Drag coefficient

Drag coefficient at normal incidencek₀

The simulated drag coefficient at normal incidencek₀ is compared with the following four empirical correlations listed in Section 2.2.2: the model by Wieghardt (Eq. 2.8), as cited in [35], the model by Ehrhardt (Eq.2.10), as cited in [35], the model by Wakeland and Keolian (Eq.2.11) proposed in [69], and the model by Brundrett (Eq.2.13) proposed in [9].

Fig.2.3to2.5compare the numerical data with model predictions at porosities of 0.6,0.7 and 0.8. Fig. 2.3 shows that the equation of Ehrhardt provides a best match for the drag coefficient at a screen porosity of 0.6, but Fig.2.4and Fig.2.5show that it does not work well at porosites of 0.7 and 0.8. For porosities of 0.7 and 0.8, it can be seen that the model of Wakeland et al and the model of Brundrett are more consistent with the simulation results. However, while these two models provide good preditions fork₀at low Reynolds number (Re_d<5), they fail at higher Reynolds numbers (5<Re_d<40). In the latter regime, Wieghardt’s model agrees much better with the numerical data. All in all, none of the models, taken by itself, provides satisfying results for the full range of Reynolds numbers and porosities of relevance to stents in aneurysms. But we notice that while the curve of Wieghardt’s model does not exhibit the expected slope, the slope of the three other model curves is in accordance with the simulation

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results in a low Reynolds number region. This slope stems from the term proportional to 1/Re_d, which is used in all the three models. To emphasize this similarity, we cast all three models into the following generic shape:

k₀(Re_d) =a/Re_d+b, (2.27)

whereaandbeither depend on porosity or are constant parameters.

It should be pointed out that the discrepancies between the empirical models and our simu-lations are not surprising, as these models have themselves been obtained through corresimu-lations with different experimental setups, which are only partly comparable to the configuration of our numerical simulation. But given the fundamental importance ofk₀(the models forkⁿand

2.4 Results 23 forBwill be based onk₀), we now propose a new model, which yields a better match with our numerical simulation than the four models from the literature. It remains to be determined in future research if our own model is also appropriate for real, 3D stents in realistic arteries, or if one of the four models from the literature has a more general range of validity for actual stent simulations.

Based on Eq.2.27, we propose a combination of terms depending on 1−β and on 1/β, inspired by the four models of the literature for the coefficients ofaandb, which leads to a good match with our data. From a least-square fit, we obtain the following law:

k₀= The match between the numerical data points and our model, Eq.2.28, is very good, as shown in Fig. 2.6, and we will therefore use this model as a starting point to derive more advanced screen-model based stent laws.

Fig. 2.6 The fitting correlation for the drag coefficientk₀at different porosities.

Drag coefficient at an incident anglekⁿ

Fig. 2.7 Compare simulatedkⁿwith two models at different incident angles

2.4 Results 25 Now that the equation fork₀ is known, we turn to computing the drag coefficient of a more general screen, whith a non-rothogonal alignment with respect to the flow. In Section2.2.2, two models for the computation of kⁿ are presented. Both of them are now tested agains numerical data, obtained with different parameters, including the angleθ₁, the porosityβ, and the Reynolds number Re_d. According to Eq.2.18,k_ncan be easily obtained from the pressure drop across the stent. The results are presented in Fig.2.7. The hollow symbols represent the simulation results, while the lines and the crosses stand for the model of Schubaueret aland the model of Reynolds respectively.

It is observed that both models accurately predictkⁿfor small angles, such as 30^◦and 45^◦, deviate slightly atθ1=60^◦, and show substantial devations at large angles (75^◦and 85^◦).

Furthermore, both models perform well at low Reynolds number Reⁿ_d, when Reⁿ_d<1, but some deviation appear when Reⁿ_d>1. The discrepancies are particularly significant at angles of 75^◦and 85^◦.

As far as porosity is concerned, at intermediate Reynolds number region (Reⁿ_d >1), the model of Schubaueret alis more accurate forβ =0.6, while the model of Reynolds is more accurate forβ =0.8, as it clearly stands out in the plots corresponding to angles of 45^◦and 60^◦.

