4.2.1. Theoretical formulation

The indicial response for an aerofoil translating at a speed $U_\infty$ and interacting with a travelling gust was derived by Miles (Reference Miles1956). In this formulation, a weighting parameter $\lambda$ is introduced such that the gust travels with a relative velocity of $\lambda ^{-1}U_\infty$ . The sign of $\lambda$ determines whether the gust moves downstream or upstream relative to the aerofoil. For the present application, $\lambda$ is evaluated for the two scenarios described previously. In the first case, the transverse gust (induced by the opposite wing) travels in the opposite direction to the aerofoil with the same magnitude of velocity, giving a relative velocity of $2U_\infty$ and, thus, $\lambda = 0.5$ . In the second case, the gust is stationary relative to the aerofoil, giving $\lambda = 1$ , where Miles’s solution approaches Kussner’s function. In the following analysis, Miles’s solution is computed for both values of $\lambda$ and compared with experimental results. It should be noted that for $\lambda = 0.5$ , the gust velocity is normalised by $U_\infty$ , whereas for $\lambda = 1$ , the normalisation is based on $2U_\infty$ . Because the relative approach speed and gap geometry evolve during cross-over, an effective $\lambda$ may vary with time; adopting a piecewise-constant or smoothly time-varying $\lambda (t)$ is a potential refinement for future modelling.

The coefficient of lift for an aerofoil encountering a travelling sharp-edge gust is expressed as

(4.1) \begin{align} c_l(\sigma ) = 2\pi \varPhi (\sigma ,\lambda ), \end{align}

where $\sigma = U_\infty t/b$ is the non-dimensional distance travelled in terms of semi-chord $b$ and $\varPhi (\sigma ,\lambda )$ is the indicial response. It can be decomposed into circulatory and non-circulatory components as

(4.2) \begin{align} \varPhi (\sigma ,\lambda ) = \underbrace {\varPhi _\circ (\sigma ,\lambda )}_{\textit{Circulation growth}} + \underbrace {\varPhi _1(\sigma ,\lambda )}_{\textit{Virtual mass}}. \end{align}

Miles (Reference Miles1956) approximated $\varPhi _\circ$ by an ascending-power expansion in $\sigma$ for small $\sigma$ and by an exponential series for large $\sigma$ . For intermediate values, interpolation is facilitated by rearranging (4.2) as

(4.3) \begin{align} \varPhi (\sigma ,\lambda ) = [\varPhi _\circ (\sigma ,\lambda )+\lambda \varPhi _1(\sigma ,\lambda )] + [(1-\lambda )\varPhi _1(\sigma ,\lambda )]. \end{align}

The virtual-mass and circulatory terms given by Miles (Reference Miles1956) are reproduced here:

(4.4) \begin{align} \varPhi _1(\sigma ,\lambda )= \begin{cases} \big(2|\lambda |\sigma -\sigma ^2\big)^{1/2}\big(\pi \lambda ^2\big)^{-1},& 0\leqslant \sigma \leqslant 2|\lambda |,\\ 0,& \sigma \gt 2|\lambda |. \end{cases} \end{align}

For $\lambda \gt 0$ , the circulatory response for $\sigma \leqslant |\lambda |$ is represented by a power-series expansion:

(4.5) \begin{align} \varPhi _\circ (\sigma ,\lambda ) &= \frac {1}{3\pi \sqrt {2}}\left (\frac {\sigma }{\lambda }\right )^{3/2} \Bigg [ 1 + \frac {1}{10}\left (1+\frac {3}{2\lambda }\right )\sigma - \frac {1}{70}\left (1-\frac {3}{4\lambda }+\frac {15}{32 \lambda ^2}\right )\sigma ^2 \nonumber \\&\quad + \frac {1}{360}\left (1-\frac {3}{7\lambda }-\frac {15}{112 \lambda ^2} + \frac {1515}{448 \lambda ^3}\right )\sigma ^3 + \mathcal{O}\big(\lambda ^{-4}\sigma ^4\big) \Bigg ] \end{align}

and

(4.6) \begin{align} \varPhi _\circ +\lambda \varPhi _1 &= \frac {\sqrt {2}}{\pi }\left (\frac {\sigma }{\lambda }\right )^{1/2} \Bigg [ 1 - \frac {1}{12\lambda }\sigma - \frac {1}{60 \lambda }\left (1-\frac {3}{8\lambda }\right )\sigma ^2 \nonumber \\&\quad - \frac {1}{420 \lambda }\left (1-\frac {3}{4\lambda }-\frac {15}{4 \lambda ^2}\right )\sigma ^3 + \mathcal{O}\big(\lambda ^{-4}\sigma ^4\big) \Bigg ]. \end{align}

