3.1. Classification of vortex patterns

We identify three distinct vortex patterns in the wake of parallel pitching foils (side-by-side configuration) for the given parametric space. Merged–separated characteristics of the wakes were taken into consideration when classifying the wake in figure 4. Here, a merged wake corresponds to the vortex topology that involves the vortex streets shed by upper and lower foils merging in mid-wake and forming a single street, which constitutes a new flow configuration. In separated wakes, on the other hand, upper and lower vortex streets do not amalgamate with each other. Separated wakes of in-phase pitching foils consist of two parallel vortex streets whereas a ‘v-shaped’ diverging configuration is observed in the separated wakes of out-of-phase pitching foils (see figure 4b,d). In both merged and separated wakes, vortex patterns are formed within several pitching cycles and their merged–separated features remain unchanged during the next oscillation cycles without altering significantly (compare $t_1=14P$ and $t_2=20P$ of figures 4(a), 4(b) and 4(d), where P is the period of the pitching cycle). Note that out-of-phase pitching foils at $St=0.5$ (not shown here for brevity) experience symmetric to asymmetric transition (Gungor & Hemmati Reference Gungor and Hemmati2020). However, its wake remains separated throughout the process. Conversely, transitional-merged wakes undergo distinct separation and merging stages, primarily transitioning from the former to the latter configuration. As explained earlier, oscillating foils produce deflected reverse BvK vortex streets at considerably high $St$. In the wake of in-phase pitching parallel foils, interaction between vortex streets shed by each foil results in the constitution of the symmetric wake, in which upper and lower wakes amalgamate around the centreline. (see figure 4c). The pitching cycle, in which the merging takes place, greatly depends on $d^*$ and $St$ (see table 2). For the sake of comparison, the merging process of the wakes occurs around the 22nd cycle for $d^*=1$ and $St=0.5$, whereas more than 75 cycles are needed for this phenomenon to occur for $d^*=2$ and $St=0.5$. These vortex patterns were gathered in a $St{-}d^*$ phase diagram in order to provide a thorough classification of the wakes of in-phase pitching parallel foils in figure 5(a). In this diagram, separated and merged wakes are observed in upper ($d^*\geqslant 1.5$) and lower ($d^*\leqslant 1$) regions of the diagram, respectively. On the other hand, transitional-merged wakes fall into the high $St$ region. It is important to note that this type of wake is formed only at sufficiently high $St$ that facilitate the formation of deflected wakes. The relation between the deflection phenomena and wake merging is further explained in the next part of this section. Furthermore, another diagram, explaining the wake topology of parallel foils for varying phase difference and Strouhal number at fixed separation distance of $1c$, is presented in figure 5(b). Vortex patterns display a strong dependence on phase difference. For ${\rm \pi} /3 \leqslant \phi \leqslant 5{\rm \pi} /6$, parallel foils constitute separated wakes for each $St$ examined here, which resemble separated wakes of out-of-phase oscillations, i.e. diverging vortex pattern. On the other hand, merged wakes at $\phi ={\rm \pi} /6$ do not fundamentally differ from those formed by in-phase oscillating foils. Results for an intermediate phase difference are not presented here for brevity, because they yields similar conclusions.

Figure 4.

Contour of spanwise vorticity ($\omega _{z}^{*}$) of parallel foils for (a) $St=0.25$ and $d^*=1$ (merged wake), (b) $St=0.3$ and $d^*=1.5$ (separated wake), (c) $St=0.5$ and $d^*=2$ (transitional-merged wake) and (d) $St=0.4$ and $d^*=1$ (separated wake) at different time instants for in-phase and out-of-phase pitching. Here, vorticity is normalized by $U_{\infty }/c$. (See supplementary movies 1, 2, 3 and 6, available at https://doi.org/10.1017/jfm.2022.785, for the entire wake evolution of (a–d), respectively.)

Figure 5.

Classification of the wake patterns of foils in a side-by-side configuration for $Re=4000$ at (a) a range of separation distances and Strouhal numbers for in-phase pitching, (b) a range of phase differences and Strouhal numbers for $d^*=1$. Dashed lines correspond to the boundary that distinguishes merged and separated wakes.

Table 2.

Streamwise location ($x/c$) and time instant ($t/P$) in which the wake merging occurs as well as the percent improvement in the cycle-averaged coefficient of thrust ($\Delta \widetilde {C_T}$) for separated and transitional-merged wake cases at $St=0.4$ and $St=0.5$.

