# Spanwise lift distribution of a T-shaped hydrofoil using lifting line theory and lagrangian particle tracking measurements

Hervé Bonnard, Ludovic Chatellier, Laurent David

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Hervé Bonnard, Ludovic Chatellier, Laurent David. Spanwise lift distribution of a T-shaped hydrofoil using lifting line theory and lagrangian particle tracking measurements. 4th Colloquium on Vortex Dominated Flows – DisCoVor 2025, Jun 2025, Bozeman, United States. hal-05335346

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# Spanwise lift distribution of a T-shaped hydrofoil using lifting line theory and lagrangian particle tracking measurements

H. Bonnard<sup>a*</sup>, L. Chatellier<sup>a</sup> and L. David<sup>a</sup>

<sup>a</sup> Institut Pprime, Poitiers, France<fnref n="1" />

*Correspondence: herve.bonnard@univ-poitiers.fr

## 1 Introduction

Hydrofoils are one of the various lifting surfaces utilised on boats. They are used to decrease the resistance of a ship by lifting its hull out of the water in order to reduce its wetted area. With sufficient speed, a hydrofoil can lift totally the hull out of the water, the ship going from a displacement mode to a foiling mode. Hydrofoils can be found in many nautical sports, such as surfing, windsurfing or sailing, as well as in industrial applications such as passenger transport ships or coastal defence frigate.

The shapes of 3D hydrofoils are vastly different from plane wings. A common shape of hydrofoil is the "inverted T-shaped hydrofoil", commonly called a T-foil. Such hydrofoils consist of a mast (a vertical shaft such as the rudder of a boat), with a horizontal wing which produces lift at its tip. In addition to the influence of the tip vortex of the wing, T-foils also have to be designed for the impact of the interaction between the wing and the mast, as well as the free-surface proximity. For the latter, a previous study [1] has highlighted the change in flow behaviour and lift performances depending on the depth-based Froude number.

The interaction between the wing and the mast, at their junction, is however ill documented and harder to observe. This study highlights this interaction between the mast and the wing of a T-foil through the lift distribution over the wing. This distribution was computed using downstream experimental data and the lifting line theory.

## 2 Experimental setup and model

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Diagram of the physical and geometrical parameters.</figcaption></figure>

<table><caption>**Table 1:** Table of the physical and geometrical parameters.</caption><thead><tr><td>Parameters</td><td>Value</td></tr></thead><tbody><tr><td>Chord c</td><td>40mm</td></tr><tr><td>Angle of attack α</td><td>6°; 12°</td></tr><tr><td>Reynolds number Re_c</td><td>6 500; 20 000</td></tr><tr><td>Depth d</td><td>1.75 c</td></tr><tr><td>Depth-based Froude number Fr_d</td><td>0.21; 0.63</td></tr><tr><td>Wing half span s</td><td>3.50 c</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Picture of the experimental setup, with the hydrofoil, the laser volume and the four cameras.</figcaption></figure>

The experiences were carried out on an open water channel from the Pprime Institute. The main geometrical and physical parameters are described by the Figure 1, for which the values can be found in the Table 1. This current work used a fixed depth hydrofoil with two inlet velocities, leading to two Reynolds numbers and two depth-based Froude numbers. Moreover, those configurations were studied for two angles of attack, one before and one after the stall of the 2D section.

Lagrangian Particle Tracking [2] measurements were made downstream of the hydrofoil using LaVision's Shake-The-Box [3] algorithm. The experimental setup, showed on Figure 2, was presented at DisCoVor 2023 [4]. The lagrangian data were regularised on a grid using the Affine Least Square Transform (ALST) method [5].

Prandtl's lifting line theory describe the distribution of circulation $Γ$ over a finite wing using horseshoes vortices, as showed in Figure 3. A variation of circulation $dΓ$ over the

<footnote n="1">¹This work was made during H. Bonnard's PhD thesis which was funded by the French Defence Innovation Agency (AID)</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 3:** Diagram of the lifting line theory. From Hospodář et al. (2021) [6]</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Lift distribution $C_L(y)$ over the wing between the outside of the wing $y = -s$ and the centre of the wing $y = 0$, for $\alpha = 6^\circ$ and $Re_c = 6500$.</figcaption></figure>

wing, such as $\Gamma(y + \Delta y) - \Gamma(y) = d\Gamma$, leads to the same circulation being shade in the wing's wake. The circulation at a spanwise position $y$, $\Gamma(y)$, can be computed by integrating the shed circulation from the outside of the wing to the position $y$ with $\Gamma(y) = \int_{-s}^{y} d\Gamma$. The Kutta-Joukowsky theorem relates circulation to lift: $C_L = 2\Gamma/V_\infty/c$. Thus the circulation distribution leads to the lift distribution over the wing.

Using the experimental data, the longitudinal vorticity $\Omega_x$ can be computed for a fixed streamwise position $x$ in a vertical plane $(\vec{y}, \vec{z})$. The shed circulation $d\Gamma$ at the spanwise position $y$ can be computed by integrating the vorticity over the vertical dimension $z$, $d\Gamma(y) = \Delta y \times \int_{-\infty}^{\infty} \Omega_x(y,z) dz$, with $\Delta y$ the cell size in the $\vec{y}$ direction. Therefore, it can be used to compute the circulation $\Gamma(y)$ over the wing's span.

### 3 Results

The previously described method was applied for the four cases presented. The Figure 4 shows the lift distribution over the wing for the configuration $\alpha = 6^\circ$ and $Re_c = 6500$.

Looking at the figure from the right, we can observe four

main parts on the spanwise lift distribution of the wing:

* for $-1.1 \le y/s \le -0.9$, there is an important increase of lift due to the wing tip vortex which represents almost the full circulation for a rectangular wing such as the one used;

* for $-0.8 \le y/s \le -0.4$, the lift increase of about 10% showing that the distribution is not fully rectangular;

* for $-0.8 \le y/s \le -0.4$, we observe an important loss of about 40% of the lift coefficient;

* for $-0.2 \le y/s \le -0.0$, the lift increase again.

The last two parts of the curve, characteristic of a T-foil, are due to the interaction between the mast and the wing of the hydrofoil. Such loss of lift close to the symmetry plane is not taken into account by the majority of analytical models. For those models, the lift distribution is supposed to be always increasing when going closer to the symmetry plane.

An analysis of the vorticity fields used to compute the distribution highlighted the presence of vortices at the junction between the mast and wing. Especially, an important vortex was rotating in the opposite direction of the tip vortex, and was the cause of the lift drop. This shows that the mast-wing interaction needs to be taken into account when designing an hydrofoil has a modification of the lift distribution can leads to a loss of total lift or an increase of induced drag.

Moreover, this method was used to compute the total lift of the wing and was then compared to data from a 6-axis load sensor. The few configurations tried showed a good approximation of the lift, especially at high angle of attack.

### References

[1] Hervé Bonnard, Ludovic Chatellier, and Laurent David. Investigation of 3D effects and free-surface proximity influence on the flow around a hydrofoil using PIV measurements. In *20th International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics*, Lisbon, Portugal, 2022.

[2] Andreas Schröder and Daniel Schanz. 3D Lagrangian Particle Tracking in Fluid Mechanics. *Annual Review of Fluid Mechanics*, 55(1):511–540, January 2023.

[3] Daniel Schanz, Sebastian Gesemann, and Andreas Schröder. Shake-The-Box: Lagrangian particle tracking at high particle image densities. *Experiments in Fluids*, 57(5):70, May 2016.

[4] Hervé Bonnard, Patrick Braud, Ludovic Chatellier, and Laurent David. LPT measurements around a hydrofoil close to the free surface. In *2nd Direct in-Person Colloquium on Vortex Dominated Flows (DisCoVor)*, Breckenridge, Colorado, United States, May 2023.

[5] Bertrand Mercier, Lionel Thomas, Benoit Tremblais, and Laurent David. A Particle Based Approach For Improved Resolution PIV And TOMO-PIV Based On The Coherent Point Drift And The Affine Least-Squares Transformation. *Proceedings of the International Symposium on the Application of Laser and Imaging Techniques to Fluid Mechanics*, 21:1–13, July 2024.

[6] Pavel Hospodář, Armand Drábek, and Aleš Prachař. Aerodynamic Design and Strength Analysis of the Wing for the Purpose of Assessing the Influence of the Bell-Shaped Lift Distribution. *Aerospace*, 9(1):13, December 2021.