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More About Lift and Drag

Lift and Drag Coefficient

The Lift Coefficient and the Drag Coefficient represent the changes in lift and drag as the angle of attack changes. CL and CD are not expressed by any physical unit, they are rather absolute numbers obtained from either wind tunnel tests or derived mathematically.
Initially both CL and CD increase as the angle of attack increases. At a certain point, the lift begins to drop while the drag increases sharply. This point is defined as the Critical Angle of Attack. If the angle of attack increases passed the Critical Angle of Attack, At one point all lift will be lost while the drag continues to increase.

(a) The relationship between the Angle of Attack and Lift Coefficient
 (b) The relationship between the Angle of Attack and Drag Coefficient


Lift and Drag Formulas

The lift and the drag formulas for a finite wing offer a valuable tool for analyzing aerodynamic relationships.

     
Lift and Drag Formulas

Where:

It should be noted that the wing area (S) and the air density () 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.

* 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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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.

Form Drag

The skin smoothness of aerodynamically structures determines the resistance of the skin to airflow. If such resistance exists, the stream line of a thin layer (also referred to as the boundary layer) is disturbed and affecting the adjacent layers. This form of drag is known as Skin-friction Drag.

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.

The effect of pressure differential on the angle of attack

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:

Where:      AR = Aspect Ratio       b = Wing Span      S = Wing Area       c = Wing Chord

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

 
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

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:

      AE =CL / CD
Where AE = L/D      CL = Lift coefficient       CD = Drag coefficient
      It follows that L/Dmax = CLmax / CDmin

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Last update May 17, 2005
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