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Centrifugal/elliptic instabilities in slowly varying channel flows

Published online by Cambridge University Press:  28 September 2020

Jitesh S. B. Gajjar Open the ORCID record for Jitesh S. B. Gajjar [Opens in a new window]  and
Philip Hall Open the ORCID record for Philip Hall [Opens in a new window]
Jitesh S. B. Gajjar
Affiliation:
Department of Mathematics, University of Manchester, ManchesterM13 9PL, UK
Philip Hall*
Affiliation:
School of Mathematics, Monash University, Clayton, Victoria 3800, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The instability of the flow in a two-dimensional meandering channel of slowly varying depth is considered. The flow is characterised by $\delta$ the typical slope of the channel walls and the modified Reynolds number $R_m$ which is the usual Reynolds number multiplied by $\delta$. The modified Reynolds number is shown to be the appropriate parameter controlling the instability of the flow to streamwise vortices periodic in the spanwise direction. In particular, channels periodic in the streamwise direction are considered and it is found that the most unstable mode can correspond to either a subharmonic or synchronous disturbance. The instability problem at finite $R_m$ is discussed first and then the inviscid and large wavenumber regimes are discussed in detail. The instability is shown to be a hybrid form of centrifugal instability having properties of both Görtler vortices and a parametric resonance usually referred to as an elliptic instability. The limiting case of small wall modulation amplitudes is investigated and the results suggest that at small amplitudes the subharmonic mode is always dominant.

