To ensure methodological consistency, the geometry of the TU Delft V3 kite was adopted from previous CFD studies  and wind tunnel experiments with a rigid scale model .

Representative airfoils were extracted as chordwise slices through the wing (see Fig.  in Appendix ). The discrete spanwise locations lie on the spanwise reference line defined by connecting the quarter-chord points, obtained at each station as 1xc/4=xLE+0.25(xTE-xLE). At each location, the section plane contains the global x direction and is oriented in such a way that its normal is perpendicular to the local surface normal at xc/4.

A formulation with planes aligned with the global x axis was selected rather than one oriented locally normal to the quarter-chord line. Although both formulations are equivalent in the linear lift regime, the latter introduces additional geometric non-linearities once Cl(α) departs from linear behaviour, which may bias the predicted load distribution .

The wing is discretized into np panels, each constituting a fundamental element of the solver. The number of panels is not required to coincide with the number of chordwise slices used to define the geometry. For instance, when adopting a uniform panel distribution, both geometric properties (leading-edge (LE) and trailing-edge (TE) coordinates) and aerodynamic properties (2D polars) defined at the chordwise slices may be interpolated to an arbitrary value of np.

Because the panels are rectangular, they do not form a geometrically continuous surface representation. Instead, the panel system serves to define the bound and trailing vortex filaments, rather than to reproduce the exact wing geometry.

Following Kelvin’s circulation theorem and Helmholtz’s vortex theorems, all vortex elements must form a closed system; vortices cannot spontaneously appear or vanish within the fluid . Accordingly, the bound vortex filament, placed at the quarter-chord line of each panel, is connected to a trailing vortex filament that extends in the chordwise direction to the TE (see Fig. ). At the TE, these are attached to wake filaments, which extend to infinity in the freestream direction u∞, in accordance with the frozen wake assumption . Aligning the wake with the undisturbed free stream maintains consistency with the linearized LL formulation . Allowing the wake direction to depend on the locally induced velocity would introduce geometric non-linearity and require a free-wake treatment; and for strongly swept or highly anhedral configurations, spanwise-varying wake directions may additionally lead to filament convergence or intersection in a straight, semi-infinite filament model.

At infinity, the wake filaments are closed by an additional filament, thereby completing the vortex system. This closing filament is not explicitly modelled, as its aerodynamic influence at infinite distance is assumed to be negligible. The resulting vortex structure is known as a horseshoe vortex. Helmholtz’s theorems further require that the circulation strength remains constant along the length of each vortex filament ; therefore, each panel has a single circulation value.

The induced velocity vector uind at an evaluation point x due to a finite, 3D-bound vortex filament or chord-aligned trailing vortex filament is computed using the Biot–Savart law : 2uind=Γ4πr1×r2‖r1×r2‖2r0⋅r1‖r1‖-r2‖r2‖, where Γ is the circulation strength, r1 and r2 are vectors from the filament endpoints to x, and r0 is the vector along the filament segment.

For the wake filaments, which are modelled as semi-infinite vortex filaments, the induced velocity at x is given by : 3uind,wake=Γ4π1+r1⋅u^∞‖r1‖r1×u^∞‖r1×u^∞‖2, where r1 is the vector from the filament origin to the evaluation point, and u^∞=u∞/‖u∞‖ is the wake-direction unit vector.

A vortex core model was employed to avoid singular behaviour by reducing the induced velocity at points located within a specified core radius, ϵ. For bound vortex filaments, the core radius is defined as a fixed fraction of the filament length; here, 5 % was adopted . For trailing and semi-infinite wake filaments, a viscous core model is adopted , with the core radius given by 4ϵ=2α0νℓ‖u∞‖, where α0= 1.25643 is the Oseen parameter, ν the kinematic viscosity, and ℓ the relevant filament length.

For bound and finite trailing filaments, core regularization is applied for d⟂<ϵ by evaluating the Biot–Savart velocity at the radially projected point xproj on the core boundary and scaling linearly: 5uind,core(x)=d⟂ϵuindxproj,d⟂<ϵ. The perpendicular distance to the filament centreline is 6d⟂=r1×r0r0, and xproj is constructed by keeping the axial projection of x onto the filament and setting its radial distance to ϵ.

For semi-infinite wake filaments, ϵ represents a viscous core of the far wake and is used only to desingularize the Biot–Savart kernel near the centreline. The corresponding perpendicular distance is 7d⟂,∞=r1×u^∞, and for d⟂,∞<ϵ, the induced velocity is evaluated at the projected point xproj without additional d⟂,∞/ϵ scaling: 8uind,core(x)=uindxproj,d⟂,∞<ϵ.

To reduce computational cost during the circulation-matching iterations, the aerodynamic influence coefficients were precomputed once. Let A∈R3np×np denote the block aerodynamic influence coefficient (AIC) matrix that maps the spanwise circulation vector Γ∈Rnp to the induced velocity at each control point. For panel i=1,…,np, the induced velocity is given by 9uind,i=∑j=1npAijΓj, where Aij∈R3×1 denotes the (i,j) block of A and represents the induced velocity at control point i due to a unit circulation on panel j.

To ensure compatibility with non-linear airfoil polars, the near-field 2D self-induced contribution of panel i is subtracted at its own control point to avoid double-counting induction effects already embedded in the sectional polar data . Define the per-unit-circulation 2D self-induced influence vector at control point i as 10aind,2D,i=12πy^p,i×xi-x0,i‖xi-x0,i‖2, where y^p,i is the local spanwise unit vector, and xi-x0,i is the displacement from the bound-vortex centre of panel i to its control point. For a given circulation Γi, the corresponding 2D self-induced velocity is 11uind,2D,i=Γiaind,2D,i. The effective induced velocity used for circulation matching is then 12ueff,i=uind,i-uind,2D,i. This subtraction is applied only to the self-induced contribution of panel i and not to the cross-induced contributions from other panels.

With the AIC matrix assembled, the spanwise circulation distribution was determined iteratively. At iteration k, the induced velocity vectors at all control points were computed collectively as 13vecUind(k)=AΓ(k), where vec(⋅) denotes the stacking of the three velocity components over all control points, and Uind(k)∈Rnp×3 is the corresponding reshaped matrix whose ith row contains uind,i(k).

