Numerical modeling of pulsatile turbulent flow in stenotic vessels.
ABSTRACT Pulsatile turbulent flow in stenotic vessels has been numerically modeled using the Reynoldsaveraged NavierStokes equation approach. The commercially available computational fluid dynamics code (CFD), FLUENT, has been used for these studies. Two different experiments were modeled involving pulsatile flow through axisymmetric stenoses. Four different turbulence models were employed to study their influence on the results. It was found that the low Reynolds number komega turbulence model was in much better agreement with previous experimental measurements than both the low and high Reynolds number versions of the RNG (renormalizationgroup theory) kepsilon turbulence model and the standard kepsilon model, with regard to predicting the mean flow distal to the stenosis including aspects of the vortex shedding process and the turbulent flow field. All models predicted a wall shear stress peak at the throat of the stenosis with minimum values observed distal to the stenosis where flow separation occurred.

Conference Paper: Geometry effects on flow pattern and wall shear stress in human internal carotid arteries
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ABSTRACT: Worldwide, stroke is the second leading cause of death, responsible for 4.4 million (9 percent) of the total 50.5 million deaths each year. More than 4 million people in the United States have survived a stroke or brain attack and are living with the aftereffects. It has been reported that the carotid siphon is the site most susceptible to atherosclerotic lesions in the human intracranial arteries, directly affecting the cerebral blood supply. The relatively complex geometry containing out of plane curves has a great impact on the associated hemodynamic, which plays a significant role on the atherogenesis. There are medical reports that indicate atherosclerotic stenoses is widely observed in different sites of internal carotid artery (ICA). Relatively few studies have examined the geometrical features of these arteries to identify regions of low or oscillatory shear stress that contributes to atherogenesis. However, there are still some conflictions about risky sites in these arteries, which require further studies. Furthermore, previous studies include some simplifications especially on the blood flow profiles that bring some uncertainties regarding their findings. Therefore, the purpose of the present study is to investigate the blood flow patterns in ICA to identify the risky sites prone to atherosclerosis. Geometrical models are made based on the clinical studies about the classification of internal carotid siphon geometry. Three different geometries of the ICA were simulated by Open source CFD software Open FOAM. In the present study, flow patterns and wall shear stresses have been examined extensively and the preferred sites of stenoses along the carotid siphon of the different geometries of the ICA are identified.Entech conference, Turkey; 12/2014 
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45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit; 08/2009  SourceAvailable from: Xinlei Wang[Show abstract] [Hide abstract]
ABSTRACT: Coexistence of laminar, transitional and turbulent flow regimes is very common in blood flows through arteries, airflows in human respiratory systems, and indoor airflow etc. Due to the complexity of flow physics involved, most Reynoldsaveraged NavierStokes (RANS) turbulence models are not suitable for these flows because they are designed primarily for high Reynolds number turbulent flows. In this paper, large eddy simulation with dynamic subgrid scale model has been applied to simulate the flow in a relaminarizing boundary layer that undergoes reverse transition from turbulent state to laminar one. Simulated mean velocities and turbulent intensities are in good agreement with corresponding experimental data at different streamwise positions where different flow regimes exist. The appropriateness of dynamic subgrid scale model for the study of relaminarizing boundary layer flow is demonstrated by the variation of subgrid constant: it is somewhat constant in fully turbulent regime, decreases in transitional regime and reaches zero in laminar regime. Comparison of the experimental data and computational results from three low Reynolds number RANS models shows that they do not adequately predict the flow relaminarization. The present study suggests the use of large eddy simulation with dynamic model in the study of complex flows where a combination of flow regimes (laminar, transitional, and turbulent) may exist.
Page 1
Sonu S. Varghese
Steven H. Frankel1
email: frankel@ecn.purdue.edu
School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
Numerical Modeling of Pulsatile
Turbulent Flow in Stenotic
Vessels
Pulsatile turbulent flow in stenotic vessels has been numerically modeled using the
Reynoldsaveraged NavierStokes equation approach. The commercially available com
putational fluid dynamics code (CFD), FLUENT, has been used for these studies. Two
different experiments were modeled involving pulsatile flow through axisymmetric
stenoses. Four different turbulence models were employed to study their influence on the
results. It was found that the low Reynolds number k–? turbulence model was in much
better agreement with previous experimental measurements than both the low and high
Reynolds number versions of the RNG (renormalizationgroup theory) k–? turbulence
model and the standard k–? model, with regard to predicting the mean flow distal to the
stenosis including aspects of the vortex shedding process and the turbulent flow field. All
models predicted a wall shear stress peak at the throat of the stenosis with minimum
values observed distal to the stenosis where flow separation occurred.
?DOI: 10.1115/1.1589774?
Introduction
Blood flow under pathological conditions, such as atherosclero
sis, involves a narrowing of the artery lumen, referred to as a
stenosis, which together with flow pulsatility can result in the
periodic generation of turbulence despite the relatively low Rey
nolds numbers of such flows ?the typical Reynolds number range
is from 1 to 4000 in small arterioles and large vessels, respectively
?1??. This can affect flow resistance and mixing rates and impact
platelet coagulation rates ?2,3?. The fact that the vessels involve
compliant walls also affects flow stability and the generation of
turbulence. Because of the difficulties in making in vivo measure
ments of local velocity profiles and wall shear stresses in pulsatile
flow, computational fluid dynamics ?CFD? has begun to play a
major role in improving our understanding of biofluid flows in
general, and blood flow in arteries in particular ?1,4?. Recent CFD
work has addressed pulsatile flows in stenotic vessels accounting
for fluidstructure interactions, nonNewtonian effects, and irregu
lar geometry ?3,5,6? but the nature and effects of turbulence have
not been addressed in these studies.
