# Role of rotational and divergence effects induced by wind turbines wakes on near-ground air warming

Paul Boumendil<fnref n="1" />,<fnref n="2" />, Pierre-Antoine Joulin<fnref n="1" />, Hiba Zahia Toumi<fnref n="1" />, Quentin Rodier<fnref n="2" /> and Valéry Masson<fnref n="2" />

E-mail: paul.boumendil@meteo.fr

## Abstract.

Various studies, involving numerical simulations, satellite measurements, and field measurement campaigns, have demonstrated that wind turbine wakes can lead to changes in near-ground air temperature. To investigate these phenomena an Actuator Disk with Rotation (ADR) is implemented within the LES non-hydrostatic atmospheric model MesONH and validated using the NewMEXICO wind tunnel measurements and the VERTEX field measurement campaign. The NewMEXICO case allows the aerodynamic validation of the approach, especially the effect of the wake on wind speed components. On the VERTEX case, meteorological effects of wind turbines are studied under various atmospheric stability conditions through the application of homogeneous sensible heat fluxes at the ground level. Results show wake-related warming and cooling near the ground for stable and unstable conditions respectively, and no significant change for neutral conditions. These results show a good tendency agreement with the VERTEX campaign's measurements. The comparison between an Actuator Disk with No Rotation (ADNR) and an ADR simulation show the dual contribution of the rotation of the wake and the growth of the stream tube on the temperature change in the wake but also on the near-ground air temperature.

## 1. Introduction

The expansion in both the size and number of turbines within wind farms raises questions about the meteorological impact of such infrastructures. Field measurements [1, 2, 3] have indicated changes in meteorological parameters downstream the farms, which can propagate toward the ground level and lead to an increase in near-ground air temperature up to 2 K. This phenomenon is primarily attributed to the characteristics of the wake created by the wind turbines, where the wind speed deficit and the increase in turbulent kinetic energy (TKE) contribute to enhanced vertical mixing [4, 5, 6]. The wind turbine displaces air masses initially close to the ground upward and vice versa. This phenomenon disturbs the natural stratification of the atmosphere which can result in variations in temperature and humidity [4, 5, 7, 8, 9]. To numerically simulate this interaction between turbines and the atmosphere, various turbine models have been developed. The choice generally depends on the scale and precision of the expected results, as well as on the cost of the simulation. Coarser methods model turbines by increasing ground roughness [10] or by using momentum sinks and turbulence sources [4, 5, 7, 11]. On the other hand, more refined modeling integrates the geometry and aerodynamic properties of wind turbines, using body-forces, also known as actuator methods [8, 9, 12].

<figure><img src="image_1.png" /></figure>

<footnote n="1">¹IFP Energies Nouvelles, 1-4 Avenue du Bois Préau, 92852 Rueil-Malmaison, France</footnote>

<footnote n="2">²Meteo France, CNRM, 42 Avenue Gaspard Coriolis, 31057 Toulouse, France</footnote>
In a effort to precisely understand the underlying phenomena, we propose to follow the approach established by Joulin 2020 [13] and employ the fine-scale atmospheric model LES Meso-NH coupled with an actuator. However, aiming to reduce computational costs, a new actuator model called Actuator Disk with Rotation (ADR) will be implemented and validated in Meso-NH. Other teams has already implemented such method in LES code like Porte Agel et al. 2011 [12] and Mirocha et al. [14] within WRF-LES. ADR will initially be validated with data from the NewMEXICO case [15] and compared with other actuator methods. Subsequently, the coupling will be used to study meteorological variations in the wake of the turbines, particularly at ground level, comparing results with those from the VERTEX measurement campaign [1, 16].

## 2. Numerical model

### 2.1. Meso-NH

Meso-NH is an atmospheric model developed by the Centre National de Recherche Météorologique (CNRM - CNRS/Météo France) and the Laboratoire d'Aérologie (LAERO - UPS/CNRS) [17]. Meso-NH is capable of simulating the atmosphere from the mesoscale to the metric scales, along with dedicated numerical schemes to represent specific phenomena such as forest fires, urban areas, pollution, and wind turbines. Meso-NH is a non-hydrostatic, anelastic model [17]. In this work, the choice is made to solve the pseudo-compressible system of equations as defined by Durran in 1989 [18]. The equations are closed by the mean of a subgrid kinetic energy which depends on a mixing length as defined by Masson and Honnert in 2021 [19]. Wind advection is treated with the explicit Runge-Kutta centered 4th order scheme for time integration and a 4th order centered spatial scheme for space.

