SIMPLEC algorithm

The steps involved are same as the SIMPLE algorithm and the algorithm is iterative in nature. p*, u*, v* are guessed Pressure, X-direction velocity and Y-direction velocity respectively, p', u', v' are the correction terms respectively and p, u, v are the correct fields respectively; Φ is the property for which we are solving and d terms are involved with the under relaxation factor. So, steps are as follows:

1. Specify the boundary conditions and guess the initial values.
2. Determine the velocity and pressure gradients.
3. Calculate the pseudo velocities.

${\displaystyle {\hat {u}}_{i,J}={\frac {\sum a_{nb}u_{nb}^{*}+b_{i,J}}{a_{i,J}}}}$

${\displaystyle {\hat {v}}_{I,j}={\frac {\sum a_{nb}v_{nb}^{*}+b_{I,j}}{a_{I,j}}}}$

1. Solve for the pressure equation and get the p.

${\displaystyle a_{I,J}p_{I,J}=a_{I-1,J}p_{I-1,J}+a_{I+1,J}p_{I+1,J}+a_{I,J-1}p_{I,J-1}+a_{I,J+1}p_{I,J+1}+b_{I,J}}$

1. Set p*=p.
2. Using p* solve the discretized momentum equation and get u* and v*.

${\displaystyle a_{i,J}u_{i,J}^{*}=\sum a_{nb}u_{nb}^{*}+(p_{I-1,J}^{*}-p_{I,J}^{*})A_{i,J}+b_{i,J}}$

${\displaystyle a_{I,j}v_{I,j}^{*}=\sum a_{nb}v_{nb}^{*}+(p_{I,J-1}^{*}-p_{I,J}^{*})A_{I,j}+b_{I,j}}$

1. Solve the pressure correction equation.

${\displaystyle a_{I,J}p'_{I,J}=a_{I-1,J}p'_{I-1,J}+a_{I+1,J}p'_{I+1,J}+a_{I,J-1}p'_{I,J-1}+a_{I,J+1}p'_{I,J+1}+b'_{I,J}}$

1. Get the pressure correction term and evaluate the corrected velocities and get p, u, v, Φ*.

${\displaystyle u_{i,J}=u_{i,J}^{*}+d_{i,J}(p'_{I-1,J}-p'_{I,J})}$

${\displaystyle v_{I,j}=v_{I,j}^{*}+d_{I,j}(p'_{I,J-1}-p'_{I,J})}$

1. Solve all other discretized transport equations.

${\displaystyle a_{I,J}\phi _{I,J}=a_{I-1,J}\phi _{I-1,J}+a_{I+1,J}\phi _{I+1,J}+a_{I,J-1}\phi _{I,J-1}+a_{I,J+1}\phi _{I,J+1}+b_{I,J}^{\phi }}$

1. If Φ shows convergence, then STOP and if not, then set p*=p, u*=u, v*=v, Φ*=Φ and start the iteration again.
   [2][3]

Pseudo velocity equations in x and y dir.

Pressure equation

Discretized momentum equations

Pressure correction equation

Velocity Correction equations in X and Y dir.

Transport equations

• The discretized pressure correction equation is same as in the SIMPLE algorithm, except for the d terms which are used in momentum equations.
• p=p*+p' which tells that the under relaxing factor is not there in SIMPLEC as it was in SIMPLE.
• SIMPLEC algorithm is seen to converge 1.2-1.3 times faster than the SIMPLE algorithm
• It doesn't solve extra equations like SIMPLER algorithm.
• The cost per iteration is same as in the case of SIMPLE.
• Like SIMPLE, a bad pressure field guess will destroy a good velocity field.[4]

• SIMPLE algorithm
• SIMPLER algorithm
• Navier–Stokes equations