We explain how the rotation of a discus makes it into a reasonably efficient airfoil generating substantial lift at a lift/drag ratio ~ 3, thus increasing the length of the throw with 5-10 meters. The rotation makes the boundary layer turbulent which delays separation at the high angle of attack in descent.
The flight of a discus is similar to the flight of a a frisbee as studied in the Knol Why a Frisbee Flies So Well.
A properly thrown discus acts like a symmetric wing generating lift with a lift/drag ratio ~ 3 at an angle of attack ~ 30 degrees, as explained in the Knol Why It Is Possible to Fly , which can increase the length of the throw by 5 meters in a head wind of 10m/s.
Robert Garret 1896 Olympics Wood discus weight: 2 kg diameter: 0.22 m thickness: 12 - 46 mm.
Assuming that lift and drag are constant during the flight and that the discus has unit mass, it follows by elementary mechanics that the time of flight T and traveled distance d are given by the following formulas:
T = (V sin(a) + sqrt(V2 sin2 (a) + 2gh) )/G
d = V cos(a) T - DT2 /2
where V is the initial speed, a is the launch angle, h the launch height, G = g - L the effective vertical force with g the gravitational force and L the vertical lift force and D the horisontal drag force.
The maximal lift coefficient at 30 degrees of angle of attack is ~ 1.0 with lift/drag ratio ~ 3 [1].
Typical values are V = 20 m/s, a = 35 degrees, h=1.5 m , G = 0.8g which gives T ~ 4 s and d ~ 80 m, see also Optimal discus trajectories.
In the following pictures we decribe how the flow of air around a wing generates large lift and small drag by a perturbation of zero lift/drag potential flow arising from a mechanism of instability at separation changing the pressure distribution around the trailing edge. The perturbed flow does not separate at the crest because the boundary layer is turbulent which in a fluid of small viscosity acts like a slip boundary condition. On the other hand, viscous flow with a laminar boundary layer separates at the crest and gives poor lift and large drag.
Sideview of velocity and pressure, and topview of streamwise vorticity of Naca0012 wing at aoa = 14 . Observe the turbulent streamwise vorticity emanating from separation instability. Computed solution of the Navier-Stokes equations with slip boundary condition [1] . It is possible that the rims (and holes of some frisbees) of a frisbee trigger transition to turbulence in the boundary layer and thus improves
the flight.
Principle of action of a wing: Potential flow (upper left) with zero lift/drag modified by low-pressure counter-rotating rolls of streamwise vorticity from instability mechanism at separation (upper right), switching the pressure on rear wing (bottom left ) to give both lift and drag (H high, L low pressure). Viscous flow separating at the crest with low lift and large drag (bottom right).
We see that lift peaks at 20 degrees angle of attack with lift/drag ratio ~ 3 .
Lift/drag ratio of a Naca0012 airfoil as function of the angle of attack
Drag as function of the angle of attack
The rotation of a discus has several effects:
- It stabilizes the flight into maintaining the launch angle, although it increases slightly due to precession as explained in Why a Frisbee Flies so Well.
- It makes the boundary layer turbulent which delays separation and maintains a useful lift/drag ratio.
The angle of attack changes during the flight since the flight direction changes, and is in fact negative at launch but increases to a positive maximum in descent, during which the lift helps to prolong the flight.
Discus launch
Assuming G = 0.8g during half of the flight increases d by 10 % or ~ 8 m from lift, while with D ~ 0.4 the
reduction is ~ 3m from drag , altogether ~ 5 m increase.
The World Record of discus throw is 74.08 m set in 1986 by Jürgen Schult (GER/GDR) , while for hammer throw it is 81 m and for javelin 98 m.
What determines if the boundary layer is turbulent (which is good) or laminar (which is bad) is the
Reynolds number = Re = UL/v where U i s a relevant speed, L is a relevant length scale and v is
(kinematic) viscosity which for air is about 0.00001. The switch from laminar to turbulent boundary layer occurs at Re ~ 100.000. The rotation increases the effective Reynolds number and helps the boundary to turn turbulent, thus improving lift and reducing drag.
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