Pressure correction
 

Hi everybody,

I am not a cfd expert, I am writing a code that can solve the flow of a lid driven cavity. I am doing this in in-compressible steady-state, using the simple algorithm. When I inverse and multiply my pressure correction matrix with my source terms, I get a near singularity. My equations are:

P'=pressure correction.
Ap*P'p=Ae*P'e+Aw*P'w+An*P'n+As*P's+Su
Su=(density*u_w-density*u_e)*area+(density*v_s-density*v_n)*area

I do not know why I keep getting near singularity.
Any help would be appreciated.

The pressure equation Div Grad p = q with proper BC.s leads to an algebric system that has indeed a singular matrix.
However, when the BC.s satisfies the compatibility relation, you get a solution apart from a constant value

Firstly thank you. Sorry I do not know to much about CFD yet, what do you mean by "when the BC.s satisfies the compatibility relation"?

Actually, this is not a CFD issue but a general mathematical aspect of linear algebra.
Consider the equation for pressure and integrate both side over the computational domain

Int [V] Div Grad p dV = Int [V] q dV

use Gauss

Int [S] n.Grad p dS = Int[V] q dV


The compatibility relation must be fulfilled by the boundary condition you prescribe on the frontier of the domain

n.Grad p = f

so that you have to fulfill the condition

Int [S] f dS = Int[V] q dV

The above relation must be satisfied in discrete sense after discretization.
Only this way, despite the singular matrix, you get one solution (apart a constant).

For further details you can see in literature, for example

https://books.google.it/books?id=lXB...matrix&f=false

I am very bad at these finite volume notations to me they all are gibberish so do excuse me :)

is this poisson equation, is [Ap] the laplace or grad.div. operator?
you may need to set a mean for your correction pressure such as setting 1 single value manually by modifying your matrix and your source (right hand side) otherwise to my knowledge your matrix system will be under determined.

In a FV formulation, A*p should be the discretization of (1/|V|) Int [S] n.Grad p dS.
However, you could find also several researchers considering Div Grad p. For the pressure equation it is not suitable to discretize the laplacian operator.

The correction to the matrix entry and to the source term is consequent to the proper setting of the BC.s in such a way they satisfy the compatibility constraint