**2. Characteristics of aeroelastic phenomena**

Aeroelasticity refers to the science of the interaction between aerodynamic, inertial and elas‐ tic effects. Aeroelastic effects occur everywhere but are more or less critical. Any phenomen‐ on that involves a structural response to a fluid action requires aeroelastic consideration. In many cases, when a large and flexible structure is submitted to a high intensity variable flow, the deformations can be very important and become dangerous. Most people are fa‐ miliar with the "auto-destruction" of the Tacoma Bridge. This bridge, built in Washington State, USA, 1.9 km long, was one of the longest suspended bridges of its time. The bridge connecting the Tacoma Narrows channel collapsed in a dramatic way on Thursday, Novem‐ ber 7, 1940. With winds as high as 65-75 km/h, the oscillations increased as a result of fluidstructure interaction, the base of Aeroelasticity, until the bridge collapsed. Recorded videos of the event showed an initial torsional motion of the structure combined very turbulent winds. The superposition of these two effects, added to insufficient structural dumping, am‐ plified the oscillations. Figure 1 below illustrates the visual response of a bridge subject to aeroelastic effects due to variable wind regimes. The simulation was performed using multi‐ physics simulation on ANSYS-CFX software. Some more details on similar aeroelastic mod‐ elling can be viewed from [1], [2], [3] and [4].

#### **Figure 1.** Aeroelastic response of a bridge

the traditional methodology of CFD aeroelastic modeling: mathematical analysis of the phe‐ nomenon, choice of software, computational domain calibration, mesh optimization and tur‐ bulence and transition model validation. An S809 airfoil will be used to illustrate the phenomenon and the obtained results will be compared to experimental ones. The diver‐ gence will be then studied both analytically and numerically to emphasize CFD capacity to model such a complex phenomenon. To illustrate divergence and related study of eigenval‐ ues, an experimental study conducted at NASA Langley will be analyzed and used for com‐ parison with our numerical modeling. In addition to domain, mesh, turbulence and transition model calibration, this case will be used to illustrate fluid-structure interaction and the way it can be tackled in numerical models. Divergence analysis requires the model‐ ing of flow parameters on one side and the inertial and structural behavior of the blade on the other side. These two models should be simultaneously solved and continuous exchange

This chapter will conclude with one of the most dangerous and destructive aeroelastic phe‐ nomena – the flutter. Analytical models and CFD tools are applied to model flutter and the results are validated with experiments. This example is used to illustrate the application of aeroelastic modeling to predictive control. The computational requirements for accurate aer‐ oelastic modeling are so important that the calculation time is too large to be applied for real time predictive control. Hence, flutter will be used as an example to show how we can use CFD based offline results to build Laplacian based faster models that can be used for predic‐

Aeroelasticity refers to the science of the interaction between aerodynamic, inertial and elas‐ tic effects. Aeroelastic effects occur everywhere but are more or less critical. Any phenomen‐ on that involves a structural response to a fluid action requires aeroelastic consideration. In many cases, when a large and flexible structure is submitted to a high intensity variable flow, the deformations can be very important and become dangerous. Most people are fa‐ miliar with the "auto-destruction" of the Tacoma Bridge. This bridge, built in Washington State, USA, 1.9 km long, was one of the longest suspended bridges of its time. The bridge connecting the Tacoma Narrows channel collapsed in a dramatic way on Thursday, Novem‐ ber 7, 1940. With winds as high as 65-75 km/h, the oscillations increased as a result of fluidstructure interaction, the base of Aeroelasticity, until the bridge collapsed. Recorded videos of the event showed an initial torsional motion of the structure combined very turbulent winds. The superposition of these two effects, added to insufficient structural dumping, am‐ plified the oscillations. Figure 1 below illustrates the visual response of a bridge subject to aeroelastic effects due to variable wind regimes. The simulation was performed using multi‐ physics simulation on ANSYS-CFX software. Some more details on similar aeroelastic mod‐

of data is essential as the fluid behavior affects the structure and vice-versa.

tive control. The results of this model will be compared to experiments.

**2. Characteristics of aeroelastic phenomena**

88 Advances in Wind Power

elling can be viewed from [1], [2], [3] and [4].

In an attempt to increase power production and reduce material consumption, wind tur‐ bines' blades are becoming increasingly large yet, paradoxically, thinner and more flexible. The risk of occurrence of damaging aeroelastic effects increases significantly and justifies the efforts to better understand the phenomena and develop adequate design tools and mitiga‐ tion techniques. Divergence and flutter on an airfoil will be used as introduction to aeroelas‐ tic phenomena. When a flexible structure is subject to a stationary flow, equilibrium is established between the aerodynamic and elastic forces (inertial effects are negligible due to static condition). However, when a certain critical speed is exceeded, this equilibrium is dis‐ rupted and destructive oscillations can occur. This is illustrated with Figure 2 where α is the angle of attack due to a torsional movement as a result of aerodynamic solicitations.

