ABSTRACT The scope of this study is the investigation of the influence of a convergent nozzle type stent on the flow field of one or two downstream located 50% stenoses. Both steady and unsteady inlet flow conditions are examined. The flow is predicted using the commercial code FLUENT, and the recirculation zones downstream of each stenosis are examined through flow visualization in a suitable experimental setup

Ch Ch

Christofidis

D S Mathioulakis

T G Papaioannou

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THE INFLUENCE OF A CONVERGENT NOZZLE ON THE FLOW FIELD OF 
DOWNSTREAM LOCATED MILD STENOSES 
 
T. G. Papaioannou 
School of Mechanical Engineering 
National Technical University of Athens 
Zografos 15773, Athens, Greece 
Ch. Ch. Christofidis 
School of Mechanical Engineering 
National Technical University of Athens 
Zografos 15773, Athens, Greece 
 
D. S. Mathioulakis  
School of Mechanical Engineering 
National Technical University of Athens 
Zografos 15773, Athens, Greece 
Proceedings of PVP2006-ICPVT-11 
2006 ASME Pressure Vessels and Piping Division Conference 
July 23-27, 2006, Vancoucer, BC, Canada 
 
PVP2006-ICPVT-11-93906 
ABSTRACT 
The scope of this study is the investigation of the 
influence of a convergent nozzle type stent on the 
flow field of one or two downstream located 50% 
stenoses. Both steady and unsteady inlet flow 
conditions are examined. The flow is predicted using 
the commercial code FLUENT, and the recirculation 
zones downstream of each stenosis are examined 
through flow visualization in a suitable experimental 
setup.      
Keywords: convergent nozzle, mild stenosis, flow 
separation, flow reattachment, wall shear stress.    
 
INTRODUCTION 
Ostial stenoses present several peculiarities and a 
higher restenosis rate compared to non aorto-ostial 
ones (Tan et al. [1]). Although several types of stents 
have been proposed for these cases (Colombo et al. 
[2]) there is yet no single design incorporating all the 
technical and hemodynamic characteristics of the 
ideal stent. The motivation of this work was to 
examine whether the implantation of a non 
conventional stent, nozzle-type, in the entrance of a 
coronary arterial branch with multiple mild stenoses 
would be beneficial by reducing the probability of 
arterial occlusion and in-stent restenosis. Previous 
studies investigating the hydrodynamic behaviour of 
multiple stenoses have been focused on pressure-flow 
relationship. Various factors are implicated in the 
“interference” and the influence of multiple stenoses 
on blood flow such as the severity of each stenosis 
(critical or mild) (Beckmann et al. [3]), the distance 
between the stenoses (Seeley and Young [4]), the 
order of two stenoses with different severity (Dodds 
and Phillips [5]) and the geometric characterisics of 
the stenosis (Tang et al. [6]). Flow characterisics 
through a stenosis have been examined in detail by 
Young & Tsai [7], [8] for different Reynolds numbers 
and stenosis geometries. rom: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms o 
 
In this study, a coronary artery is considered to 
initiate from the aorta the latter being simulated by a 
tank where fluid velocities are low, during the 
diastolic phase. Thus, a convergent nozzle is installed 
at the entrance of a circular straight tube simulating 
the artery, and also one or two axisymmetric stenoses 
are positioned within the tube. The big diameter of 
the nozzle is taken equal to two or three times the 
tube diameter. Both the experimental and 
computational part of this study emphasize on the 
flow field close to the solid walls e.g.  recirculation 
zones, wall shear stress distributions. Comparisons 
are made with the case that the nozzle is not present at 
the tube entrance. 
     
METHODS 
The examined flow field is predicted using the 
commercial code FLUENT, and an experimental rig 
is used to obtain more evidence about the 
recirculation regions downstream of each stenosis.  
 
COMPUTATIONAL PART 
The numerical integration of the governing equations, 
continuity and momentum, is performed using version 
6 of FLUENT code. The fluid is assumed Newtonian, 
incompressible and the flow laminar, isothermal  and 
axisymmetric. The fluid density is taken to be 1gr/cm3 
and its dynamic viscosity 4mPas. The computational 
domain is shown in Fig.1 for both cases, namely 
when a nozzle is installed at the entrance of the 
circular tube and for the case that this is not present.     
 
