3.1 Validation

We start by validating the results from our numerical simulations by comparisons with our experimental measurements and alternate numerical methods. We define the force and moment coefficients as $C_{N}=N/[(\unicode[STIX]{x1D70C}(c\unicode[STIX]{x1D6FA}_{max})^{2}sc)/2]$ , $C_{A}=A/[(\unicode[STIX]{x1D70C}(c\unicode[STIX]{x1D6FA}_{max})^{2}sc)/2]$ and $C_{M}=M/[(\unicode[STIX]{x1D70C}(c\unicode[STIX]{x1D6FA}_{max})^{2}sc^{2})/2]$ , where $N,A$ denote the force components normal to and along the plate chord, and $M$ the moment about the hinge. Here, $\unicode[STIX]{x1D70C}$ is the density of the fluid, $c$ and $s$ the chord and the span of the plate and $\unicode[STIX]{x1D6FA}_{max}$ the maximum angular velocity of the plate. The force components $N$ and $A$ are obtained by numerically integrating the normal and tangential stresses over the entire plate surface.

Figure 4.

(a) The plate motion profile described in terms of $\unicode[STIX]{x1D703}$ versus $t$ and $\dot{\unicode[STIX]{x1D703}}$ versus $t$ for the validation studies presented in § 3.1. (b) The non-dimensional normal and chord-wise force coefficients, $C_{N}$ and $C_{A}$ (shown in inset), plotted as a function of $t/t_{o}$ . The solid line and circle markers correspond to our simulations and viscous vortex particle method (VVPM) simulations (Das, Shukla & Govardhan Reference Das, Shukla and Govardhan2016), respectively, with $\unicode[STIX]{x1D703}_{max}=90^{\circ }$ and $Re=785.4$ for both the cases. (c) The non-dimensional moment coefficient $C_{M}$ plotted as a function of $t/t_{o}$ . The line styles correspond to the same scenarios as in (b).

Figure 5.

Vorticity field $\unicode[STIX]{x1D714}_{z}$ (non-dimensionalized by $\unicode[STIX]{x1D6FA}_{max}$ ) at $t/t_{o}=1$ based on (a) our numerical simulations (SIM.), (b) PIV measurements (EXP.) for the same plate rotation profile as in figure 4(a) with $Re=785.4$ . In (a,b) the dashed and the solid lines indicate contours of negative and positive vorticity, respectively. (c) Horizontal velocity profiles from the numerical simulations (——) and the PIV measurements (– – –) across the vertical dotted lines (passing through the respective starting vortex centres) shown in (a,b). (d–f) Same as in figures (a–c), but at time $t/t_{0}=2$ with the vertical dotted lines in (d,e) passing through the respective stopping vortex centres.

Figure 6.

(a) Trajectories of the position of maximum vorticity magnitude within the starting (bottom trajectories) and stopping (top trajectories) vortex regions from simulations (solid line with closed symbols) and experiments (open symbols) for the same plate motion profile as in figure 4(a) with $Re=785.4$ . (b) Corresponding non-dimensional circulation $\widehat{\unicode[STIX]{x1D6E4}}$ ( $=\unicode[STIX]{x1D6E4}/(\unicode[STIX]{x1D6FA}_{max}c^{2})$ ) associated with the starting (bottom curves) and stopping (top curves) vortex regions, where $\unicode[STIX]{x1D6E4}$ is calculated as the integral of the vorticity over the area encompassed by the contour at 5 % of the peak vorticity magnitude at the respective time. Open and closed symbols correspond to experiments and simulations, respectively. The area integrals of vorticity also included regions around the plate that are not advected away from the plate after the plate motion stops.

In figure 4, we plot the force and moment coefficients as a function of time for the plate motion profile shown in figure 4(a). The Reynolds number for the flow is $Re=c^{2}\unicode[STIX]{x1D6FA}_{max}/\unicode[STIX]{x1D708}=785.4$ in both our numerical simulations and the results based on a viscous vortex particle method (VVPM). The VVPM code has previously been used to simulate the flow around a pitching foil in a free stream by Das et al. (Reference Das, Shukla and Govardhan2016). As shown in figure 4(b), the normal force coefficient $C_{N}$ from our numerical simulations is in quantitative agreement with the estimate based on VVPM. Similarly, our results also exhibit excellent agreement with VVPM for $C_{A}$ (see inset of figure 4 b), which typically is significantly smaller than $C_{N}$ for these class of flows. Finally, the moment coefficient $C_{M}$ as a function of time (figure 4 c) is also observed to be predicted well by our numerical simulations.

Figure 5(a,b) shows the non-dimensional vorticity field distribution at $t/t_{o}=1$ from our numerical simulations and PIV measurements, respectively. Both figures correspond to the plate motion profile shown in figure 4(a) and a Reynolds number of $Re=785.4$ . Good qualitative agreement between our numerical simulations and PIV is found, with similar size and strength of the starting vortex in figure 5(a,b). Apart from the starting vortex, there is also a relatively smaller anti-clockwise vortex present right below the plate hinge at $t/t_{0}=1$ in our simulations. The formation of this anti-clockwise vortex near the plate hinge is discussed further in § 3.2 and appendix A. The corresponding positive vorticity region below the plate hinge is noticeably weaker in the PIV measurements. The experimental set-up had a 12 mm diameter SS rod holding the plate at the hinge throughout the depth of the tank. In the two-dimensional simulations, however, the plate hinge was modelled as a rounded corner whose diameter was around the plate thickness, i.e. 1.5 mm, thus resulting in the generation of a stronger anti-clockwise vortex below the hinge. Figure 5(c) shows the comparisons in the horizontal velocity profile across the vertical transect shown in figure 5(a,b). There is excellent quantitative agreement in the variation of $u$ . The bottom row of figure 5, i.e. figure 5(d–f), show similar comparisons at a later time of $t/t_{0}=2$ . While the qualitative nature of the overall vorticity distributions are similar between the simulations and experiments, there are noticeable differences within the starting vortex. The differences may be owing to one or more of the following factors: (i) the errors in the vorticity field arising from numerical differentiation of the velocity field, (ii) the presence of a small background flow in the experiments and (iii) possible three-dimensional effects (Meunier & Leweke Reference Meunier and Leweke2005) that are absent in our simulations. The horizontal velocity profiles across the dotted lines in figure 5(d–e), however, have a similar structure (figure 5 f) with a difference of approximately 11 % in the peak velocity.

