3.1 Comparison between the analytical and the numerical solutions

The output of our model is the circulation (2.16) and the position (2.15) of the vortex as functions of $r/c$ . These are well-defined quantities for a line vortex, but there exist many alternative definitions of a vortex when it has a diffuse core. For an objective comparison between the theoretical estimates and the CFD results, let us not restrict our attention to the vorticity in the core. Instead, since the flow over the plate at $90^{\circ }$ is symmetric, let us consider the circulation $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}(r)$ obtained by integrating the radial vorticity component $\unicode[STIX]{x1D714}_{r}$ over the entire half-cylinder of radius $r$ above the symmetry plane shown with green dashed lines in figure 1(a). When using $\unicode[STIX]{x1D714}_{r}$ obtained from the CFD, the vertical extent of the domain is truncated at $L_{y}/2=10c$ , yielding

where $y$ is the vertical coordinate and $\unicode[STIX]{x1D719}$ is the azimuthal coordinate.

Figure 2.

Comparison between the theoretical estimates and the CFD results for the circulation $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}$ over a cylinder surface of radius $r$ (a–d), and the distances between the edge and the vorticity centroids in the directions parallel and perpendicular to the plate, $d_{\Vert }$ and $d_{\bot }$ , respectively (e–h). Estimates for the point vortex circulation $\unicode[STIX]{x1D6E4}_{1}$ , the components of its distance to the edge $d_{1\bot }$ , $d_{1\Vert }$ , and the half-plate circulation $\unicode[STIX]{x1D6E4}_{half\text{-}plate}$ are added for reference. All quantities are normalized. A Matlab program that reproduces this figure is provided as supplementary material available at https://doi.org/10.1017/jfm.2017.257.

On the other hand, in the theoretical model, the line vortex substitutes for all vorticity in the entire domain with the exception of the boundary layers on the plate. The boundary layer vorticity is represented by the ‘bound’ circulation along a contour that intersects with the plate but does not encompass the point vortex in the physical fluid domain. The bound circulation of a half-plate is estimated using the values of $W$ given by (2.4) at $c/2$ distance from the edge, on the pressure and on the suction side of the plate (see appendix B for the derivation), resulting in

where $z_{1}$ is a real number, as given by (2.15). The theoretical estimate for $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}(r)$ is therefore

with the two components on the right-hand side evaluated using (2.16) and (3.2), respectively.

Figure 2(a–d) presents a comparison between $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{theory}$ and $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{CFD}$ at different flow regimes characterized by the root-based Reynolds number $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}$ in the range between 8 and 83. The equivalent Reynolds number based on the wing-tip velocity and the chord length $Re=\unicode[STIX]{x1D6FA}Rc/\unicode[STIX]{x1D708}$ is in the range $Re=50{-}500$ . All quantities are normalized. The agreement between the theoretical and the numerical results is good in all cases. The shape of the profiles makes the theoretical $4/3$ power law apparent, while the good pointwise agreement is ensured by substituting $K_{sp}$ with a fit

that minimizes the root-mean-square error, as discussed in the next section. Note that, even if $\tilde{K}_{sp}$ depends only on $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}$ , the dimensionless circulation (2.16) and position (2.15) depend on $r/c$ as well. It is straightforward, however, to derive a normalization that yields normalized $z_{1}$ and $\unicode[STIX]{x1D6E4}_{1}$ being functions of the root-based Reynolds number only: $z_{1}/(r^{2}c)^{1/3}=1.087(\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708})^{-1/3}$ and $\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6FA}(r^{2}c)^{2/3}=4.632(\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708})^{-1/6}$ . Similar expressions can be written in terms of the local spanwise Reynolds number $\unicode[STIX]{x1D6FA}rc/\unicode[STIX]{x1D708}$ .

Figure 3.

Flow over the plate at $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=83$ . (a) Vorticity isocontours superimposed with the theoretical position of the line vortex $z_{1}$ . The vorticity scale is in $\text{s}^{-1}$ . The grey rectangles highlight the integration domain used for calculation of the vortex circulation in this example. (b) Normalized circulation of the vortex as a function of the normalized spanwise distance.

