2. Formulation

We consider a two-dimensional flexible plate of chord length $c$ and thickness $\varepsilon$ immersed in a uniform fluid current of velocity $U$ in the $x$ direction and undergoing pitching motion about the leading edge with angle

(2.1)\begin{equation} \alpha(t) = \alpha_0 \sin (2 {\rm \pi}f t) \end{equation}

in relation to the $x$ axis (see figure 1), where $\alpha _0$ and $f$ are the pitch amplitude and frequency (in Hz), respectively. For a thin beam ($\varepsilon /c \ll 1$) with structural bending rigidity $E I$ and extensional rigidity characterized by structural tension $T$, the nonlinear equation of motion can be written as (Connell & Yue Reference Connell and Yue2007)

(2.2)$$\begin{gather} \rho_s \varepsilon\,\frac{\partial^2 {\boldsymbol{r}}}{\partial t^2} - \frac{\partial}{\partial s} \left[ T(s,t)\,\frac{\partial {\boldsymbol{r}}}{\partial s} \right] + E I\,\frac{\partial^4 {\boldsymbol{r}}}{\partial s^4} = {\boldsymbol{f}}(s,t) + {\boldsymbol{F}}_r\,\delta(s-s_0) - {\boldsymbol{g}}\,\delta'(s-s_0), \end{gather}$$

(2.3a,b)$$\begin{gather}T(s,t)= E \varepsilon \left[ 1- \left(\frac{\partial {\boldsymbol{r}}}{\partial s} \, \frac{\partial {\boldsymbol{r}}}{\partial s} \right)^{-1/2} \right] , \quad E I = \frac{E \varepsilon^3}{12} , \end{gather}$$

with $s$ the Lagrangian coordinate along the plate centreline, $s=0$ at the leading edge, and position vector ${\boldsymbol {r}}(s,t)$ fixed at the leading edge. Here, $\rho _s$ is the solid density and

(2.4a,b)\begin{equation} {\boldsymbol{f}} = {\boldsymbol{\tau}}^+ \boldsymbol{\cdot} {\boldsymbol{n}}^+{+} {\boldsymbol{\tau}}^- \boldsymbol{\cdot} {\boldsymbol{n}}^- , \quad {\boldsymbol{\tau}} =- p {{\boldsymbol{\mathsf{I}}}} + \mu ( {\boldsymbol{\nabla}} {\boldsymbol{u}} + ({\boldsymbol{\nabla}} {\boldsymbol{u}})^{\rm T} ), \end{equation}

is the Lagrangian force (per unit area) exerted on the plate by the surrounding fluid, where ${\boldsymbol {\tau }}$ is the fluid stress tensor, ${{\boldsymbol{\mathsf{I}}}}$ is the unit tensor, and ${\boldsymbol {n}}^+$ and ${\boldsymbol {n}}^-$ are the unit normal vectors on each side of the plate at the given point ${\boldsymbol {r}}$ (see figure 1). The fluid pressure $p$ and velocity ${\boldsymbol {u}}$ satisfy the incompressible Navier–Stokes equations,

(2.5a,b)\begin{equation} {\boldsymbol{\nabla}} \boldsymbol{\cdot} {\boldsymbol{u}} =0 , \quad \rho \left(\frac{\partial {\boldsymbol{u}}}{\partial t} + {\boldsymbol{u}} \boldsymbol{\cdot} {\boldsymbol{\nabla}} {\boldsymbol{u}} \right) = {\boldsymbol{\nabla}} \boldsymbol{\cdot} {\boldsymbol{\tau}} , \end{equation}

with $\rho$ and $\mu$ being the fluid density and viscosity, respectively. The velocity boundary conditions are ${\boldsymbol {u}}=\partial {\boldsymbol {r}}/\partial t$ on the fluid–plate moving boundary, and ${\boldsymbol {u}} = U {\boldsymbol {e}}_x$ at infinity.

Figure 1. Schematic of the flexible pitching plate (dimensional quantities).

