MRF framework for Turbomachinery

Definitions:

• Axis of rotation ($s$):

  $$s=x_2-x_1$$

  Where $x_1$ and $x_2$ are any two points on the axis.

• Rotation Vector ($\Omega$):

  $$\Omega =\omega \cdot s$$

  Where $\omega \to$ rotational speed

• Distance Vector ($r$):

  $$r=x_p-x_o$$

  Where $x_p\to$ cell centroid, $x_o\to$ axis of rotation Note: $r$ is unique to each cell in the mesh

• Relative Velocity ($U_r$):

Derivation

Velocity in a frame rotating with the blade

$$\begin{array}{rl}U& =U_r+\Omega \times r\end{array}\text{\hspace{1em}(1)}$$

Incompressible Navier-Stokes equations

$$\begin{array}{rl}\nabla \cdot (UU)& =-\frac{1}{\rho }\nabla p+\nabla \cdot (\nu \nabla U)\end{array}\text{\hspace{1em}(2)}$$

After substitution and simplification,¹ we have Single Reference Frame (Frame)

$$\begin{array}{rl}\nabla \cdot (U_r{U}_r)& =-\nabla p+\nabla \cdot (\nu \nabla U_r)-\underset{\text{Centrifugal}}{\underbrace{\Omega \times \Omega \times r}}-\underset{\text{Coriolis}}{\underbrace{2\Omega \times U_r}}\end{array}$$

• We end up with two source terms for the Centrifugal and Coriolis forces

• Unknown is now the relative velocity, $U_r$

• $U$ is calculated as a post-processing operation:

  $$U=U_r+\Omega \times r$$

Multiple Reference Frames

$$\nabla \cdot (U{U}_r)=-\nabla p+\nabla \cdot (\nu \nabla U)-\Omega \times U$$

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References

Footnotes

1. https://openfoamwiki.net/index.php/See_the_MRF_development ↩