AIAA AVIATION Forum 10.2514/6.2018-3634 June 25-29, 2018, Atlanta, Georgia 2018 Applied Aerodynamics Conference

High fidelity gust simulations over a supercritical

B. Tartinville 1 NUMECA International S.A., 189 Chaussée de la Hulpe, 1170 Brussels, Belgium.

V. Barbieux 2 NUMFLO, 7 Boulevard Initialis, 7000 Mons, Belgium

and, L. Temmerman 3 NUMECA International S.A., 189 Chaussée de la Hulpe, 1170 Brussels, Belgium

In order to investigate the nonlinear flow behaviour around a airfoil in the presence of three different gust scenarios a series of long-term high fidelity Improved Delay Detached Eddy Simulations has been performed. The computed flow structures around the airfoil and the nonlinear effects are investigated by means of a detailed analysis of the unsteady flow pattern and a precise comparison between the results coming from the three gust scenarios. It appears that the detached flow caused by shock/boundary layer interactions is significantly impacted by the passage of the gust. Indeed, shock velocity is driven by the pressure fluctuations due to the gust passage. Thus, the maximum shock displacements on both sides of this supercritical airfoil are directly influenced by the gust length.

I. Nomenclature

As = gust amplitude c = airfoil chord length Fg = flight profile alleviation factor H = gust gradient distance p = static pressure pref = static pressure from the undisturbed simulation s = axial space coordinate Tc = gust characteristic time scale U = vertical gust velocity Ug = design gust velocity Uref = reference gust velocity U∞ = free stream velocity Vs = shock velocity Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 II. Introduction HE influence of gusts in design is important since it defines the maximum loads a given configuration has to T support. Applications in which gusts are being considered include airplanes, air vehicles, helicopters, wind turbines, environmental flows, and civil engineering. The impact of gusts in these applications is not negligible. Indeed, in the case of wind turbines, they will affect the power output. For buildings and bridges, they will impinge the comfort of the users, and potentially the stability of the structure and its integrity. For all flying objects, gusts will impact their stability and the comfort and security of the passengers. They may be the cause of buffeting as well as the source of noise. Furthermore, the influence of such atmospheric disturbances should increase in future designs since their prevalence and strength are likely to be greatly affected by the ongoing anthropogenic climate change [1, 2].

1 Head of FineTM/Turbo CFD group. 2 Head of Consulting group. 3 Expert Development Engineer.

Copyright © 2018 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

The effects of gusts in the design loop of aircraft have traditionally been considered using Reduced Order Methods (ROM) [3] and not full Computational Fluid Dynamics (CFD) models for a question of computing time and cost. However, more stringent regulations require better prediction and the interest in using full CFD models starts to be felt. A challenge faced in the process of simulating the gust-aerofoil interaction is the capability of the CFD solver to accurately propagate the gust from the computational domain boundary which is typically located from ten to hundred reference lengths away from the body. Indeed, the gust can be diffused by the mesh whose cells greatly vary in size as it is typically done in the simulations of external aerodynamics. For this reason, the number of investigations by means of full order model simulation of gust interaction with an airfoil or airframe remains very small with the exception of a few research groups. For instance, the University of Maryland has conducted unsteady Euler simulations using a mesh adaptation technique to convect towards the airfoil a perturbation in the free stream flow [4]. Furthermore, the German Aerospace Center (DLR) has used a method based on moving chimera meshes to propagate a vortex [5-7]. Finally, the Air Force Research Laboratory has published numerous studies considering the gust airfoil interaction using Large Eddy Simulations (LES) on fixed meshes with a high order scheme [8-12]. It is hereafter proposed to conduct high fidelity gust simulations over a supercritical airfoil. Since the nonlinear effects could be impacted by the gust length, three different gust scenarios coming from the Federal Aviation Regulations are considered. These configurations are first described. Afterwards, the selected numerical methodology is described. It is set-up in order to be sufficiently general and to be able to accurately transport the gusts from the far-field boundary to the profile. All the numerical results are described, analysed, and compared. Based on those results, a mechanism is proposed in order to explain the impact of the gust on the airfoil loads.

III. Description of the configuration The supercritical airfoil retained for the present study is the bi-dimensional crank section of the Future Fast Aeroelastic Simulation Technologies (FFAST) used in [3] and [13]. As in the former references, the pitching moment is defined about the structural axis – which is located at 31% of the chord. A positive moment corresponds to a force acting on the blade that should induce a clockwise rotation of the rigid airfoil, i.e. equivalent to a positive incidence. The chord length is set to c = 8 m. The flow condition considered hereafter corresponds to a flight altitude of 35,000 feet with a free stream equal to 0.86.

