Francky Luddens

This paper investigates the formation number of vortex rings generated by a piston-cylinder mechanism in a confined tube. We use Direct Numerical Simulations (DNS) of axisymmetric confined vortex rings to study the influence of different parameters on the separation (or pinch-off) of the vortex ring from the trailing jet. It is shown that the structure of the vortex ring at pinch-off depends on the type of injection program (pulse dominated by either positive or negative acceleration ramps) and the confinement ratio $D_w/D$, where $D_w$ is the inner diameter of the tube and D the diameter of the cylinder). For low confinement ratios $D_w/D \leq 2$, a vortex of opposite sign generated at the lateral wall strongly interacts with the vortex ring and the pinch-off is not clearly observed. The pinch-off is observed and analysed for confinement ratios $D_w /D \geq 2.5$. DNS data are used to estimate the value of the formation time, which is the time necessary for the vortex generator to inject the same amount of circulation as carried by the detached vortex ring. The confined vortex ring at pinch-off is described by the model suggested by Danaila, Kaplanski and Sazhin [A model for confined vortex rings with elliptical core vorticity distribution, Journal of Fluid Mechanics, 811:67-94, 2017]. This model allows us to take into account the influence of the lateral wall and the elliptical shape of the vortex core. The value of the formation time is predicted using this model and the slug-flow model.

We present an Adaptive Mesh Refinement (AMR) method suitable for hybrid unstructured meshes that allows for local refinement and de-refinement of the computational grid during the evolution of the flow. The adaptive implementation of the Discontinuous Galerkin (DG) method introduced in this work (ForestDG) is based on a topological representation of the computational mesh by a hierarchical structure consisting of oct- quad- and binary trees. Adaptive mesh refinement (h-refinement) enables us to increase the spatial resolution of the computational mesh in the vicinity of the points of interest such as interfaces, geometrical features, or flow discontinuities. The local increase in the expansion order (p-refinement) at areas of high strain rates or vorticity magnitude results in an increase of the order of accuracy in the region of shear layers and vortices.

A graph of unitarian-trees, representing hexahedral, prismatic and tetrahedral elements is used for the representation of the initial domain. The ancestral elements of the mesh can be split into self-similar elements allowing each tree to grow branches to an arbitrary level of refinement. The connectivity of the elements, their genealogy and their partitioning are described by linked lists of pointers. An explicit calculation of these relations, presented in this paper, facilitates the on-the-fly splitting, merging and repartitioning of the computational mesh by rearranging the links of each node of the tree with a minimal computational overhead. The modal basis used in the DG implementation facilitates the mapping of the fluxes across the non conformal faces.

The AMR methodology is presented and assessed using a series of inviscid and viscous test cases. Also, the AMR methodology is used for the modelling of the interaction between droplets and the carrier phase in a two-phase flow. This approach is applied to the analysis of a spray injected into a chamber of quiescent air, using the Eulerian-Lagrangian approach. This enables us to refine the computational mesh in the vicinity of the droplet parcels and accurately resolve the coupling between the two phases.

The present paper proposes and analyzes an interior penalty technique using $C^0$ finite elements to solve the Maxwell equations in domains with heterogeneous properties. The convergence analysis for the boundary value problem and the eigenvalue problem is done assuming only minimal regularity in Lipschitz domains. The method is shown to converge for any polynomial degrees and to be spectrally correct.

In the context of numerical simulations of multiphysics flows, accurate tracking of an interface and consistent computation of its geometric properties are crucial. In this paper, we investigate a level set technique that satisfies these requirements, even for long times computations. The method is developed in a finite differences framework on cartesian grids. As in usual level set strategies, reinitialization steps are involved. Several reinitialization algorithms are reviewed and mixed to design an accurate and fast reinitialization procedure. When coupled with a time evolution of the interface, the reinitialization procedure is performed only when there are too large deformations of the isocontours. This strategy limits the number of reinitialization steps and shows a good balance between accuracy and computational cost. Numerical results compare well with usual level set strategies and confirm the necessity of the reinitialization procedure, together with a limited number of reinitialization steps.

