2.1 Flow visualization

To visualize the vortex ring, the driving chamber is filled with smoke. Minimizing the buoyancy effect is important for such vortex rings whose axis is horizontal and $Re$ is low. Else, the vortex ring either goes up or down due to buoyancy. This can easily be avoided in a vertically orientated set-up. However, the horizontal configuration ensures that the buoyancy effects does not influence the characteristics of the vortex ring. For flow visualization experiments, a continuous laser (1.2 W, 532 nm) is used for illuminating the flow field. A high speed camera of resolution $1028\times 1296$ pixels and frame rate of 2 kHz at full resolution is used for image capturing. The laser sheet is made to lie in a vertical plane passing through the nozzle diameter by using a cylindrical lens and images are captured at maximum 200 f.p.s.

2.2 Particle image velocimetry

Particle image velocimetry (PIV) technique is used to measure the velocity field of the vortex ring. A dual cavity, double-pulsed, Nd:YAG laser having energy of $300~\text{mJ}~\text{pulse}^{-1}$ , pulse width of 7 ns and repetition rate of 10 Hz is used for illuminating the flow field. The light is delivered from the laser head to the experimental zone by an articulated arm with a sheet forming optics. The sheet forming optics consists of two spherical lenses and a cylindrical lens. The thickness of the sheet is approximately 1 mm. The light sheet lies in the vertical diametral plane and parallel to the axis of the nozzle. A 12-bit, charge coupled device (CCD) camera of resolution $2048\times 2048$ pixels is used at 1 Hz frame rate in double frame mode. SAFEX fog fluid of density $1.353~\text{kg}~\text{m}^{-3}$ is used to generate the seeding particles. The diameter of the seeding particles is of the order of $1~\unicode[STIX]{x03BC}\text{m}$ . Note, as the camera frame rate used being 1 Hz, the images at higher time resolution are obtained by phase shifting the measurements with an appropriate delays in camera and laser with respect to piston motion in repeated experiments. The maximum error in the circulation of the fully formed vortex ring for repeated experiments is less than 2.3 %.

For tracing the flow field accurately, the particle response time should be faster than the smallest time scale of the flow. In order to estimate how well the seeding particles follow the fluid and their response to high velocity gradients, the Stokes number of the particle is calculated. Stokes number ( $S_{tk}$ ) can be defined as the ratio of the characteristic time of a particle ( $\unicode[STIX]{x1D70F}_{p}$ ) to the characteristic time of the flow ( $\unicode[STIX]{x1D70F}_{f}$ ). The particle relaxation time is given as $\unicode[STIX]{x1D70F}_{p}=(\unicode[STIX]{x1D70C}_{p}D_{p}^{2})/(18\unicode[STIX]{x1D707})$ where $\unicode[STIX]{x1D70C}_{p}$ is the particle density, $D_{p}$ is the diameter of the particle and $\unicode[STIX]{x1D707}$ is the fluid viscosity. The flow time scale is given by $\unicode[STIX]{x1D70F}_{f}=L_{f}/U_{f}$ where $L_{f}$ is characteristic length and $U_{f}$ is the characteristic velocity. The inner diameter of the nozzle is taken as the characteristic length and maximum velocity of the vortex ring, which it achieves after formation, is used as $U_{f}$ . The smaller the Stokes numbers, the better the tracing accuracy. When $S_{tk}\ll 0.1$ , then tracing accuracy errors are below 1 %. The Stokes number in the present experiments is of the order of $1.16\times 10^{-5}$ . Thus, the seeding particles can follow the velocity field of the flow accurately.

3.1 Criterion for forming a clean vortex ring

To quantify the nature of deceleration that causes the stopping vortex ring formation or to find a criterion for the formation of the stopping vortex ring one needs to look into the cause of the formation of a stopping vortex. To quantify this, a set of experiments are carried out by measuring piston velocities and looking closely at the stopping vortex. In these visualization experiments, smoke is placed on the top half of the outer wall of the nozzle. When the piston decelerates, a reverse flow region appears in the inside wall of the nozzle. This causes entrained fluid motion from the outer side in the nozzle exit wall region that rolls up and forms a stopping vortex. Thus the stopping vortex formation process is expected to be a function of the rapidity of the piston stoppage, which can be quantified as fractional impulse imparted by the piston during the deceleration time. The ratio of acceleration impulse ( $I_{acc}$ ) calculated over $t_{acc}=T_{p}-t_{dec}$ and total impulse ( $I_{T}=\int u^{2}\,\text{d}t$ ) associated with the complete piston time history shows that if $I_{acc}/I_{T}<0.85$ the stopping vortex is either not formed or is very weak. This can be confirmed from figure 6, which shows the formation of a strong stopping vortex for $I_{acc}/I_{T}\approx 0.92$ , weak stopping vortex at $I_{acc}/I_{T}\approx 0.86$ and no stopping vortex at $I_{acc}/I_{T}\approx 0.76$ .

