3.1. Instantaneous thrust, power and lift

Figure 5(a–j) shows ${C_T}$ (red line) and ${C_P}$ (blue line) distributions in a complete oscillation cycle for $\mathrm{\mathbb{T}} = 0.80$, 0.85, 0.90, 0.95, 1.00, 1.05, 1.11, 1.15, 1.20 and 1.40. The corresponding phase portrait $\dot{\theta }^{\prime}- \theta ^{\prime}$ (black lines) is superimposed on the ${C_T}$ and ${C_P}$ lines. The ${C_T}$ lines laying outside (solid line) and inside (dashed line) of the phase portrait indicate positive and negative values of ${C_T}$, respectively. The same sign convention is applied to ${C_P}$ lines. This figure also facilitates a comparison of the results between modified motion $(\mathrm{\mathbb{T}} \ne 1.00)$ and sinusoidal motion ($\mathrm{\mathbb{T}} = 1.00$, figure 5e). Here, we will discuss the aerodynamics only for phase ϕ = 0°–180° (the upper half) as each profile has rotational symmetry of order two. Recall that the first and second quadrants of the phase portrait complement the faster and slower strokes, respectively, for $\mathrm{\mathbb{T}} < 1.00$ (figure 5a–d) while the strokes swap their quadrants for $\mathrm{\mathbb{T}} > 1.00$ (figure 5f–j).

Figure 5. Instantaneous thrust (red line) and power (blue line) coefficients in the phase plane for modified sine motion with time period ratio 0.80, 0.85, …, 1.20 and 1.40 in (a), (b), …, (j), respectively. Positive values of both coefficients (solid lines) are outward (outside of the phase portrait) and negative values (dashed lines) are inward. Power coefficient is scaled down eight times the thrust coefficient. Here, St_A = 0.38.

When $\mathrm{\mathbb{T}} < 1.00$, three distinct features can be noticed in the ${C_T}$ and ${C_P}$ distributions. First, the magnitudes of maximum and minimum ${C_T}$ increase when $\mathrm{\mathbb{T}}$ is decreased from 1.00 to 0.80. The same happens for ${C_P}$. For example, with $\mathrm{\mathbb{T}}$ decreases from 1.00 to 0.80, the maximum ${C_T}$ and ${C_P}$ values are enhanced by 41 % and 65 %, respectively. Second, the occurrence of the maximum ${C_T}$ shifts from the retract stroke at $\mathrm{\mathbb{T}} = 1.00$ to the forward (faster) stroke when $\mathrm{\mathbb{T}}$ is decreased from 1.00. Specifically, the maximum ${C_T}$ for $\mathrm{\mathbb{T}} = 0.80$ is at ϕ = 62°, laying in the forward stroke (figure 5a), while that for $\mathrm{\mathbb{T}} = 1.00$ is at ϕ = 107°, early in the retract stroke (figure 5e). Third, the peak ${C_P}$ position postpones with decreasing $\mathrm{\mathbb{T}}$, laying at ϕ = 156° (late in the retract stroke) for $\mathrm{\mathbb{T}} = 1.00$ and at ϕ = 194° (early in the next forward stroke) for $\mathrm{\mathbb{T}} = 0.80$. That is, the relationship between the peak ${C_P}$ position and $\mathrm{\mathbb{T}}$ value is opposite to that between the peak ${C_T}$ position and $\mathrm{\mathbb{T}}$ value. As such, the ${C_T}$ and ${C_P}$ peaks occurring in the second quadrant for $\mathrm{\mathbb{T}} = 1.00$ move away from each other when $\mathrm{\mathbb{T}}$ is decreased to 0.80. Moreover, the positions of ${C_P} = 0$ and ${C_T} = 0$ (transition from drag to thrust) appear at ϕ = 33°–36° and 37°–52° (depending on $\mathrm{\mathbb{T}}$), respectively, both slightly decreasing with decreasing $\mathrm{\mathbb{T}}$. Alam & Muhammad (Reference Alam and Muhammad2020) found that ${C_T}$ and ${C_P}$ are maximum at ϕ ≈ 112° and 157° for $\mathrm{\mathbb{T}} = 1.00$ with St_A = 0.24–0.53, which are consistent with the present results.

When $\mathrm{\mathbb{T}}$ is increased from 1.00 to 1.40 (figure 5e–j): (i) magnitudes of maximum ${C_T}$ and ${C_P}$ grow by approximately 150 % and 145 %, respectively, between $\mathrm{\mathbb{T}} = 1.00$ and 1.20, and by 990 % and 1770 %, respectively, between $\mathrm{\mathbb{T}} = 1.00$ and 1.40; (ii) magnitudes of minimum ${C_T}$ and ${C_P}$ diminish between $\mathrm{\mathbb{T}} = 1.00$ and 1.20, and enhance for $\mathrm{\mathbb{T}} = 1.40$ as compared to $\mathrm{\mathbb{T}} = 1.00$; (iii) the instantaneous transition from drag to thrust (i.e. ${C_T} = 0$) moves towards ϕ = 90° (extreme position); (iv) position of maximum ${C_T}$ moves toward the nearest equilibrium position (ϕ = 0° and 180°) and (v) the gap between the ${C_T}$ and ${C_P}$ peaks in the retract stroke shrinks. The ${C_P}$ diminishes in the forward stroke (ϕ = 0°–90°) with an increase in $\mathrm{\mathbb{T}}$, whereas grows in the retract stroke (ϕ = 90°–180°). That is, faster stroke will result in a higher ${C_P}$ and vice versa.

Overall, the above findings suggest that maximum ${C_T}$ and ${C_P}$ prevail in the faster stroke (i.e. forward stroke for $\mathrm{\mathbb{T}} < 1.00$ and retract stroke for $\mathrm{\mathbb{T}} > 1.00$). Although maximum ${C_T}$ and maximum ${C_P}$ both rise when $\mathrm{\mathbb{T}}$ is increased or decreased from 1.00, the rise in the maximum ${C_P}$ is higher for $\mathrm{\mathbb{T}} > 1.00$ than for $\mathrm{\mathbb{T}} < 1.00$, as is that in the maximum ${C_T}$. However, for $1.00 < \mathrm{\mathbb{T}} \le 1.20$, the degree of the increase in ${C_T}$ is greater than that in ${C_P}$ and hence a higher efficiency may occur for $1.00 < \mathrm{\mathbb{T}} \le 1.20$ than for $\mathrm{\mathbb{T}} < 1.00$. The above observations raise a few questions. For example, which $\mathrm{\mathbb{T}}$ is beneficial? What are their implications on swimming motion? How does the efficiency vary with $\mathrm{\mathbb{T}}$?

3.1.1. Correspondence between ${C_P}$ and ${C_T}$

Swimming based on ${C_T}$ and ${C_P}$ can be broadly categorized into four scenarios: (I) spend energy and swim upstream, i.e. ${C_P} > 0$ and ${C_T} > 0$ (normal swimming); (II) spend energy, yet drift downstream, i.e. ${C_P} > 0$ and ${C_T} < 0$ (undesirable); (III) gain energy from the fluid and drift downstream, i.e. ${C_P} < 0$ and ${C_T} < 0$ (floating); and (IV) gain energy from the fluid and swim upstream, i.e. ${C_P} < 0$ and ${C_T} > 0$ (ideal). These scenarios essentially correspond to the four quadrants of the instantaneous thrust verus power plot (${C_{Pi}}$ and ${C_{Pf}}$) as shown in figure 6(a), where quadrant II is undesirable and quadrant IV is ideal. The ${C_{Pi}}$ (dashed red line) and ${C_{Pf}}$ (green line) plots are provided only for the first half (ϕ = 0°–180°) of the oscillation as the plots for the second half of the oscillation overlap with those for the first half, while this is not the case for ${C_L}$. The overlapping arises from their (${C_{Pi}}$ and ${C_{Pf}}$) generation mechanism, the product of the two waves, i.e. ${C_{Pi}} \propto \ddot{\theta }\dot{\theta }$ and ${C_{Pf}} \propto {M_z}\dot{\theta }$. The pressure difference between the upper and lower surfaces of the foil can readily be understood from the instantaneous lift coefficient $({C_L})$ (figure 6b). A positive ${C_L}$ means pressure is higher on the lower surface than on the upper surface, which produces a positive (counterclockwise) moment (see figure 2b). The maximum pitching angle is small, θ_max ≪ 1, and hence the trend of the pressure difference between the upper and lower surfaces can be reasonably predicted by the trend of ${C_L}$. Hereinafter, ${C_L}$ will be understood as an alibi for the pressure difference, unless explicitly stated otherwise.

Figure 6. Sinusoidal pitching foil at $\mathrm{\mathbb{T}} = 1.00$, variation of instantaneous (a) power coefficients (C_pi, owing to inertia and C_Pf, owing to fluid) with thrust coefficient and (b) lift coefficient with normalized angular position. (c–f) The wake structure for four phases namely (0°, 45°, 90° and 135°, respectively) are shown with $\varOmega _z^\ast ( = {\varOmega _z}d/{U_\infty }) = [ - 5,5]$, with red colour for +ve values and blue for negative. Here, St_A = 0.38.