For $\sigma \gt 2|\lambda |$ , $\varPhi _\circ$ is approximated exponentially as

(4.7) \begin{align} \varPhi _\circ (\sigma ,\lambda ) &= 1 - 0.165[I_\circ (0.045\lambda )+I_1(0.045\lambda )]e^{-0.045(\sigma -|\lambda |)} \nonumber \\&\quad - 0.335[I_\circ (0.3\lambda )+I_1(0.3\lambda )]e^{-0.3(\sigma -|\lambda |)}, \end{align}

where $I_\circ$ and $I_1$ denote modified Bessel functions of the first kind. For $\lambda \leqslant \sigma \leqslant 2\lambda$ , $\varPhi _\circ$ is obtained by interpolation. Miles’s solution can be generalised for an arbitrary gust profile using Duhamel superposition:

(4.8) \begin{align} \frac {c_l}{2\pi } = f(0)\varPhi (\sigma ,\lambda ) + \int _{0}^{\sigma } \varPhi (\sigma -\sigma _1,\lambda )f'(\sigma _1)\,\mathrm{d}\sigma _1. \end{align}

For cross-over of two counter-translating wings, the function $f(\sigma )$ inherits the measured spatiotemporal gust profile, which embeds the evolving gap geometry; thus, the quality of prediction depends critically on how well $f(\sigma )$ represents the induced disturbance.

The circulatory and non-circulatory indicial responses for the two $\lambda$ values are shown in figure 6. The non-circulatory response for $\lambda =0.5$ is shorter and of higher magnitude than that for $\lambda =1$ . Similarly, the circulatory response shows a phase shift, with $\lambda =0.5$ leading $\lambda =1$ . These trends are consistent with the results reported by Miles (Reference Miles1956). The influence of these differences on the cross-over loads is discussed in the next section.

Figure 6.

Comparison of the non-circulatory and circulatory indicial responses for $\lambda = 0.5$ and $\lambda = 1$ .

4.2.2. Comparison with the measurements

The induced velocity ( $v_g$ ) profile extracted from PIV data at a location $1.4c$ above the lower wing was applied to Miles’s formulation, and the resulting lift response was compared with experimental measurements. These velocity profiles are also presented in figure 4. This approach is inspired by the work of Leishman (Reference Leishman1997), in which the load response to a travelling gust was quantified by calculating the induced velocity from a Lamb–Oseen vortex on the plane of a translating wing and applying it within Miles’s theoretical framework. The comparisons for $\alpha = 0^\circ$ and $4^\circ$ , with $\lambda = 0.5$ and $1$ , are shown in figure 7. Based on the physical expectations of the two models, the travelling-gust (Miles, $\lambda = 0.5$ ) formulation is expected to better represent the rapid mutual convection between the wings during cross-over, whereas the stationary-gust (Kussner, $\lambda = 1$ ) model, with its slower response and longer effective gust duration, is expected to underpredict the amplitude and lag the phase of the observed load excursions.

Figure 7.

Comparison of measured load characteristics with predictions from Miles’s solutions for $\lambda = 0.5$ and $\lambda = 1$ at (a) $\alpha = 0^\circ$ and (b) $\alpha = 4^\circ$ . Decomposition of circulatory and non-circulatory contributions at (c) $\alpha = 0^\circ$ and (d) $\alpha = 4^\circ$ .

Figure 7. Long description

Panel A: A line graph shows the comparison of measured load characteristics with predictions from Miless solutions for lambda equals 0.5 and lambda equals 1. The x-axis is labeled s over c, ranging from -4 to 4. The y-axis is labeled delta c l, ranging from -0.05 to 0.05. The graph includes data points from experiments and two lines representing lambda equals 0.5 and lambda equals 1. Panel B: Another line graph compares measured load characteristics with predictions for wing 2 at lambda equals 0.5 and lambda equals 1. The axes are the same as in Panel A. The graph includes data points from experiments and two lines representing wing 2 at lambda equals 0.5 and lambda equals 1. Panel C: A line graph decomposes circulatory and non-circulatory contributions at lambda equals 0.5 and lambda equals 1. The x-axis is labeled s over c, ranging from -4 to 4. The y-axis is labeled delta c l, ranging from -0.05 to 0.05. The graph includes four lines representing lambda equals 0.5 (non-circulatory), lambda equals 0.5 (circulatory), lambda equals 1 (non-circulatory), and lambda equals 1 (circulatory). Panel D: Another line graph decomposes circulatory and non-circulatory contributions at lambda equals 0.5 and lambda equals 1. The axes are the same as in Panel C. The graph includes four lines representing lambda equals 0.5 (non-circulatory), lambda equals 0.5 (circulatory), lambda equals 1 (non-circulatory), and lambda equals 1 (circulatory).

