A parallel implementation of the ghost-cell immersed boundary method with application to stationary and moving boundary problems

Abstract

A modified version of the previously reported ghost-cell immersed boundary method is implemented in parallel environment based on distributed memory allocation. Reconstruction of the flow variables is carried out by the inverse distance weighting technique. Implementation of the normal pressure gradient on the immersed surface is demonstrated. Finite volume method with non-staggered arrangement of variables on a non-uniform cartesian grid is employed to solve the fluid flow equations. The proposed method shows reasonable agreement with the reported results for flow past a stationary sphere, rotating and transversely oscillating circular cylinder.

Appendix

Finite difference approximations of the partial derivatives in non-uniform grids at a point ($x_i,y_j,z_k$)

First derivatives

$$\begin{aligned} \frac{\partial \phi _i}{\partial x}=\frac{\phi _{i+1}(\Delta x_{i-1})^2-\phi _{i-1}(\Delta x_{i+1})^2-\phi _i\left( (\Delta x_{i-1})^2-(\Delta x_{i+1})^2\right) }{\Delta x_{i+1}\Delta x_{i-1}(\Delta x_{i+1}+\Delta x_{i-1})}+O[(\Delta x)^2] \end{aligned}$$

(13a)

$$\begin{aligned} \frac{\partial \phi _j}{\partial y}=\frac{\phi _{j+1}(\Delta y_{j-1})^2-\phi _{j-1}(\Delta y_{j+1})^2-\phi _j\left( (\Delta y_{j-1})^2-(\Delta y_{j+1})^2\right) }{\Delta y_{j+1}\Delta y_{j-1}(\Delta y_{j+1}+\Delta y_{j-1})}+O[(\Delta y)^2] \end{aligned}$$

(13b)

$$\begin{aligned} \frac{\partial \phi _k}{\partial z}=\frac{\phi _{k+1}(\Delta z_{k-1})^2-\phi _{k-1}(\Delta z_{k+1})^2-\phi _k\left( (\Delta z_{k-1})^2-(\Delta z_{k+1})^2\right) }{\Delta z_{k+1}\Delta z_{k-1}(\Delta z_{k+1}+\Delta z_{k-1})}+O[(\Delta z)^2]. \end{aligned}$$

(13c)

Second derivatives

$$\begin{aligned} \frac{\partial ^2 \phi _i}{\partial x^2}=2\left( \frac{\phi _{i+1}\Delta x_{i-1}+\phi _{i-1}\Delta x_{i+1}-\phi _i(\Delta x_{i+1}+\Delta x_{i-1})}{\Delta x_{i+1}\Delta x_{i-1}(\Delta x_{i+1}+\Delta x_{i-1})}\right) +O[\Delta x] \end{aligned}$$

(14a)

$$\begin{aligned} \frac{\partial ^2 \phi _j}{\partial y^2}=2\left( \frac{\phi _{j+1}\Delta y_{j-1}+\phi _{j-1}\Delta y_{j+1}-\phi _j(\Delta y_{j+1}+\Delta y_{j-1})}{\Delta y_{j+1}\Delta y_{j-1}(\Delta y_{j+1}+\Delta y_{j-1})}\right) +O[\Delta y] \end{aligned}$$

(14b)

$$\begin{aligned} \frac{\partial ^2 \phi _k}{\partial z^2}=2\left( \frac{\phi _{k+1}\Delta z_{k-1}+\phi _{k-1}\Delta z_{k+1}-\phi _k(\Delta z_{k+1}+\Delta z_{k-1})}{\Delta z_{k+1}\Delta z_{k-1}(\Delta z_{k+1}+\Delta z_{k-1})}\right) +O[\Delta z]. \end{aligned}$$

(14c)

Mixed derivatives

$$\begin{aligned} \frac{\partial ^2 \phi _{i,j}}{\partial x \partial y}=\frac{\phi _{i+1,j+1}-\phi _{i+1,j-1}-\phi _{i-1,j+1}+\phi _{i-1,j-1}}{(\Delta x_{i+1}+\Delta x_{i-1})(\Delta y_{j+1}+\Delta y_{j-1})}+O[\Delta x,\Delta y] \end{aligned}$$

(15a)

$$\begin{aligned} \frac{\partial ^2 \phi _{i,k}}{\partial x \partial z}=\frac{\phi _{i+1,k+1}-\phi _{i+1,k-1}-\phi _{i-1,k+1}+\phi _{i-1,k-1}}{(\Delta x_{i+1}+\Delta x_{i-1})(\Delta z_{k+1}+\Delta z_{k-1})}+O[\Delta x,\Delta z] \end{aligned}$$

(15b)

$$\begin{aligned} \frac{\partial ^2 \phi _{j,k}}{\partial y \partial z}=\frac{\phi _{j+1,k+1}-\phi _{j+1,k-1}-\phi _{j-1,k+1}+\phi _{j-1,k-1}}{(\Delta y_{j+1}+\Delta y_{j-1})(\Delta z_{k+1}+\Delta z_{k-1})}+O[\Delta y,\Delta z], \end{aligned}$$

(15c)

where

$\Delta x_{i+1}=x_{i+1}-x_i\, , \,\,\,\Delta x_{i-1}=x_i-x_{i-1}\, , \,\,\,\Delta y_{j+1}=y_{j+1}-y_j$

$\Delta y_{j-1}=y_j-y_{j-1}\, ,\,\,\,\Delta z_{k+1}=z_{k+1}-z_k\, ,\,\,\,\Delta z_{k-1}=z_k-z_{k-1}$.