numerical investigation on the blood flow characteristics ... · numerical investigation on the...

Korea-Australia Rheology Journal June 2009 Vol. 21, No. 2 119

Korea-Australia Rheology JournalVol. 21, No. 2, June 2009 pp. 119-126

Numerical investigation on the blood flow characteristics

considering the axial rotation in stenosed artery

Kun Hyuk Sung, Kyoung Chul Ro and Hong Sun Ryou*

School of Mechanical Engineering, Chung-Ang University, 221 HeukSuk-Dong, Dongjak-Gu,Seoul 156-756, Korea

(Received March 13, 2009; final revision received May 4, 2009)

Abstract

A numerical analysis is performed to investigate the effect of rotation on the blood flow characteristics withfour different angular velocities. The artery has a cylindrical shape with 50% stenosis rate symmetricallydistributed at the middle. Blood flow is considered a non-Newtonian fluid. Using the Carreau model, weapply the pulsatile velocity profile at the inlet boundary. The period of the heart beat is one second. In com-parison with no-rotation case, the flow recirculation zone (FRZ) contracts and its duration is reduced in axi-ally rotating artery. Also wall shear stress is larger after the FRZ disappears. Although the geometry ofartery is axisymmetry, the spiral wave and asymmetric flow occur clearly at the small rotation rate. It iscaused that the flow is influenced by the effects of the rotation and the stenosis at same time.

keywords : blood flow, axially rotating velocity, stenosis, pulsatile flow, non-Newtoninan

1. Introduction

Arteriosclerosis is one of the most widespread diseases in

human beings. It is a significant factor in the death rate

because it affects hemodynamics and reduces the flow rate

of blood to the heart and the brain. In particular, sudden

body movement causes the blood pressure fluctuations of a

person with an arterial diseases and an ischemic poverty of

blood to increase more than those of a healthy person. In a

serious case, when the blood flow rate decreases, the per-

son may fall because of vertigo and experience temporary

eyesight and hearing trouble. Thus, human body move-

ments can affect the blood flow characteristics of persons

with arterial diseases more than those of healthy persons.

In research on human acceleration, Burton et al. (1974)

studied eyesight trouble caused by a change of blood flow

in extreme gravity circumstances. Hooks et al. (1972) con-

ducted a clinical study of the side effects of body accel-

eration. In research on blood flow and acceleration, Misra

and Sahu (1988) developed a mathematical model to study

the blood flow through large arteries under the action of

periodic body acceleration. Belardinelli et al. (1989) per-

formed an experimental study of the effect of blood pressure

by shock acceleration, and Mandal et al. (2007) performed

a numerical study on the blood characteristics of a cylin-

drical blood vessel with periodic accelerations. Nakamura et

al. (1988) and Luo et al. (1992) carried out an investigation

of the blood flow characteristics of stenosed and bifurcated

blood vessels. Ro et al. (2008) performed a numerical study

on the blood characteristic of the carotid bifurcation artery

with periodic accelerations. However, none of these resear-

ches focused on the effect of body acceleration on the char-

acteristics of blood flow without rotation.

From the medical side, studies on the effects of cervical

or spinal rotation on hemodynamic in arteries have been

executed, but they concentrated on the velocity distribution

caused by the change in artery volume. There have been

only a few studies on how the rotational movement of the

human body affects blood flow characteristics.

In fluid dynamics, the flow characteristics of an axially

rotating pipe without stenosis have been studied. For

example, Imao et al. (1992) showed the flow instability

problem in axially rotating pipes at the critical ratio of the

circumferential velocity to the mean axial velocity.

Kikuyama et al. (1983) showed that the transition of flow

state from the laminar to the turbulent in axially rotating

pipe can occur at a low Reynolds number. Both of these

experiments showed that rotation caused destabilization of

the flow in an axially rotating pipe. However, these exper-

iments used water, so they did not address the characteristics

of blood, which has non-Newtonian viscosity. Generally, the

assumption of Newtonian behavior of blood is acceptable

for high shear-rate flow, but it is not valid when the shear

rate is low (0.1 s-1), as it is in small arteries or on the down-

stream side of stenosis (Chien et al., 1982). It has also been

pointed out that, in some diseased conditions e.g. patients

with severe myocardial infarction, cerebrovascular diseases*Corresponding author: [email protected]© 2009 by The Korean Society of Rheology

Kun Hyuk Sung, Kyoung Chul Ro and Hong Sun Ryou

120 Korea-Australia Rheology Journal

and hypertension blood exhibits remarkable non-Newtonian

behavior. Thus, non-Newtonian viscosity must be consid-

ered when analyzing the characteristics of flow in stenosed

arteries.

