MAE 4242 Aircraft Stability and Control

Unit 1 Course Notes

Airfoil and Wing Aerodynamics Review

First, we recall the definition of an airfoil. An airfoil is the two-dimensional cross-section of a

streamlined object designed to minimize the separation of fluid flow around the object. It has a

shape that looks something like this:

Airfoil at positive angle of attack [1].

As you can see, we clearly separate upper and lower surfaces of the airfoil by the chordline. The

chordline is a straight line drawn from the most forward part of the airfoil to the most rearward

part of the airfoil. It is often used as a reference line when discussing things like the orientation

of the airfoil in a fluid flow. In the previous diagram, the angle α is used to denote the angle of

attack, which is the angular separation between the approaching fluid velocity, V∞, and the

chordline.

Our definition of an airfoil is, in some ways, redundant. A more rigorous definition of

“streamlined” indicates that the body has a low resistance to passing fluid flow. In fact, from

aerodynamics we define a streamline to be the path that a parcel (or infinitesimal control

volume) of fluid takes as it moves around an object. In an experimental setup, we can observe

this in water (or more viscous fluids, like syrup) by dying the fluid in precise spots. Our

colloquial definition of “streamlined” is therefore something shaped like this type of fluid

behavior. We separate this concept from that of a bluff body by how much resistance (in terms

of change to the streamlines) the body gives to the fluid, measured as drag. We give drag a more

formal definition later. This difference is illustrated below:

Ideal fluid flow over two types of bodies [1].

The sharper changes to the observed pressures on the body are part of what contributes to the

greater drag of the bluff body. We call our shape an airfoil because the predominant fluid it

travels in is air, but hydrofoils and similar shapes are designed to interact with water and other

fluids. An airfoil also does not mean it is affixed to an airplane; Fans and wind generators also

use airfoils. A spoiler on a car is also an example of an airfoil. The parts of an airfoil are shown:

Definitions of airfoil parts [2].

The leading edge is the forwardmost point on the airfoil. In general, this does not necessarily

mean it is the first part of the shape contacting the fluid flow. However, by convention this is

true for an angle of attack of zero degrees, in which the local fluid velocity and the chordline are

parallel. The trailing edge is a similar point on the most rearward part of the airfoil. Airfoils are

commonly tapered or have a corner (in the mathematical sense) at the trailing edge. This is to

reduce the change in velocity needed for streamlines passing on the upper and lower edges to

come back together in laminar flow. The mean camber line is hallway between the upper and

lower surfaces, and the camber is the measurement of the difference from this line to the

chordline connecting the leading and trailing edges. The thickness, by American convention, is

measured perpendicular to the mean camber line, and is commonly not more than about 20% of

the chord measurement.

We define the following unit vectors for our two-dimensional surface:

Unit vector definitions [2].

We also let s be the arc length of a point on the airfoil, measured clockwise from the leading

edge. Aerodynamics tells us that we have two contributing factors to the exchange of force

between the air and the airfoil: the pressure of the air on the surface and the shear force of the

fluid moving past the surface. Even on objects sitting motionless, air is always exerting a

pressure. In such situations, however, the pressure is uniform on all sides of the object and

therefore does not move a rigid body because the contributions of pressure cancel out (for

symmetric objects) or are far lower than other forces (all objects). When in motion, however, the

pressure of air moving past a body can yield a substantial force. For an airfoil, a qualitative

description of the pressure distribution is shown below:

Distribution of pressure on an airfoil [2].

All fluids have some measurable viscosity. Viscosity is the measurement of how “thick” a fluid

is, whether it behaves more like water or like molasses. A more viscous fluid, all things being

equal, flows more slowly. In fact, what we have on a molecular level is friction, the interaction

between fluid particles and with objects to resist motion. Air, like other gases, has very low

viscosity. In many types of airflow we can even neglect it. We call these types of flows inviscid.

