Numerical simulation of wake interactions on a tandem wing configuration in high-speed stall conditions

In this work, the interaction of the separated wake of the front wing with the rear wing of a tandem configuration is investigated for high-speed stall conditions by means of hybrid RANS/LES simulations, using the zonal AZDES method. After a characterization of the transonic buffet on the front wing, the development of the separated turbulent wake behind the wing is investigated. The interaction of the separated wake with the rear wing is then analyzed in detail. The results reveal that there is a strong variation in the wake characteristics over the buffet cycle, caused by the varying amount of separation on the front wing. During the upstream movement of the shock, the flow is largely separated, resulting in a thick wake with strong, high-frequent fluctuations that can be attributed to large turbulent vortices. On the contrary, when the shock travels downstream, there is only a small amount of separation present, resulting in a thin wake with comparatively low fluctuations that are caused by corresponding smaller turbulent vortices. The impact of the wake of the front wing causes a strong variation in the rear wing loading. An oscillation with a comparatively low frequency can be distinguished from high-frequent fluctuations. The low-frequent oscillation is caused by the variation in the downwash behind the front wing as its lift changes during the buffet cycle. The high-frequent fluctuations are due to the impingement of the turbulent structures onto the rear wing. Because both size and frequency of those vortices vary significantly within the buffet cycle, the amplitude and frequency of the lift and surface pressure fluctuations also change accordingly.


Introduction
For the further development of commercial aircraft aimed at a reduction in fuel consumption and an increase in flight safety, it is essential to understand, reliably predict and be able to control the phenomena that occur at the borders of the flight envelope. The economic and safe flight regime of commercial aircraft is limited in terms of flight Mach number and angle of attack. At low Mach numbers and high angles of attack, massive separation (low-speed stall) occurs, while high-speed stall at high Mach numbers is characterized by the occurrence of unsteady shock waves and shockinduced separation, referred to as transonic buffet. Transonic buffet is a coupled periodic oscillation of a shock over the surface of an airfoil or wing and the corresponding shockinduced separation. This oscillation can in turn cause the structure to vibrate, known as buffeting, which endangers structural integrity and control.
Numerous numerical [1,2] and experimental [3,4] studies on the mechanisms of buffet have been carried out in the past, mostly for the two-dimensional case, i.e., for airfoils. However, open questions still remain, for instance, regarding the mechanisms of the 3D buffet phenomenon on swept wings. Likewise, only few studies exist so far on the development of the separated, unsteady wake of the buffetaffected wing and its interaction with the tail plane, on which the unsteady, turbulent flow is imposed. Proposedly, the wake affects aerodynamics, loads and can lead to a structural 1 3 excitation of the tail plane. The detailed mechanisms leading to the buffet phenomenon were published by Lee [5] for the two-dimensional case. According to Lee, the shock causes disturbances that propagate downstream and are scattered at the trailing edge, creating pressure waves that travel back to the shock, forming a feedback loop. The time it takes for the disturbances to travel to the trailing edge and back to the shock corresponds to the buffet period, which was confirmed by experiments [6]. Further investigations revealed that, in addition, pressure waves also travel around the pressure side of the airfoil, pass around the leading edge and interact with the shock [3,7]. Crouch et al. [8] performed a stability analysis of the linearized RANS equations and were able to identify a global unsteady mode associated with the shock movement. This mode also comprises disturbances traveling upstream on both sides of the airfoil. The three-dimensional case of buffet on swept wings, however, constitutes a more complex phenomenon. Experimental [9,10] and numerical investigations [11,12] showed that with increasing wing sweep, a spanwise oscillation of the shock front occurs, with the formation of so-called buffet cells in the wing root area that travel spanwise toward the wing tip. In addition, a more broadband spectrum of pressure fluctuations on the wing is reported [13]. These findings indicate a different buffet mechanism than in the two-dimensional case. The interaction of the separated wake of a wing with the tail plane was investigated numerically for a transport aircraft configuration in low-speed stall conditions by Waldmann et al. [14] and Tan et al. [15]. It was found that the impact of the turbulent wake leads to a significant oscillation of the tail plane's load. The level of load fluctuation is greatly affected by the amount of turbulent kinetic energy present in the wake. Furthermore, the wake size and downwash direction vary significantly with angle of attack, which directly influences both the mean tail plane lift and the force fluctuations experienced by its surface. A corresponding high-speed stall case was studied numerically by Illi [16]. It was shown that the fundamental frequency of the fluctuations in the wake and of the tail plane's load oscillations correlates with the frequency of the shock movement on the main wing.
