Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-05T05:22:44.920Z Has data issue: false hasContentIssue false

On the three-dimensional stability of a solid-body rotation flow in a finite-length rotating pipe

Published online by Cambridge University Press:  18 May 2016

Shixiao Wang*
Affiliation:
Department of Mathematics, Auckland University, Auckland 1142, New Zealand
Zvi Rusak
Affiliation:
Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy,  NY 12180-3590, USA
Rui Gong
Affiliation:
Department of Mathematics, Auckland University, Auckland 1142, New Zealand
Feng Liu
Affiliation:
Department of Mechanical and Aerospace Engineering, University of California at Irvine, Irvine, CA 92697-3975, USA
*
Email address for correspondence: wang@math.auckland.ac.nz

Abstract

The three-dimensional, inviscid and viscous flow instability modes that appear on a solid-body rotation flow in a finite-length straight, circular pipe are analysed. This study is a direct extension of the Wang & Rusak (Phys. Fluids, vol. 8 (4), 1996a, pp. 1007–1016) analysis of axisymmetric instabilities on inviscid swirling flows in a pipe. The linear stability equations are the same as those derived by Kelvin (Phil. Mag., vol. 10, 1880, pp. 155–168). However, we study a general mode of perturbation that satisfies the inlet, outlet and wall conditions of a flow in a finite-length pipe with a fixed in time and in space vortex generator ahead of it. This mode is different from the classical normal mode of perturbations. The eigenvalue problem for the growth rate and the shape of the perturbations for any azimuthal wavenumber $m$ consists of a linear system of partial differential equations in terms of the axial and radial coordinates ($x,r$). The stability problem is solved numerically for all azimuthal wavenumbers $m$. The computed growth rates and the related shapes of the various perturbation modes that appear in sequence as a function of the base flow swirl ratio (${\it\omega}$) and pipe length ($L$) are presented. In the inviscid flow case, the $m=1$ modes are the first to become unstable as the swirl ratio is increased and dominate the perturbation’s growth in a certain range of swirl levels. The $m=1$ instability modes compete with the axisymmetric ($m=0$) instability modes as the swirl ratio is further increased. In the viscous flow case, the viscous damping effects reduce the modes’ growth rates. The neutral stability line is presented in a Reynolds number ($Re$) versus swirl ratio (${\it\omega}$) diagram and can be used to predict the first appearance of axisymmetric or spiral instabilities as a function of $Re$ and $L$. We use the Reynolds–Orr equation to analyse the various production terms of the perturbation’s kinetic energy and establish the elimination of the flow axial homogeneity at high swirl levels as the underlying physical mechanism that leads to flow exchange of stability and to the appearance of both spiral and axisymmetric instabilities. The viscous effects in the bulk have only a passive influence on the modes’ shapes and growth rates. These effects decrease with the increase of $Re$. We show that the inviscid flow stability results are the inviscid-limit stability results of high-$Re$ rotating flows.

