Role of boundary conditions in helicoidal flow collimation: Consequences for the von Kármán sodium dynamo experiment

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flow with the PLUTO compressible MHD code to investigate influence of magnetic boundary conditions on the collimation of helicoidal motions. We use a simplified cartesian geometry to represent the flow dynamics in the vicinity of one cavity of a multiblades impeller inspired by those used in the Von-Kármán-sodium (VKS) experiment. We show that the impinging of the large-scale flow upon the impeller generates a coherent helicoidal vortex inside the blades, located at a distance from the upstream blade piloted by the incident angle of the flow. This vortex collimates any existing magnetic field lines leading to an enhancement of the radial magnetic field that is stronger for ferromagnetic than for conducting blades. The induced magnetic field modifies locally the velocity fluctuations, resulting in an enhanced helicity. This process possibly explains why dynamo action is more easily triggered in the VKS experiment when using soft iron impellers.

Figure 1

(a) Simulation domain for a portion of the flow in between two blades: $X,Y$, and $Z$ directions correspond to local azimuthal, radial, and vertical direction with $X\in [0,2],Y\in [0,4],Z\in [0,2]$. (b) Whirl location versus $\mathrm{\Gamma }$ value: $X$ coordinate (dotted line), $Z$ coordinate (dashed line), and module (solid line). (c) Magnetic field for simulations with different magnetic diffusion: ${\text{Re}}_m={10}^4$ (dash-dot line), ${\text{Re}}_m={10}^3$ (dotted line), ${\text{Re}}_m={10}^2$ (dashed line), and ${\text{Re}}_m=10$ (solid line)

Figure 2

Magnetic and velocity field stream lines at ${\text{Re}}_n=200$ and $\mathrm{\Gamma }=0.8$ in the perfect ferromagnetic (a) and conductor (b) simulations in the all domain. Isocontour of the magnetic field (0.005 T in the ferromagnetic and 0.001 T in the conductor cases). Magnetic field module in the $Y=2$ plane.

Figure 3

Angle between the current and the unitary surface vector in the Bvecn configuration (a) and the Bdotn configuration (b) in a zoomed box. The arrows indicate the orientation of the current on the blades and black lines are current stream lines nearby the upstream blade.

Figure 4

(a) Kinetic helicity. Ferromagnetic case (solid line), conductor case (dashed line), and hydro case (dotted line) (b) Current helicity. (c) Magnetic energy. (d) Total helicity. (e) Helicity of fluctuations. (f) Helicity of fluctuations in Bvecn case: kinetic helicity (solid line), current helicity (dashed line); and Bdotn case: kinetic helicity (dotted line), current helicity (dash-dot-dot line). All values are averaged in a volume localized around the whirl.

Figure 5

Angle between the velocity and the magnetic fields for (a) the Bvecn and (b) the Bdotn simulation (plane $Y=1)$ at $t=1$ s. Properties of the Poynting vector in the upstream blade: (c) Bvecn, (d) Bdotn cases. The arrows direction (resp. color) codes its orientation (respectively, its angle with the unitary vector normal to the surface). The color in the blade codes its modulus.