On the instabilities of a potential vortex with a free surface

Abstract

In this paper, we address the linear stability analysis of a confined potential vortex with a free surface. This particular flow h as b een r ecently u sed b y Tophøj et al. (Phys. Rev. Lett., vol. 110(19), 2013, article 194502) as a model for the swirling flow o f fl uid in an op en cy lindrical co ntainer, dr iven by ro tating the bottom plate (the rotating bottom experiment) to explain the so-called rotating polygons instability (Vatistas J. Fluid Mech., vol. 217, 1990, pp. 241-248; Jansson et al., Phys. Rev. Lett., vol. 96, 2006, article 174502) in terms of surface wave interactions leading to resonance. Global linear stability results are complemented by a Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) analysis in the shallow-water limit as well as new experimental observations. It is found that global stability results predict additional resonances that cannot be captured by the simple wave coupling model presented in . Both the main resonances (thought to be at the root of the rotating polygons) and these secondary resonances are interpreted in terms of over-reflection p henomena b y t he W KBJ a nalysis. F inally, w e p rovide experimental evidence for a secondary resonance supporting the numerical and theoretical analysis presented. These different methods and observations allow to support the unstable wave coupling mechanism as the physical process at the origin of the polygonal patterns observed in free-surface rotating flows.

Axisymmetric rotating flow with free surface in a cylindrical tank

“…An inviscid model for the axisymmetric base state on which these patterns grow has been proposed by Bergmann et al (2011) and later on improved by Tophøj et al (2013) and Fabre & Mougel (2014). Despite their simplicity, these models were able to capture the occurrence of patterns in terms of wave resonances (see also Mougel et al 2017). In a recent asymptotic analysis of the base flow, Iga (2017) gives an in-depth characterisation of the internal and boundary layers: velocity profiles, scaling laws.…”

Nitrogen swirl: Creating rotating polygons in a boiling liquid

“…We believe that this instability leading to the polygons is caused by the resonance between "gravity waves" (living near the almost horizontal surface at the rim) and "centrifugal waves" (living near the strongly inclined surface at the center) as developed in Refs. [1,2]. Interestingly, it is the resonance between a forward-moving gravitational wave and a backward-moving centrifugal wave which creates the instability.…”

Shape dynamics of rotating polygons on the surface of liquid nitrogen in a Leidenfrost state

“…But they also mention "instabilities in partially filled tanks" [11][12][13], the subject of the present paper; and they pose as an open question, whether the polygons seen in those two very different systems have the same origin. In fact, we do not believe that they are much related: the polygonal hydraulic jumps are seen at intermediary Reynolds numbers -of the order of 10 2 or less -and are believed to be caused by surface tension [14,15], while "rotating polygons" seen on swirling flows have Reynolds numbers in the 10 5 -regime, and are believed to be caused by wave resonances [16,17], as we shall describe in the present paper.…”

“…Simultaneously, Jerome Mougel was working for his phd under the guidance of David Fabre in Toulouse to understand instabilities of free surface, both analytically and numerically -in particular the Newton's bucket flow [21]. When we told them that our base flow was probably a line vortex flow, they quickly showed numerically that such flows are actually unstable, and that the spectrum splits nicely into "families" living near the center and near the cylinder wall, respectively, as explained in detail in [17]. The theory in [16] was mostly focused on a simpler model, the "Tophøj" or "channel" model, replacing the analysis of the entire surface with narrow connected channels as shown in Figure 4 and thus utilizing the family-structure of the spectrum.…”

“…In Figure 9a we show an amplitude diagram for another run of the experiment, where the first clearly non-spherical shape m = 5 appeared at t ≈ 40 s. In Figure 9b we have plotted the normalized angular frequenciesω = ω/ g /R for different m, where the color shows the value of m as in Figure 9a. Here solid lines are the detectedω m frequencies whereas dotted and dashed lines are the computed gravitationalω g ,m and centrifugalω c,m frequencies respectively, calculated from (17) and (18), where the plus-sign is chosen in (17) and the minus sign in (18). The detected solid lines are bold when the corresponding amplitude is significant.…”