Aeroelastic behaviour of a parameterised circulation-controlled wing

Abstract

The aeroelastic behaviour of the wing of a short take-off and landing aircraft using the Coandă effect depends on its properties and shape. An existing reduced-order model is parameterised for detailed investigations. On the one hand, varying mass due to tank level and varying overall stiffness is implemented in the reduced-order model. The influence of the mass change on the aeroelastic behaviour is reflected in the stability maps. On the other hand, two-dimensional steady and unsteady aerodynamics of different nose shapes are investigated with detailed computational fluid simulations and included in the reduced-order model. The dependence on the profile shape and the frequency is described. Their influence on the aeroelastic behaviour is reflected by the stability maps as well.

1 Introduction

To cover the continuously rising air traffic, the use of small, existing airports needs to be extended. This comes along with several problems. For one, the small airports are close to populated areas calling for emission reduction regarding fuel and noise. Furthermore, their runways are shorter than the ones of large airports. To use these short runways, the aircraft design has to be improved significantly.

The Coordinated Research Centre SFB 880 [1] is investigating an aircraft with short take-off and landing capabilities in accordance with the vision Flightpath 2050 [2]. The wing is, therefore, equipped with various technologies. At the trailing edge flap active circulation control making use of the Coandă effect is employed. The leading edge is equipped with a morphing droop nose to raise the stall angle [3].

The aerodynamic aspect of lift gain in steady-state flow with active circulation control has been investigated extensively [4,5,6,7,8]. The aeroelastic aspect ,however, was investigated to a lesser extend [9, 10]. Haas and Chopra [11] detected an additional heave flutter phenomenon occuring only with active circulation control and, therefore, named it circulation control flutter. In contrast to the quasi-elliptical profile without a flap examined by Haas and Chopra, a profile with a Coandă flap and two different leading edge shapes are subject of the current research.

A reduced-order model (ROM) is derived from a full-scale three-dimensional finite element wing model and used for detailed studies. The underlying ROM [15] already considers differing aerodynamic forces due to angle of attack $\alpha$, flap angle $\delta _{\text {fl}}$ and circulation control, represented by $c_\mu$ for the clean nose configuration.

Recently, four new parameters have been included in the ROM:

• mass variation $\eta _{\text {m}}$

• stiffness variation $\eta _{\text {k}}$

• shape of profile

• natural frequency $\omega$

The bending flutter phenomenon due to circulation control is displayed by the results. Variations of the nose shape, mass distribution and exciting frequency are considered and differences identified.

All results of numerical flow simulations presented in this paper are computed with the DLR TAU code of the German Aerospace Center [18], an unstructured finite volume code solving the Reynolds-averaged Navier–Stokes equations (RANS). The original Spalart–Allmaras model is used for turbulence modelling [19].

2 High-lift devices

In 1800, Thomas Young described the fundamental phenomenon of the Coandă effect [12]. Surrounding free stream flow is more reluctant to follow a curved surface than a thin jet. Accordingly, a thin jet is blown out upstream of the plain flaps and aileron to deflect the flow. Hence, the so-called Coandă flap may be deflected up to 80$^\circ$, causing additional lift [13, 14]. A momentum coefficient, describing the normalised jet momentum on the slot, represents the activated amount of circulation control (cf. [10]). The ratio of introduced jet momentum per time at the jet exit section $\dot{m}_\text {jet}v_\text {jet}$ is set to proportion to the freestream dynamic pressure $q_\infty$ and the wing reference area $A_\text {ref}$:

$$\begin{aligned} c_{\mu } = \dfrac{\dot{m}_\text {jet}v_\text {jet}}{q_\infty A_\text {ref}}. \end{aligned}$$

(1)

Three different flow conditions due to the active circulation control may be observed. Figure 1 shows that a momentum coefficient of $c_\mu = 0.027$ leads to a partially adjacent flow on the Coandă flap. Nevertheless, the flow seperates before reaching the trailing edge. Thus, this regime is referred to as “boundary layer control”. When the flow is attached over the whole range whilst using little power, the most efficient momentum coefficient is reached. Further increasing the momentum coefficient, e.g. $c_\mu = 0.054$, leads to a jet effect and continuous deflection of the streamlines after the trailing edge. In this “supercirculation” regime lift is still enhanced, but the ratio of power input to lift gain declines. To further increase lift gain and to reduce the power required for the high-lift device, the leading edge shape of the wing is modified into a droop nose [3]. The profile shapes are pictured in Fig. 2a.

The impact of the Coandă jet and designed droop nose on the pressure distribution of the wing can be seen in Fig. 2b. The suction peak at the leading edge of the clean nose configuration accounts for its stalling mechanism [3]. This suction peak is significantly reduced and spread over a wider range by the droop nose modification. An additional suction area is induced on the Coandă flap.

Together with the suction area at the leading edge, the Coandă jet provides for the endangered heave flutter.

Fig. 1

Flow around the trailing edge with $\delta _{\text {fl}}=65^\circ$, $\alpha =0^\circ$ and $v_\infty = 51$ m/s with boundary layer control and supercirculation

Fig. 2

Comparison of pressure distribution with $\delta _{\text {fl}}=65^\circ$, $\alpha =0^\circ$, $c_\mu = 0.054$ and $v_\infty = 51$ m/s

3 Reduced-order model

The fundamental setup of the ROM based on the strip theory is described by Krukow and Dinkler [15]. It is adapted from the three-dimensional finite element model described by Sommerwerk et al. [16]. Over the years, different reference configurations are investigated in the Coordinated Research Centre. The following ROM is based on the REF2-2013 configuration presented in Sommerwerk et al. [17], focussing on the wing without the tractor propeller engine. The engine is considered as a mass point, connected to the wing with rigid beam elements.

On the structure side, the former ROM [15] is enhanced by parameters $\eta _{\text {m}}$ and $\eta _{\text {k}}$ to allow for changes in mass and stiffness distribution.

3.1 Structure

The wing structure is described by the equation of motion

$$\begin{aligned} \mathbf {M}_{\mathbf {1}S}\varvec{\ddot{x}}_{\text {S}} + \mathbf {K}_{\mathbf {1}S}\varvec{x}_{\text {S}} = \varvec{g}_{\text {S}}, \end{aligned}$$

(2)

with $\mathbf {M}_{\mathbf {1}S}$ being the mass and $\mathbf {K}_{\mathbf {1}S}$ the stiffness matrix of the initial configuration. $\varvec{x}_{\text {S}}$ describes the displacement degrees of freedom and $\mathbf {g}_{\text {S}}$ is the structures’ net weight. Via the modal approach

$$\begin{aligned} \varvec{x}_{\text {S}}= \mathbf {X}\varvec{q}, \end{aligned}$$

(3)

with $\varvec{q}$ being the vector of generalised coordinates and $\mathbf {X}$ the modal matrix, containing the natural modes $\varvec{\hat{x}}_{\varvec{1}i}$ of the wing, Eq. (2) is decoupled. For each natural mode i results

$$\begin{aligned} \ddot{q}_i + \omega _{1i}^2q_i = \gamma _i g. \end{aligned}$$

(4)

Thus, the structure may be described by selected eigenfrequencies $\omega _{1i}$ and participation factors $\gamma _i$.

Deviating mass or stiffness distribution influences these properties. Therefore, it is desirable to parameterise the ROM. The mass distribution due to tank level or the structural stiffness due to the flap deflection angle or nose configuration are considered by parameters $\eta _{\text {m}}$ and $\eta _{\text {k}}$.