The effective relative velocity at the control point of panel i was then obtained by adding the local freestream velocity u∞,i to the induced velocity: 14urel,i(k)=u∞,i+uind,i(k). For each panel i, the sectional lift coefficient Cl,i(k) was obtained from the 2D polar using the effective angle of attack αeff,i(k): 15αeff,i(k)=arctan⁡urel,i(k)⋅z^p,iurel,i(k)⋅x^p,i, where x^p,i, y^p,i, and z^p,i denote the local chordwise (LE-to-TE), spanwise, and panel-normal (upward) unit vectors, respectively.

The circulation at each panel was updated using the Kutta–Joukowski theorem, normalized by the local reference velocity magnitude: 16Γnew,i(k)=12‖urel,i(k)×y^p,i‖2‖u∞,i×y^p,i‖ciCl,i(k), where ci denotes the local chord length.

To ensure numerical stability, under-relaxation was applied globally to the circulation array: 17Γ(k+1)=(1-frelax)Γ(k)+frelaxΓnew(k). The convergence metric δΓ was defined as the normalized maximum change in the global circulation distribution between successive iterations: 18δΓ=max⁡Γ(k+1)-Γ(k)max⁡Γ(k+1).

Once convergence was reached, the local aerodynamic force on each panel was computed as 19Fi=12ρurel,i2Cl,in^i+Cd,it^iciwi, where ρ is the fluid density, wi is the panel width, and Cl,i and Cd,i are the sectional lift and drag coefficients, respectively, evaluated at the three-quarter-chord control point. In other words, the local angle of attack evaluated at the three-quarter chord was used to obtain the aerodynamic coefficients from the 2D input data. The magnitude urel,i corresponds to the in-plane component of the relative velocity vector urel,i: 20urel,i=urel,i×y^p,i.

The force direction was set by an in-plane effective-flow direction following , , and . This differs from the approach suggested by and implemented by ; the choice is motivated by comparing simulations of two geometries with wind tunnel measurements, detailed in Appendix .

For each panel i=1,…,np, the in-plane drag-direction unit vector is defined as 21t^i=cos⁡αeff,ix^p,i+sin⁡αeff,iz^p,icos⁡αeff,ix^p,i+sin⁡αeff,iz^p,i.

The lift was oriented in the normal direction n^i, which was constructed as 22n^i=t^i×y^p,i‖t^i×y^p,i‖.

The local aerodynamic moment on panel i was defined as 23Mi=12ρurel,i2Cm,iy^p,iwici2, where Cm,i is the sectional moment coefficient; this moment represents the pitching moment about the quarter-chord point of the panel, consistent with the definition of Cm,i.

The contributions of all panels were summed and projected onto both the global coordinate directions e^x, e^y and e^z, and the body-fixed aerodynamic directions: drag e^D, aligned with u∞; lift e^L, perpendicular to drag; and side e^S, orthogonal to both drag and lift. With U∞=‖u∞‖, the resulting non-dimensional force and moment coefficients are Cj=2ρU∞2S∑i=1npFi⋅e^j,24for e^j∈{e^x,e^y,e^z,e^D,e^L,e^S},CM,j=2ρU∞2Scref∑i=1npMi+rc/4,i-r0×Fi⋅e^j,25for e^j∈{e^x,e^y,e^z}, where S is the projected planform area, cref the reference chord length, np the number of panels, rc/4,i the quarter-chord location about which the moment is defined for each panel, r0 the chosen global reference point, and e^j the projection direction. The centre of pressure rcp was determined by 26rcp=r0+F×M0‖F‖2, where F is the combined aerodynamic force vector of the wing, M0 is the combined moment vector about the reference point r0, and ‖F‖ denotes the Euclidean norm of F. The resulting vector rcp lies along the line of action of the aerodynamic force and represents the unique point where this line intersects the wing surface. This intersection defines the effective location at which the resultant force can be considered to act, and it provides a critical indicator of aerodynamic stability, especially in relation to the trim state of the kite.

The parameterization presented here extends the earlier approach by , who introduced two fundamental parameters: the circular LE tube diameter t and the camber height κ, shown in Fig. . Subsequently, informed by the experimental findings of , who observed a forward shift of the maximum camber position with increasing angle of attack, the chordwise location of maximum camber η was incorporated as an essential design parameter . Another step was to reduce the trailing-edge height from t/2 to zero, in accordance with the conventional airfoil definition . A TE reflex angle δ is introduced, as this is often used to enhance the static pitching moment Cm and reduce the risk of front stall.

The LEI kite profile is constructed by defining a circular LE tube, smoothly extending it towards the maximum camber location using a Bézier curve named the front spline, and subsequently connecting it to the TE via another Bézier curve called the rear spline (see Fig. ). Tangent continuity is enforced at both the junction between the curves and at the TE, ensuring aerodynamic smoothness. In this way, the profile shape can be characterized through six design parameters, summarized in Table .

Each Bézier curve is defined by four control points [P0,P1,P2,P3], with the parametric coordinate tb∈[0,1] controlling the interpolation: 27xy=1-tb3P0+31-tb2tbP1+31-tbtb2P2+tb3P3. The front spline connects the seam to the maximum camber point. The seam, i.e. the stitched junction at which the canopy is attached to the LE tube, is located on the tube circumference at the seam angle. Because a tensioned membrane cannot sustain a bending moment, the canopy must depart the tube surface tangentially; tangent continuity is therefore enforced at the seam. The seam angle is determined by a dynamic search algorithm that increases the angle stepwise until the apex of the resulting front spline coincides with the prescribed maximum camber point, thereby selecting the smallest seam angle for which a tangent departure is compatible with the prescribed camber. This generalizes earlier fixed seam-angle definitions , which can produce invalid geometries for weakly cambered profiles. On the physical kite, the seam additionally acts as the roughness element that trips the boundary layer, supporting the fully turbulent modelling assumption discussed in Appendix .