Stenotic flows involve low Reynolds number, pulsatile flow
through a restriction ?most resembling a convergingdiverging
nozzle in the idealized case?. They feature flow separation, strong
shear layers, recirculation, reattachment, and turbulence. It has
been observed that flow separation occurs in the expansion region
for Re?10 for a 70% ?by area? stenosis. The critical upstream Re
for turbulence is 300 and intensity levels reach about 100% of
upstream velocity values and remain high for about 1.5–6 diam
eters downstream. Pulsatility creates a periodic generation of tur
bulence, which is greatest during the deceleration of systole and
least during the upstroke of systole. A reduced wall shear stress
has been associated with this transition to turbulence. For stenoses
greater than 75%, turbulence effects are very large and create
significant flow resistance and pressure loss ?1?.
Turbulence in blood flow influences many physiological pro
cesses. In addition to flow resistance, turbulence affects the shear
stress acting on the blood vessel wall, tensile stress in the endot
helial cell membrane, mass transport from the blood to the vessel
wall, atherosclerosis, blood clot formation, blood rheology
through deformability of red blood cells, the hematocrit ?volume
fraction of red blood cells in blood?, cell division and surface cell
loss, as well as internal cell motion due to pressure and shear
stress ?7?. Of particular interest here is the effect of the shear
strain rate of the blood on the endothelial cells which line the
blood vessel walls, which is thought to be a contributing factor in
atherosclerosis ?8?.
Experimental studies of pulsatile flow through stenotic vessels
have been limited to laboratory studies for obvious reasons. Ojha
et al. ?9? studied pulsatile flow through constricted tubes using a
photochromic technique. It was found that with mild stenoses vor
tical and helical structures were observed during the deceleration
phase near the reattachment point. With moderate constrictions,
transition to turbulence associated with the breakdown of waves
and streamwise vortices occurred. Intense wall shear stress fluc
tuations were also observed near the reattachment point with in
creases in instantaneous wall shear stress as much as by a factor of
8. The vortical structures occurred due to jet instability during the
deceleration phase and were not shed from stenosis. Interactions
between the imposed flow oscillation and the turbulence in the
viscous sublayer were strongest during the deceleration phase.
Giddens and coworkers constructed a plexiglass model of an
axisymmetric stenotic blood vessel experiencing pulsatile flow
and measured instantaneous core velocity, ensembleaveraged ve
locity profiles, and wall shear stress using laser Doppler velocim
etry ?10–12?. For a 75% stenosis, turbulence was found to exist
during parts of the cycle. Issues related to velocity decomposition
to account for the timevarying mean and the presence of coherent
vortical structures via a triple decomposition were discussed, as
well as the effect of the apparent stresses. They reported that
vessel wall thickening may depend on the instantaneous wall
shear stress being low during the entire cycle rather than just the
mean.
There have been several recent numerical studies investigating
steady and pulsatile flow through stenotic vessels addressing sev
eral of the above issues. These issues include fluidstructure inter
action and rheology effects. Bathe and Kamm ?13? conducted a
finiteelement analysis to examine fluidstructure interaction of
pulsatile flow through a compliant stenotic artery. They used the
commercial software package ADINA to develop an axisymmetric
model of the flow and vessel. They observed an increase in the
pressure drop and wall shear stress associated with the flow as
1?Author to whom correspondence should be addressed.
Contributed by the Bioengineering Division for publication in the JOURNAL OF
BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Divi
sion January 28, 2002; revision received February 10, 2003. Associate Editor: J. E.
Moore, Jr.
Copyright © 2003 by ASMEJournal of Biomechanical Engineering
AUGUST 2003, Vol. 125 Õ 445
Page 2
they increased the degree of stenosis. They also reported an in
crease in the inner wall hoop stretch and compressive stresses on
the lesion, as well as additional decreases in vessel area during
peak flow times. Turbulence effects were not considered in their
study. Tang et al. ?5? also considered fluidstructure interactions of
steady flow through an axisymmetric stenotic vessel using the
commercial code ADINA. They observed complex flow patterns
and high shear stresses at the throat of the stenosis, as well as
compressive stresses inside the tube. Turbulence was not ad
dressed. Qiu and Tarbell ?14? used FIDAP to study pulsatile flow in
a compliant curved tube model of a coronary artery. They report
that in addition to the wall shear stress, the stress phase angle
between the circumferential strain in the artery wall and the wall
shear stress is important for locating possible coronary atheroscle
rosis. Buchanan et al. ?6? studied rheological effects on pulsatile
laminar flow through an axisymmetric stenosed tube and found
that they could affect wall shear stress quantities.
Many turbulent flows that occur both naturally ?such as blood
flow in arteries? and artificially ?fluid pumped through a channel
by a reciprocating device? are inherently unsteady. The unsteadi
ness may be due to an imposed boundary condition or due to
fluctuations in the driving force, or even a combination of both.
This imposition can alter the flow physics to such an extent that it
can even cause relaminarization of an initially turbulent flow.
Unsteady flow can be periodic or aperiodic in nature. If the
mean quantity is nonzero, the unsteadiness is pulsatile or else it is
purely oscillatory flow. A flow subject to a homogeneous, time
varying pressure gradient is equivalent to a flow in which the time
varying component of the pressure is replaced by an equivalent
oscillation in the boundary condition, via a transformation of vari
able ?15?. However, this is not true for aperiodic flow, as in the
case of flow accelerating at a constriction.
Many authors have investigated the effect of unsteadiness on
the structure of turbulence, but the possible effects of unsteadiness
on timemean flow characteristics have been disputed between
two groups of authors who have conducted experimental investi
gations on the structure of turbulent pulsatile flows. The results of
Ramaprian and Tu ?16,17? and Mizushina et al. ?18? indicated
small modifications of the timemean characteristics of pulsatile
turbulent flow ?such as timeaveraged velocity, Reynolds stress
distribution, etc.? when the pulsation frequency was sufficiently
close to the bursting frequency of turbulence or when the pulsa
tion amplitude was large. The second group of authors, which
include Ohmi et al. ?19–21?, Kita et al. ?22?, and Tardu et al.
