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Aerodynamics of Boomerangs


The purpose of the article is to present you with fundamental knowledge behind the aerodynamics of boomerangs and principles, which allow you to build your own boomerang constructions. The objective is to encourage you to make high quality returning boomerangs for this sport.

Ideal and Real Gases

ideal gas motion

Ideal Gases

  1. Gas consists of molecules in ceaseless motion.
  2. The size of the molecules is negligible in the sense that their diameters are much smaller than the average distance traveled between collisions.
  3. The molecules do no interact, except during collisions.

Potential Energy vs Separation Graph

Real Gases

The image to the left displays the typical international potential between molecules of real gases. Attractive Van der Waals forces have a very complex character at distances about 10^-7 cm. They decrease with distance as ~1/(r^7). At long distance, molecules attract each other. This attraction is also a response for the condensation of gases into liquids at low temperatures. At a short distance, molecules repel each other. This repulsion is a response for the definite volumes of liquids and solids, so they will not collapse to a point.

Notes and Lessons from the Ideal Gas Equation


  • For 1 mol of an ideal gas, PV/RT = 1 for all pressures.
  • In a real gas, PV/RT varies from 1 significantly.
  • The higher the pressure - the bigger deviation from the ideal behavior.
  • For 1 mol of ideal gas, PV/RT = 1 for all temperatures.
  • As temperature increases, the gases behave in a more ideal way.


The assumptions of the kinetic-molecular theory show where ideal gas behavior breaks down:


  • The molecules of gas have finite volume.
  • Molecules of gas do attract each other.
  • As the pressure on gas increases, the molecules are forced closer together.
  • As the molecules get closer together, the volume of the container gets smaller.
  • The smaller the container, the more of the total space the gas molecules occupy.
  • Therefore, the higher the pressure, the less the gas resembles the ideal gas.
  • As gas molecules get closer together, the intermolecular distances decrease.
  • The smaller the distance between the gas molecules, the more likely the attractive forces will develop between the molecules.
  • Therefore, the less probability the gas resembles the ideal gas.
  • As temperature increases, the gas molecules move faster and further apart.
  • Also, higher temperatures mean more energy available to break intermolecular forces.
  • As temperature increases, the negative departure from ideal gas behavior disappears.


Although the ideal gas model is very useful, it is only an approximation of the real nature of gases, and the equations derived from its assumptions are not entirely dependable. As a consequence, the measured properties of a real gas will very often differ from the properties predicted by calculations.

Real gases sometimes don't obey the ideal gas laws because the ideal gas model is based on some assumptions that aren't completely true. The main flaw in the ideal gas model is the assumption that gas molecules do not attract or repel each other. Attractions and repulsions are negligible when the distance between molecules is large, but they do become larger as the molecules become closer together. If you can contrive conditions that force the molecules into close contact, so that attractions and repulsions can't be neglected, you will likely see deviations from ideal behavior.

Gyroscopic Precession

Gyroscopic Precession Diagram

A gyroscope is a spinning device demonstrating the principle of conservation of angular momentum, in physics.
The traditional mathematical definition of the angular momentum of a particle about some origin is described below:
Angular Momentum Equation
Where L is the angular momentum of the particle, r is the position of the particle expressed as a displacement vector from the origin.
p = m*v is the linear momentum of the particle (more correct expression is  ).  m - Mass of the particle.
If a system consists of several particles, the total angular momentum can be obtained by adding (integrating) all the angular moments of the constituent particles. Angular momentum can also be calculated by multiplying the square of the distance to the point of rotation, the mass of the particle and the angular velocity.


Precession of Gyroscope

The device, once spinning, tends to resist changes to its orientation. If external force F1 is applied to some point which is at radius r from rotation axis of gyroscope, the spinning device begins to rotate. The motion "seems to be strange" as it does not follow the applied force direction, but moves in a perpendicular one. This rotation of spinning plane is called precession.

The simplest explanation of the phenomenon is shown below.
phase one of precession
resulting speed precession
axis is bended precession

The gyroscopic precession is fundamental phenomenon which explains why boomerang returns. See boomerang model in next chapter.

