2.1 Vortex configurations in relative equilibrium

The points labelled A–F in panel III of figure 2, together with their periodic images, correspond to relative equilibrium vortex configurations, wherein the vortices propagate along the $x$  axis without change in their relative positions. Of these, A–C are elliptic fixed points (or centres), corresponding to vortex configurations that are neutrally stable to infinitesimal perturbations in the $Z$ -plane. Thus, for these cases a perturbation in the $Z$ -plane that does not otherwise change the vortex strengths, the linear impulse and the assumed geometry generates an orbiting phase trajectory close to the corresponding fixed point. The value of the Hamiltonian is necessarily changed by this perturbation, but ${\mathcal{H}}$ remains conserved in the perturbed system. The points D–F are saddle fixed points, corresponding to relative equilibrium vortex arrangements that are unstable to infinitesimal perturbations in $Z$ . These perturbed configurations can deviate widely from the initial unperturbed vortex arrangement and typically involve a change in ${\mathcal{H}}$ . However, if the perturbation is such that the new $(X,Y)$ coordinates lie on the separatrix passing through the saddle point, then the value of the Hamiltonian is conserved by the perturbation. In such cases, the perturbed vortex configuration will approach a relative equilibrium configuration, which may be a different configuration than the original.

Figure 3.

Relative equilibrium vortex configurations and representative streamlines in the appropriate co-moving reference frame on the $(x,y)$ -plane for $\unicode[STIX]{x1D6FE}=3/7$ and ${\mathcal{P}}=-1$ ; the corresponding phase plane representation is shown in panel III of figure 2. Panels are labelled according to the corresponding fixed point on the phase plane. Representative vortices are labelled according to their strength. Each of these relative equilibrium configurations translates steadily to the left with respect to a fixed frame and without change in shape or orientation. The $x$ and $y$ axes are all shown to the same scale.

Consider the point labelled A in panel III of figure 2, for which $X=n\unicode[STIX]{x03C0}$ (with $n$ any integer) and $Y>0$ . This point corresponds to the singularity in ${\mathcal{H}}$ that occurs when the oppositely signed vortices coincide, leaving two widely spaced symmetrically reflective rows of oppositely signed vortices, or only two base vortices in each periodic strip, with strengths $\pm (\unicode[STIX]{x1D6E4}_{1}-\unicode[STIX]{x1D6E4}_{2})$ . Coincidence of the oppositely signed vortices is given by $z_{1}={z_{2}}^{\ast }\pm nL$ or $y_{1}=-y_{2}$ , and we find from (1.10) that $Y=-{\mathcal{P}}/(1-2\unicode[STIX]{x1D6FE})$ for this two-vortex relative equilibrium configuration.

Point B is also a singular case, with $X=n\unicode[STIX]{x03C0}$ (with $n$ any integer) and $Y=0$ , giving $z_{1}=z_{2}\pm nL$ . In this case the like-signed vortices coincide, and the relative equilibrium configuration consists of two rows of oppositely signed vortices having strengths $\pm (\unicode[STIX]{x1D6E4}_{1}+\unicode[STIX]{x1D6E4}_{2})$ .

Point C is one of three fixed points for which $X=(2n+1)\unicode[STIX]{x03C0}/2$ and $Y>0$ . As with points A and B, point C is a neutrally stable elliptic fixed point. In contrast to A and B, point C is not located at a singularity in ${\mathcal{H}}$ , and the corresponding vortex configuration consists of four distinct base vortices in each periodic strip, as illustrated in panel C of figure 3. This fixed point is unique, giving the only equilibrium state that is neutrally stable to perturbations on the $Z$ -plane while still preserving the four-vortex configuration. The value of $Y$ is found numerically by simultaneously solving $\unicode[STIX]{x2202}{\mathcal{H}}/\unicode[STIX]{x2202}Y=0$ and $\unicode[STIX]{x2202}{\mathcal{H}}/\unicode[STIX]{x2202}X=0$ . We consider these interesting relative equilibrium configurations in further detail in § 2.3 and § 5.1. A perturbation of these configurations that consists of varying $(X,Y)$ while maintaining the spatial symmetry and the values of $\unicode[STIX]{x1D6FE}$ and ${\mathcal{P}}$ leads to periodic relative vortex motion in the $O_{3}$ regime, as discussed in § 2.4.1.

