We numerically investigate the effect of entrance condition on the spatial and temporal evolution of multiple three-dimensional vortex pairs and the wall shear stress distribution in a curved artery model. We perform this study using a Newtonian blood-analog fluid subjected to a pulsatile flow with two inflow conditions. The first flow condition is fully developed while the second condition is undeveloped (i.e., uniform). We discuss the connection along the axial direction between regions of organized vorticity observed at various cross sections of the model and compare results between the different entrance conditions. We model a human artery with a simple, rigid curved pipe with a circular cross section and constant curvature, neglecting the effects of taper, torsion, and elasticity. Numerical results are computed from a discontinuous high-order spectral element flow solver. The flow rate used in this study is physiological. We observe differences in secondary flow patterns, especially during the deceleration phase of the physiological waveform where multiple vortical structures of both Dean-type and Lyne-type coexist. The results indicate that decreased axial velocities under an undeveloped condition produce smaller secondary flows that ultimately inhibit growth of any interior flow vortices. We highlight the effect of the entrance condition on the formation of these structures and subsequent appearance of abnormal inner wall shear stresses, which suggest there may be a lower prevalence of cardiovascular disease in curved arteries where the flow is rather undeveloped—a potentially physiologically significant result to help understand the influence of blood flow development on disease.
References
1.
P.
Davies
, “Endothelial transcriptome profiles in vivo in complex arterial flow fields
,” Ann. Biomed. Eng.
36
, 563
–570
(2008
).2.
S.
Glagov
, C.
Zarins
, D.
Giddens
, and D.
Ku
, “Hemodynamics and atherosclerosis: Insights and perspectives gained from studies of human arteries
,” Arch. Pathol. Lab. Med.
112
, 1018
–1031
(1988
).3.
A.
Mantha
, C.
Karmonik
, G.
Benndorf
, C.
Strother
, and R.
Metcalfe
, “Hemodynamics in a cerebral artery before and after the formation of an aneurysm
,” Am. J. Neuroradiol.
27
, 1113
–1118
(2006
).4.
A.
Chakraborty
, S.
Chakraborty
, V.
Jala
, B.
Haribabu
, M.
Sharp
, and R.
Berson
, “Effects of biaxial oscillatory shear stress on endothelial cell proliferation and morphology
,” Biotechnol. Bioeng.
109
, 695
–707
(2012
).5.
V.
Peiffer
, S.
Sherwin
, and P.
Weinberg
, “Computation in the rabbit aorta of a new metric-the transverse wall shear stress-to quantify the multidirectional character of disturbed blood flow
,” J. Biomech.
46
, 2651
–2658
(2013
).6.
X.
He
and D.
Ku
, “Pulsatile flow in the human left coronary artery bifurcation: Average conditions
,” J. Biomech. Eng.
118
, 74
–82
(1996
).7.
H.
Himburg
, D.
Grzybowski
, A.
Hazel
, J.
LaMack
, X.-M.
Li
, and M.
Friedman
, “Spatial comparisons between wall shear stress measures and porcine arterial endothelial permeability
,” Am. J. Physiol. Heart Circulatory Physiol.
286
, H1916
–H1922
(2004
).8.
V.
Peiffer
, S.
Sherwin
, and P.
Weinberg
, “Does low and oscillatory wall shear stress correlate spatially with early atherosclerosis? A systematic review
,” Cardiovasc. Res.
99
, 242
–250
(2013
).9.
G.
Williams
, C.
Hubbell
, and G.
Fenkell
, “Experiments at detroit, mich., on the effect of curvature upon the flow of water in pipes
,” Trans. Am. Soc. Civil Eng.
47
, 1
–196
(1902
).10.
11.
J.
Eustice
, “Experiments of streamline motion in curved pipes
,” Proc. R. Soc. A
85
, 119
–131
(1911
).12.
C.
Cox
, M.
Najjari
, and M.
Plesniak
, “Three-dimensional vortical structures and wall shear stress in a curved artery model
,” Phys. Fluids
31
, 121903
(2019
).13.
W.
Dean
, “Note on the motion of fluid in a curved pipe
,” London Edinburgh Dublin Philos. Mag. J. Sci.: Ser.
4
, 208
–223
(1927
).14.
W.
Dean
, “The stream-line motion of a fluid in a curved pipe
,” London Edinburgh Dublin Philos. Mag. J. Sci.: Ser.
5
, 673
–695
(1928
).15.
S.
Berger
, L.
Talbot
, and L.-S.
Yao
, “Flow in curved pipes
,” Annu. Rev. Fluid Mech.
15
, 461
–512
(1983
).16.
J.
Womersley
, “Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is unknown
,” J. Physiol.
127
, 553
–563
(1955
).17.
D.
Doorly
and S.
