4.1. Validation of simulation set-up

The static aeroelastic solution of the aircraft model was computed at angles of attack $\alpha = 3.0^\circ$, $3.5^\circ$, $3.7^\circ$ and $3.75^\circ$ to match the loaded wind-tunnel shape (interpolated from run 182 in the ETW campaign) and surface pressure data. Figure 2 shows the wing bending and twist deformations at $\alpha = 3.75^\circ$ on the port and starboard wings (evaluated at 50 % chord) compared with those measured using stereo pattern tracking via markers distributed over the wing surface. These markers were affixed to the lower surface of the port wing (Lutz et al. Reference Lutz, Gansel, Waldmann, Zimmermann and Schulte am Hülse2016). Overall good agreement can be observed, on par with other fluid–structure coupled simulations (Tinoco et al. Reference Tinoco2018), with a maximum bending of approximately 18 mm and a washout twist of $-1.2^\circ$ at the wing tip. Note the variability in the experimental data themselves, specifically for the twist deformation in figure 2(b), previously reported in Keye & Rudnik (Reference Keye and Rudnik2015). The figure includes error bars describing the confidence interval encompassing 90 % of all recorded values for a single data point. The relevant deformation data for our study were available only for angles of attack $\alpha =3.0^\circ$ and $4.0^\circ$ during the pitch-pause polar. Considering that the difference in the fluctuation range for these two angles of attack was quite small, linear interpolation was used for the intermediate $\alpha =3.75^\circ$ (similar to obtaining the mean deformation itself). Importantly, an asymmetry between the port and starboard wing is also clearly visible in the numerical data. The cause of this asymmetry lies in the high fidelity of the finite-element model which takes into account the different cut outs on each wing to house experimental sensors and related instrumentation as well as the asymmetric inner structure of the model itself. These features propagate to the normal mode shapes and their frequencies. For instance, the first two modes are wing bending for port (at wind-off structural frequency 39.4 Hz) and starboard (40.9 Hz), respectively, with the latter seen in figure 1(a). The corresponding surface pressure coefficient of the steady base flow can be found in figure 3 with good agreement to the experimental measurements at most spanwise stations. Herein, the interest is below and around onset conditions of a global instability which means, in our modelling framework, the time-invariant, static aeroelastic solution is approximately equal to the time-averaged mean state, which is what is shown for the experimental data. A discussion on the seemingly missing experimental data points around the mid semi-span was given in Tinoco et al. (Reference Tinoco2018) and Timme (Reference Timme2020). Differences in the surface pressure between port and starboard wing are minor and not noticeable to the naked eye. Similar levels of agreement were found in the comparison for all other angles of attack used in this study. Overall, the results are reassuring that the fluid–structure coupling has been done correctly (such as defining the mass ratio $\vartheta$ and various scalings) and the models are of sufficient fidelity.

Figure 2. Wing deformation showing ($a$) bending and ($b$) twist over dimensionless span ${\eta }$ at angle of attack ${\alpha = 3.75^\circ }$ comparing experimental data from the ETW campaign and static aeroelastic results emphasising differences on port and starboard wings.

Figure 3. Comparison of experimental and numerical surface pressure coefficient data at angle of attack ${\alpha = 3.75^\circ }$ and eight spanwise locations. Streamwise coordinate ${x}$ is normalised by corresponding local chord length ${c}$.

