14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
Time-resolved PIV measurements of oscillating flow in a
non-collapsing flexible pipe
Sebastian Große, Sebastian Burgmann, Wolfgang Schröder
Institute of Aerodynamics, RWTH Aachen University, Germany, [email protected]
Abstract The flow field in the respiratory and vascular system is known to be influenced by the flexibility
of the walls. However, up to now, most experimental biofluidic investigations have been performed in rigid
models due to the high complexity and the necessity of optical access. In the present work the oscillating
flow in a straight pipe with flexible walls is investigated. The model has been adapted such that fluidmechanical and structure-mechanical characteristics represent realistic blood flows in medium blood vessels.
That is, characteristic parameters, i.e., the Reynolds and Womersley number, as well as mechanical
properties of the flexible wall, i.e., the Young's modulus and the material compliance, have been chosen to
reasonably represent realistic flow conditions. First, a method to manufacture elastic models, which mimic
the structure-mechanical properties of vascular vessels is described. The models possess a tunable
compliance and are made of transparent polydimethylsiloxane (PDMS). High-speed PIV measurements of
oscillating pipe flow in elastic vessels at Reynolds numbers based on the non-dilated pipe diameter D and
peak velocities ReD range from 1,000 to 1,750. The Womersley number α has been set to 5-15. The
measurements are performed using a refractive-index adapted PIV system. Water/glycerine is used as flow
medium. Results of the mean flow field are presented as well as the detailed temporal analysis of both, the
evolution of the unsteady flow and the movement of the flexible wall at the different combinations of
Reynolds and Womersley numbers. The results are juxtaposed to Womersley´s analytic solution and to
experimental results of oscillating pipe flow at rigid walls.
1 Introduction
The oscillating flow in vessels with flexible walls as it is typical in the respiratory and the vascular
system is of major interest, since the elasticity of the vessel strongly affects the evolving flow field
and the shear stress evolving at the vessel wall. Especially the illness related reduction of flexibility
leads to an augmentation of the local stresses on the pathological structures. Although experimental
research on this issue has been intensively performed in the past, the fluid-structure interaction of
the oscillating flow field and the flexible vessel walls is far from being complete. An analytical
solution for the velocity distribution due an oscillating flow in rigid and slightly elastic pipes has
been introduced in [10] and [11], respectively. This simple theory by Womersley has been widely
used in the past to calculate the velocity distribution based on pressure measurements, e.g., in [5]
and [8]. To also cover larger deformations of the elastic vessel, occurring e.g. in human arteries,
Womersley´s theory has been extended by [7]. Albeit these extensive investigations the coherence
of the pressure development and the vessel deformation in real elastic vessels is still not
comprehensively clarified, as can be seen in [6]. Recently, optical measurement techniques have
been introduced to directly measure the velocity distribution in elastic vessels instead of calculating
the velocity based on pressure measurements, e.g. in [9], showing the elasticity to play a major roll
concerning the flow field by means of phase-locked laser-Doppler-velocimetry (LDV).
Furthermore, the applicability of particle image velocimetry (PIV) to capture instantaneous velocity
fields in elastic vessels has been shown in [0]. Nevertheless, a detailed description of the fluidstructure interaction in elastic vessels is still lacking, especially as a reliable database for the
validation of CFD results.
In this work the oscillating flow field in a transparent elastic tube is intensively investigated using a
refractive-index adapted PIV system at different Reynolds and Womersley numbers. An elastic tube
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
made of transparent poly-silicone is manufactured taking into account fluid-mechanical and
physiological parameters. The structure mechanical and geometrical properties of this tube are
validated by several measurement techniques. The tube is inserted into a piston-pump driven test
section, which generates a pulsatile flow field. The pumping frequency and the piston stroke can be
adjusted to achieve the desired Womersley and Reynolds number. The flow field inside the tube
and the dilatation of the tube are detected simultaneously. The results evidence a phase-lag between
the deformation and the velocity course. Furthermore, the velocity profiles are shown to accurately
match the analytical Womersley solution. Additionally, the differences between the flow field in a
rigid pipe and in the elastic tube are presented.
