2.1. Robotic fish platform

Figure 1.

(a) Robotic fish used in this study, seen in the streamwise ( $x$ )–spanwise ( $z$ ) plane. (b) Sketch of the configuration of the robotic fish in the water tunnel, seen from top in the streamwise ( $x$ )–transverse ( $y$ ) plane. The imaging section of the PIV measurements is shown as a dashed rectangle of length $180$ mm in the streamwise $x$ -direction and width $120$ mm in the transverse $y$ -direction. The free stream speed in the channel is $U_0$ , while the fish moves at frequency $f$ and with a tail amplitude $A$ .

The experimental set-up consists of a robotic fish placed at mid-depth in a laminar water tunnel (Rolling Hills Research Corporation, Model 0710). The robot, described by Gibouin et al. (Reference Gibouin, Raufaste, Bouret and Argentina2018), Sánchez-Rodríguez et al. (Reference Sánchez-Rodríguez, Celestini, Raufaste and Argentina2021) and Fu et al. (Reference Fu, Israilov, Sánchez-Rodríguez, Brouzet, Allibert, Raufaste and Argentina2023), is shown in figure 1(a). It is composed of a 3-D printed hard head (on the left, in black), a servomotor (in the middle, in blue) and a 3-D printed soft tail (on the right, in white). The fish has a total length of $L=180$ mm in the streamwise $x$ -direction, a width of $27$ mm in the transverse $y$ -direction and a height of $60$ mm in the spanwise $z$ -direction. The aspect ratio of the tail, defined as the ratio between its respective sizes in the spanwise and streamwise directions, is approximately equal to $1.7$ , which is in the same range as the objects with low aspect ratios studied in the literature (Buchholz & Smits Reference Buchholz and Smits2006, Reference Buchholz and Smits2008; Dong et al. Reference Dong, Mittal and Najjar2006; Li & Dong Reference Li and Dong2016). The servomotor drives a wheel, where two cables are attached. Each cable goes on one side of the tail. When the servomotor wheel is rotating, the two cables alternatively pull on each side of the first part of the tail, thus mimicking antagonistic muscles and deforming it. A typical deformation of the tail, seen from the top, is visible in figure 1(b). The servomotor wheel performs an oscillatory motion, leading to undulations of the fish at a frequency $f$ and with a peak-to-peak amplitude $A$ for the tail. The free stream speed of the water tunnel $U_0$ can be varied up to $110$ mm s^–1. The channel test section, with a glass plate on the top for optical access, has a length of $450$ mm, a height equal to $250$ mm and a width of $180$ mm. The end of the tunnel is also equipped with a transparent glass plate, allowing us to illuminate the test section behind the fish with a horizontal laser sheet.

Measurements were performed using two-dimensional PIV on the horizontal mid-span plane behind the robotic fish, with a camera filming from the top. The imaging section, shown as a dashed rectangle in figure 1(b), measures $180\,{\textrm{mm}} \times 120\,{\textrm{mm}}$ . The origin of the coordinate system in the $x{-}y$ plane is taken at the tip of the fish tail, when the fish is immobile. PIV acquisition and post-processing were done using a LaVision system (DaVis 10.2.0) with a Photron SA3 high-speed camera ( $1024 \times 1024$ pixels) recording images at $200$ Hz and a Litron LDY 300 PIV pulsed laser. Laser sheet width was approximately $1$  mm in the whole imaging region. Additional post-processing and analysis were done using MATLAB and the PIVMat Toolbox (Moisy Reference Moisy2007).

The fish is held fixed in the laboratory frame by its head, which is attached to a force sensor (visible as a grey rod on the left in figure 1 a). This allows us to measure the instantaneous net force $F$ exerted on the fish, combining the drag force $F_D$ of the fluid on the fish body and the thrust force $F_T$ produced by the robot, such as $F=F_T-F_D$ (Gibouin et al. Reference Gibouin, Raufaste, Bouret and Argentina2018). This configuration, where the object is fixed in a given flow, is very classical to study the wakes produced by the object. It should be noted that mimicking the situation where a fish swims at a constant speed in a quiescent fluid is possible only when the time-averaged net force $\langle F\rangle$ is equal to 0 (Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2010). If the net force is negative (drag regime) or positive (thrust regime), this means that the fish would be decelerating or accelerating in the flow. However, exploring the different wakes produced as a function of the Strouhal number allows us to fully understand the effect of this parameter on the vortex street, thus being more general.

2.2. Parameter space

While the length $L$ of the fish is constant, a large parameter space has been explored by varying the three main quantities governing the dynamics of the wake: the fish tail amplitude $A$ and frequency $f$ , and the free stream speed $U_0$ . The explored ranges of the different parameters are summarised in table 1. This parameter space can be quantified using several dimensionless quantities. The dimensionless amplitude $A/L$ has been varied within the range $0.1{-}0.3$ , which corresponds to what is observed in living fish and cetaceans (Bainbridge Reference Bainbridge1958; Hunter & Zweifel Reference Hunter and Zweifel1971; Rohr & Fish Reference Rohr and Fish2004; Saadat et al. Reference Saadat, Fish, Domel, Di Santo, Lauder and Haj-Hariri2017; Sánchez-Rodríguez et al. Reference Sánchez-Rodríguez, Raufaste and Argentina2023). The dimensionless frequency $f L / U_0$ has been varied in a wide range, between $1$ and $15$ . As in the literature (Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008, Reference Godoy-Diana, Marais, Aider and Wesfreid2009; Schnipper et al. Reference Schnipper, Andersen and Bohr2009; Andersen et al. Reference Andersen, Bohr, Schnipper and Walther2017; Saadat et al. Reference Saadat, Fish, Domel, Di Santo, Lauder and Haj-Hariri2017), these two parameters are used to represent the different experiments on a parameter space map shown in figure 2(a). However, the main control parameter of the wake is the Strouhal number based on the fish tail amplitude, defined as St $_A\equiv f A /U_0$ . This dimensionless parameter is the product of the two previously defined parameters, and combined in itself the three physical quantities varied in the experiments. Constant Strouhal curves in the diagram of figure 2(a) correspond to hyperboles. Here, we have explored a large range of Strouhal number St $_A$ , from $0.1$ to $2.2$ , while for most of living fish and cetaceans, the Strouhal number is observed to be almost constant, between $0.2$ and $0.4$ (Triantafyllou et al. Reference Triantafyllou, Triantafyllou and Gopalkrishnan1991; Gazzola, Argentina & Mahadevan Reference Gazzola, Argentina and Mahadevan2014; Saadat et al. Reference Saadat, Fish, Domel, Di Santo, Lauder and Haj-Hariri2017; Sánchez-Rodríguez et al. Reference Sánchez-Rodríguez, Raufaste and Argentina2023). We have chosen here to explore a much larger range to obtain the comprehensive scaling laws as a function of this control parameter, below and above the critical Strouhal number where the transition between the drag and thrust regimes occurs in our case (St $_A^{\star} \approx 0.48$ , see later paragraph). Note that the Reynolds number based on the robotic fish length, ${ Re}=U_0 L /\nu$ , where $\nu$ is the kinematic viscosity of the fluid, here water, is between $4500$ and $18\,000$ .