Point D is a saddle point with $X=(2n+1)\unicode[STIX]{x03C0}/2$ and $Y>0$ . The value of $Y$ is obtained numerically using the same procedure as for point C. Panel D of figure 3 shows this relative equilibrium configuration, which consists of four base vortices per period with a relatively large vertical separation between the vortex pairs. In contrast to point C, the configuration for point D is unstable to perturbations in the $Z$ -plane.

Point E is the saddle point for which $X=(2n+1)\unicode[STIX]{x03C0}/2$ and $Y$ is small but finite, as determined numerically. Panel E of figure 3 shows this configuration with four separate base vortices and a very small vertical offset between the vortex pairs. While still satisfying the assumed arrangement illustrated in figure 1(b), the small vertical offset produces a configuration in which the primary pairing is between vortices of equal (and opposite) strength that are aligned vertically and centred on the $x$ axis. Since point E is a saddle point, this configuration is unstable to perturbations in the $Z$ -plane.

Point F is located at $X=n\unicode[STIX]{x03C0}$ and $Y>0$ , with the value of $Y$ again being obtained numerically. This saddle point corresponds to the relative equilibrium configuration shown in panel F of figure 3, in which the four base vortices (per period) are all aligned vertically. This case is the only four-vortex relative equilibrium configuration with $X=n\unicode[STIX]{x03C0}$ . This configuration is also unstable to $Z$ -plane perturbations.

2.2 Lines of singularity

In addition to the separatrices joining equilibrium points, there are two horizontal lines marked as H $_{1}$ and H $_{2}$ in panel III of figure 2 that also delineate phase space behaviour. These horizontal lines correspond to the singularities that occur in ${\mathcal{H}}$ when the image vortices, namely $\{z_{1},\unicode[STIX]{x1D701}_{1}\}$ or $\{z_{2},\unicode[STIX]{x1D701}_{2}\}$ , coincide. Along H $_{1}$ , we have $z_{1}=\unicode[STIX]{x1D701}_{1}$ , so that $y_{1}=0$ . Thus, by (1.10), the line for H $_{1}$ occurs at $Y={\mathcal{P}}/[2(\unicode[STIX]{x1D6FE}-1)]$ . The corresponding vortex arrangement comprises two rows of oppositely signed vortices with circulations $\pm \unicode[STIX]{x1D6E4}_{2}$ that are vertically aligned. Similarly, along H $_{2}$ the vortices at $z_{2}$ and $\unicode[STIX]{x1D701}_{2}$ coincide, so that $y_{2}=0$ , which by (1.10) gives the location of H $_{2}$ as $Y={\mathcal{P}}/(2\unicode[STIX]{x1D6FE})$ . The corresponding vortex configuration consists of two rows of oppositely signed, symmetrically aligned vortices with circulations $\pm \unicode[STIX]{x1D6E4}_{1}$ . As $|{\mathcal{P}}|\rightarrow 0$ , both lines of singularity merge asymptotically with the $X$ axis. Note that these two horizontal lines of singularity in the phase plane do not appear in the staggered configuration we examine in Basu & Stremler (Reference Basu and Stremler2015).

2.3 A note on the stability of relative equilibria

Elsewhere in this manuscript, stability of relative equilibrium configurations is considered within the context of the assumed system constraints. That is, configurations A–C are shown, through reference to phase space diagrams such as those in figure 2, to be neutrally stable to small perturbations in the vortex locations that preserve the vortex strengths, the value of the linear impulse (1.9), and the symmetry in the relative vortex positions given by (1.6). Within this context, perturbations need not be infinitesimal. However, the significance of this (nonlinear) stability is limited by its reliance on preserving the spatial symmetry.

One can ask whether the unique configurations corresponding to point C are stable to perturbations that break the spatial symmetry. Through a number of numerical experiments, we have determined that these configurations do indeed appear to be stable to arbitrary, small perturbations that preserve the vortex strengths, the linear impulse and the $L$ -periodicity of the system. The effects of longer wavelength perturbations remain to be determined.

Figure 4.

