An LES approach combined with realistic modeling of the AWES (using an improved AL method for the wing and added modeling of the control surfaces: ailerons, rudder, and elevator) is used here. We use LES because it allows for an accurate modeling of the flow turbulence and unsteadiness, and is therefore well suited to represent realistic wind.

The LES solver is based on fourth-order finite-differences for incompressible flows. It was developed at UCLouvain . The Navier–Stokes equations are solved in their velocity-pressure formulation, truncated by the LES grid size, and supplemented with a subgrid-scale (SGS) model: 1a∇⋅v=01b∂v∂t+(v⋅∇)v=-∇P+ν∇2v+∇⋅τSGS+f, where v is the velocity field, t is the time, P=p/ρ is the kinematic pressure field, and ν=μ/ρ is the kinematic viscosity of the fluid. τSGS is the SGS stress tensor (also divided by ρ). It is modeled using the regularized variational multiscale (RVM) model . The volumetric forcing term f is used to represent the effect of AWESs. The domain is discretized using a Cartesian staggered grid. The temporal integration is handled using the second-order Adams–Bashforth scheme.

The turbulent inflow consists of either synthetic turbulent fluctuations generated using the Mann algorithm  and added to the mean wind, or a neutral atmospheric boundary layer (ABL) obtained using a co-simulation .

We stress that the LES solver, combined with improved AL modeling, has already been used in previous studies relating to wind energy. For instance, it was used to investigate the blade flexibility effects on the loads and wake of the very large IEA 15 MW horizontal-axis wind turbine in . The investigations included both cases of turbulent wind, using a Mann box (hence no mean shear) and using an ABL running as a LES co-simulation (hence also with mean wind shear).

The present study focuses on the ability of the kite and its controller to maintain stable flight in turbulent flows and under external perturbations, such as the wake of another kite. To isolate the turbulence and wake effects, it is chosen to consider a uniform mean inflow and synthetic turbulence using Mann boxes pre-generated with Hipersim .

When the kite flies with the controller, the dynamics are handled using the dynamics model from AWEbox , which is a modeling and optimization toolbox for AWESs. The AWES can be decomposed into three main components: the ground station, which hosts the generator; the tether; and the kite. In the system model, the ground station is not modeled and the kite is directly controlled using the tether jerk (third derivative of the tether length). The tether is assumed to be a straight rod of varying length, going from the ground station to the kite center of gravity (CG) and subjected to drag only. Finally, the kite itself is modeled using a 6-DOF model for its dynamics, subject to forces and moments. The computation of the forces and moments is handled using the actuator line and the control surfaces models within the flow solver, as presented in Sect. .

Two coordinate systems come into play to describe the system dynamics. The origin of the inertial coordinate system is placed at the ground station with the x axis in the flow direction, the z axis pointing upward, and the y axis sideways to form a right-hand coordinate system. The body coordinate system origin is located at the CG of the kite, with the x axis pointing backward, the y axis pointing starboard, and the z axis pointing upward, as shown in Fig. . The kite position in the inertial frame is described by its coordinates q. The orientation of the kite in this reference frame is represented using a rotation matrix R≜[ex,b;ey,b;ez,b]T that contains chord-wise, spanwise, and upwards unit vectors of the aircraft body frames, expressed in the inertial frame {ex,I;ey,I;ez,I}, and transforms the different vectors from the inertial frame to the kite body frame.

The kite state vector is the concatenation of the kite position q; translational velocity q˙; rotational velocity ω; orientation described by the rotation matrix R, control surfaces deflection δ, which is the concatenation of [δa;δe;δr] (ailerons, elevator, and rudders actuation angles, respectively) and the tether length l, tether velocity l˙, and tether acceleration l¨; and is therefore defined as 2x≜(q,q˙,ω,R,δ,l,l˙,l¨).

The control variables of the system are contained in the vector 3u≜(δ˙,l…), and is a concatenation of the control surfaces actuation rates and the tether jerk.

