Aircraft Propulsion, 2nd Edition

2.17 Friction (Drag) Coefficient C_f and D’Arcy Friction Factor f_D

Let us consider a fully developed pipe flow, where the velocity profile remains constant, that is, preserved, along the length of the pipe, as shown in Figure 2.51.

FIGURE 2.51 Schematic drawing of a fully developed pipe flow

Since the momentum of the fluid remains constant along the length of the pipe, the frictional drag on the fluid has to be in balance with the pressure forces acting on the fluid. Therefore,

(2.196)

where c is the circumference of the pipe. Dividing both sides by the dynamic pressure and introducing the friction drag coefficient C_f, we get

(2.197)

Rearranging Equation 2.197 and using the definition of hydraulic diameter D_h, we get

(2.198a)

The nondimensional term in the bracket on the RHS of Equation 2.198a is referred to as the D’Arcy friction factor, f_D. Moody (1944) developed the functional dependence of the friction factor on pipe Reynolds number for incompressible flow. A graphic presentation ...