Hereafter, we present quantification of important characteristics of the wakes and propose a model that distinguishes emerging vortex patterns. At this point, it is important to recall the dipole model by Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009), which presented a quantitative threshold for the wake deflection behind a single oscillating foil. This model also remains valid for the wake of undulating foils (Khalid et al. Reference Khalid, Wang, Dong and Liu2020). The wake of oscillating foils, consisting of a reverse BvK vortex street, is dominated by shedding of a counterclockwise (positive sign) and a clockwise (negative sign) vortex per oscillation cycle. These vortices are located slightly above and below the centreline, respectively (Koochesfahani Reference Koochesfahani1989). The structure formed by these two vortices is called a dipole. Circulations of the vortices in a dipole induce a velocity normal to the line that connects the vortex centres as described by the 2-D Bio-Savart rule (Naguib, Vitek & Koochesfahani Reference Naguib, Vitek and Koochesfahani2011). When the self-advection velocity of the dipole is strong enough, it diverts the dipole from the centreline, which is followed by the consecutive dipoles. Therefore, the model was based on the offset between advection velocity of the propulsive wake, i.e. $U_{phase}$, and self-induced translation velocity of the dipole, i.e. $U_{dipole}$. They can be mathematically defined as

(3.1)\begin{gather} U_{phase}={\rm d}X_i/{\rm d}t, \end{gather}

(3.2)\begin{gather}U_{dipole}=\varGamma / 2 {\rm \pi}\xi, \end{gather}

where $X_i$ is the $x$ coordinate of a vortex core, $\varGamma$ denotes the average of magnitudes of circulation of counter-rotating vortices and $\xi$ represents the distance between the centres of the vortices (see figure 6a). Circulation ($\varGamma$) is computed either from a line integral of the velocity field or from a surface integral of vorticity over the area bounded by a closed curve. Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009) used a rectangular frame, whose size was determined by a Gaussian fit, to extract the boundary of each vortex towards calculating $\varGamma$. However, this method has a downside of potential numerical errors due to the possibility that rectangular frames may include counter-rotating vortices, particularly in the case of structures traversing in close proximity of one another. Therefore, we use a non-predefined closed curve to accurately capture each vortex proposed earlier by Khalid et al. (Reference Khalid, Wang, Dong and Liu2020), which eliminates this error in computing $\varGamma$. The boundary of the curve is defined such that it only encompasses the region with a magnitude of vorticity ($\omega$) greater than $10\,\%$ of its value in the flow field. Then, we determine the circulation of each vortex by calculating the line integral of the velocity field around the curve using the definition

(3.3)\begin{equation} \varGamma=\oint \boldsymbol{V}\ {\cdot}\ {\rm d}\boldsymbol{l}, \end{equation}

where $\boldsymbol{V}$ is the velocity and ${\rm d}\boldsymbol{l}$ is the infinitesimal length. This method ensures that circulation is computed without any penetration by a neighbouring vortex with oppositely signed vorticity. Hence, regions in which circulations of positive and negative vortices are calculated are entirely separated from each other by non-predefined boundaries around these coherent flow structures.

Figure 6.

(a) Demonstration of the parameters used in the proposed model. (b) Effective phase velocity of the coupled vortex system with respect to radial displacement of the dipoles.

For the effective phase velocity ($U^*_{p}$), Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009) defined it in the following manner that yields positive values for deflected wakes:

(3.4)\begin{equation} U^*_{p}=(U_{phase}-U_{\infty}) \cos\alpha - U_{dipole}. \end{equation}

Here $\alpha$ is the angle between $U_{phase}$ and $U_{dipole}$ as presented in figure 6(a). We present here a model that distinguishes different classes of vortex patterns using $U^*_{p}$. Although the model of Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009) successfully predicts whether the wake is deflected behind an isolated oscillating foil, it cannot identify the nature of the vortex patterns, i.e. merged or separated, for multiple parallel foils, forming complex wakes in close proximity of one another. In order to construct an effective mathematical model, our current work focuses on differentiating merged and separated wakes and supplying information about the direction of their deflections. To illustrate it further, in-phase pitching of a transitional-merged wake at $St=0.5$ and $d^*=2$ exhibits deflection during each stage of wake development. During the separated stage at $t_1=30P$, both top and bottom wakes are deflected downwards, whereas the wake fully transitions to that of a merged configuration at $t_2=90P$. In the latter stage, the upwards deflected bottom vortex street and downwards deflected top vortex street is observed (see figure 4c). However, the model proposed by Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009) cannot distinguish the vortex patterns for these configurations since all the cases consist of deflected wakes. Similarly, a separated wake with deflected vortex streets at $St=0.4$ and $d^*=2$ (see figure 7a) and a separated wake with horizontal vortex streets at lower $St$ and $1.5 \leqslant d^* \leqslant ~2.5$ (e.g. figure 4b) are treated disparately by the model, although they are all classified as separated wakes. Thus, we introduce the term $\sin \alpha$ to the formulation to take the direction of deflection into account, because $\sin \alpha$ yields positive values for upwards wakes and negative values for downwards and non-deflected wakes. Moreover, a weight factor term, $W_i$, which yields $-1$ for diverging separated wakes and $1$ for the rest, is incorporated into the equation to distinguish the separated wakes of out-of-phase pitching parallel foils. Hence, the effective phase velocity of the coupled vortex system or $U^*_{p,sys}$ can now be defined as

