Natural convection in a bottom heated horizontal cylinder

Abstract

In this work a numerical investigation of two-dimensional steady and unsteady natural convection in a circular enclosure whose lower half is nonuniformly heated and upper half is maintained at a constant lower temperature has been carried out. An explicit finite difference method on a nonstaggered rectangular grid is used to solve the momentum and energy equations subject to Boussinesq approximation. The study is carried out for a range of Rayleigh number ͑Ra͒ varying between 10 2 and 10 6 at a fixed Prandtl number ͑Pr͒ taken as 0.71. The numerical experiments reveal that for Raഛ 8500, the flow always attains a steady state. In the steady regime, at very low Rayleigh numbers ͑RaϽ 300͒, it is shown that the velocity field is very weak and the heat transfer is predominantly by conduction. A series solution for the temperature field obtained by neglecting the fluid velocities is shown to agree well with the computed data for RaϽ 300. The convection takes place in the form of two cells with their interface aligned along the vertical diameter. As Ra is increased further, the isotherms distort to form a plume-like structure of hot fluid rising from the hottest point on the lower half of the cylinder wall. Local Nusselt number distribution over the wall shows that only a portion of the nonuniformly heated bottom half of the cylinder wall is responsible for heating the fluid. For Raജ 8900, the numerical simulations show that the steady flow looses its stability and the flow undergoes bifurcations to periodic and quasiperiodic states. On the basis of the data on the amplitude of the periodic flows obtained for a set of Ra slightly greater than 8900, it is shown that the steady flow undergoes a supercritical Hopf bifurcation at RaϷ 8830. An analysis using the proper orthogonal decomposition shows that the instability is in the form of a standing wave. The structure of the unstable mode is examined via empirical eigenfunctions obtained by the method of snapshots. In the unsteady regime ͑RaϾ 8.9ϫ 10 3 ͒, the cells start to swing their interface in an oscillatory manner with time. As Ra is increased further, the character of flow changes from periodic to quasiperiodic.

Influence of Prandtl number on the instability of natural convection flows within a square enclosure containing an embedded heated cylinder at moderate Rayleigh number

“…However, the heat transfer rate becomes correlated with (RaPr) 1/4 in the unsteady zone at low Prandtl number. Thus, at this region, Boussinesq number (RaPr = g·β ·(T s −T 0 )·L 3 α 2 ) is the proper scaling parameter. This is consistent with the similarity solutions in the laminar free-convection boundary layer, 22,23 where at the asymptotic states of Pr → ∞ and Pr → 0, the Nusselt numbers can be, respectively, expressed as Nu x ∼ Ra 1/4…”

Two-dimensional buoyancy driven thermal mixing in a horizontally partitioned adiabatic enclosure

“…The governing equations of mass, momentum, and energy that are subject to the Boussinesq approximation for a Cartesian frame 19 are as follows.…”

“…Instead, the flow pattern observed is the outcome of formation of stationary waves. 19,37 The temporal structure of the leading four modes, as represented by a ͑1͒ ͑͒ , ... ,a ͑4͒ ͑͒, is shown in Fig. 27.…”

“…13 In open literature, there are a few studies involving the application of the POD technique to unsteady laminar flows with heat transfer. [14][15][16][17][18] Recently, Hasan et al 19 applied POD to demonstrate that the oscillatory instability ͑supercritical Hopf bifurcation͒ for twodimensional thermally driven convection in a horizontal heated circular cylinder at RaϷ 8.83ϫ 10 3 takes place in the form of standing waves. In another recent study, Hasan and Sanghi 20 demonstrated that the POD analysis effectively revealed the presence and structure of the traveling waves in thermally driven flow in a horizontal rotating cylinder.…”

The convection regimes in a low-order model of fluid motion in nearly axially symmetrical ellipsoidal cavity

“…возрастает число степеней свободы, появляется много новых мод ( [12]). Работа [13], наиболее близкая к предложенной по стилю, рассматривает двумерную задачу, в которой окружность подогревалась неравномерно снизу: сильнее в нижней точке, слабее на стыке с верхней частью, державшейся при постоянной температуре. Эта работа и описывает бифуркации Хопфа на основе уравнения Ландау, фиксирует усложнение спектра колебаний по мере увеличения числа Рэлея, представляет хаос.…”