2.1. Computational set-up

We perform numerical simulations at a fixed chord-based Reynolds number of ${\textit{Re}}_c = u_\infty c/\nu =100$ , where $u_\infty$ represents the free-stream velocity, $c$ is the chord length and $\nu$ corresponds to the kinematic viscosity. Regarding the properties of the vortex gust, we explore the following (chord-based) parameters: angle of attack $\alpha$ , gust ratio $G$ , initial vertical gust position $y_0$ and gust radius $R_v$ , which represents the distance from the core at which maximum tangential velocity is achieved, and is normalised by the chord. In terms of airfoil geometry parameters, we explore the effect of airfoil thickness $\tau$ and maximum camber $\eta$ on 4-digit NACA airfoils. All variables have been non-dimensionalised such that the chord length $c=1$ and the free-stream velocity $u_\infty =1$ .

The vertical offset $y_0$ is defined with respect to the location of the leading edge, and scaled by the chord. We consider geometric angles of attack $0^\circ \leq \alpha \leq 20^{\circ }$ , gust ratios $-2 \leq G \leq 2$ , vertical offsets $-0.25 \leq y_0 \leq 0.25$ and gust radii $0.1 \leq R_v \leq 0.6$ . To systematically explore variations in airfoil geometry, we examine a range of 4-digit NACA profiles with airfoil thicknesses $0 \leq \tau \leq 0.4$ , where $\tau =0$ corresponds to a flat plate, and $\tau =1$ represents the extreme case in which the airfoil thickness matches the chord length. Regarding the airfoil camber, we consider maximum values $0.02 \leq \eta \leq 0.06$ . The position of maximum camber $\xi$ is fixed at $\xi =0.4$ . This position is chosen such that neither the leading edge nor the trailing-edge curvature is disproportionally accentuated, providing a general representation of a cambered airfoil. For all the cases considered here, the airfoil geometry, angle of attack and position remain fixed during the gust encounter, and the analysis assumes two-dimensional incompressible flow conditions.

A schematic of the parameters that determine the different airfoil configurations considered in this work can be found in figure 1. In particular, figure 1 (a) depicts a configuration that captures a vortical gust of radius $R_v$ , and strength $G$ , as it approaches an airfoil of chord length $c$ and at a geometric angle of attack $\alpha$ . Henceforth, $\alpha$ and $\alpha _e$ are used to distinguish between the geometric and effective angles of attack, respectively. According to the sign convention adopted in this work, a vortex gust characterised by a counter-clockwise rotation is considered to have a positive orientation, and is therefore indicated by a positive value of $G$ . We define the gust vortex in terms of vorticity with

(2.1) \begin{equation} \omega _r(r) = \frac {u_g}{R_v}\left (2-\frac {r^2}{R_v^2}\right )\exp {\left ( \frac {1-r^2}{2R_v^2}\right )}, \end{equation}

where $r$ represents the distance with respect to the vortex core (Taylor Reference Taira and Colonius1918). Traditionally, the gust ratio $G$ is defined as $G=u_g / u_\infty$ , that is, the relationship between the peak gust tangential velocity $u_g$ and the free-stream velocity $u_\infty$ . A diagram is provided in figure 1 (b) to depict the two variables of interest that relate exclusively to the airfoil geometry, namely: airfoil thickness $\tau$ and maximum camber $\eta$ .

Figure 1.

(a) Diagram of the flow and airfoil variables included in the parameter space explored in this work. All variables of interest are highlighted in red. (b) Schematic of the airfoil geometry parameters of 4-digit NACA profiles.

Figure 1. Long description

Panel A: A diagram showing the interaction between a gust and an airfoil. The gust is represented by a circular region with a vortex at its center, labeled with the initial position coordinates (x0, y0). The airfoil is depicted with its chord line labeled ‘c’ and the angle of attack labeled ‘α’. The flow direction is indicated by an arrow labeled ‘u∞’. The gust ratio ‘G’ is defined as the ratio of the gust velocity ‘ug’ to the free stream velocity ‘u∞’. Panel B: A schematic of the geometric parameters of a 4-digit NACA airfoil profile. The airfoil is shown with its mean camber line, maximum camber ‘m’, and the location of the maximum camber ‘p’. The thickness distribution is also depicted, with the maximum thickness ‘t’ and the location of the maximum thickness ‘ξ’. The normal vector ‘n’ and the tangential vector ‘τ’ are shown at a point on the airfoil surface.

The immersed boundary projection method is used to simulate the flow (Taira & Colonius Reference Stutz, Hrynuk and Bohl2007; Colonius & Taira Reference Colonius and Taira2008). The total extent of the computational domain is $L_x=72$ in the $x$ -direction and $L_y=40$ in the $y$ -direction. The multi-grid is formed by a total of 5 subdomains, the finest one spanning between $-4 \leq x \leq 5$ and $-2 \leq y \leq 3$ with $M=1152$ and $N=640$ cells, respectively, where all spatial lengths are non-dimensionalised by the chord $c$ . To ensure consistency across different gust encounters, the origin of our reference henceforth coincides with the leading edge (see figure 1). The time-step is chosen such that the Courant–Friedrichs–Lewy number is 0.256 for all simulations. To ensure convergence, all baselines included in this dataset (i.e. all initial conditions preceding the vortex-gust encounter) were obtained after $t=40$ convective time units since initialisation. To preserve uniformity, all vortical gusts are introduced at the same temporal instance $t_{0}=40$ , and at an initial location along the $x$ -axis corresponding to $x_0=-2$ with respect to the leading edge. Further details regarding the convergence studies performed for this mesh are given in Appendix.

2.2. Force-element analysis

To interpret the vorticity fields during a gust encounter, we incorporate the force-element analysis (Chang Reference Chang1992; Zhang Reference Zaide and Raveh2015). Let us consider the velocity potential $\phi$ . This potential is chosen to satisfy the geometric condition $\boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{\nabla }\phi _i = -\boldsymbol{n} \boldsymbol{\cdot }\boldsymbol{e}_i$ , where the right-hand side represents the projection of the surface normal onto the $i$ th basis vector of the orthonormal reference frame $\{ \boldsymbol{e}_1, \boldsymbol{e}_2, \boldsymbol{e}_3 \}$ spanning $\mathbb{R}^3$ . The use of this auxiliary velocity potential, along with the implementation of the divergence theorem on the velocity field (assumed to be solenoidal), allows us to extract the projection of the body and pressure forces onto each element $\boldsymbol{e}_i$ as

(2.2) \begin{equation} f_i = -\int _V \boldsymbol{u} \times \boldsymbol{\omega } \boldsymbol{\cdot }\boldsymbol{\nabla }\phi _i \text{d}V +\frac {1}{\textit{Re}} \int _S \boldsymbol{n} \times \boldsymbol{\omega } \boldsymbol{\cdot }(\boldsymbol{\nabla }\phi _i +\boldsymbol{e}_i)\text{d}S, \end{equation}

where the first term has been rewritten as a volume integral over $V$ involving the Lamb vector (Wu, Ma & Zhou Reference Wilder and Telionis2006), with $u=u(x,t)$ and $\omega=\omega(x,t)$ denoting the instantaneous velocity and vorticity fields, respectively. We will henceforth refer to the terms $f_D=f_1 (\boldsymbol{e}_1)$ and $f_L=f_2 (\boldsymbol{e}_2)$ as drag- and lift-force elements, respectively.

