fluid_mechanics

9.2 Boundary Layer Characteristics 487
0.014
0.012
0.010
Completely
turbulent
5 × 10 –3
ε
= 3 × 10 –3
2 × 10 –3
C Df
0.008
0.006
0.004
Turbulent
Transitional
1 × 10 –3
5 × 10 –4
2 × 10 –4
1 × 10 –4
5 × 10 –5
2 × 10 –5
5 × 10 –6
0.002
1 × 10 –6
Laminar
Turbulent
smooth plate
0
10 5 10 6 10 7 10 8 10 9
Re
F I G U R E 9.15 Friction drag coefficient for a flat
plate parallel to the upstream flow (Ref. 18, with permission).
The flat plate drag
coefficient is a
function of relative
roughness and
Reynolds number.
In general, the drag coefficient for a flat plate of length / is a function of the Reynolds number,
Re / , and the relative roughness, e/. The results of numerous experiments covering a wide range
of the parameters of interest are shown in Fig. 9.15. For laminar boundary layer flow the drag coefficient
is a function of only the Reynolds number—surface roughness is not important. This is
similar to laminar flow in a pipe. However, for turbulent flow, the surface roughness does affect the
shear stress and, hence, the drag coefficient. This is similar to turbulent pipe flow in which the surface
roughness may protrude into or through the viscous sublayer next to the wall and alter the flow
in this thin, but very important, layer 1see Section 8.4.12. Values of the roughness, e, for different
materials can be obtained from Table 8.1.
The drag coefficient diagram of Fig. 9.15 1boundary layer flow2 shares many characteristics
in common with the familiar Moody diagram 1pipe flow2 of Fig. 8.23, even though the mechanisms
governing the flow are quite different. Fully developed horizontal pipe flow is governed
by a balance between pressure forces and viscous forces. The fluid inertia remains constant
throughout the flow. Boundary layer flow on a horizontal flat plate is governed by a balance between
inertia effects and viscous forces. The pressure remains constant throughout the flow. 1As
is discussed in Section 9.2.6, for boundary layer flow on curved surfaces, the pressure is not
constant.2
It is often convenient to have an equation for the drag coefficient as a function of the Reynolds
number and relative roughness rather than the graphical representation given in Fig. 9.15. Although
there is not one equation valid for the entire Re / e/ range, the equations presented in Table 9.3
do work well for the conditions indicated.
TABLE 9.3
Empirical Equations for the Flat Plate Drag Coefficient (Ref. 1)
Equation
Flow Conditions
C Df 1.3281Re / 2 0.5
Laminar flow
C Transitional with Re xcr 5 10 5
Df 0.4551log Re / 2 2.58 1700Re /
C Df 0.4551log Re / 2 2.58 Turbulent, smooth plate
C Df 31.89 1.62 log1e/24 2.5 Completely turbulent