### 2.2. Actuator Disk models

Actuator disk models are based on body forces approaches. These techniques have the particularity of not explicitly solving the flow around the blades but applying forces to the flow in order to model its presence with drastically reduced computational costs. The Actuator Disk with No Rotation (ADNR) was theorized by Glauert and Tenot in 1947 and consists in modeling the wind turbine as a disk where a thrust force is applied using the known $c_t$ of the wind turbine. The Actuator Disk with Rotation (ADR), improves the first model by considering the characteristics of the wind turbine blades and their rotation. The blades are discretized into a series of airfoil profiles with known aerodynamic properties. Due to their nature as body forces, the Actuators models do not precisely replicate the real behavior of a wind turbine. Therefore, the tip vortices of the blades are not accurately modeled. The solution chosen is to artificially modify the forces by a factor along the radius, using the Glauert tip-loss correction [20].

### 2.3. ADR and Meso-NH coupling

The ADR and Meso-NH exchange information during the simulation. The coupling is done in its basic form: the ADR receives the wind field from Meso-NH, interpolates it into the ADR mesh. Then, it calculates the relative wind speed and angle of attack $\alpha$ for each blade segment, determines $c_l(\alpha)$ and $c_d(\alpha)$ using aerodynamic data, and utilizes these values to calculate the elementary aerodynamic forces, which are then transformed into the global reference frame of Meso-NH. As applying forces at a single point in Meso-NH can result in numerical instabilities, the forces are smeared, using a linear method [13], before being added to the momentum equations of Meso-NH. Sensitivity studies (not shown) have revealed that the ratios between the Meso-NH and ADR meshes must adhere to certain constraints to ensure the accuracy of the results: in the radial component $N_r \ge \frac{D}{2\Delta x}$; in the azimuthal component $N_\theta \ge \pi / \sin^{-1}(\frac{\Delta x}{D})$. These values align with findings in the literature [21].
### 3. Validation

#### 3.1. New MEXICO case

**3.1.1. Description of the project** The New MEXICO experiment [15] took place in the Large Low-Speed Facility, a German-Dutch wind tunnel. The wind turbine used in the project is a three-bladed turbine with a diameter of $D = 4.5$ m. The flow velocities are obtained through Particle Image Velocimetry (PIV) measurements along a line traversing the domain length and the wind turbine at $r = 1.5$ m and $\theta = 270^{\circ}$.

**3.1.2. Numerical setup** The wind tunnel conditions are simulated with Meso-NH, recreating the same conditions imposed for the Actuator Line (AL) by [13]. The study domain has dimensions of $40 \text{ m} \times 30 \text{ m} \times 30 \text{ m}$. Lateral boundary conditions are open, and the rotor is positioned at coordinates (15m, 15m, 15m). Three setups are simulated, as reported in Table 1.

<table><caption>**Table 1.** Numerical setups for AL and ADR simulations.</caption><thead><tr><th>Setup</th><th>Δx, Δy, Δz</th><th>D/Δx</th><th>N<sub>r</sub></th><th>N<sub>θ</sub></th><th>Δt</th></tr></thead><tbody><tr><td>AL-HR</td><td>0.1</td><td>45</td><td>45</td><td>-</td><td>0.001</td></tr><tr><td>ADR-HR</td><td>0.1</td><td>45</td><td>45</td><td>283</td><td>0.001</td></tr><tr><td>ADR-LR</td><td>0.25</td><td>18</td><td>36</td><td>57</td><td>0.01</td></tr></tbody></table>

**3.1.3. Results** Numerical results of the two resolutions, ie. low (LR) and high (HR), of the ADR model are compared with experimental measurements as well as the results of an AL-HR. Although validation was conducted for three wind cases, only the $10 \text{ m s}^{-1}$ case is presented for the sake of conciseness. Moreover, given the specific interest in wakes, only comparisons with wind components obtain from the PIV measurements of New MEXICO will be illustrated (Figure 1). The wind components are averaged over the last full rotation of the wind turbine.