Young (1979) showed that head loss is a nonlinear func-

tion of stenosis and that pressure losses become significant

only for stenoses greater than 50~70%. Thus, the stenosis

affects significantly the pressure distribution in the artery

and puts the person with an arterial disease in jeopardy.

The experiment showed the effect of inlet velocity on the

flow characteristics at the downstream of stenosis in the

artery (Deplano et al., 1999).

Extremely, the angular velocity is about 8 revolutions a

second when the figure skater performs a standing spin.

Although that case is rare to the common people, they

undergo rotations of body through daily exercise. So rotation

of body can affect the characteristic of blood flow such as

pressure drop, wall shear stress, and flow recirculation zone.

However, in spite of the importance of rotation effect,

there is no research on the effect of rotation on the flow

characteristics considered pulsatile velocity profile in

stenosed arteries.

Hence, for the basic study on the rotation effect to the

blood flow, we select the common carotid artery because

the artery rotates axially when people do a standing spin.

Therefore the purpose of this paper is a numerical analysis

of the effect of rotation and an unsteady pulsatile velocity

profile in a stenosed artery.

2. Numerical details

2.1. Governing equationsIn order to simulate the blood flow characteristics, mass

and momentum conservation equations are required and the

non-Newtonian viscosity and pulsatile flow must be con-

sidered. In addition, the axially rotating, centrifugal force

must be added to the momentum equation as a source term.

(1)

(2)

To simulate a non-Newtonian fluid problem, a consti-

tution equation is required for blood rheology character-

istics described by the second invariant of shear rate tensor:

(3)

where η and are apparent viscosity and shear rate.

Shear rate is represented as:

(4)

We use the Carreau viscosity model because it is more

suitable for representing blood rheology characteristics

(Cho, 1985):

(5)

where η0 is the zero shear viscosity (0.056 Pa·s), is

the infinite shear viscosity (0.00345 Pa·s), λ is the time

constant (3.313 s) and n is the power law index (0.356).

2.2. Modeling of an artery with stenosisFig. 1 shows a schematic view and grid generation of a

stenosed blood vessel. The stenosis, where is located

between 1D upstream and 1D downstream from the center

of stenosis, is modeled by Young’s model (Young, 1968),

as shown in equation (6). The diameter of the blood vessel

is 8 mm, and the minimum diameter of the stenosis is half

the size of the blood vessel. The stenosis rate is 50% with

no eccentricity.

(6)

where, a is the stenosis rate, z0 is half the length of the

stenosis ,1D , and z1 is the axial position from the starting

point of the stenosis where is 1D upstream from the center

of stenosis. The grid independent test is performed with

four different number of grid cells which are 43,587,

130,720, 434,808 and 580,320 when the angular velocity is

6 rev/s. In Fig. 2(a) and (b), the averaged wall shear stress

(WSS) with 434,808 grid cells follows that with 580,320

grid cells at each 3D and 5D downstream from the center

of stenosis. The difference of averaged WSS is within 5%

for a period of pulsatile in both cases. Thus, 434,808 hexa-

hedral grid cells is selected for numerical analysis. The

computing time for each case was about 8 hours with 8

nodes, 2.0 GHz CPU.

2.3. Boundary and initial conditionFor the numerical simulation, the 3-D time-dependent

Navier-Stokes equations were solved by the ANSYS CFX

V11.0 based on the finite volume method with the pres-

sure-based coupled solver. Fixed time step, 0.002 s, was

used with the Second Order Backward Euler scheme for

transient term. The flow is assumed to be a laminar flow,

incompressible, non-Newtonian, and the wall of artery is

∂ρ

∂t------ ∇ ρν( )⋅+ 0=

∂

∂t---- ρν( ) ∇ ρν ν⊗( )⋅+ =

pδ– µ ν ν∇( )T+∇( )+( ) ρω ω r×( )×–∇

τ ηγ· =

γ·

γ·

γ·1

2--- γij

·γji·

j

∑i

∑=

η η∞ η0 η∞–( ) 1 λγ·( )2

+[ ]n 1–( ) 2⁄

+=

η∞

R z( ) R a R 1 π z z1–( ) z0⁄cos+[ ]⋅–[ ]=

Fig. 1. The schematic view and grid generation of stenosed blood

vessel.