Federal Aviation Regulations mention that transport airplanes should be assumed to be subject to a 1-cosine shape vertical gust (FAR 25). One objective of these regulation rules is to determine the limit gust loads. The determination of these values must take into account unsteady aerodynamic characteristics. Furthermore, FAR 25 mentions that a sufficient number of gust gradient distances in the range 30 feet to 350 feet have to be investigated to find the critical response for each load quantity. According to these regulation rules, the vertical gust velocity U is defined as a function of the space coordinate s, following:

푈 휋푠 푈 = 퐴 퐹 푔 [1 − 푐표푠 ( )] for 0 ≤ s ≤ 2H (1a) 푆 푔 2 퐻 푈 = 0 for s < 0 or 2H < s (1b)

Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 H is the gust gradient distance, As is an amplitude scaling with altitude – 0.74662 at an altitude of 35,000 feet as in the present study – and Fg is the flight profile alleviation factor set to unity. The design gust velocity Ug depends on the reference gust velocity Uref:

1 퐻 ⁄6 푈 = 푈 ( ) (2) 푔 푟푒푓 350

The reference gust velocity is also specified by the regulation rules and is set to 56 feet/s (17.07 m/s). Furthermore, in Eq. (2) the gust gradient distance should be specified in feet. In order to cover the entire range of the regulations rules for gust specifications, three different gust scenarios are considered in the present study. The two extreme gust gradient distances and an intermediate value have been selected. Therefore, the sensitivity of the nonlinear unsteady characteristic of the supercritical airfoil to a gust is investigated for a short, an intermediate, and a long gust, H = 30, 150 and 350 feet (9.144 m, 45.72 m, and 106.68 m), respectively.

IV. Computational methodology The computational domain around the supercritical airfoil is a circular domain extending 20 chord lengths around the airfoil location. This means that the domain size is 320 m along both the axial and the vertical directions. This is of the same order of magnitude than the maximum gust length, i.e. about 200 m. Furthermore, as high fidelity methods are likely to produce tri-dimensional vortices, a tri-dimensional computational domain is defined. It has been set-up by extruding the bi-dimensional domain into the span-wise direction by half a chord, i.e., 4 m. This is beyond general practices for high fidelity simulations [14-16].

The flow solver that has been retained is the unstructured FINETM/Open solver from Numeca. It is a multi-block finite volume density based solver that can deal with structured and unstructured hexahedral grids with hanging nodes. A complete description of the solver can be found in [17]. The spatial discretization is performed using a second order central scheme either with scalar or matrix dissipation [18]. The second one, less dissipative, has been retained hereafter. It should be able to accurately transport the gust along the computational domain without requiring a much too fine mesh. The advantage of such a less dissipative scheme in reaching mesh convergence has been particularly highlighted by [19] and is an outcome of the participation of Numeca to the third Aerodynamic Prediction Challenge [20]. The temporal discretisation is based on a second order backward difference scheme. Each physical time step is converged using a dual-time stepping method together with a series of accelerating methods such as multigrid, local time-stepping, implicit residual smoothing, and CPUBooster. The physical time step is set to 2 10-4 s. The CFD solver includes a series of classical Reynolds Averaged Navier-Stokes (RANS) turbulence model together with more advanced Large Eddy Simulation or hybrid RANS/LES models in order to solve detached unsteady turbulent problems. The Improved Delay Detached Eddy Simulation (IDDES) hybrid model originally proposed by [21] has been retained in the present study. The improved accuracy of this approach for aerodynamic applications has been point-out in a series of research projects – see for instance [22]. A validation of hybrid RANS/LES turbulence models developed in the FINETM/Open solver can be found in the literature [23-25].

Shortest gust length

Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 Fig. 1. Superimposition of shortest gust length and upstream part of the tri-dimensional mesh. The computational mesh is generated using IGG™ software. In order to target a y+ of the order of unity, the first cell size along the airfoil is set to 8 µm thick. A O-topology is used for the mesh around the airfoil profile. One important constrain is that the cell size inside the computational domain needs to be small enough in order to capture the gusts wavelengths with accuracy. In particular, the far field cell sizes need to be small enough to allow a proper convection of the gust from the upstream to the downstream regions of the computational domain with the selected matrix dissipation scheme. Thus, at least 20 points are used in order to represent the shortest gust length as shown on figure 1. The mesh contains a total of 385 points along the blade and 321 points in the radial direction. The tri- dimensional mesh is generated by a simple extrusion of the bi-dimensional mesh by half a chord length. 65 mesh points are uniformly distributed along the span-wise direction. The total number of mesh points in the final mesh is therefore 16 million. So as to investigate the mesh convergence a first series of bi-dimensional RANS calculations has been performed using Spalart-Allmaras turbulence model [26]. The difference in computed lift and coefficients between the results from the 193x161 mesh and the one obtained on the finest 385x321 mesh is smaller than one

count. Therefore, and in accordance with aeronautic standards, the error is rather small and the results could be said as mesh-independent. The far field boundary conditions are specified along the outer boundaries of the cylindrical domain. The undisturbed free stream static pressure, temperature, and horizontal velocity are set to 23,930 Pa, 219 K, and 255 m/s, respectively. It corresponds to a flight altitude of 35,000 feet and a free stream Mach number of 0.86 without incidence. The chord based Reynolds number is 5 107. For the three simulations, the gusts are imposed along the outer boundaries and fully solved within the computational domain. The time/space evolution of vertical component of the velocity is imposed via the user-defined function implemented through the dedicated OpenLabs™ module from the FINE™/Open software. The airfoil solid boundary is assumed smooth and adiabatic. Translation periodicity is imposed along the lateral domain boundaries.