2 Experimental Setup
2.1 Mechanical Background of the Elastic Tube
An elastic tube that is supposed to model a real vessel has been manufactured taking into account
fluid-mechanical and physiological parameters. The most important ones are the vessel compliance
C, the Reynolds number Re (eq. 1), the Womersley number α (eq. 2), and the ratio of the fluid
velocity to the wave propagation speed u/c. In the following, ρ denotes the density, η the dynamic
viscosity, and u the velocity of the fluid, whereas ω is the angular frequency of the pumping
sequence.
2u ( Ra − h) ρ
Re =
(1)
η
ωρ
a =
⋅ ( Ra − h)
(2)
η
The compliance is governed by the pressure gradient defined by Re and α and can be expressed by
the rate change of the vessel cross-section A with respect to the pressure p, normalized by the nondilated vessel cross-section A0. Additionally, the compliance C is inversely proportional to the
square of the wave propagation speed c (eq. 3).
∂A ∂p 2∂ ( Ra − h) ∂p
1
C=
=
≈
(3)
2
A0
( R a − h)
ρ ⋅c
The wave-propagation speed can be expressed by the Moens-Korteweg formula (eq. 4) as a
function of the Young’s modulus E and the geometrical properties (outer radius Ra and wall
thickness h) of the vessel.
Eh
c=
(4)
2 ρ ⋅ ( R a − h)
In this experiment, corresponding to values of human arteries, the Reynolds number, which is given
in eq. 1, equals 1,750, and the ratio of the fluid velocity to the wave propagation velocity has been
chosen as u c =0.043. The outer radius of the vessel has been chosen as Ra=12mm. Using these
values and eq. 4 the required wall thickness of the vessel can be calculated and was found to be
h=1.59 mm.
2.2 Manufacturing Process of the Elastic Tube
The elastic tube is made of GE two-component silicone-rubber RTV615 (A+B). Depending on the
ratio of silicone to curing-agent the Young’s modulus can be tuned to fit the required properties
depending on the fluid mechanical and physiological parameters mentioned above. The excellent
optical properties of this material have been previously shown in PIV measurements of a realistic
transparent model of the upper human airways [0, 0]. In this work the two components of the
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
RTV615 silicone are manually mixed at a 10÷1 silicone to curing-agent weight ratio. The uncured
mixture possesses a low viscosity and can be easily cast into the mold which is a precision steel
tube with an inner radius of 12 mm. Using a syringe the necessary volume of silicone-rubber is
injected inside the steel tube to assure the final wall thickness of the elastic tube to match the
previously calculated value. The steel tube serves as a centrifugal device, which is rotated at up to
f=1,500 rpm to perfectly distribute the silicone-rubber inside the tube (see fig. 1). After up to 24
hours at 35°C the silicone-rubber has reached a curing state such that the elastic tube can be
removed from the steel tube. The final curing state is achieved after seven days.
Fig. 1: Image of the centrifugal manufacturing device
2.3 Validation of the Properties of the Elastic Tube
The geometrical and structure mechanical properties of the elastic tube have to be carefully
determined to allow reliable investigations of the flow field and the fluid structure interaction. First
of all the geometric parameters are analyzed. The wall thickness has been optically measured using
a standard CCD camera. Thin circular slices have been cut out of the tube and have been colored
with non-transparent paint. The slices have been illuminated from below and have been recorded by
the CCD camera. The contour of the slice has been detected by image processing routines.
Furthermore, the wall-thickness of the slices has been determined manually by a projection using an
overhead projector. These two measurement techniques evidence a convincing agreement showing
the wall-thickness to vary by 4-7% in the circumferential direction.