For a more detailed comparison between simulations and experiments, we plot the trajectory ( $t/t_{0}\leqslant 2$ ) of the position of peak vorticity in figure 6(a) for the plate motion profile shown in figure 4(a). The starting and stopping vortex trajectories from the experiments and simulations compare favourably with the experimental data for the starting vortex being scattered on both sides of the trajectory from the simulations. We could not perform experimental measurements beyond $t/t_{0}=2$ as the vortices started moving out of the camera’s field of view. In figure 6(b), we plot the evolution of the non-dimensional circulation ( $\widehat{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/(\unicode[STIX]{x1D6FA}_{max}c^{2})$ ) associated with the starting and stopping vortex regions. The circulation $\unicode[STIX]{x1D6E4}$ associated with the vortex regions is calculated as the integral of the vorticity over the area encompassed by the contour at 5 % of the peak vorticity magnitude at the respective time. The area integrals also included the regions in the neighbourhood of the plate that are not necessarily advected away from the plate after the plate motion stops. As shown in figure 6(b), there is excellent quantitative agreement between the simulations and experiments for both the starting and stopping vortex regions, with the difference in the saturation value being around 5 and 8 %, respectively. It is noteworthy that the values at saturation associated with the starting and stopping vortex regions almost cancel each other despite the difference in the strength of the starting and stopping vortices that are advected away from the plate. This is owing to the circulation estimate associated with the stopping vortex region also including the regions around the plate that are not advected away with the vortex pair.

In summary, the data in figures 4–6 demonstrate that our numerical simulations capture the qualitative and quantitative features of the flow with reasonable accuracy. Having established the validity of our numerical simulations, we proceed to use the numerically generated temporally and spatially well-resolved velocity fields to extract the LCSs in the flow.

3.2 Hyperbolic and elliptic LCS

Figure 7(a–d) Show the evolution in time of the bFTLE field, $\unicode[STIX]{x1D70E}_{t}^{t+T}$ (in grey scale) computed using an integration time $T=-t$ . The attracting hyperbolic LCSs extracted using the method described in § 2.3 are also shown (in red dashed line) in figure 7(a–d). The attracting LCSs in figure 7(b,c) display notable similarity with the dye visualization images shown in figure 2(b,d), including the spiral structures within the vortex pair. While the vortex pair at $t/t_{o}=2$ (figure 7 c) appears to be moving along the final plate orientation, it is oriented slightly downwards at $t/t_{o}=3$ (figure 7 d). The relative difference in the strengths between the starting and stopping vortices, which causes the vortex pair to deflect downwards, can be attributed to the vorticity associated with (i) the relatively small anti-clockwise vortex a (seen in figure 7 b,c) formed at the hinge/fixed end of the plate, and (ii) the bound circulation around the plate. This anti-clockwise vortex forms as a result of the structure of the flow that is set-up soon after the plate starts moving. As discussed in figure 17 in appendix A, the flow structure at $t=0^{+}$ is predicted well by potential flow theory. The anti-clockwise flow around the hinge separates from the plate to result in the formation of the anti-clockwise vortex to the left of the plate hinge. Finally, at $t/t_{o}=2,3$ , the Front boundary of the vortex pair seen in figure 2 seems to be correlated with the attracting LCS.

Figure 7.

The bFTLE field (shown in grey scale) $\unicode[STIX]{x1D70E}_{t}^{t+T}$ with $T=-t$ at (a) $t=0.5t_{o}$ , (b) $t=t_{o}$ , (c) $t=2t_{o}$ and (d) $t=3t_{o}$ , calculated using the velocity field from our numerical simulations. Shown as dashed red lines in each panel are the corresponding attracting hyperbolic LCSs extracted using the methods described in § 2. The plate rotates about its left tip at $(0,0)$ in the anti-clockwise direction during $0\leqslant t\leqslant t_{o}$ , with the plate motion profile described in the beginning of § 3 (shown in figure 4 a) corresponding to $Re=785.4$ . The plate is shown in hashed pink in all of the panels.

Figure 8.

The fFTLE field (shown in grey scale) $\unicode[STIX]{x1D70E}_{t}^{t+T}$ with $T=5t_{o}-t$ at (a) $t=0$ , (b) $t=t_{o}$ , (c) $t=2t_{o}$ and (d) $t=3t_{o}$ , calculated using the velocity field from our numerical simulations. Shown as dashed blue lines are the corresponding repelling hyperbolic LCSs extracted using the methods described in § 2. The plate rotates about its left tip at $(0,0)$ in the anti-clockwise direction during $0\leqslant t\leqslant t_{o}$ , with the plate motion profile described in the beginning of § 3 (shown in figure 4 a) corresponding to $Re=785.4$ . The plate is shown in hashed pink in all of the panels.