In order to be able to simulate the flow behavior around the supercritical airfoil subject to gust forcing, an undisturbed precursor simulation is first performed with a physical time step of 5 10-4 s. A total integration time of about 100 convective time-steps has been computed in order to reach a statistically representative pseudo-steady behavior. For this IDDES calculation, after a transient phase of about 30 convective time steps, a pseudo-periodic unsteady pattern takes place. The time-average values of the lift and momentum coefficients over the last two seconds of the simulation are 0.1654 and -0.0021, respectively. The unsteady fluctuations in lift coefficient are significant and correspond to ±0.034, i.e. ±20% of the time-mean value as shown on figure 2.

Fig. 2. Time evolution of lift (left panel) and momentum (right panel) coefficients for the precursor undisturbed simulation.

Mach Number Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634

Fig. 3. Instantaneous (t = 2.5 s) Mach number distribution at

mid-span and iso-surface of 2 = -5000 for the precursor undisturbed simulation.

The instantaneous Mach number distribution at mid-span is displayed on figure 3. Two shocks corresponding to a peak Mach number of 1.42 and 1.47 could be identified on the lower and the upper sides of the airfoil, respectively. The shock on the pressure side is located slightly upstream of the one along the suction side. At such a Mach number ahead of a normal shock, an incipient shock-induced boundary layer separation should occur [27]. This limit of flow separation is about 1.25 for a turbulent boundary layer along an airfoil [28]. Since the shape of the supercritical airfoil behaves as a diffuser in its rear part, the detached zones appear to be significant. These separations produce two-dimensional vortices that can be identified along both sides of the airfoil. They progressively become tri-dimensional reaching the trailing edge and merging together.

V. Sensitivity to gust length Short gust In order to control the proper convection of the gust inside the computational domain, the perturbation in vertical velocity has been checked on a series of control points ahead of the airfoil. Since this first simulation corresponds to the shortest gust, it is numerically the most difficult one to convect, and therefore, the most challenging case. Indeed, in the far field, only 20 grid cells are available per wave-length for this configuration. Four control points, labelled P1 to P4, are selected. They are located 16, 12, 8, and 4 chord lengths ahead of the airfoil leading edge. The vertical velocity time evolution is compared to a purely convected gust on figure 4. Even if the computed maximum gust velocity (8.46 m/s following Eq.(1)) is slightly damped by the convective scheme, the decrease between the computational domain inlet and the airfoil is only of 5%. Furthermore, the shape of the 1-cosine gust profile is almost preserved. However, due to the numerical scheme retained in the present study, a slight undershoot in vertical velocity is observed just downstream of the gust passage. As already noticed, because of its small wavelength, this gust scenario corresponds to the most critical gust to be convected through the domain. It is foreseen that the convection of the two other gusts will present even less discrepancy in comparison with the theoretical gust profiles.

Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634

Fig. 4. Time evolution of vertical velocity component at four control points ahead of the airfoil for short gust simulation. Dashed lines correspond to a purely convected gust.

The passage of the gust has an impact onto the airfoil loads as displayed on figure 5. The short gust is convected along the profile in about 0.1 s. The lift coefficient is first affected when the gust front reaches the shock location along the suction side, i.e. when the velocity perturbation interacts with this shock. Afterwards, the lift significantly increases until the gust centre reaches the airfoil trailing edge. Later, the lift decreases down to its minimum. This instant corresponds to an upstream displacement of the suction side shock and to a maximum computed momentum coefficient. Afterwards, both lift and pitching moment come back to their undisturbed values. Those time evolutions of lift and momentum coefficients for the short gust scenario are consistent with other results obtained for similar gust profile inputs such as in [29] using the Field Velocity Method.

The impact of the gust passage onto the flow structure around the airfoil can be summarised on figure 6a. On this figure the Mach number distribution is displayed at mid-span for three different time instants that are set as a function of a gust characteristic time scale defined as:

2퐻+푐 푇푐 = (3) 푈∞

Fig. 5: Time evolution of lift (left panel) and momentum (right panel) coefficients for the gust simulations (flat dashed line for time-mean undisturbed simulation). Gusts impact the airfoil leading edge at t = 0 s.

t/Tc=0 t/Tc=0.5 t/Tc=1 a

Time

b Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634

c

Fig. 6: Mach number distribution at mid-span at t/Tc = 0 (left panels), t/Tc = 0.5(central panels), and t/Tc = 1 (right panels) for the three different gust scenarios (upper panels for short gust (a), middle panels for intermediate gust (b), and lower panels for long gust (c)).

where U∞ is the free stream velocity set to 255 m/s. This means that at t = 0 s, the gust front is located at the airfoil leading edge. At a time instant equal to the gust characteristic time scale, the rear of the gust has reached the airfoil trailing edge, while at t/Tc = 0.5 the gust centre is located in the middle of the blade. For the short gust, this characteristic time is equal to 103 ms. Even if tri-dimensional flow structures can be identified close to the trailing edge, the shocks displacements are almost bi-dimensional and a view at mid span is representative of the major unsteady flow features. It appears that, for the short gust configuration, the locations of the shocks on both sides of the profile remain almost stable. Indeed, the shock along the suction side slowly moves upstream and its maximum displacement represents only 10% of the chord length. The pressure side shock moves downstream by about 4% of the chord length. Therefore, both shocks are getting closer and the minimal distance is reached after the gust leaves the profile. Afterwards, both shocks rapidly retrieve their original undisturbed location.

