Turbulent flow is a flow regime characterized by the following points as given below.
Shear stress in the turbulent flow
where, τv and τt = shear stress due to viscosity and turbulence.
η = eddy viscosity coefficient.
τ = ρu'v'
u' and v’ fluctuating component of velocity.
τ = ρl2(du/dy)2
where, l = Mixing length
The velocity distribution in the turbulent flow for pipes is given by the expression.
u = umax + 2.5u*logey/R
Umax = centre velocity
where, y = Distance from the pipe wall, R = radius of the pipe
u* = Shear velocity
Velocity defect is the difference between the maximum velocity (umax) and local velocity (u) at any point is given by
umax -u = 5.75u*log10R/y
where, u = velocity at any point in the turbulent flow
u* = shear velocity =
v = Kinematic viscosity of fluid
y = Distance from the pipe wall
k = Roughness factor
Velocity distribution in terms of average velocity
Common Mean Velocity Distribution Equation
[(u - u¯) / u*] = 5.75 log10 (y/R) + 3.75
(This is valid for both rough and smooth pipes, that's why it is referred to as Common Mean Velocity Distribution Equation)
Coefficient of friction f = 16/R (for laminar flow)
(for smooth pipe)
(for rough pipe)