The intermediate control points of the front spline are positioned along the tangent lines emanating from the seam and the maximum camber point, with the LE curvature parameter ϕ specifying their relative distance along these tangents. In this way, ϕ shapes the curvature distribution: larger values concentrate curvature near the centre of the curve, whereas smaller values shift it towards the endpoints. As surface curvature sets the local pressure gradient, ϕ effectively redistributes the suction along the front spline. Within the physically consistent range, however, this redistribution only slightly alters the pressure distribution and has a negligible effect on the integral coefficients , as the dominant flow features of LEI airfoils (the suction peak on the LE tube and the pressure-side recirculation) are governed primarily by t, κ, and η. For this reason, ϕ is held fixed during dataset generation (Sect. ).

The rear spline connects the maximum camber location to the TE. The intermediate control points are governed by the camber tension parameter λ, which adjusts profile smoothness, and by the TE reflex angle δ. To enhance model robustness and reduce the number of free parameters, the third control point is positioned at 20 % of the distance from the second rear control point to the TE, measured from the second rear control point.

To ensure a smooth computational mesh, the LEI kite profile geometry requires specific adaptations. In particular, the sharp corner at the junction between the lower canopy and the LE tube is smoothed using a fillet. In addition, a canopy thickness tcanopy is introduced to represent a realistic finite thickness, to distinguish top from bottom airflow, and to enable a rounded TE. The optimal fillet size and tcanopy were determined through a sensitivity study. evaluated seven leading-edge fillet sizes on two reference profiles, spanning enclosed cross-sectional areas of 0.13–45.52 cm² (conventional) and 4.22–137.12 cm² (highly cambered) for c= 1 m. The smallest numerically robust configurations, corresponding to enclosed areas of 0.98 and 4.22 cm², informed a fixed fillet construction rule that scales with the tube diameter and seam angle, and is applied uniformly to all geometries in the dataset.

A structured O-grid mesh with approximately 1.15 × 10⁵ cells, shown in Fig. , was generated for each profile using a hyperbolic-Laplace solver in Pointwise , automated via a Python script.

An O-grid topology was selected primarily for its insensitivity to angle-of-attack variations, which reduces the effort required for case setup, as only one mesh needs to be generated per profile. As the chord is normalized to unit length, the 2D mesh is extruded by a single cell over one chord length in the spanwise direction, as required by OpenFOAM's 3D formulation. In the wall-normal direction, the 201 cell layers are generated by hyperbolic extrusion with a fixed growth ratio of 1.1 following , starting from a first-layer height of 7.8 × 10⁻⁶ c, chosen to ensure ymax+< 1 over the entire airfoil at Re = 5 ×10⁶. This places the circular far-field boundary at approximately 1.4 × 10⁴ chord lengths from the profile, sufficiently far for the freestream boundary conditions not to influence the computed force coefficients . Relative to the next-finer level, the selected production resolution of 575 × 201 cells changed the lift, drag, and moment coefficients by less than 2 % across the operational angles of attack, which is taken as the remaining discretization error.

For all cases, smoothing parameters for hyperbolic extrusion in Pointwise were applied uniformly to ensure mesh quality and numerical stability. The explicit smoothing parameter was set to 8 for transverse smoothing, while the implicit parameter was set to 16.0. Kinsey–Barth smoothing, with a value of 5, was used in the wall-normal direction to address regions of strong concavity. The volume smoothing parameter was set to its default value of 0.5.

Node spacing was uniform near the LE and TE, while a hyperbolic tangent distribution was applied along the canopy surface to minimize abrupt grid transitions. Mesh independence was assessed by , who independently varied the tangential (275–775 nodes) and wall-normal (161–221 layers) resolution of the structured O-grid for representative conventional and highly cambered profiles.

The non-dimensional wall distance y+ is defined as 28y+=ρ∞yuτμ∞, where ρ∞ denotes freestream density, μ∞ the dynamic viscosity, and uτ the friction velocity.

By assuming steady-state and incompressible flow, the steady RANS equations were solved using OpenFOAMv2006, with pressure-velocity coupling managed internally by the semi-implicit method for the pressure-linked equations (SIMPLE) algorithm . For an overview of the numerical schemes, the reader is referred to . The revised k-ω SST model by was employed due to its robustness in capturing both near-wall and outer-region flow physics, supported by prior implementations in aerodynamic studies of LEI kites . All simulations were performed under the assumption of fully turbulent flow, owing to the measured turbulence intensities, the surface roughness present on both suction and pressure sides, and the high Reynolds numbers; a more elaborate discussion of this modelling choice is provided in Appendix .

The inlet velocity scalar u∞, set to 1 m s⁻¹, was projected in the direction of the angle of attack α to define the inlet velocity vector 29ui=u∞cos⁡αsin⁡α0. The turbulent kinetic energy ki is defined as 30ki=32u∞I2, where I represents the freestream turbulence intensity. The specific dissipation rate ωi at the inlet was defined as 31ωi=kiν∞νtν∞-1, where νt/ν∞ represents the eddy viscosity ratio. Values of I= 2 % and νt/ν∞= 10 were selected based on the sensitivity studies of and .

Boundary conditions for velocity u, kinematic pressure pk, turbulent kinetic energy k, specific dissipation rate ω, and eddy viscosity νt were applied as summarized in Table . Far-field boundaries employed non-reflecting types; the airfoil surface was defined as a stationary, impermeable no-slip wall with appropriate wall functions for turbulence; and spanwise boundaries were set to empty.

Convergence was monitored through both equation residuals and aerodynamic force coefficients. Specifically, residuals associated with momentum, pressure, and turbulence variables ux, uy, p, k, and ω were required to fall below 10-6 , while the force coefficients Cl, Cd, and Cm were expected to converge to within 10-4.

