I have got three questions linked with one and another, which is associated with mathematics and derivations, concerning the field of fluid dynamics simulations and general cfd. Any help and step-by-step derivation and simplification would appreciated.
a) Simplify the 2D N-S equation to a fully developed channel flow?
b) Simplify the N-S equation to the steady flow equations?
c) How the periodic BC works mathematically?
d) Explain where does the Reynolds stress term Rij come from, and how to link it with a turbulence model?
You should be able to find the answer to part (a) in most of the standard text on fluid dynamics. The idea is to use $\frac{d}{dx}$ of any quantity as zero (considering $x$ as the direction along the length of your channel. That is in a nutshell your mathematical definition of fully-developed flow.
(b) Similarly for the steady term, you have to use $\frac{d}{dt}$ of any quantity as zero i.e. in momentum equation for example, you should assume $\frac{du}{dt}$ and $\frac{dv}{dt}$ as zero to simplify further.
(c) Putting it simply, periodic boundary conditions are implemented as (if you have let's say $f_1$, $f_2$, $f_3$, .... $f_N$ data points) $$f_N = f_2$$ $$f_1 = f_{N-1}$$ That is, we are trying to tell the code/solver that the two ends reflect a similar end-conditions
(d) Reynolds stress term is a result of what is called as a Reynolds decomposition. The mathematics could be a little tricky if you are struggling with the first two parts. Any texts on turbulent flows e.g. Davidson or Pope would be able to walk you through the derivation. The idea is to decompose the instantaneous variables into a mean and a fluctuation such as: $$u = \overline{U} + u'$$ When you do the same for other velocities and pressure, terms such as $\overline{u'v'}$ appears in your Navier-Stokes equation which are called as turbulent/Reynolds stresses.
Hope this answer helps.