As $Re$ increases, $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}$ becomes smaller. This is related to the vortex becoming nearer to the edge, as shown in figure 2(e–h), in terms of the components of the distance between the edge of the plate $z_{edge}$ and the vorticity central line $z_{\unicode[STIX]{x1D6F4}}$ in the directions perpendicular and parallel to the plate, $d_{\bot }=\text{Re}(z_{\unicode[STIX]{x1D6F4}}-z_{edge})$ and $d_{\Vert }=\text{Im}(z_{\unicode[STIX]{x1D6F4}}-z_{edge})$ , respectively. The vorticity central line $z_{\unicode[STIX]{x1D6F4}}(r)$ in the CFD is calculated as

This definition is equally suitable for flows at any $Re$ , including those cases when it is difficult to identify the vortex core. Its counterpart in the line vortex model is

where $z_{half\text{-}plate}$ is calculated using the distribution of bound vorticity over the plate, as explained in appendix B. The agreement between the theoretical estimate and the results of the numerical simulation is the best over the inner central part of the plate. When $r/c>5$ , the wing-tip effects become dominant and the vorticity spreads far behind the plate in the CFD results. This effect is beyond the limitations of our theoretical model of the edge vortex, which neglects aerodynamic interactions with the wing tip.

When $Re$ is sufficiently large, the edge vortex has a distinguishable core of large axial vorticity. Let us compare its properties with the line vortex model estimate at $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=83$ . In PIV experiments as well as in numerical simulations, the circulation is usually calculated by summing up the spanwise vorticity contained in flat rectangular windows (cf. Carr, DeVoria & Ringuette Reference Carr, DeVoria and Ringuette2015). Therefore, in this example, we also use flat windows of height $c$ and width $0.5r$ , shown as grey shaded areas in figure 3(a). Sectional isolines of the vorticity component perpendicular to the integration planes reveal the vortex core. The white line superimposed on figure 3(a) shows the theoretical estimate (2.15) for the top edge vortex line. It passes through the vorticity core, which means that $z_{1}$ calculated using the line vortex model is a reasonable prediction for the apparent position of the vortex. Note that, even in two dimensions, the position of the point vortex does not exactly match the position of maximum vorticity (see Wang & Eldredge Reference Wang and Eldredge2013).

Figure 3(b) shows the normalized edge vortex circulation estimated by integration of the vorticity over the selected windows. CFD results obtained with two different discretization grids are shown: the original grid with 4.61 million cells and a refined grid with 9.96 million cells. The difference between these two results is less than 0.2 % for all $r/c<5$ , and only becomes noticeable near the tip where the wing-tip vortex enters in the integration domain. The theoretical estimate for $\unicode[STIX]{x1D6E4}_{1}$ (2.16) is in good agreement with the CFD results, with the difference being less than 17 % for all $r/c<5$ .

A remarkable property of the theoretical scaling law of $\unicode[STIX]{x1D6E4}_{1}$ with $r$ is that the exponent in (2.16) is independent of any parameters. It is therefore important to determine the best power law for fitting the CFD results. Therefore, we have carried out a two-parameter optimization of

and determined the values of $A$ and $B$ that minimize the root-mean-square deviation with respect to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{CFD}$ . The optimal values of $B$ are shown in figure 4. The mean value of $B$ over the considered range of $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}$ is 1.32, which differs by only 1 % from the theoretical estimate $4/3$ for the growth rate of $\unicode[STIX]{x1D6E4}_{1}$ with  $r$ .

Figure 4.

Optimal values of the power-law exponent in $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{fit}$ that best-fit $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{CFD}$ in the least-mean-squares sense.

