3.1 Steady aerodynamic results

In situ measurements of the steady lift, drag and moment coefficients of the flat-plate model were performed using the ATI Nano43 sensor, prior to the dynamic tests. Results are reported in figure 4 for angles of attack ranging from $0^{\circ }$ up to $90^{\circ }$ , Reynolds number $Re=1.45\times 10^{4}$ and for both the 2D and 3D (effective aspect ratio 7.88) configurations. Each point results from the average of 10 s of the force sensor signals, acquired at a sampling frequency of 1024 Hz, for a fixed angle of attack position. The pitching moment coefficients reported in figure 4(c) are defined about the mid-chord.

Figure 4.

Evolution of steady lift (a), drag (b) and moment coefficients (c) with the angle of attack in 2D and 3D (effective aspect ratio 7.88) configurations; flat-plate model of thickness-to-chord ratio 4.86 % at $Re=1.45\times 10^{4}$ ; the moment is measured about the mid-chord.

Figure 4(a) shows that the 2D model exhibits a short linear region for $\unicode[STIX]{x1D6FC}\leqslant 5^{\circ }$ , characterized by a slope $\text{d}C_{L}/\text{d}\unicode[STIX]{x1D6FC}\approx 6.2$ , which is consistent with the thin-airfoil theory (Anderson Jr Reference Anderson2010). This linear region is followed by a smooth stall behaviour for which the lift smoothly moves away from the linear evolution with neither a local maximum nor a decreasing region before $\unicode[STIX]{x1D6FC}\approx 35^{\circ }$ . The stall is more noticeable regarding the pitching moment about the mid-chord, for which a local maximum can be found for $\unicode[STIX]{x1D6FC}\approx 7^{\circ }$ .

Smooth stall behaviour at low angle of attack is well known for thin airfoils and is due to the underlying mechanism of flow separation. At low angle of attack the flow is characterized by a leading edge laminar separation bubble whose length increases gradually with the angle of attack until it completely separates from the upper surface (Wick Reference Wick1954; Gault Reference Gault1957).

The 3D model exhibits a linear lift coefficient slope $\text{d}C_{L}/\text{d}\unicode[STIX]{x1D6FC}\approx 5.5$ slightly lower than that in 2D, but higher than the one calculated using the theoretical solution for the finite wing of a general planform (Anderson Jr Reference Anderson2010).

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{d}C_{L}}{\text{d}\unicode[STIX]{x1D6FC}}}={\displaystyle \frac{a_{0}}{1+\left({\displaystyle \frac{a_{0}}{\unicode[STIX]{x03C0}\,AR}}\right)(1+\unicode[STIX]{x1D70F})}}. & \displaystyle\end{eqnarray}$$

Taking the effective aspect ratio $AR=7.88$ , $\unicode[STIX]{x1D70F}=0.05$ and $a_{0}=2\unicode[STIX]{x03C0}$ , equation (3.1) gives $\text{d}C_{L}/\text{d}\unicode[STIX]{x1D6FC}\approx 4.96$ . The tip vortex in our finite-aspect-ratio configuration then seems to be smaller than expected. This is confirmed regarding the drag coefficients in figure 4(b). Indeed, 3D results are lower than that in 2D for angle of attack $\unicode[STIX]{x1D6FC}<45^{\circ }$ . The end effect is more pronounced on the lift coefficient, where 3D values remain lower that in 2D between the stall angle of attack and $\unicode[STIX]{x1D6FC}\approx 40^{\circ }$ , suggesting that the tip vortex impacts the separated flow behaviour. The 3D effect is also more pronounced on the moment coefficient results reported in figure 4(c) since the overall values are significantly lower in 3D for $5^{\circ }<\unicode[STIX]{x1D6FC}<75^{\circ }$ .

3.2 Unsteady aerodynamic results

Unsteady tests have been done for pitch ramp motion from $0^{\circ }$ to $90^{\circ }$ , various reduced frequencies, different pivot axis locations and both 2D and 3D configurations at $Re=1.45\times 10^{4}$ . As indicated in § 2, $\unicode[STIX]{x1D6FC}_{max}=\unicode[STIX]{x03C0}/2$ radians gives the same value for the reduced frequency and the reduced pitch rate defined in (2.8). In the following part the reduced pitch rate will then be used instead of the reduced frequency.

Figure 5.

Effect of the reduced pitch rate $K_{p}$ on the lift (a,b), drag (c,d) and pitching moment (e,f) coefficients versus angle of attack for the 2D flat plate; $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ and $x_{p}=0.5$ .

3.2.1 Effect of the reduced pitch rate

The effects of the reduced pitch rate on the lift, drag and pitching moment coefficients in response to pitch ramp motion from $0^{\circ }$ to $90^{\circ }$ are illustrated in figure 5 for the 2D flat plate with a pivot axis location at mid-chord $x_{p}=0.5$ . Two major effects of $K_{p}$ on the lift coefficient are noticed. The lift coefficient slope at low angle of attack increases with $K_{p}$ as shown in the zoomed-view in figure 5(b). The maximum of the lift coefficient as well as the angle of attack at which this maximum occurs (figure 5 a) also gradually increase with $K_{p}$ . As reported by Granlund et al. (Reference Granlund, Ol and Bernal2013), the unsteady aerodynamic response for $K_{p}\leqslant 0.03$ is mainly due to stall delay. This is confirmed here for $K_{p}=0.01,0.02$ and 0.03. The lift coefficient shows no significant departure from the steady evolution at low angle of attack, and the linear region extends up to a stall angle of attack that is delayed as $K_{p}$ increases. For $K_{p}\geqslant 0.04$ the unsteady effect is more pronounced, with growing bumps in lift, drag and moment. According to previous studies (Walker et al. Reference Walker, Helin and Strickland1985; Strickland & Graham Reference Strickland and Graham1987; Granlund et al. Reference Granlund, Ol and Bernal2013), this is due to a growth of circulation associated with the development of a LEV vortex on the suction side of the ‘airfoil’. Nevertheless the impacts of the LEV on the evolutions of lift, drag and moment coefficients versus the angle of attack show distinct behaviours as $K_{p}$ increases. The maximum lift coefficient and corresponding angle of attack gradually increase but seem to saturate for $K_{p}=0.18$ , for which $C_{Lmax}\approx 3.37$ (at $\unicode[STIX]{x1D6FC}\approx 40.8^{\circ }$ ) is 3.2 times the maximum steady lift coefficient. The drag coefficient also exhibits an overall increase with the reduced pitch rate as shown in figure 5(c), while figure 5(d) highlights a distinct behaviour between low and moderate reduced frequencies. As for the lift, the drag coefficient remains close to the steady curve for $K_{p}\leqslant 0.03$ . For higher reduced pitch rate $K_{p}\geqslant 0.04$ , the drag coefficient departs early from the steady curve (figure 5 d), but the bump due to the LEV is only clearly noticeable for $K_{p}\geqslant 0.12$ (figure 5 c). For $K_{p}=0.18$ , $C_{Dmax}\approx 4.16$ (at $\unicode[STIX]{x1D6FC}\approx 62^{\circ }$ ) is more than two times the maximum steady drag coefficient for $\unicode[STIX]{x1D6FC}\approx 90^{\circ }$ .