There are many studies on sinusoidal $(\mathrm{\mathbb{T}} = 1.00)$ pitching foil. We therefore briefly iterate some important information from the literature for a comparison purpose. For $\mathrm{\mathbb{T}} = 1.00$, the foil at ϕ = 0° has negative ${C_T}$ and negative ${C_L}$ (figure 6a,b). The upper and lower surfaces act as pressure and suction surfaces, respectively, producing −M_z. As $\dot{\theta }$ is positive and maximum at ϕ = 0°, ${C_{Pf}}$ is positive but not maximum as −M_z is not necessarily maximum at ϕ = 0° but slightly before this (figure 6b). At this foil position, the resultant pressure force on the side surfaces largely contributes to the lift and negligibly to the thrust because the foil appears streamlined, having a minimum projection on the y-axis and a maximum projection on the x-axis. The pressure on the semicircular leading edge and viscous force are largely responsible for the generation of the negative thrust (Alam & Muhammad Reference Alam and Muhammad2020). As the foil moves away from the equilibrium position, the upper shear-layer separating from the foil tip continues to roll-up (figure 6c). When the foil reaches ϕ = 45°, the pressure difference across its upper and lower surfaces becomes zero (see ${C_L} = 0$ at ϕ = 45° in figure 6b). That is, the pressure on the upper surface decreases and/or that on the lower surface increases as ϕ increases from 0° to 45°. The ${C_T}$ at ϕ = 45° is higher than that at ϕ = 0°, albeit still negative. From ϕ = 0° to 45°, the magnitudes of both $\dot{\theta }$ and M_z decline, which leads to a decreased ${C_{Pf}}$. When the foil moves from ϕ = 45° to 90°, ${C_L}$ increases with ϕ (figure 6b), which suggests that the pressure on the upper surface is progressively smaller than that on the lower surface. This pressure difference contributing to the thrust makes ${C_T}$ increasingly positive (figure 6a). For the same reason, M_z is positive. From ϕ = 45° to 90°, as M_z grows and $\dot{\theta }$ declines, ${C_{Pf}}$ becomes minimum between ϕ = 45° and 90°. That is, in the second half of the forward stroke (ϕ = 45°–90°), the foil can move forward, gaining energy from the fluid (negative power input). In the forward stroke (ϕ = 0°–90°), inertia being negative $({C_{Pi}} < 0)$ also supports the motion. From ϕ = 0° to 90°, the increase in ${C_L}$ is accompanied by vortex growth and shedding from the upper side (figure 6c–e). At ϕ = 90°–135° where the foil is in the retract stroke ($\theta^{\prime}$ decreasing), ${C_L}$ keeps increasing and ${C_T}$ becoming maximum at ϕ = 107° declines with a further increase in ϕ while ${C_{Pf}}$ is positive, growing with increasing ϕ. Meanwhile, the shear layer on the lower surface starts to roll-up to form a counterclockwise vortex (red colour) behind the foil tip (figure 6f). From ϕ = 135° to 180°, both ${C_L}$ and ${C_T}$ drop while ${C_{Pf}}$ reaches a maximum at ϕ = 158°. The ${C_{Pi}}$ is positive during retract strokes, being maximum at ϕ = 135°.

Now we will discuss the case with $\mathrm{\mathbb{T}} = 1.20$ (figure 5i), where the increase in the maximum ${C_T}$ is larger than that in the maximum ${C_P}$, as compared to the case with $\mathrm{\mathbb{T}} = 1.00$ (figure 5e). Figure 7 shows ${C_{Pf}}$, ${C_{Pi}}$ versus ${C_T}$, and ${C_L}$ versus $\theta^{\prime}$. At ϕ = 0°–45°, ${C_L}$ increases slowly, i.e. the pressure difference between the upper and lower surfaces of the foil does not change much, as do not ${C_{Pf}}$ and ${C_T}$ (figure 7a,b). The ${C_{Pi}}$ magnitude increases negligibly, albeit negative. The ${C_T}$ nevertheless rapidly enhances between ϕ = 45° and 90° while ${C_{Pf}}$ is negative and small in magnitude. At ϕ = 90°, both ${C_{Pf}}$ and ${C_{Pi}}$ become zero. It is worth pointing out that because the forward stroke for $\mathrm{\mathbb{T}} = 1.20$ has a longer period than the retract stroke (figure 3a), the foil motion in the forward stroke (ϕ = 0°–90°) is relatively slow, requiring less ${C_{Pi}}$ and ${C_{Pf}}$ than that for $\mathrm{\mathbb{T}} = 1.00$ (figures 6a and 7a). Moreover, the switchover in the direction of the pressure difference is delayed as compared with that for $\mathrm{\mathbb{T}} = 1.00$, occurring at ϕ = 45° and 68° for $\mathrm{\mathbb{T}} = 1.00$ and 1.20, respectively. The foil switches to the shorter period in the retract stroke (ϕ = 90°–180°) and experiences higher $\dot{\theta }$ and $\ddot{\theta }$ magnitudes than in the forward stroke. The ${C_L}$ and ${C_{Pf}}$ both thus flare up from ϕ = 90° to 135°, yielding a maximum ${C_T}$ at ϕ = 123° (figure 5i). From ϕ = 135° to 180°, ${C_L}$ rapidly drops; as do ${C_T}$ and ${C_{Pf}}$. The ${C_{Pf}}$, however, reaches a peak at ϕ = 140° as the magnitude of $\dot{\theta }$ (−ve) increases and ${C_L}$ (+ve) decreases between ϕ = 135° and 180°. At ϕ = 180°, the ${C_{Pf}}$ for $\mathrm{\mathbb{T}} = 1.20$ is much smaller than that for $\mathrm{\mathbb{T}} = 1.00$ (figures 7a and 6a), because of lower ${C_L}$ (hence M_z) produced for $\mathrm{\mathbb{T}} = 1.20$.

Figure 7. For $\mathrm{\mathbb{T}} = 1.20$, variation of instantaneous (a) power coefficients (C_pi, owing to inertia and C_Pf, owing to fluid) with thrust coefficient and (b) lift coefficient with normalized angular position. Here, St_A = 0.38.

Figure 8 shows ${C_{Pf}}$, ${C_{Pi}}$ versus ${C_T}$, and ${C_L}$ versus $\theta^{\prime}$ for $\mathrm{\mathbb{T}} = 0.80$, which is the lowest $\mathrm{\mathbb{T}}$ studied. At the equilibrium position (ϕ = 0°), ${C_T}$ does not change appreciably between $\mathrm{\mathbb{T}} = 0.80$, 1.00 and 1.20 (figures 6–8), which indicates that ${C_T}$ at equilibrium position is not predominantly influenced by $\mathrm{\mathbb{T}}$ as it mainly arises from the pressure on the head of the foil and the viscous forces. The ${C_L}$ magnitude at equilibrium position for $\mathrm{\mathbb{T}} = 0.80$ is, however, quite high compared with that for the $\mathrm{\mathbb{T}} = 1.00$ and 1.20 (see figures 6a and 7a) cases, which results in a much higher ${C_{Pf}}$ for $\mathrm{\mathbb{T}} = 0.80$ (figure 8a). The forward stroke for $\mathrm{\mathbb{T}} = 0.80$ is the fast quarter, where $\dot{\theta }^{\prime}$ first increases and then decreases, reaching a maximum value at ϕ = 25° (see phase portrait in figure 5a or figure 3b). This increase and decrease in $\dot{\theta }^{\prime}$ are reciprocated by ${C_L}$ variations (figure 8b). There is a decrease in ${C_T}$ and an increase in ${C_{Pf}}$ immediately after the start of the forward stroke, which is followed by a large increase in ${C_T}$ and a large decrease in ${C_{Pf}}$ up to the mid (ϕ = 45°) of the forward stroke. The ${C_{Pf}}$ reaches its maximum at ϕ = 15°. The change in the ${C_L}$ sign happens at ϕ = 35° and that for ${C_{Pf}}$ takes place at ϕ = 34°. At ϕ = 45°–90°, the increase and decrease in ${C_L}$ with ϕ are accompanied by the same in ${C_T}$. The local maxima of ${C_L}$ occur at ϕ = 60°, which is followed by ${C_T}$ becoming maximum at ϕ = 62°. In the retract (slower) stroke, ${C_T}$ declines, but ${C_{Pf}}$ and ${C_L}$ increase. The general feature for $\mathrm{\mathbb{T}} \ne 1.00$ is that a higher ${C_P}$ (both ${C_{Pf}}$ and ${C_{Pi}}$) is required to make a stroke faster. It is understood from figure 6a that, on average, a higher ${C_{Pf}}$ is required in the retract stroke than in the forward stroke for $\mathrm{\mathbb{T}} = 1.00$. As an additional ${C_P}$ is required to make a stroke faster, ${C_P}$ grows more rapidly in the retract stroke for $\mathrm{\mathbb{T}} > 1.00$ (figure 5f–j) and in the forward stroke for $\mathrm{\mathbb{T}} < 1.00$ (figure 5a–d).

Figure 8. For $\mathrm{\mathbb{T}} = 0.80$, variation of instantaneous (a) power coefficients (C_pi, owing to inertia and C_Pf, owing to fluid) with thrust coefficient and (b) lift coefficient with normalized angular position. Here, St_A = 0.38.

At ϕ = 0°, the foil is streamlined, and both viscous force and pressure force on the foil head contribute to ${C_T}$ that is almost independent of $\mathrm{\mathbb{T}}$ (figures 6a–8a). However, the ${C_{Pf}}$ plays an important role in ${C_L}$ generation at ϕ = 0°. The $\dot{\theta }$ at ϕ = 0° is the same for all $\mathrm{\mathbb{T}}$ (see the slopes of curves in figure 3a, this is not clear in figure 3b, owing to ${\dot{\theta }_{max}}$ being different for each $\mathrm{\mathbb{T}}$) and hence the variation in ${C_L}$ with $\mathrm{\mathbb{T}}$ solely arises from the change in ${C_{Pf}}$ (figures 6b–8b). At ϕ = 90°, $\dot{\theta } = 0$ and hence ${C_{Pf}} = {C_{Pi}} = 0$, the ${C_L}$ at extreme position is supposed to be dependent on $\mathrm{\mathbb{T}}$, but it is not; ${C_L}$ increases from 28.0 to 29.8 (0.6 %) between $\mathrm{\mathbb{T}} = 0.80$ and 1.20, and reduces to 29.5 at $\mathrm{\mathbb{T}} = 1.40$ (not shown).

Figure 9 shows the regimes of swimming for $\mathrm{\mathbb{T}} = 0.80$, 1.00, 1.20 and 1.40 in one oscillation period. Each regime is presented in a shade of a particular colour to distinguish regimes. The shade of the same colour distinguishes the magnitude of ${C_T}$, with dull and bright shades representing low and high magnitudes of ${C_T}$, respectively. A comparison of figures 9(a) and 9(b) reflects that speeding up the forward stroke (i.e. $\mathrm{\mathbb{T}} < 1.00$) does not appreciably affect the undesirable regime (II, orange), ϕ = −20–33° and −18–35° for $\mathrm{\mathbb{T}} = 0.80$ and 1.00, respectively, laying around the equilibrium. The magnitude of ${C_T}$ is however increased $({C_T} < 0)$, with drag increasing. Nestling close to the middle of the forward stroke, floating regime (III, grey) shrinks from spanning 17° to 4° when $\mathrm{\mathbb{T}}$ is decreased from 1.00 to 0.80. The ideal regime (IV, yellow) dramatically stretches from spanning 38° $(\mathrm{\mathbb{T}} = 1.00)$ to 53° $(\mathrm{\mathbb{T}} = 0.80)$, occupying the end of the forward stroke. The ${C_T}$ in the ideal regime is also enhanced owing to the increased speed of the forward stroke (i.e. $\mathrm{\mathbb{T}} < 1.00$). The normal swimming regime (I, blue) covers most of the retract stroke, approximately 70° for $\mathrm{\mathbb{T}} = 0.80$ and 72° for $\mathrm{\mathbb{T}} = 1.00$, while the rest of retract stroke is the undesirable regime. Regimes I (normal swimming) and IV (undesirable) do not change much between $\mathrm{\mathbb{T}} = 0.80$ and 1.00; the magnitude of ${C_T}$ however decreases in the normal swimming regime (when $\mathrm{\mathbb{T}} < 1.00$), as compared to $\mathrm{\mathbb{T}} = 1.00$ (figure 9a,b).

