See the MRF development
Reynolds-Averaged Navier-Stokes formulation in the rotating frame.
Acceleration term expressed for a rotating frame around the z axis ().
Notation: I: inertial, R: rotating
General vector:
Position vector:
For the acceleration, the velocity vector is:
Eqn [1]
Navier-Stokes equations in the inertial frame
Eqn [2]
Eqn [3]
Let's look at the left-hand side of the momentum equation of Eqn [2], by taking into account Eqn [1] for the acceleration term:
Eqn [4]
since
Also, it can be noted that
Eqn [4] can be written as
Eqn [5]
Eqn [5] represents the Navier-Stokes equations in the rotating frame, in terms of rotation velocities (convection velocity and convected velocity).
Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame.
The term can be developed as:
So, the steady term of left-hand side of Eqn [5] can be written as
Eqn [5] can be written in terms of the absolute velocity:
Eqn [6]
In summary, for multiple frames of reference, the Reynolds-averaged Navier-Stokes equations for steady flow can be written