Controlling upwards and downwards gust loads on aerofoils by pitching

The fluid dynamics of vehicle-gust interactions are extremely complex. To study the fundamental effects governing this interaction, we simplify the geometries and consider only two-dimensional flow, as outlined in Fig. 2a. The wing of an aerial vehicle is represented with a thin rectangular plate that travels at constant speed U towards a high GR transverse gust. The flat plate is a common representation for wings in unsteady low Re aerodynamics, where the flow is typically separated (Eldredge and Jones 2019). The case of a wing with separated flow is frequently harder to treat in low-order models, and thus, in the present study, the separated case is useful to study the limitations of the gust mitigation approach. Other aerofoils shapes at low angles of attack are considered a simpler test scenario. Given the nature of this problem and the theory used, we consider that this work can be extrapolated to those simpler cases while its limitations emerge from the most separated flows, as discussed at the end of this paper. Another advantage of studies on flat plates is that the flow is relatively insensitive to Re variations in the \(10^5\) regime.

The position of the wing relative to the gust is given by the non-dimensional quantity \(s = Ut/c\), where \(s = 0\) corresponds to the wing leading edge entering the gust, t is time, and c is the wing chord length. The experimental gust replicates a top-hat edge shape with approximately constant transverse velocity \(v_g\) inside, and zero \(v_g\) outside. This gust profile is selected due to its extreme unsteady effects and large loads (Andreu-Angulo et al. 2020). The objective of this study is to mitigate this strong load and maintain a constant lift force for the entire wing-gust encounter, as shown in Fig. 2. The study will be performed for initial angles of attack, \(\alpha _o = \alpha (s\leqslant 0)\), of \(0^{\circ }, 10^{\circ }\), and \(20^{\circ }\) on upwards and downwards gusts. In addition, experiments on upwards gusts at \(\alpha _o = 45^{\circ }\) are performed to analyse the limits for high effective angles of attack. The initial angle of attack determines the steady-state lift, \(C_{lss}\), that is to be maintained during the gust encounter.

An example of the measured gust velocity profile is given in Fig. 3b. This gust replicates the desired top-hat shape except for a smeared transition at the edges and some fluctuations inside the gust. The wing motion used to mitigate the gust perturbation is a one-degree-of-freedom pitch about the wing mid-chord as shown in Fig. 3a. This approach provides a simple test case to explore the mitigation model. Subsequent investigations may then focus on other pitch locations, flaps, or more complex types of controls. Given that we explore large gust ratios, the mitigating pitch motions likely include large amplitudes and frequencies. This itself generates large unsteady forces, and consequently, the prediction of the required pitch profile is complicated.

The force experienced during a wing gust encounter can be divided into the effect from the wing motion and from the gust. Each of these includes a circulatory and non-circulatory force component as shown in Eq. ((1)). The circulatory force of an impulsively started pitching wing can be calculated from the first term on the right of Eq. (1), as shown by Wagner (1925) in the time domain. The circulatory lift response (assuming \(\sin \alpha = \alpha\)) for this case is, according to Gülçat (2010),This result is similar to the steady equation for \(C_l\) but with an added transient effect expressed by the Wagner function W(s). A wing starting from rest eventually reaches a steady-state lift of \(C_{lss} = 2\pi \alpha _o\). In the current study, we use an approximation of the Wagner function proposed by Garrick Garrick (1938),The Wagner solution accurately described the unsteady force experienced by small impulsive changes in angle of attack. However, we are interested in an arbitrary pitch motion. To model this, the analytical lift force due to a step change in the wing motion can be linearly superimposed to recreate any motion. The lift coefficient at the location s is calculated by summing the response of all the incremental pitch motions that have influenced the wing at previous time steps (Andreu-Angulo and Babinsky 2022). This is achieved mathematically with a Duhamel's integral (Anderson 2015),where \(\alpha (s-\tau )\) is the angle of attack at the location \(s-\tau\). Note once more that the wing pitches about the mid chord while the no through flow condition is enforced at the three quarter chord point. The term \(\frac{1}{4}\frac{d\alpha (s)}{\text{d}s}\) accounts for the rotational velocity contribution at the three quarter chord. Equation (4) is the final expression for the circulatory lift on a wing pitching about the mid chord following an arbitrary motion \(\alpha (s)\).