The numerical investigations presented here are part of subproject 4 of the research unit FOR 2895 "Unsteady flow and interaction phenomena at high-speed stall conditions" funded by the German Research Foundation (DFG). In the present work, the interaction of the separated wake of the front wing with the rear wing of a tandem configuration is investigated for in high-speed stall conditions by means of hybrid RANS/LES simulations, using the zonal AZDES method [17]. Under the investigated conditions, transonic buffet occurs on the front wing but not on the rear wing. After a characterization of the transonic buffet and the corresponding unsteady flow separation from the front wing, the development and the trajectory of the separated turbulent wake behind the wing are investigated including a spectral analysis of the pressure fluctuations. The interaction of the separated wake with the rear wing is then analyzed in detail. It is examined how the resulting surface pressure and load fluctuations of the rear wing are correlated with the fluctuations in the wake and on the front wing.

Setup and numerical method
The investigated configuration is a tandem configuration consisting of two un-tapered and un-swept wing segments, as sketched in Fig. 1, which represents a conventional arrangement of front wing and horizontal tail plane. For the front wing segment, the supercritical OAT15A airfoil is chosen, which has been studied extensively in buffet conditions both experimentally and numerically in the past [1,6]. For the rear wing segment, a NACA 0012 airfoil is used with a chord length of 40% of the front wing segment. The rear wing segment is positioned two chord lengths behind the trailing edge of the front wing segment, as shown in Fig. 1, centered in the wake of the latter at z∕c OAT = 0.05 to maximize the impact of the wake for the investigation of the interaction phenomena. The choice of the vertical position is based on the analysis of the wake trajectory under buffet conditions of the isolated front wing segment without the rear wing segment. Therefore, simulations of the isolated front wing segment have been performed in addition to the simulations of the tandem in order to investigate the undisturbed development of the wake. Both the chord ratio and the relative distance of the wing segments are roughly based on existing transport aircraft.
For the inflow conditions, a Mach number of 0.73, an inflow angle of 3.9 • , which corresponds to the angle of attack of the front wing segment, and a static temperature of 273 K are selected. The inflow density is set to 0.2129 kg/m 3 according to a Reynolds number of 3 million with reference to the front wing chord. Mach number, Reynolds number and angle of attack are chosen in accordance with the conditions used in the experimental investigations of the isolated OAT15A airfoil in buffet by Jacquin [6]. For the front wing segment, the location of laminar-turbulent transition is prescribed at 7% chord on both sides, whereas the flow around the rear wing segment is modeled as fully turbulent. For the Fig. 1 Geometry of the investigated configuration rear wing segment, an incidence of − 5.9 • relative to the front wing chord is chosen, corresponding to an incidence of − 2 • with reference to the free stream. Considering the downwash of the front wing segment, this results in a moderate negative lift coefficient of the rear wing segment of around c l = − 0.3 , representative of a trimmed cruise condition.
The simulations were performed on a hybrid mesh, consisting of structured blocks of hexahedra for the boundary layer region of both wing segments, the area of shock movement on the upper side of the front wing segment, and the wake behind the latter, as shown in Fig. 2, and unstructured blocks of triangular prisms extending to the far field boundary, which is located 80 chord lengths of the front wing away from the tandem. A free-stream boundary condition is imposed at the far field. A resolution of 0.4% of the front wing chord was chosen for the wake area to ensure that the larger turbulent structures are resolved adequately. This resolution was also selected for the spanwise discretization, resulting in almost cubic cells in the wake. In the spanwise direction, 64 cell layers are used to discretize a spanwise extension of around one quarter of the front wing chord, applying periodic boundary conditions. The upper surface of the front and the rear airfoil is discretized with 285 and 260 points, respectively. For the lower surfaces, 204 and 260 points are used. The mesh topology, the spanwise extent of the domain and the mesh resolution are chosen according to previous work of the working group of the authors, in which the Automated Zonal Detached Eddy Simulation (AZDES) method described below that is used in the investigations here has been validated for different applications, especially for buffet flow conditions [17][18][19]. The mesh of the isolated OAT15A wing segment with resolved wake consists of around 17 million points and the mesh of the tandem configuration exhibits around 22 million points in total.
