Formation number of confined vortex rings

This paper investigates the formation number of vortex rings generated by a piston-cylinder mechanism in a confined tube. We use direct numerical simulations of axisymmetric confined vortex rings to study the influence of different parameters on the separation (or pinch-off) of the vortex ring from the trailing jet. It is shown that the structure of the vortex ring at pinch-off depends on the type of injection program (pulse dominated by either positive or negative acceleration ramps) and the confinement ratio $D_w/D$, where $D_w$ is the inner diameter of the tube and $D$ the diameter of the cylinder. For low confinement ratios ($D_w/D\le 2$), a vortex of opposite sign generated at the lateral wall strongly interacts with the vortex ring and the pinch-off is not clearly observed. The pinch-off is observed and analyzed for confinement ratios $D_w/D\ge 2.5$. Direct numerical simulation data are used to estimate the value of the formation time, which is the time necessary for the vortex generator to inject the same amount of circulation as carried by the detached vortex ring. The confined vortex ring at pinch-off is described by the model suggested by Danaila et al. [I. Danaila et al., A model for confined vortex rings with elliptical core vorticity distribution, J. Fluid Mech. 811, 67 (2017)]. This model allows us to take into account the influence of the lateral wall and the elliptical shape of the vortex core. The value of the formation time is predicted using this model and the slug-flow model.

Figure 1

(a) Schematic of a vortex ring in a tube. (b) Time evolution of the piston velocity $U_p\left(\tau \right)$ for positive sloping ramp and negative sloping ramp programs.

Figure 2

Contours of vorticity $\omega$ at the early stage of formation of the vortex ring in a tube for a high confinement case ($D_w/D=2$). Dashed curves show contours of negative vorticity. Values of $\omega$ from $-6$ to 6 with increments of 1 are shown. Plots correspond to time instants (a) $\tau =8$, (b) $\tau =10$, and (c) $\tau =12$. (d) Close-up of (c) showing the vortex structures generated at the lateral confinement wall. A PS injection program was used (see Table 1).

Figure 3

Contours of vorticity $\omega$ at the early stage of formation of the vortex ring in a tube for a confinement parameter ($D_w/D=2.5$). Dashed curves show contours of negative vorticity. (a)–(c) The PS injection program and (d)–(f) the NS injection program [Fig. 1]. Three time instants are illustrated for each case: (a) and (d) $\tau =8$, (b) and (e) $\tau =10$, and (c) and (f) $\tau =12$ (the injection was stopped at ${\tau }_{\text{off}}=8$).

Figure 4

Time evolution of the maximum vorticity ${\omega }_{\text{max}}$ and streamwise position of the vorticity center $X_c$ corresponding to the maximum vorticity for (a) and (b) the PS injection program and (c) and (d) the NS injection program.

Figure 5

Vorticity domain ${\mathrm{\Omega }}_{0.05}=\{(r,x)\mid \omega (r,x)/{\omega }_{\text{max}}\ge 0.05\}$ (gray patches) used to compute integral quantities for the case of the PS injection program. Time instants (a) $\tau =8$, (b) $\tau =10$, and (c) $\tau =12$. The corresponding vorticity fields are illustrated in Figs. 3–3. The inside area of ${\mathrm{\Omega }}_{0.05}$ is remeshed using triangular finite elements [a mesh using larger triangles than in the actual postprocessing is shown for illustration in (c)]. The border of the domain ${\mathrm{\Omega }}_{0.01}=\{(r,x)\mid \omega (r,x)/{\omega }_{\text{max}}\ge 0.01\}$ is also shown (in blue).

Figure 6

Time evolution of circulation ${\mathrm{\Gamma }}_{*}$ computed from (7) for three vorticity patches (${\mathrm{\Omega }}_{*}={\mathrm{\Omega }}_{0.05}$, ${\mathrm{\Omega }}_{0.01}$, and ${\mathrm{\Omega }}_{\text{VR}}$). The vortex ring circulation ${\mathrm{\Gamma }}_{\text{VR}}$ was computed only after the pinch-off. The (a) PS injection program and (b) NS injection program were used.

Figure 7

Time evolution of the nondimensional energy $E_{\text{VR}}^{*}=E_{\text{VR}}/I_{\text{VR}}^{1/2}{\mathrm{\Gamma }}_{\text{VR}}^{3/2}$ after pinch-off. The (a) PS and (b) NS injection programs were used.

Figure 8

Contours of (a) normalized vorticity $\omega /{\omega }_{\text{max}}$ and (b) corresponding normalized stream function $\psi /{\psi }_{\text{max}}$: DNS results (solid lines) and the prediction of the vortex ring model (dashed lines) after fitting parameters $L_{ef}$ and $\beta$ (values indicated in Table 3). Values of $\omega /{\omega }_{\text{max}}$ and $\psi /{\psi }_{\text{max}}$ are shown from 0.1 to 0.9 with increments of 0.1. Case PS, $D_w/D=3$, and $\tau =14$.

Figure 9

Time evolution of the parameters describing the vortex ring geometry during the postformation phase. The parameter ${\theta }_e$ is related to the viscous length of the model (19), while $\beta >0$ describes the elongation along the longitudinal $x$ axis. All parameters are fitted from DNS data. The (a) and (c) PS and (b) and (d) NS injection programs were used for the analysis.

Figure 10

Values of ${(L/D)}_{\text{VR}}$ [see Eq. (36)] versus geometrical parameters $\beta$ and ${\theta }_e$ [cf. Eqs. (18) and (19)]. The symbols show the values taken from Table 3 for the PS (circle) and NS (square) injection programs.