In addition to the time-smoothed axial velocities for tube flow in Fig. 10.5, Laufer (1954) measured the fluctuations in all three velocity components. 

Figure 10.5

Some of those data are shown in Fig. P10.4. The ordinate is the time-smoothed value of the product u_ru_z, normalized by the square of the friction velocity. Data are shown for two values of Re. The equation of the line is

where y is the distance from the wall and R is the tube radius. Starting with the definition of the Reynolds stress, derive this equation from first principles. What conclusion can be reached from the fact that the data in Fig. P10.4 closely follow the line?

Transcribed Image Text:

Local/Centerline Velocity 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 O • Data, Re 4.1 x 104 Data, Re = 4.2 x 105 - 1/7 Power - Logarithmic y/R 0.6 0.8 1