The equations of motion of the kite are derived from Lagrangian mechanics , expressed as 4amq¨-λq=RTFe,b-m‾gez,I4bc¨+2κtc˙+κt2c=04cJω˙+ω×Jω=Me,b4dddtR=R(κR2(I-RTR)-skew(ω)).

Equation () corresponds to the translational kinematics, in which m=(mK+13mT) and m‾=(mK+12mT) are the effective inertial and effective gravitational masses (with mK and mT the mass of the kite and of the tether, respectively), Fe,b is the external forces expressed in the body frame, g is the gravitational acceleration, and λ is the Lagrangian multiplier inherent to the chosen formulation. The translational kinematics is constrained with Eq. (), in which c=12(qTq-l2)=0 is the constraints that forces the kite to navigate on a hemisphere of radius equal to the tether length; and κt is a constant stabilizing parameter. It is the Baumgarte stabilized form of the latter constraint c. Equation () refers to the rotational kinematics, in which J is the kite inertia matrix, and Me,b is the external moments. A similar stabilization is used to guarantee the orthogonality of the rotation matrix. Equation () both describes the evolution of the rotation matrix and guarantees its orthogonality cR=RTR-I=0. In this equation, κt is another constant stabilizing parameter, and skew is an operator that transforms a vector into the corresponding skew-symmetric matrix. Such stabilization also requires the initial constraint C(x0)=c(x0),c˙(x0),cR(x0)=0. The interested reader can find more details on the dynamics formulation in and .

The system dynamics are integrated in time using a 4th order Runge–Kutta scheme: 5x˙=dxdt=(q˙,q¨,ω˙,R˙,δ˙,l˙,l¨,l…).

The external forces Fe,I=RTFe,b applied to the system are the aerodynamic forces. The aerodynamic forces consist of the aerodynamic forces acting on the kite and the tether drag. Their evaluation is described in the following sections.

The tether drag is evaluated assuming the drag coefficient CD,T=1.2 for a circular cross-section. The total drag on the tether is shared between the ground station and the kite, as detailed in . The contribution that acts on the kite is evaluated by integrating along the tether using the coordinate s∈0,1. An infinitesimal tether fragment therefore has a length of dl=lds, and the integral writes 6FDT=∫01s(12ρ∥va,t(s)∥va,t(s)CD,TdT)(lds), where va,t(s) is the apparent velocity projected perpendicularly to the tether and dT is the tether diameter. The weighting factor s specifies the fraction of the total drag attributed to the kite. For each infinitesimal tether element, the drag is partitioned between the ground station and the kite according to its position along the tether: a weight of (1-s) applies to the ground station, and a weight of s applies to the kite. The other part acts on the ground station. The integration is required because the drag force is not constant along the tether, as the apparent velocity varies along its length. In AWEbox, the integration is performed numerically by dividing the tether into n elements. Here, n=5.

The aerodynamic forces that act on the kite are taken from the flow solver AL model. In this model, the main wing of the kite is modeled using an actuator line (AL) model, as in , and that line is immersed in the flow solver, meaning that it moves relative to the fixed Cartesian grid of the flow solver.

Capturing the effects of the turbulence on the kite requires that the flow solver grid size be sufficiently fine relative to the wingspan of the kite. It must therefore be sufficiently fine everywhere in the region of the flow domain through which the kite moves.

The AL method involves three main steps: flow velocity sampling, forces and moments evaluation, and forces projection into the fluid. The AL is discretized into N segments, with a control point at the center of each segment. These control points are used as reference locations for the three main operations of the method. Knowing the position and orientation of the wing, the AL is placed along the quarter-chord line, and the positions of the control points are determined. For each control point, the local flow velocity is evaluated by performing a weighted average of the flow velocity in the vicinity of the control point. Once the velocity is known, the aerodynamic forces and moments are computed on the basis of airfoil polar data. Finally, the forces are projected onto the mesh surrounding the corresponding control point. They are accounted for by the flow solver through the volumetric force term on the right-hand side of Eq. (). Velocity sampling and force distribution are performed by means of a 2D Gaussian regularization kernel. Indeed, 2D kernels have been shown to be more accurate in predicting aerodynamic forces than the 3D kernel . In practice, a planar grid with its associated weights, referred to as a template, is placed at each control point, perpendicular to the AL. Linear interpolation is then used to transfer information from the template to the flow solver grid, and vice versa. More details on the implementation of the AL method can be found in .

