Page 1

Sonu S. Varghese

Steven H. Frankel1

e-mail: 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

Reynolds-averaged Navier-Stokes 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 (renormalization-group 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 fluid-structure interactions, non-Newtonian 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 converging-diverging

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, atheroscle-rosis, 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 co-workers constructed a plexiglass model of an

axisymmetric stenotic blood vessel experiencing pulsatile flow

and measured instantaneous core velocity, ensemble-averaged 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 time-varying 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 fluid-structure inter-

action and rheology effects. Bathe and Kamm ?13? conducted a

finite-element analysis to examine fluid-structure 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 fluid-structure 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 time-mean 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 time-mean characteristics of pulsatile

turbulent flow ?such as time-averaged 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 time-mean 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 sub-grid-scale 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 near-wall

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 Navier-Stokes ?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… sharp-edged 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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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… Sharp-edged 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

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Page 4

stream of the stenosis. These studies highlight the utility of con-

ducting three-dimensional 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 two-equation 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 near-wall treatment and grid-

independence are also discussed.

Turbulence Models

The Reynolds averaged Navier-Stokes ?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

Navier-Stokes equations such as axial velocity, u, can be ex-

pressed in terms of an ensemble-averaged quantity, u ¯, and a fluc-

tuation from this average, u?. Likewise for other scalar quantities

like pressure. Applying this decomposition to the Navier-Stokes

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 time-averaged 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 two-equation 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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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

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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 Navier-Stokes 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 near-wall 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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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 Hagen-Poseiulle 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 time-dependent 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

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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

time-dependent and time-independent 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 two-equation 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 pressure-velocity 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 near-wall region is not

resolved with the near-wall 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 near-wall 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 low-Reynolds 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 time-periodic 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 post-processing 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 Sharp-Edged Stenosis.

The experimental investigation of Ojha et al. ?9? on pulsatile flow

through an axisymmetric sharp-edged 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 sharp-edged stenosis „dif-

Journal of Biomechanical Engineering

AUGUST 2003, Vol. 125 Õ 453

Page 10

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 co-workers ?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 sharp-edged 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 Sharp-Edged 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 ensemble-averaged 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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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 sharp-edged 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 sharp-edged stenosis from the low Reynolds number k–? model at different phases in the flow cycle.

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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 sharp-edged stenosis.

The models in this study predict highly inflectional velocity

profiles, especially during the deceleration phase, which may

cause Kelvin-Helmholtz-type 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 sharp-edged 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 two-equation 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 near-wall 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 sharp-edged stenosis at different phases in the flow cycle. The axial

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