Intermediate gust For the second simulation, the impact of the gust passage onto the airfoil loads is more pronounced that for the shortest gust configuration. Indeed, momentum drastically increase and lift exhibits a first increase and a sudden decrease during the first 300 ms. At that time instant – which corresponds to one convective time step – lift and momentum have reached their minimum and maximum, respectively. Afterwards, loads are coming back to their original value in an oscillating way, with a time period similar to the convective time. From figure 6b, with the gust characteristic time scale equal to 390 ms, it appears that both shocks exhibit significant displacements. For the last two instants the shock along the pressure side is located downstream of the one located along the suction side. The maximum displacement corresponds to time instant at which lift and momentum have reached their extremum. At this time, both pressure side and suction side shocks have moved by about 20% of the chord length, downstream and upstream respectively. Such a shock movement induces a modification of the boundary layer thickness downstream of the shocks and, therefore, of the position and intensity of the wake.

Long gust Finally, the simulation for the long gust configuration also exhibits significant changes in the loads caused by the passage of the gust along the profile. The behaviour is similar to the one described for the intermediate gust except that extremum are reached about 600 ms after the gust front has reached the airfoil leading edge. The maximum perturbation in momentum is also increased by 24% with respect to the results from the intermediate gust configuration. From figure 6c, one can clearly identify the impact of the shock movement on the downstream boundary layer and on the further downstream wake. Indeed, at t/Tc = 0.5 (with Tc = 868 ms), both shocks have reached their maximum displacements. This displacement corresponds to about 25% of the chord length – either upstream or downstream. Along the pressure side, the shock is located almost at the trailing edge, and only a weak detached flow can be identified. Along the other side of the profile, the shock is located at about mid-chord and a strong detached flow due to shock boundary layer interaction can be identified. Such a detached region reaches the trailing edge. This leads to an upward deflection of the downstream wake.

VI. Discussion The compared time evolutions of lift and momentum coefficients computed for all the simulations have already Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 been provided on figure 5. The time-mean lift and momentum computed from the undisturbed simulation have also been added. From this figure, it is obvious that the shortest gust has a relatively small impact on lift and momentum coefficients. Except from the first 0.1 s (when the gust impacts the profile), the time variations of lift and momentum are of the same order of magnitude than the undisturbed results as displayed on figure 2. On the contrary, the two largest gusts have a significant impact onto the airfoil loads. Indeed, for both simulations the lift becomes negative and the pitching moment coefficient becomes higher than 0.1. It is also important to notice that all the simulations produce high frequency oscillations of the lift and momentum that are similar to the ones observed for the undisturbed reference simulation. In order to further investigate the spectral content of the time evolution in lift coefficient from all simulations, a Power Spectral Density (PSD) of the lift coefficient is displayed on figure 7. For all the simulations, this plot shows significant peaks at around 35, 70, and 90 Hz. Strouhal numbers – based on the free stream velocity and on the airfoil maximum thickness – are 0.14, 0.28, and 0.36, respectively. Such high frequency oscillations are directly caused by the shock boundary layer interaction that induces an important detached flow along the latest part of the profile. Pressure within the detached region oscillates at the same frequencies. Indeed, computing the PSD of the

pressure signal at control points located along the airfoil surface downstream of both shocks reveals that this variable also exhibits significant peaks at the same frequencies. Performing the same analysis for control points upstream of the shocks does not produce any peak at a frequency higher than 10 Hz. Therefore, high frequency oscillations of the lift are directly caused by the pressure fluctuations with the detached region. For the two highest frequencies, it is interesting to note that the Strouhal number is in line with the ones associated to von Kármán streets at very high Reynolds number as described in [30]. Furthermore, such a high frequency signal is independent of the gust scenario. Therefore, one can conclude that high frequency pressure oscillations within the detached zone are not drastically affected by the passage of the gust.

Fig. 7. Power Spectral Density of lift coefficient. Since the passage of the gust has a direct impact on the location of the shocks, their time evolution at mid-span is displayed on figure 8. Shock position is computed based on the location of the maximum axial pressure gradient along the airfoil surface. As previously, time instant t=0 s corresponds to the front of the gust reaching the profile leading edge. As the tri-dimensional animations reveal that the shock displacement is almost two-dimensional, a comparison at mid-span is relevant. From this figure, it is obvious that nonlinear effects are rather important for the two largest gust scenarios. Indeed, the amplitude of the shock displacement is rather large (several tenth of percent of the airfoil chord length) and this has a direct impact on the downstream detached flow structure. It appears that the initial (first 0.1 s) shock displacements along the suction and the pressure sides are rather similar for the three gust scenarios. They are driven by the changes in incidence at the leading edge from zero to their maximum values (1.9°, 2.5° and 2.9° for the three gusts, respectively) and by the progressive passage of the gust along the profile. At the end of this time period, the shortest gust has left the airfoil. However, for this configuration, both shocks continue to move close to each other during 50 ms before coming back to their original location. For the two other gust configurations, shock displacements are much larger. Indeed, the amplitude of the shock displacement is of the order of a few tens of percent of the axial chord length and the shock along the pressure side is not systematically Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 located upstream of the shock along the suction side.

