Triadic resonances in nonlinear simulations of a fluid flow in a precessing cylinder

Abstract

We present results from three-dimensional nonlinear hydrodynamic simulations of a precession driven flow in cylindrical geometry. The simulations are motivated by a dynamo experiment currently under development at Helmholtz-Zentrum Dresden-Rossendorf in which the possibility of generating a magnetohydrodynamic dynamo will be investigated in a cylinder filled with liquid sodium and simultaneously rotating around two axes. In this study, we focus on the emergence of non-axisymmetric time-dependent flow structures in terms of inertial waves which-in cylindrical geometry-form so-called Kelvin modes. For a precession ratio (Poincaré number) Po 0 . 0 1 4 p c = W W = considered by us, the amplitude of the forced Kelvin mode reaches up to one fourth of the rotation velocity of the cylindrical container confirming that precession provides a rather efficient flow driving mechanism even at moderate values of Po. More relevant for dynamo action might be free Kelvin modes with higher azimuthal wave number. These free Kelvin modes are triggered by nonlinear interactions and may constitute a triadic resonance with the fundamental forced mode when the height of the container matches their axial wave lengths. Our simulations reveal triadic resonances at aspect ratios close to those predicted by the linear theory except around the primary resonance of the forced mode. In that regime we still identify various free Kelvin modes, however, all of them exhibit a retrograde drift around the symmetry axis of the cylinder and none of them can be assigned to a triadic resonance. The amplitudes of the free Kelvin modes always remain below the forced mode but may reach up to 6% of the of the container's angular velocity. The properties of the free Kelvin modes, namely their amplitude and their frequency, will be used in future simulations of the magnetic induction equation to investigate their ability to provide for dynamo action.

Rapidly rotating precessing cylinder flows: forced triadic resonances

“…The results of this general investigation of precessional forcing show that as viscous effects are reduced the forced response loses stability via Hopf bifurcations where the Hopf modes essentially consists of two Kelvin modes that are nearly resonant with the forced response, and this happens in all cases. We then re-analyse other results in the literature (Giesecke et al 2015;Albrecht et al 2016), and find that triadic resonance between the forced response and two free Kelvin modes is also the mechanism involved in those experimental observations.…”

“…We can illustrate this point by looking at two such studies. In Giesecke et al (2015), the forced flow is away from the K 111 Kelvin mode, and a range of aspect ratios was considered, with Γ ∈ [1.81, 2.24], whilst ω 0 was kept at a moderate value of 6500, so the resonant collapse was not observed. The maximum response was found at Γ = 1.871, away from the theoretical resonances between Kelvin modes.…”

Dynamos in precessing cubes

“…However, the energy taken from the Poincaré mode is too small to lead to experimentally manageable critical magnetic Reynolds numbers. The situation seems to be similar in cylinders, where the energy outside modes driven directly by precession does not exceed a few percent of the total energy [31]. The most promising route to a laboratory precession dynamo may therefore be the search for container shapes with a laminar mode which is already a dynamo.…”

Instability of precession driven Kelvin modes: Evidence of a detuning effect

“…These modes constitute the most unstable modes in the precessing system with aspect ratio Γ = R/H = 0.5, which is close to the exact (m = 1, n = 1, l = 1) resonance at Γ = 0.502559 and there are nearly no doubts about this interpretation. The behavior may be different for another aspect ratio, where other free Kelvin modes emerge [21], and where the forced mode is no longer resonant, which would result in a weaker amplitude and a different location in the stability diagram shown in Figure 16a.…”