See the MRF development
The section describes the development for the Reynolds-Averaged Navier-Stokes formulation in the rotating frame.
To start, we will look at the acceleration term for a rotating frame around the z axis .
Notation: I: inertial, R: rotating
For a general vector:
For the position vector:
The acceleration is expressed as:
Eqn [1]
The Navier-Stokes equations in the inertial frame are:
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 [3] can be written as
Eqn [5]
Eqn [5] represents the Navier-Stokes equations in the rotating frame, in terms of rotating 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
Frame Convected velocity RANS equations Inertial absolute velocity Rotating relative velocity Rotating absolute velocity