In addition to circulatory effects, an unsteady wing motion also experiences non-circulatory, or ‘added mass’, forces. The ‘added mass’ force is directly related to time variations of the velocity distribution along the chord due to wing motions \(\frac{\text{d}v}{\text{d}t}\) (Brennen 1982). Substituting the wing-normal flow distribution along a wing pitching about its half-chord and travelling at a constant speed results in the ‘added mass’ expression (Limacher et al. 2018; Limacher 2021; Andreu-Angulo 2022),which shows that the non-circulatory force is dependent on the angle of attack and the pitch rate. Prior to entering the gust, the pitch rate is zero, and therefore, there is no ‘added mass’ force. Once the wing starts to pitch inside the gust, this force component becomes substantial. This force component accounts for the effect of large angles of attack, which is considered the most appropriate for gust mitigation applications, as explained in (Andreu-Angulo and Babinsky 2022).

The last major contribution to the lift arises from the gust flow impinging on the wing. Küssner used a very similar approach to Wagner to find an analytical solution for a wing experiencing a transverse step gust (Küssner 1930, 1932). The main difference between this scenario and Wagner’s impulsive pitch motion is that the gust flow affects the wing progressively while wing motions have a uniform effect across the chord. This permits to readily calculate the circulatory and non-circulatory components in one term. In addition to the previous assumptions, the gust shear layer is considered rigid so that it does not deflect during the wing-gust encounter. The effects of gust shear layer deflection on the total loads have been measured to be small (Gehlert et al. 2021). The predicted lift response for a sharp-edged gust is (Leishman 2000),where \(\frac{v_g}{U}\) is effectively the angle of attack induced by the gust. This solution, therefore, diverges from the steady result by the transient Küssner function. The complex Küssner response can be approximated with a simpler function (Bisplinghoff et al. 1955),While simple, the Küssner model has been shown to accurately predict the transient wing force in a wide variety of gust encounters (Andreu-Angulo et al. 2020). The solution is however derived for a wing at zero degrees angle of attack. In the current study, large geometric angles of attack need to be modelled in order to mitigate the effects of high-amplitude gusts. This requires some adjustments to the original theory, such as the introduction of the additional term \(D_c = \frac{c}{2}(1-\cos (\alpha ))\), as described in Andreu-Angulo and Babinsky (2022). Lastly, an arbitrary gust velocity profile and wing motion can again be modelled by using a Duhamel's integral similar to equation (4). The final lift force contribution due to an arbitrary transverse gust velocity profile is,The basic Küssner additional gust force from equation (6) is not affected by having a nonzero \(\alpha _o\). However, the adjustments to include high angles of attack were initially derived assuming \(\alpha (s) \leqslant 0\), while in the current study \(\alpha (s)\) can also be positive. Fortunately, all the trigonometric terms in Eq. (8) are cosines and independent of the sign of \(\alpha\). Moreover, the \(D_c(s - \tau )\) and \(\cos (\alpha (s - \tau ))\) continue to be valid.

