Flow in a Stenosed Aorta – Influence of the Mesh and the Numerical Method

Abstract

This report looks at the flow of blood downstream the stenosis to determine the extent of flow separation, and also create a simulation to model blood flow through the aorta. The created simulations were done in the lab using the STAR CCM+ software. Provided simulations of fine, medium and coarse mesh solutions and both 1^st and 2^nd order accuracy were given for comparison between them and the created mesh simulation to see which provided the most accurate results. Provided simulations were analysed and compared at different level of mesh refinement and more comparisons were made between the created simulations and the provided ones. Findings showed finer mesh with 2^nd order accuracy provided the most accurate results due to the reduced effect of artificial viscosity. Also the provided mesh results were more accurate than the created lab simulations as a result of the provided mesh being more refined. Finally the extent to which flow separation was predicted was accurate as below the stenosis evidence of flow separation is seen.

Table of Contents

1 Introduction.......................................................................................................... 3

2 Results................................................................................................................. 3

2.1 Effect of mesh refinement and 2nd order accuracy, flow field.......................... 4

2.2 Comparison of own vs. provided simulations.................................................. 4

2.3 Accuracy of flow separation............................................................................ 4

3 Summary ..............................................................................................................4

1 Introduction

This experiment aimed to study whether or not flow separation would occur downstream the stenotic aorta. Another aim of this study was to compare the provided mesh with our created mesh… The STARCCM+ software was used to create a simulation that depicts a simplified model of the flow of blood through the aorta. For this case Blood was considered to be a steady, incompressible Newtonian fluid, with a viscosity of 0.00365Pasec.To further simplify the case the pulsatile nature of blood is ignored and flow was assumed to be laminar with an average inflow velocity of 0.132 m s in the aorta and an average outflow velocity of 0.09 m s at the outlet.

The simplified geometry model of the aorta used for the stimulation, is shown in Figure 1, Blood enters the aorta through inlet and flows through the arch and then down towards the stenosis.

At the stenosis, blood flows through a constricted area with a reduced cross-sectional area. According to Bernoulli principle, at the stenosis we should expect to see a drop in pressure gradient as cross sectional area decreases, whilst velocity increases. Flow continues downstream till the aorta bifurcates- this is when the abdominal aorta divides into the left and right outlet branches (also known as daughter vessels).As a result of bifurcation, flow separation in possible- this is evident when a region of reversed blood flow is induced by the pressure gradient within the boundary layer (Fox JA). Also, a stagnation point is seen at the small vessel off the aorta arch and where the two outlet branches intersect, this is where pressure is at a maximum and velocity flow is zero. At the aorta walls there is a ^Figure _{geometry of aorta to be} ^{1, displays} no slip condition, where fluid velocity is zero. _modelled.

In this report we will consider the three provided meshes with different levels of refinement and different order of accuracy. The three mesh levels presented are Fine, medium, and coarse mesh, we expect the fine mesh to have the highest resolution, due to having more cells. To improve accuracy of simulation results, a finer mesh with 2^nd order accurate discretisation is required, although this could reduce the stability. Artificial viscosity (A.V) is introduced in order to improve the stability of a system, however this results in viscous errors ; hence its effect needs to be minimised- this is done by reducing the mesh width until truncation errors are smaller than any other errors ( J.D Mueller), this in turn leads to mesh convergence.

2 Results

2.1 Effect of mesh refinement and 2nd order accuracy, flow field

The velocity magnitude contour plots for all three mesh levels can be seen in Figure 2, where it is clear that second order meshes have better mesh quality than first order meshes and also we can observe the higher the level of mesh refinement, the clearer the contour plots appears.

For both the 1^st order and 2^nd order accuracy simulations, we can notice as the level of mesh refinement is increased from coarse to fine, the contour lines become smoother – this implies a lower velocity gradient. We know artificial viscosity is proportional to change in velocity, therefore the more a mesh is refined the lower the presence of artificial viscosity. This is due to fact that when a mesh is refined the mesh width is reduced and mesh elements increased which in turn reduces artificial viscosity- thus explaining why the finer mesh produces the most accurate results.

The dark blue lines, seen at the walls of all meshes are regions of very low velocity or zero velocity due to the no slip condition at the walls. However, according to continuity, flow should accelerate towards the centre region of the aorta. This is evident at the centre right of the aorta, where we observe the maximum velocity value occurring in all mesh levels. These observations are most visible is the 2^nd order fine and medium meshes and less obvious in the 1^st order meshes – especially the 1^st order coarse mesh. This suggests 2^nd order simulations have higher accuracy than 1^st order ones, this could be due to the fact the cell jump in 2^nd order simulations being proportional to artificial viscosity. Therefore it follows that halving the mesh width would half the artificial viscosity for first order accuracy and decrease by a factor 4 for second order accuracy. This highlights why increasing the order of accuracy reduces the artificial viscosity.

