Difference between revisions of "See the MRF development"
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<math>\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</math> | <math>\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</math> | ||
− | <math>\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</math> | + | <math>\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</math> '''Eqn [4]''' |
− | <math>\nabla \cdot \vec | + | since <math>\nabla \cdot \vec u_R = \nabla \cdot \vec u_I = 0</math> |
− | <math>\nabla \cdot \vec u_I = \nabla \vec u_R + \underbrace{\nabla \left [ \vec \Omega \times \vec r \right ]}_{0}</math> | + | <math>\nabla \cdot \vec u_I = \nabla \cdot \left [ \vec u_R + \vec \Omega \times \vec r \right ]</math> |
+ | <math>\nabla \cdot \vec u_I = \nabla \cdot \vec u_R + \underbrace{\nabla \cdot \left [ \vec \Omega \times \vec r \right ]}_{0}</math> | ||
+ | <math>\nabla \cdot \vec u_I = \nabla \cdot \vec u_R</math> | ||
− | <math>\nabla \cdot \vec u_I = \nabla \vec u_R</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>\nabla \cdot \nabla (\vec u_I) = \nabla \cdot \nabla (\vec u_R) + \nabla \cdot \underbrace{\nabla (\Omega \times \vec r)}_{0}</math> | ||
+ | <math>\nabla \cdot \nabla (\vec u_I) = \nabla \cdot \nabla (\vec u_R )</math> | ||
+ | |||
+ | Eqn [4] can be written as | ||
+ | |||
+ | <math> | ||
+ | \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} | ||
+ | </math> '''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 frmae. | ||
+ | |||
+ | The term <math>\nabla \cdot (\vec u_R \otimes \vec u_R)</math> can be developed as: |
Revision as of 21:09, 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]
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:
Eqn [4]
since
Also, it can be noted that
Eqn [4] can be written as
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 frmae.
The term can be developed as: