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 width2h 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

The no-slip condition at the wall cannot be satisfied with a finite constant of integration. This is expected that the appropriate condition for the present problem should be that at a very small distance from the wall. Hence, Eq. (34.5) becomes

(34.7)

The distance is of the order of magnitude of the thickness of the viscous layer. Now we can write Eq. (34.7) as

(34.8)

where

, the unknown

is included in

Equation (34.8) is generally known as the universal velocity profile because of the fact that it is applicable from moderate to a very large Reynolds number.

However, the constants and have to be found out from experiments. The aforesaid profile is not only valid for channel (rectangular) flows, it retains the same functional relationship for circular pipes as well . It may be mentioned that even without the assumption of having a constant shear stress throughout, the universal velocity profile can be derived.

Experiments, performed by J. Nikuradse, showed that Eq. (34.8) is in good agreement with experimental results. Based on Nikuradse's and Reichardt's experimental data, the empirical constants of Eq. (34.8) can be determined for a smooth pipe as

(34.9)

This velocity distribution has been shown through curve (b) in Fig. 34.2
.

Fig 34.2 The universal velocity distribution law for smooth pipes

However, the corresponding friction factor concerning Eq. (34.9) is

(34.10)

the universal velocity profile does not match very close to the wall where the viscous shear predominates the flow.

Von Karman suggested a modification for the laminar sublayer and the buffer zone which are

(34.11)

(34.12)

Equation (34.11) has been shown through curve(a) in Fig. 34.2.

It may be worthwhile to mention here that a surface is said to be hydraulically smooth so long

(34.13)

where

is the average height of the protrusions inside the pipe.

Physically, the above expression means that for smooth pipes protrusions will not be extended outside the laminar sublayer. If protrusions exceed the thickness of laminar sublayer, it is conjectured (also justified though experimental verification) that some additional frictional resistance will contribute to pipe friction due to the form drag experienced by the protrusions in the boundary layer.

In rough pipes experiments indicate that the velocity profile may be expressed as:

(34.14)

At the centre-line, the maximum velocity is expressed as

(34.15)

Note that no longer appears with and . This means that for completely rough zone of turbulent flow, the profile is independent of Reynolds number and a strong function of pipe roughness .

However, for pipe roughness of varying degrees, the recommendation due to Colebrook and White works well. Their formula is

(34.16)

where

is the pipe radius

For , this equation produces the result of the smooth pipes (Eq.(34.10)). For , it gives the expression for friction factor for a completely rough pipe at a very high Reynolds number which is given by