4.2. Flutter analysis

Initially, eleven samples of the interaction matrix $S^c$ were computed using the linearised frequency-domain solver, assuming simple harmonic structural motion at reduced frequencies in the range $\omega =0$ to 3 in increments of 0.3. The sampling frequency range corresponds to the wind-off structural frequencies, once made dimensionless based on the actual wind-tunnel flow conditions. Specifically, computing a matrix $\boldsymbol{\mathsf{S}}^c(\omega )$ involves first solving linear systems with the frequency-shifted fluid Jacobian matrix, $(\boldsymbol{\mathsf{J}}_{ff}-{\rm i}\omega \boldsymbol{\mathsf{I}})$, for each of the $m$ columns of $(\boldsymbol{\mathsf{J}}_{f\eta }+{\rm i}\omega \boldsymbol{\mathsf{J}}_{f\dot {\eta }})$ and, second, the projection with the matrix $\boldsymbol{\mathsf{J}}_{sf}$. Combining the $2m$ columns of the matrix $\boldsymbol{\mathsf{J}}_{fs}$ into $m$ columns has been discussed at length in other work and is not repeated herein (Badcock et al. Reference Badcock, Timme, Marques, Khodaparast, Prandina, Mottershead, Swift, Da Ronch and Woodgate2011; Timme, Badcock & Da Ronch Reference Timme, Badcock and Da Ronch2013). This was done at three angles of attack $\alpha =3.0^\circ$, $3.5^\circ$ and $3.75^\circ$. Upon inspection of the results at the highest angle of attack, additional samples were added in regions of significant activity, bringing the total to 23. Figure 4 shows both the absolute value and phase of some selected entries of matrix $\boldsymbol{\mathsf{S}}^c$, specifically of its submatrix $\boldsymbol{\mathsf{G}}=(\boldsymbol{\varPhi} ^{T} \partial \boldsymbol {f}/\partial \boldsymbol {w}_f)(\boldsymbol{\mathsf{J}}_{ff}-{\rm i}\omega \boldsymbol{\mathsf{I}})^{-1}(\boldsymbol{\mathsf{J}}_{f\eta }+{\rm i}\omega \boldsymbol{\mathsf{J}}_{f\dot {\eta }})$, at two angles of attack. Values at angle of attack $\alpha =3.0^\circ$ were found to be almost identical to those at $\alpha =3.5^\circ$ and are not shown here for clarity. At the lower angle of attack shown in the figure, the matrix elements describe a smooth trend with respect to the frequency. At the higher angle of attack, which corresponds to a shock-buffet condition in the fluid-only analysis, the entries have significant variation in absolute value and phase in the frequency range where the band of aerodynamic modes with increased decay rate is observed (Timme Reference Timme2020), concretely between approximately $\omega =2$ and $3$, indicating, first, a strong aerodynamic response to a structural forcing and, second, a strong coupling between the fluid and structure as discussed shortly. The cause lies both in the proximity of the sampling frequency to some eigenvalues of the fluid system and in the non-normality of the governing equations (He & Timme Reference He and Timme2020). For instance, the significant peaks at a forcing frequency of approximately $\omega =2.7$ coincide with one computed eigenvalue, labelled $c'$, of the band of shock-buffet modes shown in figures 5 and 7 below. It must be noted though that interpreting a linearised frequency-domain aerodynamic response to a structural forcing in the globally unstable flow at angle of attack $\alpha =3.75^\circ$ should be done carefully. Also observe the jumps in the phase going from $-{\rm \pi}$ to ${\rm \pi}$ which is due to visualisation purposes, whereas the actual flutter calculations with the matrix $\boldsymbol{\mathsf{S}}^c$ are done with the equivalent Cartesian complex numbers instead of the modulus and phase shown in the figure.

Figure 4. Log magnitude (a–d) and phase (e–h) of complex entries of matrix $\boldsymbol{\mathsf{G}}$ at angles of attack ${\alpha = 3.5^\circ }$ and ${3.75^\circ }$ over sampling frequency ${\omega }$. Indices above the plots refer to matrix entries ${\mathsf{G}}_{ij}$. Symbols indicate sample locations.

Figure 5. Eigenvalues originating in structural system based on pk-type flutter analysis at ${\alpha = 3.0^\circ }$, $3.5^\circ$, $3.7^\circ$ and ${3.75^\circ }$ at the flow condition encountered in the wind-tunnel environment. Mode 28 at ${\alpha = 3.75^\circ }$, denoted ${\blacksquare }$, failed to converge and the proper value, computed with the Arnoldi method, is therefore also shown. A close-up view, indicated by the red box in ($a$), is shown in ($b$) for clarity. The fluid modes denoted by $b$ and $c'$ correspond to the leading buffet and destabilised modes, respectively, found with the Arnoldi method and are included to show their proximity to the structural modes.

Once the interaction matrices have been computed and assessed, they feed into the flutter stability calculation, specifically (2.11). Figure 5 shows the predicted eigenvalues from the flutter analysis at angles of attack $\alpha = 3.0^\circ$, $3.5^\circ$, $3.7^\circ$ and $3.75^\circ$ at the target flow condition encountered in the wind-tunnel environment. Since tracing the eigenmodes from the uncoupled system state is somewhat arbitrary and would depend on how the test point is reached in the wind tunnel, we chose to increase the density until the simulated flow condition matches the experiment, while keeping all other variables, specifically velocity, temperature and pressure, frozen at the target condition. The target corresponds to a density of approximately $1.53\ {\rm kg}\ {\rm m}^{-3}$. It should be emphasised that, while the chosen approach to tracing the modes is arbitrary and does not necessarily agree with how the target condition is reached in the wind tunnel, the mode tracing agrees with the direct calculation using the Arnoldi method, as shown in § 4.4. The figure itself appears rather busy, hence a moment is taken to explain it step by step. The eigenvalues are shown in the complex plane. While the focus is on the eigenmodes originating in the structural system for wind-off conditions, specifically those of the matrix $\boldsymbol{\mathsf{J}}_{ss}$, the figure also includes the shock-buffet modes, labelled $b$ and $c'$ hereafter consistent with previous work (Timme Reference Timme2020), in faint colour for angles of attack $\alpha =3.7^\circ$ (mode $b$) and $3.75^\circ$ (mode $c'$), to demonstrate the connection with the results presented in figure 7. Strictly speaking, denoting the aeroelastic modes as either structural or fluid modes would not be correct due to the coupled nature of the problem. For ease of writing, however, we will do so nevertheless, when the origin of the modes in the uncoupled case can be unambiguously traced to either the structure or the fluid, as is the case in our investigation.

