Difference between revisions of "See the MRF development"
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− | Reynolds-Averaged Navier-Stokes formulation in the rotating frame. | + | The section describes the development for the Reynolds-Averaged Navier-Stokes formulation in the rotating frame. |
− | + | To start, we will look at the acceleration term for a rotating frame around the z axis <math> (\vec \Omega)</math>. | |
Notation: I: inertial, R: rotating | Notation: I: inertial, R: rotating | ||
− | + | For a general vector: | |
<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> | ||
− | + | For the 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>\left [ \frac{d \vec r}{dt} \right ]_I = \left [ \frac{d \vec r}{dt} \right ]_R + \vec \Omega \times \vec r</math> | ||
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<math>\vec u_I = \vec u_R + \vec \Omega \times \vec r</math> | <math>\vec u_I = \vec u_R + \vec \Omega \times \vec r</math> | ||
− | + | The acceleration is expressed as: | |
<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 \vec u_I}{dt} \right ]_R + \vec \Omega \times \vec u_I</math> | ||
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<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]''' | <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 | + | The Navier-Stokes equations in the inertial frame are: |
<math> | <math> | ||
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</math> | </math> | ||
− | Eqn [ | + | Eqn [3] can be written as |
<math> | <math> | ||
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</math> '''Eqn [5]''' | </math> '''Eqn [5]''' | ||
− | Eqn [5] represents the Navier-Stokes equations in the rotating frame, in terms of | + | Eqn [5] represents the Navier-Stokes equations in the rotating frame, in terms of rotating velocities (convection velocity and convected velocity). |
Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame. | Eqn [5] can be further developed so the convected velocity is the velocity in the inertial frame. | ||
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<math> | <math> | ||
\begin{cases} | \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 | + | \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_I) \\ |
\nabla \cdot \vec u_I = 0 | \nabla \cdot \vec u_I = 0 | ||
\end{cases} | \end{cases} | ||
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| <math> | | <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 | + | \nabla \cdot (\vec u_R \otimes \vec u_I) + \vec \Omega \times \vec u_I = - \nabla (p/\rho) + \nu \nabla \cdot \nabla (\vec u_I) \\ |
\nabla \cdot \vec u_I = 0 | \nabla \cdot \vec u_I = 0 | ||
\end{cases} | \end{cases} |
Revision as of 03:24, 27 May 2009
The section describes the development for the Reynolds-Averaged Navier-Stokes formulation in the rotating frame.
To start, we will look at the acceleration term for a rotating frame around the z axis .
Notation: I: inertial, R: rotating
For a general vector:
For the position vector:
The acceleration is expressed as:
Eqn [1]
The Navier-Stokes equations in the inertial frame are:
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 [3] can be written as
Eqn [5]
Eqn [5] represents the Navier-Stokes equations in the rotating frame, in terms of rotating 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
Frame Convected velocity RANS equations Inertial absolute velocity Rotating relative velocity Rotating absolute velocity