Dear reader,

in autumn 2021, we started working on hydrofoil concepts to increase the efficiency of electrically powered ferries as part of the BMBF-funded research project E2MUT (Emission-free electromobility for maritime transport). In this article, we will introduce you to the technical basics of a hydrofoil. You will better understand the advantages of hydrofoils and learn about their field of application. In doing so, we present you with results from the research activities of ar engineers GmbH, which include results of the running resistance calculation of various hydrofoil vessel concepts.

## Retrospect on Foiling

As early as the beginning of the last century, with the development of the first aeroplanes, the idea of using hydrofoils on ships emerged (e.g. by Enrico Forlanini around 1900). The idea was to position the wings (foils or hydrofoils) below the hull and thus be able to use the dynamic lift generation of hydrofoils, which is efficient at high speeds. So the motivation was to be able to reach higher speeds with a given drive system. This motivation is also found today in sailing racing, for example in the Americas Cup, where underwater wings have been used since 2013.

The French trimaran Hydroptère, which held the speed record for sailing over one nautical mile at 41.5 knots between 2007 and 2012, can be cited as a trend-setter here. In times of global climate change, foil technology plays a central role in making urban maritime transport emission-free. The main focus is on electric drives with hydrogen or battery-based energy storage systems. In this context, the advantage of foil technology lies particularly in energy saving, as the range is more limited by modern energy storage than compared to fossil fuels. At the same time, foil technology combines the characteristics of high cruising speed and low energy consumption that are important for passenger transport. In the E2MUT research project, we would like to incorporate precisely these advantages of foil technology in order to make a contribution to an emission-free mobility future.

## Hydrostatic and hydrodynamic relations on a hydrofoil ship

A hydrofoil ship is characterised by the fact that hydrofoils are fitted in the underwater area of the ship’s hull, which generate dynamic buoyancy. The property “dynamic” describes that the lift is dependent on the inflow velocity, or more precisely, that it changes proportionally to the square of the inflow velocity. In a hydrofoil, therefore, two different types of lift generation come into play. On the one hand, the dynamic buoyancy just described and, on the other, the hydrostatic buoyancy of the ship’s hull.

In addition, a ship’s hull can also generate dynamic lift at high speed, also known as planing. The planing of a ship’s hull depends to a large extent on the shape of the hull. In the class of catamaran hulls considered here, partial planing is achieved at cruising speeds of approx. 22 knots, but this is significantly less effective than the foil systems considered. At low speeds, the share of dynamic lift of the fuselage is vanishingly small.

The forces acting on a generic hydrofoil, in this case with a tandem foil system, are shown in Figure 1. The hull generates the buoyancy *B*, the hull resistance *D _{H}* and the weight force

*W*, which also includes the weight of the foil system for the sake of simplicity. Each foil then generates a lift

*L*and drag

_{i}*D*, which depends on the relative incident flow

_{i}*v*.The propulsive force

*P*of the propulsion unit is entered at the rear foil, for example.

The total drag of the hydrofoil can be calculated if interference resistances are neglected:

calculated In order to further illustrate the relationships, it is useful to introduce so-called glide ratios, which are commonly used in aviation. A glide ratio reflects the ratio of lift and drag as a dimensionless number and can be understood as a measure of the efficiency of a means of transport. The glide ratio of the foils and the hull are defined with:

definiert. If we now assume that the hull operates in pure displacement mode, a significant difference between hull and foils can be worked out. This difference becomes clear when the dependencies of the force variables are included as arguments in the equations (lift coefficient *C _{L}*, density of water

*ρ*, drag coefficient

*C*, acceleration due to gravity

_{D}*g*, displaced water volume of the hull

*V*):

The difference now is that with the foils both the lift and the drag depend quadratically on the speed. In the case of the fuselage, only the drag is velocity dependent if the fuselage is operating in pure displacement mode or the components of dynamic lift are negligible. In fact, the glide ratio of the foils remains approximately constant as the speed increases, provided that their angle of attack remains constant. The glide ratio of the fuselage is particularly high at low speed, or goes towards infinity, at zero speed. At higher speeds the fuselage drag increases strongly, thus the glide ratio of the fuselage is continuously reduced.

In the following, the evaluation of the glide ratios of equation (3) is presented on the basis of a concept of a flying hydrofoil. The concept of the hydrofoil is shown in Figure 2. The wing area of the main foil is 9.2 m x 1.5 m and generates a lift of 50 tons from 28 kn. In total, the foil system is designed to generate 80 tonnes of lift.

To calculate the buoyancy and resistance of the foil system, a separate foil calculation software was developed which takes into account wave resistance, induced resistance, frictional and pressure resistance, as well as the effect of the water surfaces on the buoyancy coefficient. The resistance of the hull was calculated over the speed range from 6 knots to 40 knots for different displacements with a potential-theoretical CFD software by Tamsen Maritim GmbH. The performance data of the foils and the fuselage are fully automatically merged in our in-house foil calculation software and allow a calculation of the drag of the hydrofoil over the entire speed range. This approach is particularly suitable for the concept phase for the initial assessment of the performance of hydrofoils. Figure 3 shows the evaluation of the glide ratio of the foil strut system and the fuselage and the evaluation of the generated lift by the foil system. Struts are the vertical structural elements that connect the horizontal foils to the fuselage. Diese werden auch als Struts bezeichnet.

It is clear from Figure 3 that the hull’s glide ratio is very high at low speeds, making the use of foil systems at low speeds impractical. From a speed of about 15 knots, the foils then become continuously more efficient than the hull. The overall effectiveness of the foil system then still depends on the buoyancy generated, which determines how far the hull is still submerged in the water. In the design phase, the lift and the efficiency of the foils are opposing parameters, because increasing the angle of attack of the foils increases the lift but at the same time decreases the efficiency (glide ratio). The relationship just described between the glide ratio and displacement of the fuselage and the lift and glide ratio of the foils can be explained by combining equations (1) and (2) and *B* = *W* - *L* to:

where *L* is the total lift of the foil system and *E* is the glide ratio of the entire foil system. The evaluation of equation (4) is shown in figure 4 as Total drag “Full Flying Cat”. In addition, the resistance of the fuselage without foiling system is shown, the immersion depth of the fuselage with foiling system, the resistance of the fuselage with foiling system and the resistance of the pure foiling system.

## Summary

Figure 4 clearly shows the effect that foiling systems can have. Drag is reduced by 44% at a speed of 32 knots compared to the drag of the bare hull at 22 knots. From a speed of 28 knots, the hull can be lifted completely out of the water, and as the speed increases, the overall drag can even be reduced as the foils are set lower. Significant savings can also be achieved with much smaller foiling systems, which only partially lift the hull out of the water.

The hydrofoil concepts studied in the E2MUT research project will be presented in the next E2MUT blog post. Stay tuned!