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The ability of an airplane to be able to launch into the sky and keep floating is an incredible engineering achievement. In order to understand how airplanes launch and keep flying, we need to understand three important concepts, which are lift, drag, and pitch. This article will explain what they are using the airfoil theory and its application in airplanes and rotating machinery such as wind turbines.

The simplest way to understand lift generation is through an example of an airplane. Consider the following figure:

For an airplane flying, we can see the forces that are acting on the airplane (weight and drag) and generated by the airplane (thrust and lift). The airplane wing is made of several airfoils stacked along its axis. If we understand the physics for a single airfoil we can understand it for the whole structure of the wing.

Consider a streamline representing state ‘1’ before interacting with the airfoil and state ‘2’ after interacting with the airfoil. Figure 3 shows the schematics for the associated variable terminologies:

Applying Bernoulli’s equation along this streamline gives:

$$ P_2 – P_1 = – \frac{1}{2} \rho(U_2^{2} – U_1^{2}) \tag{1} $$

Where,

- \(P\) is the static pressure
- \(\rho\) is the density of the fluid medium
- \(U\) is the fluid velocity

The shape of the airfoil is designed for different purposes. This allows the airfoil to have different fluid velocities along its length. The change in velocity leads to a change in pressure relative to the local atmospheric pressure according to Bernoulli’s equation.

These corresponding pressure loads act normal to the surface and can be integrated to give the total force exerted per unit surface area.

**Top Surface**

Now, if we imagine the flow of air only over the top surface of the airfoil in 2D, we can see that the velocity accelerates around the curvature of maximum thickness. This means that \(U_2\) will be more than the freestream velocity \(U_1\) and thus the static pressure \(P_2\) will be lesser than the freestream pressure \(P_1\). This creates suction on the top surface and pulls the airfoil upwards contributing towards the lift.

**Bottom Surface**

In a similar fashion, the pressure distribution on the bottom surface can be analyzed. For this particular airfoil, the flow around the leading edge accelerates creating a suction pressure pulling it downwards while further ahead the flow decelerates causing an increase in pressure. Since this pressure is higher than the local atmospheric pressure it will act into the airfoil thus contributing to the lift.

Note that each airfoil will have a different flow pattern around it that solely depends on the shape of the airfoil.

This pressure distribution on both the top and bottom surface can now be integrated to give a total force vector \(\vec F\) whose perpendicular and parallel components to the flow give us lift and drag respectively.

Lift is the component of the total force vector \(\vec F\) that works through the center of pressure of an object and is perpendicular to the incoming flow. For a zero angle of attack, it acts opposite to the weight (see Figure 1). Lift is a mechanical force that is produced by the movement of an object through the air. Therefore, it has a magnitude and direction.

It is also important to remember that lift needs two things:

**Fluid**: Lift only generates when there is an interaction between a solid object and a fluid.**Motion**: Lift only occurs when there is a difference in velocity between the solid object and fluid. This motion also introduces drag, which is called induced drag.

$$F_l = \frac{1}{2} \rho V^2 A C_l $$

Where:

- \(F_l\) \([N]\) is the sum of forces in the specified lift direction;
- \(C_l\) is the lift coefficient;
- \(ρ\) \([kg/m³]\) is the density of the fluid;
- \(V\) \([m/s]\) is the freestream velocity;
- \(A\) \([m²]\) is the reference area.

Drag is the component of the total force vector \(\vec F\) that works through the center of pressure of an object and acts parallel to the direction of the incoming flow. For a zero angle of attack, it acts opposite to the thrust of the airplane (see Figure 1).

However, drag generates due to the velocity difference between the solid body, in this case, an airplane, and the fluid. Therefore, drag only generates when there is a relative movement between an object and a fluid. If one of those two things does not exist, then there is no drag. For a flying object there are two important drag forces, which are:

**Parasitic Drag**: Parasitic drag is a combination of form drag and skin friction drag.**Form Drag**: This type of drag depends on the shape of the object. The drag force can be calculated by integrating the local pressure and multiplying it with the surface area of the object.

**Skin Friction Drag:**This drag develops from the direct interaction between the fluid and the skin of the object. The higher the wetted area, the higher the skin friction drag is.

**Induced Drag**: Induced drag or lift-induced drag is caused by the generation of lift. In airplanes, vortices form at the wingtips producing a swirling flow that disturbs the airflow distribution around the wingspan. This reduces the wing’s ability to generate lift and thus requires a higher angle of attack for the same lift. This results in the shifting of the total aerodynamic force rearwards which increases the drag component of that force.

This phenomenon also appears in lift-based turbomachinery such as wind turbines.

Note

The generation of wingtip vortices and the trailing of eddies is a 3D phenomenon. This is one of the reasons for why drag results differ between 2D and 3D simulation studies.

$$F_d = \frac{1}{2} \rho V^2 A C_d $$

Where:

- \(F_d\) \([N]\) is the sum of forces in the specified drag direction;
- \(C_d\) is the drag coefficient;
- \(ρ\) \([kg/m³]\) is the density of the fluid;
- \(V\) \([m/s]\) is the freestream velocity;
- \(A\) \([m²]\) is the reference area.

