We numerically investigate spatial and temporal evolution of multiple three-dimensional vortex pairs in a curved artery model under a fully developed pulsatile inflow of a Newtonian blood-analog fluid. We discuss the connection along the axial direction between regions of organized vorticity observed at various cross sections of the model, extending previous two-dimensional analysis. We model a human artery with a simple, rigid 180° curved pipe with circular cross section and constant curvature, neglecting effects of taper, torsion, and elasticity. Numerical results are computed from a discontinuous high-order spectral element flow solver using the flux reconstruction scheme and compared to experimental results obtained using particle image velocimetry. The flow rate used in both the simulation and the experiment is physiological. Vortical structures resulting from secondary flow are observed in various cross sections of the curved pipe, in particular, during the deceleration phase of the physiological waveform. We provide side-by-side comparisons of the numerical and experimental velocity and vorticity fields during acceleration and deceleration, the latter during which multiple vortical structures of both Dean-type and Lyne-type coexist. Correlations and quantitative comparisons of the data at these cross sections are computed along with trajectories of Dean-type vortices. Comparing cross-sectional flow fields and vortices provides a means to validate wall shear stress values computed from these numerical simulations, since the evolution of interior flow structures is heavily dependent upon geometry curvature and inflow and boundary conditions.
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.
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
.10.
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. Civ. Eng.
47
, 1
–196
(1902
).11.
J.
Eustice
, “Flow of water in curved pipes
,” Proc. R. Soc. A
84
, 107
–118
(1910
).12.
J.
Eustice
, “Experiments of streamline motion in curved pipes
,” Proc. R. Soc. A
85
, 119
–131
(1911
).13.
W.
Soh
and S.
Berger
, “Laminar entrance flow in a curved pipe
,” J. Fluid Mech.
148
, 109
–135
(1984
).14.
W.
Dean
, “Note on the motion of fluid in a curved pipe
,” London, Edinburgh, Dublin Philos. Mag. J. Sci.
4
, 208
–223
(1927
).15.
W.
Dean
, “The stream-line motion of a fluid in a curved pipe
,” London, Edinburgh, Dublin Philos. Mag. J. Sci.
5
, 673
–695
(1928
).16.
S.
Berger
, L.
Talbot
, and L.-S.
Yao
, “Flow in curved pipes
,” Annu. Rev. Fluid Mech.
15
, 461
–512
(1983
).17.
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
).18.
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
).19.
D.
Doorly
, S.
Sherwin
, P.
Franke
, and J.
Peiró
, “Vortical flow structure identification and flow transport in arteries
,” Comput. Methods Biomech. Biomech. Eng.
5
, 261
–275
(2002
).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.
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
).22.
M.
Plesniak
and K.
Bulusu
, “Morphology of secondary flows in a curved pipe with pulsatile inflow
,” J. Fluids Eng.
138
, 101203
(2016
).23.
J.
Siggers
and S.
Waters
, “Steady flow in pipes with finite curvature
,” Phys. Fluids
17
, 077102
(2005
).24.
J.
Siggers
and S.
Waters
, “Unsteady flow in pipes with finite curvature
,” J. Fluid Mech.
600
, 133
–165
(2008
).25.
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
).26.
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
).27.
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
).28.
L.
Talbot
and K.
Gong
, “Pulsatile entrance flow in a curved pipe
,” J. Fluid Mech.
127
, 1
–25
(1983
).29.
C.
Hamakiotes
and L.
Talbot
, J. Fluid Mech.
195
, 23
–55
(1988
).30.
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
).31.
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
, 100
(2016
).32.
A.
Chorin
, “A numerical method for solving incompressible viscous flow problems
,” J. Comput. Phys.
135
, 118
–125
(1997
)33.
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
.34.
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
.35.
D.
Drikakis
and W.
Rider
, High-Resolution Methods for Incompressible and Low-Speed Flows
(Springer
, 2005
).36.
H.
Huynh
, “A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods
,” in 18th AIAA Computational Fluid Dynamics Conference
(AIAA
, Miami, FL
, 2007
).37.
H.
Huynh
, “A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion
,” in 47th AIAA Aerospace Sciences Meeting
(AIAA
, Orlando, FL
, 2009
).38.
Z.
Wang
and H.
Gao
, “A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids
,” J. Comput. Phys.
228
, 8161
–8186
(2009
).39.
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
).40.
P.
Vincent
, P.
Castonguay
, and A.
Jameson
, “Insights from von Neumann analysis of high-order flux reconstruction schemes
,” J. Comput. Phys.
230
, 8134
–8154
(2011
).41.
A.
Jameson
, P.
Castonguay
, and P.
Vincent
, “On the non-linear stability of flux reconstruction schemes
,” J. Sci. Comput.
50
, 434
–445
(2012
).42.
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
).43.
F.
Witherden
, A.
Farrington
, and P.
Vincent
, “PyFR: An open-source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach
,” Comput. Phys. Commun.
185
, 3028
–3040
(2014
).44.
D. D.
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
).45.
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
).46.
J.
Romero
, K.
Asthana
, and A.
Jameson
, “A simplified formulation of the flux reconstruction method
,” J. Sci. Comput.
67
, 351
–374
(2016
).47.
B.
Vermeire
, F.
Witherden
, and P.
Vincent
, “On the utility of GPU accelerated high-order methods for unsteady flow simulations: A comparison with industry standard tools
,” J. Comput. Phys.
334
, 497
–521
(2017
).48.
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
).49.
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
.50.
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 Fluids
79
, 1309
–1331
(2009
).51.
A.
Jameson
, “Time dependent calculations using multigrid with applications to unsteady flows past airfoils and wings
,” in 10th AIAA Computational Fluid Dynamics Conference
(AIAA
, Honolulu, HI
, 1991
).52.
E.
Turkel
, “Preconditioned methods for solving the incompressible and low speed compressible equations
,” J. Comput. Phys.
72
, 277
–298
(1987
).53.
B.
Helenbrook
, “Artificial compressibility preconditioning for incompressible flows under all conditions
,” in 44th AIAA Aerospace Sciences Meeting
(AIAA
, Reno, NV
, 2006
).54.
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
).55.
M.
Singh
, “Entry flow in a curved pipe
,” J. Fluid Mech.
65
, 517
–539
(1974
).56.
M.
Adler
, “Strömung in gekrümmten rohren
,” Z. Angew. Math. Mech.
14
, 257
–275
(1934
).57.
W.
Lyne
, “Unsteady viscous flow in a curved pipe
,” J. Fluid Mech.
45
, 13
–31
(1970
).58.
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
).59.
J.
Jeong
and F.
Hussain
, “On the identification of a vortex
,” J. Fluid Mech.
285
, 69
–94
(1995
).60.
H.
Vollmers
, “Detection of vortices and quantitative evaluation of their main parameters from experimental velocity data
,” Meas. Sci. Technol.
12
, 1199
–1207
(2001
).61.
L.
John
, P.
Pustějovská
, and O.
Steinbach
, “On the influence of the wall shear stress vector form on hemodynamic indicators
,” Comput. Visualization Sci.
18
, 113
–122
(2017
).62.
A fourth pair of secondary recirculation can appear at the inner wall during deceleration; however, we limit our description at this point to the three main pairs of circulation.
63.
The term vortex core is used interchangeably with the term vortex and is adopted herein to refer to a vortex core region following the description by Jeong and Hussain,59 and should not be confused with a vortex core axis.
64.
The coordinates x′ and z′ represent local coordinates for a given cross section, with the origin of the x′z′-axis located at the center of the pipe where r = 0 [see Fig. 18(a)].
© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.