Convection in planetary cores can generate fluid flow and magnetic fields, and a number of sophisticated codes exist to simulate the dynamic behaviour of such systems. We report on the first community activity to compare numerical results of computer codes designed to calculate fluid flow within a whole sphere. The flows are incompressible and rapidly rotating and the forcing of the flow is either due to thermal convection or due to moving boundaries. All problems defined have solutions that alloweasy comparison, since they are either steady, slowly drifting or perfectly periodic. The first two benchmarks are defined based on uniform internal heating within the sphere under the Boussinesq approximation with boundary conditions that are uniform in temperature and stress-free for the flow. Benchmark 1 is purely hydrodynamic, and has a drifting solution. Benchmark 2 is a magnetohydrodynamic benchmark that can generate oscillatory, purely periodic, flows and magnetic fields. In contrast, Benchmark 3 is a hydrodynamic rotating bubble benchmark using no slip boundary conditions that has a stationary solution. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier–finite element code. There is good agreement between codes. It is found that in Benchmarks 1 and 2, the approximation of a whole sphere problem by a domain that is a spherical shell (a sphere possessing an inner core) does not represent an adequate approximation to the system, since the results differ from whole sphere results.

We present numerical solutions of the flow in precessing spheres and spherical shells with small ($ri/ro = 0.1$) inner cores and either stress-free or no-slip inner boundary conditions. For each of these three cases we consider the sequence of bifurcations as the Reynolds number $Re = r_0^2/\nu$ is increased up to $\sim 1280$, focusing particular attention on bifurcations that break the antipodal symmetry $\boldsymbol{U}(−\boldsymbol{r}) = −\boldsymbol{U}(\boldsymbol{r})$. All three cases have steady and time-periodic symmetric solutions at smaller $Re$, and quasiperiodic asymmetric solutions at larger $Re$, but the details of the transitions differ, and include also periodic asymmetric and quasiperiodic symmetric solutions in some of the cases.

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients.

The purpose of this note is to analyze the long term stability of the Navier–Stokes equations augmented with the Coriolis force and supplemented stress boundary conditions. It is shown that spurious stability behaviors can occur depending whether the Coriolis force is active or not when the flow domain is axisymmetric.

It is numerically demonstrated by means of a magnetohydrodynamics code that a short Taylor–Couette setup with a body force can sustain dynamo action. The magnetic threshold is comparable to what is usually obtained in spherical geometries. The linear dynamo is characterized by a rotating equatorial dipole. The nonlinear regime is characterized by fluctuating kinetic and magnetic energies and a tilted dipole whose axial component exhibits aperiodic reversals during the time evolution. These numerical evidences of dynamo action in a short Taylor–Couette setup may be useful for developing an experimental device.

Numerical simulations of the kinematic induction equation are performed on a model configuration of the Cadarache von Kármán Sodium dynamo experiment. The effect of a localized axisymmetric distribution of relative permeability $\mu_r$ that represents soft iron material within the conducting fluid flow is investigated. The critical magnetic Reynolds number $Rm^c$ for dynamo action of the first non-axisymmetric mode roughly scales like $Rm^c_{\mu_r}-Rm^c_{\infty} \propto\mu_r^{-1/2}$ i.e. the threshold decreases as μr increases. This scaling law suggests a skin effect mechanism in the soft iron discs. More important with regard to the Cadarache dynamo experiment, we observe a purely toroidal axisymmetric mode localized in the high-permeability discs which becomes dominant for large μr. In this limit, the toroidal mode is close to the onset of dynamo action with a (negative) growth rate that is rather independent of the magnetic Reynolds number. We qualitatively explain this effect by paramagnetic pumping at the fluid/disc interface and propose a simplified model that quantitatively reproduces numerical results. The crucial role of the high-permeability discs in the mode selection in the Cadarache dynamo experiment cannot be inferred from computations using idealized pseudovacuum boundary conditions ($\boldsymbol{H}\times \boldsymbol{n} = 0$).