A second configuration with deviating mass and stiffness matrices

$$\begin{aligned} (\mathbf {M_1}_{\text {S}} + \varDelta \mathbf {M}_{\text {S}}) \varvec{\ddot{x}}_{\text {S}} + (\mathbf {K_1}_{\text {S}} + \varDelta \mathbf {K}_{\text {S}})\varvec{x}_{\text {S}} = 0, \end{aligned}$$

(5)

leads to altered eigenvectors $\varvec{\hat{x}}_{\varvec{2}i}$ and natural frequencies $\omega _{2i}$. To parameterise the structure, the second configuration needs to be described in terms of the initial configuration. This is achieved by describing the displacement degrees of freedom in terms of the eigenvectors of the initial configuration:

$$\begin{aligned} \varvec{x}_{\text {S}} = \varvec{\hat{x}}_{\varvec{1}i} q_i \qquad \varvec{\ddot{x}}_{\text {S}} = \varvec{\hat{x}}_{\varvec{1}i} \ddot{q}_i. \end{aligned}$$

(6)

Inserting this approach to Eq. (5) and multiplying it by the transposed eigenvectors yields

$$\begin{aligned} \begin{aligned}&\varvec{\hat{x}}_{\varvec{1}i}^T (\mathbf {M}_{\varvec{1}S} + \varDelta \mathbf {M}_{\text {S}}) \varvec{\hat{x}}_{\varvec{1}i} \ddot{q}_i \\&\quad + \varvec{\hat{x}}_{\varvec{1}i}^T (\mathbf {K}_{\varvec{1}S} + \varDelta \mathbf {K}_{\text {S}}) \varvec{\hat{x}}_{\varvec{1}i} q_i = 0, \end{aligned} \end{aligned}$$

(7)

respectively,

$$\begin{aligned} \begin{aligned}&(1 + \underbrace{\varvec{\hat{x}}_{\varvec{1}i}^T \varDelta \mathbf {M}_{\text {S}} \varvec{\hat{x}}_{\varvec{1}i}}_{\eta _{\text {m}} \varDelta m_i}) \, \ddot{q}_{i} \\&\quad + (\omega _{1i}^2 + \underbrace{\varvec{\hat{x}}_{\varvec{1}i}^T \varDelta \mathbf {K}_{\text {S}} \varvec{\hat{x}}_{\varvec{1}i}}_{\eta _{\text {k}} \varDelta \omega ^2_{i}}) \, q_i = 0 \, . \end{aligned} \end{aligned}$$

(8)

With an altered generalised mass $\varDelta m_i$ and altered generalised natural frequencies $\varDelta \omega _i$, the equation of motion may be described via the initial natural frequencies and two parameters $\eta _{\text {m}}$ and $\eta _{\text {k}}$. The natural frequencies of the second configuration approximately result in

$$\begin{aligned} \omega _{2i}^2 \approx \dfrac{\omega ^2_{1i} + \eta _{\text {k}} \varDelta \omega ^2_{i}}{1 + \eta _{\text {m}} \varDelta m_i} \, . \end{aligned}$$

(9)

The altered generalised mass and frequencies are determined with a distinction of cases. First, altering the mass $\eta _{\text {m}} = 1$ and leaving the stiffness unchanged $\eta _{\text {k}} = 0$ leads to

$$\begin{aligned} \varDelta m_i = \dfrac{\omega _{1i}^2}{\omega _{2i}^2} - 1 \, , \end{aligned}$$

(10)

whereas altering the stiffness $\eta _{\text {k}} = 1$ and leaving the mass unchanged $\eta _{\text {m}} = 0$ yields

$$\begin{aligned} \varDelta \omega _{i}^2 = \omega _{2i}^2 - \omega _{1i}^2 \, . \end{aligned}$$

(11)

Thus, stiffness and tank level in the ROM are provided by the parameters $\eta _{\text {k}}$ and $\eta _{\text {m}}$.

3.2 Structural mode shapes

The first eight structural mode shapes of the droop nose configuration with full tank $\eta _{\text {m}} = 1.3$ and a flap deflection angle of $\delta _{\text {fl}} = 65^\circ$ are displayed in Fig. 3. The first mode is the first flap bending mode, followed by another bending mode induced by the engine mass. Modes 3 and 6 are predominantly inplane modes. Since the engine and drag forces are neglected in the aerodynamic model, the corresponding modes are not considered. The second, third and fourth flap bending modes are represented by the modes four, five and eight. While mode five already covers some torsion, mode seven is the first pure torsion mode.