Figure 8 shows the fFTLE field $\unicode[STIX]{x1D70E}_{t}^{t+T}$ (in grey scale, computed with an integration time of $T=5t_{o}-t$ ) and the corresponding repelling hyperbolic LCSs (in blue dashed lines) at times $t/t_{o}=0,1,2,3$ . A complex structure is observed for the repelling LCS at $t=0$ , with such an extraction of continuous curves being possible as a result of the strainline-based method described in § 2.3. A previous study (Shadden et al. Reference Shadden, Dabiri and Marsden2006) on the flow generated by a jellyfish used the fFTLE field to identify what is called the upstream wake, thus motivating us to conjecture that the fFTLE field at $t=0$ may encompass regions that ultimately end up inside the advected fluid bulk at large times. The validity of this conjecture is discussed later in this section. The repelling LCSs in figure 8(a) demarcate several regions in the neighbourhood of the plate. At  $t/t_{o}=1$ (figure 8 b), the starting vortex with a spiral structure within is clearly seen in the repelling LCSs, whereas the stopping vortex is just beginning to emerge. While the repelling LCSs are attached to the plate at $t/t_{o}=1$ , we observe them to be just detached from the plate at $t/t_{o}=2$ (figure 8 c). For larger times, i.e.  $t/t_{o}=3$ shown in figure 8(d), the repelling LCS contains a distinct boundary that may represent the rear edge of the vortex pair that is shed from the plate.

Figure 9.

(a) Elliptic LCSs (solid lines) extracted using the fFTLE field $\unicode[STIX]{x1D70E}_{t}^{t+T}$ at $t=2t_{o}$ shown in figure 8(c). The green and the pink boundaries correspond to the starting and stopping vortices, $V_{1}$ and $V_{2}$ respectively. The background colour represents the instantaneous non-dimensionalized vorticity field at $t=2t_{o}$ . The elliptic LCSs at $t=2t_{o}$ are advected forward in time as material lines to obtain the dashed and dash-dotted curves at $t=3t_{o}$ and $4t_{o}$ , respectively. (b) Hyperbolic attracting (dashed red) and repelling (dashed blue) LCSs plotted along with the elliptic LCS (solid green and pink) at $t/t_{o}=2$ . The region enclosed by the hyperbolic LCSs form the advected bulk that moves away from the plate in a direction close to the final orientation of the plate.

Elliptic LCSs have recently been shown to extract closed boundaries within which the fluid material remains as a coherent unit over a finite period of time, thus resulting in an objective definition of a vortex (Haller & Beron-Vera Reference Haller and Beron-Vera2013). In figure 9(a), we show the elliptic LCSs associated with the starting and stopping vortices at $t/t_{o}=2$ , shown as thick green and thin pink solid lines, respectively. Indeed, the advected positions of the elliptic LCSs at later times ( $t/t_{o}=3,4$ ) shown in figure 9(a) imply that the material inside each of these elliptic LCSs advect without any filamentation near the boundaries. These regions within the elliptic LCSs therefore display Lagrangian coherence and remain as coherent blobs of fluid in time. It is also noteworthy that the elliptic LCS boundaries encompass a significantly larger region than what could be defined as a vortex based on a simple thresholding of the vorticity magnitude (shown using the background colour in figure 9 a). We plot the hyperbolic and elliptic LCSs simultaneously at $t/t_{o}=2$ in figure 9(b). While the elliptic LCS boundaries seem to overlap with the attracting or repelling LCSs in some regions, there are noticeable regions that are entrapped within the hyperbolic LCSs but lie outside the elliptic LCSs.

In an earlier study by Shadden et al. (Reference Shadden, Katija, Rosenfeld, Marsden and Dabiri2007) on vortex rings, features in the bFTLE and fFTLE fields were used to define vortex boundaries associated with the bulk fluid transport. Extending this approach of Shadden et al. (Reference Shadden, Katija, Rosenfeld, Marsden and Dabiri2007), we construct a region that is bounded everywhere by the locally outermost hyperbolic LCS and investigate the relevance of the region to describe the bulk fluid transport in our flow. Unlike the axisymmetric vortex ring studied in Shadden et al. (Reference Shadden, Katija, Rosenfeld, Marsden and Dabiri2007), the two vortices within the vortex pair of our flow differ in their strengths. As a consequence, the simple scenario of the front and the rear boundaries of the material bulk (that gets transported away from the plate) being given by the attracting and repelling LCSs, respectively, does not hold in our flow. In summary, we have identified the material bulk of the fluid which gets transported away from the plate as the region encompassed by the hyperbolic LCSs; this region will be referred to as the advected bulk in the rest of the paper. A subset of the advected bulk are two separate regions encompassed by elliptic LCS, representing the starting and stopping vortices.

We recall from figure 8 that the repelling LCSs, and possibly the stopping vortex, detach from the plate at around $t/t_{o}=2$ . An elliptic LCS extraction at any earlier time would not capture the stopping vortex in its entirety as the stopping vortex completes its formation only around $t/t_{o}=2$ . We also note that at a much later time, say $t/t_{o}=5$ , the vortex pair will include non-negligible amounts of ambient fluid entrained via diffusion, the influence of which would be insignificant on the flow dynamics at early times. In summary, while the choice of $t/t_{o}=2$ for the identification of the advected bulk is not unique, it is probably appropriate for isolating the effects of the plate motion on the vortex pair that is formed. It is also noteworthy that the advected bulk as identified by the hyperbolic LCS boundaries are not too sensitive to the integration time $T$ , whereas the elliptic LCS boundaries do depend on the specific choice of $T$ . We proceed to investigate the origin and evolution of the various regions within the advected bulk.

Figure 10.