Dynamics of the system perturbed from the equilibrium configuration in panel C of figure 3 for $\unicode[STIX]{x1D6FE}=3/7$ and ${\mathcal{P}}=-1$ using $\unicode[STIX]{x1D716}=L/1000$ , $\unicode[STIX]{x1D703}_{1}=\unicode[STIX]{x03C0}/4$ and $\unicode[STIX]{x1D703}_{2}=\sin ^{-1}[-\unicode[STIX]{x1D6E4}_{1}(\sin \unicode[STIX]{x1D703}_{1})/\unicode[STIX]{x1D6E4}_{2}]\approx -0.559$ : (a) shows the symmetric case with $\unicode[STIX]{x1D703}_{3}=-\unicode[STIX]{x1D703}_{1}$ , $\unicode[STIX]{x1D703}_{4}=-\unicode[STIX]{x1D703}_{2}$ ; (b) shows the other solution to (2.2a ),  $\unicode[STIX]{x1D703}_{3}=\unicode[STIX]{x1D703}_{1}$ , $\unicode[STIX]{x1D703}_{4}=\unicode[STIX]{x1D703}_{2}$ , which breaks the spatial symmetry. Panels in the topmost row show phase plane trajectories for the initial condition $Z+\unicode[STIX]{x0394}Z=\unicode[STIX]{x03C0}[(z_{1}-z_{2})+\unicode[STIX]{x1D716}(\text{e}^{i\unicode[STIX]{x1D703}_{1}}-\text{e}^{i\unicode[STIX]{x1D703}_{2}})]/L$ ; the unperturbed $Z$ is shown for reference. Panels in the bottom two rows show close-up views of the trajectories of the perturbed vortices in a frame of reference moving with the vortices in the equilibrium configuration, positions of which are also shown for reference. Integration time for both cases is $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}=500$ . Calculations were performed using $L=\mathbb{S}=1$ .

In the numerical experiments, vortices in an equilibrium configuration with locations $z_{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D6FC}}$ were given perturbations of the form

where the magnitude of the perturbation, $\unicode[STIX]{x1D716}$ , was assumed to be the same for each vortex. These perturbed positions were used as initial conditions in the equations of motion (1.1), which were integrated in time using Wolfram Mathematica™. Applying a perturbation that preserves the linear impulse (1.4) requires that the perturbation angles be related according to

For perturbations that maintain the reflective symmetry, $\unicode[STIX]{x1D703}_{3}=-\unicode[STIX]{x1D703}_{1}$ and $\unicode[STIX]{x1D703}_{4}=-\unicode[STIX]{x1D703}_{2}$ , so that (2.2a ) is satisfied automatically and (2.2b ) becomes

In figure 4 we show two representative examples of the dynamics that occurs when the equilibrium configuration from panel C of figure 3 is perturbed according to (2.1) and (2.2). For the sake of comparison, these examples were chosen to have the same perturbed value of $Z$ . The example in figure 4(a) preserves the spatial symmetry, and thus is clearly (neutrally) stable according to the analysis presented in the remainder of this manuscript. The example in figure 4(b) breaks this symmetry and yet still shows similar results, indicating that the system remains neutrally stable under (small) arbitrary perturbations that satisfy (2.2). All other perturbation choices that were tested produced similar behaviour (not shown). We evaluated the accuracy of the numerical solutions by monitoring variations in the time-dependent values of what should be constants of the motion, as given by

For the cases shown in figure 4, an integration time of $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}=500$ gives ${\mathcal{E}}_{{\mathcal{Q}}}<10^{-14}$ , ${\mathcal{E}}_{{\mathcal{P}}}<10^{-15}$ and ${\mathcal{E}}_{{\mathcal{H}}}<10^{-6}$ for all $0\leqslant \unicode[STIX]{x1D70F}\leqslant 500$ . During this integration time, the vortex configurations travel a distance greater than $21L$ along the $x$ axis.

2.4 Vortex motions for $\unicode[STIX]{x1D6FE}=3/7$ , ${\mathcal{P}}=-1$

As outlined in § 1, the motion of the four base vortices (and their periodic images) in the $(x,y)$ plane is given by the relative vortex motion along a level curve in panel III of figure 2, together with constrained integration of one equation from (1.1). Vortex motion within each regime of motion is qualitatively the same, and thus we can explore all possible motions for $(\unicode[STIX]{x1D6FE},{\mathcal{P}})=(3/7,-1)$ by considering one example trajectory from each regime. These nine regimes of motion can be classified as giving four different general types of vortex motion, which we refer to as orbiting motion, exchanging motion, reverse exchanging motion and mixed motion.

3.1 The special case $\unicode[STIX]{x1D6FE}=0$

Choosing $\unicode[STIX]{x1D6FE}=0$ (equivalently, $\unicode[STIX]{x1D6E4}_{1}=0$ ) leads us to the ‘restricted vortex problem’, in which the ‘vortices’ at $z_{1}$ and $\unicode[STIX]{x1D701}_{1}$ have zero circulation and thus are merely passive particles being advected by the two vortices (with non-zero strength) at $z_{2}$ and $\unicode[STIX]{x1D701}_{2}$ . The system therefore consists of just two rows of (finite strength) vortices with mirror symmetry about the centreline, horizontal (periodic) spacing $L$ between the like-signed vortices, and a vertical separation between the oppositely signed vortices of $2y_{2}={\mathcal{P}}L/\unicode[STIX]{x03C0}$ according to (1.10). As a result of this degeneracy, the Hamiltonian (1.3) is trivially a constant of the motion that gives no information regarding the relative vortex spacing  $Z$ .