In the figure, the eigenvalues at angles of attack $\alpha = 3.0^\circ$ and $3.5^\circ$ show no instabilities and the modes do not migrate significantly with increasing angle of attack. At angles of attack $\alpha = 3.7^\circ$ and $3.75^\circ$, it is evident that modes with frequencies below approximately $\omega =1.7$ have changed also very little compared with the lower angles of attack. At the same time, instabilities are found for different structural modes at these two higher angles of attack. Modes 29 and 30 are unstable at $\alpha = 3.7^\circ$, but decrease greatly in growth/decay rate at $\alpha = 3.75^\circ$. This strong decrease is also seen for modes 22, 23 and 25. In contrast, modes 19 to 21 are destabilised further to have an increased growth rate, bringing these modes into the unstable half-plane. However, mode 28 at angle of attack $\alpha =3.75^\circ$ (denoted by symbol $\blacksquare$), did not trace correctly despite all efforts and the additional samples in this frequency range to avoid heavy reliance on interpolation of the matrix $\boldsymbol{\mathsf{S}}^c$. Instead it jumped onto an odd state. This was attributed to the fact that the entries of the interaction matrix $\boldsymbol{\mathsf{S}}^c$, and therefore $\boldsymbol{\mathsf{S}}(\lambda )$ in (2.12), show extreme variation due to the shifted fluid Jacobian matrix, $(\boldsymbol{\mathsf{J}}_{ff}-\lambda \boldsymbol{\mathsf{I}})$, being close to singular due to the proximity to the mode labelled $c'$ in figure 7. Indeed, mode $c'$ crosses the imaginary axis when going from a fluid-only to the coupled system, passing in close distance to the troublesome mode 28 while increasing the value of fluid density. This provides another reason for the use of the Arnoldi method, besides the ability to compute fluid modes in the first place. It can distinguish between fluid and structural modes in highly contested regions. The correct mode 28 coming from the ARPACK calculation is therefore also included in the figure, denoted by the green square labelled ‘Mode 28’. It has been noted that the second term of matrix $\boldsymbol{\mathsf{J}}_{ss} = \boldsymbol{\mathsf{D}} + \vartheta \boldsymbol{\mathsf{E}} \boldsymbol{\varPhi} ^{T} \partial \boldsymbol {f}/\partial \boldsymbol {w}_s$ can often be discarded (Timme et al. Reference Timme, Marques and Badcock2011). This was confirmed herein by evaluating the sensitivity of the generalised forces, $\boldsymbol{\varPhi} ^{T}\boldsymbol {f}$, with respect to the modal amplitudes $\boldsymbol {\eta }$ (at fixed base-flow solution). The added term had negligible influence on the results. Viscous force contributions were also discarded for similar reasons, as outlined earlier.

To identify the composition of the structural entries of an eigenmode, we adopted and adapted the modal assurance criterion (MAC) (Allemang Reference Allemang2003), defined herein as

(4.1)\begin{equation} \text{MAC}(i,j) = \frac{|\boldsymbol{e}_i^T\boldsymbol{\hat{\eta}}_j |^2}{(\boldsymbol{e}_i^T\boldsymbol{e}_i)\,\,(\boldsymbol{\hat{\eta}}_j^H\boldsymbol{\hat{\eta}}_j)}, \end{equation}