This work considers the rigid-wing reference AWES, called MegAWES . It has a wingspan of 42.5 m and a main wing aspect ratio of 12. The complete geometry considered in the present work is detailed in Appendix . The geometry is slightly adapted to solve inconsistencies and simplify some parts.

The aircraft has two ailerons on the wing, one elevator, and two rudders. Their associated quantities are denoted with the subscripts a, e, and r, respectively. They are modeled as flat plates hinged about their aerodynamic center (i.e., at their quarter chord). Owing to their low aspect ratio, these control surfaces are not well suited for an AL representation. Moreover, their dimensions are small relative to affordable grid resolution of the flow solver, which prevents an adequate spatial discretization. Their force contribution is therefore modeled analytically as (⋄∈[e,r]) 7L⋄=12ρ∥va,⋄∥2AdCLdα|⋄α⋄, where A is the control surface area, dCLdα|⋄ is its effective lift slope coefficient, and α⋄ is the angle of attack of the measured apparent velocity relative to the chord. The lift slope coefficients were evaluated using a lifting surface method, thus properly taking into account the low aspect ratio of the control surface and the wake vorticity produced by it. As these estimates closely match those obtained from the Helmbold approximation for low-aspect-ratio wings, the latter is adopted, yielding values of dCLdα|e=1.01π for the elevator and dCLdα|r=0.50π for the rudders.

The apparent velocity va,⋄ is obtained from the kite velocity and the local flow velocity. The flow velocity is interpolated at the aerodynamic center of each control surface using an M′4 kernel . The resulting aerodynamic loads are added to the forces acting on the body, with the resulting moments. Because these loads are small compared to those generated by the main wing, their impact on the flow field is neglected (i.e., we neglect the small wake vortices that they produce), and their influence is restricted to the body dynamics.

Aileron effects are incorporated directly within the actuator line framework by adjusting the local airfoil characteristics according to the aileron deflection angle δa. For the ith wing section of chord ci, the lift per unit span is expressed as 8li(α)=12ρ∥va,i∥2ci(Cl(α)+pi((ητ)dCldα|aδa)), where τ is the aileron effectiveness factor which depends on the fraction of the chord spanned by the aileron ca/ci and η=η(δa)<1 is a correction factor to account for the reduced aileron effectiveness as the deflection angle increases – see . In this case, ca/ci=0.25 and τ=0.6. The lift slope of the ailerons dCldα|a is that of the aerodynamic profile at its zero-lift angle. The contribution of an aileron to an AL control point is weighted proportionally to the fraction pi that it spans on that AL segment. The influence of the ailerons is thus taken into account in the computation of the force distribution over the wing. A schematic of the whole AL-based kite model is depicted in Fig. .

It should be noted that, due to the variation of the wing properties (chord, twist, presence of ailerons), the number of AL control points must be chosen so that it can capture those characteristics.

The power output of an AWES is determined by its trajectory. It also depends on how well the kite can follow the planned trajectory. In the present work, the kite flies so-called “optimal trajectories” in the sense that the reference trajectories are generated through an optimization process performed using AWEbox . This is presented in Sect. . The controller is then used to fly these pre-computed optimal trajectories in the turbulent wind using flight path tracking. The control strategy is introduced in Sect. .

In this section, we investigate the requirements in terms of flow solver grid cell points and AL control points to ensure a good representation of the wing aerodynamic properties. To this end, the aircraft, without its control surfaces, is considered in steady level flight at an angle of attack of -5°. The aircraft has a zero-lift angle of -11°.

The chosen number of grid points per wingspan sets the flow solver grid cell size. The number of AL control points must be larger, yet it can be “only a bit larger” (i.e., about 1.2 times larger). This lower bound was determined such that the discretized AL takes full advantage of the grid resolution for the velocity sampling but without overkill (i.e., in the sense that using more control points does not improve the results further) – see . Concerning the mollification width σ of the Gaussian template, it is taken to be 2h, where h is the flow solver grid size .