The Young’s modulus E has been analyzed using several means. Note, the curing cycle plays a
major role on the final value of this parameter. Especially the curing temperature has to be carefully
controlled. Together with the elastic tube a tensile specimen has been manufactured. The Young’s
modulus can be easily calculated from the linear fit to the stress-strain relation due to static
extension. The same procedure has been applied to the elastic vessel itself, showing an adequate
agreement of the corresponding moduli, i.e. Especimen= 1,37 ⋅ 10 6 N m 2 and Evessel= 1,44 ⋅ 10 6 N m 2 .
Fig. 2 evidences the results of the measurements of Young’s modulus and shows the material to
possess no mechanical hysteresis.
The compliance of the vessel has been optically measured using a CCD camera. The vessel has
been stepwise pressurized using a water column connected to the vessel. The outer contour of the
vessel has been evaluated by image detection routines. The compliance has been calculated from
the linear part of the relation of the cross-section area and the applied pressure (see eq. 1) and was
found to be C=0.091%/mmHg. A plot of the cross-section area vs. pressure relation is given in fig. 3.
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
Fig. 2: Stress-strain relation for the tensile Fig. 3: Cross-section area vs. pressure for an
specimen
elastic vessel with h=1.55 mm and 28 days after
curing. Values for increasing (+) and decreasing
(•) load cycles evidence negligible hysteresis
Further information concerning the manufacturing process of the elastic vessel and the
determination of the structure mechanical properties is given in [0].
2.4 Test Facility
The measurements of the oscillating pipe flow are performed in a piston-pump driven test facility.
A sketch of the experimental setup is presented in fig. 4. The elastic tube is placed in a Plexiglas
test section which is filled with a water/glycerin mixture with 60.7 weight percent glycerin to assure
an adequate refractive-index matching to the silicone-rubber of the tube. On the right hand side the
elastic tube is connected to the piston-pump and on the left hand side to a reservoir R1 which
assures a constant static pressure level. The test section is also slightly pressurized using an
additional reservoir R2 connected to the test section to simulate the effect of the surrounding
pressure of tissue around real vessels. The piston stroke and the pumping frequency can be
independently adjusted such that several combinations of Reynolds numbers and Womersley
numbers can be investigated. In this measurement campaign Reynolds numbers of Re=1,000 and
1,750 and Womersley numbers of α =5, 10, and 15 are analyzed. The flow is investigated using a
water/glycerin mixture with 45 weight percent glycerin.
The boundary condition of the flow inside the vessel, i.e., the oscillatory laminar flow, has been
previously validated by measurements in a rigid Plexiglas tube applying the same specifications and
measurement technique.
The light sheet, generated by a Darwin Duo high-speed laser, is aligned in flow direction in the
center-plane of the elastic vessel. Perpendicular to the light-sheet a Photron Fastcam 1024 pci
camera is installed. A recording frequency of 400 Hz was applied. The pulse distance of the laser
pulses was adapted to achieve high enough particle shifts in the recorded images. A 50 mm Nikon
lens with a maximum f-number of 1.8 is used. Polyamid particles with a mean size of 11 µm are
used to seed the flow.
Due to the refractive index matching to the silicone-rubber of the tube optical access is guaranteed
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
and, furthermore, no optical distortion occurs once the vessel is completely immersed in the fluid
[0].
The PIV evaluation and post-processing is performed using the commercial VidPIV® software.
Adaptive cross-correlation routines with window shifting and window deformation and a final
resolution of 24×24px with 50% overlap yield approximately 60 velocity vectors per diameter.
Note, PIV evaluation is performed on every tenth image pair, which was considered enough for a
high temporal resolution of the results.
The outer contour of the vessel is determined by applying image-detection routines on the raw
particle images to analyze the vessel dilatation.
Fig. 4: Sketch of the experimental setup
3 The Oscillating Flow Field
3.1 The Oscillating Flow in a Rigid Pipe and Womersley Solution
In a first step, the oscillating flow in a rigid Plexiglas pipe is investigated. The pipe is mounted in
the test-section substituting the elastic tube. The measured velocity results are validated by the
analytic Womersley solution of oscillating flow. The results for some combinations of Reynolds
and Womersley numbers are shown in fig. 5, 6, and 7. The velocity profiles for different phase
angles, i.e., the phase angles of the oscillating velocity on the centerline of the pipe, are presented.
