# Basic Aerodynamic Theory

## A (brief) introduction to basic aerodynamic principles

## Table of Contents

This page is meant to be an extremely brief introduction to the complex world of aerodynamics for underclassmen who have not had much exposure to fluid mechanics. It is by no means a comprehensive guide but rather a starting point for those who want to delve deeper into flight sciences.

- Airfoil Nomenclature

- Chord
- (Geometric) Angle of Attack
- Camber
- Lift and Drag
- Normal and Axial Forces
- Lift and Drag Coefficients

- Thin Airfoil Theory

- Assumptions
- Results
- Kutta Condition
- Angle of Zero Lift

- Finite Wing Theory

- Aspect Ratio
- Trailing Vortices
- Downwash Velocity
- Induced Angle of Attack

- How do planes fly anyway?
- Summary
- Additional Materials
- References

## Airfoil Nomenclature

An airfoil is a 2D cross section of a infinitely long wing. You may wondering exactly why it matters that it's an infinite wing, but we'll get to that later. First, let's look at a labeled diagram of an airfoil, shown below in *Figure 1*.

Figure 1: A labeled diagram of an airfoil^{[1]}

The first important characteristic that we will discuss is the *chord* of the airfoil, shown in *Figure 1* as the dotted red line. The chord is the line from the *leading edge* to the *trailing edge* of the airfoil. The chord is used to define the *geometric angle of attack* (often denoted with *α*), the angle between the freestream direction and the chord. The dotted blue line in *Figure 1* shows the *camber line*, or the midpoint between the *upper surface* and *lower surface*.

As I'm sure you know, when air moves around a wing or an airfoil a force known as *lift* is generated perpendicular to the freestream direction- otherwise there would be no such thing as planes! The lift is caused by a pressure difference between the upper and lower surfaces of the wing/airfoil due to the way air moves around the body. An additional force, known as *drag*, also develops parallel to and in the same direction as the freestream. Lift always points upwards by definition, but the shape of a body can sometimes cause this perpendicular force to point downwards; an example of this is in cars where lift is actually undesirable and the force is known as *downforce*. A freebody diagram of the forces on an airfoil due to the freestream is shown in *Figure 2*.

Figure 2: A freebody diagram of the forces due to the freestream on an airfoil^{[2]}

Notice in *Figure 2* how there are two additional forces acting on the airfoil. These forces are known as the *Normal* and *Axial* forces measured relative to the chord. They are not seperate forces, but rather a way of representing the lift and drag in a different coordinate system. The relationship between these forces is shown in *equations (1) & (2)*.

Often times in aerodynamics, we want to nondimensionalize the lift and drag forces in order to get an idea of the properties of the airfoil or wing independent of the freestream velocity or the ambient pressure. We now introduce the *2D lift and drag coefficients*, shown below in *equations (3) and (4)*, respectively.

where

It is often convient to express part of the denominator as a single quantity, known as the *dynamic pressure*.

It is possible to calculate the 2D lift and drag using integral equations. However, it is beyond the scope of this tutorial and honestly not really applicable to our situation since we will mainly be using computational fluid dynamics (CFD) to calculate experimental lift and drag values. The results of the CFD analysis will then be used to calculate lift and drag coefficients and used to select airfoils/wings for our designs.

## Thin Airfoil Theory

Thin airfoil theory uses the following assumptions for it's analysis:

- "Zero" thickness:
*t*<<*c* - "Small" camber
- "Small" angles of attack: small angle approximation (
*α*< 10-15 degrees) - Inviscid (no viscosity) and incompressible flow
- Airfoil does not disturb the flow

Although some of these assumptions seem a little ridiculous, thin airfoil theory gives us a good starting point for analyzing flows. If the airfoil is symmetric, the lift coefficient can be described by *equation (5)* for small angles of attack.

If the airfoil is not symmetric, we must define the quantity known as the *angle of zero lift*. This quantity physically represents the angle of attack where the airfoil has a lift force equal to zero. The angle is usually less than zero for a cambered airfoil or equal to zero for the case of a symmetric airfoil. The angle of zero lift can be calculated from a complex integral equation, but it again is not important to our applications and often times given. If the airfoil is asymmetric, the lift coefficient is now described by *equation (6)*.

The important thing to take away from this theory is that when the airfoil is very thin, the lift coefficient increases linearly as a function of the geometric angle of attack and angle of zero lift (if asymmetric).

