3.1 Typical result from the force measurements

During each run the instantaneous force $F_{x}$ (perpendicular to the plate surface) is sampled at a rate of 10 kHz. The grey line in figure 3 represents $F_{x}$ as a function of time for an experiment with a velocity $V=0.30~\text{m}~\text{s}^{-1}$ , a plate depth of $h=100$  mm and an acceleration of $a=0.82~\text{m}~\text{s}^{-2}$ . As one can see, the 10 kHz sampled raw signal shows significant fluctuations. All calculations and analyses are performed using the unfiltered signals. However, for better readability the signal is filtered using a second-order Savitzky–Golay filter (Savitzky & Golay Reference Savitzky and Golay1964) with a filter width of 201 samples, i.e. 0.02 s. The black line in figure 3 represents the filtered signal. The force signal exhibits a clear initial peak at 0.36 s which coincides with the time when the plate reaches its maximum velocity of $V=0.30~\text{m}~\text{s}^{-1}$ ; see figure 3(b). The initial peak is due to the added mass and the acceleration of the plate, as is discussed in detail in § 3.7. The time interval $0<t<0.36$  s is called the ‘acceleration phase’ and is indicated by A in figure 3. After the peak, the force gradually decreases and finally reaches a steady value for $t\gtrsim 3.4$  s. This phase is called the ‘steady phase’ and is indicated by C. The time interval in between the initial peak and the beginning of the steady phase ( $0.36~\text{s}<t<3.4$  s) is called the ‘transition phase’ and is indicated by B. The measurement ends at $t=4.6$  s. It is noted that the force signal shows two distinct local maxima during the transition phase B, indicated as ‘peak 1’ and ‘peak 2’ in figure 3. In § 3.6 it is shown that these peaks are related to the development of large flow structures in the wake of the plate. Also, throughout the experiment, high-frequency oscillations are present in the force signal, which is due to Kelvin–Helmholtz-like instabilities in the shear layer, which is further discussed in § 3.4. Right after starting the plate we observe a local peak followed by a dip in the force signal, i.e. a step response as indicated in figure 3. This is a result of the finite stiffness of the plate, the force transducer and the streamlined strut that connects the two. This finite stiffness causes a typical response of a mass–spring–damper system to a step function; in this case the sudden acceleration of the plate causes a sudden force due to the hydrodynamic mass and mass of the plate (Meirovitch Reference Meirovitch2001).

Figure 3.

(a) A typical unfiltered force signal $F_{x}$ sampled at 10 kHz (grey) and the filtered force signal (black). Throughout the force signal high-frequency oscillations are present which are caused by Kelvin–Helmholtz-like instabilities in the shear layer as discussed in § 3.4. Right after starting the plate we observe a step response due to the finite stiffness of the plate, force transducer and the strut which connects the two. (b) The plate velocity $V$ as a function of time.

3.2 The effect of the plate depth on the steady-phase drag

The plate is moved through the tank for different immersion depths $h$ to investigate the effect of the free surface on the drag on the plate. The immersion depth $h$ is varied between 0 and 200 mm in a randomised order. In total 140 runs were carried out.

The drag force $F_{x}$ during the steady phase is expressed as

(3.1) $$\begin{eqnarray}F_{x}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}V^{2}C_{D}A,\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the fluid density, $V$ the plate velocity, $A$ the frontal area of the plate and $C_{D}$ the steady-phase drag coefficient. For each run the steady-phase drag coefficient $C_{D}$ is calculated and plotted in figure 4 using open markers, where each marker represents $C_{D}$ of a single run at a given depth  $h$ . The steady phase drag coefficient $C_{D}$ reaches a minimum value of $C_{D}=1.10$ for $h=0$ , i.e. when the top of the plate is at the surface. For larger values of  $h$ , i.e. when the plate is submerged, the drag coefficient increases to a peak value of $C_{D}=1.60$ at a depth $h=20$  mm; a relative increase of 45 %. When the plate is submerged further below the surface the drag coefficient decreases, and a constant value of $C_{D}=1.3$ is reached for large  $h$ .

Figure 4.

The steady-phase drag coefficient $C_{D}$ as function of plate depth  $h$ . For reference the dashed line gives the $C_{D}$ value for flow over a fence ( $C_{D}=1.10$ ) and the solid line the $C_{D}$ value for flow around a 1 : 2 aspect ratio plate ( $C_{D}=1.30$ ).

To the knowledge of the authors, the observed behaviour of $C_{D}$ with respect to the plate depth $h$ has not been reported previously. However, to be able to make a comparison with the literature we consider two limiting cases. The first limiting case occurs for large $h$ where the free surface does not affect the drag on the plate any more. For the deep water cases we found $C_{D}=1.30$ , which is in close agreement with a value of $C_{D}=1.26$ –1.32 for a fully submerged plate with $AR=6$ (Schubauer & Dryden Reference Schubauer and Dryden1937) and $C_{D}=1.2$ –1.26 for square plates (Bearman Reference Bearman1971). Another limiting case occurs for $h=0$  mm, where we compare the measured drag coefficient with that of a fence, as if the free surface acts as a wall such that the flow does not pass over the top of the plate. Again the found drag coefficient $C_{D}=1.10$ matches well with values found in the literature $C_{D}=1.05$ –1.12 (Jacobs Reference Jacobs1985). The assumption that the plate acts like a fence is verified by letting the plate pierce the surface such that no water flows over the plate, i.e. $h<0$ . The drag coefficients for a partially submerged plate are represented by the closed markers in figure 4. Of course, for this surface piercing case the frontal projected area of submerged part of the plate is used for calculating  $C_{D}$ .

3.3 Instantaneous force signals for selected depths

Three cases are selected for further analysis: the surface case $h=0$  mm, the maximum drag case $h=20$  mm and the case $h=100$  mm, which equals one blade height  $l_{b}$ . The surface case ( $h=0$  mm) corresponds to the neutrally buoyant position of an actual rowing oar blade, $h=100$  mm, i.e. one blade height, is the practical limit of immersion during actual rowing, and the maximum drag case ( $h=20$  mm) would be optimal for propulsion. Note that a local minimum and trend break can be seen at $h=50$  mm. Although not investigated in this study, we speculate that two drag enhancing mechanisms play a role at this depth as further discussed in § 3.6. The flow visualisations in § 3.5 show that the case of $h=100$  mm develops a symmetric wake (top–bottom) which suggests it may also be representative for a fully submerged plate. The three cases are indicated in figure 4. For the three selected depths, i.e. $h=0$  mm, $h=20$  mm and $h=100$  mm, experiments are carried out at a velocity $V=0.30~\text{m}~\text{s}^{-1}$ and an acceleration $a=0.82~\text{m}~\text{s}^{-2}$ . The instantaneous drag force profiles for these three cases are shown in figure 5. It is observed that for all three cases the plate drag is rapidly increasing during the acceleration phase (A), and the maximum drag is reached at the end of this phase. For the case $h=0$  mm, the drag force is significantly lower during most of the transition phase (B) and during the steady phase (C). For the case $h=20$  mm the drag is higher than in the other cases for $t>1.5$  s, which also follows from figure 4. Significant differences occur during the transition phase. The force signals show a very different decay to their steady-phase values. The case $h=0$  mm initially shows the slowest decay but reaches the lowest steady state drag value, while the case $h=100$  mm shows the fastest decay but eventually reaches a relatively high steady-phase drag value. Also, the peaks 1 and 2 are only present during the transition phase (B) of the case $h=100$  mm.

Figure 5.

The drag force signals $F_{x}(t)$ for the three selected plate depths, $h=0$  mm,  $h=20$  mm and $h=100$  mm. The vertical dashed lines indicate time instances $t_{1}$ , $t_{2}$ and $t_{3}$ when snapshots of the hydrogen bubble flow visualisation were taken as discussed in § 3.5.

The drag signals for the three selected cases all show a steep increase during the acceleration phase and a relatively gradual decrease during the transition phase towards the steady phase. Traditionally, the steady-phase drag force is described by (3.1). However, when dealing with an accelerating object in a quiescent fluid the effect of the added mass must be incorporated into the description of the drag force  $F_{x}$ :

(3.2) $$\begin{eqnarray}F_{x}(t)=F_{CD}(t)+F_{vm}(t)=\underbrace{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}V(t)^{2}C_{D}A}_{F_{CD}(t)}+\underbrace{\overbrace{(m_{p}+m_{h})}^{m_{v}}a(t)}_{F_{vm}(t)},\end{eqnarray}$$

where $F_{CD}$ is the steady-phase drag force and $F_{vm}$ the product of the virtual mass $m_{v}$ and the plate acceleration  $a$ . The virtual mass $m_{v}$ is the sum of the plate mass $m_{p}$ and the hydrodynamic mass  $m_{h}$ . All variables in (3.2) are known except for the hydrodynamic mass  $m_{h}$ . The plate velocity $V(t)$ and the plate acceleration $a$ are prescribed as previously shown in figure 2. For each run the value for $C_{D}$ is taken from figure 4. The plate mass $m_{p}=0.400$  kg, while the hydrodynamic mass $m_{h}$ for a submerged geometry accelerating from rest is estimated by using an empirical correlation. For a rectangular flat plate with an aspect ratio of $AR=2$ for inviscid frictionless flow the hydrodynamic mass is modelled as (Patton Reference Patton1965):

(3.3) $$\begin{eqnarray}m_{h(Patton)}=0.84\frac{\unicode[STIX]{x03C0}}{4}\unicode[STIX]{x1D70C}l_{a}l_{b}^{2}=1.3~\text{kg},\end{eqnarray}$$

where $l_{a}$ and $l_{b}$ are the major dimensions of the plate (with $l_{a}>l_{b}$ ). Alternatively, Yu (Reference Yu1945) provides an empirical correlation valid for arbitrary plate aspect ratio and plate thickness  $l_{c}$ :

(3.4) $$\begin{eqnarray}m_{h(Yu)}=\unicode[STIX]{x1D70C}\left[0.788\frac{l_{a}^{2}l_{b}^{2}}{(l_{a}^{2}+l_{b}^{2})^{1/2}}+0.0619l_{a}l_{b}l_{c}^{1/2}\right].\end{eqnarray}$$

Figure 6 shows the measured drag force $F_{x}(t)$ for $h=100$  mm, next to the theoretical drag force, equation (3.2) with $m_{h}$ according to (3.4). The predicted initial peak in drag force of 2.8 N significantly underestimates the measured peak of 3.6 N. When using expression (3.3) for the hydrodynamic mass, the initial peak is estimated to be even lower at 2.6 N. Also, the predicted force shows a sharp drop as soon as the acceleration of the plate ends at $t=0.36$  s. This is because a force is no longer required for accelerating the virtual mass  $m_{v}$ . However, in the measurements a much more gradual decrease in drag force after the initial peak at $t=0.36$  s is observed. During experiments, upon accelerating the plate for $h=0$  mm and $h=20$  mm, a vortex pair is visible at the free surface which is a viscous effect, potentially explaining the mismatch between theory and experiment since the correlations for the hydrodynamic mass are for inviscid flow only. The formation of these vortices is further addressed in § 3.5.

Figure 6.

The experimentally determined drag force signal $F_{x}(t)$ for $h=100$  mm together with the theoretical drag force and its components $F_{CD}$ and $F_{vm}$ according to (3.2) with the hydrodynamic mass $m_{h}$ according to (3.4).

3.4 Shear layer instabilities

As described in the introduction, many studies investigating accelerating flat plates encounter vortices which are formed in the shear layer between the wake of the plate and the flow separating at the plate edge, e.g. Lian & Huang (Reference Lian and Huang1989). The mechanism by which these vortices are generated is similar to that of the Kelvin–Helmholtz instability and is depicted in figure 7(a). These shear layer instabilities are very close to the plate and can be seen in the force signal; see figure 3. These are also visible in the flow visualisations discussed in § 3.5, where the hydrogen bubbles end up in the vortex cores; see figure 7(c). Not only during the acceleration phase, but also during the other phases, i.e. where the plate travels at constant velocity, the secondary vortices are generated in the shear layer at a consistent frequency. When the starting vortex has disintegrated the generated secondary vortices are simply shed into the wake, as shown in figure 7(b). The frequency at which the vortices are generated was determined based on the flow visualisations for each $h$ at $a=0.82~\text{m}~\text{s}^{-2}$ and $V=0.20^{-1}$ and $0.30~\text{m}~\text{s}^{-1}$ and was found to be 17 Hz. The power spectrum of the drag force signal consistently shows a peak at this frequency for all runs. The power spectra determined for different values of acceleration, velocity and immersion depth, were all similar, i.e. the eight most dominant frequencies all lie in the range of 13–20 Hz with the most dominant frequency at approximately 17 Hz. Minor variations between runs are observed. We hypothesise that these variations are due to perturbations introduced by the experimental apparatus that might be different for each run; see also Lian & Huang (Reference Lian and Huang1989).

