Cooling of aPr<1 fluid in a rectangular container

Abstract

The flow behaviour associated with the cooling of an initially quiescent isothermal Newtonian fluid with Prandtl number Pr less than unity in a rectangular container by unsteady natural convection with an imposed lower temperature on vertical sidewalls is investigated by scaling analysis and direct numerical simulation. The flow is dominated by two distinct stages of development, i.e. the boundary-layer development stage adjacent to the sidewall and the subsequent cooling-down stage. The first stage can be further divided into a start-up stage, transitional stage, and steady-state stage. The parameters characterizing the flow behaviour are the boundary-layer thickness, the maximum vertical velocity within the boundary layer, the time for the boundary layer to reach the steady state, the Nusselt number across the sidewall at the boundarylayer development stage, the time for the fluid in the container to be fully cooled down, and the average fluid temperature over the whole volume of the container.

An experimental study of the unsteady thermal flow around a thin fin on a sidewall of a differentially heated cavity

“…Further investigations of the properties of the transient boundary layer have since focused on isoflux boundary conditions, variation in the Prandtl number, the longer timescale development, and the interaction with the interior flow (e.g. Lin and Armfield, 2005;Xu et al, 2005;Lin et al, 2007). The relevance of the behaviour of the boundary layer on a semi-infinite plate to that on the heated wall of a cavity was identified numerically by Patterson and Armfield (1990), Schladow (1990) and Armfield and Patterson (1992), and experimentally by Schö pf and Patterson (1995) and Patterson et al (2002).…”

Scaling of unsteady natural convection boundary layers with a non-instantaneous initiation

“…The scales for the boundary layer were verified in a detailed way by [16,17,20]. Similarly, scales for the development of the boundary layer for sudden isoflux heating were described by [5], and these scales were verified by detailed comparisons with numerical simulation over a range of forcing parameters.…”

“…These parameters are defined later. For the case of a rectangular cavity, the flows also depend on the aspect ratio [20].…”

Natural convection boundary-layer adjacent to an inclined flat plate subject to sudden and ramp heating

“…If Pr w O(1), then the unsteady and viscous terms are of the same order, and thus both terms need to be included in a balance with the buoyancy term. This balance was introduced by Lin et al [16]. The unsteady term is O(u/t) and the viscous term is O(Pru/t), so these two terms together are O((1 þ Pr)u/t).…”