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**Chapter 11 Turbulent Flow**

- Velocity distribution is relatively uniform and velocity profile is much flatter than the corresponding laminar flow parabola for the same mean velocity, as shown below :

• Shear stress in turbulent flow

where,

μ = dynamic coefficient of viscosity (fluid characteristic)

h = eddy viscosity coefficient (flow characteristic)

- eddy viscosity come into picture due to turbulence effect
- Hydro-Dynamically Smooth And Rough Pipes
- If the average height of irregularities (k) is greater than the thickness of laminar sublayer (d'), then the boundary is called hydrodynamically Rough.
- If the average height of irregularities (k) is less than the thickness of laminar sublayer (d'), then the boundary is called hydrodynamically smooth.
- On the basis of NIKURADSE's EXPERIMENT the boundary is classified as :

Hydrodynamically smooth : (k/d) < 0.25

Boundary in transition :0.25 < (k/d)< 6.0

Hydrodynamically Rough :(k/d) > 6.0

- (R/K) is known as specific roughness. where ‘k’ is average height of roughness and ‘R’ is radius of the pipe.

- Velocity Distribution For Turbulent Flow in Pipes

(a) Prandtl’s universal velocity distribution equation :

where

shear or friction velocity.

y = distance from pipe wall

R = radius of pipe.

- The above equation is valid for both smooth and rough pipe boundaries.

(b) Karman - Prandtl Velocity distribution equation :

(i) Hydro Dynamically Smooth pipe

(ii) Hydro Dynamically Rough pipe

where

V_{*} = shear velocity

y = distance from pipe wall

k = average height of roughness

v = kinematic viscosity.

(c) Velocity distribution in terms of mean velocity

The above equation is for both rough and smooth pipes.

- Friction Factor

(a) Friction factor ‘f’ for laminar flow :

f = (64/Re) where Re = Reynolds number

(b) Friction factor ‘f’ for transition flow :

There exists no specific relationship between f and Re for transition flow in pipes.

(c) Friction factor (f) for turbulent flow in smooth pipes :

(d) Friction factor (f) for turbulent flow in rough pipes

This equation shows that for rough pipes friction factor depends only on R/K (Relative smoothness) and not on Reynolds number (Re)

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