Fig. 3

First eight mode shapes of model with droop nose with full tank, deformation scaled by a factor of five

Fig. 4

Natural frequencies over tank level $\eta _{\text {m}}$

Varying the nose shape and flap deflection angle has little influence on the general mode shapes. Varying the fuel mass, however, primarily influences the natural frequencies and shapes. Figure 4 shows the declining natural frequencies with increasing tank fill levels. The labels correspond to the mode shapes displayed in Fig. 3. It can be seen that modes 5 and 7 become indistinct at a certain tank fill level and trade places for small tank fill levels.

4 Aerodynamics

For a better understanding of the phenomena, the two-dimensional aerodynamics are investigated in detail. The considered approach velocity for all simulations is about $v_\infty = 51$ m/s.

The steady aerodynamics are extended by lift and moment coefficients for the droop nose configuration and compared to the clean nose configuration.

The unsteady aerodynamics for both nose shapes are investigated for a heave and pitch motion with an impulse procedure [22, 23] to account for frequency dependence.

4.1 Steady aerodynamics

The results for the clean nose are approximated by a mathematical smoothing function, depending on the local angle of attack $\alpha$, flap deflection angle $\delta _{\text {fl}}$ and momentum coefficient $c_\mu$ given in [20]. The results for the droop nose are approximated by the following smoothing functions, also depending on the local angle of attack $\alpha$ and momentum coefficient $c_\mu$ for a flap deflection angle of $\delta _{\text {fl}} = 65^\circ$:

$$\begin{aligned} c_{L,{\text {droop}}} =\,&3.35 + 33.17\,c_\mu + (7.95-83.7\,c_\mu )\,\alpha \nonumber \\&-\, (27.53-632.5\,c_\mu )\,\alpha ^2 \nonumber \\&-\, (0.318-7.58\,c_\mu )\,10^3\,\alpha ^3 \nonumber \\&+ \,(0.12-3.37\,c_\mu )\,10^4\,\alpha ^4 \end{aligned}$$

(12)

$$\begin{aligned} c_{M,{\text {droop}}} =&-\,0.65-7.53\,c_\mu -(0.39-17.7\,c_\mu )\,\alpha \nonumber \\&+\,(6.51-125.8\,c_\mu )\,\alpha ^2 \nonumber \\&+\,(0.64-14.9\,c_\mu )\,10^2\,\alpha ^3 \nonumber \\&-\,(2.64-68.9\,c_\mu )\,10^2\,\alpha ^4. \end{aligned}$$

(13)

Only lift and pitching moment are considered, described by their dimensionless coefficients

$$\begin{aligned} \begin{bmatrix} c_{\text {L}}\\ c_{\text {M}} \end{bmatrix}=&\begin{bmatrix} c_{L0}\\ c_{M0} \end{bmatrix} + \begin{bmatrix} 0&c_{L\alpha }\\ 0&c_{M\alpha } \end{bmatrix} \begin{bmatrix} h\\ \alpha \end{bmatrix}\nonumber \\&+ \dfrac{1}{v_\infty } \begin{bmatrix} c_{L\dot{h}}&l\,c_{L\dot{\alpha }}\\ c_{M\dot{h}}&l\,c_{M\dot{\alpha }}, \end{bmatrix} \begin{bmatrix} \dot{h}\\ \dot{\alpha } \end{bmatrix} \end{aligned}$$

(14)

with the pitching moment being related to the quarter chord. The aerodynamic mass inertia is ignored and the terms are made dimensionless by the chord length l and the freestream velocity $v_\infty$.

