Turbulent Flow

By Deepanshu Rastogi|Updated : November 24th, 2021

This article contains basic notes on the "Turbulent Flow"  topic of the "Fluid Mechanics & Hydraulics" subject.           

Turbulent Flow

Turbulent Flow: Turbulent flow is a flow regime characterized by the following points as given below

Shear stress in the turbulent flow

12-Elementary-turbulent-flow (1)

where, τv and τ= shear stress due to viscosity and turbulence.

η = eddy viscosity coefficient.

Turbulent shear stress by Reynolds

τ = ρu’v’

u' and v’ fluctuating component of velocity.

Shear stress in turbulent flow by Prandtl

12-Elementary-turbulent-flow (2) 

where, l = Mixing length

The velocity distribution in the turbulent flow for pipes is given by the expression.

12-Elementary-turbulent-flow (3)

Umax = centre velocity

where, y = Distance from the pipe wall, R = radius of the pipe

u* = Shear velocity 12-Elementary-turbulent-flow (4)

Velocity defect is the difference between the maximum velocity (umax) and local velocity (u) at any point is given by

12-Elementary-turbulent-flow (5)

Karman-Prandtl velocity distribution equation.

Hydro dynamically pipe

12-Elementary-turbulent-flow (6)

 

12-Elementary-turbulent-flow (7)

where, u = velocity at any point in the turbulent flow

u* = shear velocity = 12-Elementary-turbulent-flow (8)

v = Kinematic viscosity of fluid

y = Distance from pipe wall

k = Roughness factor

Velocity distribution in terms of average velocity

12-Elementary-turbulent-flow (9)

12-Elementary-turbulent-flow (10)

Common Mean Velocity Distribution Equation: (Very Important for Exams)

[(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 as Common Mean Velocity Distribution Equation) 

Coefficient of friction 12-Elementary-turbulent-flow (12) (for laminar flow)

12-Elementary-turbulent-flow (13) (for smooth pipe)

12-Elementary-turbulent-flow (14)

(for smooth pipe)

12-Elementary-turbulent-flow (15) (for rough pipe)

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