text: The basic tool required for the derivation of the RANS equations from the instantaneous Navier–Stokes equations is the Reynolds decomposition. Reynolds decomposition refers to separation of the flow variable (like velocity ) into the mean (time-averaged) component () and the fluctuating component (). Because the mean operator is a Reynolds operator, it has a set of properties. One of these properties is that the mean of the fluctuating quantity is equal to zero . Thus,
   where  is the position vector. Some authors prefer using  instead of  for the mean term (since an overbar is sometimes used to represent a vector). In this case, the fluctuating term  is represented instead by . This is possible because the two terms do not appear simultaneously in the same equation. To avoid confusion, the notation , , and  will be used to represent the instantaneous, mean, and fluctuating terms, respectively.
  The properties of Reynolds operators are useful in the derivation of the RANS equations. Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid):
  where  is a vector representing external forces.
  Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged,
  to yield:
  The momentum equation can also be written as,
  On further manipulations this yields,
  where,  is the mean rate of strain tensor.
  Finally, since integration in time removes the time dependence of the resultant terms, the time derivative must be eliminated, leaving: