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Copyright © 2012 by ASME 1
INTRODUCTION
This work is a collaborative effort between the Biomechanics and
Living System Analysis Laboratory (BIOLISYS) in Cyprus and the
Biomechanics Laboratory of IACM/FORTH in Greece. Both labs
combine interdisciplinary skills from engineering, medicine and
biology to provide solutions to clinical problems associated with
cardiovascular and other diseases. For this study, numerical flow
simulations were performed using: a) open source software VMTK
and commercial software ICEM CFD as pre-processors, b) the finite
volume based solver Fluent and c) Tecplot 360 (Amtec Inc.) for post-
processing.
METHODS
Solver type and details
We used the finite volume solver Fluent v.12.1.4 (Ansys Inc.) for the
numerical approximation of the Navier-Stokes equations. For the
steady state cases the Coupled scheme was selected for pressure
velocity coupling using the pressure-based coupled solver. For the
transient solutions the PISO scheme was selected for the pressure
velocity coupling using the pressure-based segregated solver. We
apply the second order upwind scheme to discretise the convection
terms in the momentum equations and a second order pressure
interpolation scheme. A first order iterative time advancement scheme
is applied for the transient solutions. Gradients are computed using the
Green-Gauss node based method.
Mesh and boundary conditions
We use tools from VMTK to apply cylindrical flow extensions at the
inlet (Dinlet ~ 0.56 cm) and outlet of the domain so that we prescribe a
fully developed flow boundary condition at the inlet and a traction free
boundary condition at the outlet. The length of the outflow extension
was calculated based on the approximate relation for the entrance
length for steady laminar rigid pipe flow: Le/D~0.06Re where Le is the
entrance length, D is the tube diameter and Re the Reynolds number
[1]. In our case the maximum Re is 649 corresponding to a peak
systolic flow rate of 11.42 ml/s. Based on the outlet diameter (0.44
cm) the outflow length using the above relation should be at least 17
cm. An extension of 25 cm was applied.
We used ICEM CFD v12.1 (Ansys Inc.) to discretise the
computational domain and generate an unstructured mesh. The
computational domain (excluding flow extensions) is discretised with
~2.1 106 hybrid, linear elements with an average cell center spacing of
0.25 mm. Near-wall layers of prism elements were used throughout
the domain for boundary layer refinement with a 10-2 Dinlet distance of
the center of the first element from the wall. Triangles were used to
discretise the surface of the aneurysm and quads for the extensions.
Pyramid and tetrahedral elements were used to fill the core of the
computational domain in the aneurysm. The o-grid method was used
to generate layers of hexahedral elements in the flow extensions. A
parabolic velocity profile was applied at the inlet for the steady flow
cases with a mean velocity corresponding to the required flow rate.
For the transient flow cases the velocity profiles prescribed at the inlet
at each time step were obtain from the Womersley solution
(Womersley number~3.5) based on the flow waveform provided
(scaled appropriately to generate the desired mean flow rates) .
Steady-state flow computations were obtained on a HP Z800
workstation with 4 quadratic Intel Xeon processors in parallel. The
total CPU time was around 380 hrs corresponding to total wall clock
time of 95 hrs. Time varying solutions were obtained on an Intel Xeon
X5355 @ 2.66 GHz processor based Linux cluster requiring a total
CPU time of 75 hrs per flow cycle.
Grid size and time step independence study
We refined our mesh by reducing the mean cell center distance from
0.25 to 0.18 mm thus increasing the number of elements in the
aneurysm (excluding the extensions) from ~2.1 106 to ~4 106 and
Proceedings of the ASME 2012 Summer Bioengineering Conference SBC2012
June 20-23, Fajardo, Puerto Rico, USA
CFD CHALLENGE: SOLUTIONS USING THE COMMERCIAL FINITE VOLUME SOLVER,
FLUENT
Nicolas Aristokleous1, Mohammad Iman Khozeymeh
1, Yannis Papaharilaou
2, Georgios C.