A Hamiltonian for this special case can be determined by considering that $z_{1}$ and $\unicode[STIX]{x1D701}_{1}$ represent the positions of passive particles that do not influence the behaviour of the point vortices at $z_{2}$ and $\unicode[STIX]{x1D701}_{2}$ . The vortices $\{z_{2},\unicode[STIX]{x1D701}_{2}\}$ move steadily in relative equilibrium with speed

in the $x$ direction, as determined by (1.1). The variable $Z$ represents the dimensionless position of the passive particle at $z_{1}$ in a frame of reference that moves with the equilibrium vortex configuration. Thus, level curves for the relative ‘vortex’ motion in the $Z$ plane are equivalent (for $L=\unicode[STIX]{x03C0}$ ) to streamlines of the flow generated by vortices $\{z_{2},\unicode[STIX]{x1D701}_{2}\}$ in a co-moving frame with background velocity $-{\mathcal{U}}$ (3.1). The streamfunction for the flow generated by vortices $\{z_{2},\unicode[STIX]{x1D701}_{2}\}$ in the co-moving frame is given by

where $z=x+\text{i}y$ gives the location of a passive particle advected by the vortices. The (dimensionless) Hamiltonian representing the special case $\unicode[STIX]{x1D6FE}=0$ can thus be written as

where here $Z=\unicode[STIX]{x03C0}(z_{1}-z_{2})/L=\unicode[STIX]{x03C0}(z-z_{2})/L$ .

Figure 9.

Representative level curves of $\hat{{\mathcal{H}}}$ on the $(X,Y)$ -plane ( $W=\unicode[STIX]{x03C0}$ ) and, equivalently, streamlines on the $(x,y)$ -plane ( $W=L$ ) for $\unicode[STIX]{x1D6FE}=0$ and various values of $|{\mathcal{P}}|$ . Circles show fixed points and, equivalently, vortices; heavy lines show separatrices. Panels are labelled by Roman numerals VIII–X and the corresponding values of ${\mathcal{P}}$ . Each panel shows two periods for clarity. Horizontal and vertical axes are all shown to the same scale.

Representative level curves of the Hamiltonian in (3.3) and, equivalently, representative stream lines for the streamfunction in (3.2) are shown in figure 9. Three distinct types of level curve topology exist for $\unicode[STIX]{x1D6FE}=0$ . For $|{\mathcal{P}}|=|{\mathcal{P}}_{IX}|\approx 1.3168$ , there is one degenerate saddle fixed point (per period) at $(X,Y)=(\unicode[STIX]{x03C0}/2,-{\mathcal{P}}/2)$ . For $0<|{\mathcal{P}}|<|{\mathcal{P}}_{IX}|$ , there are two saddle fixed points (per period) along the line $Y=-{\mathcal{P}}/2$ , and the level curve topology is consistently as shown in example VIII. For $|{\mathcal{P}}|>|{\mathcal{P}}_{IX}|$ , the two saddle fixed points lie instead along the vertical line $X=\unicode[STIX]{x03C0}/2$ , and panel X of figure 9 shows the level curve topology for this range of ${\mathcal{P}}$ values. The case ${\mathcal{P}}=0$ is a degenerate case in which the vortices at $z_{2}$ and $\unicode[STIX]{x1D701}_{2}$ coincide and cancel each other out.

In the limit as $\unicode[STIX]{x1D6FE}\rightarrow 0$ , the line $\text{H}_{1}$ becomes the horizontal line at $Y={\mathcal{P}}/2$ . In contrast to all other cases, with $\unicode[STIX]{x1D6FE}=0$ there is no singularity when taking $z_{1}=\unicode[STIX]{x1D701}_{1}$ , so $\text{H}_{1}$ is merely a level curve in this case. The line $\text{H}_{2}$ still corresponds to the singularity that occurs with taking $z_{2}=\unicode[STIX]{x1D701}_{2}$ , but this line is now located at $Y\rightarrow (\text{sgn}\,{\mathcal{P}})\infty$ .

