3.1. Extracting the fluid damping from ‘ring down’ experiments

We conduct ‘ring down’ experiments to measure the fluid damping experienced by elastically mounted pitching wings. In the ‘ring down’ experiment, a short-time constant-torque impulse is applied to the CPS as the perturbation, after which the system response and the amplitude decay of the wing are recorded and analysed. Figure 2(a) shows the results from a typical ‘ring down’ experiment. In this specific case, we use an unswept wing ($\varLambda =0^\circ$) which pivots around the mid-chord ($x/c=0.5$) at a frequency of $f_p=0.40\ \textrm {Hz}$. We conduct the ‘ring down’ experiment twice – once in air and once in water. The pitching amplitude of the wing decays faster in water than in air, indicating a higher total damping in water.

Figure 2.

(a) System response and amplitude decay in a typical ‘ring down’ test, where an elastically mounted unswept wing ($\varLambda =0^\circ$) pivots around the mid-chord ($x/c=0.5$) at a frequency of $f_p=0.40\ \textrm {Hz}$. The inset shows the measurements of the pitching amplitude $A_n$ and the pitching frequency $f_p$ of the $n$th peak. The fluid damping $b_f$ at $A_n$ is extracted by fitting an exponential curve (i.e. the red solid line) to the three adjacent peaks. (b) Extracted $b_f$ in air and in water. The zero value is indicated by the black dashed line. The inset compares the measured pitching frequency $f_p$ in air and in water.

To quantify this amplitude decay, the positive peaks of the system response are identified. As shown in the inset, the amplitude of the $n$th peak is denoted by $A_n$, and the corresponding pitching frequency is measured as $f_p = 2/(t_{n+1}-t_{n-1})$. To measure the total damping $b + b_f$ at amplitude $A_n$, we fit an exponential, $y=\alpha \textrm {e}^{-\gamma t}$, to the three adjacent peaks, $n-1$, $n$ and $n+1$, and extract the corresponding $\gamma$ (see (2.3)). Now the only unknown in (2.4b) is the added mass, $I_a$. After obtaining $I_a$, we calculate the fluid damping, $b_f$, using (2.4a) (Rao Reference Rao1995). Since $f_p$ and $\gamma$ are both measured, $I_a$ and $b_f$ are also measured quantities. Moreover, both $I_a$ and $b_f$ are cycle-averaged, meaning they cannot reflect the instantaneous variation of the fluid inertia and damping. The values of the measured fluid damping, $b_f$, in air and water are compared in figure 2(b). Since $\tau _f$ in (2.1) is negligible in air as compared to other forces in the equation, $b_f$ stays near zero, which is indicated by the good agreement between the red circles and the black dashed line. As shown by the green squares, $b_f$ in water is significant because of the existence of the fluid damping torque, $\tau _{b}$. It is also observed that $b_f$ in water increases non-monotonically with $A$. This nonlinear behaviour will be revisited later in § 3.4. The inset of figure 2(b) shows the measured pitching frequency, $f_p$, in both air and water. Because of the combined effect of the fluid inertia and damping, we see that $f_p$ is slightly lower in water than in air.

3.2. Frequency scaling of the fluid damping

We repeat the ‘ring down’ experiment for the unswept wing ($\varLambda =0^\circ$) pivoting at the mid-chord, $x/c=0.5$, and change the pitching frequency by tuning the virtual inertia, $I_v$, and the virtual stiffness, $k_v$, while keeping the virtual damping $b_v$ constant (Onoue & Breuer Reference Onoue and Breuer2016, Reference Onoue and Breuer2017). Figure 3(a) shows the extracted fluid damping, $b_f$. (Note that figures 3(a) and 3(b) share the same legend.) We observe that $b_f$ increases monotonically with the pitching frequency, $f_p$, and that the trend of $b_f$ remains consistent for all frequencies. This agrees with the observations of Keulegan & Carpenter (Reference Keulegan and Carpenter1958) and Shih & Buchanan (Reference Shih and Buchanan1971) for heaving rigid plates, where the fluid damping coefficient scales inversely with the oscillation period. As we discussed earlier, $b_f$ derives from the fluid damping torque $\tau _{b}$, which depends strongly on the vortex-induced forces on the wing (Kang & Shyy Reference Kang and Shyy2014). Onoue & Breuer (Reference Onoue and Breuer2016, Reference Onoue and Breuer2017) have shown that the circulation of leading-edge vortices scales with the strength of the feeding shear-layer velocity. In our case, without a free-stream flow, the feeding shear-layer velocity equals the leading-/trailing-edge velocity, which is proportional to $f_p$. Based on this, we divide $b_f$ by $f_p$ (figure 3b). It is seen that with this scaling, all of the fluid damping curves collapse nicely.

