Wake transitions and melting dynamics of a translating sphere in warm liquid

TL;DR

This work addresses how a translating solid sphere melts in a warmer liquid by resolving the coupled evolution of the flow, temperature field, and solid–liquid interface with a sharp-interface, three-dimensional method. It reveals four distinct wake/melting regimes under forced convection and shows that melting tends to homogenize surface melting rates across the particle, even through wake bifurcations. A shape-aware model incorporating an aspect-ratio correction to surface area provides a predictive description of volume evolution, with key scalings $t_f \sim Re_0^{1/2} St^{-1}$ and $Ar(t) \sim (1-\hat{t})^{-1/2}$, validated across a wide range of $\mathit{Re}_0$. Introducing buoyancy ($\mathit{Ri}\neq 0$) reorganizes wake dynamics and interface morphologies in a manner consistent with experimental observations, offering a unified framework for forced and mixed convection melting relevant to geophysical and industrial processes.

Abstract

We investigate the three-dimensional melting dynamics of an initially spherical particle translating in a warmer liquid using sharp-interface simulations that fully resolve both solid and fluid phases with the Stefan condition. A wide parameter space is explored, spanning initial Reynolds number ($Re_0$), Stefan number ($St$), and Richardson number ($Ri$). In the absence of buoyancy ($Ri= 0$), the interface evolution is governed by canonical wake bifurcations. Four regimes are identified: an axi-symmetric regime ($Re_0<212$) with a rounded front and planar rear; a steady-planar-symmetric regime ($212<Re_0<273$) with an inclined rear plane; a periodic-planar-symmetric regime ($273<Re_0<355$) where vortex shedding emerges in the wake; and a chaotic regime ($Re_0>355$) with fluctuating stagnation points and a more rounded rear. Despite these differences, all regimes exhibit a tendency toward melt-rate homogenisation over time. Besides, we introduce an aspect-ratio-based surface-area formulation that yields a predictive model, accurately capturing volume evolution across regimes. Hydrodynamic loads also reflect the coupling between shape and flow: drag follows rigid-sphere correlations only at moderate $Re_0$; planar rears enhance drag at higher $Re_0$; lift appears only in symmetry-broken regimes and reverses late in time; torque reorients the rear plane toward vertical, consistent with free-body experiments. When buoyancy is included, assisting configurations ($Ri>0$) suppress recirculation and maintain quasi-spherical shapes, whereas opposing or transverse buoyancy ($Ri<0$) destabilises wakes and promotes tilted planar rears. These results provide a unified framework for convection-driven melting across laminar, periodic, and chaotic wakes, with implications for geophysical and industrial processes.

Figures (29)

• Figure 1: Schematic of the numerical setup (not to scale). A solid sphere composed of a pure substance with initial diameter $D_0$, denoted by $\Omega^s$, with uniform initial temperature $T = T_0$, is immersed in its warmer liquid phase, $\Omega^\ell$, driven by an incoming flow. At the inflow boundary, a uniform velocity $\bm{U} = (U_\infty, 0, 0)$ and temperature $T = T_\infty > T_0$ are imposed.
• Figure 2: Parameter space $(\mathit{Re}_0, \mathit{St})$ explored in this study, with Prandtl number fixed at $Pr= 7$. (a) Range of initial Reynolds numbers, $25\leq \mathit{Re}_0\leq 1000$, covering the canonical regimes for non-melting spheres: steady axi-symmetric ($\mathit{Re}_0<212$), steady-planar-symmetric ($212<\mathit{Re}_0<273$), periodic-planar-symmetric ($273<\mathit{Re}_0<355$), and chaotic ($\mathit{Re}_0>355$), with regime boundaries from ern2012wake. (b) Representative snapshots of melting spheres and surrounding flow for each regimes, coloured by local temperature. The unfilled markers in the panel (a) highlight corresponding visualisations.
• Figure 3: Grid refinement and numerical convergence for the case $(\mathit{Re}_0,\mathit{St},\mathit{Pr},\mathit{Ri})=(1000,0.25,7,0)$. (a) Grid distribution on the mid-plane at $z=0$ at $t=30$ with $D_0/\Delta_\mathrm{min} = 204$, where $\Delta_\mathrm{min}$ denotes the finest mesh size. (b) Time evolution of normalized remaining solid volume $V(t)/V_0$ for varying spatial resolution $D_0/\Delta_\mathrm{min}$. (c) Solid-liquid interfaces on the mid-plane at $z=0$ at $t=60$ for varying spatial resolution $D_0/\Delta_\mathrm{min}$. The legend applies to both panels.
• Figure 4: Melting dynamics at $\mathit{Re}_0=180$. (a) Snapshots of the temperature field $\theta$ and streamlines on the mid-plane at $z=0$, together with the 3D melting interface coloured by the local melting rate $v_\mathit{\Gamma}$, shown at $t/t_f = 0.010,0.412,0.722$ (top to bottom). The corresponding effective Reynolds numbers $\mathit{Re}_e(t)$ are indicated in each panel, and the same notation is used in the subsequent figures illustrating the melting process.(b) Angular distribution of $v_\mathit{\Gamma}$ along the interface as a function of the polar angle $\varphi$, from $t/t_f=0.052$ to $0.845$ in increments of $\Delta t/t_f=0.072$. The inset at the lower left shows the polar angle $\varphi$; the origin is the body’s instantaneous mass centre $(x_c,0,0)$, and $\varphi$ is measured from the front stagnation point. The inset at the upper right shows the interface-averaged melting rate $\bar{v}_{\Gamma}(t)$ as a function of the normalised remaining volume $V(t)/V_0$, revealing a clear $-1/6$ scaling.
• Figure 5: Effect of the initial Reynolds number $\mathit{Re}_0$ on interface evolution in the axi-symmetric melting regime. (a–d) Time evolution of the interface on the mid-plane at $z=0$ for $\mathit{Re}_0=25, 100, 150,$ and $200$, coloured by the local melting rate $v_\mathit{\Gamma}$. The sequences span nearly the entire melting process, with frames shown at uniform time intervals up to $t/t_f \approx 0.9$. (e) Comparison of the interfaces on the mid-plane at $z=0$ when the remaining solid volume reaches $V(t)/V_0=0.1$, highlighting the progressive flattening of the rear surface as $\mathit{Re}_0$ increases.