The left hand side of this equation represents the change in mean momentum of fluid element owing to the unsteadiness in the mean flow and the convection by the mean flow. This change is balanced by the mean body force, the isotropic stress owing to the mean pressure field, the viscous stresses, and apparent stress owing to the fluctuating velocity field, generally referred to as the Reynolds stress. This nonlinear Reynolds stress term requires additional modeling to close the RANS equation for solving, and has led to the creation of many different turbulence models. The time-average operator is a Reynolds operator.
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 being 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 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,
^The true time average () of a variable () is defined by
For this to be a well-defined term, the limit () must be independent of the initial condition at . In the case of a chaotic dynamical system, which the equations under turbulent conditions are thought to be, this means that the system can have only one strange attractor, a result that has yet to be proved for the Navier-Stokes equations. However, assuming the limit exists (which it does for any bounded system, which fluid velocities certainly are), there exists some such that integration from to is arbitrarily close to the average. This means that given transient data over a sufficiently large time, the average can be numerically computed within some small error. However, there is no analytical way to obtain an upper bound on .
^Reynolds, Osborne, 1895: "On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion." Philosophical Transactions of the Royal Society of London. A, v. 186, pp. 123-164. Available online from JSTOR.
^Tennekes, H.; Lumley, J. L. (1992). A first course in turbulence (14. print. ed.). Cambridge, Mass. [u.a.]: MIT Press. ISBN978-0-262-20019-6.
^Splitting each instantaneous quantity into its averaged and fluctuating components yields,
Time-averaging these equations yields,
Note that the nonlinear terms (like ) can be simplified to,
^This follows from the mass conservation equation which gives,