3.2 The special case $\unicode[STIX]{x1D6FE}=1/2$

When all the vortex strengths are equal in magnitude, we have $\unicode[STIX]{x1D6FE}=1/2$ , and the phase space topology becomes that shown in figure 10(a). For this representative case, we have arbitrarily chosen $|{\mathcal{P}}|=0.70$ . Owing to the vortex strength equivalence, an inherent symmetry is observed in the phase domain on the $Z$ -plane. All fixed points but B and E move to $|Y|\rightarrow \infty$ in this case, giving new regimes of motion in which vortex configurations with initially finite values of $Y$ have unbounded separation. We refer to these regimes as the scattering modes $S_{1}$ and $S_{2}$ . In real space, vortex pairs consisting of oppositely signed vortices approach each other from opposite sides of the wake centreline before eventually diverging away with $Y\rightarrow \pm \infty$ . Figure 10(b) gives examples of the real space trajectories for the two scattering regimes. Similar scattering motion also occurs with staggered 2P arrangements (Stremler et al. Reference Stremler, Salmanzadeh, Basu and Williamson2011; Basu & Stremler Reference Basu and Stremler2015).

Figure 10.

(a) Selected level curves of ${\mathcal{H}}$ in the $(X,Y)$ phase space for $\unicode[STIX]{x1D6FE}=1/2$ and ${\mathcal{P}}=-0.70$ . The phase space is labelled VII according to the corresponding point in figure 11. The scattering mode regimes are labelled $S_{1}$ , $S_{2}$ . (b) Representative vortex trajectories corresponding to the $S_{1}$ ( $S_{2}$ ) regimes. Solid circles mark the initial locations of the vortices. The boundaries on the periodic strip are shown with dashed lines, and the solid lines represent the vortex trajectories. The absolute starting positions of the vortices in the $(x,y)$ -plane are chosen arbitrarily. The horizontal and vertical axes are shown to the same scale in (a) and (b), respectively.

5.1 Neutrally stable relative equilibria shed from two bluff bodies

As pointed out in § 2.1, point C in the phase plane corresponds to a relative equilibrium configuration in which the symmetric array of four vortices (per spatial period) moves without any change in relative position. The apparent stability of this configuration, discussed in § 2.3, suggests that such a vortex arrangement should be observable in experiments. As illustrated in figure 12, the vortex pattern in this equilibrium configuration is indeed similar to that observed in adjacent von Kármán wakes of two closely spaced bluff bodies. If we assume this cylinder arrangement produces a wake that is at (or close to) an equilibrium state, the point-vortex model can relate the cylinder spacing and vortex positions to the vortex circulation ratio.

Following the standard notation for a von Kármán street, let $b$ be the spanwise separation between the vortices in each pair, giving

Using (1.10), this transverse separation can be non-dimensionalized as

We assume that the cylinders lie medially spanwise between the vortices in each pair. Thus, the inter-cylinder transverse spacing is

where we have made use of equations (1.10) and (5.1). Applying this perspective to the experiment by Fayed et al. (Reference Fayed, Portaro, Gunter, Abderrahmane and Ng2011), as shown in figure 12(b), yields $(Y,B)\equiv (2.96,0.92)$ .

As discussed in § 2.1, point C is a neutrally stable fixed point with $X_{C}=(2n+1)\unicode[STIX]{x03C0}/2$ . For given values of $(\unicode[STIX]{x1D6FE},{\mathcal{P}})$ , the correspondence of point C with a relative equilibrium configuration allows for determination of $Y_{C}$ by numerical solution of

subject to the following constraints: (i) the neutral stability of C requires that the Hessian of ${\mathcal{H}}$ maintain

and (ii)  $|{\mathcal{P}}|$ must be greater than the value at which the cusp bifurcation occurs for the corresponding value of $\unicode[STIX]{x1D6FE}$ .

