12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

TURBULENCE IN SIMULATIONS OF BLOOD FLOW IN CEREBRALARTERIES WITH ANEURYSMS

Rafaello D. Luciano
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon,
SK S7N 5A9, Canada
rafaello.luciano@usask.ca

Xiongbiao Chen
Department of Biomedical Engineering
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon,
SK S7N 5A9, Canada
xbc719@mail.usask.ca

Donald J. Bergstrom
Department of Mechanical Engineering
University of Saskatchewan
57 Campus Drive, Saskatoon, SK S7N 5A9, Canada
don.bergstrom@usask.ca

ABSTRACT

porting evidence of aneurysm bruits and turbulence throughexperimental means. Appropriately evaluating temporal phenomena and turbulence is expected to be of importance to understanding the mechanobiology of aneurysm growth and rupture. Studies have discussed how accurate temporal wall shearstress gradients (White et al., 2005) and turbulence (Davies
et al., 1986) are important in the mechanobiology of endothelial cells, which line the interior of blood vessels.

In recent years, the number of studies using computational fluid dynamics to numerically simulate blood flow inarteries with aneurysms has been increasing dramatically. Although turbulent-like localized flow instabilities have been observed in studies on patients and laboratory measurements,
these instabilities were rarely reported in studies involving numerical simulations. These instabilities can be important forthe pathophysiology of aneurysms. Our understanding of what
is required to correctly simulate these flow instabilities is still
rudimentary. The present study therefore investigates OpenFOAM simulations of a realistic intracranial aneurysm geometry, looking to better understand the flow and verify the requirements, in terms of grid resolution, boundary conditions
and perturbations, for reproducing these flow instabilities. The
results reinforce that the outcomes of numerical simulations
are very sensitive to the user set-up. The assumption by researchers that the flow is laminar is misleading, leading to
the usage of methodologies that will not adequately capture
the turbulence that may be present despite the very low flow
rates. In addition, using fine grids with small timesteps is not
sufficient to adequately capture and evaluate turbulence. Finally, the results provide further evidence that turbulence may
be more prevalent in cerebral blood flow than anticipated.

Conversely, reports of instabilities are relatively rare in
the aneurysm numerical simulation literature. This can be explained by the expectation that flow will be laminar due to thelow flow rates, and the common usage of simulation setupsthat would be incapable of appropriately capturing and evaluating such instabilities. For example, Valen-Sendstad et al.
(2014) discussed how different solver numerical settings influenced the observation of flow instabilities in aneurysms. The
authors point out that the typical expectation of laminar flow inthe literature is self-fulfilling, as overlooking the possibility offlow instabilities may lead to the usage of schemes and methods that render the simulations incapable of detecting said flow
instabilities. However, the authors did not discuss what is necessary for the flow instabilities to appear in these simulations,nor discuss possible sources of numerical error or disturbanceand how they would relate to the flow instabilities observed.
Pipes are the classical engineering geometry most related
to that of arteries, and several studies, including the review of
Eckhardt et al. (2007), lead to the understanding that unless anumerical simulation is perturbed enough, it will not develop
from the laminar state into a turbulent one.

INTRODUCTION
The review of Ku (1997) discussed how there is turbulence in artery stenoses around a Reynolds number of only
300, with the pulsatility of blood flow creating a periodictransition between laminar flow and turbulence. Additionally,aneurysm bruits, which are characteristic sound frequencies
observed in patients with aneurysms, were associated in thepast to turbulence with the explanation that an aneurysm may
act as a Helmholtz resonator (Olinger and Wasserman, 1977).
Several studies with patients that had intracranial aneurysms orusing aneurysm glass models (Ferguson, 1972) provided sup-However, there are various other studies and examples ofsimulations in other fields where authors observe turbulencearising seemingly naturally, with no clear boundary or initial condition as the source of perturbation. This leads to thequestion of what the sources of perturbation triggering theseturbulence-like flow features in cerebral aneurysm simulations
are, and what is necessary in order to observe turbulence inthese simulations.

