Stability Of Axisymmetric Flow Research Articles

Stability analysis of Rayleigh-Bénard convection in a cylinder with internal heat generation.

The flow instabilities of Rayleigh-Bénard convection in a cylinder with effect of uniform internal heat source are investigated numerically. The instabilities of the static state and of axisymmetric flows are investigated by linear stability analysis. The convection threshold depends on the strength of internal heat source q and the aspect ratio of the cylinder Γ. The stability of axisymmetric flows is strongly affected by these two parameters, as well as the Prandtl number Pr. Depending on the value of q, three regimes are identified: weak internal heating, moderate internal heating, and strong internal heating regime. In a weak internal heating regime, the instability characteristics are similar to Rayleigh-Bénard convection. In a moderate internal heating regime, intense interaction of buoyancy instability and hydrodynamic instability result in complex instability curves. When q is large enough, the internal heating effect overwhelms the boundary heating effect. Specifically, the influence of Pr on instability is studied at a moderate internal heat strength q=6.4. An extremely multivalued stability curve is observed. At most five critical Rayleigh numbers can be determined for the axisymmetry-breaking instability at a certain Prandtl number. An axisymmetric unsteady instability mode is observed as well. By nonlinear simulation, the oscillatory flow patterns are obtained, and the axisymmetry-breaking bifurcation of the unsteady toroidal flow is studied.

Stratified flows with vertical layering of density: experimental and theoretical study of flow configurations and their stability

AbstractA vertically moving boundary in a stratified fluid can create and maintain a horizontal density gradient, or vertical layering of density, through the mechanism of viscous entrainment. Experiments to study the evolution and stability of axisymmetric flows with vertically layered density are performed by towing a narrow fibre upwards through a stably stratified viscous fluid. The fibre forms a closed loop and thus its effective length is infinite. A layer of denser fluid is entrained and its thickness is measured by implementing tracking analysis of dyed fluid images. Thickness values of up to 70 times that of the fibre are routinely obtained. A lubrication model is developed for both a two-dimensional geometry and the axisymmetric geometry of the experiment, and shown to be in excellent agreement with dynamic experimental measurements once subtleties of the optical tracking are addressed. Linear stability analysis is performed on a family of exact shear solutions, using both asymptotic and numerical methods in both two dimensions and the axisymmetric geometry of the experiment. It is found analytically that the stability properties of the flow depend strongly on the size of the layer of heavy fluid surrounding the moving boundary, and that the flow is neutrally stable to perturbations in the large-wavelength limit. At the first correction of this limit, a critical layer size is identified that separates stable from unstable flow configurations. Surprisingly, in all of the experiments the size of the entrained layer exceeds the threshold for instability, yet no unstable behaviour is observed. This is a reflection of the small amplification rate of the instability, which leads to growth times much longer than the duration of the experiment. This observation illustrates that for finite times the hydrodynamic stability of a flow does not necessarily correspond to whether or not that flow can be realised from an initial-value problem. Similar instabilities that are neutral to leading order with respect to long waves can arise under the different physical mechanism of viscous stratification, as studied by Yih (J. Fluid Mech., vol. 27, 1967, pp. 337–352), and we draw a comparison to that scenario.

Influence of the volume of liquid on the onset of three-dimensional flow in thermocapillary liquid bridges

The stability of axisymmetric flow in thermocapillary liquid bridges is investigated by a linear stability analysis using a finite-difference method. We consider the case when the static meniscus is non-cylindrical depending on the volume of liquid contained in the bridge. Neglecting dynamic free-surface deformations (zero-Capillary-number limit) the deformed domain occupied by the liquid is mapped onto a cylindrical volume by a coordinate transformation. We present critical Reynolds numbers, compare with previously published data, and discuss the influence of the free-surface deformation on the instability mechanism.

回転円筒内の軸対称流の安定性

A stability of axisymmetric flows in the rotating fluid annulus with respect to the wave perturbations is studied by the numerical time integration of linearized hydrodynamic equations. Firstly, two steady axisymmetric flows are established by integrating nonlinear Boussinesq equations. One (Case (C)) is supposed to be located in the upper symmetric regime and the other (Case (B)) in the wave regime. Then, the stability of the two axisymmetric flows with respect to the wave disturbance with an assumed zonal wave number is examined by integrating a system of linearized perturbation equations as an initial value problem. The axisymmetric flow for Case (B) is shown to be unstable with respect to the disturbance with a wave number 8. The developed wave is essentially similar to the classical Eady wave. Near the upper and lower boundaries, the structure of the wave is modified to form the Ekman layer balance. Energetics of the developed wave is also discussed.