[Happy11]

Hi

I am really confused about this explanation in the paper 'A Fixed-Grid Numerical Modeling Methodology for Convection-Diffusion Mushy Region Phase-Change Problems' ---''In the isothermal case due to the step change in deltaU
along with a zero velocity at the solid-liquid interface the convective part of this source term takes the value zero. In a mushy region case, however, the convective term needs to be included. ''

In my opinion, in the case of isothermal melting/solidification, the velocity at the phase change cell near the fluid side is still not zero.

Can somebody explain this a little bit, thanks!

[LuckyTran]

There is certainly some confusion that requires unraveling.

The velocity near the interface is irrelevant because the source term being discussed concerns only the infinitesimally-thin melting line. Away from the interface, you use the normal transport equations (continuity, navier-stokes, and energy transport) without sources.

On the interface, the solid is non-moving so the velocity has to approach zero. Additionally, in isothermal phase change you know there is a step change in the enthalpy, and if velocity is not zero then you've created work without any temperature difference and last time I checked that violates at least one law of thermodynamics.


Maybe you are confusing the isothermal phase change case with the non-isothermal mushy zone. Or something like that.

[Happy11]

Originally Posted by LuckyTran

There is certainly some confusion that requires unraveling.

The velocity near the interface is irrelevant because the source term being discussed concerns only the infinitesimally-thin melting line. Away from the interface, you use the normal transport equations (continuity, navier-stokes, and energy transport) without sources.

On the interface, the solid is non-moving so the velocity has to approach zero. Additionally, in isothermal phase change you know there is a step change in the enthalpy, and if velocity is not zero then you've created work without any temperature difference and last time I checked that violates at least one law of thermodynamics.


Maybe you are confusing the isothermal phase change case with the non-isothermal mushy zone. Or something like that.

Firstly I am genuinely greateful for your reply!

But as I know, even you mentioned the velocity near the interface is nearly zero, it is still not zero, isn't it? Many paper propose to apply darcy source term to make the velocity near the interface reduce to zero during the solidification process. Based on this, I think it is not correct to just simply regard the velocity near the interface as 0 all the time. I am not sure which part of my understanding going wrong, by the way, I can totally understand the convection term could not vanish in the non-isothermal mushy zone solidification process.

I really appreciate your help!

[LuckyTran]

Take the no slip condition for example. The velocity near a solid boundary approaches zero and the velocity on the solid boundary being zero are not inconsistent with one another.

Given f(x) and f(a)=0, the limit of f as x approaches a is zero doesn't mean that f(a) is not zero. At the same time f can be non-zero near the vicinity of a as long as it obeys epsilon-delta arguments for the limit to still exist (which itself is a sufficient but not necessary condition).

[Happy11]

Originally Posted by LuckyTran

Take the no slip condition for example. The velocity near a solid boundary approaches zero and the velocity on the solid boundary being zero are not inconsistent with one another.

Given f(x) and f(a)=0, the limit of f as x approaches a is zero doesn't mean that f(a) is not zero. At the same time f can be non-zero near the vicinity of a as long as it obeys epsilon-delta arguments for the limit to still exist (which itself is a sufficient but not necessary condition).

https://imgur.com/a/1aU249X
Thank you for your reply!

But as you can see from the picture (link uploaded already), if staggered grid is applied for the velocity cell, both Vn and Vs will not be zero, aren't they?

Thanks a lot

[Happy11]

https://imgur.com/a/1aU249X

Thank you for your reply!

But as you can see from the linked picutre, if staggered grids are applied, both the Vn and Vs which are function of latent heat, will not be zero, aren't they?

Thanks a lot

[LuckyTran]

Then you have the convective flux must be equal to the latent heat change which doesn't give you anything new. This is again just like the no-slip condition on a wall. Any velocity near the wall must be exactly balanced by the shear stress to arrive at zero velocity on the wall.


There is no inconsistency.