Equation for turbulent flow
Introduction: Opposite of laminar, where considerable mixing occurs, velocities are high.
Turbulent flows can be characterized and quantified using Reynolds Number
Established by Osborne Reynolds and is given as:
Where: NR – Reynolds number
V – Velocity of flow (m/s)
D – Diameter of pipe (m)
ρ – Density of water (kg/m3)
ή – Dynamic viscosity (kg/m.s)
ν – Kinematic viscosity (m2/s)
When – units are considered – NR is dimensionless.
NOTE: Reynolds number directly proportional to velocity & inversely proportional to viscosity!
NR < 2000 – laminar flow
NR > 4000 – Turbulent flow
For 2000 < NR < 4000 – transition region or critical region: flow can either be laminar of turbulent – difficult to pin down exactly.
Reynolds numbers for some real-life examples:
- Blood flow in brain ~ 100
- Blood flow in aorta ~ 1000
- Typical pitch in major league baseball ~ 200000
- Blue whale swimming ~ 300000000
Friction losses in Pipes: Energy equation can be given as:
p1/γ z1 v12/2g hA – hR – hL = p2/γ z2 v22/2g
Where hA, hR, hL are the heads associated with addition, removal and friction loss in pipes, respectively.
Note that the terms are added on the LHS of the equation!
The head loss in pipes = hL can be expressed as
hL = f * (L/D) * v2/2g
Darcy’s equation for energy loss (GENERAL FORM):
f – Friction factor
L – Length of pipe
D – Diameter of pipe
v – Velocity of flow
Similarly, another equation was developed to compute hL under Laminar flow conditions only
Called the Hagen-Poiseuille equation.
If you equate Darcy’s equation and Hagen-Poiseuille equation. Then we can find the friction factor
Thus the friction factor is a function of Reynolds’s number.