Fig. 8. Time evolution of shock location at mid-span along the pressure (dashed lines) and suction sides (continuous lines) for the three gust scenarios: blue lines for short gust, red lines for intermediate gust, and green lines for long gust.

One important finding coming out of figure 8 is that the shock along the suction side goes back to its original location in a similar way for all the simulations. Indeed, several segments of these time evolutions have an almost identical slope corresponding to a shock velocity of the order of 10 m/s. For instance, such a slope - with a positive or a negative sign - can be identified for the curve representing the shock position along the pressure side for the intermediate gust (red dashed line in figure 8) after t = 0.15 s.

As described above, a shock velocity of the order of 10 m/s can be identified either during the passage of the gust along the profile or after the passage of the gust, along both sides of the airfoil, and for the three gust scenarios. Therefore, independently of the gust length, a common mechanism should be able to explain such shock displacements. Various processes have been proposed in the literature in order to explain shock oscillations along at transonic speeds [31]. Most of them are linked to self-sustained shock oscillations. Among them, two potential mechanisms for shock oscillations are caused by upstream-moving waves coming from the region just downstream of the airfoil trailing edge - also known as Kutta waves [28, 32, 33]. Those waves can be observed in the results of all the simulations of the present study, even without any gust as shown on figure 9. Kutta waves are produced in the wake region near the trailing edge. Indeed, because of shock wave boundary-layer interactions, two- dimensional vortex shedding could be observed within the detached boundary layer along the two sides of the profile. Vortices are merging together at the trailing edge into three-dimensional structures. As experimentally observed in [33, 34], vortex formation at the trailing edge is responsible for the generation of pressure waves. Those Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 waves are emitted at the same frequency than the vortex shedding and are running upstream before reaching the shock. However, in the present study, no shock displacement is observed at the frequency of the Kutta waves. This is consistent with some observations reported in [33]. Furthermore, the characteristic time-scale associated to the shock displacement observed during the gust passage is several orders of magnitude higher than the characteristic time of vortex shedding. Therefore, the upstream running Kutta waves cannot be the cause of the observed shock motion at a velocity of the order of 10 m/s.

Fig. 9. Instantaneous (t = 2.5 s) numerical Schlieren at mid-span for the precursor undisturbed simulation.

Shock position can also be affected by the deflection of the wake as suggested in [35]. However, this mechanism responsible for self-sustain shock oscillation requires that the boundary layer becomes fully attached when the shock moves forward. Such a reattachment of the boundary layer for an upstream running shock is not observed for any of the gust scenario considered here. It is proposed in [36] that the trailing edge pressure – directly affected by the boundary layer thickness – drives the shock position and therefore its movement. Indeed, starting from the Rankine-Hugoniot equation for a normal shock, one can compute the shock velocity from downstream pressure fluctuations. This has been applied either to a transonic duct [27] or to external aerodynamics [28]. It is of particular interest to note that the maximum shock velocity measured and computed for internal flow in [27, 37] at Mach numbers 1.4 and 1.5 is about 10 m/s. This study also reveals that the shock velocity is independent of the perturbation frequency. This is consistent with the observation that the computed shock velocity is independent of the gust scenario. Furthermore, observed shock velocity derived from the results obtained for a supercritical arifoil at maximum Mach number of about 1.35 is in the range 3-5 m/s [28], i.e., slightly below the present observations. This velocity only depends on the time evolution of the pressure ratio. In order to verify that fluctuations in pressure ratio are responsible for shock displacements, the flow solution at mid-span for the intermediate gust simulation is analysed at two specific times and compared to the results of the undisturbed simulation. At t/Tc = 0.5 (t ~ 200 ms) the shock along the pressure and suction sides are moving downstream and upstream, respectively. For the same simulation, the shock goes is in the opposite direction at t/Tc = 1 (t ~ 400 ms). The fluctuating static pressure ratio at those two instants is displayed on figure 10. Fluctuating pressure ratio is defined as:

Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 푝(푡)−푝푟푒푓 푝푟푎푡 = (4) 푝푟푒푓

where p(t) is the instantaneous static pressure from the simulation with the gust and pref is the local static pressure computed for the undisturbed simulation (at t = 2.5 s). From this figure it appears that, outside of the shock region, pressure fluctuation can represent up to 10% of the undisturbed pressure. Furthermore, in the first part of the simulation, higher pressure differences are located upstream of the shocks, whereas they are on the rear part of the shock at t/Tc = 1. Using the Rankine-Hugoniot equation for an upstream perturbation of 7% in static pressure and at a shock Mach number of 1.45 (shock strength equal to 1.286), one can compute a shock velocity of 12 m/s – using a sound speed velocity of 270 m/s. This result is in accordance with our computational results and the same conclusion can be drawn for a downstream perturbation of the same intensity. Thus, the computed pressure fluctuations are in accordance with the observed shock velocities.