Type
Papers
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ash, R. L. & Khorrami, M. R. 1995 Vortex stability. In Fluid Vortices (ed. Green, S. I.), chap. 8, pp. 317372. Kluwer.CrossRefGoogle Scholar
Bayomi, Z. N., Said, M. H., Sedrak, M. F. & Sye, A. N. 2011 An Investigation on vortex breakdown phenomena in a vertical cylindrical tube. JKAU: Engng Sci. 22, 121141.Google Scholar
Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593629.Google Scholar
Beran, P. S. 1994 The time-asymptotic behavior of vortex breakdown in tubes. Comput. Fluids 23, 913937.Google Scholar
Cary, A. W., Darmofal, D. L. & Powell, K. G.1997 Onset of the spiral mode of vortex breakdown. AIAA Paper 97-0439.CrossRefGoogle Scholar
Dennis, D. J. C., Seraudie, C. & Poole, R. J 2014 Controlling vortex breakdown in swirling pipe flows: experiments and simulations. Phys. Fluids 26 (5), 053602.Google Scholar
Di Prima, R. C. & Habetler, G. J. 1969 A completeness theorem for non-selfadjoint eigenvalue problems in hydrodynamic stability. Arch. Rat. Mech. Anal. 34 (3), 218227.CrossRefGoogle Scholar
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 13851400.CrossRefGoogle Scholar
Fletcher, C. A. J. 1991 Computational Techniques for Fluid Dynamics. Springer.Google Scholar
Gallaire, F. & Chomaz, J.-M. 2004 The role of boundary conditions in a simple model of incipient vortex breakdown. Phys. Fluids 16 (2), 274286.Google Scholar
Garg, A. K. & Leibovich, S. 1979 Spectral characteristics of vortex breakdown flow field. Phys. Fluids 22, 20532064.Google Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamics and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463476.Google Scholar
Kelvin, Lord 1880 Vibrations of a columnar vortex. Phil. Mag. 10, 155168.Google Scholar
Leclaire, B. & Sipp, D. 2010 A sensitivity study of vortex breakdown onset to upstream boundary conditions. J. Fluid Mech. 645, 81119.Google Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335356.Google Scholar
Leibovich, S. 1984 Vortex stability and breakdown: survey and extension. AIAA J. 22, 11921206.Google Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65 (4), 769779.Google Scholar
Lessen, M., Singh, M. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63 (4), 753763.Google Scholar
Lopez, J. M. 1994 On the bifurcation structure of axisymmetric vortex breakdown in a constricted pipe. Phys. Fluids 6, 36833693.Google Scholar
Mattner, T. W., Joubert, P. N. & Chong, M. S. 2002 Vortical flow. Part 1. Flow through a constant diameter pipe. J. Fluid Mech. 463, 259291.Google Scholar
Meliga, P. & Gallaire, F. 2011 Control of axisymmetric vortex breakdown in a constricted pipe: nonlinear steady states and weakly nonlinear asymptotic expansions. Phys. Fluids 23, 084102.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93, 148154.Google Scholar
Ruith, M. R., Chen, P., Meiburg, E. & Maxworthy, T. 2003 Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486, 331378.Google Scholar
Rusak, Z. & Judd, K. P. 2001 The stability of non-columnar swirling flows in diverging streamtubes. Phys. Fluids 13 (10), 28352844.Google Scholar
Rusak, Z. & Wang, S. 2014 Wall-separation and vortex-breakdown zones in a solid body-rotation flow in a rotating finite-length straight circular pipe. J. Fluid Mech. 759, 321359.CrossRefGoogle Scholar
Rusak, Z., Wang, S. & Whiting, C. H. 1998 Axisymmetric breakdown of a Q-vortex in a pipe. AIAA J. 36 (10), 18481853.Google Scholar
Rusak, Z., Wang, S., Xu, L. & Taylor, S. 2012 On the global nonlinear stability of near-critical swirling flows in a long finite-length pipe and the path to vortex breakdown. J. Fluid Mech. 712, 295326.Google Scholar
Sarpkaya, T. 1971 On stationary and traveling vortex breakdowns. J. Fluid Mech. 45, 545559.Google Scholar
Sarpkaya, T. 1974 Effect of adverse pressure-gradient on vortex breakdown. AIAA J. 12 (5), 602607.Google Scholar
Spall, R. E. 1996 Transition from spiral- to bubble-type vortex breakdown. Phys. Fluids 8 (5), 13301332.CrossRefGoogle Scholar
Spall, R. E. & Gatski, T. B. 1991 A computational study of the topology of vortex breakdown. Proc. R. Soc. Lond. A 435, 321337.Google Scholar
Synge, L. 1933 The stability of heterogeneous liquids. Trans. R. Soc. Can. 27, 118.Google Scholar
Tromp, J. C. & Beran, P. S. 1996 The role of non-unique axisymmetric solutions in 3-D vortex breakdown. Phys. Fluids 9 (4), 9921002.Google Scholar
Uchida, S. N. Y. & Ohsawa, M. 1985 Experiments on the axisymmetric vortex breakdown in a swirling air flow. Trans. Japan. Soc. Aeronaut. Space Sci. 27, 206216.Google Scholar
Umeh, C. O. U., Rusak, Z., Gutmark, E., Villalva, R. & Cha, D. J. 2010 Experimental and computational study of nonreacting vortex breakdown in a swirl-stabilized combustor. AIAA J. 48 (11), 25762585.Google Scholar
Vyazmina, E., Nichols, J. W., Chomaz, J. M. & Schmid, P. J. 2009 The bifurcation structure of viscous steady axisymmetric vortex breakdown with open lateral boundaries. Phys. Fluids 21 (7), 074107.Google Scholar
Wang, S. 2008 A novel method for analyzing the global stability of inviscid columnar swirling flow in a finite pipe. Phys. Fluids 20, 074101.Google Scholar
Wang, S. & Rusak, Z. 1996a On the stability of an axisymmetric rotating flow in a pipe. Phys. Fluids 8 (4), 10071016.Google Scholar
Wang, S. & Rusak, Z. 1996b On the stability of non-columnar swirling flows. Phys. Fluids 8 (4), 10071016.Google Scholar
Wang, S. & Rusak, Z. 1997a The dynamics of a swirling flow in a pipe and transition to axisymmetric vortex breakdown. J. Fluid Mech. 340, 177223.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 1997b The effect of slight viscosity on near-critical swirling flows. Phys. Fluids 9 (7), 19141927.CrossRefGoogle Scholar
Wang, S. & Rusak, Z. 2011 Energy transfer mechanism of the instability of an axisymmetric swirling flow in a finite-length pipe. J. Fluid Mech. 679, 505543.Google Scholar
Wang, S., Rusak, Z., Taylor, S. & Gong, R. 2013 On the active feedback control of a swirling flow in a finite-length pipe. J. Fluid Mech. 737, 280307.Google Scholar
Wu, J. Z., Ma, H. Y. & Zhou, M. D. 2006 Vorticity and Vortex Dynamics. Springer.Google Scholar