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specify a reference pressure point in FD method? 

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February 26, 2008, 16:24 
specify a reference pressure point in FD method?

#1 
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Hi everyone,
I had a problem to specify a reference pressure point in finite difference method cases. We are use laplacian(P)=div(U)*rho/dt to calculate the pressure. In a fully wall bounded case, we allpied homogeneous Nuemman boundary condition at the wall for the Poisson equation. But because it is a fully wallbounded case, we cannot get a unique results without specify a reference pressure point in the domain. We can do some operation to the Laplacian coeffecient matrix to get a specified pressure value at the reference point. But how we can update the pressure value at this point? Any reply will be highly appreciated! Cheers, Rui 

February 26, 2008, 20:53 
Re: specify a reference pressure point in FD metho

#2 
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At one node on your grid, replace the finite difference approximation to the LaPlace equation with P = P_reference.
In other words, you don't update it, you freeze it. For direct methods, I think you must do this to avoid a singular coefficient matrix. For an iterative solution (relaxation for example), you'll obtain a value for the reference node, p = p_rn. Use this to correct the entire pressure field by p_node = p_reference  p_rn . You'll probably get other suggestions ... 

February 27, 2008, 04:05 
Re: specify a reference pressure point in FD metho

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Thank you very much, otd! Freeze it? You mean using the matrix, changing the diagonal entry into 1, and other entry into 0? But in this way, when I calculate the pressure gradient, I will have trouble. I need the pressure gradient to correct the velocity, which is the projection method to couple velocity and pressure.
For an iterative solution, what is it? Can you provide more details?? 

February 27, 2008, 08:13 
Re: specify a reference pressure point in FD metho

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Change the diagonal to 1, the RHS to P_reference (which could of course be 0). Then use the calculated value (P_reference) at that point for your corrections.
I'm not on top of the latest relaxation schemes (including multigrid). My experience is that relaxation is tolerant of what would be a singular matrix. You can learn a lot about these things with small 2d heat conduction simulations. Numerical analysis books (maybe online course notes) will tell you a lot about relaxation schemes, which you can try out on a PC with a relatively small investment of your time. When you're done, you'll know as much as most of the rest of us! Good luck. 

February 27, 2008, 09:04 
Re: specify a reference pressure point in FD metho

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Thank you very much!!!!!


February 28, 2008, 06:59 
Re: specify a reference pressure point in FD metho

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But the thing is I am using projection method to update velocity from pressure field, which means I calculate a intermedia velocity, U*, from momentum equation, but without pressure gradient, and then solve poisson euqation to get the pressure, afterwards use the pressure gradient to correct U* to get U(n+1).
So if I cannot update the pressure at the reference point, I cannot get an accurate U(n+1). The error will turned to be very big, even I use interpolation to update the pressure very time step. Is there any other way to update the pressure at the reference point?? Many thanks in advance! 

February 28, 2008, 10:42 
Re: specify a reference pressure point in FD metho

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In my experience, the pressure equation can be solved using iterative methods such as SOR (successive overrelaxation), ADI (alternating direction implicit) WITHOUT pinning a reference pressure. What counts is the gradient (numerically the difference in pressure between neighboring nodes).
I haven't tried multigrid but someone in the cfdonline community surely has. My level of linear algrebra experience (solution of simultaneous equations using direct solvers) requires passing a nonsingular matrix the solver (pinning one value for instance) to obtain a unique solution. Perhaps someone else on the forum has some suggestions for you. 

February 28, 2008, 10:54 
Re: specify a reference pressure point in FD metho

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Yes. But in FD, for a fully wallbounded case, if we do not specify a pressure reference point, the linear problem will be illconditioned. Poisson equation is a second derivative equation, but at the boundary, we only have dp/dn = 0, the homogeneous Neumann boundary condition. So we need a reference point to get a unique result.
I think we are talking about different things, otd. thank you very much for the suggestion!!! 

February 29, 2008, 09:48 
Re: specify a reference pressure point in FD metho

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I've assumed that you're interested in incompressible flows here  you're not solving for energy transport and an equation of state. In particular, you have a fluid/flow combination in which density variation is negligible. Is this the case?
If so, the pressure LEVEL is arbitrary. Only the pressure GRADIENT is unique. To show this, assume that you calculate a pressure field that satisfies the Poisson equation grad p = D*rho/delt. Call that solution p*(x,y). Add ANY constant to that, PP(x,y) = p*(x,y) + p_const. Because grad p_const = 0, PP(x,y) also satisfies the Poisson equation. If you are comparing to data (a tranducer reading at one point in the flow for example), use that to determine p_const. If in fact you have a compressible fluid/flow problem, that's a whole different discussion. 

February 29, 2008, 12:34 
Re: specify a reference pressure point in FD metho

#10 
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But the thing is for finite difference method, in a fully wallbounded domain, you cannot solve the Poisson equation without any Dirichlet boundary condition. So we need specify a point with a guessed pressure. The problem arises here that how we can update the pressure at this point?


August 30, 2010, 08:50 
Did you find out what to do with the reference pressure point?

#11 
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Hello,
I followed your discussions. I have exactly the same problem with the Poisson equation of the projection method for incompressible NavierStokes. @Rui: How did you solve the problem? Thanks in advance 

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