1 Introduction

Modern high-agility aircraft generate complex vortical flow fields and large areas of flow separation at subsonic speeds and medium to high angles of attack. With increasing angle of attack the flow field shows highly unsteady characteristics. Especially downstream of vortex breakdown the flow field is characterized by high turbulent intensities and distinct frequency contents [5]. The unsteady flow field leads to an excitation of the aircraft structure through pressure fluctuations on the configuration surface. The structural response following the excitation through the unsteady flow field is commonly known as buffeting [18]. Buffeting often occurs on the wings and tail planes of modern high-agility aircraft. It can lead to severe structural damage and degraded handling qualities.

Due to the severity and complexity of the phenomenon tail buffeting has long been subject of numerical and experimental research. However, with increasing computational capabilities, approaches for numerical buffeting prediction are continuously evolving. Challenges in computational buffeting prediction are (i) the complexity of the aerodynamic flow field; (ii) complex structural layouts; and (iii) modeling of the complex coupling mechanisms between flow field and structure. Partitioned fluid–structural coupling (see for example [23]) is among the most promising approaches to tackle these challenges in buffeting prediction. Separate solvers for Computational Fluid Dynamics (CFD) and Computational Structural Mechanics (CSM) are coupled to use the distinctive advantages of proven solvers in each domain.

Regarding the data exchange algorithm between the solvers, partitioned coupling methods can be divided into one-way coupling and two-way coupling methods. One-way coupling methods involve a uni-directional transfer of the aerodynamic buffet excitation from a CFD solver to the structural solver. The aerodynamic excitation is independent from the structural buffeting response, i.e., there is no feedback from the structural response to the unsteady flow field. Hence, one-way coupling approaches cannot predict complex coupling effects between structure and flow field. Examples for one-way coupling approaches can be found in [1, 15, 16].

Two-way coupling methods involve a mutual coupling of the separate solvers in time domain. Albeit being computationally expensive, these approaches can resolve aeroelastic effects with complex interaction between fluid and structure. Examples for two-way couplings can be found in [10, 12, 28, 29, 31, 32].

Despite these research efforts, a validation of new coupling methods and investigation of the basic coupling effects in buffeting with wind tunnel models are still essential. In this regard, a major challenge is the complexity of aeroelastic wind tunnel models for tail buffeting. In early developments, Davis Jr. and Huston [7] described different technologies and requirements for buffeting analysis with flexible wind tunnel models. Model damping, structural frequency requirements, model support in the tunnel as well as its instrumentation were discussed as possible influences on the buffeting analysis. Their focus was the comparison of an aeroelastically scaled wind tunnel model to real aircraft buffeting loads. Without specific scaling efforts, the structural behavior of the wing was captured well by their wind tunnel model. However, the influence of the elastic fuselage on the behavior of the horizontal tail plane (HTP) was not captured in the wind tunnel model. Furthermore, Davis Jr. and Huston [7] analyzed sting-mounted and floor-mounted concepts. Sting-mounted systems induced rigid-body vibration modes due to flexibilities in the sting support. Structural modes induced through the sting support can interfere with structural modes of the horizontal tail in a similar frequency range.

Rainey and Igoe [24] developed a 1/4-scale model of the X-1E. The modes of the dynamically scaled wing matched well with those of the actual aircraft. The HTP was scaled only geometrically and its modes were not captured by the model. However, large buffeting loads were found at the HTP. Rainey and Igoe [24] analyzed three different sources of the HTP buffeting loads: (i) wing wake flow, (ii) separated flow at the HTP, and (iii) motions of the aircraft structure transmitted to the HTP through the fuselage. In a detailed investigation, they attributed the major part of the buffeting loads to the wing wake.

A model with elastically and dynamically scaled lifting surfaces was developed by Rigby and Cornette [26]. In contrast to the model of Rainey and Igoe [24], the HTP of the model is located below the wing center line. Consequently, the wing wake only contributes to the HTP buffeting loads for negative angles-of-attack. However, the wing wake fluctuations affected the vertical tail of the model as well. Rigby and Cornette [26] also included wing weights to reduce the wing bending frequency. The altered wing motion reduced the fluctuations in its wake flow. Consequently, the velocity fluctuations aroung the HTP and the vertical tail response significantly decreased.

With the emergence of twin vertical tails the focus of buffeting research switched to vertical tails. Cole, Moss and Doggett Jr. [6] fitted an otherwise rigid 1/6-scale F-18 model with flexible vertical tails. The tails were dynamically scaled representations of the F-18 tails. Two tail versions of different mass and stiffness were build from aluminum plates covered with balsa wood. The aluminum plate was attached to the fuselage structure and equipped with strain gages. They found the main buffet response in the first bending mode of the tails [6]. Furthermore, the response of both tail versions correlated reasonably well.

Similarly, Bean and Wood [3] used a brass-spar and balsa wood elements to model a flexible fin which was excited by the unsteady flow of a delta wing. They were able to reduce the buffeting response of the fin by tangential leading edge blowing.

Ricketts [25] reviewed the status and future of aeroelastic wind tunnel testing at that time (1990). The most common structural layouts of the wind tunnel models comprised simple metal spars or plates which were covered with balsa wood forming the aerodynamic surface or pods for the mass/inertia distribution. More advanced models were constructed with fiber-composite skins and honeycomb or foam inner structures. These designs enabled anisotropic structural behavior and exact tailoring of the models’ structural characteristics. A recent example of the composite design is presented by [27, 34] for an aeroelastic flutter wind tunnel model. However, such composite structures are expensive to manufacture. Furthermore, the foam core materials are often challenging for numerical models [34].

