# No-Slip Boundary Conditions for Pressure Equation in SIMPLE Algorithm

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 November 4, 2020, 11:55 No-Slip Boundary Conditions for Pressure Equation in SIMPLE Algorithm #1 New Member   Rafael March Join Date: Mar 2020 Posts: 5 Rep Power: 4 Hello all, I have implemented the SIMPLE algorithm to solve steady-state 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 no-slip elsewhere. I have a question regarding the consistent formulation of no-slip boundary conditions. For the momentum equations, it is trivial to implement no-slip 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 NAVIER-STOKES 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
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,061
Rep Power: 64
Quote:
 Originally Posted by rafaelmarch3 Hello all, I have implemented the SIMPLE algorithm to solve steady-state 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 no-slip elsewhere. I have a question regarding the consistent formulation of no-slip boundary conditions. For the momentum equations, it is trivial to implement no-slip 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 NAVIER-STOKES EQUATIONS, GRESHO AND SANI, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 7, 1 1 1 1 - 1 145 (1987)

The BCs for the pressure equation have nothing to do with the no-slip 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
New Member

Rafael March
Join Date: Mar 2020
Posts: 5
Rep Power: 4
Quote:
 Originally Posted by FMDenaro The BCs for the pressure equation have nothing to do with the no-slip 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).
Hi Filippo, many thanks for your reply.

Yes, when I mentioned the no-slip 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 Stokes-Brinkman system, meaning that I have a term proportional to the velocity that vanishes at the boundary (no-slip). 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 well-posed system?

Thank you,
Rafael.

November 5, 2020, 06:03
#4
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,061
Rep Power: 64
Quote:
 Originally Posted by rafaelmarch3 Hi Filippo, many thanks for your reply. Yes, when I mentioned the no-slip 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 Stokes-Brinkman system, meaning that I have a term proportional to the velocity that vanishes at the boundary (no-slip). 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 well-posed system? Thank you, Rafael.

Here You can find many details about the derivativon of the pressure equation and the compatibility relation
https://www.researchgate.net/publica...ary_conditions

 Tags boundary conditions, pressure poisson, simple algorithm

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