Research

Motivated mainly by understanding the dynamics of planetary and satellite cores, my research activities focus on, but are not restricted to, fluid flows under the influence of magnetic fields and rapid rotation. To study these phenomena, I am using the mathematical tools of linear stability analysis as well a massively parallel unstructured finite-volume code, that is capable of handling arbitrarily complex three-dimensional geometries.

Rapidly rotating flows - Inertial modes
One of the most striking features of incompressible, rotating flows is their ability to sustain wave motion. The restoring force giving birth to these waves is the Coriolis force. Because this phenomenon only relies on inertia, eigenmodes of wave propagation in bounded geometries bear the name inertial modes; they can be excited and maintained by thermal and shear instabilities or orbital perturbations such as precession and libration. The Coriolis force being dominant in the force balance that governs the dynamics of numerous geo- and astrophysical processes, it is widely believed that inertial modes are fundamental concepts for our understanding of many of these bodies’ properties.

In a recent study, I have provided a solution for the inertial mode problem in a triaxial ellipsoid of arbitrary shape (Vantieghem 2014). More specifically, I have shown that the infinite-dimensional problem separates into an infinite number of finite-dimensional problems when expanding the uknown eigenmode with respect to certain bases of solenoidal cartesian polynomial vector fields. Furthermore, this approach also allows to prove rigorously some integral properties of inertial modes.

Rapidly rotating flows - Harmonic forcings
Because the orbit of a planetary satellite is elliptical, and its orbital plane is oblique with respect to its host's ecliptic, these bodies are subject to time-dependent gravitational torques, which at their turn induce harmonic variations in their rotation. The amplitude of these effects is small, and thus one would intuitively expect that these phenomena can only drive weak flows. However, these phenomena can drive inertial modes (see above), and under certain conditions, inertial mode resonances may occur.

In Vantieghem et al. (JFM 2015), we have studied two resonance mechanism that can be driven by libration in latitude. A direct resonance can take place if the libration frequency matches the spin-over frequency. Triadic resonances between the base flow and two free inertial modes on the other hand can lead to hydrodynamic instability and turbulent breakdown, as illustrated in the movie below (in .avi format).



Geodynamo modeling and simulation
The term geodynamo refers to the mechanism by which fluid planetary cores generate and sustain a magnetic field. This process being inherently dissipative, it is usually assumed that the energy to drive the dynamo is provided by thermal and compositional convection. However, as mentioned above, variations in the rotation of planets may induce hydrodynamic instabilities and subsequent turbulence in the liquid core of planets. We are currently investigating whether these flows can power a dynamo.
Investigating whether these flows are dynamo-capable and generate planetary-like magnetic fields requires that the assumption of spherical symmetry is abandoned. This however implies that conventional spectral methods (based on a development in spherical harmonics). I have developed/am developing a finite-volume numerical code that is capable of simulating of dynamos in arbitrary domains. This code was succesfully benchmarked against a solution obtained with a spherical harmonics-based codes (Jackson et al., 2014).
The animation below (in .avi format) shows a similar simulation (Vantieghem et al., in preparation) that exhibits relaxation oscillations, which are reminiscent of the solar cycle. Shown are the radial velocity component UR, the temperature field TEMP_NP1, and the theta component of the magnetic field BT.


Magnetohydrodynamics in the quasi-static limit
In most industrial applications of MHD, the coupling between flow and magnetic field is often one-way, i.e. the magnetic field acts on the flow through the effect of a Lorentz force, but the magnetic field induced by the flow remains negligible with respect to the externally imposed one. This allows a simplification of the governing equations. Nevertheless, the accurate computation of MHD flows in complex geometries (such as pipes, bends, ...) remains a challenging task, especially when the strength of the Lorentz force is large with respect to inertial and/or viscous effects. In Vantieghem et al. (2009), we reported the first numerical calculations of laminar MHD pipe flow for high magnetic field, thereby confirming previously established scaling laws, and elucidating new features of the velocity profile in the case of well-conducting walls.