112th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
are located along the artery and aneurysm centerlines.
The inlet boundary condition that is used for the transient
simulations is the time-dependent Womersley velocity profile,which comes from the theoretical solution for laminar pulsatile
flow in a straight pipe. The pulsatile flow rate used was obtained using data from real older adult patients (Hoi et al.,2010), with a cardiac cycle period of T = 0.948 seconds. The
inlet Reynolds number for the simulations ranged from 211 to
627 throughout the cardiac cycle, with an average of 379. For
the simulations that used the simplified geometry, which discarded the internal carotid artery (ICA) with the inlet then positioned at the beginning of the middle cerebral artery (MCA),
see Figure 1, the inlet flow rate was estimated using a flow rate
splitting method based on the physiological Murray law anddescribed in Chnafa et al. (2018). A pulsatility dampening factor was also applied on the inlet flow waveform to account for
the reduction in pulsatility observed in patients from the ICA
to the MCA. A pulsatility reduction or dampening of 7.4% wastherefore considered for the simplified geometry with the inlet
at the MCA, based on the research of Zarrinkoob et al. (2016)
with elderly patients.
A steady-state laminar simulation was also used for comparison purposes, and it had a Hagen-Poiseuille profile specified at the inlet, with the flow rate equivalent to the averaged
Reynolds number in the pulsatile simulation. The numericalsetup of this simulation was also more dissipative in order tosuppress the turbulence and obtain a laminar solution.
The no-slip boundary condition is used at the walls, andthe artery walls were assumed to be rigid. Wall compliance isconsidered to be of relatively low importance in cerebral blood
flow, not to mention the associated uncertainties and significantly increased computational costs (Steinman and Pereira,2019). Similarly, modeling blood as a non-Newtonian fluid isalso considered to be of less importance and associated withsignificant uncertainties, so in this study blood is represented
as a Newtonian fluid with kinematic viscosity of 3.5 cP anddensity of 1060 kg/m3 . Lastly, the outlet boundary conditions for the transient pulsatile simulations were estimated using the already mentioned flow rate splitting method based on
the Murray law (Chnafa et al., 2018).
The unrealistic initial condition that is given to the simulation significantly perturbs the initial simulation time steps.
After around one fifth to one fourth of the first cardiac cycle
had been simulated these initial transients were mostly washed
out and not as noticeable. It is relevant to note that the fluid residence time within these arteries is significantly shorter than acardiac cycle. The longest artery simulated has an estimatedmaximum residence time, from the inlet to the furthest outlet,of less than half a cardiac cycle. Due to the initial perturbation, we observed that discarding the first simulated cardiac
cycle made a noticeable difference in the energy spectra andturbulence intensity results. Discarding more than one cardiac
cycle did not make as significant of a difference, particularly
considering the uncertainty due to the reduced number of cardiac cycles being taken into account when discarding more initial cycles. Nonetheless, for this paper we discarded the firsttwo initial cycles of a total of 30 simulated, and present theresults based on the remaining 28 cardiac cycles.

Figure 1. Artery with aneurysm geometry, with dashed line
indicating the location where it was cut for the simplified geometry.

This study investigates blood flow in a realistic artery witha cerebral aneurysm, looking to evaluate the existence of turbulence, and the impact of numerical settings and perturbations on the results. Grid, cycle and phase convergence will be
evaluated, in addition to looking at artificially-added perturbations to the inlet boundary condition, geometry simplifications,turbulence intensities and the turbulence energy spectra.

NUMERICAL MODEL AND METHODS
The direct numerical simulation approach was used forthe transient simulations with OpenFOAM 7, using secondorder accurate differencing for the spatial and temporal derivatives, with the same numerical setup that we used in our previous study (Luciano et al., 2022) with pipes at higher Reynolds
numbers. Similar approaches are widely used in the literature and its quality is additionally documented, for example,
in Komen et al. (2017).
The computational grids or meshes used were composedof a hexahedral core with polyhedral elements to completeit, built using ANSYS Fluent v194. Grids used in this study
ranged from 200 thousand cells in the coarsest grid to 1.5 million in the finest grid, and grid dependency will be discussed
in the results and limitations sections. All the results shown
used the finest grid, unless otherwise specified.
The realistic geometry used for the simulations is froma 28 year old female patient and can be seen in Figure 1. This patient has a brain aneurysm in the first middle cerebral artery bifurcation. This geometry was obtained
from the Aneurisk database, ecm2.mathcs.emory.edu/
aneuriskweb, Case 78, database which is maintained by
the Departments of Mathematics and Computer Science from
Emory University, with further information available in (Sangalli et al., 2014). This specific case was chosen because there
already are reports of significant flow instabilities with this geometry (Natarajan et al., 2020). The probes shown in the figure, which were used for evaluation of the simulations results,

RESULTS AND DISCUSSION
Figure 2 shows the results on different grids for thevolume-averaged turbulence intensity within the region near
the aneurysm. These grid convergence results are based on
simulations with 28 cardiac cycles averaged and data sampled2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
15.0

60
Turbulence intensity (%)

Turbulence intensity (%)

14.8
14.6
14.4
14.2
14.0
13.8
13.6

Probe 2
Probe 1

50

40

30

20

13.4
0

1

2

3

1/Number of Grid Cells

101
102
Number of Averaged Phases

4
×10

−6

Figure 3.
sults.

