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October 16, 2015, 16:41 
Split the load of the Buoyancy Source term

#1 
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Marcel
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I am testing currently a 2d incompressible code with natural convection in a square cavity. To include the natural convection part I implemented the Boussinesq approach, indicating the addition of the source in the NSy.
As of right now I am using the SIMPLE algorithm for pressurevelocity coupling and the code seems to be working fine, for lower Rayleigh numbers. On higher Rayleigh numbers however (Ra>1e5) the convergence is tediously slow, or not converging at all. I have some ideas to handle this, but they don't include any charming solutions, and I don't want to indefinitely underrelaxate my variables. I was thinking now, maybe it is possible to linearize the source term in such a way that I divide the load of the source term in my NSy, by solving it partly explicitly and implicitly, depending on linearization of the source term. As of right now, I handle Picard's method (S = Sc + SpTp), where Sc determines completely my natural convection and thus Sp=0 and thus solving the natural convection completely explicitly. Now I would like to find out if it is possible to reduce the load on Sc by incorporating a part of the load in Sp to come up with the same solution and by that solving partly implicitly. The idea behind is that I would like to obtain a more diagonal dominant system matrix, as the source term right now is pulling it out of balance. It should contribute to more stability, although it would probably not contribute to a faster convergence. Anyone knows if this even is possible? Or has any experience with it? I would really like to find out. Any literature on this would be welcome too. 

October 16, 2015, 17:03 

#2 
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Filippo Maria Denaro
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high Ra number flows becomes transitional, the fact you don't get a steady solution is coherent to the physics of the problem.


October 16, 2015, 17:21 

#3 
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Marcel
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although I understand that its not that black and white but the transition zone is usually close to Ra~1e9 and I run simulations up to a Rayleigh of 1e8, still in the 'laminar' regime.


October 16, 2015, 17:54 

#4  
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Filippo Maria Denaro
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Quote:
yes, authors still consider laminar the flow at Ra=10^8, but also if the flow is laminar, the breakdown of the single flow structure happens at Ra=10^6 and if you run an unsteady simulation you will clearly see the onset of minor structures. Reaching an equilibrium state at such configuration need a very long time and small oscillations still are present in the small structures. A very refined grid is required to allow dissipation to act. Therefore, running a steady equation system and getting no convergence to steady solution, can be at high Ra number a physical indicator 

October 17, 2015, 20:27 

#5 
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Marcel
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ok granted that it would be a physical problem, would I be able to 'cheat' by using a first order spatial interpolation scheme, like the first order upwind, to find a steadystate solution covered with artificial numerical diffusion and then use a higher order interpolation scheme to calculate a more reasonable solution?
and out of curiosity, still the question remains of splitting the load in the source term? Anyone knows if this is possible? And/or has some literature on it? 

October 18, 2015, 04:21 

#6  
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Filippo Maria Denaro
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Quote:
well, if you use the trick of a strong dissipative scheme like firstorder scheme your steady solution will be simply not accurate and you cannot recover the accuracy by a successive interpolation... furthermore, the Bousinnesq model is already based on a linear expansion... 

October 18, 2015, 11:51 

#7 
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Marcel
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ok thanks sir!
and my code is indeed converging for a finer grid, but yeah, i need a lot of patience to finally be able to postprocess Also, I am aware the source term is linearized... excuse me, perhaps I am slow in understanding but does that really answer the question? I simly want to try to incorporate a part of the buoyancy load directly into the discretized equations and by that reducing the impact of the explicit source term. I will probably try to incorporate it one of these days to see if it is possible, but I just wanted to read a little bit before starting doing it. 

Tags 
boussinesq approximation, cavity flow, code testing, picard's method, source term 
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