M. Wanschura

Convective instability mechanisms in thermocapillary liquid bridges

The primary instability of axisymmetric steady thermocapillary flow in a cylindrical liquid bridge with non‐deformable free surface is calculated by a mixed Chebychev‐finite difference method. For unit aspect ratio the most dangerous mode has an azimuthal wavenumber m=2. The physical instability mechanisms are studied by analyzing the linear energy balance of the neutral mode. If the Prandtl number is small (Pr≪1), the bifurcation is stationary. The associated neutral mode is amplified in the shear layer close to the free surface. For large Prandtl number (Pr=4), the basic state becomes linearly unstable to a pair of hydrothermal waves propagating nearly azimuthally. Both mechanisms are compared with those previously proposed in the literature.

Flow in two-sided lid-driven cavities : non-uniqueness, instabilities, and cellular structures

The steady flow in rectangular cavities is investigated both numerically and experimentally. The flow is driven by moving two facing walls tangentially in opposite directions. It is found that the basic two-dimensional flow is not always unique. For low Reynolds numbers it consists of two separate co-rotating vortices adjacent to the moving walls. If the difference in the sidewall Reynolds numbers is large this flow becomes unstable to a stationary three-dimensional mode with a long wavelength. When the aspect ratio is larger than two and both Reynolds numbers are large, but comparable in magnitude, a second two-dimensional flow exists. It takes the form of a single vortex occupying the whole cavity. This flow is the preferred state in the present experiment. It becomes unstable to a three-dimensional mode that subdivides the basic streched vortex flow into rectangular convective cells. The instability is supercritical when both sidewall Reynolds numbers are the same. When they differ the instability is subcritical. From an energy analysis and from the salient features of the three-dimensional flow it is concluded that the mechanism of destabilization is identical to the destabilization mechanism operative in the elliptical instability of highly strained vortices.

Three-dimensional instability of axisymmetric buoyant convection in cylinders heated from below

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr , the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

Elliptic instability in two-sided lid-driven cavity flow

Abstract The recently discovered concentration of vorticity in slender vortex tubes in turbulent flow fields has motivated the investigation of a class of vortices with elliptical streamlines. As a prototype of this flow, long vortices confined in a rectangular cavity and driven by tangentially moting walls are studied. These vortices are characterized by a large rate of plane strain at the core. The quasi-two-dimensional flow is found to be unstable at small Reynolds numbers, if the eccentricity of the streamlines, i.e. the strain rate, is sufficiently large. The three-dimensional supercritical flow is found to be steady with a wavelength of the order of the vortex core diameter. The flow pattern appears in the form of rectangular cells that are very robust. Good agreement between experiment and numerical calculations is obtained. It is argued that the instability found is due to the elliptic instability mechanism.

Linear stability of thermocapillary convection in cylindrical liquid bridges under axial magnetic fields

The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic eld is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic elds cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic elds act to stabilize the basic state. The stabilizing eect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic eld also influences the azimuthal symmetry of the most unstable mode.

Energy analysis of some flow instabilities in liquid bridges

Abstract The convective stability of two-dimensional thermocapillary flow in a half zone is considered. The most dangerous three-dimensional modes of a linear analysis are subjected to an a posteriori energy analysis. This enables to identify the physical processes that are most effective in transfering energy from the basic state to the disturbance. It is found that the steady bifurcation at low Prandtl numbers is due to shear induced by the surface forcing, while the Hopf bifurcation at large Prandtl numbers is indentical with the hydrothermal wave instability for plane thermocapillary liquid layers.

Thermo- and solutocapillary convection in a cylindrical liquid bridge: Stability of steady axisymmetric flow

Abstract The convective stability of steady axisymmetric flow in a cylindrical liquid bridge of a binary fluid mixture between two rigid circular walls of different temperature is investigated numerically. As driving forces we consider thermo- and solutocapillary effects. Buoyancy forces are neglected and the free surface is assumed to be non-deformable. As numerical method we use a Chebyshev collocation approach in radial direction and a second order finite difference scheme in axial direction. This combines the advantages of fast convergence and good resolution of high gradients and leads to a band structure of the linear systems which saves storage and computing time. In a first step the steady axisymmetric flow is calculated and then its stability with respect to 3D time dependent disturbances is investigated with linear stability theory. Critical Reynolds numbers are presented for different parameter variations.

Non-Uniqueness and Instabilities of Two-Dimensional Vortex Flows in Two-Sided Lid-Driven Cavities

The flow in rectangular cavities is investigated both numerically and experimentally. The motion is driven by two facing cavity walls which move tangentially in opposite directions. For a certain range of cavity aspect ratios the two-dimensional basic flow is not unique. At low Reynolds numbers the flow consists of separate co-rotating vortices adjacent to each of the moving walls. On an increase of the wall velocities a jump transition occurs, the two vortices partially merge, and the flow pattern appears in the form of cat’s eyes. Hysteresis has been observed and investigated extensively. For high Reynolds numbers the cat’s eye flow is the preferred state in the present experiment and it becomes unstable to a steady three-dimensional cellular flow that subdivides the basic stretched vortex flow into rectangular convective cells. For even higher Reynolds numbers these cells start to oscillate.