In summary, both models are consistent with the simulation results at low incident angles for all the simulated Reynolds numbers. For large angles (θ₁≥75^◦), the two models still perform well at low Reynolds numbers (Reⁿ_d<1), but they can not accurately describekⁿin the intermediate Reynolds number region(Reⁿ_d>1). The difference between the two models depends on the porosity. The model of Schubaueret alis better for the comparatively low value ofβ =0.6 and the model of Reynolds is suitable for the comparatively high value ofβ =0.8.

The model of Schubaueret alcan be considered a good candidate for the intermediate value β =0.7, given its simplicity.

2.4.2 Deflection

The measuring of the deflection coefficient

The deflection coefficientB, defined by Eq.2.5, represents the deviation of velocity caused by the screen. According to Eq.2.5,Bcan be obtained by measuring the downstream and upstream velocities. Another method is given by Eq.2.7, which introduces a relationship between the deflection coefficientBand the tangential stressτ. The tangential component of the drag force on the cylinder struts is measured directly in the simulation, andτ is computed by dividing the tangential drag force by the cross sectional area of the stent. As a next step,Bis computed fromτ. For the sake of discussion, one may distinguish between the coefficientBin Eq.2.5, as

it is computed from the velocity (and which will therefore be calledB_v), and the coefficientB in Eq.2.7, which it is obtained from the drag force (and will therefore be calledB_f). Eq.2.7, which is of central importance to our model, is only valid when B_f is equal toB_v. Fig.2.8 verifies the consistency of these two approaches.

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Fig. 2.8 Two methods for computing the deflection coefficientB

In practice, these two methods can actually produce quite different results. It should be reminded that Eq.2.7is derived from Eq.2.4, which assume the existence of an asymptotic upstream and downstream flow far from the screen. According to our numerical experiments, B_f is very sensitive to the distances separating the screen with the inlet and the outlet boundaries.

At low Reynolds number in particular, a very long distance between the inlet and the screen is required for the model to be valid.

Moreover, Fig.2.8shows that the data points corresponding to a given porosity are strongly correlated, suggesting the existence of a simple relationship betweenkⁿandB. At low angles (and using a logarithmic axis forkⁿ), the correlation is linear, while strong deviations from this trend are observed at large angles (θ₁=75^◦and 85^◦). Consistently with previous observations, it is concluded that the hydrodynamic effect of screens obeys different laws at very large inclinations than at low and moderate inclinations.

Equations forB

As described in Section 2.2.3, many attempts have been made to establish the relationship between the drag coefficient and deflection coefficient, and two main families of correlations are proposed: one correlatesk_θ1 withB, as shown in the model of Schubauer et al (Eq.2.21) and the model of Tayler et al (Eq.2.22). The other relateskⁿtoB, as presented in the model of Elder (Eq.2.23) and the model of Gibblings (Eq.2.24).

2.4 Results 27

Fig. 2.9 Compare the existing models ofBwith simulation results

Fig.2.9compares the deflection coefficient measured in the simulation with the different model predictions. It is seen that none of the models provides a good match forB, and we there-fore need to formulate our own, more appropriate model. A qualitative comparison between Fig.2.9aand Fig.2.9bshows that the correlation betweenBandk_θ1 strongly depends on the porosity, while the correlation betweenBandkⁿexhibits a more uniform trend, independent on porosity. Based on this observation, we therefore formulate a model ofBdepending onkⁿ.

It can further be pointed out that although none of the models directly matches the numerical data, the model of Gibblings follows most closely the general trends of the numerical curve.

This observation is compatible with the findings of Laws and Livesy [37], who report that a best overall agreement is obtained with the Gibblings [23] model. It is therefore natural to use the general shape of this equation,

B=1+ak_θ1−[(ak_θ1)^b+c]^1/b, (2.29) as a basis for the new model, and to reassign the constants to better fit the numerical data.

A least-square fit leads to the following model:

B=1+ kⁿ Fig.2.10shows that the fitted model of Eq.2.30matches the numerical results very well.

Thus, with the constants adjusted to our specific 2D numerical setup, the model of Gibblings turns out to be of very high quality.

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Fig. 2.10 The fitting correlation of the deflection coefficient

In summary, we have proposed in Sections2.4.1and2.4.2two model equations that allow to compute the values ofkⁿ and B for our 2D stents. From there, Eq. 2.26 can be used to compute the body forces f_nand f_t, and apply them as a replacement for a real flow diverter in a coarse-grained fluid flow simulation.