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Fractional Step Method and SIMPLE

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Old   December 14, 2000, 08:35
Default Fractional Step Method and SIMPLE
  #1
Ha Lim Choi
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Dear Friends;

As you know well the fractional step method is usually used in DNS and LES. I want to know the difference between the SIMPLE and the fractional step method. In my knowledge, the fractional step method is explict method and neglect the nonlinearity(by taking very small time step) in solving the Navier-Stokes equation. If they take into account the nonlinearlity problem(for example, by iteration) and solving the pressure correction equation instead of pressure equation(you can do that easily), what is difference between the SIMPLE and the fractional step method? Could any one can answer clearly to me ?
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Old   December 14, 2000, 12:05
Default Re: Fractional Step Method and SIMPLE
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Ghanshyam Singh
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The Fractional time step (FTS) methods are used to enhance the stability of the solution. However, in contest to the simulation of the incompressible flow using finite element method, using a suitable FTS one can over come the Babusra Brezzi stability restriction.

No it is not an explicit method. I think you have to go a long way.

GS
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Old   December 14, 2000, 20:08
Default Re: Fractional Step Method and SIMPLE
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Ha Lim Choi
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I am talking about the fractional step method developed by Kim and Moin (1985). What is the FTS you are talking about ? Can you give me some references?

Ha Lim Choi
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Old   December 15, 2000, 10:36
Default Re: Fractional Step Method and SIMPLE
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Sebastien Perron
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1) When you use the SIMPLE family of methods you take into account

the other forces such as convection and diffusion that drive the flow. This is done by constructing the equation for pressure correction directly from the equation for momentum.

2) This family of methods as nothing to do with takin or not taking int account the nonlinearlity, nor being explicit or implicit. At first (around 1971), this method was used with an explicit sheme. But nowadays, this algorithm is used with explicit or implicit schemes.

3) It is easier to implement a fractional step method, Because you don't have to make any special routines to compute the coefficient for pressure from the equation for momentum.

4) If you do a complete splitting of the operators, as Kim and Moin do, the time step will have to be very small in order to get an accurate solution. Furthermore, the dirichlet boundary conditions for velocity will have to be changed. Beleive, it has to be done. (I have tried it)

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Old   December 15, 2000, 11:09
Default Re: Fractional Step Method and SIMPLE
  #5
Ghanshyam Singh
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Look at :

Zienkiewicz, O.C., P. Nithiarasu, R. Codina, M. Vazquez and P. Ortiz, (1999) "The Characteristic-Based-Split Procedure: An Efficient and Accurate Algorithm for Fluid Problems" IJNMF, vol. 31, pp 359-392.

and the references cited therein.

GS
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Old   December 15, 2000, 20:33
Default Re: Fractional Step Method and SIMPLE
  #6
Ha Lim Choi
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Dear Sebastien Perron;

The SIMPLE method is implicit method and taking into account the nonlinearity by iteration. So there is no explicit SIMPLE method.

Ha Lim Choi
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Old   December 16, 2000, 00:09
Default Re: Fractional Step Method and SIMPLE
  #7
Ghanshyam Singh
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Let me correct you. SIMPLE method is NOT IMPLICIT, it is SEMI-IMPLICIT. Thais why the name "SIMPLE". Here, only pressure is treated implicitly, because you have no other way to handle it.

GS
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Old   December 16, 2000, 05:05
Default Re: Fractional Step Method and SIMPLE
  #8
Ha Lim Choi
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Dear Ghanshyam Singh;

The word SEMI-IMPLICIT in the name SIMPLE have been used to acknowledge the omission of the term (Sum of AnbU'nb) in deriving the velocity correction equation in SIMPLE algorithm (Patankar's book ,127 page). Other than that the SIMPLE method is implicit method. All the momentum equations and other transport equations are solved implicitly. Only the pressure equation is treated implicitly in the fractional step method.