with $\boldsymbol {e}_i$ as linearly independent unit basis vectors to represent one free-vibration mode shape at a time and $\boldsymbol {\hat {\eta }}_j$ indicating the structural part of the coupled eigenmodes. In general terms, the MAC gives an indication of the consistency between modes (such as those measured experimentally and computed from a numerical model) and can take a value between 0 and 1, with 1 indicating fully consistent mode shapes. Figure 6 shows the MAC of the structural part, $\hat {\boldsymbol {\eta }}$, of the structural eigenmodes computed using the Schur complement method compared with the amplitudes of the wind-off normal modes for angles of attack of $\alpha =3.0^\circ$, $3.5^\circ$ and $3.75^\circ$. (Mode 28 at angle of attack $\alpha =3.75^\circ$ uses the solution from the Arnoldi method, as explained earlier.) Recall that the aircraft deformation results from $\boldsymbol {x}_s=\boldsymbol{\varPhi} \boldsymbol {\eta }$ with $\boldsymbol{\varPhi}$ as the column matrix of spatial orthogonal normal mode shapes. The structural part of the six fluid modes, specifically the pairs $a$, $b$ and $c$, computed using the Arnoldi method are included as well for the highest considered angle of attack as it is rather instructive. For the structural modes at all angles of attack, the MAC is dominated by the diagonal, indicating the largest contributions to the eigenmodes come from the corresponding normal modes. This also gives a means of identifying whether a mode originates in the fluid or structural system, as the MAC for a structural mode is likely to be dominated by a single entry. At angles of attack $\alpha = 3.0^\circ$ and $3.5^\circ$, no discernible difference is present and all off-diagonal entries are at least an order of magnitude lower, as no significant input from other modes is present. Noteworthy are the three pairs of modes, specifically modes 1/2, 14/15 and 19/20, which have a comparably high MAC between them. This can be explained by these pairs having similar mode shapes and/or close wind-off frequencies. For instance, modes 1 (39.4 Hz) and 2 (40.9 Hz) are both wing bending emphasising either port or starboard deformation. At angle of attack $\alpha = 3.75^\circ$, the MAC for modes 1 to 18 shows rather similar features compared with the two lower angles of attack. However, the structural parts of the aeroelastic eigenvectors 19 to 30 now come with a significant contribution from a wide range of normal modes. This observation agrees with the discussion around figure 5. Modes 20 to 25, the frequencies of which lie around the frequency of the leading fluid modes, seem to have a particularly salient interconnection, indicating that the aerodynamics related to the shock-buffet phenomenon can cause a strong coupling between structural degrees of freedom. In addition, a clear contribution gap of normal modes 3 to 6 and 9 to 11 to those active higher-frequency aeroelastic modes is noticeable. Upon inspection of the free-vibration mode shapes, it can be said that modes 3 to 6 are dominated by fuselage bending and modes 9 to 11 are dominant on the horizontal tail. Indeed, the shock-buffet unsteadiness on the wings (and its wake) does not heavily impact on the horizontal tail in this study. Finally, the structural parts of the shock-buffet modes ($a$ to $c$), discussed in more detail below, have contributions from most normal modes (except those aforementioned fuselage and tail modes) and, in particular, from those in the same frequency range as the shock-buffet dynamics.

Figure 6. Visualisation of modal assurance criterion correlating the structural part of the aeroelastic (structural) modes with the amplitudes of the wind-off finite-element method (FEM) modes at different angles of attack. At angle of attack $\alpha =3.75^\circ$ the corresponding values of the leading aeroelastic (fluid) modes, labelled $a$, $b$ and $c$, are included (with mode $c'$ in final column).

4.3. Aerodynamic global stability analysis

The interest now turns to elucidating the impact of the asymmetrically deformed full-span wing geometry on the aerodynamic stability characteristics at angles of attack $\alpha =3.7^\circ$ and $3.75^\circ$. For the eigenspectra computed with the shift-invert Arnoldi method, multiple shifts were used to cover the relevant frequency range. Figure 7(a) shows the eigenspectra of the fluid-only system, as computed on the perfectly symmetric case (symmetric with respect to the fuselage centre plane) and the asymmetric case from our static aeroelastic simulation, cf. the deformation presented in figure 2. The full-span symmetric data are taken from Timme (Reference Timme2020) and the labelling follows accordingly. While the symmetric, pre-deformed geometry (corresponding to deformations measured in run 182 of the experimental campaign) gives two nearly identical unstable eigenvalues at approximately $\lambda = 0.16 + 2.37i$, the asymmetric geometry from the static coupling simulation results in two visibly distinct eigenvalues. In the symmetric case, the corresponding eigenvectors revealed symmetric/anti-symmetric coherent spatial features of equal amplitudes on both wings. An interpretation is that unsteadiness on the two wings is more or less independent (leaving some coupling effects from the global flow field aside). Interestingly, for the asymmetric, statically deformed geometry, on the other hand, the coherent spatial structures of the two unstable modes, while appearing similar to the coherent features on the symmetric geometry, now dominate one wing each, as presented in figure 8. The figure shows the magnitude of the unsteady surface pressure coefficient of the two unstable global shock-buffet modes, labelled b in figure 7, and a visualisation of coherent structures through the volumetric iso-surfaces of the real part of the $x$-momentum $\widehat {\rho u}$ at values of $\pm 0.75$. Note that the mode with the highest growth rate (in figure 8a) shows activity on the port wing. Recall from Timme (Reference Timme2020) that those modes are discrete realisations of the continuous band of medium-wavelength modes reported on infinite swept wings (Crouch et al. Reference Crouch, Garbaruk and Strelets2019; Paladini et al. Reference Paladini, Beneddine, Dandois, Sipp and Robinet2019a; He & Timme Reference He and Timme2021). Hence, the fluid modes $a$ to $c$ (and additional modes visualised in Timme Reference Timme2020), linked to shock buffet on a finite swept wing, all come with similar coherent structures with a spanwise wavelength/wavenumber correlated with the frequency. Note that the distinct mode labelled $c'$ migrated significantly, approaching the unstable region, due to the increased physical realism of simulating an asymmetric static aeroelastic deformation and we will return to discussing this mode shortly.