Early aerodynamics focused on the behavior of potential flow, which assumes that the flow is

inviscid as well as irrotational (no local rotation or vortices). It is called a potential flow

because the resulting fluid behavior is governed by a form of Laplace’s equation. Solutions to

this equation are called potential functions. They have many useful properties and are widely

studied. We can derive many useful results, including fluid flow around an airfoil, using

potential theory. However, a major problem with potential theory (known as early as the 1700s)

is that it predicts zero drag for objects in the flow. However, it was known at the time that drag

(at least as far as we have discussed it – resistance to the fluid flow) was observed to occur

around boats and therefore the potential theory did not accurately describe the real world. This

discrepancy is known as d’Alembert’s paradox. It turns out that this paradox arises from the

inviscid assumption. We must have friction within the fluid to predict the drag forces observed

at the time around boats. We confine this viscous behavior to an extremely narrow region near

the surface, called the boundary layer. According to viscous theory, a fluid at an interface like a

wall (or an airfoil surface) has zero velocity. But, we also know that the streamlines show a

nonzero velocity away from the surface. Over the boundary layer, we see a sharp transition from

the zero velocity fluid at the surface to the local inviscid flow solution. The energy that this

transition takes is exerted on the body as a shear force (or, along the direction of the surface), as

shown:

Shear forces from fluid viscosity [2].

Together, the interaction of pressure and shear forces result in forces on the body, in this case an

airfoil. We call the resultant force, together with its components in any coordinate system,

aerodynamic forces.

Pressure acts normal (perpendicular) to the surface along unit vectors n, while the shear acts

tangential (parallel) to the surface along unit vectors k, as shown in our previous diagrams. We

then have a resultant force

The first term is the force due to air pressure on the body, while the second is due to friction

forces. In general, this resultant is not aligned with either axes of a coordinate system placed on

the body. As discussed before, we generally use the incoming air velocity as the horizontal of

our coordinate system for an airfoil. So, this resultant force on an airfoil at a positive angle of

attack becomes:

Resultant force on airfoil at positive angle of attack [2].

Note that movement of the airfoil through stationary air is the same as air flowing past a

stationary body. Our discussion is centered on the relative velocity between the two. By

convention, we center our reference frame (and later equations) on a fixed center of gravity for

the airfoil. The airfoil remains motionless, except for rotations about the center of mass to

change the angle of attack. We resolve the components of the resultant force into one parallel to

the air flow and one perpendicular to the air flow. The force orthogonal to the velocity, we call

lift, and the force parallel to velocity we call drag.

The terms lift and drag historically have positive and negative connotations. However, note that

our definitions are relative to the current velocity only. Lift does not mean we go further up into

the air. The lift produced by a spoiler on the back end of a high-performance car, or on the

wings of an aircraft during landing, is directed towards the ground to prevent flight. The lift

produced by an aircraft turning while upside down pulls it back towards the ground. When an

aircraft has a negative angle of attack, or when there is a relative upward wind, the drag force

may have an upward component. Drag is simply a force resisting motion in the current velocity

direction (whatever that may be), while lift is a force perpendicular to the current motion.

We note that the resultant force needs to be applied at a point. When conducting the surface

integration we previously saw, we could do that around any suitable reference point. However, it

is clear that the unit vectors do not all have a line of action passing through a given point. Each

of these differential forces will therefore also have an associated moment. Regardless of the

reference point that we use (and therefore where we locate the resulting lift and drag forces)

there will be a moment associated with that point. At some point, the resulting moment will be

zero. If we applied the lift and drag forces we calculated to the body at this point, it will exactly

match the actual loads distributed across the surface. We call this point the center of pressure. It

is not typically easy to calculate for realistic geometries, and changes with the angle of attack as

the pressure distribution changes. We therefore consider fixed points of the geometry and resign

ourselves to the existence of a moment trying to change our angle of attack. We can see this

phenomenon by sticking a piece of cardboard out the window of a moving car, or into a breeze.

