# Why is turbulence dissipation $\epsilon$ defined this way?

When deriving the turbulence kinetic energy equation $$k$$ by taking the trace of Reynolds stress transport PDE, this term appears :

$$2\nu \overline{\frac{\partial u'_i}{\partial x_k}\frac{\partial u'_j}{\partial x_k}}$$

Where $$\nu$$ is the kinematic viscosity and $$u'$$ is velocity fluctuations.

This term is turbulence kinetic energy dissipation rate $$\epsilon$$, which represents the rate at which Kolmogorov eddies energy is converted back into the flow’s internal energy.

I understand the physic of energy cascade and turbulence dissipation, but I don't understand how did we relate the above definition to turbulence dissipation?

The two factors under the Reynolds-averaging overbar (and inside the Einstein-convention-implied summation over $$k$$) are:
• $$\partial\!u_i/\partial\!x_k$$, the (quotient by dynamic viscosity of the) $$i$$ component of shear force per area on surfaces of a fluid element whose normal is in the $$k$$-direction; and
• $$\partial\!u_j/\partial\!x_k$$, the difference per $$k$$-direction size of the fluid element between the $$j$$ component of velocity on opposite faces of the fluid element with normals in the $$\pm k$$-direction.