Figure 7.

Comparison of vortex ring diameter with that of Didden (Reference Didden1979), Weigand & Gharib (Reference Weigand and Gharib1997) and Allen & Auvity (Reference Allen and Auvity2002).

4.1 Core diameter

In the literature, the core diameter of a vortex ring is defined in different ways such as that based on the vorticity distribution and the velocity distribution. It is an important parameter of the vortex ring and its accurate estimation is necessary for predicting the translational velocity theoretically. The method of calculating the core diameter ( $d_{c}$ ) in the present work is shown in figure 11. The azimuthal vorticity and the velocity vectors of the vortex ring are shown in figure 11(a) and the corresponding azimuthal vorticity distribution along the axial and the radial direction (along the lines AB and CD respectively) are shown in figures 11(b) and 11(c), respectively. The core diameter is calculated by two methods; (i) by taking the location where the vorticity drops to 10 % of its maximum value as given by Weigand & Gharib (Reference Weigand and Gharib1997) and (ii) by tracing the location of peak velocities as given by Arakeri et al. (Reference Arakeri, Das, Krothapalli and Lourenco2004). The average of the axial and radial core diameters is considered for both the methods. The vorticity distribution in figures 11(a) and 11(b) shows that the vorticity distribution in the vortex ring is nearly Gaussian. The corresponding velocity distribution along the radial and axial directions are shown in figures 11(d) and 11(e), respectively.

The non-dimensional core diameters of vortex rings for cases 2, 3 and 5 are shown in figure 12. Core diameter increases rapidly during the roll-up process when shear layer vorticity is fed into it. Its growth rate gradually decreases as the piston decelerates ( $t_{n}>4.3$ ) and reaches a nearly constant value after the vortex ring pinches off from the trailing jet shear layer ( $t_{n}>7.5$ ) and feeding of shear layer vorticity to the core stops. However, at larger time (not considered in the present study) due to viscous diffusion the $d_{c}$ should increase.

Figure 12.

Defining core radius with different models for (a) case 2, (b) case 3 and (c) case 5.

As discussed above, core radius ( $a$ ) can be measured in different ways such as from the velocity variations or from the vorticity distribution. The core radius obtained as the distance between the points where the vorticity drops to 10 % of the maximum value of the vorticity is shown in figure 12 for the three cases. It can be seen that these values are comparable with the peak velocity based values (similar to Arakeri et al. Reference Arakeri, Das, Krothapalli and Lourenco2004), as discussed above. Two other models are available for predicting core radius, Saffman (Reference Saffman1970) used $a=\sqrt{4\unicode[STIX]{x1D708}T}$ and Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) considered total time elapsed in the calculation by adding the time of flight ‘ $t$ ’ to the stroke time. Figure 12 shows that both these models underpredict the core radius from the experimental values substantially. For comparison, the core radius calculated based on 50 % of the vorticity containing fluid is also plotted in figure 12. It is important to note that at higher $Re$ , the vorticity based core radius is more than the velocity based core radius. In all subsequent analyses the core radius $a$ is referred to as the core radius based on 90 % of vorticity.

Figure 13.

Schematic showing that the fluid from the wall boundary layer forming the vortex ring core and annular shear layer zone.

4.1.1 Model for core diameter during formation stage

In the literature, the model for the core radius and its comparison with the experimental value during the formation stage of the vortex ring is not adequate, as discussed above. However, with the assumption that during the formation stage the fluid from the boundary layer region enters the core, we attempt here to establish such a model. Saffman (Reference Saffman1978) and Pullin (Reference Pullin1979) found the self-similar nature of the initial roll-up process with the assumptions that the flow has just started and the vortex sheet at the nozzle exit behaves like a two-dimensional vortex sheet. Beyond this time, the core model is absent in the literature. The data that we have are for time periods beyond the self-similar roll-up process time. A simple model, with the assumption that the boundary layer vorticity flux from the nozzle inner wall forms the torus shaped vortex core and the annular shear layer in the trailing jet, is proposed here. The mass of boundary layer fluid ejected from the nozzle is equated with the volume of the core of the vortex ring and volume of the annular shear layer region of the trailing jet, as shown in figure 13. Flow from the central part of the pipe, having velocity ( $U_{j}$ ), forms the rest of the flow field along with the entrained fluid. This part is not directly involved in the present model. Entrainment of the ambient fluid into the core is also neglected in the model. The entrainment study of Dabiri & Gharib (Reference Dabiri and Gharib2004) shows that the volume of fluid entrained in the vortex bubble is 30 %–40 % of its volume for an isolated vortex ring with $L_{eq}/D_{0}=2{-}4$ . Thus, the volume of the entrained fluid into the vortex core alone during the formation stage is expected to be considerably lower and is neglected here. The balance equation can be written as,

