Numerical simulation of the von Kármán sodium dynamo experiment

Abstract

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flows in the von Kármán sodium (VKS) set-up. The counter-rotating impellers made of soft iron that were used in the successful 2006 experiment are represented by means of a pseudo-penalty method. Hydrodynamic simulations are performed at high kinetic Reynolds numbers using a large eddy simulation technique. The results compare well with the experimental data: the flow is laminar and steady or slightly fluctuating at small angular frequencies; small scales fill the bulk and a Kolmogorov-like spectrum is obtained at large angular frequencies. Near the tips of the blades the flow is expelled and takes the form of intense helical vortices. The equatorial shear layer acquires a wavy shape due to three coherent co-rotating radial vortices as observed in hydrodynamic experiments. MHD computations are performed: at fixed kinetic Reynolds number, increasing the magnetic permeability of the impellers reduces the critical magnetic Reynolds number for dynamo action; at fixed magnetic permeability, increasing the kinetic Reynolds number also decreases the dynamo threshold. Our results support the conjecture that the critical magnetic Reynolds number tends to a constant as the kinetic Reynolds number tends to infinity. The resulting dynamo is a mostly axisymmetric axial dipole with an azimuthal component concentrated near the impellers as observed in the VKS experiment. A speculative mechanism for dynamo action in the VKS experiment is proposed.

Effect of fluctuations on mean-field dynamos

“…2006) and, as already mentioned, the geometry of the magnetic field generated by the mean flow alone strongly differs from the one that takes into account turbulent fluctuations (Nore et al. 2017). In conclusion, the laminar threshold cannot be used to predict the dynamo characteristics in the presence of turbulence.…”

“…Direct numerical simulations taking into account the boundary conditions related to iron impellers have been recently performed (Nore et al. 2016, 2017). It has been checked that increasing the magnetic permeability of the impellers (up to ) decreases the dynamo threshold.…”

“…In addition, it has been checked (Nore et al. 2017) that the magnetic field generated by the time-averaged flow alone is an equatorial dipole and therefore strongly differs from the one which is observed in direct simulations. This confirms that the VKS dynamo strongly depends on turbulent fluctuations, as already emphasized (Petrelis et al.…”

“…Much less data are available for but it is observed that the threshold of the axial dipole is decreased when is increased from to whereas the threshold of the equatorial dipole is increased (Nore et al. 2017). This can confirm that turbulent fluctuations provide an efficient dynamo mechanism for the axial dipole observed in the VKS experiment.…”

Turbulence in realistic geometries with moving boundaries: When simulations meet experiments

“…and c e = 1 in Cappanera et al [8] for flows in precessing cylinders and in Wang et al [42] for turbulent flows in a flexible pipe (notice that the parameter α, defined therein in equation (2.13), is equal to c e /2). We have used this method in Nore et al [28] to perform high Reynolds number computations in a magnetohydrodynamics version of the von Kármán experiment.…”