(a) Tracer positions at $t=0$ obtained by backward advection of material inside the elliptic LCSs (shown in figure 9 a) at $t=2t_{o}$ . The green and the pink colours correspond to the starting and stopping vortices, respectively. Shown in grey scale in the background is the fFTLE field $\unicode[STIX]{x1D70E}_{t}^{t+T}$ at $t=0$ . (b) The lobes at $t=0$ that constitute the material inside the hyperbolic LCSs at $t=2t_{o}$ . Lobes 1 (dark green) and 2 (blue) span the entire green region ( $V_{1}$ ) in (a); lobes 3 (red) and 4 (orange) span the entire pink region ( $V_{2}$ ) in (a). Lobe 5 (brown) is obtained by backward advection of the material outside the elliptic LCSs but inside the hyperbolic LCSs at $t=2t_{o}$ shown in figure 9(b).

3.3 Advected bulk – where does the fluid come from?

We start by advecting the material inside the elliptic LCSs at $t/t_{o}=2$ backward in time to $t=0$ . The backward-advected positions of the starting and stopping vortices ( $V_{1}$ and $V_{2}$ ) are shown in green and pink, respectively in figure 10(a). Also shown using the background grey scale colour in figure 10(a) is the fFTLE field at $t=0$ . There is a strong correlation between the green and pink tracer boundaries, and the ridges in the fFTLE field. There also exist small regions between the tracer boundaries and the fFTLE ridges. In figure 10(b), apart from the backward-advected tracer positions from the starting and stopping vortices, we show as region 5 (in brown) the backward-advected material from the regions outside the elliptic LCSs but within the hyperbolic LCSs at $t/t_{o}=2$ . Interestingly, the brown-coloured tracer region in figure 10(b) occupies the small regions between the green/pink tracer boundaries and the fFTLE ridges in figure 10(a). In summary, the fFTLE field of the repelling LCSs at $t=0$ provides a direct estimate of the locations of all the fluid material that constitute the advected bulk that eventually forms after the plate motion stops, including an identification of what constitutes the starting and stopping vortices. Motivated by the results from the dye visualization experiment presented in figure 2, we distinguish between the lobes to the forward and rearward sides of the plate: lobes 1 (dark green) and 2 (blue) belong to the starting vortex, and lobes 4 (orange) and 3 (red) belong to the stopping vortex (figure 10 b). The entire brown region in figure 10(b) is referred to as lobe 5.

Figure 11.

(a–d) Temporal evolution during $0<t\leqslant 3t_{o}$ of the material inside the five different lobes of fluid identified in figure 10(b). The times corresponding to each panel is indicated in the top right. Lobes 1 and 2 mix to form the starting vortex. Similarly, lobes 3 and 4 mix to form the stopping vortex. Lobe 5 (brown), which is outside of both the starting and stopping vortices, remains a part of the advected bulk.

Figure 11 shows the evolution of lobes 1–5 with time. Specifically, the advected positions of the material inside various lobes are shown at times $t/t_{o}=0.5,1,2,3$ . We recall that, at $t/t_{o}=2$ the tracer positions fit in exactly with the boundaries of the hyperbolic and elliptic LCSs shown in figure 9(b). Lobe 1, originally present on the forward side of the plate at $t=0$ , crosses over to the rearward side of the plate, also entraining fluid from lobe 2 in the process. At $t/t_{o}=2$ (figure 11 c), the fluid from lobe 2 is completely entrained into lobe 1, as a result of which lobe 1 alone is sufficient to describe the elliptic LCS boundary. Interestingly, at $t/t_{o}=2$ , the fluid from lobe 2 describes the attracting LCS boundary (figure 9 b) that is present just inside of the front edge of the starting vortex. At a later time, i.e.  $t/t_{o}=3$ (figure 11 d), the advected attracting and elliptic LCS boundaries at the front edge of the starting vortex become coincidental. We also quantify the areas contained in lobes 1 and 2 (denoted by $A_{lobe1}$ and $A_{lobe2}$ ) of the starting vortex $V_{1}$ by defining the entrainment fraction $\unicode[STIX]{x1D702}_{V1}=A_{lobe2}/(A_{lobe1}+A_{lobe2})$ , which takes a value of 26 %. We have thus precisely estimated the amount of fluid material from either side of the plate at $t=0$ that ultimately constitutes the starting vortex.

We recall from figure 1 that the starting vortex $V_{1}$ is mostly green in an experiment in which the forward and rearward sides of the plate were coated green and red, respectively. This observation is understood via the results in figure 11 where lobe 1 is clearly seen to be in contact with the forward side of the plate at $t/t_{o}=0.5$ whereas lobe 2 is prevented from touching the rearward side of the plate by the material from lobe 3 in between. At $t=0$ (figure 10 b) too, while lobe 1 (dark green) is in contact with the forward side of the plate tip, a thin filament from lobe 3 is present in between lobe 2 and the rearward side of the plate tip. While the dye visualization experiment shown in figure 1 is inconclusive about the relative amounts of fluid from either side of the plate that get into the starting vortex, our LCS analysis has conclusively identified the initial locations of all the fluid material that gets entrapped in the starting vortex.

Lobes 3 and 4, which constitute the stopping vortex, exhibit an evolution in time that is similar to that of lobes 1 and 2 of the starting vortex, but with the main difference of entrainment (of lobe 4 by lobe 3) occurring after the plate motion stops. Lobe 3, which is originally on the rearward side of the plate, acquires sufficient momentum during the plate motion to move towards the forward side and subsequently entrain the fluid in lobe 4. Also, since lobe 3 is much more in contact with the plate than lobe 4 (figures 10 b, 11 a,b), the dye visualization experiment in figure 1 produced a predominantly red-coloured stopping vortex. For $t/t_{o}\geqslant 2$ , lobe 3 alone describes the elliptic LCS boundary associated with the stopping vortex, with lobe 4 being fully entrained into it. The entrainment fraction for the stopping vortex $V_{2}$ , defined as $\unicode[STIX]{x1D702}_{V2}=A_{lobe4}/(A_{lobe3}+A_{lobe4})$ , takes a value of 18 %.

