Dynamical evolution of buoyancy-driven flow induced by submerged thermal vertical pipes

The buoyancy-driven natural convection induced by submerged thermal vertical pipes acts as a passive thermal pump without moving parts, characterised by efficient heat transfer. It comprises three key components: natural convection inside the pipe, a thermal plume above the pipe exit and a radial intrusion flow along the upper free surface. Our previous work (Nie &amp; Xu 2022 J. Fluid Mech ., vol. 937 , A29) investigated transient natural convection in a vertical pipe for fluids with a Prandtl number ( italic probability <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Pr</mml:mtext> </mml:mrow> </mml:mrow> </mml:math> ${\textit{Pr}}$ ) less than unity. In this study we extend the scaling analysis to water ( italic probability greater than 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="italic" class="MJX-tex-mathit">Pr</mml:mtext> </mml:mrow> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:math> ${\textit{Pr}}\gt 1$ ), which exhibits fundamentally different dynamics due to the dominance of viscous diffusion over thermal diffusion. We identify three distinct flow configurations, each undergoing a transient development before reaching a fully developed state: (i) the boundary layer flow; (ii) the bulk momentum flow with thermal boundary layer, characterised by a merged velocity layer but a distinct thermal layer; and (iii) the pipe flow resulting from the simultaneous merging of both the velocity and thermal boundary layers. Scaling laws governing the flow and heat transfer across these regimes are derived. Moreover, scaling analyses for the external thermal plume and the radial intrusion flow are also performed. All scaling laws are validated, for the axisymmetric and laminar regime, by extensive numerical simulations.