Figure 2. Grid convergence results for volume-averaged turbulence intensity. The region for volume-averaging is the one
highlighted with no transparency in the geometry view at the
top right corner.

103

Phase convergence of the turbulence intensity re-pled per cardiac cycle, the results obtained are seen in Figure 3. For this evaluation, the raw data was sampled at specified points (see the probes in Figure 1) at solver resolution witha sampling frequency of about 74 kHz, or 70,000 time steps
per cardiac cycle, for the 28 cardiac cycles averaged. The raw
probe data was then “sampled” at different frequencies, using
factor frequency numbers with no interpolation, to simulate asaccurately as we can what would happen if the actual simulation data had been sampled at a lower frequency. The data forprobe 3 is not shown in this plot because the results for turbulence intensity are 0% (around 10−6 ) regardless of the numberof averaged phases, due to flow being laminar in the region
around probe 3.
There is a clear convergence of the turbulence intensity
results when the number of phases being averaged increases,
with about 1,000 phases producing turbulence intensity resultsfor probes 1 and 2 within 0.01%, relative, of the converged
value and 10,000 phases with relative differences on the order of 10−8 . For the results further from convergence, witharound 20 phases, they are varying roughly within 10% of theconverged value (e.g. for probe 2, 43.2 ± 4.3%), and with 100
phases within 2% (43.2 ± 0.9%).
Now evaluating the influence of the number of cardiac
cycles being averaged, from 2 to 28, using the data sampled atsolver frequency, the results are shown in Figure 4. A linear
fit of the results is also displayed in this figure. The number ofcycles being averaged makes a significant difference in the turbulence intensity results compared to the linearly extrapolated
value, with relative differences of 10% up to around 15 cycles
and around 2% for 28 cycles.
The averaged energy spectrum periodogram, using thefull probe data at 70,000 phases per cardiac cycle (74 kHz)with the finest grid, for the velocity fluctuation component
in the x-direction, is shown in Figure 5. The x-direction isaligned with the inlet flow direction, but the three velocity
fluctuation components had the same behavior so only one ofthem is shown here. The energy spectrum was averaged using
the Bartlett method with 1000 non-overlapping segments in order to reduce the variance and facilitate the observation of theoverall trend of the data. The energy is normalized by dividing
it by the bulk velocity at the inlet squared, and the frequency
is normalized by multiplying it by the cardiac cycle period.
It is surprising to notice that, despite the very low

at a frequency of 20 phases per period (or 20 times per cardiac
cycle), with more discussion about these values in the limitations section. With the grid convergence index (GCI) methodof Roache (1997), which is based on the generalized Richardson extrapolation, using the 2 finest grids and the coarsest one,a relative uncertainty of the turbulence intensity for the finest
grid around 0.5% was estimated. A linear fit (shown in the figure) through all the grid results and extrapolated to what would
be the grid with an infinite number of cells, results in a relative difference around 1% for the finest grid compared to theextrapolated one.
One behavior that is noticeable in Figure 2 and that was
not expected is the fact that using finer grids is producing simulations that have higher turbulence intensities. This is opposite
to the behavior observed in our previous study with pipes andsomewhat higher Reynolds numbers (Luciano et al., 2022),
where more refined grids were observed to produce less turbulent, even laminar, simulations compared to coarser gridsthat were more turbulent. We believe this behavior in the previous study was due to the increase of numerical discretization
errors by using coarser grids, which perturbed the simulation.
In the present aneurysm simulations which are at very low
Reynolds numbers, it seems that the additional perturbations
introduced when using coarser grids are not enough to counteract the spatial resolution that is lost, with the resulting turbulence intensity being reduced. Another, more likely, possibility is that the aneurysm simulation was already perturbed
enough by numerical noise, even in the finest grids. Therefore,
additional perturbation from using coarser grids did not make adifference, as a perturbation threshold was reached and crossedwith additional perturbations not significantly influencing the
turbulent behavior. This behavior with the existence of a maximum perturbation threshold, and additional perturbations over
the threshold not making a difference, was also seen in theprevious study mentioned. Lastly, this also shows how sensitive the cerebral aneurysm simulations are to perturbations andnoise.
When evaluating the convergence of the turbulence intensity calculation considering the number of phases per period
being sampled and averaged, i.e. how many times data is sam-3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
10−5
Probe 1
Probe 2
Probe 3

