There are no known ways to mathematically represent turbulence. Yet, we have to work with it. One way out is to treat the state variables as a combination of an “average” value and “fluctuating” values even if the flow itself is steady-state (i.e. doesn’t change with time). In other words, a turbulent flow is considered to be a combination of a “smooth” flow with “fluctuations” superimposed on this.
There are many different ways to do this. The maths isn’t trivial, as explained here. Note that RANS means you’re “modeling” turbulence as opposed to “resolving” turbulence. This is important to remember since all models are wrong, but some are useful.
RANS is currently the most efficient and reliable way to solve CFD problems, despite its known shortcomings. You should also see URANS, LES and DNS.
