# CFX Comparison

This benchmark investigates a steady flow in a section of a human carotid artery. There exists no analytical solution for this problem: far from it, the benchmark reproduces the characteristics of a typical production environment. The geometry of the artery is provided in form of a generic surface mesh, and inlet/outlet conditions are obtained from a semi-analytical description of the full human arterial system. The problem is simulated with Palabos and with AnSys(R) CFX, and the benchmark includes a grid convergence study, and comparison of point-wise and integral quantities. With the large amount of provided accurate data, this is an ideal benchmark for validation of your own CFD code on a non-trivial problem.

## Benchmark Overview

The image on the right represents the benchmark geometry, in the form of a triangular surface mesh. It represents a section of a human carotid that undergoes a bifurcation (the main artery is split into two secondary arteries). The flow enters from the main artery, on the bottom of the image, and the outflows are located in the secondary arteries. A so-called aneurysm, a pathological, ball-shaped deformation is visible at the intersection. The rupture of such an aneurysm can be deadly, and one of the interests of a numerical study is the investigation of the flow structure and of wall stresses in the aneurysm to devise new methods to inhibit such an event.

The conditions of the benchmark are kept simple, for the sake of clarity and ease of comparison:

- The flow is incompressible, Newtonian and stationary.
- A constant flux is imposed at the inlet by means of a velocity Poiseuille profile, fully placed inside the approximate circle of the inlet channel (the velocity is exactly zero on the walls).
- A constant pressure is imposed on the outlets, of equal value for both of them.

The surface mesh was generated artificially with help of the open-source software shapes, and we would like to express our thanks to the authors of this highly useful program.

The STL file of this benchmark, as well as the source code of the Palabos implementation of the benchmark, are wrapped up in the Palabos release, and can be found in the directory examples/showCases/aneurysm.

## Measured Quantities

The following data is measured at the steady flow state:

- The average energy in the domain
- The RMS vorticity in the domain
- The pressure drop between inlet and outlets (both outlets have the same pressure)
- The velocity-norm, vorticity-norm, and pressure in each of the six points P1, ..., P6.

Furthermore, energy iso-contour lines are plotted on the three planes Plane1, Plane2, and Plane3.

The coordinates of the benchmark points and planes are listed in the following table, and are illustrated in the figure.

## Numerical method in Palabos

To simulate the flow inside the aneurysm, some of the new features of Palabos that can be found in version 1.0 were used. Specifically, a Guo off-lattice boundary condition was utilized. The aneurysm surface geometry was read from an STL file and processed appropriately. The new parallel voxelizer was used to voxelize the computational domain and distinguish between lattice nodes that belong to the interior of the artery from the ones that lay outside of it. Lastly, the aneurysm application code incorporates the concept of smooth grid refinement until convergence achieved. The word "smooth" indicates that the whole grid was doubled in each direction, and successive solutions were obtained until a grid independent solution was finally achieved.

## Numerical method in CFX

AnSys(R) CFX is a commercial software package using traditional highly efficient first- and second-order finite volume methods on a variety of different grid types. Only pure tetrahedral meshes were considered in the present benchmark. These meshes were refined close to the arterial wall and were prefered from other more complicated hybrid-type grids (like the ones that consist of prism layers near the walls) because the results they produced in this specific test were considered to be of a higher quality. All computational meshes were constructed by the AnSys(R) Icemcfd code. Concerning CFX, a fully second-order accurate finite volume solver was used for all simulations.

## Convergence Study: benchmark resolution

At which resolution must a program run to yield sufficient accuracy? And which criteria must be met so one can argue that CFX and Palabos have reached the same level of accuracy? There is no single answer to these questions, especially as the two numerical tools have a different convergence behavior, depending on the monitored quantity. In Palabos for example, the average energy appeared to converge much faster with respect to the grid resolution, while CFX showed a distinctly higher speed of convergence for the pressure. In the end, we selected the RMS vorticity as our main criteria of convergence, as it seemed to best reflect the global convergence behavior of both tools.

Performing a grid convergence study is particularly easy in Palabos, because the grid is generated automatically. Thus, a single batch job can be started to obtain the data for an array of grid sizes. For the CFX benchmark data, a fixed mesh with approximately 1 million grid points was generated, with a relative error of 3e-2 with respect to its own high-resolution grid (see next section). In Palabos, a grid convergence study (see picture on the right), showed that 700'000 grid points are required to obtain the same relative error. Again, the error is measured relative to Palabos' own high-resolution grid.

## As large as possible: reference resolution

To get reference results, the simulation was executed on the largest possible grid size. In the case of our CFX run, the maximal grid size was limited by the abilities of our (non-parallel) grid generation tool. In the case of Palabos, the size was limited by our wish to get the results within a few days on a medium-sized parallel machine.

In the following, all numerical results are displayed at both the benchmark and the reference resolution. The following table summarizes the grid points used by both methods at both these resolutions

## Results: Wall Shear Stress

In this first, qualitative part of the benchmark, a plot of the wall-shear-stress in the area of the aneurysm is shown. The picture is quite typical, as high stresses occur at entry points of an artery into the aneurysm, and at the contact of the aneurysm with a flow vortex (the vortex can be seen in the contour plots in the next section).

One should point out that while this comparison is only qualitative, it is probably the most relevant one to actual medical physics. Wall shear-stresses have direct medical implications, and obtaining the same picture with Palabos and AnSys(R) here means that both tools have converged to the same medical prediction.

## Results: Contours of the Velocity norm

The following image shows the contours of the velocity-norm on the three benchmark planes, at reference resolution.