Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder: A numerical and an analytical study

Abstract

This paper presents a combined numerical and analytical study of the impulsive axisymmetric spin-up from rest of an isothermal liquid metal in a closed cylinder. The motion of the liquid is caused by the action of a low-frequency, low-induction rotating magnetic field, whose magnetic Taylor number is in the range (0.01–0.9) Tacr3D with Tacr3D given by Grants and Gerbeth [“Linear three-dimensional instability of a magnetically driven rotating flow,” J. Fluid Mech. 463, 229 (2002)]. The computations were performed for cylindrical containers of aspect ratios (diameter/height) R equal to 0.5, 1, and 2. The numerical simulations are compared with the predictions of an analytical model, valid for small Ekman numbers E extending a former work by Ungarish [“The spin-up of liquid metal driven by a rotating magnetic field,” J. Fluid Mech. 347, 105 (1997)]. The first phase of the motion from rest is an initial adjustment: the inviscid fluid begins to rotate due to the externally forced azimuthal acceleration, and concomitantly a meridional velocity field is induced by the unbalanced centrifugal effects; this occurs during, approximately, the first revolution about the axis. Subsequently, the spin-up flow is dominated by an axially independent swirl in the core (which can be regarded as a geostrophic mode) and Bödewadt (Ekman) layers on the top-bottom boundaries. This is accompanied by inertial oscillations. Both the development of the swirl motion and the decay of the inertial mode occur on the spin-up time scale predicted by the asymptotic model. The investigation also points out the important role of the aspect-ratio parameter in the analysis of the magnetohydrodynamics-driven spin-up. The efficiency of the driving decays strongly for R>0.6 and becomes close to zero at R≈3.3. The numerical results confirm the analytical inference that the thickness of the sidewall layer for the angular velocity adjustment is E1∕4∕R, and when this parameter is larger than 0.3 the sidewall influence gains dominance over the Ekman-layer effects. The frequency of the inertial oscillations increases with R.

Measurements of transient flow fields driven by a discontinuously applied rotating magnetic field

“…We determined the duration of the initial adjustment phase, t ia , as a function of the applied magnetic field strength B 0 from our experimental data (see figure 1). Figure 2 shows a satisfying agreement between experiment and numerical calculations [7] in the subcritical Ta number range although in the case of the numerical analysis the initial adjustment time was derived from the time history of the volume-averaged meridional velocity U rz . A further increase of the magnetic field intensity reveals that tia follows a 1/B-dependency in the turbulent range.…”

“…The problem concerns a fluid being at rest or in the state of steady rotation, which experiences an increase of the angular velocity. Analytical and numerical studies concerning the spin-up of an RMF-driven flow have been published recently [6,7]. In case of an RMF spin-up from the rest the azimuthal Lorentz force initiates a rotating motion of an electrically conducting liquid.…”

“…The curve in figure 1, which has been obtained from an ensemble average of five miscellaneous reruns, reveals an oscillatory raise of the vertical velocity before a steady state is reached. Following the suggestions given by Nikrityuk et al [7] we can identify the so-called initial adjustment phase and the inertial phase, respectively. We determined the duration of the initial adjustment phase, t ia , as a function of the applied magnetic field strength B 0 from our experimental data (see figure 1).…”

“…In contrast to the case of a rotating container the secondary flow is an inherent feature of the steady state RMF flow. Nikrityuk et al [7] divided the RMF-driven spin-up into two phases: a so-called initial adjustment phase in which the secondary flow is firstly established in form of two toroidal vortices and a subsequent inertial phase showing distinct inertial oscillations until the flow reaches a steady state after the spin-up time. Figure 1 displays the evolution of the square value of the vertical velocity U z averaged along the vertical measuring line at U…”

An analogy of Taylor’s instability criterion in Couette and rotating-magnetic-field-driven flows

“…In this case, the analytical approximations 9,10 predict that, after a spin-up time interval, the RMF-driven liquid metal reaches a dominant swirling motion with angular speed XðrÞ about the axis z of the cylinder cavity (r is the radius, and the lids of the cavity are at z ¼ 0; H o ). Moreover, in typical circumstances of H 2 ½1; 6, say, there is a significant core of r-independent solid body rotation with X ¼ X ce (ce stands for central-effective), see Fig.…”