6.5.1 Reynolds Number

For an object moving through a fluid, the Reynolds number is defined as the ratio of inertial forces to viscous forces acting on the object. The Reynolds number is high if inertial forces dominate, low if viscous forces dominate. Two important properties of a fluid are its density ρ (Greek rho) and its viscosity η (Greek eta). For an object of size L moving through a fluid with speed v, the Reynolds number is

Re=inertial forcesviscous forces=ρvLη (6.15)

NOTE   The symbol for the Reynolds number is Re. It is the only symbol in this text that has two letters.

We won’t attempt to justify this expression for the Reynolds number, but it makes sense. More force is needed to deflect a denser fluid or to deflect the fluid around an object that is larger or moving faster, and these quantities all appear in the numerator. Likewise, the size of any viscous forces is related to the fluid’s viscosity, which we see in the denominator.

Notice that the Reynolds number, a ratio of forces, is a pure number with no units; it is said to be dimensionless. When Re is high, inertial drag is dominant; when Re is low, viscous drag is more important.

In SI units, density is measured in kg/m3 and viscosity in Pa s. Pa is the abbreviation for pascal, the SI unit of pressure, which is defined as 1 Pa=1 N/m2. We’ll have a lot more to say about pressure in later chapters. TABLE 6.2 lists the values of density and viscosity for some typical fluids. Viscosity is very dependent on temperature—think how quickly honey loses viscosity as you heat it—so the values shown are not appropriate at temperatures other than those listed.

TABLE 6.2 Density and viscosity

Fluid ρ (kg/m3) η  (Pa s)
Air (20°C at sea level)      1.2 1.8×10−5
Water (20°C) 1000 1.0×10−3
Water (40°C) 1000 6.5×10−4
Ethyl alcohol (20°C)  790 1.3×10−3
Olive oil (20°C)  910 8.4×10−2
Honey (20°C) 1400 10
Honey (40°C) 1400 1.7

The object’s size L is a characteristic or typical size of the object, which could be a height, width, or diameter. This seems a bit odd; for an object that is rectangular, the value of Re depends on whether you select the width or the height for L. However, in practice, how you define L makes very little difference. As you’ll see, we want to know whether Re is “high” or “low,” values that differ by factors of 1000 or more. A difference of a factor of 2 or 3 will not affect our judgment about the size of Re. Said another way, our use of the Reynolds number needs only approximate values, not precise calculations.

As a simple example, consider a 75-mm-diameter baseball moving through 20°C air at a speed of 20 m/s. We can take the ball’s characteristic size L to be its diameter. The Reynolds number for this motion is

Re=ρvLη=(1.2 kg/m3)(20 m/s)(0.075 m)1.8×10−5 Pa s=100,000

As you’ll see, this is a very high Reynolds number.