2.3. Vorticity production analysis

To further characterise the mechanisms underpinning the generation and evolution of strong vortical structures during a gust encounter, we examine the source of vorticity. Specifically, we consider the vorticity production flux following the definition first provided in Lighthill (Reference Lighthill1963), and revisited in Morton (Reference Menon and Mittal1984), Hornung (Reference Hornung1989) and Panton (Reference Panagiotou, Mitridis, Dimopoulos, Kapsalis, Dimitriou and Yakinthos2013). Let us define the vorticity flux tensor

(2.3) \begin{equation} \boldsymbol{J} \equiv -\nu \boldsymbol{\nabla }\boldsymbol{\omega }. \end{equation}

For two-dimensional flows, the vorticity transport equation is

(2.4) \begin{equation} \frac {D\boldsymbol{\omega }}{Dt} = \nu {\nabla} ^2 \boldsymbol{\omega }, \end{equation}

where the right-hand side represents the rate of viscous diffusion of vorticity. To better characterise the vorticity generation along the airfoil surface, we adopt a local curvilinear coordinate where $(\boldsymbol{s},\boldsymbol{n})$ correspond to the wall-normal and surface unit vectors, respectively, at each location along the airfoil surface. In this reference, the instantaneous velocity field is expressed as $\boldsymbol{u}(\boldsymbol{x},t) = [u_n(s,n,t),u_s(s,n,t)]$ .

We are particularly interested in the component of the vorticity flux along the wall-normal direction and evaluated at the wall ( $n=0$ ), given by

(2.5) \begin{equation} J_{n,0}=\left (\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{J}\right )_{0}=-\nu \left (\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\omega }\right )_{n=0}, \end{equation}

which quantifies the generation of vorticity with the no-slip condition. In this reference, the vorticity can be written as

(2.6) \begin{align} \omega &= \frac {\partial u_s}{\partial n} - \frac {\partial u_n}{\partial s} + \frac {u_s}{r}\nonumber\\ &= \frac {\partial u_s}{\partial n} - \frac {\partial u_n}{\partial s} + \kappa u_s, \end{align}

where $\kappa =1/r$ represents the curvature of the airfoil surface. Evaluating the projection of the vorticity gradient onto the wall-normal direction $\boldsymbol{n}$ gives

(2.7) \begin{equation} \left (\frac {\partial \omega }{\partial n}\right )_{n=0}=\left (\frac {\partial ^2 u_s}{\partial n^2} +\frac {1}{r}\frac {\partial u_s}{\partial n}\right )_{n=0}, \end{equation}

at the airfoil surface. Although not explicitly included in the vorticity transport equation, surface pressure plays a fundamental role in the process of vorticity generation at solid boundaries. The connection between the wall-normal component of the vorticity gradient and pressure is established through the momentum equation. In particular, the momentum equation in the $s$ -direction, evaluated at the wall, is reduced to

(2.8) \begin{equation} \left (\frac {\partial u_s}{\partial t}+\frac {1}{\rho }\frac {\partial p}{\partial s}\right )_{n=0}=\nu \left (\frac {\partial ^2 u_s}{\partial n^2} +\frac {1}{r}\frac {\partial u_s}{\partial n}\right )_{n=0}, \end{equation}

where $\rho$ represents the fluid density, body forces are excluded from the analysis and the no-slip condition is enforced at the airfoil surface. According to (2.8), for fluid elements at the wall ( $n=0$ ), vorticity-producing unbalanced shear stresses are balanced by both the pressure gradient and wall motion (Lighthill Reference Lighthill1963; Morton Reference Menon and Mittal1984).

The wall-normal component of the vorticity flux in (2.5) can now be written as

(2.9) \begin{align} J_{n,0} &=-\left (\frac {\partial u_s}{\partial t}+\frac {1}{\rho }\frac {\partial p}{\partial s}\right )_{n=0}, \end{align}

where the wall-normal component of the vorticity gradient in (2.7) has been replaced by the left-hand side of (2.8). Herein, we define the quantity

(2.10) \begin{equation} J_{n,0}^c=\nu \left (\frac {1}{r}\frac {\partial u_s}{\partial n}\right )_{n=0}, \end{equation}

as the contribution of the curvature term to the total wall-normal vorticity flux. This analysis is particularly meaningful in the context of gust encounters, as it provides a direct connection between the gust-induced changes in the surface pressure and modified vorticity at the airfoil surface. These changes in the vorticity production levels affect the net circulation around the airfoil, and therefore influence the temporal evolution of the lift during a gust encounter.

2.4. Baseline flows in our dataset

This subsection provides a brief overview of the baseline (unperturbed) state of the airfoil wakes. At this ${\textit{Re}}$ , all baseline flows are at steady states with no vortex shedding. The angle of attack $\alpha$ and the airfoil parameters $(\tau ,\eta )$ , however, determine the initial properties of the flow around the airfoil before the gust–airfoil interaction begins. We characterise the baseline states through the lift $C_L=L/(({1}/{2})\rho u_\infty ^2)$ and drag $C_D=D/(({1}/{2})\rho u_\infty ^2)$ coefficients, where $L$ and $D$ denote the lift and drag forces experienced by the airfoil, respectively. We consider three symmetric and two cambered airfoils. The chord-wise location of maximum camber is fixed at $\xi =0.4$ to highlight the influence of the camber.

Figure 2.

Baseline streamlines and vorticity (coloured contours) fields of three symmetric airfoils at different angles of attack $\alpha$ . Red markers indicate the bounds of the separated region, and its chordwise extent is denoted by $l$ .

Let us first discuss the influence of $\alpha$ and airfoil geometry on the baseline flows shown in figure 2, in which the velocity fields are shown with streamlines, and the vorticity fields are visualised with coloured contours. Flow separation originates near the trailing edge and develops upstream in configurations with increasing $\alpha$ , regardless of the airfoil geometry. The extent of the separated region is denoted as $l$ and represents the projection of this region onto the chord-wise direction (for which the bounds are determined as the locations at which $\partial u_s/\partial n$ changes its sign). We report the emergence of a separated region at lower incidences for thick profiles, which becomes comparable in extent across symmetric airfoils at higher angles of attack $\alpha =20^\circ$ , and decreases for cambered airfoils. The influence of the separated region is reflected in the lift $C_{L,b}$ (left) and drag $C_{D,b}$ (right) coefficients for the five baseline states at different angles of attack shown in figure 3. The first rows correspond to symmetric airfoils, while the third and fourth rows correspond to cambered airfoils.

Figure 3.

Baseline lift $C_{L,b}$ (a) and drag $C_{D,b}$ (b) coefficients of symmetric and cambered 4-digit NACA profiles at different angles of attack $\alpha$ .

Figure 3. Long description

Panel A: Symmetric airfoils. The top left graph is a line graph showing the lift coefficient (C L,b) versus the angle of attack (α) for different NACA profiles. The top right graph is a line graph showing the drag coefficient (C D,b) versus the angle of attack (α) for different NACA profiles. The bottom left graph is a line graph showing the lift coefficient (C L,b) versus the thickness ratio (τ) for different angles of attack. The bottom right graph is a line graph showing the drag coefficient (C D,b) versus the thickness ratio (τ) for different angles of attack. Panel B: Cambered airfoils. The top left graph is a line graph showing the lift coefficient (C L,b) versus the angle of attack (α) for different NACA profiles. The top right graph is a line graph showing the drag coefficient (C D,b) versus the angle of attack (α) for different NACA profiles. The bottom left graph is a line graph showing the lift coefficient (C L,b) versus the camber ratio (η) for different angles of attack. The bottom right graph is a line graph showing the drag coefficient (C D,b) versus the camber ratio (η) for different angles of attack.

Baseline lift increases with $\alpha$ for all airfoil geometries, and flattens at high angles of attack (Abbott & von Doenhoff Reference Abbott and von Doenhoff1959; Leishman Reference Leishman2006). This trend is directly related to the presence of flow separation at high angles of attack observed in figure 3. For symmetric airfoils, thick airfoils yield less lift than their thin counterparts, although the flattening of the lift curve occurs at lower angles of attack for thin geometries. For cambered airfoils, the lift increases with a slope independent of the airfoil geometry, and the flattening of the curves with increasing $\alpha$ occurs similarly for all airfoils. We observe a steady decrease in lift, with a steeper slope for larger $\alpha$ , with increasing airfoil thickness $\tau$ . On the other hand, the lift increases with $\eta$ with a slope independent of $\alpha$ for all cambered airfoils.

