Flight Science/Engineering

Want to learn about some of the actual science concepts involved in flight and aircraft design? You can get a college degree in Aerospace Engineering and go on to work for aircraft companies like Boeing/Lockheed/Airbus/etc, or spaceflight organizations like NASA/SpaceX/BlueOrigin/etc. Getting an Aerospace Engineering degree requires a lot of work (it is rocket science), but it is doable and very rewarding. There is a lot you learn in this degree. Below is an overview of some of the major concepts:

1. Overview of Engineering
2. Aerodynamics
3. Stability & Control
4. High-speed Aerodynamics
5. Engine Design
6. Structural Analysis
7. Miscellaneous Aerodynamic Phenomena
8. Aircraft Design Workflow

Overview of Engineering

“Engineering” refers to any discipline that uses science/physics to solve real-world problems. If you need to build a bridge to cross a river you will need to understand how loads are distributed through a structure and how much stress certain materials can take. If you want to develop a more efficient car engine, you will need to know the chemistry of fuel and thermodynamics. If you want to make something fly you will need to know how lift and drag are generated over a wing. All of these concepts are grounded in physics.

Major types of Engineering:

• Aeronautical/Aerospace
• Chemical
• Civil
• Computer Science/Engineering
• Electrical
• Mechanical
• Nuclear

Core concepts:

• Coding/Software/Robotics
• Statics/Dynamics – sum of forces/moments
• Thermodynamics – conduction/convection/radiation, aerothermo (ablation), propulsion (compressor/turbine),
• Guidance/Navigation/Control
• Structural/Materials
• Fluid Dynamics, CFD, Aerothermo – mach waves
• Electricity/Power
• Communications
• Calculus & DiffEq – Flow rates, integration, etc
• Physics – Newtons Laws, Gravity, Keplers Laws,
• Misc – attitude (euler angles, quats)

Aeronautical/Aerospace Engineering

While all those other types of engineering are cool, we’re here for flight! So here is an overview of foundational aerodynamic concepts used in aircraft design:

Aerodynamics

An airfoil is a cross-section of a wing. Picture a 2D slice of a 3D plane wing, like this:

The airfoil shape affects the lift properties of the wing. A wing based on a curved (or “cambered”) airfoil generally produces more lift (at a given angle of attack) than a wing based on a symmetric (zero-camber) airfoil. (Angle-of-attack is the angle between the airfoil axis (or “chord”) and the incoming air stream direction. It is denoted with the Greek letter alpha.)

An airfoil’s geometric shape causes the air stream to form a high-pressure region underneath the airfoil and a low-pressure region above the airfoil (per Bernoulli’s principle), thus creating an upward force. This upward force is called “Lift“.

To clarify, an airfoil is just a 2D shape; it doesn’t directly create lift, but it has a direct effect on the lift created by a wing based on the airfoil cross-section. So airfoils are generally characterized in terms of their “lift coefficient” (denoted as “cl”), which varies based on angle of attack. An airfoil’s cl-alpha curve shows the relationship between the lift coefficient at different angles of attack:

There are infinite airfoil shape possibilities, each with different lift characteristics. Check out airfoiltools.com to see thousands of airfoil designs and their associated cl-alpha curves.

Again, airfoils are 2D shapes with associated cl-alpha characteristic. A full 3D wing is what actually creates lift. The amount of lift force generated is given by the Equation of Lift. (Note: A real 3D wing has a finite length. Aerodynamic losses occur on the wingtips, so the lift coefficient of a real wing is slightly smaller than the airfoil it is based on, but the 2D cl value of the base airfoil is a good initial estimate for the 3D wing lift value.)

So, for a given wing with a particular lift cofficient (determined by the airfoil and angle-of-attack) and surface area, moving through air with a particular density, at a particular speed, you can compute how much lift it generates.

Now here’s the fun part! If the lift generated by an object is greater than its weight, then it will rise into the air and start to fly! A key component of aircraft design is answering the question of what size/shape of a wing is needed, and how fast will it have to go in order to generate enough lift to get the vehicle+passenger+cargo weight off the ground?