Figure  shows the distribution of the lift and drag coefficients for different numbers of grid points per wingspan. We have σ=c¯/4 for the highest resolution and 1.5c¯ for the lowest one. One can note that the accuracy is still very good close to the tip for the lowest resolution. The main differences appear near the middle of the wing, where the lowest resolution leads to an overestimated lift and underestimated drag. The total lift coefficients are 0.604, 0.619, and 0.627 for 96, 32, and 16 points per wingspan, respectively. The total drag coefficients are 0.0237, 0.0234, and 0.0228, respectively. The difference between the finest and coarsest resolutions is <4% for both lift and drag. This indicates that, despite the slight difference in loading in the middle of the wing, 16 grid points are enough to still have a fairly accurate representation of the wing aerodynamic loads. This also corresponds to the resolution used in .

When varying the number of AL control points for a given number of grid points per wingspan, here 16, the wing loading hardly changes at all, as depicted in Fig. . The difference in terms of total lift and drag is of the order of a 10th of a percent. The accuracy is therefore conserved regardless of the number of control points, as long as it is higher than the number of grid points. In the following, 1.2 times the number of grid points will be used.

To investigate the domain size, the single-loop trajectory from is flown with MPC path tracking in domains of different sizes. The trajectory diameter D is about 5b. The domain is 9D long and has a square frontal area of height H, with periodic conditions on the sides, and slip conditions on the top and bottom (ground location). The trajectory was flown, in domains with H=2D, 3D, and 6D. Assuming a circular trajectory orthogonal to the mean wind, the swept area of the trajectory corresponds to 15.7 %, 7.0 %, and 1.7 % of the total frontal area, respectively. The mean power measured during 12 power cycles after the flow is developed is 770.8, 777.7, and 775.3 kW, respectively. The greatest difference is less than 1 %, which indicates that a domain with H=2D is already enough for the flow to have negligible blockage effects. In this work, we consider larger trajectories for which the diameter is about 10b. Since the ratio of the swept area to the frontal area scales with b/D, increasing D reduces the swept-area fraction for a given domain size, expressed in terms of D.

The trajectory is a four-loop trajectory, generated using AWEbox. It is computed for a uniform inflow U∞=12 ms-1. It is depicted in Fig. . The resulting trajectory has a spatial footprint of 674m×494m×407m. We define a characteristic length D=10b, which approximately corresponds to the “characteristic diameter” of the trajectory (see Fig. ). The kite rotates clockwise when looking downstream. The kite flight velocity ranges from 16.4 to 89.4 ms-1 over the whole trajectory, with an average value of 55.6 ms-1. The kite flies faster during the reel-out phase, with an average flight velocity of 67.8 ms-1, and reaches a minimum of 50.7 ms-1 at the end of the ascending parts of the loops. The velocity then decreases drastically during the reel-in phase. The trajectory has a time period Tp=112 s, and we define the dimensionless time t* as t/Tp. A fraction of the fourth loop is used for starting the reel-in. The reel-out phase accounts for 70% of the time and the reel-in phase for 30%. The average power output is 1.38 MW, and the instantaneous power can turn negative in the ascending phase of each loop to help the kite reach the top of the loop.

It should be noted that the trajectory is optimized for the given set of parameters and would therefore differ for other wind profiles. In the case of a uniform inflow, the kite flies at the lowest feasible altitude to limit the tether length and elevation angle. For a sheared inflow, the altitude-dependent wind velocity also plays a role, and the optimal trajectory may reach higher altitudes because of the lower wind speed near the ground. However, increasing the tether length and the elevation angle is detrimental to power production.

The set of constraints is applied to the state and control variables. They are provided in Appendix , in Tables  and , along with the other constraints of the OCP and the NMPC. In addition to states and control variables, the instantaneous power is also constrained to limit the Pmax/Pavg ratio. There are also constraints on aerodynamic quantities, such as the angle of attack and the side-slip angle of the aircraft, and on the acceleration and tether force. The constraints are chosen to ensure operation within limits considered acceptable for the different components of the system. They are tightened during the trajectory generation to have some additional freedom and margin during the flight path tracking.

