[lostking18]

Hi, there

Could you kindly tell me how the the convection term in the momentum equation is discretized in the implicit temporal scheme?

Since several equations are involved in my question, I have attached a pic to state my question in details. Thank you so much for your attention!

[lostking18]

Here is the picture.

https://www.cfd-online.com/Forums/at...1&d=1529617047

[ghorrocks]

Your equation does not look right to me. You appear to have developed an explicit formulation - but CFX is implicit.

Look at the theory documentation, section 11.1 on development of the numerical approach. It also includes many things you appear to have neglected, such as the mesh element and integration points, shape functions, how all the terms are handled and pressure velocity coupling.

[lostking18]

Hi, mighty Ghorrocks:

Thank you very much for you answers! I have looked at the chapter you recommended. They are really helpful! So what I understand from the document is that the convection term always use the newest value. But I seem to find a contradiction:

On one hand, in the compressibility section, the documents say the mass flow rate uses the newest mass flow rate, m_n, by Newton-Raphson linearization.

But on the other hand, if we use the mass source in the solver, the solver will automatically add the additional source term, Sm*phi, in the Momentum and energy equation to account for the additional momentum and energy brought by the mass source.

The place I feed contradicted is that if the solver use the newest mass flow rate, then the mass flow rate, m, in the convection term, m*phi, of the Momentum and energy equation is already the mass flow rate with the source. So it has accounted for the additional momentum and energy brought by the additional mass source. Then why do we still need to add the additional source term in the Momentum and energy equation in the mass source?

What I guess to explain the contradiction is that, CFX will compute two versions of mass flow rate: the one with source term and the one without mass source. It will use the mass flow rate without mass source in the convection term. Is that right?
Now I am facing this problem because we are trying to use the source term to increase the accuracy of the cooling flow simulation. Thus I have to understand precisely how the mass source tangled with the momentum equation in order to correctly implement the source.

Thank you very much!

[ghorrocks]

Mighty?

I do not understand your question. You are talking about mass sources, but your original question was about discretisation of the convection term. So your original question had nothing to do with mass sources.

Mass sources are simply added to the end of the equations as described in the theory documentation, and allow you to add or remove mass. They do not require any special treatment, they are just another term in the equation.

Your final comment suggests you are using a mass source to increase accuracy of a cooling flow simulation. I am a bit concerned that this question is actually an XY problem (http://xyproblem.info/). How does a mass source increase accuracy of your simulation? Why can't the normal approaches resolve it?

[Opaque]

I think there is a confusion about the discretization mass flows, and the actual mass flow leaving at a face of a control volume

The continuity equation for a control volume states

Sum over the faces of CV ( mass flow_face) = mass source

The transport equation for a control volume states (only accounting for advection)

Sum over the faces of CV (massflow_face * Transported variable_face) = mass source * Transported variable_io

where

Transported_variable_io is the user specified value if incoming, or the local variable if outgoing.

The discretized massflow_face = density * velocity dot Norma Area + other terms (depending of discretization details)

Nowhere the mass source is double accounted. Otherwise, the flow balances will be obviously wrong.

[lostking18]

Originally Posted by Opaque

I think there is a confusion about the discretization mass flows, and the actual mass flow leaving at a face of a control volume

The continuity equation for a control volume states

Sum over the faces of CV ( mass flow_face) = mass source

The transport equation for a control volume states (only accounting for advection)

Sum over the faces of CV (massflow_face * Transported variable_face) = mass source * Transported variable_io

where

Transported_variable_io is the user specified value if incoming, or the local variable if outgoing.

The discretized massflow_face = density * velocity dot Norma Area + other terms (depending of discretization details)

Nowhere the mass source is double accounted. Otherwise, the flow balances will be obviously wrong.

Aha! Opaque, thanks! The answer catches exact my point! :P

Do you mean that the ‘Sum over the faces of CV( ... )’ is used to compensate the imbalance of mass, momentum and energy brought by the mass source? Then it makes perfect sense to me.

To make a metaphor, it is like the source term put out the request and the flux over the CV try to adapt itself to fulfill the order of the source. Is it right? Thank you very much!

[ghorrocks]

You could think of it that the sum of faces over the CV is used to compensate for the imbalance from the mass source. A more general way of looking at it is that the source term simply adds a term to the equation which the solver is converging to conservation on.

Yes, the fluxes on the CV will then adapt to the material added by the mass source. They will attempt to converge to conservation.