Lobe 5, which corresponds to the fluid that is present outside of the elliptic LCSs but inside of the hyperbolic LCSs at $t/t_{o}=2$ , never comes in contact with the plate and wraps around the stopping vortex as the flow evolves in time. For $t/t_{o}\geqslant 2$ , lobes 1 and 2 and lobes 3 and 4 move as individual coherent elliptic units, whereas lobe 5 gets significantly stretched and folded while still remaining a part of the advected bulk. It is worthwhile repeating that the delineations between the five different lobes are captured accurately by the repelling LCS at $t=0$ .

Figure 12.

(a) Temporal evolution of the non-dimensional circulation $\widehat{\unicode[STIX]{x1D6E4}}=\unicode[STIX]{x1D6E4}/(c^{2}\unicode[STIX]{x1D6FA}_{max})$ associated with each of the five different lobes of fluid identified in figure 10(b), where $\unicode[STIX]{x1D6E4}=\int \unicode[STIX]{x1D714}_{z}\,\text{d}A$ . (b) Corresponding non-dimensional horizontal momentum $\widehat{p_{x}}$ associated with the five lobes of fluid and the entire advected bulk, where $\widehat{p_{x}}=p_{x}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FA}_{max}c^{3})$ and $p_{x}=\int \unicode[STIX]{x1D70C}u\,\text{d}A$ . The colours of the curves in both (a) and (b) are consistent with those in figure 10(b) and the lobe numbers are marked near the curves. The solid black curves in (a,b) show the total circulation and the horizontal momentum associated with the entire advected bulk.

We define the non-dimensional circulation associated with a given lobe as $\widehat{\unicode[STIX]{x1D6E4}}=\int \unicode[STIX]{x1D714}_{z}\,\text{d}A/(c^{2}\unicode[STIX]{x1D6FA}_{max})$ , where $\unicode[STIX]{x1D714}_{z}$ is the vorticity and the integral is over the entire area occupied by the lobe. Owing to the discretized representation of a lobe, which actually is a continuous region of material fluid, the calculation of $\widehat{\unicode[STIX]{x1D6E4}}$ is non-trivial, particularly if the lobe stretches or folds significantly in its evolution. Here, $\widehat{\unicode[STIX]{x1D6E4}}$ at a given time is numerically estimated by binning an area that encompasses all the tracer particles associated with a lobe, and then identifying those bins that contain at least one tracer particle. The area integral of vorticity associated with the lobe is then performed over these identified bins. The size of the bins and the density of tracer particles were chosen such that the numerical estimate is accurate for pre-defined areas with known circulation values. More details of the numerical procedure used to calculate $\widehat{\unicode[STIX]{x1D6E4}}$ for each lobe is given in appendix B. In figure 12(a), we plot the evolution of $\widehat{\unicode[STIX]{x1D6E4}}$ associated with each of the lobes 1–5. Lobe 1 acquires negative circulation right from $t=0$ , and saturates at a nearly constant value of $-$ 1.2 at $t/t_{o}=0.6$ . In other words, lobe 1 continuously acquires negative circulation during the entire acceleration phase of the plate motion ( $0\leqslant t/t_{0}\leqslant 0.5$ ) and some time soon after ( $0.5\leqslant t/t_{0}\leqslant 0.6$ ), and acquires no further significant circulation during the remaining plate deceleration phase. It is also noteworthy that lobe 1 loses contact with the plate at some point in the interval $0.5<t/t_{0}<0.6$ , paving way for lobe 3 in the stopping vortex to gain additional circulation during the plate deceleration phase. These results build upon previous related studies (Kim & Gharib Reference Kim and Gharib2011; DeVoria & Ringuette Reference DeVoria and Ringuette2013b ) where the relation between the stopping vortex formation and the plate deceleration phase was investigated. The circulation associated with lobe 2 remains negligible till $t/t_{o}\sim 0.3$ and then increases during the interval $0.3<t/t_{o}<0.6$ before saturating at a value of $-$ 0.33. While lobe 1 acquires its vorticity from being in contact with the forward side of the plate right from $t=0$ and subsequently separates off the plate tip, lobe 2 acquires its vorticity only during entrainment by lobe 1. We recall from figure 11 that lobe 2 makes minimal contact with the plate and hence has only lobe 1 as its potential source of vorticity. The small value of the time scale ( ${\approx}0.3t_{o}$ ) over which lobe 2 acquires vorticity suggests that the intricate spiral structure inside the starting vortex significantly enhances the rate of vorticity mixing by creating additional interfacial area over which lobes 1 and 2 are in contact.

Lobe 3, which belongs to the stopping vortex, acquires vorticity right from the beginning as a result of being in contact with a significant portion of the rearward side of the plate. It is noteworthy that the stopping vortex, while appearing to emerge only after $t/t_{o}=1$ in the dye visualization experiment shown in figure 1, actually acquires vorticity right from $t=0$ . Then, at around $t/t_{o}=0.7$ , lobe 3 separates from the plate tip and hence gains additional vorticity before saturating at a value of 1 at $t/t_{o}=1.5$ . Lobe 4 attains negligible vorticity during $t/t_{o}<0.5$ before acquiring negative vorticity by being on the forward side of the plate. At $t/t_{o}=0.65$ , lobe 4 begins to get entrained by lobe 3 and hence acquires positive vorticity to reach a value of 0.15 at $t/t_{o}=2$ . Lobe 5, though being a part of the advected bulk, acquires negligible vorticity throughout. Finally, the circulation associated with the entire advected bulk (black solid curve in figure 12 a) increases in magnitude as the starting vortex forms, and then starts decreasing even before the plate motion stops.

