3.1. Aneurysm flow field topology

The contours of velocity magnitude at the aneurysm midplane are shown in figure 2 for all parent vessel flow rates and curvatures. Streamlines are used to highlight the direction of the flow inside the aneurysm with flow in the parent vessel (not shown) travelling from right to left. At the lowest $De$, upper left, the flow expands into the aneurysm at the leading (upstream) edge and exits at the trailing (downstream) edge of the aneurysmal cavity, resulting in counterclockwise rotation (co-rotating with the parent vessel flow) in the aneurysmal dome. This flow topology is similar to low-$Re$ expansion flow over a cavity, without the existence of corner recirculation eddies (Moffat eddies) due to the quasi-spherical geometry. In the work done by Higdon (Reference Higdon1985), high-$Re$ flow inside of a circular cavity with this level of depression was shown to be comprised of a single counter-rotating vortex. The fact that flow in this aneurysm is attached and co-rotates with the parent vessel flow is the result of the FDS slowing down flow inside the cavity, and subsequently acting as a barrier between the viscous-dominated aneurysm flow and high inertia of the parent vessel flow.

Figure 2.

Velocity contours and streamlines at the streamwise midplane of the aneurysm sac. Flow inside the parent vessel (not shown) is from right to left. Parent vessel flow rate (or $Re$) increases with descending rows and parent vessel curvature increases moving left to right across columns. The value of $De$ increases along the diagonal.

As $De$ increases, however, either by increasing parent vessel flow rate (down the rows) or curvature (left to right, across the columns), the flow inside the aneurysm begins to separate and two recirculation eddies form. At intermediate $De$, both clockwise and counter-clockwise rotating eddies exist inside the aneurysm, and the entrance location of flow into the aneurysm shifts downstream. Further increases in $De$ (panels along the bottom and right edge of the $4 \times 4$ image matrix) result in a single recirculating vortex with the flow entering the aneurysm at the trailing edge and exiting at the leading edge of the cavity.

In figure 3, the velocity magnitude is shown at the midplane of the parent vessel for all values of $Re$ and $\kappa$. The location of peak velocity magnitude can be seen to shift towards the outer wall with increasing $De$ (which increases with increasing parent vessel curvature and $Re$). The axial velocity profiles, plotted in figure 4(a), are computed by dotting the three-dimensional velocity measurements with the vector normal to the extraction line. In the straight aneurysm model, the axial velocity profile is parabolic, as expected at this laminar $Re$, and increases in magnitude with flow rate. In the curved models, the axial velocity profile begins to shift towards the outer wall. The construction of the aneurysm models is such that the arc length of the model shortens with increasing $\kappa$. In figure 4(a), the shift in axial velocity towards the outer wall is less pronounced in the model with $\kappa = 0.22$ mm$^{-1}$ than it is for the two other models with smaller curvature. This is because the arc length from entrance region to measurement plane is shortest for $\kappa = 0.22$ mm$^{-1}$, and the flow is still developing, a phenomenon highlighted in Bovendeerd et al. (Reference Bovendeerd, Van Steenhoven, Van De Vosse and Vossers1987).

Figure 3.

Contours of velocity magnitude measured at the midplane of the aneurysm parent vessel. Parent vessel flow rate ($Re$) increases with descending columns and parent vessel curvature increases left to right across columns. The value of $De$ increases along the diagonal.

Figure 4.

(a) The axial velocity profile extracted from each experimental condition at the midplane of the parent vessel. (b) A zoomed-in view of the axial velocity profile at the leading edge of the aneurysm neck. Velocity profiles are coloured blue if the flow inside the aneurysm has separated and is rotating clockwise and green if any counterclockwise rotation exists inside the aneurysm.

In figure 4(b), the details of axial velocity measurements near the outer wall, at the leading edge of the aneurysm, are highlighted. The velocity field is extracted along a line extending from the leading edge of the aneurysm sac to the centroid of the parent vessel. Each line in the figure is coloured by the flow topology inside the aneurysm (clockwise or counterclockwise vortices) for the corresponding flow and curvature conditions. Inspection of the parent vessel flow field, right at the leading edge, shows that the topology of the flow inside the aneurysm is a function of the streamwise velocity at the aneurysm neck (specifically the shear stress at the cavity leading edge that determines whether flow detaches or remains attached to the cavity wall), which in the case of curved vessels is parameterized by $De$.