Note that phase angles of -180° to 0° correspond to the suction mode and 0° to 180° phase angle to
the pumping mode of the piston-pump driven flow, respectively.
The shape of the profiles exhibit a strong dependence on the Womersley number α, i.e., at higher α
the velocity profile close to the wall is steeper which is also predicted by the analytic Womersley
solution. The analytically calculated profiles and the measured velocity profiles are juxtaposed in
fig. 5 to 7 and evidence excellent agreement. Hence, the piston-pump driven flow can be considered
to represent laminar oscillating pipe flow. This is a prerequisite for the further analysis of the
oscillating flow field and the fluid structure interaction in an elastic tube.
It can be evidenced from these figures that the analytic Womersley solution qualitatively describes
the flow in the elastic tube although the distension of the vessel exceeded ±10% of the mean value
and the wall thickness of the tube is about 13% of the inner radius of the vessel, which is beyond
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
the restrictions of the extended Womersley theory given in [7]. At a Womersley number of α =15,
some discrepancies between the analytic and the measured velocity profiles occur in the near wall
region. These differences cannot be explained actually and need to be investigated in detail in the
future.
Fig. 5: Rigid pipe: Velocity profiles at intervals of 45°, α=5, Re=1,000.
Measurement ( • ), Womersley solution (—)
Fig. 6: Rigid pipe: Velocity profiles at intervals of 45°, α=10, Re=1,750.
Measurement ( • ), Womersley solution (—)
Fig. 7: Rigid pipe: Velocity profiles at intervals of 45°, α =15, Re=1,000.
Measurement ( • ), Womersley solution (—)
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
3.2 The Flow Field in the Elastic Tube
In the following, the results of the PIV measurements in the elastic tube are presented. First, the
measured, normalized velocity profiles are juxtaposed to the results of the oscillating flow field in
the rigid pipe. Subsequently, the coherence of the vessel-dilatation and the oscillating flow field are
analyzed.
3.2.1 Rigid Pipe and Elastic Tube
The experimentally determined velocity profiles at representative phase angles of the sinusoidal
pumping cycle are juxtaposed to the corresponding Womersley solutions at rigid walls in fig. 8 to
10. Note, the coordinate r is related to the temporal inner radius Ri(t) of the pipe and the tube,
respectively. The shrinking and expansion of the elastic tube is axis-symmetric. The velocity
profiles given in these figures differ only slightly.
3.2.2 Vessel Dilatation and Velocity Field
In the following, the relation between the vessel dilatation, the flow rate, and the velocity inside the
vessel is investigated. Fig. 11 to 16 show details of the temporal evolution of the normalized
dilatation ε /|εmax|, the normalized flow rate Q / Q max , and the normalized centerline velocity in the
elastic vessel V0 /V0,max for some selected cases. Note, the dilatation ε is defined as the difference
between the vessel radius during the flow cycle and the radius of the relaxed vessel, i.e. at zeroflow. The dilatation is further normalized by the maximum absolute dilatation |εmax|. Note, the
results evidence the vessel to possess a stronger contraction (ε < 0) than expansion (ε > 0).
According to the results of e.g. Womersley [12] and Ling and Atabek [7] the vessel dilatation is
directly related to the pressure inside the vessel. Hence, a rising edge of the dilatation slope
corresponds to a rising pressure, whereas a falling edge denotes pressure decay, such that the rising
edge characterizes the pumping mode of the piston pump and the falling edge denotes the suction
mode, respectively. Based on the given results, this phase shift between the vessel dilatation and the
flow rate can be calculated. As can be seen, the phase shift is ϕ ≈85° at lower Womersley numbers
(α =5, 10) and ϕ ≈0° at α =15. That is, the flow rate Q / Q max and the velocity at the centerline of
the vessel lag the dilatation/pressure-curve. Furthermore, the flow rate slightly leads the velocity on
the centerline at the lower Womersley numbers. The zero-phase lag between flow rate and
centerline velocity at the highest Womersley number results from the high oscillation frequency
causing the velocity profile to possess an almost rectangular shape such that the flow rate can be
characterized by the velocity on the centerline and hence the phasing of the flow rate approximately
matches with the phasing of the velocity on the centerline of the vessel. At lower frequencies,
however, a phase shift in the velocity distribution close to the vessel wall causes a temporal shift
between the volumetric flow rate and the centerline velocity.