Example: Say you want to build a hang-glider you can wear on your back:

• Say you select the NACA M25 airfoil, which provides cl = 1.0 at 5 degrees AOA. (Ignore Reynold’s effects and 3D CL effects for now.)
• Now, say you create a rectangular wing from that airfoil with a 2 meter chord and 10 meter wingspan (wing area = 2*10 = 20 meters^2). (Most hang-gliders use triangular/delta-wing planforms, but a rectangular planform makes for a simpler example.)
  • Say you make this wing out of styrofoam wrapped in fiberglass; it would probably weigh ~35lbs (15.9kg).
• Average air density at sea-level (standard temperature and pressure) can be found online: 1.225 kg/m^3
• Say you run at 10mph into a 10mph head-wind (20mph airspeed = 8.94m/s) and jump from a hilltop with this wing attached to your back.
• The lift force generated will be: Lift = (1/2) * 1.225kg/m^3 * (8.94m/s * 8.94m/s) * 20m^2 * 1.0 = 979 Newtons (220 lbs).
• If you weigh ~165 lbs plus the 35 lbs wing (200lbs total) and the wing generates 220lbs of lift at that airspeed, then it will lift you off the ground. (Net vertical force will be 20 lbs (0.1g) upward acceleration.)
• If you want more lift you’d need to:
  • Fly in denser atmospheric conditions (i.e. lower altitudes or colder temperatures)
  • And/or utilize a higher airspeed (i.e. run faster than 10mph and/or rely on a strong headwind)
  • And/or pick an airfoil with a higher CL value (or potentially the same airfoil held at a higher AOA)
  • And/or increase the size (surface area) of this already large wing. (This is why hang-gliders are generally quite large and use a delta-wing shape to achieve different lift characteristics than a rectangular wing would provide. They are also generally made of aluminum and fabric to achieve a very light airframe weight.)

Now as you can see in the preceding equation, lift is directly proportional to air speed, which is perpendicular to the lift direction. You’ll need the aircraft to be moving pretty quickly to generate enough speed/lift to takeoff. So, controlling the aircraft’s horizontal speed is another critical piece of the puzzle. To do this you need another force: Thrust. Thrust is generally provided by propellers or jet engines mounted on the aircraft. Computing the thrust force of a particular engine is more complex than computing the lift force of a wing, but generally the engine manufacturer will provide specs on how much thrust their engine can output at a given power setting. If your thrust force exceeds your drag force then you will accelerate forwards. As you accelerate forwards your freestream velocity increases, which increases your lift force.

So the foundation of flight is the balancing of 4 forces: Lift, Weight, Thrust, and Drag.

Now what is drag? Drag is a resistive force imparted by the air on the vehicle. There are multiple components of drag force: lift-induced drag and profile/parasite drag (which itself includes pressure (form) drag due to air separation effects + skin-friction drag due to surface friction with the air + shock wave drag (for transonic aircraft only)). Minimizing drag helps with aerodynamic efficiency and fuel efficiency which is why aircraft are designed to have smooth surfaces. You will need to estimate the drag of your vehicle and ensure that the engines can provide enough thrust to overcome drag at the flight speed needed to get lift to overcome weight. 3D computational fluid dynamics (CFD) tools and wind tunnel tests can be used to estimate drag over an entire aircraft. Estimates can also be made using the drag equation:

As you can see, drag has a very similar equation to lift, but the drag coefficient (Cd) is a bit more complex. There are multiple components of this term, including the induced-drag coefficient generated by the lift force. Generally, any lift produced at non-zero angles of attack will generate some drag force as a result. You can visualize this lift-induced drag as the component of the lift force (L*sin(alpha)) that is acting against the direction of travel, like so:

The drag equation can be plotted to see the relationship between cl & cd (known as the drag polar plot).

The profile drag component is based on the shape/size/materials of the aircraft, and can vary between subsonic and supersonic airspeed regimes, or based on differing freestream effects (laminar vs turbulent), so it is generally harder to estimate and may require wind tunnel testing to get a complete picture. The y-intercept value included on an airfoil’s drag polar (such as those plotted on airfoiltools.com) is a good initial estimate. (Those plots generally include results for various Reynold’s conditions. We won’t cover the nuances of this here, but for a general overview see: Reynolds number calculator (airfoiltools.com).)

As one final note on the drag polar, the flatter the bowl-shape (i.e. the higher CL available at low CD), the better. This lift to drag ratio, or “L over D” (L/D), is a primary metric for an aircraft’s aerodynamic efficiency. Maximizing this ratio is often a major goal in aircraft design.