All simulations were conducted with the finite-volume solver TAU developed by the German Aerospace Center (DLR) [20] as hybrid RANS/LES simulations applying the AZDES method described below. A low-dissipative skew-symmetric scheme of second order by Kok [21] was used for the calculation of the numerical fluxes, whereas an implicit dual-time stepping method of second order was applied for the integration in time. For stabilization, artificial dissipation based on the scheme by Jameson, Schmidt and Turkel [22] is applied to the numerical flux in TAU via second and fourth-order diffusion terms. Here, a matrix scheme [23] is used which generally adds less dissipation than the scalar variant without sacrificing stability. For the scaling coefficient of the fourth-order term, a value of 1/128 is chosen, which is lower than the default setting in TAU (1/64) in order to further reduce dissipation. The choice of the scheme and the dissipation settings are based on previous work [17][18][19]. The physical time step was set to 1/100 of the convective time period c OAT ∕U ∞ based on previous investigations of buffet flow [17][18][19]. For the hybrid simulations, the choice of the time step is not prescribed primarily by the shock motion, which can be captured properly with a coarser temporal resolution, but by the requirements of the LES in the wake. A local CFL number in the order of one is generally recommended for LES mainly for the sake of accuracy [24][25][26]. In the present work, CFL numbers in the wake range from 1.5 to 2.5, meeting this criterion. However, a better resolution of very high-frequent fluctuations might be achieved by further reducing the CFL number down to one. For the modeling of turbulence in the RANS regions of the flow field, the SSG/LRR-Reynolds stress model [27] was chosen, serving as a sub-grid scale model in the corresponding LES areas [18].
The AZDES is a zonal hybrid RANS/LES method that was developed at the Institute of Aerodynamics and Gas Dynamics of the University of Stuttgart for the application to transonic flows [17][18][19]. Within this method, the zone division between the RANS and LES regions is based on a precursor unsteady RANS simulation. An integral turbulent length scale is calculated from the modeled turbulent quantities, which reaches high values in the areas of separated flow. These areas, where large turbulent structures are expected to be present that can be resolved by the mesh, are then marked for DDES computation in the following actual hybrid simulation, whereas attached boundary layers will be treated in RANS mode. The zone division can be adjusted by the user by specifying the cutoff length scale above which LES is activated. In addition, the shielding of the boundary layer can be tuned by enforcing RANS for all positions closer to the airfoil or body surface than a chosen distance. Similarly, LES mode can be forced for all distances greater than a set limit. For the simulations of the tandem configuration, a cutoff length scale of 8% of the chord of the front wing is chosen, and the LES mode is enforced at distances greater than 6% chord, in line with the recommendations of previous studies [18] that recommend a cut-off length scale of at least 6% chord and a DES switching distance of at least 4%, respectively, to sufficiently shield the region of the shock and the attached part of the boundary layer, so that the shock motion is covered in RANS mode. Thus, grid-induced separation, which will lead to an nonphysical damping of the buffet motion, is avoided. The choice of the cutoff length scale is also based on the grid resolution. The cutoff length scale should be set according to the size of smallest turbulent eddies that can be resolved with the grid. As the central scheme used in the present work is of second order, the number of grid points required to resolve a flow structure can generally be roughly estimated to be in the range of ten to twenty [24,28,29]. Based on the grid resolution of 0.4% chord, this corresponds to a size of 4-8% chord, which fits the chosen value. After the buffet is fully established, the precursor unsteady RANS simulations are continued for five buffet cycles to calculate the zone division and then continued in the AZDES mode for 15-20 further cycles for evaluation to ensure convergence of the flow field statistics. The zone division for the tandem simulations is shown in Fig. 3.

Wake development of the front wing segment in Buffet conditions
The lift of the front wing segment varies strongly during the buffet cycle due to the movement of the shock, as seen in Fig. 4, which shows the development of the lift coefficient over several buffet periods. The lift coefficient oscillates between c l = 0.80 and c l = 1.03 , corresponding to an amplitude of ĉ l = 0.115 , and with a frequency of 18.7 Hz or a Strouhal number of Sr = 0.0772, respectively. Compared to the experimental results of Jacquin [6], who investigated buffet on the isolated OAT15A airfoil for the same inflow conditions, displayed in Table 1, the dimensionless frequency found in the simulation compares quite well with the measured value with a difference of about 16%. However, the amplitude is significantly smaller than in the experiment, with a difference of about 30%. There are different reasons for this difference. First, the inherent numerical dissipation present in all simulations due to the finite spatial and temporal resolution tends to dampen all oscillations, leading to a smaller amplitude than expected in reality. Second, the area close to the shock lies within the URANS zone where the turbulence is modeled and not resolved. This is intended in the AZDES method to prevent grid-induced