Difference between revisions of "See the MRF development"
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In summary, for multiple frames of reference, the Reynolds-averaged Navier-Stokes equations for steady flow can be written | In summary, for multiple frames of reference, the Reynolds-averaged Navier-Stokes equations for steady flow can be written | ||
+ | ::{| cellpadding="8" cellspacing="1" border="3" | ||
− | + | |- | |
− | <math> | + | | Inertial frame (absolute velocity) |
+ | | <math> | ||
\begin{cases} | \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 \otimes \vec u_I) = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ | ||
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\end{cases} | \end{cases} | ||
</math> | </math> | ||
− | + | |- | |
− | <math> | + | | Rotating frame (relative velocity) |
+ | | <math> | ||
\begin{cases} | \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 \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) \\ | ||
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\end{cases} | \end{cases} | ||
</math> | </math> | ||
− | + | |- | |
− | <math> | + | | Rotating frame (abstolute velocity) |
+ | | <math> | ||
\begin{cases} | \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_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_R) \\ |
Revision as of 02:17, 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
Inertial frame (absolute velocity) Rotating frame (relative velocity) Rotating frame (abstolute velocity)