Figure 7. Comparison of eigenspectra showing ($a$) fluid-only results for symmetric (sym) vs asymmetric (asym) static deformation at angle of attack $\alpha = 3.75^\circ$ and ($b$) asymmetric fluid-only vs coupled aeroelastic results, all computed with the Arnoldi method. Modes are labelled according to Timme (Reference Timme2020), along with mode $c'$, which migrates into the unstable half-plane in the coupled system. For reference in ($b$), all faint-coloured fluid–structure interaction (FSI) modes are structural modes with the red-dashed box indicating the relevant region from figure 5.

Figure 8. Magnitude of unsteady surface pressure coefficient ${|\hat {C}_p|}$ and volumetric iso-surfaces of real part of ${x}$-momentum ${\widehat {\rho u}}$ at values of ${\pm 0.75}$ of $(a)$ leading and $(b)$ second unstable global modes from fluid-only stability analysis on statically deformed, asymmetric geometry at angle of attack $\alpha =3.75^\circ$. Underlying eigenvectors are scaled to unit length with respect to the inner product, specifically $\langle \hat {\boldsymbol {w}}_{f},\hat {\boldsymbol {w}}_{f}\rangle =1$. The base-flow zero-skin-friction line is also shown on the surface.

4.4. Coupled aeroelastic global stability analysis

The ramifications of including an elastic aircraft structure in the shock-buffet stability analysis are now addressed. Figure 7(b) gives the eigenspectra as computed by ARPACK for the fluid-only system and for the coupled system (and should be examined in unison with figure 5). Although this figure also shows results for angle of attack $\alpha = 3.7^\circ$ for completeness to demonstrate the impact a fluid–structure coupled approach can have in the vicinity of instability onset, the focus here is on angle of attack $\alpha =3.75^\circ$. Emphasising it again, all structural modes found with the earlier pk-type flutter method (bar the aforementioned difficulties with mode 28) discussed in § 4.2 were also identified by the Arnoldi method applied to the coupled system, further confirming that the implementation is correct. Multiple interesting features can be observed. The discussion focuses on three regions of the eigenspectrum, specifically the leading shock-buffet modes b, the unstable structural modes near the angular frequency $\omega = 1.8$, and the unstable modes 27, 28 and $c'$ at approximately $\omega = 2.7$.