The CFD data presented above are for a wing with aspect ratio 6. The wing length does not enter in our theoretical estimate, but in the numerical simulation there may be some wing-tip effects when the aspect ratio is small. We have carried out an additional numerical simulation of a wing with the chord length 2 mm, i.e. twice as wide as the original plate. The angular velocity is equal to $\unicode[STIX]{x1D6FA}=260~\text{s}^{-1}$ . Apart from that, all parameters are the same. In particular, the wing length is equal to $6$  mm. The aspect ratio is therefore equal to 3. The root-based Reynolds number is equal to $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=67$ . The comparison between the theoretical and the numerical results is shown in figure 5. The wing-tip effects are significant over the distal part of the plate between $r/c=2$ and $3$ . Importantly, the extent of that domain is similar to what we found in the case of aspect ratio 6. Over the proximal half of the plate, the agreement between the CFD results and the theory is good.

Figure 5.

A plate with aspect ratio equal to 3. Comparison between the theoretical estimates and the CFD results for (a) the circulation $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}$ over a cylinder surface of radius $r$ , and (b) the distances between the edge and the vorticity centroids in the directions parallel and perpendicular to the plate, $d_{\Vert }$ and $d_{\bot }$ , respectively. All quantities are normalized.

3.2 Estimates of the average spanwise vorticity transport coefficient

The algebraic growth rate of $\unicode[STIX]{x1D6E4}_{1}$ as $r^{4/3}$ is fully defined by the line vortex model, but the prefactor in (2.16) contains a parameter $K_{sp}$ that determines how fast the Lagrangian elements of the line vortex are transported spanwise. We therefore refer to $K_{sp}$ as the spanwise vorticity transport coefficient. The exact value of $K_{sp}$ in each case depends on the distribution of the radial vorticity and the spanwise velocity in the flow field. Consequently, it depends on $Re$ , for the reason that the structure of the edge vortex varies significantly with $Re$ . Let us first derive a quick theoretical estimate of the spanwise vorticity transport coefficient $K_{sp}$ suitable for the low end of the range of $Re$ considered in the previous section. Let $V_{r}(r,\unicode[STIX]{x1D719},y)$ be the radial velocity component in the cylindrical polar coordinates. At the plate, $\unicode[STIX]{x1D719}=0$ and the radial direction is aligned with the spanwise direction. The vorticity transport in the radial direction mainly takes place at those locations where both the radial vorticity $\unicode[STIX]{x1D714}_{r}$ and the radial velocity $V_{r}$ are large enough. To quantify it, we introduce the vorticity-weighted average radial velocity

The parameter $\unicode[STIX]{x1D6F7}\in [0,360^{\circ }]$ controls the extent of azimuthal averaging. Further, the CFD results suggest that $V_{r}$ is approximately linear in $r$ over the inner central part of the plate. We therefore propose an estimate for the spanwise vorticity transport coefficient,

which we subsequently evaluate at a representative location $r_{ref}$ . The overbar is a reminder that the estimate is based on space averaging.

Near the plate, the vorticity is confined in two shear layers that start from the edges and propagate in the downstream direction. Owing to the viscous exchange of momentum, the thickness of these vorticity sheets increases with the distance from the edges, and the peak vorticity magnitude decreases. We therefore use the one-dimensional diffusion equation in an unbounded domain to describe the evolution of the vorticity profiles with the angular distance $\unicode[STIX]{x1D719}$ from the plate. After introducing the time $t$ required for the plate to travel the angular distance $\unicode[STIX]{x1D719}$ , the vorticity is approximated as

which satisfies the diffusion equation with the diffusivity equal to $\unicode[STIX]{x1D708}$ , and the initial condition corresponding to delta distribution of the vorticity at the sharp edges.

The radial velocity is mainly driven by the centrifugal forces acting on the fluid trapped in the recirculation bubble, and it also decays with distance away from the plate due to the action of viscosity. We assume the initial condition for $V_{r}$ of the form $V_{r}(r,0,y)=V_{r\,max}(r)(1-4y^{2}/c^{2})$ , where, according to Maxworthy (Reference Maxworthy2007), $V_{r\,max}(r)=\sqrt{2}\unicode[STIX]{x1D6FA}r$ . The solution of the one-dimensional diffusion equation that satisfies the initial condition is

Figure 6.