1 Copyright © 2006 by ASME
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Do 
Fig.1 Computational domain 
 
In order to take into account the fact that the flow 
starts from a space where low velocities exist, the 
entrance upstream of the nozzle is taken to be large 
enough, that is 6 times the diameter d of the tube. The 
diameter of the nozzle at its entrance is 3d and it is 
reduced to 1d in an axial length of 1d. Next to the 
nozzle, a 50% stenosis is installed, with a length L=2d 
and 1d long neck. A similar stenosis is installed 
downstream of the previous one  in a distance of 3d. 
The exit of the computational domain is at a distance 
10d downstream of the second stenosis, where fully 
developed flow boundary conditions are imposed. On 
the solid walls no-slip conditions hold, and symmetry 
conditions at the axis of symmetry. For the steady 
case, a flat velocity profile is assumed at the entrance 
of the computational domain, whereas in the unsteady 
case a harmonically varying in time flat profile is 
prescribed.  
With regard to the used computational grid, this is 
mainly structured, excluding the nozzle region where 
this is unstructured, consisted of triangular elements. 
Fig.2 shows details of the grid in the nozzle, and Fig.3 
the grid in the divergent part of a stenosis.  
 
 
Fig.2 Grid at the nozzle 
 
Fig.3 Grid in divergent part of a stenosis  
 
2
wnloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of The closest to the wall nodes are at a distance of 
0.01d and the streamwise distance between two 
neighbouring nodes 0.02d. Care was taken so that the 
grid is thinner close to the walls where flow 
separation and reattachment take place and 
information about the wall shear can be obtained. The 
total number of nodes for the two cases, namely with 
and without the nozzle was of the order of 8.3x104.  
The flow field is predicted for various Reynolds 
numbers for the steady case and both Reynolds and 
Womersley numbers for the unsteady case. The input 
data for the computations is α flat velocity profile, 
considering that for all cases the diameter d=4mm. 
The Reynolds number is defined as Re=Ud/ν, where 
U=4Q/(πd2), with Q the volume flow rate.        
 
EXPERIMENTAL PART 
The experimental study of the recirculation zones is 
done through flow visualization. Axisymmetric 
stenosεs with a 10mm neck diameter were machined 
out of Teflon, with a length of 28mm and an outer 
diameter of 14mm. These are introduced into a 
straight transparent Plexiglas tube of the same 
diameter. In order to avoid leakage between the outer 
surface of the stenosis and the inner surface of the 
Plexiglas tube two o-rings are attached on each 
stenosis. This permitted the easier positioning of each 
stenosis in the tube, the latter being connected to a 
tank with or without a nozzle. The nozzle made out of 
Plexiglas has a 2d diameter at its entrance and d/2 
long. In order to avoid light scattering from the 
Plexiglas walls, the straight tube is submerged into 
another tank where stationary liquid exists, being the 
same as the flowing medium. This is a glycerine –
water mixture in a volume proportion 40%-60%, 
giving an index of refraction close to that of the 
Plexiglas. The flow is established by gravity between 
two tanks and its rate is adjusted by a needle valve. In 
the unsteady case, a reciprocating piston of variable 
frequency is used, changing the flow to a 
harmonically varying one, the amplitude of which is 
adjusted by two valves. Flow visualization was done 
using an Argon Ion Laser light sheet and hollow glass 
spheres of 10μm diameter as tracers. The pictures 
obtained by a CCD camera were stored in a computer 
and analyzed in order to calculate the length of the 
reciculation zones. Fig.4 is a schematic of the used 
experimental set-up.                   
 
Copyright © 2006 by ASME
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Dow 
 
[1] test section, [2] stenoses, [3] inlet section including stent 
model; a) sharp edged and b) curved-edged, [4]  plexiglas 
tank with glycerine-water, [5] reservoir-1 simulating the 
aorta, [6] high level reservoir-2, [7] peristaltic pump, [8] 
Laser,  [9] lens, [10] mirror, [11] connecting tube, [12] low 
level reservoir-2, [13] valve. 
 
Fig.4 Experimental Set up 
 
 
COMPUTATIONAL RESULTS  
 
Steady Case 
Based on the predicted velocity field, the wall shear 
stress distribution is calculated as well as the points of 
separation and reattachment (points of zero wall 
shear). For all the examined Reynolds numbers,  
which were in the range 300-800, the flow separates 
in the divergent part of the stenosis and it reattaches 
at a point which moves downstream with increasing 
Re. With regard to the separation point this moves 
towards the stenosis neck with increasing Re but with 
a much smaller pace (ten times smaller) compared to 
the downstream movement of the reattachment point. 
The latter point covers a distance of about 1.5d for the 
examined Re range. Fig.5 shows schematically a 
typical separated region downstream of the stenosis 
neck, with the separation and reattachment points and 
its axial length.    
 