Figure 7.

(a) The flow separates at the plate edge. At some distance from the plate instabilities in the shear layer evolve into Kelvin–Helmholtz-like vortices. (b) At constant velocity small vortices are generated in the shear layer and shed into the wake, similar to the observations by Prandtl (Reference Prandtl1904). (c) The smaller secondary vortices generated in the shear layer during the acceleration phase roll up in the large starting vortex in a very similar way as was observed by Lian & Huang (Reference Lian and Huang1989).

3.5 Flow visualisations

To determine what causes the discrepancy between the predicted plate drag (based on a constant added mass coefficient) and measured plate drag, the flow around the plate is visualised for the three selected plate depths, $h=0$  mm, $h=20$  mm and $h=100$  mm. The acceleration is again set to $a=0.82~\text{m}~\text{s}^{-1}$ and the velocity $V=0.30~\text{m}~\text{s}^{-1}$ for all three runs. During each run the camera moves with the plate while viewing the plate and its wake from the right-hand side. Snapshots are taken at three different time instants, i.e. (i)  $t_{1}=0.35$  s, close to the end of the acceleration phase (A), (ii) at $t_{2}=0.8$  s during the transition phase (B), (iii) and $t_{3}=4.0$  s during the steady phase (C): $t_{1}$ , $t_{2}$ and $t_{3}$ are also marked in figure 5. Flow visualisation movies are available in the supplemental material at https://doi.org/10.1017/jfm.2019.102, for $h=0$ (movie 1), 20 (movie 2) and 100 mm (movie 3).

Figure 8.

Flow visualisations using hydrogen bubbles generated at the plate surface for each selected depth  $h$ , at a plate acceleration $a=0.82~\text{m}~\text{s}^{-2}$ , and plate velocity $V=0.30~\text{m}~\text{s}^{-1}$ . The hydrogen bubbles collect in the cores of the vortices that are formed in the shear layer and wake. (a,d,g) Acceleration phase; (b,e,h) transition phase; (c,f,i) steady phase.

Figure 8(a–c) shows the development of the flow for the deep water case $h=100$  mm. Immediately after setting the plate in motion ( $t=0$  s) a vortex ring forms at the plate edges, closely trailing behind the plate until the end of the acceleration phase; see figure 8(a). During the transition phase this vortex ring deforms: the top and bottom of the ring move away from the plate, while the part of the ring that formed at the left and right edges of the plate contracts towards the centre of the plate, remaining close to the plate surface; see figure 8(b). During the transition phase the vortex ring continues to stretch and finally breaks up after which the wake gradually assumes its steady shape; see figure 8(c).

In the $h=0$  mm case, the top face of the plate coincides with the air–water interface. Therefore, during the acceleration of the plate, the formation of a closed vortex ring, as found in the deep water case $h=100$  mm, is prevented. Instead, a U-shaped starting vortex is formed, of which the free ends attach to the air–water interface and produce strong depressions in the interface; see figure 8(d). A schematic side-by-side comparison of the vortices in the $h=0$ and the $h=100$  mm cases is given in figure 9. During the transition phase the U-shaped vortex detaches and quickly loses strength, visible as the flattening of the bottom of the surface vortices, meaning that they are no longer connected to a strong vortex below; see figure 8(e). Also it appears that the vortex cores are no longer strong enough to capture the hydrogen bubbles in a well-defined core. The shifting away of the surface vortices as well as the break-up of the U-shaped vortex continues until a steady-phase wake similar to that of the deep water case is formed, except that in the case of $h=0$  mm the wake is limited by the free surface, effectively cutting off part of the wake; see figure 8(f).

Figure 9.

Vortex formation during the acceleration phase. In the case of $h=100$  mm (a) the plate is fully submerged and a closed vortex ring is formed behind the plate. In the case of $h=0$  mm (b) the top of the plate coincides with the water surface, and a closed vortex ring cannot be formed. Instead, a U-shaped vortex ring is formed, which attaches to the surface with both ends, creating surface depressions.

In the $h=20$  mm case, i.e. the case with maximum drag during the steady phase, the flow behaviour is a mixture of features observed in the flows for the deep water ( $h=100$  mm) and surface cases ( $h=0$  mm). During the acceleration phase both a vortex ring and vortices connected to the air–water interface are formed; see figure 8(g). During the transition phase the surface vortices shed more quickly than in the $h=0$  mm case, deforming the vortex ring in streamwise direction; see figure 8(h). After the vortex ring disintegrates a steady wake is formed similar to that of the $h=0$  mm case. However, the gap between the top of the plate and the air–water interface causes a flow over the plate creating a large circulation zone closely trailing the plate as indicated in figure 8(i). This creates a low pressure region on the wake side of the plate explaining the maximum drag during the steady phase found for this case.

3.6 Large flow structures

Figure 5 shows that the maximum drag is reached at the end of the acceleration phase for all three cases. During the acceleration phase the drag is mainly due to acceleration of the plate mass $m_{p}$ and the hydrodynamic mass  $m_{h}$ . One would expect the plate deepest submerged to entrain most water and therefore to have the largest hydrodynamic mass. However, it is clear that both the $h=0$  mm and the $h=20$  mm cases have a higher initial peak. This is caused by the observed strong surface vortices formed in these two cases, resulting in strong low pressure zones close to the plate, and thus creating a larger drag. One more difference is the observed peaks 1 and 2 in the drag force signal for the $h=100$  mm case, which are not observed in the force signals for the $h=0$  mm and $h=20$  mm cases. It is observed that in the case of $h=100$  mm several large vortical structures are formed and shed during the transition phase, instead of just the formation of a starting vortex. The structures are of similar size and shape as the starting vortex, although less defined, and their creation and shedding coincides with peaks 1 and 2.

