Especially in 1D cases (simple one), is it then better to use an Energy Conversation equation?
Is there any additional explanation, as I found if I include Navier-Stocks equations, then either Poisson's or continuity equation turn to be wrong? I would appreciate the opinion what is normally used for this case? Thanks in advance!
Using Poisson equation is somewhat less popular approach, but is still applicable. See the Helmholz decomposition approach in Bernard & Wallace's (2002) "Turbulent Flow - Analysis, Measurement and Prediction". (http://books.google.com.sg/books?id=N44_L3SCN6EC&lpg=PP1&pg=PA97#v=onepage&q&f=false). An implementation is in Popinet's (2003) paper (http://pages.csam.montclair.edu/~yecko/icodes/Popinet_GERRIS.pdf)
Dear Novena, yes, it is possible, as the colleague Nguyen said. I just add the the flow (Navier-Stokes equation) is as simple as the irrotational flow (Laplace equation). If both are linear, then they support the superposition.
Energy eq is used when you have energy exchange to consider such as the heat transfer and gasdynamic. Otherwise, you should not replace the momentum equation/Poisson eq/continuity eq with the energy equation.
The common issue in incompressible flow is the lack of equation that links the pressure and density. Therefore in DNS on incompressible flows, the pressure Poisson's equation is solved in connection with continuity equation to ensure that the flow is divergence free ( the Poisson's equation can be obtained by splitting pressure from NS equation ). But It is not always possible to get the solutions with very small errors.
The Poisson equation for the pressure can be obtained from the momentum equation by enforcing continuity (take the divergence of the momentum balance equation), and, as such, replaces the continuity equation div(u)=0. There is no need to solve both as they are not independent. (Then the actual solution depends on the numerical discretization used).
For solving time-dependent incompressible fluid-flow problems, the projection method use a poisson equation for the pressure term to decouple the momentum equation. It replaces the continuity equation.
I am thankful for all your help!! Dear Nguyen, thank you for the sent links. These papers will surely help. Actually, I wanted to avoid the effect that my velocity depends on the chosen time step, which surely happens when NS equations for 1D are introduced. Now I have the idea on my mind to cope with. Also, thanks to you, Michaele and Shiuh-Hwa for the explanation and suggestion!
For Incompressible flows As you Can seen from above comments, NO. Since in incompressible flows continuity equation is just div(V)=0 and poisson equation is a mathematical way to close the system of equation and enforce velocity field to be divergence free. Also when in incomp. flows rho=rho(X,Temp,t) the variations is included in poisson equations ( see for example sec 8.7 of "an introduction to computational fluid dynamics .. by versteeg and malalasekra).
Thanks Mohsen!
I have already implemented NSE + Poisson, and Bernoulli + Continuity equation. And it works just fine ibn both cases!
Nevertheless, I am interessed, if someone of you did the simulation with Newtonian and non-Newtonian fluid (concerning shear stress in NSE) and if yes, if the difference is big?
@Ph. Mandin: What do you mean by equation with UDS?
the link to Popinet's (2003) paper is not working for me and I am wondering if you can give me the full name of the paper or its citation so I can find it.
@ Reza: Popinet S., Geriss: a tree-based adaptive solver for the incompressible euler equations in complex geometry, J. comput. phys. 190 (2003) 572-600
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