 
Fig.5 Recirculation zone downstream of a stenosis 
 
Fig.6 shows the length of the separated region non 
dimensionalized by the diameter d, versus Re for the 
first and the second stenosis with and without the 
nozzle. According to this figure, the influence of the 
nozzle upon the length of the separated region is more 
l
3
nloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Uevident in the first stenosis reducing its length by at 
most 8%d. However, the second stenosis seems 
practically not to be affected by the presence of the  
 
Fig.6 Length of separation regions versus Re for the two 
 
nozzle. 
With regard to the wall shear stress level, the major 
differences between the two cases appear as expected 
upstream of the stenosis, namely inside the nozzle. In 
Fig.7 the streamwise wall shear stress without the 
nozzle takes, for Re=800, some negative values due 
to a short separation bubble close to its entrance and 
in general much smaller values (up to 250%)  
Fig.7 Wall shear stress upstream of stenosis 
 
compared to the nozzle case.  
Calculating the spatially averaged wall shear stress 
for the same region for various Re, this takes higher 
values for the nozzle case, the difference increasing 
with increasing Re (see Fig.8). The maximum Copyright © 2006 by ASME
se: http://www.asme.org/about-asme/terms-of-use
Downloadifference is 50% compared with the non nozzle case 
and it appears at Re=800.        
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
300 400 500 600 700 800
Re number
m
ea
n 
m
ag
ni
tu
de
 W
SS
 (P
as
ca
l) 
  .
Cylindric Stent
Nozzle-stent
 
Fig.8 Mean wall shear stress upstream of stenosis versus 
Re 
 
Higher wall shear stresses appear as well in the 
divergent part of the first stenosis for the nozzle case 
but of smaller difference (at most 7%). This is 
depicted in Fig.9 where the mean magnitude of the 
wall shear stress is shown in the space between the 
two stenoses as a function of Re. It is noticeable that 
the influence of Re is not significant for this region.     
 
0.002
0.003
0.004
0.005
0.006
0.007
0.008
300 400 500 600 700 800
Re number
m
ea
n 
m
ag
ni
tu
de
 W
SS
 (P
as
ca
l) 
   
.
Cylindric Stent
Nozzle-stent
 
Fig.9 Mean wall shear stress between two stenosis versus 
Re 
 
The influence of the nozzle on the flow field of a 
more severe stenosis like for example a 75% stenosis 
is null. This is clearly shown in Fig.10 where the 
separation flow region covers the whole space 
between the two stenoses for both cases without any 
evident difference between them. It is clear that in a 
severe stenosis the flow loses its “memory”, its 
beginning being the neck of each stenosis.  
 
 
nozzle case 
 
no-nozzle case 
 
Fig.10 Flow separation for a 75% stenosis  
 
Unsteady Case 4
ded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of UThe velocity profile at the inlet of the computational 
space is flat, varying in time according to the formula: 
u=u0(1-cosωt). Several cases were run changing the 
amplitude u0 and keeping constant the period of the 
flow T=1sec, which corresponds to Womersley 
number W=d/2(ω/ν)1/2=2.56.  The wall shear stress 
distribution for Remax=500 is shown in Fig.11 for four 
time instants, namely t/T=0.2, 0.4, 0.6 and 0.8 when  
Fig.11 Wall shear stress at four time instants 
 
the nozzle is present.  
According to Fig.11, in the beginning of each cycle 
that flow acceleration takes place, the flow is attached 
(positive values of wall shear stress, WSS). Later on 
and before the flow rate maximizes, flow separation 
takes place downstream of each stenosis (negative 
WSS), the length of each recirculation zone being 
time dependent. It is characteristic that during 
deceleration, the reattachment point is further 
downstream compared with that of the acceleration 
phase although the velocity at the examined time 
instants is the same. This is shown in Fig.11 at  
t/T=0.4 and 0.6 for which although the instantaneous 
inlet velocity is the same, the curves deviate in the 
divergent part of each stenosis. Apparently, it is the 
positive pressure gradient of the deceleration phase 
which moves the reattachment point further 
downstream and therefore the two curves with respect 
to each other. Comparing the length of the 
recirculation regions for both cases with and without 
the nozzle, the difference was found to be at most 4% 
smaller for the nozzle case. This is shown in Fig.12 
for the first stenosis since in the second one the 
differences are even smaller.   
Fig.12 Length of separation region versus time 
 
Also, the values of WSS differ between the two cases 
only up to the end of the neck of the first stenosis, Copyright © 2006 by ASME
se: http://www.asme.org/about-asme/terms-of-use
Downloadednamely only where the flow is attached, taking higher 
values for the nozzle case (Fig.13).   
 
