3.1. Evaluation of kinematics

The kinematics of oscillating foil in combined heaving and pitching motion is described within the range $0^\circ \le \phi \le 270^\circ$ at $0.16\le St_c \le 0.48$. Instantaneous variations in trailing and leading edge amplitudes ($a_{T}^+$, $h_{L}^+$), and effective angle of attack ($\alpha$), are depicted in figure 6 for one oscillation cycle, whereas the effect of increasing $\phi$ on $a_{T}^+$ and $\alpha$ is particularly observable in figures 6(a,c). Figure 6(d) depicts further variations in the peak angle of attack ($\alpha _{o}$) with respect to increasing phase offset and $St_c$. The effective angle of attack for a foil with combined heaving and pitching motion is the resultant of both pitch- and heave-induced angle of attack (Lagopoulos et al. Reference Lagopoulos, Weymouth and Ganapathisubramani2019). This is expressed mathematically as

(3.1)\begin{equation} \alpha={-}\theta-\arctan \left(\dot{h} / U_{\infty}\right), \end{equation}

where $\dot {h}$ is the heave velocity of the foil. The variation of $\phi$, corresponding to the peak angle of attack (i.e. $\phi = 90^\circ$), is consistent for increasing $St_c$.

Figure 6. Temporal variation of (a) trailing edge amplitude ($a_{T}^+$), (b) leading edge (heave) amplitude ($h_{L}^+$), and (c) instantaneous $\alpha$, within one single oscillation cycle. (d) Variation of peak $\alpha$ with respect to $\phi$ and $St_c$.

The maximum $a_{T}^+$ as observed from figure 6(a) is attained for $\phi = 0^\circ$, which then decreases with $\phi$ approaching $180^\circ$. Increasing $\phi$ from $180^\circ$ to $270^\circ$ results in a higher peak $a_{T}^+$ within one oscillation cycle. On comparing the ratio of peak $a_{T}^+$ and $h_{L}^+$ with changing $\phi$, it can be deduced that a smooth shift from pitch- to heave-dominated kinematics occurs with the phase offset approaching $180^\circ$. Then it reverts back to the pitch-dominated motion as $\phi$ increases beyond $180^\circ$. Our understanding about the influence of variations in such kinematics on evolution of wakes is still incomplete and therefore needs more attention. Moreover, the variation in $\alpha$ also depicts an increase in its peak, which then decreases to a minimum at $\phi = 270^\circ$. This is also observed in figure 6(d). The peak angle of attack holds considerable importance in terms of governing the dynamics of leading edge vortex and stall characteristics of the oscillating foil (Visbal Reference Visbal2009; Carr Reference Carr2012). In order to investigate these aspects, specifically when a transition occurs from pitch- to heave-dominated coupled motion of a foil, we now discuss newly identified important observations with respect to evolution of wake that coincides with the aforementioned transition in kinematic regimes.

3.2. Evaluation of wake mode transition at $Re= 1000$

To understand the configuration of wake and its transition for a range of phase offset, amplitude and $St_c$ at $Re= 1000$, we draw representative wake maps shown in figure 7. Specifically, figures 7(a) and 7(b) correspond to phase spaces characterized by $\phi$ versus $St_{c}$ and $A_{c}$ versus $St_{c}$, respectively. Figure 7(b), in particular, allows a comparison of transitions in wake topology with studies conducted on oscillating foils in pure pitching motion (Schnipper et al. Reference Schnipper, Andersen and Bohr2009). The nomenclature of wake modes is also aligned with the studies conducted by Williamson & Roshko (Reference Williamson and Roshko1988) and Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), which considered the number of shed vortical structures in one oscillation cycle for a cylinder and teardrop foil, respectively. For $St_c$ below 0.32 in figure 7(a), it is observed that three wake modes corresponding to wavy Bénard–von Kármán ($wBvK$) (Lentink et al. Reference Lentink, Muijres, Donker-Duyvis and van Leeuwen2008), $2P+2S$ and $2P$, (Schnipper et al. Reference Schnipper, Andersen and Bohr2009) are prominent at $Re= 1000$. More details of the wake categorization and transition at $Re=1000$ can also be found in Verma & Hemmati (Reference Verma and Hemmati2022). For $St_c=0.4$ and $Re= 1000$, the heave-dominated kinematics corresponding to $90^\circ \le \phi \le 180^\circ$ also depicts the dominance of $2P$ mode along with a characteristic bifurcation (marked by $B$ in figure 7a) noticed between the two shed vortex pairs of $2P$, particularly at $\phi = 180^\circ$. A transition from $rBvK$ to $BvK$, followed by the similar bifurcation of $2P$ pairs, is further observed at $St_{c}= 0.48$.

Figure 7. Phase map representation of identified wake modes at increasing $St_c$, $\phi$ and $A_{c}$ for $Re= 1000$.

It is worthwhile to observe if such wake mode transitions in terms of number of shed vortex structures, or characteristic bifurcations of shed $2P$ pairs, could coincide with a threshold value of $St_A$, as investigated by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), who focused particularly on purely pitching foils. Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) observed that the transition from $2P$ to standard von Kármán ($2S$) mode coincided with the iso-line represented by $St_A= 0.11$. Furthermore, the transition of $2S$ to $rBvK$ mode coincided with the line for $St_A= 0.18$ (Schnipper et al. Reference Schnipper, Andersen and Bohr2009). In order to evaluate such coinciding transitions of wake modes with iso-lines of constant $St_A$ in our study, we explain the wake modes at $Re= 1000$ on an $A_{c}\unicode{x2013}St_{c}$ phase space diagram shown in figure 7(b). Here, $A_{c}$ is the normalized trailing edge amplitude calculated as $2A_{T}/c$. The product of $A_c$ and $St_c$ yields lines of constant $St_A$ on the phase space represented in figure 7(b). The two iso-lines drawn on the phase diagram correspond to $St_A= 0.1$ and 0.2, respectively. The range between the observed iso-lines coincides with $2P$ mode. The region beyond $St_A= 0.2$ marks a transition towards $rBvK$ mode. Such findings therefore reveal a close resemblance to the observations reported by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009). The transition from $2P$ mode to $rBvK$ also goes through a $BvK$ type or standard von Kármán wake mode as observed at $St_{c}= 0.48$. Further, the bifurcation of $2P$ pairs is found to lie below the line of $St_A= 0.1$, whereas increasing $St_A$ beyond 0.1 does not reveal any bifurcations on the phase diagram. Besides the similarity observed in the trends of transition in wake mode configuration, with increasing $St_A$, the bifurcation of $2P$ pairs observed at some kinematic settings also reveals a qualitative similarity with the visualizations of $2P$ mode by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) and Andersen et al. (Reference Andersen, Bohr, Schnipper and Walther2017). However, specific details and features of the topological transitions in spatial arrangements of vortex structures, constituting the $2P$ mode, were not identified or investigated. We will therefore provide in-depth qualitative and quantitative descriptions of the characteristics of evolution while the $2P$ mode undergoes transitions in spatial topologies and encounters bifurcation of shed vortex pairs at certain kinematic settings observed in figure 7.