To investigate the dynamical relevance of the various lobes in the advected bulk, we calculate the linear momentum (  $\boldsymbol{p}=\int \unicode[STIX]{x1D70C}\boldsymbol{u}\,\text{d}A$ ) associated with each of them. The numerical procedure used to calculate $\boldsymbol{p}$ is the same as that used for $\widehat{\unicode[STIX]{x1D6E4}}$ , as detailed in appendix B. We recall from our discussion in § 1 that turning manoeuvres result in the fish moving along its final orientation, which is along the $x$ -axis in our studies. Furthermore, we have shown that the generated vortex pair moves roughly along the direction of the final plate orientation. We therefore restrict our dynamical relevance studies to the horizontal momentum $p_{x}$ only. In figure 12(b), we plot the temporal evolution of non-dimensional horizontal momentum, $\widehat{p_{x}}$ defined as $\widehat{p_{x}}=p_{x}/(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FA}_{max}c^{3})$ for each of lobes 1–5 and the sum total associated with the entire advected bulk. Lobes 1 and 2 start acquiring horizontal momentum right from $t=0$ , with lobe 2 gaining $\widehat{p_{x}}$ faster. Lobe 2, which gets entrained by lobe 1 to form the starting vortex, attains maximum $\widehat{p_{x}}$ at around $t/t_{0}=0.55$ , after which its horizontal momentum decays to a small value ( ${\approx}0.05$ ) at $t/t_{0}=2$ . Lobe 1 acquires $\widehat{p_{x}}$ faster after $t/t_{0}=0.5$ before saturating at a value of around $\widehat{p_{x}}=0.2$ at $t/t_{0}=2$ . Lobes 3 and 4 start acquiring horizontal momentum at around $t/t_{0}=0.3$ , with lobe 4 possessing a small negative $\widehat{p_{x}}$ at $t/t_{0}=1$ . Lobe 3 attains a maximum $\widehat{p_{x}}$ of around 0.2 at $t/t_{0}=0.8$ . At $t/t_{0}=2$ , $\widehat{p_{x}}$ for lobe 3 flattens at around 0.09, whereas lobe 4 settles at around $\widehat{p_{x}}=0.04$ , nearly the same as lobe 2.

Strikingly, lobe 5, while being outside of the elliptic LCSs, acquires significant horizontal momentum, settling at a value of around $\widehat{p_{x}}=0.09$ at $t/t_{0}=2$ . Furthermore, lobe 5 acquires its almost steady horizontal momentum as early as $t/t_{0}=0.7$ , and contributes to 20 % of the horizontal momentum associated with the entire advected bulk at $t/t_{0}=2$ . It is therefore important that the region encompassed by the hyperbolic LCSs, which includes the elliptic LCSs and lobe 5, is considered to identify the advected bulk. Finally, the total horizontal momentum of the entire advected bulk (solid line in figure 12 b) is observed to attain a maximum value of $\widehat{p_{x}}=0.55$ at $t/t_{0}=0.8$ before reaching an almost steady value of $\widehat{p_{x}}=0.46$ at $t/t_{0}=1.5$ .

Figure 13.

(a) Temporal evolution of the total horizontal impulse ( $I_{x}(t)=\int _{0}^{t}F_{x}\,\text{d}t$ , where $F_{x}$ is the $x$ -component of the instantaneous force experienced by the plate) experienced by the plate (solid curve) and the horizontal momentum associated with the entire advected bulk ( $p_{x1-5}$ , dashed curve). (b) Temporal evolution of the ratio $p_{x1-5}/I_{x}$ . In (b), the ratio for times $t/t_{0}<0.1$ is not shown as $I_{x}$ is very small for $t/t_{0}<0.1$ .

We proceed to investigate the relation between the horizontal momentum associated with the advected bulk and the total horizontal impulse felt by the plate. As shown by the solid curve in figure 13(a), the plate experiences significant horizontal force till around $t/t_{0}=0.8$ , before the total impulse (calculated as $I_{x}(t)=\int _{0}^{t}F_{x}\,\text{d}t$ , where $F_{x}$ is the horizontal force estimated from numerical simulations) saturates at around $I_{x}=0.035$ . The evolution of $p_{x}$ associated with the entire advected bulk (dashed line) follows a similar trend as that of $I_{x}$ , suggesting that the advected bulk contains a constant fraction of the total impulse experienced by the plate. Indeed, as shown in figure 13(b), the ratio $p_{x1-5}/I_{x}$ is reasonably steady at a value of around $0.5$ for the entire duration of $0.1\leqslant t/t_{0}\leqslant 2$ . Furthermore, based on a control volume analysis, one can argue that $I_{x}$ is equal to the horizontal momentum associated with the entire flow around the plate (Epps Reference Epps2010).

