Modeling the aeroelastic behavior of a tethered soft kite flying in a turbulent wind environment poses a number of challenges. In this section, we introduce the relevant assumptions to simplify the physical problem and develop a simulation approach.
2.1. Simulation Setup–Virtual Wind Tunnel and Model Assumptions
The aerodynamic and structural characteristics of the parafoil, such as the glide ratio, the tether and bridle line forces, and the mechanical stresses are determined in a virtual wind tunnel environment that reproduces realistic flight conditions. The power harvesting envelope of a SkySails Power parafoil consists of four phases: the power, transfer, retraction, and restart phases [
21]. During the power phase, the kite is flown in crosswind figureeight maneuvers at relatively lowelevation angles to produce a maximum tether tension. At the same time, the winch reels out the tether, modulates the speed, and also controls the apparent wind speed at the kite and the tether force. To avoid modeling a whole figureeight trajectory with steering input and transient aerodynamic and structural effects, a steady flight state is assumed between two turning maneuvers when the parafoil reaches a constant flight velocity and inflow angle, as depicted in
Figure 1a. This steadystate flight is defined by an equilibrium of all forces acting on the kite: the distributed aerodynamic load, the gravitational forces, and the tether force. The equilibrium state during steadystate flight provides valuable insight into the coupled aerodynamic and structural characteristics of the kite.
Figure 1b depicts the influence of gravity on the airborne system components, which cause the tether to sag toward the ground and require a small part of the lift force of the kite to compensate for the effect of gravity on the tether and kite.
Consequently, the tension in the tether is slightly reduced, and the tether elevation angle at the kite is slightly larger than the elevation angle at the ground, with both values deviating from the geometric elevation angle [
6]. For an AWE system utilizing soft kites, the weight of the parafoil and tether is less than 1% of the average tether force during the power phase, and, therefore, the influence of gravity can be neglected during the power phase, which allows for the use of a symmetry condition as explained later. On the other hand, tether sag due to aerodynamic drag is accounted for in the model.
The apparent wind velocity vector experienced by the kite is defined as follows:
where
${\mathbf{V}}_{\mathrm{w}}$ is the wind velocity and
${\mathbf{V}}_{\mathrm{k}}$ is the flight velocity of the kite. The latter can be further decomposed into a component
${\mathbf{V}}_{\mathrm{k},\tau}$ perpendicular to the tether and a component
${\mathbf{V}}_{\mathrm{k},\mathrm{r}}$ aligned with the tether, which for a straight tether, is identical to the tether reeling velocity
${\mathbf{V}}_{\mathrm{t}}$ at the ground [
22]. The velocities acting on the kite during crosswind flight are illustrated in
Figure 2a.
The angle
$\vartheta $ is measured between the wind velocity vector
${\mathbf{V}}_{\mathrm{w}}$ and the tether, while the angle
$\varphi $ is measured in the
$yz$plane of the wind reference frame between the vertical and the orthogonal projection of the kite position.
Figure 2b shows the definition of the apparent wind velocity, the angle of attack, and the aerodynamic forces acting on the kite. For a kite performing crosswind maneuvers in a uniform wind field, the apparent wind speed along the tether varies linearly [
23]. However, a realistic wind field is always nonuniform, with the magnitude and orientation of the velocity vector varying in both space and time. Also, the flight speed of the kite depends on the resultant glide ratio of all airborne components, which varies with the paidout length of the tether and the flight altitude of the kite. For a simplified analysis of crosswind flight, the inflow velocity
${\mathbf{V}}_{\mathrm{a}}$ is assumed to be linearly varying along the tether.
Figure 3 depicts the initial condition of the virtual wind tunnel setup with a horizontal, linearly varying inflow and a vertical tether connecting via a bridle line system to the undeformed parafoil.
