# A Highly Scalable Direction-Splitting Solver on Regular Cartesian Grid to Compute Flows in Complex Geometries Described by STL Files

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Governing Equations

#### 2.2. Numerical Algorithm: Direction Splitting

- Similar to the fractional time stepping technique, we first predict the intermediate pressure (${p}^{*}$) at ${t}^{\mathit{n}+1/2}$ written as:$${p}^{*,n+\frac{1}{2}}={p}^{n-\frac{1}{2}}+{\varphi}^{n-\frac{1}{2}},$$
- Then, we use the predicted pressure and Laplacian approximation to estimate the updated velocity (${u}^{n+1}$).$$\begin{array}{cc}\hfill {\rho}_{f}\frac{{\mathbf{\xi}}^{n+1}-{\mathit{u}}^{n}}{\Delta t}-{\mu}_{f}({\partial}_{xx}{\mathbf{\zeta}}^{n}+{\partial}_{yy}{\mathbf{\eta}}^{n}+{\partial}_{zz}{\mathit{u}}^{n})& ={\mathit{f}}_{b}^{n+\frac{1}{2}}-\nabla {p}^{*,n+\frac{1}{2}}-{\rho}_{f}NL({\mathit{u}}^{n-1},{\mathit{u}}^{n}),\hfill \\ \hfill {\rho}_{f}\frac{{\mathbf{\zeta}}^{n+1}-{\mathbf{\xi}}^{n+1}}{\Delta t}& =\frac{{\mu}_{f}}{2}{\partial}_{xx}({\mathbf{\zeta}}^{n+1}-{\mathbf{\zeta}}^{n}),\hfill \\ \hfill {\rho}_{f}\frac{{\mathbf{\eta}}^{n+1}-{\mathbf{\zeta}}^{n+1}}{\Delta t}& =\frac{{\mu}_{f}}{2}{\partial}_{yy}({\mathbf{\eta}}^{n+1}-{\mathbf{\eta}}^{n}),\hfill \\ \hfill {\rho}_{f}\frac{{\mathit{u}}^{n+1}-{\mathbf{\eta}}^{n+1}}{\Delta t}& =\frac{{\mu}_{f}}{2}{\partial}_{zz}({\mathit{u}}^{n+1}-{\mathit{u}}^{n}).\hfill \end{array}$$The non-linear advective term ($NL({\mathit{u}}^{n-1},{\mathit{u}}^{n})$) is explicitly approximated with a second-order Adam-Bashforth discretization with conditional stability of $|u\Delta t/\Delta x|<0.35$.$$NL({\mathit{u}}^{n-1},{\mathit{u}}^{n})=\frac{3}{2}{\mathit{u}}^{n}\xb7\nabla {\mathit{u}}^{n}-\frac{1}{2}{\mathit{u}}^{n-1}\xb7\nabla {\mathit{u}}^{n-1}.$$
- Now, we project the updated velocity to a divergence-free space and solve the Poisson problem. However, the Laplacian approximation makes the solution non-divergence-free locally near the fluid-solid interface. The correction in the pressure ${\varphi}^{n+\frac{1}{2}}$ is calculated using following equations:$$\begin{array}{cc}\hfill \theta -{\partial}_{xx}\theta & =-\frac{{\rho}_{f}{\lambda}_{\mathrm{min}}}{\Delta t}\nabla \xb7{\mathit{u}}^{n+1},\hfill \\ \hfill \psi -{\partial}_{yy}\psi & =\theta ,\hfill \\ \hfill {\varphi}^{n+\frac{1}{2}}-{\partial}_{zz}{\varphi}^{n+\frac{1}{2}}& =\psi .\hfill \end{array}$$The parameter ${\lambda}_{min}$ is considered to avoid the instabilities caused by the density jumps near the fluid-solid interface, computed as:$${\lambda}_{min}=min\left\{1,\underset{\forall \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\Omega}{min}\left\{\frac{{\rho}_{s,i}}{{\rho}_{f}}{|}_{i\in [1,N]}\right\}\right\},$$
- Finally, the corrected pressure is taken to update the pressure ${p}^{n+\frac{1}{2}}$ as follows:$${p}^{n+\frac{1}{2}}={p}^{n-\frac{1}{2}}+{\varphi}^{n+\frac{1}{2}}-\frac{{\mu}_{f}\chi}{2}\mathbf{\nabla}\xb7({\mathit{u}}^{n+1}+{\mathit{u}}^{n}),$$

- We first check the state of the computational grid node inside or outside the fluid domain (${\Omega}_{f}$), defined by an Indicator function (I):$$I\left(\mathbf{x}\right)=\left\{\begin{array}{cc}1\hfill & \mathbf{x}\in {\Omega}_{f}\hfill \\ 0\hfill & \mathbf{x}\in {\Omega}_{s}\hfill \end{array}\right.,$$
- The presence of the interface is detected by the indicator function I. It is the location between two consecutive nodes where one node belongs to ${\Omega}_{f}$ ($I=1$) and the other node belongs to ${\Omega}_{s}$ ($I=0$), or vice versa. The accurate intersection distance of the neighbouring fluid node from $d\Omega $ is further estimated depending on the type of rigid body.

## 3. Influence of Complex Geometries on Spatial Discretization

#### 3.1. Indicator Function

Algorithm 1 Computation of the indicator function (I) on a grid of size ${N}_{x}\times {N}_{y}\times {N}_{z}$. | ||

1: | for i = 1:${N}_{x}$ do | |

2: | for j=1:${N}_{y}$ do | |

3: | for k = 1:${N}_{z}$ do | |

4: | ${\mathbf{x}}_{{R}_{1}}\leftarrow (x\left(i\right),y\left(j\right),z\left(k\right))$ | ▹ we build the two extremities of the ray |

5: | ${\mathbf{x}}_{{R}_{2}}\leftarrow (x\left(i\right),0,z\left(k\right))$ | |

6: | for s∈Tdo | ▹ assuming T is the set containing the triangles |

7: | if intersection(s,${\mathbf{x}}_{{R}_{1}}$,${\mathbf{x}}_{{R}_{2}}$) then | |

8: | ${n}_{int}\leftarrow {n}_{int}+1$ | ▹ number of intersections |

9: | end if | |

10: | end for | |

11: | if ${n}_{int}$ is even then | |

12: | $I(i,j,k)\leftarrow 0$ | |

13: | else if ${n}_{int}$ is odd then | |

14: | $I(i,j,k)\leftarrow 1$ | |

15: | end if | |

16: | ${n}_{int}\leftarrow 0$ | |

17: | end for | |

18: | end for | |

19: | end for |

#### 3.2. Fluid-Solid Interface Distances

#### 3.3. Optimization of the Method

- The rays used to compute the indicator function are by definition parallel to the y axis.
- For a given pair of nodes, the node-triangle distance is computed along a line either parallel to the x, y or z axis.

## 4. Numerical Tests in Complex Geometries Described by STL Files

#### 4.1. Poiseuille Flow in a Pipe

#### 4.2. Flow in a Wavy Channel

#### 4.3. Flow in a Porous Medium: Computation of the Permeability Coefficient in a Sandstone

^{3}/s), k the permeability of the porous medium (m

^{2}), A the cross-sectional area m

^{2}, $({P}_{out}-{P}_{in})$ the pressure drop across the medium (Pa), ${\mu}_{f}$ the fluid viscosity (Pa·s) and L the domain length (m). The permeability coefficient is obtained by setting a pressure drop and measuring the induced volumetric flow rate.

**Figure 18.**Profiles of ${u}_{x}$,${u}_{y}$ and ${u}_{z}$ for $h/\Delta x=24,48,96$ along the wall-normal direction, profile set at $({x}_{3},{z}_{3})$.

^{3}. We simplify the STL file provided by [34] by consequently reducing its size: we only need to ensure that the criteria established in Section 4.1 $\Delta {c}_{t}<\Delta x$ is satisfied, there is no requirement to choose $\Delta {c}_{t}$ very small compared to $\Delta x$. Given our framework the initial triangulation was unnecessarily highly refined, leading to a high computing time for the pre-step at which the indicator function is computed. Figure 19 shows the STL triangulation used for the simulations, and we can observe that the triangles are relatively large on the front and back faces orthogonal to the z axis (Figure 19a) which is of no importance since there are located on a planar surface. The triangles mapping the pores of the sandstone are evenly sized (Figure 19b) and verify for all simulations $\Delta {c}_{t}<\Delta x$.

^{−3}. The corresponding Reynolds number is $Re=0.002$.