Figure 3.

(a) Extracted fluid damping $b_f$ at different pitching frequencies for $x/c=0.5$. (b) A frequency scaling for the fluid damping which collapses $b_f$ at different $f_p$ into one curve. Note that panels (a) and (b) share the same legend.

We extend this frequency scaling to unswept wings with different pivot axes (figure 4a) and to swept wings with different sweep angles (figure 4b). For comparison, in figure 4(a,b) we include the previous results (figure 3b) using purple circles. Note that each symbol shape in figure 4 contains five different pitching frequencies: $f_p = 0.20$, 0.28, 0.40, 0.56 and 0.78 Hz.

Figure 4.

(a) Values of $b_f/f_p$ for an unswept wing ($\varLambda =0^\circ$) pivoting at $x/c=0$ to $x/c=1$ with a step size of 0.125. The pivot location for each dataset is shown by the inset. (b) Values of $b_f/f_p$ for swept wings with $\varLambda =0^\circ$, $10^\circ$, $15^\circ$, $20^\circ$ and $25^\circ$. The inset shows side views of the five swept wings, and the dashed line indicates the pivot axis. The colours of the wings correspond to the colours of the $b_f/f_p$ curves in the figure. The purple circles in (a,b) are replotted from figure 3(b). Note that each dataset in (a,b) includes five different $f_p$.

For the unswept wing ($\varLambda =0^\circ$), we change the pivot axis from $x/c=0$ to $x/c=1$ with a step size of 0.125 (see the inset of figure 4a). We observe that $b_f/f_p$ increases as the pivot axis is moved away from the mid-chord, $x/c=0.5$. For pivot axes that are symmetric with respect to the mid-chord (i.e. $x/c=0.375$ & 0.625, 0.25 & 0.75, 0.125 & 0.875 and 0 & 1), the values of $b_f/f_p$ roughly overlap. The slight inconsistency between the $b_f/f_p$ values for $x/c>0.5$ and $x/c<0.5$ comes from the asymmetry of the NACA 0012 wing geometry with respect to the mid-chord; we see that the scaled damping, $b_f/f_p$, is always slightly higher for $x/c<0.5$. In these cases, the damping at the trailing edge dominates because of the higher velocity and longer moment arm, and it is stronger than in the cases when $x/c>0.5$, where the leading-edge damping dominates. We will show in § 3.4 that this is due to differences in the vortex structures generated by the sharp and rounded geometries.

This frequency scaling, $b_f/f_p$, also holds for three-dimensional swept wings (figure 4b). Again, each curve includes data from five pitching frequencies. Here, the pivot axes of swept wings are kept as a vertical line passing through the mid-chord of the mid-span plane (see the inset of figure 4b). As $\varLambda$ increases, the average pivot axes of the top and the bottom portion of the swept wing move away from the mid-chord, leading to the increase of the scaled damping, $b_f/f_p$, in a manner similar to that observed for unswept wings with different pivot locations (figure 4a). This argument will be revisited in the next section.

3.3. Universal fluid damping scaling for unswept and swept wings

Figure 4(a) indicates that the pivot axis plays an important role in determining the fluid damping of unswept wings. We extend the frequency scaling of $b_f$ to take into account this effect. First, we divide the wing into two parts, the fore part from LE to the pivot axis with a chord length of $c_{LE}$, and the aft part from the pivot axis to TE with a chord length of $c_{TE}$ (see the inset of figure 5 for an example when the wing pivots at $x/c=0.5$). The Morison equation (Morison et al. Reference Morison, Johnson and Schaaf1950) indicates that the fluid damping force $F$ scales with $0.5 \rho U^2 sc$, where $\rho$ is the fluid density, $U \sim \dot {\theta } c$ is the characteristic velocity and $sc$ is the wing area. We can express the total fluid damping torque as the sum of the torque exerted on the fore and aft portions of the wing,

(3.1)\begin{equation} \tau_b \sim K_{LE} F_{LE} c_{LE} + K_{TE} F_{TE} c_{TE}, \end{equation}

where the subscripts $LE$ and $TE$ refer to the leading- and trailing-edge contributions, and $K_{LE}$ and $K_{TE}$ are empirical factors that account for the subtle differences in the damping associated with the specific geometries of the leading and trailing edges (figure 4a). Since the differences are small, $K_{LE}$ and $K_{TE}$ should be close to one, and for consistency, their average value must equal one ($(K_{LE} + K_{TE})/2=1$).

