Gallery

Figures and films from my work on sharp-interface cut-cell methods: first, recent simulations of bodies moving through a fluid and of fronts that melt and freeze; then how the discretisation behaves on adaptive meshes, how interface conditions are imposed, and what the resulting accuracy looks like.

Moving bodies and phase change

Each body and each front below is a sharp boundary cutting through a Cartesian grid, moved with space-time cut cells. The films are drawn from the solver's own field dumps.

Melting and Rayleigh–Bénard convection. A liquid layer heated from below melts its way into the solid above. The solid is colder than its melting point and conducts heat in its own right, as in the configuration of Favier, Purseed & Duchemin (2019), so the front gains heat from the liquid and loses part of it to the solid. At first heat crosses the liquid by conduction alone and the flow is only a weak seed. Once the layer is deep enough, near an effective Rayleigh number Ra_e ≈ 3000, Rayleigh–Bénard rolls set in and take over. They carry heat unevenly, so the front scallops into arches, and the solid's isotherms, flat until then, take on its waviness. By t = 2, Ra_e has reached about 12 600. The colour is the liquid's temperature, the white lines are streamlines, and the grey-blue shading and thin lines are the solid's temperature and isotherms. The front is a sharp interface moved by the Stefan condition, both temperature fields are solved together, and the periodic box is drawn twice side by side.

Left: eighteen disks settling together. Eighteen disks, 1 % denser than the fluid, are released in a loose lattice in a closed box. Every disk is a moving cut boundary, and all of them are coupled to the flow at once. As they fall, each disk sheds a pair of shear layers (blue clockwise, red anticlockwise, with dashed contours where the vorticity is negative), and the wakes of one row run into the next. The grid has 12.8 points per diameter.

Middle: drafting, kissing, tumbling. Two disks, 1 % denser than the fluid, released one above the other in a channel. The trailing disk falls into the leading one's wake and catches up with it (drafting, then kissing). Falling in tandem is unstable, so the pair tumbles and the disks separate. The window follows them as they fall.

Right: a cold particle settling. A particle colder than the fluid falls down a long vertical channel. The cooled fluid trails behind it as a plume (the colour), and the black lines are contours of the wake's vorticity (dashed where it turns clockwise). At Grashof number 4500 the wake goes unsteady and the particle swings from side to side across the channel instead of keeping a fixed path, the regime reported by Yu et al. Heat and motion are solved together on the same cut-cell grid, at 16 points per diameter, and the window follows the particle.

To bounce or not to bounce. The same disk, eight times denser than the fluid, is dropped onto a wall in two fluids. Before it can touch, the fluid trapped underneath has to drain out of a thinning film. In the viscous fluid (Stokes number 3.8) the film absorbs all of the disk's momentum and the disk comes to rest on the wall. In the thin fluid (Stokes number about 150) it bounces back. Over a sweep of viscosities the restitution coefficient follows the trend seen in experiments (Gondret et al. 2002; Legendre et al. 2006): no rebound below St ≈ 10, then a rise towards the dry value.

Freezing water, and its density maximum at 4 °C. Water at 11 °C lies under a plate held at −10 °C, with an ice layer between them. Water is densest at 4 °C, so below that temperature colder water is lighter. The convection rolls therefore fill only the water warmer than 4 °C, below the dashed isotherm. Between them and the ice, a stably stratified layer from 0 to 4 °C (in blue) insulates the front, which still carries the imprint of the rolls. The ice conducts too: its temperature is solved together with the water's.

Cut-cell methods

Cut-cell Poisson with interface unknowns, on an adaptive quadtree. The solution of −Δu = 4 with a homogeneous condition on the immersed boundary Γ, over a mesh spanning levels 6 to 9 (25 015 leaves). Refinement concentrates where it is needed, along the interface. The zoom shows the cell-wise error against the exact solution: it stays below 4 × 10⁻⁵, and the cut cells are no worse than the bulk, which is the point of carrying unknowns on the interface itself.

Two-phase Poisson across a curved interface. Here both phases are solved at once, with a conductivity ratio κ₁/κ₂ = 2, coupled by scalar and flux jump conditions written directly on the interface rows of the system. The radial profile makes the coupling visible: the solution is continuous through Γ while its gradient kinks, exactly as continuity of κ ∂_nu requires. The measured L2 error is 4.7 × 10⁻⁶, at observed order 1.92.

Interface concentration around a rising bubble. A sharp two-fluid solver for mass transfer, where the concentration is discontinuous at the interface: the liquid and gas traces are related by Henry's law, c_γ1 = α c_γ2. Plotting both traces cell by cell around the bubble shows the condition is imposed to 3.2 × 10⁻⁶ relative, over all 67 interface cells, rather than smeared over a transition band.

Shape inversion by automatic differentiation. An inverse problem: recover the shape of an inclusion from sensor measurements around it. The gradient is obtained by differentiating straight through the whole solver, including the geometric volume-fraction computation and the multigrid solve, so the 16 shape parameters can be driven by a standard descent. Over 39 iterations the misfit falls from 2.5 × 10⁻⁵ to 1.9 × 10⁻¹¹, and the recovered mode amplitudes settle onto their true values.