Transition from classical to ultimate melting

Abstract

Melting is omnipresent in nature and technology, with applications ranging from metallurgy, biology, food science, and latent thermal energy storage to oceanography, geophysics, and climate science, and occurring on all scales from sub-millimeter to global scales. The key objective is to understand the rate at which an object melts as a function of its size and of the ambient conditions. To achieve this it is important to be able to extrapolate from small scale experiments and observations to large or even global scales. This is done by scaling laws. However, these are only meaningful if there is no transition from one scaling relation to another one. Here we show, however, that for both fixed and freely-advected melting objects immersed in a turbulent flow a melting transition does exist, namely from slow melting at the small scales to fast melting at the large scales. We do so by controlled melting experiments and corresponding direct numerical simulations, covering four orders of magnitude in scale. The transition corresponds to the transition from a laminar-type boundary layer around the melting object to a turbulent-type boundary layer, i.e., from so-called classical turbulence to ultimate turbulence, with its enhanced transport properties. Our results thus provide a quantitative understanding of the flow physics of the melting process and thereby enable a better extrapolation and prediction of melt rates on large scales such as relevant in geophysics, oceanography, and climate science.

Figures (3)

• Figure 1: Present experimental and numerical setups (a) Experimental setup: twenty independently-controlled electric motors mounted on the vertices of a $\approx$210 dodecahedral water tank drive propellers to generate HIT in the tank center. (b) Volume contours of instantaneous temperature in our present DNS at $\textit{Re}_{\lambda} \approx 50$, $\textit{Re}_{D_0} \approx 80$ for a Lagrangian ball at $50\%$ of its initial volume. The melting ball is represented by an immersed boundary zhong2025, as visualized in the left panel.
• Figure 2: Reynolds number controls melting rate of ice balls in both experiments and DNS Snapshots of a freely-advected melting ice ball at two different times at fixed $\textit{Re}_\lambda$ and fixed $\textit{Re}_{D_0}$ (as noted) and temporal evolution of the volume-equivalent radius for various $\textit{Re}_{D_0}$. The dark curve corresponds to the case shown in the two left panels. (a) Experimental results (b) DNS results.
• Figure 3: A scaling transition from slow to fast melting (a) Nusselt number compensated by $\textit{Pr}^{1/3}$ for all the presently considered experiments and DNS and data from the literature machicoane2013mccutchan2024guo2026. The dashed lines indicate $\textit{Nu} \propto \textit{Re}_{D_0}^{1/2}$, $\textit{Nu} \propto \textit{Re}_{D_0}^{0.8}$ and $\textit{Nu} \propto \textit{Re}_{D_0}$ scaling relations. The symbol legend indicates the $\textit{Re}_\lambda$ considered, Stefan number $\textit{Ste}$, Prandtl number $\textit{Pr}$, and whether Eulerian (E) or Lagrangian (L) melting is considered. The grey-shaded region indicates a range where the laminar-to-turbulent boundary-layer transition takes place. In the two sketches, the boundary-layer thickness is not to scale. (b) Data in (a) compensated by $\textit{Re}_{D_0}^{1/2} \textit{Pr}^{1/3}$ to highlight the laminar boundary-layer scaling for $\textit{Re}_{D_0}\lesssim 4000$. (c) Data in (a) compensated by $\textit{Re}_{D_0}$ highlighting the ultimate regime scaling. The dashed lines in (b,c) indicate $\textit{Nu} \propto \textit{Re}_{D_0}^{1/2}$, $\textit{Nu} \propto \textit{Re}_{D_0}$, $\textit{Nu} \propto \textit{Re}_{D_0}^{0.8}$ scalings.