?23,24?, indicated that there was no such effect of unsteadiness on
the timemean characteristics of pulsatile turbulent flows for full
ranges of the flow parameters such as the pulsation frequency,
amplitude, or Reynolds numbers. Most recently, He and Jackson
?25? provided a thorough review of the literature on experimental
studies of turbulence in transient pipe flow. In their study, they
find that the turbulence intensity is attenuated during the acceler
ating phase of the transient flow and increased during the decel
erating phase. This behavior was attributed to time delays in the
response of turbulence production and energy redistribution, as
well as propagation of turbulence radially.
Recently, Scotti and Piomelli ?15? conducted direct numerical
simulations ?DNS? and large eddy simulations ?LES? of pulsatile
turbulent channel flow subjected to an unsteady pressure gradient.
In DNS, numerically accurate and complete resolution of all spa
tial and temporal flow scales is required and no turbulence model
is used. In LES, numerically accurate resolution of the large scales
is required and the effect of the small scales, which are not sup
ported by the computational grid, on the resolved scales is mod
eled using a subgridscale turbulence model. Both techniques re
sult in simulations that should enable capture of unsteady scale
dependent vortex dynamics, transition, and turbulence. Scotti and
Piomelli ?15? showed that fluctuations generated in the nearwall
region did not propagate beyond a certain distance from the wall,
which depended on the imposed oscillation frequency. Because of
the instantaneous nature of the simulations, coherent flow struc
tures, including streaks and spots, were able to be identified. In a
related study, Scotti and Piomelli ?26? employed the Reynolds
averaged NavierStokes ?RANS? equations to assess the capabili
ties of three turbulence models for the same flow. In the RANS
approach, only the mean flow is computed, with the effect of all
the turbulent eddies being modeled. The authors found that all the
turbulence models were in reasonable agreement with the DNS
and LES for the velocity profiles over the cycle, but the models
differed when it came to other turbulence parameters.
Fig. 1
Ahmed and Giddens †11‡, where LÄ4.0? and DÄ2.0?, and the
„b… sharpedged stenosis of Ojha et al. †9‡, where LÄ1.5 mm
and DÄ5.0 mm. For both geometries, the normalized distance
from the stenosis center is given by ZÄZ?ÕD.
Stenosis geometries for the „a… smooth stenosis of
Table 1Closure coefficients for the k–? models
Model
C?
?k
??
C?1
C?2
Standard
RNG
Low Re RNG
0.09
0.085
1.0
0.72
0.72
1.3
0.72
0.72
1.44
1.42
1.42
1.92
1.68
1.68
Table 2
k–? model
Closure coefficients for the low Reynolds number
Model
?k
??
??*
1.0
?0*
0.024
Rk
??
?0
R?
?0*
?0
Low Re k–?
2.0 2.06.0 0.52
1
9
2.95
9
100
9
125
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Transactions of the ASME
Page 3
Bluestein et al. ?27? correlated platelet deposition in a model
stenosis with flow dynamics using a steady flow numerical model
at different Rynolds numbers. They employed the two equation
standard k–? turbulence model and observed a single closed vor
tex within the recirculation region. In a later study, Bluestein et al.
?2? used the two equation low Reynolds number k–? turbulence
model ?similar to that used in the present study? to simulate steady
flow through a stenosis. In this case they observed the process of
vortex shedding in the separated flow region downstream of the
stenosis which was also confirmed by digital particle image ve
locimetry. They proposed that the development of vortical struc
tures in the flow field near the walls for high Reynolds numbers
leads to enhanced platelet deposition.
More recently, Mallinger and Drikakis ?28? and Mittal et al.
?29? have conducted DNS and LES, respectively, of pulsatile
stenotic flow. Drikakis et al. observed asymmetries in the form of
helical disturbances develop in the distal region of the stenosis,
while Mittal et al. found evidence of turbulence further down
Fig. 2
Reynolds number RNG k–? model and normalized by the mean inlet velocity.
The distances are in normalized units „normalized by diameter…. Only a third of
the actual number of grid points used is shown and profiles are offset by 5
units in the x direction. „a… Smooth stenosis. The profiles were obtained during
peak inlet flow. „b… Sharpedged stenosis. The profiles were obtained during
minimum flow conditions at the inlet.
Grid refinement tests. The velocity profiles were obtained using the low
Fig. 3
distal to the smooth stenosis with the experimental profiles of Ahmed and
Giddens †10‡, for steady flow at inlet ReÄ500
Comparison of computed velocity profiles at different axial locations
Table 3Key flow parameters used for the two stenotic flow simulations
Case
U0(m/s)
Um(m/s)
? ?rad/s?
D (m)
? ?kg/m3? ? ?Pa–s?
Ahmed and Giddens
Ojha et al.
0.04254
0.2178
0.02808
0.1364
0.314
17.05
0.0508
0.005
1000
755
3.6014 ? 10?3
1.43 ? 10?3
Journal of Biomechanical Engineering
AUGUST 2003, Vol. 125 Õ 447
Page 4
stream of the stenosis. These studies highlight the utility of con
ducting threedimensional transient simulations of pulsatile
stenotic flows to identify important flow features not captured in
more traditional RANS studies of the type reported here. Our own
recent DNS studies have also revealed the development of asym
metries in the distal region ?30?.
In this study, the pulsatile stenotic flow experiments of Giddens
et al. ?11? and Ojha et al. ?9? are simulated, within the framework
of twoequation RANS turbulence models. The capabilities of
four different models in predicting the flow features that were
observed in the experiments are assessed. Various issues involved
in the modeling, such as nearwall treatment and grid
independence are also discussed.
Turbulence Models
The Reynolds averaged NavierStokes ?RANS? equations rep
resent transport equations for the mean flow quantities only, with
the effect of all the turbulent eddies being modeled. In time
dependent simulations, such as those performed during this study,
a computational advantage is that the time step is determined by
the global unsteadiness in the mean flow, rather than by the tur
bulence ?31?.
