varied and their effect onto temporal and spatiotemporal stability properties is considered. Particularly, the limit between absolute (A) and convective (C) instability is investigated as a function of the different parameters. For a given confinement, there exists an optimal value of the shear layer thickness for which the absolute instability is maximal. The absolute frequency and complex wavenumber of the mode at the C/A transition are discussed. Furthermore, the continuous profiles are approximated by means of piecewise broken-line profile, with similar spatiotemporal properties. As a typical application, a few nonparallel base flows, computed by direct numerical simulation at Re = 500, are analyzed on a weakly nonparallel basis by plotting the locus of the local velocity profile in the (δw,Λ) plane. The absolute frequency at the C/A transition point is seen to predict accurately the frequency prevailing in the nonlinear direct numerical simulations. These results further help interpreting the influence of confinement on the Strouhal number measured experimentally in the wake of confined cylinders.

2011

Rayleigh-Taylor instability under a spherical substrate

Gioele Mariano Nicolò Balestra, François Gallaire, David Minh-Phuc Nguyen

We investigate the Rayleigh-Taylor instability of a thin viscous film coating the inside of a spherical substrate. The aim of this work is to find and characterize the instability pattern in this spherical geometry. In contrast to the Rayleigh-Taylor instability under a planar substrate, where the interface is asymptotically unstable with respect to infinitesimal perturbations, the drainage induced by the component of gravity tangent to a curved substrate stabilizes the liquid interface, making the system linearly asymptotically stable. By performing a linear optimal transient growth analysis we show that the double curvature of a spherical substrate yields a critical Bond number, prescribing the ratio between gravitational and capillary forces, before an initial growth of perturbations is possible two-times larger than for a circular cylindrical substrate. This linear transient growth analysis however does not yield any selection principle for an optimal azimuthal wave number and we have to resort to a fully nonlinear analysis. By numerically solving the nonlinear lubrication equation we find that the most amplified azimuthal wave number increases with the Bond number. Nonlinear interactions are responsible for the transfer of energy to higher-order harmonics. The larger the Bond number and the farther away from the apex of the sphere, the richer the wave-number spectrum.

2018

Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets

Gioele Mariano Nicolò Balestra, François Gallaire, Pier Giuseppe Ledda, Gaëtan Pierre Clément Lerisson

We study the pattern formation of a thin film flowing under an inclined planar substrate. The phenomenon is studied in the context of the Rayleigh-Taylor instability using the lubrication equation. Inspired by experimental observations, we numerically study the thin film response to a streamwise-invariant sinusoidal initial condition. The numerical response shows the emergence of predominant streamwise-aligned structures, modulated along the direction perpendicular to the flow, called rivulets. Oscillations of the thickness profile along the streamwise direction do not grow significantly when the inclination is very large or the liquid layer very thin. However, for small inclinations or thick films, streamwise perturbations grow on rivulets. A secondary stability analysis of one-dimensional and steady rivulets reveals a strong stabilization mechanism for large inclinations or very thin films. The theoretical results are compared with experimental measurements of the streamwise oscillations of the rivulet profile, showing a good agreement. The emergence of rivulets is investigated by studying the impulse response. Both the experimental observation and the numerical simulation show a marked anisotropy favouring streamwise-aligned structures. A weakly nonlinear model is proposed to rationalize the levelling of all but streamwise-aligned structures.

2020

Critical inclination for absolute/convective instability transition in inverted falling films

Nicolas Igor Jacques Kofman

Liquid films flowing down the underside of inclined plates are subject to the interaction between the hydrodynamic and the Rayleigh-Taylor (R-T) instabilities causing a patterned and wavy topology at the free surface. The R-T instability results from the denser liquid film being located above a less dense ambient gas, and deforming into an array of droplets, which eventually drip if no saturation mechanism arises. Such saturation mechanism can actually be provided by a fluid motion along the inclined plate. Using a weighted integral boundary layer model, this study examines the critical inclination angle, measured from the vertical, that separates regimes of absolute and convective instability. If the instability is of absolute type, growing perturbations stay localized in space potentially leading to dripping. If the instability is of convective type, growing perturbations move downwards the inclined plate, forming waves and eventually, but not necessarily, droplets. Remarkably, there is a minimum value of the critical angle below which a regime of absolute instability cannot exist. This minimum angle decreases with viscosity: it is about 85 degrees for water, about 70 degrees for silicon oil 20 times more viscous than water, and reaches a limiting value for liquid with a viscosity larger than about 1000 times the one of water. It results that for any fluid, absolute dripping can only exist for inclination angle (taken from the vertical) larger than 57.4 degrees. Published by AIP Publishing.

Amer Inst Physics2016

Ultralow Interfacial Tension Measurement through Jetting/Dripping Transition

Laura Augello, François Gallaire

In this paper, we present a dynamic microfluidic tensiometer able to perform measurements over more than four decades and which is suitable for high throughput experimentations. This tensiometer is able to withstand hard conditions such as high pressure, high temperature, high salinity, and crude oil. It is made of two coaxial capillaries in which two immiscible fluids are injected. Depending on the. flow rate of each phase, either droplets or jetting will be obtained. The transition between these two regimes relies on the Rayleigh Plateau instability. This transition can be theoretically computed thanks to a linear analysis based on the convective and absolute instabilities theory. From this model, the interfacial tension between the two phases can be calculated.

American Chemical Society2017

Fiber bundle model under fluid pressure

Internal fluid pressure often plays an important role in the rupture of brittle materials. This is a major concern for many engineering applications and for natural hazards. More specifically, the mechanisms through which fluid pressure, applied at a microscale, can enhance the failure at a macroscale and accelerate damage dynamics leading to failure remains unclear. Here we revisit the fiber bundle model by accounting for the effect of fluid under pressure that contributes to the global load supported by the fiber bundle. Fluid pressure is applied on the broken fibers, following Biot's theory. The statistical properties of damage avalanches and their evolution toward macrofailure are analyzed for a wide range of fluid pressures. The macroscopic strength of the new model appears to be strongly controlled by the action of the fluid, particularly when the fluid pressure becomes comparable with the fiber strength. The behavior remains consistent with continuous transition, i.e., second order, including for large pressure. The main change concerns the damage acceleration toward the failure that is well modeled by the concept of sweeping of an instability. When pressure is increased, the exponent beta characterizing the power-law distribution avalanche sizes significantly decreases and the exponent gamma characterizing the cutoff divergence when failure is approached significantly increases. This proves that fluid pressure plays a key role in failure process acting as destabilization factor. This indicates that macrofailure occurs more readily under fluid pressure, with a behavior that becomes progressively unstable as fluid pressure increases. This may have considerable consequences on our ability to forecast failure when fluid pressure is acting.

Amer Physical Soc2016

Non-linear modelling of saturated internal and external MHD instabilities in tokamaks

Andreas Kleiner

\Delta'

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q_{\text{edge}} \lesssim m/n

EPFL2019