# 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

**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.**

## 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

**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

**20**degrees angle of attack with lift/drag ratio

**~ 3**.

## Flight of a Discus

- 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.

**G = 0.8g**during half of the flight increases

**d**by

**10**% or ~

**8**m from lift, while with

**D ~ 0.4**the

**~ 3m**from drag

**,**altogether

**~ 5**m increase.

**74.08**m set in 1986 by

**Jürgen Schult**(GER/GDR)

*,*

**while for hammer throw it is**

**81**m and for javelin**98**m.**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.**