In OpenFOAM, each outer iteration reports two residuals per variable: the initial residual, evaluated on the newly assembled equation before the inner linear solve and thus measuring how well the previous iterate satisfies it; and the final residual, which only reflects the inner linear-solver settings. Outer-loop convergence is therefore monitored through the initial residual, on which the automatic stopping criterion acts. Because this criterion considers residuals only, a simulation could terminate before the force coefficients satisfy the 10-4 criterion. To prevent such early termination and ensure consistency across all cases, the stopping threshold was set slightly below the nominal residual criterion, at 8 × 10⁻⁷ for all flow variables. Each simulation was limited to a maximum of 5000 iterations. Cases not reaching the required thresholds within this limit were classified as non-converged; these typically occurred at high or negative angles of attack, where steady RANS solutions become unreliable owing to increased unsteadiness, e.g. flow separation.

For improved numerical stability, the velocity relaxation factor was set to 0.8, which dampened residual oscillations and improved convergence consistency. Representative histories for a typical LEI profile at α= 6° are shown in Fig. , where final flow residuals fell below 10-6 and force residuals just below 10-4.

Training data for the ML model were generated by varying the profile parameters, as shown in Table , over a range that encompasses physically viable profile designs. As varying ϕ within the marginal physically consistent range was found to have minimal aerodynamic influence, it was fixed at ϕ= 0.65, allowing greater resolution of the remaining parameters. The angle-of-attack range and sampling density were chosen to optimize accuracy in the lift polar and to maintain computational efficiency, with fewer points in the linear region (2° ≤α≤ 10°) and negative α, and a finer step of 2° near stall. With ϕ held fixed, the sampled input space is six-dimensional (five varied-shape parameters and the angle of attack), analysed separately for three Reynolds numbers: Re = 1 × 10⁶, representative of small kites or static flight; Re = 5 × 10⁶, corresponding to typical in-flight operation; and Re = 2 × 10⁷, characteristic of larger kites and/or higher inflow velocities .

Sampling Re at discrete levels is consistent with the fully turbulent modelling assumption (Appendix ). With transition fixed by the seam roughness elements and the elevated freestream turbulence encountered in operation, the regime changes associated with free transition are excluded. The force coefficients then vary weakly, smoothly, and monotonically with Re between the sampled levels, in line with turbulent skin-friction scaling, as quantified in Appendix .

This is corroborated numerically by the 3D RANS results of , which showed a negligible effect on the integral forces over a factor-two change in Re. It was also corroborated experimentally by the wind tunnel measurements presented in the companion paper , in which the force coefficients varied only gradually and without abrupt regime changes across Reynolds numbers from 1.3 × 10⁵ to 6.1 × 10⁵. Only the lowest measured Reynolds number indicated an earlier stall onset, consistent with transitional effects at Reynolds numbers well below the fully turbulent regime considered here.

In preprocessing, the profiles were filtered out if the camber tension control point λ was positioned too far aft, i.e. η+λ≥ 0.8, or if the camber height κ was excessively large, i.e. κ> 2.5t. After filtering, 3464 configurations remained per angle of attack, resulting in 45 032 simulations per Reynolds number.

Case generation was fully automated: for each profile-parameter combination in the design space, the scripts executed geometry construction, mesh generation, and case setup for the required angles of attack and Reynolds numbers without manual intervention. The resulting RANS CFD simulations were performed on the HPC-12 cluster at Delft University of Technology, with each case decomposed over four cores, requiring a total of 1.4 × 10⁵ core hours, corresponding to an average of approximately 1 core hour per simulation. A limited number of cases did not converge within 5000 iterations, predominantly at high angles of attack beyond stall, and were subsequently removed from the data.

Additionally, a z-score filtering approach , 32z=x-μσ, where μ represents the mean and σ the standard deviation, was used to identify and remove statistical outliers. Specifically, for each α and force coefficients, any data point with an absolute z score greater than 3 standard deviations from the mean was classified as an outlier and removed, illustrated in Fig. .

Profiles with a maximum camber smaller than tube radius, i.e. κ<t2, are defined as camber-limited profiles and represent 0.23 % of the training dataset. For these cases, the number of parameters affecting the front and rear splines was reduced to t and δ, as the removed parameters had little to no effect. The camber-limited profiles exhibit distinct aerodynamic characteristics, producing lower lift and stalling earlier at approximately α= 10°. Consequently, they exhibit relatively high drag coefficients post-stall and increased pitching moments, indicating a stronger nose-up tendency in pre-stall flight conditions.

Building a predictive model within this design space requires the definition of both training and test datasets. A separate test dataset was constructed at different angles of attack from those used in the training set to verify the predictions at angles on which the model was not trained, shown in Table . After removing non-converged cases and statistical outliers, as summarized in Table , the final dataset comprises 42 183 training samples and 5016 test samples, corresponding to a test–train split of 10.6 %.

The test dataset exhibited a higher overall convergence rate, which was attributed to its narrower angle-of-attack bounds, making convergence less prone to failure near the limits, i.e. at negative α or under stall conditions.

A separate ML model was trained for each Re to predict Cl, Cd, and Cm from any given set of the design parameters and the angle of attack, resulting in a model with a 7D input space (six shape parameters and α) and a 3D output space. Because α enters as a continuous input feature, the model can be evaluated at any requested angle of attack, not only at the values contained in the training data.

By contrast, Re was deliberately not included as a model input: with only three sampled levels, a tree-based regressor would represent the Re-dependence as piecewise-constant steps between the training values. Instead, a separate model was trained per Re, and for intermediate Reynolds numbers the sectional polars are obtained by interpolating between the bracketing models in log⁡ Re, which yields a smooth and monotonic Re-dependence. The resulting surrogate model is applicable within the trained range Re ∈[1×106,2×107] under the fully turbulent assumption (Appendix ); transitional flow at substantially lower chord-based Reynolds numbers falls outside its scope.

The quantities Cl, Cd, and Cm were selected for the output because they represent key aerodynamic metrics and allow for the determination of the chordwise centre of pressure xcp, which is a crucial stability indicator. The centre of pressure is determined through 33xcp=xref-CmcCl2+Cd2sin⁡tan⁡-1ClCd+α, which yielded values within 1 % of those obtained by pressure integration .

Model development began with standardizing the filtered training datasets, after which several regression algorithms were evaluated using their default settings . The selected models were chosen for their strong baseline performance reported by , and linear regression was included for reference. The full ML workflow is described in Appendix  and illustrated in Fig. .