Figure 5.

Non-dimensional fluid damping coefficient $B_f^*$ versus pitching amplitude $A$ for unswept wings pivoting at $x/c=0$ to $x/c=1$ and swept wings with sweep angles $\varLambda =0^\circ$ to $\varLambda =25^\circ$. The inset shows the definition of the leading-edge chord $c_{LE}$ and the trailing-edge chord $c_{TE}$, with black dashed lines indicating the pivot axes. The black dotted line indicates the small-amplitude prediction for a drag coefficient of $C_D=2.8$.

Since the damped oscillations are observed to be near-sinusoidal (figure 2a), the average angular velocity is given by $4 f_p A$. Simplifying, we arrive at an expression for the fluid damping:

(3.2)\begin{equation} b_f \sim 2 \rho f_p A s (K_{LE} c_{LE}^4 + K_{TE} c_{TE}^4), \end{equation}

or, in non-dimensional form,

(3.3)\begin{equation} B_f^* \equiv \frac{b_f}{2 \rho f_p s (K_{LE} c_{LE}^4 + K_{TE} c_{TE}^4)} \propto A. \end{equation}

For swept wings, because the pivot axis passes through $x/c=0.5$ at the mid-span, the top half of the wing has an average pivot axis $x/c>0.5$, while the bottom half has an average pivot axis $x/c<0.5$. Ignoring three-dimensional effects, we approximate the swept wing by two ‘equivalent’ unswept wing segments. We choose not to divide the wing into a large number of narrow ‘blade elements’ (Glauert Reference Glauert1983), because the pivot axis of some elements near the wing root/tip for large sweep angles may lie outside the range $x/c = [0,1]$, where our scaling has not been tested. The inset of figure 5 shows how these two unswept wing segments are configured (rectangles with red dotted lines). Based on the wing geometry, we see that

(3.4)\begin{equation} \left.\begin{gathered} c_{LE,top} = c_{TE,bot} = \frac{c}{2}+\frac{s}{4}\tan{\varLambda},\\ c_{TE,top} = c_{LE,bot} = \frac{c}{2}-\frac{s}{4}\tan{\varLambda}. \end{gathered}\right\}\end{equation}

Following the same analysis as for the unswept wing, and adding the fluid damping of the top and the bottom wing segments together, we find that the fluid damping for the full swept wing is given by

(3.5)\begin{equation} b_f \sim \rho f_p A s (K_{LE} c_{LE,top}^4 + K_{TE} c_{TE,top}^4 + K_{LE} c_{LE,bot}^4 + K_{TE} c_{TE,bot}^4). \end{equation}

If we define an effective leading-edge chord $c_{LE} = c_{LE,top} = c_{TE,bot}$ and an effective trailing-edge chord $c_{TE} = c_{TE,top} = c_{LE,bot}$, this scaling reduces to (3.2) with $K_{LE}$ and $K_{TE}$ cancelled out. This cancellation results because the effective pivot axes of the top and the bottom segments are symmetric about $x/c=0.5$ at the mid-span, which averages out the slight differences in fluid damping experienced by the top and the bottom segments. For the same reason, $K_{LE}$ and $K_{TE}$ also cancel out in (3.3) for swept wings.

Figure 5 shows the non-dimensional fluid damping, $B_f^*$, as a function of the pitching amplitude, $A$, for unswept and swept wings. Here, we have used $K_{LE} = 0.95$ and $K_{TE} = 1.05$. We see that all of our measurements collapse remarkably well under the proposed scaling, especially for $A<1.57\ (90^\circ )$, despite the wide range of pitching frequencies ($f_p = 0.20$ Hz to $f_p = 0.78$ Hz), pivot axes ($x/c=0$ to $x/c=1$) and sweep angles ($\varLambda = 0^\circ$ to $\varLambda = 25^\circ$) tested in the experiments. In the small-amplitude limit ($A<0.5$), $B_f^*$ scales linearly with $A$, with a slope that corresponds to the drag coefficient, $C_D$. We note that $C_D \approx 2.8$, which is comparable to that of an accelerated normal flat plate (Ringuette, Milano & Gharib Reference Ringuette, Milano and Gharib2007). At higher pitching angles ($A>0.5$), however, the linear approximation no longer holds and we see a decreasing slope of $B_f^*$ as a function of $A$. This is presumably because the shed vortices no longer follow the rotating wing and the fluid force becomes non-perpendicular to the wing surface as $A$ increases. For $A>1.57\ (90^\circ )$, the scaling works reasonably well except for the case $\varLambda =0^\circ$, $x/c=0.5$, where a decreasing $B_f^*$ is observed. In the next section, we will use insights from the velocity fields to explain this non-monotonic behaviour.

