[Farhad9]

Hello everyone,

It's my first time posting here, so I am sorry if I am not posting in the proper forum or for any other mistakes I may have made.
So, my question is about the theory of the QUICK scheme. While reading Versteeg and Malalasekera's book "An Introduction to Computational Fluid Dynamics: The Finite Volume Method", I have encountered a problem with the handling of the diffusion boundary terms. The explanation for the equation seems a bit vague. I have attached two pictures. It seems like eq. 5.54 for diffusion flux at the boundaries emerged out of nowhere. I have tried to find online about this issue, but with no success. I would appreciate it if someone could explain the reasoning behind that formula.

Thank you in advance

[FMDenaro]

Originally Posted by Farhad9

Hello everyone,

It's my first time posting here, so I am sorry if I am not posting in the proper forum or for any other mistakes I may have made.
So, my question is about the theory of the QUICK scheme. While reading Versteeg and Malalasekera's book "An Introduction to Computational Fluid Dynamics: The Finite Volume Method", I have encountered a problem with the handling of the diffusion boundary terms. The explanation for the equation seems a bit vague. I have attached two pictures. It seems like eq. 5.54 for diffusion flux at the boundaries emerged out of nowhere. I have tried to find online about this issue, but with no success. I would appreciate it if someone could explain the reasoning behind that formula.

Thank you in advance

Not sure about but it seems a derivative computed from a quadratic interpolation on 3 nodes at different step sizes, that is A and P at h/2 and P and E at h. You could check that

[FMDenaro]

Check that
http://web.media.mit.edu/~crtaylor/calculator.html

[Farhad9]

Originally Posted by FMDenaro

Check that
http://web.media.mit.edu/~crtaylor/calculator.html

Thank you very much

[hchzty]

use the boundary point, first and second internal nodes for quadratic interpolation, then calculate derivative at the boundary point

[nipinl]

Taking the boundary, Phi_P and Phi_E points gives the required formula 5.54. One can check by entering 0,1,3 in the link provided by Denero.





However, considering boundary, Phi_P and Phi_e values, gradient estimate to be (25*Phi_P-22*Phi_A-3*Phi_e)/8

[Josh54]

This question is really difficult for me. I can't solve it either. Thanks to your post I already know the answer. Thanks!

[mmehta1973]

I have not been able to derive equation (5.54) as already questioned by Farhad9 earlier, few posts above. Can someone please share the derivation. Thanks.