In unsteady turbulent flows, instantaneous flow variables in the
NavierStokes equations such as axial velocity, u, can be ex
pressed in terms of an ensembleaveraged quantity, u ¯, and a fluc
tuation from this average, u?. Likewise for other scalar quantities
like pressure. Applying this decomposition to the NavierStokes
equations yields the RANS equations. The incompressible form of
these equations can be written in the Cartesian form as:
?u ¯i
?xi?0,(1)
Du ¯i
Dt??1
?
?p ¯
?xi?
?
?xj???
?u ¯i
?xj??u ¯j
?xi???
?
?xj??ui?uj??, (2)
where D( )/Dt??( )/?t?uj??( )/?xj? and ? is the kinematic vis
cosity.
As seen above, averaging the equations of motion gives rise to
new terms such as, ?ui?uj?. Also known as the Reynolds stress
tensor, it is the timeaveraged rate of momentum transfer due to
turbulence ?32?. It is expressed in terms of ?T, the kinematic eddy
viscosity, and the mean strain rate tensor, Sijas:
?ui?uj???ij??T?
?u ¯i
?xj??u ¯j
?xi??2?TSij. (3)
A turbulence model can be used to model the cross correlations
which arise from the nonlinear terms in the governing equations
and thereby close the equations. Four different twoequation tur
bulence models, readily available in FLUENT were used in this
study. The models use different transport equations for turbulent
kinetic energy ?TKE?, k, and dissipation rate, ?, or pseudovortic
ity, ?. The turbulent transport equation for the TKE can be de
scribed as:
Dk
Dt??ij
?u ¯i
?xj?D?
?
?xj??k
?k
?xj?.(4)
Fig. 4
used by Ahmed and Giddens is also shown. A flat inlet velocity profile was
used for the simulation. „b… Comparison of computed velocity profiles at differ
ent axial locations distal to the stenosis with the experimental profiles of
Ahmed and Giddens †11‡ for pulsatile flow with mean inlet ReÄ600. Velocity
profiles are compared during peak inlet flow conditions „T2….
„a… Flow inlet waveform for the smooth stenosis. The inlet waveform
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Page 5
For the k–? models, the Boussinesq approximation is used to
determine the eddy viscosity as ?T?C?(k2/?) and in Eq. ?4?,
D??,(5)
?k????T
?k.(6)
The transport equation for ? is:
D?
Dt?C?1
?
k?ij
?u ¯i
?xj??C?2?R??2
k?
?
?xj???
??
?xj?, (7)
where ??????T/??. The closure coefficients for the different
models are given in Table 1.
The value of R in Eq. ?7? for the standard k–? is 0. The stan
dard model is widely used despite known limitations, especially in
the prediction of complex flows involving strong pressure gradi
Fig. 5
inlet velocity. The distances are in normalized units „by diameter away from the center of the stenosis… and the
profiles are offset by 5 units in the x direction.
Velocity profiles for the smooth stenosis at different phases in the flow cycle, normalized by the mean
Journal of Biomechanical Engineering
AUGUST 2003, Vol. 125 Õ 449
Page 6
ents, separation, and strong streamline curvature. The RNG k–?
model is similar to the standard k–? model and is derived from
the instantaneous NavierStokes equations, using a mathematical
technique called renormalization group ?RNG? theory ?33–35?. It
features slightly modified modeling constants and includes an ad
ditional term in the ? equation ?Eq. ?7??, introduced by R as:
C??3?1?
1???3
R?
?
?0?
, (8)
where ??(k/?)?2SijSji, ?0?4.38, and ??0.012. This term im
proves the accuracy for rapidly strained flows.
For the low Reynolds number variation of the RNG k–? model,
?k??????? ˆ in Eqs. ?4? and ?7?. A differential equation is used
for the turbulent viscosity:
d?
?? ˆ3?1?C?
k
?????1.72
? ˆ
d? ˆ,(9)
where C??100 and ? is given by the relation
?
??1.393
1?1.393?
0.6321?
??2.393
1?2.393?
0.3679
?1
? ˆ. (10)
During the computation, Eq. ?9? is integrated to accurately obtain
the variation of effective turbulent transport with the effective
Reynolds number, allowing the model to better predict low Rey
nolds number and nearwall flow behavior. In the high Reynolds
number limit, the model reverts to the high Reynolds number
form of the RNG model ?31?.
The fourth turbulence model used for this study was the low
Reynolds number variation of the k–? model. The standard k–?
model is an empirical model based on modeled transport equa
tions for the turbulent kinetic energy k as given in Eq. ?4? with
D??0*f?*k?, (11)
?k????T
?k,(12)
f?*??
1
?k?0
1?680?k
1?400?k
2
2
?k?0,
?k?
1
?3
?k
?xj
??
?xj, (13)
Fig. 6
The stream function increment is 0.0015 sÀ1. The axial distance is indicated in normalized units Z „normalized by diameter….
Streamlines for the smooth stenosis from the low Reynolds number RNG k–? model at different phases in the flow cycle.
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Transactions of the ASME
Page 7
and the specific dissipation rate ?(??/k) ?32?. The transport
equation for ? is
D?
Dt???
k?ij
?u ¯i
?xj??0f??2?
?
?xj???
??
?xj?,(14)
where ??????T/??. The eddy viscosity for the low Reynolds
number variation is computed as ?T??*(k/?). The auxiliary re
lations are described as
?*???*?
?*?
?0*?Ret/Rk
1?Ret/Rk?,
1?Ret/R??,
(15)
????
?0?Ret/R?
(16)
Ret?
??0*??3?,
k
??, (17)
f??1?70??
1?80??,
????
?ij?jkSki
?ij?1
2?
?u ¯i
?xj??u ¯j
?xi?.
