The stratorotational instability of Taylor-Couette flows with moderate Reynolds numbers

Abstract

In view of new experimental data the instability against adiabatic nonaxisymmetric perturbations of a Taylor-Couette flow with an axial density stratification is considered in dependence of the Reynolds number (Re) of rotation and the Brunt-Väisälä number (Rn) of the stratification. The flows at and beyond the Rayleigh limit become unstable between a lower and an upper Reynolds number (for fixed Rn). The rotation can thus be too slow or too fast for the stratorotational instability. The upper Reynolds number above which the instability decays, has its maximum value for the potential flow (driven by cylinders rotating according to the Rayleigh limit) and decreases strongly for flatter rotation profiles finally leaving only isolated islands of instability in the (Rn/Re) map. The maximal possible rotation ratio µ max only slightly exceeds the shear value of the quasi-uniform flow with U φ ≃ const.Along and between the lines of neutral stability the wave numbers of the instability patterns for all rotation laws beyond the Rayleigh limit are mainly determined by the Froude number Fr which is defined by the ratio between Re and Rn. The cells are highly prolate for Fr > 1 so that measurements for too high Reynolds numbers become difficult for axially bounded containers. The instability patterns migrate azimuthally slightly faster than the outer cylinder rotates.

Viscous and inviscid strato-rotational instability

“…A similar closed unstable domain phenomenon is seen in figure 6 of Rüdiger et al (2017). This figure, derived numerically, shows a closed unstable domain in the (Re, Rn)-parameter space for η = 0.52 and m = 1.…”

“…Here, we have also defined the vorticity Z = (1/r)∂ r (r 2 Ω) which is constant. For viscous flows, we define our Reynolds number Re to be consistent with the work of Shalybkov & Rüdiger (2005), Le Bars & Le Gal (2007), Ibanez et al (2016), Leclercq et al (2016) and Rüdiger et al (2017) Re = r in Ω in d ν .…”

Instability of vertically stratified horizontal plane Poiseuille flow

“…Our study follows recent investigations on instabilities of stratified rotating or non-rotating shear flows: the strato-rotational instability in Taylor–Couette flows (Le Bars & Le Gal 2007; Park & Billant 2013; Rüdiger et al. 2017) or plane Couette flows (Vanneste & Yavneh 2007; Facchini et al. 2018), and the radiative instability in boundary layer flows (Candelier, Le Dizès & Millet 2012; Chen, Bai & Le Dizès 2016), rotating flows (Riedinger, Le Dizès & Meunier 2011) or vortices (Le Dizès & Billant 2009; Riedinger, Le Dizès & Meunier 2010).…”

“…According to the studies of Basovich & Tsimring (1984) and Bakas & Farrell (2009a,b), horizontal shear flows can indeed excite internal gravity waves in the presence of a vertical stratification. Our study follows recent investigations on instabilities of stratified rotating or non-rotating shear flows: the strato-rotational instability in Taylor-Couette flows (Le Bars & Le Gal 2007;Park & Billant 2013;Rüdiger et al 2017) or plane Couette flows (Vanneste & Yavneh 2007;Facchini et al 2018), and the radiative instability in boundary layer flows (Candelier, Le Dizès & Millet 2012;Chen, Bai & Le Dizès 2016), rotating flows (Riedinger, Le Dizès & Meunier 2011) or vortices (Le Dizès & Billant 2009;Riedinger, Le Dizès & Meunier 2010). All of these examples belong to a class of instabilities caused by the resonant interaction of Doppler-shifted internal gravity waves (Molemaker, McWilliams & Yavneh 2001;.…”

Higher-order compact scheme for high-performance computing of stratified rotating flows

“…Only turbulence can achieve such a large angular momentum transport. Recent studies mentioned that, at fixed value of the Froude number, F r = i /N, with N = gα∂ T /∂ z the buoyancy frequency and i the angular velocity of the inner cylinder, the system is more stable when increasing the Reynolds number [30,31] . The final objective is to eventually explore the not yet understood turbulent flows, to get closer to actual astrophysical applications, by taking advantage of optimized HPC.…”