What happens? In most cases, when allowed to rotate freely, the cardboard tries to flip over in

one direction or the other. The same happens to an airfoil. In fact, when allowed to rotate,

neither the cardboard or our airfoil are very useful for an extended period of time. That is what

this class is all about.

By convention, moments that tend to increase angle of attack are positive. Two points we

commonly refer to are the moment at the leading edge, MLE, and the moment at the quarter-

chord, or 25% from the leading edge towards the trailing edge, Mc/4. In general both of these

may change as a function of the angle of attack. In theory, there exists a single fixed point that

has a moment independent of the angle of attack, called the aerodynamic center. We will

discuss the aerodynamic center in further detail later.

Historically, aerodynamics has made much of dimensionless systems. This has made it easier to

exchange information between countries but, more importantly, allows for scale testing such as

that done in a wind tunnel. As long as 2 special parameters match, the aerodynamic flows are

dynamically similar. This means that the results obtained in one case (for instance in a wind

tunnel) can be scaled and compared to another (like the real world). One of these parameters is

the local mach number, which is the ratio of the fluid velocity, V∞, to the speed of sound in the

fluid, a∞:

The second parameter that must match for two flows to be dynamically similar is the Reynolds

number, Re. The Reynolds number is also dimensionless, and relates the momentum energy of a

flow to the viscous energy of the flow. It requires us to use a characteristic size, which we

choose to be the chord length, c. Then, using the local fluid density, ρ∞, and the fluid viscosity,

μ∞, we have:

We can then compare for two different problems, not the lift and drag forces directly, but

dimension coefficients that represent those forces. We first define the local dynamic pressure as

Note that dynamic pressure has the same units as pressure, or force per unit area. Given a

suitable reference area, we can then nondimensionalize our system. The commonly chosen

example is the wing planform area, S. This is the projected area of the wing onto a plane.

Our coefficient of lift is therefore computed as

While the drag coefficient is

Historically, we know that for most airfoils the drag coefficient is strong function of the lift

coefficient. In fact, in most cases we can express this relationship as a quadratic, using two

parameters CD,0 and K:

This relationship is often called the drag polar, and is shown in the figure below:

Drag polar diagram [2].

This breaks up the components of drag into two main categories, static drag and induced drag. A

constant amount of the drag is produced independent of lift generated. The quadratic term due to

lift that we add to this constant is often called induced drag, because it is an effect induced by lift

production.

Similar to the lift and drag coefficients, the nondimensional measure of our airfoil moment is the

moment coefficient:

Where we typically use the chord, c, as the needed relative length measurement. Again, the value

of this moment will vary depending on the location around which it is measured. We will have

various moment coefficients possible, such as that at the leading edge, the coefficient at the

quarter chord point (why is this point important?), and the moment at the aerodynamic center.

Earlier, we discussed the concept of the aerodynamic center. This is the point on the airfoil

where the moment is independent of angle of attack. To find this point, we first need a point of

reference. For convenience, we choose the quarter chord point. Wherever it is, we can find the

moment at the aerodynamic center using a transformation of the moment from the quarter-chord

by applied the force(lift) and the distance between the two points:

Conversion to the coefficients gives

Now, if we take the derivative with respect to the angle of attack, we know that it must be zero at

the aerodynamic center:

So we can then find the location of the aerodynamic center. If the lift coefficient and the

moment coefficient at the quarter-chord are linear functions of angle of attack, with slopes m0

and a0 respectively, then the location in terms of the chord is:

Thin airfoil theory predicts that the lift coefficient of a flat plate is

And that the moment at the quarter chord is zero. Therefore, the aerodynamic center is the

quarter-chord point. We can compute using the same process, but measured from the leading

edge. The same theory predicts that the coefficient of the moment at the leading edge is

From which we can calculate the distance of the aerodynamic center from the leading edge to be

Below is a projection of a typical straight wing:

Geometry of straight wing [2].