Numerical investigation on the blood flow characteristics considering the axial rotation in stenosed artery


rigid with no slip conditions.

For the study of the effect of the pulsatile flow on the

blood flow characteristics in an artery, the idealized pul-

satile velocity profile of the common carotid artery is used

as the inlet boundary condition (Gijsen et al., 1999). Fig. 3

shows the pulsatile velocity profile which is dimensionless

by the period of pulsatile cycle, tp, which is 1 s. The time-

averaged dynamic viscosity for a period of pulsatile is

0.007865 . The peak of velocity is 0.20 m/s at inlet in

pulsatile and the density of blood is 1100 kg/m3. The pres-

sure boundary condition is used in the outlet of the artery.

The initial velocity through whole domain is equal to the

start of systole, t/tp =0, in pulsatile. For the study on the

effect of rotation on the blood flow characteristics, we use

a MRF (Multiple Reference Frame) method for application

to the rotating effect of blood vessel (Luo et al., 1994).

Through the MRF method, the only wall is rotated.

3. Results and discussion

For the validation of our numerical method, the numer-

ical results are compared with the experiment in axially

rotating pipe (Imao et al., 1992). The angular velocity of an

axially rotating artery is varied as 1, 2, 4 and 6 revolutions

per a second (rev/s), and the results are compared with

those of the no-rotation case.

We compare the axial velocity profiles of Newtonian

fluid flow to that of non-Newtonian fluid flow for 6 rev/s.

Consequently, the Newtonian fluid flow is more unstable

than another due to the magnitude of viscosity. Overall, the

viscosity of non-Newtonian fluid is larger than that of

Newtonian fluid. For unsteady flow, the simulations are

executed over at least three cycles to achieve a periodic

solution. The velocity variation after two cycles is less than

1% at test points behind the stenosis and results are saved

for the final cycle. In this section, the blood flow char-

acteristics such as axial velocity profiles, pressure distri-

bution, flow recirculation zone (FRZ) and wall shear stress

(WSS) distribution are presented. The blood flow char-

acteristics are obtained for the entire flow domain at four

different instants (t / tp = 0.14, 0.16, 0.40, 0.78) in pulsatile.

3.1. ValidationsDue to the instability problem of flow in the experiment

Pa s⋅

Fig. 2. Averaged axial WSS on circumferential lines at two dif-

ferent locations from the stenosis in pulsatile for various

number of grid cells. ((a) At 3D downstream from the

stenosis, (b) At 5D downstream from the stenosis).

Fig. 3. The pulsatile inlet velocity profile.



(Imao et al., 1992), the spiral wave appears as the ratio of

circumferential velocity to the axial velocity increases and

it is clear when the rotation rate is 3 in axially rotating pipe.

Thus, we have been performed the numerical analysis

when the rotation rate is 3. In the experiment, the Reynolds

number is 500 and the ratio of circumferential velocity to

the axial velocity is 3. The circumferential velocity of a

pipe and the flow rate of water are given as constant. Fig.

4(a) shows the circumferential velocity which is dimen-

sionless by the wall velocity and Fig. 4(b) shows the axial

velocity which is dimensionless by the inlet velocity in axi-

ally rotating pipe.

The numerical results follow the experimental results

except the axial velocity profile at Z=120. The profile of

measured axial velocity tends to become the turbulent at

Z=120. This reason is that the flow state changes from

laminar to turbulent by the rotation effect in experiment.

However, the circumferential velocity profile approaches

the solid-body rotation gradually as the numerical simu-

lation predicts.