Coming back to the observed trend break at $h=50$  mm in figure 4, we speculate that at this depth two drag enhancing mechanisms play a role as stated in § 3.3. Firstly, a circulation zone is formed close to the plate, as is discussed in § 3.5. This circulation zone increasingly enhances plate drag from $h=0$ to $h=20$  mm where this effect reaches its maximum. We further speculate that as the gap between the top of the plate and the free surface increases, from $h=20$ to $h=50$  mm, the circulation zone weakens and consequently the plate drag decreases. The second drag enhancing mechanism is the growth of the wake size, i.e. the wake height, with increasing plate depth. Since the separation points on the flat plate are well defined due to the sharp edges of the plate, unlike e.g. the separation points on a cylinder which vary with Reynolds number (Williamson Reference Williamson1996), the drag on the flat plate and the wake size of the flat plate, i.e. the wake height, are positively correlated. At larger plate depths a smaller part of the wake is clipped by the free surface, and consequently, a larger mass of water is entrained in the plate wake thus increasing the plate drag. Figure 4 indicates that the drag during the steady phase for the $h=0$  mm case is significantly lower than that for the deep water case ( $h=100$  mm). The wake size of the surface case ( $h=0$  mm), is only approximately 75 % of the size of the deep water case ( $h=100$  mm); see figures 8(c) and 8(f). The ratio of drag coefficients $C_{D0mm}/C_{D100mm}\approx 0.8$ reflects the ratio of the wake sizes in the vertical direction ( ${\approx}0.75$ ), which is in accordance with our hypothesis that the drag and wake size are positively correlated. Finally, in figure 4 a local maximum at $h=100$  mm can be seen, although less pronounced than the maximum at $h=20$  mm. We hypothesise that this local maximum or trend break is caused by weakening interactions between the plate wake and the free surface.

In the next sections (§§ 3.7 and 3.8) the instantaneous force signal during the acceleration and transition phase is discussed. For a further analysis of the large structures on the basis of PIV measurements we refer the reader to §§ 3.9 and 3.10.

3.7 Alternative modelling of the hydrodynamic mass

As observed in the flow visualisations the entrained mass in the wake of the plate during the acceleration phase is not constant, but grows larger over time, and so does the plate drag force. This suggests an increasing hydrodynamic mass $m_{h}$ over time, instead of a constant hydrodynamic mass as suggested by Patton (Reference Patton1965) and Yu (Reference Yu1945). Moreover, due to the acceleration strong vortices are formed close to the plate that increase the drag on the plate even further through their pressure fields. To investigate the variation of the hydrodynamic mass over time, experiments are carried out for different combinations of plate velocity, acceleration and plate depth. Each combination of velocity ( $V=0.20$ , 0.25, 0.30, 0.35, $0.40~\text{m}~\text{s}^{-1}$ ) and acceleration ( $a=0.41$ , 0.62, 0.82, 1.02 1.23, 1.44, $1.64~\text{m}~\text{s}^{-2}$ ) is tested for each depth ( $h=0$ , 20, 100 mm) resulting in 105 measurements. Figure 10 shows the effects of variations in acceleration $a$ and target velocity $V$ on the drag force $F_{x}$ as function of time; note that from here on, all force signals are filtered to improve readability using a second-order Savitzky–Golay filter with a 0.1 s window. During the steady phase the plate drag scales with $F_{x}\sim V^{2}$ as is expected from (3.2); see figure 10(b). However, during the acceleration phase, the plate drag appears to increase linearly with time $F_{x}\sim t$ which for constant acceleration is identical to $F_{x}\sim V$ (figure 10 b), and is not in agreement with (3.2). Also, the increase of the plate drag with respect to time, i.e. the rate of change $\text{d}F/\text{d}t$ , appears to scale with acceleration $\text{d}F_{x}/\text{d}t\sim a$ ; see figure 10(a).

Figure 10.

Drag force signals $F_{x}(t)$ measured at plate depth $h=100$  mm. (a)  $F_{x}(t)$ for various accelerations towards a fixed plate velocity $V=0.30~\text{m}~\text{s}^{-1}$ . (b)  $F_{x}(t)$ for a fixed acceleration of $a=0.82~\text{m}~\text{s}^{-2}$ towards various plate velocities $V$ .

To investigate the increase in force with time during the acceleration phase, the common force decomposition (3.2), is rewritten such that a residual force due to the acceleration of the hydrodynamic mass $m_{h}$ is defined:

(3.5) $$\begin{eqnarray}F_{mh}=F_{x}-\underbrace{m_{p}a(t)}_{F_{mp}}-\underbrace{{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}V(t)^{2}C_{D}A}_{F_{CD}}.\end{eqnarray}$$

This modelling choice effectively isolates the increase in drag due to unsteady effects. For small movements starting from rest, the residual force $F_{mh}$ matches the product of the acceleration $a$ and the hydrodynamic mass $m_{h}$ from (3.4) as proposed by Yu (Reference Yu1945); see the markers in figure 11(a). Note that a small offset in time, equal to $t_{sr}=0.07$  s, takes into account the step response. Since the behaviour of the force signal during the acceleration phase is independent of the target velocity $V$ (figure 10 b), we use the measurements with the highest target velocity of $V=0.40~\text{m}~\text{s}^{-1}$ to fit a model, as these data contain the longest acceleration time. Figure 11(a) shows linear fits through the theoretical added mass as proposed by Yu (Reference Yu1945), with a fit for each signal defined as:

(3.6) $$\begin{eqnarray}F_{mh}=\frac{\text{d}F_{mh}}{\text{d}t}(t-t_{sr})+m_{h(Yu)}a,\end{eqnarray}$$

where the only free fitting parameter is the rate of change of the residual force $\text{d}F_{mh}/\text{d}t$ . In figure 11(b) all found rates of change of force $\text{d}F/\text{d}t$ are plotted as a function of acceleration  $a$ . The markers show the values of $\text{d}F/\text{d}t$ for different accelerations for the three selected depths, while the lines indicate the clear linear behaviour of $\text{d}F/\text{d}t$ with respect to acceleration. To give a physical interpretation to this, it is assumed that the residual force $F_{mh}$ during the acceleration phase is due to the acceleration of the hydrodynamic mass in the wake of the plate. Note that this assumption does not necessarily capture the intricate phenomena of the formation of the ring vortex or the subsequent deformation of the free surface, which possibly increase the plate drag via other mechanisms than entrainment. However, this assumption does enable us to model the enhanced plate drag in a simple and convenient manner. We define the residual force as:

(3.7) $$\begin{eqnarray}F_{mh}=m_{h}a,\end{eqnarray}$$

where the hydrodynamic mass $m_{h}$ is time dependent since the acceleration $a$ is constant for each measurement, while $F_{mh}$ is increasing in time. This leads to the definition of the entrainment rate of mass in the wake of the plate:

