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April 16, 2012, 13:11 
SIMPLE algorithm for twophase

#1 
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I'm writing a computer code to model a twophase flow. I have read Patankar's book, "Numerical Heat Transfer and Fluid Flow", which discusses the SIMPLE algorithm. It was a great read, but he doesn't talk about twophase flows at all. So far, I have setup my momentum equations in order to obtain my guessed velocity field, but I'm having trouble on what to do to setup the pressure correction equation. For a onephase flow, you use the momentum equation to put the velocity correction, u', in terms of the pressure correction, p'. Then you put this into the continuity equation along with your guessed velocity field results in order to form a pressure matrix for your mesh that can be solved to get the corrected pressures. But for my twophase flow, I have 2 continuity equations instead of one, which are coupled together by the void fraction. I assume that pressure is the same in the two phases. So now how do I arrive at the pressure correction? Do I need to solve both continuity equations simultaneously, using the olditeration void fractions for my convection terms, in order to simultaneously calculate the pressure corrections and void fractions? And what about the energy equations? They also have the void fraction terms present, so do I also need to solve them simultaneously with the continuity equations as well? I would appreciate any advice and also any information on books dealing with applying SIMPLE to twophase flow because I've had trouble finding anything of the sort.


April 17, 2012, 09:22 

#2 
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To form the pressure correction equation, I considered total volume continuity, so that I only have one continuity equation. In other words, I consider all mass convection into the cell, be it vapor or liquid, and the change in total mass over time (meaning change in vapor AND liquid mass). If you consider total volume, there is no source term due to evaporation or condensation. This allowed me to form the pressure correction equation. I use density and void fraction explicit from the last iteration to form the pressure correction. This allows me to correct the pressure field and the velocity field, but now I need to solve for the remaining dependent variables, which are the void fraction and the phase enthalpies. I'm stuck here because I can see that I have 2 energy equations and 2 continuity equations. However, I only have 3 dependent variables  void, liquid enthalpy, and vapor enthalpy. So it seems like an overdefined problem to me.
I picked up a paper from the 2000 Proceedings of the ASME Fluids Engineering Division, Summer Meeting, entitled "A Phase Coupled Method for Solving Multiphase Problems on Unstructured Meshes", which explains an algorithm, PCSIMPLE (phasecoupled SIMPLE). It confirms my use of total volume continuity to form the pressure correction equation, but I'm still at a loss on how I should now get my void fraction and enthalpies out of the remaining conservation equations. 

April 17, 2012, 11:53 

#3 
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My searching has lead me to a paper by Oliveira and Issa, "Numerical aspects of an algorithm for the Eulerian Simulation of TwoPhase Flows". It discusses two methods for handling the "overdefinition" that I discussed. The first method is to solve just one of the continuity equations using the updated (corrected) velocities and then simply use (1\alpha) to get the void fraction of the other phase that the continuity equation was not solved for. The second method is to solve both equations, but for two new, introduced variables \alpha_l* and \alpha_v*, which are preliminary void fractions for liquid and vapor phases. The actual liquid and vapor voids are then obtained using a correction factor, f, as follows:
\alpha_l = f x \alpha_l* \alpha_v = f x \alpha_v* The f correction factor is defined as f = 1/(\alpha_l* + \alpha_v*). This allows the physics of both phases to be captured, but also ensures that the void fractions add up to 1 at each iteration. When convergence of the solution is achieved, f should be equal to 1. 

April 20, 2012, 05:45 

#4 
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Yes what you mention is a very tricky problem !!! and papers on this topic are very rare.
Especially when you deal with 3 different phases.. Check the serie of papers from Darwish and Moukalled. They have proposed several approaches to deal with such problem Good luck ;) 

April 20, 2012, 09:02 

#5 
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I looked up some of their papers and they look like they're right along the lines of what I was looking for. Thanks for the suggestion.


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simple algorithm, twophase flow 
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