For the simplified steadystate flight condition with neglected gravitational effects, only half of the wing is modeled, which exploits the symmetry of the FSI problem with respect to the center rib of the parafoil. This also means that no steering input can be applied in the simulation. During the coupling iterations, the kite translates and pitches from this initial position and orientation until it eventually assumes an equilibrium state, where the aerodynamic forces are balancing the tether force. This final state corresponds to the trim condition of the parafoil with the angle of attack at its trim value. To illustrate this,
Figure 4 compares the initial configuration and shape of the kite for two different positions
${\delta}_{1}$ and
${\delta}_{2}$ of the bridle point with the shapes in steadystate flight, resulting in two different trim values
${\alpha}_{1}$ and
${\alpha}_{2}$ of the angle of attack [
18].
To avoid a front stall of the parafoil in the first coupling iteration, which would produce a negative lift and a collapse due to a low angle of attack, a minimum angle of attack ${\alpha}_{0}$ is set as an initial value.
The distributed aerodynamic load on the kite is computed with the open source tool
APAME, which is a potential flow solver based on the panel method [
24]. For computing the deformation of the inflated membrane structure and the connected bridle lines, the open source tool
mem4py is used [
19,
20]. This inhouse FEM solver was specifically developed for fabric membrane structures and is based on the kinetic dynamic relaxation method introduced in [
25,
26]. To correct the compressive stress state in each membrane element, a wrinkling model valid for orthotropic materials is used [
27].
2.2. Force Coefficients and Glide Ratio
The structural force distribution in the kite can be characterized by the tether and steering belt forces,
${\mathbf{F}}_{\mathrm{tether}}$,
${\mathbf{F}}_{\mathrm{belt},\mathrm{left}}$ and
${\mathbf{F}}_{\mathrm{belt},\mathrm{right}}$, respectively [
9].
Figure 5 shows a photo of the SkySails Power system during takeoff and a schematic closeup of the control pod, indicating where these three forces are measured with load cells.
Since the control pod is not modeled, the resultant force acting in the bridle lines that are not connected to the steering belt can be determined as follows:
From the dimensional values, the following nondimensional coefficients are derived
where
$\rho $ is the air density, and
S is the projected wing surface area.
The resultant aerodynamic lift and drag forces acting on the kite are determined by integrating the aerodynamic surface pressure distribution
${p}_{\mathrm{a}}$, overall membrane elements, and the aerodynamic line drag over all bridle line elements. The glide ratio
E can then be evaluated as the ratio
$L/D$ of lift and drag forces. It should be noted that the surface pressure
${p}_{\mathrm{i}}$ on the inside of the membrane elements is not computed with the potential flow solver but assumed to be uniform and identical to the stagnation pressure at the inflow opening of the wing. This is a reasonable assumption for ramair wings because the flow speeds in the enclosed volume are relatively low since the air can escape only through the permeable fabric, seams, and other leakage options. With
${p}_{\mathrm{i}}={p}_{\infty}+1/2\rho {V}_{\mathrm{a}}^{2}$, the resultant pressure coefficient can then be computed as follows:
where
${C}_{\mathrm{p},\mathrm{a}}$ is the pressure coefficient determined by the potential flow solver and
${p}_{\infty}$ the static pressure in the far field.
2.3. Coupling Algorithm
The interface coupling of fluid and structure solvers is outlined with Algorithm 1.
The coupling tool
preCICE [
28] initializes the communication between the solvers and in each iteration, interpolates all field variables between the two interfaces. Due to the steady flight assumption, the coupling iterations occur in a single pseudo time step, and either explicit or implicit coupling methods can be used to find the equilibrium state. In case of explicit coupling, no information of the previous iteration is used, the pressure is given to the FEM, and the displacements are handed back to the panel code until convergence in the displacement is reached. In case of implicit coupling, the IQNILS approached proved to perform better for some kite geometries than did the explicit method. In the implicit approach, an implicit leastsquares method (ILS) is used to approximate the relation between pressure and displacement within a single time step. The difference between the implicit and explicit coupling is that for the implicit approach, the displacement vector is scaled by the leastsquares method (see line 13 in Algorithm 1), whereas for the explicit method, no scaling is applied, and the first iteration of both approaches is equivalent. A growing difference between both interfaces defines divergent coupling behavior, and it usually results in an wing flying upside down, which is indicated by a negative lift coefficient. Hence, the simulation is aborted if
${C}_{\mathrm{L}}<0$. For the kite geometries used in this work, the total number of coupling iterations varied between 10 and 60. It was observed that a higher mesh resolution also resulted in more coupling iterations because the kite would show more local deformation details that had an influence on the convergence. Also, the kite model provided by SkySails would converge only when using the implicit coupling approach.
Algorithm 1 FSI coupling procedure. 
 1:
Read structure and fluid mesh.  2:
Initialize displacement field $\mathbf{u}$ and surface pressure field p.  3:
Set iteration counter $i=1$.  4:
while ${\mathbf{u}}_{i}$ − ${\mathbf{u}}_{i1}$>${\epsilon}_{\mathrm{FSI}}$ do.  5:
if ${C}_{\mathrm{L}}$ < 0 then  6:
break  7:
end if  8:
Run APAME on geometry with deformation field ${\mathbf{u}}_{i}$.  9:
Correct ${C}_{\mathrm{p}}$ distribution with internal pressure using Equation ( 5).  10:
Pass ${C}_{\mathrm{p}}$ distribution to mem4py.  11:
Run mem4py with current pressure distribution.  12:
$i=i+1$.  13:
Pass ${\mathbf{u}}_{i}$ back to APAME. For implicit coupling scale ${\mathbf{u}}_{i}$.  14:
end while  15:
Postprocessing

2.4. Solver Verification
For verification, the developed FSI solver was used to simulate the ramair kite with an 160 m
${}^{2}$ flat (laidout) wing surface area that was investigated in [
18].
Figure 6 illustrates the computed shapes of the halfkite for typical apparent wind speeds
${V}_{\mathrm{a}}$ = 10, 20, and 30 m/s.
While a small change in the profile shape could already be observed when the speed was increased from 10 to 20 m/s, an increase to 30 m/s led to a strong shape deformation of the leading edge, which in return affected the aerodynamics of the wing.
In a next step, the computed aerodynamic coefficients and pressure distributions were compared with the results presented in [
18]; it is important to note that in this earlier study, only the aerodynamic solver was different, with the CFD solver
OpenFOAM being used instead of the panel method
APAME. The FEM solver and coupling procedure were identical.
Figure 7a,b show the lift and drag coefficients as functions of the angle of attack at the center rib.
It can be seen that both coefficients exhibited a similar trend until larger angles of attack, where the RANS solution (Folkersma et al 2020 [
18]) started to drop in lift as a result of flow separation. As expected, the potential flow stayed attached to the wing without reduction of the lift at high angles of attack. The drag coefficient showed a good match between the two solutions for angles of attack smaller than 15 degrees when both curves started to deviate.
Figure 7c,d show the resultant force coefficient and glide ratio as functions of the angle of attack. The
${C}_{\mathrm{R}}\left(\alpha \right)$ and
${C}_{\mathrm{L}}\left(\alpha \right)$ functions are very similar, which emphasizes the importance of a high lift for generating a high pulling force as the
${C}_{\mathrm{L}}$ contribution to
${C}_{\mathrm{R}}$ dominates by a factor of 5 to 10 depending on the angle of attack. For the comparison of the glide ratio of both approaches, a good match except for low angles of attack of around 5 degrees can be seen despite the differences in lift and drag.
In
Figure 8, the resultant pressure coefficient along the chord of the center rib is presented, computed for a trim configuration with
$\delta =42\%$.
Despite an identical bridle configuration, the trim angle of attack in the RANS simulation is 12.4 degrees, whereas the value computed with the panel method is 14.8 degrees. The ${C}_{\mathrm{p}}$distributions have a similar appearance. The panel method exhibits a steeper slope over the first 10% of the chord, and the pressure peak is only half of that observed in the RANS simulation. Also, the line attachment points create a kink in the parafoil shape to which both solvers react with a local pressure drop. This drop is particularly pronounced for the panel method, leading to larger angles between neighboring panels.