(3.5)\begin{equation} U^*_{p,sys}=\frac{U^*_{p,upper} \sin{\alpha}_{upper}}{U^*_{p,lower} \sin{\alpha}_{lower}} \, W_i. \end{equation}

Here, $W_i$ is the wake weighting function defined as

(3.6)\begin{equation} W_{i} = \begin{cases} 1, & \text{small $\beta$ ($|\beta| < \beta^*/\epsilon$)}\\ \sin(\beta)/|\sin(\beta)|, & \text{large $\beta$ ($|\beta| \geqslant \beta^*/\epsilon$)}, \end{cases} \end{equation}

where $\beta =\alpha _{lower}-\alpha _{upper}$ and $\beta ^*=|\alpha _{lower}|+|\alpha _{upper}|$. Here, $\epsilon$ denotes a positive number, which helps setting up a threshold for different classes of wakes under a broad range of kinematic parameters. We examine the performance of this weight factor term with $\epsilon = 5$, $7.5$ and $10$, and all these values serve the purpose very well.

Figure 7.

Contour of spanwise vorticity ($\omega _{z}^{*}$) of in-phase pitching parallel foils at $d^*=2$ for (a) $St=0.4$ (separated wake) and (b) $St=0.5$ (transitional-merged wake) at different time instants. Here, vorticity is normalized by $U_{\infty }/c$. (See supplementary movies 3 and 4 for the entire wake evolution of (b,a), respectively.)

Equation (3.5) yields negative $U^*_{p,sys}$ values for merging wakes, whereas separated wakes produce positive $U^*_{p,sys}$. The model requires $U^*_{p}$ and $\sin {\alpha }$ for the upper and lower vortex dipoles (see figure 7a) that are shed in the same pitching cycle. Although it can be calculated at a certain location, it is preferred to trace these dipoles as they move in the downstream direction. It helps demonstrate that the model is not limited to a specific location but is valid for the whole domain. Figure 6(b) shows variations in $U^*_{p,sys}$ for separated, merged and transitional-merged wakes with respect to the radial displacement given by $r=\sqrt {(X_1-X_0)^2+(Y_1-Y_0)^2}$, where $X_1$ and $Y_1$ define an instantaneous location of a vortex core. Moreover, $X_0$ and $Y_0$ provide the location of the vortex core just after its detachment process from the foils is completed. Note that a geometric mean of the respective quantities for the counter-rotating vortices is used as the location of the dipole. Because the counter-rotating vortices are shed alternatively from each foil, it is also important to mention that each dipole is formed by those two counter-rotating vortices that have a smaller distance between their centres. It is evident from the plot in figure 6(b) that the proposed model successfully differentiates between separated and merged wakes for the given parametric space.

In this model, separated and merged stages of transitional-merged wakes are treated individually and marked with different colours, since these stages are contradictory to one another in terms of their vortex configuration. It is important to note that merged wake cases are tracked for a relatively short radial displacement, i.e. $r/c\leqslant 3$. This is because their upper and lower vortex streets merge at mid-wake, which inhibits further tracking. However, dipoles of the separated wakes are traceable until circulation of the vortices shrink to negligible values due to the viscous diffusion around $r/c=5$. To clarify the working mechanism of the model, $U^*_{p}$ of merged wakes (see figures 4(a) or 4(c) at $t_2=90$) yields positive values for both bottom and top vortex streets as they are deflected upwards and downwards, respectively. Furthermore, $\sin {\alpha }$ for top and bottom wakes switch signs ($\sin {\alpha }<0$ for top and $\sin {\alpha }>0$ for bottom), which results in $U^*_{p,sys}<0$. On the other hand, horizontal vortex streets in the separated wakes (see figure 4b) have $U^*_{p}<0$ and $\sin {\alpha }<0$, thus leading to $U^*_{p,sys}>0$. The separated wake, whose vortex streets are deflected (see figure 7a), or a transitional-merged wake at the separated stage (see figure 4(c) at $t_1=30P$) yield $U^*_{p}>0$ and $\sin {\alpha }<0$, which translates to $U^*_{p,sys}>0$. Finally, diverging separated wakes whose vortex streets are deflected in opposite directions yield positive values for $U^*_{p}$ and opposite signs for $\sin {\alpha }$ ($\sin {\alpha }>0$ for top and $\sin {\alpha }<0$ for bottom). However, it gets $-1$ from the weight factor since $\beta =\alpha _{lower}-\alpha _{top}<0$, which translates into $U^*_{p,sys}>0$.