(18)
Closure coefficients for this model can be found in Table 2. The
low Reynolds number variation of the k–? model has been
shown to demonstrate superior performance for wall bounded low
Reynolds number flows. Wilcox ?32? also showed that the low
Reynolds number k–? model has potential to predict transitional
flows quite well, though the model has tended to predict excessive
and early separation ?31?. More details on all of the discussed
models can be found in books on turbulent flows and modeling
?32,36?.
Numerical Modeling
Analytical solutions of the governing equations of fluid dynam
ics are rare, especially as the dimensionality of the problem in
creases. Analytic solutions for laminar pipe flow are available in
the form of the HagenPoseiulle solution. Many investigators have
conducted theoretical studies of both pulsatile and oscillatory pipe
flows. Womersley ?37? introduced a nondimensional frequency
parameter, ?, and proposed a solution for the timedependent axial
velocity for fully developed pulsatile laminar flow in a circular
pipe which is a function of radial location and time and takes the
following form:
Fig. 7
streamfunction increment is 0.0018 sÀ1.
Streamlines for the smooth stenosis from the low Reynolds number k–? model at different phases in the flow cycle. The
Journal of Biomechanical Engineering
AUGUST 2003, Vol. 125 Õ 451
Page 8
Fig. 8
phases in the flow cycle. The distances are in normalized units „by diameter
away from the center of the stenosis… and the profiles are offset by 5 units in
the x direction.
Turbulence intensity profiles for the smooth stenosis at different
Fig. 9
distance is indicated in normalized units, Z.
Axial wall shear stress profiles for the smooth stenosis case at different phases in the flow cycle. The axial
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u?r,t???
iP
???
1?J0?i3/2?r
a?
J0?i3/2???exp?i?t?,(19)
where a is the pipe radius, J0( ), is the Bessel function of type 0,
? is the angular frequency of the pulsation, ? is a nondimensional
parameter called the Womersley number where ??a?(?/?), P is
the pressure gradient, ? is the density, and ? is the kinematic
viscosity.
Investigators focusing on turbulent pulsatile flow have concen
trated on the effect of pulsation frequency in terms of ? ?or the
frequency? with the effect of turbulence being modeled using both
timedependent and timeindependent eddy viscosity concepts
?38?. The RANS equations represent transport equations for the
mean flow quantities only, with the effect of all the turbulent
eddies being modeled, as discussed in the previous section. How
ever, finding an analytic solution is difficult and so numerical
solutions are usually relied upon.
In this study, the commercially available CFD code FLUENT,
Version 6.0, was used to solve the incompressible RANS equa
tions together with a twoequation turbulence model. The dis
cretized equations were solved in a segregated manner using the
PISO ?pressure implicit splitting of operators? algorithm, which is
useful for stability in unsteady flow calculations and to achieve
proper pressurevelocity coupling. The grids used for the calcula
tions was generated using GAMBIT, the preprocessor for FLUENT.
An important consideration for turbulence modeling is near
wall treatment and this can be carried out in FLUENT by using the
wall function approach or the enhanced wall treatment. In the wall
function approach, the viscosity affected nearwall region is not
resolved with the nearwall mesh being relatively coarse. Here,
the solution is determined by empirically based wall functions and
it is important that the first grid point off the wall not get into the
wall layer. Wall functions become much less reliable in the pres
ence of separated flow ?32?. The enhanced wall treatment gener
ally requires a very fine nearwall mesh capable of resolving the
viscous sublayer, with the first grid point off the wall being within
the region y??1, the distance being measured in wall units,
y?(??u?y/?) ?where u?is the friction velocity?. The latter ap
proach is more suited for lowReynolds number flows though it
obviously requires a greater amount of computational resources.
The enhanced wall treatment approach was employed for all the
cases reported in this study.
For the time periodic calculations, the time period, T, was cal
culated in terms of the angular frequency, ?, as T?2?/?. The
time step was then calculated by dividing the time period with the
number of iterations required for convergence during each step. In
FLUENT, an unsteady formulation was used and a temporally peri
odic boundary condition was applied to the inlet of the tube by
creating a user defined function ?UDF? for the sinusoidal velocity
of the form
U?U0?Umsin??t?,(20)
where U0is the mean velocity and Umis taken as a fraction of
U0. This resulted in a timeperiodic calculation and by varying
the total number of iterations during each run, flow characteristics
at different positions on the periodic cycle were obtained during
the postprocessing step.
The turbulence intensity and hydraulic diameter were also
specified as inlet boundary conditions according to:
I?
u?
uavg?0.16?ReDH??1/8, (21)
ReDH??U0D
?
,(22)
where ReDHis the Reynolds number based on hydraulic diameter
?the main vessel diameter, D, in the present stenotic flow cases?, ?
and ? the fluid density and viscosity, respectively. More details on
turbulent flow parameters as well as modeling guidelines can be
found in Refs. ?31,32,36?.
Description of Modeled Experiments
Pulsating Turbulent Flow Through a Smooth Stenosis.
experimental investigation of Ahmed and Giddens ?11? consider
ing pulsatile flow through an axisymmetric smooth stenosis was
studied here. The geometry used was similar to theirs with the
stenosis shape given by a cosine function as shown in Fig. 1a. If
r0is the radius of the nonstenotic part of the tube, S(x), gives the
shape of the stenosis as:
The
S?x??s0r0?1?cos?2??x?x1?/?x2?x1???/2,
where s0is the % stenosis severity with x1and x2(x1?x?x2)
specifying the position and length of the stenosis. A 75% axisym
metric stenosis was used in this model. Pulsatile flow was studied
at a mean Reynolds number of 600 and a Womersley number of
??D/2?(?/?)?7.5, similar to the experiments. Other param
eters used in Eqs. ?20? and ?21? can be found in Table 3.
(23)
Pulsating Turbulent Flow Through a SharpEdged Stenosis.
The experimental investigation of Ojha et al. ?9? on pulsatile flow
through an axisymmetric sharpedged stenosis was studied here.
