Hydrofoil wing design basics: lift and drag

The basics of hydrofoil wing design come down to one idea: a good wing carries the load you want, at the speed you choose, for the smallest possible force: we measure that quality by the lift-to-drag ratio. Tip losses degrade it, and the shorter the wing, the worse they get. That is why a high aspect ratio, a well-placed lift distribution and, where needed, winglets all pay off. A propeller blade obeys the same laws: a propeller can be regarded as a rotating wing.

This page recalls the basic notions you need to build a wing, a hydrofoil or a propeller. We make no distinction here between a wing and a foil: apart from cavitation, the principle is the same.

On this page:

Lift, drag and the lift-to-drag ratio
The lift and drag formulas
The blade or wing cut into elements
Losses at the wing tip or blade tip
Aspect ratio and wing shape
Winglets
Choosing the dimensions in Heliciel

1 : Lift, drag and lift-to-drag ratio of a wing

Understanding what happens on a wing, and making it fly, took humanity a few centuries, and we are still far from doing it with as little energy as nature does. The best quality criterion remains the wing's first purpose: carrying a load from one point to another. Moving an anvil 10 metres with a ten-megaton charge would be impressive, but a little expensive.

Mushroom cloud of a nuclear explosion at sea

One way to move an anvil. It lacks finesse.

The goal is therefore to make the heaviest possible object fly with the least possible energy. Transport speed matters too: taking ten days to cover 10 metres is something even an oyster can do.

An open oyster

A Normandy oyster, champion of low speed.

A quality wing therefore carries the desired load, at the chosen speed, with the minimum force applied to it. To estimate that quality, we compare the force the wing produces, the lift, with the force it consumes, the drag, at the intended operating speed: lift-to-drag ratio = lift / drag.

Lift (newtons) is the friendly force, perpendicular to the fluid stream, that carries the wing and generates the propeller's thrust. Lift is good!
Drag (newtons) is the nasty force that pulls the wing along with the flow and stops the wing from advancing, or the propeller from turning. Drag is bad! It is the drag that you must compensate by supplying energy to the propeller shaft.

Wind-tunnel or tank tests (numerical or experimental) give two dimensionless coefficients for each airfoil. They are the basic data for computing the forces on wings, foils and propeller blades:

Cd, the drag coefficient (Cx in French) ;
Cl, the lift coefficient (Cz in French).

The lift-to-drag ratio of an airfoil is Cl/Cd. A high ratio lifts a heavy load with little energy. By default, Heliciel applies to the blades the angles that give the maximum lift-to-drag ratio.

These coefficients hold for a wing of infinite length: losses and vortex phenomena are not included. We speak of 2D performance (two dimensions) for the infinite wing, and of 3D performance for a real wing of finite, known length.

2 : Lift and drag formula for a wing

Lift and drag are written the same way, with Cl for one and Cd for the other:

Lift (2D) = Cl × S × ρ × V² / 2

Drag (2D) = Cd × S × ρ × V² / 2

with forces in newtons (N), S the wing area (m²), ρ the fluid density (kg/m³) and V the fluid speed (m/s). Both forces grow with the square of the speed: doubling the speed multiplies lift by four, and drag as well.

3D prototype of a propeller in the Heliciel interface: the blade elements are drawn as coloured bars, with drag and lift histograms below the view

Drag and lift on a propeller blade, shown element by element in Heliciel (earlier version of the interface).

3 : Blade element calculation for wings and blades

A wing could be defined by a single airfoil of constant width. But to hold together mechanically and to perform well, real shapes are more complex. To analyse them, it is practical to cut the wing or the propeller blade into slices, from the root to the tip. The sections of these slices are the airfoils: they describe the shape of the blade in cross-section.

A portion framed by two airfoils is a blade or wing element. Heliciel shows this division from its start screen.

The elements of the wing or blade are the parts lying between two airfoils.

The known performance of the airfoils that frame an element lets us compute its drag and lift. This blade-element analysis is the basis of propeller calculation by the BEM (Blade Element Momentum) method. It uses 2D performance, so it needs a correction for the length of the wing, the third dimension, the one of leaks. A page covers the blade element theory for thrust and propulsion propellers.

4 : Wing tip losses and tip vortices

Drag and lift computed with Cd and Cl are theoretical values: they ignore the losses of the third dimension. An ideal blade with no leaks would have the efficiency of the 2D performance of its airfoils. These theoretical values, measured in a wind tunnel or computed by numerical methods, correspond to a wing of infinite span.

Lift results from the pressure difference between the underside and the top of the wing. The larger this difference, the more the fluid wants to cross to the other side by the shortest path it can find. That path is the tip of the wing or blade.

Front view of a wing: overpressure below, low pressure above, and fluid flowing around the wing tip, shown by two curved arrows

Overpressure under the wing, low pressure above: the fluid flows around the wing tip.

Front photo of a crop-duster aircraft: a visible vortex curls up at each wing tip

The photo speaks for itself: a vortex forms at each wing tip.

The moral comes in two lessons:

Keep the lift away from the tip. The more lift there is near the wing tip, the more leakage there is. If the lift is spread as far from the tip as possible, the fluid has a longer way to go and the leaks decrease. In theory, the wing shape that spreads lift this way is the ellipse: that is why some aircraft, such as the Spitfire, adopted elliptical wings. In practice, a trapezoidal shape is enough to come close to the optimum.

A Spitfire seen from above in a turn, with its elliptical wing

The Spitfire's elliptical wing keeps the lift away from the tip.

Lengthen the wing. The shorter the blade or wing, the more leakage there is. We therefore benefit from spreading the area over a large length and a small width. The notion of aspect ratio measures this distribution.

5 : Wing aspect ratio and wing shape

For a rectangular wing, the aspect ratio is the ratio of the span (length of the wing) to the chord (width of the wing). One formula holds for all wing shapes and uses the area:

Aspect ratio = span² / wing area

A high aspect ratio spreads the lift over a large span: the pressure difference between the lower and upper surfaces then gives the fluid less urge to cross to the other side. That is very good. The aspect ratio page details its effect.

A glider in flight, with a very long and narrow wing

A glider: a long, narrow wing, hence a high aspect ratio.

To compute the performance of a wing including its vortices and tip losses, we must know their effect on the 2D performance of the airfoils. We do it by computing the deflection of the fluid induced by the vortices: these induced velocities correct the angles each airfoil sees, and we deduce the performance of the finite-span wing (3D). The calculation rests on vortex theory, which links lift to vortices. To learn more: the finite-span wing theory based on the distribution of vortices.

Here is the influence of shape and aspect ratio: vortex sheets and tip losses of four wings, shown schematically in the Heliciel software.


Heliciel video, episode 14: designing and shaping a hydrofoil (8 min 21). Lengthening the foil from 1 m to 1.3 m raises its lift-to-drag ratio from 28.5 to 30.3; chords, profile and generatrix complete the study. Episode page	Rectangular wing, 9 metres long. Aspect ratio = 18. Lift-to-drag ratio = 32

Tapered wing, 2 metres long. Aspect ratio = 5. Lift-to-drag ratio = 19.	Tapered wing, 9 metres long. Aspect ratio = 33. Lift-to-drag ratio = 36

The lift-to-drag ratio rises from 16 to 32 for the rectangular wing as the aspect ratio goes from 4 to 18, and from 19 to 36 for the tapered wing as it goes from 5 to 33. Lengthening the wing and tapering it toward the tip therefore gains lift-to-drag ratio.

6 : Winglets and elliptical wings against tip losses

Another device reduces tip leakage: a barrier placed at the wing tip, the winglet. That is very good. Combining a high aspect ratio, a good lift distribution and winglets is very, very, very good.

Solar-powered aircraft with a very large span seen in night flight: very long wings, tapered toward the tips and ending in winglets

A wing designed for optimum efficiency: high aspect ratio, taper toward the tip to keep the lift away from it, and winglets as a barrier against flow around the tip.

The Solar Impulse project aims to make an aircraft powered exclusively by solar energy take off and fly autonomously, day and night, up to a round-the-world flight without fuel or pollution. And that is very, very, very good!

7 : Choosing wing or foil dimensions in Heliciel

With these principles, you can guide your choices when you set the geometry of your wing, your foil or your propeller in Heliciel. You decide the span, the chord and the shape; Heliciel computes the corresponding performance.

In Heliciel
In the 2: Blade geometry tab, 2.1: Blade dimensions sub-tab, a series of sliders shapes the blade or wing in all its dimensions: length, root chord, tip chord, thickness. Presentation of the software.

The sliders of the Blade dimensions tab, here for a propeller (earlier version of the interface).

One phenomenon is specific to hydrofoils and marine propellers: cavitation.

To go further, we can:

study the influence of aspect ratio;
compare the calculation methods for wings and propellers;
go deeper into the finite-span wing theory, based on the distribution of vortices;
build a wing step by step with the introductory tutorial.

The low pressure on the upper surface and the acceleration of the air show up in simulation: the CFD wing case shows a real wing at three mesh fineness levels.