Difference between revisions of "See the MRF development"
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<math>\left [ \frac{d \vec A}{dt} \right ]_I = \left [ \frac{d \vec A}{dt} \right ]_R + \vec \Omega \times \vec A</math> | <math>\left [ \frac{d \vec A}{dt} \right ]_I = \left [ \frac{d \vec A}{dt} \right ]_R + \vec \Omega \times \vec A</math> | ||
| + | |||
| + | Position vector: | ||
| + | |||
| + | <math>\left [ \frac{d \vec r}{dt} \right ]_I = \left [ \frac{d \vec r}{dt} \right ]_R + \vec \Omega \times \vec r</math> | ||
| + | |||
| + | <math>\vec u_I = \vec u_R + \vec \Omega \times \vec r</math> | ||
| + | |||
| + | For the accelration, 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> | ||
| + | |||
| + | <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 + 2 \vec \Omega \times \vec u_R + \vec \Omega \times \vec \Omega \times \vec r</math> | ||
Revision as of 20:03, 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:
![\left [ \frac{d \vec A}{dt} \right ]_I = \left [ \frac{d \vec A}{dt} \right ]_R + \vec \Omega \times \vec A](/images/math/d/d/7/dd7f88256c509a0be7a15c53f866d116.png)
Position vector:
![\left [ \frac{d \vec r}{dt} \right ]_I = \left [ \frac{d \vec r}{dt} \right ]_R + \vec \Omega \times \vec r](/images/math/4/1/3/4131b2e6b777a32980f1ba8735800c00.png)
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](/images/math/7/f/2/7f20a49ddaaf0f5f75839a3418ed3021.png)
![\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 ]](/images/math/e/a/d/ead6fc46c5c666b0e14e890a9bca6578.png)
![\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](/images/math/9/0/5/905c94c42b2830a357980001da1b3fbc.png)
![\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](/images/math/8/c/2/8c25e2e6b1588e6fb319dce79cabaddf.png)