Up to today, the development of flexible wind tunnel models is demanding. Common structural layouts are still using simple metal spars with covers of different material or expensive composite structures. In the paper at hand we present the development of a flexible wind tunnel model for tail buffeting analysis using rapid prototyping technologies. Rapid prototyping materials as polylactide (PLA) provide sufficient structural flexibility to highlight buffeting effects in a wind tunnel. At the same time, they promise sufficient structural strength to withstand the high structural loads during buffeting. With current 3D-printing technologies complex structural layouts can be manufactured to tailor the structural characteristics to specific requirements. Furthermore, they are cheap to manufacture. They have been applied in wind tunnel research in several areas, e.g., for rigid models (e.g., [19]) or for flutter wind tunnel models in aeroelasticity (e.g., [21]). To the best of the authors’ knowledge PLA or similar rapid prototyping materials have not been used for aeroelastic buffeting wind tunnel models yet. The presented development provides a first basis for the application of 3D-printed PLA in buffeting wind tunnel models from which more complex structural layouts can be derived.

The paper is further organized as follows. Section 2 gives a general overview over the design concept of the developed wind tunnel model. Sections 3 and 4 present the unsteady flow field characteristics and the FE-modeling of the structural layout, respectively. The unsteady flow field is predicted with high-fidelity detached eddy simulations (DES). To represent the structural characteristics, a normal-modes dynamic FE model and a static FE model for stress prediction of the configuration were developed. The unsteady flow field was used as to excite the dynamic FE model in a random-response vibration prediction. The predicted buffet response and loads of the model for the intended wind tunnel conditions are shown in Sect. 5. To prevent any damage for the model and the wind tunnel, the maximum structural stresses in the model are predicted in Sect. 6. For stress prediction maximum dynamic load cases of the buffeting response were used as a loading for the static FE model. Section 7 presents the results of the manufacturing process, while Sect. 8 provides some insights into sensor integration, some preliminary measurement results and future measurements. The presented paper is based on results presented at the Deutscher Luft- und Raumfahrtkongress (DLRK) 2020 [17].

2 Design concept

The objectives of the wind tunnel tests pose unique requirements to the wind tunnel model design:

  1. Size:

    The model is sized and designed for experimental investigation in the wind tunnel A (WTA) facility of the Technical University of Munich (TUM), including the applicable flow conditions and range of angle-of-attack.

  2. Buffeting effects:

    The model has to exhibit and highlight tail buffeting effects in complex vortical flow fields. The model should, therefore, allow to properly investigate aeroelastic coupling effects during tail buffeting.

  3. Controllable dynamics:

    Dynamic effects should not introduce any additional complex aeroelastic effects through the model design.

  4. Model strength:

    The model strength should be high enough (including safety factor) to sustain high loads and stresses during buffeting.

  5. Modularity:

    The model parts should be easily accessible and interchangeable.

  6. Representativeness:

    The model should represent current high-agility aircraft configurations.

The developed initial model concept to comply with these requirements is presented in Figs. 1 and 2. The aeroelastic wind tunnel model (AWTM) is designed as a half-model configuration to avoid additional dynamic effects typically seen on sting-mounted wind-tunnel models. The model consists of a fuselage structure with wing and HTP structures attached to it. The fuselage was chosen to represent current high-agility configurations and includes a chine. A peniche is used to place the model outside of the wind tunnel floor boundary layer. The peniche height was set according to previous experiences in the chosen wind tunnel (see for example [14]). The wing is designed as a \(76^\circ \)/\(40^\circ \)-double-delta wing with a NACA64A005 airfoil (Fig. 1). The wing thickness is reduced to \(3\%\) in the inner wing part. The selected wing configuration is based on a standard wing planform which has been subject of many numerical and experimental research efforts (e.g., [8]). The HTP structure is mounted downstream of the wing. Its center line is placed at \(z_{\text {HTP}}/c_{r,W}=0.025\) above the wing center line to place it directly in the unsteady turbulent flow field generated by the wing. In addition, the rear fuselage is drawn outward. The HTP is designed as a \(15^\circ \)/\(-5^\circ \)-trapezoidal structure with NACA64A005 airfoil (Fig. 1).

Fig. 1
figure 1

Basic dimensions of the AWTM configuration

The modular design of the model allows different wing and HTP structures to be attached to the fuselage (Fig. 2). This concept allows not only to test different geometries of wing and HTP, but also to determine the influence of different structural characteristics on the buffeting phenomenon. The fuselage is manufactured from aluminum and is regarded rigid. Wing and HTP structures are available in two versions. The flexible versions are manufactured from 3D-printed polylactide (PLA) with 100% fill density. Some material characteristics of PLA can be found in [9]. On the one hand, PLA gives the wing and HTP structure enough flexibility to exhibit buffeting effects in the wind tunnel. On the other hand, PLA promises to provide enough structural strength to withstand the high structural loads and stresses during buffeting. Modern 3D printing technologies allow to print complex structures. The desired structural behavior can be tailored in detail to fulfill scaling requirements to an actual aircraft. Furthermore, complex sensor assemblies can be easily integrated.

Stiff versions of wing and HTP structures are manufactured as aluminum structures. The stiff component versions provide a basis for comparison between a (quasi-)rigid and a flexible wind tunnel model. Thus the effects of the structural response of wing and HTP on the buffeting phenomenon can be investigated in different configurations.