separation. However, the solution, especially the location of the separation, is then dependent on the turbulence model, in contrast with LES or DNS which do not require this additional modeling. Thus, certain deviations from the reality are to be expected. This issue can naturally not be overcome in hybrid methods, only by switching to a full LES or DNS with the accompanying significant increase in computational demand. Furthermore, wind tunnel effects are not considered in the simulation. It seems likely that the deviations from the experiment in terms of the range of shock motion at least partly originate from the wind tunnel setup, which is not simulated in this work as free stream conditions are imposed at the far field boundary. Deviating shock positions different from the experimental results of [6] are consistently observed in several simulations and in experiments [30][31][32][33][34]. A slightly different effective angle of attack of the airfoil in the wind tunnel caused by the wind tunnel walls may be a possible explanation, as the shock position is rather sensible toward the wind tunnel setup [33]. Despite the smaller amplitude, however, the peak level of the pressure fluctuations on the surface of the wing segment compares well with the measurements, as shown in Fig. 5. Figure 6 displays the distribution of the pressure coefficient at four moments in time during one buffet period: when the shock reaches its most downstream position (I), during the upstream movement of the shock (II), when the shock reaches its most upstream position (III), and during the downstream movement of the shock (IV), showing the range of the shock motion. It is found that the shock position moves between x∕c = 0.38 and x∕c = 0.52 . In addition, the small gradient of the pressure behind the shock close to the trailing edge during the upstream movement of the shock (II) and at the most forward shock position (III) indicates flow separation, whereas the continuous increase in pressure toward the trailing edge suggests only small separation or attached flow at the other times. This is confirmed by Fig. 7, which shows the Mach number in the flow field around the rear section of the wing segment together with streamlines close to its surface for the four moments in time introduced above. It is notable that the amount of separation behind the shock varies strongly within the buffet cycle. When the shock reaches its most downstream location (I), there is only a small amount of separation present due to a comparatively small shock strength. During the upstream movement of the shock, however, the flow is largely separated (II). As the shock moves upstream, the velocity of the fluid relative to the shock is increased, which further increases the shock strength and thus moves the point of separation forward. When the shock temporarily stops at its most upstream position, this effect vanishes, reducing the shock strength and amount of separation (III). Then, during the downstream  movement of the shock, the velocity of the fluid relative to the shock is decreased. Therefore, the resulting reduced shock strength leads to a reattachment of the flow during this phase (IV). This strong variation in the amount of separation during the buffet cycle has a great impact on the characteristics of the wake and its interaction with the rear wing segment, which is discussed later. The fluctuations of the pressure coefficient on the upper surface of the wing segment caused by the buffet are displayed in Fig. 5 in terms of root mean square (rms) values over the chord position. The highest levels of fluctuation are found between x∕c = 0.35 and x∕c = 0.55 , which corresponds to the range of shock movement. A comparison with measured values [6], shown here as black triangles, suggests that both the maximum of the fluctuations and the most downstream position of the shock are matched quite well by the simulation, whereas the shock travels further upstream to x∕c = 0.25 in the experiment. The smaller range of shock movement in the simulation could explain the shorter buffet period or higher buffet frequency, respectively, noted above. Consistent with the mechanism proposed by Lee [5], a smaller distance between shock and trailing edge means a shorter time needed by the pressure disturbances generated at the shock to complete their propagation to the trailing edge and back to the shock, resulting in a shorter buffet period. The pressure fluctuations upstream of the shock are comparatively small. Downstream of the shock, however, elevated values are found that suggest strong pressure disturbances or turbulent fluctuations in the separated flow.
The time-averaged wake behind the OAT15A wing segment is shown in Fig. 8, which displays the mean axial Mach number in the flow field of a simulation without the rear wing segment. Here, the position of the latter for the corresponding tandem configuration is indicated, for reference, by a small star symbol. The largely separated wake is visible as an area of reduced flow velocity behind the wing segment. Notably, close to the trailing edge, the wake is orientated almost parallel to the x-axis or the chord of the wing segment, despite the incidence of 3.9 • of the free inflow, due to the downwash created by the wing segment. At distances greater than roughly one chord away from the trailing edge ( x∕c > 2 ), the trajectory of the wake is slightly curved upwards in z-direction as the downwash decreases and the wake becomes continuously more aligned with the inflow. In addition, an increase in the mean velocity in the wake with increasing distance from the trailing edge is visible, which indicates a progressive dissipation of the wake.
