Secondary flows in a laterally heated horizontal cylinder

Abstract

In this paper we study the problem of thermal convection in a laterally heated, finite, horizontal cylinder. We consider cylinders of moderate aspect ratio (height/diameter approximate to 2) containing a small Prandtl number fluid (sigma < 0.026) typical of molten metals and molten semiconductors. We use the Navier-Stokes and energy equations in the Boussinesq approximation to calculate numerically the basic steady states, analyze their linear stability, and compute some nonlinear secondary flows originated from the instabilities. All the calculated flows and the stability analysis are characterized by their symmetry properties. Due to the confined cylindrical geometry, -presence of lateral walls and lids-, all the flows are completely three dimensional even for the basic steady states. In the range of Prandtl numbers studied, we have identified four different types of instabilities, either oscillatory or stationary. The physical mechanisms, shear or buoyancy, of the corresponding flow transitions have been analyzed. As the value of the Prandtl number approaches sigma = 0.026 the scenario of bifurcations becomes more complicated due to the existence of two different stable basic states originated in a saddle-node bifurcation; a fact that had been overlooked in previous works. (C) 2014 AIP Publishing LLC.Postprint (published version

Natural convection in a horizontal cylinder with axial rotation

“…Notice that in these cross sections, the maximum transverse velocity is bigger than the azimuthal velocity at the walls, v = R/l = 0.65. Due to the rotation of the wall, the reflection invariance with respect to the vertical plane y = 0 that the basic flow manifested in the case of = 0 [12] is now broken, and we observe a tilt in the longitudinal circulation with respect to this plane, as can be appreciated in the contour plots of the axial velocity field. In Fig.…”

“…Our present study incorporates rotation to the system analyzed in Ref. [12]. We have characterized the symmetric basic states and analyzed their stability when the temperature gradient increases for rotation rates in the range ∈ (0,8), in which the two driving effects, natural convection and rotation, have been found to be comparable.…”

“…All the bifurcations that have been identified are supercritical in all the cases that have been considered. In this section, we perform an energy analysis that allows us to characterize the physical mechanisms of each instability, as we had done in a previous work in a nonrotating cylinder [12].…”

“…A bounded cylinder can be used to represent realistically the melt zone in the horizontal Bridgman crystal growth process. Such a system was studied by Vaux et al [3] for a fixed value of the Prandtl number, σ = 0.026, and different cell sizes, and was recently analyzed in the work of Mercader et al [12] for a cylinder of aspect ratio = H/2R = 2 and Prandtl numbers in the range σ < 0.026. However, the superposition of rotation along the cylinder axis has received less attention, despite providing a possible mechanism for the stabilization and homogenization of the flow, which could be relevant in different crystal growth techniques involving horizontal cylindrical geometries.…”

“…[12] and focuses on a range of rotation rates where natural convection is not yet dominated by rotational effects. Our aim is to check to what extent rotation can be used as a new method in crystal growth processes.…”