The axial velocity profiles extracted from conditions that resulted in a single recirculating vortex inside the aneurysm are coloured in blue and those with any counterclockwise rotation are highlighted in green. A clear separation is seen in the slope of the axial velocity at the leading edge of the aneurysm cavity (the shear at the wall) for the two flow topologies. The critical $De$, above which flow at the leading edge separates and the sense of rotation inside the aneurysm is entirely opposite to the bulk flow direction (clockwise), is ${\approx }180$.

At a sufficiently low $\textit {Re}$, the flow inside the aneurysm remains attached to the wall and travels in the same direction as the bulk flow (upper left corner in figure 2). In the straight model, increasing $\textit {Re}$ eventually leads to flow separation, recirculation and a downstream shift in the entrance location of parent vessel flow into the aneurysm. As the curvature of the parent vessel increases, this transition from attached, forward-rotating flow to fully counter-rotating circulation flow in the aneurysmal sac happens at lower values of parent vessel $\textit {Re}$.

3.2. Aneurysm velocity characterization

The velocity vectors at the neck of the aneurysm are extracted along curves just above the FDS, at each measurement plane. The vectors extracted from the midplane are shown in figure 5, coloured by the magnitude of the component normal to the aneurysmal neck surface (${\boldsymbol {u}}\boldsymbol {\cdot } {\boldsymbol {n}}$). Again, parent vessel $\textit {Re}$ increases with descending rows and parent vessel curvature increases across columns from left to right. The direction of the flow entering the aneurysm is highlighted here with non-separated, counterclockwise rotation occurring at the lowest $De$ (upper left) and separated, clockwise rotation occurring with increasing $De$ (bottom right).

Figure 5.

Velocity vectors extracted from the aneurysm neck at the midplane. Vectors are coloured by the magnitude of $\boldsymbol {u} \boldsymbol {\cdot } \boldsymbol {n}$.

Aneurysm inflow velocity, $\bar {u}$, is plotted against flow rate in figure 6(a) (Part 1, (2.1)). The average aneurysm inflow velocity increases with increasing parent vessel flow rate and with parent vessel curvature for all models. Figure 6(b) shows the aneurysm inflow Reynolds number ($Re_{neck} = \bar {u} D_{neck}/\nu$) plotted against parent vessel $De$. For all models with curvature, the $Re_{neck}$ values collapse onto a single curve, indicating that the amount of flow that enters the cerebral aneurysm sac, treated with FDS, is a function of parent vessel $De$ only. The shift in axial velocity towards the outer wall in a curved vessel is accompanied by secondary vortices transporting fluid along the centre plane from the inner to the outer wall. The magnitude of the outward radial velocity component at the midplane of the parent vessel increases with increasing $De$ and, as a result, the magnitude of flow entering the aneurysm also increases.

Figure 6.

The average aneurysm inflow velocity for each value of parent vessel curvature and flow rate shown in (a) dimensional and (b) non-dimensional units.

In the straight model, the radial velocity component is zero upstream of the aneurysm. At the leading edge, flow expands into the aneurysm, creating a radial velocity component which is a function of axial velocity, aneurysm geometry and stent porosity. Increases in parent vessel inertia lead to separated flow with the point where the parent vessel flow enters the aneurysmal sac moving downstream (but never reaching the trailing edge). Because the vessel is not curved, there are no increases in parent vessel radial velocity due to secondary vortices, so the increase in aneurysm inflow velocity with parent vessel $\textit {Re}$ is limited, as seen in figure 5.