Table 1.

Range of dimensional parameters and Strouhal numbers explored in this study.

Figure 2.

(a) Kinds of vortex streets observed in the parameter space $(A/L,fL/U_0$ ). The colour of each point corresponds to the free stream speed $U_0$ (orange $U_0=26\,{\textrm{mm s}}^-{^1}$ , red $U_0=53\,{\textrm{mm s}}^-{^1}$ , black $U_0=103\,{\textrm{mm s}}^-{^1}$ ), while its shape represents the observed form of the wake () for 2S, () for 2P, () for 2S + 2P and () for 4P wakes), described in § 2.3. The empty (respectively filled) symbols indicate that the jets formed behind the fish have a time-averaged streamwise velocity component below (respectively above) the free stream speed $U_0$ . The horizontal green band between $A/L=0.1$ and $0.3$ marks the limit observed for different fish species, while the grey region delimits the Strouhal number range explored in this study, between $St_A=0.1$ and $2.2$ . The solid curve represents the critical Strouhal number $St_A^{\star} =0.48$ , for which drag–thrust transition occurs for the robotic fish used in this study, and the dashed curve corresponds to the vortex or jet inversion at $St_A^{\textit{inv}}=0.36$ . (b) Time-averaged force $\langle F \rangle$ measured by the force sensor as a function of the Strouhal number $St_A$ , for experiments of series 2 (see table 1). A vertical solid line indicates the drag/thrust transition at $St_A^{\star} \approx 0.48$ and a vertical dashed line shows the vortex or jet inversion at $St_A^{\textit{inv}}=0.36$ . The red curve represents the fit given by (2.1). The symbols are similar to those used in the other figures when the fish is moving, while the point at $St_A=0$ , taken when the fish is immobile, is represented by a cross (x).

The time-averaged force $\langle F \rangle$ , measured by the force sensor, is shown in figure 2(b) as a function of the Strouhal number. Its trend is consistent with the curves shown in the literature (Bohl & Koochesfahani Reference Bohl and Koochesfahani2009; Tytell et al. Reference Tytell, Borazjani, Sotiropoulos, Baker, Anderson and Lauder2010; Mackowski & Williamson Reference Mackowski and Williamson2015). At low Strouhal number, the fish is in the drag regime with $\langle F \rangle \lt 0$ . As the Strouhal number increases, the force also increases and becomes positive for a critical Strouhal number. This means that the averaged thrust $\langle F_T \rangle = C_T \rho A^2 f^2 L^2$ produced by the fish is larger than the fluid drag $\langle F_D \rangle = \rho C_D U_0^2 L^2$ . Here, $C_T$ is the a priori unknown proportionality constant between the real thrust and its scaling law (Gazzola et al. Reference Gazzola, Argentina and Mahadevan2014; Gross et al. Reference Gross, Roux, Raufaste and Médéric2021), while $C_D$ is the drag coefficient of the fish (Gibouin et al. Reference Gibouin, Raufaste, Bouret and Argentina2018; Sánchez-Rodríguez et al. Reference Sánchez-Rodríguez, Celestini, Raufaste and Argentina2021). Note that the drag coefficient $C_D$ is assumed to be the same when the robot is immobile and when it is moving, even though this coefficient is slightly higher when the fish is moving (Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2010; Gibouin et al. Reference Gibouin, Raufaste, Bouret and Argentina2018; Gross et al. Reference Gross, Roux, Raufaste and Médéric2021). At the critical Strouhal number $St_A^{\star}$ , the averaged fish thrust balances the fluid drag and the force $\langle F \rangle$ cancels. Therefore, the force $\langle F \rangle$ can rewritten as

(2.1) \begin{equation} \langle F \rangle = \langle F_T \rangle -\langle F_D \rangle =\rho C_D L^2 U_0^2 \left ( \left (\frac {{St}_A}{{St}_A^{\star} }\right )^2 - 1\right ), \end{equation}

with $St_A^{\star} =\sqrt {C_D/C_T}$ . Note that this equation is fully compatible with the evolution of the time-averaged force $\langle F \rangle$ with the Strouhal number, as shown by the fit of the experimental data in figure 2(b). This fit allows us to determine the value of critical Strouhal number for our experiments: $St_A^{\star} \approx 0.48$ . This value is consistent with the critical Strouhal number reported for a very similar fish, built on the same principle, in a previous paper (Gibouin et al. Reference Gibouin, Raufaste, Bouret and Argentina2018). Note that this value is higher than the typical Strouhal numbers for which biological fish are found to swim ( $St_A=0.2{-}0.4$ ; Triantafyllou et al. Reference Triantafyllou, Triantafyllou and Gopalkrishnan1991). We attribute this to both the relatively blunt and rigid head of the robotic fish, which increases drag, and to tail kinematics that do not fully replicate the efficiency of biological propulsion. Nonetheless, all these values are of the same order of magnitude, and we here vary the Strouhal number in a large range around the critical Strouhal number.

2.3. Observed wakes

Figure 3.

Vorticity, velocity and wakes observed. Each row represents a typical wake: (a) 2S; (b) 2P; (c) 2S + 2P; and (d) 4P. The left column shows the phase-averaged vorticity (in colour) and velocity (with arrows and where $U_0 \boldsymbol{e}_x$ is subtracted) fields when the tail is at its maximum amplitude. The middle column represents the normalised time-averaged streamwise velocity field, obtained as $(\langle U \rangle -U_0)/U_0$ , where $\langle U \rangle$ is the time-averaged streamwise velocity field. The trajectories of the vortices are represented on top of the normalised streamwise velocity, with red and blue dots. The colour of the points represent clockwise (blue) or counter-clockwise (red) vortices. The position of the vortices visible in the left panels are marked by blue or red disks in the middle panels. At $x=100$ mm, the time-averaged streamwise velocity profiles are plotted as black lines, illustrating deviations from the free stream speed (dashed line), which indicate the presence of one or multiple jets. The right column shows sketches of the corresponding wakes, with the definitions of the notation given in panel (b): $\theta$ is the wake angle, $\alpha$ the orientation angle of the main vortex pair with the transverse $y$ -direction, $\xi$ the pair diameter, i.e. the distance between the two vortices of the same pair and $\pm \Gamma$ the circulation of each vortex.

