Question

Why this prandtl velocity distribution is called universal

Answer 1

Universal Velocity Distribution Law And Friction Factor In Duct Flows For Very Large Reynolds Numbers

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

.

Fig 34.1 Distibution of universal velocity defect law of Prandtl in a turbulent channel flow

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

where