It can be concluded that the panel method can approximate the pressure field fairly well compared to the RANS simulation. Care should be taken for large deformations that can cause strong and even nonphysical local pressure drops. As an initial design tool for parafoils, the panel method allows for efficient and relatively accurate determination of the aerodynamic surface pressure distribution without the generally extensive preprocessing effort to generate highquality volume meshes for CFD and the long simulation times. Aerodynamic parameters like lift and drag are approximated within a 30% difference for a range of angles of attack. Nevertheless, it should be emphasized that the definition of an angle of attack for soft kites is not trivial due to possible chord deformation and rotation [
29]. A better way to compare the solutions is the trim position defined on the undeformed kite geometry (see
Figure 4).
2.5. Measurement Setup, Data Acquisition, and Postprocessing
The instrumentation of the control pod is detailed in
Figure 5. Next to force and apparent wind speed at the kite, the tether angles
$\vartheta $ and
$\varphi $, as defined in
Figure 2a, were measured with a LIDAR, as was the vertical wind profile. The instrumentation of the system was similar to the setup described in Erhard et al. [
30].
The tensile forces in the bridle line system, ${F}_{\mathrm{tether}}$, ${F}_{\mathrm{belt},\mathrm{left}}$, and ${F}_{\mathrm{belt},\mathrm{right}}$, were measured with three load cells. These data were recorded at a frequency of 10 Hz and were synchronized during flight. The instruments were calibrated before the flight using a tension test machine and proved to be reliable for many flight hours.
The apparent wind speed at the kite was recorded with two vane anemometers attached to the sides of the control pod, as indicated in
Figure 5b. Because the anemometers were aligned with the roll axis of the kite, they in fact only measured the component of the velocity along this axis. In
Figure 2b, this measured velocity component can be identified as
${V}_{\mathrm{a},\tau}$ and is related to the apparent wind speed
${V}_{\mathrm{a}}$ as follows:
This fundamental equation is based on the geometric similarity of the velocity and force triangles, linking the kinematics of the relative flow at the kite to the decomposition of the resultant aerodynamic force [
22]. Equation (
6) can be used to correct for the misalignment of the anemometer with the local inflow vector:
where
E is the parafoil’s glide ratio. During the retraction phase and turning maneuvers, the angle of attack can vary significantly, which leads to erroneous deviations of
${V}_{\mathrm{a}}$ because the misalignment correction uses a constant glide ratio. To reduce such inaccurate measurements, only the power and transfer phases were considered for validation when the kite was not steered, and the tether sag was small due to a high apparent velocity and pulling force. The manufacturer calibrated the vane anemometers, and before the flight, a new set of propellers was attached. Not measuring the angle of attack and the apparent wind speed directly is a clear disadvantage of the described method. Also, the control pod influences the airflow around the vane anemometers, such that the measured apparent wind speed deviates to some degree from the actual speed. Because of this disturbance effect and the error introduced when using the misalignment correction with a constant glide ratio, uncertainty remains in determining
${V}_{\mathrm{a}}$ during flight.
The tether angles
$\vartheta $ and
$\varphi $, as defined in
Figure 2a, were measured at the ground station with line angle sensors. Together with the recorded length of the reeled out tether, this allowed for the estimation of the position of the kite during the power phase. It was also possible to measure the angle between the tether and the control pod and use this to improve the
$\vartheta $estimate. The control pod was also equipped with an inertial measurement unit (IMU) to determine the accelerations in three axes such that both the orientation and turn rates of the kite system could be derived.
The glide ratio of the kite, defined as the assembly of the wing, bridle line system, and control pod, was not directly measured but estimated from the quasisteady theory of tethered flight, using the following equation: [
30]
where
$\vartheta $ is the angle between the wind velocity vector and the tether, as indicated in
Figure 2a. Equation (
8) is valid for a negligible effect of gravity and a glide ratio
$E\gg 1$, which is common for AWE applications.
The following data filtering and selection procedures were applied to the raw flight test data:
The data were smoothed with a 3 s moving average to level out shortterm fluctuations using the Matlab function movmean. It was found that this procedure provided a good smoothing behavior without influencing the measurement over a power cycle.
The symmetry assumption of the parafoil was enforced by selecting only data points during the straight flight when the steering belt position was not further than 5 cm from the neutral position in both directions.
To enforce the quasisteady flight assumption, the magnitude of the measured acceleration during the flight was used to select data points which satisfied a $0.1$ g offset.