Within the explored parameter space, four different types of wakes have been quantitatively observed, whose typical examples are given in figure 3. It is important to note that, since these wakes own an obviously 3-D structure due to the low aspect ratio of the tail, we observe only the 2-D footprint of these wakes in the mid-span plane behind the fish with the 2-D PIV technique. This technique is obviously limited and cannot fully described the 3-D vortex structures and interactions. Nevertheless, most of the wake structures reported here have already been described and observed in 3-D within the literature. In addition, we mainly focus here on tracking these structures in the wake, to measure the relevant physical parameters of their trajectories. For this, the 2-D PIV technique is fully sufficient and allows us to classify the observed wakes in different categories. The first one corresponds to 2S wakes (figure 3 a), already documented in 3-D by Buchholz & Smits (Reference Buchholz and Smits2006, Reference Buchholz and Smits2008). It is composed of 3-D horseshoe vortices, whose footprint in the mid-plane corresponds to two single vortices emitted per period. In the figures of the paper, the symbol representing this type of wake is a square. The second type of wake, the most commonly observed in our experiments, corresponds to 2P wakes (figure 3 b), with 3-D vortex rings emitted on each side of the fish (Dong et al. Reference Dong, Mittal and Najjar2006; Buchholz & Smits Reference Buchholz and Smits2008). Its signature in the mid-plane corresponds to two pairs of contra-rotating vortices emitted per period. This wake is represented by a circle in the following. Note that 2S and 2P wakes have already been observed for fish in several situations (Tytell et al. Reference Tytell, Borazjani, Sotiropoulos, Baker, Anderson and Lauder2010), such as PIV experiments behind living fish or numerical simulations with a realistic fish body. The two other types of wakes are more complex, since they are composed by two groups of three or four vortices emitted per period in the mid-plane (figures 3 c and 3 d). Their 3-D structure has not been described in the literature, but their 2-D footprints in the mid-span plane correspond to a combination of the two first types of wakes. They are therefore named respectively 2P + 2S and 4P, and represented respectively by a diamond and a triangle. We expect these 3-D wakes to be the simple combinations of 2S and 2P wakes, i.e. that 2S + 2P wakes are composed of a vortex ring (2P) and a horshoe vortex (2S), while 4P wakes are composed of two adjacent vortex rings (2P). However, detailed geometries of the vortex structures and their interactions cannot be really described without 3-D wake information.

Despite their differences in structure and number of vortices, one can clearly see in figure 3 that both vortex trajectories and time-averaged velocity field form a V-shaped or bifurcated wake behind the fish, whatever the type of wake. As already observed in the literature for 2S and 2P wakes, the time-averaged velocity field exhibits jets behind the fish. Here, we have between one and four jets, depending on the wake type. For each wake, the time-averaged streamwise velocity within the jets can be smaller (represented by open symbols in figure 2) or larger (solid symbols in figure 2) than the free stream speed $U_0$ . In all cases, these jets are delimited by the vortex trajectories that typically mark the transition where the time-averaged streamwise velocity field is equal to $U_0$ (white lines with the colourmap used). This clearly shows that the jets in the time-averaged velocity field are created by the periodic passing of the vortex structures, inducing a net momentum in the fluid (Dong et al. Reference Dong, Mittal and Najjar2006).

Such bifurcated wakes with jets in the time-averaged velocity field are expected for 3-D wakes of low aspect ratio flapping objects (Buchholz & Smits Reference Buchholz and Smits2006, Reference Buchholz and Smits2008; Dong et al. Reference Dong, Mittal and Najjar2006) and fish (Tytell et al. Reference Tytell, Borazjani, Sotiropoulos, Baker, Anderson and Lauder2010). However, the evolution of the wake structure and vortex properties with respect to the Strouhal number, and their detailed connections to jet velocities and thrust production, have never been studied in the literature.

3.1. Measurement of vortex properties

Vortex characteristics are measured by following the different vortical structures in the time-dependent phase-averaged vorticity field, using a method proposed by Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009). For each experiment, the positions of the main vortical structures in the wake are tracked, using a search of local extrema in the vorticity field. Examples of vortex detection are shown in figure 3 for different wake types. Once the vortical structures are identified, we extracted several relevant physical quantities, such as their positions in the $(x,y)$ plane and their circulation $\Gamma$ . For pairs, we also measured the orientation angle $\alpha$ of the vortex pairs with respect to the transverse $y$ -direction and the distance $\xi$ between the two vortices of the pair. These quantities, defined in figure 3(b), are then followed in time. An example of the time evolution of such quantities is illustrated in figure 4, for the 2P wake shown in figure 3(b).

The vortex mean velocities in the streamwise $x$ -direction, $U_{{vortex}}$ , and in the transverse $y$ -direction, $V_{{vortex}}$ , are then obtained by a linear fit of the time evolution of the vortex positions, as illustrated by the dashed lines in the two first columns in figure 4. Note that the fitted values obtained for each vortical structure are then averaged to obtain a mean value and an errorbar, for each experiment. The time evolution of the angle $\alpha$ , illustrated in the third column in figure 4, is also derived from the vortex positions in the $(x,y)$ -plane. A mean value and an errorbar are then obtained for each experiment by combining the two panels and performing a conditional averaging in the intermediate region where the vortices are neither too close nor too far from the fish tail, typically for $x$ between $50$ and $125$ mm.

Figure 4.

Example of vortex characteristics measured as a function of time, for the 2P wake shown in figure 3(b). Panel (a) represents the data obtained for the vortex pair in the top-half of the image, while panel (b) represents the data for the vortex pair in the bottom-half. Red points correspond to counter-clockwise vortices and blue points to clockwise ones, as the colourcode used in figure 3. The two first columns show the vortex position in the $(x,y)$ -plane, from where the vortex velocities are extracted using a linear fit. The fitted slopes are shown with dashed lines. The two other columns represent the time evolution of the angle $\alpha$ and of the absolute value of the circulation $\Gamma$ measured for each vortex. The averaged quantities are shown using dashed horizontal lines.

The circulation $\Gamma$ has been calculated by integrating the spanwise vorticity field over the surface of each vortex in the mid-span plane. The typical size $a$ of the vortex, and therefore its surface, has been determined by a Gaussian fit of the vorticity field in the $x$ - and $y$ -directions, around the centre of the vortex (Godoy-Diana et al. Reference Godoy-Diana, Marais, Aider and Wesfreid2009). The time evolution of the circulation $\Gamma$ for a 2P wake is shown in the forth column in figure 4. First, since the two vortices of each pair form a single vorticity ring in 3-D which is cut in half by the mid-span PIV plane, one expects the absolute values of the circulation to be equal, which is the case. Second, one can clearly see a strong decay of the circulation as the vortices get away from the tail, particularly after they have been emitted by the tail. This is clearly an overlap effect, since close to the tail, two vorticity rings are very close to each other: the one that just got ejected by the tail and the one that is currently under formation (Buchholz & Smits Reference Buchholz and Smits2008). This situation is visible in the vorticity snapshot of figure 3(b), with two red vortices close to each other in the vicinity of the tail. One belongs to the vortex ring which is going to move in the positive transverse direction, with the blue vortex already formed, while the other belongs to the vortex ring which is going to move in the negative transverse direction, with the blue vortex currently forming at the tail. The decay of the circulation far from the tail is more moderate. As for the angle $\alpha$ , a mean value and an errorbar are obtained for the circulation $\Gamma$ thanks to a conditional averaging in the intermediate region where $x$ is between $50$ and $125$ mm. Note that due to the moderate decay of the circulation in this region, the errorbars obtained may be rather large.