The last two terms on the right-hand side of (2.2) are included to account for the point reaction ${\boldsymbol {F}}_r$ (per unit span) that fixes the leading edge at the origin of coordinates, and for the point torque at the leading edge, modelled here as a couple of opposite point forces $\pm {\boldsymbol {g}}$ that provide the necessary torque ${\boldsymbol {M}}_r$ to generate the pitching motion (2.1). Mathematically, the localized force is modelled by Dirac's delta function $\delta$, and the torque by its derivative $\delta '$ (Fernandez-Feria & Alaminos-Quesada Reference Fernandez-Feria and Alaminos-Quesada2021), both applied at point $s=s_0$ that is chosen very close to the leading edge, $s_0 \to 0^+$, but not exactly at $s=0$ for structural (and mathematical) reasons.

The boundary conditions at the constrained leading edge are

(2.6)\begin{equation} {\boldsymbol{r}}={\boldsymbol{0}} \quad \text{and} \quad \frac{\partial {\boldsymbol{r}}}{\partial s} = \cos \alpha(t)\,{\boldsymbol{e}}_x + \sin \alpha(t)\,{\boldsymbol{e}}_y \equiv {\boldsymbol{e}}_\alpha (t)\quad \text{at } s=0 , \end{equation}

where ${\boldsymbol {e}}_\alpha (t)$ is the unit vector tangent to the aerofoil at the leading edge, while at the free trailing edge of the plate,

(2.7)\begin{equation} -T\,\frac{\partial {\boldsymbol{r}}}{\partial s} + E I\,\frac{\partial^3 {\boldsymbol{r}}}{\partial s^3}={\boldsymbol{0}} , \quad \frac{\partial^2 {\boldsymbol{r}}}{\partial s^2} ={\boldsymbol{0}}\quad \text{at } s =s_f \simeq c , \end{equation}

where $s_f(t)$ is the length of the plate at the instant $t$, which for large extensional rigidity is practically equal to $c$.

Integrating (2.2) between $s=0$ and $s=s_f$, using integration by parts and applying the above boundary conditions, one obtains the reaction ${\boldsymbol {F}}_r$ at the leading edge (actually at $s_0 \simeq 0$) as

(2.8) \begin{align} {\boldsymbol{F}}_r(t) &=- {\boldsymbol{F}}(t) +\rho_s \varepsilon \int_0^{s_f} \frac{\partial^2 {\boldsymbol{r}}(s,t)}{\partial t^2} \, {\rm d} s -T(0,t)\, {\boldsymbol{e}}_\alpha(t) +EI \left[ \frac{\partial^3 {\boldsymbol{r}} }{\partial s^3} \right]_{s=0} , \notag\\ {\boldsymbol{F}} (t) &\equiv \int_0^{s_f} {\boldsymbol{f}}(s,t) \, {\rm d} s ,\end{align}

where ${\boldsymbol {F}}$ is the total force (per unit span) exerted by the fluid on the plate. On the other hand, multiplying (2.2) vectorially by ${\boldsymbol {r}}$ and integrating between $s=0$ and $s=s_f$, using again integration by parts and the boundary conditions, the torque at the leading edge can be written as

(2.9)\begin{equation} {\boldsymbol{M}}_r(t) \equiv {\boldsymbol{e}}_\alpha \wedge {\boldsymbol{g}}(t) =- {\boldsymbol{M}}(t) + \rho_s \varepsilon \int_0^{s_f} {\boldsymbol{r}} \wedge \frac{\partial^2 {\boldsymbol{r}}(s,t)}{\partial t^2} \, {\rm d} s - EI \left[ \frac{\partial {\boldsymbol{r}}}{\partial s} \wedge \frac{\partial^2 {\boldsymbol{r}}}{\partial s^2} \right]_{s=0},\end{equation}

where

(2.10)\begin{equation} {\boldsymbol{M}}(t) \equiv \int_0^{s_f} {\boldsymbol{r}} \wedge {\boldsymbol{f}}(s,t) \, {\rm d} s \end{equation}

is the total bending moment (per unit span) with respect to the leading edge that the fluid exerts on the aerofoil.