Line plots of the radial components of the vorticity $\unicode[STIX]{x1D714}_{r}$ (red) and the velocity $V_{r}$ (blue), sampled on vertical line segments between $y=-c/2$ and $y=c/2$ , at a constant radial location $r_{ref}=3c$ and five different angular distances from the plate, $\unicode[STIX]{x1D719}=7.64^{\circ }\times n$ , where $n=1,\ldots ,5$ . The solid lines show the analytical profiles (3.10) and (3.11). The dashed lines show the CFD results. Vorticity scale is per second and velocity scale is metres per second.

Sample profiles of $\unicode[STIX]{x1D714}_{r}$ and $V_{r}$ are shown in figure 6(a) and compared with the CFD data. They correspond to a plate revolving with angular velocity $\unicode[STIX]{x1D6FA}=260~\text{s}^{-1}$ , i.e. $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=17$ . The profiles are calculated at the radial location $r=3c$ . The parameter $\unicode[STIX]{x1D6FE}(r)$ is set to $2.7~\text{m}~\text{s}^{-1}$ when evaluating (3.10), but it cancels out in the subsequent calculation of $\overline{K}_{sp}$ . The analytical profiles adequately describe the peaks of $\unicode[STIX]{x1D714}_{r}$ and $V_{r}$ as they flatten with distance away from the plate. It should be remembered, however, that the analytical profiles do not account for the dynamic coupling between $\unicode[STIX]{x1D714}_{r}$ and $V_{r}$ and for many three-dimensional effects that may influence the rate of decay at larger $\unicode[STIX]{x1D719}$ . In the following, we use them to obtain a rough order-of-magnitude approximation to $\overline{K}_{sp}$ that does not rely on any data from the CFD. On the other hand, to evaluate $\overline{K}_{sp}$ accurately, it is critical to account for the spatial distribution of $\unicode[STIX]{x1D714}_{r}$ and $V_{r}$ in all detail available from the CFD.

After substituting (3.10) and (3.11) into (3.8), performing numerical integration and substituting the result in (3.9), we obtain the desired theoretical estimate for $\overline{K}_{sp}$ . The result does not depend on $r$ because $V_{sp\,max}(r)$ is linear in $r$ . Figure 7(a) compares the values of $\overline{K}_{sp}$ calculated using the profiles (3.10) and (3.11) with $\overline{K}_{sp}$ evaluated using $\unicode[STIX]{x1D714}_{r}$ and $V_{r}$ from the CFD at $r_{ref}=3c$ . The plots are shown in a range of $\unicode[STIX]{x1D6F7}$ to explore the sensitivity to this parameter. All values are within the interval between 0.23 and 0.4 when the linear distance from the plate is greater than $c$ , i.e. $\unicode[STIX]{x1D6F7}>19^{\circ }$ . The general trend is a slow decrease with $\unicode[STIX]{x1D6F7}$ . The CFD result saturates at $\unicode[STIX]{x1D6F7}>240^{\circ }$ when the numerator and the denominator in $\overline{V}_{r}$ in (3.8) attain their finite maximum values. The sudden drop at $\unicode[STIX]{x1D6F7}=360^{\circ }$ is explained by the inward spanwise velocity on the pressure side of the plate previously reported by Kolomenskiy, Elimelech & Schneider (Reference Kolomenskiy, Elimelech and Schneider2014).

Figure 7.

Estimates of the spanwise vorticity transport coefficient $K_{sp}$ : (a) as a function of $\unicode[STIX]{x1D6F7}$ at $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=17$ , $\unicode[STIX]{x1D6F7}$ being the upper limit of integration in (3.8); and (b) as a function of the root-based Reynolds number $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}$ , with $\unicode[STIX]{x1D6F7}=360^{\circ }$ . (c) Spanwise distribution of the CFD-based estimate of the vorticity transport coefficient $\overline{K}_{sp}$ , calculated as given by (3.8) and (3.9) with $\unicode[STIX]{x1D6F7}=360^{\circ }$ .