The lift coefficients of the clean and droop nose for different momentum coefficients $c_\mu$ are compared in Fig. 5. While all momentum coefficients relate to the supercirculation regime, an increase in maximum lift $c_{L,\max }$ is still observed. For the clean nose configuration, this is accompanied by a decrease in corresponding angle of attack $\alpha _{c_{\text {L}},\max }$ to 1$^\circ$ with a momentum coefficient $c_\mu = 0.054$. Increasing the angle of attack beyond $\alpha _{c_{\text {L}},\max }$ leads to a decrease in lift; however, the flow is not seperated. This behaviour is referred to as $c_\mu$–$\alpha$-stall [11]. An increasing boundary layer thickness leads to less effective circulation control in this regime [7].

Whereas the suction peak at the leading edge of the clean nose (s. Fig. 2) causes large losses of momentum, the droop nose configuration is less sensitive to momentum variation. The thickness of the boundary layer is smaller and remains almost constant with varying momentum coefficients [3]. $\alpha _{c_{\text {L}},\max }$ is shifted by 10$^\circ$. The maximum lift of the clean nose is reached with the droop nose at 8$^\circ$ with a momentum coefficient of $c_\mu = 0.039$ (s. filled marks Fig. 5). Therefore, the droop nose needs less energy to reach the same lift.

The moment coefficients (Fig. 6) are shifted to higher angles of attack as well. The results for comparable lift coefficients show a decreasing moment coefficient for the droop nose configuration (s. filled marks Fig. 6).

Fig. 5

Lift coefficient for clean nose and droop nose with $\delta _{\text {fl}} = 65^\circ$

Fig. 6

Moment coefficient for clean nose and droop nose with $\delta _{\text {fl}} = 65^\circ$

The steady derivatives $c_{L\alpha }$ and $c_{M\alpha }$ are directly known from the steady-state simulations. The mathematical smoothing functions for the clean nose configuration are given in [20]. The smoothing functions for the droop nose configuration result from Eqs. (12) and (13) to

$$\begin{aligned} c_{L\alpha ,{\text {droop}}}&= \,(7.95-83.7\,c_\mu ) - (55.06-1265\,c_\mu )\,\alpha \nonumber \\&\quad -\, (0.954-22.74\,c_\mu )\,10^3\,\alpha ^2\nonumber \\&\quad +\, (0.48-13.48\,c_\mu )\,10^4\,\alpha ^3, \end{aligned}$$

(15)

$$\begin{aligned} c_{M\alpha ,{\text {droop}}} =&\,-(0.39-17.7\,c_\mu ) + (13.02-251.6\,c_\mu )\,\alpha \nonumber \\&+\,(1.92-44.7\,c_\mu )\,10^2\,\alpha ^2\nonumber \\&-\,(10.56-275.6\,c_\mu )\,10^2\,\alpha ^3. \end{aligned}$$

(16)

The discrete results and corresponding smoothing functions are displayed in Figs. 7 and 8.

Fig. 7

Lift derivative $c_{L\alpha }$ for clean nose and droop nose with $\delta _{\text {fl}} = 65^\circ$

Fig. 8

Moment derivative $c_{M\alpha }$ for clean nose and droop nose with $\delta _{\text {fl}} = 65^\circ$

4.2 Unsteady aerodynamics

For the derivatives referring to heave $\dot{h}$ and pitch $\dot{\alpha }$ motion, unsteady simulations are necessary.

The former approach of using harmonic pitching oscillations to determine the derivatives for rotational speed $\dot{\alpha }$ is described by Krukow and Dinkler [15]. A frequency of $f = 5.84 \, \mathrm {Hz}$ close to the first torsional frequency of the former structure was investigated with this approach [21].

Since eigenfrequencies vary with deviating properties and loads, it is useful to cover a range of frequencies. This can be achieved using an impulse motion [22, 23]. A Fourier transform is used to transform the system responses into the frequency domain. This results in a very high resolution of the entire frequency range.