The overall evolution of the wind components demonstrates good consistency between the computations from Meso-NH and experimental measurements. The induction effect upstream the rotor is well captured by the model. Only the *v* component is slightly underestimated upstream of the rotor by all numerical simulations. Downstream of the rotor, slight differences between measurements and numerical configurations can be noted. For each configuration the norm of *w* component is overestimated. In detail, disparities are observed between the ADR configurations. ADR-HR is close to AL-HR but shows oscillations on the *v* and *w* components. ADR-LR is close to experimental measurements on the *u* and *v*. ADR-LR model provides a good wake evaluation with a calculation time reduced by a factor of 415 compared to ADR-HR and by a factor of 774 compared to AL-HR. Therefore, this configuration will be used for the next part of our study.

**3.1.4. Vertical movement induced by wind turbines** The New MEXICO wind tunnel case has the advantage of allowing the visualization of the wake in a idealized inflow (ie. no shear, no veer and no turbulence). It allows to easily underline the main characteristics of the wake, such as the velocity deficit, the rotation of the wake and particularly the vertical movements. In Figure 2, the tangential velocity field near the position of the ADR is plotted. The projection contours of the ADR are represented by a black dashed lines. Two distinct zones are observed: on the periphery of the ADR, the velocity field diverges as the thrust of the wind turbine induces a spreading of the streamlines. In the center of the ADR, the wake rotates counterclockwise, opposite to the
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Comparison of mean velocity components *u*, *v*, and *w* along a line traversing the domain (in orange on the sketch) for different actuator configurations with experimental measurements of the New MEXICO case. The wind turbine is positioned at *x* = 0 m and represented by a dashed black line.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Tangential wind speed at 0.25*D* behind a ADR.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Vertical wind component at 0.25*D* behind a ADR.</figcaption></figure>

rotation of the wind turbine, following the principle of action and reaction. As mentioned in the introduction, the research conducted by [4, 5, 8] identified mixing and vertical movements as the potential cause of temperature variations associated with wake effects. To highlight these
vertical movements, the vertical component of the wind velocity $w$ has been isolated in Figure 3. A Yin-Yang shape becomes apparent, matching previous observations [14]. This distinctive shape arises from the two effects mentioned earlier. When viewed from upstream, the lower-left portion of the wake exhibits downward movements, while the upper-right part experiences upward movements.

## 3.2. VERTEX case

### 3.2.1. Description of the measurement campaign
The VERTEX (VERTical Enhanced miXing) measurement campaign [1, 16] took place during the summer of 2016 in the wake of a single Gamesa G90-2MW wind turbine located on the east coast of the United-States in a marsh along the Atlantic Ocean. This campaign aims at verifying whether vertical mixing near the ground is enhanced by the wind turbine. While this mechanism has been widely held responsible for the meteorological variations observed in the wake [6, 5], it has never been demonstrated close to the ground.

### 3.2.2. Numerical setup
The VERTEX case is replicated in an idealized way in Meso-NH. A cyclic domain of 4.5 km × 1 km × 1 km is defined with a horizontal resolution of $\Delta x = \Delta y = 5$ m, which will correspond to the “LR” setup (see Table 1) when the wind turbine is added. The vertical resolution $\Delta z$ gradually transitions from 4 m where the wind turbine is located to 20 m after reaching an altitude of 175 m. Simulations are decomposed into different steps:

(i) An initial neutral atmosphere is defined by specifying the wind and potential temperature profiles with an inversion zone of 5K between 450 m and 500 m of altitude which also sets the top of the boundary layer. To counteract the impact of the Earth's rotation on the wind profile and to ensure a wind direction of 0° at the wind turbine hub, the atmosphere is forced with a geostrophic wind empirically determined. The neutral atmosphere is spun up for 8 hours.

(ii) From this neutral atmosphere and by changing the homogeneous surface sensible heat fluxes, different atmospheric stability conditions are generated. The infrared cooling of the ground to establish a stable atmosphere is modeled by imposing a negative sensible heat flux near the ground (–14 W m$^{-2}$). The unstable atmosphere is modeled by imposing a positive sensible heat flux (25 W m$^{-2}$ and 50 W m$^{-2}$).

(iii) In order to prevent the return of disturbances induced by a wind turbine placed within the cyclic main domain, a nesting is established with a son domain with identical resolution, featuring one-way boundary conditions.

(iv) As the data for the Gamesa G90-2MW turbine were not available for modeling, a comparable turbine scaled at a diameter of $D = 90$ m is added to the son domain. Its rotation speed was scaled (considering an iso tip-speed ratio) and its twist angle was adjusted to achieve a $c_t$ and a power coefficient $c_p$ near classical value in our range of wind speed. The rotation speed, as well as the pitch angle of the blades, are set according to wind conditions. The disk is discretized using $N_{\theta} = 71$ and $N_r = 46$. The first 10 minutes allow the wind turbine wake to initialize. The output fields are averaged over 10 minutes.

(v) Step (iv) is replicated without a wind turbine, in order to have a control simulation.

### 3.2.3. Meteorological profiles
In Figure 4, few meteorological profiles of the neutral atmosphere generated in step (i), as well as stable and unstable atmospheres generated in step (ii), are plotted. The neutral atmosphere generated in the first step clearly shows the features of a conventionally neutral boundary layer: a constant potential temperature profile in the lower part of the atmosphere. The potential temperature $\theta$ is the temperature that an air mass would
have if it were adiabatically brought to sea level. The stable atmosphere generated by a negative ideal flux, shows a cooling of the temperature near the ground over time. The TKE decreases, as the turbulence is damped by buoyancy. The initiation of a low-level jet is observable at 200 m of altitude. Finally, the unstable atmosphere exhibits higher potential temperatures near the ground and an increased TKE.

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Space-averaged profile of horizontal wind, TKE and potential temperature at different times for 4 simulations with different prescribed surface sensible heat-flux (–14,0,25, et 50 W m⁻²). The wind turbine hub and rotor dimension are represented by solid and dashed black lines.</figcaption></figure>

3.2.4. *Temperature variation in the wake* In each of these atmospheres, an ADR is added, and the temperature field differences $\Delta T_{ADR} = T_{ADR} - T$ with the same simulation without ADR are computed (step v). Firstly, very different results are observed for the three cases studied. The wake of the stable wind turbine (Figure 5a) shows temperature variations up to +0.29K. These temperature variations seem to occur at two locations. The first starts before even reaching the wind turbine, in the so-called induction zone, below the level of the rotor. This positive temperature anomaly (referred later as local warming) due to the wind turbine appears until $x \approx 550$ m. This effect takes place outside the boundary of the rotor disk where the flow diverges (as illustrated in Figure 2) as the thrust of the wind turbine induces a spreading of the streamlines. The second temperature variation occurs at the level of the rotor (inside the dashed circles) when the flow goes through the wind turbine ($x > 550$ m) where the wake is rotated (as illustrated in Figure 2). The combined effects of rotation and divergence results in vertical movements as illustrated in Figure 3. Assuming that those vertical downward movements lead to adiabatic compression, the temperature of these air masses follows the adiabatic lapse rate. For a stable atmosphere, the near-ground lapse rate is lower, so the air masses reaching the ground from above have a higher potential temperature than the ambient air masses. The opposite phenomenon happens for air masses subject to upward movements. A lateral advection of air masses above and below the wind turbine hub is also noted, attributable to the *v* wind component influenced by Coriolis force. This results in the stretching of the wake and temperature variations. Consequently, the maximum temperature variation (Figure 8) is not axially aligned with the wind turbine, as found by Wu, Archer and Mirocha 2023
[8]. In the unstable simulation (Figure 5b), a cooling is observed just after the wind turbine, occurring between 500 m < *x* < 700 m. This is caused by the same phenomenon as before, with the difference that in this case the near-ground lapse rate is higher than the adiabatic one, so the air has a colder potential temperature above the wind turbine than near the ground. The vertical movements created by the wind turbine led to a cooling near the ground and warming above the wind turbine. In the far wake *x* = 1040 m ⇔ *x* = 6 D, only slight cooling anomalies remain. As an unstable atmosphere is much more turbulent, these local temperature variations are quickly mixed and attenuated. This is the same phenomenon that makes the wake length of wind turbines much shorter in unstable atmospheres compared to stable ones [12]. Finally, in the case of a neutral atmosphere, the potential temperature profile is constant in the lower layer of the atmosphere. The vertical movements of the wind turbine lead to a adiabatic compression at the same rate than the natural stratification of the atmosphere leading to no temperature change (not shown).

<figure><img src="image_6.png" /><figcaption>**Figure 5.** Average temperature difference between a simulation with and without ADR (a) for a stable atmosphere (9 h, -14 W m⁻²) (b) for an unstable atmosphere (8 h, 50 W m⁻²). The position of the ADR is represented by a solid circle and its projection by a dotted circle.</figcaption></figure>

3.2.5. *Comparison of temperature variation in the wake between ADR and ADNR* The role played by the rotation of the wake and vertical movements has already been noted by Xie 2017 [9] and Wu in 2023 [8]. However, the distinct role of wake rotation and divergence has never been addressed according to the authors’ knowledge. To verify this hypothesis, the temperature variation in the stable wake is compared between an ADR (Figure 5a) and an ADNR (Figure 6). The ADNR do not take rotation into account but only considers wake widening (divergence effects). It is parameterized with a trust coefficient *c*<sub>*t*</sub> = 0.745, identical to the one extracted from the ADR simulation. Since rotation is not considered, no significant
temperature variation occurs within the boundaries of the ADNR represented by a dotted circle in Figure 6. However, below the ADNR, the divergence effect leads to temperature increase. To highlight the difference between the results of ADR and ADNR, $\Delta T_{ADNR} - \Delta T_{ADR}$ is plotted in figure 7. The differences are primarily contained within the rotor area as rotation is not considered anymore. In the far wake, negative differences are observed at ground level, indicating an underestimation of temperature variations by the ADNR. This is confirmed in Figure 8, where regardless of the stability, the temperature difference at the $z = 2$ m level is systematically lower for the ADNR model. Although flow divergence near the wind turbine induces ground-level temperature variations, flow rotation plays a crucial role in this phenomenon.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Average temperature difference for a stable atmosphere (9 h, -14 W m⁻²) between a simulation with and without ADNR.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 7.** Average temperature difference for a stable atmosphere (9 h, -14 W m⁻²) between a simulation with an ADNR and with an ADR.</figcaption></figure>

### 3.2.6. *Temperature variation on near-ground air*
The temperature variation $\Delta T_{ADR}$ is computed in the wake near the ground at the same distance that the VERTEX campaign (Figure 8). The maximum warming along the y axis is then plotted (Figure 9) as a function
of the lower-rotor lapse rate between 49 m and 10 m as defined by Wu and Archer 2021 [16] (Equation 1). If $\Gamma_{LR}$ is above the dry adiabatic lapse rate $\Gamma_d = 0.0098 \text{ K}^{-1} \text{ m}$, the atmosphere is defined as unstable. If $\Gamma_{LR}$ is below, the atmosphere is defined as stable. When $\Gamma_{LR}$ is close to $\Gamma_d$ with a margin of $\pm 0.002$, the atmosphere is defined as neutral [16].

$$ \Gamma_{LR} = - \frac{T_{49 \text{ m}} - T_{10 \text{ m}}}{49 - 10} \qquad (1) $$

<figure><img src="image_9.png" /><figcaption>Figure 8. Mean temperature difference in the wake along the y axis downstream an ADR and an ADNR at $z = 2 \text{ m}$ and $x = 6 \text{ D}$ for different stable atmospheres.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 9. (a) Temperature profile for several stability cases (b) Temperature difference at $x = 6 \text{ D}$ behind the ADR and $z = 2 \text{ m}$ for several stability cases depending on the lower-rotor lapse rate between 10 and 49 m (equation 1). Experimental data from Wu and Archer 2021 [16].</figcaption></figure>

For stable and unstable atmospheres, a similar trend to the VERTEX campaign is observed: an increase in temperature variation near the ground as the atmosphere becomes more stable or unstable. However, the amplitude of these variations in the simulation is slightly lower than the experimental values. In the case of a neutral atmosphere, no notable temperature variation in the wake has been found. The underestimation of temperature variations near the ground is a recurring issue in simulations combining the LES method and actuators [8, 9]. The common point among all these simulations is the idealized aspect of the imposed ground heat flux for generating an atmosphere. Therefore, the need for faithful modeling of the surface and its interactions with the atmosphere appears necessary. The difference between numerical and experimental results can also be explained by the imposed ground flux that can have an influence on the temperature measured at ground level itself. Furthermore, the stability calculation method in the VERTEX
case is very limited as it only indicates static stability between two levels without considering
elsewhere. Moreover, the stability calculated in the VERTEX campaign uses virtual temperature
to account for the ambient humidity of the marsh while our atmospheres are dry.