3.2. Comparison between vortex and jet velocities

The middle column of figure 3 shows that the vortex trajectories completely surround the jets in the time-averaged velocity field. Indeed, the travel of the vortices in the wake is responsible for the creation of these jets in the mean velocity fields, since the vortices regularly pass in the wake and locally induce a different speed than the free stream speed in a different direction. To fully understand the relationship between the jets and the vortex structures, the velocities of the two objects are compared quantitatively in figure 5. The jet velocities are measured along $y$ profiles at given $x$ coordinates, as shown in the middle column in figure 3. In these profiles, the jet velocity is averaged in the region delimited by the trajectories of the vortices. Indeed, as one can see in the middle column of figure 3, the vortex trajectories almost correspond spatially to the region where the time-averaged streamwise velocity field is equal to the free stream speed (in white with the colourmap used). The location of these profiles is varied in an intermediate region, where $x \in [50,125]$ mm. This thus leads to a mean jet velocity and an errorbar for each experiment. For 2S + 2P and 4P wakes, we have measured the velocity of the jets formed by the main vortex pairs only. Figures 5(a) and 5(b) show the comparison between the time-averaged jet velocities and the vortex velocities, in both streamwise $x$ - and transverse $y$ -directions. For the streamwise velocity, the agreement between the two is good, except for the wakes with the highest normalised vortex velocities ( $U_{{vortex}}/U_0\gt 2$ ), where the vortex velocity is significantly larger than the jet velocity. For the transverse velocity, the trend is relatively good but a slight offset is visible, since $V_{{vortex}}$ is systematically higher than $\langle V_{{jet}} \rangle$ . Again, the largest difference is for the wakes with the highest normalised vortex velocities ( $V_{{vortex}}/U_0\gt 1$ ). From figure 6, it appears that these points with the highest normalised vortex velocities correspond to those with the highest Strouhal number, i.e. with $St_A\gt 1$ . A careful examination of these 2P wakes at high Strouhal number shows that the vortex pairs are still identifiable but weaker in vorticity. This may be due to the fact that the wake acquires a more complex 3-D structure, as also observed by Buchholz & Smits (Reference Buchholz and Smits2006, Reference Buchholz and Smits2008). Thus, the pairs are less efficient to generate jets in the horizontal mid-plane, but still travel at a high speed. Despite this phenomenon at high Strouhal number, the agreement between vortex and jet velocities is rather good for wakes at Strouhal number smaller than 1.

Figure 5.

Jet versus vortex velocities. In $y$ -coordinates, panels (a) and (b) show the mean jet velocities in the streamwise and transverse directions, respectively, while panels (c) and (d) show the root-mean-square jet velocities. All quantities are normalised by the free stream speed $U_0$ . The dashed line in all panels shows the unit line and is used to guide the eyes.

However, it is necessary to pay attention to the way the jet velocities are averaged in time. Indeed, since vortical structures are emitted periodically by the tail, the velocity field owns both a mean and a fluctuating component. These two components have to be taken into account for the time average, since they both influence the average pressure field (Bohl & Koochesfahani Reference Bohl and Koochesfahani2009). Therefore, we now compare the root-mean-square jet velocities ( $\langle U_{{jet}}^2 \rangle ^{1/2}$ and $\langle V_{{jet}}^2 \rangle ^{1/2}$ ) with the vortex velocities. Figures 5(c) and 5(d) show that these velocities are very close to the vortex velocities, except for the wakes at high Strouhal numbers. In addition, the differences between panels (a) and (c) and panels (b) and (d) in figure 5 show that the fluctuating component of the velocity field is mainly negligible in the streamwise direction, while it cannot be neglected in the $y$ -direction. This can be explained by the fact that the vortex pairs are mainly oriented towards the $y$ -direction, i.e. with $\alpha$ close to $0$ (see figure 3), thus having a significant effect on the pulsating component in this direction. In the following, we therefore consider that

(3.1) \begin{equation} U_{{vortex}}=\sqrt {\big\langle U_{{jet}}^2 \big\rangle } \approx \langle U_{{jet}} \rangle \,\,\,\,\textrm {and}\,\,\,\,V_{{vortex}}=\sqrt {\big\langle V_{{jet}}^2 \big\rangle }. \end{equation}

Note that (3.1) for the streamwise velocity implies that the jet inversion, defined as $\langle U_{{jet}} \rangle =U_0$ , is therefore equivalent to $U_{{vortex}}=U_0$ .

3.3. Vortex velocities as a function of the Strouhal number

Since jet and vortex velocities are well connected, we now focus on the evolution of the normalised vortex velocities with the Strouhal number, as shown in figure 6. All points collapse well on two master curves, one for the streamwise velocity $U_{{vortex}}/U_0$ (at the top) and one for the transverse velocity $V_{{vortex}}/U_0$ (at the bottom). This clearly shows that the Strouhal number is the relevant control parameter for the vortex velocities. In addition, two points (one for each velocity direction) have been added from the literature (as magenta triangles at $St_A=0.23$ ), since we were able to estimate the vortex velocities from a time series of images from Buchholz & Smits (Reference Buchholz and Smits2006). These literature points are consistent with our experimental points and lie well on the two master curves.

Figure 6.

Normalised streamwise ( $U_{{vortex}}/U_0$ ) and transverse ( $V_{{vortex}}/U_0$ ) vortex velocities as a function of the Strouhal number $St_A$ . The colours and shapes of the points are the same as those in figure 2, while the magenta triangles at $St_A=0.23$ correspond to data from Buchholz & Smits (Reference Buchholz and Smits2006). The two solid curves show the fits of the two vortex velocities, using (3.2) and (3.8). The vertical solid line at $St_A^{\star} =0.48$ indicates the drag–thrust transition measured with the force sensor, while the vertical dashed line at $St_A^{\textit{inv}}=0.36$ shows the jet inversion where $ U_{{vortex}} = U_0$ .

In figure 6, the transverse vortex velocity $V_{{vortex}}/U_0$ seems to be proportional to the Strouhal number. Such behaviour can be easily understood since the typical dimensional scaling for $V_{{vortex}}$ is expected to be given by tail amplitude times tail frequency, such as $V_{{vortex}} \sim Af$ . This leads to $V_{{vortex}}/U_0 \sim {St}_A$ , as clearly visible in figure 6. All the experimental points collapse along a line, passing through the origin. A linear fit constrained to the origin of these points leads to

(3.2) \begin{equation} \frac {V_{{vortex}}}{U_0} = K_v\,{St}_A, \end{equation}

with $K_v \approx 0.82$ , and is shown by a line in figure 6.

We now focus on the streamwise vortex velocity $U_{{vortex}}$ . In figure 6, $U_{{vortex}}/U_0$ is first almost constant and slightly below unity at very low Strouhal numbers, before increasing at high Strouhal numbers. This behaviour is explained in the following model. Since we have shown that the vortex and the time-averaged jet velocities are strongly coupled, we perform a momentum balance on the mean velocity field, on a control surface surrounding the fish (Batchelor Reference Batchelor1967; Koochesfahani Reference Koochesfahani1989; Anderson et al. Reference Anderson, Streitlien, Barrett and Triantafyllou1998; Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008; Bohl & Koochesfahani Reference Bohl and Koochesfahani2009). This surface is chosen as the surface of a cuboid encompassing the fish and the wake behind it in all dimensions. Therefore, its width is chosen to be equal to the width of the imaging section in the $y$ -direction, its length to be equal to the sum of the fish length $L$ and the length of the imaging section in the streamwise $x$ -direction and its thickness $H$ represents the typical scale of the layer where the 3-D vortex structures are confined within the spanwise direction. Following Batchelor (Reference Batchelor1967), the time-averaged resultant in the $x$ -direction of the different fluid forces is given by

(3.3) \begin{equation} \langle F \rangle = \left\langle \int \left(p_1 + \rho u_1^2-p_2-\rho u_2^2 \right)\,\textrm {d}\Sigma -\rho \int u (\boldsymbol{u}\cdot \boldsymbol{n})\,\textrm {d}S \right\rangle , \end{equation}

where $u_1$ , $p_1$ , $u_2$ and $p_2$ are the values of the $x$ -component of the velocity field or the pressure at the upstream and downstream faces, respectively. Here, $\Sigma$ is the area of these two faces, perpendicular to the streamwise direction, while $S$ is the area of the other faces of the cuboid, perpendicular to the $y$ - or $z$ -directions. On these latter faces, it is assumed that the flow is very close to free stream conditions, i.e. that $\boldsymbol{u}=U_0 \boldsymbol{e}_x$ . This cancels the integral on the surface $S$ in (3.3).

By applying Bernoulli’s theorem (Batchelor Reference Batchelor1967; Bohl & Koochesfahani Reference Bohl and Koochesfahani2009) and neglecting the velocity in the spanwise direction, (3.3) becomes

(3.4) \begin{equation} \langle F \rangle = \left\langle \frac {1}{2}\rho \int \left(u_1^2-v_1^2 - u_2^2 + v_2^2 \right)\textrm {d}\Sigma \right\rangle , \end{equation}

where $v_1$ and $v_2$ are the values of the $y$ -component of the velocity field at the upstream and downstream faces. We now assume that the velocity passing through the upstream face is $U_0$ , i.e. that $\langle u_1^2 \rangle =U_0^2$ and $\langle v_1^2 \rangle =0$ . At the downstream face, we consider that the velocity is also equal to the free stream speed, i.e. that $\langle u_2^2 \rangle =U_0$ and $\langle v_2^2 \rangle =0$ , except in the jets forming the V-shaped wake, where $\langle u_2^2\rangle =\langle U_{{jet}}^2\rangle$ and $\langle v_2^2\rangle =\langle V_{{jet}}^2 \rangle$ . The temporal average to consider for the mean field is therefore the one that takes into account both mean and fluctuating components of the velocity field (Bohl & Koochesfahani Reference Bohl and Koochesfahani2009), as demonstrated in § 3.2. Note that this simplified mean velocity field does not conserve mass in the mid-span plane, while mass conservation is classically used in the literature in such momentum balance (Batchelor Reference Batchelor1967; Anderson et al. Reference Anderson, Streitlien, Barrett and Triantafyllou1998; Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008). However, as the wake structure is fully 3-D in our case, it is not an issue to not conserve mass in a plane. Mass is of course conserved in the volume considered, but the velocity fields in the top-span and bottom-span planes, at the upper and lower limits of the fish tail in the spanwise direction, are probably very different from that considered here as a simplification. These assumptions lead to

(3.5) \begin{equation} \langle F \rangle =\frac {1}{2}\rho \big(U_0^2-\big\langle U_{{jet}}^2\big\rangle +\big\langle V_{{jet}}^2\big\rangle \big)2\ell H, \end{equation}

with $\ell$ the width of the jets along the $y$ -direction, defined in figure 3(b). Since the vortex velocities are equal to the root-mean-square jet velocities (3.1), except for the wakes at high Strouhal number, and with $V_{{vortex}}/U_0 = K_v {St}_A$  (3.2), one can therefore write that

(3.6) \begin{equation} \langle F \rangle =\rho \ell H U_0^2\left (1- \left (\frac {U_{{vortex}}}{U_0}\right )^2+ K_v^2 {St}_A^2\right ). \end{equation}

Since the fish position is fixed in the laboratory frame, the force $\langle F \rangle$ is that measured by the force sensor, also given in (2.1). Combining (2.1) and (3.6) leads to

(3.7) \begin{equation} \frac {U_{{vortex}}}{U_0} =\sqrt {1-\frac {C_D L^2}{\ell H}+\left (K_v^2+\frac {C_D L^2}{\ell H \left( {St}_A^{\star} \right)^2}\right ) {St}_A^2}. \end{equation}

Here, all quantities are known except the ratio $C_D L^2/(\ell H)$ . One therefore fits the experimental points for $U_{{vortex}}/U_0$ in figure 6 with (3.7) using a single fit parameter. This leads to $C_D L^2/(\ell H) \approx 0.19$ and to

(3.8) \begin{equation} \frac {U_{{vortex}}}{U_0} = \sqrt {K_0^2+K_u^2\, {St}_A^2}, \end{equation}

with $K_0^2 =1- C_D L^2/(\ell H)\approx 0.81$ and $K_u^2 = K_v^2+C_D L^2/(\ell H ( {St}_A^{\star} )^2)\approx 1.50$ . This fit describes well the trend observed experimentally, as shown by the curve in figure 6. In particular, it nicely captures that $U_{{vortex}}/U_0$ is almost constant at low St $_A$ , close to the value where the fish is not moving at all. With a measured drag coefficient $C_D L^2=1.1 \times 10^{-3}$ m $^2$ (Gibouin et al. Reference Gibouin, Raufaste, Bouret and Argentina2018), a width of the jet of $\ell \approx 40$ mm and a value of the effective thickness $H\approx 60$ mm, estimated as the dimension of the fish tail in the spanwise direction, one expects $C_D L^2/(\ell H)\approx 0.45$ . This represents the good order of magnitude, with a slight difference with the fit value. This shows that the effective thickness $H$ is larger than the fish dimension in the spanwise direction. Such an effective thickness is expected to account for 3-D effects in the wake, thus depending on the fine 3-D structure of the wake. However, it is remarkable that the shape of the vortex velocity curve as a function of the Strouhal number is recovered with this quasi-2-D momentum balance, by measuring the velocity field in the mid-span plane only.