Fig. 2
figure 2

Modular concept of the AWTM

The wing is attached to the fuselage through a plate structure (see Fig. 2). The plate is an integral part of the fuselage. The wing shows a corresponding cutout and is attached to the plate by several fasteners.

The HTP attachment should enable an angular deflection \(\delta _{\text {HTP}}\) of the HTP depending on the angle-of-attack. The use of actuators might induce additional dynamic and damping effects which are difficult to control and simulate. Therefore, the HTP is attached to the fuselage through a connector element (HTP Connector Rod, see Fig. 2). The connector has a cylindrical segment on the fuselage side and a plate segment on the HTP side. The HTP is attached to the connector plate with fasteners similar to the wing attachment. The cylindrical part of the connector is clamped between the main fuselage and the rear fuselage cover. The connector can be rotated to any angle to adjust the HTP deflection. Its position and rotation angle is secured through the fastening mechanism between the main fuselage and the rear fuselage cover.

Regarding the mechanical design the attachments of wing and HTP are critical parts regarding the buffeting loads in the wind tunnel. A computational stress analysis for the applied loads is described in Sect. 6.

3 Unsteady flow field

The unsteady flow field generated by the wind tunnel model is computed with the flow solver DLR-TAU (c.f. [30]) on unstructured, hybrid grids. An Airbus Defence and Space inhouse meshing software has been used to generate the CFD grids. Based on a triangulated surface grid, the boundary layer is resolved with 35 prism cell layers with a stretching factor of 1.25. The first layer cell height is chosen as \(h = 1.0\cdot 10^{-3}\) mm to obtain \(y^{+}\le 1\). In the simulations \(y^{+}\le 0.8\) was reached on the whole model for the investigated flow conditions. The remaining volume has been meshed with tetrahedral elements. The volume mesh has been significantly refined in the whole region over wing and tail which is the primary region of interest regarding the unsteady flow field. Based on the results of [15] an Improved Delayed Detached-Eddy-Simulation (SA-IDDES, [33]) turbulence approach with Spalart–Allmaras (SA) basis has been chosen with dual time stepping and an implicit Backward–Euler scheme with Lower–Upper Symmetric Gauss–Seidel (LUSGS) algorithm. A second-order central scheme with matrix dissipation is used for spatial discretization with a single-grid approach. Low-dissipation low-dispersion (LD2) settings were set in the SA-IDDES simulation to reduce the artificial damping of the resolved flow structures and improve the accuracy of the computation. However, such settings support the potential for numerical instabilities in the solver. Consequently, a blending function was used to restrict the LD2-scheme to the Large-Eddy Simulation (LES) regions. The Reynolds-Averaged Navier–Stokes (RANS) and LES regions established by the solver were verified after the simulations. Based on a combined study regarding grid refinement and physical time step size of a simplified configuration (cf. [15, 16]) a medium grid (78.1 M grid cells, see Fig. 3) and a time-step size of \(\varDelta t = 4.0\cdot 10^{-5}\) s have been chosen. The computations converged well within 200 inner iterations per physical time step. The flow field was computed for a time-span of \(t_{\text {av}} = 0.5\) s. Based on the wind tunnel characteristics and the buffeting loads computed in a preliminary study (see [16]) the wind tunnel conditions of interest are given in Table 1.

Fig. 3
figure 3

Medium CFD grid, overall grid break-up and detailed view on the AWTM

Table 1 Selected wind tunnel conditions for the CFD runs

At the flow conditions of interest, the configuration’s wing generates two primary vortices at the leading edges of the double-delta wing. Due to the mutual induction the vortices move towards each other, interact and burst over the wing. Details of the vortical flow field development over the angle of attack are presented for a simplified configuration at Ma\(_\infty =0.2\) in [16]. Figure 4a, b shows the vortical flow field over the wind tunnel model for the mean flow and a single time instant. The results show the development of both primary vortices at the leading edges of the double-delta wing. An additional vortex is generated at the chine of the fuselage.

For \(\alpha =25^\circ \), high axial velocities in the core area of the first vortex can be observed (see Fig. 4a). Subsequent to the interaction with the second vortex, the vortex system bursts over the center area of the wing. The flow field downstream of the vortex breakdown is characterized by turbulent structures which lead to a highly unsteady flow field. At the same time the vorticity of the flow field decreases downstream of the vortex breakdown (Fig. 5a). For \(\alpha =35^\circ \), the vortex breakdown position moves upstream and is located in the front part of the wing (Fig. 4b). Turbulent flow structures can be observed over the whole wing and towards the downstream HTP. At this angle of attack, large areas of the wind tunnel model are affected by these turbulent structures. The vorticity in the flow field decreases rapidly for both vortices downstream of vortex breakdown (Fig. 5b).

Fig. 4
figure 4

Time-averaged and instantaneous flow field around the wind tunnel model visualized with the \(\lambda _2\)-criterion; \({\text {Ma}}_\infty =0.15\), \({\text {Re}}_{/1m}=3.5\cdot 10^6\)

Fig. 5
figure 5

Vorticity slices and iso-surfaces for the wind tunnel model; \({\text {Ma}}_\infty =0.15\), \({\text {Re}}_{/1m}=3.5\cdot 10^6\)

The flow field over the wind tunnel model leads to pressure fluctuations on the surface of the configuration. The development of \(c_{\text {p,rms}}=p_{\text {rms}}/q_\infty =\sqrt{\overline{\left( p-\overline{p}\right) ^2}}/q_\infty \) over the angle-of-attack range of \(10^\circ \le \alpha \le 35^\circ \) for a simplified version of the wind tunnel model (double-delta wing with HTP) has been investigated in [16]. Figure 6 shows the rms-values of the pressure coefficient \(c_{\text {p,rms}}\) on the surface of the wind tunnel model.