1/5 1/4

1/3

1/2

1/Number of Averaged Cycles

3

1/∞ 1/20 1/10

7/

25

−

30

x

Probe 2
Probe 1

35

10−7
∝

Pressure energy spectrum (p0 2/U 4 )40yTurbulence intensity (%)

45

10−9

10−11

10−13

Figure 4.
sults.

Cycle convergence of the turbulence intensity re-

10−6
Velocity energy spectrum (u0x 2/U 2 )102

Probe 1
Probe 2
Probe 3Figure 6.
tions.

103
104
Frequency (f T )Averaged energy spectrum of the pressure fluctua-

10−9cause turbulence regardless of the existence of other sourcesof perturbations, through the introduction of energy at solver
resolution. Saqr et al. (2022) also observed a significant inverse energy cascade, though at much lower frequencies thanthe ones being evaluated in the present paper.
The averaged energy spectrum of the pressure fluctuations, Figure 6, is also surprisingly close but conversely has
a slightly slower roll-off than the − 73 exponent that would be
expected for high Reynolds number flows and high frequencies (George et al., 1984). Also noticeable is the previously
mentioned inverse energy cascade at the highest frequencies,
but now in all three probes.
Two additional topics were evaluated: (i) the additionof random normally-distributed perturbations with intensity
equivalent to 5% turbulence intensity superimposed on the
Womersley inlet velocity boundary condition, similar to theperturbations previously used in Luciano et al. (2022), and (ii)the simplification of the geometry by cutting off the upstream
arteries from the aneurysm down to the middle cerebral artery,as schematized by the cutting line in Figure 1. Both were evaluated through the volume-averaged turbulence intensity within
the aneurysm (see Figure 2), with the resolution of the finest
grid and with 100 phases being sampled per cycle.
Regarding (i), when adding random perturbations to theinlet there was no significant difference in the aneurysm turbulence intensity results, with the relative difference on the orderof 10−4 . This shows that the inlet in the full geometry is far
away enough that these perturbations have the time to dissipate before reaching the aneurysm, therefore not significantly
influencing the turbulence intensity inside the aneurysm. Thisbehavior may have been different if all sources of numericalnoise were sufficiently reduced, and the only source of perturbation was the perturbation at the inlet. In this case, we expect
the added perturbation would be necessary in order to trigger
turbulence in this flow.
As for (ii), the simplification of the geometry by cutting

10−12

y∝
10

−15

x − 5/ 3

10−18

10−21

102

Figure 5.
tions.

103
104
Frequency (f T )Averaged energy spectrum of the velocity fluctua-

Reynolds numbers and no attempt being made to identify theinertial range, the averaged roll-off of the velocity energy spectrum at all probes is only slightly faster (exponent more negative) than the Kolmogorov − 53 spectrum. Saqr et al. (2020)
observed much faster roll-offs in intracranial aneurysms in theoretical calculations and patient measurements, however they
only looked at frequencies up to 50 Hz.
In addition, there is the evidence of an inverse energy cascade at the highest frequencies for probe 3 in Figure 5. Asprobe 3 is located in a region where the flow is laminar, we believe this inverse energy cascade may be simply due to numerical error as these frequencies are approaching the solver time
stepping resolution. This may also be related to how infinitesimal numerical errors or noise can influence the simulation and412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
dersson and Karlsson (2021) have evaluated up to 80 cardiac
cycles with 100 phases.
Related to the previous point, additional cardiac cycles tothe 30 we simulated for this paper would be important not only
in order to better evaluate the number of cycles required foran acceptable solution, but also to provide additional data on
whether discarding more than two of the initial cardiac cycles
is beneficial at all. This could be done while using probes sampling data at solver time step frequency in order to eliminate
any concerns with the sampling frequency.
The time step dependence (i.e. the temporal convergence)of the results was also not evaluated in detail. We maintained
the time step sizes low enough so that the Courant-FriedrichLewy (CFL) number was always lower than 1, which resulted
in the time step frequency of 70,000 times per cardiac cycle.
We did not evaluate whether lowering them even further or
raising them above CFL = 1 made any difference in the results.
With that said, the time step sizes should definitely be significantly smaller than the equivalent to 100 time steps per cycle
as that would be close to the least required for an acceptablephase-average, never mind the temporal resolution of the simulation itself. For reference, the results in Luciano et al. (2022)show that time steps as low as CFL 0.1 could potentially makea difference in the results of relatively low Reynolds number
simulations.
Lastly, the turbulence intensity was the main output variable evaluated in this study. Simultaneously evaluating othervariables such as the wall shear stress could be useful particularly towards clinical practice, though that leads into other uncertainties and discussions such as what hemodynamics variables are actually important for the pathophysiology of this
vascular disease.