Consequently, at $\alpha = 0^\circ$ , the phase and magnitude of the first load peak are captured more accurately by the $\lambda = 0.5$ case than by the $\lambda = 1$ case. In particular, both the experimental data and the $\lambda = 0.5$ prediction exhibit a sharper rise in load leading to the first peak, whereas the $\lambda = 1$ case shows a more gradual increase. Following the first peak, the subsequent drop in load is also sharper for the $\lambda = 0.5$ case, in closer agreement with the measured response. Although the second peak differs slightly in phase and amplitude, its predicted magnitude remains closer to the experimental value than that obtained using $\lambda = 1$ . To interpret these differences, the total lift was decomposed into circulatory and non-circulatory components (figure 7 b). The first load peak is dominated by the circulatory contribution, whereas the second peak is primarily associated with the non-circulatory component. The circulatory effects here likely arise from the induced velocity primarily due to the wing thickness that changes the effective angle of attack which was discussed in § 4.1. The non-circulatory response for $\lambda = 0.5$ is larger than for $\lambda = 1$ , consistent with the indicial behaviour shown in figure 6. This distinction can be further interpreted using the impulse-theorem expression for lift (Sears Reference Sears1938):

(4.9) \begin{align} \begin{aligned} \underbrace {L}_{\text{Lift}} = \underbrace {-\rho \frac {\text{d}}{\text{d}t}\int _{-1}^{1}\gamma _\circ (\sigma )\sigma \, \text{d}\sigma }_{\textit{Non-circulatory}} + \underbrace {\rho U_\infty \varGamma _\circ }_{\textit{Quasi-steady}} + \underbrace {\rho U_\infty \int _{1}^{\infty }\gamma (\zeta )\frac {\text{d}\zeta }{\sqrt {\zeta ^2-1}}}_{\textit{Wake effects}}, \end{aligned} \end{align}

where the integrations are performed over the aerofoil chord ( $-1 \leqslant \sigma \leqslant 1$ ) and into the wake ( $\zeta \gt 1$ ). The first term represents the non-circulatory contribution, the second the quasi-steady lift (circulatory effects) and the third the wake-induced effects associated with temporal changes in bound circulation. Here, the quasi-steady lift arises instantaneously at each time step from the change in effective angle of attack caused by the induced velocities from the other wing, with quasi-steady assumption. In a travelling-gust scenario, the time rate of change of the non-circulatory vorticity $\gamma _\circ$ is enhanced by the higher relative velocity between the aerofoil and gust, captured through the parameter $\lambda$ in Miles’s formulation. Consequently, the sharp velocity gradient beyond $s/c\gt 0$ yields a larger non-circulatory component for $\lambda = 0.5$ , improving agreement with the measured loads. The dominance of the non-circulatory term in the second peak is consistent with a high reduced-frequency response, where added-mass effects outpace the development of circulation.

Although Miles’s solution provides a reasonable prediction of the force history, some discrepancies remain, particularly in the magnitude and phase of the second peak and the post-cross-over oscillations. These deviations likely arise from differences in boundary conditions between the idealised gust-interaction model and the actual two-wing configuration, especially the non-penetration condition imposed as the wings align. As demonstrated in § 4.1, this constraint introduces an additional inward-directed force that advances the onset of the second load peak relative to purely induced effects. This additional inward force arises from the increased dynamic pressure on the suction side of the bottom wing and the pressure side of the top wing due to the flow constriction introduced by wing alignment. Despite these limitations, Miles’s formulation remains valuable for predicting preliminary load trends during cross-over, particularly under conditions dominated by circulatory effects.

The next analysis extends this comparison to cases with circulatory boundary conditions at $\alpha = 4^\circ$ . For $\alpha = 4^\circ$ , the $\lambda = 0.5$ case again shows better agreement with the experimental data. While the $\lambda = 1$ prediction follows the overall trend, it underestimates the peak magnitudes. Force decomposition (figure 7 d) indicates that the non-circulatory component remains nearly unchanged from the $\alpha = 0^\circ$ case and the higher lift peaks primarily result from circulatory effects. Comparing the velocity profiles for $\alpha = 0^\circ$ and $4^\circ$ (figures 4 c and 4 d) reveals only minor differences in velocity gradients, explaining the minimal change in non-circulatory contributions. However, the transverse induced gust is stronger at $\alpha = 4^\circ$ , producing larger circulatory lift forces.