JFM classification

Instability: Absolute/convective instability Instability: Parametric instability Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A27
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Banks, C. 2015 Instabilities in high Reynolds number flows. PhD Thesis, Imperial College.Google Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.CrossRefGoogle ScholarPubMed
Bergles, A. E. & Webb, R. L. 1985 A guide to the literature on convective heat transfer augmentation. In Advances in Heat Transfer 1985 (ed. Shenkman, S. M., O'Brien, J. E., Habib, I. S. & Kohler, J. A.), vol. 43, pp. 81–90. ASME Symposium, HTD.Google Scholar
Cho, J. K., Kim, M. U. & Shin, H. D. 1998 Linear stability of two-dimensional steady flow in wavy-walled channels. Fluid Dyn. Res. 23, 349–370.CrossRefGoogle Scholar
Choudhari, M., Hall, P. & Street, C. 1994 On spatial evolution of long wavelength Görtler vortices governed by a viscous-inviscid interaction. Part 1. The linear case. Q. J. Mech. Appl. Maths 47, 207–230.CrossRefGoogle Scholar
Dean, W. 1928 Fluid motion in a curved channel. Proc. R. Soc. Lond. A 121, 402–412.Google Scholar
Denier, J., Hall, P. & Seddougui, S. O. 1991 On the receptivity problem for Görtler vortices: vortex motions induced by wall roughness. Phil. Trans. R. Soc. Lond. A 334, 51–85.Google Scholar
Eagles, P. M. & Weissman, M. 1975 On the stability of slowly varying flow: the divergent channel. J. Fluid Mech. 69, 241–262.CrossRefGoogle Scholar
Floryan, J. M. 2002 Centrifugal instability of flow over a wavy wall. Phys. Fluids 14, 301–322.CrossRefGoogle Scholar
Floryan, J. M. 2015 Flow in a meandering channel. J. Fluid Mech. 770, 52–84.CrossRefGoogle Scholar
Floryan, J. M. & Saric, W. S. 1982 Stability of Görtler vortices in boundary layers. AIAA J. 20 (3), 316–324.CrossRefGoogle Scholar
Gaster, M. 1974 On the effects of boundary layer growth on flow stability. J. Fluid Mech. 66, 465–480.CrossRefGoogle Scholar
Görtler, H. 1941 Instibiltat laminarer Grenzchichten an Konkaven Wanden gegenber gewissen dredimensionalen Storungen. Z. Angew. Math. Mech. 21, 250–252.CrossRefGoogle Scholar
Gschwind, P., Regele, A. & Kottke, V. 1995 Sinusoidal wavy channels with Taylor–Görtler vortices. Exp. Therm. Fluid Sci. 11, 270–275.CrossRefGoogle Scholar
Hall, P. 1982 a Taylor–Görtler vortices in fully-developed or boundary layer flows. J. Fluid Mech. 124, 475–494.CrossRefGoogle Scholar
Hall, P. 1982 b On the nonlinear evolution of Görtler vortices in growing boundary layers. J. Inst. Maths Applics. 29, 173–196.CrossRefGoogle Scholar
Hall, P. 1983 The linear development of Görtler vortices in growing boundary layers. J. Fluid Mech. 130, 41–58.CrossRefGoogle Scholar
Hall, P. 1990 Görtler vortices in growing boundary layers: the leading edge receptivity problem, linear growth and the nonlinear breakdown stage. Mathematika 37, 151–189.CrossRefGoogle Scholar
Hall, P. 2013 On the breakdown of Rayleigh's criterion for curved shear flows: a destabilization mechanism for a class of inviscidly stable flows. J. Fluid Mech. 734, 36–82.CrossRefGoogle Scholar
Hall, P. 2020 An instability mechanism for channel flows in the presence of wall roughness. J. Fluid Mech. (submitted).CrossRefGoogle Scholar
Hall, P. & Horseman, N. 1991 The linear inviscid secondary instability of longitudinal vortex structures in boundary layers. J. Fluid Mech. 232, 357–375.CrossRefGoogle Scholar
Hall, P. & Lakin, W. D. 1988 The fully nonlinear development of Görtler vortices in growing boundary layers. Proc. R. Soc. Lond. A 415, 421–444.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34, 83–113.CrossRefGoogle Scholar
Kowaleski, T. A., Szumbarski, J. & Blonski, S. 2008 Low Reynolds number instability of the laminar flow between wavy walls. In Proceedings of the Sixth International ASME Conference on Nanochannels, Microchannels and Minichannels ICNMM2008. ASME.CrossRefGoogle Scholar
Li, F. & Malik, M. 1995 Fundamental and subharmonic secondary instabilities of Görtler vortices. J. Fluid Mech. 297, 77–100.CrossRefGoogle Scholar
Ligrani, P. M., Oliveira, M. M. & Blaskovich, T. 2003 Comparison of heat transfer augmentation techniques. AIAA J. 41 (3), 337–362.CrossRefGoogle Scholar
Loh, S. A. & Blackburn, H. M. 2011 Stability of steady flow through an axially corrugated pipe. Phys. Fluids 23, 111703.CrossRefGoogle Scholar
Mclachlan, N. W. 1947 Theory and Application of Mathieu Functions. Oxford University Press.Google Scholar
Nishimura, K., Yoshino, T. & Kawamura, Y. 1987 In stability of flow in a sinusoidal wavy channel with narrow spacing. J. Chem. Engng Japan 20, 102–104.CrossRefGoogle Scholar
Nishimura, T., Murakami, S., Arakawa, S. & Kawamura, Y. 1990 a Flow observations and mass transfer characteristics in symmetrically wavy-walled channels at moderate Reynolds numbers for steady flow. Intl J. Heat Mass Transfer 33, 835–845.Google Scholar
Nishimura, T., Yano, K., Yoshino, T. & Kawamura, Y. 1990 b Occurrence and structure of Taylor–Görtler vortices induced in two-dimensional wavy channels for steady flow. J. Chem. Engng Japan 23, 697–703.CrossRefGoogle Scholar
Park, D. S. & Huerre, P. 1995 Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate. J. Fluid Mech. 283, 249–272.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.CrossRefGoogle Scholar
Smith, A. M. O. 1955 On the growth of Taylor–Görtler vortices along highly concave walls. Quart. J. Math. 13, 213–262.Google Scholar
Smith, F. T. 1979 On the nonparallel flow stability of the Blasius boundary layer. Proc. R. Soc. Lond. A 366, 91–109.Google Scholar
Swearingen, J. D. & Blackwelder, R. F. 1987 The growth and breakdown of streamwise vortices in the presence of a wall. J. Fluid Mech. 182, 255–290.CrossRefGoogle Scholar
Taylor, G. 1923 Stability of viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223, 289–293.Google Scholar
Wu, X., Zhao, D. & Luo, J. 2011 Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances. J. Fluid Mech. 682, 66–100.CrossRefGoogle Scholar