The character of fluid flow depends on a dimensionless quantity, the Reynolds number. Named for Belfast-born scientist Osborne Reynolds, it determines whether the flow is laminar (smooth) or turbulent (rough). Normally the drag force increases with speed.

The Reynolds number is defined as Re = *VL*/*ν* where *V* is the flow speed, *L* the length scale and *ν* the viscosity coefficient. The transition from laminar to turbulent flow occurs at a critical value of Re which depends on details of the system, such as surface roughness.

**The Curious Drag Crisis**

Normally the drag force increases with speed. In 1912, Gustave Eiffel – of Eiffel Tower fame – made a remarkable discovery, known as the *drag crisis*. Studying flow around a smooth sphere, he found a drop in the drag force as the flow speed increased above a critical Reynolds number 200,000 and continued to drop until about Re = 300,000 (see figure).

This is extraordinary: if we consider a spherical ball fixed in a wind tunnel, there is a point at which the drag force on the sphere actually decreases as the flow speed is increased.

The drag crisis was explained by Ludwig Prandtl in terms of his boundary layer theory. Reynolds had found that, as the speed increases, the flow changes from laminar to turbulent. The transition point occurs at the critical Reynolds number.

Prandtl argued that the turbulence mixes rapidly-moving external air into the boundary layer, increasing the range over which it adheres to the surface, making the trailing low pressure wake smaller and thereby reducing the drag force.

**Sporting Consequences**

The drag crisis extends the range of a rapidly travelling ball such as a golf ball. For a smooth ball, the critical speed is well above the practically attainable range. By roughening the surface of the ball, the critical Reynolds number is reduced to about 50,000. The resulting reduction in drag can double the distance flown compared to what is possible with a smooth ball.

The figure here shows that there is a range of Reynolds numbers for which the drag on a rough ball is substantially less than that on a smooth ball for the same flow conditions.