(4.1) $$\begin{eqnarray}\int _{R_{0}-\unicode[STIX]{x1D6FF}}^{R_{0}}(2\unicode[STIX]{x03C0}tur\,\text{d}r)=(2\unicode[STIX]{x03C0}R_{0}\unicode[STIX]{x1D6FF}-\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FF}^{2})U_{j}t+2\unicode[STIX]{x03C0}R\unicode[STIX]{x03C0}a^{2}.\end{eqnarray}$$

Here, the exit velocity at the central zone is $U_{j}$ , the vortex ring radius is $R$ , the boundary layer thickness is $\unicode[STIX]{x1D6FF}$ and $u$ is the velocity inside the boundary layer. All these quantities are functions of time. It will be shown in the next section that $D(t)$ follows the similarity law of Pullin (Reference Pullin1979) during the formation stage also, and hence $R=k_{s}t^{2/3}$ , where $k_{s}$ is the proportionality constant. According to Das (Reference Das1998), $\unicode[STIX]{x1D6FF}\approx 3.14\sqrt{\unicode[STIX]{x1D708}t}$ and from Shusser et al. (Reference Shusser, Gharib, Rosenfeld and Mohseni2002) and Krueger (Reference Krueger2005)

(4.2) $$\begin{eqnarray}U_{j}=U_{s}\left(1+8\sqrt{(\unicode[STIX]{x1D708}t)/(\unicode[STIX]{x03C0}D_{0}^{2})}\right).\end{eqnarray}$$

The velocity both in the central region and inside the boundary layer in the pipe can be determined exactly from Das & Arakeri (Reference Das and Arakeri1998), however for simplicity, $u$ is assumed to be a linear function of $r$ inside the boundary layer. Substituting these values in (4.1) one can easily find that,

(4.3) $$\begin{eqnarray}a=\sqrt{\frac{k_{1}^{2}U_{s}}{6\unicode[STIX]{x03C0}k_{s}}t^{4/3}+\frac{k_{1}^{2}k_{2}}{6\unicode[STIX]{x03C0}k_{s}}t^{11/6}}+\unicode[STIX]{x1D6FF}_{c}.\end{eqnarray}$$

Here, $K_{1}=3.14\sqrt{\unicode[STIX]{x1D708}}$ and $k_{2}=8U_{s}\sqrt{(\unicode[STIX]{x1D708})}/(\sqrt{\unicode[STIX]{x03C0}}D_{0})$ . The proportionality constant, $k_{s}$ is obtained in the next section for cases 2, 3 and 5 as 0.152, 0.164 and 0.146, respectively. The $\unicode[STIX]{x1D6FF}_{c}$ is the growth of the core due to viscous diffusion of the boundary layer fluid that rolls up and forms the core. Accurate estimation of the growth of the curved vortex sheet during unsteady roll up is not available. It is assumed to be the same as the Stokes first problem of impulsively started flow and hence, $\unicode[STIX]{x1D6FF}_{c}\approx 4\sqrt{\unicode[STIX]{x1D708}t}$ is used in present calculation.

Figure 14.

Comparison of the predicted core diameter from the model with the experimental values during formation stage.

The values of $a$ during the formation stage i.e. for times less than 200 ms in the present study, are plotted from (4.3) and compared with the experimental data in figure 14. A reasonable comparison is achieved with this simple model until a time of approximately $t\simeq 150~\text{ms},t_{n}\simeq 5$ as after this time the piston motion decelerates (see figure 8).