Figure 7(b) displays $K_{sp}$ as a function of the root-based Reynolds number $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}$ . In addition to the estimate $\overline{K}_{sp}$ obtained by integration of $V_{r}$ and $\unicode[STIX]{x1D714}_{r}$ from the CFD, figure 7(b) shows the values of $\tilde{K}_{sp}$ that best-fit the theoretical estimate to the CFD data in the least-mean-squares sense, i.e.

A power-law fit of those values leads to the empirical formula (3.4) that we used in the previous section. The agreement between these different estimates is good except for large $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}$ and $r_{ref}$ , when the discrepancy of up to 50 % is caused by the vortex core structure becoming more complex and necessitating further investigation. Apart from that, the estimate $\overline{K}_{sp}$ is consistent with $\tilde{K}_{sp}$ that matches the observed circulation and location of the vortex.

Two sample spanwise distributions of $\overline{K}_{sp}$ , obtained from the numerical simulations at $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=17$ and $83$ , are shown in figure 7(c). In both cases, as postulated earlier, $\overline{K}_{sp}$ is roughly constant over the central part of the plate. Variation only becomes large near the ends of the plate, i.e. $r_{ref}<1c$ or $r_{ref}>4.5c$ . Between these ends, the profile of $\overline{K}_{sp}$ depends on the Reynolds number: $\overline{K}_{sp}$ is monotonically decreasing when the Reynolds number is small, but it has a local maximum when the Reynolds number is large. Despite this small variability, the values sampled at $r_{ref}=1$ and $3$ , used in figure 7(b), are representative of the average vorticity transport coefficient $\overline{K}_{sp}$ over the inner central part of the plate that we need for the edge vortex circulation and position estimates.

Let us conclude this section with a comment on the physical mechanisms that drive the spanwise flow. This question has been extensively studied in past research, and several different mechanisms have been proposed. Our objective is not to describe all the factors that may have a certain influence on the spanwise velocity $V_{r}$ , but to quantify the role of $V_{r}$ in the vorticity dynamics. Our theoretical estimate (3.11) is based on the model proposed by Maxworthy (Reference Maxworthy2007), who postulated that the centrifugal force and the outwards pressure gradient in the conical vortex core are the two equally important drivers of $V_{r}$ . Other effects, such as the Coriolis acceleration and the wing-tip vortex-induced velocity, that are not accounted for in our model are likely to have less influence on $K_{sp}$ compared with the two main effects postulated above. For instance, the CFD computations by Garmann & Visbal (Reference Garmann and Visbal2014) with the centrifugal term eliminated from the Navier–Stokes equations show a dramatic decrease of the outwards spanwise velocity over the plate. Even though the peak outwards spanwise velocity in the vortex core is positive and may be an order of magnitude greater than the average (Garmann & Visbal Reference Garmann and Visbal2014; Limacher et al. Reference Limacher, Morton and Wood2016), it is the average velocity that apparently matters for $K_{sp}$ and for the edge vortex dynamics, as we infer from the overall good agreement between $\overline{K}_{sp}$ and $\tilde{K}_{sp}$ for the conditions examined.

3.3 Time evolution of the edge vortices

The solution derived in § 2.2 is steady. However, the wing rotation starts from rest in our numerical simulations, as in many practical situations (such as the experiments by Carr et al. (Reference Carr, DeVoria and Ringuette2015) using rectangular wings operating at $45^{\circ }$ angle of attack and $Re$ of order several thousand). In addition to that, the flow may become unsteady due to hydrodynamic instabilities. It is therefore important to consider the time evolution of the edge vortex.

Figure 8.

Instantaneous isosurfaces of $\unicode[STIX]{x1D706}_{2}=-10^{8}$  s $^{-2}$ , coloured according to the sign of the spanwise vorticity component, at four different time instants after startup. The root-based Reynolds number is equal to $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=83$ .