The troubleshooting of errors related to the setting up of lift and drag coefficients in SimScale is discussed in the following knowledge base article:

Pitch is an up and down movement of the nose of an aircraft about an axis. This motion will heavily influence the lift generated by the wings of the airplane.

From the figure below we can imagine an axis in the plane of the trajectory going from one end of the wing to the end of the other wing through the center of gravity and the motion of the airplane based on that axis.

Upward pitching will lead to an increase in the angle of attack thus increasing the lift component of the total force (see Figure 10 below). This is because of the added downward deflection leading to an accelerated flow over the airfoil. The more upward movement, the larger the lift that the wings generate, but only up to the point of stall (discussed below).

With reference to an airfoil the schematics for the relation between pitching and angle of attack are shown below:

For an airfoil, the angle of attack is the angle between the incident freestream fluid and the chord line extending from the leading edge to the trailing edge. The pitch angle, on the other end, is the angle made by the chord line with any reference plane. This reference plane can be a flat ground for a flying object or the plane of the rotor disk of a turbine.

Depending on the reference plane, the angle of attack may be greater than, less than, or the same as the pitch angle.

An increase in the angle of attack causes the ratio of lift force to drag force to increase up to a certain point. Increasing the angle of attack beyond this point leads to a sudden decrease in lift and a sharp increase in drag entering into a state of stall.

Aircrafts should avoid stall at all costs as this would mean an inadequate amount of lift force to balance the weight. The phenomenon of stalling can be observed in compressors too causing non-uniform blade rotation slowing down the rotor while also causing blade failure.

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Horizontal or vertical axis rotating machinery such as wind turbines, jet engines, centrifugal pumps, and compressors consist of a rotor or impeller with a set of blades arranged in a symmetric fashion. Like the airplane wings, these blades are also constructed from a set of airfoils.

Each airfoil has a different function. The ones closer to the root ensure structural rigidity while the ones in the middle and towards the tip contribute mostly to the lift.

A major difference between airplanes and rotating machinery is that in the latter the airfoil receives the wind/fluid velocity from two components: freestream fluid and rotation of the blades.

Let us consider a horizontal axis wind turbine blade being observed from the top in the plane of the rotor when the blade is at the bottom-dead-center. The wind approaches horizontal to the ground and the rotation is clockwise. The schematics for this scenario, in top view, is shown in the figure below:

Each section of the blade has a different set of airfoils meaning that each section serves a different purpose as previously discussed. The skeleton clearly shows how each airfoil has a different pitch angle. The ones close to the root/hub are heavily pitched while the ones near the tip require less pitching. The explanation for this is given in the following section.

Flow angle \(\phi\) is the angle made by the resultant incident wind speed and the plane of the rotation. This flow angle will become lesser and lesser as we move from the root of the blade to the tip. This is because the tangential velocity near the root is lower than at the tip.

$$ \vec v = \vec \omega \times \vec r $$

Where:

- \(\vec v\) = tangential velocity
- \(\vec \omega\) = angular velocity
- \(\vec r\) = radial vector in the direction away from the root/hub

A simple observation from Figure 16 tells us that this flow angle is a sum of the the angle of attack and the pitch angle. Therefore,

$$ \phi = \alpha + \beta $$

Where,

- \(\phi\) is the flow angle
- \(\alpha\) is the angle of attack
- \(\beta\) is the pitch angle

As discussed above, an increase in the angle of attack leads to stall and therefore it becomes paramount to keep \(\alpha\) within the prescribed limits especially at the root where \(\phi\) is large. This is done by excessively pitching the airfoils near the root by increasing the pitch angle \(\beta\).

Did you know?

We can achieve increased efficiency in a Francis water turbine by reducing the separation region around the blade airfoils.

Check the following image that shows velocity characteristics on a plane cut through the spiral casing of a Francis turbine:

Observe the velocity vectors and magnitude contours in the turbine on the right. Changing the blade angle of the outer blades (stator) results in the reduction of the effective angle of attack (AoA). This reduces the separation (blue regions with low velocities) and the flow becomes attached.

To get more insights into this simulation refer to our demo and discussion series here.

SimScale allows its users to calculate lift and drag forces and lift, drag, and moment coefficients on specific surfaces of the geometry. This requires defining certain parameters like the reference length and area, freestream velocity, pitch axis, and some other important parameters. The setup is well documented in the following article

Important Information

If none of the above suggestions solved your problem, then please post the issue on our forum or contact us.

References

- Beginner’s Guide to Aerodynamics”, Grc.nasa.gov, 2021.
- “Fluid Mechanics 101”, Youtube.com, 2018.
- J.F. Manwell, J.G. McGowan and A.L. Rogers, Wind Energy Explained. Chichester : John Wiley & Sons, 2011

Last updated: April 20th, 2022

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