# Turbulence intensity

## Definition

The turbulence intensity is defined as:

$Tu \equiv \frac{u'}{U}$

Where $u'$ is the root-mean-square of the turbulent velocity fluctuations and $U$ is the mean velocity (Reynolds averaged).

If the turbulent energy, $k$, is known $u'$ can be computed as:

$u' \equiv \sqrt{\frac{1}{3} \, ( u_x'^2 + u_y'^2 + u_z'^2 )} = \sqrt{\frac{2}{3}\, k}$

$U$ can be computed from the three mean velocity components $U_x$, $U_y$ and $U_z$ as:

$U \equiv \sqrt{U_x^2 + U_y^2 + U_z^2}$

## Estimating the turbulence intensity

When setting boundary conditions for a CFD simulation it is often necessary to estimate the turbulence intensity on the inlets. To do this accurately it is good to have some form of measurements or previous experince to base the estimate on. Here are a few examples of common estimations of the incoming turbulence intensity:

1. High-turbulence case: High-speed flow inside complex geometries like heat-exchangers and flow inside rotating machinery (turbines and compressors). Typically Tu is between 5% and 20%
2. Medium-turbulence case: Flow in not-so-complex devices like large pipes, ventilation flows etc. or low speed flows (low Reynolds number). Typically Tu is between 1% and 5%
3. Low-turbulence case: Flow originating from a fluid that stands still, like the flow across cars, submarines and aircrafts. Very high-quality wind-tunnels can also reach really low turbulence levels. Typically Tu is very low, well below 1%. In this case Tu is normally not used directly to set the inlet conditions for a CFD simulation. Instead a typical eddy viscosity ratio is estimated.

### Fully developed pipe-flow

For fully developed pipe flow the turbulence intensity at the core can be estimated as:

$Tu = 0.16 \; Re_d^{-\frac{1}{8}}$

Where $Re_d$ is the Reynolds number based on the pipe diameter.