The results of the lift and moment derivatives for heave ($c_{L\dot{h}}$, $c_{M\dot{h}}$) and pitch ($c_{L\dot{\alpha }}$, $c_{M\dot{\alpha }}$) over a range of natural frequencies $\omega$ at a trimm angle of attack $\alpha _\text {T}$

Fig. 9

Lift derivative $c_{L\dot{h}}$ over natural frequency for clean nose and droop nose with $\alpha _\text {T} = 6^\circ$ and $\delta _{\text {fl}} = 65^\circ$

Fig. 10

Moment derivative $c_{M\dot{h}}$ over natural frequency for clean nose and droop nose with $\alpha _\text {T} = 6^\circ$ and $\delta _{\text {fl}} = 65^\circ$

Fig. 11

Lift derivative $c_{L\dot{\alpha }}$ over natural frequency for clean nose and droop nose with $\alpha _\text {T} = 0^\circ$ and $\delta _{\text {fl}} = 65^\circ$

Fig. 12

Moment derivative $c_{M\dot{\alpha }}$ over natural frequency for clean nose and droop nose with $\alpha _\text {T} = 0^\circ$ and $\delta _{\text {fl}} = 65^\circ$

are shown in Figs. 9, 10, 11 and 12. The natural frequencies $\omega _i$ for the considered modes i of the wing with a full tank ($\eta _{\text {m}} = 1.3$) are marked by vertical lines. Different momentum coefficients $c_\mu$ are investigated. Positive derivatives indicate possible destabilisations and the corresponding areas are highlighted grey in the figures.

For the heave motion, a trimm angle of attack of $\alpha _\text {T} = 6^\circ$ is displayed (s. Figs. 9, 10). According to Fig. 5 the lift coefficient in this area has a negative gradient for the clean nose, also displayed in Fig. 7, which indicates instability. Figure 9 shows that this, however, only applies to natural frequencies $\omega$ up to 30 rad/s. Accordingly, only the first eigenmode is affected. The corresponding moment derivative $c_{M\dot{h}}$ (Fig. 10) shows positive results for the clean nose configuration for natural frequencies $\omega$ from 45 to 130 rad/s. Therefore, here the modes 4 and 5 are concerned.

For the pitch motion at a trimm angle of attack of $\alpha _\text {T} = 0^\circ$, the clean nose shows positive values for the lift derivative $c_{L\dot{\alpha }}$ (Fig. 11) and negative values for the moment derivative $c_{M\dot{\alpha }}$ (Fig. 12) over the whole frequency range. The results of the droop nose with a momentum coefficient of $c_\mu = 0.033$ are comparable to the equivalent results of the clean nose. The results with greater momentum coefficients, however, indicate possible instabilities concerning the moment derivative $c_{M\dot{\alpha }}$ (Fig. 12) and stabilty for the lift derivative $c_{L\dot{\alpha }}$ (Fig. 11). These areas are limited to natural frequencies $\omega$ smaller than 40 rad/s. In the displayed case solely, the frequency of the first natural mode is located in the shaded area of the moment derivative $c_{M\dot{\alpha }}$.

For a better understanding of the behaviour depending on the angle of attack, Figs. 13, 14, 15 and 16 show the results over the angle of attack $\alpha$ for the frequency of the first natural mode ($\omega = \omega _1$). The results are again approximated by mathematical smoothing functions, based on the angle of attack $\alpha$, the momentum coefficient $c_\mu$ and the natural frequency $\omega$.

The lift derivative $c_{L\dot{h}}$ displayed in Fig. 13 shows a sign change for both nose shapes comparable to Fig. 7. Angles of attack beyond $\alpha _{c_{\text {L}},\max }$, therefore, lead to instabilities for the first natural frequency, whereas smaller angles of attack are stable. The moment derivative $c_{M\dot{h}}$ displayed in Fig. 14 shows no sign change.

Fig. 13

Lift derivative $c_{L\dot{h}}$ for clean nose and droop nose with $\omega = \omega _1$ and $\delta _{\text {fl}} = 65^\circ$

Fig. 14

Moment derivative $c_{M\dot{h}}$ for clean nose and droop nose with $\omega = \omega _1$ and $\delta _{\text {fl}} = 65^\circ$

The lift derivative $c_{L\dot{\alpha }}$ displayed in Fig. 15 shows a sign change most notably for the clean nose configuration at high momentum coefficients $c_\mu$ and angles of attack greater than 0$^\circ$. The moment derivative $c_{M\dot{\alpha }}$, however, indicates instabilities for high momentum coefficients $c_\mu$ and angles of attack smaller than 5$^\circ$ for the droop nose and, respectively, smaller than approximately $-\,2^\circ$ for the clean nose configuration.