Figure 9. Regimes of swimming for (a) $\mathrm{\mathbb{T}} = 0.80$, (b) $\mathrm{\mathbb{T}} = 1.00$, (c) $\mathrm{\mathbb{T}} = 1.20$ and (d) $\mathrm{\mathbb{T}} = 1.40$ for St_A = 0.38. The blue, yellow, grey and orange colours represent the regimes I, II, III and IV, respectively. The shades of each colour represent the magnitude of C_T.

The slowing down of the forward stroke (i.e. $\mathrm{\mathbb{T}} > 1.00$), as expected, stretches undesirable (IV) and floating (III) regimes to a span of 75° and 22°, respectively, at $\mathrm{\mathbb{T}} = 1.20$ from a span of 53° and 17° at $\mathrm{\mathbb{T}} = 1.00$. The magnitude of ${C_T}$ (in forward stroke) also decreases for $\mathrm{\mathbb{T}} > 1.00$ as compared to $\mathrm{\mathbb{T}} = 1.00$ (${C_T} < 0$, decreased drag). Ideal regime (II) contracts from a span of 38° at $\mathrm{\mathbb{T}} = 1.00$ to 17° at $\mathrm{\mathbb{T}} = 1.20$, with ${C_T}$ also decreasing (figure 9b,c). In the faster retract stroke $(\mathrm{\mathbb{T}} > 1.00)$, the span of normal swimming regime (I) diminishes from 72° at $\mathrm{\mathbb{T}} = 1.00$ to 66° at $\mathrm{\mathbb{T}} = 1.20$, but the maximum ${C_T}$ is enhanced twofold. As such, undesirable regime (IV) expands from $\mathrm{\mathbb{T}} = 1.00$ to 1.20, laying mostly in the forward stroke (figure 9b,c).

The pattern modifies significantly when the forward stroke is further slowed down to the case of $\mathrm{\mathbb{T}} = 1.40$, where the floating regime (III) nestling around the equilibrium position now spans most of the forward stroke i.e. from −30° to 50° (figure 9d). The undesirable regime (IV) shrinks and shifts further towards the end of the forward stroke, spanning 50°–77°. One more floating regime (III) and ideal regime (II) are observed at the end of the forward stroke, both being very narrow. The normal swimming regime (I) further shrinks from 66° for $\mathrm{\mathbb{T}} = 1.20$ to 59° for $\mathrm{\mathbb{T}} = 1.40$, followed by an infinitesimal undesirable (IV) regime spanning only 2°. The floating regime (III) covers the rest of the retract stroke and expands much further into the next forward stroke. The magnitude of ${C_T}$ is much larger for $\mathrm{\mathbb{T}} = 1.40$ than for $0.80 \le \mathrm{\mathbb{T}} \le 1.20$. Hence, figure 9(d) uses a scale four times the scale for figure 9(a–c).

The following points can be distilled from the above discussion. The undesirable and ideal modes are fashionably opposite to each other; the former provides power but obtains no net thrust while the latter receives both power and thrust. The forward stroke (ϕ = 0° to 90°) for $\mathrm{\mathbb{T}} = 1.00$ generally envelops undesirable and ideal regimes separated by a narrow floating regime, while the retract stroke largely provides normal swimming (figure 9b). When $\mathrm{\mathbb{T}}$ is increased from 0.80 to 1.20, the spans of the undesirable and floating regime (both involving ${C_T} < 0$) in the forward stroke dramatically grow and that of the ideal regime (involving ${C_T} > 0$) contracts. That is, the corresponding change in the forward stroke essentially degrades thrust. However, with $\mathrm{\mathbb{T}}$ increasing from 0.80 to 1.20, the spans of regimes I $({C_T} > 0)$ and II $({C_T} < 0)$ in the retract stroke do not change much (figure 9), but the magnitude of thrust is radically enhanced in regime I (see figure 5 or figure 9), which is the main reason for enhancement of the time-averaged thrust at $\mathrm{\mathbb{T}} = 1.20$. The two scenarios (decreasing and increasing thrust in the forward and retract strokes with increasing $\mathrm{\mathbb{T}}$) further point to the occurrence of the minimum ${\bar{C}_T}$ between $\mathrm{\mathbb{T}} = 0.80$ and 1.20. For $\mathrm{\mathbb{T}} = 1.40$, ${C_T} < 0$ in regimes III and IV of the forward stroke further degrades when compared with that for $\mathrm{\mathbb{T}} = 1.20$. The retract stroke for $\mathrm{\mathbb{T}} = 1.40$, unlike the same for $\mathrm{\mathbb{T}} = 1.20$, has a floating regime $({C_T} < 0)$. The ${C_T}$ in the normal swimming regime is very large, which makes the ${\bar{C}_T}$ large for $\mathrm{\mathbb{T}} = 1.40$.

Figure 10 shows the individual contributions ${\bar{C}_{TF}}$ and ${\bar{C}_{TR}}$ of the forward and retract strokes, respectively, to ${\bar{C}_T}$ for St_A = 0.38, where ${\bar{C}_{TF}}$ and ${\bar{C}_{TR}}$ are the thrust force coefficients integrated over the forward and retract strokes of a complete oscillation cycle, respectively, i.e.

(3.1a,b)\begin{align}{\bar{C}_{TF}} = \frac{2}{{\rm \pi} }\left[ {\int_0^{{\rm \pi} /2} {{C_T}\,\textrm{d}\phi } + \int_{\rm \pi} ^{3{\rm \pi} /2} {{C_T}\,\textrm{d}\phi } } \right]\quad \textrm{and}\quad {\bar{C}_{TR}} = \frac{2}{{\rm \pi} }\left[ {\int_{{\rm \pi} /2}^{\rm \pi} {{C_T}\,\textrm{d}\phi } + \int_{3{\rm \pi} /2}^{2{\rm \pi} } {{C_T}\,\textrm{d}\phi } } \right].\end{align}

As discussed above, the ${\bar{C}_{TF}}$ and ${\bar{C}_{TR}}$ contributions to ${\bar{C}_T}$ decrease and increase, respectively, when $\mathrm{\mathbb{T}}$ is increased. The magnitude of increase with $\mathrm{\mathbb{T}}$ is however greater than that of the decrease, which explains why ${\bar{C}_T}$ grows with $\mathrm{\mathbb{T}}$. The figure also indicates that ${\bar{C}_T}$ would be a minimum at $\mathrm{\mathbb{T}} = 0.83$.

Figure 10. Strokewise averaged thrust dependence on $\mathrm{\mathbb{T}}$ for St_A = 0.38.

There are four general locomotion behaviours for swimming fish observed in nature: steady swimming, kick and glide (intermittent swimming), fast start (including C-start and S-start) and braking (e.g. Jayne, Lozada & Lauder Reference Jayne, Lozada and Lauder1996; Shadwick & Lauder Reference Shadwick and Lauder2006). The overview of figure 9 echoes some interesting observations in natural biological swimming. First, for a fast start that is typically associated with the escape response of a fish, high power is required to hastily bend the fish body to its limit, and the thrust should be generated in-phase with the body movement (power). This is the case for $\mathrm{\mathbb{T}} > 1.00$ (figure 9c,d), where ${C_T}$ and ${C_P}$ are almost in-phase and thrust is generated near the extreme position of the foil (figure 5f–j). In addition, the fast-start response depending on the fish body curvature and the escape route is characterized as C-start (fish escapes from its wake) or S-start (fish escapes side-ways). This is achieved by high lateral forces or moments achieved when the foil tail is near its extreme position ($\mathrm{\mathbb{T}} > 1.00$, see figure 7b showing high ${C_L}$ near the extreme position). Second, for steady swimming, the steady motion of the foil tail guarantees the impulse of thrust that sustains steady swimming, just as in the case of $\mathrm{\mathbb{T}} = 1.00$ (figure 9b). Third, for braking, the drag should be achieved near the equilibrium position which is the case for $\mathrm{\mathbb{T}} = 0.80$, where ${C_T} < 0$, i.e. undesirable regime concentrated near the equilibrium position (figure 9a). Also, the fact that for $\mathrm{\mathbb{T}} = 0.80$, the maximum drag is generated when the most power is applied (i.e. ${C_T}$ and ${C_P}$ are out of phase, see figure 5a) further shows the similarity of $\mathrm{\mathbb{T}} < 1.00$ motion to the braking phenomenon.

3.2. Effects of θ′, $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ on instantaneous thrust, power and lift

Figure 11 shows the contour plots of instantaneous thrust, power and lift on the $\mathrm{\mathbb{T}}- t^{\prime}$ plane for St_A = 0.38, overlaid with solid, dashed and dash-dotted lines representing extremes of θ′, $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ ($= \ddot{\theta }/{\ddot{\theta }_{max}}$, where ${\ddot{\theta }_{max}}$ is the maximum angular acceleration, rads⁻²), respectively, with maximum and minimum values represented by green and black lines, respectively. Here, the subscripts ‘max’ and ‘min’ denote maximum and minimum, respectively. The dotted line marks the half time-period (end of the retract stroke), while the extremes of θ′ mark the ends of the forward strokes. The contours are coloured such that shades of red and blue represent positive and negative values of the contour variable. When $\mathrm{\mathbb{T}} < 1.00$, the extremes (positive and negative) of $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ shift from the stroke boundaries (θ′ = 0 and $\theta ^{\prime} = {\theta ^{\prime}_{max}}$) to the fast forward stroke (figure 11). In contrast, when $\mathrm{\mathbb{T}} > 1.00$, the extremes of $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ fall in the fast retract stroke.

