Aerodynamics of Discus Throw

Abstract

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.

A Discus Acts Like a Wing

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: 2kg diameter: 0.22m thickness: 12 - 46mm. 

Elementary Calculus 

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(V2sin2(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. 

Shortcut to the Action of a Wing

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).

                                      

                                       Lift/drag ratio of a Naca0012 airfoil as function of the angle of attack

      Lift (and circulation) as function of the angle of attack                                                       

                                                                                          

                                                                                        

                Drag as function of the angle of attack

                                          

We see that lift peaks at 20 degrees angle of attack with lift/drag ratio ~ 3.

Flight of a Discus

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  ~ 8m from lift, while with D ~ 0.4 the

reduction is ~ 3m from drag , altogether ~ 5m increase.

The World Record of discus throw is 74.08m set in 1986 by Jürgen Schult (GER/GDR), while for hammer throw it is 81m and for javelin 98m.

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 is 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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