Stability & Control

Ok, so we now know the forces acting on an aircraft. These forces cause it to move, but how do we keep it stable? How do we maintain controlled flight? You can strap a wing and propeller on your back and let it rip, but without stability and control you’ll quickly crash. To address this piece requires the balancing of moments!

Aircraft have 3 primary rotation axes, referred to as roll, pitch, and yaw:

Stability and control of all 3 of these rotations are critical, and are all achieved in different ways. An aircraft is said to be stable if it encounters a rotational perturbation about one of these axes and automatically restores itself to the original orientation. (e.g. If a gust of wind hits your nose from underneath and causes you to pitch up, a stable aerodynamic design will cause the vehicle to automatically level out.) An aircraft is said to be neutrally stable if it encounters such a perturbation and remains at the perturbed orientation (e.g. it stays pitched up). An aircraft is said to unstable if it encounters a perturbation and continues rotating farther away from the original orientation (e.g. continues pitching up even higher).

In the early days of flight the pilot’s joystick was physically linked to the control surfaces and the pilot had to manually maintain level flight. It was imperative then that the vehicle be designed with a high level of stability otherwise it was very difficult for the pilot to control. Nowadays, with fly-by-wire aircraft (control surfaces driven by motors and electrical signals rather than mechanical linkages to the joystick), aircraft can be designed unstable and controlled by continuous commands to the control surfaces from an onboard flight computer. Fighter jets are often designed this way for the maneuverability benefits that come with an unstable design.

Let’s unpack this a bit more, starting with pitch stability. For an aircraft to be pitch-stable it must restore itself to its original orientation after a pitch perturbation. Similar to an airfoil’s lift coefficient (cl) which affects the lift force (Newtons), there is also a pitch moment coefficient (cm) which affects the pitch moment (Newton*meters). Like cl, cm varies with angle of attack. You can frequently find the cm-alpha curves in the same dataset with the cl-alpha curves for a given airfoil. A pitch-stable aircraft has a negative-slope Cm-alpha curve, as shown:

A stable aircraft will always return to the trim angle. The negative-slope ensures that. For example, if you’re flying at 7 degrees angle-of-attack, and you get pitched up to 10degrees by a gust of wind (or by deflecting and releasing the elevator surfaces) you will move to the right on the curve (higher alpha) which puts you down into the negative cm area; negative cm means you’ll get a negative (nose-down) pitching moment which will drive you back towards 7degrees alpha. As you are pitching down if you overshoot 7degrees and reach 5 degrees alpha, you’ll be in the positive cm area which will yield a positive (nose-up) pitching moment that will drive you back up to 7 degrees. So you will tend back to your trim angle when flying a stable aircraft. NOTE: Many airfoils are unstable on their own (i.e. positive cm-alpha trends); what counts is the effective Cm-alpha of the entire aircraft (including the wing Cm, tail Cm, etc). When these are combined the slope of the net Cm-alpha trend will determine the overall pitch stability.

If you have to fight with the stick to get the airplane to depart from the trim angle, then it might be too stable. So stability and control are both important. The degree of stability is driven in large part by the physical relationship between the aircraft’s Center of Gravity (CG) and the aircraft’s Aerodynamic Center (AC). The center of gravity is the fulcrum point on the aircraft where the weight force acts. (If you supported the aircraft at this point it would balance like a seesaw.) The aerodynamic center is the point on the aircraft where the aerodynamic moments are zero (Cm,ac = 0). (Think of it like the point on the aircraft through which all the lift force is acting.) The AC of a traditional wing surface is along the quarter-chord (1/4 of the way back on the chord line). In a pitch-stable aircraft the CG is located in front of the AC, as shown below. The distance between the CG and AC is referred to as the Static Margin (SM). The farther ahead the CG is relative to the AC (positive SM), the more stable it is. If the CG is co-located with the AC (SM = 0), the vehicle is neutrally stable. If the CG is behind the AC (negative SM), the vehicle is unstable.

You’ll need a net positive (upward) force to rise into the air, but once at the desired altitude you’ll want the sum of the forces and moments on the aircraft to equal zero in order to achieve steady, level flight for cruising. The angle that the wing and horizontal tail surfaces are mounted onto the fuselage (along with their respective sizes (areas) and airfoils) determines the lift forces they will produce at cruise speed. You can adjust these in the design-phase in order to ensure the sum of the forces and moments will be zero at that speed, and to tailor your trim angle.