#### 4.4. Motion of a Rigid Spherical Particle in a Curved Pipe

## 5. Massively Parallel Computing: Flow through a Random Array of Spheres

## 6. Conclusions and Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Definition of the current framework. The fluid domain and the solid domain are defined by ${\Omega}_{f}$ and ${\Omega}_{s}$, respectively. $d\Omega $ denotes the fluid-solid interface.

**Figure 2.**STL file of a blood vessel positioned in a cuboid computational domain. The blue dots correspond to the field (i.e., ${u}_{x},{u}_{y},{u}_{z},p$) nodes.

**Figure 3.**Ray-tracing approach employed for two nodes located at $({x}_{1},{y}_{1},{z}_{1})$ and $({x}_{2},{y}_{2},{z}_{2})$: (

**a**) front and (

**b**) top-left views of the blood vessel.

**Figure 4.**Computation of the distances between the grid node and the intersecting triangles in both x and y directions denoted respectively ${d}_{x}$ and ${d}_{y}$.

**Figure 5.**(

**a**) Subdivision of the computational domain along y into 5 subdivisions: ${H}_{{x}_{1}},\dots ,{H}_{{x}_{5}}$. If a node belongs to ${H}_{{x}_{3}}$, the loop over the set of triangles is reduced to the triangles belonging to ${H}_{{x}_{3}}$. (

**b**) Subdivision of the computational domain along x into 3 subdivisions: ${H}_{{y}_{1}},\dots ,{H}_{{y}_{2}}$. If a node belongs to ${H}_{{y}_{2}}$, the loop over the set of triangles is reduced to the triangles belonging to ${H}_{{y}_{2}}$.

**Figure 7.**Influence of the grid refinement ${D}_{c}/\Delta x=6,12,24,48$ and 96 on the convergence of the velocity profiles: (

**a**) ${D}_{c}/\Delta {c}_{t}=6$, (

**b**) ${D}_{c}/\Delta {c}_{t}=24$, (

**c**) ${D}_{c}/\Delta {c}_{t}=96$.

**Figure 8.**Influence of the triangle characteristic size $\Delta {c}_{t}$ on the convergence of the velocity profiles: (

**a**) ${D}_{c}/\Delta x=6$, (

**b**) ${D}_{c}/\Delta x=24$, (

**c**) ${D}_{c}/\Delta x=48$.

**Figure 9.**Combined influence of the ratios ${D}_{c}/\Delta {c}_{t}$ and ${D}_{c}/\Delta x$ on the convergence of the method.

**Figure 10.**(

**a**) STL triangulation corresponding to the semi-wavy configuration in x and y directions considered for the simulations, (

**b**) computational domain used in Nek5000.

**Figure 11.**Contours of ${u}_{x}/\left|\right|{u}_{{x}_{Nek}}{\left|\right|}_{\infty}$ at the steady state on a plane located at $z=1.19$ orthogonal to the z axis. Left: Nek5000, right: Direction Splitting, ${h}_{s}/\Delta x=96$.

**Figure 12.**Contours of ${u}_{y}/\left|\right|{u}_{{y}_{Nek}}{\left|\right|}_{\infty}$ at the steady state on a plane located at $y=0.19$ orthogonal to the z axis. Left: Nek5000, right: Direction Splitting, ${h}_{s}/\Delta x=96$.

**Figure 13.**Contours of ${u}_{x}/\left|\right|{u}_{{x}_{Nek}}{\left|\right|}_{\infty}$ at the steady state on a plane located at $y=0.15$ orthogonal to the y axis. Left: Nek5000, right: Direction Splitting, ${h}_{s}/\Delta x=96$.

**Figure 14.**Contours of ${u}_{z}/\left|\right|{u}_{{z}_{Nek}}{\left|\right|}_{\infty}$ at the steady state on a plane located at $y=0.15$ orthogonal to the y axis. Left: Nek5000, right: Direction Splitting, ${h}_{s}/\Delta x=96$.

**Figure 16.**Profiles of ${u}_{x}$,${u}_{y}$ and ${u}_{z}$ for $h/\Delta x=24,48,96$ along the wall-normal direction, profile set at $({x}_{1},{z}_{1})$.