Sherwin
, “Geometry and flow
,” in Cardiovascular Mathematics, Modeling and Simulation of the Circulatory System, Modeling, Simulation and Applications
, edited by L.
Formaggia
, A.
Quarteroni
, and A.
Veneziani
(Springer
, 2009
), Vol. 1
, Chap. 5, pp. 177
–209
.18.
F.
Smith
, “Pulsatile flow in curved pipes
,” J. Fluid Mech.
71
, 15
–42
(1975
).19.
K.
Sudo
, M.
Sumida
, and R.
Yamane
, “Secondary motion of fully developed oscillatory flow in a curved pipe
,” J. Fluid Mech.
237
, 189
–208
(1992
).20.
O.
Boiron
, V.
Deplano
, and R.
Pelissier
, “Experimental and numerical studies on the starting effect on the secondary flow in a bend
,” J. Fluid Mech.
574
, 109
–129
(2007
).21.
B.
Timite
, C.
Castelain
, and H.
Peerhossaini
, “Pulsatile viscous flow in a curved pipe: Effects of pulsation on the development of secondary flow
,” Int. J. Heat Fluid Flow
31
, 879
–896
(2010
).22.
J.
Alastruey
, J.
Siggers
, V.
Peiffer
, D.
Doorly
, and S.
Sherwin
, “Reducing the data: Analysis of the role of vascular geometry on blood flow patterns in curved vessels
,” Phys. Fluids
24
, 031902
(2012
).23.
S.
van Wyk
, L.
Wittberg
, K.
Bulusu
, L.
Fuchs
, and M.
Plesniak
, “Non-Newtonian perspectives on pulsatile blood-analog flows in a 180° curved artery model
,” Phys. Fluids
27
, 071901
(2015
).24.
C.
Krishna
, N.
Gundiah
, and J.
Arakeri
, “Separations and secondary structures due to unsteady flow in a curved pipe
,” J. Fluid Mech.
815
, 26
–59
(2017
).25.
M.
Najjari
, C.
Cox
, and M.
Plesniak
, “Formation and interaction of multiple secondary flow vortical structures in a curved pipe: Transient and oscillatory flows
,” J. Fluid Mech.
876
, 481
–526
(2019
).26.
D.
Ku
, “Blood flow in arteries
,” Annu. Rev. Fluid Mech.
29
, 399
–434
(1997
).27.
Y.
Chen
, X.
Yang
, A.
Iskander
, and P.
Wang
, “On the flow characteristics in different carotid arteries
,” Phys. Fluids
32
, 101902
(2020
).28.
J.
Vetel
, A.
Garon
, and D.
Pelletier
, “Lagrangian coherent structures in the human carotid artery bifurcation
,” Exp. Fluids
46
, 1067
–1079
(2009
).29.
A.
Seetharaman
, H.
Keramati
, K.
Ramanathan
, M.
Cove
, S.
Kim
, K.
Chua
, and H.
Leo
, “Vortex dynamics of veno-arterial extracorporeal circulation: A computational fluid dynamics study
,” Phys. Fluids
33
, 061908
(2021
).30.
N.
Freidoonimehr
, R.
Chin
, A.
Zander
, and M.
Arjomandi
, “Effect of shape of the stenosis on the hemodynamics of a stenosed coronary artery
,” Phys. Fluids
33
, 081914
(2021
).31.
M.
Nagargoje
, D.
Mishra
, and R.
Gupta
, “Pulsatile flow dynamics in symmetric and asymmetric bifurcating vessels
,” Phys. Fluids
33
, 071904
(2021
).32.
M.
Najjari
and M.
Plesniak
, “Evolution of vortical structures in a curved artery model with non-Newtonian blood-analog fluid under pulsatile inflow conditions
,” Exp. Fluids
57
, 1
–16
(2016
).33.
L.
Talbot
and K.
Gong
, “Pulsatile entrance flow in a curved pipe
,” J. Fluid Mech.
127
, 1
–25
(1983
).34.
W.
Soh
and S.
Berger
, “Laminar entrance flow in a curved pipe
,” J. Fluid Mech.
148
, 109
–135
(1984
).35.
C.
Hamakiotes
and L.
Talbot
, “Fully developed pulsatile flow in a curved pipe
,” J. Fluid Mech.
195
, 23
–55
(1988
).36.
M.
Plesniak
and K.
Bulusu
, “Morphology of secondary flows in a curved pipe with pulsatile inflow
,” J. Fluids Eng.
138
, 101203
(2016
).37.
D.
Holdsworth
, C.
Norley
, R.
Frayne
, D.
Steinman
, and B.
Rutt
, “Characterization of common carotid artery blood-flow waveforms in normal human subjects
,” Physiol. Meas.
20
, 219
–240
(1999
).38.
A.
Chorin
, “A numerical method for solving incompressible viscous flow problems
,” J. Comput. Phys.
135
, 118
–125
(1997
).39.