It is clear that not all wings must have the same chord throughout. Typically, we calculate

design values based off of the mean chord length, . The aspect ratio is a measurement of the

proportions of the wing, and depends on the overall wingspan, b, and the planform area.

A finite span wing differs in aerodynamic behavior from an airfoil, due to the vortices that are

generated at the wing tips and leading edge. The following diagram shows typical airfoil wind

tunnel data:

From this type of data, we get our airfoil lift coefficient derivative, a0. These are often corrected

for the effects of having a finite length using the following formulas using Prandtl’s lifting line

theory (note that the use of a variable as a subscript has the usual meaning of differentiation):

Where e1 is an efficiency parameter usually 0.90-0.95. For low aspect ratio wings, we can use

the formula called Helmbold’s relation:

References:

[1] B.N. Pamadi, Performance, Stability, Dynamics, and Control of Airplanes, AIAA, 2004.

[2] J.D. Anderson, Aircraft Performance and Design, McGraw-Hill, 1999.

Notation Systems and Terminology Review

First consider the diagram of the aircraft below:

Basic aircraft components [1].

The key components of interest are the wings, the horizontal and vertical stabilizers (together the

tail), the fuselage (also called the body), and the nacelles (which house the engines). Next,

consider the diagram of control surfaces below:

Aircraft control surfaces [1].

To control the aircraft, we have the rudder, elevators, and ailerons. To trim the aircraft, we have

the flaps as well as tabs, which will be discussed later in the course. By trim we generally mean

to stabilize current motion. The flaps are also used to generate additional lift during takeoffs and

landings. By extending the flaps from the wing, we change the effective camber of the airfoil

presented to the airstream, thereby changing the lift curve slope. The other paired control

surfaces operate using the same mechanism. The ailerons control rotations about the lengthwise

axis of the aircraft, known as roll motion. By moving only one aileron, we can increase the lift

of one side of the aircraft, generating a moment about the roll axis. We can get a similar effect

by lowering one aileron and raising the other. The ailerons function changing the lift balance

between the right and left sides of the aircraft. The elevators affect rotational motion around an

axis parallel to the wings, known as the pitch axis. They work in uniform motion on both sides

of the aircraft to generate additional (or less) lift on the tail, generating a moment that allows the

aircraft to pitch down (or up). The rudder works like a conventional rudder in a boat, but we can

also think of it as an airfoil. When neutral, the rudder generates no lift. When moved to one

side, it alters the camber of the airfoil, increasing lift towards the opposite side. This causes a

moment about the vertical axis, known as yaw motion, bringing the aircraft nose around in the

direction of the rudder.

These three types of rotational motion bring the total number of system variables to 6 (including

3 translation components). We typically call this type of system 6-DOF for 6 degrees of

freedom. To set a system of notation, we choose to follow the designation in the text.

Notice we have position and velocity along 3 axes, plus angle of rotation and rate of rotation.

Even though we have 6 degrees of freedom, we typically need 12 variables to solve the 2nd

order

equations of motion that result. Also note that, in this fixed body reference frame, rotations of

the craft about the x axis are roll motion, rotations about y are pitch motion, and rotations about z

are yaw motion.

In this course, we assume the Earth to be an inertial (nonaccelerating) reference system. We use

FE to denote this reference frame.

The velocity of the aircraft relative to the stationary Earth is

denoted by the vector VE, called the groundspeed. The velocity of the air above the Earth (the

wind) is denoted by the vector W, while the velocity of the aircraft with respect to the air is the

vector V, known as the airspeed. Clearly, the airspeed is responsible for the aerodynamic forces

generated by the aircraft. The relationship between these reference frames is

Unless otherwise noted, we can assume that W = 0 for most derivations since we are interested

in the behavior of the aircraft, not its position relative to the ground. The reference frame fixed

to the body is denoted by FB. In this frame, the air approaches the fixed center of gravity for the

airplane with a velocity of –V. Consider motion in the two coordinate planes shown in the

figure below:

Definitions of relative angles and motion in two dominant planes [2].