For example, t
he angular momentum from turning bicycle wheels makes them act like gyroscopes to help stabilize the bicycle. This gyroscopic action also helps to turn the bicycle.

biker torque

Stability of free-hand biking: Take a look for yourself to see how the front wheel turns to the left/right when you shift your center of mass (and bend the bike) to the left/right side of a bicycle.
stability of free-hand biking 

Air Foils

Fluid stream through air foil

Note 1. The Fluid Stream Front is a fluid stream front
Note 2. There are 2 types of friction:
  • Friction between the lowest air stream and the foil. The friction bends the layer front into foil surface. This results in "dynamic sticking to the surface".
  • Friction between different layers of streams (dynamic viscosity). The lower layer of the stream drags and bends down the upper neighboring layer.
Note 3. The stream layers have velocities gradient along the normal to foil surfaces.
Note 4. The figure is simplified - the foil is considered to be very thin. Air flow is laminar. The stream bending (and little compression for air only) is exaggerated in order to show it.
Note 5. The Coanda effect has a dynamic origin - it appears at nonzero relative to fluid and foil speed.

Coanda Effect Upon Air Foil


Boundary Layer. Reference

When the air hits the airfoil leading edge, it will separate into the upper and lower airstream; which meets again at the trailing edge. It is obvious that the air, very close to the airfoil, "rubs" against the solid surface and is slowed down. In other words, starting downstream of the impact point, the air loses some of its momentum, or velocity. In addition, it loses more and more momentom as we follow it along the path close to the solid airfoil. We can see that friction creates an area where there is less speed. The reduced speed area, just outside of the airfoil, becomes thicker and thicker as we follow it from the leading edge to the trailing edge. This area is called the boundary layer. Its thickness is increasing as described, and is defined as the thickness at which the local free stream speed is finally reached. A typical boundary layer thickness is 1/2" near the trailing edge. The friction, which obviously is a loss, results in the friction drag of the airfoil.

Laminar versus Turbulent air flow
Ref. Glenn Research Center

Again the theory of fluid dynamics shows that there are two possible types of stable boundary layers Ref. : The first to build up is called 'laminar," because the flow is nice and steady while the friction drag is relatively low.
The second is called 'turbulent," because the flow is rather rough and the friction drag is higher.

Unfortunately, the "laminar boundary layer" will automatically become turbulent (with associated higher drag) close to the leading edge of the airfoil unless very special precautions are taken. These precautions are:

  • A very smooth airfoil surface: Slight construction defects (or bugs as they stick to the airfoil leading edge) will change the laminar boundary layer into a turbulent one. Unless you have a perfect airfoil and keep it this way, forget about the possible gain with a laminar flow! 
  • A special shape of the airfoil: The pressure distribution on the airfoil is related to the airfoil shape. Today, we can calculate (with high speed computers) airfoils, which maximize the length of the laminar boundary layer. Still, what is mentioned in a) applies. But, do not get desperate. The friction drag of the airfoil with a laminar boundary layer is .08, whereas in turbulent flow it becomes .12. Sure, this is a 50% increase but only on the friction drag of the airfoil.

Lift force of airfoil

Lift depends on the density of the air, the square of the velocity, the air's viscosity and compressibility, the surface area over which the air flows, the shape of the body, and the body's inclination to the flow. In general, the dependence on body shape, inclination, air viscosity and compressibility are very complex.
One way to deal with complex dependencies is to characterize the dependence by a single variable. As for the lift, this variable is called the lift coefficient, designated "Cy." The lift equation states that lift L is equal to the lift coefficient Cy times the density rho times half of the velocity V squared times the wing area A.

  • Lift force= 1/2*Cy * A * rho * V^2

For given air conditions, shape and inclination of the object, we have to determine a value for Cy to determine the lift. For some simple flow conditions, geometries and low inclinations, aerodynamicists can determine the value of Cy mathematically. But, in general, this parameter is determined experimentally. The combination of terms "density times the square of the velocity divided by two" is called the dynamic pressure.

