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Low Reynolds Number Flight

Low Reynolds Number Flight

Paper airplanes are smaller and fly slower than most other aircraft. So how does that affect their aerodynamics? Back in 1883 Osborne Reynolds, professor of engineering at the University of Manchester (England) carried out experiments to determine why fluid forces through pipes changed for different conditions. Basically what he discovered is how viscosity affects the way fluids behave. All fluids (a fluid is anything that flows - air, water, maple syrup, …) have some viscosity, or stickiness, to them. As a fluid flows over a surface, the fluid molecules closest to the surface cling to the microscopic roughness of the surface. As you move away from the surface there is a small transition distance where the fluid's viscosity limits the change in speed of the adjacent molecules, until at a certain distance the fluid is at full speed. The narrow region near the surface where the fluid is less than full speed is called the boundary layer. All boundary layers start as "laminar" where the molecules travel in a straight line, with a smooth transition in fluid velocity from the surface to the outer edge of the boundary layer. Further downstream disturbances and waves form in the boundary layer and transition the smooth orderly laminar boundary layer into a "turbulent" boundary layer. Turbulent boundary layers have a laminar sub-layer next to the surface, but are mainly characterized by swirling random eddies throughout the boundary layer.

A number was devised which gives the relative importance of viscosity in fluid flow. It is called the Reynolds Number, and it is the ratio of momentum forces to viscous forces in a fluid. The bigger the number, the less influential the viscosity. The viscosity is essentially a constant for a fluid (it changes a bit with temperature), but momentum is proportional to the speed of a fluid over a surface times the distance it has traveled over the surface. For air it is roughly:


Re=K*V*L
Re=Reynolds number (non-dimensional)
K=9340
V=Velocity relative to surface (miles per hour)
L=Length over surface fluid has traveled (feet)
So for a paper airplane (remember, this is about paper airplanes) Re=9340*10*.4=37,000

By comparison the wings of a four passenger airplane have a Reynolds Numbers of up to about 6,000,000. Also, remember the transition from laminar to turbulent? That happens at a Reynolds number of no less than about 10,000, so the first ½ to ¼ of the flow over a paper airplane's wing is laminar. Since the Reynolds Number is much less than for full sized airplanes, this means viscosity is much more dominant, resulting in more drag, and more difficulty in creating lift.

The low Reynolds Number of paper airplanes also means thin wings are best. As wings get thicker, the air has to work harder to make it around the airfoil. At high Reynolds numbers with turbulent boundary layers this is easy. At low Reynolds Numbers and laminar boundary layers, this is very difficult. If a thick (say 10% of chord or more) airfoil is used on a paper airplane, the air cannot make it around the airfoil and separates about midway across the wing resulting in huge amounts of drag, and little lift - the paper airplane won't fly. I try to keep my wings no thicker than about 3% to 5% of the chord length, so its important to fold your wings nice and flat. Mother nature knows this. Birds fly faster than paper airplanes, and they have thick curved airfoils. Insects are closer in Reynolds number to paper airplanes, and they have thin flat wings - look at a butterfly's wings some time.

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