Chaotic nature
Chaotic mixing
Pulsation velocity additives
Chaotic nature of flow
Empirical turbulence theories
Turbulent mixing intensity
Prandtl's theory
Far from the wall
Laminar sublayer
Mixing length
Mixing length
Empirical
Sharply
Far from the walls
Shape of roughness protrusions
Dip characteristic
Radius of the pipe
Relative smoothness
Quadratic resistance law
Re < 2300 or lg Re < 3.6
Increases
Sub-quadratic zone
Becomes independent of Re
Height of roughness protrusions
Grainy roughness
Larger values
Reynolds number and relative roughness (ε)
Head losses and velocity distribution
Pipe diameter, fluid viscosity, flow velocity, and wall roughness
Proportional to velocity