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 ().
Notation: I: inertial, R: rotating
General vector:
Position vector:
For the accelration, the velocity vector is:
Eqn [1]
Navier-Stokes equations in the inertial frame
Eqn [2]