The classification and mathematical modelling of the vortex patterns presented here is developed for $Re=4000$. However, it relies on kinematic quantities of coherent structures in the wake, such as the angle between vortex cores, the circulation of vortices and the phase velocity of dipoles. Therefore, it is expected to work well for low and medium $Re$ ranges considering all the flow topologies, i.e. von Kármán street, reverse von Kármán street and deflected von Kármán street, observed for $10 \leqslant Re \leqslant 2000$ (Das et al. Reference Das, Shukla and Govardhan2016), already covered in the analysis. It is noteworthy to state that the coherent structures remain the same even though the wake transitions to 3-D at $Re=8000$ (Verma & Hemmati Reference Verma and Hemmati2021). A similar argument can be made for the range of oscillation amplitudes. Patterns of the coherent structures behind oscillating foils for a range of oscillation amplitudes (Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008; Das et al. Reference Das, Shukla and Govardhan2016) do not fundamentally differ from those considered in the current study. This suggests that wakes of oscillating foils at different oscillating amplitudes can be classified and mathematically modelled using the presently proposed procedure. In an effort to test this, we also simulate cases for $\theta _0=5^{\circ }, 11^{\circ }$ and $14^{\circ }$ at a range of $St$ for $d^*=1.5c$. Their wakes display characteristics of one of the three vortex patterns examined above: separated wake, merged wake or transitional-merged wake. Furthermore, (3.5) performs excellently in distinguishing different topologies of the wake. Nevertheless, these results are not shown here for brevity.

3.2. Mechanism of wake merging

After establishing a mathematical model to quantitatively identify and characterize different wake patterns behind pitching foils in a side-by-side (parallel) configuration, we focus our attention to identify and explain the mechanism of wake merging. To this effect, we analyse primary transitions in the wake of in-phase pitching foils by associating it with the production and dynamics of secondary vortex structures. When an oscillating single foil produces a deflected wake, secondary structures appear from the primary vortex street to move away from the direction of deflection (Gungor et al. Reference Gungor, Khalid and Hemmati2021). Such secondary structures were also observed in the experiments of Godoy-Diana et al. (Reference Godoy-Diana, Aider and Wesfreid2008) and Jones et al. (Reference Jones, Dohring and Platzer1998) and numerical simulations of Liang et al. (Reference Liang, Ou, Premasuthan, Jameson and Wang2011). But no further analyses were conducted for this important feature of the wake dynamics. These structures are considerably weaker in their relative strength compared with those in the main street, which is why they have not received adequate attention in the literature. We hypothesize that secondary structures play a key role in the merger of upper and lower vortex streets behind parallel oscillating foils. Figure 7(a) shows a separated wake of the foils for the case of $St=0.4$ and $d^*=2$ at $t=60P$. Even though secondary structures are present in the wake, structures from the lower wake are convected in the downstream direction before reaching the upper wake. At $St=0.5$ and $d^*=2$, in which wakes are merged, however, these secondary structures from the lower foil deflect upward to interact with the primary street of the upper foil, traversing downward (see figure 7b). These observations hint that this interaction triggers the wake merging process, because constructive or destructive interference of secondary vortices with the bigger coherent structures change their strengths in terms of circulation (Zhu et al. Reference Zhu, Wolfgang, Yue and Triantafyllou2002; Akhtar et al. Reference Akhtar, Mittal, Lauder and Drucker2007; Khalid et al. Reference Khalid, Wang, Akhtar, Dong, Liu and Hemmati2021a,Reference Khalid, Wang, Akhtar, Dong, Liu and Hemmatib). Furthermore, onset of the complete wake merging appears located very close to the point of interaction between secondary structures and the primary street. For instance, secondary structures shed by the lower foil at $St=0.5$ and $d^*=2$ catch the upper main vortex street at $x/c \approx 4$, which coincides with the location for the merging of wakes (see figure 4c). Likewise, both the interactions between secondary structures and the upper wake as well as the merging of upper and lower wakes occur at $x/c \approx 3.6$ for $St=0.5$ and $d^*=2$ (see the supplementary movie 3 and table 2). The alignment of the spatial merging location and that of the vortex interactions strengthens our argument on the role of these smaller structures in initiating and facilitating the wake merging process.

