Application of the boundary-layer theory

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

The velocity of the fluid may he assumed to obey the Prandtl one-seventh power law, given by equation 11.26. If the boundary layer thicloiess is replaced by the pipe radius r, this is then given by  

The relation between the mean velocity and tihe velocity at the axis is derived using this expression in Chapter 3. There, the mean velocity u is shown to be 0.82 times the velocity Us at the axis, although in fliis calculation the thickness of the laminar sub-layer was neglected and the Prandtl velocity distribution assumed to ply over die whole cross-section. The result therefore is strictly amicable only at very high Reynolds numbers where the thickness of the laminar sub-layer is very small. At lower Reynolds numbers the mean velocity will be rather less than 0.82 times die velocity at the axis. 

The expressions for the shear stress at die wads, die diickness of the lamiimr sub-layer, and the velocity at the outer edge of die laminar sub-layer may be plied to the turbulent flow of a fluid in a pipe. It is convenient to express these relations in terms of the mean velocity in the pipe, the pipe diameter, and the Reynolds group with respect to the mean velocity and diameter. 

The shear stress at the walls is given by the Blasius equation (11,23) as  

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