• For flows in a rectangular channel at very large Reynolds numbers the laminar sublayer can practically be ignored. The channel may be assumed to have a width 2h and the x axis will be placed along the bottom wall of the channel.
  
  
• Consider a turbulent stream along a smooth flat wall in such a duct and denote the distance from the bottom wall by y, while u(y) will signify the velocity. In the neighbourhood of the wall, we shall apply

• According to Prandtl's assumption, the turbulent shearing stress will be

(34.1)

At this point, Prandtl introduced an additional assumption which like a plane Couette flow takes a constant shearing stress throughout, i.e

(34.2)

          where denotes the shearing stress at the wall.

• Invoking once more the friction velocity , we obtain

(34.3)

(34.4)

On integrating we find

(34.5)

• Despite the fact that Eq. (34.5) is derived on the basis of the friction velocity in the neighbourhood of the wall because of the assumption that = constant, we shall use it for the entire region. At y = h (at the horizontal mid plane of the channel), we have . The constant of integration is eliminated by considering

Substituting C in Eq. (34.5), we get

(34.6)

Equation (34.6) is known as universal velocity defect law of Prandtl and its distribution has been shown in Fig. 34.1

.

Here, we have seen that the friction velocity is a reference parameter for velocity.Equation (34.5) can be rewritten as