The 75% axisymmetric stenosis case in their study was modeled
and the geometry used is shown in Fig. 1b. The Womersely pa
rameter was ??7.5. The mean and modulation Reynolds numbers
were 575 and 360, respectively, matching the experiments. The
other parameters used in Eqs. ?20? and ?21? can be found in
Table 3.
Results
Grid Refinement Study.
formed for the cases reported in this study. Figure 2 shows two
typical grid independence studies that were conducted for both
stenosis models using the low Reynolds number RNG k–? model.
The profiles were obtained during minimum flow conditions at the
inlet. Grids consisting of 30, 60, and 120 points in the radial
direction were used in this study ?the number of points in the
Grid refinement studies were per
Fig. 10
ferent from the waveform used for the smooth stenosis…. A flat
inlet velocity profile was used for the simulation.
Flow inlet waveform for the sharpedged stenosis „dif
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AUGUST 2003, Vol. 125 Õ 453
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streamwise direction was correspondingly varied in order to avoid
skewness of the elements? and the profiles obtained on the two
finer grids, corresponding to 60 and 120 radial points, fall on top
of each other. The results shown in the following sections ?for
both the stenosis models? were all obtained using the grid with 60
points in the radial direction. Simulations with the low Reynolds
number k–? model took approximately 16 h to converge with
two parallel processors. Computational times for simulations with
the k–? models were approximately 19 to 22 h with the low
Reynolds number k–? model requiring the maximum.
Pulsating Turbulent Flow Through the Smooth Stenosis.
FLUENT calculations were also performed for the 75% axisymmet
ric stenosis model of Giddens and coworkers ?10–12?. As part of
Fig. 11
mean inlet velocity. The distances are in normalized units „by diameter away from the center of the stenosis… and
the profiles are offset by 5 units in the x direction.
Velocity profiles for the sharpedged stenosis at different phases in the flow cycle, normalized by the
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a validation process, steady flow computations were performed at
a mean inlet Reynolds number of 500 and the velocity profiles
distal to the stenosis were compared to the data in Ref. ?10?,
shown in Fig. 3. A parabolic inlet velocity profile was specified
2.5 diameters upstream of the stenosis throat for this case. The
figure shows the low Reynolds number k–? model performing
better than its counterparts in the region immedietly downstream
of the stenosis. Far downstream, all the models show poor agree
ment with the experimental data, though the low Reynolds num
ber k–? model better predicts the jetlike flow at the center of the
vessel.
The previously described pulsatile flow conditions were chosen
to mimic the experiments of Ahmed and Giddens ?11?. The flow
input waveform and the times at which the results will be pre
sented are indicated in Fig. 4a along with the waveform used in
the experiments. A flat inlet velocity profile was specified 2.5
diameters upstream of the stenosis throat. Computed velocity pro
files were compared with the experimental data during peak inlet
flow as shown in Fig. 4b. All the four models show poor agree
ment with the experiments especially in predicting flow separation
far downstream of the stenosis during this time. Although, as for
the steady flow case, the low Reynolds number k–? model better
predicts the jetlike flow behavior further downstream from the
stenosis. Part of the discrepancy in the profiles may be due to the
differences between the inlet waveforms, highlighted in Fig. 4a,
as well as the nature and location of the inlet velocity profile
specified for these computations. In the experiments the entrance
length of the tube prior to the constriction was 96 diameters ?11?.
Figure 5 shows the velocity profiles computed by all the turbu
lence models at various distal locations ?distances being measured
from the stenosis throat?. The low Reynolds number k–? model
predicts more jetlike behavior immediately downstream of the
stenosis throat than any of the three k–? models. Similar to the
experiments, a permanent recirculation region is not observed in
any of the present results. The profiles during acceleration ?time
T1) and peak flow (T2) indicate that the flow is mostly in the
forward direction during these times. During deceleration (T4)
and minimum flow (T5), the profiles are more inflectional in
nature, which could lead to instabilities further downstream.
The streamlines computed by the low Reynolds number RNG
model and the low Reynolds number k–? model in Figs. 6 and 7,
respectively, indicate a region of flow reversal upstream of the
stenosis during parts of the cycle, consistent with the reverse flow
observed byAhmed and Giddens. The low Reynolds number k–?
model predicts strong vortex formation in the jetlike region during
the deceleration phase, when an adverse pressure gradient exists
across the vessel, similar to the vortex formation observed in the
experiments. The three k–? models fail to predict this behavior.
Turbulence intensity profiles shown in Fig. 8 once again high
light the differences between the three k–? models and the low
Reynolds number k–? model. The low Reynolds number k–?
model predicts lower turbulence intensities than the other models
immediately downstream of the stenosis during both peak and
minimum flow (T2 and T5, respectively?. Further downstream,
around Z?4.0, the low Reynolds number k–? model predicts
higher intensity, due to the breakdown of vortices in that region.
However, all the turbulence models predict low levels of turbu
lence intensities across the distal section of the stenotic vessel. In
the experiments, high turbulence levels were observed at Z
?4.0.
Axial profiles of wall shear stress ?WSS? in the vicinity and
distal to the stenosis are plotted in Fig. 9 at different phases of the
flow cycle and the levels can be seen to be fluctuating throughout
the cycle. Again, peak WSS levels are observed at the stenosis,
consistent with the high velocities associated with flow accelera
tion through the restriction. The highest values are observed dur
ing the acceleration phase and the early part of the deceleration
phase. The maximum value occurs at time T2 ?peak flow? similar
to experimental observations at the same phase. Negative values
of the WSS can be observed during the deceleration phase consis
tent with the presence of flow reversal. Consistent with the experi
ments, the magnitude of shear stress is relatively insignificant af
ter Z?4.0.
Pulsating Turbulent Flow Through the SharpEdged Steno
sis.
