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Nonlinear axisymmetric Taylor–Couette flow in a dilute gas: multiroll transition and the role of compressibility

Published online by Cambridge University Press:  08 December 2020

Pratik Aghor  and
Meheboob Alam Open the ORCID record for Meheboob Alam [Opens in a new window]
Pratik Aghor
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore560064, India
Meheboob Alam*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur P.O., Bangalore560064, India
*
†Email address for correspondence: [email protected]
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Abstract

The compressible Navier–Stokes–Fourier equations are numerically solved for axially bounded Taylor–Couette flow (TCF) of an ideal gas, with rotating inner cylinder and stationary outer cylinder, to understand the roles of compressibility and finite aspect ratio on the genesis of nonlinear Taylor vortices and the related bifurcation scenario. Restricting to the axisymmetric case in the wide-gap limit, the aspect ratio $\varGamma =h/\delta$ (where $\delta$ is the annular gap between two cylinders and $h$ is the height of the cylinders) is changed quasistatically by varying the height of the cylinders, following the experimental protocol of Benjamin (Proc. R. Soc. Lond. A, vol. 359, 1978b, pp. 27–43). The symmetric even-numbered ($2,4,6,\ldots$) vortices are found at $\varGamma \geq 2$, whereas the asymmetric 2-roll modes (called the single-roll or ‘anomalous’ modes), that break the midplane $Z_{2}$-symmetry, are uncovered at $\varGamma \leq O(1)$. The phase boundaries of both symmetric and asymmetric rolls and the coexisting regions of different number of rolls are identified in the ($\varGamma , Re$)-plane. These phase diagrams, along with related bifurcation diagrams and various patterns, at finite Mach numbers ($Ma$) are contrasted with their incompressible ($Ma\to 0$) analogues. It is shown that the ‘$1\leftrightarrow 2$’-roll transition in small aspect ratio cylinders is subcritical in compressible TCF in contrast to the supercritical nature of bifurcation in its incompressible counterpart. In general, the gas compressibility has a stabilizing effect on nonlinear Taylor vortices as the underlying phase diagrams in the ($\varGamma , Re$)-plane are shifted to larger values of $Re$ with increasing $Ma$. The stabilizing role of compressibility can be tied to the weakening of the outward jets, which is further aided by the strengthening of the Ekman vortices with increasing Mach number.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Nonlinear Dynamical Systems: Pattern formation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A24
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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