Moments are computed relative to a point. The pitch moment about the aircraft’s center of gravity (CG) can be computed as such:

We’ve been focusing here on pitching moment coefficients and pitch (longitudinal) stability. There are also coefficients for roll and yaw moments, which affect the roll (lateral) stability and yaw (directional/”weather-cock”) stability. These are more nuanced, so we won’t dive into these here, but in general, the horizontal tail influences the pitch stability characteristics, the main wing influences the roll stability and the vertical tail influences yaw stability. Additionally, the horizontal tail surfaces (elevators) offer pitch control, the wing surfaces (ailerons) provide roll control, and the vertical tail surface (rudder) offers yaw control. Remember earlier when we defined the “camber” (or degree of curvature) of an airfoil/wing? The larger the camber, (generally) the more lift at a given angle of attack. Deflections of the elevator/ailerons/rudder effectively increases the camber of those respective lifting surface, thus increasing the lift force. For example, deflecting the rudder to the left increases the camber of the vertical tail, which creates a lift force to the right, which yields a moment about the CG which causes the vehicle to yaw left. Deflecting the elevators down increases the camber of the horizontal tail, which creates an upward lift force, which yields a moment that causes the vehicle to pitch down. Deflecting the left aileron up and the right aileron down causes the left wing to develop a negative lift force and the right wing to develop a positive lift force, which yields a moment that causes the vehicle to roll left. Note: There is generally a coupling between the moments, such that rolling left and pitching up will cause you to do a banked turn left without using any rudder deflection. If you also deflect the rudder in this maneuver this will allow you to perform a level, coordinated turn.

All the lifting surfaces will contribute skin-friction drag. The lift generated by these surfaces will create lift-induced drag. The drag forces will yield additional moments on the vehicle, as will the thrust force (assuming the engines are offset from the center-of-gravity axis). Clearly there are a lot of factors in designing an aircraft to have the desired aerodynamic performance characteristics. Tools like XFLR allow you to assess the lift/moment characteristics of a design and make tweaks to dial-in the general performance you want.

The overall aerodynamic behavior of an aircraft is governed by the Navier-Stokes equations. These are non-linear equations with no single closed-form solution, but Computational Fluid Dynamics (CFD) is a field focused on computing approximate solutions that allow you to estimate with high accuracy the flight characteristics of an aircraft in any scenario. CFD tools (like Ansys) are used to model such effects.

Once you have an initial design, you can also create a scaled model and test it in a wind tunnel, which is a mechanical system that can generate a constant airflow at a given speed and measure the forces/moments that are generated on a given object mounted inside the tunnel. Wind tunnel tests are helpful for validating the aerodynamic characteristics of a design, but CFD tools are becoming so powerful and accurate now that one day we may not need wind tunnel tests at all.

This concludes the basic concepts relevant for aircraft dynamics and design. There are many additional concepts pertinent to theoretical aerodynamics (Reynolds number, compressibility, laminar/turbulent flow, lifting line theory, circulation, vortex interactions, etc), but we will not delve into these here. For now, let’s move to some other fun topics.

High Speed Aerodynamics

Some aircraft are designed to fly at supersonic speeds, or above the speed of sound. The speed of sound is the speed at which sound waves travel in a medium. Speed of sound can be computed using the following equation:

The standard atmosphere is a tabulated set of pressure/density/temperature values of atmospheric air at various altitudes. These tables can be found online from various sources. Actual pressure/density/temperature at a given altitude varies with geographical location, time of day, weather, season, etc, so you’ll want to factor in your actual operating conditions into your designs, but the standard values in the table represent the average with these local factors excluded.

As you can see, the speed of sound varies with altitude, or more specifically with pressure/density or temperature. (Either form of the above equation can be used to compute speed of sound; they are identical thanks to the Ideal Gas Law.) So speed of sound at sea level is much different than speed of sound at 40000 meters, or at the edge of space. In general, atmospheric pressure and density (and thus speed of sound) decrease as you go up in altitude.