Figure 11. Effect of instantaneous $\theta ^{\prime},\dot{\theta }^{\prime},\;\; \textrm{and},\;\ddot{\theta }^{\prime}$ on (a) C_T, (b) C_Pf, (c) C_Pi and (d) C_L for St_A = 0.38. The straight, dashed and dash–dotted lines represent extremes of $\theta ^{\prime}$, $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$, with green and black colours of lines for maximum and minimum values. FS and RS stands for Forward and Retract Stroke, respectively.

The ${C_T}$ for $\mathrm{\mathbb{T}} = 1.00$ is minimum in the forward stroke at t′ ≈ 0.062 (ϕ ≈ ${\rm \pi}$/8) and maximum in the retract stroke t′ ≈ 0.31 (ϕ ≈ 5${\rm \pi}$/8) (figure 5e), see also Alam & Muhammad (Reference Alam and Muhammad2020). With $\mathrm{\mathbb{T}}$ decreasing from 1.00, ${C_T}$ remains minimum at t′ ≈ 0.062 while the maximum ${C_T}$ gradually shifts from the retract stroke to the preceding forward stroke, as does ${\ddot{\theta }^{\prime}_{min}}$. The reason for the occurrence of the maximum ${C_T}$ following ${\ddot{\theta }^{\prime}_{min}}$ is the fluid inertia lagging the foil inertia, which generates a large pressure difference between the two surfaces of the foil (Alam & Muhammad Reference Alam and Muhammad2020). The relationship between fluid inertia, foil motion and fluid forces is detailed by Alam & Muhammad (Reference Alam and Muhammad2020). When $\mathrm{\mathbb{T}}$ is increased from 1.00, both ${\ddot{\theta }^{\prime}_{min}}$ and ${\dot{\theta }^{\prime}_{min}}$ come into being in the retract stroke, and the maximum ${C_T}$ lagging ${\ddot{\theta }^{\prime}_{min}}$ approaches ${\dot{\theta }^{\prime}_{min}}$ at $\mathrm{\mathbb{T}} = 1.40$ (figure 11a). Particularly for $1.30 \le \mathrm{\mathbb{T}} \le 1.40$, the $\dot{\theta }^{\prime}$ in the retract stroke is much higher (see the slope for $\mathrm{\mathbb{T}} = 1.40$ in figure 3a), which makes it difficult for the fluid inertia to keep up with the foil, resulting in a more time delay (t′) between ${\ddot{\theta }^{\prime}_{min}}$ and maximum ${C_T}$, compared with that for $1.00 < \mathrm{\mathbb{T}} < 1.30$.

The ${C_{Pf}}$ at the start of the forward stroke is positive and negative for $0.80 \le \mathrm{\mathbb{T}} \le 1.30$ and $1.30 < \mathrm{\mathbb{T}} \le 1.40$, respectively (see also ϕ = 0° in figures 5j and 8a), linked to the motion state at the end of the earlier retract stroke, which will be discussed later. The ${C_{Pf}}$ for $\mathrm{\mathbb{T}} > 1.30$ is positive in the middle of the slow forward stroke, which results from the slow-motion foil pushing the fluid. The ${C_{Pf}}$ switches to negative before the end of the forward stroke (i.e. at $t^{\prime} \approx t^{\prime}({\theta ^{\prime}_{max}}) - 0.125$) because of the decelerating foil resisting the fluid inertia. The ${C_{Pf}}$ becomes zero at the boundary between the forward and retract strokes (see ${\dot{\theta }^{\prime}_{max}}$ line in figure 11b) as $\dot{\theta }^{\prime} = 0$ (see (2.9)). The region of the retract stroke largely consists of positive ${C_{Pf}}$, except for $1.30 < \mathrm{\mathbb{T}} \le 1.40$ in which the ${C_{Pf}}$ is negative beyond ${\dot{\theta }^{\prime}_{max}}$ (figure 11b). The change in the behaviour of ${C_{Pf}}$ with t′ in the retract stroke for $\mathrm{\mathbb{T}} > 1.30$ is attributed to the change in the foil acceleration from negative to positive at ${\dot{\theta }^{\prime}_{min}}$ at which the fluid (negatively accelerated before ${\dot{\theta }^{\prime}_{min}}$) bumps into the foil that moves at a slower pace than the fluid.

Being the product of $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ (2.9), ${C_{Pi}}$ is zero when one of them is zero, i.e. at $\dot{\theta }^{\prime} = 0$ (${\theta ^{\prime}_{max}}$ and ${\theta ^{\prime}_{min}}$ lines) and at $\ddot{\theta }^{\prime} = 0$ (${\dot{\theta }^{\prime}_{max}}$ and ${\dot{\theta }^{\prime}_{min}}$ lines). The ${C_{Pi}}$ for $\mathrm{\mathbb{T}} > 1.00$ is negative in the forward stroke because of the foil deceleration. However, for $\mathrm{\mathbb{T}} < 1.00$, the ${C_{Pi}}$ is initially positive before ${\dot{\theta }^{\prime}_{max}}$ and becomes negative after ${\dot{\theta }^{\prime}_{max}}$where $\dot{\theta }^{\prime}$ diminishes. The retract stroke is engulfed with positive ${C_{Pi}}$ as $\dot{\theta }^{\prime}$ increases in magnitude until ${\dot{\theta }^{\prime}_{min}}$ (figure 11c). For $\mathrm{\mathbb{T}} = 1.00$, the ${C_{Pi}}$ is negative and positive in the forward and retract strokes, respectively, dictated by negative and positive $\ddot{\theta }^{\prime}$ at the corresponding strokes. In constrast, for $\mathrm{\mathbb{T}} \ne 1.00$, the sign of ${C_{Pi}}$ is contingent on $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ in the individual strokes, as explained above.

Because θ_max ≪ 1, the ${C_L}$ is directly linked to the difference in pressure between the upper and lower surfaces of the foil. Because the foil undergoes forced motion, the ${C_L}$ will typically acts opposite to the motion with some phase lag because of the fluid inertia (Alam & Muhammad Reference Alam and Muhammad2020). The ${C_L}$ is essentially positive (see figure 2b for sign convention) for most of the downward stroke including retract and forward strokes between ${\theta ^{\prime}_{max}}$ and ${\theta ^{\prime}_{min}}$, with ${C_L}$ lagging θ′ (see figure 11d). The ${C_L}$ largely follows the trend of ${C_{Pf}}$ variation with t′ in the retract stroke for $\mathrm{\mathbb{T}} > 1.30$, with ${C_L}$ becoming momentarily negative after ${\dot{\theta }^{\prime}_{min}}$. As expected, the amplitude of ${C_L}$ increases as the magnitude of $\dot{\theta }^{\prime}$ enhances, i.e. with the retract stroke getting shorter. The $\dot{\theta }^{\prime}$, $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ all affect ${C_T}$, ${C_{Pf}}$, ${C_{Pi}}$ and ${C_L}$. The ${C_{Pi}}$ is influenced by $\dot{\theta }^{\prime}$ and $\ddot{\theta }^{\prime}$ only, following (2.9), whereas ${C_T}$, ${C_L}$ and ${C_{Pf}}$ all have a strong dependence on $\mathrm{\mathbb{T}}$.

3.3. Effect of $\mathrm{\mathbb{T}}$ on time-averaged thrust, power coefficient and efficiency

Very complex scenarios have been observed in the instantaneous ${C_T}$ and ${C_P}$ when $\mathrm{\mathbb{T}}$ is increased or decreased from 1.00. It would be interesting to see the dependence of time-averaged hydrodynamic parameters on $\mathrm{\mathbb{T}}$. Figure 12 shows dependence of ${\bar{C}_T}$, ${\bar{C}_P}$ and η on $\mathrm{\mathbb{T}}$ for the three selected cases with St_A = 0.23, 0.36 and 0.43 lying within the St_A range of flying and swimming animals reported in the literature (see e.g. Triantafyllou, Triantafyllou & Grosenbaugh Reference Triantafyllou, Triantafyllou and Grosenbaugh1993; Taylor et al. Reference Taylor, Nudds and Thomas2003). At a given $\mathrm{\mathbb{T}}$, ${\bar{C}_T}$ escalates with increasing St_A. With $\mathrm{\mathbb{T}}$ increasing from 0.80 to 1.40, ${\bar{C}_T}$ first declines and then grows, being minimum at $\mathrm{\mathbb{T}} = 0.85$ for all St_A values (figure 12a). This however is not in conflict with the earlier finding (from figure 10), because the simulated $\mathrm{\mathbb{T}}$ value closest to 0.83 is 0.85 and hence we see minimum ${\bar{C}_T}$ occurring at $\mathrm{\mathbb{T}} = 0.85$. The border of the minimum ${\bar{C}_T}$ coincides with that separating 2P and 2S wakes, which will be discussed in the later sections. For sinusoidal motion, ${C_T}$ is large and small when the foil is close to the extreme and equilibrium, respectively ($\mathrm{\mathbb{T}} = 1.00$, figure 6a). As the $\mathrm{\mathbb{T}}$ is increased (say from 0.83 to 1.40), the increased contribution of the retract stroke to ${\bar{C}_T}$ is slightly higher than the decreased contribution of the forward stroke to ${\bar{C}_T}$ (figure 10). Hence the rate of increase in ${\bar{C}_T}$ with $\mathrm{\mathbb{T}}$ grows with increasing $\mathrm{\mathbb{T}}$ (figure 12a). The wider 2P wake forming for $\mathrm{\mathbb{T}} \le 0.85$ (shown later) results in ${\bar{C}_T}$ growing for $\mathrm{\mathbb{T}} \le 0.85$ (figure 12a). This can also be explained in terms of the shift of the ${C_T}$ peak toward the extreme position (figure 5a). When appearing near the extreme, the peak becomes broad because of the small velocity magnitude of the foil. When $\mathrm{\mathbb{T}}$ is increased from 0.85 to 0.95, the ${C_T}$ peak declines in height but shifts toward the extreme (ϕ = 90°) (figure 5). The effects of two scenarios cancel each other; the ensuing effect on ${\bar{C}_T}$ is thus very small between $\mathrm{\mathbb{T}} = 0.85$ and 0.95 (figure 12a). However, for $\mathrm{\mathbb{T}} = 0.95- 1.40$, although the ${C_T}$ peak moves away from the extreme, its height grows rapidly, such that the value of ${C_T}$ at ϕ = 90° does not change appreciably (figure 5). The ${\bar{C}_T}$ increase with the increase in $\mathrm{\mathbb{T}}$ is thus rapid for $\mathrm{\mathbb{T}} > 0.95$ (figure 12a).