After an initial model run with default settings, hyperparameters were systematically fine-tuned using grid-search cross-validation. In this context, validation refers to the internal process of developing the regression model across different hyperparameter combinations using subsets of the training data before evaluating the final model's performance on an independent test dataset. The coefficient of determination R2 , defined as 34R2=1-∑i=1nyi-y^i2∑i=1nyi-y‾2, was selected as the scoring metric, where yi denotes the true value for each data point, y^i the corresponding model prediction, y‾ the mean value of the observed data, and n the number of points in the dataset. The hyperparameter search utilized a k-fold cross-validation approach, dividing the training set into k= 10 subsets or folds, with 90 % of the data used for training and 10 % for validation in each iteration. This was feasible because the dataset was sufficiently large to provide ample training data for improving model accuracy, measured here by R2, while still reserving a large enough test set to ensure reliability, as quantified by a low standard deviation.

Tree-based regressors yield piecewise-constant responses, so although the model returns a value for any requested α, the raw prediction varies in fine steps rather than continuously. Where a continuously differentiable polar is required, e.g. for derivative-based analyses or within iterative solvers, the model output is sampled at the training angles of attack. These samples are then fitted with a piecewise cubic Hermite interpolating polynomial (PCHIP) spline, whose shape-preserving, monotone formulation guarantees C1 continuity without introducing spurious oscillations. For a representative profile, raw predictions queried at arbitrary α deviate from the PCHIP-fitted polar by less than 0.03 in Cl (RMS 0.006), well below the regression error itself, confirming that the spline is a smoothness refinement rather than a required postprocessing step.

After tuning was finalized, the performance of each model was assessed on the test dataset using three metrics: R2, the percentage root mean square error 35RMSE%=1n∑i=1nyi-y^i2〈|ytest|〉×100, and the percentage standard deviation 36σ%=1n-1∑i=1nri-r¯2〈|ytest|〉×100, where n is the number of test samples, yi and y^i are the true and predicted values, 〈|ytest|〉 is the mean absolute value of the test targets, and ri and r¯ are the residuals and their mean, respectively.

Table  shows the averaged results per model, from which extremely randomized trees (ERTs) were selected as they outperform the other algorithms, achieving the highest R2, and lowest RMSE% and σ%.

As shown in Table , the ERT exhibited high predictive accuracy with R2> 0.98 across all Re. The standard deviation σ% decreased with increasing Re, indicating greater prediction consistency and improved alignment with CFD predictions at higher Re.

To further evaluate the model's performance, three distinct and representative profiles were selected from the test dataset, detailed in Table . The third profile is notable as it represents a camber-limited configuration. The prediction metrics for these profiles are summarized in Table  and illustrated in Fig.  for Cl, Cd, and Cm.

In the plots, the ML model is evaluated at the angles of attack present in the training data, and the PCHIP spline introduced in Sect.  provides a C1-continuous polar through these predictions, from which the coefficients are interpolated at the test angles of attack for comparison with the true test data.

Overall, profile 2, with t= 0.086 close to the LE tube thickness of the mid-span profile of the V3 kite, exhibited the highest R2 values and the lowest RMSE% values, indicating the greatest predictive certainty. No consistent trend was observed in a specific force coefficient; for example, R2 and RMSE% were highest either for Cd or for Cm. In general, a lower R2 corresponded to a higher RMSE, with the RMSE% showing larger absolute changes when mispredictions occurred. For instance, in the drag polar of profile 2, two clear mispredictions increased the RMSE% by a factor of 2 compared with Cl, where only a single misprediction was present. The drag polars of profiles 1 and 3 revealed a similar pattern: in profile 1, one large misprediction resulted in an RMSE% approximately four times greater.

The flow topology at α= 2° and α= 8° is shown in Fig. , indicating the pressure-coloured streamlines around the V3 kite mid-span airfoil and the corresponding surface distributions of Cp and Cf. From the pressure-side streamlines, it is evident that a recirculation region is present, splitting into two zones. The first is a corner eddy close to the tube; the second, larger zone is more sensitive to inflow angle variations, decreasing in size as α increases. The rotation direction of the pressure-side recirculation zone can be inferred from the friction coefficient; a negative friction coefficient indicates counterclockwise rotation.

At α= 8°, the Cp indicates a comparatively larger negative pressure peak at the suction side and shows a distinct upward pressure peak at the centre of the recirculation zone. Where the recirculation zone stretches to the airfoil's surface, the friction coefficient Cf displays a peak in the opposite (downwards) direction. Increasing α from 2 to 8° leads to a more concentrated and more negative peak. Furthermore, double values are plotted for both Cp and Cf in the region where, along a vertical line, one encounters three pressure-side surfaces, a distinct feature of an LEI airfoil.

To assess the feasibility of inviscid methods for generating aerodynamic polars, NeuralFoil, a physics-informed surrogate model trained on tens of millions of XFOIL simulations and analytical solutions, was utilized .

Although XFOIL, on which NeuralFoil was trained, is not a purely inviscid method but a viscous–inviscid interaction solver, its integral boundary layer formulation assumes a thin shear layer coupled to an outer potential flow. Once separation becomes extensive and the flow no longer resembles a thin boundary layer attached to the surface, the mathematical foundation of the model is violated, and the predicted lift, drag, and pressure distributions can no longer be considered physically reliable . As LEI airfoils exhibit persistent pressure-side separation and recirculation that contravene these assumptions, the inclusion of NeuralFoil permits a structured assessment of the limitations of such viscous–inviscid approaches when applied to geometries dominated by extensive separated flow.

In this study, the largest model size, xxxlarge, was used; it achieves maximum errors of less than 0.3 % in the lift, drag, and moment coefficients relative to XFOIL . The boundary layer transition was set to x/c= 0 to mimic the fully turbulent assumption used in the deployed 2D RANS CFD simulations.

To put the LEI airfoil aerodynamic trends into perspective, the lift and drag wind tunnel measurements of a flat plate were used . Because moment measurements were not available, the moment polar was computed following classical flat-plate airfoil theory: 37Cm=-π4sin⁡2α, where α is in radians.