The wake formations at increasing $\phi$ are depicted in figure 8, at $Re=1000$ and $St_c= 0.4$. A dominant shedding of counter-rotating vortex pairs in each half oscillation cycle is observed commonly for all the cases presented here. However, the changes in spatial configurations of paired structures are observed for increasing $\phi$. At $\phi = 0^\circ$, a $2P$ mode (Williamson & Roshko Reference Williamson and Roshko1988; Schnipper et al. Reference Schnipper, Andersen and Bohr2009) appears, which is characterized by horizontally pointed pairs. The counter-rotating vortex structures TEV$_{ac}$ and LEV$_{c}$, marked on figure 8(a), represent a $2P$ vortex pair. Here, subscripts $ac$ and $c$ denote the anticlockwise and clockwise rotating vortical structures, respectively. Another shed structure, LEV$_{ac}$, undergoes mutual interaction with the leg of TEV$_{ac}$. It is important to mention here that the abbreviations TEV$_s$ and LEV$_s$ denote the coherent structures formed at the trailing and leading edge of the foil, respectively. For the pitch-dominated motion profile at $\phi = 0^\circ$ (see figure 6), TEV$_{ac}$ is stronger than LEV$_{ac}$ based on their circulation strength and size in figure 8(a). It results eventually in the merger of similar sign LEV$_s$ and TEV$_s$ of the neighbouring pairs, traversing downstream in the wake. A similar $2P$ configuration was observed and discussed previously by Williamson & Roshko (Reference Williamson and Roshko1988), Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) and Hultmark, Leftwich & Smits (Reference Hultmark, Leftwich and Smits2007) in the wake of a transversely oscillating cylinder, a pitching foil and flexible robotic fins in sinusoidal oscillations, respectively. These $2P$ modes have also presented similarities in spatial configurations of thrust-producing wakes by swimming fish (Muller et al. Reference Muller, van den Boogaart and van Leeuwen2008; Schnipper et al. Reference Schnipper, Andersen and Bohr2009).

Figure 8. Wake modes at increasing $\phi$ and $Re= 1000$. Panels (a–f) represent $\phi$ corresponding to $0^\circ$, $45^\circ$, $90^\circ$, $120^\circ$, $180^\circ$ and $225^\circ$, respectively, while $St_c=0.4$.

In order to elaborate further on the effects of increasing $\phi$, we observe that at $\phi = 45^\circ$, $2P$ vortex pairs are tilted and point in the upstream direction. No noticeable merger or interaction is observed between the neighbouring structures of these pairs, as shown in figure 8(b). These pairs also advect symmetrically along the wake centreline. The shed TEV possesses higher strength of circulation, in a qualitative sense, similar to the observations made at lower $\phi$. Quantitative assessments for circulation are, however, presented in the next subsection for understanding the effects of changing $\phi$ on generation of vorticity. Moreover, wake formation at increasing $\phi$ up to $90^\circ$ depicts shedding of counter-rotating vortices with a decreased separation distance ($\zeta$) between their centres compared to the cases with lower values of $\phi$. A similar observation can be made for $\phi = 120^\circ$, where resemblance to dipole-like $2P$ pairs now become more vivid. We refer to such vortex pairs as $2P^D$ pairs in order to highlight them as dipole-like paired configurations, which appear in stark contrast to the $2P$ pairs observed at $\phi = 0^\circ$ and $45^\circ$, respectively. Also, the wake still possesses a symmetry with respect to its centreline until $\phi = 120^\circ$. Beside mutual interaction and merger of vortices, we do not observe any bifurcation of the shed pairs and dipole-like ($2P^D$) structures observed at increasing $\phi$. However, two vortex streets constituting fully developed $2P^D$ pairs diverge away from the wake centreline with $\phi$ approaching $180^\circ$ (see figure 8e). Structures forming such fully developed dipole-like configurations possess similar strengths of circulation and size, which was also reported by Cleaver et al. (Reference Cleaver, Wang and Gursul2012), Calderon et al. (Reference Calderon, Cleaver, Gursul and Wang2014), He & Gursul (Reference He and Gursul2016) and Verma & Hemmati (Reference Verma and Hemmati2021b). Experimental studies of Cleaver et al. (Reference Cleaver, Wang and Gursul2012) and Calderon et al. (Reference Calderon, Cleaver, Gursul and Wang2014) discussed the formation of dipoles and couples behind oscillating foils in purely heaving motion and further explained their prominent association and coincidence with deflected wake modes (i.e. Mode A or Mode B). Here, the dual vortex streets seen at $\phi = 180^\circ$ in figure 8(e) retain a symmetry along the wake centreline. This observation could indicate qualitatively that the entrainment effects of shed dipole-like structures (Calderon et al. Reference Calderon, Cleaver, Gursul and Wang2014) are equally balanced, thus preventing the overall wake deflection in the upward or downward direction as noted in previous studies (Cleaver et al. Reference Cleaver, Wang and Gursul2012; Calderon et al. Reference Calderon, Cleaver, Gursul and Wang2014). Increasing $\phi$ to $225^\circ$, as shown in figure 8(f), depicts an absence of diverging vortex streets, where trailing edge vortices are more stretched in the streamwise direction. An additional single vortex highlighted as $S$ in figure 8(f) is also shed with a coincident shedding of a $2P$ pair. This mode therefore falls in the category of $2P+2S$ (Williamson & Roshko Reference Williamson and Roshko1988; Schnipper et al. Reference Schnipper, Andersen and Bohr2009). It is now evident that while the bifurcation of $2P^D$ pairs resembles the observed $2P$ mode configuration of Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), an elaborate evaluation is still warranted in order to understand the transitioning spatial topologies of $2P$ mode shown in figure 8. Furthermore, the kinematics investigated by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) followed a pure pitching motion rather than a combined heaving and pitching motion assumed in our current study.