The kite controller is based on NMPC, as described in Sect. . The prediction horizon Th of the NMPC corresponds to 20 NMPC sampling periods Tc=0.10 s. During one sampling period, the dynamics are integrated 20 times, each dynamics time step being 0.005 s.

The size of the domain is 5.4D×2.0D×1.36D. This leads to a swept area corresponding to 11.5% of the domain frontal area, which is shown to be sufficient in Sect. . The domain is discretized using 1024×384×256 grid cells, which corresponds to a uniform spatial resolution of 2.25 m. This leads to 19 grid cells per wingspan. We use periodic conditions on the sides, and slip conditions on the top and bottom. The domain is shown in Fig. . The lower boundary of the domain corresponds to the ground, and the z coordinate therefore denotes the altitude. The first kite trajectory is located so that it stays at least 3 wingspans away from the inlet. When there is also a second kite, it is placed as close as possible, ensuring that the tethers do not intercept (straight tethers). As the study focuses on the turbulent nature of the inflow and the kite's flight, we chose a uniform inflow of 12 ms-1, with a synthetic turbulence obtained using the Mann algorithm. The turbulence intensity is 6% and is representative of offshore ABL conditions. The pre-computed box of turbulence has a spatial resolution of 4.5 m and is therefore linearly interpolated at the domain inlet. The box is long enough to last six power cycles. The time step used for the flow solver is 0.020 s. The investigations are carried out after the flow and the wake have developed during two power cycles (which corresponds to flowing 1.17 times the entire length of the domain) and during six additional power cycles.

When the kite flies alone, it is only subject to the inflow velocity wind and its turbulence. In the case of a uniform turbulent inflow with 6% turbulence, the kite follows the trajectory without particular difficulty. The different quantities studied are represented as the average of what is performed by the kite during the six power cycles analyzed. A shaded area surrounds that average, and it corresponds to plus and minus 1 standard deviation. In the different plots, the time is made dimensionless using Tp, the reference trajectory period, which is 112 s. Figure  shows the results for position and attitude (Euler angles). One can see that those are very close to the reference and that the inter-cycle variation is almost nil. The RMS error in position is at most 1.1b. As previously noted, the flight altitude in a uniform inflow is low and reaches the specified lower bound.

The actuation of the control surfaces is presented in Fig. . Although it remains close to the references, some deviations result from the controller mitigating the turbulence of the inflow. The average deflection of the ailerons remains fairly close to the reference, while it varies significantly from one cycle to another. The elevator actuation shows fast variations, yet up to a maximum of 6°, and the rudder is a bit less used at maximal positive deflection than planned. The reel-in phase, where the kite will roll and pitch to come back facing the wind, is well identifiable after t/Tp=0.7.

The force and moment coefficients in the body frame are provided in Fig. . The forces from the simulations are close to those from the reference. The vertical force coefficient Cz remains quite constant during the reel-out phase. Regarding the moments, the rolling and pitching moments are quite noisy. This likely indicates a larger mismatch between the simplified model and the AL-based model for these quantities. The yawing moment is more precisely reproduced. The moments show a high variability due to the encountered perturbations and the actuation of the control surfaces used to mitigate them.

The tether acceleration and traction, and the power output are very well followed, as seen in Fig. . The averaged cycles are close to the reference, and the inter-cycle variation is very low. One can see that the tether speed turns negative at the end of each loop. This is what makes the power negative and helps the kite to go up at the end of the ascending phase of a loop, as pointed out in Sect. . The reel-in phase can be identified as the tether speed becomes highly negative at the end of the power cycle. The tether traction is always positive, and it reaches maxima when the kite is at the bottom of the loop with maximum velocity. It is interesting to note that the tether traction is here null during the reel-in phase, corresponding to zero power consumption. The fact that it is zero is due to the simplified hypotheses of the AWEbox model (there is no model for the generator and the drum, and the tether is considered straight). During the reel-in phase, the kite essentially flies like a glider. It compensates for both its weight and that of the tether and is continuously descending – all of that without energy consumption. Here, the kite only consumes energy at the end of each loop, during the ascending phase.