First, the two unstable shock-buffet modes, denoted b, are also observed in the coupled system, albeit at slightly decreased frequencies and increased growth rates. This suggests that including an elastic structure could destabilise an otherwise stable fluid-only mode at a reduced angle of attack (Nitzsche et al. Reference Nitzsche, Ringel, Kaiser and Hennings2019). The results indicate that this change in onset angle of attack is small (particularly when compared with other typical factors having an impact on this type of simulation, such as turbulence modelling). However, it is important to point out that the migration of the leading buffet-related modes for the current test case is rather rapid (cf. figure 6 in Timme Reference Timme2020). Specifically, the leading modes $b$ only emerge from the dense cloud of spurious non-descript modes, to become identifiable from a global stability analysis, at an angle of attack of approximately $\alpha =3.6^\circ$. Also observe, as exemplified for the near-onset angle of attack $\alpha =3.7^\circ$ in our test case, that the leading shock-buffet modes $b$ are influenced in their migration by structural mode 26. These shock-buffet modes again emphasise a separate wing each, as pictured in figures 9(a) and 9(b), and the surface flow characteristics (not included in the plot for reasons of clarity) and the volumetric iso-surfaces are very similar to their fluid-only counterparts in figure 8. An interpretation could be that the shock-buffet unsteadiness remains the dominant physics even when including the elastic wing structure. Having said this, the structural part of the eigenvectors shows highest activity (i.e. deformation) on the same wing as the coherent spatial flow features, possibly revealing a coupling effect. To be clear, the unsteady wing deformation is visualised through the relation $\hat {\boldsymbol {x}}_s={\boldsymbol{\varPhi} }\,\hat {\boldsymbol {\eta }}$ and the real part of the dominant $z$-component (predominantly normal to the wing surface) is shown in the figure. To aid reproducibility of the results, when plotting the structural part of eigenvectors in physical space herein, the underlying vector $\hat {\boldsymbol {\eta }}$ was scaled so that the phase of its modal degree-of-freedom with the highest magnitude was forced to be zero. In addition, the plot of the MAC (cf. figure 6) describing the composition of the structural part of these fluid modes indicates a strong contribution from, and hence coupling with, most of the wind-off structural mode shapes (bar the aforementioned fuselage and empennage modes). The largest contributions come from modes 19 to 24, having wind-off frequencies close to the lower shock-buffet frequency range. Consequently, the structural motion described by these coupled fluid modes is a mix of several wind-off structural mode shapes that combine in a non-trivial manner. When visualised over a period of oscillation via $\tilde {\boldsymbol {x}}_s(t) = \boldsymbol{\varPhi} \,\hat {\boldsymbol {\eta }}\, {\rm e}^{\lambda t}+\text {c.c.}$ (with c.c. denoting the complex conjugate eigensolution, cf. figures 9(a) and 10a), the wing deformation resembles a stationary oscillation predominantly along the trailing edge towards the outboard region of the large scale coherent flow structures. The differences in deformation magnitude between port and starboard wings must be interpreted together with the static deformation in figure 2. Likewise, the adjoint modes (shown as a representative slice at constant span for the leading eigenmode in figure 9d) are again very similar to their fluid-only counterpart. As is often found with adjoint modes in external aerodynamics, coherent features reveal both a strong upstream support compared with their corresponding direct mode (in figure 9c) and little spatial overlap. A triangular structure is visible, resembling the observations for the span-periodic modes on infinite wings (Paladini et al. Reference Paladini, Beneddine, Dandois, Sipp and Robinet2019a; He & Timme Reference He and Timme2020). Similarly, for the two-dimensional adjoint aerofoil mode (and accordingly the span-uniform mode on the infinite wing) it was argued that the oblique lines, impinging near the shock foot and therefore likely to be important in the global dynamics, coincide with so-called characteristic lines (Sartor et al. Reference Sartor, Mettot and Sipp2015).

Figure 9. Visualisation of ($a$) leading and ($b$) second shock-buffet modes of fluid–structure coupled system (labelled b in figure 7) at angle of attack $\alpha =3.75^\circ$. Surface contours show real part of deformation in ${z}$-direction derived from structural part ${\hat {\boldsymbol {\eta }}}$ of coupled eigenvector, while volumetric iso-surfaces illustrate real part of $x$-momentum ${\widehat {\rho u}}$ at values of ${\pm 0.75}$. Underlying direct eigenvectors are unit length with respect to the inner product, specifically $\langle \hat {\boldsymbol {w}},\hat {\boldsymbol {w}}\rangle =1$, whereas the adjoint eigenvector additionally satisfies bi-orthonormality, specifically $\langle \check {\boldsymbol {w}},\hat {\boldsymbol {w}}\rangle =1$. Slices of ($c$) direct mode at dimensionless span $\eta = 0.66$, ($d$) adjoint mode at $\eta = 0.55$ and ($e$) momentum-only wavemaker at $\eta = 0.576$ are also shown. The inset of ($e$) shows iso-surfaces of the wavemaker at values of $\theta _{f}=5\times 10^2$, $1\times 10^3$ and $1\times 10^4$. All slices include the base-flow sonic line (solid black). The base-flow zero-skin-friction line is also shown in ($a$), ($b$) and inset of ($e$).

Figure 10. Visualisation of structural part of leading shock-buffet mode $b$ showing imaginary part of deformation in $z$-direction for ($a$) direct (cf. the real part in figure 9a) and ($b$) adjoint mode. Corresponding structural wavemaker is given in ($c$). The base-flow zero-skin-friction line is also included for orientation.

Figure 9(e) visualises the fluid wavemaker (computed with only the momentum components of the direct and adjoint solution vectors) for the leading shock-buffet mode as a slice at constant span alongside three volumetric iso-surfaces over the wing surface. Discernible sensitivity to localised feedback can be pinpointed in (and above) the separation region behind the shock foot, having relatively little activity around the shock wave itself, with quite a narrow span extent overall. Close inspection of the inset plot shows the highest value of the wavemaker, $\theta _{f}=1\times 10^4$, right at the shock foot where shock-induced boundary-layer separation initiates. This is an important result of our study, showing the wavemaker related to shock buffet on a finite wing for the first time and extends the insight gained from previous aerofoil and infinite-wing studies (Paladini et al. Reference Paladini, Beneddine, Dandois, Sipp and Robinet2019a). This also reinforces recent arguments trying to establish the active localities involved in the instability. Paladini et al. (Reference Paladini, Marquet, Sipp, Robinet and Dandois2019b) identified the shock foot as the core of the instability and the separated boundary layer and the shock front as essential in the dynamics. Moise et al. (Reference Moise, Zauner and Sandham2022) bring arguments that feedback loops involving wave propagation mechanisms, as suggested by Lee's model (Lee Reference Lee1990), are not at the heart of aerofoil shock buffet and question the role of the shock wave as either passive or active in combination with the separated boundary layer. The structural wavemaker is shown in figure 10(c) with the direct and adjoint structural deformations given in figure 10(a,b). To be precise, for visualisation purposes, what is shown in figure 10(c) is a modification of the structural wavemaker, defined earlier for modal coordinates in (2.10), using physical Cartesian coordinates ${\boldsymbol {x}}_s=(x_{s},y_{s},z_{s})$ instead, specifically,

