On the mechanisms of icicle evolution

Abstract

We present a study of a cylinder of ice melting in warm air in order to quantify the heat-transfer mechanisms controlling the evolution of its shape, which are inherent in a range of phenomena involving phase change and fluid flow. Motivated by the initial melting at the top of a flat-topped cylinder of ice, we analyse laminar, natural convection above a cooled, finite, horizontal plate (or below a heated, finite, horizontal plate) and show that, to a very good approximation, the partial-differential, boundary-layer equations can be separated with self-similar vertical profiles scaled by the boundary-layer thickness. We find that the horizontal evolution of the boundarylayer thickness is governed by equations describing a steady, viscous gravity current fed by diffusive entrainment, and therefore describe such flows as diffusive gravity currents. We first use the predictions of our model to examine previous experimental results in two dimensions. Our experimental results relating to the melting of ice in air are then compared with predictions based on our analysis of the axisymmetric thermal boundary layer. This comparison confirms the vertical thermal structure and shows that melting is governed in roughly equal measure by heat transfer from the air, the latent heat of condensation of water vapour, and the net radiative heat transfer from the surroundings to the ice.

On the origin and evolution of icicle ripples

“…Any topography present on the ice surface is reflected in the shape of the water-air interface in a manner controlled by the surface tension γ . Because the thermal conductivity of air is much less than that of water, the temperature difference T across the thin film is much smaller than the total temperature difference driving the growth [11,20]. In our experiments, the latter was ∼10 • C, while T < 0.1 • C. The latent heat released at the ice-water interface as ice is formed is diffused and advected through the flowing water film and carried away by the surrounding stirred air.…”

“…The relevant dimensionless parameter characterizing the relative importance of advection and diffusion in the water film is the Péclet number Pe = U h/κ ∼ 5, where κ is the thermal diffusivity of water. It has been shown that, in addition to diffusion and advection, both thermal radiation and water evaporation with vapor advection by the air make significant contributions to the heat transport outside the icicle [20].…”

“…The familiar elongated form of a ripply icicle is the result of a subtle interplay between the solidification dynamics of ice [17,18] and the gravity-driven flow of the thin supercooled liquid water film flowing over its evolving surface [16,19]. The latent heat released by freezing, which controls ice growth, is advected by the water film and ultimately carried away by the surrounding sub-zero air, which is also flowing [16,20]. Ripples are presumed to be due to a morphological instability of ice growth in the presence of a thin flowing water film.…”

Do Dissolving Objects Converge to a Universal Shape?

“…Clearly a more detailed analysis of the convection−diffusion− dissolution dynamics is needed, perhaps along the lines of recent work on icicle melting. 13 …”

“…Despite major differences in the physical and chemical processes, ideal dripping icicles obey the same power law. 12, 13 The mathematical analysis of these two free-boundary problems leads to the same nonlinear ordinary differential equation for the asymptotic shape, but for icicles, atmospheric heat transfer qualitatively takes the place of CO 2 exchange. In addition, both structures feature ripples around their circumference, which in the case of icicles result from dissolved ionic impurities.…”