Accurately predicting the melting of encapsulated phase-change materials (PCMs) is essential for optimising thermal energy storage (TES) systems, especially when natural convection dominates at high-Rayleigh-number conditions. This study conducts a pore-scale study on the constrained melting of spherical PCM capsules, using a multiple-relaxation-time lattice Boltzmann method for the thermal flow, combined with an immersed boundary method for the solid–liquid interface. A novel ray-based phase identification scheme is introduced to resolve concave phase boundaries under strong convection, thereby improving the model accuracy in high-Rayleigh-number simulations. The model is validated against analytical, numerical and experimental benchmarks, showing superior capability and accuracy. For constrained PCM melting, the melting behaviour is reproduced, and effects of boundary temperature ($T_b$), initial subcooling ($\Delta T_s$) and capsule size ($l_z$) are examined with a fixed Prandtl number ($\textit{Pr}=59.76$). Higher $T_b$ accelerates melting, whereas $\Delta T_s$ has only minor effects. Reducing $l_z$ shortens the melting time due to the smaller PCM volume, but increases the dimensionless melting time by suppressing natural convection and shifting the melting process from convection- to conduction-dominated regimes. Accordingly, a critical capsule size $l_{z,c}$ is identified, below which conduction governs the melting process. A unified Rayleigh number of $Ra_c\approx 1.9\times 10^4$ is obtained for all $l_{z,c}$ under varying $T_b$, serving as a universal threshold between the two melting regimes. For predicting liquid fraction evolutions in both conduction- and convection-dominated regimes, two empirical correlations are proposed via dimensional analysis. These findings advance the understanding of constrained PCM melting and support TES system optimisation across diverse operating conditions.

Near-space hypersonic vehicles encounter significant rarefaction effects during the flight through the atmosphere, causing the classical Navier–Stokes–Fourier (NSF) equations to break down and posing challenges for the evaluation of surface drag and heat flux. In this paper, the nonlinear momentum and heat transfer in a hypersonic transitional boundary layer are analysed based on the generalized hydrodynamic equations (GHE), and the generality of the derived formulae is also discussed. The leading transport relations are obtained by estimating the relative orders of the various terms in GHE according to the hypersonic flow and boundary-layer requirements. Local non-equilibrium parameters characterising the shear non-equilibrium effect ($K_\sigma$) and thermal-gradient non-equilibrium effect ($K_q$) are introduced, and a set of correlation formulae for local surface pressure, shear stress and heat flux are proposed as corrections to continuum-based solutions. The correction function depends only on the non-equilibrium parameters $K_\sigma$ and $K_q$, and the continuous solutions can be either analytical formulae or NSF simulation results. This enables us to predict the surface aerothermodynamics with enhanced accuracy while still using the solutions of the NSF equations. The proposed formulae are carefully verified by comparing with direct simulation Monte Carlo (DSMC) results of different hypersonic rarefied flows, including flat-plate, sharp-wedge, cylinder and blunt-cone flows, and partial experimental data are also given. The results demonstrate that the proposed formulae can significantly enhance the accuracy of the continuum-based solutions, and show good agreement with DSMC simulations and experimental measurements in the near-continuum regime.

The evolution of the flow structure around an impulsively stopped sphere is investigated in an incompressible viscous fluid under a transverse magnetic field. The study focuses on the wake structure and drag force over the range of Reynolds numbers $60 \leqslant {\textit{Re}}_{\!D} \leqslant 300$ and $ {\textit{Re}}_{\!D}=1000$, with the interaction parameters $0 \leqslant N \leqslant 10$, where $N$ characterises the strength of the magnetic field. The wake is fully developed before the impulsive stop, after which it moves downstream and interacts with the sphere under the influence of a transverse magnetic field. The complex flow structures are characterised by skin friction lines on the downstream side of the sphere and categorised into five regimes in the $\{N, {\textit{Re}}_{\!D}\}$ phase diagram based on nearly 200 cases. The drag force generally decays over time following the impulsive stop. A drag decomposition model based on the vorticity diffusion scale is proposed, attributing the drag decay to three components: the original Stokes contribution, an inertia correction at high Reynolds numbers and a magnetohydrodynamic (MHD) correction, where the inertia and MHD effects both contribute a temporal power-law decay with an exponent of $-1/6$. Temporal scaling laws of the drag decay are derived by coupling these three different effects, considering flow structures at short and long time scales, as well as the dependence on ${\textit{Re}}_{\!D}$ and $N$. The prediction results are consistent with present simulations. Furthermore, the proposed drag decomposition model is successfully extended to complex vortex flow past a sphere at ${\textit{Re}}_{\!D}=1000$, to an anisotropic ellipsoidal particle and to different magnetic field orientations.