The last terms in (2.8) and (2.9) evaluated at $s=0$ come from the fact that the leading edge is not a free end like the trailing edge, but it is constrained with the pitching motion (2.6). If the stretching stiffness is very large – more precisely, if the non-dimensional stretching stiffness is $K= E \varepsilon /(\rho U^2 c) \gg 1$ – in first approximation $|\partial {\boldsymbol {r}}/\partial s|= 1$, $T=0$, $s_f=c$, and one can neglect the second term in (2.2). Consequently, the term in (2.8) containing the tension $T$ at the leading edge can be neglected. On the other hand, according to the condition (2.6), the position vector close to the leading edge (i.e. for small $s$) can be written as

(2.11)\begin{equation} {\boldsymbol{r}}(s,t) = s\,{\boldsymbol{e}}_\alpha (t) + \left[ \frac{\partial^2 {\boldsymbol{r}}}{\partial s^2} \right]_{s=0} \frac{s^2}{2} + \left[ \frac{\partial^3 {\boldsymbol{r}} }{\partial s^3} \right]_{s=0} \frac{s^3}{6} + \cdots , \end{equation}

which allows the computation of the derivatives at $s=0$ of the last terms in (2.8) and (2.9) proportional to the bending rigidity $EI$ from the knowledge of ${\boldsymbol {r}}$ near the leading edge, calculated either numerically or experimentally. Clearly, these terms vanish for a rigid foil, where ${\boldsymbol {r}}(s,t) = s\,{\boldsymbol {e}}_\alpha (t)$ everywhere (see below). But even for flexible foils, in practical applications the bending rigidity has to be sufficiently large for efficient propulsion (e.g. Alben Reference Alben2008; Moore Reference Moore2014), and though the flexural deflection may become large in some parts of the foil, at the leading edge the curvature cannot be large if the actuating torque has to generate an efficient pitching motion of the foil. Thus these terms evaluated at the leading edge are usually small compared with the inertial terms in most practical applications (see § 3), and will be neglected in what follows.

Equations (2.8) and (2.9) provide the force and torque ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M_r}}(t)$ that one has to apply at the leading edge to generate the desired pure pitching motion (2.1) in terms of the fluid force and moment, ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$, and the kinematics ${\boldsymbol {r}}(s,t)$ of the flexible aerofoil of known structural parameters and geometry. Alternatively, these equations provide the fluid force and moment, ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$, if the reactions ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M_r}}(t)$ are measured experimentally and the instantaneous position ${\boldsymbol {r}}(s,t)$ of the flexible aerofoil is obtained either numerically or experimentally. Here, ${\boldsymbol {r}}(s,t)$ is obtained numerically along with ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$, and the resulting ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M_r}}(t)$ will be compared with experimental measurements in a wind tunnel. But the expressions can be used the other way around, to obtain ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$ from experimental measurements of ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M_r}}(t)$, and simultaneous video recording of the aerofoil motion to determine ${\boldsymbol {r}}(s,t)$.

For a rigid aerofoil, i.e. when the non-dimensional stiffness ratio is $S=E \varepsilon ^3/(\rho U^2 c^3) \gg 1$, one has ${\boldsymbol {r}}(s,t)=s\,{\boldsymbol {e}}_\alpha (t) = s [ \cos \alpha (t)\,{\boldsymbol {e}}_x + \sin \alpha (t)\, {\boldsymbol {e}}_y]$. Thus ${\boldsymbol {r}}$ is known if $\alpha (t)$ is known, and the inertial and structural terms in (2.8) and (2.9) can be obtained analytically (all the structural terms at $s=0$ vanish):

(2.12)$$\begin{gather} {\boldsymbol{F}} =- {\boldsymbol{F}}_r + \frac{m c}{2}\,[ -( \dot{\alpha}^2 \cos \alpha + \ddot{\alpha} \sin \alpha ) {\boldsymbol{e}}_x + ( -\dot{\alpha}^2 \sin \alpha + \ddot{\alpha} \cos \alpha ) {\boldsymbol{e}}_y ] , \end{gather}$$

(2.13)$$\begin{gather}{\boldsymbol{M}} =- {\boldsymbol{M}}_r + \frac{m c^2}{3}\,\ddot{\alpha} {\boldsymbol{e}}_z , \end{gather}$$

where $m=\rho _s \varepsilon c$ is the mass of the aerofoil per unit span, and the dots denote time derivatives. These expressions can be used to determine ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$ just measuring $\alpha (t)$ synchronously with ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M_r}}(t)$.