Figure  plots the lift, drag, and moment coefficients as a function of α for the mid-span airfoil of the V3 kite (shown in Appendix ), comparing classical flat-plate theory, flat-plate wind tunnel measurements, Breukels' regression model, NeuralFoil, 2D RANS CFD results, and wind tunnel measurements .

The predictions generated by Breukels’ regression model were considered invalid outside the range ‖α‖≤ 20°. At these higher angles of attack, the extrapolated polars exhibited unphysical behaviour, including lift and drag discontinuities near α=±20° and negative drag coefficients for α<-20°, which were attributed to insufficient smoothing between the empirical expressions used to extrapolate the model beyond the CFD-simulated range. Furthermore, assuming symmetry and antisymmetry about α= 0° was considered inappropriate due to the airfoil's inherent geometric asymmetry.

In terms of lift, the NeuralFoil predictions align well with wind tunnel measurements and 2D CFD predictions up to approximately α= 13°. Beyond this range, however, NeuralFoil overpredicts lift and delays stall. For drag, between α=-5 and 10°, the NeuralFoil and Breukels' predictions exhibit similar trends to those of CFD simulations and wind tunnel measurements but are off in magnitude. At α> 20°, the Cd predictions of NeuralFoil and the CFD approach the flat-plate wind tunnel measurement in slope and magnitude.

Among all numerical approaches, 2D CFD simulations exhibit the closest agreement with wind tunnel measurements. Within the linear regime, from approximately α=-1° to α= 10°, the average percentage errors are 4 % and 16 % for lift and drag, respectively. The largest discrepancies occur outside the linear regime, in the negative-lift regime and in the prediction of the stall angle, where 2D CFD indicates a slightly delayed stall onset.

The largest discrepancies among the model predictions arise in the moment coefficient, for which wind tunnel measurements were unavailable for both the LEI airfoil and the flat plate. Breukels' model yields the greatest deviation from all other approaches. From α=-3 to 3° or so, the CFD results closely follow the flat-plate prediction for Cm. Beginning around α= 3°, the CFD-predicted Cm diverges from the flat-plate trend and instead flattens, maintaining an almost constant negative value from α= 5° to α= 12°. Beyond this point, a sharp decrease in Cm is observed, coinciding with the onset of stall. NeuralFoil is unable to capture this behaviour, and also deviates strongly with negative angles of attack.

The VSM permits arbitrary spanwise panel distributions; for instance, the area between existing strut locations may be divided into equal parts, arranged with a uniform distribution or spaced according to a cosine-based scheme – commonly used in LL models to concentrate panels near the tips . However, for the V3 kite, the cosine distribution proved ineffective and failed to converge when the number of panels np exceeded 50, whereas a uniform panel distribution yielded the most robust and reliable convergence as np increased.

Within this paper, the simulations were considered converged when the metric δΓ< 1 × 10⁻⁶, and a relaxation factor of frelax= 0.01 was employed to ensure numerical stability.

Figure  demonstrates the convergence of aerodynamic coefficients by showing the percentage error relative to the np= 150 case for α= 2, 8, and 16°. The np= 150 case was selected as the endpoint because the VSM often failed to converge for larger values of np. The results were computed using airfoil data obtained from CFD simulations fitted with PCHIP, and the used V3 kite airfoils are shown in Fig.  in Appendix .

The convergence analysis demonstrated that convergence deteriorated with increasing α, with markedly less smooth trends at α= 16°, consistent with VSM's post-stall behaviour, discussed in Sect.  and . Because the solution converged with increasing np, the largest panel count – np= 150 – was adopted throughout the remainder of the paper.

To place the VSM results with different airfoil inputs in context, wind tunnel measurements and accompanying confidence interval (CI)  and 3D RANS CFD simulations  are plotted in Fig. , all at Re = 5 × 10⁵ except for the 3D CFD simulations, which are at Re = 1 × 10⁶. The 3D CFD simulations demonstrated that both the inclusion of struts and the difference in Reynolds number between Re = 1.0 × 10⁶ (with struts) and Re = 5.0 × 10⁵ (without struts) had a negligible effect on the integral forces . The plotted 3D CFD results differ from those presented in , as they were corrected for a geometric offset of Δα= 1.02°; further details are provided in the appendix of the companion paper .

The VSM results, generated with 2D RANS CFD PCHIP-fitted data, differ slightly from the geometry used in the wind tunnel and 3D CFD, as struts are omitted and differences between the profiles' pressure sides emerge near the tips, an artefact of relying on the parameterization scheme and an automatic meshing script (see Fig.  in Appendix ). The VSM results based on NeuralFoil, meanwhile, use the geometries with struts. The pitching moment CM,y was defined with respect to the tow point, defined as the location where the bridle lines converge beneath the wing, corresponding to the attachment point of the tether or kite control unit in AWE systems. The tow point of the V3 kite is located at x= 1.16 m, y= 0 m, and z= -11 m, relative to the mid-span LE point. The results are presented in Fig. .

The V3 kite operates at α≈ 1° during reel-in and α≈ 8° during reel-out , placing the nominal operating conditions largely within the near-linear portion of the polar. Accordingly, comparisons are assessed over α∈[-1,10]° to bracket the operational range and its associated variability.

Within this band, all results shown in Fig.  exhibited a consistent linear increase in lift, except for the wind tunnel measurements, which showed a change in slope at around α= 9°. At negative α, the VSM results based on 2D CFD data aligned most closely with the measurements. At higher α, where LL models are known to be inaccurate, the VSM 2D CFD predictions surprisingly agreed more closely with the measurements than the 3D CFD data. In drag, the sensitivity of the VSM predictions to the airfoil-level inputs was also evident: the VSM 2D CFD case – the only one without struts – yielded the lowest drag. Considering both lift and drag, the VSM with 2D CFD input reproduces the measurements more closely than the 3D CFD predictions. Averaged over α∈[-1,10]°, the respective mean absolute errors are 9 % for lift and 13 % for drag.