(3.8) $$\begin{eqnarray}\frac{\text{d}m_{h}}{\text{d}t}=\frac{1}{a}\frac{\text{d}F_{mh}}{\text{d}t},\end{eqnarray}$$

shown as a function of the acceleration $a$ in figure 11(c). When the plate is at the free surface ( $h=0$  mm) the entrainment rate $\text{d}m_{h}/\text{d}t$ is strongly enhanced when compared to the deeper immersed plates; at lower accelerations this almost doubles. Also, we show that for all three selected plate depths the entrainment rate is gradually increasing with acceleration up to $a\approx 1~\text{m}~\text{s}^{-2}$ , while for higher accelerations the entrainment rate has a constant value. At a higher entrainment rate the wake grows more rapidly over time thus faster increasing the plate drag force. The maximum force experienced by the plate during uniform acceleration is the sum of the steady phase drag force $F_{CD}$ and the residual force  $F_{mh}$ , which are both dependent on immersion depth $h$ as shown in figures 4 and 11(c), respectively. The maximum force achieved at some depth $h$ is then dependent on the set target velocity $V$ and acceleration  $a$ . Apparently a trade-off exists between steady-phase drag and entrainment, e.g. the case $h=0$  mm has the lowest steady-phase drag coefficient but has the highest entrainment rate.

Figure 11.

(a) The solid line represents the residual force $F_{mh}$ as function of time ( $t$ ) for different accelerations towards a fixed plate velocity $V=0.40~\text{m}~\text{s}^{-1}$ at plate depth $h=0$  mm. The markers represent the force due to hydrodynamic mass as proposed by Yu (Reference Yu1945). The dashed line represents a linear fit through the force signal $F_{mh}(t)$ during the acceleration phase. The vertical dotted line marks the time offset $t_{sr}$ to account for the step response. (b) The rate of change of force $\text{d}F/\text{d}t$ as function of the accelerations $a$ for the selected plate depths. (c) The entrainment rate $\text{d}m_{h}/\text{d}t$ as function of acceleration $a$ for the three selected plate depths.

3.8 Force during the transition phase

Figure 11(a) shows that the residual force $F_{mh}$ does not vanish immediately after the plate reaches its target velocity (i.e. when $a$ reaches $0~\text{m}~\text{s}^{-2}$ ), although this would be expected from the general definition; see (3.7). Instead, the force $F_{mh}$ appears to drop sharply to a non-zero value and then gradually decays to zero, after which the total force $F_{x}$ equals the steady-phase drag force  $F_{CD}$ . The force which vanishes immediately upon $a$ reaching 0 we call a ‘force due to added mass’, since it relates to the acceleration of the plate. The gradually decreasing force we call a ‘history force’, which is due to the developing wake caused by the past acceleration of the plate, which exists long after the plate reaches its target velocity. This is analogues to the virtual mass force for a spherical particle: $F=(\unicode[STIX]{x1D70C}_{c}V_{p}/2)(\text{D}\boldsymbol{u}/\text{D}t-\text{d}\boldsymbol{v}/\text{d}t)$ (Crowe et al. Reference Crowe, Schwarzkopf, Sommerfeld and Tsuji2011), where $\unicode[STIX]{x1D70C}_{c}V_{p}/2$ is the displaced fluid mass, $\text{d}\boldsymbol{v}/\text{d}t$ the particle acceleration (added mass term) and $\text{D}\boldsymbol{u}/\text{D}t$ the material derivative of the fluid (history term).

We assume that the plate reaches the constant target velocity at time $t_{c}$ and location  $x_{c}$ . As discussed in § 3.5 the acceleration and transition phase are dominated by the formation and shedding of vortices. As discussed in § 1.3, a suitable dimensionless time to describe these flow phenomena is  $t^{\ast }$ , i.e. the ‘formation time’, defined as

(3.9) $$\begin{eqnarray}t^{\ast }=\frac{1}{l_{b}}\int _{0}^{t}V(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F},\end{eqnarray}$$

where $V(t)$ is the instantaneous velocity at time $t$ , and $l_{b}$ a characteristic length, which here is chosen to be the plate height. The non-dimensional time $t^{\ast }$ is then effectively the number of plate heights travelled, which is identical to the non-dimensional time used in the simulations by Koumoutsakos & Shiels (Reference Koumoutsakos and Shiels1996).

Figure 12.

The residual force in dimensional form $F_{mh}$ (a,c) and non-dimensional form $F^{\ast }$ (b,d) as a function of the post-acceleration time $t-t_{c}$ or formation time $t^{\ast }-t_{c}^{\ast }$ for different accelerations towards a fixed plate velocity $V=0.30~\text{m}~\text{s}^{-1}$ (a,b) or towards different velocities at a fixed acceleration $a=0.82~\text{m}~\text{s}^{-2}$ (c,d) for plate depth $h=100$  mm. In (a,b) the dashed line indicates the dimensionless step response time $t_{sr}^{\ast }$ . In (d) the dotted lines indicate the locations of peak 1 and peak 2 which for different velocities coalesce in non-dimensional time. (a) Residual force as function of post-acceleration formation time for $h=100$  mm, velocity $V=0.30~\text{m}~\text{s}^{-1}$ and varying accelerations. (b) Normalised residual force as function of post-acceleration formation time for $h=100$  mm, velocity $V=0.30~\text{m}~\text{s}^{-1}$ and varying accelerations. (c) Residual force as function of post-acceleration time for $h=100$  mm, acceleration $a=0.82~\text{m}~\text{s}^{-2}$ and varying velocities. (d) Normalised residual force as function of post-acceleration formation time for $h=100$  mm, acceleration $a=0.82~\text{m}~\text{s}^{-2}$ and varying velocities.

Figure 12 shows the drag force signal as a function of the post-acceleration formation time $t^{\ast }-t_{c}^{\ast }$ during the early transition phase for a plate depth $h=100$  mm and a velocity $V=0.30~\text{m}~\text{s}^{-1}$ . Note that $t^{\ast }$ at constant velocity linearly increases with dimensional time  $t$ . As shown in figure 12(a) the force signals $F_{mh}$ quickly decrease when the acceleration ends and converge after $t^{\ast }-t_{c}^{\ast }\approx 1$ , i.e. when the plate has travelled at constant velocity $V$ over a distance of a single plate height  $l_{b}$ . This implies that the varying accelerations at which the plate accelerated to its target velocity do not cause lasting effects in the flow beyond the travelled distance of a single plate height  $l_{b}$ . Next, we introduce the normalised force $F^{\ast }$ defined as

(3.10) $$\begin{eqnarray}F^{\ast }=\frac{F_{mh}}{F_{mh}(t_{c})},\end{eqnarray}$$

where the residual force $F_{mh}$ is normalised by its maximum value at time  $t_{c}$ . In figure 12(b) it can be seen that the normalised force $F^{\ast }$ immediately after the acceleration ends, thus within the step response time of the force signal due to the abrupt change in force on the plate (indicated by the dashed line), the force has decreased by 50–70 %. The initial decrease appears to scale with acceleration, i.e. force signals based on a higher acceleration have a larger initial decrease, while for lower acceleration the initial decrease is smaller. This implies that the added mass force is dominant for higher accelerations, while for lower accelerations $F_{mh}$ is more like a history force.