For \(\alpha =25^\circ \), vortex breakdown causes high levels of pressure fluctuations in the breakdown region for the center wing area. The resulting unsteady flow field downstream of breakdown causes pressure fluctuations from the rear wing area up to the HTP. High \(c_{\text {p,rms}}\) values can also be observed for the front parts of the wing. Since no such fluctuations could be observed for the simplified configuration without the fuselage (see [16]) they likely result from the influence of the chine vortex and the fuselage geometry. For \(\alpha =35^\circ \), high \(c_{\text {p,rms}}\) values can be observed in large parts of the front wing area, since the vortex breakdown location is located upstream compared to \(\alpha =25^\circ \). Due to the upstream vortex breakdown, the vortical flow field is not shifted towards the outer wing parts. Consequently, lower levels of pressure fluctuations can be observed in the outer and center parts of the wing. However, the impact of the center fuselage structure leads to higher \(c_{\text {p,rms}}\) values in the inner wing area.

High levels of pressure fluctuations can be observed at the HTP leading edge and on its upper surface. They result from (i) the unsteady flow field of the wing and (ii) vortex generation and separation due to locally increased angle of attack through the wing flow.

Fig. 6
figure 6

Rms-values of the surface pressure coefficient \(c_{\text {p,rms}}\) on the upper side of wing and HTP for \(\alpha =25^\circ \) and \(\alpha =35^\circ \); \({\text {Ma}}_\infty =0.15\), \({\text {Re}}_{/1m}=3.5\cdot 10^6\)

4 Structural dynamics

The Finite-Element (FE) model of the wind tunnel model was generated with Hyperworks Hypermesh. The computation of normal modes and model stresses is performed with MSC Nastran [20]. For simplicity and reduction of computational effort, the fuselage structure is assumed to be rigid. The wing with its connection plate and the HTP with its connector are modeled using finite elements. The HTP deflection is zero in the FE models (\(\delta _{\text {HTP}}=0^\circ \)).

The FE model of the wind tunnel model consists of around 240.000 second-order tetra-elements for wing, HTP and both connector components to the fuselage (see Fig. 7). The connectors are clamped with SPC conditions (connection to the rigid fuselage). The grid was significantly refined around the drill holes and the attachment areas to adequately capture the stresses. The material properties of PLA (see Table 2) are assigned to the wing and HTP elements, while aluminum properties (see Table 2) are assigned to the connector elements. The PLA parts are modeled with isotropic, full-material characteristics.

The model was used for dynamic analysis and stress analysis. For dynamic analysis the contact between the components was modeled as a permanently glued contact in MSC Nastran. This form of contact blocks a relative movement of contacting nodes in all six degrees of freedom (DoFs) and is often used for dynamic FE models. For stress analysis contact surfaces between wing and plate as well as between HTP and connector are modeled as general contact surfaces with a friction coefficient of \(\mu _{\text {f,Al-PLA}}=0.2\) (see [35]). The connection between the components is established by fasteners. Fasteners are modeled as CBAR structures with the material properties of steel (see Table 2) which are connected to the components through RBE3 elements.

Fig. 7
figure 7

Finite-element modeling for the wind tunnel model

Table 2 Selected material properties

MSC Nastran was used to compute the first 20 normal modes (mass-normalized) of the wind tunnel model. The first six modes of the configuration are presented in Fig. 8.

Fig. 8
figure 8

First six normal modes of the wind tunnel model

5 Buffeting loads

5.1 Buffeting prediction approach

The buffeting loads of the wind tunnel model are predicted using a stochastic one-way coupling approach. The CFD solution of the desired physical time span at the fluid–structure boundary interface is transferred to the structural solver. The structural solver computes the structural response to the given aerodynamic excitation. The response computation is treated as a stochastic problem using Power-Spectral Densities (PSDs) or Cross-Power Spectral Densities (CSDs). The stochastic approach enables a straightforward estimation of maximum and minimum responses within defined confidence intervals. The linear dynamic equations of motion of the aeroelastic system in frequency domain can be written in modal space as

$$\begin{aligned}&\left[ -\omega ^2\mathbf {M}_\mathrm{gen}+\right. i\omega \mathbf {D}_\mathrm{gen} +\mathbf {K}_{gen} \nonumber \\&\quad \left. -q_\infty \mathbf {GAF}(\omega )\right] \varvec{\xi }(\omega ) =\varvec{\varPhi }^T\mathbf {F}_B(\omega ), \end{aligned}$$
(1)

with the modal amplitudes \(\varvec{\xi }(\omega )\) and the modal matrix \(\varvec{\varPhi }\). \(\mathbf {M}_\mathrm{gen} = \varvec{\varPhi }^T\mathbf {M}\varvec{\varPhi }\) describes the generalized mass matrix, whereas \(\mathbf {D}_\mathrm{gen} = \varvec{\varPhi }^T\mathbf {D}\varvec{\varPhi }\) and \(\mathbf {K}_\mathrm{gen} = \varvec{\varPhi }^T\mathbf {K}\varvec{\varPhi }\) are the generalized damping and stiffness matrices, respectively. The self-induced generalized aerodynamic forces are defined as \(\mathbf {GAF}(\varvec{\xi }(\omega ))=\varvec{\varPhi }^T\mathbf {F}_\mathrm{SI}(\varvec{\xi }(\omega ))\).