The impact of the reduced pitch rate on the moment coefficient is reported in figure 5(e,f). As before, the results for $K_{p}\leqslant 0.03$ are close to the steady curves at low ( $\unicode[STIX]{x1D6FC}<7^{\circ }$ ) and high ( $\unicode[STIX]{x1D6FC}>40^{\circ }$ ) angle of attack and exhibit an overall increase at moderate angle of attack due to a delay in stall. The maximum moment coefficient and its associated angle of attack also increase with $K_{p}$ . The same kind of behaviour is observed up to $K_{p}=0.12$ . For $K_{p}>0.12$ , the unsteady effects are more pronounced. $C_{M}$ starts to decrease at low angle of attack, but a big overshoot is observed at high angle of attack. For $K_{p}=0.18$ , $C_{Mmax}\approx 0.7$ (at $\unicode[STIX]{x1D6FC}\approx 55^{\circ }$ ) is more than five times the maximum steady moment coefficient. In summary, the reduced pitch rate $K_{p}$ have a strong impact on the unsteady aerodynamic loads, particularly for $K_{p}>0.03$ , for which the impact of LEV formation produces overshoot in $C_{L}$ , $C_{D}$ , $C_{M}$ that increases globally with $K_{p}$ .

Figure 6.

Comparison of the 2D and 3D (effective aspect ratio 7.88) configuration on the lift (a), drag (b) and pitching moment (c) coefficients versus angle of attack for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ , $x_{p}=0.5$ and different reduced pitch rates.

3.2.2 2D versus 3D configuration

Figure 6 compares the results obtained for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ , a pivot axis located at mid-chord $x_{p}=0.5$ and a selection of reduced pitch rates for the 2D and 3D configurations. 3D curves are systematically lower than that in 2D, suggesting that the end effects reduce the motion-induced impact of the LEV. The maximum lift coefficient in the 2D configuration reaches 1.32, 2.11, 3.04 and 3.37 for $K_{p}=0.02$ , 0.06, 0.14 and 0.18, respectively. In contrast, the corresponding maximum lift coefficient in the 3D configuration reduces to 1.22, 2.00, 2.88 and 3.10, i.e. a reduction close to $92\,\%\sim 95\,\%$ . A decrease of the same amount is also observed in the maximum drag coefficient (figure 6 b). Regarding the moment coefficient (figure 6 c), the 3D impact is more pronounced and increases with the reduced pitch rate. For $K_{p}=0.18$ the small negative bump at low angle of attack is significantly reduced, and the maximum overshoot that occurs close to $\unicode[STIX]{x1D6FC}\approx 55^{\circ }$ is reduced by 20 %.

Three-dimensional flow is accompanied by tip vortices (Green Reference Green1995). Based on flow visualizations carried out over a flat plate with an effective aspect ratio of 2 performing a smoothed linear pitch from $0^{\circ }$ to $90^{\circ }$ at $K_{p}=0.5$ and $Re=20\,000$ , Granlund, Ol & Bernal (Reference Granlund, Ol and Bernal2011b ) showed that tip vortices evolved more slowly than leading edge vortices, and remained coherent and nearly attached to the wingtip during the leading edge vortex process. Furthermore, they found that the LEV for the 3D configuration was less coherent than that in 2D. In the present study, steady aerodynamic measurements (figure 4) suggest that the tip vortices associated with our 3D flat plate, with effective aspect ratio close to 8, are rather small. Nevertheless, the 3D effect observed on the unsteady lift, drag and moment for reduced pitch rates lower than 0.2 are significant. In particular, the reduction of the overshot induced by the LEV in 3D is consistent with the findings of Granlund et al. (Reference Granlund, Ol and Bernal2011b ) and Son et al. (Reference Son, Cetiner, Stevens, Babinsky, Manar, Mancini, Jones, Ol and Gozukara2016).

Figure 7.

Effect of the pivot axis location on the lift, drag and pitching moment coefficients versus angle of attack for $K_{p}=0.06$ (a–c) and $K_{p}=0.16$ (d–f); 2D flat plate, $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ .

3.2.3 Effect of the pivot point location

The effect of pivot axis location on the lift, drag and moment coefficients are reported in figure 7 for $K_{p}=0.06$ and $K_{p}=0.16$ . For low reduced pitch rate ( $K_{p}=0.06$ ), no considerable changes can be observed in the lift and drag coefficients in figures 7(a) and 7(b) even though the lift results suggest an overall increase with the absolute distance between the pivot point location and the $3/4$ -chord location. Distinct behaviours can be observed for the moment coefficient defined about the pivot axis location in figure 7(c). When the pivot location is at the leading edge ( $x_{p}=0$ ), the moment coefficient decreases from 0 at $\unicode[STIX]{x1D6FC}=0^{\circ }$ to $-1.21$ at $\unicode[STIX]{x1D6FC}=54.6^{\circ }$ and then increases slightly as the angle of attack further increases. When the pivot location moves rearwards to $x_{p}=0.25$ , the $C_{M}$ decreases monotonically from 0 at $\unicode[STIX]{x1D6FC}=0^{\circ }$ to $-0.48$ at $90^{\circ }$ . The moment coefficient turns to positive for $x_{p}\geqslant 0.5$ . It firstly increases with the angle of attack and then decreases slowly. As expected, $C_{Mmax}$ increases with $x_{p}$ , reaching 1.36 at $x_{p}=1$ . When the reduced pitch rate increases to $K_{p}=0.16$ , more pronounced effects of the pivot axis location can be observed on the dynamic lift and drag force. The slope of $C_{L}$ at small angle of attack decreases from 10.97 to 5.96 as $x_{p}$ moves from the leading edge to the trailing edge (figure 7 d). The maximum $C_{L}$ decreases from 3.98 for $x_{p}=0$ to 2.63 for $x_{p}=0.75$ , and $C_{L\,max}$ increases to 2.97 for $x_{p}=1$ . Figure 7(d) also suggests a gradual increase of the angle of attack at which the maximum lift coefficient occurs when the pivot point moves towards the trailing edge. For $x_{p}=0{-}0.75$ the drag coefficient evolution does not change for $\unicode[STIX]{x1D6FC}\leqslant 20^{\circ }$ (figure 7 e). For higher angle of attack the bump due to the LEV is slightly reduced when the pivot point moves towards the $3/4$ -chord position. Figure 7(e) also shows a singular behaviour for $x_{p}=1$ , with a drag coefficient evolution significantly lower up to $\unicode[STIX]{x1D6FC}\approx 50^{\circ }$ and a bump over the one observed for $x_{p}=0.75$ . The moment coefficient for $K_{p}=0.16$ (figure 7 f) remains lower than that for $K_{p}=0.06$ (figure 7 c) at low angle of attack $\unicode[STIX]{x1D6FC}\leqslant 21^{\circ }$ . In contrast, larger absolute values of maximum moment coefficients are obtained for every $x_{p}$ at high angle of attack, reaching $C_{Mmax}\approx -2$ at $\unicode[STIX]{x1D6FC}\approx 50^{\circ }$ for $x_{p}=0$ and $C_{Mmax}\approx +2$ at $\unicode[STIX]{x1D6FC}\approx 65^{\circ }$ for $x_{p}=1$ .

Figure 8.

Evolution of the mean slope of lift coefficient at small angle of attack $\unicode[STIX]{x1D6FC}<5^{\circ }$ with the reduced pitch rate for the 2D configuration and different pivot axis locations.

A summary of the effects of reduced pitch rate and pivot axis location on the lift coefficient is presented in figures 8 and 9. Figure 8 surveys the evolution of the lift coefficient slope $\text{d}C_{L}/\text{d}\unicode[STIX]{x1D6FC}$ at low angle of attack as a function of $K_{p}$ and different pivot axis locations for the 2D flat-plate configuration. Fitting curves using linear regression have been added to highlight the general trends. Note that the slope values can be dispersed, so the linear fitting lines can only be used for a qualitative analysis. $\text{d}C_{L}/\text{d}\unicode[STIX]{x1D6FC}$ slightly decreases with $K_{p}$ for $x_{p}=1$ . For $x_{p}=0.75$ , the slope varies little with further increase of $K_{p}$ and remains approximately $2\unicode[STIX]{x03C0}$ . For $x_{p}\leqslant 0.5$ , $\text{d}C_{L}/\text{d}\unicode[STIX]{x1D6FC}$ increases with the reduced pitch rate and grows more rapidly as the pivot axis location moves towards the leading edge. 3D effects on the lift coefficient can be appreciated in figure 6. Results show that, for each reduced pitch rate, the lift coefficient slope at low angle of attack is 10–15 % lower for the 3D flat-plate configuration.