Ojha et al. ?9? described a moderate stenosis to be one for
which turbulence is generated in a large portion of the region
distal to the stenosis, but which does not result in a physiologi
cally significant reduction in blood flow. The 75% axisymmetric
stenosis geometry they studied falls within this category. Their
experiments also indicated transition to turbulence for the same
case. FLUENT was used here to model this experiment with param
eters described in the previous section. The flow input waveform
shown in Fig. 10 is different from the one used in the computa
tions for the smooth stenosis. A flat inlet velocity profile was
specified 2.5 diameters upstream of the stenosis throat. In the
experiments the test section was located 160 diameters down
stream from the entrance ?9?.
Velocity profiles at different phases of the flow cycle, as calcu
lated by all the models, are presented at various axial locations
distal to the stenosis ?the distance being measured from the center
of the stenosis? in Fig. 11. The profiles predicted by the three
variations of the k–? model are almost the same throughout the
cycle. However, the low Reynolds number variation of the k–?
model predicts a more jetlike flow immediately downstream of the
stenosis with higher velocities at the center of the tube, similar to
the experiments. All the models show that towards the end of the
acceleration phase, at peak flow (T2), the velocity profiles be
come more inflectional in nature, which can cause instabilities in
the flow. As the flow becomes more unstable, a region of reversed
flow starts to develop immediately downstream of the stenosis.
Variations in the size of the flow separation regions distal to the
stenosis are also observed between the turbulence models, espe
cially during the deceleration phase ?time T4) and minimum flow
(T5). It should be noted that a direct comparison with the photo
chromic measurements of Ojha et al. ?9?, which represent instan
taneous velocity profiles, is not possible here because our predic
tions are of the ensembleaveraged flow field.
Fig. 12
edged stenosis. Waveforms computed by the four turbulence
models are compared to the experimental data.
Centerline velocity waveforms at ZÄ0.6 for the sharp
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AUGUST 2003, Vol. 125 Õ 455
Page 12
Figure 12 compares computed centerline velocities, obtained
during one cycle at Z?0.6, with data from the experiments. None
of the turbulence models are able to predict the variations in the
centerline velocity at that location which is close to the stenosis
throat.
Streamlines computed by the low Reynolds number variation of
the RNG model and the low Reynolds number k–? model at
different phases of the flow cycle during one time period are
shown in Figs. 13 and 14, respectively. Streamlines computed by
the other k–? models are not shown here due to their similarity
with the low Reynolds number RNG model. The streamline plots
show that a region of reversed flow starts to develop immediately
downstream of the stenosis at peak flow. Later in the deceleration
phase, and into the early part of the acceleration phase, this re
versed flow region continues to grow, until it almost completely
disappears during the middle of acceleration. Both models predict
a permanent region of flow reversal immediately distal to the
stenosis as well as a recirculation region in the region proximal to
the stenosis during deceleration when an adverse pressure gradient
results across the model, similar to that observed by Ojha et al.
The reattachment point as well as the size of the separation
zone fluctuated throughout the cycle. During deceleration, around
time T4, the low Reynolds number k–? model predicts the size
of the separation zone to be between 0.35R and 0.4R ?where R is
the radius of the vessel? from the exit of the stenosis to around
Z?2.5, beyond which it starts to taper. This corresponds well to
the separation zone thickness of 0.38R measured in the experi
ments. At the same time, the k–? models predict separation zones
that are smaller in size ?about 0.2R). In the experiments, as the
flow accelerated, the thickness of the separation zone was found
to reduce along the tube in a wavelike manner from the edge of
the stenosis, quite similar to computations by the k–? model.
During the middle of acceleration, when conditions are least
favourable for separation ?around time T1), the model again was
in better agreement, predicting a thickness value of approximately
0.25R within Z?2.0, while the k–? models predict a smaller
zone that is only about half the size. Towards the end of the
acceleration phase, near peak flow, the separation boundary moves
closer to the wall and eventually disappears between Z?3.0 and
Z?4.0, while at the same time, the k–? model predicts the gen
eration of waves and streamwise vortices in the high shear layer,
between Z?1.0 and Z?1.6, agreeing very well with the experi
mental values of Z?3.7 and Z?1.4, respectively. This is not the
case for the other models.
Turbulence intensity profiles at peak and minimum flow times
are shown in Fig. 15 ?profiles at other times are not shown here
Fig. 13
cycle. The stream function increment is 0.000425 sÀ1. The axial distance is indicated in normalized units Z „normalized by
diameter….
Streamlines for the sharpedged stenosis from the low Reynolds number RNG k–? model at different phases in the flow
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due to their similarities with profiles in corresponding phases of
the cycle?, further highlighting the differences between the models
used for the calculation. While the k–? models predict higher
intensities in the jetlike region immediately downstream of the
stenosis, the low Reynolds number k–? model predicts higher
values further downstream after approximately Z?2.8, which cor
responds to the region in which the structures start to breakdown
and transition to turbulence occurs. The latter is more consistent
with the experimental observations. However, high turbulence
levels observed in the region between Z?4.5 and Z?7.5 are not
predicted by any of the models.
All the turbulence models predict peak wall shear stress ?WSS?
levels at the stenosis, consistent with the high velocities associated
with flow acceleration through the restriction as seen in Fig. 16.
The maximum value occurs at time T2 when there is maximum
flow into the tube and the minimum value occurs at time T5,
during minimum flow conditions. Due to the geometry of the
constriction employed in this study, the WSS increases as the fluid
enters the constriction and then decreases as the fluid flows
through the region of minimum area only to increase again upon
entering the diverging section, resulting in the two peaks observed
in the figures. The low Reynolds number k–? model again differs
from the results of the other models, especially in the region 0.5
?Z?3.3, where flow separation and reattachment occurs, as was
observed for the smooth stenosis.