3.3. Conclusion

In an effort to better understand the interactions between wind turbines and the atmosphere, the ADR model has been implemented into the LES Meso-NH atmospheric model. This coupling has been firstly validated using the New MEXICO measurement campaign, confirming the accurate modeling of wake characteristics and the effects induced on the flow by the wind turbine. A sensitivity study has been conducted to determine the optimal discretization for the ADR that balances fidelity and computational cost. Subsequently, the ADR method was used to investigate the impact of wind turbines on temperature variations, particularly near the ground. Simulations confirmed the significant role played by the atmospheric stability and vertical movements induced by the wind turbines. Temperature variations of up to +0.25K and -0.076K were respectively found at ground level at x = 6 D downstream of the turbine for stable and unstable atmospheres, while no significant change was found for neutral conditions. The dual origin of vertical movements and temperature variation by wind turbine was also highlighted by comparing the results of the ADR and ADNR. The divergence mainly induces temperature variations above and below the wind turbine, while the rotation of the wake induces temperature variations not only at the turbine height but also at ground level in the far wake. In future work, modeling of the surface and its interactions with the atmosphere will be conducted to alleviate the use of ideal surface heat flux. This could be achieved by utilizing reanalysis data to conduct meteorological analysis on the VERTEX case. Later on, the cumulative effects induced by wind farms will be studied.

Acknowledgments

New MEXICO data used have been supplied by the consortium which carried out the EU FP5 project MEXNEXT. Authors would like to express their gratitude to Dr. Christina Archer for allowing the reuse of data from the VERTEX campaign and article [16].

References

[1] Archer, C., Wu et al. *Journal Of Turbulence*. **20**, 64-92 (2019,1).

[2] Rajewski et al. *Bulletin Of The American Meteorological Society*. **94**, 655-672 (2013,5).

[3] Smith, C., Barthelmie, R. & Pryor, S. *Environmental Research Letters*. **8**, 034006 (2013,9).

[4] Baidya Roy et al. *Journal Of Geophysical Research: Atmospheres*. **109**, 2004JD004763 (2004,10).

[5] Baidya Roy, S. & Traiteur, J. *Proceedings Of The National Academy Of Sciences*. **107**, 17899-17904 (2010,10).

[6] Armstrong et al. *Environmental Research Letters*. **11**, 044024 (2016,4).

[7] Baidya Roy, S. *Journal Of Wind Engineering And Industrial Aerodynamics*. **99**, 491-498 (2011,4).

[8] Wu, S., Archer, C. & Mirocha, J. *Wind Energy*, 1-22 (2023).

[9] Xie, S. & Archer, C. *ABoundary-Layer Meteorology*. **165**, 87-112 (2017,10).

[10] Keith et al. *Proceedings Of The National Academy Of Sciences*. **101**, 16115-16120 (2004,11).

[11] Fitch et al. *Monthly Weather Review*. **140**, 3017-3038 (2012,9).

[12] Wu, Y. & Porté-Agel, F. *Boundary-Layer Meteorology*. **138**, 345-366 (2011,3).

[13] Joulin, P. et al. *Frontiers In Earth Science*. **7** (2020).

[14] Mirocha et al. *Journal Of Renewable And Sustainable Energy*. **6**, 013104 (2014,1).

[15] Schepers et al. *Journal Of Physics: Conference Series*. **555** pp. 012089 (2014,12).

[16] Wu, S. & Archer, C. *Monthly Weather Review*. **149**, 879-898 (2021,3).

[17] Lac et al. *P.Geoscientific Model Development*. **11**, 1929-1969 (2018,5).

[18] Durran, D. R. *Journal of Atmospheric Science*. **46**, 1453–1461 (1989).

[19] Honnert, R., Masson, V., Lac, C. & Nagel, T. *Frontiers In Earth Science*. **8** pp. 582056 (2021,1).

[20] Glauert, H. Airplane Propellers. *Aerodynamic Theory*. pp. 169-360 (1935).

[21] Moens, M. Thesis, École Polytechnique de Louvain (2018).