**Figure 17.**Profiles of ${u}_{x}$,${u}_{y}$ and ${u}_{z}$ for $h/\Delta x=24,48,96$ along the wall-normal direction, profile set at $({x}_{2},{z}_{2})$.

**Figure 19.**STL triangulation of the sandstone rock used in the simulations: (

**a**) $xy$ view and (

**b**) $yz$ view showing a single triangle in the z direction.

**Figure 20.**Sketch of the flow configuration: (

**a**) upstream part of the domain without any buffer region and (

**b**) full domain with inlet and outlet buffer regions and a thin layer removed in the y direction.

**Figure 21.**Contours of ${\left|\right|\mathbf{u}\left|\right|}_{2}$ (in m/s) at the steady state. The STL triangulation is also shown here in light gray. (

**a**) Whole computational domain, (

**b**) zoomed view of the purple square as shown in (

**a**).

**Figure 24.**Snapshot of ${\left|\right|\mathbf{u}\left|\right|}_{2}/{\left|\right|\mathbf{u}\left|\right|}_{\infty}$ for $Re=115$ and $({x}_{p}/{R}_{c},{y}_{p}/{R}_{c},{z}_{p}/{R}_{c})=(1,0.5,0)$ as the initial position of the particle in the pipe.

**Figure 25.**Trajectory of the particle in the $xy$ plane, sequenced as periods (colored lines). (

**a**) $Re=12$, (

**b**) $Re=115$. The red cross indicates the initial particle position, the bold black lines correspond to the pipe interface, and the dashed black line correspond to the centerline of the pipe.

**Figure 26.**Trajectory of the particle in the $xy$ plane, sequenced as periods (colored lines), case $Re=113$. The red cross indicates the initial particle position, the bold black lines correspond to the pipe interface, and the dashed black line correspond to the centerline of the pipe.

**Figure 27.**(

**a**) Computational domain showing all rigid spheres. (

**b**) Sub-domain on a single computing core showing well-resolved spheres with a grid resolution of 24 grid points per diameter.

**Figure 28.**Flow visualization in multiple sub-domains at the steady state. The streamlines and rigid spheres for each sub-domain are shown in the sub-images.

**Table 1.**Wall-time ($sec$) of the pre-step for small, medium and large triangulations, and an increasing number of subdivisions, ranging from ${N}_{s}=1$ to ${N}_{s}=20$.

Number of Triangles | ${\mathit{N}}_{\mathit{s}}=1$ | ${\mathit{N}}_{\mathit{s}}=5$ | ${\mathit{N}}_{\mathit{s}}=10$ | ${\mathit{N}}_{\mathit{s}}=20$ |
---|---|---|---|---|

20 × 10${}^{3}$ | 300.2 | 18 | 5.2 | 2.2 |

100 × 10${}^{3}$ | 550.7 | 45.1 | 35.2 | 15.2 |

200 × 10${}^{3}$ | 920.9 | 108 | 78.4 | 55.2 |

**Table 2.**Weak scaling results in the flow through a random array of spheres at $Re=19.2$ and $\varphi =0.15$.

Number of Nodes | Number of Cores | Number of Cells | Number of Spheres | Run Time per Time Step (s) |
---|---|---|---|---|

1 | 40 | 40,000,000 | 849 | 9.279 |

8 | 320 | 320,000,000 | 6792 | 9.924 |

170 | 6800 | 6,800,000,000 | 144,327 | 12.18 |

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## Share and Cite

**MDPI and ACS Style**

Morente, A.; Goyal, A.; Wachs, A.
A Highly Scalable Direction-Splitting Solver on Regular Cartesian Grid to Compute Flows in Complex Geometries Described by STL Files. *Fluids* **2023**, *8*, 86.
https://doi.org/10.3390/fluids8030086

**AMA Style**

Morente A, Goyal A, Wachs A.
A Highly Scalable Direction-Splitting Solver on Regular Cartesian Grid to Compute Flows in Complex Geometries Described by STL Files. *Fluids*. 2023; 8(3):86.
https://doi.org/10.3390/fluids8030086

**Chicago/Turabian Style**

Morente, Antoine, Aashish Goyal, and Anthony Wachs.
2023. "A Highly Scalable Direction-Splitting Solver on Regular Cartesian Grid to Compute Flows in Complex Geometries Described by STL Files" *Fluids* 8, no. 3: 86.
https://doi.org/10.3390/fluids8030086