Figure 9.

Time evolution of the circulation in the case $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=83$ . (a) Values obtained from (3.1) using the CFD data. (b) Theoretical values of $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}$ as given by (3.22). Matlab programs that reproduce this figure are provided as supplementary material.

Let us discuss only the largest Reynolds-number case, $\unicode[STIX]{x1D6FA}c^{2}/\unicode[STIX]{x1D708}=83$ , which illustrates well different kinds of unsteady effects. The aspect ratio of the plate is equal to 6. We select the time instants at $0.12T$ , $0.25T$ , $0.5T$ and $0.75T$ for the flow visualization, where $T=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D6FA}=4.8332\times 10^{-3}$  s. The time development of the vortex structure is illustrated by isosurfaces of the $\unicode[STIX]{x1D706}_{2}$ -criterion in figure 8. In addition, we plot the normalized circulation as a function of the normalized spanwise distance in figure 9(a). The vortices over the proximal part of the plate, $r<3c$ , reach steady state by the time  $t=0.25T$ . At the same time instant one can see a symmetric pair of counter-rotating vortices shed from the distal part of the plate, $r>4.5c$ . Later, the flow becomes nominally steady over $r<4.5c$ , but the wing-tip vortex is unsteady and small-scale eddies develop at this large Reynolds number.

Let us now amend our analysis to account for the gradual buildup of the edge vortex after the beginning of rotation. Let $t$ be physical time with $t=0$ at startup. We extend the time profile of the plate angular velocity to negative $t$ as

Negative $t$ is the time before startup, when the plate and the surrounding fluid are at rest. Large positive $t>t_{ac}$ is when the plate revolves steadily. Note that, even though our solution is defined for any arbitrary large $t$ , we are only interested in $t<T$ before the plate encounters its own wake from the previous revolution.

In the following analysis, the main difference with respect to the steady case is that now we track vortex particles over a physical time interval from the startup until a set time instant. The radial position $r$ of a tracer satisfies the evolution equation

Hence, the radial position of the tracer with the initial condition $r(0)=r_{0}$ can be written as

where

From the definition of $\tilde{\unicode[STIX]{x1D70F}}$ (2.11) we obtain

where the subscript $(\cdot )_{s}$ stands for the derivative. Integration by parts yields

We use a Taylor series approximation $\text{e}^{x}\approx 1+x$ for the exponential under the integral sign, and express $r_{0}$ in terms of $r$ using (3.15). We thus obtain

where

The rest of the derivation is similar to the steady case. We finally obtain the position of the vortex

and its circulation

The half-plate circulation $\unicode[STIX]{x1D6E4}_{half\text{-}plate}$ is calculated with the same formula as in the steady case (see appendix B), but using the time-dependent $\unicode[STIX]{x1D6E4}_{1}$ (3.22).

The sum circulation $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{theory}=\unicode[STIX]{x1D6E4}_{1}+\unicode[STIX]{x1D6E4}_{half\text{-}plate}$ is shown in figure 9(b), for the same values of the aspect ratio and the Reynolds number as in the numerical simulation, and using $K_{sp}$ as given by (3.4). The trend of $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}$ increasing in time until it saturates is similar to what we observe in the numerical simulation, but the theory predicts slightly smaller growth, and it does not account for the overshoot at $t=0.25T$ and $0.5T$ near the tip of the plate. For small $t$ , the vortex circulation $\unicode[STIX]{x1D6E4}_{1}$ is small, and the largest contribution to $\unicode[STIX]{x1D6E4}_{\unicode[STIX]{x1D6F4}}^{theory}$ is from the linear term $\sqrt{2}r/c$ in the half-plate bound circulation $\unicode[STIX]{x1D6E4}_{half\text{-}plate}$ (3.2). As $t$ becomes large, the $r^{4/3}$ power law becomes dominant. Similar trends were found in the experiments by Carr et al. (Reference Carr, DeVoria and Ringuette2015).