The force $F_{mh}$ as a function of $t-t_{c}$ does not converge for different velocities when $a$ remains constant (see figure 12 c). The normalised force $F^{\ast }$ as a function of $t^{\ast }-t_{c}^{\ast }$ does not converge for different velocities either, but the low-frequency peaks, indicated as peak 1 and peak 2, now nicely align in dimensionless time (see figure 12 d). Those peaks are associated with the formation and shedding of large vortical structures in the wake, see § 3.6. It appears that the formation time  $t^{\ast }$ , as proposed by Gharib et al. (Reference Gharib, Rambod and Shariff1998) and applied to a vortex ring generated by a jet, also is a universal time scale for the development of the vortical structures in the wake of the plate. Also, the lower velocities show a larger initial decrease than the higher velocities, indicating that for lower velocities the residual force $F_{mh}$ acts more like a force due to added mass than like a history force.

Figure 13(a) shows that, also for the case of $h=0$  mm for different accelerations towards a fixed plate velocity of $V=0.30~\text{m}~\text{s}^{-1}$ , the force signals converge after the plate has travelled at constant velocity over a distance of a few plate heights, i.e. the plate drag force is no longer affected by the initial acceleration by which the plate reached its target velocity. However, when considering the normalised force $F^{\ast }$ for the case $h=0$  mm, shown in figure 13(b), it is obvious that the initial decrease, i.e. the decrease within the step response time $t_{sr}^{\ast }$ , is much smaller than for the case of $h=100$  mm (see figure 12 b); 20 %–40 % for $h=0$  mm, compared to 50 %–70 % for $h=100$  mm. This indicates that the effect of added mass is far less pronounced for the surface case ( $h=0$  mm) than it is for the deeper immersed plate ( $h=100$  mm) and that the residual force $F_{mh}$ for $h=0$  mm acts more like a history force than a force due to added mass. Despite the difference in the initial decrease of the residual force and the absence of peak 1 and peak 2 in the force signal for the case of $h=0$  mm, the force signal is similar to that of the $h=100$  mm case with respect to changing acceleration: the relative initial decrease is proportional to acceleration.

Figure 13.

The residual force in dimensional form $F_{mh}$ (a,c) or non-dimensional form $F^{\ast }$ (b,d) as a function of the post-acceleration time $t-t_{c}$ or formation time $t^{\ast }-t_{c}^{\ast }$ for different accelerations towards a fixed plate velocity $V=0.30~\text{m}~\text{s}^{-1}$ (a,b) or towards different velocities at a fixed acceleration $a=0.82~\text{m}~\text{s}^{-2}$ (c,d) for plate depth $h=0$  mm. In (a,b) the dashed line indicates the dimensionless step response time $t_{sr}^{\ast }$ . (a) Residual force as function of post-acceleration formation time for $h=0$  mm, velocity $V=0.30~\text{m}~\text{s}^{-1}$ and varying accelerations. (b) Normalised residual force as function of post-acceleration formation time for $h=0$  mm, velocity $V=0.30~\text{m}~\text{s}^{-1}$ and varying accelerations. (c) Residual force as function of post-acceleration time for $h=0$  mm, acceleration $a=0.82~\text{m}~\text{s}^{-2}$ and varying velocities. (d) Normalised residual force as function of post-acceleration formation time for $h=0$  mm, acceleration $a=0.82~\text{m}~\text{s}^{-2}$ and varying velocities.

Figure 13 shows the change in force signals during the transition phase for different velocities and a fixed acceleration of $a=0.82~\text{m}~\text{s}^{-2}$ for the case $h=0$  mm. Comparison of figures 13(b) and 12(b) shows that the initial decrease for all velocities is again smaller for the case of $h=0$  mm than for the deeper immersed plate $h=100$  mm. However, in each case the initial decrease is largest for the smallest velocity and becomes smaller with increasing velocity. Contrary to the deeper immersed plate $h=100$  mm, the force signal of the $h=0$  mm case shows no peaks during its decrease in magnitude from acceleration phase to steady phase. Not unexpectedly due to the absence of characteristic features during the transition phase, introducing the formation time $t^{\ast }$ does not seem to better align the force signals in time during the transition phase.

When considering the force–time signal of the entire run at a depth of $h=0$  mm for different accelerations towards a target velocity of $V=0.30~\text{m}~\text{s}^{-1}$ , shown in figure 14(a), we see that the signals do not converge until $t=1.5$  s. However, when introducing the formation time $t^{\ast }$ we clearly see that all force signals collapse on a single curve during the transition phase; see figure 14(b). Different target velocities each appear to have their own unique curve to which force signals from different accelerations collapse, as shown in figure 15. The curve to which all signals collapse are of the form $c_{1}/(x-c_{2})^{2}$ with $c_{1}$ and $c_{2}$ increasing with increasing target velocity $V$ . The fitting coefficients are shown in the legend of figure 15. The shapes of these curves appear to be determined in full by the target velocity. The formation number from which a force signal starts following this curve depends on the acceleration only. We consider a scaling argument to interpret the behaviour of the force signal by $1\sim 1/x^{2}$ . From the definition of the formation time  $t^{\ast }$ , equation (3.9), it is obvious that $t^{\ast }$ is identical to the plate travel distance $x(t)$ expressed in the number of plate heights  $l_{b}$ . The flow visualisation showed that the acceleration phase is dominated by the generation of a large starting vortex which is shed upon reaching the target velocity; see § 3.5. Upon shedding this starting vortex quickly becomes irrotational since it is no longer subject to an external force. Let $U$ be the flow velocity of such an irrotational vortex that scales with the inverse of the distance to this vortex $U\sim 1/x$ . Since dynamic pressure $q$ scales with velocity squared $q\sim U^{2}$ we can state that the dynamic pressure at the wake side of the plate due to the shed vortex scales with $q\sim 1/x^{2}$ . For the sake of simplicity we assume that we can use Bernoulli’s principle for incompressible irrotational flow such that the total pressure $P$ is constant throughout the flow, i.e. $P=p+q$ . We state that the static pressure $p$ increases over plate travel distance as $q$ decreases by ${\sim}1/x^{2}$ . Since the plate drag is the difference in static pressure over the plate, an increase in static pressure at the wake side of the plate implies a decrease in drag force on the plate. This scaling argument matches the observed behaviour $F_{mh}\sim 1/x^{2}$ .