We proceed with providing another quantitative explanation to the impact of secondary structures on the overall wake dynamics. In this manner, the locations and circulation of vortex dipoles are traced in the wake to provide evidence for the impact of secondary structures. Figure 8 exhibits the change of $\varGamma$ for negative vortices associated with separated ($St=0.4$ and $d^*=2$) and transitional-merged ($St=0.5$ and $d^*=2$) wakes, as dipoles move downstream. For the transitional-merged wake (figure 8b), circulation of the negative vortices of the upper and lower wakes overlap in the near wake region. However, proximity in this sense is broken in favour of the upper wake, after which there is constructive interference of secondary structures with negative vorticity at $r/c \approx 3.5$. This observation remains valid at different time instants. Non-dimensional circulation at $t_1=44P$ and $t_2=55P$ is presented in figure 8(b). On the contrary, there exists no significant difference in circulation of the upper and lower wakes for the separated wake (figure 8a), because the secondary structures have no influence on the upper wake. Similarly, wakes at $t_1=40P$ and $t_2=50P$ show that this trend is independent of the wake evolution and time. It is apparent that enhancement of the negative vorticity of the upper wake due to the influence of the secondary structures causes alterations in deflection of the vortex street (see (3.2) and (3.4)). It eventually results in the primary wake transition, i.e. merger of the wakes. We also note that relatively weaker and smaller secondary structures compared with the primary vortices could be the reason for the gradual merging of the vortex streets in transitional-merged wakes. As illustrated in figures 8 and 9, they have a delicate but significant effect on the upper vortex street, which may result in the crawling merging process (see supplementary movie 3 for the entire process). Therefore, the dynamics of secondary structures only account for the merging mechanism and physical process, while the discussion in § 3.1 outlines the mathematical model to effectively categorize the wake patterns. Furthermore, circulation of negative vortices of the upper and lower wakes is computed for other separated wakes and transitional-merged wakes to support this explanation for this mechanism. It is evident from figure 9 that our illustration stays valid for all transitional-merged wake cases investigated in this study.

Figure 8.

Magnitude of non-dimensional circulation of negative vorticity of upper and lower vortex streets at $d^*=2$ for (a) $St=0.4$ (separated wake) and (b) $St=0.5$ (transitional-merged wake before the merger) at different time instants for in-phase pitching.

Figure 9.

Magnitude of non-dimensional circulation of negative vorticity of upper and lower vortex streets for transitional-merged wakes before the merger and separated wakes for in-phase pitching.