Figure 9.

Evolution of the maximum lift coefficients (a), and associatedangle of attack (b) with the reduced pitch rate for different pivot axis locations.

The evolutions with $K_{p}$ of the maximum lift coefficients and the angle of attack at which this maximum occurs are reported in figure 9. For $K_{p}\leqslant 0.08$ , figure 9(a) shows that $C_{Lmax}$ increases almost linearly with $K_{p}$ for all pivot point locations, with a mean rate of increase close to 19. For $K_{p}>0.08$ , $C_{Lmax}$ keeps increasing almost linearly with $K_{p}$ for $x_{p}=0$ . For $x_{p}=0.25$ , $C_{Lmax}$ slightly departs from the linear line and at $K_{p}=0.18$ , $C_{L\,max}$ is 12 % lower than that for $x_{p}=0$ . For $x_{p}=0.5$ , the rate of increase of $C_{Lmax}$ with $K_{p}>0.08$ is significantly reduced (it is close to 7.5), while $C_{Lmax}$ seems to saturate beyond $K_{p}=0.08$ for $x_{p}=0.75$ and $K_{p}=0.1$ for $x_{p}=1$ , respectively. As pointed out previously, at moderate reduced frequency ( $K_{p}>0.08$ ) for which the unsteady response is dominated by the LEV effect, the minimum $C_{L\,max}$ is observed for $x_{p}=0.75$ and gradually increases with the absolute distance between the pivot point location and this neutral point.

Angles of attack at which the maximum lift coefficients are attained are plotted in figure 9(b). Due to the smooth stall behaviour of the flat plate, an exact evaluation of the angle of attack associated with $C_{Lmax}$ is difficult at low reduced pitch rate. Errors bars have been added when necessary. Despite the small scattering of those results, an overall increase with $K_{p}$ can be pointed out. This progressive delay of the peak lift to higher angle of attack has been reported by previous studies; see, for example, Granlund et al. (Reference Granlund, Ol and Bernal2013). At low reduced pitch rate it is mainly due to a delay in stall, which increases with increasing pitch rate (Sheng, Galbraith & Coton Reference Sheng, Galbraith and Coton2006). For higher $K_{p}$ it can be related to the delay in the LEV formation, which also increases with the pitch rate (Helin & Walker Reference Helin and Walker1985). As reported in Granlund et al. (Reference Granlund, Ol and Bernal2013), a saturation of the angle of attack associated with the peak lift coefficient was also observed. It is close to $35^{\circ }$ for $x_{p}=0.25$ and 0.75, close to $40^{\circ }$ for $x_{p}=0.5$ , and close to $45^{\circ }$ for $x_{p}=1$ . No real saturation is observed for the case with the pivot axis at the leading edge. Figure 9(b) also shows that the effect of pivot point location is more dominant for $K_{p}>0.08$ . In this reduced pitch rate regime, the gradual reduction of $C_{Lmax}$ when the pivot point moves from the leading edge to the trailing edge (figure 9 a) is associated with a gradual increase of the associated angle of attack (figure 9 b). This is in accordance with Granlund et al. (Reference Granlund, Ol and Bernal2013) and Yu & Bernal (Reference Yu and Bernal2013), who pointed out a increasing delay of the formation and growth of the LEV as the pivot point location moved downstream. Only the case whose pivot point is at the $3/4$ -chord shows a distinct behaviour, because the angle of attack at which peak lift is attained is lower than that for $x_{p}=0.5$ .

3.2.4 Effect of maximum angle of attack in pitch-up and down motion

Additional unsteady 2D tests have been performed for pitch-up and pitch-down ramp motion with different maximum angles of attack, pivot point location $x_{p}=0.5$ and various reduced frequencies at the same Reynolds number, $Re=1.45\times 10^{4}$ . Results are reported here to highlight the effect of the maximum angle of attack, as well as to focus on the hysteresis response in the pitch-up and pitch-down kinematic. Here the downstroke occurs immediately at the end of the upstroke using an exact symmetric pitch profile. Figure 10 illustrates the evolutions of $C_{L}$ and $C_{D}$ for different maximum pitch angles $\unicode[STIX]{x1D6FC}_{max}=30^{\circ },45^{\circ },60^{\circ },90^{\circ }$ at the same reduced frequency $K=0.06$ and pivot axis location $x_{p}=0.5$ . The solid lines represent the force coefficients during the pitch-up motion and the dashed lines are for the pitch-down motion. The evolutions of $C_{L}$ with angle of attack overlap during the first ‘linear’ part of the pitch-up motion. Both the $C_{Lmax}$ and the angle of attack at which the maximum is reached increase with $\unicode[STIX]{x1D6FC}_{max}$ . Beyond this maximum the lift coefficient decreases gradually and collapses to the steady curve at the end of the upstroke. During the pitch-down motion the lift coefficients decline below the steady lift rather than going back along the upstroke path, forming hysteresis loops that get larger when increasing $\unicode[STIX]{x1D6FC}_{max}$ . Similar hysteresis loops are found for $C_{D}$ , as shown in figure 10(b). The unsteady response of drag departs further away from the steady curve in pitch-up than that in pitch-down and the overall area of the hysteresis path increases with the maximum angle of attack.

Figure 10.

Effects of maximum pitch angle and hysteresis in pitch-up and pitch-down kinematic for $K=0.06$ and $x_{p}=0.5$ .

A synthesis of the effects of the reduced frequency and maximum angle of attack is proposed in figure 11. The maximum lift coefficients and the angles of attack at which those maxima are attained are reported as a function of the reduced pitch rate $K_{p}$ for various maximum angles of attack $\unicode[STIX]{x1D6FC}_{max}=30^{\circ },45^{\circ },60^{\circ },90^{\circ }$ . Note that, due to the limitation of the motor, $K_{P}$ is restricted to lower values for smaller maximum pitch angles.

A first look reveals that the curves for different $\unicode[STIX]{x1D6FC}_{max}$ nearly collapse, suggesting that the reduced pitch rate plays a dominant role. A closer look shows that the rate of increase of the maximum lift coefficient with $K_{p}$ is higher for $\unicode[STIX]{x1D6FC}_{max}=30^{\circ },45^{\circ }$ and $60^{\circ }$ up to $K_{p}=0.04$ . For higher pitch rate the maximum lift coefficients continue to increase, but seem to saturate at $C_{Lmax}\approx 2.5$ beyond $K_{p}=0.08$ for $\unicode[STIX]{x1D6FC}_{max}=30^{\circ }$ and at $C_{Lmax}\approx 3$ beyond $K_{p}=0.12$ for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ and $60^{\circ }$ . Regarding the angles of attack at which the maximum lift coefficients are attained, the curves for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ show similar behaviour up to $K_{p}=0.1$ . Results for $\unicode[STIX]{x1D6FC}_{max}=30^{\circ }$ follow the same rate of increase with $K_{p}$ , but with an overall reduction close to $5^{\circ }$ .

In the present study, every pitch ramp motion is done using an acceleration–deceleration profile for which the maximum pitch rate is obtained at the mid-ramp (see § 2). As a consequence, for $\unicode[STIX]{x1D6FC}_{max}=30^{\circ }$ , $45^{\circ }$ and $60^{\circ }$ , the maximum pitch rate occurs at angles of attack that can enhance the dynamic stall delay. This can explain the higher values of maximum lift coefficient observed for $\unicode[STIX]{x1D6FC}_{max}\leqslant 60^{\circ }$ . Further investigation would be necessary to clarify that point.