4.2 Diameter of the vortex ring and vortex bubble

The diameter of the vortex ring is measured as the distance between the core centre observed in the cross-sectional plane view of the flow visualization images. The diameter of the vortex rings for all five cases are plotted in figure 15. Oscillations in the vortex ring diameter are due to the rotation of the core, which is not purely circular, as can be observed in figure 9. The vortex ring radius obtained from the flow visualization experiments is compared with the prediction from the theoretical model available. Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) equated the volume swept out by the piston to the volume of fluid in the inner spheroid of the vortex ring and obtained the radius of the vortex ring as,

(4.4) $$\begin{eqnarray}R=\left(\frac{3R_{0}^{2}L}{4\unicode[STIX]{x1D6FE}}\right)^{1/3},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ is the ratio of semi-minor to semi-major axes of the bubble and is taken as 0.6 by Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008).

Figure 15.

Variation of vortex ring diameter with time.

The vortex atmosphere seals the vortex ring from oncoming flow and the fluid inside the vortex bubble is isolated from the rest of the jet. The bubble radius is calculated from the PIV experiments (figure 16) and is taken as the reference for the comparison of the theoretical model available for the estimation of the bubble radius. The model that is available in the literature for determining the vortex bubble radius is again by Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) who used mass and momentum balance to reach at the following result for the bubble radius,

(4.5) $$\begin{eqnarray}R_{b}=\left(\frac{9\unicode[STIX]{x03C0}R_{0}^{2}L}{4\unicode[STIX]{x1D6FE}^{2}\left(\ln {\displaystyle \frac{8R}{a}}-0.558\right)(1+k)}\right).\end{eqnarray}$$

Here, $k$ is the Bessel added mass factor which is taken as 0.65 (Sullivan et al. Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008).

Figure 16.

Bubble streamlines used to calculate bubble radius for case 3 at $t_{n}=10.14$ .

The above mentioned models for the three cases discussed here are shown in figures 17(a), 17(b) and 17(c), respectively. The vortex ring radius and the bubble radius (figure 17 a–c) as calculated by the models by Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) are high compared to the experimentally observed vortex ring radius and the bubble radius. The reason for such a huge discrepancy is probably due to the inadequate knowledge of the entrainment effect, which is not included in the model. Also, the model is unable to account for the trailing jet fluid accurately. These models balance either exit mass or momentum of the fluid that is coming out of the nozzle with that of vortex bubble. Actually, some part of the jet fluid goes into the vortex atmosphere and some part of the ambient fluid enters the core, which are not accounted for in the model.

The vortex ring radius is also predictable during the initial roll-up process from the similarity solution of Saffman (Reference Saffman1978) and Pullin (Reference Pullin1979). During the initial spiralling process, substantial radial movement of the spiral centre is observed. However, once the radial velocity diminishes, the diameter does not change with time, unlike the core diameter. This can be observed from figures 12 and 15 where the vortex ring diameter at approximately $t_{n}=5.8$ becomes constant whereas the core diameter continues to grow beyond this time. Thus, it is expected that the core dynamics is more complex compared to the inviscid roll-up process that determines the ring diameter. The data up to $t_{n}=4$ for all the cases are fitted as $D\sim t^{q}$ and it is found that $q$ varies from 0.6 to 0.7. Thus the similarity model for which $q=2/3$ predicts the experimentally obtained $D$ values reasonably well during the formation stage. The proportionality constant ( $k_{s}$ ) from the experimental fit data for case 2, case 3 and case 5 are 0.152, 0.164 and 0.146, respectively.

Figure 17.

Vortex ring radius and bubble radius obtained from different models for (a) case 2, (b) case 3 and (c) case 5.

4.3 Impulse

The impulse associated with the formation of a vortex ring is the external force that is applied to the slug of fluid that forms the vortex ring and is defined by Lamb (Reference Lamb1932) (Akhmetov Reference Akhmetov2008). The impulse ( $P$ ) is calculated using (1.4) and the non-dimensional impulse ( $P_{n}$ ), shown in figure 18, is calculated as,

(4.6) $$\begin{eqnarray}P_{n}=\frac{P}{\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}R^{2}\unicode[STIX]{x1D708}}.\end{eqnarray}$$

Note that $P_{n}=Re_{\unicode[STIX]{x1D6E4}}$ , a local Reynolds number of the vortex ring. It is also noted that the time when $Re_{\unicode[STIX]{x1D6E4}}$ becomes constant is the time when the vortex ring is fully formed.

Figure 18.

Variation of non-dimensional impulse with non-dimensional time.