Figure 7. Comparison of wall shear stress magnitude resultsfor (a) steady-state laminar simulation and (b) time-averaged
turbulent simulation.

it at the beginning of the MCA does make a significant difference by relatively increasing the turbulence intensity within
the aneurysm by more than 10%. This is despite the fact thatdampening was added to the flow rate waveform of the MCA
inlet in the simulation with the cut geometry.
The results of the steady-state laminar simulation arecompared to the turbulent simulation in Figure 7. The steadystate laminar and transient turbulent aneurysm wall shear
stress magnitude results are completely different. The turbulent results show averaged wall shear stress values that areabout 5 times higher than the laminar solution around thecenter of the aneurysm wall in the figure, and this is basedon the time-average and not the transient peaks which will
be even higher in the turbulent simulation. Needless to say,the assumption that flow is laminar is completely misleading.
In fact, as shown in the previous images (figures 2-4), turbulence intensity values approaching 50% are observed, andwhen volume-averaged through the entire aneurysm region arearound 15% indicating a highly turbulent flow.

CONCLUSIONS
Not only is there turbulence in the artery with ananeurysm simulated here, this is in reality a highly turbulent
flow, despite the very low Reynolds numbers involved.
The common assumptions in the literature that blood flowis laminar and that the evaluation of a few cardiac cycles isenough are misleading. Appropriately capturing the very complex turbulence in the flow could be very important for physicians and researchers. In addition, in order to save computational costs, researchers often set the inlet of the computational
artery geometry close to where the aneurysm is, but our simulations show that this simplification can also make a significant
difference in the turbulence intensity within the aneurysm.
As general recommendations for the simulation of cerebral aneurysms, based on this study as lower bounds it would
be suggested that (i) at least 30 cardiac cycles are simulated
while discarding the first cycle due to initial transients, (ii)data is sampled at least 100 times per cardiac cycle for phaseaveraging and (iii) as much of the real geometry is used as
feasible. Lastly, all of these should be evaluated on a case-bycase basis, as we believe the requirements may vary significantly and it is always important to be aware of the associateduncertainties.

Limitations
Several limitations had to be imposed on the simulations
and post-processing in order to enable the feasibility of thisstudy in the time-frame available, limitations which will affectthe results to varying degrees.
The grid dependency evaluation is planned to be redone
based on the aggregate results obtained from this study, withadditional grids. Even though the general trend may have been
captured correctly, increasing the number of phases, i.e. sampling frequency, was shown to be important (see Figure 3).
Only 20 phases were used for the preliminary grid evaluation
discussed here, which means that the different grid results arealmost within each others’ uncertainties. Note that this gridevaluation was the first step in the present study. Ideally, atleast 100 phases should be used due to the associated uncertainties observed (even better using probes at solver time step
resolution as done for other simulations in the present study),
also with significantly more than 30 averaged cardiac cycles,
to limit the propagation of uncertainties. For reference, An-

REFERENCES
Andersson, M. and Karlsson, M. 2021 Model Verification andError Sensitivity of Turbulence-Related Tensor Characteristics in Pulsatile Blood Flow Simulations. Fluids 6(1) 11.
Chnafa, C., Brina, O., Pereira, V. M. and Steinman, D. A. 2018
Better Than Nothing: A Rational Approach for Minimizing