Georgiou3, Andreas S. Anayiotos
1
1Department of Mechanical
Engineering and Materials Science
and Eng., Cyprus University of
Technology, Limassol, 3503, Cyprus
Proceedings of the ASME 2012 Summer Bioengineering Conference SBC2012
June 20-23, Fajardo, Puerto Rico
SBC2012-80691
1Department of Mechanical
Engineering and Materials Science
and Eng., Cyprus University of
Technology, Limassol, 3503, Cyprus
2Institute of Applied and Computational
Mathematics, Foundation for Research and
Technology – Hellas, Heraklion, Crete,
71110, Greece
3Department of Mathematics and
Statistics, University of Cyprus,
Nicosia, 1678, Cyprus
Copyright © 2012 by ASME 2
repeated our steady flow computations for the maximum flow case
(Q=11.42 ml/s). Our results indicated a maximum difference of less
than 1% in the computed centerline pressures between the coarse and
refined mesh. The initial mesh was thus considered sufficient.
A time step of 2.5 10-4 s was selected based on the expected peak
streamwise velocity and the mean element spacing in the aneurismal
sac. This corresponds to 3960 time-steps per flow cycle. To ensure
time step independence we repeated our transient computation for
Case 1 with a time step of 1.25 10-4 s or ½ the initial time step. The
maximum difference in the computed centerline pressures was less
than 1%.
Time periodicity and solution convergence
To exclude flow transients that appear during the early stages of the
numerical computation from our results we solve for 4 consecutive
flow cycles and present the results of the last cycle. We compared
peak systolic centerline pressures for the 3rd and 4th flow cycle and
found a maximum difference of less than 1% indicating that a time
periodic solution has been achieved in the fourth cycle. The total force
integrated over the aneurysm surface was also used as a measure to
verify time periodicity of the transient solution and convergence for
the steady flow case.
RESULTS
Figure 1 presents 3D plots of peak systolic velocity magnitude (using
color-coded streamlines) and surface pressure distributions for the two
pulsatile cases. In Figure 2, pressure values extracted at points along
the centerline applying inverse distance interpolation in Tecplot for all
cases are shown. Inlet pressure was set to 90 and 120 mmHg for the
cycle averaged and peak systole results respectively.
DISCUSSION
Our results indicate that the presence of the stenosis just proximal to
the aneurysm ostium causes a peak systolic pressure drop of ~20
mmHg and a cycle averaged pressure drop of ~7 mmHg for case 1. For
case 2 these values are ~14mmHg and ~5 mmHg respectively. It is of
note that there are only minor differences in the centerline pressures
between the time varying and steady state computations at peak
systole flow rates (Fig. 2a). Thus, in this case the quasi steady
assumption is acceptable in predicting both the peak systolic and the
time averaged pressure drop along the centerline.
From the streamlines shown in Fig. 1 we note that as blood flows
through the highly tortuous inlet conduit it obtains a spiral motion
which is then further modulated by the lumen stenosis creating an
asymmetric flow injection in the aneurismal sac. This asymmetric
influx strongly influences the vortex structures that develop within the
aneurismal expansion and the stress distribution on the sac wall. We
should also note that by comparing the results for the various steady
and pulsatile flow cases shown in Fig. 2, the pressure drop across the
stenosis, as expected, approximately scales with the flow rate.
ACKNOWLEDGMENTS
The project has been partially funded by a grant
IPE/YGEIA/DYGEIA/0609/11 and IPE/PLIRO/0609(BE)/11
(GPGPUS) from the Research Promotion Foundation, Nicosia,
Cyprus.
REFERENCES
[1] F. M. White, “Fluid Mechanics”, Fourth Edition, McGraw-Hill.
(a)
(b)
(c)
(d)
Figure 1. Velocity streamlines and pressure distribution at peak systole for pulsatile case 1 (a,c) (Qmean=6.41 ml/s) and case 2 (b,d) (Qmean=5.13 ml/s) respectively.
98
102
106
110
114
118
122
0 1 2 3 4 5 6
Abscissa (cm)
Cen
terl
ine
P (
mm
Hg)
SS Q9.14 SS Q11.42 PeakSys Q5.13 PeakSys Q6.41
(a)
82
84
86
88
90
92
0 1 2 3 4 5 6
Abscissa (cm)
Cen
terli
ne P
(m
mH
g)
SS Q5.13 SS Q6.41 CycAv Q5.13 CycAv Q6.41
(b)
Figure 2. (a) Peak systolic and corresponding steady state centerline pressure for the two cases (Q = 9.14 & 11.42 ml/s). (b) Cycle-averaged and corresponding steady state centerline pressure for the two cases (Q = 6.41 & 5.13 & ml/s)