 
 
Fig.13 Wall shear stress distribution for nozzle and no-
nozzle cases at t/T=0.4 
 
EXPERIMENTAL RESULTS 
 
Using flow visualization, the separation regions 
downstream of each stenosis are investigated since 
the stenoses are non transparent. Therefore, only the 
reattachment point could be observed. This was 
defined by observing the recorded video of each case 
and specifying the location where the flow changes 
direction close to the wall. It should be stressed that 
the followed procedure is not very accurate, 
supplying however qualitative evidence of the flow in 
the examined regions. In Fig.14 the stenosis 
downstream end is at the right part of the picture and 
the flow direction is from right to left. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig.14 Typical picture of the flow downstream of a 
stenosis 
 
The lower part of the picture corresponds to the wall 
region where the observation was focused in order to 
define the reattachment point. Here, the fluid 
velocities are much smaller, that’s why the seeding 
particle images are depicted as dots, compared to the 
long streak lines far from the wall where the 
velocities are high.    
Running  several steady cases for Re between 210 and 
420 with and without the nozzle it was found that 
Tube center 
Recirculation zone 
5
 From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Uusing the nozzle the length of the separation region of 
the first stenosis is shorter by about 6%.  
With regard to the unsteady case, several flow rates 
were examined so that the mean Re varied between 
200 and 600. The flow rate amplitude was between 70 
and 80% of the mean flow rate and Womersley 
number 4.2. Due to the unsteadiness of the flow field, 
the reattachment point (namely the point downstream 
of which the velocity next to the wall was positive at 
a given instant of a cycle) was travelling back and 
forth, making its observation not an easy task. 
However, what it was systematically measured is an 
increased flow rate by 10% at most for the same 
experimental settings when the rounded nozzle was 
used.   
  
CONCLUSIONS 
The influence of a convergent nozzle upon the flow 
field of two downstream located 50% stenoses seems 
to be evident practically only on the first stenosis by 
reducing the extent of its recirculation zone and 
increasing the wall shear stresses, especially in the 
attached part of the flow. Similar trends appear both 
for steady and unsteady inlet conditions. These effects 
induced by the new stent configuration could be 
important for the delay of restenosis processes and 
should be further investigated in-vivo. 
 
ACKNOWLEDGEMENTS  
The present work was co-funded by the European 
Social Fund (75%) and National Resources (25%) –
Operational Program for Educational and Vocational 
Training II (EPEAEK II) and particularly the Program 
PYTHAGORAS. 
 
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Percutaneous transluminal coronary angioplasty of 
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16:631–639, 1995.  
 
[2] Colombo A., Stankovic G., and Moses J. W. 
Selection of coronary stents. J. Am. Coll. Cardiol. 
40:1021–1033, 2002. 
 
 [3] Beckmann CF, Levin DC, Kubicka RA, 
Henschke CI. The effect of sequential arterial 
stenoses on flow and pressure. Radiology 1981; 
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[4] Seeley BD, Young DF. Effect of geometry on 
pressure losses across models of arterial stenoses. J 
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[5] Dodds SR, Phillips PS. The haemodynamics of 
multiple sequential stenoses and the criteria for a 
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26:348-353. 
Copyright © 2006 by ASMEse: http://www.asme.org/about-asme/terms-of-use
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[6] Tang D, Yang C, Kobayashi S, Zheng J, Vito RP. 
Effect of stenosis asymmetry on blood flow and 
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[7]Young DF, Tsai FY. Flow characteristics in 
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[8] Young DF, Tsai FY. Flow characteristics in 
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6 Copyright © 2006 by ASME
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THE INFLUENCE OF A CONVERGENT NOZZLE ON THE FLOW FIELD OF DOWNSTREAM LOCATED MILD STENOSES

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THE INFLUENCE OF A CONVERGENT NOZZLE ON THE FLOW FIELD OF DOWNSTREAM LOCATED MILD STENOSES

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