# Introduction to turbulence/Homogeneous turbulence

## A first look at decaying turbulence

Look, for example, at the decay of turbulence which has already been generated. If this turbulence is homogeneous and there is no mean velocity gradient to generate new turbulence, the kinetic energy equation reduces to simply:

 $\frac{d}{dt} k = - \epsilon$ (1)

This is often written (especially for isotropic turbulence) as:

 $\frac{d}{dt} \left[ \frac{3}{2} u^{2} \right] = - \epsilon$ (2)

where

 $k \equiv \frac{3}{2} u^{2}$ (3)

Now you can't get any simpler than this. Yet unbelievably we still don't have enough information to solve it. Let's try. Suppose we use the extanded ideas of Kolmogorov we introduced in Chapter 3 to related the dissipation to the turbulence energy, say:

 $\epsilon = f \left( Re \right) \frac{u^{3}}{l}$ (4)