The Coupled System of Equations

MissCFD · May 31, 2016, 00:07

Can somebody explain the notation of the system of equations written in CFX theory guide ?

$\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i$

I don't understand the double notation, subscrit and superscript ...

MissCFD · May 31, 2016, 00:15

Quote:

Originally Posted by MissCFD

Can somebody explain the notation of the system of equations written in CFX theory guide ?

$\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i$

I don't understand the double notation, subscrit and superscript ...

Opaque · May 31, 2016, 11:50

nb stand for (N)eigh(B)ors

i stand for equation_i

the equation reads as "sum around the neighbors of its influence coefficients on i times its variable value is equal to b_i"

Those coefficients include the advection, diffusion and other physics interacting between "control volume_i" and its neighbors..

Note the equation may be written in multiple forms where phi can be the value of the solved variable, or the iterative correction to the variable. Based on the interpretation of phi, b can be the source or the residual of the equation.

Hope the above gives some context.

MissCFD · May 31, 2016, 15:47

Quote:

Originally Posted by Opaque

nb stand for (N)eigh(B)ors

i stand for equation_i

the equation reads as "sum around the neighbors of its influence coefficients on i times its variable value is equal to b_i"

Those coefficients include the advection, diffusion and other physics interacting between "control volume_i" and its neighbors..

Note the equation may be written in multiple forms where phi can be the value of the solved variable, or the iterative correction to the variable. Based on the interpretation of phi, b can be the source or the residual of the equation.

Hope the above gives some context.

Thank you for your response !

I'm not sure to visualize, so can you tell me if this notation means the same ?

$a_i\,\phi_i + \sum_{nb} a^{nb}\,\phi^{nb} =b_i$

Does this correspond to system equations for one node

?

Opaque · May 31, 2016, 22:04

Assuming "i" stands for node number, and "nb" includes all the other influencing "i"s (say j) around "i", your are correct.

MissCFD · May 31, 2016, 23:43

Quote:

Originally Posted by Opaque

Assuming "i" stands for node number, and "nb" includes all the other influencing "i"s (say j) around "i", your are correct.

Thank you, so are these two equations equivalent ?

$\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i$ (1) & $a_i\,\phi_i + \sum_{nb} a^{nb}\,\phi^{nb} =b_i$ (2)

And, this equations (1 or 2) is written for all the nodes of mesh, so we obtain i times this equation and this is what forms the matricial system ?

Opaque · June 1, 2016, 15:12

I would read them as equivalent

May 31, 2016, 00:07	The Coupled System of Equations	#1
MissCFD Senior Member Join Date: Aug 2014 Posts: 160 Rep Power: 12	Can somebody explain the notation of the system of equations written in CFX theory guide ? $\sum_{nb\,i} a^{nb}_i\,\phi^{nb}_i =b_i$ I don't understand the double notation, subscrit and superscript ... Last edited by MissCFD; May 31, 2016 at 09:28.

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May 31, 2016, 11:50		#3
Opaque Senior Member Join Date: Jun 2009 Posts: 1,869 Rep Power: 33	nb stand for (N)eigh(B)ors i stand for equation_i the equation reads as "sum around the neighbors of its influence coefficients on i times its variable value is equal to b_i" Those coefficients include the advection, diffusion and other physics interacting between "control volume_i" and its neighbors.. Note the equation may be written in multiple forms where phi can be the value of the solved variable, or the iterative correction to the variable. Based on the interpretation of phi, b can be the source or the residual of the equation. Hope the above gives some context.

May 31, 2016, 22:04		#5
Opaque Senior Member Join Date: Jun 2009 Posts: 1,869 Rep Power: 33	Assuming "i" stands for node number, and "nb" includes all the other influencing "i"s (say j) around "i", your are correct.

June 1, 2016, 15:12		#7
Opaque Senior Member Join Date: Jun 2009 Posts: 1,869 Rep Power: 33	I would read them as equivalent