Baseline drag, on the other hand, increases with $\alpha$ for all geometries, coinciding with the growth of the separated region that emerges at high incidences. We observe a steeper rate of growth on the drag curves for thin airfoils, and a less dramatic increase for thick airfoils. This coincides with the slower rate of growth of the length of the separated region observed for thick geometries. Moreover, the baseline drag also increases with $\tau$ , for which the separated region originates at lower incidences. In contrast, the slope of the baseline drag against $\alpha$ is similar across the cambered geometries, and increases for cambered airfoils across all incidences.

4.1. Effect of gust ratio

The effect of the gust ratio $G$ on the behaviour of the flow around the airfoil during a gust encounter is determined by both the direction and strength of the vortex gust. Let us consider an example of a NACA 0018 profile with $(R_v,y_0,\alpha )=(0.25,-0.1,5^\circ )$ . The main results of this analysis are presented in figure 5.

Two main observations are drawn from the lift curves: the amplitude of the lift response $|\Delta {C}_{L}|=|\max {(C_L)}-\min {(C_L)}|$ (or lift fluctuation) increases with $|G|$ , and the sign of the first lift peak matches that of $G$ (i.e. we observe a first positive peak for $G\gt 0$ , and a first negative peak for $G\lt 0$ ). As the vortex gust approaches the airfoil, an increased value of $|G|$ will lead to an accentuated deviation of the streamlines and a stronger local flow acceleration near the leading edge (see velocity and kinetic energy fields for $G=\{1,2\}$ in figure 5). As a result, the vorticity production flux at the leading edge will be enhanced, amplifying the net circulation around the airfoil, and therefore yielding a stronger lift response. At this ${\textit{Re}}$ , the flow remains fully attached for $G=1$ . We also report an increase in the extent of the LSB that emerges near the leading edge for $G=2.5$ ( $l\approx 1$ ) against the case shown in figure 4 for $G=2$ . Moreover, for $G=2.5$ , we observe a region of strong flow reversal coming from the bottom surface, bending around the trailing edge, and moving upstream along the upper surface that reaches the end of the LSB. The separated region that emerges from the trailing edge is reminiscent of near-stall conditions, and a reflection of the considerable magnitude of the vortical disturbance. In this case, we report only the total extent of the separated region $l$ , since it includes both the LSB and the trailing-edge recirculation zone.

Figure 6.

Influence of $G$ on the temporal evolution of the volumetric, surface and total lift (a) and drag (b) elements during a gust–airfoil interaction with a gust of $(R_v,y_0,\alpha )=(0.25,-0.1,5^\circ )$ by a NACA 0018 airfoil. Instantaneous volumetric lift (c) and drag (d) elements at $t=(t_0,t_2)$ for $G=(-2.5,2.5)$ .

Let us turn our attention to the case in which $G\lt 0$ . As the gust approaches the airfoil, downward local momentum is induced at the leading edge, shifting the stagnation point to the upper surface. This flow redistribution leads to the generation of positive vorticity at the leading edge, shifting the sign of the total circulation around the airfoil, and a first negative lift peak as a result. The influence of the gust ratio can be directly seen on the instantaneous volumetric lift elements shown in figure 6, which depict lift elements near the leading edge that match the signs of the peaks shown in figure 5. In this case, we observe a region of high kinetic energy above the leading edge, and an LSB near the leading edge on the lower surface of extent $l=0.37$ .

We report a negative trend in the transient drag for all values of $G$ , and the differences in the drag curves between cases of equal $|G|$ are attributed to the asymmetries introduced by the non-zero angle of attack $\alpha$ and the offset in the initial vertical position of the gust $y_0$ with respect to the leading edge. We associate the negative trend for all gust ratios, regardless of the direction of the vortex gust, with the transient thrust provided by the projection of the gust-induced newly generated vorticity at the leading edge along the streamwise direction, which remains the same despite the sign of the gust ratio. This is revealed by the volumetric drag element shown in figure 6, which depicts a negative element near the leading edge at both $t=t_0$ and $t=t_2$ for both positive and negative gust ratios.

The influence of $G$ follows the same trends post-impingement shown in figure 5. For $G\gt 0$ , the lower portion of the gust leads to a more prominent thickening of the boundary layer and a more acute acceleration of the flow on the lower surface for larger gust ratios. As a consequence, the positive lift element on the lower surface, and therefore the secondary lift peak, is also more pronounced for higher values of $|G|$ . At $t=t_1$ , the extent of the separated region has increased to $l=0.82$ , which is also larger than the separated region observed for $G=2$ in figure 4. Notably, we do not report a separated region at this stage for $G=1$ . Moreover, we identify the formation of a coherent LEV (Eldredge & Jones Reference Eldredge and Jones2019) for $G=2.5$ , as a result of the transient near-stall conditions induced by the gust. The post-impingement dynamics for $G=-2.5$ leads to the expansion of the LSB to a length $l=0.62$ on the lower surface. Moreover, in this case, the upper portion of the vortex gust is responsible for the thickening of the boundary layer on the upper surface, coinciding with the positive volumetric lift element shown in figure 6 and yielding a positive secondary lift peak.

The principal trends observed in this section agree with previously established findings: the magnitude of the lift fluctuations increases with $G$ (Küssner Reference Küssner1936; Corkery et al. Reference Corkery, Babinsky and Harvey2018; Martínez-Muriel & Flores Reference Mallik and Raveh2020), and the direction of rotation of the vortex gust determines the sign of the first lift peak (Viswanath & Tafti Reference Taylor2010). Nonetheless, we attribute an increase in the magnitude of the lift fluctuations to accentuated vorticity production levels at the leading edge, signalled by enhanced lift elements in this region. Moreover, we report an increase in the magnitude of the drag fluctuation with $|G|$ , attributed to negative pressure drag responses that are also accentuated by high levels of vorticity fluxes.

4.2. Effect of size of the vortical gust

Let us now turn our attention to the influence of the vortex size on the dynamics observed during a vortex-gust encounter. To illustrate the effect of this parameter, we consider the case with a NACA 0018 profile with a gust of $(G,y_0,\alpha )=(2,-0.1,5^\circ )$ . The results of the analysis on the influence of $R_v$ can be observed in figure 7. In this analysis, the net circulation of the vortex remains the same with $R_v$ , and the vorticity gradient in the radial direction varies with $R_v$ .

Figure 7.

Influence of $R_v$ on $C_L(t)$ and $C_D(t)$ , kinetic energy $k$ , velocity (streamlines), vorticity $\omega$ and volumetric lift-element $f_{L,V}$ fields, observed during a vortex-gust encounter with a gust of $(G,y_0,\alpha )=(2,-0.1,5^\circ )$ by a NACA 0018 airfoil.

Figure 7. Long description

Panel A: The top left graph is a line graph showing the lift coefficient (C_L) over time (t). The x-axis represents time (t) with units in seconds, and the y-axis represents the lift coefficient (C_L) with units in unspecified values. Multiple lines represent different Reynolds numbers (R_v) with specific time points marked (t = t_L1, t = t_0, t = t_1, t = t_2). Panel B: The top right graph is a line graph showing the drag coefficient (C_D) over time (t). The x-axis represents time (t) with units in seconds, and the y-axis represents the drag coefficient (C_D) with units in unspecified values. Multiple lines represent different Reynolds numbers (R_v) with specific time points marked (t = t_L1, t = t_0, t = t_1, t = t_2). Panel C: The middle and bottom rows contain visualizations for different Reynolds numbers (R_v = 0.375, R_v = 0.5, R_v = 0.6) at different time points (t = t_0, t = t_1). Each column represents a different Reynolds number, and each row represents a different time point. The visualizations include kinetic energy (k), vorticity (ω), and volumetric lift-element fields (f_LV). The kinetic energy visualizations use a color scale from 0 to 3, the vorticity visualizations use a color scale from -8 to 8, and the volumetric lift-element fields use a color scale from -4 to 4. The visualizations show the interaction of the airfoil with the vortex-gust encounter at different stages.