Furthermore, the dilatation of the vessel is in phase with the flow rate due to the strong fluctuation
of high pressure forces that outweigh the inertia forces of the fluid.
The double-peak in the distributions of the vessel dilatation at α=10 shown in fig. 13 and fig. 14
need further investigation. Note, these peaks are only visible in the distribution of the vessel
dilatation and cannot be observed in the velocity and volumetric flow rate distributions.
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
Fig. 8: Elastic tube and rigid pipe: Velocity profiles at intervals of 45° at α=5, Re=1,000.
Elastic wall ( • ), Womersley solution for rigid walls (—)
Fig. 9: Elastic tube and rigid pipe: Velocity profiles at intervals of 45° at α=10, Re=1,750.
Elastic wall ( • ), Womersley solution for rigid walls (—)
Fig. 10: Elastic tube and rigid pipe: Velocity profiles at intervals of 45° at α=15, Re=1,000.
Elastic wall ( • ), Womersley solution for rigid walls (—)
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
Fig. 11: Elastic tube: Phasing of the normalized
velocity (----), the flow rate (─┼─) and the
corresponding dilatation (thin black line) at α =5,
Re=1,000.
Fig. 12: Elastic tube: Phasing of the normalized
velocity (─┼─), the flow rate (─┼─) and the
corresponding dilatation (thin black line) at α =5,
Re=1,750.
Fig. 13: Elastic tube: Phasing of the normalized
velocity (─┼─), the flow rate (─┼─) and the
corresponding dilatation (thin black line) at
α =10, Re=1,000.
Fig. 14: Elastic tube: Phasing of the normalized
velocity (─┼─), the flow rate (─┼─) and the
corresponding dilatation (thin black line) at
α =10, Re=1,750.
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
Fig. 15: Elastic tube: Phasing of the normalized
velocity (─┼─), the flow rate (─┼─) and the
corresponding dilatation (thin black line) at
α =15, Re=1,000.
Fig. 16: Elastic tube: Phasing of the normalized
velocity (─┼─), the flow rate (─┼─) and the
corresponding dilatation (thin black line) at
α =15, Re=1,750.
4 Conclusion
In this work the temporal evolution of fluid flow in an elastic tube and the corresponding vessel
dilatation are presented based on the results of the standard and high-speed PIV measurements. The
velocity is measured in the cross-section of the vessel at high temporal and spatial resolution.
Comparisons of the velocity distribution in the elastic vessel and the analytic Womersley solution in
rigid pipes evidence the analytic solution to be able to qualitatively describe the temporal velocity
field. The fluid structure interaction is analyzed based on measurements of the temporal vessel
dilatation and the corresponding flow rate. At low Womersley numbers (α <15) the dilatation is
shown to lead the flow rate and the velocity inside the vessel, whereas at Womersley numbers of α
≥15 the flow rate and the dilatation are in phase.
Further investigations are planned to clarify the influence of the mechanical properties of the vessel,
e.g., Young´s modulus and compliance, on the fluid-structure interaction. Measurements of flow in
vessels with larger compliance are under preparation, allowing to investigate to which level
Womersley analytic solution will still show reasonable agreement with the experimental findings
even at higher levels of dilatation in the order of 20-30%.
Pressure taps will be inserted in the wall of the vessel to directly measure the pressure, the velocity,
and the dilatation of an elastic vessel.
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14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
References
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