The dynamics of a viscous soap film with soluble surfactant

International audienceNearly two decades ago, Couder (1981) and Gharib and Derango (1989) used soap films to perform classical hydrodynamics experiments on two-dimensional flows. Recently soap films have received renewed interest and experimental investigations published in the past few years call for a proper analysis of soap film dynamics. In the present paper, we derive the leading-order approximation for the dynamics of a flat soap film under the sole assumption that the typical length scale of the flow parallel to the film surface is large compared to the film thickness. The evolution equations governing the leading-order film thickness, two-dimensional velocities (locally averaged across the film thickness), average surfactant concentration in the interstitial liquid, and surface surfactant concentration are given and compared to similar results from the literature. Then we show that a sufficient condition for the film velocity distribution to comply with the Navier-Stokes equations is that the typical flow velocity be small compared to the Marangoni elastic wave velocity. In that case the thickness variations are slaved to the velocity field in a very specific way that seems consistent with recent experimental observations. When fluid velocities are of the order of the elastic wave speed, we show that the dynamics are generally very specific to a soap film except if the fluid viscosity and the surfactant solubility are neglected. In that case, the compressible Euler equations are recovered and the soap film behaves like a two-dimensional gas with an unusual ratio of specific heat capacities equal to unity

The role of boundary conditions in a simple model of incipient vortex breakdown

International audienceWe consider incipient vortex breakdown and describe how infinitesimal perturbations may destabilize a columnar swirling jet. The framework is axisymmetric and inviscid following Wang and Rusak's [J. Fluid Mech. 340, 177 (1997)] analysis. The goal of the present study is to relate the local properties of swirling flows in infinite pipes to their global stability properties in pipes of finite length. A spatial linear stability analysis is pursued which gives a complementary point of view to the subcritical/supercritical concept introduced by Benjamin [J. Fluid Mech. 14, 593 (1962)]. In contrast to supercritical flows which exhibit two neutral spatial branches traveling downstream and two counterpropagating evanescent spatial branches, subcritical flows exhibit a frequency range where all spatial branches are neutral, three of which travel downstream and one upstream. By using global energy budget arguments and monitoring how the upstream wave is reflected into the downstream waves and conversely, the inlet and outlet conditions are shown to drive the instability in the limit of long but finite pipes. Various inlet and outlet conditions are proposed that stabilize or destabilize the flow, depending on their ability to supply energy. The analysis demonstrates therefore that the global instability accounting for incipient vortex breakdown in Wang and Rusak's model may arise from the combination of a locally neutral flow and suitable inlet and outlet conditions. © 2004 American Institute of Physics

Instability mechanisms in swirling flows

International audienceWe investigate the stability of the screened Rankine vortex with added plug flow where the azimuthal velocity decreases abruptly outside the core of the vortex. The jump in circulation is known to induce centrifugal and azimuthal Kelvin-Helmholtz instabilities. Their effect on the stability of the different azimuthal wave number m is discussed using physical considerations associated with asymptotic expansions and numerical computations of the dispersion relation. It is shown that the axial shear and centrifugal instability are active for all m, and that modes with m = 2 are also destabilized by azimuthal shear. In contrast, the bending modes m = ± 1 are stabilized by a coupling with Kelvin waves in the core. Effects of rotation on the absolute/convective transition are also discussed. The absolute instability of positive helical modes is seen to be promoted by centrifugal instability and azimuthal shear. © 2003 American Institute of Physics