The total lift force during a pitching wing-gust encounter becomes a combination of the circulatory component of the Wagner model, the ‘added mass’ term due to the wing pitch motion, and the Küssner force adjusted for a wing at high angles of attack. These components arise from different wing-normal velocity distributions and can be linearly combined due to the direct relation between \(C_l\) and v / \(v'\) shown in Eq. (1). These first order terms account for most of the wing loads during gust mitigation, even for high amplitudes, as concluded in Andreu-Angulo and Babinsky (2022). The result for the unsteady model used in this work is consequently a combination of equations (4), (5), and (8). This equation is used to calculate the pitch profile, \(\alpha (s)\), that results in a constant lift value. The constant value is given by the initial lift, \(C_{lss} = 2 \pi \alpha _o\), which can be substituted in order to obtain,Equation (9) provides a direct relation between the lift, the angle of attack, and the transverse gust velocity. The desired pitch kinematics to achieve a target lift force can now be determined using an iterative approach, which is shown schematically in Fig. 4. Here, the lift coefficient is calculated for one time step t and compared with the desired lift (\(2 \pi \alpha _o\)). If \(C_l\) is within a designed error (set to 0.01 in our case) of \(C_{l}^{desired}\), the computation progresses to the next step. Otherwise, \(\alpha\) is corrected by a value proportional to the difference \(- (C_l - C_{l}^{desired})\), which converges the results due to the direct relation between \(\alpha\) and \(C_{l}\). A new \(C_l\) is calculated for the corrected \(\alpha\) and the process is repeated until \(C_l\) is within 0.01 of \(C_{l}^{desired}\), at which point the calculation progresses to the next step \(t+dt\). For the current study, a convergence study showed that a dt of 0.02 s produced accurate results.

To achieve the desired pitch motions, the wing mid-chord is bolted into a shaft connected to a Maxon EC-45 brushless motor, as shown in Fig. 6a. Two bearings on either side of the shaft are used to reduce vibrations. The motor shaft is attached to a 74:1 gearbox which increases the torque and gives greater control during the pitch manoeuvres. The motion is measured with an encoder hosted inside the motor that has a precision greater than 0.01 degrees. A positional EPOS-4 controller is used to actuate the motor to the desired pitch profile.

The measured gust velocity profile shown in Fig. 3b is substituted into Eq. (9) to calculate the unsteady pitch profiles. The results for the three initial angles of attack, \(\alpha _o\), are shown in Fig. 6b for upwards and downwards gusts with a GR = 0.5. In addition, this figure also includes the pitch profiles that would maintain a zero effective angle of attack. The difference between the coloured and grey curves effectively visualises the wing \(\alpha _{eff}\) for the pitching cases. All pitch profiles present a maximum magnitude at s/c = 2.2 and have a relatively similar shape for different \(\alpha _o\)s.

The stream-normal and streamwise wing load components are acquired using a two-component force balance at a sampling frequency of 1 kHz. Each of the two components is measured by a load cell with a range of ± 50 N and a resolution of 0.01 N. The stream-normal force data are normalized by the dynamic pressure and the wing area in order to calculate the coefficient of lift. The raw signal exhibits large amplitude noise at 19 Hz, as seen in Fig. 7a. Therefore, a low-pass bidirectional filter with a cut-off frequency of 18 Hz is applied to the signal. Afterwards, the result from five runs is ensemble averaged. The lift coefficient curves resulting from the various levels of filtering are shown in Fig. 7b. The force uncertainty due to cell cross-talk and offset loading is determined below 2\(\%\) when ensemble averaging data from 5 runs (Corkery 2018).

Planar time-resolved PIV data are acquired using the two camera set-up shown in Fig. 5. A high-speed Nd:YLF 527nm laser creates a beam pulsing at a frequency of 0.2 kHz that is transformed into a laser sheet through a set of optics. The laser sheet has a thickness of about 2 mm and illuminates titanium dioxide seeding particles placed in the flow. The entire flow field around the wing can be resolved thanks to the use of glass, which permits the laser light to pass through the model. At the same time, surface reflections are reduced. The particle data are collected by two cameras with a resolution of \(1280\times 800\) pixels that are synchronised with the laser using a programmable timing unit PTU-X from Davis (LaVision). The field of view of the first camera resolves the data on the wing upper surface, while the second camera is positioned to measure the data on the lower surface. Using the DaVis software, the images from both cameras are later processed and stitched together to obtain the final velocity fields.