Furthermore, the agreement between measurement and prediction ( $\lambda = 0.5$ ) improves for $\alpha = 4^\circ$ , although deviations appear during the post-cross-over recovery phase ( $s/c \gtrapprox 2$ ). As before, these discrepancies likely stem from boundary-condition differences that grow with increasing pitch angle, where the assumption of small angles becomes less valid and the linear model less accurate. In addition, at finite incidence, the induced gust not only alters the effective angle of attack, but also perturbs the timing of vortex-sheet roll-up; such feedback on vortex shedding is outside the scope of linear indicial models and likely also contributes to phase discrepancies during recovery.

4.2.3. Interpretation of model’s discrepancy

Improved modelling approaches could adopt a hybrid framework that combines Miles’s formulation with empirical corrections throughout the cross-over event, augmented by a gap-contraction pressure term calibrated from experiments. Such an approach may be particularly valuable for the $\alpha = 0^\circ$ case, where larger discrepancies are observed. As discussed earlier, the initial stages of cross-over ( $s/c \lt 0$ ) are dominated by circulatory effects, whereas the near-alignment phase ( $s/c \approx 0$ ) is dominated by non-circulatory contributions. The discrepancy between Miles’s prediction and the experimental measurements suggests that the formulation does not account for an additional non-circulatory component associated with the non-penetration boundary condition. During wing alignment, flow constriction between the two surfaces produces rapid pressure variations within the gap region, generating additional force contributions. These pressure variations are expected to increase the rate of change of non-circulatory vorticity (Leishman Reference Leishman1997), which contributes directly to the non-circulatory lift term in (4.9).

To further examine whether the discrepancies between the Miles model and the experimental measurements are associated with additional non-circulatory contributions, we refer to Leishman’s exponential approximation of the travelling sharp-edged gust response (Leishman Reference Leishman1997), given as

(4.10) \begin{align} c_l = c_{l_\alpha }(M) \left ( 1 - \sum _{i=1}^{N} G_i \, \mathrm{e}^{-g_i s} \right ) + G_{N+1} \, \mathrm{e}^{-g_{N+1} s} - G_{N+1} \, \mathrm{e}^{-g_{N+2} s}, \end{align}

where $c_{l_\alpha }(M)$ is the lift curve slope, and $G$ and $g$ are constants, and $s$ is the non-dimensional distance.

The first term here is associated primarily with the asymptotic circulatory buildup, whereas the latter exponential pair represents the localised transient response, associated with non-circulatory contribution, that becomes important during rapidly varying gust encounters. The circulatory buildup is represented by a bilateral exponential term, analogous to the Jones-pole approximation of the Wagner indicial function (Jones Reference Jones1940). Owing to its symmetric nature, this term cannot capture the localised and asymmetric pressure variations associated with wing alignment during cross-over. The effect of the non-penetrating boundary condition is therefore better represented by a difference of two exponential terms, following the structure of Leishman’s indicial formulation (Leishman Reference Leishman1997), which produces an asymmetric pulse-like response. The presence of such an asymmetric non-circulatory term suggests that the rapid pressure changes associated with the non-penetrating boundary condition introduce additional non-circulatory forces that contribute to the discrepancies observed between the Miles prediction and the experimental measurements, particularly near wing alignment. Motivated by this, a non-circulatory correction was introduced to the Miles formulation to improve agreement with the measurements. The correction applied is as follows:

(4.11) \begin{align} (c_l)_{\textit{Total}} = (c_l)_{\textit{Miles}} + \sigma (s;\,s_{0,\textit{nc}},\,\varepsilon )\, B_{\textit{nc}} \left [ e^{-\beta _1 (s - s_{0,\textit{nc}})} - e^{-\beta _2 (s - s_{0,\textit{nc}})} \right ]\!. \end{align}

Here, the correction was applied only within the range $-0.05\lt s/c\lt 1.4$ , where the non-circulatory component becomes dominant (refer to figure 7). The sigmoid function $\sigma (s;\,s_{0,\textit{nc}},\,\varepsilon )$ enforces causality by ensuring the correction activates smoothly at the gust-front crossing location $s_{0,\textit{nc}}$ over a transition width $\varepsilon$ . The exponential function consists of a localised asymmetric pulse whose amplitude ( $B_{\textit{nc}}$ ), onset location ( $s_{0,\textit{nc}})$ , and decay rates and width ( $\beta _1$ and $\beta _2$ ) were determined empirically from the present experimental data ( $B_{\textit{nc}} = -0.041$ , $\beta _1 = 1.2$ and $\beta _2 = 6.5$ ). Physically, this term is intended to represent the transient pressure buildup generated by flow constriction between the wings as they align.