The steady value of $p_{x1-5}/I_{x}$ of around 0.5 for $t/t_{0}>1$ can be partly understood in terms of how the overall fluid momentum is distributed between the advected bulk and the rest of the flow. The advected bulk is bound by the hyperbolic LCSs, which can be approximated as material lines in the flow (Peng & Dabiri Reference Peng and Dabiri2007; Haller Reference Haller2011) on which no normal flow is possible. One can therefore model the rest of the flow as a potential flow created by a bluff body whose size, shape and momentum are the same as that of the advected bulk. As seen in figure 11, the entire advected bulk can be reasonably represented by an equivalent deformed elliptical shape, whose aspect ratio is close to unity at both $t/t_{0}=2$ and $t/t_{0}=3$ . For an ellipse of major and minor axes $2a$ and $2b$ moving along its major axis, the added mass coefficient associated with its direction of motion is given by $\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}b^{2}$ , where $\unicode[STIX]{x1D70C}$ is the fluid density (Batchelor Reference Batchelor2000). Thus, the overall horizontal momentum of the potential flow around the ellipse moving along $x$ with a velocity $U$ is given by $\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}b^{2}U$ . Since the ellipse models the advected bulk filled with the fluid of density $\unicode[STIX]{x1D70C}$ , its horizontal momentum is $\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}abU$ . The ratio between the horizontal momentum of the ellipse and the overall horizontal momentum of the flow (ellipse plus the potential flow around it) is therefore $r=1/(1+b/a)$ . For a circular shape, i.e.  $a=b$ , $r=0.5$ , which is very close to the value we observe for $p_{x1-5}/I_{x}$ at $t/t_{0}=2$ in figure 13(b). For a relatively wide range of $0.75\leqslant (b/a)\leqslant 1.25$ , $r$ varies from 0.57 to 0.44, suggesting that the ratio $p_{x1-5}/I_{x}$ may remain close to 0.5 as long as the aspect ratio of the advected bulk is around unity, and does not depend too strongly on either the size or the momentum imparted to the advected bulk. It is important to note that the advected bulk at $t/t_{0}=1$ (figure 11 b) is far from being idealized as an elliptic shape of aspect ratio close to unity; the relevance of our physical arguments pertaining to added mass is therefore unclear for times closer to $t/t_{0}=1$ . Finally, it is worthwhile to note that the simple model of dividing the entire flow into what occurs inside the advected bulk and the potential flow around it is possible only via the identification of the advected bulk by hyperbolic LCSs, which encompass one continuously connected coherent region of fluid.

4.1 Repelling LCSs at $t=0$

Here, we show results from dye visualization injection experiments that were designed to directly demonstrate the relevance of the five different lobes of fluid identified at $t=0$ in § 3. Specifically, we release dye in different regions of the flow and verify if the tracer evolution is consistent with the LCSs predicted based on the numerical simulations. To recall our results from figure 10(b) in § 3, lobes 1 (dark green) and 2 (blue) belong to the starting vortex, whereas lobes 3 (red) and 4 (orange) belong to the stopping vortex. Lobe 5 (brown), while being outside the starting and stopping vortices, still is a part of the advected bulk. In figure 14, each row corresponds to a different experiment. In all the figures in the first column (corresponding to $t=0$ in each experiment) of figure 14, lobes 1–5 at $t=0$ identified using numerical simulations is superimposed on the experimentally acquired image.

Figure 14.

Experimental visualization of the temporal evolution of dye (shown in black) released in (a) lobe 1 (dark green), (b) lobe 2 (blue), (c) lobe 3 (red), (d) lobe 4 (orange), (e) outside the advected bulk and (f) on the boundary dividing lobe 5 (brown) and the region outside the advected bulk. In each row, the dye visualization is shown at $t=0$ , $t=t_{o}$ and $t=2t_{o}$ in columns 1, 2 and 3, respectively. Lobes 1–5 identified at $t=0$ using numerical simulations are superimposed on all the $t=0$ images with the same colour scheme as in figure 10(b). Circles around the initial dye location (and arrows pointing to them) are drawn in the $t=0$ images to serve as a visual aid. For the $t=t_{o}$ and $t=2t_{o}$ images in the last row, the repelling LCS at the corresponding times are shown in blue.

As shown in figure 14(a), the dye was carefully released within lobe 1 in the experiment corresponding to figure 14(a). Indeed, at $t/t_{o}=1$ and $t/t_{o}=2$ , the dye is observed to be present entirely within the starting vortex which moves away from the plate after the plate motion stops. Similarly, figure 14(b) demonstrates that the dye released within lobe 2 is also contained within the starting vortex at later times. Figure 14(c,d) provides corresponding experimental evidence for the dye released in both lobes 3 and 4 ending up in the stopping vortex. In figure 14(e), at $t=0$ , we released the dye on the forward side of the plate where none of the five lobes are present. As shown in the second and third columns in figure 14(e), the dye does not enter the advected bulk and instead remains (without any significant stretching or folding) in the neighbourhood of the plate even at $t/t_{o}=2$ . Finally, in figure 14(f), we released a patch of dye on the repelling LCS boundary at $t=0$ , thus being partly inside lobe 5 and partly outside the advected bulk. The normal repulsion of the repelling LCS boundary to the left of the starting vortex is directly evident in the second and third panel in figure 14(f) (where the repelling LCS are shown in blue dashed lines), with the portion of the dye outside the advected bulk getting left behind. The portion of the dye inside lobe 5 (brown region) gets significantly stretched and occupies the region in between the starting and stopping vortices, which is fully consistent with our conclusions in § 3. To the best of our knowledge, figure 14(f) represents the first ever experimental evidence for the normal repulsion characteristic of repelling LCSs in any flow.

In summary, figure 14 has provided detailed experimental evidence for the kinematic relevance of each of lobes 1–5 at $t=0$ in determining the tracer patterns in the flow at later times. We highlight here that we have performed several dye visualization experiments other than what are shown in figure 14. While the placement of the dye patch at desired locations was a challenge owing to small background flow that was present before the plate motion starts, the subsequent evolution of the dye patch for $t>0$ was always consistent with what the LCS analysis predicts for the initial location of the dye at the instance when the plate motion starts. It is intriguing that despite differences being present between the experiments and simulations (§ 3.1), the five different lobes in the experiment are very accurately captured by the LCS calculations based on the numerical velocity fields. This is, however, consistent with an earlier study by Haller (Reference Haller2002) that showed that hyperbolic LCSs in model velocity fields containing noise can be reliable predictors for the LCSs in the corresponding true velocity fields.