Another thing that is worth mentioning is the Kutta Condition. This principle is often best expressed through *equation (7)*.

This basically states that in order for there to be lift, there must be some circulation (denoted by *Γ*) around the body. This is not super important to our application but worth mentioning.

## Finite Wing Theory

Great! We've made it to finite wings. The first thing that is important to mention is the *aspect ratio (AR)* of a wing. It is a dimensionless quantity that measures the ratio of the square of the *wingspan (b)* to the *surface area (S)* of the wing. This is explicitly outlined in *equation (8)*.

The greater the aspect ratio, the higher the coefficient of lift. This is usually why unmanned vehicles (our focus!) usually have really long, thin wings. However, loading on longer wings cause more of a moment on the fuselage so they're often avoided in many situations, like commercial aircraft.

Have you ever looked out the window of an aircraft and seen how the wings flare up on the end? These are known as winglets and they're not just there for looks. They actually help reduce what are known as *trailing vortices*, a phenomenon that appears with finite wings. The vortices are caused by high pressure underneath the wing "leaking out" and moving upwards over the wing. As a result, these vortices actually induce a downwards velocity, known as the *downwash velocity*, which combines with the freestream to actually push downwards on the wing. The angle between the resulting velocity vector and the original freestream vector is known as the *induced angle of attack* and cause the geometric angle of attack to appear smaller than it actually is. This is shown in *Figure 3*.

Figure 3: The downwash velocity and induced angle of attack from the trailing vortices^{[3]}

Using what we know from thin airfoil theory, a smaller geometric angle of attack leads to a smaller lift coefficient. Therefore, a finite wing actually has *less lift* than an airfoil with the same cross-section at the same conditions. There is also now an induced drag force that forms on the wing due to this down wash velocity, also shown in *Figure 3*. The lift and drag coefficients can now be redefined for the 3D finite wing case, shown in *equations (9) & (10)*.

## How do planes fly anyway??

Now the question that every aerospace engineer wants to know: how the hell do planes fly? You may have heard of a theory known as the *Equal Transit Theory*. This theory states that the airflowing around a wing separates at the leading edge and meets back up at the trailing edge. However, since the path on top of the wing is longer than the path underneath the wing, the velocity of air on top is higher and thus causes a pressure drop due to Bernoulli's principle. The pressure difference then causes an upwards force to attack on the wing, pushing it upwards.

**However, the equal transit theory is false!** Although Bernoulli's principle does state that an increase in velocity causes a decrease in pressure (granted other conditions stay the same), it is simply proven false that air meets back up at the trailing edge. So what is the real reason? The answer has to do with the shape of the airfoil and its ability to push into the flow. Let's look at a pressure distribution of an airfoil at several angles of attack, shown in *Figure 4*.

Figure 4: The pressure distribution on an airfoil subjected to different angles of attack^{[4]}

If you look at *Figure 4*, you see that the pressure distribution changes with the angle of attack. At a negative angle of attack less than the angle of zero lift, the lift on the airfoil is actually pointing downwards (downforce), but increases as the angle of attack increases (up until stalling occurs). This is because of how the airfoil "pushes" into the airflow. At a positive angle of attack, the flow pushes into the bottom of the wing but flows away from most of the top of the wing. The "pushing" of the air on the bottom of the wing causes high pressure on the bottom of the wing and the lack of a body to push on the top causes low pressure. This then produces an upwards net force. Think of sticking your hand out the window in a car when you're going very fast. When you tilt your fingers upwards, the air pushes you up but pushes you down if you angle your fingers downwards. It's pretty much the same principle! This also explains why a symmetric airfoil will have zero lift at 0 angle of attack- the pressure contributions to lift cancel each other out.

## Summary

The important thing to take away from all this is that we can (and will) use CFD for complex airfoil and wing shapes to generate pretty accurate lift and drag values. We can then nondimensionalize these values in order to compare and contrast wing/airfoil shapes and select the appropriate one for our design. It also is important to recognize what happens in the finite wing cases and how trailing vortices decrease the lift and increase drag when compared to the theoretical 2D case. And that the equal transit theory is BS

## Additional Materials

There are tons of resources out there if you're interested in learning more. Here are a few that I recommend:

*Fundamentals of Aerodynamics, 6th Edition*By John Anderson (MAE 150B Textbook)*Aerodynamics, Aeronautics, and Flight Mechanics, 2nd Edition*By B.W. McCormick (MAE 154S Textbook)