The convective Stefan problem: shaping under natural convection

Abstract

Abstract

Anomalous convective flows carve pinnacles and scallops in melting ice

“…3(a)-(d) comparisons of the shape progression as measured in experiments and computed in simulations for T ∞ = 4 • C and 8 • C. The strong agreement across all times (dark to light blue) serves as a cross-validation of the simulations and experiments and demonstrates the robustness of the pinnacle form. These pinnacles are reminiscent of those recently observed for bodies dissolving in natural convective flows [15][16][17], for which a boundary layer theory analysis predicts that the pinnacle apex sharpens via a power law growth of curvature: κ 0 (t) = κ 0 (0)(1 − t/t s ) −4/5 [17,20]. Here κ 0 (0) is the initial tip curvature and t s is the blow-up time for the predicted singular dynamics, which were shown to accurately describe the initial stages of sharpening during dissolution.…”

“…Here the solid-liquid interface recedes due to heat conduction along temperature gradients, and the energy released during phase change in turn modifies the temperature field in the fluid. In many situations, these temperature changes cause density variations that drive gravitational convective flows, which also feed back on the interface motion [14,15]. This convective Stefan problem has recently been studied in the related context of solids dissolving into liquids, where the effects of flows due to solutal convection can be seen in fine-scale surface features and overall forms [16][17][18].…”

“…This work identifies three shape motifs of melting ice and their physical origins in natural convective flows, shape-flow coupling, and the density anomaly. Sharplypointed pinnacles directed downwards for T ∞ 5 • C and upwards for T ∞ 7 • C are formed by persistent, directional boundary layer flows, which are analogous to the flows that form pinnacles in dissolution under solutal convection [15][16][17][18]. These observations may also relate to the pinnacle morphology of icebergs, which has been attributed to natural convective flows but has not previously been validated through models or simulations, nor realized in lab experiments [11,41].…”

Morphological attractors in natural convective dissolution

“…Initial numerical evidence and scaling analysis of the SE suggested shock formation and finite-time blowup of the tip curvature [23]. Likewise, similarity solutions of a matched-asymptotic approximation predict unbounded growth of tip curvature for certain initial conditions [25,26].…”

“…( 2), but with the GT regularization required to maintain numerical stability [23]. Other studies employed a matched-asymptotic expansion, but with approximation error that may grow large with time [25,26]. In contrast, we introduce a method to directly propagate characteristics of Eq.…”