Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-03T20:23:32.073Z Has data issue: false hasContentIssue false
  • English
  • Français

Helicity effects on inviscid instability in Batchelor vortices

Published online by Cambridge University Press:  23 June 2020

Toshihiko Hiejima Open the ORCID record for Toshihiko Hiejima [Opens in a new window]
Toshihiko Hiejima*
Affiliation:
Department of Aerospace Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Nakaku, Sakai, Osaka599-8531, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigate the instability properties of Batchelor vortices with a large swirl number and a fixed axial velocity deficit. In particular, it elucidates the effect of the helicity profile on the instability of the vortices as swirling wakes. In a linear stability analysis, a negative helicity profile destabilised a vortex with a large swirl number; the name given to this instability is ‘helicity instability’. Note that helicity instability is qualified for the case of axial flow with wake. In contrast, a conventional Batchelor vortex was stable at swirl numbers above a value of circulation, which is determined by the axial velocity deficit. The instability was related to a parameter $D$ proportional to the square of the inverse azimuthal vorticity thickness. Decreasing this helicity-profile parameter increased the growth property of the vortex. Such unstable features (helicity effects) were also studied in direct numerical simulations of vortices subjected to small random disturbances at Mach numbers 2.5 and 5.0. The instability based on the vorticity thickness originally grew at the outer edge of the vortex, whereas the instability waves in a conventional Batchelor vortex originate inside the vortex core. The simulation results support the results of the linear stability analysis on the helicity profile when the parameter $D$ is small. Because of the helicity instability, the nonlinear developments yielded a large fluctuation field with many small scales and high radial spreading rates. Even at the Mach number of 5.0, negative helicity exerted a much greater destabilisation effect than a zero entropy gradient. Therefore, the investigated novel effect established a reasonably powerful instability in compressible fluids, which is favourable for supersonic mixing.

JFM classification

Vortex Flows: Vortex instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 897 , 25 August 2020 , A37
Copyright
© The Author(s), 2020. Published by 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