3.2. Results of numerical analysisThe axial velocity profiles are presented in a plane con-

taining the axis of artery because the geometry is axi-

symmetric. Fig. 5(a), (b), (c) and (d) show axial velocity

profiles and FRZs when the artery is not rotating at dif-

ferent time phases in pulsatile. The contour of zero velocity

indicates the boundary of FRZ, in which the flow is either

stagnant or reversed. The jet velocity profile like a piston

shape appears behind the stenosis in flow acceleration

phase. Due to the decrease of the area at the stenosis and

reverse flow near the vessel wall, axial velocity of the center

line of artery increases in order to satisfy the flow rate con-

servation law at 1D downstream from the stenosis. The FRZ

appears at all time in pulsatile because the stenosis causes

disturbance to the flow and its size varies due to the change

Fig. 4. Dimensionless velocity profile ((a) The dimesionless cir-

cumferential velocity, (b) The dimensionless axial velocity).

Fig. 5. Axial velocity profiles and flow recirculation zone at dif-

ferent time phase in pulsatile. The position of profile is

presented as the multiple of diameter distal to stenosis.

((a) No rotation model, t/tp=0.14, (b) No rotation model,

t/tp=0.16, (c) No rotation model, t/tp=0.40, (d) No rotation

model, t/tp=0.78, (e) 6 rev/s model, t/tp=0.40, (f) 6 rev/s

model, t/tp=0.78).



of magnitude of velocity in pulsatile. The FRZ grows and

expands to the downstream of stenosis as flow accelerates

and extends up to the whole downstream at t / tp=0.4. After

that time, FRZ contracts rapidly and its size becomes con-

stant because the gradient of velocity is zero in pulsatile.

Due to the centrifugal force by rotating of vessel, the pres-

sure along axial line decreases but the pressure near wall

increases at the same time as shown in Fig. 6(a). Conse-

quently, the iso-pressure lines become more convex as the

angular velocity increases like Fig. 6(b). This causes the

friction increases and this phenomenon suppresses the heart

due to the increase of pressure drop at the rotating artery.

Fig. 5(e) and (f) show axial velocity profiles and FRZs

when the angular velocity is 6 rev/s. In comparison with the

no-rotation case, the FRZs are reduced remarkably due to

the increase of radial velocity from the centrifugal force at

t/tp =0.4. Also, axial velocity profiles and FRZs are asym-

metry although the geometry of artery is axisymmetry.

Fig. 7(a) shows the axial velocity contours when the

angular velocity is 6 rev/s at four different time phase in

pulsatile. Asymmetric contour with three or four protru-

sions appears at t/tp=0.4 and possesses the eccentricity . In

Fig. 6, the protrusions constitute the spiral wave as the

angular velocity is 4 rev/s. Fig. 7(b) shows axial velocity

contours as variation on the angular velocity at t/tp=0.40.

As the fluid flows to the downsteram, the four protrusions

of contour become dim at 6 rev/s. On the other hand, the

shape of four protrusions is maintained at 4 rev/s. The

asymmetric coutours appear after 3D downstream from the

stenosis at the whole rotating cases and after that location,

3D, the shape of protrusions rotates slowly.

Imao et al. (1992) investigated the structure and char-

acteristic of a spiral wave with the flow visualization tech-

nique in axially rotating pipe. In experimental study, the

spiral wave occurs as the variation of rotation rate which is

the ratio of circumferential velocity to the axial velocity.

As the rotation rate is 3, the spiral wave is most amplified

and is reduced with a greater rotation rate.

Fig. 8 shows the Iso-contour of the axial velocity at four

different instants in pulsatile. As the angular velocity is

4 rev/s, the spiral wave occurs at t/tp=0.4. The spiral wave

appears and disappears as the variation of rotation rate due

to the pulsatile inlet velocity. The stenosis affects the sta-

bility of flow in the artery. Tang et al. (1999) showed the

asymmetric flow patterns occur and become unstable in an

axisymmetric geometry with the stenosis. Buchanan et al.

(1998) showed in their numerical study that two co-rotat-

ing vortices occurred in a 75% (area reduction) axisym-

metrical stenosed model when the flow started to decel-

erate in pulsatile. In our numerical analysis, the spiral wave

and asymmetric flow occur clearly although the rotation rate

is smaller than 3. It is caused by the flow is influenced by

Fig. 6. Pressure distribution at t/tp=0.40 ((a) Pressure distribution

along the axial line at t/tp=0.40, (b) Pressure contour at yz-

plane as x=0).