3.3. Formation mechanism of $2P^D$ pairs and symmetric bifurcation

We now examine the formation process of $2P^D$ pairs at $Re=1000$ by assessing the temporal evolution of shed vortical structures for the case $St_{c}= 0.4$ and $\phi = 180^\circ$. This parameter space is selected based on observations in figures 7(a) and 8(e), which allows for a better examination and characterization of bifurcations observed in $2P$ mode. The snapshots representing contours of spanwise vorticity ($\omega _z^+$), at each quarter phase of an oscillation cycle, are illustrated in figure 9. At the beginning of an oscillation cycle (i.e. at $t=0$) in figure 9(a), a developed LEV1$_c$ is located close to the trailing edge, where TEV1$_{ac}^\prime$ is formed in the previous half oscillation cycle. At $t=T/4$ in figure 9(b), LEV1$_{c}$ is detached from the trailing edge, where a new anticlockwise TEV$1_{ac}$ starts forming behind the foil. Also, a new LEV2$_{ac}$ appears developed at approximately the mid-chord region. LEV1$_c$ forms a dipole-like paired configuration with TEV$1_{ac}$ at the end of the half oscillation cycle (i.e. $t=T/2$) in figure 9(c), while LEV2$_{ac}$ is on the brink of detachment from the trailing edge. With the oscillations proceeding to $t=3T/4$ in figure 9(d), the first dipole-like pair ($D1$) is observed, which consists of vortical structures LEV1$_c$ and TEV1$_{ac}$. Simultaneously, a new trailing edge structure (TEV$2_{c}$) begins to form and pairs with LEV2$_{ac}$, which is now detached from the trailing edge. By the end of one oscillation cycle in figure 9(e), TEV2$_c$ and LEV$2_{ac}$ form a second dipole-like pair ($D2$). Both $D1$ and $D2$ are marked in figure 9(e). This description of the formation mechanism of $2P^D$ pairs is quite similar to that of a $2P$ pair observed in the wake of oscillating cylinders (Williamson & Roshko Reference Williamson and Roshko1988) and foils (Schnipper et al. Reference Schnipper, Andersen and Bohr2009; Andersen et al. Reference Andersen, Bohr, Schnipper and Walther2017) with a single degree of freedom motion. Such $2P$ pairs were also seen in wakes corresponding to $\phi = 0^\circ$ in this study, as observed in figures 8(a) and 13(a). However, what distinguishes the $2P^D$ pair observed at $\phi = 180^\circ$ from the previously discussed $2P$ pairs is the approximate equality of circulation strength and size of the fully developed leading and trailing edge vortical structures ahead of their complete detachment from the foil's trailing edge. On account of these features, the mutual induced velocity of the dipole-like pair contributes to its movement away from the wake centreline, thus leading to the bifurcated dual vortex streets observed in figure 8(e). The dipole-like formation is also different from the dual branch Mode-1 described by Cleaver et al. (Reference Cleaver, Wang, Gursul and Visbal2011) for purely heaving foils. There, the dipole configuration constituted a weak trailing edge structure and another collective structure that was formed on account of a merger between strong leading and trailing edge vortices. This collective structure was formed and shed within one half oscillation cycle. In the case of $2P$ mode characterized by a fully developed $2P^D$ pair, we do not observe the formation of any collective structure similar to those reported by Cleaver et al. (Reference Cleaver, Wang, Gursul and Visbal2011). The above described process, therefore, hints at a spatial wake topology that is unique to foils with a combined heaving–pitching motion with different characteristics in terms of the vortex pair configuration, despite similarities in their formation process.

Figure 9. Temporal snapshots representing contours of $\omega _z^+ = \omega _z c/U_{\infty }$, and depicting formation of a $2P^D$ pair and bifurcation within the $2P$ mode at $St_c= 0.4$, $\phi = 180^\circ$ and $Re= 1000$.

The above mentioned details of formation mechanism further allow us to relate the correspondence of the $2P^D$ pairs to the heave-dominated regime at $Re= 1000$ (see figure 7a). As observed here, only heave-dominated cases at $Re= 1000$ present a viable scenario, where the generated trailing and leading edge vorticities attain equal circulation strength and size after getting detached from the foil's trailing edge. It results in a net induced velocity of the dipole-like pair, pushing it away from the wake centreline and thus leading to bifurcations. As the kinematics transition to the pitch-dominated regime, the trailing edge vorticity gains enough strength relative to the leading edge vorticity that an eventual merger between similar sign structures occurs (see figure 8a). Hence it leads to a symmetric $rBvK$ wake profile. Quantitative assessment described in the next subsection provides further evidence for our qualitative observations described above.

3.4. Quantitative assessment of evolution characteristics

The quantitative assessment of circulation strength ($\varGamma ^+=\varGamma /U_{\infty }c$) and separation distance between the centres of vortices ($\zeta ^+=\zeta /c$) can provide deeper insights into changes in evolution characteristics of the shed vortex structures with increasing $\phi$ at $Re= 1000$. These would also help us to understand particular features associated with changes in spatial topology of $2P$ wake, and appearance of bifurcations of $2P^D$ pairs. Specifically, we focus on the computed values of $\varGamma ^+$ and $\zeta ^+$ just at the instance (shown in figure 8) when the counter-rotating vortex structures form pairs ahead of the foil's trailing edge. The estimation process for $\varGamma$ involves calculating the surface integral of vorticity on a rectangular contour window using

(3.2)\begin{equation} \varGamma = \iint_{S}(\boldsymbol{\nabla} \times \boldsymbol{V}) \boldsymbol{\cdot} \boldsymbol{dS}. \end{equation}

To calculate $\zeta$, the location of maximum vorticity magnitude, within the rectangular window surrounding the vortex, is evaluated for each structure that form either a co-rotating or counter-rotating pair. The coordinates of centres $(X,Y)$ are then used to find $\zeta$:

(3.3)\begin{equation} \zeta = \sqrt{\left(X_{2}-X_{1}\right)^{2}+\left(Y_{2}-Y_{1}\right)^{2}}, \end{equation}

where subscripts $1$ and $2$ indicate the centre coordinates of two similar-sign or counter-rotating vortical structures, respectively. Figure 8 depicts three windows around vortex structures TEV$_{ac}$, LEV$_c$ and LEV$_{ac}$. TEV$_{ac}$ and LEV$_c$ form a pair of counter-rotating vortices, which eventually transitions to a dipole-like configuration as $\phi$ increases to $180^\circ$. TEV$_{ac}$ and LEV$_{ac}$ also undergo mutual interaction on account of their similar sense of rotation. These aspects further motivate us to understand the evolution of marked vortex structures, quantitatively.

Figure 10 shows the variation of circulation ($\varGamma ^+$) and separation distance ($\zeta ^+$) with increasing $\phi$ for the heave-dominated kinematic settings ($\phi =45^\circ \unicode{x2013}180^\circ$), since the major focus is on characterizing the bifurcated vortex street observed for $2P$ mode in this range of $\phi$. As seen from figure 10(a), $\varGamma ^+$ for TEV$_{ac}$ drops consistently as $\phi$ increases from $45^\circ$ to 180$^\circ$. However, the paired vortex structure LEV$_c$ depicts an opposite trend compared to TEV$_{ac}$, although the variation of $\varGamma ^+$ is not as significant as in case the of TEV$_{ac}$. Despite the observed variations for these paired vortex structures, the newly shed LEV$_{ac}$ depicts no changes with respect to increasing $\phi$. This signifies clearly that the dominant evolution characteristics of a leading edge vortex remain approximately similar as it attains a paired configuration with the TEV, while kinematics transition from a pitch-based motion setting to a heave-dominated oscillation. The trailing edge vorticity, however, is influenced greatly by the changes in trailing edge amplitude that were discussed previously for figure 6(a). The variation of $\zeta _1$ and $\zeta _2$, corresponding to the distance calculated between counter-rotating ($LEV_{c}$ and TEV$_{ac}$) and co-rotating ($LEV_{ac}$ and $TEV_{ac}$) vortex structures, is depicted further in figure 10(b). Similar to the qualitative discussion provided with respect to figure 8, we observe a decrease in $\zeta _1$ as $\phi$ increases from $45^\circ$ to $180^\circ$. On the other hand, $\zeta _2$ shows an increase up to $\phi = 90^\circ$, followed by a slightly decreasing trend towards $\phi = 180^\circ$. The birfurcated vortex streets observed at $\phi = 180^\circ$, which were characterized qualitatively by a fully developed dipole-like configuration, do possess a reduced separation distance between the counter-rotating vortex structures, along with their equal magnitude of $\varGamma ^+$. Comparing these quantitative characteristics to the dipoles observed in deflected wakes of purely heaving and pitching foils (Cleaver et al. Reference Cleaver, Wang and Gursul2012; Calderon et al. Reference Calderon, Cleaver, Gursul and Wang2014; Deng et al. Reference Deng, Sun and Shao2015; He & Gursul Reference He and Gursul2016), the dipole-like $2P^D$ pairs constituting bifurcated wake streets also resemble similar evolution characteristics. However, the resultant wake still possessed a symmetry along the wake centreline, which therefore differs from the deflected wake modes observed previously (Cleaver et al. Reference Cleaver, Wang and Gursul2012; Calderon et al. Reference Calderon, Cleaver, Gursul and Wang2014; Deng et al. Reference Deng, Sun and Shao2015).

Figure 10. Variation of (a) circulation strength ($\varGamma ^+$) and (b) separation distance ($\zeta ^+$) between counter-rotating and co-rotating vortex structures (highlighted in figure 8), with respect to $\phi$.

To understand further the transition from a paired configuration of counter-rotating structures with unequal circulation to the dipole-like arrangement characterized by equal strength structures ($2P^D$ pairs), we model mathematically the circulation estimates of a single leading edge vortex and trailing edge vortex at the instant of their shedding from the foil's trailing edge. This will also highlight any existing dependence of vortex evolution characteristics on $\phi$. A single leading edge vortex structure is formed as a result of boundary layer roll-up near the leading edge, on one side of the foil. Hence its circulation estimation ($\varGamma _{LEV}$) can be obtained by calculating the vorticity transport in the boundary layer (Schnipper et al. Reference Schnipper, Andersen and Bohr2009). This transport also assumes no-slip conditions on the foil, while considering contributions from the freestream ($U_\infty$) and heave velocity ($\dot {h}$) at the leading edge (see figure 1). Similar assumptions were also in place for the models developed by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), although their analysis was limited to pitching motion and did not include the dependence on heave velocity. Based on the estimation of vorticity production in the boundary layer, given by Tietjens, Prandtl & Rosenhead (Reference Tietjens, Prandtl and Rosenhead1957), we can calculate $\varGamma ^+$ for a leading edge vortex just at the instant of its shedding in one half oscillation cycle using the relation

(3.4)\begin{equation} \varGamma_{LEV}=\tfrac{1}{2} \int_{0}^{T / 2} (U_{\infty}^{2}+\dot{h}^2)\, \mathrm{d} t. \end{equation}

Here, $U_{\infty }$ represents the freestream velocity, while the heave velocity ($\dot {h}$) could be evaluated using the time derivative of (2.1). The integral can be subdivided as

(3.5)\begin{gather} \varGamma_{L E V}=\frac{1}{2}\left[\frac{U_{\infty}^{2} T}{2}+\int_{0}^{{T}/{2}} h_{o}^{2} 4 {\rm \pi}^{2} f^{2} \cos ^{2}(2 {\rm \pi}f t) \,{\rm d}t\right], \end{gather}

(3.6)\begin{gather} \varGamma_{L E V}=\frac{1}{2}\left[\frac{U_{\infty}^{2} T}{2}+h_{o}^{2} 4 {\rm \pi}^{2} f^{2} \frac{1}{8}\left(\frac{\sin (2 {\rm \pi}f T)}{{\rm \pi} f}+2 T\right)\right]. \end{gather}

The $\sin$ term evaluates to zero in the above equation, which therefore simplifies the expression to

(3.7)\begin{equation} \varGamma_{L E V}=\frac{U_{\infty}^{2}}{4 f}+\frac{h_{o}^{2} {\rm \pi}^{2} f}{2}. \end{equation}

Here, $f$ represents the frequency of oscillation (i.e. $f= 1/T$). Further, using the foil diameter ($D$) and diameter-based Strouhal number ($St_D= f D/U_{\infty }$), we can again rewrite the last expression as

(3.8)\begin{equation} \varGamma_{L E V}=\frac{D^{2} U_{\infty}+2\,St_{D}^{2}\,U_{\infty} h_{o}^{2} {\rm \pi}^{2}}{4 D\,St_{D}}. \end{equation}

Now, the circulation for the trailing edge vortex ($\varGamma _{TEV}$) can be estimated in a way similar to that followed by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), where they assumed that the freestream contribution to the vorticity is negligible. The expression can be written as

(3.9)\begin{equation} \varGamma_{T E V}=\tfrac{1}{2}\left(\int_{0}^{{T}/{2}} U_{T E}^{2}(t)\,{\rm d} t\right) \approx \tfrac{1}{2} {\rm \pi}^{2} A_{T}^{2} f. \end{equation}

In terms of $St_{D}$ and $A_D = 2A_{T}/D$, we can further rewrite the above expression as

(3.10)\begin{equation} \varGamma_{T E V}\approx\tfrac{1}{2} {\rm \pi}^{2} A_{D}^{2}\,S t_{D}\,D U_{\infty}. \end{equation}

Now, in order to obtain an approximate estimate for the ratio of $\varGamma$ for the leading and trailing edge vortex structures, we can divide (3.10) by (3.8). This ratio can be represented in its final form:

(3.11)\begin{equation} \frac{\varGamma_{T E V}}{\varGamma_{L E V}}\approx\left[\dfrac{2{\rm \pi}^{2} A_{D}^{2}}{\dfrac{1}{S t_{D}^{2}}+\dfrac{2 h_{o}^{2} {\rm \pi}^{2}}{D^{2}}}\right]. \end{equation}

Here, the second term in the denominator is a constant value since the heave amplitude is not varying in our study. We can denote it as $\kappa =2h_{o}^2{\rm \pi} ^2/D^2$. Also, after making some more simplifications to the above expression, we can also obtain a dependence on the amplitude-based Strouhal number ($St_A= 2fA_{T}/U_{\infty }$), thus

(3.12)\begin{equation} \frac{\varGamma_{T E V}}{\varGamma_{L E V}}\approx\frac{1}{2}\left[\frac{{\rm \pi}^{2}\,S t_{A}^{2}}{1+\kappa\,S t_{D}^{2}}\right]. \end{equation}

The above expression clearly highlights that at a constant $St_D$, dependence of the ratio $\varGamma _{TEV}/\varGamma _{LEV}$ on $St_A$ is quite similar to the case of a pitching foil developed by (Schnipper et al. Reference Schnipper, Andersen and Bohr2009), although with a difference in constant of proportionality. We can still modify the above relationship to further obtain a dependence on the heave-based Strouhal number, $St_h= 2fh_{o}/U_{\infty }$. This formulation can be applied for cases where the variation of heave amplitude is also taken into account. This relationship becomes

(3.13)\begin{equation} \frac{\varGamma_{T E V}}{\varGamma_{L E V}}\approx\left[\frac{{\rm \pi}^{2}\,S t_{A}^{2}}{2+ {\rm \pi}^{2}\,S t_{h}^{2}}\right]. \end{equation}

By relating (3.13) to the qualitative depiction of wake modes on the $A_{c}\unicode{x2013}St_{c}$ phase diagram in figure 7(b), we identify that there is a wake mode transition from $2P$ to $rBvK$ type with increasing $St_A$ from 0.1 to 0.2 and beyond. This is consistent with the observations of Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), who found that a similar rise in relative strength of trailing edge structures was responsible for transition of wake from $2P$ to an $rBvK$ mode. We can also relate this derived model to our quantitative details of kinematics (figure 6) and circulation estimates depicted in figure 10(a), at constant $St_h= 0.2$ and $St_c= 0.4$. Specifically, the kinematics indicated that with increasing $\phi$ from $45^\circ$ to $180^\circ$, the peak trailing edge amplitude decreases to a minimum. This would result in a decrease in $St_A$, which, based on (3.12) and (3.13), should lead to a decrease in $\varGamma _{TEV}$. Ahead of $\phi = 45^\circ$, the variation of $\varGamma _{TEV}$ in figure 10 confirms this qualitative trend based on the strength of fully developed TEV$_{ac}$. Note that the parameters $St_D$ and $St_h$ are kept as constants in (3.12) and (3.13), respectively. The bifurcation of dipole-like vortex pairs ($2P^D$ pairs) within $2P$ mode can be also be linked to (3.13). As stated, when $St_A$ reduces on account of increasing $\phi$, the relative trailing edge circulation strength ($\varGamma _{TEV}$) also decreases. At the specific $St_A$ (corresponding to $\phi = 180^\circ$ and $St_c= 0.4$ in our phase space diagram), $\varGamma _{TEV}/\varGamma _{LEV}$ at the instance of shedding is such that the fully developed $2P^D$ pairs, ahead of their complete detachment from foil's trailing edge, constitute vortex structures of equal $\varGamma ^+$ and lowest $\zeta$. This was confirmed quantitatively in figure 10(a) wherein we see an equal magnitude of $\varGamma _{TEV_{ac}}^+$ and $\varGamma _{LEV_{c}}^+$ at $\phi = 180^\circ$, $St_c= 0.4$ and $St_h= 0.2$. This contributes towards a transverse induced velocity leading to advection of the $2P^D$ pairs away from the wake centreline, thus causing bifurcation.

It is clear that the approximations for circulation ratio obtained in Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) for purely pitching motion and (3.13) are quite comparable when $St_h$ is constant. However, (3.13) still provides a valid and easy extension of the previous approximation (Schnipper et al. Reference Schnipper, Andersen and Bohr2009) for an oscillating foil in coupled heaving and pitching motion, as considered here. Thus far, the validity of the proposed model is qualitatively interpreted only in terms of its accuracy in predicting the trend between relative circulation strength ($\varGamma _{TEV}/\varGamma _{LEV}$) and $St_A$ for cases corresponding to heave-dominated kinematics at $Re= 1000$ and $St_h= 0.2$ (figure 7). In order to evaluate quantitatively the model's predictive capability at a wider parameter space, an approximate relationship is derived for the relative circulation strength in terms of $\phi$, by transforming $St_A$ into heave-based ($St_h$) and pitch-based ($St_{\theta }$) Strouhal numbers. This is achieved by evaluating the vector addition of $St_h$ and $St_{\theta }$ (Van Buren et al. Reference Van Buren, Floryan and Smits2019), and incorporating the phase offset $\phi$ between the pitch and heave motion. This is represented as

(3.14)\begin{equation} St_{A}^{2}=St_{h}^{2}+St_{\theta}^{2}+2\,St_{h}\,St_{\theta} \cos\phi. \end{equation}

Here, $St_{\theta }$ is defined as $2fc\theta _{o}/U_{\infty }$. Substituting (3.14) in (3.13), we obtain

(3.15)\begin{equation} \frac{\varGamma_{T E V}}{\varGamma_{L E V}}\approx\left[\frac{{\rm \pi}^{2}\left(S t_{h}^{2}+S t_{\theta}^{2}+2\,S t_{h}\,S t_{\theta} \cos \phi\right)}{2+{\rm \pi}^{2}\,S t_{h}^{2}}\right]. \end{equation}

Now we introduce coefficients ($c_{n}$) in each term of (3.15), to account for the multiplicative constants similar to the approach followed by Van Buren et al. (Reference Van Buren, Floryan and Smits2019) and Verma, Freeman & Hemmati (Reference Verma, Freeman and Hemmati2022a). The resulting expression is therefore written as

(3.16)\begin{equation} \frac{\varGamma_{T E V}}{\varGamma_{L E V}} = \left[\frac{c_{1}\,S t_{h}^{2}+c_{2}\,S t_{\theta}^{2}+c_{3}\,S t_{h}\,S t_{\theta} \cos \phi}{c_{4}+c_{5}\,S t_{h}^{2}}\right]. \end{equation}

The coefficients $c_{n}$ are determined by performing linear regression over the entire parameter space represented in the phase maps (see figure 7). The ranges of $St_h$, $St_\theta$ and $\phi$ correspond to $0.08 \le St_{h} \le 0.24$, $0.05 \le St_{\theta } \le 0.17$ and $0^\circ \le \phi \le 180^\circ$. Figure 11 depicts plots for calculated estimates of $\varGamma _{TEV}/\varGamma _{LEV}$, determined through numerical simulations, and output of the expression represented by right-hand side of (3.16). The coefficients correspond to $c_1 = 2.02$, $c_2 = -0.51$, $c_3 = 0.97$, $c_4 = 0.11$ and $c_5 = -0.96$. As is evident in figure 11, we observe a collapse of $\varGamma _{TEV}/\varGamma _{LEV}$ and the model outputs (3.16), which therefore indicates a reasonable accuracy of the model at an increasing range of $St_h$, $St_\theta$ and $\phi$. We also depict vertical iso-lines in figure 11 that correspond to the model output $\approx 0.4$ and 1.5. These further mark the approximate bounds where a particular wake topology will dominate for a specific combination of $St_h$, $St_{\theta }$ and $\phi$. Kinematic setting corresponding to the left of iso-line 1 shows a dominant $2P+2S$ wake mode, based on the observations from phase maps depicted in figure 7. Between iso-lines 1 and 2, the $2P$ mode dominates the wake. The specific bounds associated with the $2P$ wake mode with vortex bifurcation ($2P^D$ pairs) occur within the region bounded by iso-lines 1 and 2. The region to the right of iso-line 2 depicts the dominance of the $rBvK$ wake mode while the $2P$ mode is entirely absent. Overall, (3.16) constitutes a novel relationship that best describes the transition of wake topology for an oscillating foil with combined motion, based on most critical kinematic parameters, including pitching ($St_\theta$) and heaving ($St_h$) Strouhal numbers, and phase offset ($\phi$).

Figure 11. Scaling of estimated $\varGamma _{TEV}/\varGamma _{LEV}$ and model (3.16) outputs at increasing $St_h$, $St_\theta$ and $\phi$.

3.5. Evaluation of wake mode transition at $Re= 4000$

To identify the topological transition features of wake modes, the validity of the mathematical model (i.e. (3.13) and (3.15)) and the existence of bifurcations in $2P$ mode at increased $Re$, we now present the phase maps in $\phi \unicode{x2013}St_{c}$ and $A_{c}\unicode{x2013}St_{c}$ phase spaces at $Re= 4000$ in figure 12. We observe that at low $St_c < 0.24$, $wBvK$ mode is supplemented by the shedding of two additional single vortical structures, thus representing the configuration as a $wBvK+2S$ mode. At $St_c > 0.32$, existence of bifurcation is revealed, although an additional $rBvK$ vortex street also co-exists with $2P$ mode. Thus the resulting wake mode is categorized as $rBvK+2P$. The $rBvK+2P$ mode also appears at an increased $St_c$ of 0.48.

