Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number

Abstract

International audienceThe paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the cross-section aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleigh-type steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized

Secondary flows in a laterally heated horizontal cylinder

“…This result differs from that obtained in a square cross section channel subjected to a horizontal temperature gradient. 6 The authors obtain that the dominant instability is steady and maintains the symmetry with respect to the vertical plane, which is equivalent to our R 1 symmetry. Our results for very small Prandtl number also differ from the work of Henry and BenHadid 15 in 3D rectangular cavities.…”

“…The basic state in this configuration is a parallel flow known as Hadley circulation, 4 in which the fluid moves in the direction of the temperature gradient in the lower part of the channel, and in the opposite direction in the upper part, leading to a stable temperature stratification. This problem has been studied both in two dimensions, 5 and in three dimensional rectangular 6 and cylindrical 7 geometries. In this problem, the stability of the basic flow is considered against perturbations periodic in the streamwise direction.…”

Multiscale Modeling of Multiphase Fluid Flow

“…Moreover, the flow and heat transfer process are also affected by the aspect ratio and boundary conditions of the enclosure. Therefore, many studies of natural convection in a cavity focus on the effects of relative strength of buoyancy and viscous forces [230], various fluids properties [231][232][233], aspect ratio [230,234,235], and boundary conditions [236][237][238].…”

Natural convection in a horizontal cylinder with axial rotation

“…In this latter problem, crystals grown from the melt can present inhomogeneities in the form of striations, caused by oscillatory variations of the concentration in the solidification front [1]. For this reason, many studies have focused on the study of the oscillatory threshold in low-Prandtl-number fluids in different geometrical configurations [2][3][4][5][6][7]. The system is also interesting from a fundamental fluid dynamics point of view, since it exhibits a rich nonlinear behavior that leads to complex spatiotemporal dynamics [8][9][10][11].…”