In the curved vessels, the aneurysm inflow velocity is heavily influenced by the radial velocity generated by the pressure gradient associated with the sudden expansion, and the secondary (Dean's) vortices in the parent vessel created by the curvature and subsequent inertial instability. Since aneurysm inflow velocity is approximately equal for all cases at the lowest parent vessel $\textit {Re}$, the radial velocity that arises due to secondary vortices is small compared to that due to the pressure gradient at the expansion, for the lowest value of parent vessel $\textit {Re}$. However, with increasing parent vessel inertia, the aneurysm inflow velocity in all curved models rapidly becomes larger than that in the straight vessel, indicating that the radial velocity in the aneurysm generated by the curvature-induced flow in the parent vessel becomes the driving component in aneurysm inflow velocity at higher $De$.

4.1. Flow structure across the cardiac cycle

In figure 7, the aneurysm flow field is plotted at five instants during the cardiac cycle for the condition $\kappa =0.0625\ \textrm {mm}^{-1}$, $Re = 320$ and $Wo = 2.6$. In each snapshot of this representative case, streamlines at the aneurysm midplane are overlaid on contours of the $y$ component of velocity (component of velocity entering and exiting the aneurysm), $u_{aneurysm}$. At the beginning of the cardiac cycle (A), as the parent vessel flow starts to accelerate, the flow entering the aneurysm remains attached to the leading edge of the aneurysm cavity. Towards the end of the accelerating phase (B), the flow remains attached, yet the primary vortex in the aneurysm begins to shift towards the distal edge of the aneurysm, and the flow entering the aneurysm at the leading edge no longer rotates with the primary aneurysm vortex. Right at peak systole (C), the aneurysm flow separates at the leading edge. This counter-rotating vortex then continues to grow until it encompasses the entire aneurysm dome during the deceleration phase (D). Towards the end of the cardiac cycle (E), the counter-rotating vortex begins to break down and flow at the distal edge of the aneurysm once again rotates with the primary direction of the parent vessel flow.

Figure 7.

Velocity contours and streamlines at the aneurysm midplane for the condition $\kappa = 0.0625\ \textrm {mm}^{-1}$, $Re = 320$ and $Wo = 2.6$ at five locations in the cardiac cycle. Velocity contours are of the vertical component of velocity, $u_{aneurysm}$.

The structure of the flow at the leading edge of the aneurysm is primarily controlled by two pressure gradients, both functions of the flow rate entering the aneurysm, $Q(t)$. As flow expands into the aneurysm, an adverse pressure gradient develops at the leading edge of the aneurysm wall which is proportional to $Q(t)^2$. A second pressure gradient exists due to the acceleration of the fluid which is proportional to $\partial Q(t)/\partial t$. As soon as flow begins to enter the aneurysm, the adverse pressure gradient begins to grow. At the beginning of the cardiac cycle, however, when $\partial Q(t)/\partial t$ is positive, the acceleration-based gradient is also positive and acts to keep the flow attached to the leading edge. As $Q(t)$ increases, however, and the cardiac cycle moves through systole, $\partial Q(t)/\partial t$ decreases while the expansion-based pressure gradient increases in magnitude. Eventually, the magnitude of the expansion-based pressure gradient becomes larger than the acceleration-based gradient, and flow at the leading edge begins to separate. Throughout diastole, with the acceleration-based gradient becoming negative, the separated aneurysm vortex continues to grow until it fills the entire aneurysm cavity.

4.2. Inertial effects (Dean and Reynolds numbers)

In figure 8, velocity contours and streamlines are plotted for the same aneurysm model ($\kappa =0.22\ \textrm {mm}^{-1}$) at increasing values of parent vessel $\textit {Re}$. Each panel shows the flow field at the same instant in the cardiac cycle. At the lowest values of parent vessel $\textit {Re}$, the flow remains completely attached to the aneurysm wall, rotating in the direction of the bulk flow. As $\textit {Re}$ increases, the flow topology shifts, from attached forward-rotating to separated counter-rotating flow at the same point in the cardiac cycle. In other words, the flow begins to separate at an earlier point in time as $\textit {Re}$ is increased.

Figure 8.

Velocity contours and streamlines at the aneurysm midplane for the condition $\kappa =0.22$ mm$^{-1}$, $Wo = 2.6$. Each panel highlights a snapshot from the same location in the cardiac cycle (halfway through acceleration) for increasing values (left to right) of cycle-averaged $Re$. Velocity contours are of the vertical component of velocity, $u_{aneurysm}$.

