Lift and Drag Formulas
The lift and the drag formulas for a finite wing offer
a valuable tool for analyzing aerodynamic relationships.
Where:
* The area of a wing is the product of the wing span and the Mean Aerodynamic Chord (MAC). The mean aerodynamic chord is an imaginary chord line that is derived from the length of the chord line at various locations of the wing.
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It should be noted that the wing area (S) and the air density (
= Air density
V = Velocity
) are constant for any given
altitude while both CL/ CD and the velocity (V }are variables. The air velocity is a major
contributor to lift and drag because both are proportional to square of the velocity.
From the lift and drag formulas, it follows that the velocity and the angle of attack
(represented by either CL or CD) are inversely proportional. For example, an increase an
angle of attack at a constant power will decrease the speed. Conversely, High speed at a
constant power will require lower angle of attack.
Total Drag
Parasite Drag
The drag on the airfoil is only a part of the total drag of an airplane. Reducing drag is essential for flight efficiency.
The total drags on an airplane consist of all the drag contributing elements.
It is customary to refer to drag caused by the airplane parts which are not lift producers
as Parasite Drag. To minimize the parasite drag it is desired to design in airfoil shape
all aircraft parts such as struts, wheel fairing, etc.
The two major contributors to parasite drag are the form drag and the skin-friction
drag. The shape, or form of objects being exposed to airflow determines the magnitude of
drag. The flow around round objects is smoother than around square objects and the airflow
around a symmetric airfoil is almost ideal. The form drag results from the applied pressure
on moving objects and depends largely on the generation of wake. To reduce the parasite drag
aircraft parts that come in contact with the airflow have an airfoil design.
Interference Drag
It is not enough to add all the form and the skin-friction drag values of parts that where separately tested to obtain the total parasite drag. Interference drag is obtained from testing parts assembled rather testing them separately. As this test is conducted, the wake of one part may affect the drag of the another. This effect may be favorable as well as unfavorable.
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Induced Drag
From Bernoulli's Principle we know that the pressure below an airplane wing is higher than the pressure above it. As a result, there is a constant tendency of air to flow from bottom to top(a). Since the airplane is constantly moving the air is forced up at the wing tips.
Considering an airfoil at each wing tip, as demonstrated in (b), the airflow (V) is
deflected upwards (w) resulting in airflow (Vres), thus increasing the angle of attack. Angle
of attack 
is greater than angle of attack 
. as a result, both lift and drag are increased
at the wing tips section.
The original explanation of lift and drag assumed an ideal airflow. Induced drag results from
imperfection in the airflow caused by lift. Two theories offered here to explain the induced
drag. As explained earlier, the pressure below an airplane wing is higher than the pressure
above it. As a result, there is a constant tendency of air to flow from bottom to top. Since
the airplane is constantly moving the air is forced up at the wing tips(spillage).
The Formation of a Downwash Field Behind a Finite Wing
Prandtl* Theory -
As a finite wing moves through air, vortices are generated around each wing tip (as shown above).
The wing tip vortices are so powerful that they affect the entire airflow as it departs
behind the wing.
The strong vortex produced by the wingtip spillage is called Vortex Sheet. The Bound Vortex
describes actually the airflow as it follows the airfoil boundary. A Starting Vortex is a
product of circulation around an airfoil.
A combination of these vortices generates an area of Downwash Field which is trailing behind
the wing.
The illustration to the right demonstrates how the downwash changes in the airflow and
contributes to drag. The airflow (V) is deflected down by the down wash in an angle 
resulting in actual airflow (VREF). Vector 
represents the vertical flow caused by the
downwash. Because lift is perpendicular to the direction of the airflow, it is demonstrated
that the actual lift (Lmod) is deflected in an angle from the original lift(L). The
vector D shows an additional drag force that would not otherwise present. This additional
drag is called induced drag.
Wing Tip Theory - As shown earlier, the angle of attack at the area near to the wing tips
increases as a result of the spillage of air around them.
The spillage produces an upward component of airflow in addition to
the original airfoil (V)resulting in actual airflow Vres. 
The new angle of attack is greater than angle of the original
angle of attack thus increasing both lift and drag at the wingtips
area. This additional drag occurs without an apparent change of the angle of attack but by
an induced change. This drag is affecting a limited portion of the wing. However, with a
short wing span its effect becomes more significant.
*Ludwig Prandtl, German engineer and professor (1875-1953)
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Induced Drag Reduction
Induced drag is inversely proportional to the speed (velocity) of the air. As explained
earlier, the angle of attack (represented by CL and CD), is inversely proportional to the
air velocity. As a result, flight at higher speed requires smaller angle of attack. The
decrease in angle of attack reduces the pressure differentials on the airplane's wing thus
reducing the air spillage and the induced drag. It should be noted the under flight
conditions that require a reduction of speed, one can expect higher angle of attack, larger
wing tip vortices and greater induced drag.
Reduced drag can also be minimized by design. This can be accomplished by high Aspect Ratio
or by mechanically limiting the air spillage around the wing tips.
Aspect Ratio: Aspect Ratio is defined as the quotient of the wing span squared and the area of the wing. When the wing is rectangular, the Aspect Ratio is quotient of the wing span and the chord line.
The definition can be expressed mathematically by:
or 
when the wing is rectangular.
Increasing the Aspect Ratio, i.e., maximizing the wing span while minimizing the wing's chord, reduces the down wash and the effect of the air spillage around the wing tips. Though increasing the aspect ratio decreases the induced drag, it imposes other limitations on performance specifically at high speeds.
Wing Tip Solutions: A significant interference with the air spillage around the wing tip
limits the vortex. Such interference is achieved by constructing a wing tip the is either
diverting the higher pressure under the wing upward or downward. This limits the flow from
high pressure to low pressure and reduces the downwash. In other words it reduces the induced
drag. The following figure demonstrates the Win Tip Winglets, a common method to reduce induced
drag in modern airplanes.
The Total Drag vs. Speed
Aerodynamic Efficiency: An optimum efficiency performance of an airplane requires maximum
lift at minimum drag. The aerodynamic efficiency is defined as the ratio between the lift
coefficient and the drag coefficient. It is expressed in the mathematically form:
On the left is a graphic illustration of the total drag on an airplane taking into
consideration the by parasite drag, form and friction drag and induced drag.
The magenta curve represents the sum of the parasite, form and friction drag. The
drag increases proportionally to the square speed as the air velocity increases. The orange
curved line represents the induced drag. As shown previously, induced drag is inversely
proportional to the velocity of the air. The green line is obtained from combining the two
other graphs and represents the total drag on the airplane. This information is vital for
airplane performance analysis and will be discussed later on.
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Lift-Drag Relationship
- It follows that L/Dmax = CLmax / CDmin
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