- Access by Xinjiang University
Stability of a horizontal vortex in weakly stratified flow
Phys. Rev. Fluids 6, 104802 – Published 19 October, 2021
DOI: https://doi.org/10.1103/PhysRevFluids.6.104802
Abstract
A trailing vortex behind a wing moving through a stratified fluid will act to twist the density field into a pattern where the density profile continuously overturns along the axis of the vortex. This configuration is approximated here with a density field that is overturning periodically along the axis of the vortex. The matching velocity field is found approximately assuming weak stratification and constant axial vorticity at leading order and confining the flow to a fixed radius. This base flow is shown to be unstable to a wave triad consisting of two disturbance waves and a component of the base flow as the third “wave.” Three components of this base flow lead to instability: (1) the twirling component, (2) the streaming component, and (3) the rolling component. All three instabilities depend strongly on the axial length for the density field to overturn (the pitch). The twirling and rolling instabilities are important when the pitch is small and they also depend on the Froude number. The streaming instability is dominant when the pitch is large and is independent of the Froude number.
Physics Subject Headings (PhySH)
Article Text
References (37)
- J. F. Garten, S. Arendt, D. C. Fritts, and J. Werne, Dynamics of counter-rotating vortex pairs in stratified and sheared environments, J. Fluid Mech. 361, 189 (1998).
- F. M. Hill, A numerical study of the descent of a vortex pair in a stably stratified atmosphere, J. Fluid Mech. 71, 1 (1975).
- F. Holzapfel and T. Gerz, Two-dimensional wake vortex physics in the stably stratified atmosphere, Aerosp. Sci. Technol. 3, 261 (1999).
- K. K. Nomura, H. Tsutsui, D. Mahoney, and J. W. Rottman, Short-wavelength instability and decay of a vortex pair in a stratified fluid, J. Fluid Mech. 553, 283 (2006).
- R. E. Robins and D. P. Delisi, Numerical simulations of three-dimensional trailing vortex evolution, AIAA J. 35, 1552 (1997).
- P. R. Spalart, On the motion of laminar wing wakes in a stratified fluid, J. Fluid Mech. 327, 139 (1996).
- S. Shaw and J. P. McHugh, Evolution of a line vortex in stratified flow, Phys. Rev. Fluids 4, 064803 (2019).
- T. Sarpkaya, Trailing vortices in homogeneous and density-stratified media, J. Fluid Mech. 136, 85 (1983).
- S. C. Crow, Stability theory for a pair of trailing vortices, AIAA J. 8, 2172 (1970).
- D. W. Moore and P. G. Saffman, in Aircraft Wake Turbulence and Its Detection, edited by J. H. Olsen, A. Goldburg, and M. Rogers (Springer, Boston, 1971), pp. 339–354.
- D. W. Moore and P. G. Saffman, The motion of a vortex filament with axial flow, Philos. Trans. R. Soc. London 272, 403 (1972).
- S. E. Widnall, D. B. Bliss, and C. Y. Tsai, The instability of short waves on a vortex ring, J. Fluid Mech. 66, 35 (1974).
- D. W. Moore and P. G. Saffman, The instability of a straight vortex filament in a strain field, Proc. R. Soc. London 346, 413 (1975).
- C. Y. Tsai and S. E. Widnall, The stability of short waves on a straight vortex filament in a weak externally imposed strain field, J. Fluid Mech. 73, 721 (1976).
- R. R. Kerswell, Elliptic instability, Annu. Rev. Fluid Mech. 34, 83 (2002).
- M. Lessen, N. V. Deshpande, and B. Hadji-Ohanes, Stability of a potential vortex with a non-rotating and rigid-body rotating top-hat jet core, J. Fluid Mech. 60, 459 (1973).
- T. Loiseleux, J. M. Chomaz, and P. Huerre, The effect of swirl on jets and wakes: Linear instability of the rankine vortex with axial flow, Phys. Fluids 10, 1120 (1998).
- L. Lacaze, A. Birbaud, and S. Le Dizès, Elliptic instability in a rankine vortex with axial flow, Phys. Fluids 17, 017101 (2005).
- C. Eloy and S. Le Dizès, Three-dimensional instability of burgers and lamb-oseen vortices in a strain field, J. Fluid Mech. 378, 145 (1999).
- D. Sipp and L. Jacquin, Widnall instabilities in vortex pairs, Phys. Fluids 15, 1861 (2003).
- S. Le Dizès and F. Laporte, Theoretical predictions for the elliptical instability in a two-vortex flow, J. Fluid Mech. 471, 169 (2002).
- S. Ortiz, C. Donnadieu, and J. M. Chomaz, Three-dimensional instabilities and optimal perturbations of a counter-rotating vortex pair in stratified flows, Phys. Fluids 27, 106603 (2015).
- G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders, Philos. Trans. R. Soc. London Ser. A 223, 289 (1923).
- P. Billant, Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations, J. Fluid Mech. 660, 354 (2010).
- P. Billant and J. M. Chomaz, Experimental evidence for a new instability of a vertical columnar vortex pair in a strongly stratified fluid, J. Fluid Mech. 418, 167 (2000).
- H. P. Greenspan, The Theory of Rotating Fluids (Breukelen, New York, 1990).
- P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, 1992).
- W. Thomson, Vibrations of a columnar vortex, Philos. Mag. 10, 155 (1880).
- J. J. Kobine, Inertial wave dynamics in a rotating and precessing cylinder, J. Fluid Mech. 303, 233 (1995).
- R. Lagrange, P. Meunier, F. Nadal, and C. Eloy, Precessional instability of a fluid cylinder, J. Fluid Mech. 666, 104 (2011).
- R. Manasseh, Breakdown regimes of inertia waves in a precessing cylinder, J. Fluid Mech. 243, 261 (1992).
- A. D. McEwan, Inertial oscillations in a rotating fluid cylinder, J. Fluid Mech. 40, 603 (1970).
- A. Mashayek and W. R. Peltier, The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 1 Shear aligned convection, pairing, and braid instabilities, J. Fluid Mech. 708, 5 (2012).
- A. Mashayek and W. R. Peltier, The ‘zoo’ of secondary instabilities precursory to stratified shear flow transition. Part 2 The influence of stratification, J. Fluid Mech. 708, 45 (2012).
- A. Mashayek, C. P. Caulfield, and W. R. Peltier, Role of overturns in optimal mixing in stratified mixing layers, J. Fluid Mech. 826, 522 (2017).
- Y. Hattori and Y. Fukumoto, Modal stability analysis of a helical vortex tube with axial flow, J. Fluid Mech. 738, 222 (2014).
- O. M Phillips, On the dynamics of unsteady gravity waves of finite amplitude. Part I. The elementary interactions, J. Fluid Mech. 9, 193 (1960).
