AERODYNAMICS
LIFT AND DRAG
Lift is the force due to the difference in the pressure between the lower and upper surfaces, multiplied by the planform area of the surface.
The pressure difference between the upper and lower surfaces is adjusted by adjusting the surface geometry and attitude to the oncoming flow. In low-speed flows of air (<0.3 times the speed of sound, or Mach 0.3), there are 3 main ways of creating such a pressure difference:
In each case shown above, the flow moves more rapidly at some places than at others. In these regions of high velocity, the pressure is lower. The relation between pressure and velocity in low-speed flow is given by the Bernoulli equation:
, or 
This equation is derived from Newton's Second Law of Motion, which expresses "Conservation of Momentum".
p0 is called the Stagnation pressure, or total pressure. It is the pressure you feel at the center of your hand if you stick it out, palm facing the flow, from the window of a car (DO NOT STICK YOUR HAND OUT OF A MOVING CAR: ITS VERY DANGEROUS!)
because the flow is brought to a stop at the center of your hand (it can't decide which way to go around your hand).
P is called the Static pressure.
is called the dynamic pressure, also denoted as "q".
The pressure coefficient is a way to express the pressure with respect to some reference pressure, as a "dimensionless" quantity.
.
Cp = 0 indicates the undisturbed freestream value of static pressure.
Cp = 1 indicates a stagnation point.
Cp < 0 indicates a suction region.
Lift is related to freestream velocity by
where 
is the lift coefficient, and S is the planform area of the wing.
For an airfoil (infinite span, no tips) at angle of attack, the lift coefficient 
varies with angle of attack 
. If the airfoil is cambered, the lift coefficient is positive even at zero angle of attack, and reaches zero only at some negative value of angle of atack: this is called the "zero-lift angle of attack", 
. As the camber is increased, 
becomes more negative. Thus airfoil lift coefficient is
varies with angle of attack . If the airfoil is cambered, the lift coefficient is positive even at zero angle of attack, and reaches zero only at some negative value of angle of atack: this is called the "zero-lift angle of attack", . As the camber is increased, becomes more negative. Thus airfoil lift coefficient is
. The lift-curve slope is 
, and a very useful result from low-speed aerodynamics is that
, where 
is in radians.LIFT-INDUCED DRAG and ASPECT RATIO
At the ends of the wings, the pressure difference between the upper and lower sides is lost, as the flow rolls up into a vortex. This has two effects:
1. The overall lift is reduced, relative to the airfoil lift value predicted for a section of an infinite wing.
2. The lift vector is tilted back, so that an "induced drag" is created.
Both of these (usually undesirable) effects are reduced by increasing the Aspect Ratio of the wing. The Aspect Ratio is defined as:
where b is the wing span and S is the wing planform area.
The drag is given by:
The drag coefficient in low-speed flow is composed of 3 parts: 
. where
. where
is the parasite drag, which is independent of lift. It is usually due to the losses of stagnation pressure which occur when part of the flow separates somewhere along the wing or body surface. In high speed flight, the effect of shocks and wave drag must be added to this, and becomes the dominant source of drag.
Most aircraft are designed to minimize 
, and aerodynamics experts have become quite good at this, although the need to place huge antennae, externally-carried tanks and missiles, and the constraints imposed by "stealth" designs make this part of design very challenging. To see the effectiveness of aerodynamic design in reducing profile drag, consider that the profile drag of an airfoil of chord 1unit is about the same as that of a circular cylinder whose diameter is only 0.005 units. This is a remarkable result. One simple way to reduce profile drag is to ensure that the airfoil has a sharp trailing edge, so that the streamlines come smoothly off the upper and lower surfaces, without leaving a blunt edge behind which some flow can "hang around" and get dragged along with the airfoil.
