Meshing of porous foam structures on the micro-scale

Abstract

A semi-automatic block-structured grid generation technique for hexahedral meshing of porous open cell Kelvin foam structures for investigation of the pore scale fluid flow is presented. The performance of the algorithm is compared with a tetrahedral full automatic Delaunay meshing technique. In the first part of the paper the meshing strategies are explained. In the second part grid generation times, simulation times and the mesh quality are evaluated. For this Computational Fluid Dynamics (CFD) simulations for both a diffusion-dominated case (Re = 0.129) and a convection-dominated case (Re = 129) are carried out and analysed on four different cell resolutions of each mesh type. For the quality evaluation three different a posteriori error estimates are studied for the two mesh types on the different mesh sizes. The results are: the block-structured grid generation technique is about 10–20 times faster than the tetrahedral full automatic technique. While the mean field error estimates are comparable for both meshes, the maximum field error estimates for the block-structured meshes are only half of those for the tetrahedral meshes. Reaching simulation results of the same quality the hexahedral mesh needs about 20–40% less iterations with comparable mesh sizes. The time per iteration for the hexahedral meshes are up to 94% smaller than for the tetrahedral meshes. This makes the semi-automatic block-structured grid generation technique especially suitable for parameter studies and for the investigation of micro-scale flows in foam structures consisting of large quantities of Kelvin cells.

Appendix

1.1 Residual definition

The residual res(u) and res(p) are deduced from the momentum predictor and the pressure corrector of the SIMPLE algorithm. Let’s recall the discretised form of the momentum equation:

$$ a_{P}{\bf U}_{P} =H\left({\bf U}\right)-\frac{1}{\rho}\nabla p $$

(9)

with H(U) =  −∑ _N a _N U _N representing the matrix coefficients (a _N ) of the neighbouring cells multiplied by their velocity (U _N ), the matrix coefficient a _p and the velocity U _P at the cell of cell of interest indexed by P. As mentioned before, the pressure gradient is defined as $\nabla p=\nabla P+\frac{\triangle\tilde{p}}{L}\hat{e}$ for our periodic problem. The cell values for the velocity residual⁹ is defined by:

$$ {\text{res}}_{P}(U_{x,y,z}) =\left|\left({\bf U}_{P}\cdot\nabla\right){\bf U}_{p}+\frac{1}{\rho}\nabla p+\nabla\cdot\left(\nu\nabla{\bf U}_{P}\right)\right|_{x,y,z} $$

The $\left|\,\right|_{xyz}$ operator denotes the componentwise magnitude of the vector. Dividing (9) by a _P , taking the divergence and with the continuity equation (Eq. 1) the pressure correction is obtained:

$$ \nabla\cdot{\bf U}_{P} = \nabla\cdot\left(\frac{H\left({\bf U}\right)}{a_{P}}\right)-\nabla\cdot\left(\frac{1}{a_{P}}\frac{1}{\rho}\nabla p\right)=0 $$

(10)

which gives a corrected pressure field which is in accordance to the predicted velocity field. The cell values for the pressure residual are calculated from this pressure corrector by:

$$ {\text{res}}_{P}(p) = \left|\nabla\cdot\left(\frac{H\left({\bf U}\right)}{a_{P}}\right)-\nabla\cdot\left(\frac{1}{a_{P}}\frac{1}{\rho}\nabla p\right)\right| $$

(11)

All the cell residuals are summed up over the whole domain and normalised to

$$ res(U_{x,y,z}) = \frac{\sum_{P}\left({\text{res}}_{P}(U_{x,y,z})\right)}{F_{\text{norm}}(U_{x,y,z})} $$

(12)

and

$$ res(p) = \frac{\sum_{P}\left({\text{res}}_{P}(p)\right)}{F_{\text{norm}}(p)} $$

(13)

with the normalisation factors F _norm(U _x,y,z ) for the velocity residuals res _P (U _x,y,z )

$$ \begin{aligned} F_{\text{norm}}(U_{x,y,z}) & = \left|\sum_{P}\left(a_{P}{\bf U}_{\text{cell}}-a_{U,P}U_{\text{ref}}\right)\right|_{x,y,z} \\ & \quad + \left|\sum_{P}\left(H\left({\bf U}\right)-\frac{1}{\rho}\nabla p-a_{U,P}U_{\text{ref}}\right)\right|_{x,y,z}+10^{-20}\frac{m}{s} \end{aligned} $$

(14)

and the normalisation factors F _norm(p) for res(p)

$$ \begin{aligned} F_{\text{norm}}(p) &= \left|\sum_{P}\left(a_{p,P}p_{P}-a_{p,P}p_{\text{ref}}\right)\right| \\ & \quad + \left|\sum_{P}\left(\nabla\cdot\left(\frac{H\left({\bf U}\right)}{a_{U,P}}\right)-a_{p,P}p_{\text{ref}}\right)\right|+10^{-20}\frac{m^{2}}{s^{2}}. \end{aligned} $$

(15)

The reference values are defined by $U_{\text{ref}}=\frac{\sum_{P}\left(U_{x,y,z}\right)}{N_{P}}$ and $p_{\text{ref}}=\frac{\sum_{P}\left(p\right)}{N_{P}}.$ Note that a _p,P in F _norm(p) are the matrix coefficients of the pressure corrector (Eq. 10). The last terms on the RHS of (14) and (15) are introduced to prevent division by zero in (12) and (13).