**Figure 2.** Airfoil model to illustrate aerodynamic flutter

If we consider an angle of attack sufficiently small such that *cos*α ≈1 and *sin*α≈ α, and writ‐ ing the equilibrium of the moments, M, with respect to the centre of the rotational spring, we have:

∑ M=0

$$\mathbf{\color{red}{L}L\mathbf{e} + Wd\mathbf{\color{red}{L}} - K\_{\alpha}\alpha = 0} \tag{1}$$

Where the lift L is:

$$\mathbf{L} = \mathbf{q} \mathbf{S} \mathbf{C}\_l = \mathbf{q} \mathbf{S} \mathbf{M}\_0 \mathbf{a} \tag{2}$$

S, surface area of the profile, Cl is the lift coefficient, M0 is the moment coefficient. This leads to an angle of attack at equilibrium corresponding to:

$$\alpha = \frac{\text{Wd}}{\text{K}\_a \cdot \text{q} \text{SM}\_0 \text{e}} \tag{3}$$

For a zero flow condition, the angle of attack αz, is such that:

$$
\alpha\_x = \frac{\text{Wd}}{\text{K}\_a} \tag{4}
$$

Divergence occurs when denominator in equation (3) becomes 0 and this corresponds to a dynamic pressure, qD expressed as:

$$\mathbf{q}\_{\rm ID} = \frac{\mathbf{K}\_n}{\mathbf{e} \mathbf{S} \mathbf{M}\_0} \tag{5}$$

Therefore:

$$
\alpha = \frac{\alpha\_x}{\left[1 - \left(\frac{q}{q\_D}\right)\right]} \tag{6}
$$

When velocity increases such that dynamic pressure q approaches critical dynamic pressure qD, the angle of attack dangerously increases until a critical failure value – divergence. This is solely a structural response due to increased aerodynamic solicitation due to fluid-struc‐ ture interaction. This is an example of a *static* aeroelastic phenomenon as it involves no vi‐ bration of the airfoil. Flutter is an example of a *dynamic* aeroelastic phenomenon as it occurs when structure vibration interacts with fluid flow. It arises when structural damping be‐ comes insufficient to damp aerodynamic induced vibrations. Flutter can appear on any flexi‐ ble vibrating object submitted to a strong flow with positive retroaction between flow fluctuations and structural response. When the energy transferred to the blade by aerody‐ namic excitation becomes larger than the normal dynamic dissipation, the vibration ampli‐ tude increases dangerously. Flutter can be illustrated as a superposition of two structural modes – the angle of attack (pitch) torsional motion and the plunge motion which character‐ ises the vertical flexion of the tip of the blade. Pitch is defined as a rotational movement of the profile with respect to its elastic center. As velocity increases, the frequencies of these oscillatory modes coalesce leading to flutter phenomenon. This may start with a rotation of the blade section (at t=0 s in Figure 3). The increased angle amplifies the lift such that the section undertakes an upward vertical motion. Simultaneously, the torsional rigidity of the structure recoils the profile to its zero-pitch condition (at t=T/4 in Figure 3). The flexion ri‐ gidity of the structure tends to retain the neutral position of the profile but the latter then tends to a negative angle of attack (at t = T/2 in Figure 3). Once again, the increased aerody‐ namic force imposes a downward vertical motion on the profile and the torsional rigidity of the latter tends to a zero angle of attack. The cycle ends when the profile retains a neutral position with a positive angle of attack. With time, the vertical movement tends to damp out whereas the rotational movement diverges. If freedom is given to the motion to repeat, the rotational forces will lead to blade failure.

**Figure 3.** Illustration of flutter movement

<sup>α</sup><sup>=</sup> Wd

<sup>α</sup><sup>z</sup> <sup>=</sup> Wd Kα

qD <sup>=</sup> <sup>K</sup><sup>α</sup> eSM0

<sup>α</sup><sup>=</sup> <sup>α</sup><sup>z</sup> <sup>1</sup> - ( <sup>q</sup> qD

Divergence occurs when denominator in equation (3) becomes 0 and this corresponds to a

When velocity increases such that dynamic pressure q approaches critical dynamic pressure qD, the angle of attack dangerously increases until a critical failure value – divergence. This is solely a structural response due to increased aerodynamic solicitation due to fluid-struc‐ ture interaction. This is an example of a *static* aeroelastic phenomenon as it involves no vi‐ bration of the airfoil. Flutter is an example of a *dynamic* aeroelastic phenomenon as it occurs when structure vibration interacts with fluid flow. It arises when structural damping be‐ comes insufficient to damp aerodynamic induced vibrations. Flutter can appear on any flexi‐ ble vibrating object submitted to a strong flow with positive retroaction between flow fluctuations and structural response. When the energy transferred to the blade by aerody‐ namic excitation becomes larger than the normal dynamic dissipation, the vibration ampli‐ tude increases dangerously. Flutter can be illustrated as a superposition of two structural modes – the angle of attack (pitch) torsional motion and the plunge motion which character‐ ises the vertical flexion of the tip of the blade. Pitch is defined as a rotational movement of the profile with respect to its elastic center. As velocity increases, the frequencies of these oscillatory modes coalesce leading to flutter phenomenon. This may start with a rotation of the blade section (at t=0 s in Figure 3). The increased angle amplifies the lift such that the section undertakes an upward vertical motion. Simultaneously, the torsional rigidity of the structure recoils the profile to its zero-pitch condition (at t=T/4 in Figure 3). The flexion ri‐ gidity of the structure tends to retain the neutral position of the profile but the latter then tends to a negative angle of attack (at t = T/2 in Figure 3). Once again, the increased aerody‐ namic force imposes a downward vertical motion on the profile and the torsional rigidity of the latter tends to a zero angle of attack. The cycle ends when the profile retains a neutral position with a positive angle of attack. With time, the vertical movement tends to damp out whereas the rotational movement diverges. If freedom is given to the motion to repeat, the

For a zero flow condition, the angle of attack αz, is such that:

dynamic pressure, qD expressed as:

rotational forces will lead to blade failure.

Therefore:

90 Advances in Wind Power

<sup>K</sup><sup>α</sup> - qSM0e (3)

) (6)

(4)

(5)