In summary, the investigations indicate instabilities in particular at low frequencies and angles of attack beyond the respective $\alpha _{c_{\text {L}},\max }$.

Fig. 15

Lift derivative $c_{L\dot{\alpha }}$ for clean nose and droop nose with $\omega = \omega _1$ and $\delta _{\text {fl}} = 65^\circ$

Fig. 16

Moment derivative $c_{M\dot{\alpha }}$ for clean nose and droop nose with $\omega = \omega _1$ and $\delta _{\text {fl}} = 65^\circ$

5 Aeroelastic coupling

The strip theory delivers the aerodynamic loads

$$\begin{aligned} \varvec{L} = \varvec{L_0} + \mathbf {A_0} \varvec{x}_A + \mathbf {A_1} \varvec{\dot{x}}_A, \end{aligned}$$

(17)

at each strip, with a constant load vector $\varvec{L_0}$, the aerodynamic stiffness $\mathbf {A_0}$ and aerodynamic damping $\mathbf {A_1}$ matrix. Equation (14) shows the involved degrees of freedom, vertical displacement h and torsional rotation $\alpha$, which differ from the discretization of the finite element model. Thus, the structural modes $\hat{\mathbf {x}}_i$ are transformed to be described solely by vertical displacement h and torsional rotation $\alpha$. With the transformed modes $\hat{\mathbf {x}}_{Ai}$, the modal approach in Eq. (3) accordingly leads to

$$\begin{aligned} \ddot{q}_i + \omega _i^2q_i = \gamma _i g + \varvec{\hat{x}}_{Ai}^T \, \varvec{L}(q,\dot{q}). \end{aligned}$$

(18)

Due to the additional loads on the right handside, the natural frequencies and natural modes of the system alter. The ROM stays based on the initial natural modes displayed in Fig. 3. The natural frequencies, however, are updated and the loads are calculated correspondingly to the adapted chosen frequency and calculated deformations.

The stability boundaries for different configurations are compared in Figs. 17, 18 and 19. Here, a range of angles of attack, measured on the y-axis, and momentum coefficients, measured on the x-axis, is evaluated. All other parameters remain constant for the respective stability map. For each examined condition, it is checked whether a damped or undamped state is present. The transition from a damped to an undamped state is marked by a line, hereinafter referred to as the stability boundary. Critical areas are highlighted in grey. The basic flutter phenomenon is explained in detail in [20].

Figure 17 shows the stability boundaries for the clean nose configuration with different mass distributions ($\eta _{\text {m}} = 0$ and $\eta _{\text {m}} = 1.3$) and an exciting frequency equivalent to the first natural frequency ($\omega = \omega _1$). As pictured in Fig. 4, the first natural frequency of the lighter structure is slightly higher than the first natural frequency of the heavier structure. This is taken into account in the respective investigation.

The observable stability boundary for the heavy configuration results from the lift derivatives $c_{L\dot{h}}$ shown in Fig. 13, corresponding to the sign change of the derivative $c_{L\dot{h}}$. A coupling of different modes is not observed, thus a pure first bending mode is present.

Fig. 17

Stability boundaries for clean nose with varying mass distribution with $\omega = \omega _1$

However, the interaction of the lighter structure and aerodynamic forces leads to coupling of the first three bending modes and, therefore, a higher first natural frequency. As pictured in Fig. 9, the derivative $c_{L\dot{h}}$ indicates a more stable state with increasing frequencies. Therefore, the corresponding unstable area is significantly smaller.

Figure 18 shows the stability boundaries for the clean nose configuration at a full tank level ($\eta _{\text {m}} = 1.3$) at two different excitation frequencies ($\omega = \omega _1$ and $\omega = \omega _4$). Hence, different modes are addressed.