One can also note that the tether traction, which is closely related to the vertical force on the kite, fluctuates a lot during the power cycle. During the reel-out phase, its standard deviation is 43% of the mean value. In contrast, the vertical force coefficient is more or less constant during the reel-out, with a standard deviation of only 5% of its mean value during reel-out. The kite flight speed experiences large fluctuations, and so does the relative air velocity, resulting in large variations in vertical force magnitude. The averaged power is also close to the reference, despite some deviation during the first and fourth loops. The inter-cycle variation is quite low. The power production reaches a plateau during each loop due to the maximal power constraint. The total average power production is 1.39 MW, to be compared to the 1.38 MW planned by AWEbox. The RMS error on the mean power output of the six flown power cycles is 0.8% of the reference power output.

The angle of attack and wing loading along the span are displayed in Fig. , where fa stands for the aerodynamic force per unit length. Let us first note that the loading is almost symmetric during the reel-in phase because the kite comes back in a quasi-level flight toward its anchor point and with a reduced loading. However, it is slightly skewed toward the starboard side of the wing during the reel-out phase due to the rotation. In terms of force coefficient, the average loading barely varies during the reel-out phase, as does the angle of attack. However, as already pointed out, it varies greatly in terms of magnitude due to the large variation of the flight velocity. The effect of the ailerons (0.31≤∥y/b∥≤0.47) is well marked. The port (left) aileron is deflected down, increasing lift; and the starboard (right) aileron is deflected up, decreasing lift. This counteracts the asymmetric loading induced by the rotation, i.e., induced roll, which would otherwise make it roll toward the interior of the loop.

A vertical longitudinal cut of the mean streamwise velocity field is displayed in Fig. . Three key features can be identified. First, the velocity deficit is predominantly located in the lower part of the trajectory. This behavior has also been reported by and , who also observed that the lower portion of the wake spreads more rapidly in the radial direction. In the present case, the weaker velocity deficit in the upper part of the wake can be partly attributed to the fact that the kite reaches different altitudes in the upper parts of the loops, whereas the lower parts reach the lower altitude bound and therefore occur at nearly the same altitude. However, this lower velocity deficit on the upper part is also reported for purely circular single-loop trajectories in . Second, the wake is deflected downward. This deflection results from the tilted rotation plane and the presence of a vertical component in the aerodynamic force acting on the kite. A similar effect is observed for tilted wind turbines . Finally, the magnitude of the normalized velocity deficit is fairly low and goes only up to 0.06U∞. For comparison, the velocity deficit of wind turbines can be on the order of 0.3U∞, even a few diameters downstream of the turbine; see for instance .

Figure  shows the velocity deficit in cross-flow planes. The averaged force density, computed from the forces predicted by AWEbox and derived from the analogy with an actuator annulus, is plotted in Fig. a. The first step in establishing this analogy is to obtain a force distribution along the kite trajectory that is constant in time, such that the integral of this force distribution equals the time-averaged streamwise (axial) force: 141Tp∫0TpFx(t)dt=∫0Lfxsds, where Fx denotes the instantaneous total axial force acting on the kite, fx is the axial force distribution per unit length along the trajectory, and s is the curvilinear coordinate along the trajectory. The trajectory has a total length L and a time period Tp. The force distribution fx(s) can be expressed by introducing the change of variable s=s(t), so that ds=‖q˙(t)‖dt, and by imposing equality of the integrands: 1Tp∫0TpFx(t)dt=∫0Tpfxs(t)‖q˙(t)‖dt15⇒fx(s(t))=Fx(t)‖q˙(t)‖Tp. The force per unit trajectory length fx (in Nm-1) is then distributed across the span to obtain a surface force density fx/b (in Nm-2). This force density is projected onto a single plane to obtain the equivalent actuator annulus. Figure a shows that the aerodynamic forces are mainly concentrated in the lower-left part of the trajectory, between the lowest point and the first half of the ascending phase. This is also what is observed in Fig. , which shows the norm of the total aerodynamic force Fa along the trajectory.

A prediction of the time-averaged wake velocity can then be obtained from the momentum conservation. In a reference frame moving at the reel-out velocity Ur,x (here taken as the streamwise projection of the tether speed), the momentum conservation yields 16fx=ρ∫-ξξU¯(U∞-Ur,x)-U¯dr, where the integration bounds ±ξ delimit a control volume across the annulus, and U¯=U¯(r) is the wake velocity profile at its downstream boundary, where it is assumed that the pressure has recovered to p∞. Further adopting the simplified Betz approach (i.e., assuming that Bernoulli's equation applies between the force application point and the wake plane) and using a top-hat profile of width bw for the wake velocity deficit leads to 17fx≃ρUw*(U∞-Ur,x)-Uw*bw, where Uw*=Uw-Ur,x is the wake velocity in the moving frame. Using the mass conservation gives b(U∞+Uw*)/2=bwUw* and taking the wake velocity back in the fixed reference frame, one obtains the following estimation: fxb≃12ρ(U∞-Ur,x)2-(Uw-Ur,x)218⇔Uw=(U∞-Ur,x)2-fxb2ρ+Ur,x.

This equation is only valid if the term inside the square root is positive. This justifies the use of the streamwise projection of the tether speed as the approximate reel-out velocity instead of the local streamwise velocity of the kite. This velocity deficit estimation is shown in Fig. b. The axial force representation already provides insight into the wake shape. The velocity deficit is concentrated in the lower half of the loop and is slightly rotated clockwise, corresponding to the regions where the aerodynamic forces are largest. Overall, the estimated velocity deficit slightly overpredicts the measured wake deficit. This discrepancy is due to the simplifying assumptions introduced above. In particular, all forces were collapsed onto a single plane, whereas the trajectory spans a wide range in x, and the high ambient turbulence rapidly mixes the wake. However, the distribution of the deficit is well predicted, as depicted in Fig. c–f, and a follower kite will therefore experience a lower wind speed when flying in that area.

The deficit decreases in intensity as it is advected downstream, while its shape remains essentially unchanged. The low averaged velocity deficit is due to the nature of the wake produced by such four-loops trajectory. Indeed, the loading of the kite is very low during the reel-in phase, which accounts for 30% of the trajectory time period, and the kite mainly produces a wake during the reel-out phase. The wake is therefore discontinuous in space and time, and released in batches, resulting in a low time-average velocity deficit.

Those discontinuities can be observed in Fig. , which shows an instantaneous streamwise-velocity deficit. On the left, a longitudinal (a) cut shows the instantaneous velocity deficit just after the kite lowest position in its fourth loop. The passage of the kite left traces around z≃0.2D at x≃2.2, 2.5, 2.8, and 3.1D. Those traces have already decreased in intensity since the time they were released and are no longer stronger than the ambient turbulence. The maximum deviation from the inflow velocity in that cut is 33%, and it will become even lower by the time it reaches the flight area of the second kite. However, in the vicinity of the kite, the induced velocity can be quite significant. The transversal cut is taken when the kite is near its maximal loading, at the bottom part of the second loop. The position and near wake of the kite is clearly identifiable, and the velocity deficit in that plane goes as low as 85% of U∞ (out of the color map range). It corresponds to the maximal velocity induced by the tip vortices. This observation is consistent with the kite's flight velocity at this stage of its trajectory, which reaches 89.4 ms-1. Relative to its flight velocity, the deficit is only of 11%. This is of the order of magnitude of the downwash that can be expected for such aircraft when flying at a high angle of attack.

For the cases with two kites, both kites try to fly the same optimal reference trajectory. Two scenarios are investigated. For the first case, the second kite flies in-phase with the first kite. In the second case, the second kite trajectory is phase-shifted relative to the first kite's trajectory. In order to isolate the effect of the wake of the first kite, we perform additional simulations of the second kite only. Those simulations, hereafter referred to as “2nd only”, will assess the effect of the ambient turbulence on kite 2.