Air flow across an air foil

Actually, the coefficient Cy hides a mechanism of lift (also physics of aerodynamics), but this allows us to collect all the effects, simple and complex, into a single equation. The question, what kind of mechanisms convert the drag force into the lift one, still remains under discussion.


  • The amount of air diverted by a wing is proportional to the speed of the wing and the air density. 
  • The vertical velocity of the diverted air is proportional to the speed of the wing and the angle of attack. 
  • The lift is proportional to the amount of air diverted times the vertical velocity of the air. 


The Coanda Effect

If a stream of water is flowing along a solid surface which is curved slightly away from the stream, the water will tend to follow the surface. This is an example of the Coanda effect and is easily demonstrated by holding the back of a spoon vertically under a thin stream of water from a faucet. If you hold the spoon so that it can swing, you will feel it being pulled toward the stream of water. The effect has limits: if you use a sphere instead of a spoon, you will find that the water will only follow a part of the way around. Further, if the surface is too sharply curved, the water will not follow but will just bend a bit and break away from the surface.

The Coanda effect works with any of our usual fluids, such as air at usual temperatures, pressures, and speeds. I make these qualifications because (to give a few examples) liquid helium, gasses at extremes of low or high pressure or temperature, and fluids at supersonic speeds often behave rather differently. Fortunately, we don't have to worry about all of those extremes with model planes.

A stream of air, such as what you'd get if you blow through a straw, goes in a straight line
straight line stream of air
A stream of air alongside a straight surface still goes in a straight line
stream of air parallel to a flat surface
A stream of air alongside a curved surface tends to follow the curvature of the surface. Seems natural enough.
Air Stream following curvature of surface upwards
Strangely, a stream of air alongside a curved surface that bends away from it still tends to follow the curvature of the surface. This is the Coanda effect.
Air stream follows curvature downwards

Another thing we don't have to wonder about is why the Coanda effect works; we can take it as an experimental fact. But I hope your curiosity is unsatisfied on this point and that you will seek further.

Coanda and Flight

Many scientists have recently begun using the Coanda effect to at least partially explain how planes fly.

For a long time many people believed (and many people still do) that lift during flight is achieved due to something called the Bernoulli Effect. This theory suggest that air moving across a wing moves more quickly over the top than underneath. This creates an area of lower pressure on top of the wing in comparison to the underside of the wing. Thus, less pressure pushes down on the wing and more pressure pushes up and consequently lifts the craft into the air.

However, many scientists disagree with this explanation! Is it time to say good bye to Bernoulli's principle while speaking about a lift of wing?

Some scientists have suggested recently that due to the shape of a planes wing, air moving along it due to the Coanda effect will be deflected downwards as it leaves the wing and thus push the craft up into the air (due to Newton's Third Law of Motion) and consequently assist with lift.

What is the Coanda effect?

Very simply, the Coanda (or ‘wall attachment’) effect is the tendency for a moving fluid (either liquid or gas) to attach itself to a surface and flow along it.

coanda effect explanation with soldiers diagram Coanda Effect explained with soldier part 2

Displayed above is an image of a girl and a few soldiers. Soldiers are marching forward in straight line. Each soldier holds hand of his neighbor. Suddenly outsider soldier caches a hand of a girl standing on a sidewalk.

See what happens. This is simplest explanation of Coanda effect.

Note: it is assumed, that "soldiers" are some fluid elements, not single molecules.

One way of explaining this effect is to understand that as a fluid moves across a surface, and certain amount of friction (called skin friction) occurs between the two surfaces (friction is that force that slows down or prevents two surfaces from moving across each other). This friction tends to slow down the fluid as it moves across the other surface. This resistance to the flow of the fluid will then pull the fluid towards the other surface, making it stick to it… even as it bends around corners! 

