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Nature of Pressure Correction Equation

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Old   January 5, 2001, 01:35
Default Nature of Pressure Correction Equation
  #1
Abhijit Tilak
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hi all,

my question is about nature of boundary conditions for pressure correction for simple class of methods. P' equation is solved with Neumann (derivative) boundary conditions ie dp'/dx=dp'/dy=0. with Neumann boundary condition on all boundaries. isn't this an ill-posed problem ? you must have at least one dirichlet (value specified) B.C on atlest one of the boundaries. even with this so called ill-posedness the p' equation has a unique solution. How is it so? can anyone throw some light on physical/mathematical interpretation on this.

thanks Abhijit tilak.
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Old   January 5, 2001, 07:33
Default Re: Nature of Pressure Correction Equation
  #2
Sebastien Perron
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1) If normal velocity is specified on all boundaries, than you end up with dp'/dn=0 everywhere. For this case, The system of equations has more than one solution and can't be solved with a direct solution method. But, this is not a problem for a an iterative solver, it will converge to one of the solutions. Furthermore, for this case, it not a good idea to specify p' at one point, it will incease the solution cost.

2) Dont forget that for incompressible flow, the solution for p is only seaken to additive constant. If p' is specified at on point, the constant will the be specified.

It is not the first time I give this article as a reference, but it is a must read for those who want some info about the simple (simplec) algorithm, read it:

J.P. Van Doormaal and D.D. Raihtby. Enhancements of the simple method for predicting incompressible fluid flows.

Numerical heat transfer, vol 7 pp147-163, 1984.

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Old   January 5, 2001, 13:36
Default Re: Nature of Pressure Correction Equation
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kalyan
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I thought that the pressure boundary condition at the outflow is Dirchlet in SIMPLE type solvers. Correct me if this is incorrect since I have never used SIMPLE based codes.

Even with all Neumann conditions, the elliptic equation would converge to within an additive constant. So, you have check for the convergence of the pressure gradient field rather than the pressure itself. So p(i) (i=some spatial location) can change rapidly during iterations, but if p(i-1) changes as rapidly, then the gradient [p(i)-p(i-1)]/dx would converge if the solver is convergent.
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Old   January 5, 2001, 14:01
Default Re: Nature of Pressure Correction Equation
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Sebastien Perron
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You don't always have an outflow boundary. For exemple, the liddriven flow in a square cavity is a problem with only neumann boundary conditions for p'.
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Old   January 5, 2001, 14:52
Default Re: Nature of Pressure Correction Equation
  #5
Adrin Gharakhani
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> I thought that the pressure boundary condition at the outflow is Dirchlet in SIMPLE type solvers. Correct me if this is incorrect since I have never used SIMPLE based codes.

Maybe I misunderstand the question that was raised, but p' (pressure correction) is not the same as p (the pressure itself). Check the book by Patankar, for example, for reference.

Adrin Gharakhani
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Old   January 5, 2001, 15:43
Default Re: Nature of Pressure Correction Equation
  #6
kalyan
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p' is not the same as p but the BC for p' are determined by the pressure BC. If a Dirchlet BC is used for p, then p' would also have a Dirchlet BC.

There isn't necessarily an outflow boundary is some problems as Sebastien pointed out and hence one is left with solving for p' with all Neumann conditions.
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Old   January 5, 2001, 20:20
Default Re: Nature of Pressure Correction Equation
  #7
Ha Lim Choi
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There are generally two types of boundary conditions, pressure boundary condition and velocity boundary condition. When there is a pressure boundary condition, the pressure correction equation becomes Dirchlet type since p' is zero there. When velocity boundary conditions are specified at all boundaries, the pressure correction equation becomes Neuman type. As you explain well in Neuman boundary condition case, the solution of pressure correction equation cause no problem either cases.

Best regards

Ha Lim Choi
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Old   January 8, 2001, 00:55
Default Re: Nature of Pressure Correction Equation
  #8
K.Senthil
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hello all, now after going thro'u the disscussion iam slightly confussed. Iam solving flow over an obstacle in a channel.i use velocity b.cs only.ie., no slip at the upper and lower walls of the channel and on all the walls of the obstacle with convective out flow condition.

regarding pressure i donot fix pressure any where and also donot make the pressure derivative zero at the channel walls.i get good results with this. will the solution improove by putting some kind of pressure b.c. thanks in advance. senthil
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Old   January 8, 2001, 00:57
Default Re: Nature of Pressure Correction Equation
  #9
K.Senthil
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hello all, fogot to specify i use MAC algorithm. any way i need clarification . can any on e help. byee senhtil
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Old   January 8, 2001, 12:04
Default Re: Nature of Pressure Correction Equation
  #10
Jim Park
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You should check the MAC documentation,

F. H. Harlow, J. E. Welch, J. P. Shannon, and B. J. Daly, "The MAC Mechod," Los Alamos Scientific Lab, Rep. No. LA-3425 (1965),

or the corresponding journal article,

Francis H. Harlow and J. Eddie Welch, "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface," The Physics of Fluids, v. 8, no. 12, p. 2182-2189 (1965).

The pressure boundary conditions are completely determined through the conditions imposed on the velocity as specified in the documents.

The Los Alamos group quickly discovered that the computations could be simplifed greatly by using the SMAC (Simplified MAC) modification of MAC. That is described in

Anthony A. Amsden and Francis H. Harlow, "The SMAC Method: A Numerical Technique for Calculating Incompressible Fluid Flows," Los Alamos Scientific Laboratory, report LA-4370 (1970),

and its journal equivalent,

Anthony A. Amsden and Francis H. Harlow, "A Simplified MAC Technique for Incompressible Fluid Flow Calculations," Journal of Computational Physics, v. 6, p. 322-325 (1970).

To learn more about this work and the 30 years of incompressible flow work at Los Alamos since MAC, check http://gnarly.lanl.gov/Home.html.
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