, and aerodynamics experts have become quite good at this, although the need to place huge antennae, externally-carried tanks and missiles, and the constraints imposed by "stealth" designs make this part of design very challenging. To see the effectiveness of aerodynamic design in reducing profile drag, consider that the profile drag of an airfoil of chord 1unit is about the same as that of a circular cylinder whose diameter is only 0.005 units. This is a remarkable result. One simple way to reduce profile drag is to ensure that the airfoil has a sharp trailing edge, so that the streamlines come smoothly off the upper and lower surfaces, without leaving a blunt edge behind which some flow can "hang around" and get dragged along with the airfoil.
is the skin friction drag, which is due to viscosity. This becomes important in two limits: one where the size of the wing, or the speed of the flow, is extremely small, as might be the case for an insect-sized aircraft. This is called the "low-Reynolds number" limit. We will see later what this "Reynolds number" is. The other limit is that of high-speed flight, where the skin friction can be severe enough to heat up the wing surface to melting point. In the case of ordinary low-speed aircraft of usual-size airplanes, this skin-friction drag is a very small quantity, and is usually lumped together with the profile drag.
is the Induced Drag. In low-speed flight, this is the largest cause of drag, because you have to have lift to fly, and this drag is caused by lift.
. Here the quantity e is called the "spanwise efficiency factor". It is the answer to the question: How does this wing rate compared to the ideal wing for this aspect ratio? Its value is usually close to 1, perhaps as high as 0.99.
Note that:
, so that 
. Also, 
as 
. So to minimize induced drag, one should design wings with the largest possible aspect ratio, but also provide enough surface area so that you need only a small angle of attack to provide the necessary lift even at low speed. Of course when you increase aspect ratio (increase span) or increase wing area, the weight of the aircraft goes up, and probably the skin friction drag goes up. Aircraft designed for high effiency at low speed, such as the Pathfnder soloar-powered aircraft, the Post glider and the U-2 high-altitude, long-endurance reconnaissance aircraft shown in the pictures below, have very large aspect ratios, maybe reaching above 40).
Vortex-Induced Lift and Delta Wings
Now there is a third way to generate lift. The vortex generated at the wing tip is generally bad news, because it means lift loss and drag rise. However, being a vortex, it has regions of high velocity and low pressure. If we can make the vortex go close to the upper surface of the wing, this low pressure can provide the suction we need to generate lift. This principle is used on aircraft which, for other reasons, must hav wings with extremely low aspect ratio. In fact most aircraft designed for high-speed flight and high maneuverability have wings of small aspect ratio, with highly swept wing leading edges. The wing sweep is so high than we can think of the entire leading edge as the wing tip. Even at small angles of attack, a vortex forms along this edge (called, obviously, the Leading Edge Vortex), and this provides much of the lift of such wings when the aircraft is flying at low speed (even supersonic aircraft need to land, fairly slowly). When vortex lift is used, the wings can be very thin, and have sharp leading edges, which are good to minimize shocks and wave drag in high speed flight.
The vortex lift-curve slope is very small compared to the ideal lift curve slope of 2p per radian. However, vortex lift can be obtained upto large angles of attack, sometimes up to 30 degrees angle of attack. So adequate lift can be obtained by going to high angles of attack during landing and low-speed flight. The North American XB-70 supersonic bomber and the British Aerospace - Aerospatiale Concorde (shown against the sun, below) and the Soviet Tupolev Tu-144supersonic jetliners are examples of delta-winged aircraft. The delta wings are good for supersonic flight. When the aircraft comes in for a landing, it does so at a high angle of attack where the wings produce vortex lift.
Speed for Minimum Drag
For convenience, we will lump the friction drag with the profile drag, so that the total drag is composed of a part which depends on lift, and one that does not.
Total drag is thus, 
Thus,
Let us consider what it takes to keep L = W, i.e., provide enough lift to maintain steady level flight.
.So 
. Substituting,
. Now, as 
increases, the first term (profile drag) increases, but the second term (induced drag) decreases. This is obvious when you think about it: as the speed increases, the dynamic pressure increases as the square of the speed. You need less lift coefficient (i.e., smaller angle of attack) to create the lift required to balance the weight.
To find the speed for minimum drag, we can either plot the total drag for various speeds, or find the answer using calculus. We differentiate the expression for drag with respect to dynamic pressure, and set the result to zero, and solve for the speed. This should be either the speed for a minimum or maximum. Strictly, to know if its a maximum or minimum, we should also see the sign of the second derivative (+ for minimum, - for maximum), but here we will take it for granted that it is in fact a minimum.