The experimental error associated with PIV is calculated in the DaVis software, which uses the approach described in Wieneke (2015) and Sciacchitano et al. (2015). The maximum uncertainty in the region of interest is calculated to be below 8\(\%\) for one run. The results are averaged over 5 runs to reduce this uncertainty to \(U/\sqrt{N}\) 3.5\(\%\). This uncertainty calculation does not account for systematic errors, such as “peak-locking”. When “peak-locking” occurs, measured velocities are biased towards integer values in terms of pixels per second (Raffel et al. 2007). To verify that “peak-locking” is avoided, the probability density function is computed to ensure that particle velocities corresponding to fractions of a pixel are measured. The results are shown in Fig. 8, for which the peak at 1.3 pixels/second corresponds to the gust velocity. The absence of spikes at non-zero integer values confirms that peak locking does not occur.

As seen in Fig. 10, when a wing at \(\alpha = 20^{\circ }\) enters an upwards gust, the vorticity shed from the leading edge rolls into a LEV on the wing upper surface. The LEV separates from the wing surface relatively quickly. At the same time, vorticity shed from the TE remains planar until the effect of the LEV separation pushes it closer to the wing upper surface. During gust exit, s/c = 3.0, the flowfield observed prior to the encounter does not resume immediately. Instead, at s/c = 4.5, the flow is mostly attached. Farther away from the gust, at s/c = 6, there is evidence that the wing sheds a new starting TE vortex and a secondary LEV. These vortices are expected to eventually advect away allowing the steady-state flow observed in Fig. 9 to re-establish. This flow behaviour is qualitatively similar to that described for a wing at zero degrees incidence in a previous study (Andreu-Angulo and Babinsky 2022). The main differences are that, in the \(\alpha _o = 0^{\circ }\) case, the primary LEV stays attached longer and the secondary LEV does not emerge.

The flat plate at \(\alpha = 20^{\circ }\) encountering a downwards gust experiences a decrease in its effective angle of attack. This causes a significantly different flow evolution compared to the upwards gusts, as shown in Fig. 10. During gust entry, the downwards gust flow impinges on the separated upper surface of the wing, ’pushing’ the separated wake away from the wing. The gust influence then reduces vorticity shedding from the wing due to the lowering of \(\alpha _{eff}\). For the \(\alpha _o = 20^{\circ }\) and GR = 0.5 case, the effective angle of attack inside the gust is only \(6.5^{\circ }\). Consequently, the flow is predominantly attached. After exiting the gust, the vorticity shedding does not resume straight away. Instead, similar to the upwards gust case, a LEV (shown in Fig. 10) develops on the upper surface and eventually sheds.

The flow for the \(\alpha _o = 0^{\circ }\) case encountering a downwards gust is the inverted version of the upwards gust in Fig. 11. Here, the effective angle of attack is \(-26.5^{\circ }\) and thus the flow separates and a LEV is shed from the wing surface. In addition, the wing does not experience a secondary LEV.

The lift responses for upwards wing-gust encounters at four angles of attack are compared in Fig. 12. Prior to the gust, the wing experiences a constant steady-state lift, \(C_{lss}\), whose value depends on the angle of attack. At low \(\alpha _o\), the magnitude of \(C_{lss}\) is consistent with the theoretical result \(2 \pi \alpha\), which assumes attached flow. However, at high \(\alpha\), where the flow is clearly separated, the steady-state lift is below \(2 \pi \alpha\). As the wing enters the gust, the lift magnitude increases for all incidences. The lift response produces a spike centred around s/c = 1.5, after which the lift decreases. After gust exit, the lift magnitude eventually returns to the steady-state value. The maximum lift value does not exceed \(C_l = 3.2\) for any angle of attack.

The results can be compared further by considering exclusively the additional lift produced by the gust \(\Delta C_l = C_l-C_{lss}\), as shown in Fig. 13a. In addition, the loads are normalised by the gust ratio in order to facilitate future GR comparisons. There is a lower and earlier maximum additional lift for greater angles of attack. The value of \(\Delta C_{lmax}/GR\) decreases from 4.6 to 2.6 as \(\alpha _o\) increases from \(0^{\circ }\) to \(45^{\circ }\). Another large difference in lift for the four angles of attack is observed after gust exit, where the \(\alpha _o = 20^{\circ }\) and \(45^{\circ }\) cases experience a large secondary lift peak.

