## 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/m^{3})

ή – Dynamic viscosity (kg/m.s)

ν – Kinematic viscosity (m^{2}/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:

p_{1}/γ z_{1} v_{12}/2g h_{A} – h_{R} – h_{L} = p_{2}/γ z_{2} v_{22}/2g

Where h_{A}, h_{R}, h_{L} 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 = h**_{L }can be expressed as

** h**_{L} = f * (L/D) * v_{2}/2g

**Darcy’s equation for energy loss (GENERAL FORM):**

Where

f – Friction factor

L – Length of pipe

D – Diameter of pipe

v – Velocity of flow

**Similarly,** another equation was developed to compute h_{L} 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. **