The effect of the gust radius $R_v$ on the lift and drag curves is obvious: the amplitude of both lift and drag fluctuations is accentuated by larger gust sizes. As $R_v$ increases, its effective area of influence on the surrounding flow expands, as indicated by the streamlines and kinetic energy distributions in figure 7. Moreover, larger gust sizes correspond to longer interactions, in which vorticity strongly accumulates at the leading edge for longer time periods (Barnes & Visbal Reference Barnes and Visbal2020). This naturally translates as stronger vortical structures of increasing strength and spatial extent, as revealed by the vorticity and volumetric lift-element distributions, and the enhanced kinetic energy at the leading edge. Notably, as the gust size increases, flow separation emerges from the trailing edge and its extent grows along the upper surface. While for $R_v=0.25$ only an LSB was reported on the upper surface, we observe both an LSB and a massive separated region on the upper surface for $R_v\gt 0.25$ . Indeed, for larger gust sizes, $R_v=\{0.5,0.6 \}$ , we observe full separation on the upper surface ( $l\approx 1$ ). Moreover, greater recovery periods are attributed to larger gust sizes, as reported in Barnes & Visbal (Reference Barnes and Visbal2020).

Figure 8.

Influence of gust size on the temporal evolution of the volumetric, surface and total lift (a) and drag (b) elements during a gust–airfoil interaction with a gust of $(G,y_0,\alpha )=(2,-0.1,5^\circ )$ by a NACA 0018 airfoil. Instantaneous volumetric (c) and surface (d) drag elements at $t=t_2$ for $R_v=(0.1,0.375,0.6)$ .

All configurations showcased in figure 7 exhibit a decreased drag upon impingement ( $t=t_0$ ) with respect to the undisturbed case ( $t=t_{-1}$ ). This is attributed to an increase in the projection of the wall-normal vorticity flux onto the streamwise direction for larger gust sizes, increasing the amplitude of the volumetric component $f_{D,V}$ in the form of negative pressure drag responses, as shown in figure 8. Notably, the magnitude of the lift variation at $t=t_0$ increases significantly from $R_v=0.1$ to $R_v=0.375$ , after which it flattens. We attribute this trend to the similarity across the extent of the recirculation region on the upper surface for all cases $R_v\geqslant 0.375$ . Note that $l\approx 1$ for all configurations and time instances showcased in figure 7.

We observe a distinct LEV of increasing size and coherence for larger gust sizes. We attribute this to a prolonged time window of vorticity generation and accumulation at the leading edge for large $R_v$ , which precipitates the formation of a stronger LEV via vortex roll-up upon impingement. At this stage, the entirety of the upper surface is part of the recirculation region, as shown by the streamlines in figure 7. Moreover, we observe a secondary vortical structure at the trailing edge that becomes more prominent as $R_v$ increases. This structure has a positive contribution toward lift, as revealed by force-element analysis, which presumably mitigates the amplitude of the secondary lift peak.

The growing size and strength of the vortical structures identified above the upper surface at $t=t_1$ lead to a more negative drag value at the time of the secondary lift peak. At this temporal instance, we observe flow acceleration on the lower surface, as well as a massive region of recirculation on the upper surface, enhanced by increasing values of $R_v$ . Notably, while the transient pressure drag term becomes increasingly negative at $t=t_2$ for larger gust sizes, the surface component experiences significant negative responses as well. This is reflected by the surface drag elements shown in figure 8: a negative drag element of negative sign is observed on the upper surface for $R_v=0.6$ that coincides with the large recirculation region that arises in this case, and that provides additional temporary thrust in the form of viscous drag.

Regarding the influence of the vortex-gust size, the findings discussed in this section align with the current understanding of its influence: larger gust sizes translate as prolonged exposure times to unsteady conditions (Martínez-Muriel & Flores Reference Mallik and Raveh2020), allowing for larger vorticity injection from the leading edge, and higher aerodynamic responses (Barnes & Visbal Reference Barnes and Visbal2020). In addition, we observe larger vorticity production levels at the leading edge for large gust sizes, leading to the formation of a distinct LEV at high values of $R_v$ . The size and influence of this LEV also increase with $R_v$ , and so does the magnitude of the drag fluctuations post-impingement. We attribute this to a strong recirculation pattern observed on the upper surface that provides temporary thrust in the form of viscous drag. The rapid accumulation of vorticity near the leading edge and its subsequent roll-up into a distinct LEV is reminiscent of the dynamics observed in dynamic stall (McCroskey Reference Martínez-Muriel and Flores1981; Eldredge & Jones Reference Eldredge and Jones2019). In both cases, a rapid increase in the effective angle of attack delays the onset of stall and facilitates the formation of a coherent LEV that drives the larger aerodynamic responses. Our observations for large core sizes $R_v$ share close similarities with the behaviour reported during the upstroke of pitching airfoils, in which the growth of the dynamic stall vortex is associated with large aerodynamic responses (Carr Reference Carr1988; Mulleners & Raffel Reference Morton2012).

4.3. Effect of vertical offset of the vortical gust

Here, we discuss the influence of the initial vertical position of the vortex gust $y_0$ with a gust of $(G,R_v,\alpha )=(2,0.25,5^\circ )$ for a NACA 0018 airfoil. The results of the analysis for various gust initial vertical positions $y_0$ are summarised in figure 9. The lift curves in figure 9 highlight a dual sensitivity to both the magnitude and sign of $y_0$ : the amplitude of the first lift peak decreases with $|y_0|$ (the lift fluctuations with $|y_0|=0.1$ surpass those observed for $|y_0|=0.25$ ); and the peak associated with $y_0=0.1$ is greater than the peak corresponding to $y_0=-0.1$ . Similarly, the lift peak observed for a given $y_0\gt 0$ is larger than its negative counterpart.

Figure 9.

Influence of $y_0$ on $C_L(t)$ and $C_D(t)$ , kinetic energy $k$ , velocity (streamlines) and vorticity $\omega$ fields, observed during a vortex-gust encounter with a gust of $(G,R_v,\alpha )=(2,0.25,5^\circ )$ by a NACA 0018 airfoil.

Figure 9. Long description

Panel A: Two line graphs depict the lift coefficient (C_L) and drag coefficient (C_D) over time (t). The lift coefficient graph shows multiple lines representing different initial positions (y_0) with markers indicating specific time points (t = t_0, t = t_1, t = t_2). The drag coefficient graph similarly shows multiple lines for different initial positions with markers at specific time points. Panel B: Nine visualizations show the kinetic energy (k) and vorticity fields (ω) for different initial positions (y_0 = -0.25, y_0 = 0, y_0 = +0.25) at different time points (t = t_0, t = 0.50, t = 0.26, t = 0.31, t = 0.66, t = 0.59). Each visualization includes streamlines and color scales indicating the magnitude of kinetic energy and vorticity.

