Abstracts - KTH Mechanics

Prandtl-Batchelor channel flows pastplatesatnormal
incidence
F. Gallizio ∗ , L. Zannetti ∗ , A. Iollo †
The classical problem of 2-D steady inviscid flow past bluff bodies is addressed.
Prandtl-Batchelor flows are solution of the Navier-Stokes equations in the limit of
the Reynolds number going to inifinity. In particular we consider the problem of the
Prandtl-Batchelor 1 flow past wall-mounted plates in a channel. Inviscid separated
flows may present multiple solutions according to multiple allowable values of vorticity
inside the recirculating regions and of the Bernoulli constant jump on their boundaries.
The Prandtl-Batchelor solution is one of these solutions with the property of being
the limit of the viscous solution for the Reynolds number going to infinity. Briefly, the
solutions here considered have zero Bernoulli constant jump, constant vorticity and
are completely or partly embedded in an external potential flow. Their multiplicity
is related to different allowable values of area and vorticity.
Particular cases of Prandtl-Batchelor flows are the Finite Area Vortex Regions
(FAVR) 2 . If these vortex regions are stable, they don’t change the shape within the
flow-field in which they are introduced.
As an example, let consider the symmetric inviscid flow past a circular cylinder as
studied by Elcrat et al. 3 . By examining the half plane problem, the simplest solution
is offered by a standing point vortex. This solution is relevant to a vanishing area
vortex region. There is an infinity number of possible standing point vortices and
their locus is defined by the generalized Föppl curve 4 . Finite area solutions can be
obtained as accretions of point vortex solutions.
Thus, the point vortex solution is interesting as a seed of FAVRs solutions. Moreover,
there is the strong numerical suggestion that if there is not a point vortex
solution, there is not a FAVR solution either.
These considerations can be extended to the flow inside a parallel wall channel. It
can be shown that a flat plate orthogonal to the flow in a channel does not allow a
standing vortex. This conjecture may explain the difficulties in reproducing Turfus’
result 5 for the flow past a flat plate in a channel. Therefore several numerical and
analytic evidences seem not to be in accordance with the solutions bifurcation branch
hypothesis for the large Re wakes past an infinite row of flat plates 6 .
∗ DIASP, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
† MAB-INRIA, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence, France
1 Batchelor, J. Fluid Mech. 1, 177 (1956).
2 Deem and Zabusky, Phys. Rev. Lett. 40, 859 (1978).
3 Elcrat et al., J. Fluid Mech. 409, 13 (2000).
4 Zannetti, J. Fluid Mech. (submitted) (2005).
5 Turfus, J. Fluid Mech. 249, 59 (1993).
6 Turfus and Castro, Fluid Dynamics Research 249(3), 181 (2000).
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