Collective effects of neighbouring melting ice objects

TL;DR

Two identical ice blocks are immersed in freshwater and melted under 2D direct numerical simulations with a phase-field model to isolate collective melting effects. By exploring vertical and horizontal alignments across a range of Rayleigh numbers, the study reveals that the bottom block's melting is strongly modulated by the inter-object distance: close proximity yields slow melting due to shielding, while larger separations enhance melting via plume spreading, with a non-monotonic dependence on $\text{Ra}$. The bottom melting behavior follows mixed convection and can be collapsed onto a single curve using a scaled heat-transfer metric based on $\text{Re}^{1/2}\,\text{Pr}^{1/3}$; the formation of a persistent bottom cavity correlates with the enhanced melting and with wider plume spread from the top object. The findings offer fundamental insight into collective phase-change dynamics relevant to natural ice systems and latent-heat technologies, and point to future 3D validation and extensions to more realistic ambient conditions.

Abstract

We present a study on the melting dynamics of neighbouring ice bodies by means of idealised simulations, focusing on collective effects, with the goal of obtaining fundamental insight into how collective interactions influence the melting of ice. Two neighbouring (vertically or horizontally aligned), square-shaped, and equally sized ice objects (size on the order of centimetres) are immersed in quiescent fresh water at a temperature of 20°C. By performing two-dimensional direct numerical simulations, and using the phase-field method to model the phase change, the collective melting of these objects is studied. When the objects are horizontally aligned, no significant influence of the neighbouring object on the melting time is observed. On the other hand, when vertically aligned, though the melting of the upper object is mostly unaffected, the melting time and the morphology of the lower ice body strongly depends on the initial inter-object distance. We report that the melting of the bottom object can be enhanced by more than 10%, or delayed more than 20%, displaying a non-monotonic dependence on the initial object size. We show that this behaviour results from a non-trivial competition between layering of cold fluid, which lowers the heat transfer, and convective flows, which favour mixing and heat transfer. For this melting in mixed convection, we were able to collapse our data onto a single curve.

Figures (10)

• Figure 1: (a) Simulation setup. Two ice objects of size $L$ are placed one on top of the other at a distance $D$. The dashed lines indicates the zoomed-in region for visualisation. (b) Region of parameter space explored in this work. The colour code corresponds to $\text{Ra}$.
• Figure 2: Zoomed-in evolution (see figure \ref{['fig:setup']}a) of the melting of the top and bottom objects with initial size $L = 5~\text{cm}$ ($\text{Ra} \approx 2.3\times 10^7$). The dotted lines indicate the initial contours. The instantaneous snapshots of the non-dimensional temperature field $\theta$ are shown for different times, in units of the reference case melting time $\mathcal{T}_\text{ref}$. The colour-bar indicates $\theta$ within the liquid, and the region corresponding to ice is shown in white. The top row corresponds to an initial displacement $D/L = 0.2$ ($D/\delta_\text{th} \approx 14$), where the bottom melts slower than the top. The bottom row is for $D/L = 1.0$ ($D/\delta_\text{th} \approx 69$), where the object below melts faster.
• Figure 3: (a) Melting times $\mathcal{T}_\text{top}$ and $\mathcal{T}_\text{bot}$ of the top (open, pointing-up markers) and bottom objects (filled, pointing down markers), respectively, in units of the reference melting time $\mathcal{T}_\text{ref}$, as a function of the vertical distance between objects $D$, for different $\text{Ra}$ (i.e. for different initial object sizes), indicated by the different colours. The distance is normalised by the estimation of the thermal boundary layer based on the initial object size $\delta_\text{th} \propto L~\text{Ra}^{-1/4}$. (b) 2D map of the ratio of bottom to top melting times. The shaded grey region indicates the critical distance $D^\text{crit}$ where the top and bottom melting times are equal. The blue (red) symbols correspond to distances where $\mathcal{T}_\text{bot}$ is greater (smaller) than $\mathcal{T}_\text{top}$. (c) Critical distance $D^\text{crit}$ and its uncertainty, normalised by $\delta_\text{th}$, as a function of $\text{Ra}$. Colours as in panel (a).
• Figure 4: Average Nusselt number of the bottom object normalised by the forced convection heat transfer scaling. The points are coloured according to the ratio of melting times $\mathcal{T}_\text{bot}/\mathcal{T}_\text{top}$, indicated by the colour bar. The typical error bar is of the size of the marker. The line corresponds to a fit of the data with the function $y = b\, x^a$, yielding $a = 0.84 \pm0.02$, and $b = 1.07\pm0.01$ (95% confidence intervals).
• Figure 5: (a) Representative contours of the ice morphology, for cases without and with a cavity on the lower face of the object ("bottom" cavity). The profiles corresponds to $\text{Ra} \approx 5.0 \times 10^6$ at $t\approx 0.43~\mathcal{T}_\text{ref}$ for $D/\delta_\text{th} \approx 85.2$. The dashed lines indicate the initial ice contour, shown for reference. (b) Percentage of the total bottom object's evolution where a cavity on its bottom face is present. (c) Typical horizontal amplitude (in units of the object size $L$) spanned by the plume of the reference case at a given distance $D$. In panels (b) and (c) the grey shaded region indicates the distance where $\mathcal{T}_\text{bot} = \mathcal{T}_\text{top}$. The regions where $\mathcal{T}_\text{bot}$ is larger and smaller than $\mathcal{T}_\text{top}$ are also indicated.