For a flexible aerofoil, the inertial and structural terms on the right-hand sides of (2.12) and (2.13) are more complex and cannot be obtained analytically, in general. They have to be computed by integrating numerically the second terms on the right-hand sides of (2.8) and (2.9), and computing the derivatives of ${\boldsymbol {r}}(s,t)$ at $s=0$ for the additional terms associated with the rigidities of the aerofoil at the leading edge, once the kinematics of the foil ${\boldsymbol {r}}(s,t)$ is obtained experimentally or numerically. In either case, one may discretize the aerofoil initial centreline into $N$ segments of length ${\rm \Delta} s=c/N$ to follow the time evolutions of the positions of each segment central point: ${\boldsymbol {r}}_i(t)=x_i(t)\,{\boldsymbol {e}}_x+y_i(t)\,{\boldsymbol {e}}_y$, $i=1, \ldots, N$. Then the terms containing ${\boldsymbol {r}}$ in (2.8) and (2.9) are approximated by finite differences. Using, for instance, first-order finite differences and neglecting, as mentioned before, the structural terms at $s=0$, one obtains the following approximate relations between [${\boldsymbol {F}}_r(t)$, ${\boldsymbol {M}}_r(t)$] and [${\boldsymbol {F}}(t)$, ${\boldsymbol {M}}(t)$]:

(2.14)$$\begin{gather} {\boldsymbol{F}} \simeq- {\boldsymbol{F}}_r + \sum_{i=1}^N m_i \ddot{{\boldsymbol{r}}}_i , \end{gather}$$

(2.15)$$\begin{gather}{\boldsymbol{M}} \simeq- {\boldsymbol{M}}_r + \sum_{i=1}^N m_i {\boldsymbol{r}}_i \wedge \ddot{{\boldsymbol{r}}}_i , \end{gather}$$

where $m_i = \rho _s \varepsilon ({\rm \Delta} s)$ is the mass (per unit span) of each element of the plate. For a rigid aerofoil, ${\boldsymbol {r}}_i=(i-1/2)({\rm \Delta} s) {\boldsymbol {e}}_\alpha$, and one recovers exactly the analytical expressions (2.12) and (2.13), knowing that $m= \sum _{i=1}^N m_i=N m_i$ and $c=N ({\rm \Delta} s)$. The effect of flexibility on inertia force and moment can be obtained by substituting ${\boldsymbol {r}}_i (t)= (i-1/2)({\rm \Delta} s)\,{\boldsymbol {e}}_\alpha (t) + {\boldsymbol {d}}_i(t)$ into (2.14) and (2.15) and subtracting (2.12) and (2.13), respectively.

Although the approximations made in (2.14) and (2.15) have errors of the order of ${\rm \Delta} s = c/N$, we will show in § 3 that $N$ need not be very large to get an excellent approximation of the inertial terms if the bending stiffness is not too low.

3. Numerical method and results

The flow–structure interaction (FSI) problem is solved numerically using the coupled fluid and solid equations of motion discretized in both the fluid and plate domains separated by a moving boundary. In addition to (2.5a,b) for the fluid, more general differential structural equations inside the plate are used instead of the thickness-averaged nonlinear equation of motion (2.2), as described in Sanmiguel-Rojas & Fernandez-Feria (Reference Sanmiguel-Rojas and Fernandez-Feria2021), but now for the pitching motion (2.1) imposed at the leading edge instead of a heaving motion. For both fluid and solid, the finite-volume-based solver Ansys Fluent is used, with the well known $k\unicode{x2013}\omega$ SST turbulence model for the fluid, and the intrinsic FSI algorithm to simulate the fluid–structure coupling. For the Reynolds numbers considered (approximately $10^5$; see § 4), the $k\unicode{x2013}\omega$ SST turbulence model implemented in the solver package Ansys Fluent has been sufficiently validated experimentally for transitional flows in similar aerodynamic problems (e.g. Gharali & Johnson Reference Gharali and Johnson2013). All simulations are started from rest. The harmonic motion (2.1) is emulated by a user-defined function. Actually, to avoid spurious initial oscillations, $\alpha (t)=[(1 + \tanh ((t-t_0)/t_1))/2] \alpha _0 \sin (2 {\rm \pi}f t)$ is used in place of (2.1), with $t_0=0.1/f$ and $t_1=0.5/f$.