Figure 12. Phase map representation of identified wake modes at increasing $St_c$, $\phi$ and $A_c$ for $Re= 4000$.

On the $A_{c}\unicode{x2013}St_{c}$ phase space shown in figure 12(b), we observe a transition from $wBvK+2S$ to $2P$ mode that coincides roughly with the line of $St_A= 0.1$. Beyond $St_A= 0.2$, the $2P$ mode transitions to either $rBvK$ or $rBvk+2P$ mode. We also observe that the bifurcations in $rBvK+2P$ mode do not coincide or fall within any threshold $St_A$, which is in contrast to the observations at $Re= 1000$ (see figure 7b). These bifurcations are noticeable at increased $St_c$ and in the entire range of $A_c$ assumed in our study. On account of the above noted observations in $A_{c}\unicode{x2013}St_{c}$ phase space, it can be deduced further that the mathematical relationship outlined in (3.13) and (3.15) may not provide an accurate prediction of wake transition at a higher $Re$ of 4000. The limitation of our model, as stated above, could also indicate that there are other possible effects on vorticity production at higher $Re$, besides strengthening of the trailing edge vortex structures with increasing $St_A$ (as suggested by (3.13)). This would then make the approximation of leading and trailing edge vortex strengths inaccurate in (3.4) and (3.9). These relationships used in Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) have been validated only at $Re$ close to 1000, which were further confirmed in our study as well. Thus an extensive analysis is still needed to address this limitation in the application of our mathematical model over a range of $Re$. It is also important to note that studies and models concerning phase space representation of wake modes and their transition (Godoy-Diana et al. Reference Godoy-Diana, Aider and Wesfreid2008, Reference Godoy-Diana, Marais, Aider and Wesfreid2009; Deng et al. Reference Deng, Sun and Shao2015; Sun, Deng & Shao Reference Sun, Deng and Shao2018; Zheng et al. Reference Zheng, Xie, Zheng, Ji and Zhu2019) were also limited to $1000 < Re< 1500$. The mechanism of transitioning wake topology and bifurcations of vortex pairs in $2P$ mode at $Re= 4000$ could further reveal differences in wake evolution that could hint at weakness of the mathematical model in predicting the changes in spatial topologies of $2P$ mode.

In order to highlight the transitions in wake topology and presence of bifurcations in $rBvK+2P$ mode at $Re= 4000$, figure 13 depicts qualitatively the modes in the range $0^\circ \le \phi \le 180^\circ$ at $St_{c} = 0.4$. The wake is dominated by a $2P$ mode at $\phi = 0^\circ$, which is similar to the observations at $Re= 1000$ (see figure 8a). The paired trailing and leading edge vortex structures retain coherence only for a short while, after which the similar-sign neighbouring structures merge to form a single coherent structure shown in figure 13(a). At $\phi = 45^\circ$, an $rBvK$ mode is observed in figure 13(b), which is characterized by shedding of strong trailing edge structures that advect close to the wake centreline. In addition, we see dipole-like paired counter-rotating structures ($2P^D$ pairs) that possess weaker strength compared to the trailing edge structures characterizing the $rBvK$ mode. Further, one of the vortical structures constituting this $2P^D$ pair also undergoes stronger stretching in comparison to its counterpart. This $2P^D$ pair, however, does not diverge away from the wake centreline as seen in figure 13(b). On account of the combined presence of features characterizing $rBvK$ and $2P$ modes, the resulting wake configuration was termed $rBvK+2P$ in figure 12(b).

Figure 13. Wake modes at increasing $\phi$ and $Re= 4000$. Panels (a–f) represents $\phi$ corresponding to $0^\circ$, $45^\circ$, $90^\circ$, $120^\circ$, $180^\circ$ and $225^\circ$, respectively, while $St_c=0.4$.

Vortex streets that depict clear bifurcation in addition to the advecting $rBvK$ structures are observed along the wake centreline with increasing $\phi$ to 90$^\circ$ (figure 13c). The counter-rotating vortical structures constituting these streets also resemble the dipole-like configuration observed previously for $2P^D$ pairs at $Re= 1000$. We see further that the vortical structures associated with $rBvK$ mode have more circulation strength and larger size compared to the dipole-like paired structures. This characteristic, however, undergoes a reverse transition at $\phi = 120^\circ$, where the dipole-like paired vortices possess qualitatively a greater strength and size compared to $rBvK$ structures. We see further that during this transition at $\phi = 120^\circ$, the similar-sign vortices associated with bifurcated vortex streets and the $rBvK$ street, respectively, merge and form single coherent structures approximately around the middle (streamwise) of the wake. A bifurcation of dipole-like pairs is again visible while increasing $\phi$ to 180$^\circ$ (figure 13e), although the weaker $rBvK$ street along the wake centreline is now more vivid compared to $\phi = 120^\circ$. The transition and appearance of bifurcated vortex streets in the wake associated with higher $Re$ therefore show different wake features compared to the wakes identified at lower $Re$ in figure 8. Here, we first observe formation of dipole-like $2P^D$ pairs in conjunction with $rBvK$ structures at $\phi = 45^\circ$. Typically, the individual vortices forming the $2P^D$ pair possessed lower strength and size compared to the vortical structures that constitute the $rBvK$ street. However, the $2P^D$ pairs undergo a consistent increase in their strength and size compared to the $rBvK$ structures with increasing $\phi$ from $45^\circ$ to $180^\circ$. The diverging vortical streets comprising the $2P^D$ pairs are observed at $\phi = 90^\circ$, which then appeared to resist their movement away from the wake centreline as $\phi$ increases to $120^\circ$. A re-emergence of a bifurcated vortex street was then observed at $\phi = 180^\circ$. We believe that this dynamic change in the behaviour of bifurcated dipolar streets could be associated directly with the consistent increase, and decrease, in the strength and size of structures, which constitute bifurcated vortex streets and $rBvK$ wake, respectively. A more closer attention to the formation mechanism will reveal further details of the transitions of modes observed here. Finally, at $\phi = 225^\circ$ (figure 13f), the wake again represents a $2P+2S$ mode, which was even observed and discussed in figure 8(f) at $Re=1000$. Thus the unique observations and characterization of bifurcations in $2P$ mode still hold for a higher Reynolds number, although there are additional complexities observed in terms of a combined $rBvK+2P$ mode.