4.4 Circulation

Circulation  $\unicode[STIX]{x1D6E4}$ is a measure of the strength of the vortex ring and is a very important parameter to describe the dynamics. Exact values of the core radius, $a$ ( $=d_{c}/2$ ) and $\unicode[STIX]{x1D6E4}$ are required to find the theoretical value of translational velocity of the vortex ring and can be easily obtained from PIV measurements. A polygon (shown in figure 19) encircling the core region is used to calculate $\unicode[STIX]{x1D6E4}$ and is denoted as the vortex circulation. The polygon is chosen manually such that the jet shear layer circulation is eliminated in the estimation of core circulation. The total circulation, $\unicode[STIX]{x1D6E4}_{T}$ is calculated using the velocity along the centreline. Variations of $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}_{T}$ are shown in figure 20 for three of the cases for which PIV measurements are carried out. Both $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}_{T}$ are non-dimensionalized by $U_{s}D_{0}$ (Weigand & Gharib Reference Weigand and Gharib1997). $\unicode[STIX]{x1D6E4}$ increases with time during the formation stage and gradually becomes constant as the feeding of shear layer vorticity to the core stops. $\unicode[STIX]{x1D6E4}$ does not change after the vortex ring pinches off from the jet shear layer and $\unicode[STIX]{x1D6E4}_{T}$ becomes constant when the flow at the nozzle exit stops. As Reynolds number increases, the difference between $\unicode[STIX]{x1D6E4}$ and $\unicode[STIX]{x1D6E4}_{T}$ increases, which is an effect of the strength of the shear layer vorticity. To establish the initial roll-up process, the value of $\unicode[STIX]{x1D6E4}$ is fitted with time and it is observed that the variation of $\unicode[STIX]{x1D6E4}$ does not follow $\unicode[STIX]{x1D6E4}\sim t^{1/3}$ , as given by Pullin (Reference Pullin1979), during the formation process beyond the time of initial spiralling of the vortex sheet.

Figure 19.

The polygon used for the calculation of circulation, $\unicode[STIX]{x1D6E4}$ of the vortex ring core.

Figure 20.

Variation of circulation with time for (a) case 2, (b) case 3 and (c) case 5.

However, beyond the self-similar roll-up time, the data show rapid augmentation of $\unicode[STIX]{x1D6E4}$ . This is clear from the nature of the variation of $\unicode[STIX]{x1D6E4}$ with time, obtained from experiments and shown in figure 20. The nature of the curve at the formation stage in figure 20 is similar to that obtained by Dabiri & Gharib (Reference Dabiri and Gharib2004). Figure 20 also shows that $\unicode[STIX]{x1D6E4}_{T}$ is marginally greater than $\unicode[STIX]{x1D6E4}$ during the initial roll-up process. At the spiralling stage, the shear layer $\unicode[STIX]{x1D6E4}$ is nearly absent as the trailing jet is not established and the contribution of circulation due to overpressure $\unicode[STIX]{x1D6E4}_{p}$ to $\unicode[STIX]{x1D6E4}_{T}$ , as discussed by Krueger (Reference Krueger2005), plays a role. The equation for variation of $\unicode[STIX]{x1D6E4}$ with time can be written as,

(4.7) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}t}=\int _{0}^{\infty }u\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D703}}\,\text{d}y=\int _{0}^{\infty }\left(\frac{\text{d}v}{\text{d}x}-\frac{\text{d}u}{\text{d}y}\right)u\,\text{d}y.\end{eqnarray}$$

On the right-hand side of (4.7), the first term is the variation of $\unicode[STIX]{x1D6E4}_{p}$ and the second term is the variation of slug model $\unicode[STIX]{x1D6E4}_{s}$ with time. These terms are directly calculated from the measured PIV data and their sum is compared with the variation of $\unicode[STIX]{x1D6E4}$ obtained through the integration of velocity around a contour shown in figure 19 and are tabulated in table 2 for $t=100~\text{ms}$ . Estimation from the PIV data show that the term for overpressure is small for all these cases. The value of $\text{d}\unicode[STIX]{x1D6E4}/\text{d}t$ (sum of both the terms) is close to the corresponding $\text{d}\unicode[STIX]{x1D6E4}/\text{d}t$ values estimated from contours. Thus it gives us a direct measurement and shows that contribution of overpressure circulation is less than 2.5 % in the present experiments.

Table 2.

Comparison of theoretical $\text{d}\unicode[STIX]{x1D6E4}/\text{d}t$ ( $\text{m}^{2}~\text{s}^{-2}$ ) and $U_{j}$ ( $\text{m}~\text{s}^{-1}$ ) with experiments.