An asymptotic expansion for the vortex-induced vibrations of a circular cylinder

International audienceThis paper investigates the vortex-induced vibrations (VIV) of a spring-mounted circular cylinder. We compute analytically the leading-order equations describing the nonlinear interaction of the fluid and structure modes by carrying out an asymptotic analysis of the Navier-Stokes equations close to the threshold of instability of the fluid-only system. We show that vortex-shedding can occur at subcritical Reynolds numbers as a result of the coupled system being linearly unstable to the structure mode. We also show that resonance occurs when the frequency of the nonlinear limit cycle matches the natural frequency of the cylinder, the displacement being then in phase with the flow-induced lift fluctuations. Using an extension of this model meant to encompass the effect of the low-order added-mass and damping forces induced by the displaced fluid, we show that the amount of energy that can be extracted from the flow can be optimized by an appropriate choice of the structural parameters. Finally, we suggest a possible connection between the present exact model and the empirical wake oscillator model used to study VIV at high Reynolds numbers. We show that for the low Reynolds numbers considered here, the effect of the structure on the fluid can be represented by a first coupling term proportional to the cylinder acceleration in the fluid equation, and by a second term of lower magnitude, which can stem either from an integral term or from a term proportional to the third derivative of the cylinder position. © 2011 Cambridge University Press

Breaking of rotational symmetry in a swirling jet experiment

International audienceIn this experimental investigation, the dynamics of symmetry-breaking instabilities in swirling jets is analyzed for swirl parameters S in the pre-breakdown range O=S=Sc, where Sc=1.3 is the critical swirl value for the appearance of vortex breakdown determined by Billant, Chomaz, and Huerre [J. Fluid Mech. 376, 183 (1998)]. As S is increased, three distinct dynamical regimes have been identified in the streamwise region extending to the end of the potential core. In the low swirl range S < 0.6, the evolution is governed by the same instability mechanisms as in a nonswirling jet. The shear in axial direction generates axisymmetric vortex rings at a Strouhal number independent of the swirl S. As S increases, the amplitude of the axisymmetric mode decreases in magnitude. Concurrently, co-rotating streamwise vortices form in the braids connecting the rings due to a secondary instability mechanism. The advection by the mean rotation of these secondary structures generates an azimuthal wave propagating cyclonically when compared to the imposed rotation, at a phase velocity proportional to swirl. When swirl reaches the transitional swirl level S ~0.6, no azimuthal or standing wave is observed, and the swirling jet is completely dominated by the development of the axisymmetric mode into ring vortices. In the intermediate swirl range 0.6 < S =1, vortex rings form concurrently with several interacting helical cyclonic waves of azimuthal wave number 2. The mean phase velocity of the resulting propagating wave increases at a constant rate with swirl, and much more rapidly than in the low swirl regime. In this swirl range, azimuthal and axisymmetric deformations are of comparable high levels. In the high swirl range 1 < S < 1.3, another step toward complexity is reached, and there is a strong interaction between the azimuthal waves and the ringlike structures. The most striking feature of this flow regime is the emergence of a bending mode m=1 propagating with a high negative phase velocity. © 2003 American Institute of Physics

Propagating Pattern Selection and Causality Reconsidered

International audiencePattern selection, occurring when a nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous basic state is reconsidered by application of the causality principle. In particular, the nonlinear marginal stability criterion that determines the selection of a nonlinear front solution is replaced by an exact general necessary condition that has never been considered before. The demonstration is based on the causal signaling problem derived in the context of plasma physics

Global modes in a confined impinging jet: application to heat transfer and control

We investigate the stability and control of a plane, laminar jet impinging on a flat plate in a channel, a geometry used to cool down a hot wall with a cold air jet in many industrial configurations. The global stability analysis indicates that, even for a strong confinement, the two-dimensional (2-D) steady flow is unstable to three-dimensional (3-D), steady perturbations. In the simplest limit case where dilatation effects are neglected, we show that the development of the instability induces a significant spanwise modulation of the heat flux at the impacted wall. To control the leading global mode, we propose adjoint-based 3-D harmonic and 2-D steady forcing in the bulk or at the wall. We show for instance that the unstable mode is controllable using a spanwise uniform blowing at the upper wall, in a specific domain corresponding to the footprint of the upper recirculating bubble. These techniques are applied to a novel open-loop control, in which we introduce into the flow a small airfoil, modelled by the lift force it exerts on the flo