1st order	2nd order

Before and after the stenosis of the aorta we can predict the flow to be fully developed, as the contour lines are parallel to the aorta walls. This observation is apparent in all mesh levels and both 1^st and 2^nd order accuracy, however it is clearer in the 2^nd order finer meshes with better resolution. The most obvious change can be seen when using the same 2^nd order but going from a coarse mesh to a medium one, there's not much of a noticeable difference in mesh resolution when going from a 2^nd order medium to fine mesh suggesting mesh convergence

Figure 2, shows the contour plots for velocity magnitude for 1st and 2nd order at stenosis, starting at top with coarser mesh refinement then at the bottom finest mesh.

Further evidence for the effect of mesh refinement and order of accuracy is seen in figure 3, where we see peak negative yvelocity increases as the level of mesh refinement increases and becomes finer and also the order of accuracy from 1^st order to 2^nd order. However the difference between both 2^nd order medium and fine refinement is trivial and we can consider the medium level mesh refinement as mesh converged. Figure 3, confirms what is shown in the velocity contour plots in figure 2, as we can see the graphs show lower velocity at the aorta walls and as it goes towards the centre the velocity increases.

Flow separation becomes evident due to the negative velocity Figure 3 showing velocity profile graph of all provided cases which we observe across all plots, the change in velocity which suggest reverse flow is most evident in the 2^nd order simulations.

1st order	2nd order

According to Bernoulli principle, we should expect to see a drop in pressure gradient at this stenosis. This is evident in Figure 4, where we see a higher pressure region in green just before the stenosis then a reduced pressure region at the stenosis in blue. Due to the lower pressure gradient at the stenosis, velocity should increase. This corresponds to the high velocity region in red in Figure 2 and also the high velocity peaks in figure 3 at centre of the stenosis. From Figure 4, we can observe the 1^st order simulation especially in the coarse mesh have contour lines that are more parallel compared to other finer mesh- suggesting fully developed flow before the stenosis is more evident in lower order less refined mesh. For looking at pressure contours coarse mesh refinement might be the better option.

Figure 4 Showing pressure contour plots of different mesh refinements. Coarse level refinement at top and gets finer downwards

Further evidence for effect of 2^nd order accuracy is seen figure 5, where we see two residual plots, of both 1^st order and 2^nd order accuracy for the provided medium mesh. We know the 2^nd order simulations are more accurate due to the truncation error being proportional to h² therefore refining the mesh by a factor of 2 will reduce the error by a factor of 4, hence a smaller error means a more accurate solution. However second order simulations take longer to converge as can be seen in figure 5, where we see the 2^nd order simulation needed about 100 more iterations than the 1^st order simulation before converging. The longer time taken by 2^nd order have consequences such as high computational costs; in order to make the 2^nd order simulation more computationally efficient, the convergence threshold should be set at the level which the simulation stops this will not only reduce costs but also increase accuracy. Although the 1^st order simulation is much faster than 2^nd order it may cause the solver to produce errors due to instability leading to less accurate simulation results when compared to 2^nd order.

Figure 5, showing residual plots for both 1^st order and 2^nd order provided medium mesh case.

2.2 Comparison of own vs. provided simulations

In this section, we will discuss how the simulation was set up and consider a provided 2^nd order medium mesh, alongside my own 2^nd order medium mesh and explore the possible reasons why the mesh differs, even though both contains the same number of cells.

Firstly we started by creating a 1^st order simulation then ran it a second time to produce a 2^nd order simulation. In order to produce a solution that models the behaviour of blood flow in aorta, some assumptions were made and certain flow parameters applied. To do this we set time discretisation of the model to steady flow and segregated. The fluid was also incompressible so a constant density equation of state was chosen, and finally flow was laminar. The solution stopping criteria was set to run for a maximum of 1000 steps and a minimum of 1x10-4.This was done to make sure the solver stops when the residual drops below the minimum value. To ensure steadiness of solution, the stopping criteria was further lowered to 5x10^-5 where convergence was reached, and the solution was now stable as changes in data no longer occurred. To reach steady state with zero residual is near impossible on a computer and even if it was possible it wouldn't be computationally efficient.

From figure 6, we can observe both 2^nd order simulations have a structured mesh with regular quadrilateral cells, however it is clear that the provided mesh seem to have a smaller mesh width with smaller jump between cells thus we can assume the provided mesh is more refined with a higher resolution than my provided solution. The smaller mesh width implies artificial viscosity will be reduced therefore providing a more accurate solution in the provided case.