Fig. 10: Mid-span distribution of fluctuating static pressure ratio for the intermediate gust scenario at t/Tc = 0.5 for left panel, and t/Tc = 1 for right panel.

A mechanism for shock displacement associated to the passage of the gust could therefore be proposed. Indeed, a gust reaching the leading edge of the airfoil induces an increase in the incidence and therefore a modification of the pressure distribution on both sides of the airfoil. This progressive modification of the pressure ratio leads to a downstream shock movement along the pressure side and a reverse movement along the suction side. As shown on figure 8, both shock velocities progressively increase as the perturbation in pressure increase. The shock velocity is directly linked to the pressure ratio. When the maximum velocity deflection has passed the shock locations, pressure progressively retrieves its undisturbed value upstream of the shock and evolves downstream of the shock. This conducts to the reverse motion of both shocks that slowly come back to their initial location.

Such a proposed mechanism is of importance in order to better understand the impact of nonlinear effects on the response of an aircraft to gusts. As shown in [3], in transonic regime, nonlinearities due to shock motion are more pronounced for gusts of large amplitude. As far as the shock velocity is almost independent of the gust length, it is rather easy to anticipate that above a certain limit the shock displacement will be large enough to promote nonlinear effects. Indeed, one can compare the characteristic length associate to the shock displacement during the gust passage to the chord length. This is done by computing the factor

푉 2퐻 휑 = 푠 (5) 푈∞푐

where Vs si the shock velocity. Nonlinear effects associated to shock displacement are of importance if this factor is of the same order or greater than unity. For the three gust scenarios investigated in the present study, going for the shortest to the longest one, the factor  is 0.09, 0.45, and 1.04. This confirms that only the results of the first simulations are not affected by strong shock movement. This observation is in line with results obtained by other participants of the AEROGUST4 project by comparing computed time evolution of lift and pitching moment to results obtained by a linear ROM. Furthermore, gust can behave as a source for undesirable self-sustained oscillatory motion such as transonic buffeting [38]. Indeed, a series of phenomena that have been described above are also observed during buffeting. In Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 both cases shocks are located on the rear part of the profile, while an out-of-phase shock displacement along both sides of the airfoil can be observed. Furthermore, such a shock movement leads to a strong modification of the structure of the boundary layer behind the shock (from fully attached to detached) and of the downstream wake. In both cases Kutta waves traveling from the trailing edge towards the shocks are observed. It is likely that at a free stream Mach number of 0.86, the supercritical crank airfoil retained for the present study is at the limit of buffet onset. Several partners of the AEROGUST project have also reported such a potential buffeting. Indeed, the numerical prediction of buffet onset is rather difficult to simulate because it is highly sensitive to modelling and numerical parameters [39-41]. It is believed that only high fidelity simulations being able to capture the above mentioned phenomena can accurately capture buffet onset.

4 www.aerogust.eu

VII. Conclusions A series of high fidelity IDDES gust simulations have been conducted over a supercritical airfoil for three different gust scenarios. The analysis of the results of those simulations reveals that the loads are affected by a series of flow phenomena. First, independently of the gust scenario, high frequency oscillations of the lift at 35, 70, and 90 Hz can be observed. They are due to pressure fluctuations within the detached region behind shocks along both pressure and suction sides. Those pressure fluctuations reaching the airfoil trailing edge are responsible for upstream running Kutta waves. However, no shock displacement associated to those pressure waves has been identified. Secondly, the gust passage induces a shock movement with a velocity that is almost the same for the three gust scenarios, of the order of 10 m/s. It appears that such a displacement is directly driven by the Rankine-Hugoniot equation and only depends on the pressure ratio between both sides of the shock. The passage of the gust can explain the observed deficit in pressure along the suction side and its excess along the pressure side. Because of their impact on the size of the detached regions behind shocks, nonlinear effects are mainly driven by the location of the shocks during their displacement. Therefore, the loads along the supercritical airfoil are more impacted by nonlinear effect for long gusts than for short ones. From our three high fidelity gust simulations, it appears that only the shortest one does not exhibit any significant downstream shock displacement along the pressure side that leads to an almost attached boundary layer.

Acknowledgments The research leading to this work was supported by the AEROGUST project which has been funded from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 636053. The partners in AEROGUST are: University of Bristol, INRIA, NLR, DLR, University of Cape Town, NUMECA, Optimad Engineering S.r.l., University of Liverpool, Airbus Defence and Space, Dassault Aviation, Piaggio Aerospace and Valeol.