The introduced non-circulatory function significantly improves the agreement between the measurements and Miles’s formulation (refer to figure 8 a), particularly in the vicinity of wing alignment where the largest phase and amplitude discrepancies are observed. This improved agreement supports the interpretation that additional non-circulatory effects, associated with the non-penetration boundary condition during wing alignment, contribute to the measured load response, but are not captured by the original travelling-gust formulation and can be accounted for by introducing non-circulatory corrections.

Figure 8.

Comparison of experimental measurements ( $\alpha =0^\circ$ ) with Miles’s formulation and (a) non-circulatory correction to the Miles’s formulation and (b) non-circulatory and wake decay corrections.

Figure 8(a) also shows that the decay characteristics for $s/c \gtrapprox 0.75$ differ between the Miles prediction and the experimental measurements. This discrepancy is likely associated with interactions between each wing and the shed wake of its counterpart. In the Miles formulation, wake effects are limited to the self-induced wake response and therefore do not account for mutual wake interactions between the two wings. To examine this effect, an additional bilateral exponential correction is introduced to account for the mutual wing–wake decay in the post-cross-over region. This term is analogous in form to the Jones-pole approximation of the Wagner indicial function (Jones Reference Jones1940), which models the gradual buildup and decay of circulatory lift associated with wake development. The updated equation including this correction is as follows:

(4.12) \begin{align} (c_l)_{\textit{Total}} &= (c_l)_{\textit{Miles}} + \sigma (s;\,s_{0,\textit{nc}},\,\varepsilon )\, B_{\textit{nc}} \left [ e^{-\beta _1 (s - s_{0,\textit{nc}})} - e^{-\beta _2 (s - s_{0,\textit{nc}})} \right ] \nonumber \\&\quad + \sigma (s;\,s_{0,w},\,\varepsilon )\, B_{w}\,e^{-\beta _{w}\,|s - s_{0,w}|}. \end{align}

Here, $B_{w}$ scales the magnitude of the wake decay correction, while $\beta _{w}$ controls its decay rate. These parameters were determined from the present experimental data, and were found to be $B_{w} = 0.016$ and $\beta _{w} = 0.65$ . The resulting comparison is shown in figure 8(b). The inclusion of this correction term improves the agreement between the model and the measurements during the decay phase, suggesting that wake-induced interactions between the two wings contribute to the post-cross-over load response as the wings move apart. This observation is consistent with the hypothesis that mutual wake effects become increasingly important once the direct cross-over interaction has subsided.

The following section examines the influence of key non-dimensional parameters, including Reynolds number, wing spacing and aerofoil characteristics, on the cross-over dynamics and evaluates the robustness of Miles’s solution with $\lambda = 0.5$ across a wider operating range.

4.3.1. Reynolds number effects

The Reynolds number is defined using the chord. It can be varied by changing either the chord length or the translation velocity. To evaluate sensitivity to both, we consider the reduced time $s_r$ (Leishman Reference Leishman2006):

(4.17) \begin{align} s_r = \frac {2}{c}\int _{0}^{t} u \, \mathrm{d}t. \end{align}

For the cross-over, we define $t$ as the time for the wings to move from leading-edge alignment to trailing-edge alignment, i.e. one chord. With $t = c/u$ , (4.17) gives $s_r = 2$ , independent of $u$ and $c$ . Thus, the temporal scaling of the event is the same across cases, but this does not imply identical Reynolds-number effects. Instead, any variation in $\delta c_l$ should be interpreted as arising from changes in ${\textit{Re}}$ , rather than from differences in the non-dimensional timing of the cross-over itself. Equivalently, the reduced frequency $k\sim \pi f c/U_\infty$ associated with the cross-over impulse remains $O(1)$ across cases, indicating similar unsteady time scales even though the flow response may still vary with ${\textit{Re}}$ .

We tested this by varying both $u$ and $c$ while holding other parameters fixed: thickness ratio $12\,\%$ , initial bound circulation zero ( $\alpha =0^\circ$ ) and spacing $d_N=1.4$ . The test matrix (table 2) spans $c=0.036,\,0.072\,\mathrm{m}$ and $u=0.2$ – $0.97\,\mathrm{m\,s^{-1}}$ , giving ${\textit{Re}}=1.5\times 10^4$ – $5\times 10^4$ ; each condition was repeated three times.

Figure 9.

(a) Lift coefficient across Reynolds numbers; shading denotes standard deviation. Insets show box plots of $(\delta c_l)_{\textit{max}}$ and its location $(s/c)_{(\delta c_l)_{\textit{max}}}$ . Data down-sampled for clarity. (b) Induced velocity profile on the companion plane. Flow fields at (c) ${\textit{Re}}=15\,000$ , (d) ${\textit{Re}}=25\,000$ , (e) ${\textit{Re}}=35\,000$ .

Figure 9(a) shows the ensemble-mean $c_l$ histories with small standard deviations across ${\textit{Re}}$ . The maxima $(\delta c_l)_{\textit{max}}$ and their locations are similar across cases. The induced-velocity profiles (figure 9 b) and representative flow fields (figure 9 c–e) are likewise consistent, including the stagnation region formation at alignment. We therefore conclude that, over the present range, Reynolds-number effects on the lift excursions are minimal. Any residual ${\textit{Re}}$ -dependence is more likely to appear in drag (skin friction and pressure) and in the post-impulse relaxation rather than in the primary lift peaks.

Although laminar-separation bubbles and wake shedding can occur within this ${\textit{Re}}$ range (Mueller & Batill Reference Mueller and Batill1982), their influence on the dominant lift excursions is limited here. This invariance suggests the results may extend to a broader ${\textit{Re}}$ window; confirmation would require additional testing. By contrast, drag is expected to be more ${\textit{Re}}$ -sensitive due to both pressure and skin-friction components. Our load cells were not sufficiently sensitive to resolve small drag variations at low incidence, motivating future measurements with higher sensitivity. Targeted PIV of the near-wake thickness and spectral content during alignment would help quantify whether ${\textit{Re}}$ modulates the recovery dynamics even when peak loads remain invariant.

4.3.2. Effects of aerofoil thickness to chord ratio

Unsteady loads are known to depend on aerofoil thickness and shape in pitching and gust problems (Lysak et al. Reference Lysak, Capone and Jonson2013; Stangfeld et al. Reference Stangfeld, Rumsey, Mueller-Vahl, Greenblatt, Nayeri and Paschereit2015; Motta et al. Reference Motta, Guardone and Quaranta2015). Motta et al. (Reference Motta, Guardone and Quaranta2015) found NACA0004 responses close to Theodorsen’s thin-plate theory, with deviations increasing with thickness due to circulatory changes. In the present cross-over, as thickness ratio increases, the volume displaced by a translating wing increases, potentially strengthening upwash on the companion wing and modifying its circulatory load. In potential-flow terms, the thickness field increases the local doublet strength, raising the induced transverse velocity felt by the neighbouring wing; to leading order, this scales with $\beta _h$ and decays with spacing.

Lysak et al. (Reference Lysak, Capone and Jonson2013) also showed that leading-edge curvature in a NACA 65 series aerofoil smooths the step-gust response compared with a flat plate, producing earlier onset. Similar behaviour may occur in cross-over. Here, curvature effectively advances the ‘gust encounter’ by altering how the incoming disturbance is turned around the nose, yielding earlier but smoother peaks relative to sharp-edged sections.

Figure 10.

(a) Excursions $\delta c_l$ for flat plate, NACA0012 and NACA0018 (averaged over ${\textit{Re}}=15\,000, 25\,000,\,35\,000$ ). (b, c) Flow fields for (b) NACA0018 and (c) flat plate. (d) Peak magnitude $(\delta c_l)_{\textit{max}}$ and (e) its location. (f) Induced-velocity profiles for the three sections.

Figure 10. Long description

Panel A: A line graph shows excursions for flat plate, NACA0012, and NACA0018 wings. The x-axis is labeled s/c and the y-axis is labeled δc_l. Different colored lines represent different wing profiles, with visible trends indicating variations in excursions. Panel B: A vector field plot for NACA0018 shows z-vorticity with the x-axis labeled s/c and the y-axis labeled y/c. The color scale indicates z-vorticity values, with red and blue regions highlighting areas of positive and negative vorticity, respectively. Panel C: A similar vector field plot for a flat plate shows z-vorticity with the same axis labels and color scale as Panel B. Panel D: A box plot compares the peak magnitude of excursions for NACA0012, NACA0018, and flat plate wings. The x-axis labels the wing profiles and the y-axis labels (δc_l)_max. Panel E: Another box plot shows the location of the peak magnitude with the x-axis labeling the wing profiles and the y-axis labeled s/c(δc_l)_max. Panel F: A line graph depicts induced-velocity profiles for the three wing sections. The x-axis is labeled s/c and the y-axis is labeled v_g/U_∞. Different colored lines represent different wing profiles, showing variations in induced velocities.

