Abstracts - KTH Mechanics

110
Modelling of optimal vortex ring formation using the Stokes
approximation
F. Kaplanski ∗ and Y. Rudi ∗
Previous studies show that vortex rings generated in the laboratory can be optimized
for efficiency, based on the jet length-to-diameter ratio (L/D), with peak
performance occurring 3.5 < L/D < 4.5 1 . The objective of the present research is to
develop predictive model of the optimal vortex ring formation. For this purpose we
considered an analytical solution describing the evolution of a vortex ring from thin
to thick-cored form in the Stokes approximation. The proposed model agrees with
the reported theoretical and experimental results referring to the formation and to
the decaying stages of ring development. The obtained class of rings can be classified
in terms of the parameter τ, representing the ratio of the ring radius R0 to the
time-dependent core radius ℓ. The present contribution extends analysis reported in 2 ,
which only considers the scale ℓ equal to √ 2νt, (ν is the kinematic viscosity). It is
found new similarity variables and functional forms using the invariance of the governing
equations, boundary conditions and vorticity impulse. The obtained circulation,
kinetic energy and translation velocity are compared with the results for Norbury
vortices and are used to give predictions for the normalized energy and circulation
describing the flow behaviour near the experimentally discovered critical value of L/D
(”formation number”). The predicted values match very well with the experimental
data. The experimental results 1 indicate that the vorticity ω extends to the symmetry
axis when L/D=4. This feature is employed for evaluating ”formation number”
parallel with the method of the entrainment diagrams 3 . The obtained results for both
approaches are shown in figure 1(a) and(b).
∗ Tallinn University of Technology, Akadeemia tee 23A Tallinn 12618, Estonia.
1 Gharib et. al, J. Fluid Mech. 360, 121 (1998).
2 Kaplanski and Rudi, Phys. Fluids 17, 087101 (2005).
3 Cantwell J. Fluid Mech. 104, 369 (1981).
S/R 0
1
0.75
0.5
0.25
(a)
2 4 6 8
L/D
Re 0
600
300
(b)
2 3 4 5
L/D
Figure 1: (a) Distance between the contour lines of vorticity 0.01 ωmax along a line
connecting two cores of a vortex ring versus L/D. (b) Boundaries for the flows
consisting of a wake andavortexringintheparameterspace(L/D and Re0).