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Nikuradse experiment | variation of frictional factor (f) for laminar and turbulent flow

 Nikuradse', a German Engineer, He by gluing uniform sand grains on the inner side of the pipe wall to artifically roughened the pipe conducted a series of well- planned experiments on pipes.


Here we choose the pipe of different diameter (D) and by changing the size of sand grain which gives (Roughness height= k),

We can observe from his experiment that value of (k/D) varies from about "1/1014 to 1/30"

 Since from dimensional analysis 'f is the function of Reynold's no VD/ν and ratio of 'k/D'



Where,

k =Average roughness height of pipe wall, D= Diameter of pipe,  ν= Kinematic viscosity of flowing fluid, Re
=VD/ν = Reynold's no, k/D = Relative roughness. 
Sometimes, "k/D" is also replaced by "R/k"
Where, R= Radius of pipe and (R/k)= Relative smoothness whose value varies from "15 to 507''.


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Change(Variation) of friction factor for laminar flow (Re<2000)


Head loss in laminar flow (i.e. Hagen-Poisseullie equation)= 32μvL/
γD2
Head loss in turbulent flow (i.e. Darcy-Weisbach equation) = fLV2/
2gD
Equation head losses in laminar and turbulent flow, We get
Hence, f = φ(Re). This shows that 'f ' depends on Reynolds number only and it is independent of the relative roughness (k/D).

Variation of friction factor (f)for transition flow (2000<Re<4000)

There is no specific relationship between .'f' and 'Re'

Variation of friction factor (f) for turbulent flow (Re>4000)


In the fully developed turbulent flow, the friction factor (f) depends on either 'Re' or (k/D) or both, depending on whether the boundary is hydrodynamically smooth or rough or it is intermediate roughness. Friction factor (f) Vs Reynolds number (Re) exists for every relative roughness (k/D) and the horizontal
lines in the Figure confirm that "Roughness" is more important than Reynolds number (Re) for determining friction factor (f).


Variation of friction factor (f) for turbulent flow in smooth pipes




However, on the basis of Nikuradse's experimental data for turbulent flow in smooth pipes, it has been indicated that the experimental results, instead of following trend of above equation follow closely the trend of the following equation, 



Hence, f = φ(Re). This shows that 'f' depends on Reynolds number only and it is independent of the relative roughness (k/D).




Variation of friction factor (f) for turbulent flow in rough pipes



From the equation of turbulent flow for the rough pipe,



Again, Nikuradse's experimental data on turbulent flow in rough pipes has shown that the experimental results instead of following the trend of above equation , follow closely the trend of the following equation,



Hence, f =φ ( k/ D ). This shows that 'f' depends on Relative roughness only and it is independent of the Reynolds number. So, in turbulent flow in rough pipes viscosity of fluid has no significant role.


Now,
For rough pipe, 1/√f= 2 log(R/k) + 1.74 (Karman - Prandtl resistance equation)

Variation of friction factor (f) for turbulent flow in intermediate rough pipes





Nikuradse's was not able to find out the exact formula to determine the friction factor (f). But, from the Nikuradse's analysis of rough and smooth pipe it is conformed that the friction factor (f) depends on both Reynolds number (Re) and Relative roughness (k/D).
f=φ(Re,k/D)




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