This is further visualized in figure 9, where the the area-averaged vorticity, or aneurysm circulation, $\varGamma (t)$, is plotted over the cardiac cycle, normalized by the circulation calculated at peak systole, $\varGamma _s$, for the same conditions shown in figure 8. The dashed vertical line indicates the snapshot in time where the four panels in figure 8 are shown. At the lowest value of parent vessel $\textit {Re}$, the aneurysm circulation becomes negative at the beginning of the cardiac cycle and eventually transitions to positive (indicating primarily separated flow) about one-fifth of the way through the cardiac cycle. As $\textit {Re}$ is increased, this transition from attached to separated flow with the average vorticity becoming positive occurs at an earlier point in the cardiac cycle. The time it takes for separation to occur, from the beginning of the cardiac cycle, is defined as the separation time, $T_s$.

Figure 9.

Aneurysm averaged vorticity plotted across a single cardiac cycle for $\kappa = 0.22\ \textrm {mm}^{-1}$, $Wo = 2.6$. Aneurysm vorticity is normalized by peak systolic vorticity.

Increasing the parent vessel $Re$ results in an increase of flow entering the aneurysm (figure 6a). As the expansion-based pressure gradient is proportional to $Q(t)^2$, and the acceleration-based pressure gradient is only proportional to $\partial Q(t)/\partial t$, increasing $\textit {Re}$ results in faster growing and dominating adverse pressure gradients. Thus, for the same geometry, an increase in $\textit {Re}$ results in separation that occurs earlier in the cardiac cycle.

The transition from attached to separated flow during the cardiac cycle has been previously documented in studies of untreated (Asgharzadeh & Borazjani Reference Asgharzadeh and Borazjani2016) and FDS-treated (Lieber et al. Reference Lieber, Livescu, Hopkins and Wakhloo2002) aneurysms, but only in straight idealized aneurysms. The dependence of the time of separation on inertia and the topology of the flow over a spherical cavity was highlighted by Sobey (Reference Sobey1980), where an increase in free-stream $Re$ results in lower separation times. Here, we extend that work and also explore the effect of free-stream geometry by varying the curvature and including $De$ in the analysis.

In figure 10, velocity contours and streamlines are plotted for four values of parent vessel curvature, at the same value of $Re$ (increasing $De$). All four panels represent the same instant in the cardiac cycle: halfway through the acceleration phase. As $De$ increases, the structure of the flow, at the same instant of time, evolves: the flow remains attached at the lowest value of $De$, while it is fully separated at the highest value of $De$.

Figure 10.

Velocity contours and streamlines at the aneurysm midplane for the condition $Re=360$, $Wo = 2.6$. Each panel highlights a snapshot from the same instant in the cardiac cycle (halfway through acceleration) for increasing values (left to right) of curvature, and thus increasing $De$. Velocity contours are of the vertical component of velocity, $u_{aneurysm}$.

This is the same behaviour seen in figure 8. At the same instant in the cardiac cycle, vessels with larger curvature, or $De$, will have more flow entering the aneurysm cavity (figure 6b). With $\textit {Re}$ and the waveform fixed for all cases in figure 10, the magnitude of the acceleration gradient is approximately equal at the same moment in time. The adverse pressure gradient, proportional to the flow rate entering the aneurysm, however, will have a larger value at the same instant in time, resulting in flow separation at an earlier time in the cardiac cycle.

In figure 8, by increasing $\textit {Re}$ while maintaining the vessel curvature, $De$ is also increased. This begs the question as to whether flow separation is predominately a $\textit {Re}$ or $De$ effect. In figure 11 each snapshot pair shows streamlines and velocity contours plotted at the same $De$, but different $\textit {Re}$, at the same instant in the cardiac cycle. Here $De$ is matched at different $\textit {Re}$ by selecting vessels with different curvature. At both values of $De$, the flow patterns are nearly identical despite having different values of parent vessel $\textit {Re}$. This suggests that the flow topology inside the aneurysm and the separation time are truly a function of $De$ only.

Figure 11.