Since $G\gt 0$ , the stagnation point is pushed toward the bottom surface, and new (negative) vorticity accumulates at the leading edge. The vertical offset of the vortex gust, however, determines the position of the stagnation point. For instance, we observe that the stagnation point moves aft along the bottom surface for $y_0=-0.25$ . In contrast, in the case with $y_0=+0.25$ , the stagnation point is significantly closer to the leading edge (see vorticity contours in figure 9). According to the perspective of the effective angle of attack $\alpha _e$ (Perrotta & Jones Reference Peng and Gregory2017; Biler, Badrya & Jones Reference Biler, Badrya and Jones2019; Menon & Mittal Reference McCroskey2020), the significant recession of the stagnation point toward the leading edge for $y_0=0.25$ can be interpreted as a negative gust contribution toward the effective angle of attack $\alpha _e$ , such that $\alpha _e\lt \alpha$ . This would explain the reduced lift response observed for $y_0=0.25$ with respect to $y_0=0$ or $y_0=-0.1$ . Meanwhile, while the stagnation point moves aft for $y_0=-0.25$ , which would in theory signal an increase in the effective angle of attack $\alpha _e\gt \alpha$ , the kinetic energy of the flow travelling around the leading edge is suppressed compared with the configurations shown in figures 4 and 9, resulting in a reduced lift response. The schematic shown at the bottom of figure 10 provides further clarity: as the region of high vorticity production changes with the location of the vortex gust, so does the projection of the wall-normal surface vector onto the $x$ - and $y$ -directions, therefore affecting the lift and drag responses. The projection of the wall-normal vorticity flux onto the $y$ -direction is larger for $y_0=0.25$ compared with $y_0=-0.25$ , and results in a larger volumetric lift element (both in extent and magnitude) throughout the interaction (see panel a and panel f row in figure 10). As a result, we observe a boost in the magnitude of the volumetric lift elements in this case as well. The direction of the streamlines is substantially affected by $y_0$ as well: they tightly wrap around the leading edge with no visible LSB for $y_0=-0.25$ , while the streamlines lift away from the surface and an LSB of $l=0.26$ is observed for $y_0=+0.25$ . Notice that an LSB was observed near the leading edge with $l=0.35$ for $y_0=-0.1$ (see figure 4), and that the extent of this region expands as $|y_0|$ further decreases. In particular, it expands to $l=0.50$ for $y_0=0$ .

Variations of $y_0$ yield a rich set of post-impingement dynamics. For instance, we do not observe vortex roll-up for $y_0=-0.25$ . On the other hand, we observe the formation of a LEV for $y_0=0$ , indicative of a stronger vorticity accumulation at the leading edge. While vorticity shedding from the upper surface is observed post-impingement for $y_0=0.25$ , it leads to the formation of a vortical structure with an elongated shape. Moreover, we observe a small separated region $l=0.31$ with $y_0=-0.25$ at $t=t_1$ . In the other two cases, the initial LSB has shifted downstream on the upper surface and towards the trailing edge. The extent of the separated region becomes $l=0.66$ for $y_0=0$ , and reaches the trailing edge with $l=0.59$ for $y_0=0.25$ . In terms of lift-element analysis, we observe a faint positive lift element at the trailing edge for $y_0\leq 0$ that becomes weaker as $y_0$ increases. Moreover, the influence of $y_0$ on the magnitude of the second lift peak is less pronounced than for the first peak.

Figure 10.

Influence of $y_0$ on the temporal evolution of the volumetric, surface and total lift (a) and drag (b) elements during a gust–airfoil interaction with a gust of $(G,R_v,\alpha )=(2,0.25,5^\circ )$ by a NACA 0018 airfoil. Instantaneous volumetric lift element (c), volumetric (d) and surface (e) drag elements and instantaneous wall-normal vorticity fluxes (f) at $t=t_1$ for $y_0=(-0.25,0,0.25)$ .

The influence of $y_0$ is reflected in the drag curves as well. Notably, we observe the largest negative drag fluctuation for $y_0=-0.25$ , the largest positive fluctuation for $y_0=0.25$ , and the rest of the drag curves fall in between. The drag-element decomposition shown in figure 10 at $t=t_1$ provides further insight into the nature of these trends. The volumetric drag element appears to be the primary driver of both lift and drag responses, in line with the results observed thus far. However, in terms of the surface drag elements, we observe a negative response for $y_0=-0.25$ , and a positive response for $y_0=+0.25$ . We attribute this to the alignment between the direction of the streamlines around the airfoil before the interaction, and that of the local velocity around the vortex: for $y_0=-0.25$ , the vortex induces a local motion below the airfoil that opposes the direction of the free stream, resulting in a negative surface drag element; for $y_0=+0.25$ , the vortex gust induces a local deviation of the streamlines that is aligned with the direction of the free stream, boosting the viscous drag levels on the upper surface. Both of these trends concur with the surface drag elements shown in figure 10. Nonetheless, the surface drag elements only paint a partial picture of the nature of the drag response, since the primary trends are attributed to pressure drag. The volumetric drag elements in figure 10 depict a negative drag element near the leading edge for $y_0=-0.25$ , and an increasing number of both negative and positive drag elements in the vicinity of the leading edge as $y_0$ becomes increasingly positive. We postulate that the increasing number of volumetric drag elements, and in particular positive elements, in this region yield the net positive drag response observed in the evolution of the $f_D(t)$ curves in the top right plot in figure 10.

As previously reported in the literature, the magnitude of the aerodynamic responses is accentuated for small $|y_0|$ (Wilder & Telionis Reference Weingaertner, Tewes and Little1998; Barnes & Visbal Reference Barnes and Visbal2018; Chen & Jaworski Reference Chen and Jaworski2020; Martínez-Muriel & Flores Reference Mallik and Raveh2020), and the magnitude of the lift fluctuation follows an asymmetric distribution with respect to $y_0=0$ that is dependent on the sign of $G$ (Horner et al. Reference Horner, Saliveros, Kokkalis and Galbraith1993; Peng & Gregory Reference Peng and Gregory2017): e.g. we observe a larger lift fluctuation for $y_0=+0.25$ than $y_0=-0.25$ . Indeed, we observe higher kinetic energy values at the leading edge for $y_0\gt 0$ , which lead to enhanced vorticity production levels in this region and a larger lift response than their negative counterpart. Lift-element analysis also reveals a stronger positive volumetric lift element at the leading edge for $y_0=+0.25$ when impingement occurs, which decreases in size and strength for $y_0=-0.25$ . We additionally provide insight regarding the influence of the parameter $y_0$ on the magnitude and sign of the drag fluctuations, which are governed by the pressure drag components, and significantly enhanced by their surface drag counterparts.

4.4. Effect of angle of attack

Let us now examine the influence of the angle of attack $\alpha$ on the dynamics observed during a vortex-gust encounter. Representative cases for a NACA 0018 profile with a gust of $(G,R_v,y_0)=(2,0.25,-0.1)$ are presented in figure 11 at different angles of attack. At first glance, the transient component of the lift curves showcases a modest dependence on $\alpha$ , although the maximum lift peak increases with this parameter. This suggests that a gust of the same properties, namely strength, size and vertical offset, will provide the airfoil with additional circulation of similar magnitude in all cases. In other words, the unsteady effective angle of attack $\alpha _e$ increases at higher incidences, but the offset with respect to $\alpha$ remains fairly consistent across configurations. We do report, however, a flattening in the magnitude of the first lift peak past $\alpha =10^\circ$ , signalling the occurrence of dynamic stall.

Figure 11.

Influence of $\alpha$ on $C_L(t)$ and $C_D(t)$ , kinetic energy $k$ , velocity (streamlines), vorticity $\omega$ and lift-element $f_L$ fields, observed during a vortex-gust encounter with a gust of $(G,R_v,y_0)=(2,0.25,-0.1)$ by a NACA 0018 airfoil.

Figure 11. Long description

Panel A: Two line graphs depict the lift coefficient (C_L) and drag coefficient (C_D) over time (t). The x-axis represents time (t) with units in non-dimensional time, and the y-axis represents the coefficients (C_L and C_D) with units in non-dimensional values. Different colored lines represent different angles of attack (α) ranging from -5° to 20°. Key time points (t0, t1, t2, t3) are marked on the graphs. Panel B: Nine visualizations show the kinetic energy (k) and vorticity (ω) fields around a NACA 0018 airfoil at different angles of attack (α = 0°, 10°, 20°) and different time points (t = t0, t ≈ 0.19, t = 1). The x-axis represents the non-dimensional horizontal distance, and the y-axis represents the non-dimensional vertical distance. The color scales indicate the magnitude of kinetic energy (k) and vorticity (ω). Streamlines and vorticity contours illustrate the flow patterns around the airfoil.

At $t=t_0$ , we observe a small LSB with $l=0.19$ for the case $\alpha =0^\circ$ . At higher incidences, $\alpha \geqslant 10^\circ$ , we the airfoil experiences full separation ( $l\approx 1$ ). We also report larger values of kinetic energy of the flow moving around the leading edge at high angles of attack, leading to an enhanced accumulation of vorticity at the leading edge, and a stronger positive lift element.

As the vortex gust progresses past the leading edge, the initial LSB shifts downstream. At low angles of attack, in which no secondary separated region was observed, the LSB grows in size and reaches a length of $l=0.46$ for $\alpha =0^\circ$ at $t=t_1$ . For higher angles of attack, however, the extent of the separated region reaches a value of $l=0.83$ for $\alpha =10^\circ$ and $l=0.85$ for $\alpha =20^\circ$ at $t=t_1$ . Moreover, at $\alpha =20^\circ$ , the flow still experiences considerable acceleration at the leading edge, as revealed by the kinetic energy distribution. As the angle of attack increases, the negative vortical structures diffused from the leading-edge post-impingement become increasingly more compact, and evolve into a distinct LEV at $\alpha =20^\circ$ . Moreover, we report the emergence of a positive volumetric lift element near the trailing edge that arises at high incidences, and is present both before and after impingement (see figure 12).

Figure 12.

Influence of $\alpha$ on the temporal evolution of the volumetric, surface and total lift (a) and drag (b) elements during a gust–airfoil interaction with a gust of $(G,R_v,y_0)=(2,0.25,-0.1)$ by a NACA 0018 airfoil. Instantaneous volumetric lift (c) and drag (d) elements, and instantaneous wall-normal vorticity fluxes (c,d) at $t=(t_0,t_2)$ for $\alpha =(0^\circ ,10^\circ ,20^\circ )$ .

Regarding the evolution of the drag curves in figure 11, we observe the largest negative drag peak for $\alpha =-5^\circ$ and the largest positive one for $\alpha =20^\circ$ . The pressure drag elements shown in figure 12 for $\alpha =\{0^\circ ,20^\circ \}$ provide further insight into the drag trends. First, the influence of the surface drag element decreases at higher incidences, as revealed by the instantaneous $f_D(t)$ curves. Second, the volumetric drag element becomes positive at $\alpha =20^\circ$ . Notably, we observe a single dominant negative pressure drag element at $t=t_0$ and $\alpha =0^\circ$ , while some positive elements emerge near the leading and trailing edges at $\alpha =20^\circ$ , resulting in a net positive drag response. In particular, the positive drag element at the trailing edge is observed consistently throughout the interaction, and is counteracted by negative drag elements of considerable extent at $\alpha =20^\circ$ , resulting in a slight negative variation with respect to the baseline at $t=t_2$ .

As was previously reported in the literature, unsteady stall conditions can be reached during a VAI (Barnes & Visbal Reference Barnes and Visbal2020). Moreover, all lift curves in figure 11 appear to be self-similar (Engin et al. Reference Engin, Aydin, Zaloglu, Fenercioglu and Cetiner2018), and the duration of the VAI remains fairly consistent across $\alpha$ (Martínez-Muriel & Flores Reference Mallik and Raveh2020). Although we report higher kinetic energy levels at the leading edge and high angles of attack, which would usually result in larger vorticity production levels and lift values, the lift fluctuations remain consistent at high incidences $\alpha \gt 10^\circ$ . We associate this observation with the occurrence of dynamic stall. This aligns with the findings of Mallik & Raveh (Reference Zhu, Jiang, Zhao, Tao and Lei2019), who observed an attenuation in lift fluctuations for airfoils at high angles of attack compared with their low-incidence counterparts under the same gust conditions. We attribute this trend to the massively separated region observed at baseline at these high incidences. While a lift overshoot is still observed, the capacity of the airfoil to enhance the magnitude of the lift fluctuation is limited with respect to low incidences. This saturation in the response is characteristic of a deep dynamic stall regime, which occurs when the interaction is rapid enough to trigger the deviation of the aerodynamic forces from linear trends (McCroskey Reference Martínez-Muriel and Flores1981). The angle of attack has a significant influence on the drag response as well, for which we observe positive peaks at high angles of attack, driven by the emergence of positive pressure drag elements near the leading and trailing edges, and negative peaks at low angles of attack, for which we observe a large negative drag element near the leading edge.

4.5. Effect of the airfoil geometry

This subsection presents a characterisation of the influence of the airfoil thickness and camber on the dynamics observed in a vortex-gust encounter. The dynamics discussed so far has been centred around a symmetric NACA 0018 profile. We first discuss the effect of varying the airfoil thickness $\tau$ on the trends discussed so far for symmetrical airfoils in § 4.5.1. The influence of camber $\eta$ is then discussed in § 4.5.2 for thin airfoils ( $0.06$ thickness) to highlight the influence of the camber.

4.5.1. Effect of airfoil thickness

We consider symmetric airfoils of varying thickness $\tau$ at $\alpha =5^\circ$ with $G=2$ , $R_v=0.25$ and $y_0=-0.1$ , as shown in figure 13. Two main trends are drawn from these results: an increasing airfoil thickness attenuates the lift response, and we observe negative drag peaks for all airfoils of increasing magnitude for increasingly thick profiles.

Let us first discuss the vortex dynamics preceding vortex impingement. First, we observe that the kinetic energy decreases at the leading edge for thick airfoils. An LSB of extent $l=0.50$ emerges near the leading edge for $\tau =0.06$ , and is significantly reduced to $l=0.16$ for $\tau =0.24$ (NACA 0024). Nonetheless, an additional separated region arises in this case from the trailing edge on the upper surface and has an extent of $l=0.31$ . As airfoil thickness further increases, we observe a separated region arising from the trailing edge with a length of $l=0.48$ .

Figure 13.

Influence of $\tau$ on $C_L(t)$ and $C_D(t)$ , kinetic energy $k$ , velocity (streamlines) and vorticity $\omega$ , observed during a vortex-gust encounter with a gust of $(G,R_v,y_0,\alpha )=(2,0.25,-0.1,5^\circ )$ .

Figure 13. Long description

Panel A: Two line graphs depict the lift coefficient (C_L) and drag coefficient (C_D) over time (t) for different NACA airfoil profiles. The lift coefficient graph shows multiple lines representing NACA 0001, NACA 0006, NACA 0012, NACA 0018, NACA 0024, NACA 0030, NACA 0036, and NACA 0040. The drag coefficient graph similarly shows multiple lines for the same profiles. Key time points t_{-1}, t_0, t_1, and t_2 are marked on both graphs. Panel B: Three columns of visualizations show kinetic energy (k) and vorticity (ω) for NACA 0006, NACA 0024, and NACA 0040 at different time points (t = t_0 and t = t_1). Each column includes two rows of visualizations, with the top row showing kinetic energy and the bottom row showing vorticity. The visualizations use color gradients to represent the intensity of kinetic energy and vorticity, with streamlines indicating flow direction. Specific values of l are noted for each visualization.

The high accumulation of vorticity at the leading edge for thin profiles leads to a higher diffusion flux of vorticity to enter this flow, facilitating the development of a LEV, as observed for $\tau =0.06$ (NACA 0006). This is not observed for thicker profiles, as there is not enough vorticity accumulation at the leading edge to promote the development of a LEV. As the lower and upper portions of the gust advance downstream, the LSB observed for $\tau =0.06$ expands to a length of $l=0.57$ . Conversely, for thick profiles, the extent of the secondary separated region increases as well post-impingement, reaching a length of $l=0.72$ for $\tau =0.40$ (NACA 0040). Notably, for $\tau =0.24$ , the LSB appears to overlap with the secondary recirculation region at this time, and the total extent of the separated region amounts to $l=0.85$ .

The vorticity production levels observed at impingement for symmetric airfoils are shown in figure 14(b). We report significantly higher vorticity production levels from the leading edge for thinner airfoils, in agreement with the heightened lift response recorded for such profiles. A linear trend is observed between the lift fluctuation $\Delta C_L$ and $\tau$ in figure 14(c), further underscoring the relevance of the curvature term $\kappa$ on the total vorticity production (see second term in (2.8)). This drastic reduction in $\kappa$ observed for thicker airfoils, following $R_c=1.1019 \tau ^2$ in Abbott & von Doenhoff (Reference Abbott and von Doenhoff1959) for four-digit NACA airfoils, translates into a mitigated influence of the curvature term in the vorticity production flux, leading to a more modest lift response.

The dampening of the magnitude of the vorticity production levels as a result of an increased thickness is observed throughout the entire VAI. See the net vorticity production integrated over the airfoil surface as the interaction progresses over time in figure 14(a). This result agrees with the trends observed for the lift and drag coefficients, and can be explained by the change in the projection of the wall-normal vector along the surface when airfoil thickness increases. Note that the length of the region with a significant contribution of the horizontal component is limited to a few locations at the leading edge for a NACA 0006 airfoil, while its extent increases significantly as the bluntness of the leading edge increases. Hence, in the case of thick airfoils, due to the projection of the wall-normal vector near the leading edge onto the horizontal component being larger, the magnitude of the drag fluctuations becomes more prominent. We report a decrease in the projection of the wall-normal vorticity production onto the $y\hbox{-}$ axis with airfoil thickness at $t=t_0$ in the bottom row of figure 14, which in turn enhances the magnitude of the drag response (see drag curves in figure 13). This trend is consistent throughout the interaction, as portrayed in the top middle and right plots in figure 14, which represent the net projection of the instantaneous vorticity production onto the $x$ - and $y$ -directions, respectively.

Figure 14.

(a) Net vorticity production and its $x$ and $y$ projections over time for multiple NACA 4-digit airfoils during a gust–airfoil interaction with a gust of $(G,R_v,y_0,\alpha )=(2,0.25,-0.1,5^\circ )$ . (b) Distribution of wall-normal vectors (black) and their $x$ (red) and $y$ (blue) projections across three representative geometries. Spatial distribution of the instantaneous wall-normal vorticity flux (black) and their $x$ (red) and $y$ (blue) projections at $t = t_0$ . (c) Lift fluctuation $\Delta C_L$ against airfoil thickness $\tau$ .

Figure 15.

Influence of $\tau$ on the temporal evolution of the volumetric, surface and total lift (a) and drag (b) elements during a gust–airfoil interaction with a gust of $(G,R_v,\alpha )=(2,0.25,5^\circ )$ by a NACA 0018 airfoil (c). Instantaneous volumetric lift element (top row), volumetric drag element (bottom row) at $t=t_1$ for $y_0=(-0.25,0,0.25)$ .

Figure 15. Long description

Panel A: A line graph shows the temporal evolution of the volumetric, surface, and total lift elements. The x-axis represents time (t) and the y-axis represents the lift function (f_L(t)). Three different airfoil profiles (NACA 0006, NACA 0018, and NACA 0040) are compared, each represented by different colored lines. The graph shows peaks and troughs indicating changes in lift over time. Panel B: Another line graph illustrates the temporal evolution of the drag elements. The x-axis represents time (t) and the y-axis represents the drag function (f_D(t)). Different line styles and colors represent the combined drag elements (f_D,S + f_D,V), surface drag (f_D,S), and volumetric drag (f_D,V). The graph shows variations in drag over time. Panel C: Visual representations of the instantaneous volumetric lift and drag elements for three airfoil profiles (NACA 0006, NACA 0024, and NACA 0040) are shown. The top row displays the volumetric lift element with a color scale ranging from -4 to 4. The bottom row shows the volumetric drag element with a color scale ranging from -2 to 2. Each airfoil profile is depicted with color gradients indicating the intensity of lift and drag elements.

This redistribution of the aerodynamic response is reflected in the lift and drag trends shown in figure 13. Examination of the contribution of the volumetric (pressure) and surface (viscous) components of the lift and drag coefficients in figure 15 further supports this claim. Note that the volumetric component is the primary driver of the lift fluctuations, as discussed in § 3, regardless of airfoil thickness. The morphology of the volumetric component of the lift $f_{L,V}$ remains consistent across airfoil geometries at $t=t_0$ , although its magnitude decreases with airfoil thickness. Furthermore, while we report variations in the distribution of the surface components $f_{L,S}$ across the different airfoil geometries, the contribution of this term remains inferior to that of the volumetric component. In terms of drag, we observe significant differences in the morphology of the volumetric (pressure) and surface (viscous) elements across the geometries. Overall, the contribution of the pressure term $f_{D,V}$ dictates the trend of the drag fluctuation for all geometries, but this contribution is accentuated for thick airfoils (see top right plot in figure 15). In particular, we observe an attenuation of the viscous element $f_{D,S}$ for thick airfoils, likely attributed to an attenuation of the velocity gradients around the airfoil (see figure 13). On the other hand, we observe a negative volumetric drag element $f_{D,V}$ at the leading edge whose extent increases with airfoil thickness as a result of the redirection of the wall-normal surface vector at the leading edge toward the horizontal axis.

Previous studies had hinted at the possibility of mitigating the lift response in gusty environments using blunt leading edges (Gementzopoulos et al. Reference Gementzopoulos, Sedky and Jones2024), or thick airfoils in general (Zhong et al. Reference Zhang2019). Notably, Chen & Jaworski (Reference Chen and Jaworski2020) demonstrated that increased airfoil thickness significantly alters the trajectory and deformation of the incident vortex as it passes the leading edge. Here, we have examined the influence of airfoil geometry from the perspective of vorticity production fluxes and determined that the gust-induced vorticity fluxes at the leading edge are effectively attenuated for low-curvature airfoils. In summary, the observed dampening of lift fluctuations with increasing airfoil thickness is driven by two complementary mechanisms. First, the reduction in leading-edge curvature for thicker sections decreases the net vorticity production over time, as shown in figure 14. Second, the geometry of thick airfoils promotes the spatial redirection of the transient aerodynamic forces, particularly the volumetric (pressure) components, onto the $x$ -direction. While the wall-normal surface vectors of thin airfoils are almost exclusively aligned with the vertical axis, increased thickness introduces a significant horizontal projection near the leading edge. This shift simultaneously enhances drag fluctuations and mitigates the amplitude of the lift peaks.

4.5.2. Effect of airfoil camber

Next, let us characterise the influence of the airfoil camber. Here, we examine the differences in the aerodynamic response of the following profiles: $\{$ NACA 0006, NACA 2406, NACA 4406, NACA 6406 $\}$ . All airfoils share the same airfoil thickness $\tau =0.06$ , and the location of maximum camber is fixed at $\xi =0.4$ . This dataset is thus parametrised according to the camber magnitude $\eta$ , with values $\eta =\{0,0.02,0.04,0.06\}$ . The corresponding results are presented in figure 16.

Figure 16.

Influence of $\eta$ on $C_L(t)$ and $C_D(t)$ , kinetic energy $k$ , velocity (streamlines), vorticity $\omega$ and lift-element $f_L$ fields, observed during a vortex-gust encounter with a gust of $(G,R_v,y_0,\alpha ;\xi )=(2,0.25,-0.1,5^\circ ;0.4)$ .

Camber effects seem to have little influence on the magnitude of the lift fluctuations in this case, while the influence on the drag is significantly more noticeable. The flow states observed at impingement reveal increased levels of kinetic energy at the leading edge for increasing camber. On the one hand, we observe an LSB for all three profiles at $t=t_0$ . In particular, the extent of this region increases from $l=0.50$ to $0.61$ between $\eta =0$ and $\eta =0.02$ ( $\eta =0$ shown in figure 13). Moreover, while the extent of the LSB reduces for $\eta \gt 0.02$ , it is compensated by a growing separated region that originates from the trailing edge with a positive lift contribution (as revealed by volumetric lift elements, which dominate the lift fluctuations in this case as well, as shown in the top left plot in figure 17).

After impingement, at $t=t_1$ , the separated region that originated from the trailing edge collapses (although the positive lift element prevails at the trailing edge), and the LSB moves downstream along the upper surface. Interestingly, the extent of this separated region decreases with respect to $\eta =0$ for all cambered airfoils. We report the formation of a LEV post-impingement that becomes weaker as the magnitude of the camber decreases, signalling a strong accumulation of vorticity at the leading edge pre-impingement for airfoils with high camber. We also observe boundary-layer thickening on the pressure side that becomes more concentrated at the trailing edge with increasing $\eta$ .

Figure 17.

Evolution of the volumetric, surface and total lift (a) and drag (b) force elements during a gust–airfoil interaction with a gust of $(G,R_v,y_0,\alpha )=(2,0.25,-0.1,5^\circ )$ for different cambered airfoils. Instantaneous lift (first two columns) and drag (third and fourth columns) elements at $t=t_0$ .

Figure 17. Long description

Panel A: The first graph is a line graph showing the evolution of lift force elements over time for different cambered airfoils. The x-axis represents time (t) and the y-axis represents the lift force (f_L(t)). Three lines represent different airfoil types: NACA 2406, NACA 4406, and NACA 6406. The graph shows a peak in lift force around t = 0 and variations in the lift force over time for each airfoil type. Panel B: The second graph is a line graph showing the evolution of drag force elements over time for the same airfoil types. The x-axis represents time (t) and the y-axis represents the drag force (f_D(t)). Three lines represent different components of the drag force: f_D,S + f_D,V, f_D,S, and f_D,V. The graph shows fluctuations in drag force over time for each component. Panel C: The third set of visuals shows instantaneous lift and drag elements at two different time points (t = t_0 and t = t_2) for the three airfoil types (NACA 2406, NACA 4406, and NACA 6406). Each visual depicts the airfoil and the surrounding flow field, with color gradients indicating the magnitude of the force elements.

The volumetric (pressure) component of the drag response is predominant in this case as well, as revealed by the top-right plot in figure 17. Specifically, the most cambered airfoil exhibits a more pronounced (negative) pressure drag contribution near the leading edge, which explains the magnitude of the initial peak at $t=t_0$ . Nonetheless, this trend reverses at $t=t_2$ , where the second peak is mitigated in the high-camber case by a positive drag contribution at the trailing edge. While we observe a similar negative drag element near the leading edge for all three airfoils, the counteracting effect of the positive element at the trailing edge increases for highly cambered airfoils, suppressing the magnitude of the second drag peak.

5.1. Combined effect of angle of attack and airfoil thickness

Several factors can trigger the development of a wide separated region that originates from the leading edge and grows upstream along the upper surface. Prominent examples of this behaviour include vortex gusts of large size ( $R_v$ ), high angles of attack ( $\alpha$ ) and thick airfoils. The combined influence of thickness and angle of attack on this separated region, previously characterised under steady conditions in § 2.4, is illustrated in figure 18. For the thinnest profile, that is $\tau =0.06$ , we observe a separated region at $\alpha =5^\circ$ corresponding to an LSB that originated near the leading edge before impingement. At higher angles of attack, the extent of the LSB increases, and at $\alpha =20^\circ$ we observe the overlap of a secondary separated region near the trailing edge, and the LSB, indicating stall-like conditions. Meanwhile, for the thickest profile, that is a $\tau =0.40$ , we observe a separated region emerging from the trailing edge at $\alpha =5^\circ$ whose extent also increases at higher incidences.

Figure 18.

Kinetic energy ( $k$ ) and velocity (streamlines) fields at $t=t_1$ with a gust of $(G,R_v)=(2,0.25)$ for a set of symmetrical airfoils of various thicknesses $\tau$ at different angles of attack $\alpha$ .

In general, thick airfoils experience reduced (or mild) lift fluctuations during a gust encounter due to the suppression of the vorticity production levels at the leading edge (see figure 14 b). Thin airfoils, on the other hand, experience a larger lift fluctuation because the high curvature at the leading edge promotes high vorticity production levels. Moreover, a linear relationship was found between the magnitude of the lift fluctuation and airfoil thickness in § 4.5.1 for $(G,\alpha )=(2,5^\circ )$ . Nonetheless, this relationship persists at different angles of attack and gust ratios, as indicated in figure 19. On the other hand, we observe a decrease in the magnitude of the lift fluctuations after $\alpha =10^\circ$ for all airfoils, signalling the occurrence of dynamic stall at higher incidences.

Figure 19.

Magnitude of the lift fluctuation $|\Delta C_L|$ with a gust of $(G,R_v,y_0)=(2,0.25,-0.1)$ against $\alpha$ and $\tau$ .

5.2. Combined effect of angle of attack, gust initial position and gust ratio

This section examines the influence of the angle of attack on the trends observed in § 4.3 for various $y_0$ . At low incidences, we observed the development of an LSB for $y_0\gt -0.25$ whose extent increased as the core of the vortex gust is initialised closer to the leading edge (i.e. small $|y_0|$ ). As $y_0$ becomes increasingly positive, the location of this separated region is found further aft on the upper surface. We identified an instance in which the LSB extended up to the trailing edge post-impingement for $y_0=+0.25$ .

Figure 20.

Kinetic energy ( $k$ ) and velocity (streamlines) fields at $t=t_1$ at different angles of attack $\alpha$ and vertical offsets $y_0$ with a gust of $(G,R_v)=(2,0.25)$ for a NACA 0018 airfoil.

The influence of both $\alpha$ and $y_0$ on the extent and location of the separated region at $t=t_1$ is presented in figure 20. For all $y_0$ , we observe a consistent increase in the size of the separated region in configurations with increasing angles of attack. This increase, however, is significantly less dramatic for $y_0=0.25$ . Notably, the sharpest increase is observed for $y_0=-0.25$ , for which the recirculation region grows from $l=0$ at $\alpha =0^\circ$ to a full separation $l\approx 1$ at $\alpha =20^\circ$ . Moreover, at $\alpha =20^\circ$ , we observe a large separated region for all $y_0$ that originates at the trailing edge and expands upstream. The extent of this region, however, is the largest for $y_0=-0.25$ , and progressively decreases as $y_0$ becomes more positive.

Figure 21.

Magnitude of the lift fluctuation against $y_0$ at different angles of attack for $G=\{-2,2 \}$ with $R_v=0.25$ for a NACA 0018.

The magnitude of the lift fluctuation $|\Delta C_L(t)|$ at different angles of attack against $y_0$ is shown in figure 21. Notably, the curves become increasingly symmetric with respect to $y_0=0$ in configurations at higher angles of attack and positive gust ratios. Meanwhile, we observe the opposite trend for $G\lt 0$ : the curve becomes more asymmetric at high incidences. In fact, this asymmetric behaviour has been previously reported in Horner et al. (Reference Horner, Saliveros, Kokkalis and Galbraith1993) and Peng & Gregory (Reference Peng and Gregory2017) for $\alpha =0^\circ$ . Furthermore, these trends can be mirrored to negative angles of attack, although not shown here, such that the curve for $G=-2$ becomes symmetric about $y_0=0$ for $\alpha =-20^\circ$ , and the curve for $G=2$ becomes a mirror against $y_0=0$ of the curve for $G=-2$ at $\alpha =20^\circ$ . Notably, these trends hold for the present choice of $|G|$ , and would change for other gust ratios and/or parameter combinations.