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The section describes the development for the incompressible Navier-Stokes formulation in the rotating frame.

1 Accelerations

To start, we will look at the acceleration term for a rotating frame $(\vec \Omega)$.

Notation: I: inertial, R: rotating

For a general vector:

$\left [ \frac{d \vec A}{dt} \right ]_I = \left [ \frac{d \vec A}{dt} \right ]_R + \vec \Omega \times \vec A$

For the position vector:

$\left [ \frac{d \vec r}{dt} \right ]_I = \left [ \frac{d \vec r}{dt} \right ]_R + \vec \Omega \times \vec r$

$\vec u_I = \vec u_R + \vec \Omega \times \vec r$

The acceleration is expressed as:

$\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \vec u_I}{dt} \right ]_R + \vec \Omega \times \vec u_I$

$\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \left [ \vec u_R + \vec\Omega \times \vec r \right ] }{dt} \right ]_R + \vec \Omega \times \left [ \vec u_R + \vec \Omega \times \vec r \right ]$

$\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \vec u_R}{dt} \right ]_R + \frac{d \vec \Omega}{dt} \times \vec r + \vec \Omega \times \underbrace{ \left [ \frac{d \vec r}{dt} \right ]_R }_{\vec u_R} + \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r$

$\left [ \frac{d \vec u_I}{dt} \right ]_I = \left [ \frac{d \vec u_R}{dt} \right ]_R + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r$ Eqn [1]

2 Navier-Stokes equations in the inertial frame with absolute velocity

The incompressible Navier-Stokes equations in the inertial frame with constant molecular viscosity are:

$\begin{cases} \frac {D \vec u_I}{D t} = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$ Eqn [2]

$\begin{cases} \frac {\partial \vec u_I}{\partial t} + \vec u_I \cdot \nabla \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$

$\begin{cases} \frac {\partial \vec u_I}{\partial t} + \nabla \cdot (\vec u_I \otimes \vec u_I) - \underbrace{( \nabla \cdot \vec u_I )}_{0} \vec u_I= - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$

$\begin{cases} \frac {\partial \vec u_I}{\partial t} + \nabla \cdot (\vec u_I \otimes \vec u_I) = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$ Eqn [3]

3 Navier-Stokes equations in the relative frame with relative velocity

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:

$\frac {D \vec u_I}{D t} = \frac{D \vec u_R}{Dt} + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r$

$\frac {D \vec u_I}{D t} = \frac {\partial \vec u_R}{\partial t} + \vec u_R \cdot \nabla \vec u_R + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r$

$\frac {D \vec u_I}{D t} = \frac {\partial \vec u_R}{\partial t} + \nabla \cdot (\vec u_R \otimes \vec u_R) + \frac{d \vec \Omega}{dt} \times \vec r + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r$ Eqn [4]

since $\nabla \cdot \vec u_R = \nabla \cdot \vec u_I = 0$

$\begin{alignat}{2}
 \nabla \cdot \vec u_I & = \nabla \cdot \left [ \vec u_R + \vec \Omega \times \vec r \right ] = 0 \\
 & = \nabla \cdot \vec u_R + \underbrace{\nabla \cdot \left [ \vec \Omega \times \vec r \right ]}_{0} = 0 \\
 & = \nabla \cdot \vec u_R = 0
 \end{alignat}$

Also, it can be noted that

$\begin{alignat}{2}
\nabla \cdot \nabla (\vec u_I) & = \nabla \cdot \nabla \left [ \vec u_R + \vec \Omega \times \vec r \right ] \\
 & = \nabla \cdot \nabla (\vec u_R) + \nabla \cdot \underbrace{\nabla (\Omega \times \vec r)}_{0} \\
 & = \nabla \cdot \nabla (\vec u_R )
 \end{alignat}$

Eqn [3] can be written as

$\begin{cases} \frac {\partial \vec u_R}{\partial t} + \frac{d \vec \Omega}{dt} \times \vec r + \nabla \cdot (\vec u_R \otimes \vec u_R) + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ \nabla \cdot \vec u_R = 0 \end{cases}$ Eqn [5]

Eqn [5] represents the incompressible Navier-Stokes equations in the rotating frame, in terms of rotating velocities (convection velocity and convected velocity).

4 Navier-Stokes equations in the relative frame with absolute velocity

Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame.

The term $\nabla \cdot (\vec u_R \otimes \vec u_R)$ can be developed as:

$\begin{alignat}{2} 
\nabla \cdot (\vec u_R \otimes \vec u_R) & = \nabla \cdot ( \vec u_R \otimes \left [ \vec u_I - \vec \Omega \times \vec r \right ] ) \\
 & = \nabla \cdot (\vec u_R \otimes \vec u_I) - \underbrace{\nabla \cdot \vec u_R}_{0} (\vec \Omega \times \vec r) - \underbrace{\vec u_R \cdot \nabla(\vec \Omega \times \vec r)}_{\vec \Omega \times \vec u_R} \\
 & = \nabla \cdot (\vec u_R \otimes \vec u_I) - \vec \Omega \times \vec u_R
 \end{alignat}$

So, the steady term of left-hand side of Eqn [5] can be written as

$\begin{alignat}{2} 
\nabla \cdot (\vec u_R \otimes \vec u_R) + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r & = \nabla \cdot (\vec u_R \otimes \vec u_I) - \vec \Omega \times \vec u_R + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r \\
 & = \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r \\
 & = \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times ( \vec u_R + \vec \Omega \times \vec r ) \\
 & = \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I
\end{alignat}$

Eqn [5] can be written in terms of the absolute velocity:

$\begin{cases} \frac {\partial \vec u_R}{\partial t} + \frac{d \vec \Omega}{dt} \times \vec r + \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$ Eqn [6]

5 Summary

In summary, for multiple frames of reference, the incompressible Navier-Stokes equations for steady flow can be written

Frame	Convected velocity	Steady incompressible Navier-Stokes equations
Inertial	absolute velocity	$\begin{cases} \nabla \cdot (\vec u_I \otimes \vec u_I) = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$
Rotating	relative velocity	$\begin{cases} \nabla \cdot (\vec u_R \otimes \vec u_R) + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ \nabla \cdot \vec u_R = 0 \end{cases}$
Rotating	absolute velocity	$\begin{cases} \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ \nabla \cdot \vec u_I = 0 \end{cases}$

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