3.6. Formation mechanism of $rBvK+2P$ mode and symmetric bifurcations

The wake modes observed for increasing $St_c$ and $\phi$ at $Re= 4000$ were characterized by an $rBvK+2P$ configuration, where the $2P$ mode observed at $Re= 1000$ is supplemented with an $rBvK$ vortex street. Figure 14 shows the formation and temporal evolution of wake structures constituting the $rBvK+2P$ mode, along with observed bifurcation of $2P^D$ pairs, at $Re= 4000$. As the oscillation cycle begins (i.e. at $t=0$) in figure 14(a), we observe a very similar shedding pattern near the trailing edge compared to $Re= 1000$ (see figure 9). The counter-rotating leading and trailing edge structures form an approximate dipole-like paired configuration in each half shedding cycle, which then detaches from the trailing edge. Two dipole-like structures, marked as $D1^\prime$ and $D2^\prime$, are shown in figure 14(a). The dipole-like $2P^D$ pair then moves away from the wake centreline on account of the mutual induced velocity of the pair. However, as the dipole-like structures advect downstream, we observe that an elongated leg of the trailing edge vortex shed in the previous half shedding cycle splits into two structures. This splitting phenomenon was not observed for the $2P$ mode formation at $Re= 1000$. At $t=T/4$ in figure 14(b), we observe vortical structures TEV$1_{ac}^s$ and S$_{ac}$, which form due to the splitting of the vortex leg shown in figure 14(a). It is further seen that TEV1$_{ac}^s$ constitutes a dipole-like configuration of diverging vortex street, while S$_{ac}$ eventually becomes a part of the $rBvK$ wake. A similar splitting process again starts at the end of the first half oscillation cycle, shown in figure 14(c). However, the sense of rotation for the trailing edge vortex leg is opposite to that of the first half shedding cycle. The resulting split structures TEV2$_c^s$ and S$_c$ are shown in figure 14(d). These structures contribute further to the formation of $rBvK+2P$ mode. This characterization constitutes a new wake mode that has not been observed previously for bluff bodies or oscillating foils. It also accounts for a very different wake dynamics observed for oscillating foils with combined heaving and pitching motion, compared to those with either pure heaving or pure pitching kinematics.

Figure 14. Temporal snapshots representing contours of $\omega _z^+ = \omega _z c/U_{\infty }$, and depicting formation of $rBvK+2P$ mode with bifurcation at $Re= 4000$.

The shear splitting phenomenon that explained reasonably the formation process of $rBvK+2P$ and coinciding bifurcation could also provide insights into the disappearance and re-emergence of the birfurcated dipole streets observed and highlighted in the previous subsection. As $\phi$ increases from $45^\circ$ to $180^\circ$, we observed (figures 13b–e) that a clear bifurcation of dipole-like pairs appears in conjunction with the $rBvK$ structures for $\phi = 90^\circ$. However, at $\phi = 120^\circ$, the bifurcation was not imminent. Further increase in $\phi$ to 180$^\circ$ depicted a reappearance of bifurcation for the dipole-like pairs. As indicated earlier, there is a consistent decrease in strength of trailing edge vortical structures relative to the structures forming at the leading edge. This change has a direct consequence on the shear splitting phenomenon where a delay in splitting of the trailing edge vortex leg occurs as $\phi$ increases from 45$^\circ$ to $180^\circ$. We provide qualitative evidence in figure 15, which depicts the shear splitting process for wake observed at $Re= 4000$, and $\phi = 45^\circ$, $90^\circ$, $120^\circ$ and $180^\circ$, respectively. At $\phi = 45^\circ$ (figure 15a), we see that as the trailing edge vortex separates from the trailing edge, the dipole-like pair ($D$) is well detached, devoid of any mutual interaction with the trailing edge vortex leg. As $\phi$ increases to $90^\circ$ (see figure 15b), the dipole-like pair ($D$) now appears close to the detached trailing edge vortex while also undergoing mutual interaction with its leg. Thus a delay is observed in the splitting process with the weakening of the trailing edge vortex. Further increasing $\phi$ to $120^\circ$, the splitting is further delayed compared to $\phi = 90^\circ$. The weaker trailing edge vortex strength compared to vortical structures constituting $D$ is also observed in figure 15(c). At $\phi = 180^\circ$ (see figure 15d), only the dipole-like pair $D$ appears well developed, with the splitting process undergoing an even greater delay compared to the cases at lower $\phi$. The bifurcation of dipole streets that occurs at $\phi < 120^\circ$ could be attributed to the rapid splitting process that happens on account of the longer trailing edge vortex development. However, bifurcation observed at $\phi > 120^\circ$ largely could be a consequence of higher induced velocity of the dipole-like configuration, which further triggers the shear splitting ahead in the wake.

Figure 15. Changes in the dynamic shear splitting process with increasing $\phi$.

The lack of bifurcated dipole-like streets at $\phi = 120^\circ$ also needs closer attention. Figure 16 depicts the temporal evolution of the trailing edge vortex and dipole-like pair $D$. Even after the shear splitting (at approximately $t= 0.45T$ in figure 16) occurs, the short separation distance between the clockwise vortex, constituting the dipole pair ($D$), and the trailing edge vortex promotes a merger rather than movement of $D$ away from the wake centreline. Thus we also see in figure 16(c) that the split trailing edge vortices eventually merge with the similar-sign vortical structures of neighbouring dipoles that further undergo a combined interaction near the wake centreline. Besides the determination of critical $\phi$ (i.e. $\phi = 120^\circ$) where the bifurcation disappears and dipole-induced velocity begins to dominate the wake dynamics, a more detailed investigation is underway in a separate study, which will model quantitatively the above transition process.

Figure 16. Temporal evolution of wake splitting and merging at $\phi = 120^\circ$.

The splitting observed at $Re= 4000$ provides additional explanations for the potential limitation of the proposed mathematical model in (3.13) and (3.15) for higher Reynolds numbers. The observations of Schnipper et al. (Reference Schnipper, Andersen and Bohr2009) indicated no splitting in their wake structures near the foil trailing edge at $Re\approx 1200$. The approximation of downward transport of vorticity, as employed in our study similar to that of Schnipper et al. (Reference Schnipper, Andersen and Bohr2009), was based on estimating the vorticity production from Tietjens et al. (Reference Tietjens, Prandtl and Rosenhead1957), which does not account for any effect of splitting mechanism in the wake of bluff bodies. As also detailed in the study by Ramesh et al. (Reference Ramesh, Ke, Gopalarathnam and Edwards2012), with regard to unsteady effects on leading edge vortex shedding with increasing $Re$, the vorticity distribution and thereby the circulation is inherently dependent on the wake-induced downwash velocity in parallel and normal directions to the foil chord. This wake-induced velocity at $Re= 4000$ will not be similar to the approximation (Ramesh et al. Reference Ramesh, Ke, Gopalarathnam and Edwards2012) that holds at $Re= 1000$ due to the onset of unsteady splitting mechanisms in the near wake. This attributes directly to the inaccuracy in estimating circulation of the leading edge, or boundary layer, vortices (i.e. (3.4)). A similar observation holds for the study of Schnipper et al. (Reference Schnipper, Andersen and Bohr2009). This provides a reasonable explanation for why our newly proposed mathematical model does not work well in predicting the transition in wake topological characteristics at higher Reynolds numbers, e.g. $Re=4000$.

The higher complexity of wake dynamics observed at $Re= 4000$ also hints at a reason for the lack of correspondence between the $rBvK+2P$ mode (and the observed bifurcation) and the pitch- or heave-dominated kinematics. The qualitative and quantitative observations of $2P$ mode at $Re= 1000$ suggested that the equalization of strength and size of leading and trailing edge vortices are the only prominent contributing aspects for the observed bifurcated streets at higher Reynolds numbers, which coincided only with heave-dominated kinematics at low $Re$. However, the qualitative transition and formation mechanisms at $Re= 4000$ revealed a shear splitting process that was not seen at $Re= 1000$. This phenomenon also depicted varying temporal characteristics as the kinematics switch from pitch- to heave-dominated kinematics based on $\phi$. We believe that since the dominance of this splitting process (responsible for formation of bifurcated vortex streets) at $Re= 4000$ is not limited within certain regimes of prescribed kinematics for the foil, such bifurcations in $rBvK+2P$ wake mode can appear in different kinematic settings that induce similar effects, such as $St_c= 0.48$ at $Re=4000$ (see figure 12b).

