PISO法

PISO (Pressure-Implicit with Splitting of Operators) is a pressure-velocity coupling algorithm proposed by Issa in 1986. While the SIMPLE method converges by running many outer iterations, the PISO method performs two pressure corrections within one time step to ensure time accuracy without outer iterations.

Exactly. For unsteady flows (LES, DNS, moving mesh problems, etc.), the continuity equation must be strictly satisfied at each time step. PISO has a higher computational cost per time step, but since it doesn't require outer iterations, it is often more efficient overall.

The PISO method proceeds with the following steps.

Step 1: Momentum Predictor

Solve the momentum equation using the pressure from the previous time step $p^n$ to obtain a tentative velocity $\mathbf{u}^*$:

Step 2: First Corrector

Correct the velocity: $\mathbf{u}^{**} = \frac{H(\mathbf{u}^*)}{a_P} - \frac{1}{a_P}\nabla p'$

Step 3: Second Corrector

Final velocity: $\mathbf{u}^{n+1} = \frac{H(\mathbf{u}^{**})}{a_P} - \frac{1}{a_P}\nabla(p' + p'')$

Precisely, the second correction in PISO reflects the update of the neighbor cell coefficients $H$, which improves the approximation omission in SIMPLE. As a result, relaxation factors become unnecessary and time accuracy is maintained.

Let's organize the essential differences between the two.

The time discretization itself is implicit, but to sufficiently reduce the splitting error with PISO's two corrections, the CFL number needs to be kept somewhat low. Practically, CFL < 1 is safe, and a maximum of CFL < 5 is recommended. If the CFL number is too large, accuracy deteriorates.

It's an expression often seen in the OpenFOAM context. Discretizing the momentum equation gives:

Here, define $H(\mathbf{u}) = -\sum_N a_N \mathbf{u}_N + \mathbf{b}$. Then:

In this form, velocity can be expressed as a function of the pressure gradient. Substituting this into the continuity equation yields the pressure Poisson equation.

nCorrectors is the number of PISO pressure correction steps. The default is 2, but it can be set to 3-4 for higher CFL numbers or when higher accuracy is desired.

nNonOrthogonalCorrectors are additional iterations to correct mesh non-orthogonality. For non-orthogonal meshes, non-orthogonal contributions arise in the pressure Laplacian, which need to be corrected iteratively.

```

PISO

{

nCorrectors 2;

nNonOrthogonalCorrectors 1;

}

```

The PIMPLE method is a PISO and SIMPLE hybrid implemented in OpenFOAM. It performs outer iterations (SIMPLE-like loops) within a time step, finishing with PISO corrections.

```

PIMPLE

{

nOuterCorrectors 2; // SIMPLE-like outer iteration

nCorrectors 1; // PISO correction

nNonOrthogonalCorrectors 1;

}

```

The advantage of the PIMPLE method is that it can significantly relax the CFL number limit. With sufficient nOuterCorrectors, stable calculation is possible even for CFL >> 1.

However, increasing nOuterCorrectors increases the cost per time step. If CFL < 1, pure PISO (nOuterCorrectors=1) is more efficient.

Let's compare common time discretizations combined with the PISO method.

For LES calculations, the 2nd order backward or CrankNicolson should be used. For pseudo-transient RANS steady-state calculations, Euler is sufficient.