D.
Elsworth
and E.
Toro
, “Riemann solvers for solving the incompressible Navier-Stokes equations using the artificial compressibility method
,” Technical Report No. 9208
(Cranfield Institute of Technology
, 1992
).40.
D.
Elsworth
and E.
Toro
, “A numerical investigation of the artificial compressibility method for the solution of the Navier-Stokes equations
,” Technical Report No. 9213
(Cranfield Institute of Technology
, 1992
).41.
D.
Drikakis
and W.
Rider
, High-Resolution Methods for Incompressible and Low-Speed Flows
(Springer
, 2005
).42.
H.
Huynh
, “A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods
,” AIAA Paper No. 2007-4079, 2007
.43.
H.
Huynh
, “A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion
,” AIAA Paper No. 2009-403, 2009
.44.
P.
Vincent
, P.
Castonguay
, and A.
Jameson
, “A new class of high-order energy stable flux reconstruction schemes
,” J. Sci. Comput.
47
, 50
–72
(2011
).45.
A.
Jameson
, P.
Castonguay
, and P.
Vincent
, “On the non-linear stability of flux reconstruction schemes
,” J. Sci. Comput.
50
, 434
–445
(2012
).46.
D.
Williams
, P.
Castonguay
, P.
Vincent
, and A.
Jameson
, “Energy stable flux reconstruction schemes for advection-diffusion problems on triangles
,” J. Comput. Phys.
250
, 53
–76
(2013
).47.
D.
De Grazia
, G.
Mengaldo
, D.
Moxey
, P.
Vicent
, and S.
Sherwin
, “Connections between the discontinuous Galerkin method and high-order flux reconstruction schemes
,” Int. J. Numer. Methods Fluids
75
, 860
–877
(2014
).48.
G.
Mengaldo
, D. D.
Grazia
, P.
Vincent
, and S.
Sherwin
, “On the connections between discontinuous Galerkin and flux reconstruction schemes: Extension to curvilinear meshes
,” J. Sci. Comput.
67
, 1272
–1292
(2016
).49.
J.
Romero
, K.
Asthana
, and A.
Jameson
, “A simplified formulation of the flux reconstruction method
,” J. Sci. Comput.
67
, 351
–374
(2016
).50.
C.
Cox
, W.
Trojak
, T.
Dzanic
, F.
Witherden
, and A.
Jameson
, “Accuracy, stability, and performance comparison between the spectral difference and flux reconstruction schemes
,” Comput. Fluids
221
, 104922
(2021
).51.
H.
Huynh
, “Discontinuous Galerkin via interpolation: The direct flux reconstruction method
,” J. Sci. Comput.
82
, 75
(2020
).52.
C.
Cox
, C.
Liang
, and M.
Plesniak
, “A high-order method for solving unsteady incompressible Navier-Stokes equations with implicit time stepping on unstructured grids
,” AIAA Paper No. 2015-0830, 2015
.53.
C.
Cox
, C.
Liang
, and M.
Plesniak
, “A flux reconstruction solver for unsteady incompressible viscous flow using artificial compressibility with implicit dual time stepping
,” AIAA Paper No. 2016-1827, 2016
.54.
C.
Cox
, C.
Liang
, and M.
Plesniak
, “A high-order solver for unsteady incompressible Navier-Stokes equations using the flux reconstruction method on unstructured grids with implicit dual time stepping
,” J. Comput. Phys.
314
, 414
–435
(2016
).55.
C.
Cox
, “Development of a high-order Navier-Stokes solver using flux reconstruction to simulate three-dimensional vortex structures in a curved artery model
,” Ph.D. thesis (The George Washington University
, Washington, DC
, 2017
).56.
C.
Geuzaine
and J.-F.
Remacle
, “GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities
,” Int. J. Numer. Methods Eng.
79
, 1309
–1331
(2009
).57.
M.
Najjari
and M.
Plesniak
, “Secondary flow vortical structures in a 180° elastic curved vessel with torsion under steady and pulsatile inflow conditions
,” Phys. Rev. Fluids
3
, 013101
(2018
).58.
M.
Zamir
, The Physics of Pulsatile Flow, Biological Physics
(Springer
, 2000
).59.
M.
Zamir
, Hemo-Dynamics, Biological and Medical Physics, Biomedical Engineering
(Springer
, 2016
).60.
M.
Singh
, “Entry flow in a curved pipe
,” J. Fluid Mech.
65
, 517
–539
(1974
).61.
W.
Lyne
, “Unsteady viscous flow in a curved pipe
,” J. Fluid Mech.
45
, 13
–31
(1970
).62.
J.
Jeong
and F.
Hussain
, “On the identification of a vortex
,” J. Fluid Mech.
285
, 69
–94
(1995
).© 2021 Author(s). Published under an exclusive license by AIP Publishing.
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