Velocity contours and streamlines for two models with approximately matching $De$: (a) $De\approx 180$ and (b) $De\approx 230$. For each snapshot pair, models are selected with different $Re$ and $R_c$ but matching $De$. Velocity contours are of the vertical component of velocity, $u_{aneurysm}$.

4.3. Transient inertial effects (Womersley number)

In figure 12(a), the separation time is plotted against parent vessel $De$ for both values of $Wo$. The separation time is shown to decrease with increasing $De$, and the functional decay of separation time with $De$ is the same for both values of $Wo$: at a given $De$, separation occurs at the same physical time in the cardiac cycle, regardless of the flow pulsatility (for the range investigated in this paper). Separation time is again plotted against $De$ in figure 12(b) but normalized by the time it takes to reach peak systole, $T_s$. The horizontal dashed line in figure 12(b) represents the transition in the cardiac cycle between the systolic and diastolic phase of the cardiac cycle. For a given $De$, increasing $Wo$ causes separation to occur at a later relative phase in the cardiac cycle. By looking at the intersection of the data points and the line $t_s/T_s=1$, we can estimate a critical $De$ for which separation occurs in the accelerating or decelerating phase of the cardiac cycle. As an increasing $Wo$ pushes separation into a later phase of the cardiac cycle, the critical $De$ for separation occurring during systole also increases: it takes the values $\approx$75 and 150 for $Wo= 2.6$ and 3.6, respectively.

Figure 12.

(a) Separation time plotted against Dean number and (b) separation time normalized by peak systole ($T_s)$ for all cases.

To investigate the effect of $Wo$, non-dimensional parent vessel flow rate $Q(t)/Q_{sys}(t)$ and acceleration $\partial Q(t)/\partial t$ are plotted against time, both in dimensional form and non-dimensionalized by the length of the cardiac cycle, for $Re=200$ and both values of $Wo$ (figure 13). The occurrence of separation is primarily dependent on the interplay between the two pressure gradients which are proportional to these curves. Increasing $Wo$ shortens the cardiac cycle which, at the beginning of the cardiac cycle, increases both the magnitude of the instantaneous flow rate and the acceleration as seen in figure 13(a,c). Separation occurs at the same time for both values of $Wo$ (at a given $De$) because the ratio between the expansion-based and acceleration-based pressure gradients remains the same.

Figure 13.

Parent vessel flow rate non-dimensionalized by systolic flow rate $\bar {Q}$ and plotted against time (a) and time non-dimensionalized by cardiac cycle length (b). Time derivative of flow rate (acceleration) plotted against dimensional time (c) and non-dimensional time (d).

In figure 13(b,d), the same curves are plotted against time, non-dimensionalized by the cardiac cycle period. In the acceleration phase, it can be seen that the magnitude of flow rate is nearly identical for both values of $Wo$. However, the magnitude of the acceleration is larger for the larger $Wo$ case. This suggests that flows with a higher $Wo$ are able to remain attached for longer and separate at a later instant in the cardiac cycle, as is seen in figure 12(b). As flow attachment is maintained by this acceleration-based pressure gradient, an increase in $Wo$ will also result in a higher critical $De$ for separation to occur in the acceleration phase.

In the work by Sobey (Reference Sobey1980), it was shown that separation time depends on both $Re$ and the frequency of bulk flow pulsatility or Strouhal number ($St$). An increase in $St$ was shown to increase separation time (non-dimensionalized by period length) for the same value of $Re$. The same trend is shown here except that $Wo$ replaces $St$ and $De$ replaces $Re$ as we have added the influence of curvature. Sobey (Reference Sobey1980) also discusses a critical $Re$ below which separation occurs only in the deceleration phase of the the pulsatile flow phase. This critical separation behaviour is shown to be a function of $St$ and, above $St$ of 0.1, flow is only able to separate in the deceleration phase of the cycle, regardless of $Re$. Values of $St$ of the flow waveforms studied here are below 0.1 ($St = 0.007$–$0.03$), but the trend in figure 12(b) suggests that further increases in $Wo$ would eventually result in a condition where separation only occurs in the deceleration phase of the cardiac cycle.

