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Weakly nonlinear versus semi-linear models of the nonlinear evolution of the centrifugal instability

Published online by Cambridge University Press:  20 February 2023

Eunok Yim Open the ORCID record for Eunok Yim [Opens in a new window] ,
Paul Billant Open the ORCID record for Paul Billant [Opens in a new window]  and
François Gallaire Open the ORCID record for François Gallaire [Opens in a new window]
Eunok Yim
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
Paul Billant*
Affiliation:
Laboratoire d'Hydrodynamique (LadHyX), CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France
François Gallaire
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We carry out a weakly nonlinear analysis of the centrifugal instability for a columnar vortex in a rotating fluid, and compare the results to those of the semi-linear model derived empirically by Yim et al. (J. Fluid Mech., vol. 897, 2020, A34). The asymptotic analysis assumes that the Reynolds number is close to the instability threshold so that the perturbation is only marginally unstable. This leads to two coupled equations that govern the evolutions of the amplitude of the perturbation and of the mean flow under the effect of the Reynolds stresses due to the perturbation. These equations differ from the Stuart–Landau amplitude equation or coupled amplitude equations involving a mean field that have been derived previously. In particular, the amplitude does not saturate to a constant as in the supercritical Stuart–Landau equation, but decays afterwards reflecting the instability disappearance when the mean flow tends toward a neutrally stable profile in the direct numerical simulations (DNS). These equations resemble those of the semi-linear model except that the perturbation in the weakly nonlinear model keeps at leading order the structure of the eigenmode of the unperturbed base flow. The predictions of the weakly nonlinear equations are compared to those of the semi-linear model and to DNS for the Rossby number $Ro=-4$ and various Reynolds numbers and wavenumbers. They are in good agreement with the DNS when the growth rate is sufficiently small. However, the agreement deteriorates and becomes only qualitative for parameters away from the marginal values, whereas the semi-linear model continues to be in better agreement with the DNS.

JFM classification

Instability: Instability Vortex Flows: Vortex Flows Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 957 , 25 February 2023 , A18
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

REFERENCES

Antkowiak, A. & Brancher, P. 2004 Transient energy growth for the Lamb–Oseen vortex. Phys. Fluids 16, L1–L4.CrossRefGoogle Scholar
Barthelet, P. & Charru, F. 1998 Benjamin–Feir and Eckhaus instabilities with Galilean invariance: the case of interfacial waves in viscous shear flows. Eur. J. Mech. B/Fluids 17 (1), 1–18.CrossRefGoogle Scholar
Batchelor, G.K. & Gill, A.E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14 (4), 529–551.CrossRefGoogle Scholar
Bayly, B.J. 1988 Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows. Phys. Fluids 31 (1), 56–64.CrossRefGoogle Scholar
Billant, P. & Gallaire, F. 2005 Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities. J. Fluid Mech. 542, 365–379.CrossRefGoogle Scholar
Carnevale, G.F., Kloosterziel, R.C., Orlandi, P. & Van Sommeren, D.D.J.A. 2011 Predicting the aftermath of vortex breakup in rotating flow. J. Fluid Mech. 669, 90–119.CrossRefGoogle Scholar
Charru, F. 2011 Hydrodynamic Instabilities. Cambridge University Press.CrossRefGoogle Scholar
Coullet, P. & Fauve, S. 1985 Propagative phase dynamics for systems with Galilean invariance. Phys. Rev. Lett. 55 (26), 2857.CrossRefGoogle ScholarPubMed
Davey, A. 1962 The growth of Taylor vortices in flow between rotating cylinders. J. Fluid Mech. 14 (3), 336–368.CrossRefGoogle Scholar
Di Prima, R.C. & Swinney, H.L. 1981 Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H.L. Swinney & J.P. Gollub), Topics in Applied Physics, vol. 45, pp. 139–180. Springer.CrossRefGoogle Scholar
Drazin, P.G. 1970 Kelvin–Helmholtz instability of finite amplitude. J. Fluid Mech. 42 (2), 321–335.CrossRefGoogle Scholar
Gallaire, F., Boujo, E., Mantic-Lugo, V., Arratia, C., Thiria, B. & Meliga, P. 2016 Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake. Fluid Dyn. Res. 48 (6), 061401.CrossRefGoogle Scholar
Gibbon, J.D. & McGuinness, M.J. 1981 Amplitude equations at the critical points of unstable dispersive physical systems. Proc. R. Soc. Lond. A 377 (1769), 185–219.Google Scholar
Hecht, F. 2012 New development in FreeFEM++. J. Numer. Maths 20 (3–4), 251–265.Google Scholar
Kloosterziel, R.C., Carnevale, G.F. & Orlandi, P. 2007 Inertial instability in rotating and stratified fluids: barotropic vortices. J. Fluid Mech. 583, 379–412.CrossRefGoogle Scholar
Manneville, P. 1990 Dissipative Structures and Weak Turbulence. Academic.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113, 084501.CrossRefGoogle ScholarPubMed
Meliga, P. 2017 Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description. J. Fluid Mech. 826, 503–521.CrossRefGoogle Scholar
Meliga, P., Chomaz, J.-M. & Sipp, D. 2009 Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. J. Fluid Mech. 633, 159–189.CrossRefGoogle Scholar
Orlandi, P. & Carnevale, G.F. 1999 Evolution of isolated vortices in a rotating fluid of finite depth. J. Fluid Mech. 381, 239–269.CrossRefGoogle Scholar
Pedlosky, J. 1970 Finite-amplitude baroclinic waves. J. Atmos. Sci. 27 (1), 15–30.2.0.CO;2>CrossRefGoogle Scholar
Renardy, M. & Renardy, Y. 1993 Derivation of amplitude equations and analysis of sideband instabilities in two-layer flows. Phys. Fluids A 5 (11), 2738–2762.CrossRefGoogle Scholar
Siggia, E.D. & Zippelius, A. 1981 Pattern selection in Rayleigh–Bénard convection near threshold. Phys. Rev. Lett. 47 (12), 835–838.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Smyth, W.D. & McWilliams, J.C. 1998 Instability of an axisymmetric vortex in a stably stratified, rotating environment. Theor. Comput. Fluid Dyn. 11 (3–4), 305–322.CrossRefGoogle Scholar
Stuart, J.T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4 (1), 1–21.CrossRefGoogle Scholar
Stuart, J.T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9 (3), 353–370.CrossRefGoogle Scholar
Watson, J. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. J. Fluid Mech. 9 (3), 371–389.CrossRefGoogle Scholar
Yim, E., Billant, P. & Gallaire, F. 2020 Nonlinear evolution of the centrifugal instability using a semilinear model. J. Fluid Mech. 897, A34.CrossRefGoogle Scholar