The computational domain, which reproduces the dimensions of the wind tunnel test section described in § 4, together with the different regions into which the computational domain is decomposed to ease the meshing procedure, is given in figure 2(a). Figure 2(b) shows the dimensions of the computational domain and the aerofoil (not to scale). With larger computational domains, the results practically did not change even in the most demanding cases. The aerofoil, of chord length $c = 0.3 \ {\rm m}$, consists of a steel plate with thickness $\varepsilon = 0.4 \ {\rm mm}$ attached to an aluminium head of diameter $12 \ {\rm mm}$. These dimensions correspond to the aerofoil used in the experiments described in § 4. The boundary conditions are: a flat velocity profile at a length $3.5 c$ upstream of the foil; non-slip walls on the foil and tunnel walls; and an outlet boundary condition with pressure $p=0$ at a length $10 c$ downstream of the aerofoil. Air is used in all the computations ($\rho =1.225$ kg m$^{-3}$, $\mu =1.789 \times 10^{-5}$ kg m$^{-1}$ s$^{-1}$). Details about the mesh and the grid convergence study are given in the Appendix.

Figure 2. (a) Computational domain and mesh grid regions (to scale). (b) Dimensions used in the computations and the wind tunnel experiments, not to scale. LE, leading edge; TE, trailing edge.

Once the FSI problem is solved numerically for a given aerofoil and given values of $U$, $\alpha _0$ and $f$, the force ${\boldsymbol {F}}(t)$ and moment ${\boldsymbol {M}}(t)$ that the fluid exerts on the flexible aerofoil at each instant $t$ are computed. In addition, the position ${\boldsymbol {r}}(s,t)$ of the foil centreline is also recorded at the same instants of time. With this information, one may compute the reactions at the leading edge ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$ from (2.14) and (2.15) by discretizing the aerofoil centreline into $N$ segments and using the time evolution of their central points ${\boldsymbol {r}}_i(t)$.

Interestingly, it is found that even with a relatively small number of elements, the approximation made in (2.14) and (2.15) works quite well. This is shown in figure 3, where the two components of ${\boldsymbol {F}}_r(t)$ and the only component of ${\boldsymbol {M}}_r(t)$ are compared as the number $N$ of plate elements used in the approximations (2.14) and (2.15) is increased, once ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$ are computed numerically. In particular, the results obtained with $N=3$, 6, 11 and $21$ plate elements are compared for one of the most unfavourable cases considered in the experimental measurements reported in § 4, with a flexural deflection at the trailing edge of approximately $0.15 c$ (see figure 4). The results are practically indistinguishable. (In figure 3, ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$ are also shown, with dotted lines, for reference, showing that they are quite different from ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$.) This property makes this method very suitable for experimentation, for ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$ may be computed from (2.14) and (2.15), measuring ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$, and simultaneously recording the position of just a few marker points ${\boldsymbol {r}}_i(t)$ along the flexible plate chord length. It is also checked that the contributions of the structural terms at $s=0$ in (2.8) and (2.9), neglected in (2.14) and (2.15), are very small, less than $1.2 \times 10^{-3}$ N and $4.5 \times 10^{-4}$ N m, respectively, in the case considered in figure 3.

Figure 3. Comparison of ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$ computed with different numbers $N$ of plate elements for the inertial terms, as indicated, for a steel plate with $c=0.3 \ {\rm m}$, $\varepsilon =0.4 \ {\rm mm}$, span $0.31 \ {\rm m}$, $U=7 \ {\rm m}\ {\rm s}^{-1}$, $\alpha _0=3^\circ$ and $f=5 \ {\rm Hz}$. Dotted lines correspond to ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$.

Figure 4. Flexural deflection at the trailing edge $|y_t^f-y_t^r|/c$ (dashed line) for the same case plotted in figure 3, where $y_t^f$ is the trailing edge position of the flexible foil (blue line), and $y_t^r$ is that of an otherwise identical rigid foil (red line).

The method can in principle be applied to any flexural deflection of the plate, with no maximum deflection limit (or minimum stiffness limit). Obviously, the reactions would not converge with just a few points if the bending rigidity is small. However, as mentioned above, the bending rigidity has to be large enough in practical applications for efficient propulsion. Just for reference sake, the non-dimensional bending stiffness $EI/(\rho U^2 c^2)$ of the case considered in figure 3 is approximately unity, with a flexural deflection at the trailing edge that is not too small, as shown in figure 4. In spite of that, the reactions converge with only three points.