3.3. Effect of wake merging on the propulsive performance of the system

We now focus on the relationship between propulsive performance of in-phase pitching parallel foils (side-by-side configuration) and primary transitions in the wake by assessing the cycle-averaged performance metrics. Figure 10 shows temporal variations of the coefficients of thrust and power, as well as efficiency at a range of $St$ and $d^*$ for the system (average of foils 1 and 2) and an isolated foil. The performance parameters of this dynamical system are examined for a large number of oscillation cycles until all cases reach quasi-steady conditions in their propulsion performance. This translates into 40 cycles for isolated cases, 65 cycles for cases with $St=0.4$, 90 cycles for $d^*=1$ and $St=0.5$ and $d^*=1.5$ and $St=0.5$, and 120 cycles for $d^*=2$ and $St=0.5$ and $d^*=2.5$ and $St=0.5$. Figure 10 covers all transitional-merged wakes observed in the current study as well as separated wakes, i.e. $St=0.4$ and $d^*=2$, $St=0.4$ and $d^*=2.5$, $St=0.5$ and $d^*=2.5$, and isolated foils’ wakes. In this section we aim at establishing the impact of unsteady alterations in wake dynamics on the propulsive performance of the system. Thus, parameters for cases with lower $St$, i.e. $St \leqslant 0.3$, are not presented here as they display no significant wake transitions. Previously, it was demonstrated by Gungor & Hemmati (Reference Gungor and Hemmati2020) that parallel foils generated highly quasi-steady performance and wake characteristics for lower $St$ at $d^*=1$. With this background, it is further noted in the present study that the same attributions persist for other separation distances examined here. Power requirements of the system marginally vary in time, which translates into resembling trends for the cycle-averaged coefficient of thrust and efficiency. The percent improvement in thrust ($\Delta \widetilde {C_T}$) is given in table 2 together with the location and time instant of the wake merging. Here $\Delta \widetilde {C_T}$ is calculated between the fifth cycle and the cycle in which the thrust coefficient reaches quasi-steady conditions, e.g. 90th cycle for $d^*=1$ and $St=0.5$, 120th cycle for $d^*=2$ and $St=0.5$. The first five cycles are disregarded, considering that the performance metrics of steady cases require five cycles to converge (Gungor & Hemmati Reference Gungor and Hemmati2020). It is evident from figure 10(a) that the generated thrust by transitional-merged wakes improves with time and reaches a steady state after the primary transition in the wake is established. This indicates that the wake merging process could be a contributing factor for thrust enhancement. Furthermore, it is important to note that the separated wakes presented here ($St=0.4$ and $d^*=2$, $St=0.4$ and $d^*=2.5$, and $St=0.5$ and $d^*=2.5$) experience trivial alterations in propulsive performance parameters, which further supports our argument. Thus, we affirm that the merging of these wakes improves the propulsive thrust of the system by increasing thrust generation through amplification of the circulation associated with amalgamated vortices around the mid-wake. This results in the formation of a high-momentum jet on the centreline.

Figure 10.

The variation of cycle-averaged (a) thrust and (b) power coefficients, as well as (c) efficiency of the system (averaged using foil 1 and foil 2), and the isolated foil in time at a range of $St$ and $d^*$ for in-phase pitching.

There are two consequential inferences from figure 10 and table 2, which strengthens our argument. First, the time instance of peaks in thrust variation lags the instant of the wake merging process. For example, thrust generation for the case of $St=0.5$ and $d^*=1$ has its maximum at $t/P=29$, whereas its wake merging occurs at $t/P=22$. Second, thrust enhancement in the system decreases as the streamwise location of the onset of wake merging moves downtream. This is due to the reduction in circulation of the dipoles as they travel downstream of the wake (see figures 8 and 9). When upper and lower vortex streets merge in closer proximity to the foils, amalgamated vortices have greater circulation. This causes an increased momentum jet that is induced by the vortex street. This is consistent with insignificant improvements in thrust for the case of $St=0.4$ and $d^*=1.5$, whose wake merging occurs farther in the wake, i.e. increased distance from the foils.

Propulsive performance metrics of isolated pitching foils are also presented in figure 10. These data are tracked for $40$ oscillation cycles, negligible alterations are witnessed after $20$ oscillation cycles. Note that a single foil with $St=0.4$ and 0.5 forms deflected wakes with quasi-steady characteristics although it is not demonstrated here for brevity. This range of $St$ agrees well with the threshold $St$ for wake deflection shown in other studies (Jones et al. Reference Jones, Dohring and Platzer1998; Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008; Deng et al. Reference Deng, Sun and Shao2015, Reference Deng, Sun, Teng, Pan and Shao2016; Das et al. Reference Das, Shukla and Govardhan2016). Pitching foils in side-by-side configurations exhibit superior performance compared with a solitary foil. Although parallel foils with all separation distances produce less thrust compared with a foil in isolation, they require substantially less power. This results in improved efficiency for parallel foils. Similar findings are also presented by Huera-Huarte (Reference Huera-Huarte2018) and Dewey et al. (Reference Dewey, Quinn, Boschitsch and Smits2014). It is worth mentioning that efficiency of an isolated foil at $St=0.4$ does not substantially differ from the case with separated wakes ($d^*=2$ and $d^*=2.5$). Likewise at $St=0.5$, an isolated foil attains similar efficiency with a transitionally merged wake ($d^*=2$) when it is not merged ($t/P \leqslant 78$) and a separated wake ($d^*=2.5$). On the other hand, efficiency of the two parallel foils, in which the primary wake transition occurs, is greater than an isolated foil for the same $St$. These observations imply that interactions of wakes favourably impact the performance of these complex dynamical system.