3.4. Insights obtained from velocity fields

To gain more insight regarding the nonlinear behaviour of $B_f^*$, we conduct two-dimensional PIV experiments to measure the surrounding flow fields of an unswept wing ($\varLambda =0^\circ$) with a prescribed pitching motion, $\theta = A \sin {(2 {\rm \pi}f_p t)}$. The results are shown in figure 6. The pitching frequency is kept at $f_p=0.5$ Hz for all the cases, and the pitching amplitude is varied from $A=0.52\ (30^\circ )$ to $A=2.09\ (120^\circ )$ with a step size of $0.52\ (30^\circ )$. Two pivot axes are tested, $x/c=0.5$ (figure 6a–d) and $x/c=0.25$ (figure 6e–h). Note that the flow fields shown in figure 6 are not sequential. Instead, all the snapshots are taken right before $t/T=0.25$ for different pitching amplitudes, where $T$ is the pitching period. This specific time instant is chosen because it best reflects the difference in dynamics associated with the different pitching amplitudes and pivot axes.

Figure 6.

The PIV flow field measurements for an unswept wing undergoing prescribed sinusoidal pitching motions in quiescent water. (a–d) Pivot axis (shown by green dots) $x/c=0.5$, pitching frequency $f_p=0.5$ Hz, pitching amplitude $A=0.52\ (30^\circ )$, $1.05\ (60^\circ )$, $1.57\ (90^\circ )$ and $2.09\ (120^\circ )$. (e–h) Same as (a–d), except that the pivot axis is at $x/c=0.25$. All the velocity fields are phase-averaged over 20 cycles. Only every fifth velocity vector is shown. Spanwise vorticity $\omega$: positive (red), counterclockwise; negative (blue), clockwise. See supplementary material, available at https://doi.org/10.1017/jfm.2021.578, for the full movie.

For both pivot locations ($x/c=0.5$ in figure 6(a–d) and $x/c=0.25$ in figure 6(e–h)), the spanwise vorticity of the pitch-generated leading-edge vortex (LEV) and trailing-edge vortex (TEV) increases with the pitching amplitude, $A$. This can be explained by the increase in the feeding shear-layer velocities associated with the higher pitching amplitudes (Onoue & Breuer Reference Onoue and Breuer2016). The boundary vortices near the wing surface, which are related to the added mass effect (Corkery et al. Reference Corkery, Babinsky and Graham2019), also become more prominent because of the increase of the angular acceleration. When the wing pivots at $x/c=0.5$ (figure 6a–d), the leading-edge velocity equals the trailing-edge velocity. As a result, the LEV and TEV are fairly symmetric about the pivot axis, with some subtle differences caused by the rounded and sharp edges, respectively. This confirms the arguments given earlier for the differences between the values of $b_f/f_p$ for $x/c>0.5$ and $x/c<0.5$ (figure 4a). For $x/c=0.25$ (figure 6e–h), however, the TEV is much more prominent than the LEV because of the higher trailing-edge velocity. Because of the low leading-edge velocity and the pitch-induced rotational flow, the sign of the LEV even reverses and becomes negative for $A=1.05$ to $A=2.09$ (figure 6f–h).

For both pivot locations, owing to the absence of a convective free stream and the existence of the pitch-induced rotational flow, the LEV and TEV (only the TEV for $x/c=0.25$) are entrained closer to the wing surface as $A$ increases. For $x/c=0.5$, as shown in figure 6(c,d), the LEV moves towards the aft portion of the wing and the TEV moves towards the fore portion of the wing when $A \geqslant 1.57\ (90^\circ )$. The torque generated by these two vortices, which counteracts the wing rotation for small $A$, now assists the rotation as the wing pitches up towards higher angular positions. This assist reduces the fluid drag experienced by the wing and thus lowers the fluid damping. This effect accounts for the non-monotonic behaviour of $B_f^*$ for $x/c=0.5$ (figure 5). For $x/c=0.25$ (figure 6g,h), a similar scenario is observed, in which the TEV moves towards the fore portion of the wing and gets closer to the wing surface as $A$ increases. However, because of the existence of a counter-rotating LEV, the TEV is not able to approach the wing surface as closely as in the $x/c=0.5$ case. This explains why a flattening behaviour, rather than a non-monotonic trend of $B_f^*$, is observed for $x/c=0.25$ and presumably for other pivot locations at high pitching amplitudes.