, i.e, 
. . This is a remarkable result: It means that:
To fly an airplane of a given weight, straight and level, the condition for minimum drag (maximum lift-to-drag ratio) is that the profile drag coefficient is the same as the induced drag coefficient.
Example:
The GT2010 aircraft will have a wing loading (W/S) of 130 pounds per square foot (6233N/m2), aspect ratio of 7.667, and wing span of 60.96m. We'll assume that its spanwise efficiency factor will be 0.99. Let's assume that the profile drag coefficient is given by
where Mcr is the Mach number at which shocks start forming. Here Mcr is taken as 0.85.
For the moment, let's take 
.
.
Thus, for maximum Lift-to-Drag ratio (minimum drag, and the lift is always equal to the weight for straight and level flight),
, so that the corresponding CL is calculated as 0.598, and the dynamic pressure is 10423N/m2. At an altitude of 11,000 meters in the Standard Atmosphere, the density is approximately 0.36kg/m3, so that the flight speed is 240.64 m/s. At this altitude, the Standard temperature is 216.7K, so that the speed of sound is 295 m/s. So the flight Mach number for minimum drag (or best lift-to-drag ratio) is 0.8 at 11,000 meters for this aircraft.
NOTE: The above calculation is not quite correct. We should not use formulae for CL and CD obtained for low-speed flows, to calculate results at Mach 0.8, which is close to the speed of sound. Some inaccuracy will be involved. We'll correct and refine these results later.
AERODYNAMICS SUMMARY
There are 3 ways of generating lift (meaning force perpendicular to the flow direction, due to pressure differences across surfaces):
a) angle of attack
b) camber
c) vortex-induced lift.
For symmetric airfoils at low Mach number, the center of pressure, and the aerodynamic center, are both at 0.25 times the chod from the leading edge.
Camber causes a nose-down pitching moment. For cambered airfoils, the center of pressure is at a chordwise station downstream of 0.25 (x/c >0.25), but the aerodynamic center is still at 0.25.
An infinite (2-dimensional) wing is entirely described by its airfoil section.
Finite wings have less lift than corresponding span-lengths of an infinite wing at the same angle of attack, and also have lift-induced drag.
The total drag is composed of profile drag, which does not vary with lift, and induced drag, which rises as the square of the lift coefficient.
To fly an airplane of a given weight, straight and level, the condition for minimum drag (maximum lift-to-drag ratio) is that the profile drag coefficient is the same as the induced drag coefficient.
WING LOADING AND CRUISE DESIGN POINT
The wing loading, W/S, is a decision to be made by the designer. If this is too low, then the aircraft will be efficient in low-speed flight, and perhaps have lower aerodynamic noise in high speed flight, but the structure may weigh too much (large wings) and the skin friction drag will be high in high-speed flight. Also, the aircraft will be more responsive to gusts, and hence will get bounced around a lot. Low wing loading is good for gliders and for aircraft intended for long endurance.
The wing loading goes up as the expected design speed goes up. The empirical data on this is shown in the figure from Tennekes et al. We see that for an aircraft in the class of the GT2010, a typical wing loading value is around 130 to 140 psf. We will select 130 psf.
Now, knowing the takeoff weight, we can calculate the wing area. Lets choose a wing span of nearly 200 feet, because that's about as large as wing spans get on large aircraft. If we go any higher, there may be problems getting in and out of the airport gates and hangars and parking spaces. Thus, b= 200 ft. This lets us calculate the aspect ratio AR=b2/S. For such an aircraft, we can design a wing to nearly the best efficiency, so that we can take the spanwise efficiency factor to be 0.99. Given these values, and the preceding discussion on aerodynamics, we can compute the best speed for each altitude, and also vary this as the weight of the aircraft changes (i.e., as the payload varies and the fuel is burned. In selecting the best altitude, we will also try to stay just below the Drag divergence Mach number: this avoids unnecessary rise in profile drag.
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