Figure 14.

Force versus time $t$ and the formation time $t^{\ast }$ for varying accelerations towards $V=0.30~\text{m}~\text{s}^{-1}$ at depth $h=0$  mm. (a) Dimensional time  $t$ . (b) Formation time  $t^{\ast }$ .

Figure 15.

Force signal $F_{mh}$ versus formation time $t^{\ast }$ during the transition phase for different accelerations towards different velocities (solid lines). Fits of the form $c_{1}/(t^{\ast }-c_{2})^{2}$ through each set of force signals corresponding to a single velocity (dashed lines).

3.9 Vorticity

The flow field was analysed using PIV to provide a quantitative insight and to reveal the more intricate structures in the flow that were not captured by the visualisations. Although the flow is highly three-dimensional, as is evident from figure 8, two-component planar PIV still reveals interesting differences, and similarities, between the three selected cases $h=0$ , 20 and 100 mm.

The PIV measurements are done in the horizontal mid-plane of the plate for all three selected plate depths $h=0$ , 20 and 100 mm, at an acceleration of $a=0.82~\text{m}~\text{s}^{-1}$ , and a target plate velocity $V=0.30~\text{m}~\text{s}^{-1}$ . The chosen parameters correspond with the three force measurement results shown in figure 5 and discussed in § 3.3. The PIV results are represented through the vorticity $\unicode[STIX]{x1D714}_{z}$ based on the local flow circulation by an 8-point estimation (Luff et al. Reference Luff, Drouillard, Rompage, Linne and Hertzberg1999). A discrete representation is used (Adrian & Westerweel Reference Adrian and Westerweel2011) since it allows for different spacing between vectors in the $x$ - and $y$ -directions. In the here presented results the dimensionless vorticity

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{z}^{\ast }=\frac{\unicode[STIX]{x1D714}_{z}l_{b}}{V}\end{eqnarray}$$

is used, similar to Ringuette et al. (Reference Ringuette, Milano and Gharib2007). The locations $x$ and $y$ are made dimensionless by $l_{b}$ resulting in $x^{\ast }$ and  $y^{\ast }$ , respectively, such that the plate location $x^{\ast }$ is identical to the formation time  $t^{\ast }$ , i.e. $x^{\ast }(t^{\ast })=t^{\ast }$ . The order of presentation of the different depths is the same as in the results of the flow visualisation in § 3.5, i.e. $h=100$  mm, followed by $h=0$  mm, and $h=20$  mm.

Figure 16.

Dimensionless vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }$ for different instances in time $t$ at the three selected depths (a)  $h=100$  mm, (b)  $h=0$  mm and (c) 20 mm. The plate location $x^{\ast }$ matches the formation time  $t^{\ast }$ , i.e. $x^{\ast }(t^{\ast })=t^{\ast }$ .

Figure 16 shows the vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }$ for all three selected depths for different times  $t$ , including the instances in time corresponding to those of the flow visualisations in figure 8. Note that a movie of the time evolution of all three cases, side-by-side, is available in the supplemental material (movie 4). For all three cases the formation of a vortex pair during the acceleration phase, i.e. up to $t=0.35$  s appears very similar and matches to the cross-section of the vortex ring at the mid-plane of the plate, as shown in figure 8(a,d,g).

When the plate approaches $x^{\ast }=2$ at $t=0.80$  s the wakes start to differ. The vortex pair of the case $h=100$  mm is still very close and concentrated to the mid-plane of the plate, as is also seen in figure 8(b), while in both cases $h=0$ and $h=20$  mm the vortices start to travel away from the plate and become less defined. This matches the behaviour shown in figure 8(e,h).

For $t^{\ast }>4$ at $t=1.80$  s, i.e. well into the transition phase, each wake becomes very distinct. The vortices in the case of $h=100$  mm are still attached to the plate, but have moved inwards, i.e. the vortex ring has contracted in the $y$ -direction, and two large circulation zones around the vortex pair are formed in which the secondary vortices formed in the shear layer are shed. During this contraction process the two highly concentrated circulation zones move from the plate edge towards the centre of the plate creating a strong and large low pressure zone. We speculate that this is associated with a characteristic peak in the drag force at $t^{\ast }=2.8$ , e.g. shown as peak 1 in figure 12(d). At the same plate depth $h=100$  mm, some time after the situation at $t=1.80$  s shown in figure 16, the two well-defined vortices trailing closely behind the centre of the plate touch and collapse which may be associated with a second peak in the drag force signal, e.g. shown as peak 2 in figure 12(d). In the other two cases the circulation zones are not so concentrated. The large circulation zones in the case of $h=0$  mm stretch in streamwise direction and detach, while the circulation zones in the case of $h=20$  mm already disintegrate into a chaotic wake.

Well into the steady phase, i.e. at $t=4.00$  s where the plate passes $x^{\ast }=11.5$ , the wakes also look very distinct. The case of $h=100$  mm shows an asymmetric wake with what appears to be an oscillating tail, which is a well-known phenomenon for steady flow over a plate (Fage & Johansen Reference Fage and Johansen1927; Hemmati, Wood & Martinuzzi Reference Hemmati, Wood and Martinuzzi2016). The cases $h=0$  mm and $h=20$  mm both show a symmetric wake of similar size, but the latter has a large amount of vorticity very close to the plate which, we conjecture, (partially) explains the maximum steady-phase drag $C_{D}$ found for this plate depth ( $h=20$  mm), see figure 4. Also, we note the difference in wake angle. Whereas the wake angles in the vertical plane are quite similar, clearly visible in the visualisation (figure 8), the wake angles in the horizontal plane differ substantially. It appears that the case with the highest drag ( $h=20$  mm) has the smallest wake angle, while the case with the lowest drag ( $h=0$  mm) has the largest wake angle, see figure 16.