Solving for $Y=Y_{C}$ using (5.3) with given values of $(\unicode[STIX]{x1D6FE},{\mathcal{P}})$ determines $b$ (5.1) and $h$ (5.2). However, as we have already seen, $b$ and $h$ are the parameters most easily observed in a wake experiment. It is thus the inverse problem that is the more interesting: given values of $h$ and $b$ , and therefore fixing $Y$ , what value of $\unicode[STIX]{x1D6FE}$ satisfies (5.3)? By (1.12) and (5.3), determining $\unicode[STIX]{x1D6FE}$ for a given value of $Y$ also determines ${\mathcal{P}}$ . For the experiment in figure 12(b) with $(Y,B)\equiv (2.96,0.92)$ , or more precisely $(2.9569,0.9202)$ , the solution of this inverse problem gives $\unicode[STIX]{x1D6FE}=0.4845$ , thereby suggesting a circulation ratio of $\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}\approx 0.94$ for the wake vortices. In figure 12(c) we present level curves in the $(Y,B)$ plane along which equation (5.3a ) and the associated constraints are satisfied for constant values of $\unicode[STIX]{x1D6FE}$ . The lower bound on each curve is determined by occurrence of the cusp bifurcation. As $\unicode[STIX]{x1D6FE}\rightarrow 0$ , the contours tend to coalesce together; as $\unicode[STIX]{x1D6FE}\rightarrow 1/2$ , C moves to $Y\rightarrow \infty$ , causing the contours to become more and more separated. This diagram promises to be a convenient tool to gauge the relative strengths of the wake vortices in the case of similar stable co-existence of two neighbouring von Kármán streets shed by two closely separated bluff bodies.

5.2 On the connection between the model trajectories and an experimental wake

The rich variety of motions displayed by this point-vortex model provides motivation for determining the extent to which this behaviour can be observed in an experimental wake. We explore the connection between our model and real wakes by focusing on the experimental wake shown in figure 1(b). This wake is generated by two counter-rotating cylinders with equal diameter $d=0.635~\text{cm}$ in a water tunnel at a Reynolds number $Re=U_{e}d/\unicode[STIX]{x1D708}=200$ , corresponding to a mean flow speed $U_{e}\approx 3.15~\text{cm}~\text{s}^{-1}$ .

The point-vortex model is not a direct representation of an experimental wake with finite length. For mathematical tractability, the modelled motion of the $N=4$ base vortices is tied to interaction with their spatially periodic images, and each successive set of four vortices simultaneously depicts the same dynamics. This infinite collection of point vortices moves about in an inviscid fluid that is otherwise at rest. In a real wake, in contrast, successive clusters of four vortices are shed from the bluff bodies at regular intervals, and these vortices exhibit both spatial and temporal evolution as they are swept downstream by a flow moving with speed $U_{\infty }$ . Vorticity is concentrated in distinct structures, but the distribution of vorticity is continuous, not singular. Viscosity leads both to vortex formation and to dissipation of vorticity with increasing time.

A number of assumptions must be made in order to compare the dynamics in these two systems. First, we represent the locations of each experimental vortex as a point at the estimated centre of the vortex core, as illustrated in figure 1(b). The successive sets of four vortices we refer to as cluster $k=1,2,\ldots ,K$ , with $k=1$ representing the first distinct cluster adjacent to the bluff bodies and with $K=4$ clusters in this system. The position of each experimental vortex in the laboratory frame we label $\hat{z}_{\unicode[STIX]{x1D6FC},k}=\hat{x}_{\unicode[STIX]{x1D6FC},k}+\text{i}{\hat{y}}_{\unicode[STIX]{x1D6FC},k}$ ( $\unicode[STIX]{x1D6FC}=1,\ldots ,4;k=1,\ldots ,K$ ) according to the assumed sign and relative magnitude of the vortex circulation, with

following (1.6). While the directions of the circulations are clear from the experimental image, the relative circulation magnitudes are less certain, so the fitting procedure is performed for both possible choices of $\hat{z}_{1,k}$ and $\hat{z}_{2,k}$ . The average spacing of the vortices is used to determine the effective spatial period of the experimental system,

which we calculate to be $L_{e}=2.82~\text{cm}$ .

Figure 13.