The Mach Number is a measure of a vehicle’s speed in multiples of the speed of sound. (e.g. Mach 2 means you are traveling at twice the speed of sound.) It is computed as follows: Mach = vehicle speed / speed of sound. Mach number is a useful quantity for conveying the magnitude of a vehicle’s speed, but because speed of sound changes with altitude, so does Mach Number, so it’s a relative term. (Mach 1 at sea level is faster than Mach 1 at an altitude of 40000 meters.)

When you fly slower than the speed of sound (< Mach 1), you are said to be traveling at subsonic speeds. When you exceed Mach 1, you have “broken the sound barrier” and are then traveling at supersonic speeds. At these speeds you are compressing the air in front of you so intensely that it forms a shock wave. Observers on the ground might hear a sonic boom when the shock wave passes them. The shock wave is a thin, high-pressure boundary that moves with the vehicle through the air. As upstream air particles pass through the shock wave the air pressure and temperature decrease rapidly. Sometimes you can visually see vapor condensing behind the shock wave when this happens.

Air speeds up as it passes around an airfoil. So even if an aircraft itself is traveling at subsonic speeds, the airspeed over the wings might exceed Mach 1, in which case shock waves will form on just the wings. Another location where shock waves often form is on the tips of engine propellers, which move very quickly. And of course, jet engines typically generate supersonic exhaust, so shock waves form inside the exhaust column, which can often be seen visually:

The drag on a vehicle increases tremendously as it approaches Mach 1. In the early days of flight aerodynamicists thought drag would increase to infinity, thus forming an unbreakable speed limit; hence the name “sound barrier”. But eventually they built aircraft capable of overcoming the large (but finite) drag forces. On 10/14/1947, Chuck Yeager, piloting the Bell X-1 (“Glamorous Glennis”), was the first to break the sound barrier. Today, spacecraft and fighter jets regularly fly at speeds much higher than Mach 1. Supersonic aircraft are often designed with swept wings so that the wing surfaces reside within the shock wave to help minimize drag. Some aircraft, like the F-14 Tomcat, have actuated wings that can change their sweep angle during flight as they speed up and slow down.

When you exceed Mach 5 you enter a regime known as hypersonic speeds. This is where the vehicle is moving so fast that when it makes contact with nearby (quasi-stationary) air molecules it rips them apart into individual atoms. When space capsules re-enter Earth’s atmosphere they can be moving around 20000mph (~Mach 27). At these speeds the air gets super-heated and rips apart, turning to plasma as it passes over the shock wave in front of the capsule. Capsule heat shields are made from an ablative material, meaning they are designed to burn away to dissipate heat. The plasma generation and heatshield ablation are what give the fireball appearance to these vehicles during entry:

Engine Design

There are multiple engine types used in aircraft, as illustrated on Flying Things. The major types include propeller systems and jet engines. They vary in terms of efficiencies and certain types are often more suitable than others for a given aircraft design based on the intended flight speed and power requirements. These devices provide the thrust to enable the aircraft to move forward, generating speed and airflow over the wing, which generates lift.

In the early days of flight, aircraft all used propeller systems, in which a traditional engine-driven gear system turned a shaft connected to a propeller. Propeller blades are essentially twisted wing surfaces oriented perpendicular to the air stream. As they spin, they draw air through them and generate a horizontal lift force (thrust).

Eventually came jet engines, which operate on the premise of combustion reactions. There are various types of jet engines (turbojets/turboprops/turbofans/ramjets) but the basic premise is shown below:

In a jet engine assembly the compressor blades spin around which sucks in air and compresses it to a high pressure. The pressurized air then enters the combustion chamber where it is mixed with aerosolized jet fuel (via a fuel-injector system). The high pressure air/fuel mixture is then ignited (via a spark mechanism), generating very hot, fast-moving gas. The gas then passes through the turbine blades to decrease the pressure, before passing through the nozzle which channels the gas, further increasing the flow speed. The hot, fast gas then exits the nozzle and fires out the back of the engine as exhaust, generating thrust to drive the aircraft forward. In some cases, jet engines are also outfitted with an afterburner which is a system that creates a second combustion after the turbine to further increase power output of the engine, at the expense of fuel efficiency.

