Mixed baroclinic convection in a cavity

Abstract

We study the convective patterns that arise in a nearly semi-cylindrical cavity fed in with hot fluid at the upper boundary, bounded by a cold, porous semi-circular boundary at the bottom, and infinitely extended in the third direction. While this configuration is relevant to continuous casting processes that are significantly more complex, we focus on the flow patterns associated with the particular form of mixed convection that arises in it. Linear stability analysis and direct numerical simulations (DNS) are conducted, using the spectral element method to identify observable states. The nature of the bifurcations is determined through Stuart-Landau analysis for completeness. The base flow consists of two counter-rotating rolls driven by the baroclinic imbalance due to the curved isothermal boundary. These are however suppressed by the through-flow, which is found to have a stabilising influence as soon as the Reynolds number Re based on the through-flow exceeds 25. For a sufficiently high Rayleigh number, this base flow is linearly unstable to three different modes, depending on Re. For Re ≤ 75, the rolls destabilise through a supercritical bifurcation into a travelling wave. For 100 ≤ Re ≤ 110, a subcritical bifurcation leads to a standing oscillatory mode, whereas for Re ≥ 150, the unstable mode is non-oscillatory and grows out of a supercritical bifurcation. The direct numerical simulations confirm that in all cases, the dominant mode returned by the linear stability analysis precisely matches the topology and evolution of the flow patterns that arise out of the fully nonlinear dynamics. *The second ingredient is the through-flow fed in at the upper boundary of the cavity and escaping at the lower boundary through solidification. Through-flows are found in two types of mixed convection problems of some relevance to our configuration: at the boundary of a heated cavity [38], or through a heated conduit (see [25] and [30] for reviews), whether a pipe [49], a duct [37] or a channel [19]. In all conduit configurations, the shear associated to the through-flow acts as a source of instability. For example, in channels Tollmien-Schlichting waves are favoured in this way [45] and the mixed convection patterns result from a competition between buoyancy-driven and hydrodynamically-driven instabilities. Whether one or the other dominates depends on the ratios of buoyancy, and inertia to viscous forces, respectively measured by the Rayleigh number Ra = βg∆T h 3 (να) and the Reynolds number Re = U0h ν (β, ∆T , h, g, U0 are the fluid's thermal expansion coefficient, a temperature difference between boundaries, a domain diameter, gravitational acceleration, and a fluid inlet velocity). In the simplest configuration of the Rayleigh-Bénard-Poiseuille problem, transversal rolls dominate at low Reynolds numbers while Tollmien-Schlichting waves characteristic of the Poiseuille flow problem dominate for Re > 140 [18]. In rectangular cavities, by contrast, natural convection sets in through an oscillatory mode if the hot wall is...

Isi Taşinimi İçi̇n Yapisal Olmayan Ağlarda Bi̇r Hizlandirilmiş Süreksi̇z Galerki̇n Metdu: Formülasyonu Ve Doğrulanmasi

“…Hossain et al (Hossain et al, 2021) developed a spectral/hp element method for the Direct Numerical Simulation (DNS) of incompressible thermal convective flows by considering Boussinesq type thermal bodyforcing with periodic boundary conditions and enforcing a constant volumetric flow rate. In Kumar and Pothérat (2020), the authors studied the convective patterns that arise in a nearly semi-cylindrical cavity placed in a fluid heated at the upper boundary and bounded by a cold and porous semicircular boundary. They used spectral element method to obtain results and performed a linear stability analysis of the fluid flow.…”

Off-centre gravity induces large-scale flow patterns in spherical Rayleigh–Bénard convection

“…This results in a large-scale circulation pattern as shown in figure 2( c ). In steady flows, large-scale circulations induced by baroclinic torque are also reported for side-heated cavity (Jiang, Sun & Calzavarini 2019), stratified flow in a semicylindrical cavity (Kumar & Pothérat 2020), and natural convection in spherical annuli (Scurtu, Futterer & Egbers 2008).

Figure 2.Baroclinic imbalance induces a regular large-scale circulation due to the offset gravity centre towards the south (, see ) for a steady flow with .

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