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Reasons for decoupling the pressure and velocity computations 

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April 9, 2017, 01:21 

#21 
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Lucky
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Hey folks, some food for thought...
To begin with, this pressurevelocity coupling problem starts exactly because we try to decompose the stress tensor into a deviatoric part and nondeviatoric part. Put another way, why do we even solve the NavierStokes equations and not the generalized Cauchy momentum transport equation? In even more obvious language, why do we solve for pressure & velocity and not just velocity and then reconstruct the pressure field from the nondeviatoric part of the stress tensor? We already implicitly tried to decompose/decouple pressure out of the stress tensor (for a very good reason, because we need pressure for many things). We tend to treat the pressure/velocity fields as potentials for each other only in the inviscid case. And we say that unless it is a potential flow, that pressure & velocity are always coupled. However, don't forget that pressure can be interpreted as just a part of the stress tensor. This becomes even more painfully obvious if you drop NavierStokes and try to include surface tension. In a way, we have set up the problem such that it is extremely favorable in many situations to be solved in a segregated manner, because we are looking for this decomposed part of the stress tensor (the pressure). 

April 9, 2017, 04:28 

#22 
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Filippo Maria Denaro
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To tell the truth, I am not sure that the Newton law can be still extended to the isotropic part in such a way to get a relation only with the velocity.
In any case, for incompressible flow, we denote with "pressure" something that has philosophically nothing to do with the extraction of the isotropic part of the stress tensor. It could be denoted as a,b,c,d functions... The only reason to introduce the gradient of such function is to take into account of the divergencefree constraint and the additional scalar function acts as a lagrangian multiplier. 

April 9, 2017, 05:21 

#23  
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Lucky
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April 9, 2017, 05:34 

#24 
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Filippo Maria Denaro
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but still, we have a coupling with this additional function.... The only way I could think is to project the momentum along the velocity vector field, integrate over the domain and only if Int[S] n.vp dS=0 you get the weak form where the "pressure" disappear...


April 9, 2017, 12:02 

#25 
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Michael Prinkey
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We are in the weeds, kind folks. 8)
My view is that incompressible flow is a nonphysical, but truly functional, approximation to a compressible flow with high speed of sound. I think of it as an asymptotic expansion with rho(p) => rhoConstant and c > inf. In my mind, incompressible flow...for all of its simplicity, it not truly fundamental in the same way that, say, the Boussinesq approximation is also simple and useful, but also not fundamental. Compressible flow is properly local, as required by phenomena described by differential equation. By introducing an infinite speed of sound, we've broken locality in a significant way. Now, minute deviations in the divergencefree velocity field in one area can instantly effect the flow in the entire rest of the domain. When systems are no longer local, I view them as fundamentally requiring an integrodifferential expression to model them. And the incompressible pressure term is just such an expression. It can be viewed as the Green's function for the Laplacian times div(u) integrated over the entire domain. We don't evaluate it that way for efficiency reasons, but one could write the incompressible momentum equation without pressure using that form, and I'd argue that it is irreducible from there without some explicit simplifying assumptions. But, I am truly just a humble CFD coder. My mathematical analysis day are two full decades behind me. 

April 9, 2017, 12:12 

#26 
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Filippo Maria Denaro
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yes, you are right about the mathematical modelling that simplifies the real case of a very low compressible flow
In my mind, the role of the coupling with the "pressure" is of mathematical nature. If you work with the pointwise differential form you need two PDE equations and two unknowns to close the problems (v,p). Using the weak form you can eliminate the pressure fro the resulting integrodifferential equation but, while we know that a weak solution could exist, what about the unicity? From the weak form is a classical issue to determine multiple solutions... 

April 9, 2017, 12:39 

#27 
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Michael Prinkey
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The boundary conditions on our "pressure" variable are baked into the Green's function. That Green's integral result gives us the "pressure," so the momentum equation will use the gradient of it. That eliminates any gauge constant (for the case of no fixedpressure boundary conditions). The resulting momentum equation is still subject to our normal velocity boundary conditions. That should lead to a properly closed system and a unique solution. Again, this is based more instinct than hard analysis. I am happy to be corrected.


April 9, 2017, 12:58 

#28 
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Filippo Maria Denaro
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At the best of my knowledge, the weak solution is not unique, despite fixing the BC.s
Actually, that should be a general problem of the NS equations also for the compressible formulation. In a very poor example, you cannot find a unique function that provide the known value of the Int[a,b] f(x) dx even if you know f(a) and f(b). 

April 9, 2017, 13:04 

#29 
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Michael Prinkey
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Indeed, you are right.
http://www.claymath.org/sites/defaul...vierstokes.pdf If you solve it, the Clay Institute has a million dollars for you. 