A combined experimental and direct numerical simulation (DNS) investigation is undertaken to study the laminar boundary-layer (BL) flow adjacent to a melting vertical ice face at two far-field water salinities ($S_\infty =0$ and 34 ‰) and a range of far-field temperatures ($T_\infty$). Wall-normal distributions of vertical velocity and temperature within the BL are measured by a modified molecular tagging velocimetry and thermometry technique. Experimental data match with DNS only when a nonlinear equation of state (EoS) for density is used rather than a linear EoS. For all $S_\infty =0$, i.e. freshwater cases, the flow remains uni-directional, although the flow reverses direction at $T_\infty =4^{\,\circ} \text{C}$. A bi-directional flow, however, exists for $S_\infty =$ 34 g kg−1, where an inner salinity-driven upward flow of fresher water is accompanied by a downward-flowing temperature-driven outer flow. Although the contribution of temperature to density relative to salinity is small $({\approx}1/40)$, the thermal BL region is larger owing to higher diffusivity. This results in increased total buoyancy force when the buoyancy is integrated across the BL, which combined with effects of wall shear stress on salinity BL and a freer thermal BL growth reveals that buoyancy from temperature contributes almost equally to the overall flow. Melt rates ($V$) also show differing features in uni- and bi-directional flows. The uni-directional flows exhibit the standard scaling of increasing velocity magnitude and BL thickness, and decreasing $V$ with distance along the flow direction. Such scalings are not followed in the bi-directional flows. These show a more uniform $V$ with height, which is attributed to the counteracting effects of an upward-growing salinity BL and a downward-growing temperature BL, combined with the necessity of maintaining salinity and temperature flux balance at the ice–water interface.

Numerical simulations of turbulent flows at realistic Reynolds numbers generally rely on filtering out small scales from the Navier–Stokes equations and modelling their impact through the subgrid-scale stress tensor ${\tau }_{\textit{ij}}$. Traditional models approximate ${\tau }_{\textit{ij}}$ solely as a function of the filtered velocity gradient, leading to deterministic subgrid-scale closures. However, small-scale fluctuations can locally exhibit instantaneous values whose deviation from the mean can have a significant influence on the flow dynamics. In this work, we investigate these effects by employing direct numerical simulations combined with Gaussian filtering to quantify subgrid-scale effects and evaluating the local energy flux in both space and time. The mean performance of the canonical Clark model is assessed by conditioning the energy flux distributions on the invariants of the filtered velocity gradient tensor, $Q$ and $R$. The Clark model captures to a good degree the mean energy flux. However, the fluctuations around these mean values for given ($Q,R$) are of the order of the mean, displaying fat-tailed distributions. To be more precise, we examine the joint distributions of true energy flux and the predictions from both the Clark and the Smagorinsky models. This approach mirrors the strategy adopted in early stochastic subgrid-scale models. Clear non-Gaussian characteristics emerge from the obtained distributions, particularly through the appearance of heavy tails. The mean, the variance, the skewness and the flatness of these distributions are quantified. Our results emphasise that fluctuations are an integral component of the small-scale feedback onto the large-scale dynamics and should be incorporated into subgrid-scale modelling through an appropriate stochastic framework.