[Ryan LI]

Hi CFDers

I just stepped into the mathematics of the RANS model and I got stuck. When I read the Boussinesq hypothesis, also called the eddy viscosity hypothesis, I saw this equation and the name of every term. Such as



where denotes the eddy viscosity, the strain rate tensor and the turbulent kinetic energy.

But I still have concerns,

After some research, here are what I know so far:
the strain rate tensor describes the rate of change of the deformation of the fluid in the neighborhood of a certain point. The turbulent momentum transfer results in the deformation of the fluid, and the deformation gives rise to viscous forces in its interior against the deformation, due to friction between adjacent fluid elements, and due to the internal stress tensor.

And the turbulent kinetic energy defines the energy of motion carried by fluid flow. And it dissipates down the turbulence energy cascade.

Here are what I still don't know：
Why the kinetic energy term in the above equation? I don't know how the turbulent kinetic energy is related to the stress tensor.
Why and in the above equation? I don't know whether this is pure empirical or if there is a theory behind these factors.

Thank you