The development of the wake properties is displayed in more detail in Fig. 9, which shows vertical profiles of the  At the most upstream position, the wake exhibits a sharp velocity deficit of about 50 m/s centered on z∕c ≈ 0 . This velocity deficit decreases significantly to about 28 m/s at a position of x∕c = 3 , and the wake velocity profile widens significantly due to dissipation effects. For the most downstream position of x∕c = 4 , a further decrease in the velocity deficit is found; however, the reduction is considerably smaller than from x∕c = 2 to x∕c = 3 , indicating that the rate of dissipation is slowed down due to the decreasing velocity gradient in the mean wake profile. In addition, the vertical shift of the center of the wake discussed above is also evident. Furthermore, a high level of velocity fluctuations can be seen behind the trailing edge, with a corresponding rms value of about 40 m/s at x∕c = 2 . The velocity fluctuations in the wake generally decrease with increasing distance from the trailing edge as well. However, opposed to the overall trend, an increase in the fluctuations in axial direction is observed from x∕c = 2 to x∕c = 3 . This increase is followed by decreasing levels downstream, which obey the overall trend again. It should be noted that further upstream, the fluctuations also decrease as expected, as shown in Fig. 10. The velocity fluctuations in vertical direction, however, follow the general trend and do not exhibit a comparable increase. On the contrary, a significant reduction is found between x∕c = 2 and x∕c = 3 . At x∕c = 2 , the fluctuations in vertical direction are much higher than those in horizontal direction, but notably smaller at x∕c = 3 . This suggests that the initial strong anisotropy of the velocity fluctuations evident at x∕c = 2 is reduced, and the fluctuations are driven toward a more isotropic distribution. The strong fluctuations The development of the wake over the buffet cycle is shown in Fig. 11, which displays the instantaneous Mach number field around and behind the wing segment at the four moments in time introduced above. It can be seen that large turbulent vortices are generated in the separated flow behind the trailing edge, evolving from the shear layers at the boundaries of the wake. These vortices can be visualized more clearly applying the 2 -criterion as shown in Fig. 12, zooming into the part of the wake between x∕c = 2 and x∕c = 4 . It is also evident that the wake characteristics change significantly within the buffet cycle. When there is only a small amount of separation present on the wing segment or no separation at all, the wake appears thin, and thus, only small vortices can be generated, which is the case for the most downstream shock position (I) and during the downstream movement of the shock (IV). On the other hand, when the flow behind the shock is largely separated, large vortices are generated from the thick separated wake, which happens during the upstream movement of the shock (II) and for the most upstream position (III). Notably, the largest structures are present when the shock has reached its most upstream position (III), despite the fact that the largest amount of separation is found during its upstream movement (cf. Fig. 7). This is due to the time needed by the flow to travel downstream from the trailing edge to the location considered. Thus, the further downstream the location, the larger the time shift between the situation at the wing segment and the appearance of the corresponding vortices. In addition, it can be seen that the vortices form further upstream when a large separation is present on the wing segment and the wake is thick, e.g., at around x∕c = 1.2 (II), and further downstream when the amount of separation is small and the wake is thin, e.g., at around x∕c = 1.8 (I). Moreover, the vortex pattern changes within the buffet cycle. For both the phase of minimal separation (I) and the phase of maximal separation (III), pairs of alternating vortices are found in the wake. Yet, additional small vortices can be identified for the latter phase. However, during the transitional phases (II) and (IV) the pattern seems to be more irregular. The vortices begin to dissipate and the pattern becomes increasingly chaotic further downstream at around x∕c = 3 to x∕c = 3.5 , depending on the buffet phase, corresponding to a position of around 2 to 2.5 chord lengths behind the trailing edge.
The presence of large vortices in the wake suggests a high level of fluctuations, which is confirmed by Fig. 13 that shows the development of the horizontal and vertical velocity over time at a position of x∕c = 2.5 . Especially when looking at the vertical component, it becomes evident that periods with a comparatively low level of velocity fluctuations alternate with periods of high fluctuations in the order of 100 m/s during one buffet cycle. During periods with lower levels of fluctuation, small vortices are present as the amount of separation is small and the wake is thin, and during periods with strong fluctuations, large vortices are present as the amount of separation is large. This alternation of fluctuation levels can also be observed for the pressure fluctuations, as shown in Fig. 14 for two positions at x∕c = 2 and x∕c = 3 , respectively. Here, the change in amplitude is even greater than for the velocity. Comparing both positions, a decrease in the amplitude with increasing distance is found for the period of high fluctuation levels; yet, this is not the case for the period of smaller fluctuation levels, which shows a slight increase in the observed amplitudes. Additionally, the phase shift between the two positions noted above is also visible. The two corresponding pressure spectra are displayed in Fig. 15 in terms of the power spectral density of the pressure coefficient. These reveal that the pressure fluctuations during one buffet cycle are dominated mainly by the oscillation at the buffet frequency of 18.7 Hz itself and-to a lesser extent-by its first four to five higher harmonics, indicated by the peaks in the spectra in the low-frequency range. Furthermore, elevated values in the high-frequency range can be observed, mainly between 800 and 1000 Hz or Strouhal numbers of around 3-4, and between 200 and 400 Hz or Strouhal numbers of around 0.8-1.6, which correspond to the turbulent structures and vortices in the wake. The broadband characteristic in the spectrum mirrors the variation in the size and frequency of those structures. Comparing the spectra at the two positions, an overall decrease in the spectral density with increasing distance is found for almost the whole frequency range, in particular for the high-frequent turbulent oscillations around 800-1000 Hz, which can be attributed to dissipation effects. However, the fluctuations in the range of 200-400 Hz increase slightly from x∕c = 2 and x∕c = 3 . The reason behind this is not fully clear yet. The turbulent structures corresponding to this frequency range might be still developing further at these distances from the trailing edge, extracting energy from the mean shear flow. This is consistent with the sharp decrease of the velocity deficit between x∕c = 2 and x∕c = 3 and, accordingly, of the velocity gradient in the shear layer described above (cf. Fig. 9). Additionally, an increase of the axial component of the velocity fluctuation is also observed in this region. Furthermore, Figs. 11 and 12 indicate a growth especially for the smaller structures downstream of around x∕c = 2.5 . This dispersion effect could contribute to a reduction in the oscillation frequency and would also explain the significant reduction in the levels at the higher frequencies around 800-1000 Hz.