Figure 11.

Evolution of the maximum lift coefficients (a) and associated angle of attack (b) with the reduced pitch rate for $x_{p}=0.5$ .

4.1 The normal velocity model (NVM)

Using the Wagner function (Wagner Reference Wagner1925), which accounts for the circulatory lift due to a step change in the angle of attack in the linear regime, the unsteady circulatory response to arbitrary changes in angle of attack can be obtained through the superposition of indicial lift responses via the Duhamel integral. A general formulation for the circulatory lift response to an arbitrary pitching motion with low amplitude can be found in Fung (Reference Fung2002):

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle C_{L}^{C}(t^{\ast })={\displaystyle \frac{2\unicode[STIX]{x03C0}}{U}}\left[w_{3/4}(t^{\ast }\rightarrow 0)\,\unicode[STIX]{x1D719}(t^{\ast })+\int _{0}^{t^{\ast }}{\displaystyle \frac{\text{d}w_{3/4}(\unicode[STIX]{x1D70E})}{\text{d}\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D719}(t^{\ast }-\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D70E}\right], & \displaystyle\end{eqnarray}$$

where $C_{L}^{C}$ is the circulatory lift coefficient, $\unicode[STIX]{x1D719}$ is the Wagner’s function, $t^{\ast }=2Ut/c$ is the reduced time and $w_{3/4}$ is the downwash velocity at the $3/4$ -chord point:

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle w_{3/4}=U\sin \unicode[STIX]{x1D6FC}+2U(0.75-x_{p}){\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}t^{\ast }}}. & \displaystyle\end{eqnarray}$$

In the downwash velocity formulation (4.6), $U\sin \unicode[STIX]{x1D6FC}$ is generally replaced by $U\unicode[STIX]{x1D6FC}$ in low-amplitude linear formulations and the pitch rate term accounts for the induced-camber effect due to the pitch-rate-induced normal velocity distribution along the chord (Fung Reference Fung2002; Leishman Reference Leishman2006).

To account for higher angle of attack the same formulation can be used for the normal coefficient, instead of the lift. Using the exact normal velocity definition at the $3/4$ -chord point (4.6), the NVM is then built as follows:

(4.7) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{NVM}=C_{N}^{\mathit{Trans\_NVM}}+C_{N}^{Rot}+C_{N}^{Cen}+C_{N}^{In}, & \displaystyle\end{eqnarray}$$

where the $C_{N}^{\mathit{Trans\_NVM}}$ is the circulatory part of the response, associated with the ‘translational’ $U\sin \unicode[STIX]{x1D6FC}$ term of the normal velocity:

(4.8) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{\mathit{Trans\_NVM}}=2\unicode[STIX]{x03C0}\left[\sin (\unicode[STIX]{x1D6FC}(t^{\ast }\rightarrow 0))\,\unicode[STIX]{x1D719}(t^{\ast })+\int _{0}^{t^{\ast }}{\displaystyle \frac{\text{d}\sin \unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70E})}{\text{d}\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D719}(t^{\ast }-\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D70E}\right]. & \displaystyle\end{eqnarray}$$

The term $C_{N}^{Rot}$ accounts for the circulatory part of the response, associated with the normal velocity, which is proportional to the pitch rate and to the relative distance between the pivot point location and the $3/4$ -chord point.

(4.9) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{Rot}=2\unicode[STIX]{x03C0}\left[2(0.75-x_{p}){\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}t^{\ast }}}(t^{\ast }\rightarrow 0)\unicode[STIX]{x1D719}(t^{\ast })+\int _{0}^{t^{\ast }}{\displaystyle \frac{\text{d}\left[2(0.75-x_{p}){\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}t^{\ast }}}\right]}{\text{d}\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D719}(t^{\ast }-\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D70E}\right]. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

This term is zero for $x_{p}=0.75$ , i.e. the above-mentioned rear neutral point.

To account for general motions, non-circulatory terms must be added. It is done here with both instantaneous centrifugal and inertia terms as expressed in Fung (Reference Fung2002). The instantaneous centrifugal force is a normal force acting at the $3/4$ -chord. It is proportional to the apparent mass $\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}bc^{2}/4$ , to the pitch rate and to the velocity $U$ . In non-dimensional time, it is expressed as follows:

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{Cen}=\unicode[STIX]{x03C0}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}t^{\ast }}}. & \displaystyle\end{eqnarray}$$

The instantaneous apparent mass term is a normal force acting at the mid-chord, equal to the apparent mass $\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}bc^{2}/4$ times the vertical acceleration at the mid-chord. In non-dimensional time it is given by

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{In}=\unicode[STIX]{x03C0}(\text{1-2}x_{p}){\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FC}}{{\text{d}t^{\ast }}^{2}}}. & \displaystyle\end{eqnarray}$$

We recall here the main assumption of the NVM: (1) the airfoil acts as a linear dynamical system and the principle of linear superposition can be used in the Duhamel integral; (2) the normal force response to an increment in normal velocity is proportional to $2\unicode[STIX]{x03C0}$ , i.e. the slope of the normal coefficient in the linear regime; (3) the Wagner indicial function can be used for high angle of attack; and (4) non-circulatory terms are linearly additive and one can use the formulation taken from the linear unsteady airfoil theory.

4.2 The steady curve model (SCM)

The second assumption of the NVM can be partially removed, modifying the Duhamel formulation to account for the nonlinear variation of the steady normal force coefficient. This was first proposed in Taha et al. (Reference Taha, Hajj and Beran2014) and taken up here to build the so-called SCM, which only differs from the NVM in its translational circulatory term.

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{SCM}=C_{N}^{Trans}+C_{N}^{Rot}+C_{N}^{Cen}+C_{N}^{In}. & \displaystyle\end{eqnarray}$$

Following the idea of Taha et al. (Reference Taha, Hajj and Beran2014), the steady normal force coefficient curve $C_{N}^{S}(\unicode[STIX]{x1D6FC})$ is used to build a time-varying input function:

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{Trans}=C_{N}^{S}(\unicode[STIX]{x1D6FC}(t^{\ast }\rightarrow 0))\unicode[STIX]{x1D719}(t^{\ast })+\int _{0}^{t^{\ast }}{\displaystyle \frac{\text{d}C_{N}^{S}(\unicode[STIX]{x1D6FC})}{\text{d}\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D719}(t^{\ast }-\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D70E}. & \displaystyle\end{eqnarray}$$

4.3 The artificial circulation model (ACM)

As reported in § 3, a motion-induced leading edge vortex is responsible for an intense buildup of the circulation for $K_{p}>0.03$ . Both the NVM and SCM fail to predict this additional circulatory effect. To correct this, an additional circulation term is added to the SCM to build the so-called ACM:

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{ACM}=C_{N}^{SCM}+C_{N}^{AC}. & \displaystyle\end{eqnarray}$$

Results for $0^{\circ }{-}90^{\circ }$ pitch ramp motions also showed that the growth of circulation is well correlated to the temporal evolution of the pitch rate, if a small delay is introduced. The additional circulation term is then built as follows:

(4.15) $$\begin{eqnarray}\displaystyle & \displaystyle C_{N}^{AC}=A\unicode[STIX]{x03C0}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}t^{\ast }}}(t^{\ast }\rightarrow 0)\unicode[STIX]{x1D719}(t^{\ast })+\int _{0}^{t^{\ast }}{\displaystyle \frac{\text{d}A\unicode[STIX]{x03C0}{\displaystyle \frac{\text{d}\unicode[STIX]{x1D6FC}}{\text{d}t^{\ast }}}}{\text{d}\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D719}(t^{\ast }-\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D70E}, & \displaystyle\end{eqnarray}$$

introducing an amplitude coefficient $A$ , and keeping the Duhamel integral formulation along with the Wagner function used for the other circulatory terms, to account for small delay. The optimization solver ‘fminbnd’ in Matlab was used to find the optimal $A$ that minimizes the objective function, defined as the sum of the squared errors of theoretical $C_{N}^{ACM}$ (4.14) and a set of experimental data. The solver is based on golden section search and parabolic interpolation (Forsythe, Moler & Malcolm Reference Forsythe, Moler and Malcolm1977).