We compare three sections of equal chord $c=0.036\,\mathrm{m}$ : a $6\,\%$ -thick flat plate, NACA0012, and NACA0018. Figure 10(a) shows similar qualitative behaviour, outward lift prior to alignment and inward lift after, consistent with the stagnation zone observed in flow fields (figures 10 b and 10 c). Differences appear in peak magnitudes and phase. Symmetric NACA sections show earlier onset, consistent with gust studies (Lysak et al. Reference Lysak, Capone and Jonson2013). Increasing thickness increases the first peak: the mean $(\delta c_l)_{\textit{max}}$ approximately doubles from $0.019$ (NACA0012) to $0.045$ (NACA0018). Flat-plate excursions are comparable to NACA0012 despite different maximum thickness, likely due to similar induced effects. This suggests that in addition to thickness ratio, nose curvature controls the onset timing, while peak magnitude correlates more directly with displaced volume (and thus $\beta _h$ ).

Peak location trends (figure 10 e) also reflect curvature. For NACA0012/0018, $(\delta c_l)_{\textit{max}}$ occurs before $s/c=0$ with little change between the two, consistent with their symmetric shapes (max thickness at $0.3c$ ). For the flat plate, $(\delta c_l)_{\textit{max}}$ occurs after $s/c=0$ , indicating a curvature effect on timing. During alignment, both thickness and curvature matter (figure 10 a): the second peak is similar for NACA0012/0018, but lower for the flat plate, plausibly due to weaker curvature-driven contraction dynamics.

Induced gust profiles (figure 10 f) rationalise these trends. The first induced-velocity peak is largest for NACA0018, and similar for NACA0012 and the flat plate, matching the initial lift excursions. The phase offset between NACA sections and flat plate mirrors a phase delay in the flat-plate induced profile. Second induced-velocity peaks for NACA0012/0018 are similar, consistent with their similar second lift peaks. The velocity-induced-load phase offset remains small but finite, reflecting unsteady response lag; quantifying this as $\Delta (s/c)$ across sections would facilitate model–data comparisons.

Figure 11.

(a, b) Miles ( $\lambda =0.5$ ) predictions versus measurements for (a) NACA0018 and (b) flat plate.

Comparisons with Miles’s model ( $\lambda =0.5$ ) for NACA0018 and the flat plate (figure 11) capture overall trends, but show discrepancies in peak magnitude and phase, consistent with non-penetration effects near alignment that leads to additional non-circulatory effects. Thus, while Miles provides preliminary estimates, thickness and curvature modulate details that the linear model does not fully represent. Additional corrections, similar to those in the previous section, can be applied to improve these estimates. Next, we consider the influence of pitch angle that sets the steady-state lift coefficient.

4.3.3. Effects of steady-state lift coefficient

Section 4.1 argued that finite bound circulation increases load excursions during cross-over. We quantify this by extracting the peak excursions $\delta c_{l,\textit{peak}}$ and their locations $(s/c)_{\delta c_{l,\textit{peak}}}$ for multiple pitch angles (figure 12; filled symbols, first peak $s/c\lt 0$ ; open symbols, second peak $s/c\gt 0$ ). The first-peak magnitude increases approximately linearly with $\alpha$ , consistent with increasing bound circulation, and remains comparable between the two wings up to stall at $\alpha \approx 12^\circ$ . The first-peak location shifts from $\alpha =0^\circ$ to $4^\circ$ and then varies weakly with further $\alpha$ , indicating growth in upwash magnitude without a large extension of its spatial reach. This supports a picture where $\alpha$ primarily scales the strength of mutual induction rather than the spatial footprint of the interaction.

Figure 12.

(a) Peak excursions in $c_l$ and (b) their locations versus angle of attack. Shaded markers, pre-cross-over peaks; open markers, post-cross-over peaks.

Figure 12. Long description

The image contains two scatter plots labeled (a) and (b). Panel A: A scatter plot shows peak excursions in delta C L versus angle of attack (alpha). The x-axis represents the angle of attack (alpha) in degrees, ranging from 0 to 14 degrees. The y-axis represents the peak excursions in delta C L, ranging from -0.4 to 0.4. The plot includes data points for four different wing configurations: Wing 1 (d/c = 1.4), Wing 2 (d/c = 1.4), Wing 1 (lambda = 0.5), and Wing 2 (lambda = 0.5). Shaded markers represent pre-cross-over peaks, while open markers represent post-cross-over peaks. Panel B: A scatter plot shows the locations of peak excursions versus angle of attack (alpha). The x-axis represents the angle of attack (alpha) in degrees, ranging from 0 to 14 degrees. The y-axis represents the locations of peak excursions in s/c, ranging from -2 to 4. Similar to Panel A, this plot also includes data points for the same four wing configurations, with shaded markers for pre-cross-over peaks and open markers for post-cross-over peaks.

Figure 13.

Excursions in (a,b) lift, (c,d) drag and (e,f) moment coefficients for wings 1 and 2 at $\alpha =8^\circ ,\,10^\circ ,\,12^\circ$ . Insets indicate operating conditions and force/moment sign conventions.