Regarding the jet inversion, which is here characterised by $U_{{vortex}}=U_0$ (see 3.1), (3.7) shows that it occurs for a Strouhal number given by

(3.9) \begin{equation} {St}_A^{\textit{inv}} = \frac {{St}_A^{\star} }{\sqrt {1+\frac {K_v^2 \ell H}{C_D L^2}\left({St}_A^{\star} \right)^2}}. \end{equation}

For $K_v$ non null, (3.9) shows that the critical Strouhal number $St_A^{\star}$ for the drag–thrust transition is always larger than the Strouhal number $St_A^{\textit{inv}}$ where the jet inversion occurs. This is consistent with what has been observed in the literature of 2-D wakes (Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008, Reference Godoy-Diana, Marais, Aider and Wesfreid2009; Bohl & Koochesfahani Reference Bohl and Koochesfahani2009; Das et al. Reference Das, Shukla and Govardhan2016; Andersen et al. Reference Andersen, Bohr, Schnipper and Walther2017) and here highlights the importance of the vortex velocities in the transverse $y$ -direction. Using the values of $K_v\approx 0.82$ and $C_D L^2/{\ell H}\approx 0.19$ obtained from (3.2) and (3.8), and $St_A^{\star} \approx 0.48$ , we obtain $St_A^{\textit{inv}}\approx 0.36$ . This value is shown as a dashed line in figure 6, while the drag–thrust transition based on the force sensor at $St_A^{\star} \approx 0.48$ is represented by a solid line, and is indeed slightly higher. At this latter Strouhal number, (3.8) gives $U_{{vortex}}^\star /U_0 \approx 1.1$ , as observed in figure 6.

4.1. Wake structure model

We further extend our model, assuming that each vortex pair behaves like a self-propelled dipole, advected by the free stream flow and its own induced velocity (Couder & Basdevant Reference Couder and Basdevant1986; Dong et al. Reference Dong, Mittal and Najjar2006; Buchholz & Smits Reference Buchholz and Smits2008). The resulting trajectory defines the wake geometry. This formulation thus directly connects vortex dynamics, discussed in § 3, and wake topology, discussed in this section.

Let us consider the sketch of a 2P wake shown in the horizontal mid-plane behind the fish, as shown in figure 3(b). Each vortex pair corresponds to the 2-D trace of a 3-D vortex ring, has an orientation with respect to the $y$ -direction given by the angle $\alpha$ , and a circulation $\pm \Gamma$ for each vortex. Once generated, the pair has a self-propelled speed, noted $U_{{dipole}}$ , that depends on its circulation $\Gamma$ and on its geometry (Guyon, Hulin & Petit Reference Guyon, Hulin and Petit2001). In our case, the fish position is fixed in the laboratory frame but a free stream flow speed $U_0$ is imposed. Therefore, the vortex pair velocity in the laboratory frame results from the vector combination of the free stream velocity and the self-induced velocity of the vortex pair. Even though the formation of the vortical structures in the vicinity of the tail is a very complex phenomenon (Dong et al. Reference Dong, Mittal and Najjar2006; Buchholz & Smits Reference Buchholz and Smits2006, Reference Buchholz and Smits2008; Li & Dong Reference Li and Dong2016), we assume for simplicity that a vortex pair is shed at time $t$ at the end of the fish tail, in position $A$ . At time $t+ \Delta t$ , this pair has moved to position $B$ , driven by the free stream speed $U_0$ in the streamwise direction and by its self-induced speed $U_{{dipole}}$ in an oblique direction at an angle $\alpha$ . Therefore, its trajectory $AB$ can be decomposed as $AC = U_0 \Delta t$ and $CB = U_{{dipole}} \Delta t$ . By repeating this operation for different $\Delta t$ , all vortex pairs align on two symmetrical straight lines at an angle $\theta$ with respect to the axis of the wake (Couder & Basdevant Reference Couder and Basdevant1986; Dong et al. Reference Dong, Mittal and Najjar2006). The angle $\theta$ is given by

(4.1) \begin{equation} \tan \theta = \frac {V_{{vortex}}}{U_{{vortex}}}, \end{equation}

where the vortex velocities in the laboratory frame can be expressed through the combination of advection and self-induced velocities:

(4.2) \begin{eqnarray} U_{{vortex}} = U_0+U_{{dipole}} \sin \alpha , \end{eqnarray}

(4.3) \begin{eqnarray} V_{{vortex}} = U_{{dipole}} \cos \alpha , \end{eqnarray}

respectively in $x$ - and $y$ -directions.

Equations (4.1)–(4.3) show that the wake angle and vortex properties are indeed intimately all interconnected. In the remainder of the section, we demonstrate that this simplified framework is valid for the 3-D wakes observed in this work and in the literature, by comparing (4.1)–(4.3) to experimental or literature data. Note that this description has been performed for a 2P wake, but most of the considerations and definitions can be at least extended to the 2S + 2P and the 4P wakes observed and described in this work. Indeed, for these two types of wakes, one can see that there is a main pair of vortices, propagating along the oblique thick lines, and an extra vortex (for 2S + 2P) or an extra pair (for 4P). As illustrated in figure 3, one can clearly see that these extra vortical structures travel together at a similar speed and same oblique angle $\theta$ as the main pair, within the observation window used in this study. By applying the definitions mentioned previously to the main pair, we can generalise (4.1)–(4.3) to 2S + 2P and 4P wakes. For the 2S wake, composed of horseshoe vortices (and not vortex rings) that interact with each other with their vorticity in the streamwise direction (Buchholz & Smits Reference Buchholz and Smits2006), this view cannot be entirely applied and remains limited to the generic equation (4.1) between the wake angle and vortex speeds.