4. Validation with experimental measurements in a wind tunnel

As mentioned above, to validate this technique for computing ${\boldsymbol {F}}(t)$ and ${\boldsymbol {M}}(t)$ when ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$ are measured, we proceed in an indirect way, without measuring ${\boldsymbol {r}}(s,t)$ simultaneously to ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$, but solving the problem numerically for ${\boldsymbol {F}}(t)$, ${\boldsymbol {M}}(t)$ and ${\boldsymbol {r}}(t)$, and applying (2.14) and (2.15) to compute ${\boldsymbol {F}}_r(t)$ and ${\boldsymbol {M}}_r(t)$, which then are compared with the reaction force and torque at the leading edge measured experimentally in a wind tunnel.

For the experiments, we use the subsonic, closed-circuit wind tunnel in the Aero-hydrodynamics Laboratory at the University of Malaga, with a test section of $1\times 1 \ {\rm m}^2$ cross-section that is $4 \ {\rm m}$ long. An aerofoil of chord length $c=0.3$ m, consisting of a thin steel ($\rho _s=7864$ kg m$^{-3}$, $E=2.1 \times 10^{11}$ Pa) plate of thickness $\varepsilon = 0.4$ mm, with its leading edge embedded into an aluminium rod of diameter $12$ mm, is mounted vertically onto the force/torque (F/T) sensor installed on the base of the wind tunnel test section $2$ m from its inlet, with resolution $0.006$ N and $0.001$ N m. The steel plate span is $0.31$ m, but an endplate on its top, with a gap of $5$ mm, ensures the two-dimensionality of the flow along the major part of the aerofoil, and reduces end effects (see figure 5; the effective test section is $1\times 0.31$ m$^2$).

Figure 5. Experimental set-up in the wind tunnel test section.

The inlet velocity $U$, measured with a hot wire anemometer (KIMO CTV210), ranges between $0$ and $7$ m s$^{-1}$ in the present experimental runs, well below the maximum velocity provided by the wind tunnel of approximately $40$ m s$^{-1}$. The free stream turbulence intensity is always less than $3\,\%$. The aerofoil is attached through the aluminium rod at its leading edge to a rotational servo motor, with position feedback provided by a high-resolution rotary encoder. The rotation is transmitted from the motor to the aluminium rod through two pulleys and a timing belt, with transmission relation 10 : 3. Thus an accurate harmonic pitching motion $\alpha (t)$ can be transmitted to the aerofoil's leading edge, with amplitude ranging between $2$ and $6$ degrees, and frequency between $3$ and $6$ Hz in the present experimental runs. The velocity $U$ was varied between $0$ and $7$ m s$^{-1}$.

Experimental data of the three force and torque components – pitch angle, and free stream wind velocity and temperature – are synchronized and recorded in a computer through a data acquisition card controlled by the software LabVIEW, with sampling frequency $1000$ Hz. Noise from mechanical vibrations and electronic devices is filtered with a seven-order low-pass Butterworth filter, with cut-off frequency $2f+2$ Hz for $F_{rx}$, $3f+3$ for $F_{ry}$, and $f+2$ for $M_r$. Every experimental run, with a measurement duration of more than $15$ s, is repeated at least three times, from which the standard deviation of each experimental measurement is also recorded.

The aerofoil and F/T sensor rotate together, and an alignment of the sensor coordinates with the free stream is necessary before each experimental run for given $\alpha _0$ and $U$. This is made by measuring the forces on the static aerofoil at different angles between $-\alpha _0$ and $+\alpha _0$ to find the angle at which the modulus of the measured force, $F_r = \sqrt {F_{rx}^2 + F_{ry}^2}$, reaches a minimum. The corresponding angle is then summed to $\alpha (t)$ in the projections of ${\boldsymbol {F}}_r$.

Among the many experimental data recorded for different values of $\alpha _0$, $U$ and $f$ in the ranges mentioned above, figures 6 and 7 show two examples of the forces and torque measured experimentally through the F/T sensor, $F_{rx}(t), F_{ry}(t)$ and $M_r(t)$, compared with the results of the numerical simulations under identical conditions of $\alpha (t)= \alpha _0 \sin (2 {\rm \pi}f t)$ and $U$. Note that time is scaled in figures 6 and 7 with the period $T=1/f$. The corresponding Reynolds number and the reduced frequency, defined as $Re = {\rho c U}/{\mu }$, $k= {{\rm \pi} f c}/{U}$, respectively, are also included in the figure captions for reference. Both $\rho$ and $\mu$ are computed using the instantaneous temperature measurements of the air stream at the inlet of the wind tunnel test section. Also included in the figures are the force and moment exerted by the fluid on the aerofoil, $F_x(t)$, $F_y(t)$ and $M(t)$, as they are obtained numerically. Actually, numerically one obtains ${\boldsymbol {F}}$ and ${\boldsymbol {M}}$ by solving the FSI problem as described in § 3, and then computes ${\boldsymbol {F}}_r$ and ${\boldsymbol {M}}_r$ from (2.14) and (2.15). As mentioned above, the experimental values of $F_{rx}(t)$, $F_{ry}(t)$ and $M_r(t)$, plotted with black dashed lines in figures 6 and 7, correspond to the mean values of at least three experimental runs, while the uncertainty bands are obtained from the standard deviations of these measurements. The uncertainty bands are noticeable only in figure 6, being almost indistinguishable from the mean values for the case plotted in figure 7.

