Turbulent Flow in Pipes and Closed Channels

In flow of fluid through a dosed channel turbulence cannot exist permanently at the boundary between the solid and the flowing fluid. The velocity at the interface is zero because of the adherence of the fluid to the solid, and (except very-infrequently) velocity components normal to the wall do not exist. Within a thin volume immediately adjacent to the wall, the velocity gradient is essentially constant and the flow is viscous most of the time. This volume is called the viscous sublayer. Formerly it was assumed that this sublayer had a definite thickness and was always free from eddies, but measurements have shown velocity fluctuations in the sublayer caused by occasional eddies from the turbulent fluid moving into this region. Very close to the wall, eddies are infrequent, but there is no region that is completely free of eddies. Within the viscous sublayer only viscous shear is important, and eddy diffusion, if present at all, is minor. 

The viscous sublayer occupies only a very small fraction of the total cross section. It has no sharp upper boundary, and its thickness is difficult to define. A transition layer exists immediately adjacent to the viscous sublayer in which both viscous shear and shear due to eddy diffusion exist. The transition layer, which is sometimes called a buffer layer, also is relatively thin. The bulk of the cross section of the flowing stream is occupied by entirely turbulent flow called the turbulent core. In the turbulent core viscous shear is negligible in comparison with that from eddy viscosity. 

VELOCITY DISTRIBUTION FOR TURBULENT FLOW. Because of the dependence of important flow parameters on velocity distribution, considerable study, both theoretical and experimental, has been devoted for determining the velocity distribution in turbulent flow. Although the problem has not been completely solved, useful relationships are available that may be used to calculate the important characteristics of turbulence and the results of theoretical calculations check with experimental data reasonably well. 

A typical velocity distribution for a newtonian fluid moving in turbulent flow in a smooth pipe at a Reynolds number of 10,000 is shown in Fig. 5.3. The figure also shows the velocity distribution for laminar flow at the same maximum velocity at the center of the pipe. The curve for turbulent flow is clearly much flatter than that for laminar flow, and the difference between the average velocity and the maximum velocity is considerably less. At still higher Reynolds numbers the curve for turbulent flow would be even flatter than that in Fig. 5.3. 

In turbulent flow, as in laminar flow, the velocity gradient is zero at the centerline. It is known that the eddies in the turbulent core are large but of low intensity, and those in the transition zone are small but intense. Most of the kinetic-energy content of the eddies lies in the buffer zone and the outer portion of the turbulent core. At the centerline the turbulence is isotropic. In all other areas of the turbulent-flow regime, turbulence is anisotropic otherwise, there would be no shear. 

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