Wake interactions for the tandem configuration
The impingement of the unsteady turbulent wake of the front wing segment in buffet causes strong variations in the rear wing segment loading, as shown in Fig. 16, which displays the development of the lift coefficient of the rear wing segment over several buffet cycles. A low-frequency oscillation of the loading with a period equal to the buffet period of the front wing segment can be distinguished from superposed high-frequency oscillations of varying amplitude. The lowfrequency oscillation with a variation in the lift coefficient between c l = −0.5 and c l = −0.1 , corresponding to an amplitude of ĉ l = 0.2 , dominates the load fluctuation and is caused by the change of the intensity of the downwash behind the front wing segment within the buffet cycle. As the lift of the front wing segment oscillates, so does the vertical velocity induced by the corresponding circulation, which leads to a variation in the effective angle of attack for the rear wing segment. The high-frequency oscillations, on the other hand, are caused by the impingement of the turbulent structures or vortices in the wake identified above upon the rear wing segment. This is visualized in Fig. 17, which shows the instantaneous Mach number in the flow field around the tandem configuration. At the depicted moment in time, a turbulent vortex, indicated by an area of low Mach number, impinges on the leading edge of the rear wing segment. In addition, another turbulent structure is visible on the lower side, i.e., suction side, of the rear wing segment close to the trailing edge that appears stretched and distorted due to the interaction with the wing segment. The impingement of the turbulent vortices causes a variation in the effective angle of attack and the effective inflow velocity for the rear wing segment, resulting in a change in lift. Moreover, the smaller turbulent structures can distort the local pressure distribution on the wing surface as they pass the rear wing segment, contributing to the load fluctuation. Notably, the amplitudes of the vortex-induced oscillations appear to be comparatively high, as shown in Fig. 16, reaching up to ĉ l = 0.14 , which roughly corresponds to two thirds of the amplitude of the low-frequency oscillations. It is also evident that the frequencies and especially the amplitudes of the high-frequency oscillations vary significantly within the buffet cycle. Periods with low levels of fluctuation alternate with periods of high fluctuation levels, which is caused by the alternation of comparatively weak fluctuations of velocity and pressure in the wake, when only small vortices are present, and strong fluctuations, respectively, accompanied by the presence of large vortices, as discussed above.
