Skip to main content Accessibility help

Hollow wakes past arbitrarily shaped obstacles

  • H. TELIB (a1) and L. ZANNETTI (a1)


An analytical solution is presented for steady inviscid separated flows modelled by hollow vortices, that is, by closed vortex sheets bounding a region with fluid at rest. Steady flows past arbitrary obstacles protruding from an infinite wall are considered. The solution is similar to that of the vortex patch model; it depends on two free parameters that define the size of the hollow vortex and the location of the separation point. When a sharp edge constrains the separation point (Kutta condition), the solution depends on a single parameter. As with the vortex patch model, families of growing vortices exist, which represent the continuation of desingularized point vortices. Numerical results are presented for the flows past a semicircular bump, a Ringleb snow cornice and a normal flat plate. The differences from the previous results found in the literature are analysed and discussed with the present solutions for the flow past a normal flat plate.


Corresponding author

Email address for correspondence:


Hide All
Batchelor, G. K. 1956 On steady laminar flows with closed streamlines at large Reynolds number. J. Fluid Mech. 1, 177190.
Birkoff, G. & Zarantonello, E. H. 1957 Jets, Wakes, and Cavities. Academic.
Chernyshenko, S. I. 1998 Asymptotic theory of global separation. Appl. Mech. Rev. 51, 523536.
Elcrat, A., Fornberg, B., Horn, M. & Miller, K. 2000 Some steady vortex flows past a circular cylinder. J. Fluid Mech. 409, 1327.
Gallizio, F. 2004 Modello di Prandtl–Batchelor per il flusso normale ad una placca piana posta all'interno di un canale: studio numerico dell'esistenza e unicità della soluzione. Dissertazione di Tesi Laurea, Ingegneria Aerospaziale, aa 2003/2004, Politecnico di Torino, Turin, Italy.
Gallizio, F., Iollo, A., Protas, B. & Zannetti, L. 2010 On continuation of inviscid vortex patches. Physica D 239, 190201.
Gilbarg, D. 1960 Jets and cavities. In Handbuch der Physik (ed. Truesdell, C.), vol. 9, pp. 311445. Springer.
Gurevich, M. I. 1965 Theory of Jets in Ideal Fluids. Academic.
Hicks, W. M. 1883 On the steady motion of a hollow vortex. Proc. R. Soc. Lond. 35, 304308.
Ives, D. C. 1976 A modern look at conformal mapping, including multiply connected regions. AIAA J. 14, 10061011.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
Lavrentiev, M. A. 1962 Variational Methods for Boundary Value Problems for Systems of Elliptic Functions. Noordhoff.
Levi-Civita, T. 1907 Scie e leggi di resistenza. Rend. Circ. Mat. Palermo 23, 137.
Lin, A. & Landweber, L. 1977 On a solution of the Lavrentiev wake model and its cascade. J. Fluid Mech. 79, 801823.
Nehari, Z. 1975 Conformal Mapping. Dover.
Pocklington, H. C. 1895 The configuration of a pair of equal and opposite hollow straight vortices, of finite cross-section, moving steadily through fluid. Proc. Camb. Phil. Soc. 8, 178187.
Ringleb, F. O. 1961 Separation control by trapped vortices. In Boundary Layer and Flow Control (ed. Lachman, G. V.), pp. 265294. Pergamon.
Smith, J. H. B. & Clark, R. W. 1986 Nonexistence of stationary vortices behind a two-dimensional normal plate. AIAA J. 13 (8), 11141115.
Tanveer, S. A. 1984 Topics in 2-D separated vortex flows. PhD thesis, California Institute of Technology.
Tanveer, S. A. 1986 A steadily translating pair of equal and opposite vortices with vortex sheets on their boundaries. Stud. Appl. Maths 74, 139154.
Tricomi, F. 1951 Funzioni Ellittiche. Zanichelli.
Turfus, C. 1993 Prandtl–Batchelor flow past a flat plate at normal incidence in a channel – inviscid analysis. J. Fluid Mech. 249, 5972.
Zannetti, L. 2006 Vortex equilibrium in the flow past bluff bodies. J. Fluid Mech. 562, 151171.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Hollow wakes past arbitrarily shaped obstacles

  • H. TELIB (a1) and L. ZANNETTI (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed