In this article, we delve into the fundamental concepts of lift, drag, and pitch dynamics, exploring their significance in both aerodynamics and hydrodynamics. Using airplane wings and wind turbine blades as examples, we explain how these forces interact and influence the performance and stability of various vehicles and machinery. Detailed diagrams and explanations make complex theories accessible, enhancing your understanding of these critical forces in flight and rotating systems.

**Lift and Drag: Understanding the Aerodynamics on Moving Airfoils and Bubbles**

Lift and drag are aerodynamic forces acting on airfoils moving through air, as well as forces acting on bubbles moving through water. Using an airplane as an example (similar principles apply to bubbles in water), let’s explore how lift, drag (or drag force), and pitch movement affect the airplane’s motion in the air.

### 1. How Do Wings Generate Lift?

A wing refers to the shape that generates lift when an object moves through the air. The cross-section of an airplane wing is known as an airfoil, which generates lift by creating a pressure gradient between the upper and lower surfaces of the wing.

**Refer to the airplane illustration below.**

**Figure 1:** Overview of forces acting on an airplane. Lift must balance weight, and thrust must balance drag for the airplane to maintain flight.

For an airplane in flight, the forces acting on it include weight pulling it downward and drag pulling it backward. Simultaneously, the airplane generates thrust to move forward and lift to counteract the weight, enabling it to continue flying.

Airplane wings are composed of multiple airfoils stacked along an axis. By understanding the physics of a single airfoil, we can grasp the principles governing the entire wing structure. **Figure 2** shows the force distribution on a single airfoil. Note the direction and magnitude of each force. In this state, the airplane can accelerate and climb because thrust and lift exceed weight and drag, respectively.

**Figure 2:** Force distribution on a single airfoil of an airplane wing.

### 2. Pressure Distribution on an Airfoil

Consider a streamline, with state “1” representing the state before interacting with the airfoil, and state “2” representing the state after interaction. **Figure 3** illustrates the terminology of the relevant variables:

**Figure 3:** Diagram of the streamline between state 1 and state 2, representing before and after interaction with the airfoil.

Applying Bernoulli’s equation along this streamline yields:

\(P\) is the static pressure, \(\rho\) is the fluid density, and \(U\) is the fluid velocity.

The shape of an airfoil is designed for various purposes, causing different fluid speeds along its length. According to Bernoulli’s equation, changes in velocity lead to pressure variations relative to local atmospheric pressure. These corresponding pressure loads act normal to the surface, and integrating them yields the total force per unit area.

**Comparing the Top and Bottom Surfaces of the Airfoil**

If we envision airflow only on the top surface of a two-dimensional airfoil, we can observe that velocity accelerates near the region of maximum thickness curvature. This means velocity \(U_2\) will be greater than free-stream velocity \(U_1\). Thus, static pressure \(P_2\) will be less than free-stream pressure \(P_1\). This creates a suction on the top surface, pulling the airfoil upwards and generating lift (**see Figure 4**).

**Figure 4:** Airflow acceleration near the curvature of maximum thickness reduces local pressure, generating lift.

Similarly, we can analyze the pressure distribution on the bottom surface. For this specific airfoil, airflow accelerates around the leading edge, creating downward suction. Further forward, the airflow decelerates, causing a pressure increase. As this pressure is higher than local atmospheric pressure, it acts on the airfoil, producing lift (**see Figure 5**).

**Figure 5:** Downward suction near the leading edge and upward pull in the middle part contribute to the total lift. Different flow patterns around each airfoil depend entirely on its shape.

By integrating the pressure distributions on the upper and lower surfaces, we obtain the total force vector \( \vec{F} \), whose perpendicular and parallel components to the airflow provide lift and drag, respectively.

**Figure 6:** The total force acting on the airfoil has a vertical component called lift and a parallel component called drag.

### 3. What is Lift?

Lift is a component of the total force vector \( \vec{F} \), acting through the center of pressure of the object and perpendicular to the incoming airflow. For zero angle of attack, it acts opposite to weight (**see Figure 1**). Lift is a mechanical force generated when an object moves through the air, possessing both magnitude and direction.

Two conditions are necessary for lift:

1. **Fluid:** Lift is generated only when a solid object interacts with a fluid.

2. **Motion:** Lift is produced only when there is a velocity differential between the solid object and the fluid, i.e., when the object is moving through the fluid. This motion also produces drag, known as induced drag.

**Lift Equation:** Lift is a function of fluid density, free-stream velocity, and the reference area of the wing. It also involves a dimensionless quantity called the lift coefficient, used to compare the performance of different wings at varying shapes or speeds. Essentially, the lift coefficient helps measure how the shape, inclination, and flow conditions of a wing affect its lift.

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

### 4. What is Drag?

Drag is a component of the total force vector \( \vec{F} \), acting through the center of pressure of the object and parallel to the direction of incoming airflow. At zero angle of attack, it acts opposite to the thrust of the airplane (**see Figure 1**). Drag is produced due to the velocity differential between the solid object and the fluid, thus, only arising when there is relative motion between the two. If there is no such motion, there will be no drag.

For flying objects, there are two important types of drag:

1. **Parasitic Drag:** A combination of form drag and skin friction drag.

– **Form Drag:** This drag depends on the shape of the object. It is calculated by multiplying the local pressure by the object’s surface area.

– **Skin Friction Drag:** This drag arises from the direct interaction between the fluid and the object’s surface. The larger the wetted area, the greater the skin friction drag.