The pressure fluctuations on the surface of the rear wing segment due to the interaction with the wake are displayed in Fig. 18, which shows the development of the pressure coefficient over several buffet periods for three locations on the suction side of the wing segment, x∕c = 0.05 , x∕c = 0.2 and x∕c = 0.6 . Both the low-frequency and the high-frequency components noted before are visible. In addition, it is evident that the intensity of the fluctuations is decreased at the most downstream position, as the turbulent structures are dampened and dissipated by the interaction with the rear wing segment. The spectral analysis of the pressure fluctuations at the three locations is shown in Fig. 19 in terms of the power spectral density (PSD) of the pressure coefficient. It can be seen that the fluctuations are dominated by oscillations at the frequency of the buffet on the front wing segment, which are related to the variation in the downwash described above, and at the first higher harmonics of the buffet frequency. Furthermore, the increased levels in the high-frequency range that extent over a large range of frequencies from 200 to 1200 Hz correspond to the impingement of the turbulent vortices of varying size and frequency. Here, two frequency ranges can be identified, one at around 200-400 Hz ( Sr = 0.8 … 1.6 with regard to the chord of the front wing) and the other at 800-1200 Hz ( Sr = 3 … 5 ), which are related to the interaction with the larger and, respectively, smaller sized structures in the wake. This is also supported by the comparison of the pressure spectrum on the surface with the corresponding spectrum in the wake (cf. Fig. 15). The spectra appear very similar and exhibit the same features, i.e., the peaks in the low-frequency range at the buffet frequency and its higher harmonics and the plateau or plateaus in the high-frequency range corresponding to the turbulent fluctuations. This clearly is to be expected as the turbulent structures themselves cause the pressure fluctuations on the rear wing segment in the first place; yet, it reveals and underlines the direct correlation between the wake fluctuations and the variation in the rear wing segment loading. Comparing the spectra at different chordwise locations, a decrease in the spectral density over the whole spectrum can be seen for increasing chord position, which indicates the dissipation of the turbulent structures due to the interaction with the rear wing segment. This effect also becomes apparent in the decreasing rms of the pressure fluctuations with increasing downstream position, as shown in Fig. 20. As the vortices impinge on the leading edge of the rear wing segment, the highest fluctuation levels are found there, continuously decreasing toward the trailing edge, in general. However, for the suction side, a peak of the pressure fluctuations, located around x∕c = 0.25 , is observed, which is explained later. Furthermore, the effect of the wake on the pressure side is a smaller than the impact on the suction side, as both the flow velocity and the rate of change of the pressure with angle of attack are higher on the suction side. In the following, the interaction with the wake is analyzed in more detail for two periods within the buffet cycle, denoted as intervals J and K in Fig. 16. In interval J, the lift coefficient of the rear wing segment reaches its minimum value of about c l = −0.5 , corresponding to the strongest (negative) loading, as seen in Fig. 21. At this point, a region of supersonic flow is present on the suction side of the rear wing segment terminated by a shock at around x∕c = 0.25 , as shown in Figs. 22 and 23. During this period, the lift oscillates with a moderate amplitude of about Δc l = 0.08 and a high frequency of up to 1200 Hz (Sr = 5). This frequency falls into the uppermost range of the significant fluctuations in the wake, which is linked to the smaller turbulent structures, as described above. Figure 22 displays the Mach number in the flow field around the rear wing segment for two successive moments of one oscillation, at the time of maximum (left) and minimum (right) lift, respectively, as indicated in Fig. 21. Here, the vortices responsible for the fluctuation can be seen impinging on the leading edge of the rear wing segment. The two corresponding instantaneous pressure distributions are shown in Fig. 23. The impact of the turbulent vortex causes a pressure drop and velocity increase in the supersonic region at the front of the wing segment, which increases lift. In addition, a distortion of the shock front further away from the surface is visible in the flow field. There is also a slight variation in the shock position during the interaction with the vortex, which contributes to the high surface pressure fluctuations at x∕c = 0.25 noted above. Despite the small range of movement, high levels of fluctuations are plausible, as an observer at a fixed location experiences a sudden, strong increase or drop of the pressure when the shock passes its location. Yet, the dominant part of the fluctuation is more likely generated by the formation and subsequent breakdown of the shock at this location caused by the alternation of high and low load conditions at the rear wing segment during the buffet cycle, as the shock moves forward and almost vanishes completely when the loading decreases (cf. Fig. 26). It should be noted that no indication of a self-sustaining shock movement like in the case of buffet has been found for the rear wing segment. However, it is imaginable that buffet might be triggered at an even higher loading, i.e., at a more negative lift coefficient.
In interval K, the mean lift coefficient is higher, or less negative, than in interval J, at around c l = −0.25 , corresponding to a lower loading, as shown in Fig. 24. At this point, the front wing segment generates less lift and thus produces a weaker downwash, resulting in a higher effective angle of attack for the rear wing segment. Consequently, the flow around the latter is largely subsonic without strong   Fig. 16 shocks, as illustrated in Figs. 25 and 26, in contrast to the transonic case in interval J. During this period, however, the lift oscillates with the highest amplitude found during the buffet cycle of about Δc l = 0.24 and with a frequency of around 800 Hz (Sr = 3). These oscillations are caused by the large vortices present in the wake at that time, which can be seen in 25 that displays the flow field, again for two successive moments of one oscillation, at the time of maximum (left) and minimum (right) lift, respectively. The corresponding instantaneous pressure distributions are shown in Fig. 26. The impingement of the vortex leads to the formation of a suction peak on the suction side of the rear wing segment and an accompanied small supersonic region, and its successive breakdown. Thus, high levels of surface pressure fluctuation are observed at the leading edge. Moreover, a modulation of the pressure in the rear section of the rear wing segment caused by the passing, weakening vortex can be seen. The strong fluctuations of the rear wing segment loading caused by the interaction with the wake can become relevant for the structural integrity. Despite the fact that these are smaller in amplitude than the static loading, the pressure oscillations might cause the structure to vibrate with a comparatively high frequency, potentially contributing to an accelerated material fatigue.