The resulting stability map for mode 1, adressed by $\omega = \omega _1$, equals the map displayed in Fig. 17 for $\eta _{\text {m}} = 1.3$.

Mode 4, adressed by $\omega = \omega _4$, enters no coupling and remains a clean second bending mode. The corresponding area, therefore, indicates bending flutter. Again, with increasing frequencies the lift derivative $c_{L\dot{h}}$ in Fig. 9 indicates more stability. Therefore, the displayed stability boundary for the second bending mode is rather comparable to the stability boundary of the light structure ($\eta _{\text {m}} = 0$) in Fig. 17.

Fig. 18

Stability boundaries for clean nose with varying excitation frequency with $\eta _{\text {m}} = 1.3$

The largest differences can be seen in Fig. 19. Here, clean nose and droop nose with a full tank level ($\eta _{\text {m}} = 1.3$) and an excitation frequency equal to the respective first natural frequency ($\omega = \omega _1$) are compared.

Fig. 19

Stability boundaries for clean nose and droop nose with $\omega = \omega _1$ and $\eta _{\text {m}} = 1.3$

For the droop nose, the stability boundary can be traced back to the lift derivatives $c_{L\dot{h}}$, displayed in Fig. 13. Since the maximum lift is shifted to higher angles of attack (s. Fig. 5), the change of sign of the lift derivative $c_{L\alpha }$ is shifted to the respective angles of attack (s. Fig. 7). Accordingly, the lift derivative $c_{L\dot{h}}$ for the first natural frequency (s. Fig. 13) shows a change of sign at comparatively high angles of attack.

In comparison with the clean nose, the stability boundary of the droop nose configuration is shifted to higher angles of attack. However, there is no coupling of modes in case of the droop nose either. A pure first bending mode is observed here as well. The bending flutter phenomenon is driven by the pressure distribution shown in Fig. 2 and the resultant lift derivatives $c_{L\dot{h}}$.

6 Conclusion and future work

The paper outlines the investigation of aeroelastic effects of a circulation-controlled wing. A ROM, based on a full-scale three-dimensional finite element wing model, is extended by a parameterisation of essential properties. Since the natural frequencies and eigenvectors depend on the mass and stiffness distribution, the parameters $\eta _{\text {m}}$ and $\eta _{\text {k}}$ are introduced.

Steady and unsteady aerodynamics are based on two-dimensional flow simulations. An impulse method is used to determine the frequency dependency of the derivatives due to heave and pitch motion. Therefore, the ROM includes the natural frequency $\omega$ as a parameter. The forces are mapped onto the wing structure by strip theory.

Two profiles, clean and droop nose, are examined. Comparing the two profiles shows an increased angle of attack of maximum lift coefficient $\alpha _{c_{\text {L}},\max }$ by 10$^\circ$ for the droop nose. The needed blowing power for comparable lift coefficients is reduced significantly by including the droop nose.

Results of the frequency-dependent derivatives $c_{L\dot{h}}$ and $c_{M\dot{h}}$ show comparable values for the clean and droop nose depending on the shifted $\alpha _{c_{\text {L}},\max }$. In particular, the results of the moment derivative $c_{M\dot{\alpha }}$ indicate increasing areas of possible instabilities for the droop nose.

Aeroelastic coupling yields an additional flutter phenomenon due to circulation control. Bending flutter occurs corresponding to the maximum lift at low frequencies. Since $\alpha _{c_{\text {L}},\max }$ is increased with the droop nose, the bending flutter is also shifted to higher angles of attack. It is shown, that varying the mass distribution affects the flutter boundary through changing frequencies. Differing the excitation frequency basically addresses different modes. Here, the second bending mode is adressed, which also shows pure bending flutter.

As aircrafts usually fly beneath well below $\alpha _{c_{\text {L}},\max }$, the bending flutter is not likely to occur. Further studies on the impact of diverging properties of the wing structure are necessary to evaluate possible instabilities due to the moment derivative $c_{M\dot{\alpha }}$.