From Eq. 1 the random response problem can be written in form of (see [1, 22])

$$\begin{aligned} \mathbf {S}_{\varvec{\xi }}(\omega )=\mathbf {H}(\omega )\varvec{\varPhi }^T\mathbf {S}_{F_\mathrm{B}}(\omega )\varvec{\varPhi }\mathbf {H}^H(\omega ), \end{aligned}$$
(2)

Equation 2 is only valid for wide-sense stationary random processes [22]. \(\mathbf {H}^H\) is the hermitian of the frequency response function \(\mathbf {H}(\omega )\):

$$\begin{aligned} \mathbf {H}(\omega )= & {} \left[ -\omega ^2\mathbf {M}_{gen} \right. +i\omega \mathbf {D}_\mathrm{gen} \nonumber \\&\left. +\mathbf {K}_\mathrm{gen}-q_\infty \mathbf {GAF}(\omega )\right] ^{-1} \end{aligned}$$
(3)

The response loads can be computed using the mode displacement method (c.f. [4]) as

$$\begin{aligned} \mathbf {S}_{\mathbf {Y}}(\omega )=\mathbf {K}\varvec{\varPhi }\mathbf {S}_{\varvec{\xi }}(\omega )\varvec{\varPhi }^T\mathbf {K}^T. \end{aligned}$$
(4)

The nodal loads in Eq. 4 can be integrated to obtain monitor station loads. Using the integration matrix \(\mathbf {T}_i\) for a monitor station i we obtain

$$\begin{aligned} \mathbf {S}_{\mathbf {Y}_{MS,i}}(\omega )= \;& {} \mathbf {T}_i \mathbf {S}_{\mathbf {Y}}(\omega )\mathbf {T}_i^T\nonumber \\= \;& {} \mathbf {T}_i \mathbf {K}\varvec{\varPhi }\mathbf {S}_{\varvec{\xi }}(\omega )\varvec{\varPhi }^T\mathbf {K}^T\mathbf {T}_i^T. \end{aligned}$$
(5)

As developed by [16], Proper Orthogonal Decomposition (POD) is used in the stochastic buffeting prediction. POD is a convenient approach to reduce the size and associated computational effort of the buffet excitation. The original buffet excitation signals can be expressed through a small number of POD-modes \(\varvec{\varPhi }_\mathrm{POD}\) with limited losses of accuracy. The approach allows to use the full spectral density matrix of the reduced buffet excitation signals. It does neither require any prior investigation of the excitation correlations and nor any predefined arrangement of panels in the CSD matrix as other approaches (cf. [2]). To incorporate the POD in the random buffet response computation the time dependent surface forces from a CFD computation \(\mathbf {F}_\mathrm{B}(t)\) are decomposed with the POD to obtain \(\mathbf {F}_{\mathrm{B},\mathrm{POD}}(t)\) and the corresponding spectral density matrix \(\mathbf {S}_{F_{\mathrm{B}, \mathrm{POD}}}(\omega )\). The integration of the reduced spectral density matrix into the random buffeting response equation (Eq. 2) reads

$$\begin{aligned}&\mathbf {S}_{\varvec{\xi }}(\omega )=\nonumber \\&\quad \mathbf {H}(\omega )\varvec{\varPhi }^T\varvec{\varPhi }^T_{\mathrm{POD}}\mathbf {S}_{F_{\mathrm{B}, POD}}(\omega )\varvec{\varPhi }_{\mathrm{POD}}\varvec{\varPhi }\mathbf {H}^H(\omega ), \end{aligned}$$
(6)

To obtain the buffeting excitation data set \(\mathbf {F}_{\mathrm{B}}(t)\) for the wind tunnel model the time-accurate CFD surface forces described in Sect. 3 are splined onto a set of 1391 structural points on the configuration surface for each angle of attack. The use of thin-plate splines (TPS) ensures the (global) conservation of forces and moments. The direct method of the POD is used to reduce the data set as described above. The cumulative signal energy is plotted over the number of POD modes in Fig. 9. The number of modes required to reach a certain energy level decreases with increasing angle-of-attack (Fig. 9). Similar observations were made in [16]. The first three POD modes are presented in Fig. 10 for \(\alpha =25^\circ \) and \(\alpha =35^\circ \). The characteristics of the flow field described in Sect. 3 can be observed in the POD modes. The force fluctuations over the central wing area due to vortex breakdown are clearly resembled in the decomposition for \(\alpha =25^\circ \). For \(\alpha =35^\circ \), the first POD mode represents the highly unsteady characteristics at the wing apex. The higher modes show the force fluctuations due to vortex breakdown. Compared to the modes of \(\alpha =25^\circ \), the affected areas are located in the inner wing area close to the fuselage. Force fluctuations on the HTP are partly visible for the first three modes. They are more strongly represented by higher POD modes.

