UDS Diffusion to another gradient

CFDJonas · January 6, 2023, 15:56

Hello,

I am currently implementing a pseudo-binary mixture by making use of User-Defined Scalars.

What does pseudo-binary mixture mean?
In the species model of Fluent I have defined 2 species, let's call them A and B. The thermophysical data is fully described by mixing rules by just these two components.

Additionally, component A consists of 4 sub-components, let's call them A1, A2, A3 and A4. Therefore it is valid to say for the mass-fractions:
$w_A = w_{A1} + w_{A2} + w_{A3} + w_{A4}$
The sub-component's mass fractions are modeled as User-Defined Scalars. I know that in this explicit example, I am able to fully use the normal species model of Fluent. But other things prevent me from doing so. I have to use UDS for the sub-components.

Diffusion
Now it get's complicated: I want to implement the diffusion of both, the whole component A and additionally the sub-components. Mathematically this is quite easy to write down, for example for the UDS transport equation of component A1:

$\frac{\partial \rho w_{A_1}}{\partial t} + \nabla \left( \rho \vec{v} w_{A1} - \left( \rho \frac{w_{A1}}{w_A} D_{\text{binary}} \nabla w_{A} + \rho w_{A} D_{\text{subdiffA}} \nabla \frac{w_{A1}}{w_{A}} \right) \right) = S_{w_{A1}}$

There the problem rises: The diffusion term is not proportional to the gradient of the species itself ( $\nabla w_{A1}$ ) but instead to either the "summed" gradient $\nabla w_{A}$ or the quotient $\nabla \frac{w_{A1}}{w_A}$ .

This lead me to the conclusion that I am not able to use the UDS_DIFFUSIVITY for this use case, as this would just modify the diffusion coefficient and multiply it by $\nabla w_{A1}$ . The only thing I could do is put everything before the gradient into the diffusion coefficient and divide by the scalar's gradient. But this seems a little bit fishy (or doesn't it?).

The other possibility that came to my mind was to use DEFINE_UDS_FLUX and put everything into the convection vector.
I am able to construct the needed vectors with C_YI_G of component A for $\nabla w_{A}$ and on the other hand $\nabla \frac{w_{A1}}{w_A}$ is constructable by both, C_YI_G of component A and C_UDSI_G of component A1 and the quotient derivative rule. I have set up a UDF that does exactly that by calculating the diffusion flux term at a face. However this leads to very unstable results and I'm having a hard time debugging it. It also seems very "fudged" doing it like that with all the gradient juggling...

Therefore the question: Is there an easier opportunity to achieve the described problem, is there something important to know about dealing with gradients, or modifying the flux term of a UDS? Would it be better to put it into the diffusion coefficient (which would then need to be anisotropic, because I have to modify each gradient vector's components...)? Stability-wise, are there any modifications to the solver I could do to ensure diffusion of UDS work better?

Thank you very much for your kind help!

January 6, 2023, 15:56	UDS Diffusion to another gradient	#1
CFDJonas New Member Join Date: May 2018 Posts: 29 Rep Power: 7	Hello, I am currently implementing a pseudo-binary mixture by making use of User-Defined Scalars. What does pseudo-binary mixture mean? In the species model of Fluent I have defined 2 species, let's call them A and B. The thermophysical data is fully described by mixing rules by just these two components. Additionally, component A consists of 4 sub-components, let's call them A1, A2, A3 and A4. Therefore it is valid to say for the mass-fractions: $w_A = w_{A1} + w_{A2} + w_{A3} + w_{A4}$ The sub-component's mass fractions are modeled as User-Defined Scalars. I know that in this explicit example, I am able to fully use the normal species model of Fluent. But other things prevent me from doing so. I have to use UDS for the sub-components. Diffusion Now it get's complicated: I want to implement the diffusion of both, the whole component A and additionally the sub-components. Mathematically this is quite easy to write down, for example for the UDS transport equation of component A1: $\frac{\partial \rho w_{A_1}}{\partial t} + \nabla \left( \rho \vec{v} w_{A1} - \left( \rho \frac{w_{A1}}{w_A} D_{\text{binary}} \nabla w_{A} + \rho w_{A} D_{\text{subdiffA}} \nabla \frac{w_{A1}}{w_{A}} \right) \right) = S_{w_{A1}}$ There the problem rises: The diffusion term is not proportional to the gradient of the species itself ( $\nabla w_{A1}$ ) but instead to either the "summed" gradient $\nabla w_{A}$ or the quotient $\nabla \frac{w_{A1}}{w_A}$ . This lead me to the conclusion that I am not able to use the UDS_DIFFUSIVITY for this use case, as this would just modify the diffusion coefficient and multiply it by $\nabla w_{A1}$ . The only thing I could do is put everything before the gradient into the diffusion coefficient and divide by the scalar's gradient. But this seems a little bit fishy (or doesn't it?). The other possibility that came to my mind was to use DEFINE_UDS_FLUX and put everything into the convection vector. I am able to construct the needed vectors with C_YI_G of component A for $\nabla w_{A}$ and on the other hand $\nabla \frac{w_{A1}}{w_A}$ is constructable by both, C_YI_G of component A and C_UDSI_G of component A1 and the quotient derivative rule. I have set up a UDF that does exactly that by calculating the diffusion flux term at a face. However this leads to very unstable results and I'm having a hard time debugging it. It also seems very "fudged" doing it like that with all the gradient juggling... Therefore the question: Is there an easier opportunity to achieve the described problem, is there something important to know about dealing with gradients, or modifying the flux term of a UDS? Would it be better to put it into the diffusion coefficient (which would then need to be anisotropic, because I have to modify each gradient vector's components...)? Stability-wise, are there any modifications to the solver I could do to ensure diffusion of UDS work better? Thank you very much for your kind help!

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