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NoSlip Boundary Conditions for Pressure Equation in SIMPLE Algorithm 

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November 4, 2020, 11:55 
NoSlip Boundary Conditions for Pressure Equation in SIMPLE Algorithm

#1 
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Rafael March
Join Date: Mar 2020
Posts: 5
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Hello all,
I have implemented the SIMPLE algorithm to solve steadystate Stokes Equations in 3D structured grid. My Geometry is a cube. My boundary conditions are: 1) Fixed pressure at right and left boundary faces (call it x and x+) and noslip elsewhere. I have a question regarding the consistent formulation of noslip boundary conditions. For the momentum equations, it is trivial to implement noslip conditions: simply set ux and uy=0 at the y+, y, z+ and z boundary faces. However, what's the correct boundary condition for pressure in these boundary faces? Right now I'm simply assuming dP/dn = 0, which means a zero pressure gradient in at these boundary sides. Is this the correct assumption? I have seen some references e.g. [1] suggesting to project the momentum equation in the normal direction to get a boundary condition for pressure. What do you think? REFERENCES: [1] ON PRESSURE BOUNDARY CONDITIONS FOR THE INCOMPRESSIBLE NAVIERSTOKES EQUATIONS, GRESHO AND SANI, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 7, 1 1 1 1  1 145 (1987) 

November 5, 2020, 04:44 

#2  
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Filippo Maria Denaro
Join Date: Jul 2010
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Quote:
The BCs for the pressure equation have nothing to do with the noslip condition. They are derived from the mass conservation, for this reason the Neuman condition for the pressure is prescribed. The condition dp/dn=0 can be applied provided that the source term is congruently modified, this way the compatibility relation will be fulfilled and a solution exists (apart a constant). 

November 5, 2020, 05:24 

#3  
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Rafael March
Join Date: Mar 2020
Posts: 5
Rep Power: 4 
Quote:
Yes, when I mentioned the noslip condition, I wanted to make sure the compatibility relation is being fulfilled. Right now I don't have a source term in the pressure Equation. So I assume my set of BCs are not consistent. I notice that my code converges or diverges depending on the initial condition, which made me suspicious on my choice of BCs for the momentum and pressure equations. I would implement the consistent conditions for pressure like in the equations below. Note that I have a StokesBrinkman system, meaning that I have a term proportional to the velocity that vanishes at the boundary (noslip). I would apply these boundary conditions for pressure using the velocity values in the previous iteration. I guess my concern is that this condition does not have any information on the wall shear (tau_x = dux/dy). So I'm not entirely sure this is the proper way. https://imgur.com/ktaJoOY Could you please point me to a reference on how to find the proper compatible boundary conditions for the pressure equation to have a wellposed system? Thank you, Rafael. 

November 5, 2020, 06:03 

#4  
Senior Member
Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,095
Rep Power: 66 
Quote:
Here You can find many details about the derivativon of the pressure equation and the compatibility relation https://www.researchgate.net/publica...ary_conditions 

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boundary conditions, pressure poisson, simple algorithm 
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