(4.2)\begin{equation} \theta_{s,i} = \|\boldsymbol{\mathsf{R}}_i\check{\boldsymbol{x}}_s\|_2 \, \|\boldsymbol{\mathsf{R}}_i\hat{\boldsymbol{x}}_s\|_2 + \|\boldsymbol{\mathsf{R}}_i\check{\boldsymbol{u}}_s\|_2 \, \|\boldsymbol{\mathsf{R}}_i\hat{\boldsymbol{u}}_s\|_2, \end{equation}

where the diagonal matrix $\boldsymbol{\mathsf{R}}_i$ extracts the direct and adjoint structural solution at a given spatial location $i$, and the vector 2-norm is indicated by $\|{\cdot }\|_2$. Considering the values of the respective wavemakers, this would suggest that the buffet phenomenon can be most effectively influenced by providing feedback through the fluid degrees of freedom. This would agree with the observation that the shock-buffet unsteadiness does not require structural vibration to initiate or self-sustain. Having said this, the spatial distribution of the structural wavemaker is interesting in that it emphasises the trailing edge region of the wing approximately where the large scale coherent flow perturbations are located. However, trying to relate the wavemaker to the stiffness and mass properties of the underlying wing structure is a rather intricate discussion. Nevertheless, our identification of the fluid and structural wakemakers could also be useful for realising effective control strategies for the buffet phenomenon, and the interested reader is referred to e.g. D'Aguanno, Schrijer & van Oudheusden (Reference D'Aguanno, Schrijer and van Oudheusden2019) and Sartor et al. (Reference Sartor, Minervino, Wild, Wallin, Maseland, Dandois, Soudakov and Vrchota2020).

Second, in figures 5 and 7(b), for the group of modes near angular frequency $\omega = 1.8$, the Arnoldi method does indeed identify three unstable structural modes, specifically modes 19, 20 and 21, also found by the earlier flutter analysis. The coherent flow features of these modes’ eigenfunctions (not explicitly shown herein for reasons of brevity) closely resemble those of the shock-buffet modes. Having said this, as noted earlier, a correlation between the frequencies of shock-buffet modes and their spatial amplitudes (specifically the spanwise wavelength of coherent flow features) was reported in Timme (Reference Timme2020), which is also seen in the flow features of the structural modes. For instance, the spanwise wavelengths are larger than those of the shock-buffet modes b, in accordance with their lower frequencies. In figure 6, the entries of the MAC for these modes are largest in their respective underlying wind-off structural modes, as is expected, while also revealing a strong coupling activity with neighbouring modes.

