SA, why not include density in material derivative?

[bigfooted]

At the nasa website,
https://turbmodels.larc.nasa.gov/spalart.html

they mention that you can write SA in the form

$\frac{\partial \rho \nu}{\partial t} + \frac{\partial \rho u_j \nu}{\partial x_j} = ...$

So why do we not use this formulation instead of the formulation without density?

[pcarruscag]

It just looks like more work TBH, for example we would need the gradient of density for that version.