4.2. Wake angle

As shown in (4.1), the wake angle $\theta$ is defined as the angle formed between the vortex trajectories and the streamwise $x$ -direction. Combining the two fits (3.2) and (3.8) obtained for $U_{{vortex}}/U_0$ and $V_{{vortex}}/U_0$ as a function of St $_A$ into (4.1) for the angle $\theta$ , one obtains

(4.4) \begin{equation} \tan \theta = \frac {K_v {St}_A}{\sqrt {K_0^2+K_u^2\,{St}_A^2}}. \end{equation}

Since the vortices are responsible for the formation of the jets within the wake (see figure 3), the jets also exhibit the same angle $\theta$ with respect to the streamwise direction. The wake angle $\theta$ is therefore directly measured using the time-averaged velocity field from 2-D PIV data. For each experiment, the angle of each jet behind the fish is measured, from both sides of the jet. The angles obtained for the different jets are then averaged and used to estimate error bars. Another method to measure the wake angle based on vortex trajectories led to very similar results, but with slightly larger errorbars. The tangent of $\theta$ is plotted in figure 7 as a function of the Strouhal number. While three different physical parameters have been varied in large ranges and four different wake patterns have been observed, all data collapse well on a master curve, showing again that the Strouhal number is the relevant dimensionless parameter that controls the wake properties. One nicely recovers the master curve by plotting $\tan \theta$ defined in (4.4) in figure 7. The master curve exhibits two limit regimes, shown by the two dashed lines. At low Strouhal number, $\tan \theta$ is proportional to ${St}_A$ since $U_{{vortex}}/U_0$ is almost constant and $V_{{vortex}}/U_0$ is linear with the Strouhal number. At large Strouhal number, the angle saturates to an asymptotic value, as both normalised velocities are linear with the Strouhal number. Using (4.4), the proportionality and saturation constants are equal to $K_v/K_0$ and $K_v/K_u$ , respectively, estimated as $0.91$ and $0.67$ from the fits (3.2) and (3.8). Note that Buchholz & Smits (Reference Buchholz and Smits2006) have already noticed that, at low Strouhal number, the angle of the wake increases proportionally with the Strouhal number.

Figure 7.

Tangent of the wake angle $\theta$ as a function of the Strouhal number St $_A$ . The colours and shapes of the points are the same as those in figure 2, while the magenta triangles correspond to literature data. The dashed lines indicate $\tan \theta = K_v/K_0\, {St}_A\approx 0.91\, {St}_A$ and $\tan \theta = K_v/K_u\approx 0.67$ , while the curve is defined by (4.4).

In addition to our experimental measurements, several points extracted from the literature (when a 3-D wake is visible and can be easily measured) have been added. These points are plotted as magenta triangles (pointing left) in figure 7 and overlap well with the master curve, independently of their configuration: numerical simulations of fish (Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2008) or moving panels (Blondeaux et al. Reference Blondeaux, Fornarelli, Guglielmini, Triantafyllou and Verzicco2005; Dong et al. Reference Dong, Mittal and Najjar2006; Li & Dong Reference Li and Dong2016), experiments with pitching panels (Buchholz & Smits Reference Buchholz and Smits2006) or with another robotic fish (Epps et al. Reference Epps, Valdivia y Alvarado, Youcef-Toumi and Techet2009). This seems to show that this master curve is universal, and is mainly independent of other parameters, such as Reynolds number, exact shape, aspect ratio or swimming mode of the moving objects.

4.3. Orientation of the vortex rings

We now focus on the orientation of the vortex pairs with respect to the $y$ -direction. In this section, we restrict ourselves to the main vortex pair of the 2P, 2S + 2P and 4P wakes, since it is possible to define and measure the proper orientation of such a vortex pair in these wakes.

Using (4.2) and (4.3), the tangent of the angle $\alpha$ can easily be obtained as

(4.5) \begin{equation} \tan \alpha = \frac {U_{{vortex}}-U_0}{V_{{vortex}}}. \end{equation}

This expression is tested in figure 8(a). It shows that (4.5) is well verified for the 2P and 4P wakes. The three points (black diamonds) corresponding to a 2S + 2P wake appear as slightly shifted from the remainder of the points. On each oblique direction of this wake, there is a main vortex pair and a single vortex that travel together at the same speed. They are probably the 2-D footprint of a vortex ring and a horseshoe vortex. The horseshoe vortex interacts directly with the vortex ring due to the short distance between them, but may also interact with other structures located at a larger distance through its streamwise leg. This may influence the orientation of the vortex ring and affect its trajectory. Note that the secondary vortex ring in the 4P wake does not seem to produce such difference, maybe because it is a vortex ring and thus does not own a streamwise leg, in contrast to horseshoe vortices.

Figure 8.

Tangent of the angle $\alpha$ as a function of (a) the combination of $U_{{vortex}}/U_0$ and $V_{{vortex}}/U_0$ given in (4.5) and (b) the Strouhal number $St_A$ . Only the experiments exhibiting a 2P, 2S + 2P or a 4P wake are plotted. The lines show the prediction of (4.5) (panel a) and (4.6) (panel b). Magenta triangles in panel (b) correspond to the points given by Li & Dong (Reference Li and Dong2016). In panel (b), the vertical solid line indicates the drag–thrust transition at $St_A^{\star} =0.48$ , while the vertical dashed line at $St_A^{\textit{inv}}=0.36$ shows the vortex inversion where $ U_{{vortex}} = U_0$ .

With the fit expressions (3.2) and (3.8) for the normalised vortex velocities as a function of the Strouhal number, $\tan \alpha$ can easily be expressed as a function of the Strouhal number, through (4.5):

(4.6) \begin{equation} \tan \alpha = \frac {\sqrt {K_0^2+K_u^2\, {St}_A^2}-1}{K_v {St}_A}. \end{equation}

Figure 8(b) shows the comparison between our experimental measurements of $\tan \alpha$ , the data from Li & Dong (Reference Li and Dong2016) (magenta triangles) and (4.6), as a function of the Strouhal number. Again, all data collapse well on the expected behaviour. In particular, it nicely shows that the angle $\alpha$ of the wake increases from negative to positive value with increasing Strouhal number. According to (4.5), the angle $\alpha$ is equal to $0$ when $U_{{vortex}}=U_0$ , i.e. when the Strouhal number reaches the inversion Strouhal number $St_A^{\textit{inv}}\approx 0.36$ , defined in (3.9). This corresponds to the vortex or jet inversion, where the vortex pair orientation changes from negative to positive or where the mean streamwise component of the jet velocity is equal to the free stream speed. As shown by the vertical dashed dotted lines at $St_A^{\textit{inv}}= 0.36$ in figures 6 and 8(b), $U_{{vortex}}=U_0$ and $\alpha =0$ occur at the same Strouhal number. Since the inversion Strouhal number is only slightly smaller than the drag–thrust transition Strouhal number, the orientation of the vortex pairs is a good indicator of the propulsion regime (drag or thrust), as already noticed in the literature for 2-D wakes (Andersen et al. Reference Andersen, Bohr, Schnipper and Walther2017) and 3-D wakes (Dong et al. Reference Dong, Mittal and Najjar2006). However, the drag–thrust transition does not occur exactly for $\tan \alpha =0$ but for $\tan \alpha \approx 0.09$ here, i.e. $\alpha \approx 5^\circ$ , using (4.6).