Values of the model constants (a)  ${\mathcal{P}}_{k}$ and (b)  ${\mathcal{H}}_{k}$ based on the locations of the experimental vortices in figure 1(b) for the circulation ratio $\unicode[STIX]{x1D6FE}=0.455$ . Values on the $x$  axis indicate cluster number. Error bars are determined by considering maximum fluctuations of $d/4$ from the assumed point-vortex locations. (c) Model phase space representation for $\unicode[STIX]{x1D6FE}=0.455$ and ${\mathcal{P}}=\bar{{\mathcal{P}}}_{\unicode[STIX]{x1D6FE}=0.455}=-0.578$ . Experimental data points $Z_{i}$ are shown with hollow circles and a dashed line; the thick solid line marks the best-fit model trajectory along ${\mathcal{H}}(Z;\unicode[STIX]{x1D6FE},\bar{{\mathcal{P}}}_{\unicode[STIX]{x1D6FE}})=\bar{{\mathcal{H}}}_{\unicode[STIX]{x1D6FE}}=-7.99\times 10^{-3}$ . The beginning and end of the model trajectory, corresponding to projections of $Z_{1}$ and $Z_{4}$ onto the model curve, are shown with a solid and hollow square, respectively. Arrows indicate direction of the phase space evolution. (d) Model vortex trajectories (solid lines) and experimental vortex positions at $z_{\unicode[STIX]{x1D6FC},k}$ (hollow circles connected by dashed lines) in a co-moving frame of reference with zero background velocity for $\unicode[STIX]{x1D6E4}_{1}\approx 2.81~\text{cm}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D6E4}_{2}\approx 3.36~\text{cm}^{2}~\text{s}^{-1}$ . Solid squares mark the initial locations of the model vortices, and solid circles mark the model vortex locations at times corresponding to the experimental shedding period. (e) Model predictions of the vortex locations (solid circles), with the added displacement provided by the free stream, superimposed on the estimated vortex locations from figure 1(b) (hollow circles). The horizontal and vertical axes in (d–e) are shown to the same scale.

Next we assume a value for $\unicode[STIX]{x1D6FE}$ and use the experimental vortex locations to determine corresponding values for the constants of motion needed in the model. For each cluster $k$ , the positions of vortices $\hat{z}_{1,k}$ and $\hat{z}_{2,k}$ in the co-moving frame are used in (1.9c,d ) with $L=L_{e}$ to determine corresponding values

as shown in figure 13(a). The average value,

is taken as the representative experimental value for ${\mathcal{P}}$ , which specifies the phase plane topology in the corresponding model system for the selected value of $\unicode[STIX]{x1D6FE}$ . Error bounds on the ${\mathcal{P}}_{k}(\unicode[STIX]{x1D6FE})$ are determined by considering the error associated with defining the locations of the experimental vortices. For the errors shown in figure 13(a), the vortex locations were varied over the circles shown in figure 1(b), each with diameter $d/2$ . A similar procedure is used with equation (1.12) to determine

such as illustrated in figure 13(b), where we have

from applying (1.7). The average of the ${\mathcal{H}}_{k}(\unicode[STIX]{x1D6FE})$ ,

gives the representative experimental value for ${\mathcal{H}}$ , which determines the relevant level curve in phase space from the model, as determined by ${\mathcal{H}}(Z;\unicode[STIX]{x1D6FE},\bar{{\mathcal{P}}}_{\unicode[STIX]{x1D6FE}})=\bar{{\mathcal{H}}}_{\unicode[STIX]{x1D6FE}}$ . The experimental phase space data and example model phase space curves are shown in figure 13(c).

The optimal value of $\unicode[STIX]{x1D6FE}$ is obtained by minimizing the differences between the experimental and model positions in phase space. Initial and final model positions are first determined by projecting $Z_{1}$ and $Z_{4}$ , respectively, onto the ${\mathcal{H}}(Z;\unicode[STIX]{x1D6FE},\bar{{\mathcal{P}}}_{\unicode[STIX]{x1D6FE}})=\bar{{\mathcal{H}}}_{\unicode[STIX]{x1D6FE}}$ curve, as illustrated in figure 13(c) by the square symbols. The system of equations (1.15) is then integrated between these two positions from $\unicode[STIX]{x1D70F}=0$ to $\unicode[STIX]{x1D70F}={\mathcal{T}}$ to determine the model integration time ${\mathcal{T}}$ and model positions $Z(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE})$ at times $\unicode[STIX]{x1D70F}=(k-1)\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}$ , $k=1,\ldots ,K$ , with $\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}={\mathcal{T}}/(K-1)$ . We use the notation $Z(\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE})$ to emphasize that these model positions in phase space are dependent on the choice of $\unicode[STIX]{x1D6FE}$ . The error in the phase space fit between model and experiment is quantified by

where we have chosen to normalize by the phase space period, $\unicode[STIX]{x03C0}$ . We have determined that for this example $\mathbb{E}_{p}(\unicode[STIX]{x1D6FE})$ is minimized by taking $\unicode[STIX]{x1D6FE}=0.455$ with the vortex assignments shown in figure 1(c) (see also figure 13 d); this choice gives ${\mathcal{T}}=12.1$ , $\bar{{\mathcal{P}}}_{\unicode[STIX]{x1D6FE}}=-0.578$ , $\bar{{\mathcal{H}}}_{\unicode[STIX]{x1D6FE}}=-7.99\times 10^{-3}$ and $\mathbb{E}_{p}=0.0474$ . As shown in figure 13(c), the model-predicted vortex dynamics belongs to the $O_{3}$ regime (cf. § 2.4.1). Note that this phase space comparison assumes spatial symmetry in the experiment, so that only the positions of the vortices with positively signed circulation are used in the analysis.