Conclusion
Hybrid RANS/LES simulations with the AZDES method have been performed for a tandem configuration in highspeed stall conditions. Under the investigated conditions, transonic buffet occurs on the front wing but not on the rear wing. The development of the turbulent wake behind the front wing segment in buffet and its impact on the rear wing segment have been investigated. The results reveal that there is a strong variation in the characteristics of the wake over the buffet cycle, caused by the varying amount of separation on the front wing segment. During the upstream movement of the shock, the flow behind it is largely separated, resulting in a thick wake with strong, high-frequent fluctuations of pressure and velocity that can be attributed to large turbulent vortices developing from the shear layers at the edges of the wake. On the contrary, when the shock travels downstream, there is only a small amount of separation present on the Fig. 22 Flow field around the rear wing segment in interval J, at the moment of minimum loading (left, marked red in Fig. 21), and at the moment of maximum loading (right, marked green in Fig. 21) wing segment as the point of separation is far downstream, resulting in a thin wake with comparatively low fluctuations that are caused by corresponding smaller turbulent vortices. This leads to an alternation between higher and lower fluctuation levels and between larger and smaller vortices in the wake during one buffet cycle. The significant turbulent fluctuations of varying vortex size and frequency can be identified in the pressure spectrum in the high-frequency range. Nevertheless, the pressure variation in total is dominated by the buffet frequency and its first higher harmonics. Due to dissipation effects, the fluctuation levels slowly decrease with increasing distance from the trailing edge of the wing segment. Notably, close to the wing segment, the velocity fluctuations are significantly anisotropic as they are stronger in the vertical direction, becoming more isotropic between around one to two chord lengths behind the trailing edge.
The impact of the wake of the front wing segment causes a strong variation in the rear wing segment loading. Here, an oscillation of the lift coefficient with a comparatively low frequency can be distinguished from high-frequent fluctuations. The low-frequent oscillation is caused by the variation in the downwash behind the front wing segment as its lift changes during the buffet cycle, resulting in a varying effective angle of attack of the rear wing segment. The high-frequent fluctuations, however, are due to the impingement of the turbulent structures or vortices onto the rear wing segment, which change the effective inflow velocity and angle of attack as well as distort the pressure distribution on the rear wing segment's surface. Because both size and frequency of those vortices vary significantly within the buffet cycle, dependent on the current phase of the wake, the amplitude and frequency of the lift and surface pressure fluctuations also change accordingly. The vortex-induced fluctuations of the surface pressure are visible in the corresponding pressure spectrum in the high-frequency range. Again, the highest level of fluctuation is found, however, for the buffet frequency and its first higher harmonics. The spectrum as a whole is very similar to the spectrum of the pressure fluctuations in the wake, which underlines the direct correlation between the wake fluctuations and the variation in the rear wing segment loading due to the turbulent vortices. Because the latter impinge onto the rear wing segment's leading edge, the strongest fluctuations are observed there, generally decreasing in intensity with increasing chord position, with the exception of a local maximum at x∕c = 0.25 on the suction side. This extremum is caused by the formation and subsequent breakdown of a shock as the loading increases and decreases again within one buffet cycle.
Analyzing the interaction with the wake more closely for the time interval of high loading, lift oscillations of moderate amplitude are observed at a high frequency. These are due to the impingement of smaller vortices that lead to a pressure drop and velocity increase in the supersonic region at the front of the wing segment and a slight variation in the shock position. During the interval of lower loading, which corresponds to the period of lower lift and larger separation on the front wing segment, larger and stronger vortices impinge on the rear wing segment, leading to a greater amplitude of the lift oscillations. Here, the vortices lead to the formation of a suction peak on the suction side with an accompanied Fig. 25 Flow field around the rear wing segment in interval K, at the moment of minimum loading (left, marked red in Fig. 24), and at the moment of maximum loading (right, marked green in Fig. 24)   Fig. 26 Pressure distributions of the rear wing segment in interval K, at the moment of minimum and maximum loading (marked red and green in Fig. 24) small supersonic region and its successive breakdown. The strong fluctuations of the rear wing segment loading caused by the interaction with the wake can become relevant for the structural integrity as the pressure oscillations might cause structural vibrations, potentially contributing to fatigue.