Difference between revisions of "See the MRF development"

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Reynolds-Averaged Navier-Stokes formulation in the rotating frame.

Acceleration term expressed for a rotating frame around the z axis ( $\vec \Omega$ ).

Notation: I: inertial, R: rotating

General vector:

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

Position vector:

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

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

For the acceleration, the velocity vector is:

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

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

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

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

Navier-Stokes equations in the inertial frame

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

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 [4] can be written as

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

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

@@ Line 15: / Line 15: @@
 <math>\vec u_I = \vec u_R + \vec \Omega \times \vec r</math>
-For the accelration, the velocity vector is:
+For the acceleration, the velocity vector is:
 <math>\left [ \frac{d \vec u_I}{dt} \right ]_I  = \left [ \frac{d \vec u_I}{dt} \right ]_R  + \vec \Omega \times \vec u_I</math>
@@ Line 21: / Line 21: @@
 <math>\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 ]</math>
-<math>\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{\vec r}{dt} \right ]_R }_{\vec u_R}  + \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math>
+<math>\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</math>
 <math>\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</math>  '''Eqn [1]'''
@@ Line 43: / Line 43: @@
 <math>
 \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) \\
+\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}
@@ Line 65: / Line 65: @@
 since <math>\nabla \cdot \vec u_R = \nabla \cdot \vec u_I = 0</math>
-  <math>\nabla \cdot \vec u_I = \nabla \cdot \left [ \vec u_R + \vec \Omega \times \vec r \right ]</math>
+  <math>
- <math>\nabla \cdot \vec u_I = \nabla \cdot \vec u_R + \underbrace{\nabla \cdot \left [ \vec \Omega \times \vec r \right ]}_{0}</math>
+ \begin{alignat}{2}
- <math>\nabla \cdot \vec u_I = \nabla \cdot \vec u_R</math>
+ \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}
+ </math>
 Also, it can be noted that
-  <math>\nabla \cdot \nabla (\vec u_I) = \nabla \cdot \nabla \left [ \vec u_R + \vec \Omega \times \vec r \right ]</math>
+  <math>
-  <math>\nabla \cdot \nabla (\vec u_I) = \nabla \cdot \nabla (\vec u_R) + \nabla \cdot \underbrace{\nabla (\Omega \times \vec r)}_{0}</math>
+ \begin{alignat}{2}
-  <math>\nabla \cdot \nabla (\vec u_I) = \nabla \cdot \nabla (\vec u_R )</math>
+\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}
+ </math>
 Eqn [4] can be written as
@@ Line 86: / Line 94: @@
 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 frmae.
+Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame.
 The term <math>\nabla \cdot (\vec u_R \otimes \vec u_R)</math> can be developed as:
+ <math>
+ \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}
+ </math>
+So, the steady term of left-hand side of Eqn [5] can be written as
+ <math>
+ \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}
+ </math>