We present numerical results on dynamo action in a flow driven by an azimuthal body force localized near the end of an elongated cylindrical container. The analysis focuses on the central region of the cylinder, where axial variations in the flow are relatively weak, allowing the magnetic field to be represented as a helically traveling wave. Four magnetic impeller configurations and multiple forcing intensities are examined. In all cases, the velocity profiles in the central region display a similar \propto r^{-2} dependence across a wide range of Reynolds numbers and forcing region widths. The magnetic field is found to start growing under conditions similar to those of the Riga dynamo. However, the growing modes exhibit a substantial nonzero group velocity, indicating that the associated instability is convective: the flow can amplify an externally applied magnetic field but cannot sustain it autonomously. We outline several approaches for overcoming this limitation in order to realize a working laboratory dynamo based on an internally unconstrained swirling jet-type flow.
Screw-like motion of an electrically conducting fluid is arguably the simplest flow capable of sustaining a magnetic field through dynamo action. 1 The most prominent example is the Ponomarenko dynamo, 2 which can generate a magnetic field at relatively low flow velocities attainable in laboratory conditions. 3 This was first demonstrated in the celebrated Riga dynamo experiment 4,5 , where the flow of liquid sodium was kinematically constrained and guided to mimic the solid-body helical motion of the original Ponomarenko dynamo. An even more constrained configuration was used in the Karlsruhe dynamo experiment, which, however, relied on a different dynamo model, the Roberts-Busse dynamo. 6 In both cases, the flow had severely restricted freedom to respond to the growing electromagnetic force once the magnetic field reached significant strength. This nonlinear interaction between the flow and the magnetic field -responsible not only for magnetic field saturation but also for potentially complex temporal behavior -is arguably the most scientifically challenging aspect of fluid dynamos.
The aim of the present study is to numerically explore the feasibility of a laboratory screw-type dynamo driven by an impeller in a cylindrical vessel, whose only constraints are the external walls. Previous analysis based on Wentzel-Kramers-Brillouin (WKB) approximation suggests that a smooth swirling-jet flow, which is often referred to as the smooth Ponomarenko dynamo, 7 may be able to generate a magnetic field at significantly lower velocities than the solid-body Ponomarenko dynamo. 8 However, the accuracy of the underlying asymptotic solution is uncertain, and the analysis relies on highly idealized velocity profiles.
In the present study, we numerically solve the one-dimensional induction equation for velocity profiles that approximate a concentrated vortex driven by a small-diameter impeller in a finite cylinder. A striking and well-known example of such a flow is produced by a laboratory magnetic bar stirrer. 9 A similar vortex flow can also be generated by azimuthal electromagnetic body forces arising from a rotating permanent magnet placed coaxially near the cylinder’s end wall. 10 We use direct numerical simulations to compute several representative realizations of such flows, which then serve as input for the induction equation within the smooth Ponomarenko dynamo framework. The flow described above may enable a technically simple implementation of a liquidmetal laboratory dynamo using a commercially manufactured large sodium storage tank, which can contain up to 22 m 3 of liquid sodium. In the simplest realization, the flow could be driven in an unmodified tank by a rotating permanent magnet. Alternatively, the flow could be generated by a mechanically driven impeller, actuated either through a sealed shaft or by magnetic coupling.
The considered configuration shares strong similarities with the Riga dynamo. 4,5 The primary distinction is the absence of internal walls, which results in smooth radial velocity profiles. Eliminating these walls simplifies the experiment by removing the requirement for reliable electrical contact across them. It also relaxes kinematic constraints on how the flow can respond to the generated magnetic field. Consequently, the proposed configuration may be advantageous for studying strongly nonlinear regimes well above the dynamo threshold.
Another important difference concerns the direction of the axial flow relative to the agitator. In the Riga dynamo, a propeller pushes the liquid sodium axially. In contrast, in our setup, the impeller produces a centrifugal radial jet while simultaneously drawing axial flow toward itself. At the opposite end of the cylinder, a flow topology forms that resembles a tornado or, to a lesser extent, an accretion disk with polar jets. These distinctions, however, do not alter the fundamental generation mechanism of the magnetic field by the helical vortex core.
The flow in the von Kármán sodium (VKS) dynamo experiment 11 is driven by two relatively large-diameter impellers. The liquid-metal vessel has a diameter nearly equal to its height, and the impellers counter-rotate. The design philosophy of the VKS experiment emphasizes strong turbulence rather than screw-like coherent motion. As a result, both the flow structure and the overall dynamo concept differ substantially from the Ponomarenko-type dynamo considered here.
The paper is organized as follows. In the next section, we introduce the mathematical model and briefly describe the numerical method, which is based on the Chebyshev-tau approximation. Results of the direct numerical simulations of the impeller-driven flow are presented in Sec. III A, and these velocity fields are then used in Sec. III B to determine the threshold of dynamo action by numerically integrating the induction equation. The paper concludes with a discussion of the results in Sec. IV.
We consider a swirling flow of an inco
This content is AI-processed based on open access ArXiv data.