The effect of unsteady flow on Eulerian metrics is investigated by calculating the Reynolds number based on the circulation inside the aneurysmal sac, $Re_{\varGamma } = \varGamma _{Aneurysm} /\nu$. In figure 14, $Re_{\varGamma }$ is plotted against $De$ for all values of $Wo$ investigated here, including steady flow ($Wo=0$). For pulsatile cases, $Wo = 2.6$ and $3.6$, the circulation inside the aneurysm is phase-averaged over the cardiac cycle. First, it can be seen that the strength of the vorticity inside the aneurysm is a function of $De$, with the strength of vorticity increasing with increasing $De$. The cycle-averaged vortex strength for both values of $Wo$ is also shown to be equivalent for a given $De$, with the phase-averaged values for pulsatile cases equalling the corresponding steady flow values ($Wo=0$). This suggests that pulsatility has little influence on the phase-averaged values of vortex strength inside the aneurysm for the parameter range studied here.

Figure 14.

Aneurysm vortex Reynolds number as a function of Dean number for all values of Womersley number.

In a numerical analysis, Asgharzadeh & Borazjani (Reference Asgharzadeh and Borazjani2016) investigated the effect of $Re$ and $Wo$ on the vortex structure in untreated idealized and patient-specific aneurysm models. It was shown that $Wo$ has little effect on the structure and characteristics of the primary aneurysm vortex, but that it did affect the location of the vortex at different instances of the cardiac cycle. This is similar to the findings presented here where the primary structure of the aneurysm flow field and cycle-averaged metrics are not impacted by $Wo$ (in the range studied here) but the temporal evolution of the aneurysm vortex during the cardiac cycle is affected.

Due to the FDS, the aneurysm vortex Reynolds number is small ($Re_{\varGamma }<1$). Thus, the viscous-dominated aneurysm flow behaves similarly to a linear Stokes flow where cycle-averaged metrics for pulsatile flows are equal to the steady flow values with the same mean free-stream flow. At higher values of aneurysm $Re$, we would expect that the curves shown in figure 14 would begin to diverge but, for the parameter range studied here that is representative of the physiological values in FDS-treated cerebral aneurysms, $Wo$ is shown to have little influence on the characteristics of the flow, and $De$ is the driving non-dimensional parameter (centrifugal inertia in the parent vessel injecting flow into the aneurysm cavity is the main mechanism).

5.1. Steady free-stream flow

The primary function of FDS treatment is to restrict flow into the aneurysmal cavity and create an environment inside the aneurysmal sac that promotes the growth of a stable thrombus, isolating the aneurysmal wall from mechanical stresses associated with the flow of blood and, thus, reducing the risk of rupture. In figure 15, the Reynolds number computed from the average inflow velocity measured at the neck of the aneurysm, $Re_{neck}$, is plotted as a function of parent vessel $De$ for treated and untreated cases under steady free-stream flow conditions. Across the entire range of $De$ studied, the magnitude of velocity entering the aneurysm is approximately an order of magnitude lower in the treated than in the untreated cases. Additionally, the degree or magnitude of flow reduction is nearly constant across all values of $De$, as is evident in figure 15.

Figure 15.

Cavity inflow Reynolds number as a function of Dean number for treated and untreated models.

This significant reduction in aneurysmal flow results in markedly different flow topologies in the aneurysmal cavity. From figure 3 in Part 1 (Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021), the flow is shown to be separated at the cavity leading edge, and rotating in the direction opposite to the free-stream flow for all vessels (with different curvatures) and all values of $Re$. However, in figure 3 in the present paper, the flow field remains attached to the aneurysmal wall at low $De$ and begins to separate forming a counter-rotating vortex as $De$ increases. A comparison between three treated (bottom) and untreated (top) flow fields for the same conditions is shown in figure 16. The first two cases are for straight vessels ($\kappa = 0.0$ mm$^{-1}$) at $Re$ of 120 and 360. For both untreated cases, the flow is fully separated with a single counter-rotating vortex inside the cavity. For the treated model, the flow remains fully attached at the lowest $Re$, but begins to separate at the higher $Re$ (in the second panel). The third panel shows the flow fields for a higher-curvature vessel ($\kappa = 0.14$ mm$^{-1}$) and $Re$ of 140. At this higher $De$, both flow fields exhibit the same topology, but the magnitude of velocity and circulation is much higher in the untreated case.

Figure 16.

Streamlines plotted at the aneurysm midplane for untreated (top) and treated (bottom) cases.

Separation in the aneurysm cavity occurs when the adverse pressure gradient that results as flow expands into the aneurysm is greater than the inertia in the parent vessel near the wall, at the aneurysmal leading edge. For the flow to remain attached at the leading edge, the flow near the aneurysmal wall must be sufficiently slow to enable viscous forces to dominate over inertia. This condition is only achieved, in the flow regimes relevant for cerebral aneurysm, when a FDS is placed across the neck of the aneurysm and flow entering the sac is significantly restricted.

5.2. Unsteady flow

For unsteady free-stream flow conditions, the trends are similar to those for the steady free-stream conditions: the magnitude of flow entering the aneurysm is decreased for treated cases resulting in large differences in the topology of the flow that develop across the duration of a cardiac cycle. In Part 1 (Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021), the flow in the aneurysm was shown to be counter-rotating for the entirety of the cardiac cycle, for all conditions. The flow vorticity inside the aneurysm increases and decreases throughout the cardiac cycle but the primary structure of the aneurysm flow remains unchanged with a single counter-rotating vortex persisting in the aneurysm cavity. In the present Part 2, it is shown that at low values of $De$ the flow in the aneurysm cavity is attached at the beginning of the cardiac cycle, and separates only during the deceleration phase of the cardiac cycle. At higher values of $De$, the instant at which the flow becomes separated and counter-rotates shifts earlier towards the start of the systolic phase, but all cases start the cardiac cycle in the co-rotating, attached flow topology.

In figure 17(a), the circulation inside the aneurysm is normalized by peak circulation and plotted throughout the cardiac cycle for three conditions, for both treated and untreated cases. For all of the treated vessels, at the start of the cardiac cycle the vorticity is negative indicating attached flow. Circulation becomes positive, indicating primarily separated flow inside the aneurysm, during the first third of the cardiac cycle. As $De$ increases, this transition point, from co-rotating to counter-rotating flow, happens earlier in the cardiac cycle (figure 12). For the untreated cases, the circulation is positive (separated flow) throughout the entire cardiac cycle and the baseline magnitude of circulation as a percentage of peak circulation increases with increasing $De$. The higher the value of $De$, the more momentum is maintained from the previous acceleration phase, resulting in higher values of circulation at the start of a new cardiac cycle.

Figure 17.

(a) Aneurysm circulation normalized by peak circulation (systole) plotted throughout the cardiac cycle for three representative models ($Wo = 2.6$ for all models). (b) The magnitude of peak circulation plotted as a function of Dean number for treated and untreated models at $Wo= 2.6$ and 3.6.

In figure 17(b), the peak circulation is plotted as a function of $De$ for treated and untreated cases at $Wo = 2.6$ and 3.6. For both treated and untreated cases, the value of peak circulation increases with increasing $De$. Again, there exists an approximate 10-fold reduction in the magnitude of peak circulation between the untreated and treated cases. The exception is the circulation at the lowest values of $De$, where there is a 1000-fold reduction in circulation for the treated case, compared to the untreated. For treated cases, the flow starts the cardiac cycle attached to the wall and separates during the deceleration phase. This late transition to counter-rotating flow delays the development of a counter-rotating aneurysm vortex, and prevents the vortex from reaching a significant strength, as it does for conditions that start the cardiac cycle in the separated flow topology. While the effect of $Wo$ on values of peak circulation is negligible for the untreated cases and the treated cases with higher $De$, there is larger variation in peak circulation between $Wo = 2.6$ and $Wo = 3.6$ at low values of $De$ for treated cases. This subset of cases, treated with small $De$, is most significantly affected by flow switching (from attached to separated flow). Thus, for a given $De$, the vorticity in cases with a higher $Wo$, or shorter cardiac cycle, has less time to develop and tends to have a smaller peak circulation value than cases with a smaller $Wo$.