[daylen]

Hi, im having trouble understanding the simple algorithm, specifically the discretization step of the momentum equation, in the link a matrix form of the momentum equation is showed, how do you get it?



https://www.openfoam.com/documentati...olver-momentum


or the semi-discretization in this other link: https://openfoamwiki.net/index.php/O...hm_in_OpenFOAM


again, how do you arrive to such equations?

[LuckyTran]

The discretization is not shown and you need to do it. What is given is a very high level procedure. However, given that there is some discretization, you should arrive at a linear system with some coefficient matrix.

First you need to integrate the governing equations over the comutational cell and apply the gauss-divergence theorem. Then you need to discretize term by term.

The things you need to discretize are:

1. the temporal discretization the du/dt term (1st or 2dt order euler, which is just upwinding)
2. the advection term (i.e. 1st, 2nd order upwind etc)
3. the diffusion term (usually central difference)

After you do all of this (it is a lot of work), then you should get a linear system. If you don't, then you did something wrong. Go back and check every step. The particular coefficient matrix that you end up with depends on the actual discretization scheme you use, whether you apply limiters, and all the little details. That's why the presentation omits it.

[daylen]

i was confused by the non linear terms of the momentum equations, but from what i read (versteeg) and your post they can be linearized, obtaining a linear system.


Thanks a lot.

[LuckyTran]

The way it works is you trade one non-linear PDE valid everywhere into system of algebraic equations that you solve over each cell. You linearize the non-linear thingy and achieve order reduction at the expense of increasing the order of the number of equations you have to solve from 1 to N (the number of cells). Perfectly balanced as all things should be.