2. **Induced Drag:** Induced or lift-induced drag is caused by lift generation. On airplanes, wingtip vortices create swirling flows that disturb the airflow distribution around the wingspan. This reduces the wing’s ability to generate lift, requiring a greater angle of attack to achieve the same lift, thereby increasing the drag component.

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

**Drag Equation:** Drag is also a function of fluid density, free-stream velocity, and the reference area of the wing. It involves another dimensionless quantity called the drag coefficient, which helps measure the drag experienced by the wing in a fluid environment.

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

### 5. What is Pitch?

Pitch movement refers to the up and down motion of an airplane’s nose around an axis. This motion significantly affects the lift generated by the airplane wings. Using **Figure 7**, imagine a line running from one wingtip to the other, passing through the center of gravity. Consider the airplane’s motion around this axis.

**Figure 7:** Description of pitch movement, with the axis running from wingtip to wingtip.

Pitching up increases the angle of attack (defined below), thereby increasing the lift component of the total force (**see Figure 10**). This is because the increased downward deflection accelerates the airflow over the wing. The more upward motion, the greater the lift generated by the wing. However, this is sustainable only up to a certain point, beyond which stall occurs (discussed below).

**Figure 8** illustrates the relationship between pitch angle and angle of attack for the wing. Note how lift and drag magnitude change with increasing angle of attack. Both forces increase, but not equally. Since lift increases faster than drag, the lift-to-drag ratio also increases.

**Figure 8:** Increasing the angle of attack generates more lift and drag; however, the lift-to-drag ratio increases.

### 6. Angle of Attack and Pitch Angle

For an airfoil, the angle of attack is the angle between the incoming free-stream fluid and the chord line extending from the leading edge to the trailing edge. The pitch angle is the angle between the chord line and any reference plane. The reference plane could be the flat ground for a flying object or the rotor plane for a turbomachine.

**Figure 9:** Highlighting the difference between angle of attack and pitch angle. The angle of attack can be greater than, less than, or equal to the pitch angle.

Depending on the reference plane, the angle of attack can be greater than, less than, or equal to the pitch angle.

**Stall:** Increasing the angle of attack increases the lift-to-drag ratio to a certain extent. Beyond this point, further increasing the angle of attack results in a sudden decrease in lift and a sharp increase in drag, leading to a stall. This means the airplane cannot generate enough lift to support its weight, causing it to descend.

**Figure 10:** Lift and drag coefficients of a NACA 0012 airfoil; Re: Reynolds

### Stall: Understanding Lift, Drag, and Pitch in Aerodynamic and Hydrodynamic Environments

Stall should be avoided at all costs in aircraft as it means insufficient lift to balance the weight. Stalling can also be observed in compressors, causing uneven blade rotation, slowing down the rotor, and potentially leading to blade failure.

### Lift, Drag, and Pitch Movement in Rotating Machinery

Horizontal or vertical axis rotating machinery consists of a rotor or impeller and a set of symmetrically arranged blades. This includes wind turbines, jet engines, centrifugal pumps, and compressors. Like aircraft wings, these blades are also composed of a set of airfoils.

**Figure 12:** Wind turbine blade wireframe sketch showing different airfoil sections and their functions. Each airfoil section has a different function.

Each airfoil section serves a different function. Airfoils near the root ensure structural rigidity, while those at the middle and tip primarily generate lift.

### Pitch in Rotating Machinery

A major difference between aircraft and rotating machinery is that in rotating machinery, the airfoils experience wind speed/flow from two components: free-stream fluid and blade rotation.

**Figure 13:** Aircraft wings experience wind only from the free-stream component, while airfoils in rotating machinery experience an additional rotational component.

Consider observing the horizontal axis wind turbine blade from the top of the rotor plane, with the blade at the bottom dead center. The wind approaches horizontally from the ground and rotates clockwise. The top view diagram in **Figure 14** illustrates this. Note the different inclination angles of the airfoils across the blade span.

**Figure 14:** Wind turbine blade skeleton showing different airfoil sections inclined at various angles across the span.

Each part of the blade has a set of different airfoils, meaning each part has different functions as mentioned earlier. The skeleton clearly shows that each airfoil has a different pitch angle. The pitch angle is larger near the blade root/hub, while the pitch angle near the tip is smaller. This will be explained below.

**Flow Angle:** Flow angle \(ϕ\) is the angle between the incoming wind speed and the plane of rotation. As we move from the root to the tip of the blade, this flow angle becomes smaller. This is because the tangential velocity near the root is lower than at the tip.

\(v⃗ =\) Tangential velocity, \(ω⃗ =\) Angular velocity, \(r⃗ =\) Radial vector away from the root/hub.

**Figure 15:** Flow angle is the angle between the resultant wind speed and the reference plane.

In wind turbines, this reference plane is the plane of rotation. Near the root, the flow angle is larger compared to the tip. **Figure 16** simply shows that this flow angle is the sum of the angle of attack and the pitch angle.

**Figure 16:** Flow angle is the sum of the angle of attack and the pitch angle.

\(ϕ\) is the flow angle, \(α\) is the angle of attack, \(β\) is the pitch angle.

An increase in the angle of attack leads to stall; hence, the angle of attack \(α\) must be controlled within specified limits, especially near the root where the flow angle \(ϕ\) is larger. This can be managed by increasing the pitch angle \(β\) to over-pitch the airfoils near the root.

Using pitch angle to reduce drag, we can improve the efficiency of mixed-flow turbines by reducing the separation regions around the blade airfoils. The diagram below shows the velocity characteristics in a plane passing through the volute of a mixed-flow turbine:

**Figure 17:** In the modified design, reducing the blade angle of the stator blades can decrease the angle of attack and flow separation.

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