The lift characteristics can be explained by some of the flow features. The smaller and earlier \(\Delta C_l\) peak for the greater \(\alpha _o\) is associated with a faster detachment of the LEV from the wing surface. This LEV detachment results in a decrease in lift. The flow evolution also explains the wing lift response after exiting the gust. The lower \(\alpha\) cases experience attached flow before and after the gust and show only small lift variations for \(s/c>\) 5, but remaining close to the steady-state values. The \(\alpha = 20^{\circ }\) case presents separated flow before entering the gust attached flow soon after exiting. Attached flow is expected to produce higher lift and thus Fig. 13a shows higher \(\Delta C_l\) after exiting the gust (\(s/c>\) 5) than before entering. Eventually, the flow around the wing sheds a LEV and separates, causing the lift to return to its steady-state value.

Normalised additional lift results for the downwards gust case are displayed in Fig. 13b. The state of the wing before entering the gusts is independent of gust direction, and therefore, the steady-state lift on the wing is the same for upwards and downwards gusts. However, as the wing enters the downwards gust, it experiences a decrease in lift, or a negative \(\Delta C_l\) spike. The magnitude of this spike, \(|\Delta C_{lmax} |\), decreases with increasing angle of attack. Afterwards, all cases present a lift increase near gust exit, at around s/c = 2.2. When the wing exits the gust, the lift continues to increase back to the steady-state value. Similar to the upwards gusts, the lift for the \(\alpha _o = 20^{\circ }\) case experiences a secondary lift peak near \(s/c = 5\) due to the unsteady shedding of vorticity.

In order to maintain constant lift for the upwards gust encounter, the wing starts to pitch down as it enters the gust, as shown on the left column of Fig. 14. Nevertheless, the wing continues to experience a relatively large \(\alpha _{eff}\) throughout the gust encounter, approximately \(26.5^{\circ }\) at s/c = 1.0. This results in significant vorticity shedding from the leading edge. The wing also sheds vorticity from the trailing edge, which rolls into the upper surface. After exiting the gust, the wing returns to the initial angle of attack at around s/c = 4.5. At this time, the wing undergoes a LEV shedding process that eventually develops into the detached flow seen prior to gust entry, similar to the gust-only cases.

Figure 14 also shows the flow around a wing that pitches upwards to mitigate a downwards gust. The increase in \(\alpha\) produces a small LEV on the wing’s upper surface, which stays attached to the wing during the entire gust encounter. The wing pitches back down when exiting the gust at s/c = 2.2, producing more leading-edge vorticity. Consequently, at s/c = 4.5, a LEV is developed and starts to separate. This contrasts with the upwards and gust-only results, where the LEV at this time is just starting to form.

The normalised additional lift curves for these mitigation experiments are plotted together with the gust-only results in Fig. 15a. Similar to previous studies (Andreu-Angulo and Babinsky 2022), the results generally demonstrate adequate mitigation of the gust lift spike. The lift variations about \(C_{lss}\) are small and relatively random within the gust region. However, after exiting the gust, the secondary lift spike is still present for \(\alpha _o = 20^{\circ }\). For the mitigation experiments, the pitch action has shifted the secondary lift peak to s/c = 5.5 but it did not alleviate the magnitude. The mitigation model ignores LE vorticity shedding and thus the secondary lift peak, which is affected by this shedding, is not treated.

The mitigation approach is also successful in alleviating the primary downwards gust force spike, as shown in Fig. 15b. The lift stays within \(-0.5< C_l < 0.5\) of the target value when the wing is inside the gust. After exiting the gust, the mitigation pitch motion is once again unsuccessful at removing the secondary lift peak in the \(\alpha _o = 20^{\circ }\) case. However, there is a shift to earlier s/c in the appearance of this feature and a slight reduction in its magnitude. This earlier peak is attributed to the earlier development and shedding of the LEV after gust exit. Nevertheless, the dangerous negative lift spike caused by downwards gusts is well mitigated by this model-based approach.

