Difference between revisions of "See the MRF development"
Line 87: | Line 87: | ||
<math> | <math> | ||
\begin{cases} | \begin{cases} | ||
− | \frac {\partial \vec u_R}{\partial t} | + | \frac {\partial \vec u_R}{\partial t} + \frac{d \vec \Omega}{dt} \times \vec r + \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 (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ |
\nabla \cdot \vec u_R = 0 | \nabla \cdot \vec u_R = 0 | ||
\end{cases} | \end{cases} | ||
Line 116: | Line 116: | ||
\end{alignat} | \end{alignat} | ||
</math> | </math> | ||
+ | |||
+ | Eqn [5] can be written in terms of the absolute velocity: | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \frac {\partial \vec u_R}{\partial t} + \frac{d \vec \Omega}{dt} \times \vec r + \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> '''Eqn [6]''' | ||
+ | |||
+ | In summary, for multiple frames of reference, the Reynolds-averaged Navier-Stokes equations for steady flow can be written | ||
+ | |||
+ | |||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \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> | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \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 (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ | ||
+ | \nabla \cdot \vec u_R = 0 | ||
+ | \end{cases} | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | \begin{cases} | ||
+ | \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ | ||
+ | \nabla \cdot \vec u_I = 0 | ||
+ | \end{cases} | ||
+ | </math> |
Revision as of 02:12, 26 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 acceleration, 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 frame.
The term can be developed as:
So, the steady term of left-hand side of Eqn [5] can be written as
Eqn [5] can be written in terms of the absolute velocity:
Eqn [6]
In summary, for multiple frames of reference, the Reynolds-averaged Navier-Stokes equations for steady flow can be written