3.7. Mean flow development in bifurcated dual vortex street wakes

In order to characterize the bifurcated dual vortex streets in $2P$ and $rBvK+2P$ wake modes, we now evaluate the qualitative and quantitative time-averaged streamwise velocity ($\overline {U_{x}^+}$) distribution in the wake. Figure 17 shows the contours of $\overline {U_{x}^+}$ for increasing $\phi$ at $St_{c}= 0.4$ and $Re= 1000$. Although we depict the contours corresponding to only cases $\phi = 0^\circ$, $90^\circ$, 180$^\circ$ and 225$^\circ$, the quantitative profiles in figure 18 are shown for all $\phi$ values ranging from pitch- to heave-dominated motion kinematics. These profiles are further obtained at three streamwise locations corresponding to $X^+= 1.5$, 3.5 and 5.5, respectively, as shown in figure 17(a). Due to similarities in the mean characteristics for some cases (e.g. $\phi = 90^\circ$ and 120$^\circ$), contours for only some cases are shown, for brevity.

Figure 17. Contours of $\overline {U_{x}^+}$ at (a) $\phi = 0^\circ$, (b) $\phi = 90^\circ$, (c) $\phi = 180^\circ$, and (d) $\phi = 225^\circ$. Solid red lines depict the streamwise locations where cross-stream profiles of $\overline {U_{x}^+}$ are extracted and shown in figure 18.

Figure 18. Profiles of $\overline {U_{x}^+}$ along the cross-stream ($Y^+$) direction at streamwise locations corresponding to(a) $X^+= 1.5$, (b) $X^+= 3.5$, and (c) $X^+= 5.5$.

The mean velocity excess at the wake centreline ($Y^+= 0$) is maximum in the case of $\phi = 0^\circ$ and 45$^\circ$, at all three $X^+$ locations (see figure 18). However, a mean velocity deficit region is also observed for $\phi = 45^\circ$, which strengthens further with increasing $X^+$. These deficit regions are also observed alongside the mean thrust jet developing in the wake.

Earlier, we discussed the upstream tilted vortex pairs depicted for $\phi = 45^\circ$ in figure 8(b), which did not show a significant merger between the neighbouring vortex structures with a similar sense of rotation. These tilted pairs promote creation of a resultant induced velocity that acts against the freestream. Therefore, it results in the dual velocity deficit regions observed for the case $\phi = 45^\circ$. Such tilted pairs, however, were not seen at $\phi = 0^\circ$ in figure 8(a). Increasing $\phi$ to 90$^\circ$ showed a transition of counter-rotating paired vortex structures to a dipole-like formation in figure 8(c). This coincides clearly with a reduction in the peak magnitude of the mean thrust jet, compared to the lower $\phi$ cases observed in figure 18. This jet is also shortened in terms of its streamwise extent, as shown in figure 17(b). This is also confirmed by the sharp drop in the peak velocity at $Y^+= 0$ in figure 18(c) for the wakes of $\phi \ge 90^\circ$. Despite this decrease along the wake centreline, the neighbouring minima decrease further as $\phi$ increases to 120$^\circ$. This is observed specifically for the streamwise locations corresponding to $X^+= 1.5$ and 3.5, respectively. For $X^+= 5.5$, however, the velocity deficit regions flatten with increasing $\phi$ to 120$^\circ$. These observations also coincide with the vortex interaction and merger observed for $\phi = 90^\circ$ and 120$^\circ$ in figures 8(c) and 8(d), respectively. Since these merged structures advect closely along the wake centreline in figures 8(c,d), resultant induced velocity must be insignificant. Thus it leads to flat velocity profiles along $Y^+$.

At $\phi = 180^\circ$, the cross-stream profiles for $\overline {U_{x}^+}$ show a different characteristic time-averaged wake. Near the foil, a pair of velocity minima is observed in figure 18(a), with a relatively minor drop compared to $U_{\infty }$, which is close to the wake centreline. Assessment at increasing streamwise distance from the foil reveals further that the minima locations disperse away from the wake centreline. This feature coincides with the diverging dual vortex streets seen in figure 8(e). This is confirmed qualitatively in figure 17(c), which identified the presence of diverging velocity deficit streams along the streamwise direction. As $\phi$ is increased further to 225$^\circ$, we observe two velocity maxima or peaks that neighbour the wake centreline. This feature, observed from the quantitative profiles depicted in figure 18, is in definite contrast to the characteristics observed at $\phi = 180^\circ$. However, the twin peaks move away from the wake centreline in a fashion similar to the minima locations observed at $\phi = 180^\circ$. This feature is also noticeable in figure 17(d), where we observe two streams of relatively high mean velocity in the wake. From the above observations, it is seen that the formation of diverging vortex streets at $\phi = 180^\circ$ reflects a critical threshold characteristic, since the mean wake features deviate and follow different spatio-temporal development as $\phi$ is switched away from 180$^\circ$. These dual dipole-like vortex streets are also suggestive of a relatively low spatial disturbance near the wake centreline region, compared to mean wake features associated with $\phi$ values higher or lower than 180$^\circ$.

Figure 19(a) depicts further the contours of $\overline {U_{x}^+}$ for $\phi = 225^\circ$ at $f^*=0.48$ and $Re=1000$. This kinematic and flow configuration corresponded to the $2P$ mode in which bifurcations were evident, as identified in figure 7(a). It is observable that the mean spatial development coinciding with the bifurcated dual vortex street remains largely unaffected with increased $St_c$. The diverging velocity deficit streaks appear comparatively similar to the $\overline {U_{x}^+}$ development observed in figure 17(c), which corresponds to the case $\phi = 180^\circ$ and $f^*= 0.4$. A spatial uniformity in the cross-stream distribution of $\overline {U_{x}^+}$ is also retained at this higher $St_c$ setting. Figure 19(b), however, depicts the $\overline {U_{x}^+}$ distribution at a higher $Re$ of 4000, while $\phi$ and $St_c$ correspond to 90$^\circ$ and 0.4, respectively. This setting also coincides with the $rBvK+2P$ mode identified in figure 12(b), which underwent bifurcation and whose formation was also discussed in a previous subsection. A relatively strong thrust jet is seen along the wake centreline on account of the $rBvK$ vortex street observed in figure 13(c). We also see two bifurcated streaks that possess a relatively higher velocity compared to the surrounding flow. Such bifurcation in mean flow is again suggestive of an implication posed by dual vortex streets in $rBvK+2P$ mode. However, these weak jet streaks still appear in contrast to the velocity deficit streaks observed at lower $Re$, and which corresponded to only a $2P$ mode (see figures 17c and 19a).

Figure 19. Contours of $\overline {U_{x}^+}$ at (a) $\phi = 225^\circ$, $f^*= 0.48$, $Re= 1000$, and (b) $\phi = 90^\circ$, $f^*= 0.4$, $Re= 4000$.