Note that, in defining the angle of attack as we did earlier, we use the projection of the velocity

vector on the xz plane. This plane is the plane of symmetry for the vehicle and motion in this

plane we refer to as longitudinal motion. The plane on the right contains the y axis and the

velocity vector, and is therefore not necessarily the xy plane. Motion out of the plane of

symmetry changes the direction of the velocity vector in two directions and is referred to as

lateral motion (or directional motion). If the components of the airspeed velocity are u, v, and w

then we have the following definitions for the angle of attack, α, and the sideslip angle, β:

References:

[1] J.D. Andersen, Introduction to Flight, McGraw-Hill, 2012.

[2] B. Etkin and L.D. Reid, Dynamics of Flight: Stability and Control, John Wiley & Sons, 1996.

Aircraft Performance Review

Consider the following sketch of an aircraft in flight:

Sketch of forces affecting aircraft performance.

The four main forces affecting this flight are lift, drag, weight, and thrust. The thrust can be

considered offset from the current velocity vector by the thrust offset vector, ε. In general, if the

velocity vector is not aligned with the local horizontal, we measure the inclination of the craft

using the flight path angle, γ, as drawn. Note that this angle is the same angle between the

velocity vector and the horizontal, so a flight path angle of zero indicates levelFro flight.

Motion like this, in which the velocity vector lies within the plane of symmetry, is also called

symmetric motion, and is the default case for the aircraft. We can, in most cases, assume ε to be

small or zero. One nominal case of aircraft performance is level flight. From the above diagram,

we write the equations of motion as

Assuming that ε = γ = 0 results in

Which are our basic relationships for steady, level flight. Steady implies that the conditions are

unchanging, since both components of velocity, parallel and perpendicular to the current

direction, are not being changed. In this case, the thrust required of our engine must exactly

match the drag produced, and the lift produced must exact balance the weight of the aircraft.

With these met, the velocity remains unchanged and the aircraft does not change elevation.

We can find out some things about this case. Since lift depends on angle of attack, we assume

the following relationship

Where α0 is the angle of attack producing zero lift. We then have, from the balance with weight:

And the angle of attack required is

Since a typical maximum angle of attack before stall is approximately 10 degrees, we naturally

have a limit to the weight of the aircraft. Similarly, thrust required is

We can then note that there is a minimum thrust required for each velocity, and that thus thrust

increases relative to the square of the aircraft weight. Another type of performance condition is

steady, gliding flight. This condition would have a nonzero γ and zero thrust. In this case, we

start with the same equations

Assuming T = 0 results in

We can find the glide angle from

So to minimize the glide angle (and therefore our time in the air from a fixed height) we need a

maximum L/D ratio (small D/L). The problem is exactly the same for a steady climb, except

now we add thrust to the equations (assuming ε = 0):

From this, we get information regarding the angle of the climb:

before, for steady level flight, we had T = D. Now, we have a certain amount of thrust above

and beyond that amount. From trigonometry, we know that the rate of climb will be V sin γ,

resulting in:

Since force times velocity is power, the rate of climb is the ratio of excess power (that available

above and beyond the amount needed to balance drag) to weight. Note that, for the case of T=0

and negative flight path angle, this reduces to the same result for a steady glide.

Consider the sketch of the next scenario, in which an aircraft undergoes a steady turn:

Sketch of aircraft in steady turn.

To undertake the turn, the aircraft banks through a roll angle, ϕ. We must still have a vertical

balance of forces to avoid change of altitude. We now have an imbalance of forces in the plane

of the turn perpendicular to the velocity direction. This comes from the component of lift that is

now not balanced by weight or drag. In cylindrical coordinates, we note that the normal

acceleration towards the inside of a turn is

Where R is the radius of curvature, as shown. Considering the fact that we must maintain a

constant velocity, we now have a balance of forces of: