When an object moves through a fluid, there is a drag force which retards its motion. Its magnitude is given by
Fdrag ≈ CρAv2 (1)
where C (= 0.2) is a constant, ρis the density of the fluid, A is the cross-sectional area of the object normal to the flow direction, and v is its velocity relative to the fluid.
Equation (1) is valid only if the fluid flow develops whirls and eddies, that is, approaching the onset of turbulence. (If the fluid is essentially undisturbed, then the drag force is actually greater than the value given in equation .) The extent to which a fluid is disturbed is determined by a dimensionless constant called the Reynolds number, defined by
Re = ρvl/η (2)
where l is the linear size of the object and η is the viscosity of the fluid, a measure of its stickiness. A table of densities and viscosities is shown below.
substance ρ(kg/m3) η (kg/m s)
air 1.29 1.8 x 10–5
water 1.0 x 103 1.0 x 10–3
mercury 1.36 x 104 1.6 x 10–3
If Re is greater than about 100, then equation (1) for Fdrag is fairly accurate. The Reynolds number also determines when turbulence begins. If Re is greater than about 2 x 105, then the fluid develops whirls and eddies that break off from the flow in an essentially unpredictable manner, i.e., turbulence.
(Note: g = 9.8 m/s2.)
How fast would a car have to be going for turbulence to develop behind it?