We now proceed with implementing the finite-core vortex array model to our wake data, following Naguib et al. (Reference Naguib, Vitek and Koochesfahani2011). This model suggests that velocity profiles in the wake can be determined by superimposing a finite amount of vortex cores onto a uniform flow. Streamwise and transverse velocity profiles that are induced by superposition of $N$ vortices can be determined using

(3.7)\begin{gather} \boldsymbol{u}(x,y)=U_{\infty}-\sum_{i=1}^{N}\frac{\varGamma_i (r_i)}{2{\rm \pi}}\frac{(y-y_{ci})}{r_i^2}, \end{gather}

(3.8)\begin{gather} \boldsymbol{v}(x,y)=\sum_{i=1}^{N}\frac{\varGamma_i (r_i)}{2{\rm \pi}}\frac{(x-x_{ci})}{r_i^2}, \end{gather}

where $r_i$ is the radial distance from the $i$th vortex centre to the point of calculation, and $x_{ci}$ and $y_{ci}$ are the streamwise and transverse location of the centre of the $i$th vortex, respectively. The model requires that the number of vortices be greater than or equal to 10 in order to converge (Naguib et al. Reference Naguib, Vitek and Koochesfahani2011). To this end, locations and circulations of vortices within the range of $3 \leqslant x/c \leqslant 7$ are measured for $St=0.5$ and $d^*=1$ at $t_1=13P$ (separated stage) and at $t_2=50P$ (merged stage). The $St$ and $d^*$ of the flow are selected, considering it yields the highest percent improvement in thrust production (see figure 10 and table 2). Moreover, the range of execution is determined considering that the wake merging occurs following $x/c=2.5$ and that the circulation of the dipoles diminishes after $x/c \geqslant 7$. Mean streamwise velocity profiles calculated using the vortex array model are plotted for $x/c=4$ and $x/c=6$ in figure 11. High-momentum-surfeit regions are observed around the centreline ($y/c=0$) for the merged wake ($t_1=50P$), whereas two distinct peaks associated with the jets created by the upper and lower vortex streets are visible in the separated wake ($t_1=13P$). The excessive momentum at the outlet profiles ($x/c=4$ and $x/c=6$) created by the single velocity peak is greater than the combination of the two peaks. It clearly shows impact of the wake merging on the formation of high-momentum jets, which results in improving thrust production. Furthermore, we compare velocity profiles obtained from the vortex array model with contours of mean horizontal velocity from figure 12. The model accurately captures the general trend and locations of the velocity peaks. However, it underestimates the magnitude of the peaks in velocity profiles. This limitation may be due to the sampling space given we only consider the region after wake merging in our calculations for the model. This focus on a particular region is driven by our emphasis on the influence of wake merging. A finite-core vortex array model was also developed and utilized by Dewey et al. (Reference Dewey, Quinn, Boschitsch and Smits2014) to reproduce wake structures and time-averaged velocity fields around parallel foils for in-phase ($\phi =0$), out-of-phase ($\phi ={\rm \pi}$) and mid-phase ($\phi ={\rm \pi} /2$) oscillations. Even though they successfully demonstrated merging of wakes for in-phase pitching foils, they were not able to observe transitional characteristics during evolution of the wake, because the Strouhal number of their study was limited to $St = 0.25$.

Figure 11.

Cycle-averaged streamwise velocity ($u$) profiles of in-phase pitching foils, normalized by $U_{\infty }$, obtained from the finite-core vortex array model for $St=0.5$ and $d^*=1$ at different time instants ($t_1=13P$ and $t_2=50P$) and streamwise locations ($x/c=4$ and $x/c=6$).

Figure 12.

Cycle-averaged streamwise velocity ($u$) contours normalized by $U_{\infty }$ of in-phase pitching parallel foils for $St=0.5$ and $d^*=1$ at (a) $t_1=13P$ and (b) $t_2=50P$.