Figure 12. Dependence on $\mathrm{\mathbb{T}}$ of (a) thrust coefficient, (b) input power coefficient and (c) efficiency. To avoid the clutter, only three different St_A values are plotted.

The ${\bar{C}_P}$ follows the same trend except that it becomes a minimum at $\mathrm{\mathbb{T}} = 1.00$. Recalling figure 5, one can see that when $\mathrm{\mathbb{T}}$ is increased from 0.80 to 0.95, the ${C_P}$ peak occurring in the first quadrant moves away from the extreme, with the peak height diminishing. Both cause ${\bar{C}_P}$ to decline with $\mathrm{\mathbb{T}}$ (figure 12b). The scenario is opposite for $\mathrm{\mathbb{T}} > 1.00$, the peak budging toward the extreme with the peak height growing (figure 5). The ${\bar{C}_P}$ therefore grows, with a minimum ${\bar{C}_P}$ prevailing at $\mathrm{\mathbb{T}} = 1.00$ (figure 12). The η for St_A = 0.36 increases monotonically with $\mathrm{\mathbb{T}}$ while that for St_A = 0.43 does the same for $0.80 \le \mathrm{\mathbb{T}} \le 1.20$ but declines for $\mathrm{\mathbb{T}} > 1.2$ (figure 12c). The η decreases to a minimum at $\mathrm{\mathbb{T}} = 0.90$ for the drag cases (e.g. St_A = 0.23 in figure 12c) then increases rapidly with increasing $\mathrm{\mathbb{T}}$. Interestingly, the drag–thrust boundary can be crossed at a smaller St_A by changing the foil kinematics $\mathrm{\mathbb{T}}$ (figure 12a). Because we are not interested in drag-creating conditions, no further examination will be made for this.

Figures 13(a)–13(c) presents the full spectrum of ${\bar{C}_T}$, ${\bar{C}_P}$ and η on the $\mathrm{\mathbb{T}}- S{t_A}$ plane. The solid line represents the drag-thrust boundary. The dependence of ${\bar{C}_T}$, ${\bar{C}_P}$ and η on St_A at $\mathrm{\mathbb{T}} = 1.00$ is well known and also can be seen here, all increasing with St_A. The increase is, however, sensitive to $\mathrm{\mathbb{T}}$. A greater increase in ${\bar{C}_T}$ and ${\bar{C}_P}$ with St_A is observed when $\mathrm{\mathbb{T}}$ is increased from 1.00, shifting the drag–thrust boundary to a smaller St_A. The same happens when $\mathrm{\mathbb{T}}$ is decreased from 1.00 but the increase in ${\bar{C}_T}$ is smaller than that for $\mathrm{\mathbb{T}} = 1.00$. As such, a significant enhancement in η is observed with increasing $\mathrm{\mathbb{T}}$ from 1.00 while the opposite is the case when $\mathrm{\mathbb{T}}$ is decreased from 1.00. For St_A > 0.40, an increase in $\mathrm{\mathbb{T}} > 1.20$ leads to η declining. The benefit of the modified motion starts to decrease as the St_A is increased above ~0.45. The maximum efficiency is achieved at St_A = 0.63 for sinusoidal motion $\mathrm{\mathbb{T}} = 1.00$. The minimum ${\bar{C}_T}$ occurs at $\mathrm{\mathbb{T}} = 0.85$, coinciding with the 2P–2S wake boundary (figure 13a). The minimum ${\bar{C}_P}$ occurs at $\mathrm{\mathbb{T}} = 1.00$ for all St_A (figure 13b). When $\mathrm{\mathbb{T}}$ is decreased below 1.00, the magnitude of maximum and minimum value of ${C_P}$ increase (figure 5a–d). In contrast, for $\mathrm{\mathbb{T}} > 1.00$, the increase in $\mathrm{\mathbb{T}}$ leads to an increase and decrease in the maximum and minimum magnitudes of ${C_P}$ values, respectively (figure 5f–j); however, the increase in ${C_P}$ maximum values is larger than the decrease in the ${C_P}$ minimum values. This also leads to an increase in the overall ${\bar{C}_P}$ with an increase in $\mathrm{\mathbb{T}}$, but at a rate slightly higher than when $\mathrm{\mathbb{T}}$ in decreased below 1.00 (figure 12b). The η increases with $\mathrm{\mathbb{T}}$ and St_A for 0.3 ≤ St_A < 0.4 (Figure 13c). When St_A is further increased, the maximum η slowly shifts from $\mathrm{\mathbb{T}} = 1.40$, St_A = 0.4 to $\mathrm{\mathbb{T}} = 1.00$, St_A = 0.53. The decrease in η for $\mathrm{\mathbb{T}} > 1.00$ at high St_A arises from the large $\dot{\theta }$ in the retract stroke (figure 3), with ${\bar{C}_P}$ increasing more rapidly than ${\bar{C}_T}$.

Figure 13. Contours of average (a) thrust coefficient, (b) input power coefficient and (c) efficiency. The black solid line is the drag–thrust boundary. The dashed lines represents the minimum values of power or thrust against St_A. (d) Detailed wake map on the $S{t_A}- \mathrm{\mathbb{T}}$ plane.

3.4. Effect of foil inertia on power and efficiency

The non-elastic systems are affected by the foil inertia since ${C_P} < 0$ cannot be used. The foil inertia thus plays an important role in defining the efficiency. It should be noted that we use inertia and moment of inertia interchangeably in this section. We investigate only a single geometry with a uniform density, thus the increase in mass (i.e. density) is reflected in the moment of inertia of the foil. We consider three different moments of inertia, denoted by materials I, II and III. The foil of material I has a very small moment of inertia $(J_{zz}^I = 1 \times {10^{ - 5}}\;\textrm{kg - }{\textrm{m}^2})$, and the resulting inertial component of the power is negligible as compared to the fluid component of the power i.e. $\bar{C}_{Pi}^ +{\ll} \bar{C}_{Pf}^ + $. The foil of material II has a large moment of inertia $(J_{zz}^{II} = 10J_{zz}^I)$, making the inertial and fluid components of the power to be almost equal to each other i.e. $\bar{C}_{Pi}^ +{\approx} \bar{C}_{Pf}^ + $. The foil of material III has a very large moment of inertia $(J_{zz}^{III} = 10J_{zz}^{II})$, such that the fluid component of the power becomes negligible as compared to the inertial component of the power i.e. $\bar{C}_{Pi}^ +{\gg} \bar{C}_{Pf}^ + $. Figure 14 shows $C_P^ + $ (black line), $C_{Pf}^ + $ (blue line) and $C_{Pi}^ + $ (red line) over a complete pitching cycle in the case of St_A = 0.38 for materials I (first column), II (second column) and III (third column), and for $\mathrm{\mathbb{T}} = 0.80$ (first row), 1.00 (second row) and 1.20 (third row). The $C_{Pi}^ + $ and $C_{Pf}^ + $ both are zero at ϕ = 90° and 270° (for all materials and $\mathrm{\mathbb{T}}$), i.e. at extreme positions of the foil due to $\dot{\theta }^{\prime} = 0$. When foil moves away from the extreme positions, $C_{Pi}^ + $ and $C_{Pf}^ + $ both increase from zero (see e.g. $t^{\prime}$ = 0.20, 0.25 and 0.30 for $\mathrm{\mathbb{T}} = 0.80$, 1.00 and 1.20, respectively, in figure 14 for all materials), having positive value owing to $\dot{\theta }$, $\ddot{\theta }$ and −M_z having the same direction (see (2.9) and (2.13)). For all materials, the first peak of $C_{Pi}^ + $ occurs at $t^{\prime}$ = 0.02, 0.375 and 0.39 for $\mathrm{\mathbb{T}} = 0.80$, 1.00 and 1.20, respectively, while that of $C_{Pf}^ + $ occurs at $t^{\prime}$ = 0.035, 0.44 and 0.40 for $\mathrm{\mathbb{T}} = 0.80$, 1.00 and 1.20, respectively. That is, $C_{Pi}^ + $ leads $C_{Pf}^ + $ (Alam & Muhammad Reference Alam and Muhammad2020). The first peak of $C_P^ + $ for materials I, II and III respectively occurs at $t^{\prime}$≈ 0.035, 0.025 and 0.02 for $\mathrm{\mathbb{T}} = 0.80$, at $t^{\prime}$≈ 0.435, 0.405 and 0.38 for $\mathrm{\mathbb{T}} = 1.00$ and at $t^{\prime}$≈ 0.40, 0.395 and 0.39 for $\mathrm{\mathbb{T}} = 1.20$ (figure 14), which indicates that $C_{Pi}^ + $ leads $C_P^ + $, and $C_P^ + $ leads $C_{Pf}^ + $. It should be appreciated that $C_{Pi}^ + $ could be negative whereas $C_P^ + $ by definition is always positive or zero, thus a positive $C_{Pf}^ + $ must accompany a negative $C_{Pi}^ + $ such that $C_P^ + \ge 0$. A similar trend can be observed for $\mathrm{\mathbb{T}} = 1.00$ and 1.20 in figures 14(d–f) and 14(g–i), respectively. The peak-to-peak phase difference between $C_{Pi}^ + $ and $C_P^ + $ decreases with the increase in the moment of inertia.

Figure 14. $C_P^ + $ (black line), $C_{Pf}^ + $ (blue line) and $C_{Pi}^ + $ (red line) over a complete pitching cycle in the case of St_A = 0.38 for materials I (first column), II (second column) and III (third column), and for $\mathrm{\mathbb{T}} = 0.80$ (first row), 1.00 (second row) and 1.20 (third row).

