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Hydrodynamically smooth and rough boundaries | Velocity distribution for turbulent flow

 Hydrodynamically smooth and rough boundaries

Fig: Definition of smooth and rough boundaries

In general, a boundary with irregularities of large average height 'K' on its surface is considered to be a rough boundary and one with smaller 'K' value is considered as a smooth boundary.
✓ However, for a proper classification of smooth and rough boundaries, the flow and fluid
characteristics are required to be considered in addition to the boundary characteristics.
✓ As the flow outside the laminar sub-layer is turbulent, eddies of various sizes are present which try to penetrate through the laminar sublayer. But, due to greater thickness of laminar sub-layer, eddies can't reach the surface irregularities and thus the boundary acts as a smooth boundary. Such a boundary is termed as " Hydro-dynamically Smooth Boundary"
✓ With the increase in Reynold's no (Re), the thickness of the laminar sub-layer 
decreases and it's can even become much smaller than the average roughness height 'K'. The irregularities will then project through the laminar sub-layer and the laminar sub-layer is completely destroyed. The eddies will thus come in contact with the surface irregularities and large amount of energy loss will take place. Such a boundary is termed as "Hydrodynamically Rough Boundary".

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Velocity distribution for turbulent flow 

 Prandtl universal velocity distribution law 


From the Prandtl mixing length theory of turbulent shear stress of equation


For turbulent flow in pipe Prandtl assumed that the mixing length (/) is a linear function of distance 'y' from pipe wall (i.e. lαy)
:. l = ky, Where k = Karman universal constant = 0.4
Now, above equation becomes,


Since, turbulent shear stress also varies linearly with radius,


Where, τo = Turbulent shear stress at the pipe boundary (i.e. at y = 0 or r = R)

However, for small values of y' i.e. close to the pipe boundary, the shear stress 'τ' ,may be assumed to be
constant being approximately equal to τ0 So, from the equation 


So, integrating equation (XVIII) We get
u = u*/ klogeY + C
This equation indicates the velocity distribution in the turbulent flow.
Boundary condition,


This is the required 'Prandtl universal velocity distribution equation' for turbulent flow in pipe. This equation is applicable for both smooth and rough pipe boundaries.
Modification of equation as,


Hence, the velocity distribution curve given by equation 
appears to be independent of the nature of boundary.


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