Vortex rings have been characterised in numerous ways, it is currently under- stood that the following three non-dimensional parameters provide a reasonable characterisation of a given vortex ring: non-dimensional formation time, non- dimensional core diameter and Reynolds number. In the case of vortex rings generated under background rotation (for which few studies have been performed, c.f. section 3.4), the Rossby number is also used to quantify the effects of rotation rate.
3.2.1
Vortex ring formation time
The formation time of a vortex ring gives a measure of the time during which impulse is added to the vortex ring and Gharib et al. (1998) describe it as “the non-dimensional time taken for vortex ring formation.” Typically it is expressed as a ratio of lengths, as in equation (3.2):
ToUo
Do
= Lo
Do
It has been shown that forLo/Do &4 and without extra forcing, single vortex
rings are no longer formed (Tarasov and Yakushev, 1973; Gharib et al., 1998), instead a jet is formed (c.f. section 3.3.2.3).
3.2.2
Slenderness ratio
The slenderness ratio provides a non-dimensional estimate of vortex ring core diameter. The core diameter is typically non-dimensionalised using, vortex ring diameter. Saffman (1978) provides a discussion on practical assessment of the vortex radius, suggesting that the true radius lies at some distance from the vortex centre a, at which the vorticity ω = 0. However in practice this will not be well defined, and recommends the use of either δ or ae, the internal and
effective radii respectively. He further defines δ as being the radial extent from the vortex centre at which the tangential velocity is maximised. ae is defined
according to equation (3.3), giving an equivalent core size for a uniform vorticity vortex ring, this is illustrated and compared to δ in figure 3.3. One can see that due to the assumption of uniform vorticity,ae is necessarily symmetric about the
centre of the core. δ however is defined by the extremum in the velocity field.
UT = Γ 2πD ln4D ae − 1 4 (3.3)
3.2.3
The Reynolds number
The Reynolds number, as throughout fluid dynamics, quantifies the ratio of in- ertial forcing to viscous damping and further the propensity of fluid to become turbulent. Several variations on Reynolds number have been employed with the purpose of applying this ratio to vortex rings, each with its own merits.
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 h/D 0 u ρ | ω φ (%) u ρ ωφ δ a e
Figure 3.3: Illustration of the differences between δ and ae
1977), equation (3.4). This formulation has the distinct advantage that for ex- perimental generation, the parameters used to define the vortex ring are normally known and so the calculation is straightforward. The disadvantage with charac- terisation by Reo is that the actual inertial forcing experienced by the fluid is
further a function of the formation time, To, and outlet geometry. Both of these
factors are of lesser importance (at least for small To) than Uo and Do, however
the implication is nonetheless that Reo does not completely describe the inertial
dynamics of the vortex ring.
Reo =
UoDo
ν (3.4)
Perhaps of more relevance to the fluid dynamics however, is a definition from the vortex ring’s circulation immediately subsequent to generation, Γ0, as given
in equation (3.5a) (Glezer, 1988). As this is arrived at from an actual measure of the fluid’s inertia, it provides a good measure of the vortex ring dynamics, as
it encapsulates the entirety of the formation process. Circulation measurement requires a relatively complex measurement of the fluid flow field, as an integral of the vorticity field must be evaluated, this may be simplified through the use of Stoke’s theorem, as in equation 3.5b (Kundu and Cohen, 2002). Here the velocity field u is integrated on a surface S bounded by the line l.
ReΓ= Γ0 ν (3.5a) Γ = Z S ∇ ×u.dS= I l u.dl (3.5b)
Similarly Stanaway et al. (1988) used a description arrived at from hydrody- namic impulse, equation (3.6a) for numerical simulations of vortex rings. Whilst providing the same advantages as ReΓ, computation of ReI from experimen-
tal data is significantly more difficult, this is because the vorticity integral in equation (3.6b) cannot be simplified through Stoke’s theorem. An accurate field measurement of the vorticity field is therefore required, which poses a significant challenge. ReI = p Ih/ρ νp(t) (3.6a) Ih =πρ Z ∞ 0 Z ∞ −∞ ωϕr2dhdr (3.6b)
For a theoretical consideration of the instabilities formed, Saffman (1978) used a strain rate formulation, Res, as in equation (3.7), whereδ is the inner core
radius (c.f. 3.2.2).
Glezer (1988) showed that through application of the slug flow model (c.f. Section 3.3.2.1) one can estimate Γ0 from simple parameters such as those used
in equation (3.4). Equivalence is shown in figure 3.4. The corresponding Reynolds number is shown in equation (3.8). Note that any outlet geometry effects will be neglected from this estimate of circulation, however inertial addition due to To is
included through Lo.
Figure 3.4: Comparison between ReJ and ReΓ (Glezer, 1988)
ReJ =
UoLo
2ν (3.8)
3.2.4
Rossby number
The Rossby number provides a relationship between the inertial timescale of the fluid and the rotating timescale of the fluid. It is most commonly given as in equation (3.9), where Ω is the magnitude of background rotation (Johnson, 1966; Verzicco et al., 1996).
RoD =
Uo
DoΩ
(3.9)