References [1] Cheng, C.S., Lopes, E., Fu, C., and Huang, Z., “Possible impact of climate change on wind gusts under downscaled future climate conditions: updated for Canada,” J. Climate, Vol. 27, February 2014, pp. 1255-1270. doi: 10.1175/JCLI-D-13-00020.1 [2] Williams, P.D., “Increased light, moderate, and severe clear-air turbulence in response to climate change,” Advance in Atmospheric Sciences, Vol. 34, No. 5, May 2017, pp. 576-586. doi: 10.1007/s0037 [3] Wales, C., Gaitonde, A., and Jones, D., “Reduced-order modeling of gust responses”. Journal of Aircraft, Vol. 54, No. 4, 2017, pp. 1350-1363. doi: 10.2514/1.C033765 [4] Tang, L., Baeder, J. D., “Adaptive Euler simulations of airfoil-vortex interaction,” Int. J. Numer. Meth. Fluids, Vol. 53, 2007, pp. 777-792. doi: 10.1002/fld.1306 [5] Wolf, C., “A Chimera simulation method and detached eddy simulation for vortex-airfoil interactions,”. PhD Thesis, Göttingen University. Göttingen, Germany, 2010. [6] Radespiel, R., François, D.G., Hoppmann, D., Klein, S., Scholz, P., Wawrzinek, K., Lutz, T., Auerswald, T., Bange, J., Knigge, C., Raasch, S., Kelleners, P., Heinrich, R., Reuss, S., Probst, A., Knopp, T, ‘Simulation of wing ,” 43rd AIAA

Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 Fluid Dynamics Conference. San Diego. AIAA 2013-3175, 2013. [7] Kelleners, P., and Heinrich, R., “Simulation of interaction of aircraft with gust and resolved LES-simulated atmospheric turbulence,” Advances in simulation of wing and nacelle stall, edited by R. Radespiel et al., Springer 2016, pp. 203-221. [8] Golubev, V.V., Dreyer, B.D., Golubev, N.V., and Visbal, M.R., “High-accuracy viscous simulations of gust interaction with stationary and picthing wing sections,” 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition. Orlando, Florida. AIAA 2009-11, 2009. [9] Golubev, V.V., Hollenshade, T.M., Nguyen, L., and Visbal, M.R., “Parametric viscous analysis of gust interaction with SD7003 airfoil,” 48th AIAA Aerospace Sicences Meeting Including the New Horizon Forum and Aerospace Exposition. Orlando, Florida. AIAA 2010-928, 2010. [10] Golubev, V.V., Nguyen, L., and Visbal, M.R., “High-accuracy low-Re simulations of airfoil-gust and airfoil-vortex interactions,” 40th Fluid Dynamics Conference and Exhibit. Chicago, Illinois. AIAA 2010-4868, 2010. [11] Barnes, C.J., Visbal, M.R., and Gordnier, R.E., “Analysis of streamwise-oriented vortex interactions for two in close proximity,” Physics of Fluids, Vol. 27, No. 1, 2015. doi: 10.1063/1.4905479 [12] Garmann, D.J., Visbal, M.R., “Streamwise-oriented vortex interactions with a NACA 0012 wing,” 53rd Aerospace Sciences Meeting . Kissimmee, Florida. AIAA 2015-1066, 2015.

