Numerical simulation of the Von-Kármán-Sodium dynamo experiment
Résumé
We present hydrodynamic and magnetohydrodynamic (MHD) simulations of
liquid sodium flows in the Von-Karman-Sodium (VKS)
setup. 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.
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