The sensitivity of the VSM predictions is most evident in the pitching-moment coefficient CM,y, for which no 3D CFD data were available. The VSM results deviated from the measured trend at negative α and became less negative at higher angles. Within the mentioned α-band, all simulations predicted a continually decreasing CM,y, whereas the measurements showed a nearly constant and near-zero CM,y.

Overall, it is concluded that the VSM predicted the most consistent trends when using 2D CFD-derived aerodynamic input, primarily because the associated lift and drag polars capture stall behaviour more accurately.

Rigid-body stability derivatives were obtained by perturbing the apparent velocity angles and the body rates about a trimmed baseline and evaluating the resulting changes in aerodynamic force and moment coefficients. Angle derivatives with respect to the angle of attack α and sideslip β were reported per radian, whereas rate derivatives with respect to the body rates p, q, and r (roll, pitch, and yaw) were non-dimensionalized to hat-rates using 38p^=pb2U∞,q^=qcMAC2U∞,r^=rb2U∞, where cMAC=S/b= 2.346 m represents the mean aerodynamic chord with projected surface area S= 19.413 m² and projected span b= 8.274 m. Sensitivity ∂C/∂ξ was then approximated using a central finite-difference scheme: 39∂C∂ξ≈Cξ+Δξ-Cξ-Δξ2Δξ, with perturbation steps Δα=Δβ= 0.005 rad and Δp=Δq=Δr= 0.1 rad s⁻¹. The reported derivatives were confirmed to lie close to the central-difference plateau: varying the angular step over 0.03–1.1° and the rate step over 0.005–0.1 rad s⁻¹ altered every derivative by less than 0.3 % and 1 %, respectively, with the exception of the two yaw-moment derivatives ∂CM,z/∂β and ∂CM,z/∂p^, which varied by up to 0.4 % and 3 %.

The perturbation steps were applied around the baseline trim angle, the angle at which CM,y= 0 and ∂CM,y/∂α< 0, further discussed in Sect. . The trim angle was found at αtrim= 7.469°, with ∂CM,y/∂α=-2.708 rad⁻¹, indicating local pitch stability. Moments were computed about the tow-point, i.e. the junction where the bridle lines connect beneath the wing, as this represents the physical point about which the kite rotates.

The resulting derivatives, reported in the aircraft reference frame in Table , represent instantaneous rigid-body stability characteristics; they do not capture unsteady aerodynamic effects or the influence of deformation on the dynamic response. As deformation was not included, control effects are likewise absent, because the kite wing relies on deformation for actuation. Despite this limitation, the aerodynamically derived stability derivatives still provide insight into the kite’s intrinsic stability characteristics, particularly given the scarcity of comparable data in the literature and the reference-model status of the V3 kite.

The signs of the static derivatives – ∂CM,x/∂β<0, ∂CM,y/∂α<0, and ∂CM,z/∂β>0 – indicated static stability in roll, pitch, and yaw, respectively . The negative diagonal rate derivatives – ∂CM,x/∂p^, ∂CM,y/∂q^ and ∂CM,z/∂r^ – indicate aerodynamic damping in roll, pitch, and yaw, with yaw damping an order of magnitude weaker, consistent with the absence of a vertical stabilizing surface. Aerodynamic damping alone does not guarantee dynamic stability; resolving the dynamic modes requires a full modal analysis, including inertia and unsteady effects.

VSM, like other LL methods, is sensitive to implementation details, and inconsistent force evaluation can lead to non-physical performance improvements when these models are embedded in optimization loops.

For this reason, the present work has prioritized internal consistency and documented limitations over incremental extensions whose theoretical implications are not fully validated.

Examples of excluded incremental extensions include accelerating convergence by initializing the solver with a previously converged circulation distribution. This strategy was omitted because the solution exhibited sensitivity to the initial condition, particularly in the vicinity of stall.

Extensions proposed for swept and sideslip configurations were explored but not adopted. The first concerned a sweep-consistent kinematic projection, in which the sectional kinematics were evaluated in the plane normal to the local spanwise direction when coupling the 3D induced velocity to 2D polars, rather than in the global projection plane used in the present VSM formulation; this modification produced only modest changes in the predicted loads. The second involved a modified near-wake treatment, in which the trailing vorticity was initialized locally perpendicular to the lifting line before aligning with the far-wake direction , instead of being directly aligned with the free stream. This modification reduced the predicted lift of the V3 kite by nearly 40 % and was therefore not retained.

At high angles of attack, convergence issues may arise when the lift curve becomes non-bijective, such that multiple values of α correspond to the same Cl, particularly in post-stall regimes . Similar discontinuities were observed for the V3 kite in post-stall. To address potential spanwise discontinuities, a simple smoothing algorithm was implemented to damp sharp local gradients in the circulation field during iteration. However, no consistent results were obtained, as the smoothing effectiveness varied with both the number of panels and the geometry. It was therefore decided to omit the proposed algorithm. The model of was also tested, but it likewise did not yield consistent results and was therefore excluded.

Discrepancies in drag, most pronounced when comparing numerical results to wind tunnel measurements , may arise from several sources. One notable factor is the absence of a spanwise viscous drag contribution in the baseline model. A correction for this effect, proposed by , accounts for spanwise flow by decomposing the local velocity vector into spanwise and perpendicular components. The correction was omitted from the current VSM implementation due to its empirical nature and limited impact, as it increased drag by approximately 1 %.

To aid convergence, smooth sectional polars are recommended. Computational cost can be reduced by using fewer panels or by switching from the open-source Python version to the open-source Julia version, which is about 30 to 50 times faster, and runs in approximately 3 ms with 36 panels and in 25 ms with 150 panels on a 12th Gen Intel^® Core™ i7-1265U CPU. The 3D RANS CFD simulations required 19 394 s of CPU time , indicating a speed-up on the order of 10⁶, although the processor specifications for the 3D RANS CFD computations were not documented.

