Aerodynamics of Boomerangs. Chapter 3

by Saulius Pakalnis

Click here if you don't see all 7 chapters

Main specifications of airfoil. Lift and drag coefficients dependences on the angle of attack

The angle of attack is the angle at which relative wind meets an airfoil. It is the angle that is formed by the chord of the airfoil and the direction of the relative wind or between the chord line and the flight path. The angle of attack changes during a flight as the pilot changes the direction of the aircraft.

Lift and drag coefficients dependences on the angle of attack


Lift and drag coefficients Vs angle of attack of symmetrical airfoil. At angle of attack  less than 12 degrees Lift coefficient increases significantly, while Drag coefficient remains at small value. The anomaly of Lift coefficient enables birds and planes to fly with low energy losses.


An airfoil is fairly efficient. Asymmetry of curvatures of upper and lower airfoil surfaces (upper surface is more curved) increases the lift coefficient Cy.

A lot of lift (Cy) is obtained without much drag until you get to about 12 degrees angle of attack (a) on this curve. Then drag (Cx) goes way up, without creating much more lift. As the angle of attack increases from 12 to 19 degrees for this particular design, there is not much increase in lift but you have much  more drag. We say the wing is stalled due to turbulence when lift decreases at increasingly higher angles of attack.

What kind of mechanisms are responsible for such anomaly which lets to fly? Theory of streams (one more advanced idea..?)

Unfortunately, in literature I didn't find any reasonable explanation which would be understandable to my mother in law.  Instead of long talks about Nevier-Stokes or Boundary layer equations I present my interpretation, which is based on several statements:
1. There are static fluid layers and dynamic fluid layers. Dynamic layers exhibit extra fluid features. So they can be considered as streams.
If different layers of fluid flow get different speeds, then is assumed that the flow generates streams.
3. Stream is a dynamic fluid layer, which is separated from other layers by velocity gradient. 
4. Shear-stress produces velocity gradient, which turns stream front perpendicular to stream direction. Two neighboring streams dynamically bend into each other. The shear stress also forces stream to stick to solid body surface. The attractive force has a dynamic origin and may be called  Coanda force.
5. Stream can be either laminar or turbulent.

Let's consider an airfoil which is moving in fluid. The airfoil pushes fluid in front of it. This results in air pillow with increased dynamic pressure.

Extra pressure energizes the nearest fluid  layers making them to move relatively to neighbors. The dynamic layer now can be considered as a stream.   The stream, which is the closest to airfoil surface experiences shear stress (surface - stream) which accelerates the stream relative to neighboring layers. The stream moves relative to airfoil with different speed, so it experiences a gradient of velocity. The velocity gradient produces Coanda force, which bends the stream front towards the surface. The stream dynamically sticks to the surface.

The stream also produces shear stress with neighbouring  fluid layer. The shear stress accelerates the neighbouring layer. Then the second stream is born and dynamically sticks to the first one. This second stream also accelerates its neighbour... Let's call this process a "generation of streams".

Then every next stream is produced by the previous stream. Following several iterations of the streams formation mechanism, we obtain the boundary layer, having the features:
1. Velocity gradient - as every stream (in surface normal direction) moves more slowly as its neighbour.
2. Coanda force which dynamically stick streams. The Coanda force bundles streams and bends them to follow around the curvature of airfoil surface.

Note: Even if streams are moving along the streamlines, there is a physical difference between the streams and streamlines: streams are generated as dynamic objects and exist only in flow gradients. Having no energy input from flow gradients, they disappear as quickly as they were created. Streamlines are lines which show some marker trajectory.

Visualization of Prandtl's boundary layer

1. Leading shock wave front. Fluid doesn't "feel" airfoil and airfoil doesn't feel fluid on left 1-A-1 line.
2. Fluid pillow: the area of increased dynamic pressure. The streams generation starts here.
3. Streams. The envelope of the streams corresponds to theBoundary area. It is assumed that stream having less than 5 percent of stream maximum speed can be neglected. The threshold option permits to hide very slow streams in trailing edge, which do not affect airfoil lift, but are responsible for energy dissipation only.
4. Trailing edge of shock wave. Streams separate from trailing edge of airfoil near 4-C-4 line. The separation relaxes stress, which generates the trailing edge shock wave. No lift force is generated on right side of this line. The right (CD) area corresponds to stream tension relaxation and the region of their inertive energy dissipation.
5. Reduced pressure air bubble increases airfoil drag, but has no effects on the lift force. It starts at a separation point and follows the airfoil, sucking fluid from the nearest streams. The bubble corresponds the turbulent region.

AE - corresponds the airfoil (red one) trajectory. CD is direction of streams after the airfoil has passed. ABC is the area where airfoil and streams interaction mechanism turns the Boundary layer fluid mass around the curvature of airfoil, thus making a lift.

According to Lift equation
Lift force = 1/2· Cy·A·rho·v2,
Lift is proportional to wing area A,
where wing area A= wing_length· Chord_length,
Lift dependence on chord length is not evident. The mechanism how thickness of deflected layers depends on chord length is shown in picture below.

Lift force is proportional to the volume of deflected down air. The volume of deflected air layers corresponds to Boundary layer thickness at the point, where it separates from airfoil surface. The boundary layer thickness gradually increases along the chord length. Perfect situation is obtained then boundary layer separates at trailing edge of the airfoil. Unfortunately, separation position depends on many factors, and boundary layer separation occurs somewhere between leading and trailing edges.

Criterion of Boundary layer separation

The closest to airfoil surface stream is accelerated by friction of airfoil and the stream. The speed gradient of the stream is responsible for Coanda force witch bends stream into airfoil surface, thus producing the stream sticking to the surface. The stream follows the curvature of surface until the stream (relative to speed of airfoil) becomes very small. On the other hand the stream relative speed to neighbour stream increases. This Coanda force sucks the stream away from airfoil surface. At point C relative airfoil-stream speed becomes so small that the relation is satisfied:

and the stream separates from the airfoil surface. Boundary layer too. Total view is described in the figure below.

Do not confuse circulation in airfoil wake with generation of the lift force. Lift force is generated in range from 1-A-1 to 4-C-4. The area on right from 4-C-4 corresponds the drag and energy relaxation area. The air bubble reduced pressure area 5 gives some lift, but it also produces a significant drag.

This model gives a criterion to maximize the lift and minimize the drag. From the engineering point of view the task is (for fixed angle of attack) to shift point C toward the airfoil trailing edge as much as possible. No separation condition means, that relative airfoil - stream velocity of the nearest to airfoil surface stream should never be zero or even small value. The problem is that at trailing edge region the stream relative speed is continuously decreasing. So, if chord length is long enough (an example, for delta wing), this necessary happens.

There are some tricks to escape the boundary layer separation. First idea is to decrease the chord length. Second idea is to energize the lowest stream of upper airfoil surface. Lowest surface of airfoil at non-zero angle of attack is energized by dynamic pressure, but upper airfoil surface needs special tricks.

What about this construction of airfoil?
Turbulent flow versus laminar flow. Mini turbulence breaks separation condition, as different mini vortexes have different velocities and sign (rotation direction). Actually, the turbulent layer thickness continuously increases along chord and separation also happens. But separation point is delayed and shifted toward trailing edge.
Small dimples between B and trailing edge of airfoil. The small fluid vortexes may work as spinning balls, thus reducing friction. The nearest laminar stream is rolling on them without large relative velocity reduction. Thus separation point shifts to trailing edge. The case of small turbulent bubbles (vortexes) gives less drag than big bubble at the trailing edge of the airfoil.