April 9, 2017, 14:30 

#30  
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Lucky
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Uniqueness is another million dollar problem.
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Once again the Cauchy momentum equation gives you one equation with two unknowns, the velocity u and the stress tensor. Introduction of the constitutive Newtonian/Stokes fluid model allows you to rewrite the stress tensor in terms of grad(U). So you can say that now you have one equation with one unknown u. The continuity equation is a constraint on u (really the momentum). The incompressible condition is yet another constraint. I am not arguing that pressure is decoupled. It should be obvious, that if you try to do this decomposition on u into u & p that p must definitely be coupled to the momentum equation because that's where it came from. My point is, you can say that p is coupled to the momentum equation, or you can take a step back and just say we have a momentum equation. So why did I even bother mentioning any of this? So the original question was why is a predictorcorrector approach a good idea & why is it a bad idea to solve the coupled problem? My analogy, which has already opened a can of worms, is that well why was it a good idea to decompose the stress tensor into a deviatoric and nondeviatoric part in the first place? Why don't we just solve the more general Cauchy momentum equation? We want to solve for a p and that gives rise to certain problems that we must now deal with because of our own bias. 

April 9, 2017, 14:41 

#31  
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Arjun
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The one variable one equation scenario is lovely except that we don't know practical way of solving this one equation. I guess the discussion is about the practical part of solving these equations other wise we shall look into quantum aspects too and see if this one equation is really one equation or in some case manifests as multiple equations at those scales. Even after so many things solving navier stokes is not cheap, one shall remember turbulence models are another addition for our convenience and don't really part of navier stokes and flow. They are there to help getting solution that could be used in practical sense. 

April 9, 2017, 14:46 

#32 
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Arjun
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I wish to add the reasons why one should invest in coupled solver because this was not mentioned before.
The main case for coupled solver is in its robustness and it is in the fact that it is less sensitive to under relaxation factors. Segregated solvers are nice and cheap but sometimes you end up with cases difficult enough that segregated approach does not cut it. There you have no choice but to go with coupled solvers. There are many such problems where one can end up. This is why coupled solvers remain important class of solvers. One should know where to use which solver and should not dismiss one for other. 

April 9, 2017, 18:05 

#33 
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Filippo Maria Denaro
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Again, we should not forget that the decoupling between velocity and pressure leads to generate an error that is not present in the coupled counterpart system. And this error is irrespective of the order of spacetime discretization but is inherent to the splitting. Often, this error is not sufficiently highlighted but can affect the boundary layer resolution.


April 10, 2017, 00:39 

#34 
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Michael Prinkey
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And because I cannot stop myself from further muddying the waters, I would like to point out that there is a third way that has not seen a lot of attention here. That is using a segregated/fractional step method to precondition a proper NewtonKrylov iteration of the entire nonlinear system instead of just a linearized coupled form. The preconditioner may suffer from all of the missing crossterms and the splitting errors that they always do, but the Newton correction is always driven by the full nonlinear residual projected into that Krylov space.
Incompressible flow seems like it might be a better match for NK schemes than fully compressible flows. Newton methods are less forgiving when converging to solutions with real discontinuities like shocks. Incompressible flow solutions are usually safely differentiable, at least up to the second derivative. I don't want this to evolve into a "why don't we use NK instead of SIMPLE/PISO" discussion, but I did want to point out that there is another algorithmic approach that is *potentially* more coupled in a nonlinear sense than linearized [u,v,w,p] coupled matrix solvers. And NK is *potentially* able to offer quadratic convergence rates for periteration costs only slightly higher than SIMPLE/PISO. 

April 10, 2017, 01:47 

#35  
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Arjun
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I believe others have also tried it and there is literature around it. One can note that if one uses unpreconditioned gmres then to solve this schur all one need is matrix vector product. Which can be done without actually constructing it. If then one wishes to precondition this gmres then all one need is preconditioner that is also based on matrix vector product. This is what is going to be my viscoelastic flow solver for wildkatze when i implement. Its on card so in an year you would see it in action too. 

April 10, 2017, 02:01 

#36  
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Michael Prinkey
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I wasn't really trying to tout NK as the best of the bunch. I think it still has potential, but I fully recognize that the residual calculation and orthgonalization costs could STILL make it slower than good old SIMPLE. I just think that it belongs in the conversation with linear coupled solvers. 

April 10, 2017, 11:41 

#37  
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Lucky
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Btw, Paolo hinted at it but I would like to really highlight it even further. Parallelization and scalability is a very important problem in computing or applied math, and you don't really see this problem when you focus on only the theoretical mathematics. Divide & conquer approaches are naturally wellsuited for parallel computations. 

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poisson equation, pressure and velocity, pressure correction 
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