Abid, M. 2008 Nonlinear mode selection in a model of trailing line vortices. J. Fluid Mech. 605, 19–45.CrossRefGoogle Scholar
Alekseenko, S. V., Kuibin, P. A. & Okulov, V. L. 2007 Theory of Concentrated Vortices. An Introduction. Springer.Google Scholar
Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.CrossRefGoogle Scholar
Billant, P. & Gallaire, F. 2013 A unified criterion for the centrifugal instabilities of vortices and swirling jets. J. Fluid Mech. 734, 5–35.CrossRefGoogle Scholar
Bogdanoff, D. 1983 Compressibility effects in turbulent shear layers. AIAA J. 21, 926–927.CrossRefGoogle Scholar
Broadhurst, M. S. & Sherwin, S. J. 2008 Helical instability and breakdown of a Batchelor trailing vortex. In Prog. in Industrial Mathematics at ECMI 2006. Mathematics in Industry, pp. 191–195. Springer.Google Scholar
Cattafesta, L. N. & Settles, G. S.1992 Experiments on shock/vortex interaction. AIAA Paper 92–0315.CrossRefGoogle Scholar
Clemens, N. T. & Mungal, M. G. 1995 Large-scale structure and entrainment in the supersonic mixing layer. J. Fluid Mech. 284, 171–216.CrossRefGoogle Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.CrossRefGoogle Scholar
Delbende, I. & Rossi, M. 2005 Nonlinear evolution of a swirling jet instability. Phys. Fluids 17, 044103.CrossRefGoogle Scholar
Deng, X. & Zhang, H. 2000 Developing high-order weighted compact nonlinear schemes. J. Comput. Phys. 165, 22–44.CrossRefGoogle Scholar
Di Pierro, B. & Abid, M. 2010 Instabilities of variable-density swirling flows. Phys. Rev. E 82, 046312.Google ScholarPubMed
Di Pierro, B. & Abid, M. 2011 Energy spectra in a helical vortex breakdown. Phys. Fluids 23, 025104.CrossRefGoogle Scholar
Di Pierro, B. & Abid, M. 2012 Rayleigh–Taylor instability in variable density swirling flows. Eur. Phys. J. B 85, 69.Google Scholar
Dimotakis, P. E. 1991 Turbulent free shear layer mixing and combustion. In High Speed Flight Propulsion Systems (ed. Murthy, S. & Curran, E.), vol. 137, chap. 5, pp. 265–340. Progress in Astronautics and Aeronautics, AIAA.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Duck, P. W. & Foster, M. R. 1980 The inviscid stability of a trailing line vortex. Z. Angew. Math. Phys. 31 (4), 524–532.CrossRefGoogle Scholar
Duraisamy, K. & Lele, S. K. 2008 Evolution of isolated turbulent trailing vortices. Phys. Fluids 20 (3), 035102.CrossRefGoogle Scholar
Eckhoff, K. S. & Storesletten, L. 1978 A note on the stability of steady inviscid helical gas flow. J. Fluid Mech. 89, 401–411.CrossRefGoogle Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl numbers. J. Fluid Mech. 500, 239–262.CrossRefGoogle Scholar
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11 (4), 723–727.CrossRefGoogle Scholar
Govindaraju, S. P. & Saffman, P. G. 1971 Flow in a turbulent trailing vortex. Phys. Fluids 14 (10), 2074–2080.CrossRefGoogle Scholar
Green, S. I. 1995 Fluid Vortices. Kluwer.CrossRefGoogle Scholar
Heaton, C. J. 2007 Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex. J. Fluid Mech. 576, 325–348.CrossRefGoogle Scholar
Heaton, C. J., Nichols, J. W. & Schmid, P. J. 2009 Global linear stability of the non-parallel Batchelor vortex. J. Fluid Mech. 629, 139–160.CrossRefGoogle Scholar
Heaton, C. J. & Peake, N. 2006 Algebraic and exponential instability of inviscid swirling flow. J. Fluid Mech. 565, 279–318.CrossRefGoogle Scholar
Heaton, C. J. & Peake, N. 2007 Transient growth in vortices with axial flow. J. Fluid Mech. 587, 271–301.CrossRefGoogle Scholar
Hiejima, T. 2013 Linear stability analysis on supersonic streamwise vortices. Phys. Fluids 25, 114103.CrossRefGoogle Scholar
Hiejima, T. 2014 Spatial evolution of supersonic streamwise vortices. Phys. Fluids 26, 074102.CrossRefGoogle Scholar
Hiejima, T. 2015a A factor involved in efficient breakdown of supersonic streamwise vortices. Phys. Fluids 27, 034103.CrossRefGoogle Scholar
Hiejima, T. 2015b Stability of compressible streamwise vortices. Phys. Fluids 27, 074107.CrossRefGoogle Scholar
Hiejima, T. 2017 Streamwise vortex breakdown in supersonic flows. Phys. Fluids 29, 054102.CrossRefGoogle Scholar
Hiejima, T. 2018 Onset conditions for vortex breakdown in supersonic flows. J. Fluid Mech. 840, R1.CrossRefGoogle Scholar
Hiejima, T. 2019 Compressibility effects of supersonic Batchelor vortices. Phys. Rev. F 4, 093903.Google Scholar
Holm, D. D. & Kerr, R. M. 2007 Helicity in the formation of turbulence. Phys. Fluids 19 (2), 025101.CrossRefGoogle Scholar
Jacquin, L. & Pantano, C. 2002 On the persistence of trailing vortices. J. Fluid Mech. 471, 159–168.CrossRefGoogle Scholar
