A Comment on Drag Variation with Velocity

Beginning with Chapter 1, indeed beginning with the most elementary studies of fluid dynamics, the point is usually made that the aerodynamic force on a body immersed in a flowing fluid is proportional to the square of the flow velocity. For example, from Section 1.5,

L = poaVl0SCL and D = poaVl0SCD

As long as Ci and CD are independent of velocity, then clearly L <x and D <x V^. This is the case for an inviscid, incompressible flow, where С/, and Co depend only on the shape and angle of attack of the body. However, from the dimensional analysis in Section 1.7, we also discovered that С/, and Co in general are functions of both Reynolds number and Mach number,

Ci = f (Re, Mgo) Co = /2(Re, MTO)

Of course, for an inviscid, incompressible flow, Re and are not players (indeed,

for inviscid flow, Re —>■ 00 and for incompressible flow, —»• 0). However for all

other types of flow, Re and are players, and the values of Ci and С о depend not only on the shape and angle of attack of the body, but also on Re and For this reason, in general the aerodynamic force is not exactly proportional to the square of the velocity. For example, examine the results from Example 18.1. In part (a), we calculated a value for drag to the 175.6 N when = 100 m/s. If the drag were proportional to V^, then in part (b) where = 1000 m/s, a factor of 10 larger, the drag would have been one hundred times larger, or 17,560 N. In contrast, our calculations in part (b) showed the drag to be considerably smaller, namely 5026 N. In other words, when V^_ was increased by a factor of 10, the drag increased by only a factor of 28.6, not by a factor of 100. The reason is obvious. The value of Cf decreases when the velocity is increased because: (1) the Reynolds number increases, which from Equation (18.22) causes Cf to decrease, and (2) the Mach number increases, which from Figure 18.8 causes Cf to decrease.

So be careful about thinking that aerodynamic force varies with the square of the velocity. For cases other than inviscid, incompressible flow, this is not true.

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