We introduce a description of passive scalar transport based on a (deterministic and hyperbolic) Liouville master equation. Defining a noise term based on time-independent random coefficients, instead of time-dependent stochastic processes, we circumvent the use of stochastic calculus to capture the one-point space–time statistics of solute particles in Lagrangian form deterministically. To find the proper noise term, we solve a closure problem for the first two moments locally in a streamline coordinate system, such that averaging the Liouville equation over the coefficients leads to the Fokker–Planck equation of solute particle locations. This description can be used to trace solute plumes of arbitrary shape, for any Péclet number, and in arbitrarily defined grids, thanks to the time reversibility of hyperbolic systems. In addition to grid flexibility, this approach offers some computational advantages as compared with particle tracking algorithms and grid-based partial differential equation solvers, including reduced computational cost, no Monte-Carlo-type sampling and unconditional stability. We reproduce known analytical results for the case of simple shear flow and extend the description of mixing in a vortex model to consider diffusion radially and nonlinearities in the flow, which govern the long time decay of the maximum concentration. Finally, we validate our formulation by comparing it with Monte Carlo particle tracking simulations in a heterogeneous flow field at the Darcy (continuum) scale.

Dense granular flows exhibit both surface deformation and secondary flows due to the presence of normal stress differences. Yet, a complete mathematical modelling of these two features is still lacking. This paper focuses on a steady shallow dense flow down an inclined channel of arbitrary cross-section, for which asymptotic solutions are derived by using an expansion based on the flow’s spanwise shallowness combined with a second-order granular rheology. The leading-order flow is uniaxial with a constant inertial number fixed by the inclination angle. The streamwise velocity then corresponds to a lateral juxtaposition of Bagnold profiles scaled by the varying flow depth. The correction at first order introduces two counter-rotating vortices in the plane perpendicular to the main flow direction (with downwelling in the centre), and an upward curve of the free surface. These solutions are compared with discrete element method simulations, which they match quantitatively. This result is then used together with laboratory experiments to infer measurements of the second-normal stress difference in dense dry granular flow.

The stability of free jets is one of the fundamental problems that has driven the development of new theoretical and numerical methods in fluid mechanics. Extensive research has focused on the convective instabilities that characterise their elusive dynamics. However, in real-world configurations, free jets are often confined by solid walls which may exhibit different degrees of flexibility. The present paper presents, for the first time, evidence that even slightly flexible nozzles can lead to global instabilities. To show it, we adopted the classical tools of linear stability analysis, solving the fluid–structure interaction (FSI) problem by an arbitrary Lagrangian–Eulerian method, formulating a monolithic three-field problem. The investigation of the base flow properties reveals the effect of the Reynolds number, based on the bulk velocity and channel height, in the range $[50,200]$ and of the plate stiffness on the nozzle deformation and on the jet flow development. Exploiting an idea first proposed by Luchini and Charru, we develop an ad hoc quasi-one-dimensional model capable of predicting the displacement of elastic boundaries even for large displacements. The stability and sensitivity analysis shows that the interaction of the flow with the flexible structure leads to two categories of globally unstable modes: sinuous (in-phase) modes and varicose (out-of-phase) modes. All the results presented have been cross-checked with direct numerical simulations of the nonlinear FSI system, revealing that the instabilities correspond to supercritical bifurcations. This work has significant implications for many natural and industrial phenomena where a jet is produced by a compliant nozzle.

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.

In this paper, we perform a Floquet-based linear stability analysis of the centrifugal parametric resonance phenomenon in a Taylor–Couette system subjected to a time-quasiperiodic forcing where both the inner and outer cylinders are oscillating with the same amplitude and different angular velocities given respectively by $\varOmega _0 \cos (\omega _1t)$ and $\varOmega _0 \cos (\omega _2t)$. In this context, the frequencies $\omega _1$ and $\omega _2$ are incommensurate, where the ratio $\omega _2/\omega _1$ is irrational. Taking into account non-axisymmetric disturbances, a new set of partial differential equations is derived and solved using the spectral method along with the Runge–Kutta numerical scheme. The obtained results in this framework show that this forcing triggers new and numerous reversing and non-reversing Taylor vortex flows arising via either synchronous or period-doubling bifurcations. A rich and complex dynamics is found owing to strong mode competition between these modes that alters significantly the topology of the marginal stability curves. The latter exhibit a multitude of small and condensed parabolas, giving rise to several codimension-two bifurcation points, discontinuities and cusp points in the stability diagrams. Furthermore, a proper tuning of the frequency ratio leads to a significant control of both the instability threshold and the axisymmetric nature of the primary bifurcation. Moreover, using a local quasi-steady analysis when the cylinders are slowly oscillating, intermittent instabilities are detected, characterised by spike-like behaviour in the stability diagrams with several successive growths, dampings and periods of quietness. In this limit case, the inner cylinder drive becomes the responsible forcing of the Taylor vortices’ formation where the calculated critical instability parameters correspond to those of the inner oscillating cylinder case with fixed outer cylinder. The potentially unstable regions between the cylinders are determined on the basis of the Rayleigh discriminant, where an excellent agreement with the linear stability analysis results is pointed out.