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December 23, 2015, 16:40 
Pressure correction

#1 
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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 incompressible steadystate, 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_wdensity*u_e)*area+(density*v_sdensity*v_n)*area I do not know why I keep getting near singularity. Any help would be appreciated. 

December 23, 2015, 16:46 

#2 
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Filippo Maria Denaro
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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 

December 23, 2015, 17:00 

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December 23, 2015, 17:15 

#4  
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Filippo Maria Denaro
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Quote:
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 

December 25, 2015, 18:49 

#5  
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Kaya Onur Dag
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Quote:
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? Quote:


December 26, 2015, 04:13 

#6  
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Filippo Maria Denaro
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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 

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