Ha Lim Choi
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Old   December 17, 2000, 09:33
Default Re: Fractional Step Method and SIMPLE
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Sebastien Perron
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The simple algorithm has nothing to do with the scheme being explicit or implicit. The simple algorithm is a projection algorithm to calculate a divergence free velocity.

Your can use Gear scheme, Euler implicit, Euler explicit Crank-Nicholson, etc. with the simple family of algorithm. I have an article from Doormall and Raithby who proposed the simplec algorithm, they use an explicit discretisation (similar to Patankar). An implicit (or semi-implicit) discretisation of the momentum equations could also be used.

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Old   December 17, 2000, 22:11
Default Re: Fractional Step Method and SIMPLE
  #10
Ha Lim Choi
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Dear Sebastien Perron;

In SIMPLE algorithm we first solve the momentum equation using the pressure and velocity field from the previous time step values. Since the calculated velocity field may not satisfy the continuity equation, we sove the pressure correction equation and update velocity field and pressure. Then, using these updated velocity field and pressure, we solve the momentum equation again. Then, we solve the pressure correction equation and update the velocity components and pressure again. These procedures continue iteratively until the velocity field satisfies the momentum equation and continuity equation together within one time step. If you use the explicit method in which the velocity components are calculated only from the previous time step values, how you can do this iterative procedure?

Halim Choi
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Old   December 18, 2000, 06:55
Default Re: Fractional Step Method and SIMPLE
  #11
Sebastien Perron
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Dear Halim Choi

1) This iterative procedure is a false step procedure. If you read this article:

J.P. Van Doormall and G.D. Raithby. Enhancements of the simple method for predicting incompressible fluid flows. Numerical Heat transfer, vol 7 pp147-163, 1984.

you'll find out that this iterative procedure is nothing else than a false step method to solve permanent flow. The relaxation parameter can be assosciated with a local time step.

2) When you "solve" the momentum equation, you can use an implicit scheme on an explicit scheme. Actually the equations are often written in this manner:

(ap/alpha)*phi_p=sum (anb*phi_nb) +b + ((1-alpha)/alpha)*ap*phi_p^n

where alpha is a relaxation parameter and ap (and anb) the matrix coefficients.

which is very natural for an explicit scheme. If the phi_nb are evaluated at time t=t^(n) this scheme is explicit, If the phi_nb are evaluated at time t=t^(n+1) this scheme is implicit.

3)when you use the simple family of methods, you are seldom fully implicit (I have never seen so). Usually The convective terms are splitted in this manner: u^n*u^(n+1) or u^n*u^(n) in order to be linearised.

4) I say again, the simple's family of methods is only a manner of making a projection to calculate a divergence free velocity.

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Old   December 18, 2000, 21:50
Default Re: Fractional Step Method and SIMPLE
  #12
Ha Lim Choi
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Dear Sebastien Perron;

(1) If you use the explicit scheme, the discretised momentum equations become as follows;

(Ap/alpha)*phi_p^(n+1)=sum (Anb*phi_nb)^(n) +b^(n) + ((1-alpha)/alpha)*Ap*phi_p^(n)

where phi are the velocity components. After you solve the momentum equation, you solve the pressure correction equation and correct the velocity components and pressure to satisfy the continuity equation, then you must proceed to next time step, since if you return to above equation, you get the same results (the previous time step values are already calculated and fixed). That is not the procedure of the conventional SIMPLE (If you call it SIMPLE, that is OK). That is a kind of procedure of the fractional step method or the predictor-corrector method.

(2) You mention that the convective terms are splitted (linearized) in the manner u^(n)*u^(n+1) in SIMPLE. In SIMPLE procedure, (n) and (n+1) stand for the iteration levels during the iteration at one time step and should be written in the manner, (l), (l+1).

Ha Lim Choi

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Old   December 20, 2000, 13:33
Default Re: Projection Method and SIMPLE
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Gobbly-goop
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Projection method? Hey, so what is the advantages of using the simple(/r,/c) schemes to say a typical projection method scheme like Chorin,Bell,etc... to varying density flows????

Can you refer me to any papers on the comparisons between each technique???

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