The formation, growth and interaction of LEVs can be an important mechanism that impacts the propulsive performance of oscillating foils (Anderson et al. Reference Anderson, Streitlien, Barrett and Triantafyllou1998; Pan et al. Reference Pan, Dong, Zhu and Yue2012), while also impacting the wake development (Hemmati, Van Buren & Smits Reference Hemmati, Van Buren and Smits2019). To this effect, we now look at how alterations in the formation and growth of LEVs around the foils influence their propulsive performance during a separated-to-merged wake transition. The LEVs are formed when the angle of attack is high enough that a separation bubble is formed on the foils. Their presence and evolvement on the surface of a fin or wing is responsible for a large part of thrust and lift generation in aquatic locomotion (Borazjani & Daghooghi Reference Borazjani and Daghooghi2013; Bottom Ii et al. Reference Bottom Ii, Borazjani, Blevins and Lauder2016; Liu et al. Reference Liu, Ren, Dong, Akanyeti, Liao and Lauder2017; Xiong & Liu Reference Xiong and Liu2019) and insect flight (Ellington et al. Reference Ellington, Van Den Berg, Willmott and Thomas1996; Birch & Dickinson Reference Birch and Dickinson2001; Bomphrey et al. Reference Bomphrey, Lawson, Harding, Taylor and Thomas2005), respectively. Unsteady thrust variations of the foils in the transitional-merged wake ($St=0.5$ and $d^*=1$) throughout the separated and merged time ranges is presented in figure 13, which clearly demonstrates that higher peaks and lower troughs are achieved after the wake merging. Contours of vorticity focusing on surfaces of the foils at time instants that correspond to the highest ($\theta =8^{\circ }$) and the lowest ($\theta =0^{\circ }$) angles of attack are shown in figure 14. This enables for the comparison of this process to the evolution of LEVs around the foils. Note that these instants roughly overlap with the times at which the foils yield their highest and lowest thrust generation, as marked in figure 13. Negative (clockwise rotating) and positive (counterclockwise rotating) LEVs are formed on the upper and lower surface of the foils at the separated stage of the wake evolution, respectively. On the other hand, constituting positive LEVs on the lower surface of foil 1 and negative LEVs on the upper surface of foil 2 is significantly suppressed when the wake is fully merged. Furthermore, it is evident from contour plots in figure 14 that stronger LEVs are formed after the wake merging, which hints at a potential factor for thrust enhancement. These observations are valid for the other transitional-merged wake cases with recognizable thrust enhancement as well, although they are not shown here for brevity. Note that profiles of unsteady thrust have two peaks and two troughs per oscillation cycle. The impact of LEVs on thrust generation can be explained through the low pressure region (suction) formed by vortices. The LEVs attached close to the anterior part of the foil favourably affect thrust by dropping the pressure in this region, whereas their influence is adverse if located around the posterior part of the foil. For example, foil 1 at $t_7=50.25P$ has an enhanced LEV around the front edge compared with foil 1 at $t_3=13.25P$. However, the distribution of LEVs on the rear surfaces of the foils are matching, which translates to thrust enhancements for foil 1. Likewise, the thrust of foil 2 at $t_5=49.75P$ is considerably larger than that of foil 1. This is due to stronger LEVs formed close to the forehead of foil 2, while a LEV with large negative vorticity is present on the rear part of the foil 1.

Figure 13.

Variations in the unsteady thrust coefficient of foil 1 and foil 2 for $St=0.5$ and $d^*=1$ (transitional-merged wake) for in-phase pitching. The separated stage ($12 \leqslant t/P \leqslant 14$) of the wake evolution is illustrated in black and the merged stage ($49 \leqslant t/P \leqslant 51$) of the wake evolution is illustrated in red.

Figure 14.

Contour of spanwise vorticity ($\omega _{z}^{*}$) around in-phase pitching parallel foils for $St=0.5$ and $d^*=1$ (transitional-merged wake) at various time instants during the separated stage: (a) $t_1=12.5P$, (b) $t_2=13P$, and merged stage: (c) $t_3=49.5P$, (d) $t_4=50P$. Here, vorticity is normalized by $U_{\infty }/c$. (See supplementary movie 5 for the entire wake evolution).

It was previously demonstrated by Gungor & Hemmati (Reference Gungor and Hemmati2020) that merging of vortex streets was accompanied by the restoration of wake symmetry for parallel foils. Both performance and wake characteristics exhibit symmetric behaviour with a delay of a half-pitching period. This lag between the foils is due to the formation process of vortex dipoles. Although TEV$_1$ and TEV$_2$ are shed at the same time (e.g. $t_5=49.75P$), TEV$_2$ establishes a dipole with TEV$_0$ that has been shed half a period prior to these TEVs, whereas the coupling of TEV$_1$ and TEV$_3$ are delayed by half a cycle. Therefore, the distribution of LEVs around foil 1 and foil 2 at $t_5=49.75$ in figure 14(e) is a mirror image symmetric with switched signs of vorticity with that around foil 2 and foil 1 at $t_7=50.25P$ in figure 14(g), respectively. Besides, shifting $C_T$ values for foil 1 by a half-period in either direction results in a perfect overlap with those for foil 2 during the merged stage of the wake evolution, or vice versa (see figure 13). Contrarily, there is neither a coherent similarity of the arrangement of LEVs around the foils between different time steps nor a half-period lag between performance parameters of foil 1 and foil 2 during the separated stage of wake evolution.