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deepblue17 January 27, 2009 16:16

Hello, I've derived the mom

I've derived the momentum equation for the relative frame of reference formulated with the absolute velocity here. The result

is coincident with the literature but not with the formulation in the fluent manual. The second term on the left hand side is different, whereas the rest is equal. It can be shown that the tensor 2. rank from the dyadic product of (vec{c} \otimes vec{w}) is not symmetric, if you write it down component-by-component in cartesian coordinates, you will see that (vec{c} \otimes vec{w} \neq vec{w} \otimes vec{c}). So there seems to be a bug in the fluent manual.

That's why I think, the formulation of the UEqn in the MRFSimpleFoam solver is also not valid. There is fvm::div(phi, U), in this case phi denotes to w_i (relative velocity) for each spatial direction. Am I right?

deepblue17 June 13, 2009 11:59

There are two different derivations of Navier-Stokes equations for so called "Multiple Reference Frames" (MRF) or "Multiple Frames of Reference" (MFR):
  • The first notation is equation 6 form here, which is also implemented in the MRF library:
    \dfrac{\partial^\prime \left( \varrho \vec{w}\right)}{\partial t} + \nabla \bullet \left( \varrho \vec{w} \otimes \vec{c}\right) + \dfrac{d^\prime \left( \varrho \vec{\omega}\right)}{d t} \times \vec{r^\prime}+ \vec{\omega} \times \left( \varrho \vec{c} \right)
= -\nabla p + \nabla \bullet \tau + \varrho \vec{k}
  • The second notation is equation 21 form here:
    \dfrac{\partial^\prime \left( \varrho \vec{c}\right)}{\partial t} + \nabla \bullet \left( \varrho \vec{c} \otimes \vec{w}\right) + \vec{\omega} \times \left( \varrho \vec{c} \right)
= -\nabla p + \nabla \bullet \tau + \varrho \vec{k}
Both momentum equations are sharing the same idea, they are solving for the absolute velocity in the relative frame of reference. The main difference are mainly the first two terms on the left hand side.

If anybody wants to use the actual implementation of the MRF library within a transient solver (that includes all compressible transsonic solvers), one has to be certain, that the local time derivation is calculated with the relative velocity (\vec{w}) and not with the absolute velocity (\vec{c}) ! As this could be a little bit tricky, I would prefer either the second notation of the MRF or using the formulation with the relative velocity in the relative frame of reference, as this is done in the SRF library, with an appropriate rotor-stator interface.

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