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March 13, 2005, 12:29 
How to enforce the continuity equation?

#1 
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1)Q1:
For pure implicit method,the simple method is always employed to decouple pressure and velocity, mass conservation is enforced only at the end of the outer iteration. My question is : When the convective term is treated explicitly (so there is no nonlinearity here),and viscous term implicitly,then how to enforce the continuity equation? Is the following right? a)calculate u* form the previous pressure p0. b)solve pressure correction p' from u* c)correct p=p0+p' d)use p0=p go to 1) again until the convergent solution is reached. Is that right? 2)Q2: Rungekutta methods are always explicit,how to enforce continuity using RK method? Any comment or referrence is greatly appreciated. 

March 13, 2005, 13:25 
Re: How to enforce the continuity equation?

#2 
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You forgot two things:
1) After solving the pressure correction equation, correct the face fluxes with the calculated pressure gradient  it is the pressure equation that guaranteen that your fluxes will be conservative 2) You will need some underrelaxation  explicit for p and built into the equation for U (if steady, for p alays the same). Enjoy, Hrv 

March 13, 2005, 20:59 
Re: How to enforce the continuity equation?

#3 
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thanks dear Hrv But since the convective term is treated explicitly,so the coeff of this term is determined only by the face fluxes of previous time step,while the viscous term doesn't use fluxes at all. So why do we correct the face fluxes ? It seems we don't use it at all.


March 14, 2005, 02:55 
Re: How to enforce the continuity equation?

#4 
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OK, you are doing Finite Volume, right? Therefore, the convection term is evaluated by multiplying the face flux by the face value. So:
 for the passive scalar theta it would flux F times theta on the face: F*\theta_f  for velocity it would be flux times the velocity on the face: F*\U_f In both cases the "on the face" bit is calculated using your favourite differencing scheme but the face flux is always used! Implicit and explicit:  for implicit, you will express theta on the face as alpha times theta left plus beta times theta right (and maybe fiddle with it a bit more) and then get the matrix coefficients as alpha times the flux and beta times the flux: \theta_f = \alpha*\theta_P + \beta*\theta_N a_P = \alpha*F a_N = \beta*F (typically \alpha + \beta = 1)  for explicit, you will evaluate the flux and put the whole thing into the source/residual instead of the matrix, but the idea is the same, i.e. you always need the flux. Better? Hrv 

March 14, 2005, 03:47 
Re: How to enforce the continuity equation?

#5 
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yes ,I use finite volume. I know we always need to use the flux,what I mean is the following: 1)For explicit convection term, the flux is evaluated from the previous time step(and put into the source), and so there is no need to correct the flux after the P' equation is solved and the velocity need not be corrected either. The P' is only used for pressure correction. 2)For implicit convection term, the flux and velocity are corrected after the P' is solved.
Is that right? thanks dear hrv dragon 

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