Supersonic aircraft (like fighter jets) have jet engines designed to produce supersonic exhaust. In these systems, the exhaust which is initially subsonic is accelerated by decreasing the flow column area in the nozzle (like pinching a hose to speed up the water speed). Once the flow reaches Mach 1 an interesting phenomenon occurs: In order to further increase the flow speed, the nozzle column area must be increased. Hence supersonic engine nozzles initially close down and then widen again, taking the exhaust from subsonic to sonic and beyond. The section of the nozzle with the smallest area is called the throat, which is the location where the flow transitions to supersonic. Sometimes you can see shock waves inside the exhaust column as shown below. The shock waves reflect back and forth inside the exhaust column forming a characteristic diamond pattern, hence the name diamond shocks. Oftentimes, these jets have diagonal-slanted inlets to help control the behavior of the shock wave entering the engine.

Turbojets are the general form described above. Turbofans are an updated version of the turbojet and work the same way but they have a large fan assembly before the compressor which directs some of the flow around the engine to cool the engine, increase thrust, and quiet the engine. (Turbofans are common among airliners.) Turboprops are propeller systems driven by a jet engine. Turboshaft engines are similar to turboprops but are typically used in helicopters to drive the shaft attached to the large helicopter blades.

Some supersonic aircraft use ramjet or scramjet engines, which have no moving/rotating parts. These systems operate in different speed regimes but rely on the speed of the aircraft to self-compress the air internally so it can be combusted automatically inside the engine. Of course, in order to reach these high speeds the aircraft needs some initial thrust, which can be provided by traditional jet engines.

Rocket engines function on a similar premise to jet engines, however since combustion reactions require fuel and oxygen, unlike the jet engine which uses the oxygen in the air, rockets must carry oxygen tanks (and fuel tanks) onboard so they can operate in space where there is no air.

In recent years there has been a surge in the development of electric aircraft which don’t use these fuel-based engines at all. Electric aircraft have propeller systems driven by electric motors which are powered by battery systems, much like you would find in a toy quadcopter. These systems enable the use of a few large propellers or a myriad of small propellers, which can each be driven by their own motors installed all over the aircraft. This distributed electric propulsion concept is not possible with fuel-based engines which each require large combustion systems.

Structural Analysis

Structural engineering is an entire field and is essential in all aircraft projects. For the sake of brevity here, the premise is that if you build a long, thin wing and then you subject it to hundreds of pounds of lifting force, it will bend. The material you choose and the way you manufacture the wing can affect how strong it ends up being and whether or not it will break under normal loads. You’ll want your aircraft to be able to withstand several G’s of load so you can make tight banking turns, etc. So in addition to designing your wing to provide sufficient lift, you also need to design space inside the wing to allow for beams (wing spars) to provide structural support. But keep in mind how this affects the weight of your design, which impacts the amount of lift you need, which impacts the shape of your wing, etc. Also keep in mind that wing bending essentially alters the dihedral of your lifting surface, which can affect the stability and control of your design. It’s all connected.

Mechanics of materials and structural analysis classes teach you the equations for determining how much load a particular surface (made from a particular material) can withstand. Finite Element Analysis (FEA) software is used to perform these analysis over large, complex bodies, like aircraft. Some CAD programs, like SolidWorks, include FEA features and even CFD features all-in-one.

Miscellaneous Aerodynamic Phenomena

Aircraft Design Workflow

Consider the following process for designing a new aircraft:

• Conduct a trade study of already-existing aircraft with similar size and operational profile (e.g. similar payload weight, or range requirement, or speed requirement, etc.) Take note of the wing area, wingspan, etc for these aircraft and use this as the baseline for your new design.
• Size the aircraft wing based on the weight requirement. Size the tail for stability and control requirements. (You can use XFOIL or XFLR for low reynolds number designs (i.e. RC aircraft).)
• Select a motor with known thrust specifications.
• Draw up the full system with some sort of CAD software, placing the structural support components, passenger seats, landing gear, fuel tanks, etc. Determine the CG location based on the component masses and adjust the tail location/size to optimize the stability. Iterate on wing design.
• Use CFD software to assess lift/drag performance of the model and to estimate range/endurance/speed. Iterate on wing design.
• Use FEA software to assess structural integrity of the aircraft.
• Perform wind-tunnel testing with a small-scale model to confirm lift/drag results predicted by CFD. (One day soon, CFD may be so good that wind-tunnels won’t be needed as much.)
• Build a prototype and conduct test-flights.
• Deploy operational aircraft.