Fig. 7. The Axial velocity contours ((a) Axial velocity contours

when the angular velocity is 6 rev/s at four different time

phase in pulsatile, (b) Axial velocity contours as variation

on the angular velocity at t/tp=0.40).



the effects of the rotation and the stenosis at the same time.

Variations of averaged axial WSS on circumferential

lines are shown in Fig. 9 at four different downstream loca-

tions through the whole pulsatile. Considering for the

asymmetric flow due to the rotation and stenosis of artery,

WSS are averaged on circumferential line of each location.

The intersections of WSS curves and the horizontal axis

are two points at which WSS changes signs and hence cor-

respond to the flow separation point (WSS changes from

positive to negative) and reattachment point (WSS changes

from negative to positive). At 1D downstream from the

stenosis, there is no intersection point and the sign of WSS

is negative all the time. Those indicate FRZ occurs through

the whole cycle. After 3D downstream from the stenosis, it

is notable that the difference of negative WSS caused by

the reverse flow. The WSS of no-rotation case is about

40% and nearly twice smaller than that of rotating case due

to the stronger reverse flow at 5D and 7D downstream

from the center of stenosis, respectively. The centrifugal

force caused by rotation effect suppresses the blood flow

toward wall and this pheonomenon decreases the reverse

Fig. 8. ISO-contour of the axial velocity which is 0.065 m/s when

the angular velocity is 4 rev/s.

Fig. 9. Averaged axial WSS on circumferential lines at four different downstream locations in pulsatile ((a) 1D downstream from the

stenosis, (b) 3D downstream from the stenosis, (c) 5D downstream from the stenosis, (d) 7D downstream from the stenosis).



flow and the FRZs. But, the WSS of rotation is larger after

the FRZs disappear because the circumferential gradient of

velocity is producted by the rotation of artery. This causes

the new vascular disorder by injuring endothelium of artery

such as arteroscelrosis at the downstream of stenosis (Fry,

1972). Also, when the artery is rotating, the existence time

of FRZs shortens further from the stenosis. In comparison

with no-rotation case, the existence time is smaller about

12.5% at 7D downstream from the center of stenosis.

4. Conclusion

In this paper, the rotation effects has been studied numer-

ically on a stenosed blood vessel with the pulsatile inlet

velocity profile.

The FRZs occur through the whole pulsatile because the

stenosis causes disturbance to the flow and its size varies

in pulsatile. In comparison with the no-rotation case, the

size and existence time of FRZs are reduced remarkably

due to the increase of radial velocity from the centrifugal

force in axially rotating artery. But, the friction increases

due to the increase of pressure by the centrifugal force.

This phenomenon suppresses the heart due to the increase

of pressure drop at the rotating artery.

Also, the WSS of no-rotation case is about 40% and

nearly twice smaller than that of rotating case due to the

stronger reverse flow at 5D and 7D downstream from the

stenosis, respectively. But the WSS of rotation is larger after

the FRZs disappear because the circumferential gradient of

velocity is producted by the rotation of artery. This causes

the new vascular disorder by injuring endothelium of artery

such as arteroscelrosis at the downstream of stenosis.

Although the geometry of artery is axisymmetry, the spi-

ral wave and asymmetric flow occur clearly in spite of the

small rotation rate. The contour of axial velocity is asym-

metric with four or three protrusions at t/tp=0.4 and pos-

sesses the eccentricity after 3D downstream from the

stenosis at the whole rotating cases. And it is caused that

the flow is influenced by the effects of the rotation and the

stenosis at the same time.

Acknowledgement

This research was partially supported by the Chung-Ang

University Grant in 2009.

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List of symbols

R [m] radius of blood vessel

D [m] the maximum artery diameter

t [s] simulation time

tp [s] a period of pulsatile

z0 [m] half the length of the entire stenosis region, 1D

z1 [m]the axial position from the starting point of the

stenosis region

a [-] the rate of stenosis

v [-] dimensionless circumferential velocity, v’/R ω

r [-] dimensionless radial distance, r’/R

w [-] dimensionless axial velocity

ρ [kg/m3] density

[m/s] blood velocity tensor

η [Pa s] apparent viscosity

[1/s] shear rate

[rev/s] angular velocity

Subscripts

0 zero shear, amplitude

infinite shear

Superscripts

( )’ Dimension value

ν

γ·

ω

∞



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