Figure 12.

Evolution of the ‘optimal’ amplitude coefficient $A$ (a) and ‘optimal’ LEV centre of pressure $x_{\text{LEV}}$ (b) with the reduced pitch rate for 2D and 3D configurations with a pivot point $x_{p}=0.5$ . Results in (b) have been obtained using the average optimal coefficient associated with the 2D and 3D results for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ : $A_{2D,90^{\circ }}=5.7$ and $A_{3D,90^{\circ }}=4.67$ .

Figure 12 summarizes the optimal $A$ found for the various pitch ramp tests associated with figure 11, i.e. pivot point location $x_{p}=0.5$ ; 2D flat plate for $\unicode[STIX]{x1D6FC}_{max}=30^{\circ },45^{\circ },60^{\circ },90^{\circ }$ and 3D flat plate for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ . For $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ in 2D, the optimal value for $A$ is closed to 4.4 for $K_{p}=0.01$ ; it then increases with $K_{p}$ up to $A\approx 7$ for $K_{p}=0.04$ , before gradually decreasing to $A\approx 4.4$ for $K_{p}=0.18$ . For $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ in 3D, the optimal $A$ follows the same trend, but with slightly lower values, which is consistent with the 3D effect pointed out in § 3.2.2. The average optimal coefficient for the 2D results ( $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ ) is $A_{2D,90^{\circ }}\approx 5.7$ . For the 3D flat plate (effective ratio 7.88) it is $A_{3D,90^{\circ }}=4.67$ .

For moderate-amplitude pitch ramp motion $\unicode[STIX]{x1D6FC}_{max}=30^{\circ },45^{\circ },60^{\circ }$ (2D configuration), the evolution of the optimal $A$ with $K_{p}$ is different. Starting from high values close to 13–14, the optimal $A$ almost linearly decreases with $K_{p}$ down to 4.5 at $K_{p}=0.08$ for $\unicode[STIX]{x1D6FC}_{max}=30^{\circ }$ . For $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ and $60^{\circ }$ it also decreases with $K_{p}$ , but with a smaller rate beyond $K_{p}\approx 0.02$ . $A$ reaches a value close to 4.9 at $K_{p}=0.12$ for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ , while $A$ is close to 6 for $\unicode[STIX]{x1D6FC}_{max}=60^{\circ }$ at the same reduced pitch rate. This peculiar behaviour of the optimal $A$ for $\unicode[STIX]{x1D6FC}_{max}\leqslant 60^{\circ }$ probably results from the increased delay between the bump in the normal force coefficient and the pitch rate evolution with time when the maximum angle of attack decreases (keeping the reduced pitch rate constant). Using the standard Wagner function, the ACM formulation proposed in (4.15) does not seem adapted for $\unicode[STIX]{x1D6FC}_{max}\leqslant 60^{\circ }$ , leading to a wrong estimation of the amplitude coefficient $A$ . This will be discussed in § 5.2.

4.4 Moment coefficient formulations

Like the normal force coefficient, the formulation of the ACM for the moment is built with the sum of five terms:

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{ACM}=C_{M}^{Trans}+C_{M}^{Rot}+C_{M}^{Cen}+C_{M}^{In}+C_{M}^{AC}, & \displaystyle\end{eqnarray}$$

where the translational term is now based on the static moment coefficient curve, delayed using the Wagner function in the Duhamel integral:

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{Trans}=C_{M}^{S}(t^{\ast }\rightarrow 0)\unicode[STIX]{x1D719}(t^{\ast })+\int _{0}^{t^{\ast }}{\displaystyle \frac{\text{d}C_{M}^{S}}{\text{d}\unicode[STIX]{x1D70E}}}\unicode[STIX]{x1D719}(t^{\ast }-\unicode[STIX]{x1D70E})\text{d}\unicode[STIX]{x1D70E}. & \displaystyle\end{eqnarray}$$

Following the classical unsteady airfoil theory (Fung Reference Fung2002), it is assumed that the normal force due to the circulatory induced-camber effect acts at the quarter chord. The associated ‘rotational’ moment can then be expressed as

(4.18) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{Rot}=C_{N}^{Rot}(x_{p}-0.25). & \displaystyle\end{eqnarray}$$

Linear unsteady airfoil theory also shows that the centrifugal effect induces a normal force acting at the $3/4$ -chord, therefore,

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{Cen}=C_{N}^{Cen}(x_{p}-0.75). & \displaystyle\end{eqnarray}$$

Regarding the acceleration terms, one has to consider the impact of a normal force acting at the mid-chord and an added inertia term (Fung Reference Fung2002):

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{In}=C_{N}^{In}(x_{p}-0.5)-{\displaystyle \frac{\unicode[STIX]{x03C0}}{16}}{\displaystyle \frac{\text{d}^{2}\unicode[STIX]{x1D6FC}}{{\text{d}t^{\ast }}^{2}}}. & \displaystyle\end{eqnarray}$$

In order to obtain a simple expression for the added circulatory moment one can introduce a centre point of pressure associated with the added circulatory normal force. Following the idea that this additional circulatory contribution is due to the impact of a leading edge vortex, this centre of pressure point is called $x_{LEV}$ and the added circulatory moment is defined as

(4.21) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{AC}=C_{N}^{AC}(x_{p}-x_{LEV}). & \displaystyle\end{eqnarray}$$

From what is known about the LEV process it seems difficult to define a unique and straightforward value for $x_{LEV}$ . Flow-field analyses of Granlund et al. (Reference Granlund, Ol and Bernal2013) and Yu & Bernal (Reference Yu and Bernal2013) suggest that the formation, length and core position of the vortex during the pitch-up motion depend on the reduced pitch rate, pivot point location and amplitude of motion. $x_{LEV}$ should then depend on $K_{p}$ , $x_{p}$ , $\unicode[STIX]{x1D6FC}_{max}$ and the reduced time. However, in the present study an attempt to define the best unique value for $x_{LEV}$ is made.

Using the average optimal coefficient $A_{2D,90^{\circ }}\approx 5.7$ and $A_{3D,90^{\circ }}=4.67$ , found previously for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ (figure 12 a), optimal $x_{LEV}$ are found for each value of the reduced pitch rate $K_{p}$ . The optimization procedure is the same as that the one used for $A$ : find the optimal $x_{LEV}$ that minimizes the objective function defined as the sum of the squared errors of theoretical $C_{M}^{ACM}$ in (4.16) and a set of experimental data. Results reported in figure 12(b) show that optimal $x_{LEV}$ are nearly constant over the pitch rate range. Average values can then be identified; it is close to 0.39 for the 2D flat-plate configuration and close to 0.4 for the 3D configuration. Those values are in good agreement with Mancini et al. (Reference Mancini, Manar, Granlund, Ol and Jones2015), who quantified the LEV trajectory induced by a surging flat plate at $\unicode[STIX]{x1D6FC}=45^{\circ }$ . They noted that the relative distance of the LEV core to the leading edge increases quickly from zero to 0.4, and then remains close to 0.4 although the normal distance of the LEV to the flat plate keeps rising.