Figure 15 shows variations of $\bar{C}_{Pf}^ + $, $\bar{C}_{Pi}^ + $, $\bar{C}_P^ + $ and η ⁺ in $\mathrm{\mathbb{T}}- S{t_A}$ plane for materials I, II and III in the left, middle and right columns, respectively. The variation of $\bar{C}_{Pf}^ + $ is similar to that of ${\bar{C}_P}$. There are however two distinctions. First, $\bar{C}_{Pf}^ + $ has a slightly larger magnitude than ${\bar{C}_P}$. Second, the minimum $\bar{C}_{Pf}^ + $ for a given St_A occurs at $\mathrm{\mathbb{T}} = 1.05$, 1.15, and 1.10 for materials I, II and III, respectively, while ${\bar{C}_P}$ is minimum at $\mathrm{\mathbb{T}} = 1.00$ (figure 13b). The increase in the moment of inertia decreases $\bar{C}_{Pf}^ + $, as can be seen by comparing the colours at any particular St_A and $\mathrm{\mathbb{T}}$ in figure 15a–c). The $\bar{C}_{Pi}^ + $ undoubtedly increases with increasing moment of inertia (see the colourmap ranges for figure 15d–f). The $\bar{C}_{Pi}^ + $ first decreases with the increase in $\mathrm{\mathbb{T}}$ from 0.80, reaches a minimum at $\mathrm{\mathbb{T}} = 1.20$ for material I, and at $\mathrm{\mathbb{T}} = 1.00$ for materials II and III, and then increases rapidly with further increase in $\mathrm{\mathbb{T}}$. This implies that when $\bar{C}_{Pi}^ + \mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel> \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }\bar{C}_{Pf}^ + $, $\bar{C}_{Pi}^ + $ for the non-elastic system for a particular St_A will be minimum at $\mathrm{\mathbb{T}} = 1.00$ (see figure 15e,f). The $\bar{C}_P^ + $ for materials I and III follows the trends of $\bar{C}_{Pf}^ + $ and $\bar{C}_{Pi}^ + $, respectively, as $\bar{C}_{Pi}^ +{\ll} \bar{C}_{Pf}^ + $ (making $\bar{C}_P^ +{\approx} \bar{C}_{Pf}^ + $) for material I and $\bar{C}_{Pi}^ +{\gg} \bar{C}_{Pf}^ + $ (making $\bar{C}_P^ +{\approx} \bar{C}_{Pi}^ + $) for material III. For material II, because $\bar{C}_{Pi}^ +{\approx} \bar{C}_{Pf}^ + $, $\bar{C}_P^ + $ follows neither $\bar{C}_{Pi}^ + $ nor $\bar{C}_{Pf}^ + $. The $\bar{C}_P^ + $ decreases with $\mathrm{\mathbb{T}}$ increasing from 0.80, becoming minimum at $\mathrm{\mathbb{T}} = 1.05$ for materials I and II and at $\mathrm{\mathbb{T}} = 1.00$ for material III, and then increases with a further increase in $\mathrm{\mathbb{T}}$. The η ⁺ for 0.30 < St_A ≤ 0.45 increases with $\mathrm{\mathbb{T}}$ for $0.80 \le \mathrm{\mathbb{T}} \le 1.20$, reaching a maximum at $\mathrm{\mathbb{T}} = 1.20$, and then decreases with a further increase in $\mathrm{\mathbb{T}}$ for $1.20 < \mathrm{\mathbb{T}} \le 1.40$. The η ⁺ for St_A > 0.45, however, reaches its maximum at $\mathrm{\mathbb{T}} = 1.00$. The variations of η⁺ in the $\mathrm{\mathbb{T}}- S{t_A}$ plane for materials II and III are similar to that for material I, except that the peak η ⁺ along $\mathrm{\mathbb{T}}$ axis for 0.30 < St_A < 0.45 occurs at $\mathrm{\mathbb{T}} = 1.10$ for material II and at $\mathrm{\mathbb{T}} = 1.05$ for material III. For non-elastic storage systems, it can thus be concluded that the moment of inertia should be as low as possible, such that the inertial component of the power becomes negligible in comparison to the fluid power. Moreover, η⁺ can be enhanced with $1.05 < \mathrm{\mathbb{T}} \le 1.20$ for moderate St_A (i.e. 0.30 < St_A ≤ 0.45), while $\mathrm{\mathbb{T}} = 1.00$ is beneficial in terms of η ⁺ for St_A > 0.45.

Figure 15. Variations of $\bar{C}_{Pf}^ + $, $\bar{C}_{Pi}^ + $, $\bar{C}_P^ + $ and ${\eta ^ + }$ in the $\mathrm{\mathbb{T}}- S{t_A}$ plane for materials I (left column), II (middle column) and III (right column). The colour code bars for $\bar{C}_{Pi}^ + $ and $\bar{C}_P^ + $ each have three scales for the three materials; left most, material I; middle, material II; and right most, material III. The black line in ${\eta ^ + }$ maps corresponds to the drag–thrust transition.

The inertia does not affect ${\bar{C}_P}$ but plays an important role in determining $\bar{C}_P^ + $. This is because ${\bar{C}_{Pi}}$ is always zero but $\bar{C}_{Pi}^ + $ is positive, its magnitude increasing with increasing the inertia of the foil (figure 15d–f). The effect of inertia is then reflected in efficiency with η ⁺ always smaller than η for the respective case. Because ${\bar{C}_T}$ is not affected by the inertia, the η ⁺ decreases with the increase in inertia as $\bar{C}_P^ + > {\bar{C}_P}$ i.e. $\bar{C}_{Pi}^ + > 0$.

3.5. Effect of $\mathrm{\mathbb{T}}$ on the wake structure

Intuitively, an increase in pitching amplitude would result in a broader wake as the foil tip excursion is wide, whereas an increase in pitching frequency, owing to centrifugal force (Alam & Muhammad Reference Alam and Muhammad2020), would increase the streamwise velocity in the wake. The focal parameter involved in this study is the kinematics (i.e. $\mathrm{\mathbb{T}}$). Then, a question naturally arises, what is the effect of $\mathrm{\mathbb{T}}$ on the wake? The answer to this question will be given here. Figure 13(d) shows the wake map for the $S{t_A}- \mathrm{\mathbb{T}}$ plane. There are only two basic wake patterns, namely 2S for $0.85 < \mathrm{\mathbb{T}} \le 1.40$ and 2P for $0.80 \le \mathrm{\mathbb{T}} \le 0.85$, including their deflected patterns in small regions ($\mathrm{\mathbb{T}} < 1.00$ for St_A ≥ 0.43 and for $\mathrm{\mathbb{T}} = 0.80$ only when St_A = 0.42. The wake deflection will be discussed in a separate section later. We will focus on 2P and 2S patterns only. Figure 16 illustrates the novel 2P wake pattern for $\mathrm{\mathbb{T}} = 0.80$ and St_A = 0.38. The spanwise non-dimensional vorticity $\varOmega _z^\ast ( = {\varOmega _z}d/{U_\infty })$ snapshots are shown for ϕ = 90°, 120°, …, and 0° in figure 16(a–h), overlapped with instantaneous relative streamwise velocity $u_r^\ast $ ($= (u - {U_\infty })/{U_\infty }$, dashed line) at a streamwise distance $x_{TE}^\ast ( = {x_{TE}}/d) = 1.0$, where ${x_{TE}}$ is measured from the foil tip at the equilibrium position. The $u_r^\ast $ for the $\mathrm{\mathbb{T}} = 1.00$ (solid line) is also shown as the baseline. For $\mathrm{\mathbb{T}} = 0.80$, during the slower retract stroke (ϕ = 90°–180°), a positive (counterclockwise) vortex ${\rm A}^{\prime}$ impinges along the lower surface, and another positive vortex A grows from the foil's tip (figure 16a,b). The formation of vortex A stems from the foil tip inertia (or can be said to arise from the flow separation at the tip) while that of vortex ${\rm A}^{\prime}$ originates from the flow separation at the leading edge. However, a negative vortex ${\rm B}^{\prime}$ originating from the leading edge impinges on the upper surface. During this slower retract stroke (ϕ = 90°–180°, figure 16a–d), the tip vortex A grows and sheds above the wake centreline (y* = 0) while the vortices ${\rm A}^{\prime}$ and ${\rm B}^{\prime}$ roll down over the lower and upper surfaces of the foil, respectively. The $u_r^\ast $ strengthens around the wake centreline when the foil moves from ϕ = 90° to 180° while $u_r^\ast $ around vortex A is negative (figure 16c,d).

Figure 16. Contours of $\varOmega _z^\ast $ showing generation and evolution of 2P wake (St_A = 0.38, $\mathrm{\mathbb{T}} = 0.80$) at ϕ = 90°, 120°, …, 270° in (a–g) and ϕ = 0° in (h). The $\varOmega _z^\ast{\in} [ - 5,5]$, with positive $\varOmega _z^\ast $ shaded in red and negative $\varOmega _z^\ast $ shaded in blue. The corresponding $u_r^\ast = (u - {U_\infty })/{U_\infty }$ profiles (dashed line) are given at $x_{TE}^\ast = ((x - {x_{TE}})/d) = 1$. The solid line represents $u_r^\ast $ for $\mathrm{\mathbb{T}} = 1.00$. The horizontal and vertical lines represent y* = 0 and $u_r^\ast \; = 0$, respectively.

The forward stroke (ϕ = 180°–270°, figure 16d–g) is very fast (lowest $\mathrm{\mathbb{T}} = 0.80$ examined), hence generating a very high acceleration. The foil tail thus leaves the wake centreline (equilibrium position) very fast, leaving vortex A above the wake centreline. This phenomenon is akin to the magic trick that when a sheet of paper on a table, with a coin on the top of the sheet, is pulled horizontally with a quick snap, the coin remains on the table, rather than moving with the paper. In the meanwhile, vortex ${\rm A}^{\prime}$ arriving at the foil tip reforms itself as the forward stroke is faster (ϕ = 180°–270°, figure 16d–g). Vortex ${\rm A}^{\prime}$ lies abreast of vortex A, closer to the foil tip, merging with the braid shear-layer. Vortices A and ${\rm A}^{\prime}$ bundle together and form a paired vortex (figure 16g). Vortex ${\rm A}^{\prime}$, however, has a higher streamwise velocity than vortex A as ${\rm A}^{\prime}$ forms in the faster stroke (higher centrifugal force) and lies close to the wake centreline. The velocities of A and ${\rm A}^{\prime}$ could be understood from the corresponding $u_r^\ast $ values around them. The return of the foil (ϕ = 270°–0°, figure 16(g,h) further accelerates vortex ${\rm A}^{\prime}$. The merging line connecting A–${\rm A}^{\prime}$ centres thus rotates anticlockwise, see the $u_r^\ast $ profile and snapshots at ϕ = 270° and 0°. The same rotation signs of the individual vortices and of merging line facilitate the merging of the two vortices into one vortex in the next two strokes, which can also be seen from the evolution of the negative (blue coloured) paired vortices from ϕ = 180° to 0° (figure 16d–h). Similarly, vortex B forms as a tip inertia vortex from ϕ = 270° to 0°. Overall, the paired vortices during their downstream evolution evolve into a single vortex in about two vortex-shedding cycles. The wake thus features 2P vortices up to 5d downstream from the foil tip and 2S vortices afterward.

