Chapter 14 Turbulence

In Sec. 12.6.6, we derived the Poiseuille formula for the ow of a viscous uid down a pipe by assuming that the ow is laminar, i.e. that it has a velocity parallel to the pipe wall. We showed how balancing the stress across a cylindrical surface led to a parabolic velocity pro le and a rate of ow proportional to the fourth power of the pipe diameter, d. We also de ned the Reynolds number; for pipe ow it is R vd=, where v is the mean speed in the pipe. Now, it turns out experimentally that the ow only remains laminar up to a critical Reynolds number that has a value in the range 103 105 depending upon the smoothness of the pipe's entrance and roughness of its walls. If the pressure gradient is increased further (and thence the mean speed v and Reynolds number R are increased), then the velocity eld in the pipe becomes irregular both temporally and spatially, a condition we call turbulence. Turbulence is common in high Reynolds number ows. Much of our experience of u-ids involves air or water for which the kinematic viscosities are 105 and 106 m2 s1 respectively. For a typical everyday ow with a characteristic speed of v 10 m s1 and a characteristic length of d 1m, the Reynolds number is huge: R = vd= 106 107. It is therefore not surprising that we see turbulent ows all around us. Smoke in a smokestack