Discussion
Several previous studies have employed numerical models to
predict pulsatile flows in stenotic vessels ?5,6,13?. None of these
studies addressed the issue of turbulence. The two main experi
mental studies which considered pulsatile flow through stenotic
vessels clearly observed the presence of transitional or turbulent
flow distal to the stenosis ?9,11?. Previous experimental studies of
pulsatile turbulent flow through unrestricted channels or tubes
have shown that turbulence levels are highest during the decelera
tion phase of the cycle ?25?. Taken together these previous studies
suggest that numerical studies of turbulent pulsatile flow are war
ranted. Our own studies that were carried out for nonseparated
flows ?not discussed here? have shown that the turbulence inten
sity is indeed greater during deceleration, while it diminishes dur
ing acceleration. The characteristics which are observed for
simple confined pulsatile flows exist even in the presence of a
Fig. 14
The stream function increment is 0.00054 sÀ1.
Streamlines for the sharpedged stenosis from the low Reynolds number k–? model at different phases in the flow cycle.
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Page 14
stenosis, where the free jet that forms combines with these to
create a highly unstable flow resulting in cyclic and localized
turbulence.
For both stenosis geometries considered here, the turbulence
models predicted that the flow starts to become unstable around
peak flow, with velocity profiles becoming more inflectional in
nature. A region of reversed flow continues to grow throughout
the deceleration phase. The flow remains quite stable throughout
the acceleration phase. The low Reynolds number k–? model
was found to be in better agreement with the experiments as to the
prediction of the location and sizes of reversed flow regions that
occured distal to the stenosis throughout the cycle, than any of the
k–? models, especially in the case of the sharpedged stenosis.
The models in this study predict highly inflectional velocity
profiles, especially during the deceleration phase, which may
cause KelvinHelmholtztype instabilities to be generated at dif
ferent locations distal to the stenosis. These disturbances could
grow during parts of the cycle that are conducive to their growth
leading to localized turbulence at these times. The low Reynolds
number k–? model appears to be able to capture the generation
of streamwise vortices in the high shear layer and the resulting
unsteady vortex shedding distal to the stenosis. This in turn led to
the model predicting higher turbulence intensities at locations fur
ther downstream of the stenosis due to vortex interactions during
the deceleration phase, consistent with experiments, while its
counterparts predicted higher turbulence levels only in the jetlike
region immediately distal to the stenosis. Unsteady behavior along
the tube wall is interesting due to their possible interactions with
the vessel wall ?11?. The vortex shedding process was also ob
served by Bluestein et al. in their studies of steady flow and the
vortical structures were related to increased platelet deposition
?2?. In this context, unsteady vortex shedding observed in this
study would be very relevant to mural platelet deposition along a
stenotic vessel and warrants a thorough investigation.
The turbulence models predict maximum values of wall shear
stress at the stenosis during acceleration when the velocity is high
est at this location, followed by regions of negative WSS when
separated flow occurs. The WSS levels fluctuate throughout the
cycle and as in the experiments, the WSS remains low at locations
where higher turbulence levels are predicted, especially during the
deceleration part of the cycle. This is important in the context of
studying the effect of turbulence on wall shear stress and the
resulting arterial wall thickening. The results also show that de
tails of the stenosis geometry also affect the nature of variations in
the wall shear stress along the stenosis and distal to it.
In this study, enhanced wall functions were employed along
with the turbulence models in order to better resolve the near wall
region. These are more suited for low Reynolds number flows
with complex near wall phenomena. However, this approach is
more computationally intensive because it generally requires a
very fine near wall mesh, that can resolve the viscous sublayer.
Grid resolution requirements were found to be demanding for
these type of turbulence model calculations as opposed to the
standard wall function approach with turbulence models. The lat
ter approach utilizes relatively less computational resources, as
the viscosity affected near wall region is not resolved and the
Fig. 15
are in normalized units „by diameter away from the center of the stenosis… and the profiles are offset by 25 units in the
x direction.
Turbulence intensity profiles for the sharpedged stenosis at different phases in the flow cycle. The distances
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mesh is relatively coarse. A point to note here is that while em
ploying standard wall functions, the grid must not get into the wall
layer. However, standard wall functions generally perform poorly
for low Reynolds number type of flows ?32,39?.
Large discrepancies were observed when comparing the com
puted results with available experimental data. Part of this may be
attributed to the choice of upstream location and shape of inlet
velocity profile specified for the computations, factors which
could play an important role in such studies. The flow physics,
required accuracy, computational resources and turnaround time
constraints also play a major role in the selection of turbulence
model and near wall treatment as well as the computational grid
employed for these type of computations. Improvements in the
capabilities of turbulence models could also be made by perform
ing DNS or LES of pulsatile, stenotic flows ?28,29?.
Conclusion
Numerical predictions for turbulent pulsatile flow through two
different axisymmetric stenoses were obtained within the frame
work of twoequation turbulence models. The low Reynolds num
ber k–? model was found to be in better qualitative and quanti
tative agreement with the two experiments studied, in contrast to
the standard k–? model, the RNG k–? model or even the low
Reynolds number RNG k–? model. Many of the recent numerical
studies conducted in the biomechanical engineering community
have employed commercial CFD codes. Most commercial CFD
codes offer a variety of turbulence models from which one can
choose. Therefore, it is important to provide some guidance to the
community as to which models are appropriate for use in model
ing pulsatile turbulent stenotic flows as well as other issues in
volved in the modeling such as nearwall treatment. The results
obtained in this study could be useful in developing a viable low
Reynolds number turbulence model for predicting transition and
intermittency, the latter being the fraction of time that the flow is
turbulent, in stenotic flows. For now, it appears that the low Rey
nolds number k–? model seems to be better capable of predicting
such flows with its ability to predict low Reynolds number tran
sitional flows. Various factors such as computational resources,
flow physics and accuracy required will also have to be consid
ered for such studies.
Acknowledgments
Financial support for this work was provided through the B.F.S.
Schaeffer Award.
Fig. 16
distance is indicated in normalized units, Z.
Axial wall shear stress profiles for the sharpedged stenosis at different phases in the flow cycle. The axial
Journal of Biomechanical Engineering
AUGUST 2003, Vol. 125 Õ 459
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