Chris Howland

In so-called Lagrangian melting problems, a solid immersed in a fluid medium is free to rotate and translate in tandem with its phase-change from solid to liquid. Such configurations may be classified as a fluid-solid interaction (FSI) problem coupled to phase-change. Our present work proposes a numerical method capable of simulating these Lagrangian melting problems and adopts a front-tracking immersed-boundary (IB) method. We use the moving least squares IB framework, a well-established method for simulating a diverse range of FSI problems and extend this framework to accommodate melting by additionally imposing the Stefan condition at the interface. In the spirit of canonical front-tracking methods, the immersed solid is represented by a discrete triangulated mesh which is separate from the Eulerian mesh in which the governing flow equations are solved. A known requirement for these methods is the need for comparable Eulerian and Lagrangian grid spacings to stabilise interpolation and spreading operations between the two grids. For a melting object, this requirement is inevitably violated unless interventional remeshing is introduced. Our work therefore presents a novel dynamic remeshing procedure to overcome this. The remeshing is based on a gradual coarsening of the triangulated Lagrangian mesh and amounts to a negligible computational burden per timestep owing to the incremental and local nature of its operations, making it a scalable approach. Moreover, the coarsening is coupled to a volume-conserving smoothing procedure detailed by Kuprat et al., ensuring a zero net volume change in the remeshing step to machine precision. This added feature makes our present method highly specialised to the study of melting problems, where precise measurements of the melting solid’s volume is often the primary predictive quantity of interest.

Turbulent shear flows driven by a combination of a pressure gradient and buoyancy forcing are investigated using direct numerical simulations. Specifically, we consider the set-up of a differentially heated vertical channel subject to a Poiseuille-like horizontal pressure gradient. We explore the response of the system to its three control parameters: the Grashof number \(Gr\), the Prandtl number \(Pr\), and the Reynolds number \(Re\)of the pressure-driven flow. From these input parameters, the relative strength of buoyancy driving to the pressure gradient can be quantified by the Richardson number \(Ri=Gr/Re^2\). We compare the response of the mixed vertical convection configuration to that of mixed Rayleigh–Bénard convection, and find a nearly identical behaviour, including an increase in wall friction at higher \(Gr\), and a drop in the heat flux relative to natural convection for \(Ri=O(1)\). This closely matched response is despite vastly different flow structures in the systems. No large-scale organisation is visible in visualisations of mixed vertical convection – an observation that is confirmed quantitatively by spectral analysis. This analysis, combined with a statistical description of the wall heat flux, highlights how moderate shear suppresses the growth of small-scale plumes and reduces the likelihood of extreme events in the local wall heat flux. Vice versa, starting from a pure shear flow, the addition of thermal driving enhances the drag due to the emission of thermal plumes.

We consider the process of convective dissolution in a homogeneous and isotropic porous medium. The flow is unstable due to the presence of a solute that induces a density difference responsible for driving the flow. The mixing dynamics is thus driven by a Rayleigh–Taylor instability at the pore scale. We investigate the flow at the scale of the pores using Hele-Shaw type experiment with bead packs, two-dimensional direct numerical simulations and physical models. Experiments and simulations have been specifically designed to mimic the same flow conditions, namely matching porosities, high Schmidt numbers and linear dependency of fluid density with solute concentration. In addition, the solid obstacles of the medium are impermeable to fluid and solute. We characterise the evolution of the flow via the mixing length, which quantifies the extension of the mixing region and grows linearly in time. The flow structure, analysed via the centreline mean wavelength, is observed to grow in agreement with theoretical predictions. Finally, we analyse the dissolution dynamics of the system, quantified through the mean scalar dissipation, and three mixing regimes are observed. Initially, the evolution is controlled by diffusion, which produces solute mixing across the initial horizontal interface. Then, when the interfacial diffusive layer is sufficiently thick, it becomes unstable, forming finger-like structures and driving the system into a convection-dominated phase. Finally, when the fingers have grown sufficiently to touch the horizontal boundaries of the domain, the mixing reduces dramatically due to the absence of fresh unmixed fluid. With the aid of simple physical models, we explain the physics of the results obtained numerically and experimentally. The solute evolution presents a self-similar behaviour, and it is controlled by different length scales in each stage of the mixing process, namely the length scale of diffusion, the pore size and the domain height.

Motivated by the ablation of vertical ice faces in salt water, we use three-dimensional direct numerical simulations to investigate the heat and salt fluxes in two-scalar vertical convection. For parameters relevant to ice-ocean interfaces in the convection-dominated regime, we observe that the salinity field drives the convection and that heat is essentially transported as a passive scalar. By varying the diffusivity ratio of heat and salt (i.e., the Lewis number \(Le\)), we identify how the different molecular diffusivities affect the scalar fluxes through the system. Away from the walls, we find that the heat transport is determined by a turbulent Prandtl number of \(Pr_t ≈1\) and that double-diffusive effects are practically negligible. However, the difference in molecular diffusivities plays an important role close to the boundaries. In the (unrealistic) case where salt diffused faster than heat, the ratio of salt-to-heat fluxes would scale as \(Le^{1/3}\), consistent with classical nested scalar boundary layers. However, in the realistic case of faster heat diffusion (relative to salt), we observe a transition towards a \(Le^{1/2}\) scaling of the ratio of the fluxes. This coincides with the thermal boundary layer width growing beyond the thickness of the viscous boundary layer. We find that this transition is not determined by a critical Lewis number, but rather by a critical Prandtl number \(Pr ≈10\), slightly below that for cold seawater where \(Pr=14\). We compare our results to similar studies of sheared and double-diffusive flow under ice shelves, and discuss the implications for fluxes in large-scale ice-ocean models. By coupling our results to ice-ocean interface thermodynamics, we describe how the flux ratio impacts the interfacial salinity, and hence the strength of solutal convection and the ablation rate.