A more quantitative assessment of the load mitigation can be calculated from the maximum lift values with and without pitch control. The percentage mitigation value is calculated from the difference between the baseline and the control lift peak,where \(C_{lmax}^g\) is the maximum lift in the gust-only case and \(C_{lmax}^m\) is the maximum lift in the mitigation case. The mitigation effectiveness of the primary lift peak for different cases is presented in Table 1. good mitigation of around 80–90%—a reduction in peak load by a factor of 5–10—in all cases. When including the secondary lift peak, \(M_{\%}\) for \(\alpha _o = 20^{\circ }\) decreases to 58 % and 56 % for upwards and downwards gusts, respectively.

The effectiveness of the mitigation approach for upwards and downwards gusts is analysed further by comparing the experimental and theoretical streamlines. The theoretical streamlines are those predicted by the model from equation (9) used for mitigation. The first row in Fig. 16 shows the results around a wing pitching to mitigate a GR = 1.0 downwards gust. The results for the GR = 0.5 case are very similar and therefore omitted in the interest of brevity. The streamlines present a close match between the experiments and the theory. In both, the flow travels parallel to the wing due to the small effective angle of attack.

The second and third rows in Fig. 16 display a wing encountering an upwards GR = 0.5 and GR = 1.0 gust respectively. The experimental streamlines for both cases present a clear LEV that causes a similar amount of curvature in the flow. However, at s/c=1.0, the LEV in the GR = 1.0 case is about 0.2 chords farther away from the wing surface. At s/c = 2.0, the streamline curvature produced by the LEV remains relatively close to the wing for the GR = 0.5 case. The match between experiments and theory is worse than in the downwards case, and this is accompanied by a decrease in mitigation effectiveness to 79 %. For GR = 1.0, the difference between the theoretical and experimental streamlines is greater than for GR = 0.5 case due to the LEV advecting upwards and leaving the field of view. This causes a further reduction of the mitigation effectiveness to 58 %. These results demonstrate that it takes a significant divergence of the flow physics from the theoretical assumptions before the mitigation effectiveness reduces below 80 %.

The mitigation errors at large \(\alpha _{eff}\) are expected due to the assumptions embedded in the model, such as small perturbations and no shedding from the leading edge. A pitching wing-gust encounter with characteristics very distant from the model assumptions is shown in Fig. 17b. Here, the wing travels initially at \(45^{\circ }\) and pitches following the unsteady model to mitigate a GR = 0.5 transverse gust. Taking into account \(\alpha _o\) and \(\alpha _{ind}\), the maximum effective angle of attack for the gust-only case is \(71.5^{\circ }\). The flow is highly separated throughout the gust encounter for the unmitigated and mitigated cases shown in Fig. 17. This flow is significantly different to other mitigation cases, where a small LEV stays close to the wing’s surface and the wing pitches to negative angles of attack.

The pitch motion for this extreme case does not mitigate the gust loads, as shown in Fig. 18a, only reducing the maximum lift by 17 \(\%\). Interestingly, the most significant difference between the gust-only and mitigation cases occurs after exiting the gust. The mitigation lift response does not present the negative peak at s/c = 3.7. The lift results indicate that the mitigation approach is inappropriate for this wing-gust encounter.

In order to explore the limits of the mitigation approach, the mitigation effectiveness \(M_{\%}\) of the unsteady pitch profile is plotted in Fig. 18b for the cases from this study and those in Andreu-Angulo and Babinsky (2022). These results are plotted versus the absolute maximum \(\alpha _{eff}\) for the gust-only case. It can be seen that the mitigation is strong (typically above 85 \(\%\)) for cases where \(|\alpha _{eff} |<\) \(60^{\circ }\). The mitigation strategy is therefore robust for a large range of angles of attack, but it drops sharply for \(|\alpha _{eff} |> 60^{\circ }\). The equivalent of \(M_{\%}\) where a significant secondary peak was observed the mitigation effectiveness for this feature is also shown by black markers. This confirms that the approach is not well suited to mitigation of this feature, and the secondary peak is responsible for the greatest errors when \(|\alpha _{eff} |\) remains below 60 degrees.