4.4. Circulation and diameter of the vortex rings

We now discuss the measurements of the vortex ring circulation $\Gamma$ . We limit ourselves to the 2P wakes observed in this study, since having more vortices in the wake may significantly change the circulation magnitude of the main vortex pair. Buchholz, Green & Smits (Reference Buchholz, Green and Smits2011) have proposed a scaling for the circulation in the case of a pitching panel with low aspect ratio, i.e. in the presence of a 3-D wake. This kinematic scaling originates from the production of vorticity by the motion of the panel, having a transverse speed $Af$ on a typical time $1/f$ . They have found experimental evidence that the circulation $\Gamma$ mainly scales as $f A^2$ , with extra corrections depending on the ratio between the amplitude $A$ and the height of the panel in the spanwise direction. In their experiments, they have varied these two parameters and the free stream speed $U_0$ , but the oscillation frequency $f$ has been kept constant. In our case, the circulation has been measured for all experiments, where the three main parameters $(A,f, U_0)$ have been varied significantly while keeping the geometry of the tail constant. This has allowed us to test how the circulation varies with these three parameters with respect to several hypotheses. Figure 9(a) shows the vortex ring circulation $\Gamma$ normalised by the scaling $f A^2$ as a function of the Strouhal number $St_A$ , for the 2P wakes observed in this study. One clearly observes that the normalised circulation decreases with the Strouhal number, while it is expected to be constant according to the scaling of Buchholz et al. (Reference Buchholz, Green and Smits2011).

Figure 9.

(a) Dimensionless circulation $\Gamma /(fA^2)$ and (b) pair diameter $\xi /A$ as a function of the Strouhal number $St_A$ , for 2P wakes. The solid curve in panel (a) shows the fit of the data by (4.7), while that in panel (b) represents (4.10).

In our case, it is therefore necessary to consider another scaling for the circulation. Indeed, even if the motion of the tail clearly contributes to the creation of vorticity, one can also consider that vorticity can be created when the tail is at a fixed angle in a streamwise flow. Even though the angle of the tail is on average equal to zero, the tail spends a significant amount of time close to its extremum position during the oscillation. One can therefore expect to have two contributions for the scaling of the circulation $\Gamma$ : a kinematic scaling $fA^2$ (Buchholz et al. Reference Buchholz, Green and Smits2011), independent of the streamwise speed $U_0$ , and a stationary scaling $U_0 A$ , independent of the frequency motion $f$ . When normalising the circulation by the kinematic scaling as in figure 9(a), this leads to

(4.7) \begin{equation} \frac {\Gamma }{fA^2}=\gamma _d + \frac {\gamma _s}{ {St}_A}, \end{equation}

where $\gamma _d$ and $\gamma _s$ are two dimensionless coefficients, reflecting the contribution of the kinematic and stationary scalings, respectively. The data represented in figure 9(a) can be nicely fitted by (4.7), leading to $\gamma _d=0.21$ and $\gamma _s=0.45$ . These two coefficients are of the same order of magnitude but, for a Strouhal number smaller than $1$ , the dominant contribution comes from the stationary scaling. In the experiments of Buchholz et al. (Reference Buchholz, Green and Smits2011), the stationary contribution appears negligible. This suggests that the coefficients $\gamma _d$ and $\gamma _s$ depend strongly on the swimming mode as well as swimmer shape and flexibility. For instance, for a flexible propulsor, the combination of pitching and heaving motions has been found to be more efficient than the pitching motion alone (Quinn, Lauder & Smits Reference Quinn, Lauder and Smits2015). Since, the undulating motion of the robotic fish tail differs significantly from the rigid pitching used by Buchholz et al. (Reference Buchholz, Green and Smits2011), we can thus expect differences in vorticity production.

The circulation $\Gamma$ therefore seems to scale depending on a stationary configuration, while its magnitude clearly affects the wake structure and the thrust production that are related to the instationarity of the wake. Indeed, the circulation $\Gamma$ is connected to the vortex velocities through the pair self-advecting speed $U_{{dipole}}$ and the pair diameter $\xi$ . A comprehensive calculation (Guyon et al. Reference Guyon, Hulin and Petit2001), assuming that the vortex owns a cylindrical core of size $a$ in uniform rotation, gives

(4.8) \begin{equation} U_{{dipole}}=\frac {\Gamma }{2\pi \xi }\left (\ln \frac {4\xi }{a}-\frac {1}{2}\right ), \end{equation}

and is consistent with the typical simplified scaling $U_{{dipole}}\sim \Gamma /\xi$ .

The distance $\xi$ , normalised by the tail amplitude $A$ , is plotted in figure 9(b) as a function of the Strouhal number, showing a decreasing trend with increasing values of this parameter. To understand this trend, one can express the self-induced speed from (4.2) and (4.3):

(4.9) \begin{equation} U_{{dipole}}=\left (\left (U_{{vortex}}-U_0\right )^2+V_{{vortex}}^2\right )^{1/2}, \end{equation}

and combine the fits (3.2), (3.8) and (4.7) of the normalised vortex velocities and circulation as a function of the Strouhal number together in the simplified scaling of (4.8):

(4.10) \begin{equation} \frac {\xi }{A} \approx \frac {\Gamma }{A U_{{dipole}}}=\frac {\Gamma \!{St}_A}{fA^2}\frac {U_0}{ U_{{dipole}}}=\frac {\gamma _d {St\!}_A+ \gamma _s}{\left (\left (\sqrt {K_0^2+K_u^2\, {St}_A^2}-1\right )^2+K_v^2 {St}_A^2\right )^{1/2}}. \end{equation}

The prediction of (4.10), shown as a solid curve in figure 9(b), captures rather well the experimental trend. At small Strouhal number, i.e. $St_A\lt 0.5$ , (4.10) leads to $\xi /A \approx \gamma _s /(K_v {St}_A)$ , consistent with the decrease of $\xi /A$ for small Strouhal in figure 9(b). At large Strouhal number, i.e. $St_A\gt 1$ , (4.10) leads to the convergence of $\xi /A$ towards a constant $\gamma _s/\sqrt {K_u^2+K_v^2}\approx 0.3$ , also consistent with figure 9(b) at large Strouhal number. Note that (4.10) is obtained by using the simple scaling law $U_{{dipole}}=\Gamma /\xi$ , instead of using the full (4.8). Indeed, the factor $\ln ((4\xi /a)-1/2 )/(2\pi )$ has a weak dependence on Strouhal number, and has been measured to be approximately $0.4$ in our experiments. A factor of value $1$ better adjusts our experimental measurements for $\xi$ , but the order of magnitude is the same and (4.8) assumes a vortex ring with a cylindrical core in uniform rotation, which may be not the case for the vortices in the wake.