There are several noticeable differences between the $u_r^\ast $ profiles for $\mathrm{\mathbb{T}} = 0.80$ and 1.00. First, at ϕ = 90° (figure 16a), the braid shear-layer for $\mathrm{\mathbb{T}} = 1.00$ (figure 6e) is replaced by a vortex for $\mathrm{\mathbb{T}} = 0.80$, as can be seen from higher $u_r^\ast $ for $\mathrm{\mathbb{T}} = 0.80$ than for $\mathrm{\mathbb{T}} = 1.00$ above the wake centreline. When the foil moves from the upper extreme to the lower extreme, $u_r^\ast $ at y* < 0 (lower side of the wake) is greater for $\mathrm{\mathbb{T}} = 0.80$ than for $\mathrm{\mathbb{T}} = 1.00$ (figure 16b–g). Similarly, when the foil travels from the lower extreme to the upper (figure 16g,h), the $u_r^\ast $ at y* < 0 is enhanced for $\mathrm{\mathbb{T}} = 0.80$.

The wake for $\mathrm{\mathbb{T}} = 1.20$, however, is simply characterized by 2S vortices (figure 17), in a very similar fashion to that for $\mathrm{\mathbb{T}} = 1.00$ (figure 6c–f). The tip vortex A rapidly grows from ϕ = 90° to 180° (faster stroke, figure 17a–d). Because of the faster foil rotation, vortex A is better concentrated and closer to the foil tip than its counterpart for = 0.80 in ϕ = 90° to 180° (slower stroke, see figure 16a–d). The leading-edge vortex ${\rm A}^{\prime}$ flattens and dies out when the foil travels from the upper extreme to the equilibrium. It further dies out in the forward stroke (figure 17d–g) and could not reach vortex A (figure 17f). It, however, contributes to vorticity in the braid shear-layer (figure 17f,g). The vortex ${\rm A}^{\prime}$ does not present itself in the wake. A single vortex is thus born in the wake when the foil travels from the upper extreme to the lower extreme. Similarly, another single vortex B comes into being when the foil returns to the upper extreme i.e. ϕ = 270° to 0° to 90° (figure 17a,g,h). Comparing the snapshots for ϕ = 180°–270° between $\mathrm{\mathbb{T}} = 0.80$ (figure 16d–g) and 1.20 (figure 17d–g), one can understand that vortex ${\rm A}^{\prime}$ for $\mathrm{\mathbb{T}} = 0.80$ can reform itself in the wake, induced by the faster movement of foil tip (faster stroke) while that for $\mathrm{\mathbb{T}} = 1.20$ cannot reform as the foil tip moves slowly. The $u_r^\ast $ profile for $\mathrm{\mathbb{T}} = 1.20$ closely mimics that for $\mathrm{\mathbb{T}} = 1.00$ for most part of the cycle, except at ϕ = 210°–270° (just before reaching the extreme position). It can be seen that the vortex in the wake (e.g. vortex A in figure 17f) for $\mathrm{\mathbb{T}} = 1.20$ is stronger than its counterpart for $\mathrm{\mathbb{T}} = 1.00$, which accompanies a sudden burst of instantaneous $u_r^\ast $. Figure 18 shows wake structures for St_A = 0.38 at $\mathrm{\mathbb{T}} = 1.40$, where the $u_r^\ast $ profiles at $x_{TE}^\ast = 1$ for $\mathrm{\mathbb{T}} = 1.00$ and 1.40 are overlaid onto the vorticity contours as solid and dashed lines, respectively. Like $\mathrm{\mathbb{T}} = 1.20$, the $\mathrm{\mathbb{T}} = 1.40$ also exhibits 2S wake, but with higher peak vorticity of the vortices (not shown). In addition, the vortices (e.g. A in figure 18d) form closer to the trailing edge when compared to the case of a smaller $\mathrm{\mathbb{T}}$ (e.g. compare vortex A in figures 14d, 15d and 16d). The $u_r^\ast $ profiles are qualitatively similar to those for $\mathrm{\mathbb{T}} = 1.20$.

Figure 17. Contours of $\varOmega _z^\ast $ showing generation and evolution of 2S wake (St_A = 0.38, $\mathrm{\mathbb{T}} = 1.20$) at ϕ = 90°, 120°, …, 270° in (a–g) and ϕ = 0° in (h). The $\varOmega _z^\ast{\in} [ - 5,5]$, with positive $\varOmega _z^\ast $ shaded in red and negative $\varOmega _z^\ast $ shaded in blue. The corresponding $u_r^\ast = (u - {U_\infty })/{U_\infty }$ profiles (dashed line) are given at $x_{TE}^\ast = ((x - {x_{TE}})/d) = 1$. The solid line represents $u_r^\ast $ for $\mathrm{\mathbb{T}} = 1.00$. The horizontal and vertical lines represent y* = 0 and $u_r^\ast \; = 0$, respectively.

Figure 18 Contours of $\varOmega _z^\ast $ showing generation and evolution of 2S wake (St_A = 0.38, $\mathrm{\mathbb{T}} = 1.40$) at ϕ = 90°, 120°, …, 270° in (a–g) and ϕ = 0° in (h). The $\varOmega _z^\ast{\in} [ - 5,5]$, with positive $\varOmega _z^\ast $ shaded in red and negative $\varOmega _z^\ast $ shaded in blue. The corresponding $u_r^\ast = (u - {U_\infty })/{U_\infty }$ profiles (dashed line) are given at $x_{TE}^\ast = ((x - {x_{TE}})/d) = 1$. The solid line represents $u_r^\mathrm{\ast }$ for $\mathrm{\mathbb{T}} = 1.00$. The horizontal and vertical lines represent y* = 0 and $u_r^\ast \; = 0$, respectively.

Indeed, observing the wake structures in the entire range of St_A and $\mathrm{\mathbb{T}}$, we identified novel 2P reverse Kármán wake (figure 16), 2S reverse Kármán wake (figures 15 and 16) and asymmetric wake. The 2S Kármán or 2S aligned wake was not observed even at the lowest St_A (= 0.23) investigated here; this is in accordance with the wake map from the experimental work of Schnipper et al. (Reference Schnipper, Andersen and Bohr2009). The novel 2P reverse Kármán wake observed here is quite different from the 2P wake observed by Schnipper et al. (Reference Schnipper, Andersen and Bohr2009). The pairs observed in the present study have the same signed vortices such as A-${\rm A}^{\prime}$, contrary to the opposite signed vortices (see figure 3 of Schnipper et al. Reference Schnipper, Andersen and Bohr2009). Presently, the paired vortices form a single vortex downstream as having the same sign of vorticity. The paired vortices having different signs of vorticity do not merge, the wake appearing 2P far downstream (Schnipper et al. Reference Schnipper, Andersen and Bohr2009).

The time-averaged streamwise $\bar{u}_r^\ast $ profiles in the wake for $\mathrm{\mathbb{T}} = 0.80$, 1.00, 1.20 and 1.40 shown in figure 19 as dashed, continuous, dash–dotted line and dotted lines, respectively display that $\bar{u}_r^\ast $ is largely negative for |y*| = 0.65–1.25 (depending on $\mathrm{\mathbb{T}}$). This suggest a buffer (velocity deficit) zone between the wake jet (velocity excess) and the free stream, induced by the positive and negative vortices above and below the wake centreline. The $\bar{u}_r^\ast $ around the wake centreline is very high (e.g. at $x_{TE}^\ast = 1.00$; figure 19a), the maximum $\bar{u}_r^\ast = 0.82$, 1.05, 1.24 and 1.45 for $\mathrm{\mathbb{T}} = 0.80$, 1.00, 1.20 and 1.40, respectively, i.e. the maximum averaged streamwise velocity reaches 182 %, 205 %, 224 % and 245 % of the freestream velocity, respectively, which indicates a jet-like flow. With the increase in $\mathrm{\mathbb{T}}$, the $\bar{u}_r^\ast $ peak on the wake centreline extends at both $x_{TE}^\ast $ values, while the wake width $(\bar{u}_r^\ast > 0)$ narrows significantly at $x_{TE}^\ast = 1.00$ but negligibly at $x_{TE}^\ast = 6.00$ (figure 19b). The increased peak $\bar{u}_r^\ast $ with increasing $\mathrm{\mathbb{T}}$ is, to some extent, linked to stronger vortices for the higher $\mathrm{\mathbb{T}}$ (figure 18f).

Figure 19. The cross-stream variation of non-dimensionalized time-average streamwise wake velocity $(\bar{u}_r^\ast = (\bar{u} - {U_\infty })/{U_\infty })$ with $\mathrm{\mathbb{T}}$ at (a) $x_{TE}^\ast = ((x - {x_{TE}})/d) = 1$ and (b) $x_{TE}^\ast = 6$ measured from the foil tip for St_A = 0.38.

Increasing A* means increasing foil tip excursion; one thus expects an increased wake width. The flow centrifugal force arising from the foil rotation results from the maximum tip velocity that grows with increasing St_d or A* or both (see (2.4)). The maximum tip velocity is thus connected to the streamwise velocity around the wake centreline. This raises a question, what is the effect of $\mathrm{\mathbb{T}}$ on the wake width and wake velocity when changing $\mathrm{\mathbb{T}}$ that changes neither A* nor St_d? These simple intuitive arguments have not been addressed in the literature. We would like to shed some light on this point. Figure 20 shows the dependence of wake width w* (= w/d) on the $\mathrm{\mathbb{T}}$, A* and St_d, where w is the jet width at $x_{TE}^\ast = 1.00$, defined by the cross-stream length for $\bar{u}_r^\ast > 0$. Figure 20(a–c) shows the dependence of w* on $\mathrm{\mathbb{T}}$ and A* at different St_d values, whereas figure 20(d–f) shows the same on $\mathrm{\mathbb{T}}$ and St_d at different A* values.