The real space model vortex trajectories are then fitted to the experimental vortex locations in the co-moving frame, which includes determining the value of $\mathbb{S}$ and the optimal background flow speed, $U_{\infty }$ . We assume $U_{\infty }\approx U_{e}$ but consider uncertainty in the experimental background velocity as a free parameter by taking

For the example discussed here we explored $0.6\leqslant c\leqslant 1.5$ with a resolution of $\unicode[STIX]{x0394}c=10^{-4}$ .

A key step in the real-space fitting process involves transforming the spatially developing wake data into the temporal motion of four vortices. We assume that the experimental vortex shedding is exactly periodic in time, with a shedding frequency of  $f_{e}$ . We assume a Strouhal number

for this experimental system (Kumar, Cantu & Gonzalez Reference Kumar, Cantu and Gonzalez2011a ). The cluster $k$ will have moved to the position of cluster $k+1$ after an elapsed time $\unicode[STIX]{x0394}t=1/f_{e}$ owing to vortex interactions and a displacement of $U_{\infty }\unicode[STIX]{x0394}t$ caused by the background flow. The experimental vortex positions in a frame moving with the background flow are given by

These co-moving vortices can be viewed as showing temporal evolution of the four vortices initially from cluster  $k=1$ . The motion of vortices from cluster 1 to cluster 4 is estimated to require an elapsed time of $T=(K-1)\unicode[STIX]{x0394}t$ . Comparing the model integration time  ${\mathcal{T}}$ and the experimental elapsed time  $T$ using equation (1.14) gives a scaled value for $\mathbb{S}$ in the experiment; together with the selected $\unicode[STIX]{x1D6FE}$ value, this $\mathbb{S}$ value gives an estimate for the experimental values of the vortex strengths.

Changing the assumed value of $U_{\infty }(c)$ by varying $c$ changes the experimental vortex positions in the co-moving frame, $z_{\unicode[STIX]{x1D6FC},k}$ ; the experimental integration time, $T$ ; and the dimensional value of $\mathbb{S}$ . For each value of $U_{\infty }(c)$ considered in our analysis, we quantify the difference between the experimental positions $z_{\unicode[STIX]{x1D6FC},k}$ , which depend on $c$ , and the corresponding model positions $z_{\unicode[STIX]{x1D6FC}}(t)=z_{\unicode[STIX]{x1D6FC}}((k-1)\unicode[STIX]{x0394}\unicode[STIX]{x1D70F})$ by computing

in which we have used the notation $z_{3}(t)=\unicode[STIX]{x1D701}_{1}(t)$ , $z_{4}(t)=\unicode[STIX]{x1D701}_{2}(t)$ . We determine the optimal value of $c$ by minimizing the error $\mathbb{E}_{r}(c)$ .

Figure 13(d) shows the resulting best-fit vortex trajectory comparison between the model predictions and the experiment from figure 1(b) in a co-moving frame of reference. As determined by our fitting procedure, in this comparison we have used a background flow speed $U_{\infty }=3.00~\text{cm}~\text{s}^{-1}$ (corresponding to $c=0.9537$ , or ${\sim}5\,\%$ difference from the initial $U_{e}$ value), giving an experimental elapsed time $T=3.17~\text{s}$ . By (1.14), we thus have $\mathbb{S}=6.17~\text{cm}^{2}~\text{s}^{-1}$ for the experiment; together with $\unicode[STIX]{x1D6FE}=0.455$ , this gives experimental values $\unicode[STIX]{x1D6E4}_{1}\approx 2.81~\text{cm}^{2}~\text{s}^{-1}$ and $\unicode[STIX]{x1D6E4}_{2}\approx 3.36~\text{cm}^{2}~\text{s}^{-1}$ . For this comparison, $\mathbb{E}_{r}=0.0747$ . Note that the comparison in real space does not assume spatial symmetry in the experiment, and the error $\mathbb{E}_{r}$ accounts for all of the experimental vortex positions. The translatory displacement owing to the background flow can also be superimposed on the model vortex evolutions, thereby enabling a comparison with the experiment in the laboratory frame of reference, as depicted in figure 13(e). Comparisons in both reference frames show a strong correspondence between the modelled vortex dynamics and the experimental data.