Third, in figure 7(b), for the group of eigenmodes near angular frequency $\omega = 2.7$, one must first be able to identify and distinguish the different modes unambiguously, considering that the pk-type Schur complement method was unable to trace mode 28 (cf. the discussion surrounding figure 5). The coherent flow features of eigenmodes 27 and 28 are similar to the shock-buffet modes b (and indeed modes $c$ and $c'$), specifically the one emphasising the port wing, but have smaller spanwise wavelengths, as is expected according to their respective higher frequencies (Timme Reference Timme2020). The entries of the MAC in figure 6 are dominated by the corresponding wind-off modes. This strong diagonal dominance in the MAC for the structural eigenmodes no matter the angle of attack has been observed throughout, e.g. it is also the case for the second group of interesting (unstable) modes near angular frequency $\omega = 1.8$. Hence, even though the Schur complement method failed for one mode, when traced from wind-off conditions, the success using the Arnoldi method clearly identifies it as the missing mode 28, originating in the structural system. Mode 28 is visualised in figure 11(a). Besides the now familiar coherent flow features, the structural contribution in the eigenvector mainly describes, in agreement with the discussion on the MAC, the wind-off structural mode seen in figure 1(e) that accounts for wing twist in the outer span stations. Mode 27 (not shown in figure 1) comes with a strong deformation on the horizontal tail plane. For complementary insight into these unstable structural modes such as a frequency syncing, the reader is referred to the unsteady time-marching fluid–structure coupled simulations on the same test case, while focusing both on the initial growth of disturbances and the nonlinear saturation, in Belesiotis-Kataras & Timme (Reference Belesiotis-Kataras and Timme2021). Turning the attention to the last unstable mode in the third group of interesting modes, the entries of the MAC for mode $c'$ (last column in figure 6) are largest for wind-off structural modes 28 and 29, suggesting it is indeed a fluid mode. However, its proximity to the structural modes gives it a notably different behaviour in the MAC than the other fluid modes, with a larger emphasis on the nearby structural modes. Compared with the other fluid modes, which all show a non-trivial combination of wind-off modes to isolate the structural activity where the flow unsteadiness sits (cf. figure 9a,b), mode $c'$ reveals dominant deformation from mode 28 on the starboard wing and a somewhat stronger deformation from mode 29 on the port side, as seen in figure 11(b). Note that wind-off structural mode 29 appears anti-symmetric to mode 30 shown in figure 1( f). The fluid mode $c'$, which has now migrated into the unstable half-plane due to strong fluid–structure interaction, has coherent flow features around the port wing and these are very similar to those of its fluid-only counterpart (not shown herein). Besides our earlier statement that the coupling of aerodynamics with an elastic wing structure could destabilise the global flow field earlier, a second finding in this regard is the ability to destabilise additional, otherwise stable, fluid modes.

Figure 11. Visualisation of (a) structural mode 28 and (b) unstable mode $c'$ of the coupled system. Variables and plotting styles are identical to those in figure 9.

To further quantify the strength of fluid–structure coupling in the different direct eigenmodes, we took inspiration from a visualisation idea presented in Negi, Hanifi & Henningson (Reference Negi, Hanifi and Henningson2021). In figure 12, the coupling ratio defined as ${\langle \hat {\boldsymbol {w}}_s,\hat {\boldsymbol {w}}_s\rangle }/\langle \hat {\boldsymbol {w}},\hat {\boldsymbol {w}}\rangle$ (with ${\langle \hat {\boldsymbol {w}}_s,\hat {\boldsymbol {w}}_s\rangle }= \hat {\boldsymbol {x}}_s^H \boldsymbol{\mathsf{K}}\hat {\boldsymbol {x}}_s + \hat {\boldsymbol {u}}_s^H \boldsymbol{\mathsf{M}} \hat {\boldsymbol {u}}_s$, cf. § 2.2) is plotted for a selection of eigenmodes in the shock-buffet frequency range. The ratio defines the weight of the structural and fluid components in the eigenvector and can also be used to identify whether a mode originates in the structural or fluid system. Generally, if its value is low, it is a fluid-dominated mode. Vice versa, if it is high, the structural components dominate suggesting an origin in the structural system. Importantly, considering the variety of possible vector norms that can be used, the coupling ratio is best interpreted for a chosen norm while looking at the trend for a changing significant parameter, such as angle of attack in our study. The figure reveals interesting features for the different angles of attack presented. Note, results for angle of attack $\alpha = 3.7^\circ$ are similar to those at $\alpha = 3.75^\circ$ and therefore not shown here. First, at angles of attack $\alpha = 3.0^\circ$ and $3.5^\circ$, both below buffet onset, all relevant modes in the spectrum have relatively high values in the coupling ratio, confirming that they are indeed structural modes and no strong coupling is present. Second, near buffet onset, all structural modes have approximately two to three orders of magnitude lower values in the coupling ratio due to more weight in the fluid entries. The physically relevant fluid modes $a$ to $c$ (only discernible for angle of attack $\alpha =3.75^\circ$) have the lowest coupling ratio, indicating they are indeed fluid modes with relatively lower weight in the structural entries. Since the projection of the unstable shock-buffet (fluid) eigenmodes onto the structural degrees of freedom is nevertheless non-trivial, one can expect an aeroelastic response. Third, fluid mode $c'$ at angle of attack $\alpha =3.75^\circ$ appears in the ratio similar to its neighbouring structural modes, distinguishing it from the other fluid modes. The strong coupling of fluid and structure, which eventually causes the destabilisation of this mode, further shows itself in this visualisation. Also, while the applied vector norm must be assumed, comparing with the incompressible pitching aerofoil results at transitional Reynolds numbers presented in Negi et al. (Reference Negi, Hanifi and Henningson2021) that indicate several orders of magnitude separation in the ratio between high-frequency fluid and low-frequency aeroelastic modes, our results would suggest stronger fluid–structure coupled interactions with fluid and structural modes in close frequency proximity.

Figure 12. Coupling ratio for coupled modes at various angles of attack. Fluid and structural modes are denoted with half- and fully filled markers, respectively.