For the other two models, namely the NVM and the SCM, the rotational, centrifugal, and inertia terms are the same (see (4.18)–(4.20)). The translational term of the SCM is also the same as that in the ACM (see (4.17)). Finally, it is assumed that the normal force due to the translational circulatory effect acts at the quarter chord for the NVM, so we used

(4.22) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}^{\mathit{Trans\_NVM}}=C_{N}^{\mathit{Trans\_NVM}}(x_{p}-0.25). & \displaystyle\end{eqnarray}$$

5.1 High-amplitude $0^{\circ }{-}90^{\circ }$ pitch ramp motion

In the present section the three models are compared with experiments for the 2D flat-plate configuration in high-amplitude pitch ramp motion ( $0^{\circ }{-}90^{\circ }$ ). For the ACM model, the average optimal values found in the previous section for the amplitude parameter $A$ and $x_{LEV}$ have been used: $A=5.7$ and $x_{LEV}=0.39$ .

The measured motion has been used to calculate the angle of attack and its derivatives at each time step. An approximate expression of the Wagner’s function attributed to Jones (Reference Jones1940) has been used:

(5.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D719}(t^{\ast })=1-0.165\text{e}^{(-0.0455t^{\ast })}-0.355\text{e}^{(-0.3t^{\ast })}. & \displaystyle\end{eqnarray}$$

The recurrence algorithm D-2 proposed by Beddoes (Reference Beddoes1982) was employed to calculate the Duhamel integral. Refer to Leishman (Reference Leishman2006) for a full description of the recurrence method.

Figure 13.

Comparison of the NVM, SCM and ACM models with experiment for the normal force (a) and pitching moment (c) coefficient evolution versus reduced time; contribution of the six terms of the ACM model in the normal force (b) and pitching moment (d) prediction; for a pitch ramp motion with $K_{p}=0.02$ , $x_{p}=0.25$ and $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ .

Figures 13, 14 and 15 compare the $C_{N}$ and $C_{M}$ predicted by the NVM, SCM and ACM with experiments for $K_{p}=0.02$ , 0.08 and 0.16, $x_{p}=0.25,\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ . At low reduced pitch rate ( $K_{p}=0.02$ ), the measured $C_{N}$ increases almost quadratically with $t^{\ast }=0\sim 42$ . After a small bump, the increase is more linear with $t^{\ast }=43$ –90. The maximum $C_{Nmax}\approx 2$ is reached at $t^{\ast }\approx 99$ . Finally, $C_{N}$ slightly decreases to reach its $90^{\circ }$ value at $t^{\ast }\approx 200$ . The NVM predicts the first stage for $t^{\ast }<40$ , but it retains a quadratic evolution with $t^{\ast }$ for too long and $C_{N}$ saturates at an unrealistic value $C_{N}=6.3$ at $t^{\ast }=140$ . In contrast, the SCM provides a reasonable prediction of the experimental $C_{N}$ at both the first stage $t^{\ast }<42$ and final stage $t^{\ast }>108$ , while the stage between them is underestimated. The ACM corrects that, but induces an overestimation of the maximum $C_{Nmax}\approx 2.3$ with a small delay to $t_{max}^{\ast }\approx 110$ , in comparison with experiment. The contributions of the different terms of the ACM for $C_{N}$ are shown in figure 13(b). The translational part mainly contributes to the response. The artificial circulation term is secondary, but it suggests that a small amount of circulation is necessary to account for the impact of the stall delay at this low reduced pitch rate regime. The rotational, centrifugal and inertial terms are marginal in that case. Figure 13(c,d) display the moments. The experimental $C_{M}$ first decreases slowly for $t^{\ast }<22$ , and then slumps almost linearly to $-0.39$ at $t^{\ast }\approx 117$ . Finally, $C_{M}$ slightly increases to reach its $90^{\circ }$ value at $t^{\ast }\approx 200$ . Because the translational term of the NVM is set at the quarter chord, it is not surprising that the NVM remains very low and fails to predict the dynamic moment response. Comparatively, the translational term of $C_{M}$ based on the combination of a static moment and the Wagner function in the Duhamel integral formulation allows the SCM to better fit the experiments, as shown in figure 13(c). As for the normal coefficient, the artificial circulation term narrows the gap between the SCM and the experimental results. Figure 13(d) also highlights that the centrifugal term shows a similar trend to that of the artificial circulation term, but with a lower amplitude. As expected, the rotational term is zero at the $1/4$ -chord pivot location, and the inertial term is negligible at this low reduced pitch rate.

Figure 14.

Comparison of the NVM, SCM and ACM models with experiment for the normal force (a) and pitching moment (c) coefficient evolution versus reduced time; contribution of the six terms of the ACM model in the normal force (b) and pitching moment (d) prediction; for a pitch ramp motion with $K_{p}=0.08$ , $x_{p}=0.25$ and $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ .

Figure 15.

Comparison of the NVM, SCM and ACM models with experiment for the normal force (a) and pitching moment (c) coefficient evolution versus reduced time; contribution of the six terms of the ACM model in the normal force (b) and pitching moment (d) prediction; for a pitch ramp motion with $K_{p}=0.16$ , $x_{p}=0.25$ and $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ .

For intermediate reduced pitch rate ( $K_{p}=0.08$ ) at the same pivot location $x_{p}=0.25$ and maximum pitch angle $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ (figure 14), the first stage where $C_{N}$ increases almost quadratically with $t^{\ast }$ yields to a maximum $C_{Nmax}\approx 3$ at $t^{\ast }\approx 18$ . A second bump also appears at $t^{\ast }\approx 26$ , probably due to the impact of the LEV. Once the effect of the LEV vanishes, the normal force coefficient quickly decays to recover the steady value $C_{N}\approx 2$ at $90^{\circ }$ , as shown in figure 14(a). The NVM depicts the first stage of the increasing process of $C_{N}$ quite accurately. However, it fails to capture the overshoot and saturates to a non-realistic static value. This final stage is better predicted by the SCM, but the intermediate range in which the impact of the LEV is dominant is vastly underestimated. The ACM corrects that point and better fits the experimental results. Nevertheless, it overestimates the $C_{Nmax}$ , which now occurs at $t^{\ast }\approx 25$ , i.e. close to the second bump. For this intermediate reduced pitch rate, the aerodynamic response is the consequence of the combined effect of a delay in stall, which is not considered in the ACM model, and that of the LEV, which is modelled here via the added circulatory term. $C_{M}$ results are reported in figure 14(c). As before, the NVM fails to predict the response. Both the SCM and ACM converge towards the correct steady value of $C_{M}$ , but only the ACM satisfactorily follows the envelope of the experimental results. However, the minimum of the moment coefficient is underestimated. A possible explanation is that the relative distance of the LEV centre of pressure is slightly underestimated. In the present study the mean optimal value $x_{LEV}=0.39$ (§ 4.4) has been used. By increasing this value to 0.41, the ACM model predicts the right minimum moment coefficient for the experimental value. Figure 14(b,d) show that the translational term provides the basic bone curve of the ACM. Rotational, centrifugal and artificial circulation terms are responsible for the bump in $C_{N}$ , while the bump in $C_{M}$ is only due to the centrifugal and added circulation terms. At this intermediate reduced pitch rate, added mass and inertia contributions remain negligible.