A model of circulation can easily be written neglecting the overpressure term from (4.7) as,

(4.8) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}t}=\frac{U_{j}^{2}}{2}.\end{eqnarray}$$

Again writing $U_{j}$ from (4.2), $\unicode[STIX]{x1D6E4}$ can be written as,

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}={\textstyle \frac{1}{2}}U_{j}^{2}t+{\textstyle \frac{1}{4}}k_{2}^{2}t^{2}+{\textstyle \frac{2}{3}}U_{j}k_{2}t^{3/2}.\end{eqnarray}$$

The nature of the variation of $\unicode[STIX]{x1D6E4}$ with time does indicate the above trend during the formation stage. However, due to overprediction of $U_{j}$ in the model given in (4.2), the circulation prediction from (4.9) is not accurate. It overpredicts the circulation considerably. Experimental $U_{j}$ values obtained at $t=100~\text{ms}$ in PIV shows 37 %–83 % difference with that obtained from (4.2) (table 2).

The vortex ring circulation and total circulation measured from the PIV measurements after it is fully formed are also compared with other available theoretical models. Models that are available in the literature for determining the circulation of a vortex ring are the slug model and the impulse based model. The slug model given by Maxworthy (Reference Maxworthy1977) and Didden (Reference Didden1979) assumes that the vortex ring is formed from a cylindrical ‘slug’ of fluid of length $L$ ejected from the exit of the nozzle over a stroke time $T$ . The slug model gives an expression for circulation as (see (1.7)), $\unicode[STIX]{x1D6E4}_{S}=L^{2}/2T$ . Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) assumed that the impulse of the ring is close to the momentum given to the fluid by the piston and gives the expression for the circulation as,

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}=\frac{R_{0}^{2}L^{2}}{R^{2}T},\end{eqnarray}$$

where $L$ is the equivalent stroke length, $R_{0}$ is the nozzle exit radius, $R$ is the vortex ring radius and $T$ is the stroke time.

The circulation values obtained from the above mentioned models (valid for a fully formed vortex ring) are compared for the cases 2, 3 and 5 and shown in figures 20(a), 20(b) and 20(c), respectively. From figure 20(a–c), it can be observed that both the slug model and the model by Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) overpredict the circulation value. However, the model by Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) predicts it better than the slug model. From these figures, it can be observed that the difference with the experimental data increases with the increase of Reynolds number after the vortex ring has fully formed. This is due to the trailing jet circulation which is not accounted in these models. From the above discussion it is clear that a better model that includes the effects of the jet circulation and its entrainment into the vortex ring is required. Time resolved velocity measurements with high spatial accuracy are necessary to arrive at a better model for circulation.

4.5 Translational velocity of the vortex ring

The translational velocity, $V$ , of the vortex ring is calculated by tracing the core centre of the vortex ring at 200 Hz and is shown in figure 21 for all five cases considered. Five point central differencing is used to calculate the velocity from the measured positions. The translational velocity does not show any stopping vortex or piston vortex effects and gradually increases to fully formed $V$ for all the cases.

Figure 21.

Variation of translational velocity of vortex ring with time.

Figure 22.

Comparison of predicted translational velocity from different models with experiments for (a) case 2, (b) case 3 and (c) case 5.

We could obtain salient parameters of the vortex ring such as $D$ , $d_{c}$ and $\unicode[STIX]{x1D6E4}$ through high speed visualization and PIV measurements. Various existing theoretical models and empirical relations are available in the literature for predicting different parameters of a vortex ring, barring translational velocity, and have already been discussed. The limitations of the existing theoretical models that predict some of these parameters, such as core diameter, ring diameter and circulation, have been illustrated. Comparison of the experimentally obtained $V$ with the theoretical models are discussed in this section to illustrate their validity. Such detailed comparisons are not available in the literature and are essential in view of the several theoretical and empirical results available in the vast literature on vortex ring studies. Surprisingly, very few efforts exist on the comparison of theory with experiments when the question of translational velocity is addressed, particularly during the formation stage. The reason could be the effects of the piston vortex and stopping vortex. In figure 22(a), (b) and (c), different available theoretical models of translational velocity are compared with the experimentally obtained $V$ for case 2, case 3 and case 5, respectively. The details of the theoretical models considered can be broadly classified into two categories, (i) circulation based inviscid and viscous models and (ii) energy and impulse based models.