[13] Mowat, A.G.B., Malan, A.G., van Zyl, L.H., and Meyer, J.P., “Hybrid finite-volume reduced-order model method for nonlinear aeroelastic modelling,” Journal of Aircraft, Vol. 51, No. 6, 2014, pp. 1805-1812. doi: 10.2514/1.C032524 [14] Mellen, C.P., Fröhlich, J., and Rodi, W., “Lessons from the European LESFOIL project on LES of flow around an airfoil,” AIAA Paper 2002-0111, 2002. [15] Deck, S., “Numerical simulation of transonic buffet over a supercritical airfoil,” AIAA Journal, Vol. 43, No. 7, July 2005, pp. 1556-1566. doi: 10.2514/1.9885 [16] Grossi, F., Braza, M., and Hoarau, Y., “Delayed detached-eddy simulation of the transonic flow regime around a supercritical airfoil in the buffet regime,” Progress in hybrid RANS-LES modelling, edited by S. Fu, S.-H. Peng and D. Schwamborn, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 117, Springer-Verlag, Berlin Heidelberg, 2012, pp. 369-378. [17] Patel, A., “Development of an adaptive RANS solver for unstructured hexahedral meshes,” PhD Thesis, Université Libre de Bruxelles, Belgium, 2003. [18] Jameson, A., Schmit, W., and Turkel, E., “Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes,” AIAA Paper 81-1259, 1981. [19] Brown, D.A. and Zingg, D.W., “Performance of a Newton-Krylov-Schur algorithm for solving steady turbulent flows,” AIAA Journal, Vol. 54, No. 9, 2016, pp. 2645-2658. doi: 10.2514/1.J054513 [20] Botella Calatayud, J., Léonard, B., Temmerman, L., and Hirsch C., “Sensitivity to turbulence models and numerical dissipation of the CRM Drag prediction,” AIAA Aviation conference, Atlanta, Georgia. 2018. [21] Shur, M.L., Spalart, P.R., Strelets, M., and Travin, A., “A hybrid RANS-LES approach with delayed-DES and wall- modelled LES capabilities,” Int. Journal of Heat and Fluid Flow, Vol. 29, No. 6, December 2008, pp. 1638-1649. doi: 10.1016/j.ijheatfluidflow.2008.07.001 [22] Reuß, S., Knopp, T., and Schwamborn, D., “Hybrid RANS/LES simulations of a three-element airfoil,” Progress in hybrid RANS-LES modelling, edited by S. Fu, S.-H. Peng and D. Schwamborn, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 117, Springer-Verlag, Berlin Heidelberg, 2012, pp. 357-367. [23] Temmerman, L. and Hirsch, C., “Towards a successful implementation of DES strategies in an industrial RANS solvers,” Advance in hybrid RANS-LES modelling, edited by S.-H. Peng and W. Haase, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 97, Springer-Verlag, Berlin Heidelberg, 2008, pp. 232-241. [24] Temmerman, L., Tartinville, B., and Hirsch, C., “URANS investigation of the transonic M219 cavity,” Progress in hybrid RANS-LES modelling, edited by S. Fu, S.-H. Peng and D. Schwamborn, Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 117, Springer-Verlag, Berlin Heidelberg, 2012, pp. 471-481. [25] Temmerman, L., Lestriez, R., Mehdizadeh, O., Tartinville, B., Léonard, B., and Hirsch, C., “Turbulence modelling in industrial applications and how it matters,” Proceedings of the 6th European Congress on Computational Methods in Applied Sciences and Engineering. ECCOMAS, 2012b. [26] Spalart, P.R. and Allmaras, S.R., “A one-equation turbulence model for aerodynamic flows,” AIAA Paper 92-0439, 1992. [27] Bruce, P.J.K. and Babinski, H., “Unsteady shock wave dynamics”. J. Fluid Mech., vol. 603, May 2008, pp. 463-473. doi: 10.1017/S0022112008001195 [28] Tijdeman, H., “Investigation of the transonic flow around oscillating airfoils,” NLR TR 77090 U. National Aerospace Laboratory, The Netherlands, 1977. [29] Dong, G., Min, X., and Shilu, C., “Nonlinear gust response analysis of free flexible aircraft,” I. J. Intelligent Systems and Applications, January 2013, 02, pp. 1-15. doi: 10.5815/ijisa.2013.02.01 [30] Lienhard, J.H., “Synopsis of lift, drag, and vortex frequency data for rigid circular cylinders,” Technical report, Washington State University, Pullman, WA, 1966. Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634 [31] Lee, B.H.K., “Self-sustained shock oscillations on airfoils at transonic speeds,” Progress in Aerospace Sciences, Vol. 37, No. 2, February 2001, pp. 147-196. doi: 10.1016/S0376-0421(01)00003-3 [32] Lee, B.H.K, “Oscillatory shock motion caused by transonic shock boundary-layer interaction,” AIAA Journal, Vol. 28, No. 5, 1990, pp. 942-944. doi: 10.2514/3.25144 [33] Alshabu, A. and Oliver, H., “Unsteady wave phenomena on a supercritical airfoil,” AIAA Journal, Vol. 46, No. 8, 2008, pp. 2066-2073. doi: 10.2514/1.35516 [34] Seiler, F. and Srulijes, J., “Vortices and pressure waves at trailing edges,” Proceedings of the 16th International congress on high speed photography and photonics, Strasbourg, France, 1984, pp. 794-799. [35] Gibb, J., “The cause and cure of periodic flows at transonic speeds,” Proceedings of the 16th Congress of the International Council of the Aeronautical Sciences, Jerusalem, Israel, 1988, pp. 1522-1530. [36] Stanewsky, E. and Basler, D., “Experimental investigation of buffet onset and penetration on a supercritical airfoil at transonic speeds,” AGARD CP-483, 1990, pp. 4.1-4.11.

[37] Bruce, P.J.K., Babinski, H., Tartinville, B., and Hirsch, C., “Experimental and numerical study of oscillating transonic shock waves in ducts”. AIAA Journal, Vol. 49, No. 8, August 2011, pp. 1710-1720. doi: 10.2514/1.J050944 [38] Giannelis, N.F., Geoghegan, J.A., and Vio, G.A., “Gust response of supercritical aerofoil in the vicinity of transonic shock buffet,” Proceedings of the 20th Australian fluid mechanics conference, Perth, Australia, 2016. [39] Deck S., “Numerical simulation of transonic buffet over a supercritical airfoil,” AIAA Journal, Vol. 43, No. 7, July 2005, pp. 1556-1566. doi: 10.2514/1.9885 [40] Barakos, G, and Drikakis D., “Numerical simulation of transonic buffet flows using various turbulence closures,” Int J. Heat Fluid Flow 21, 2000, pp. 620-626. doi: 10.1016/S0142-727X(00)00053-9 [41] Thiery, M., and Coustols, E., “Numerical prediction of shock induced oscillations over a 2D airfoil: influence of turbulence modelling and test section walls,” Int J. Heat Fluid Flow 27, 2006, pp. 661-670. doi: 10.1016/j.ijheatfluidflow.2006.02.013

Downloaded by BRISTOL UNIVERSITY on July 12, 2018 | http://arc.aiaa.org DOI: 10.2514/6.2018-3634