Figure 15 displays the results for higher reduced frequency $K_{p}=0.16$ . Experimental $C_{N}$ exhibits one single maximum $C_{Nmax}\approx 4.6$ at $t^{\ast }\approx 11$ , followed by a decreasing regime where the normal force recovers its steady value. Experimental $C_{M}$ follows the same trend, but the impact of the LEV is characterized by a minimum $C_{Mmin}\approx -0.72$ at $t^{\ast }\approx 12$ . As before, the NVM overlaps the experiments in the increasing stage of $C_{N}$ but saturates at a higher and unrealistic value. The SCM prediction is correct at the beginning and final stage of the response, but fails to predict the impact of the LEV. The benefit of the additional circulation is evident regarding the ACM prediction of both the maximum $C_{N}$ and minimum $C_{M}$ . However, one can notice that the magnitudes of the aerodynamic loads are overestimated in the first stage of the ramp motion $t^{\ast }=0{-}10$ . A possible explanation is an underestimation of the time delay effect in the additional circulation term. The present model accounted for the time history effect using the Wagner function. Associated time scales could then be adjusted for a better prediction at high reduced pitch rate.

Regarding the experimental results for $K_{p}=0.08$ and 0.16, one can also notice some oscillations in both the normal force and moment in the recovery region. Those oscillations are due to small vibrations of the set-up at the end of the pitch motion, due to the combined effects of inertia and feedback control when the rotation stops. Therefore, only the relaxation stage of the aerodynamic coefficients was slightly affected. As shown in figure 15, one can also attest that despite those added oscillations the experimental data remain reliable regarding the model identification process performed in the present work.

The theoretical added inertia contribution for both the lift coefficient (4.11) and moment coefficient (4.20) is also worth noting. By definition, inertia effects increase with the pitch rate, but also strongly depend on the smoothing parameter $\unicode[STIX]{x1D70E}$ of the ramp definition (2.1). Results reported in figures 15(b) and 15(d) show that the inertia contribution remains very low in the present study, due to the smooth ramp definition ( $\unicode[STIX]{x1D70E}=0.16$ ) and moderate pitch rate range ( $K_{p}\leqslant 0.18$ ).

Figure 16.

Comparison of the ACM time-dependant model with the semiempirical models of Strickland & Graham (Reference Strickland and Graham1987) and Granlund et al. (Reference Granlund, Ol and Bernal2013) for pitch ramp motion experiments of $K_{p}=0.16$ , $x_{p}=0.25,\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ .

A comparison of the prediction of the ACM with the models of Strickland & Graham (Reference Strickland and Graham1987) and Granlund et al. (Reference Granlund, Ol and Bernal2013) is represented in figure 16. The responses of the lift and drag coefficients are plotted versus the angle of attack. Equations (4.1) and (4.2) with the experimentally determined values of $C_{L\,max}$ and $C_{D\,max}$ for $K_{p}=0.16$ , $x_{p}=0.25$ , and $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ are used to in the model of Strickland & Graham (Reference Strickland and Graham1987). Equations (4.3) and (4.4) calculate the lift and drag coefficients according to Granlund et al. (Reference Granlund, Ol and Bernal2013) using the experimental values of $C_{L\,max}^{x_{p}=0.75}$ and $C_{D\,max}^{x_{p}=0.75}$ for $x_{p}=0.75$ . Figure 16 highlights the benefits and deficiencies of the ACM. The main benefit is clearly a better prediction of the response at high angle of attack $\unicode[STIX]{x1D6FC}>45^{\circ }$ . This is particularly evident regarding the drag coefficient, which is overestimated at high angle of attack with both the semiempirical algebraic models, while the ACM captures the saturation and drop of $C_{D}$ . Nevertheless, the $C_{Dmax}$ and its corresponding angle of attack are slightly overestimated. In contrast, the lift coefficient $C_{Lmax}$ is underestimated in comparison with experiments. Figure 15 shows that the $C_{Nmax}$ predicted by the ACM model is slightly underestimated. It is also slightly delayed in time, and thus appears at a higher angle of attack. The cosine projection that reduces the $C_{L}$ value at high angle of attack can then explain the underestimation of the $C_{Lmax}$ in figure 16, while the sine projection enlarges the $C_{Dmax}$ value.

Figure 17.

Lift (a), drag (b) and pitching moment (c) coefficient evolution with the angle of attack for $x_{p}=0.25,\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ and $K_{p}=0.02$ , 0.08 and 0.16; lift (d), drag (e) and pitching moment (f) coefficient evolution with the angle of attack for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ , $K_{p}=0.16$ and five pivot axis positions $x_{p}=0$ , 0.25, 0.5, 0.75, 1; comparison of the ACM model with experiment.

ACM predictions for the lift, drag and moment coefficient as a function of the angle of attack are compared with experiment in figure 17(a–c), for three reduced pitch rates $K_{p}=0.02$ , 0.08 and 0.16, $x_{p}=0.25$ , and $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ . Qualitatively the ACM captures the main effect of the reduced pitch rate: increase of the overshoot in $C_{L}$ , $C_{D}$ and $C_{M}$ along with a coherent shifting of the associated angle of attack with $K_{p}$ . Quantitatively, the main problem of the ACM is the above-mentioned delayed angle of attack at which $C_{Nmax}$ is attained, which leads to the underestimation of $C_{Lmax}$ after the cosine projection. One can also notice an overestimation of the first-stage response in $C_{L}$ and $C_{M}$ when $K_{p}$ increases. As discussed previously, regarding the temporal results in figure 15, this could be the consequence of a prediction of the first-stage response slightly in advance in comparison with experiment, probably due to a failure to predict the delay between the LEV formation and the pitch rate in the time-dependent formulation.

Figure 17(d–f) shows the ability of the ACM to predict the effect of the pivot point location on the responses of the lift, drag and moment coefficients with the angle of attack. ACM results are plotted with experiments for $x_{p}=0,0.25,0.5,0.75$ and 1, $K_{p}=0.16$ , and $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ . The qualitative agreement is good. As pointed out previously, the $C_{Lmax}$ is underestimated but the ACM captures its decrease as the pivot point location moves towards the trailing edge. Not surprisingly, the ACM does not predict the singular behaviour experimentally observed for $x_{p}=1$ . Indeed, the pivot point-dependent term is proportional to the relative distance between the pivot point and the $3/4$ -chord location. The ACM also predicts an increase of the angle of attack at which $C_{Lmax}$ and $C_{Dmax}$ occur when the pivot point moves towards the trailing edge, but lower than the one observed in experiments. Finally, the general trend of the $C_{M}$ evolution with the angle of attack is also rather convincing, but a better prediction of the first stage of the dynamic response could be done.

Figure 18.

Lift (a), drag (b) and pitching moment (c) coefficient evolution with the angle of attack for $x_{p}=0.5$ , $K_{p}=0.08$ , and three maximum pitch angle $\unicode[STIX]{x1D6FC}_{max}=45^{\circ },60^{\circ },90^{\circ }$ . The amplitude coefficient $A=5.7$ has been used for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ . It has been adjusted according to figure 12(a) for $\unicode[STIX]{x1D6FC}_{max}<90^{\circ }$ : $A=8.24$ for $\unicode[STIX]{x1D6FC}_{max}=60^{\circ }$ and $A=7.49$ for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ . A unique value for the LEV centre of pressure, $x_{LEV}=0.39$ , has been used for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ },60^{\circ },90^{\circ }$ .

5.2 Moderate-amplitude pitch ramp motion

In figure 18, ACM predictions are compared with the experimental results for pitch ramp motions with pivot point location $x_{p}=0.5$ , moderate pitch rate $K_{p}=0.08$ , and three different maximum angles of attack, $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ . For $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ , the average amplitude coefficient $A=5.7$ , determined in § 4.3, has been used. The amplitude coefficient has been adjusted for $\unicode[STIX]{x1D6FC}_{max}<90^{\circ }$ according to figure 12(a): $A=8.24$ has been used for $\unicode[STIX]{x1D6FC}_{max}=60^{\circ }$ and $A=7.49$ for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ . For the moment coefficient model, a unique value for the LEV centre of pressure, $x_{LEV}=0.39$ , has been used for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ .

Results for the drag are in good agreement with experiments for $\unicode[STIX]{x1D6FC}_{max}=60^{\circ }$ and $90^{\circ }$ . However, for $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ , the overshoot in drag is underestimated. For the lift coefficient, the gap between the ACM prediction and experiment increases for lower values of $\unicode[STIX]{x1D6FC}_{max}$ . As pointed out previously, the ACM fails to predict the correct delay between the circulation buildup and the pitch rate in the time-dependent formulation. As a consequence, the magnitude of the lift is overestimated at the beginning of the ramp motion and the model predicts an overshoot at lower angle of attack. Moment coefficient results also highlight significant discrepancies. For $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ the ACM model underestimates the response for angle of attack below $45^{\circ }$ and fails to predict the maximum experimentally observed close to $30^{\circ }$ . The model prediction is slightly better for $\unicode[STIX]{x1D6FC}_{max}=60^{\circ }$ . For $\unicode[STIX]{x1D6FC}_{max}=45^{\circ }$ a singular experimental behaviour is observed. The moment coefficient first increases, reaches a plateau for $20^{\circ }<\unicode[STIX]{x1D6FC}<30^{\circ }$ and exhibits a rather sharp bump with a maximum close to $\unicode[STIX]{x1D6FC}=42^{\circ }$ . This peculiar behaviour can be due to the mechanical vibrations of the set-up. The ACM fails to predict that, and largely overestimates the moment coefficient evolution with the angle of attack.

5.3 Comparison with experiments for sharper ramp motions and higher reduced pitch rates

The aim of this section is to evaluate the proposed ACM for sharper ramp motions and higher pitch rates. Such experimental results can be found in Granlund et al. (Reference Granlund, Ol and Bernal2013) for a flat-plate model of thickness-to-chord ratio 2 %, round leading and trailing edge, and a physical aspect ratio of 6. Granlund and co-authors’ experiments were conducted in a water tunnel at a Reynolds number $Re=2\times 10^{4}$ , for a 2D (i.e. wall-to-wall) flat-plate model experiencing $0^{\circ }{-}90^{\circ }$ pitch ramp motions. In the present section the ACM is compared with their experimental results for a pivot location at the leading edge, sharp ramp motion characterized by a smoothing parameter of $\unicode[STIX]{x1D70E}=0.9$ , and reduced pitch rates $K_{p}$ up to 0.5.

Figure 19.

Evolution of the lift (a) and drag (b) coefficient with the angle of attack for $x_{p}=0,\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ and $K_{p}=0.1,0.2,0.3$ and 0.5; comparison of the ACM model with experimental results in Granlund et al. (Reference Granlund, Ol and Bernal2013); 2D flat plate of thickness-to-chord ratio 2 %, round leading and trailing edges, sharp pitch ramp motion ( $\unicode[STIX]{x1D70E}=0.9$ ) at $Re=2\times 10^{4}$ .

The ACM is implemented according to the experimental conditions in Granlund et al. (Reference Granlund, Ol and Bernal2013). The pitch ramp motion is calculated using the revised hyperbolic-cosine function (2.1) with a smoothing parameter $\unicode[STIX]{x1D70E}=0.9$ . The translational circulatory term $C_{N}^{Trans}$ (4.13) has been calculated using the steady aerodynamic curve provided in Granlund et al. (Reference Granlund, Ol and Bernal2013). Based on a set of four unsteady experimental results, i.e. $K_{p}=0.1,0.2,0.3$ and 0.5 for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ and $x_{p}=0$ , an average amplitude coefficient $A\approx 3.1$ for the additional circulation term (4.15) has been identified using the optimal method described in § 4.3. Figure 19 compares the lift and drag coefficients predicted by the ACM with the experimental results of Granlund et al. (Reference Granlund, Ol and Bernal2013). Qualitatively the ACM model is in good agreement with experiments. However, quantitatively one can notice a few discrepancies. Thanks to the inertia term (4.11), the ACM predicts the impact of fast non-circulatory load, i.e. initial spike in lift and final decline in drag, due to the large initial acceleration and final deceleration of the sharper pitch ramp motion. However, it overestimates the initial spike in lift (figure 19 a), and for $K_{p}=0.5$ the predicted maximum lift coefficient due to the initial acceleration is more than 50 % higher than that for experiment. However, the effect of the strong deceleration that is responsible for the abrupt decrease of the drag coefficient at the end of the ramp motion is satisfactorily predicted by the ACM (figure 19 b). The extension of the constant-pitch-rate stage for sharper ramp motions also leads the ACM to overestimate and flatten the lift coefficient for $\unicode[STIX]{x1D6FC}<30^{\circ }$ . As a consequence, the maximum lift coefficient is hard to identify with the model. This is due to the formulation of the additional circulation term, for which the time-varying input function is proportional to the pitch rate. This formulation also affects the prediction of the drag coefficient, which is overestimated with the ACM for $\unicode[STIX]{x1D6FC}>60^{\circ }$ .

Regarding the amplitude coefficient value, $A=3.1$ , which has been identified using the set of pitch ramp experiments plotted in figure 19 (i.e. for $K_{p}=0.1,0.2,0.3$ and 0.5), one can make a few comments. It is lower than the value identified in § 4.3 – $A=5.7$ based on the experiments that have been conducted in a wind tunnel for a 2D flat plate at moderate pitch rate $K_{p}\leqslant 0.18$ . Figure 20 compares the evolution of the maximum lift coefficient observed in Granlund et al. (Reference Granlund, Ol and Bernal2013) for $K_{p}=0.01{-}0.5$ , $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ and $x_{p}=0$ , with the one observed in the present study for $K_{p}\leqslant 0.18$ . For $K_{p}\leqslant 0.1$ our results are close to the ones observed by Granlund and co-authors. They depart significantly for $K_{p}\geqslant 0.2$ . What is more, those results suggest that the maximum lift coefficient evolution with the reduced pitch rate has two quasi-linear regimes. The first one, for $K_{p}\leqslant 0.1$ , where a linear regression using our wind tunnel results gives a slope close to 21. And the second one, for $K_{p}>0.1$ , for which the linear slope decreased as shown in figure 20. For $0.18\geqslant K_{p}>0.1$ a linear regression using the wind tunnel results gives a slope close to 15.4. Using the water tunnel results for $K_{p}=0.1{-}0.5$ the slope is lower, with a value close 10.4. Anyway, those results suggest that the amplitude coefficient should be adjusted for $K_{p}>0.1$ , and an amplitude coefficient, $A=3.1$ , has been identified using the unsteady results of Granlund et al. (Reference Granlund, Ol and Bernal2013), as plotted in figure 19. In this figure the ACM prediction for $K_{p}=0.1$ has also been plotted for $A=5.7$ . It shows that the amplitude coefficient associated with the lower pitch rate regime still works well to capture the maximum lift coefficient at this intermediate reduced pitch rate.

Figure 20.

Evolution of the maximum lift coefficient with the reduced pitch rate for $\unicode[STIX]{x1D6FC}_{max}=90^{\circ }$ and $x_{p}=0$ ; (squares) present study, 2D flat plate of thickness-to-chord ratio 4.86 %, sharp edges, smooth pitch ramp motion ( $\unicode[STIX]{x1D70E}=0.16$ ) at $Re\approx 1.45\times 10^{4}$ ; (circles) Granlund et al. (Reference Granlund, Ol and Bernal2013), 2D flat plate of thickness-to-chord ratio 2 %, round leading and trailing edges, sharp pitch ramp motion ( $\unicode[STIX]{x1D70E}=0.9$ ) at $Re=2\times 10^{4}$ .