Hendrik C. Kuhlmann

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.

Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem

The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of three-dimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basic-state eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugal-in...

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.

Hydrodynamic instabilities in cylindrical thermocapillary liquid bridges

The hydrodynamic stability of steady axisymmetric thermocapillary flow in a cylindrical liquid bridge is investigated by linear stability theory. The basic state and the three-dimensional disturbance equations are solved by various spectral methods for aspect ratios close to unity. The critical modes have azimuthal wavenumber one and the most dangerous disturbance is either a pure hydrodynamic steady mode or an oscillatory hydrothermal wave, depending on the Prandtl number. The influence of heat transfer through the free surface, additional buoyancy forces, and variations of the aspect ratio on the stability boundaries and the neutral mode are discussed.

Three-dimensional numerical simulation of thermocapillary flows in cylindrical liquid bridges

The dynamics of thermocapillary flows in differentially heated cylindrical liquid bridges is investigated numerically using a mixed finite volume/pseudo-spectral method to solve the Navier–Stokes equations in the Boussinesq approximation. For large Prandtl numbers ( Pr = 4 and 7) and sufficiently high Reynolds numbers, the axisymmetric basic flow is unstable to three-dimensional hydrothermal waves. Finite-amplitude azimuthally standing waves are found to decay to travelling waves. Close to the critical Reynolds number, the former may persist for long times. Representative results are explained by computing the coefficients in the Ginzburg–Landau equations for the nonlinear evolution of these waves for a specific set of parameters. We investigate the nonlinear phenomena characteristic of standing and pure travelling waves, including azimuthal mean flow and time-dependent convective heat transport. For Pr [Lt ] 1 the first transition from the two-dimensional basic flow to the three-dimensional stationary flow is inertial in nature. Particular attention is paid to the secondary transition leading to oscillatory three-dimensional flow, and this mechanism is likewise independent of Pr . The spatial and temporal structure of the perturbation flow is analysed in detail and an instability mechanism is proposed based on energy balance calculations and the vorticity distribution.

The two-sided lid-driven cavity : experiments on stationary and time-dependent flows

The incompressible fluid flow in a rectangular container driven by two facing sidewalls which move steadily in anti-parallel directions is investigated experimentally for Reynolds numbers up to 1200. The moving sidewalls are realized by two rotating cylinders of large radii tightly closing the cavity. The distance between the moving walls relative to the height of the cavity (aspect ratio) is Γ = 1.96. Laser-Doppler and hot-film techniques are employed to measure steady and time-dependent vortex flows. Beyond a first threshold robust, steady, three-dimensional cells bifurcate supercritically out of the basic flow state. Through a further instability the cellular flow becomes unstable to oscillations in the form of standing waves with the same wavelength as the underlying cellular flow. If both sidewalls move with the same velocity (symmetrical driving), the oscillatory instability is found to be tricritical. The dependence on two sidewall Reynolds numbers of the ranges of existence of steady and oscillatory cellular flows is explored. Flow symmetries and quantitative velocity measurements are presented for representative cases.

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.

Particle accumulation on periodic orbits by repeated free surface collisions

The motion of small particles suspended in cylindrical thermocapillary liquid bridges is investigated numerically in order to explain the experimentally observed particle accumulation structures (PAS) in steady two- and time-dependent three-dimensional flows. Particles moving in this flow are modeled as perfect tracers in the bulk, which can undergo collisions with the free surface. By way of free-surface collisions the particles are transferred among different streamlines which represents the particle trajectories in the bulk. The inter-streamline transfer-process near the free surface together with the passive transport through the bulk is used to construct an iterative map that can describe the accumulation process as an attraction to a stable fixed point which represents PAS. The flow topology of the underlying azimuthally traveling hydrothermal wave turns out to be of key importance for the existence of PAS. In a frame of reference exactly rotating with the hydrothermal wave the three-dimensional flo...

Nonlinear three-dimensional flow in the lid-driven square cavity

The three-dimensional flow in a lid-driven cuboid is investigated numerically. The geometry is an extension to three dimensions of the lid-driven square cavity by translating the two-dimensional lid-driven cavity parallel to the third orthogonal direction. The incompressible Navier-Stokes equations are discretized by a pseudospectral Chebyshev-collocation method. The singularities caused by the discontinuous velocity boundary conditions are reduced by including asymptotic analytical solutions in the solution ansatz. The flow is computed for Reynolds numbers above the critical onset of Taylor-Gortler vortices. Nonlinear Taylor-Gortler cells are calculated for periodic and for realistic no-slip endwall conditions. For periodic boundary conditions the bifurcation is either sub- or supercritical, depending on the wavenumber. The limiting tricritical case arises near the critical wavenumber of the linear-stability problem. On the other hand, no-slip endwall conditions have a significant effect on the supercritical three-dimensional flow. In agreement with recent experimental results we find that Taylor-Gortler vortices are suppressed near no-slip endwalls.