When the core radius is much smaller than the vortex ring radius ( $a\ll R$ ), it is shown by Saffman (Reference Saffman1970) that the vortex ring moves forward with velocity, $V$ given by (1.3). The translational velocity, given by Saffman (Reference Saffman1970), can alternatively be written with the slug model circulation, based on the stroke length (§ 1.2) and the stroke time of the piston (table 1), (Maxworthy Reference Maxworthy1977; Didden Reference Didden1979) as,

(4.11) $$\begin{eqnarray}V_{S}=\frac{L_{eq}^{2}}{8\unicode[STIX]{x03C0}RT}\left(\ln \frac{8R}{a}-\unicode[STIX]{x1D6FD}\right),\end{eqnarray}$$

where $L_{eq}$ is the equivalent stroke length of the piston, as defined in § 1.2. It is evident from the properties of vortex rings discussed in § 4.1 that ‘ $a$ ’ as $\sqrt{4\unicode[STIX]{x1D708}T}$ (Saffman Reference Saffman1970) or $\sqrt{4\unicode[STIX]{x1D708}T+t}$ (Sullivan et al. Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) underpredict the core radius and hence are not shown in figure 22(a–c). Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) accounted for the viscous effects on the motion of the vortex ring in a viscous fluid and modified the relation of Saffman (Reference Saffman1970) as,

(4.12) $$\begin{eqnarray}V_{fm}=\frac{\unicode[STIX]{x1D6E4}}{4\unicode[STIX]{x03C0}R}\left\{\ln \frac{8R}{\sqrt{4\unicode[STIX]{x1D708}T}}-0.558-3.6716\frac{\unicode[STIX]{x1D708}t}{R^{2}}\right\}.\end{eqnarray}$$

The term ( $(3.6716\unicode[STIX]{x1D708}t)/R^{2}$ ) is the viscous effect term and the negative sign indicates the slowing of the vortex ring due to viscosity. $V$ calculated from Saffman’s (1970) model and the model of Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) should differ marginally as the contribution of viscous term ( $(3.6716\unicode[STIX]{x1D708}t)/R^{2}$ ) is usually small unless we are looking for the $V$ at a much later time, when viscous diffusion becomes important. However, figure 22(a–c) show considerable differences between the two models. One can see from these figures that formula given by Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) predicts the velocity to be closer to the experimentally observed $V_{exp}$ compared to (1.3). The implication of this observation could be misleading as it indicates a substantial viscous effect at this stage. The actual reason for this is different. In (1.3), $a$ and $\unicode[STIX]{x1D6E4}$ are taken from experiments whereas in (4.12), a wrong and smaller $a=\sqrt{4\unicode[STIX]{x1D708}T}$ is used and hence the velocity predicted is higher. Tung & Ting (Reference Tung and Ting1967) provided a formula for the translational velocity of a finite core viscous vortex ring as,

(4.13) $$\begin{eqnarray}V_{TT}=\frac{\unicode[STIX]{x1D6E4}}{4\unicode[STIX]{x03C0}R}\left(\ln \frac{8R}{\sqrt{4\unicode[STIX]{x1D708}t}}-0.688\right),\end{eqnarray}$$

where the core radius is taken as $\sqrt{4\unicode[STIX]{x1D708}t}$ . Equation (4.13) overpredicts $V_{exp}$ as the core radius taken is smaller than the actual $a$ . However, if the experimental $a$ is substituted into (4.13), it underpredicts the experimental value (not shown here). This observation probably reopens the question of what should be the value of the constants in (4.12) and (4.13).

Another class of formulas that are derived for predicting translational velocity is based on energy and impulse of the vortex ring. Shusser & Gharib (Reference Shusser and Gharib2000) gave an expression for the translational velocity of thick vortex rings using impulse and the vortex circulation as,

(4.14) $$\begin{eqnarray}V_{sg}=0.5352\sqrt{\frac{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6E4}^{3}}{\unicode[STIX]{x03C0}P}}.\end{eqnarray}$$

Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) replaced slug model circulation in the expression given by Shusser & Gharib (Reference Shusser and Gharib2000) and modified the translational velocity as,

(4.15) $$\begin{eqnarray}V_{sg1}=0.5352\frac{\unicode[STIX]{x1D6E4}_{s}}{\unicode[STIX]{x03C0}R_{0}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6E4}_{s}$ is slug model circulation, calculated based on $L_{eq}$ . In comparison with the present data, only $V_{sg1}$ is plotted as $V_{sg}$ underpredicts the translational velocity substantially. According to Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008), the kinetic energy of a vortex ring with $R\gg a$ ( $a$ obtained from PIV experiment is used here) is given by equation 1.2. Hamilton’s equation relates the energy and the impulse (given in (1.4)) of the vortex ring to the translational velocity as,

(4.16) $$\begin{eqnarray}V_{g1}=\frac{\unicode[STIX]{x2202}E}{\unicode[STIX]{x2202}P}.\end{eqnarray}$$

