cbcflow.schemes.official.ipcs module

This incremental pressure correction scheme (IPCS) is an operator splitting scheme that follows the idea of Goda [1]. This scheme preserves the exact same stability properties as Navier-Stokes and hence does not introduce additional dissipation in the flow.

The idea is to replace the unknown pressure with an approximation. This is chosen as the pressure solution from the previous solution.

The time discretization is done using backward Euler, the diffusion term is handled with Crank-Nicholson, and the convection is handled explicitly, making the equations completely linear. Thus, we have a discretized version of the Navier-Stokes equations as

\[\begin{split}\frac{1}{\Delta t}\left( u^{n+1}-u^{n} \right)-\nabla\cdot\nu\nabla u^{n+\frac{1}{2}}+u^n\cdot\nabla u^{n}+\frac{1}{\rho}\nabla p^{n+1}=f^{n+1}, \\ \nabla \cdot u^{n+1} = 0,\end{split}\]

where \(\tilde{u}^{n+\frac{1}{2}} = \frac{1}{2}\tilde{u}^{n+1}+\frac{1}{2}u^n.\)

For the operator splitting, we use the pressure solution from the previous timestep as an estimation, giving an equation for a tentative velocity, \(\tilde{u}^{n+1}\):

\[\frac{1}{\Delta t}\left( \tilde{u}^{n+1}-u^{n} \right)-\nabla\cdot\nu\nabla \tilde{u}^{n+\frac{1}{2}}+u^n\cdot\nabla u^{n}+\frac{1}{\rho}\nabla p^{n}=f^{n+1}.\]

This tenative velocity is not divergence free, and thus we define a velocity correction \(u^c=u^{n+1}-\tilde{u}^{n+1}\). Substracting the second equation from the first, we see that

\[\begin{split}\frac{1}{\Delta t}u^c-\nabla\cdot\nu\nabla u^c+\frac{1}{\rho}\nabla\left( p^{n+1} - p^n\right)=0, \\ \nabla \cdot u^c = -\nabla \cdot \tilde{u}^{n+1}.\end{split}\]

The operator splitting is a first order approximation, \(O(\Delta t)\), so we can, without reducing the order of the approximation simplify the above to

\[\begin{split}\frac{1}{\Delta t}u^c+\frac{1}{\rho}\nabla\left( p^{n+1} - p^n\right)=0, \\ \nabla \cdot u^c = -\nabla \cdot \tilde{u}^{n+1},\end{split}\]

which is reducible to a Poisson problem:

\[\Delta p^{n+1} = \Delta p^n+\frac{\rho}{\Delta t}\nabla \cdot \tilde{u}^{n+1}.\]

The corrected velocity is then easily calculated from

\[u^{n+1} = \tilde{u}^{n+1}-\frac{\Delta t}{\rho}\left(p^{n+1}-p^n\right)\]

The scheme can be summarized in the following steps:

1. Replace the pressure with a known approximation and solve for a tenative velocity \(u^{n+1}\).
2. Solve a Poisson equation for the pressure, \(p^{n+1}\)
3. Use the corrected pressure to find the velocity correction and calculate \(u^{n+1}\)
4. Update t, and repeat.

[1]	Goda, Katuhiko. A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows. Journal of Computational Physics 30.1 (1979): 76-95.

Classes

class cbcflow.schemes.official.ipcs.IPCS(params=None)

Bases: cbcflow.core.nsscheme.NSScheme

Incremental pressure-correction scheme.

classmethod default_params()

solve(problem, update)