Figure 20. Contour maps of normalized wake width w* = (w/d) at $x_{TE}^\ast = 1$ for (a) St_d = 0.21, (b) St_d = 0.27, (c) St_d = 0.33, (d) A* = 1.1, (e) A* = 1.3 and (f) A* = 1.6. The black contour lines in panels (c)–(d) show w* = A*.

The w* grows with increasing A* for a given $\mathrm{\mathbb{T}}$ but declines with increasing $\mathrm{\mathbb{T}}$ for a given A* (figure 20a–c). That is, the $\mathrm{\mathbb{T}}$ effect on w* is opposite to the A* effect. The $\mathrm{\mathbb{T}}$ effect on w* although looking straightforward, a deeper explanation can be provided, which links with fluid angular momentum that is essentially a representation of ${C_{Pf}}$. Indeed, the forward stroke is more connected to w* than the retract stroke as the forward stroke diverges fluid away from the wake centreline while the retract stroke brings fluid toward the wake centreline, albeit not at the same scale. For $\mathrm{\mathbb{T}} = 1.00$, the angular momentum varies similarly in the forward and retract strokes (figure 6a). When $\mathrm{\mathbb{T}}$ is decreased from 1.00, the faster forward stroke causes a large variation in ${C_{Pf}}$ in the forward stroke (figure 8a), diverging the fluid away from the wake centreline with a large angular momentum. The w* thus widens with decreasing $\mathrm{\mathbb{T}}$ from 1.00. In contrast, when $\mathrm{\mathbb{T}}$ is increased from 1.00, the forward stroke becomes slower and has a smaller variation in ${C_{Pf}}$ than the retract stroke (figures 7a and 8a). The smaller angular momentum during the forward stroke thus corresponds to a narrower w*. Another feature is that the decrease in w* with increasing $\mathrm{\mathbb{T}}$ is not linear, rather faster at lower $\mathrm{\mathbb{T}}$, as can be observed from steeper slopes of the contour lines at lower $\mathrm{\mathbb{T}}$ (figure 20a–c). At a constant A*, the w* boosts with increasing St_d particularly for $\mathrm{\mathbb{T}} > 1.00$ (figure 20d–e) as a higher oscillation frequency produces more angular momentum. The effect of St_d on w*, however, diminishes with increasing A* (figure 20d–f). The peak-to-peak amplitude A* instead of the half-amplitude is adopted in most of the literature as it is assumed to represent wake width as well. A scrupulous observation on w* contours suggests that w* is larger than A* for $\mathrm{\mathbb{T}} = 1.00$ and is also a function of St_d (see figure 20d–f).

The jet produced in the wake is expected to be highly connected to ${\bar{C}_T}$. The maximum jet velocity in the wake is a measure of the jet strength. It is worth studying the dependence of jet velocity on A*, $\mathrm{\mathbb{T}}$ and St_d. We measured the maximum jet velocity as the maximum time-average streamwise velocity $\bar{u}_{r,max}^\ast ( = max(\bar{u}_r^\ast ))$ at $x_{TE}^\ast = 1.00$. The dependence of $\bar{u}_{r,max}^\ast $ can be understood from the contour plots of $\bar{u}_{r,max}^\ast $ on the ${A^\ast }- \mathrm{\mathbb{T}}$ and $S{t_d}- \mathrm{\mathbb{T}}$ planes in figure 21. An increase in $\mathrm{\mathbb{T}}$ and/or A* results in a monotonic increase in $\bar{u}_{r,max}^\ast $ (figure 21a–c). This is consistent with the fact that ${\bar{C}_T}$ enhances with increasing $\mathrm{\mathbb{T}}$ and/or A*. At a given $\mathrm{\mathbb{T}}$, an increase in St_d leads to an increased $\bar{u}_{r,max}^\ast $ (figure 21d–f). Overall, increasing $\mathrm{\mathbb{T}}$ from 1.00 strengthens the jet (increasing $\bar{u}_{r,max}^\ast $), which narrows the w*, whereas decreasing $\mathrm{\mathbb{T}}$ from 1.00 widens the jet (increasing w*), which weakens the jet. It is worth noting that the jet characteristics can be modified by changing $\mathrm{\mathbb{T}}$ only for constant St_d and A*. For a sinusoidal pitching foil, increasing A* is capable of both strengthening and widening the jet. However, increasing St_d largely fortifies the jet, with negligible influence on the jet width. The effects of $\mathrm{\mathbb{T}}$ on St_d, A*, St_A, ${\bar{C}_T}$, ${\bar{C}_P}$, w, and $\bar{u}_{r,max}^\ast $ are summarized in table 2.

Figure 21. Contour maps of normalized maximum wake velocity $\bar{u}_{r\textrm{,}max\; }^\ast $ at $x_{TE}^\ast = 1$ for (a) St_d = 0.21, (b) St_d = 0.27, (c) St_d = 0.33, (d) A* = 1.1, (e) A* = 1.3 and (f) A* = 1.6.

Table 2. Effect of $\mathrm{\mathbb{T}}$ on ${\bar{C}_T}$, ${\bar{C}_P}$, η, w*, and $\bar{u}_{r,max}^\ast $.

Table 3. Effect of St_d, A* and St_A on ${\bar{C}_T}$, ${\bar{C}_P}$, η, w* and $\bar{u}_{r,max}^\ast $.

When $\mathrm{\mathbb{T}}$ is increased from 0.80 to 1.40, both ${\bar{C}_T}$ and ${\bar{C}_P}$ first decrease and then increase, having minimum at $\mathrm{\mathbb{T}} = 0.85$ and 1.00, respectively. However, η for 0.3 < St_A < 0.4 increases monotonically with $\mathrm{\mathbb{T}}$ while that for St_A > 0.4 first increases and then decreases with $\mathrm{\mathbb{T}}$. For the drag regime, η has a minimum at $\mathrm{\mathbb{T}} = 0.90$, with η decreasing and increasing for $0.80 \le \mathrm{\mathbb{T}} < 0.90$ and $0.90 < \mathrm{\mathbb{T}} \le 1.40$, respectively (table 2). The w*, however, monotonically declines and $\bar{u}_{r,max}^\ast $ grows with increasing $\mathrm{\mathbb{T}}$. The parameters St_d, A* and St_A are linked to one another in a fashion of $S{t_A} = S{t_d} \times {A^\ast }$. Hence, a change in St_d and/or A* modifies St_A. When St_A is increased by increasing St_d or A*, the ${\bar{C}_T}$ and ${\bar{C}_P}$ increase exponentially and $\bar{u}_{r,max}^\ast $ increases linearly. The w* increases with St_A if St_d is held constant and if A* is constant, then w* decreases with St_A from $0.80 \le \mathrm{\mathbb{T}} < 0.90$ and it increases with St_A for $0.90 < \mathrm{\mathbb{T}} \le 1.40$ (table 3).

3.5.1. Deflected wakes and $\mathrm{\mathbb{T}}$ effect on wake asymmetry

The deflected (asymmetric) wake was found at small $\mathrm{\mathbb{T}}$ and large St_A, as seen in figure 13(d) (upper-left corner). The averaged wake jet deflection angle θ_w was measured by a linear fit of the cross-stream position of $\bar{u}_{r,max}^\ast $ in the near wake $(1 \le x_{TE}^\ast \le 10)$ as shown in figure 22(a). This method of estimating θ_w was proposed by Godoy-Diana et al. (Reference Godoy-Diana, Marais, Aider and Wesfreid2009). The θ_w for St_A = 0.43 against $\mathrm{\mathbb{T}}$ is shown in figure 22(b). The negative value of θ_w indicates the wake deflecting downward. The wake is symmetric for $\mathrm{\mathbb{T}} = 1.00- 1.40$ (i.e. θ_w = 0°) and becomes asymmetric when $\mathrm{\mathbb{T}}$ is decreased from 1.00, θ_w = −9.3° at the lowest $\mathrm{\mathbb{T}}$ ( = 0.80) examined. As shown in figure 13(d), at St_A = 0.429, asymmetric wake is only seen at $\mathrm{\mathbb{T}} = 0.80$, however, when the St_A is increased slightly (i.e. St_A = 0.432), the asymmetric wake appears at higher $\mathrm{\mathbb{T}}$ ($0.80 \le \mathrm{\mathbb{T}} < 1.00$, see figure 13d). The metamorphosis of the symmetric wake into the deflected wake at $\mathrm{\mathbb{T}} < 1.00$ arises from the slower retract stroke generating vortices further away from the foil tip (e.g. compare vortex A in figure 16d to that in figure 17d). Thus, the distance between the two completely shed nearest counter-rotating vortices in the near wake shrinks. For example, vortices ${\rm A}^{\prime}$ and B in figure 16(h) $(\mathrm{\mathbb{T}} = 0.80)$ are closer to each other than vortices A and B in figure 17(h) $(\mathrm{\mathbb{T}} = 1.20)$. This makes them susceptible to forming a dipole vortex pair with a vortex of the opposite sign of rotation shed in the previous half-cycle. The formation of the vortex pair triggers the wake asymmetry (Marais et al. Reference Marais, Thiria, Wesfreid and Godoy-Diana2012). This is clearly depicted in figure 22(c–f) where the vorticity contours behind the foil pitching for $\mathrm{\mathbb{T}} = 1.40$, 1.20, 1.00 and 0.80 are shown, respectively. The distance (normalized with d) between the first two completely shed vortices (opposite signed) is measured and marked in figure 22(c–e). The distance is measured between the locations of maximum and minimum vorticities. It can be seen that a decrease in $\mathrm{\mathbb{T}}(0.80 \le \mathrm{\mathbb{T}} \le 1.20)$ is accompanied by decreasing distance between two opposite sign vortices, which leads to increasing the chance for the asymmetric wake formation.

Figure 22. (a) The wake deflection angle measured by linear fit of maximum wake velocity points (circles) in near wake (for $\mathrm{\mathbb{T}} = 0.80$, St_A = 0.432), dash–dotted line and dashed line depict y* = 0 and least square fit for the near wake deflection angle $(1 \le x_{TE}^\ast \le 10)$. (b) The wake deflection angle against $\mathrm{\mathbb{T}}$. Vorticity contours for $\mathrm{\mathbb{T}} = 1.40$, 1.20, 1.00, and 0.80 in (c), (d), (e) and (f), respectively. All data in panels (a–f) are for St_A = 0.432.