Nonlinear evolution of the centrifugal instability using a semi-linear model

Eunok Yim (1) , P. Billant (1) , F. Gallaire (1)

1 LadHyX - Laboratoire d'hydrodynamique

We study the nonlinear evolution of the axisymmetric centrifugal instability developing on a columnar anticyclone with a Gaussian angular velocity using a semi-linear approach. The model consists in two coupled equations: one for the linear evolution of the most unstable perturbation on the axially averaged mean flow and another for the evolution of the mean flow under the e↵ect of the axially averaged Reynolds stresses due to the perturbation. Such model is similar to the self-consistent model of Mantič-Lugo et al. (2014) except that the time averaging is replaced by a spatial averaging. The non-linear evolutions of the mean flow and the perturbations predicted by this semi-linear model are in very good agreement with DNS for the Rossby number Ro = 4 and both values of the Reynolds numbers investigated: Re = 800 and 2000 (based on the initial maximum angular velocity and radius of the vortex). An improved model taking into account the second harmonic perturbations is also considered. The results show that the angular momentum of the mean flow is homogenized towards a centrifugally stable profile via the action of the Reynolds stresses of the fluctuations. The final velocity profile predicted by Kloosterziel et al. (2007) in the inviscid limit is extended to finite high Reynolds numbers. It is in good agreement with the numerical simulations.

Mots clés: Centrifugal instability; vortex; semi-linear model; stability analysis

Domaines: Milieux fluides et réactifs

Eunok Yim, P. Billant, F. Gallaire. Nonlinear evolution of the centrifugal instability using a semi-linear model. Journal of Fluid Mechanics, 2020. ⟨insu-03025061⟩