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Starting vortices generated by an arbitrary solid body with any number of edges

Published online by Cambridge University Press:  14 May 2024

Edward M. Hinton Open the ORCID record for Edward M. Hinton [Opens in a new window] ,
A. Leonard ,
D.I. Pullin Open the ORCID record for D.I. Pullin [Opens in a new window]  and
John E. Sader Open the ORCID record for John E. Sader [Opens in a new window]
Edward M. Hinton
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
A. Leonard
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
D.I. Pullin
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
John E. Sader*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA Department of Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
*
†Email address for correspondence: jsader@caltech.edu
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Abstract

The starting vortex generated at the trailing edge of a flat plate, that is impulsively translated at fixed angle of attack, is a widely studied canonical problem. Recent work that examined the effect of plate rotation on this starting vortex found that two new and distinct vortex sheet types can arise. We generalise this work to study the starting vortex generated at any sharp and straight edge of an arbitrary body under a general time-dependent two-dimensional motion. The dimensionless velocity field of the attached flow near any sharp edge is assumed to take the form, $\hat {z}^{-1/2} f(T) + g(T) + o (1)$, where $\hat {z}$ is the dimensionless position referenced to the edge, $f(T)$ and $g(T)$ are functions of dimensionless time, $T$, associated with the local flow perpendicular and parallel to the edge, respectively. This enables starting vortices to be generally calculated and their types related by simply inspecting the forms of $f(T)$ and $g(T)$. We elucidate the physics underlying all three vortex types and show that these vortices are generated by pure translation of the sharp edge. Several case studies are explored, including the leading/trailing edge vortices of a flat plate which can simultaneously be of different type (relevant to low-speed aircraft), the vortex formed by translation of a semi-infinite flat plate and the trailing-edge vortex of Joukowski aerofoils. With the ability to calculate the vortices at all edges, the theory is used to develop a general formula for the lift force of a flat plate which can find application in practice.

JFM classification

Vortex Flows: Vortex dynamics Vortex Flows: Vortex shedding
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 987 , 25 May 2024 , A11
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Alexander, R.C. 1971 Family of similarity flows with vortex sheets. Phys. Fluids 14 (2), 231–239.CrossRefGoogle Scholar
Baddoo, P.J. & Ayton, L.J. 2018 Potential flow through a cascade of aerofoils: direct and inverse problems. Proc. R. Soc. Lond. A 474 (2217), 20180065.Google ScholarPubMed
Baddoo, P.J., Kurt, M., Ayton, L.J. & Moored, K.W. 2020 Exact solutions for ground effect. J. Fluid Mech. 891, R2.CrossRefGoogle Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor instability. In Proceedings of Symposia in Applied Mathematics, vol. 13, pp. 55–76. American Mathematical Society.CrossRefGoogle Scholar
Cortelezzi, L. & Leonard, A. 1993 Point vortex model of the unsteady separated flow past a semi-infinite plate with transverse motion. Fluid Dyn. Res. 11 (6), 263–295.CrossRefGoogle Scholar
Crighton, D.G. 1985 The Kutta condition in unsteady flow. Annu. Rev. Fluid Mech. 17, 411–445.CrossRefGoogle Scholar
Eldredge, J.D. 2007 Numerical simulation of the fluid dynamics of 2D rigid body motion with the vortex particle method. J. Comput. Phys. 221 (2), 626–648.CrossRefGoogle Scholar
Gharib, M., Rambod, E. & Shariff, K. 1998 A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140.CrossRefGoogle Scholar
Graham, J.M.R. 1983 The lift on an aerofoil in starting flow. J. Fluid Mech. 133, 413–425.CrossRefGoogle Scholar
Jones, A.R. & Babinsky, H. 2010 Unsteady lift generation on rotating wings at low Reynolds numbers. J. Aircraft 47 (3), 1013–1021.CrossRefGoogle Scholar
Jones, M.A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.CrossRefGoogle Scholar
Kaden, H. 1931 Aufwicklung einer unstabilen unstetigkeitsfläche. Ing.-Arch. 2 (2), 140–168.CrossRefGoogle Scholar
Krasny, R. & Nitsche, M. 2002 The onset of chaos in vortex sheet flow. J. Fluid Mech. 454, 47–69.CrossRefGoogle Scholar
Luchini, P. & Tognaccini, R. 2002 The start-up vortex issuing from a semi-infinite flat plate. J. Fluid Mech. 455, 175–193.CrossRefGoogle Scholar
Milne-Thomson, L.M. 1996 Theoretical Hydrodynamics. Courier Corporation.Google Scholar
Moore, D.W. 1975 The rolling up of a semi-infinite vortex sheet. Proc. R. Soc. Lond. A 345 (1642), 417–430.Google Scholar
Pitt Ford, C.W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280–313.CrossRefGoogle Scholar
Prandtl, L. 1921 Applications of modern hydrodynamics to aeronautics. NACA Tech. Rep. 116, pp. 157–215.Google Scholar
Pullin, D.I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88 (3), 401–430.CrossRefGoogle Scholar
Pullin, D.I. & Perry, A.E. 1980 Some flow visualization experiments on the starting vortex. J. Fluid Mech. 97 (2), 239–255.CrossRefGoogle Scholar
Pullin, D.I. & Sader, J.E. 2021 On the starting vortex generated by a translating and rotating flat plate. J. Fluid Mech. 906, A9.CrossRefGoogle Scholar
Pullin, D.I. & Wang, Z.J. 2004 Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid Mech. 509, 1–21.CrossRefGoogle Scholar
Rott, N. 1956 Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1 (1), 111–128.CrossRefGoogle Scholar
Sader, J.E., Hou, W., Hinton, E.M., Colonius, T. & Pullin, D.I. 2024 The starting vortices generated by bodies with sharp and straight edges in a viscous fluid. J. Fluid Mech. (submitted).Google Scholar
Saffman, P.G. 1995 Vortex Dynamics. Cambridge University Press.Google Scholar
Schmitz, F.W. 1941 Aerodynamics of the Model Airplane. Part 1. Airfoil Measurements. Redstone Scientific Information Center.Google Scholar
Tchieu, A.A. & Leonard, A. 2011 A discrete-vortex model for the arbitrary motion of a thin airfoil with fluidic control. J. Fluids Struct. 27 (5–6), 680–693.CrossRefGoogle Scholar
Wagner, H. 1925 Über die entstehung des dynamicshen auftriebes von tragflügeln. Z. Angew. Math. Mech. 5, 17–35.CrossRefGoogle Scholar
Winslow, J., Otsuka, H., Govindarajan, B. & Chopra, I. 2018 Basic understanding of airfoil characteristics at low Reynolds numbers ($10^4$–$10^5$). J. Aircraft 55 (3), 1050–1061.CrossRefGoogle Scholar
Xu, L. & Nitsche, M. 2015 Start-up vortex flow past an accelerated flat plate. Phys. Fluids 27 (3), 033602.CrossRefGoogle Scholar