Difference between revisions of "See the MRF development"

From OpenFOAMWiki
Line 23: Line 23:
 
<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{\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>
+
<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]'''
 +
 
 +
Navier-Stokes equations in the inertial frame
 +
 
 +
<math>
 +
\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}
 +
</math>
 +
 
 +
<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) \\
 +
\nabla \cdot \vec u_I = 0
 +
\end{cases}
 +
</math>
 +
 
 +
<math>
 +
\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}
 +
</math>
 +
'''Eqn [2]'''

Revision as of 20:19, 25 May 2009

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 accelration, 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{\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 {\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 [2]