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
sualization 8(2) 182–193.
Olinger, C. P. and Wasserman, J. F. 1977 Electronic stethoscope for detection of cerebral aneurysm, vasospasm andarterial disease. Surgical Neurology 8(4) 298–312.
Roache, P. J. 1997 Quantification of uncertainty in computational fluid dynamics. Annual Review of Fluid Mechanics
29(1) 123–160.
Sangalli, L. M., Secchi, P. and Vantini, S. 2014 AneuRisk65: A
dataset of three-dimensional cerebral vascular geometries.
Electronic Journal of Statistics 8(2) 1879–1890.
Saqr, K. M., Tupin, S., Rashad, S., Endo, T., Niizuma, K.,
Tominaga, T. and Ohta, M. 2020 Physiologic blood flow is
turbulent. Scientific Reports 10(1) 15492.
Saqr, K. M., Kano, K., Rashad, S., Niizuma, K., Kaku, Y.,
Iwama, T. and Tominaga, T. 2022 Non-Kolmogorov turbulence in carotid artery stenosis and the impact of carotid
stenting on near-wall turbulence. AIP Advances 12(1)015124.
Steinman, D. A. and Pereira, V. M. 2019 How patient specific are patient-specific computational models of cerebral
aneurysms? An overview of sources of error and variability. Neurosurgical Focus 47(1) E14.
Valen-Sendstad, K., Piccinelli, M. and Steinman, D. A.
2014 High-resolution computational fluid dynamics detects
flow instabilities in the carotid siphon: Implications foraneurysm initiation and rupture? Journal of Biomechanics 47(12) 3210–3216.
White, C. R., Stevens, H. Y., Haidekker, M. and Frangos,
J. A. 2005 Temporal gradients in shear, but not spatial gradients, stimulate ERK1/2 activation in human endothelial
cells. American Journal of Physiology-Heart and Circulatory Physiology 289(6) H2350–H2355.
Zarrinkoob, L., Ambarki, K., Wåhlin, A., Birgander, R., Carlberg, B., Eklund, A. and Malm, J. 2016 Aging alters thedampening of pulsatile blood flow in cerebral arteries. Journal of Cerebral Blood Flow & Metabolism 36(9) 1519–
1527.

the Impact of Outflow Strategy on Cerebrovascular Simulations. American Journal of Neuroradiology 39(2) 337–343.
Davies, P. F., Remuzzi, A., Gordon, E. J., Dewey, C. F. andGimbrone, M. A. 1986 Turbulent fluid shear stress induces
vascular endothelial cell turnover in vitro. Proceedings of
the National Academy of Sciences of the United States ofAmerica 83(7) 2114–2117.
Eckhardt, B., Schneider, T. M., Hof, B. and Westerweel, J.
2007 Turbulence Transition in Pipe Flow. Annual Review
of Fluid Mechanics 39(1) 447–468.
Ferguson, G. G. 1972 Physical factors in the initiation, growth,and rupture of human intracranial saccular aneurysms.
Journal of Neurosurgery 37(6) 666–677.
George, W. K., Beuther, P. D. and Arndt, R. E. A. 1984 Pressure spectra in turbulent free shear flows. Journal of Fluid
Mechanics 148 155–191.
Hoi, Y., Wasserman, B. A., Xie, Y. J., Najjar, S. S., Ferruci,
L., Lakatta, E. G., Gerstenblith, G. and Steinman, D. A.
2010 Characterization of volumetric flow rate waveforms
at the carotid bifurcations of older adults. Physiological
Measurement 31(3) 291–302.
Komen, E. M. J., Camilo, L. H., Shams, A., Geurts, B. J. andKoren, B. 2017 A quantification method for numerical dissipation in quasi-DNS and under-resolved DNS, and effectsof numerical dissipation in quasi-DNS and under-resolved
DNS of turbulent channel flows. Journal of Computational
Physics 345 565–595.
Ku, D. N. 1997 Blood Flow in Arteries. Annual Review of
Fluid Mechanics 29(1) 399–434.
Luciano, R. D., Chen, X. B. and Bergstrom, D. J. 2022 Discretization and perturbations in the simulation of localized
turbulence in a pipe with a sudden expansion. Journal of
Fluid Mechanics 935 A20.
Natarajan, T., MacDonald, D. E., Najafi, M., Coppin, P. W.
and Steinman, D. A. 2020 Spectral decomposition andillustration-inspired visualisation of highly disturbed cerebrovascular blood flow dynamics. Computer Methods in
Biomechanics and Biomedical Engineering: Imaging & Vi-

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