4.2 Attracting LCS

In this section, we present a direct comparison between the tracer patterns observed in dye visualization coating experiments shown in figure 1 and the attracting LCS (shown in figure 7) at the corresponding times. Figure 15(a,d) shows the attracting LCS at $t/t_{o}=1,2$ , with the corresponding dye visualization coating images shown in figure 15(b,e). There is striking qualitative similarity between the tracer patterns and the attracting LCS, including the intricate spiral structure (connected to the Front) within the starting and stopping vortices, at both $t/t_{o}=1$ and 2. The similarities between the dye visualization images and the attracting LCS are understood in terms of the kinematic properties of the attracting LCS. An initial blob of passive tracers tends to move towards and then stretches along neighbouring attracting LCSs (Voth et al. Reference Voth, Haller and Gollub2002; Mathur et al. Reference Mathur, Haller, Peacock, Ruppert-Felsot and Swinney2007), thus resulting in the blob eventually marking attracting LCSs. These results represent the first experimental study to demonstrate the relevance of strainline-based attracting LCSs in describing tracer advection, which is directly visualized without any numerical effort. It is noteworthy that earlier LCS extraction algorithms based on either FTLE thresholding (Shadden et al. Reference Shadden, Dabiri and Marsden2006) or ridge extraction (Mathur et al. Reference Mathur, Haller, Peacock, Ruppert-Felsot and Swinney2007) are not capable of resolving the spirals inside the vortices as continuous curves, thus reaffirming the benefits of a strainline-based approach.

Figure 15.

(a) Attracting LCS at $t=t_{o}$ based on numerical simulations (same as in figure 7 b), (b) image obtained at $t=t_{o}$ in the dye coating experiment (same as in figure 2 b) with the front portion of the attracting LCS in (a) shown as the white dashed line, (c) the variation of $\log (r)$ with $\unicode[STIX]{x1D719}$ associated with the spiral (connected to the Front) inside the starting vortex. Dashed and dash-dotted lines in (c) correspond to numerically obtained attracting LCS and experimental dye coating image, respectively, at $t=t_{o}$ . The definition of $r$ and $\unicode[STIX]{x1D719}$ is shown in the inset of (c). (d–f) Same as (a–c) but at the later time of $t=2t_{o}$ . The slopes of the best-fit straight lines in (c) are $-$ 0.67 and $-$ 0.8, respectively for the attracting LCS and the dye coating experiment. The corresponding values for (f) are $-$ 0.41 and $-$ 0.37.

In figure 15(b,e) we also plot the front attracting LCS boundary on top of the dye visualization images. While the point of intersection between the tracer boundary and the plate agrees well with that of the attracting LCS boundary, there is noticeable difference between the tracer boundary and the attracting LCS away from the plate. Possible reasons for this discrepancy include errors associated with the calibration of the camera, the effects of the small background flow that is present in the tank, and small differences in the plate motion profile (especially at the beginning and the end when the plate velocities are small) between the simulations and the experiments. The camera calibration errors emerge from the spatially non-uniform distortion of the image, and the camera’s line of view not being exactly orthogonal to the laser illumination plane. The agreement between the attracting LCS boundary and the Front edge presented by the dye is relatively better at a later time, i.e.  $t/t_{o}=2$ as shown in figure 15(e). Finally, we infer that the Front boundary from dye coating experiments can be used as an approximate estimate for the front edge of the advected bulk if detailed velocity field measurements are not available. Furthermore, if we assume that the starting and stopping vortices are of similar size and strength, a reflection of the Front boundary about the axis passing through the vortex centres can be used to construct the back boundary of the advected bulk too, as suggested by Olcay & Krueger (Reference Olcay and Krueger2008) for a piston-generated symmetric vortex ring.

We now proceed to characterize the spiral structure within the starting vortex in more detail. As shown in the inset of figure 15(c), we define a $r-\unicode[STIX]{x1D719}$ coordinate system, with the origin for $\unicode[STIX]{x1D719}$ located on the outer boundary of the spiral structure in the starting vortex. Plotted in figure 15(c,f) are the variation of $\log (r)$ with $\unicode[STIX]{x1D719}/2\unicode[STIX]{x03C0}$ at $t/t_{o}=1$ and 2. The red dashed curves are based on the attracting LCSs obtained using numerical simulations, and the blue dotted curves correspond to the dye patterns in the dye visualization coating experiments. The $r$ versus $\unicode[STIX]{x1D719}$ variation is observed to be well represented by a exponential fit at both times, i.e.  $r\sim \text{e}^{m(\unicode[STIX]{x1D719}/2\unicode[STIX]{x03C0})}$ , with the estimates for $m$ based on numerical simulations and dye visualization experiments being in reasonable quantitative agreement. Specifically, as indicated in the caption of figure 15, the values of $m$ from the experiment and the simulation differ by approximately 16 and 10 % at $t/t_{0}=1$ and $t/t_{0}=2$ , respectively. Interestingly, the logarithmic nature of the spiral structure within vortices seems to be prevalent in other systems as well, as reported by Everson & Sreenivasan (Reference Everson and Sreenivasan1992) for dye visualizations in turbulent jets, Kelvin–Helmholtz instabilities and vortex rings. In summary, we conclude that attracting LCSs represent objective counterparts to patterns that are observed in dye visualization experiments in any flow. Furthermore, LCSs may represent a useful complementary tool to characterize spiral structures inside vortices in general (Moffatt Reference Moffatt1993; Lepage, Leweke & Verga Reference Lepage, Leweke and Verga2005), and subsequently address questions related to passive tracer and vorticity mixing within them (Meunier & Villermaux Reference Meunier and Villermaux2003).