Newbie: advantages/disadvantages of "penalty formulation" incompressible fluid flow

bzz77 · October 30, 2011, 14:32

Hello all:

I'm a chemist, interested in getting into some fluid flow modeling (e.g., advection-diffusion). I'm a newbie, so please forgive any silly questions.

Today I came across something new (the "penalty formulation") and I'd really appreciate some basic help or directions to online references in understanding (1) how it works and (2) what the advantages/disadvantages are compared to the method I'm familiar with--using conservation of fluid mass to derive an expression for the fluid pressure p and then getting the velocity profile from that.

Background:
From what I've read, using the penalty method allows elimination of pressure from the momentum equation so that the mass and momentum equations can be combined. The result is a convection-diffusion equation where velocity can then be calculated directly.

This seems kind of weird to me though. Is it a bit hokey to do this? What approximations are involved? I deal mainly with multiphase phenomena, but I so want to work on some single-phase flow too... Could it be that the penalty formulation stuff might be better suited to single-phase flow?

The first step in using the penalty method is to replace the continuity equation by:
1/e + grad u ...

I don't understand what e is. Where does it come from? The book I have just calls it a "coefficient with large values."

I guess what I'm getting at is, what ARE the advantages of the penalty method? I see that someone I respect has used it and I'm wondering why he would have chosen it over the pressure calculation method. Any advice would be very much appreciated.

praveen · October 31, 2011, 23:39

I have seen formulations in which

div(u) = 0

is replaced with

eps*p + div(u) = 0

Here eps is a small number. I think the advantage is that you dont have to solve a pressure poisson equation. You eliminate p from the momentum equation, solve for velocity, and then get p as

p = -(1/eps)*div(u)

For an analysis of the error introduced in this method see

http://www.math.purdue.edu/~shen/pub/Penalty.pdf

I am told that purists frown at such a method.

bzz77 · November 1, 2011, 03:32

Hi Praveen:

Thanks a lot for your help!

It is very helpful to know that not solving the pressure equation is why this method is used, and that purists don't like it.

Thanks very much for the reference.

October 30, 2011, 14:32	Newbie: advantages/disadvantages of "penalty formulation" incompressible fluid flow	#1
bzz77 New Member Join Date: Jan 2010 Posts: 28 Rep Power: 16	Hello all: I'm a chemist, interested in getting into some fluid flow modeling (e.g., advection-diffusion). I'm a newbie, so please forgive any silly questions. Today I came across something new (the "penalty formulation") and I'd really appreciate some basic help or directions to online references in understanding (1) how it works and (2) what the advantages/disadvantages are compared to the method I'm familiar with--using conservation of fluid mass to derive an expression for the fluid pressure p and then getting the velocity profile from that. Background: From what I've read, using the penalty method allows elimination of pressure from the momentum equation so that the mass and momentum equations can be combined. The result is a convection-diffusion equation where velocity can then be calculated directly. This seems kind of weird to me though. Is it a bit hokey to do this? What approximations are involved? I deal mainly with multiphase phenomena, but I so want to work on some single-phase flow too... Could it be that the penalty formulation stuff might be better suited to single-phase flow? The first step in using the penalty method is to replace the continuity equation by: 1/e + grad u ... I don't understand what e is. Where does it come from? The book I have just calls it a "coefficient with large values." I guess what I'm getting at is, what ARE the advantages of the penalty method? I see that someone I respect has used it and I'm wondering why he would have chosen it over the pressure calculation method. Any advice would be very much appreciated.

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October 31, 2011, 23:39		#2
praveen Super Moderator Praveen. C Join Date: Mar 2009 Location: Bangalore Posts: 342 Blog Entries: 6 Rep Power: 18	I have seen formulations in which div(u) = 0 is replaced with epsp + div(u) = 0 Here eps is a small number. I think the advantage is that you dont have to solve a pressure poisson equation. You eliminate p from the momentum equation, solve for velocity, and then get p as p = -(1/eps)div(u) For an analysis of the error introduced in this method see http://www.math.purdue.edu/~shen/pub/Penalty.pdf I am told that purists frown at such a method.

November 1, 2011, 03:32		#3
bzz77 New Member Join Date: Jan 2010 Posts: 28 Rep Power: 16	Hi Praveen: Thanks a lot for your help! It is very helpful to know that not solving the pressure equation is why this method is used, and that purists don't like it. Thanks very much for the reference.