Looking at the velocity profiles of both my own solution and the provided solution, evidence of reduced artificial viscosity is seen in the provided case due to the presence of smoother contour lines, unlike my own case where the contour lines nearer to centre as seen in figure 6, are less smooth and jagged. The smoother contour lines alongside reduced mesh width of the provided case suggest the provided simulation is better resolved than my simulation and proves to be the more accurate solution.

Figure 6, Shows both 2nd order plots, provided (right) and own (left) medium mesh simulations.

 

 

Figure 7, velocity profile of own and provided simulation.

2.3 Accuracy of flow separation

Furthermore, looking at both velocity magnitudes contour plots (with same range of velocity magnitude from 0.0 -0.2 m/s), we notice in both cases velocity is low at walls and increases towards centre, however in the provided case it reaches a higher peak velocity than my own case with the extremely small red region shown at the centre right. This is supported by the no slip condition at the wall and viscous effects which slows down flow near walls and reduces velocity, whilst continuity ensures flow accelerates in the central region with a peak at the centre. Figure 7, provides further support for this theory as we see velocity is lowest at walls and rises to highest peak at the centre right, with the provided case reaching a slightly higher velocity value of – 0.181 m/s.

Flow separation occurs post- stenosis due to the increase in cross-sectional area, and adverse pressure gradient forms in direction of flow causing the velocity near the walls to drop to nearly zero, As a result of this the flow separates from the walls and streamlines of slow negative velocity forms. This is evident in all simulations, due to the region of reverse flow seen at the walls in figure 8, Here we can observe reverse flow in the coarse solution is not resolved, whereas in the fine mesh of both order accuracy its better resolved and more visible.

Also looking at both my 2^nd order medium mesh and the provided 2^nd medium mesh, it can be noticed that they yield slightly different results, this could be due to the reduced mesh width in the provided case which lead to reduction of artificial viscosity as mentioned earlier.

Figure 8, Velocity vector plots showing extent of flow separation below the stenosis. From top (finest mesh) to bottom (coarser mesh) for both 1st order and 2nd order accuracy, alongside my 2nd order medium mesh (far left).

Further evidence for the accuracy of flow separation below the stenosis is seen in figure 9, where we notice in the velocity magnitude contour a region of low velocity suddenly experience a rise in velocity, this is evident in the small lighter blue area that appears in the very dark blue low velocity region. This corresponds with the region of pressure drop with extremely low WSS values at the walls (area of darker blue in pressure contours) in the same region reverse flow seem to be happening with arrows in opposite direction in image c)., Therefore providing evidence for flow separation post-stenosis where there's an abrupt change in velocity due to the adverse pressure gradient.

Figure 9, evidence of flow separation: a) velocity magnitude contour, b) pressure contour below stenosis c) velocity vector showing flow separation.

Finally, further evidence to support the accuracy of flow separation is seen in figure 10, at the left and right walls of the plane below the stenosis. Here we see for all mesh types and both 1^st and 2^nd order accuracy we can observe a region of recirculation near the walls with the 2nd order simulations having a more negative y- velocity suggesting a slower recirculation than the 1^st order solutions.

Also flow separation at left and right walls should be the same, but as we can see this is not the case as it differs slightly, this could be due to discrepancies and errors such as artificial viscosity that could've been introduced in the simulation. This is further supported by the velocity vector plots seen in figure 8, where we see for all simulations flow separation seem to be stronger at the right wall. The presence of errors like artificial viscosity reduces the accuracy of results.

Figure 10, velocity profile of all provided plots with red circles around region of flow separation.

3 Summary

To conclude, the second order accurate of the highest mesh refinement (fine) from the provided solutions seemed to provide the most accurate solution due to the lower effect of artificial viscosity and the reduced truncation errors when using a fine mesh of 2^nd order accuracy. This was evident in the smoothness of the fine mesh contour lines for 2^nd order. Also after comparing the provided case with my own simulation of the same medium mesh level, it is clear the provided case was the more accurate solution with a more refined mesh showing smaller mesh width, thus reduced artificial viscosity which leads to a more accurate solution.

Furthermore, CFD is a method used by engineers to solve complicated fluid flow problems, however the user should bear in mind this may not get the most accurate results due to errors within the system that form depending on level of refinement or order of accuracy. Also from my results I noticed the 1^st order simulation provided faster results, though less accurate. Whereas the 2^nd order stimulation though slower and more computationally expensive, yet provided the most accurate results. Finally, CFD can have applications in engineering, if a simplified model of fluid flow is needed for lower cost and less time.

4 References

Fox JA, Hugh AE: Static zones in the internal carotid artery: Correlation with boundary layer separation and stasis in model flows. BrJ Radiol 1970;43:370-3

Müller, J.-D. (2015). Essentials of Computational Fluid Dynamics, CRC Press.