Competition between the centrifugal and strato-rotational instabilities in the stratified Taylor–Couette flow

Abstract

The stably stratified Taylor–Couette flow is investigated experimentally and numerically through linear stability analysis. In the experiments, the stability threshold and flow regimes have been mapped over the ranges of outer and inner Reynolds numbers: $-2000<Re_{o}<2000$ and $0<Re_{i}<3000$, for the radius ratio $r_{i}/r_{o}=0.9$ and the Brunt–Väisälä frequency $N\approx 3.2~\text{rad}~\text{s}^{-1}$. The corresponding Froude numbers $F_{o}$ and $F_{i}$ are always much smaller than unity. Depending on $Re_{o}$ (or equivalently on the angular velocity ratio $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D6FA}_{o}/\unicode[STIX]{x1D6FA}_{i}$), three different regimes have been identified above instability onset: a weakly non-axisymmetric mode with low azimuthal wavenumber $m=O(1)$ is observed for $Re_{o}<0$ ($\unicode[STIX]{x1D707}<0$), a highly non-axisymmetric mode with $m\sim 12$ occurs for $Re_{o}>840$ ($\unicode[STIX]{x1D707}>0.57$) while both modes are present simultaneously in the lower and upper parts of the flow for $0\leqslant Re_{o}\leqslant 840$ ($0\leqslant \unicode[STIX]{x1D707}\leqslant 0.57$). The destabilization of these primary modes and the transition to turbulence as $Re_{i}$ increases have been also studied. The linear stability analysis proves that the weakly non-axisymmetric mode is due to the centrifugal instability while the highly non-axisymmetric mode comes from the strato-rotational instability. These two instabilities can be clearly distinguished because of their distinct dominant azimuthal wavenumber and frequency, in agreement with the recent results of Park et al. (J. Fluid Mech., vol. 822, 2017, pp. 80–108). The stability threshold and the characteristics of the primary modes observed in the experiments are in very good agreement with the numerical predictions. Moreover, we show that the centrifugal and strato-rotational instabilities are observed simultaneously for $0\leqslant Re_{o}\leqslant 840$ in the lower and upper parts of the flow, respectively, because of the variations of the local Reynolds numbers along the vertical due to the salinity gradient.

Impact of centrifugal buoyancy on strato-rotational instability

“…The helical instability mode experimentally observed by Flór et al (2018) when the Froude number Fr < 1 in a stratified Taylor-Couette flow with a very wide gap annulus of unit aspect ratio is very different to the instabilities reported in previous experimental investigations of stratified Taylor-Couette flow (e.g. Boubnov et al 1995Boubnov et al , 1996Caton et al 2000;Le Bars & Le Gal 2007;Ibanez et al 2016;Rüdiger et al 2017;Seelig et al 2018;Park et al 2018). Those other experiments were mostly conducted in the narrow gap regime, with radius ratios in the range η ≈ 0.8-0.9, with a few considering η as small as 0.3.…”

“…So far, the theoretical studies of stratified Taylor-Coutte flow (e.g. Hua, Le Gentil & Orlandi 1997;Shalybkov & Rüdiger 2005;Gellert & Rüdiger 2009;Park & Billant 2013;Leclereq, Nguyen & Kerswell 2016;Rüdiger, Seelig, Schultz, Gellert, Egbers & Harlander 2017;Park, Billant, Baik & Seo 2018) have considered the axial direction to be periodic and the base state to be the unidirectional flow of Taylor (1923)…”

Horizontal shear instabilities in rotating stellar radiation zones

“…To understand the effect of thermal diffusion, we perform an asymptotic analysis at two limits: Pe → ∞ (i.e., no thermal diffusion) and Pe → 0 (i.e., high thermal diffusivity). It is noticeable that the diffusion is often neglected by taking Pe → ∞ to understand geophysical flows in the atmosphere and oceans of the Earth (Yavneh et al 2001;Park et al 2018), while the high thermal diffusivity case with Pe → 0 is studied in astrophysical context for shear instability and mixing in stellar interiors (Lignières et al 1999;Prat & Lignières 2014). In the following subsections, we provide explicit expressions of asymptotic dispersion relations for the inertial instability in stratified and rotating fluids in both limits of Pe.…”

Onset of centrifugal instability at a rotating cylinder in a stratified fluid

“…Stratified Taylor-Couette flows have also been investigated for outer and inner cylinders rotating. 15,16 The centrifugal instability occurred in the counter rotating case, and in the co-rotating case, the stratorotational instability was found. [17][18][19][20] When only the inner cylinder rotates, this latter instability cannot occur (see Ref.…”