Figure 6. The two components of the reaction force ${\boldsymbol {F}}_r$ and the torque $M_r$ measured experimentally (black dashed lines, with bands accounting for the experimental uncertainty) compared with numerical results obtained with mesh #$1$ and $N=21$ (red lines) for $\alpha _0=3^\circ$, $U=7$ m s$^{-1}$ and $f=5$ Hz ($Re=1.37 \times 10^5$ and $k=0.675$). Also shown with continuous blue lines are the components of the force ${\boldsymbol {F}}$ and the moment $M$ that the fluid exerts on the plate obtained numerically.

Figure 7. As in figure 6, but for $\alpha _0=5^\circ$, $U=5$ m s$^{-1}$ and $f=4$ Hz ($Re=9.57 \times 10^4$ and $k=0.752$).

Although the components of ${\boldsymbol {F}}$ and ${\boldsymbol {M}}$ are quite different from the components of ${\boldsymbol {F}}_r$ and ${\boldsymbol {M}}_r$ in the two cases shown in figures 6 and 7 (and this is so for all the other experiments made), indicating that inertial forces are usually very important in wind tunnel force and torque measurements, the components of ${\boldsymbol {F}}_r$ and ${\boldsymbol {M}}_r$ obtained numerically practically coincide with those measured experimentally. The agreement remains as good as that shown in figures 6 and 7 in most of the more than 40 cases investigated in the ranges of $\alpha _0$, $U$ and $f$ mentioned above. Figure 8 summarizes the comparison between reaction forces and moments measured experimentally and computed numerically for all the cases with $U=5$ m s$^{-1}$, which is by far the velocity most used in the wind tunnel experimental measurements. In particular, we plot the peak-to-peak oscillation amplitudes of the two reaction force components, denoted by ${\rm \Delta} F_{rx}$ and ${\rm \Delta} F_{ry}$ in figures 8(a) and 8(b), and the reaction torque amplitude, ${\rm \Delta} M_r$ in figure 8(c), all of them obtained as an average from the last ten cycles in each case, measured or computed, for five different pitch amplitudes and for frequencies between $3$ and $6$ Hz. Notice that all the force and torque amplitudes show a pronounced peak at approximately $f=4$ Hz, roughly corresponding to the first resonance frequency of the present steel plate in an air current with velocity $U=5$ m s$^{-1}$: $f_r \simeq 3.8$ Hz from linear potential flow theory (Fernandez-Feria & Alaminos-Quesada Reference Fernandez-Feria and Alaminos-Quesada2021), and $f_{r0} \simeq 3.65$ Hz in vacuum, a frequency used in some of the plotted data. The largest discrepancies between measured and computed forces and moments occur near this resonant frequency for the largest pitch amplitude plotted, $\alpha _0=6^\circ$, especially for ${\rm \Delta} M_r$, mostly due to limitations in the F/T sensor for these large torques and forces. For $\alpha > 6^\circ$, the measurements near the resonant frequency were out of the sensor range.

Figure 8. Comparison between the oscillation amplitudes of the two reaction force components, ${\rm \Delta} F_{rx}$ and ${\rm \Delta} F_{ry}$, and the reaction moment, ${\rm \Delta} M_r$, measured experimentally (red squares) and computed numerically (blue circles) for $U \simeq 5$ m s$^{-1}$ and different pitch amplitudes versus the frequency $f$. The different lines connecting the symbols (continuous for the experimental values and dashed for the numerical ones) correspond to, from bottom to top, $\alpha _0=2^\circ$, $3^\circ$, $4^\circ$, $5^\circ$ and $6^\circ$.