Figure 17.

Dimensionless vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }$ for different instances in time $t$ at a selected depth of $h=100$  mm. The flow behaviour appears to be very similar for both realisations. Only in the steady phase $t=4.00$  s the tail of the wake sweeps up in one realisation (a) and down in the other (b) which matches the characteristic oscillating wake as was already observed by Fage & Johansen (Reference Fage and Johansen1927). The dashed line around the outer border shows the area of integration $S$ for which the circulation is obtained. The symmetry line through $y^{\ast }=0$ is also marked and shows the dividing line between integration surfaces $S_{-}$ and  $S_{+}$ .

To test the reproducibility of the PIV measurements, the experiments were performed three times for each plate depth  $h$ . The realisations at the same depth are highly reproducible. Only at a plate depth of $h=100$  mm around $t=4.00$  s a clear difference between realisations is found, as shown in figure 17. The two different realisations are very similar except that the tail of the wake of the realisation shown in the top figure sweeps up, while in the figure at the bottom the tail sweeps down. A sweeping tail which starts in an arbitrary direction shows that this is not due to imperfections in the experimental set-up.

3.10 Circulation and shedding events

In previous studies (Gharib et al. Reference Gharib, Rambod and Shariff1998; Ringuette et al. Reference Ringuette, Milano and Gharib2007) the total circulation $\unicode[STIX]{x1D6E4}$ was used to identify vortex shedding events. The total circulation is obtained by integrating the local vorticity over a surface, i.e.

(3.12) $$\begin{eqnarray}\unicode[STIX]{x1D6E4}=\int _{S}\unicode[STIX]{x1D714}_{z}\,\text{d}S.\end{eqnarray}$$

The dimensionless circulation $\unicode[STIX]{x1D6E4}^{\ast }$ is obtained by integration of $\unicode[STIX]{x1D714}_{z}^{\ast }$ (defined in (3.11)) over the dimensionless flow field area $S$ formed by the entire range of $x^{\ast }$ and  $y^{\ast }$ . Kelvin’s circulation theorem states that integration over $S$ should yield a net circulation of $\unicode[STIX]{x1D6E4}^{\ast }=0$ if the integration boundary is along a material contour (Cohen & Kundu Reference Cohen and Kundu2007) as long as the flow is symmetric, i.e. produces no lift. The flow conditions for the theorem to hold, i.e. inviscid and barotropic flow, are fulfilled by choosing the material contour along the boundaries of the field of view of the PIV measurements, where the flow is at rest, indicated by the dashed line along the outer border in figure 17(b).

The grey lines in figure 18 show the circulation $\unicode[STIX]{x1D6E4}^{\ast }$ as a function of the formation time $t^{\ast }$ for $h=100$ , 0 and 20 mm, respectively. For all three plate depths, throughout each realisation, the total circulation $\unicode[STIX]{x1D6E4}^{\ast }$ is very close to zero. However, at the intervals $4.8<t^{\ast }<6$ and $t^{\ast }>9.4$ the measured circulation $\unicode[STIX]{x1D6E4}^{\ast }$ does not appear to be conserved perfectly. This deviation between measurements and theory is explained by measurement errors (random noise). The non-zero total circulation based on $S$ coincides with the stitching seams of the flow field as indicated by the dashed vertical lines in figure 18, showing that the flow field is not stitched seamlessly. In the case of $h=20$  mm the trailing vortex, as shown in figure 8(i), might cause large out of plane motion which also affects the measurement accuracy (Adrian & Westerweel Reference Adrian and Westerweel2011).

As was noted before, the flow field maintains symmetrical well into the steady phase. To quantify this symmetry and to identify vortex shedding events, the flow field area $S$ is subdivided into two areas along the symmetry plane of the plate, $S_{-}$ and  $S_{+}$ , corresponding to $y^{\ast }<0$ and $y^{\ast }>0$ , respectively (figure 17 b). Integration of the vorticity over each area results in the total circulation $\unicode[STIX]{x1D6E4}^{\ast }$ of each region, as shown in figure 18. Throughout all runs, i.e. for all cases and each realisation, the circulation is $\unicode[STIX]{x1D6E4}^{\ast }>0$ for area $S_{+}$ and $\unicode[STIX]{x1D6E4}^{\ast }<0$ for area  $S_{-}$ . Clearly, for $t^{\ast }<4.8$ all cases behave very similar with virtually no difference between realisations.

The ability of the total circulation $\unicode[STIX]{x1D6E4}^{\ast }$ to identify vortex shedding becomes clear at $6<t^{\ast }<8$ for the case $h=100$  mm, where a temporary decrease in circulation occurs due to the two touching vortices as described in § 3.9. The same mechanism is described by Ringuette et al. (Reference Ringuette, Milano and Gharib2007) as an ‘inward’ pinch-off. They obtained a circulation profile with ‘inward’ pinch-off occurring at $t^{\ast }\approx 4$ where in this study the pinch-off occurs at $t^{\ast }\approx 6$ . Their circulation profile is very similar in shape compared to the one presented here, even though the plate in their work pierces the surface and has different dimensions, i.e. $l_{a}=0.127$  m and height $l_{b}=0.0635$  m. In the case of $h=20$  mm an ordinary pinch-off occurs at $t^{\ast }=8$ , identified by an interval of constant total circulation (Gharib et al. Reference Gharib, Rambod and Shariff1998). For the case of $h=0$  mm shedding events were not observed in the vorticity field, nor were indications of those events present in the measured circulation.

Apart from identifying shedding events, the total circulation in $S^{-}$ and $S^{+}$ in figure 18 quantitatively shows that the experiments are well reproducible, although the circulation increasingly deviates for larger times  $t^{\ast }$ . This deviation is smallest for the case $h=0$  mm, which suggests that the free surface suppresses this deviation.

Figure 18.

The grey lines show the circulation $\unicode[STIX]{x1D6E4}^{\ast }$ as a function of formation time $t^{\ast }$ based on the total flow field  $S$ , while the black lines show the circulation $\unicode[STIX]{x1D6E4}^{\ast }$ based on the top half-area $S_{+}$ and bottom half-area $S_{-}$ as indicated in the figure. The three different realisations at each depth are represented by different line styles. A break-up event in the case $h=100$  mm is clearly visible around $t^{\ast }=7$ , as is a pinch-off event around $t^{\ast }=8$ in the case $h=20$  mm. The vertical dashed lines show the stitching seams of the flow field.