A pronounced asymmetry appears in the second peak: the top wing exhibits larger magnitudes and peaks closer to $s/c=0$ than the bottom wing. As the wings align, mutual upwash weakens; simultaneously, the bottom wing’s effective incidence is reduced and dynamic pressure on the top wing’s pressure side increases (§ 4.1), with the bottom-to-top influence dominant in the data. Consistent with this, the quarter-chord moment on the top wing shows a stronger pitch-down then pitch-up sequence at higher $\alpha$ , indicating transient offloading then rapid reloading as the inter-wing gap re-expands.

Comparisons of peak magnitudes and locations with Miles’s model ( $\lambda =0.5$ ) show reasonable agreement at low $\alpha$ , with growing deviations at higher $\alpha$ as separation and wake deflection increase, beyond the assumptions of the linear theory.

Induced stall effects. Figure 13 shows excursions in lift, drag and quarter-chord moment for $\alpha =8^\circ ,\,10^\circ ,\,12^\circ$ . Lift exhibits only modest differences across these $\alpha$ , but the top wing at $\alpha =12^\circ$ shows a distinct drag increase before $s/c=0$ and a stronger pitch-down followed by a larger pitch-up after $s/c=0$ , consistent with a momentary stall-like response. Bottom-wing drag changes are small at $\alpha =8^\circ$ and $10^\circ$ , increasing at $\alpha =12^\circ$ ; the associated pitch-down after $s/c=0$ is also larger at $\alpha =12^\circ$ . These signatures suggest induced stall effects are more pronounced for the top wing, with weaker evidence for the bottom wing; higher-sensitivity measurements would help resolve the latter definitively. Phase-averaged PIV around the top wing nose at $\alpha =12^\circ$ would reveal whether a short-lived LEV or leading-edge suction collapse coincides with the drag/moment excursions.

4.3.4. Effects of wing spacing

The spacing between the wings determines the strength and extent of their aerodynamic coupling. As the gap increases, the induced velocity field generated by one wing weakens before reaching the other, reducing mutual interference. Figure 14 compares the lift-coefficient excursions for two spacing ratios, $d/c=1.4$ and $3$ , at $\alpha =0^\circ$ and $8^\circ$ . In both cases, the primary load features persist, the first outward force peak followed by an inward recovery, but their magnitudes diminish with increasing spacing. The locations of these peaks, however, remain nearly unchanged.

At $\alpha =0^\circ$ , the pre-cross-over excursion primarily originates from upwash and downwash induced by the finite thickness of the wings. This effect decays with spacing as the induced velocity field diffuses and weakens across the gap. Because the timing of this excursion is geometrically linked to the alignment of the quarter-chord positions, its phase remains nearly invariant with $d/c$ . At $\alpha =8^\circ$ , the same overall trend is observed, though the magnitudes of both peaks are larger owing to the stronger bound circulation. The reduction in mutual influence with spacing is again evident, confirming that induced effects dominate the near-field interaction.

Despite the weakening of the primary impulse, the temporal location of the second, post-cross-over peak remains nearly fixed, underscoring that the non-penetration constraint and subsequent gap-pressure recovery operate at a time scale determined by the kinematic alignment rather than aerodynamic diffusion. At large spacing, the induced field becomes sufficiently weak that each wing approaches quasi-isolated behaviour, marking a transition from strongly coupled to weakly coupled unsteady regimes. This trend implies that geometric attenuation, rather than viscous dissipation, governs the reduction in induced coupling over the tested range.

Figure 14.

Blade-load excursions at (a) $\alpha =0^\circ$ and (b) $\alpha =8^\circ$ for two spacings.

Figure 14. Long description

Two line graphs depict blade-load excursions for different wing spacings. Panel A: The line graph shows the variation of delta C L with respect to s over c. The x-axis is labeled s over c and ranges from -4 to 4. The y-axis is labeled delta C L and ranges from -0.05 to 0.05. The graph includes data for four different wing configurations: Wing 1 (d over c equals 1.4), Wing 2 (d over c equals 1.4), Wing 1 (d over c equals 3), and Wing 2 (d over c equals 3). Each configuration is represented by different symbols and colors. The data points show fluctuations around the zero line with some variations among the different wing configurations. Panel B: The line graph shows the variation of delta C L with respect to s over c. The x-axis is labeled s over c and ranges from -4 to 4. The y-axis is labeled delta C L and ranges from -0.10 to 0.10. Similar to Panel A, this graph also includes data for the same four wing configurations. The data points show more pronounced fluctuations and variations among the different wing configurations compared to Panel A.