Jameson, A., Schmidt, W. & Turkel, E.1981 Numerical simulation of the Euler equations by finite volume method using Runge–Kutta time stepping schemes. AIAA Paper 81–1259.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Joly, L., Fontane, J. & Chassaing, P. 2005 The Rayleigh–Taylor instability of two-dimensional high-density vortices. J. Fluid Mech. 537, 415–431.CrossRefGoogle Scholar
Kalkhoran, I. M., Smart, M. K. & Wang, F. Y. 1998 Supersonic vortex breakdown during vortex/cylinder interaction. J. Fluid Mech. 369, 351–380.CrossRefGoogle Scholar
Khorrami, M. & Chang, C.1997 Linear and nonlinear evolution of disturbances in supersonic streamwise vortices. AFOSR–TR–97.CrossRefGoogle Scholar
Khorrami, M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 97–212.CrossRefGoogle Scholar
Leblanc, S. & Le Duc, A. 2005 The unstable spectrum of swirling gas flows. J. Fluid Mech. 537, 433–442.CrossRefGoogle Scholar
Leibovich, S. & Stewartson, K. 1983 A sufficient condition for the instability of columnar vortices. J. Fluid Mech. 126, 335–356.CrossRefGoogle Scholar
Lele, S. K. 1994 Compressibility effects on turbulence. Annu. Rev. Fluid Mech. 26, 211–254.CrossRefGoogle Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769–779.CrossRefGoogle Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763.CrossRefGoogle Scholar
Mao, X. & Sherwin, S. J. 2012 Transient growth associated with continuous spectra of the Batchelor vortex. J. Fluid Mech. 697, 35–59.CrossRefGoogle Scholar
Mayer, E. W. & Powell, K. G. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.CrossRefGoogle Scholar
Moffatt, H. K. 2017 Helicity-invariant even in a viscous fluid. Science 357 (6350), 448–449.CrossRefGoogle Scholar
Moffatt, H. K. & Tsinober, A. 1992 Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech. 24 (1), 281–312.CrossRefGoogle Scholar
Morkovin, M. V. 1992 Mach number effects on free and wall turbulent structures in light of instability flow interactions. In Studies in Turbulence (ed. Gatski, S. S. T. B. & Speziale, C. G.), pp. 269–284. Springer.CrossRefGoogle Scholar
Naughton, J. W., Cattafesta, L. N. & Settles, G. S. 1997 An experimental study of compressible turbulent mixing enhancement in swirling jets. J. Fluid Mech. 330, 271–305.CrossRefGoogle Scholar
Nonomura, T. & Fujii, K. 2013 Robust explicit formulation of weighted compact nonlinear scheme. Comput. Fluids 85 (suppl. C), 8–18.CrossRefGoogle Scholar
Olendraru, C., Sellier, A., Rossi, M. & Huerre, P. 1999 Inviscid instability of the Batchelor vortex: absolute-convective transition and spatial branches. Phys. Fluids 11, 1805–1820.CrossRefGoogle Scholar
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer experimental study. J. Fluid Mech. 197, 453–477.CrossRefGoogle Scholar
Parras, L. & Fernandez-Feria, R. 2007 Spatial stability and the onset of absolute instability of Batchelor’s vortex for high swirl numbers. J. Fluid Mech. 583, 27–43.CrossRefGoogle Scholar
Pouquet, A. & Mininni, P. D. 2010 The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics. Phil. Trans. R. Soc. Lond. 368 (1916), 1635–1662.CrossRefGoogle ScholarPubMed
Qadri, U. A., Mistry, D. & Juniper, M. P. 2013 Structural sensitivity of spiral vortex breakdown. J. Fluid Mech. 720, 558–581.CrossRefGoogle Scholar
Ragab, S. & Sreedhar, M. 1995 Numerical simulation of vortices with axial velocity deficits. Phys. Fluids 7, 549–558.CrossRefGoogle Scholar
Rayleigh, Lord 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 93 (648), 148–154.Google Scholar
Sahoo, G., Bonaccorso, F. & Biferale, L. 2015 Role of helicity for large-and small-scale turbulent fluctuations. Phys. Rev. E 92 (5), 051002.Google ScholarPubMed
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sipp, D., Fabre, D., Michelin, S. & Jacquin, L. 2005 Stability of a vortex with a heavy core. J. Fluid Mech. 526, 67–76.CrossRefGoogle Scholar
Stewartson, K. & Brown, S. N. 1985 Near-neutral centre-modes as inviscid perturbations to a trailing line vortex. J. Fluid Mech. 156, 387–399.CrossRefGoogle Scholar
Stott, J. A. K. & Duck, P. W. 1994 The stability of a trailing-line vortex in compressible flow. J. Fluid Mech. 269, 323–351.CrossRefGoogle Scholar
Uberoi, M. S. 1979 Mechanisms of decay of laminar and turbulent vortices. J. Fluid Mech. 90 (2), 241–255.CrossRefGoogle Scholar
Viola, F., Arratia, C. & Gallaire, F. 2016 Mode selection in trailing vortices: harmonic response of the non-parallel Batchelor vortex. J. Fluid Mech. 790, 523–552.CrossRefGoogle Scholar
Wada, Y. & Liou, M.-S.1994 A flux splitting scheme with high-resolution and robustness for discontinues. AIAA Paper 94–0083.CrossRefGoogle Scholar
Wang, F. Y. & Sforza, P. M. 1997 Near-field experiments on tip vortices at Mach 3.1. AIAA J. 35 (4), 750–753.CrossRefGoogle Scholar