Faraday instability on a sphere: numerical simulation

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate-prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry

Optimal transient growth in thin-interface internal solitary waves

The dynamics of perturbations to large-amplitude Internal Solitary Waves (ISW) in two-layered flows with thin interfaces is analyzed by means of linear optimal transient growth methods. Optimal perturbations are computed through direct-adjoint iterations of the Navier-Stokes equations linearized around inviscid, steady ISWs obtained from the Dubreil-Jacotin-Long (DJL) equation. Optimal perturbations are found as a function of the ISW phase velocity $c$ (alternatively amplitude) for one representative stratification. These disturbances are found to be localized wave-like packets that originate just upstream of the ISW self-induced zone (for large enough $c$) of potentially unstable Richardson number, $Ri < 0.25$. They propagate through the base wave as coherent packets whose total energy gain increases rapidly with $c$. The optimal disturbances are also shown to be relevant to DJL solitary waves that have been modified by viscosity representative of laboratory experiments. The optimal disturbances are compared to the local WKB approximation for spatially growing Kelvin-Helmholtz (K-H) waves through the $Ri < 0.25$ zone. The WKB approach is able to capture properties (e.g., carrier frequency, wavenumber and energy gain) of the optimal disturbances except for an initial phase of non-normal growth due to the Orr mechanism. The non-normal growth can be a substantial portion of the total gain, especially for ISWs that are weakly unstable to K-H waves. The linear evolution of Gaussian packets of linear free waves with the same carrier frequency as the optimal disturbances is shown to result in less energy gain than found for either the optimal perturbations or the WKB approximation due to non-normal effects that cause absorption of disturbance energy into the leading face of the wave.Comment: 33 pages, 22 figure

Unsteady draining of a fluid from a circular tank

Three-dimensional draining flow of a two-fluid system from a circular tank is considered. The two fluids are inviscid and incompressible, and are separated by a sharp interface. There is a circular hole positioned centrally in the bottom of the tank, so that the flow is axially symmetric. The mean position of the interface moves downwards as time progresses, and eventually a portion of the interface is withdrawn into the drain. For narrow drain holes of small radius, the interface above the centre of the drain is pulled down towards the hole. However, for drains of larger radius the portion of the interface above the drain edge is drawn down first, rather than the central section. Non-linear results are obtained with a novel spectral technique, and are also compared against the predictions of linearized theory. Unstable Rayleigh-Taylor type flows, in which the upper fluid is heavier than the lower one, are also discussed

On the control and suppression of the Rayleigh-Taylor instability using electric fields

It is shown theoretically that an electric field can be used to control and suppress the classical Rayleigh-Taylor instability found in stratified flows when a heavy fluid lies above lighter fluid. Dielectric fluids of arbitrary viscosities and densities are considered and a theory is presented to show that a horizontal electric field (acting in the plane of the undisturbed liquid-liquid surface), causes growth rates and critical stability wavenumbers to be reduced thus shifting the instability to longer wavelengths. This facilitates complete stabilization in a given finite domain above a critical value of the electric field strength. Direct numerical simulations based on the Navier-Stokes equations coupled to the electrostatic fields are carried out and the linear theory is used to critically evaluate the codes before computing into the fully nonlinear stage. Excellent agreement is found between theory and simulations, both in unstable cases that compare growth rates and in stable cases that compare frequencies of oscillation and damping rates. Computations in the fully nonlinear regime supporting finger formation and roll-up show that a weak electric field slows down finger growth and that there exists a critical value of the field strength, for a given system, above which complete stabilization can take place. The effectiveness of the stabilization is lost if the initial amplitude is large enough or if the field is switched on too late. We also present a numerical experiment that utilizes a simple on-off protocol for the electric field to produce sustained time periodic interfacial oscillations. It is suggested that such phenomena can be useful in inducing mixing. A physical centimeter-sized model consisting of stratified water and olive oil layers is shown to be within the realm of the stabilization mechanism for field strengths that are approximately 2 × 104  V/m

Solitary waves on a ferrofluid jet

The propagation of axisymmetric solitary waves on the surface of an otherwise cylindrical ferrofluid jet subjected to a magnetic field is investigated. An azimuthal magnetic field is generated by an electric current flowing along a stationary metal rod which is mounted along the axis of the moving jet. A numerical method is used to compute fully-nonlinear travelling solitary waves and predictions of elevation waves and depression waves by Rannacher & Engel (2006) using a weakly-nonlinear theory are confirmed in the appropriate ranges of the magnetic Bond number. New nonlinear branches of solitary wave solutions are identified. As the Bond number is varied, the solitary wave profiles may approach a limiting configuration with a trapped toroidal-shaped bubble, or they may approach a static wave (i.e. one with zero phase speed). For a sufficiently large axial rod, the limiting profile may exhibit a cusp

Coherent structures, instabilities, and turbulence in interfacial and magnetohydrodynamic flows

The present summary of publications outlines several theoretical and numerical investigations of instabilities and complex nonlinear dynamics in fluid flows. The summary mostly presents the motivation of the different studies and the essential physics of the systems under consideration. Main results are also described. Details of the mathematical models and numerical methods are given in the publications.The contributions of the author to the individual papers are described in a separate appendix. The flows under study are thermal convection in single and immiscible double fluid layers, two-phase shear layers, and channel flows of electrically conducting fluid with an imposed magnetic field. These configurations are motivated by a variety of applications, e.g. interfacial heat and mass transferin chemical engineering processes, liquid atomization for combustion, or electromagnetic pumps and brakes for the processing of materials in metallurgy. The complexity of the real applications has been reduced considerably in order to examine fundamental mechanisms and properties. The geometric and conceptual simplicity achieved this way is also useful for the numerical studies since it typically allows one to use specialized but very efficient simulation methods. The results of such investigations can improve our understanding of flow physics and can also serve as benchmarks for the verification of more general computational approaches. Thermal convection was considered in several configurations.The first is purely surface-tension driven convection in a single liquid layer, for which the flow structure and the heat flux scaling was studied by two-dimensional and three-dimensional simulations.The second is a system of two layers with heating from below or from aboveand different combinations of immiscible liquids. Two different two-layer setups were studied by three-dimensional numerical simulations in the nonlinear regime with a focus on the transformation of the convective patterns with the thermal forcing. On the topic of two-phase mixing layers two linear stability studies based on coupled Rayleigh/Orr-Sommerfeld equations were performed, and a verification of the nonlinear simulation code SURFER by means of the viscous linear stability results. The novelty in the linear stability calculations consisted in a direct comparison of viscous and inviscid results for geometrically equivalent configurations, and in the identification of a specific viscous instability mechanism in the parameter range of experiments on air/water atomization. Finally, on the topic of channel flows of electrically conducting fluid with an imposed magnetic field the author has been involved in numerical studies of transition and turbulence in conducting channel flows with uniform magnetic field. A nonlinear transition mechanism for subcritical Reynolds numbers was investigated for a spanwise magnetic field, and the properties of magnetohydrodynamic turbulence were studied for both wall-normal and spanwise magnetic field