To quantify the sensitivity of kite aerodynamic loads to airfoil geometry, the ML model was deployed to systematically vary the geometric parameters κ, η, t, δ, and λ. The resulting airfoil polars were subsequently used as input for the VSM to evaluate the kite aerodynamic loads. All the airfoil slices, shown in Appendix , were uniformly varied by the same percentage relative to the V3 kite reference, while ensuring that all parameter values, shown in Table , remained within the bounds used for ML model training. As the V3 kite had a reflex angle of δ= 0°, percentage-based variation was not feasible; therefore, each profile was modified instead by an equal absolute value. The resulting variations are plotted in Fig. , with the exception of λ, which, within the parameter bounds used for training the ML model, had the least visible profile change.

The parameter study revealed that the effect on aerodynamic forces is more pronounced than the effect on the pitching moment. It furthermore showed that trends in 2D and 3D aerodynamic performance are not always aligned. For example, while the 2D ratio Cl3/Cd2 exhibits a clear peak near α= 6°, the CL3/CD2 gradually increases beyond this angle. Likewise, variations in t affected Cm differently depending on α; the 2D case exhibited a near constant trend at a 10° angle of attack, whereas the corresponding impact on the 3D CM showed a more uniform, decreasing trend across the investigated range.

For AWE, the aerodynamic design objective is to maximize the system ratio CL3/CD2, evaluated at the corresponding trim angle. Compared to the V3 kite, the results indicate that an optimal design would have smaller values of κ, larger values of η, smaller values of t, and a small δ, although for the latter, the opposite holds for α> 8°.

The effect of varying the wing planform geometry was analysed by simulating three distinct planforms. A cubic Bézier curve was fitted to the quarter-chord line of the V3 kite wing using four control points (P1, P2, P3, and P4) that preserved the endpoint positions at tip and root, as illustrated in Fig. . Two shape parameters, namely the distances from P1 to P2 and from P3 to P4, were employed to capture arc fullness and tip curvature. These distances were initially obtained by non-linear least-squares minimization of the total point-to-curve distance between measured quarter-chord points. The anhedral angle ϕan, also shown in Fig. , was defined as the angle between the line from P1 to P2 and the line from P1 to P4.

When reconstructing the V3 using cubic Bézier curves, an aspect ratio of AR= 5 was found, with AR defined as 40AR=b2S, where b= 11 m denotes the flat wingspan, not identical to the projected wingspan and measured as the quarter-chord arc length in the (y,z) plane when laid out flat.

A second planform with the same non-dimensional arc shape, but with a 25 % larger aspect ratio, was generated by keeping the shape parameters fixed, and scaling span and chord. A third planform was produced by increasing the anhedral angle by 25 % to ϕan= 41.73°. The flattened surface areas of the reference model and enlarged AR planform were nearly identical, where a reduction of less than 0.5 % was observed for the planform with enlarged ϕan.

The aerodynamic effects are presented for both α and β sweeps in Fig. . The trim angles, computed about the tow point, were αt= 6.89° for the baseline configuration, αt= 6.23° for the AR-scaled configuration, and αt= 7.20° for the ϕan-scaled configuration. As the maximum difference of 0.97° was not considered substantial in view of the applied shape changes, and to isolate the effect of the planform change, the β sweeps of all configurations were evaluated at the baseline trim angle of 6.89°.

Increasing the aspect ratio AR, in line with classical aircraft aerodynamic theory, resulted in higher CL3/CD2 values. This was accompanied by a more positive yaw-moment derivative ∂CM,z/∂β. From this, one could conclude that a kite with a larger AR would generate a greater restoring yaw moment in response to an offset in β, thereby enhancing yaw static stability.

Increasing the anhedral angle ϕan reduced CL3/CD2.

In addition, the lateral–directional coefficients exhibited stronger coupling: the side-force coefficient CS increased in magnitude for a given β, indicative of enhanced turning capability.

The results demonstrated a fundamental trade-off in kite design. Increasing AR yielded a higher lift-to-drag ratio, whereas increasing the ϕan lowered CL3/CD2 but enhanced the turning ability.

The sensitivity of the kite to variations in tow-point position was analysed by displacing its location in the x and z directions, as illustrated in Fig. , which also includes a V3 kite mid-span slice showing the LEI airfoil at the design distance, spanning precisely the figure x limits. For each tow-point configuration, the trim angle αt and its corresponding CL3/CD2 ratio were determined using the VSM. Since the VSM tends to lose accuracy in post-stall conditions, the analysis was restricted to the range α∈[-3,13°].

The figures include a grey-shaded area indicating where ∂CM,y/∂α>0, i.e. where the kite is not statically stable in pitch and no physically meaningful trim angle can be found. Additionally, the centre of gravity (CG) and the range of centre of pressure (CP) positions corresponding to the obtained trim angles were calculated and plotted.

The results indicate that the trim angle αt is more sensitive to variations in x than in z. The same trend is observed for CL3/CD2, ∂CM,y/∂α, and ∂CM,z/∂β. The largest trim angles αt, corresponding to the largest values of CL3/CD2, occur when the tow point is positioned aft of the CP. However, positioning the tow point excessively close to the TE should be avoided, as it may lead to static pitch instability. More restrictive is the yawing-moment derivative ∂CM,z/∂β, which shows that parts of the region yielding the highest CL3/CD2 coincide with ∂CM,z/∂β<0, implying static instability in yaw.

The ∂CM,x/∂β is the only parameter that is more sensitive to variations in z than in x, and it indicates static roll stability at the design tow-point location. The results show that raising the tow point can eliminate the restoring roll moment. This implies the existence of a boundary in tow-point height at which the effective roll moment arm about the wing changes sign, rendering the kite statically unstable in roll. As the CG lies above this boundary, placing the tow point at the CG location would lead to static roll instability. This behaviour is consistent with expectations for an anhedral wing in conventional aircraft , and it underscores a key distinction: kites that rotate about a sufficiently low tow point require anhedral for roll stability, whereas aircraft that rotate about the CG require dihedral.

The tow-point variation analysis shows that to avoid static pitch and yaw instability, the tow point must be placed sufficiently far forward, which becomes more restrictive at higher z values. The stability gains of placing the tow point forward come at the cost of a reduced trim angle and, thereby, a lower aerodynamic performance.