The model of Sullivan et al. (Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008) is the simplified model of Fraenkel (Reference Fraenkel1972), who derived the second-order formulae for the impulse and energy of vortex rings with finite but small values of $\unicode[STIX]{x1D700}$ ( $=a/R$ ) as,

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle E_{\unicode[STIX]{x1D700}}=\frac{1}{2}\unicode[STIX]{x1D70C}R\unicode[STIX]{x1D6E4}^{2}\left(\ln \frac{8}{\unicode[STIX]{x1D700}}-\frac{7}{4}+\frac{3}{16}\unicode[STIX]{x1D700}^{2}\ln \frac{8}{\unicode[STIX]{x1D700}}\right) & \displaystyle\end{eqnarray}$$

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle P_{\unicode[STIX]{x1D700}}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6E4}\unicode[STIX]{x03C0}R^{2}\left(1+\frac{3}{4}\unicode[STIX]{x1D700}^{2}\right). & \displaystyle\end{eqnarray}$$

Using Hamilton’s equation, the translational velocity is calculated as,

(4.19) $$\begin{eqnarray}V_{g2}=\frac{\unicode[STIX]{x2202}E_{\unicode[STIX]{x1D700}}}{\unicode[STIX]{x2202}P_{\unicode[STIX]{x1D700}}}.\end{eqnarray}$$

The core radius used for calculating energy and impulse is the one obtained from PIV experiments. In the above two models, when the velocity is recalculated by replacing the core radius based on $a=\sqrt{4\unicode[STIX]{x1D708}T}$ or $a=\sqrt{4\unicode[STIX]{x1D708}(T+t)}$ , both models overpredict the translational velocity as the core radius is underpredicted using these relations (see figure 12).

It is also observed that the model of Tung & Ting (Reference Tung and Ting1967) calculates $V$ more accurately than the model of Saffman (Reference Saffman1970) (see figure 22 a–c). For all the cases, it can be seen that $V$ calculated from the model of Fraenkel (Reference Fraenkel1972), matches the experimentally measured $V$ to a good extent.

Other than the model of Fraenkel (Reference Fraenkel1972), figure 22(a–c) show that at large $t_{n}$ , the results from the models of Tung & Ting (Reference Tung and Ting1967) and Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) match with $V_{exp}$ reasonably well, as at this stage the vortex ring has a finite core and viscous effects become gradually important. During the initial formation stage, the model of Tung & Ting (Reference Tung and Ting1967) appears best for all Reynolds numbers. The model of Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) also appears to predict the translational velocity at this stage. However, the models of Tung & Ting (Reference Tung and Ting1967) and Fukumoto & Moffatt (Reference Fukumoto and Moffatt2000) suffer from the drawback of using an underpredicted core diameter and overpredicted circulation. To predict the translational velocity at this stage one needs to consider the dynamics of a spiralling shear layer and formation of the core with Stokes layer type viscous diffusion of the shear layer fluid, as discussed in § 4.1.1 (4.3) which is beyond the scope of this work.

From the detailed comparisons of the data presented here and those of earlier studies, clear variations are observed in different quantities that are modelled. Although a great deal of experimental data are necessary to find empirical correlations for these quantities, the present work highlights the limitations and validity of existing models with the measured quantities through a novel experimental arrangement. In summary, the following important characteristics are noted for a vortex ring. The measured values of the core radius are substantially different from the values obtained in theoretical models, used by different investigators. The estimation of the core radius requires detailed velocity field measurements, which are limited in the literature, and the models used to determine its value always underpredict it. Similarly, the vortex ring radius and circulation differ from the available models considerably and this indicates the need of better models. The core radius and circulation, obtained from the respective models, are used in the models for the translational velocity (see (1.3), (4.12) and (4.13)) and it appears that the error due to using the underpredicted core radius and overpredicted circulation have resulted in the constant values of translational velocity. However, the present study clearly indicates that these constants need to be corrected when one uses the correct core radius and circulation values. The ratio of translational velocity of a fully formed vortex ring and exit velocity does not have a constant value and varies from 0.46 to 0.65 in the different cases considered here, as can be obtained from table 1. This is consistent with the observations of previous investigators (Sullivan et al. Reference Sullivan, Niemela, Hershberger, Bolster and Donnelly2008). However, in the present observations, we could identify the best model for prediction of the translational velocity as the model of Fraenkel (Reference Fraenkel1972). A detailed study of the effects of Reynolds number and the slowing down of motion can be obtained in the future.