Air Streams

Viscosity of air

"It seems natural to see the origin of viscosity in terms of the attractive and repulsive forces between molecules. However, gases have substantial viscosity even though their inter-molecular forces are weak, suggesting some other mechanism. Viscosity in gases arises principally from the molecular diffusion that transports momentum between layers of flow. A lot of fluid dynamics is concerned with in-viscid flow, but the role of viscosity is crucial to understanding some of the most important fluid phenomena, such as lift produced by a wing." (Author: Fred Senese senese@antoine.frostburg.edu). 

Touch a lift force in water 

Lift Force as shown with water and a inner spoon

Inner surface

The water "reflects" from spoon surface. According to the Newton's third law ("For every action, there is an equal and opposite reaction") the spoon moves to the opposite direction. The motion is shown by arrows.

Lift Force as shown with water and a outer spoon

Outer surface

The water follows the surface of the spoon. According to Newton's third law, the spoon moves to the opposite direction.

Why does water stick to the surface of spoon?

  • You can replace the surface tension's attractive force by a repulsive one. Just put oil or fat film on the spoon's surface. You will be able to see that the static forces (as capillarity) make no changes to water stream. It still  attracts the spoon. The effect has a dynamic origin; It is explained by the Coanda effect.


Peculiarities of dynamic and static pressure



    Dynamic pressure is the component of fluid pressure that represents fluid kinetic energy (i.e., motion), while static pressure represents hydro-static effects.
    total pressure formula
    Static pressure is isotropic - the same in all x,y,x directions. In air, it is equal to the atmospheric pressure and does not depend on the wing speed.
    static pressure
    Dynamic pressure of a fluid stream with density and speed u is given by dynamic pressure formula 
    Dynamic pressure represents the fluid stream motion at a certain direction. Air speed u is evaluated as a relative speed of the wing.

    dynamic pressure across air foil

    Note1. Pillow phenomenon may explain why air flow speed near the curved upper surface is accelerated and is higher when compared to the air flow  near the lower surface.
    Note2: Air stream "sticks" to surface and bends down near the upper surface of the wing, which is not caused by the Pillow phenomenon but the Coanda effect.
    Note3: Do not confuse Dynamic pressure with a pressure near airfoil surfaces. Air stream interaction with both airfoil surfaces has a complex dependence and gives some relative pressure, which may be applied to the normal of surface.
    Misinterpretations of Bernoulli's Law

    Misinterpretations of Bernoulli's Law

    Static pressure in a free air stream

    Static pressure is the pressure inside the stream measured by a manometer moving with the flow. At the same time, the static pressure is the pressure which is exerted on a plane parallel to the flow. Thus the static pressure within an air stream has to be measured carefully using a special probe. A thin disk must cover the probe except for the opening. The disk must be positioned parallel to the streaming flow, so that the flow is not interfered with.

    If the static pressure is measured in the way outlined above within a free air stream generated by a fan or a hair dryer it can be shown that the static pressure is the same as in the surrounding atmosphere. Bernoulli's law cannot be applied to a free air stream because friction plays an important role. It may be noted that the situation is similar to the laminar flow of a liquid with viscosity inside a tube. The different velocity of the stream layers is caused by viscosity.

    The static pressure is the same throughout the whole cross-section. A free air stream in the atmosphere is exclusively decelerated by friction. If static pressure in a free air stream is equal to atmospheric pressure, some of the striking lecture demonstrations are interpreted incorrectly since the effects observed are not caused by Bernoulli's law.
    Measurement of Static Pressure within a free stream

    Measurement of static pressure within a free stream

    A sufficiently sensitive manometer can be produced easily if not available in the lab. A fine pipe of glass is bent at one side to dip in a cup and to be fixed according to figure 7. The meniscus must be positioned in the middle of the pipe. The suitable inclination should be 1:15 - 1:30. A rubber tube connects the glass pipe with a probe. As has been pointed out before a flat disk must be glued on top of the probe leaving the opening free. The disk has to be held parallel to the streaming. If the static pressure is measured in such a way it can be shown that it is equal to the pressure in the environmental atmosphere. 


    Information courtesy of Saulius Pakalnis, Research Support Technologies