Fig. 9
figure 9

Cumulative energy of the original signal covered by different numbers of POD modes

Fig. 10
figure 10

POD modes 1–3 on the upper surface of the wind tunnel model for \(\alpha =25^\circ \) and \(\alpha =35^\circ \) (splining points in black)

The spectral density matrix \(\mathbf {S}_{F_{\mathrm{B},\mathrm{POD}}}(\omega )\) was created from the decomposed data set \(\mathbf {F}_{\mathrm{B},\mathrm{POD}}(t)\) for every angle of attack using windowing and a logarithmic frequency space with 700 frequency steps. It contains all auto- and cross-covariances of the decomposed surface force signals. Based on observations of [16] and the energy coverage in Fig. 9, \(N_{\mathrm{POD}}=150\) was selected. With 150 modes 96.7–\(97.5\%\) of the original signals’ energy is contained in the reduced data set. The spectral density matrices were then used as the buffeting excitation of the response problem in Eq. 6 to determine the buffeting response \(\mathbf {S}_{\varvec{\xi }}(\omega )\) and the integrated structural loads of the configuration.

5.2 Buffeting results

The root bending moment \(M_{x,\mathrm{root}}\) and the vertical tip acceleration \(a_{z, \mathrm{tip}}\) of wing and tail are selected as the most representative parameters regarding the buffeting loads on the wind tunnel model. Their root-mean-square values and spectra are evaluated in detail for the wing and HTP. Maximum loads and responses can be obtained from the mean and rms values assuming a Gaussian distribution of the parameters.

The response and loads for two monitor stations at the wing of the wind tunnel model are shown in Fig. 11a at different angles of attack. The PSDs of the wing tip acceleration and the root bending moment show the excitation of multiple eigenmodes of the configuration. For both angles of attack, the predominant excitation occurs for the first and second wing bending as well as the first wing torsion. For \(\alpha =35^\circ \), a mode at nearly \(k=20\) is strongly excited. The PSDs and their peaks show similar trends for \(\alpha =25^\circ \) and \(\alpha =35^\circ \). Increased PSD amplitudes can be observed for the first peak (first wing bending) of the root bending moment PSD at \(\alpha =35^\circ \) compared to \(\alpha =25^\circ \). Consequently, higher rms loads of the root bending moment occur at \(\alpha =35^\circ \). The first peak of the wing tip acceleration PSD also shows an increase between \(\alpha =25^\circ \) and \(\alpha =35^\circ \). However, the PSD amplitudes of the remaining modes are lower at \(\alpha =35^\circ \). Therefore, the rms values of the wing tip acceleration decrease from \(\alpha =25^\circ \) to \(\alpha =35^\circ \).

The response and loads for two monitor stations of the HTP of the wind tunnel model are presented in Fig. 11b. Similar to the wing response multiple modes of the HTP are excited. The predominant excitation occurs for the first three HTP modes, namely the first and second bending as well as the first torsional mode. It can be observed that the highest response peaks occur at frequencies of the first and second HTP bending modes. The trends over the angle-of-attack are similar to those of the wing. Increased PSD amplitudes for the first peak can be observed for \(\alpha =35^\circ \) leading to higher rms values of the root bending moment. The rms values of the HTP tip acceleration slightly decrease from \(\alpha =25^\circ \) to \(\alpha =35^\circ \).

Fig. 11
figure 11

Dynamic buffeting loads and responses on the wind tunnel model; \({\text {Ma}}_\infty =0.15\), \({\text {Re}}_{/1m}=3.5\cdot 10^6\)

6 Structural stresses

6.1 Stress prediction approach

The stresses in the attachment mechanism of the model are of great importance to qualify the flexible wind tunnel model for the wind tunnel tests. The maximum stresses with a safety factor S (usually \(S=1.5\) is chosen) can not exceed the maximum strength of the material (see Table 2) to maintain the structural integrity and prevent damage to the model and the wind tunnel.

The process to determine the model stresses is presented in Fig. 12. The basis for the computation is the dynamic modal buffeting response \(\varvec{\xi }_\mathrm{dyn}\) which has been discussed in Sect. 5 along with the integrated monitor station loads. For the monitor station loads at the wing root and and HTP root, three load envelopes are created based on \(F_z\), \(M_x\), and \(M_y\). With POD, the elliptical load envelopes can be expressed as functions of their respective main axes. By adequate choice of the POD amplitudes, the maximum points of the envelopes can be defined for each main axis. Thus, for each monitor station six severe dynamic responses \(\varvec{\xi }_\mathrm{sev}\) can be defined as modal response on the model. When adding the mean modal response \(\varvec{\xi }_0\) (which is also obtained during the dynamic loads process of Sect. 5) one obtains the maximum load cases \(\varvec{\xi }_\mathrm{max}\). With the mode displacement method (c.f. [4]), the corresponding nodal forces and moments acting on the FE model can be determined.

Fig. 12
figure 12

Structural stress prediction process

To determine the loads described above, the structural model with glued contacts has been used. However, to determine the structural stresses in the critical areas, the maximum loads have to be applied to the structural model with fasteners. Consequently, the nodal loads from the glued model are transferred to a set of load application points on the configuration surface. Around 460 points have been selected on the FE-model surface similar to the splining points used in Sect. 5. Since a standard spline interpolation did not lead to sufficient results, a Guyan Reduction [13] was chosen to condense the nodal loads to the load application points. The condensed loads for the different load cases (LC) are applied to the FE model with fasteners and a static computation is performed in MSC Nastran (SOL101).

6.2 Stress results

Stress distributions in the HTP attachment area are presented for some severe load cases in Fig. 13. The maximum allowable stress is the tensile strength of PLA (see Table 2), reduced by a safety factor of \(S=1.5\). Cells exceeding this stress limit are colored in gray. The stresses exceed the limit stresses in large parts of the attachment area. A redesign of the attachment area is necessary regarding material thickness and fastener positioning. Similar observations were made in the rear wing attachment area. However, critical stress limits were only exceeded in small areas.

Fig. 13
figure 13

HTP attachment stresses (von-Mises) for selected load cases of HTP Configuration 1; \({\text {Ma}}_\infty =0.15\), \({\text {Re}}_{/1m}=3.6\cdot 10^6\), \(\alpha =25^\circ \), \(\delta _{\text {HTP}}=16^\circ \)

6.3 Model modifications

Based on the stress analysis described above modifications to the structural layout are necessary to ensure that the structural stresses in the model stay within the allowable stress limits for the material. To reduce the structural stresses in the wing attachment area, the relative airfoil thickness was increased in the inner part of the double-delta wing. Furthermore, this leaves enough space to include additional shells between the fasteners and the wing structure to reduce stresses in the contact area. The number of fasteners was reduced to six fasteners. The fasteners were placed exclusively on the lower side of the wing and the fastener positions were adapted according to the stress distribution.

The relative thickness of the HTP airfoil was increased from to provide enough space for fasteners (including shells) and to reduce the stresses in the attachment mechanism. The diameter of the cylindrical connector part as well as the thickness of the connector plate were increased.

FE models were created for the modified structure of the wind tunnel model in the same manner as described in Sect. 4. Due to the increased material thickness, the eigenfrequencies of the model are increased. The structural stresses of the modified model are computed with the new FE models in the same way as for the original model. However, the aerodynamic loads were not recomputed with the modified model shape. For the efficiency of the design process it is assumed that the aerodynamic forces do not show massive force increases or significantly different frequency content due to the shape modifications. Therefore, the results of Sect. 3 were splined onto the new FE model as described for the original model in Sect. 5. The dynamic response as well as the load cases of the modified model were computed following the process described for the original model. The stresses in the attachment mechanisms of the HTP are presented for some severe load cases in Fig. 14.

Fig. 14
figure 14

HTP attachment stresses in the modified model structure for selected load cases; \({\text {Ma}}_\infty =0.15\), \({\text {Re}}_{/1m}=3.6\cdot 10^6\), \(\alpha =25^\circ \), \(\delta _{\text {HTP}}=16^\circ \)

A significant reduction of the structural stresses can be observed compared to the original model. The stresses do not exceed the structural strength (including safety factor) in any critical area. The modifications had the same effect on the stresses in the wing attachment area.

7 Manufacturing

The fuselage and the stiff components were milled from aluminum alloy. The surfaces which are subjected to the flow were polished to provide a suitable surface finish after the milling process. The manufactured model is shown in Fig. 15a. Holes and ducts for sensor equipment and wiring were integrated. Sensor positions were located in pockets on wing and HTP. The pockets are accessible through screwed caps. The assembly of the components was checked. The attachment mechanisms of wing and HTP provided adequate fastening of the components. The connection between the main fuselage and the fuselage cover provide enough force to clamp the HTP connector element. Remaining holes and ducts will be filled to provide a smooth surface.

Fig. 15
figure 15

Fully manufactured wind tunnel model

The flexible parts of the model were 3D-printed from PLA with the BigRep One printer available at Premium Aerotec. The selected printer enables printing of large-scale parts as a single piece. Assembling a PLA structure from multiple smaller parts would require some sort of fastening or gluing which introduces additional dynamic effects in the structural behavior. With the structures being printed as a single piece we ensure a consistent material structure of the flexible parts. The parts are printed with a \(0.5 \, mm\)-diameter nozzle and a layer thickness of \(0.25 \, \)mm. The layers are arranged in y-direction of the model. The printing took 87 h and 35 h for the wing and the HTP, respectively. The wing is printed with some support structure at the wing-fuselage intersection due to the wing’s non-planar shape in this area. After printing any additional support material is removed and the parts are sanded (Fig. 15b, 15(b)). According to [19] sufficient macroscopic and microscopic accuracy can be obtained after sanding of the PLA parts.

8 Instrumentation and measurements

Fig. 16
figure 16

Kulite surface pressures on the wing for different angles of attack

Fig. 17
figure 17

Kulite surface pressures on the HTP for different angles of attack

The wind tunnel model will be tested in the WTA facility of the Technical University of Munich. The Göttingen type wind tunnel has an open test section of 1.8 m   \(\times \)   2.4 m   \(\times \)   4.8 m (height \(\times \) width \(\times \) length). Tests will be conducted at low speed conditions of \({\text {Ma}}_\infty =0.15\) and \({\text {Re}}_{1/m} = 3.5\cdot 10^6\) [1/m]. The model is placed on a turn table for angle-of-attack adjustments.

The analysis of buffeting effects will be performed with measurements of aerodynamic forces and moments, unsteady flow field velocities, unsteady surface pressures and unsteady structural deformations. Aerodynamic forces and moments will be measured with an external balance below the test section. Velocity fluctuations in the flow field will be obtained with hot-wire anemometry (HWA) and Stereo Particle Image Velocimetry (PIV). Both enable a detailed analysis of vortex development, breakdown and unsteady velocity fluctuations in the flow field.

The buffet excitation of the model structure is measured with unsteady Pressure Sensitive Paint (iPSP). Recent developments have shown that unsteady iPSP is capable of measuring pressure fluctuations down to 5 Pa at 1318 Hz [11]. Thus, unsteady PSP is applicable for the intended wind tunnel tests considering the pressure fluctuations analyzed above at wind tunnel flow conditions. Unsteady PSP is especially useful for investigation of aeroelastic wind tunnel models, because the structural layout and characteristics require no major changes for sensor integration. However, one Kulite is integrated in every wing or HTP component to scale the iPSP results to the correct pressure levels. The surface pressure fluctuations obtained from unsteady PSP can be directly compared for all model configurations. Furthermore, the surface results can be used as the buffet excitation input to one-way coupling computations as described in Sect. 5. Some preliminary results of the unsteady Kulite surface pressure measurements are presented in Figs. 16 and 17 (Kulite positions see Fig. 15; Wing: \(x/c_{r,W}=0.954\), \(y/s_{W}=0.67\); HTP: \(x/c_{r,W}=1.49\), \(y/s_{W}=0.65\)). The mean surface pressure at the wing Kulite reach a suction peak at \(\alpha =25^\circ \) before vortex breakdown impacts large areas of the flow field over the wing and reduces the suction peak for higher angles of attack. Due to vortex breakdown the surface pressure fluctuations at the Kulite position increase strongly between \(20^\circ \le \alpha \le 30^\circ \) before reaching a plateau at \(\alpha =30^\circ \) (Fig. 16). Resulting from the vortex breakdown, the pressure fluctuations increase on the HTP from \(\alpha =15^\circ \) to higher angles of attack (Fig. 17). The highest increase can be observed between \(\alpha =20^\circ \) and \(\alpha =25^\circ \). From \(\alpha =25^\circ \) to \(\alpha =35^\circ \) the pressure fluctuations remain on similar levels, since the whole flow field over the configuration is affected by vortex breakdown for these angles of attack (see predictions in Fig. 5b). A full comparison between computational predictions of the surface pressures and the experimental data will be performed in the next step when the full iPSP data set is available.

Fig. 18
figure 18

Instrumentation and model installation for the ground vibration test (GVT)

Fig. 19
figure 19

Adapted FE model of the wind tunnel model

The dynamic structural response of the model will be obtained using optical deformation measurements. Results will be compared to measurements of accelerations obtained at the tips of wing and HTP. To correctly represent the structural characteristics of the wind tunnel model, the FE model of Sect. 4 is tuned with the results of a ground vibration test (GVT). The GVT was performed with the model fully installed in the wind tunnel to obtain the same conditions as during the actual wind tunnel tests. A shaker was used to excite the different model components over a frequency range. The response of the model was measured with accelerometers (Fig. 18). The frequency response function as well as the eigenmodes of the components could be extracted with the model response and the excitation. The fuselage was added to the FE model of Sect. 4 with CBARs (Fig. 19). The fuselage bars are connected to the wing, HTP and their connection plates through CBUSH elements. The CBUSH stiffness was adapted to match the mode shapes and frequencies of the model components obtained in the GVT. The adapted FE model was excited in a SOL111 (response) computation in Nastran with the same force and over the same frequency range as in the GVT. From the frequency response functions mode shapes and frequencies were extracted. Exemplary, the results for the PLA HTP are presented herein. The frequencies of the component obtained from the GVT and the adapted FE model are presented in Fig. 20. The predictions of the mode frequencies of the adapted FE model agree well with the GVT results. All predicted modes are within \(5\%\) of the GVT results over the whole frequency range. The mode shapes also agree well between simulation and experiment. As an example, Fig. 21a shows the 3rd eigenmode of the PLA HTP obtained from the GVT and from Nastran. Figure 21a presents the modal assurance criterion (MAC) for the predicted mode shapes w.r.t. the GVT mode shapes. The MAC shows high levels of agreement between the mode shapes of experiment and simulation even in high frequency ranges. Overall, the adapted FE model closely agrees with the experimental analysis of the wind tunnel model in the GVT and is, therefore, suitable for use in computational buffet predictions for the final wind tunnel model.

Fig. 20
figure 20

Comparison of the mode frequencies for the GVT and the Nastran FE results; reduced frequency \(k=f\cdot c_r/U_\infty \)

Fig. 21
figure 21

Comparison of mode shapes from GVT and Nastran

9 Conclusion

A flexible wind tunnel model has been developed for experimental tail buffeting analysis using 3D-printed polylactide parts. The model was developed using high-fidelity methods of CFD and CSM. The developed half-model configuration creates a highly vortical flow field. A system of two interacting vortices bursts over the wing and leads to an unsteady flow field impacting on the downstream HTP. It has been shown that significant pressure fluctuations can be observed on the wing and HTP surfaces. The pressure fluctuations excite the structure and lead to buffeting vibration of the model structure. The vibrations have been predicted in detail using stochastic buffeting prediction (one-way coupling). The magnitude of the vibrations makes them easily observable and measurable. However, static computations revealed that the structural strength of the initial model is not sufficient to withstand the high loads during buffeting in the attachment mechanisms. Modifications were made to the structure to increase the structural strength in critical areas. The modified model showed sufficient strength to withstand the buffeting loads. The model’s modular design enables a comparison of buffeting loads between different structures. The model was successfully manufactured with state-of-the-art CNC milling and 3D printing. A ground vibration test showed good agreement between the structural characteristics of the manufactured model and the adapted FE-model predictions.

The successful development of the model is a further step towards detailed buffeting analyses with flexible wind tunnel models. In subsequent developments wind tunnel test campaigns will be performed and compared to the computational predictions.