PIV Measurements of Flow in a
Centrifugal Blood Pump:
Time-Varying Flow
Steven W. Day1
e-mail: sday@alumni.virginia.edu
James C. McDaniel
The University of Virginia, Charlottesville, VA
22903
Measurements of the time-varying flow in a centrifugal blood pump operating as a left
ventricular assist device (LVAD) are presented. This includes changes in both the pump
flow rate as a function of the left ventricle contraction and the interaction of the rotating
impeller and fixed exit volute. When operating with a pulsing ventricle, the flow rate
through the LVAD varies from 0 – 11 L / min during each cycle of the heartbeat. Phaseaveraged measurements of mean velocity and some turbulence statistics within several
regions of the pump, including the inlet, blade passage, exit volute, and diffuser, are
reported at 20 phases of the cardiac cycle. The transient flow fields are compared to the
constant flow rate condition that was reported previously in order to investigate the
transient effects within the pump. It is shown that the quasi-steady assumption is a fair
treatment of the time varying flow field in all regions of this representative pump, which
greatly simplifies the comprehension and modeling of this flow field. The measurements
are further interpreted to identify the effects that the transient nature of the flow field will
have on blood damage. Although regions of recirculation and stagnant flow exist at some
phases of the cardiac cycle, there is no location where flow is stagnant during the entire
heartbeat. 关DOI: 10.1115/1.1865190兴
Introduction
Left ventricular assist devices 共LVADs兲 have proven clinical
efficacy, but there are currently no devices capable of providing
long-term 共10 or more years兲 of support. The newest generation
VADs are magnetically suspended rotary blood pumps and have
the potential to replace diaphragm pumps. They offer the advantages of requiring significantly fewer moving parts, no contacting
moving surfaces, no flexible materials, and no valves. All of these
increase the mechanical life of the device and decrease the potential blood damage caused by the pump. The fluid dynamics within
the pump both contribute to damage caused to the blood that it is
pumping and determine the pump performance and efficiency. An
understanding and ability to predict the flow within these devices
is critical.
In addition to the three-dimensional geometry of the pump,
transient effects complicate the flow. Transient effects include
both the interaction of the blades with the fixed geometry of the
pump and the time-varying flow rate due to pulsing of the native
left ventricle. Although the pump operates at a constant rotational
speed, the flow rate varies as a result of the ventricle contraction
changing the pump inlet pressure. When implanted, the flow rate
varies from the high-flow to low-flow condition during each heartbeat, so that the changes in flow rate from low to high occur in a
short period of time 共⬃0.25 s兲. The fact that the flow through the
pump is pulsed may have some physiological advantages 共because
blood flow delivered form the heart is normally pulsed兲, but it also
leads to a very complicated time-dependent flow within the pump.
The use of particle-based optical techniques for studying fluid
dynamics of blood pumps is common. These majority of fluid
dynamics studies of rotary pumps present qualitative 关1–4兴 or
quantitative 关5–8兴 measurements of the flow within one or more
regions of the pump at one operating flow rate.
The transient effects of the interaction of the rotating impeller
1
Corresponding author. Current address: Section of Evolution & Ecology, University of California, One Shields Avenue, Davis, CA 95616
Contributed by the Bioengineering Division for publication In the JOURNAL OF
BIOMECHANICAL ENGINEERING. Manuscript received October 7, 2003. Final manuscript
received September 20, 2004. Associate Editor: C. Ross Ethier.
254 / Vol. 127, APRIL 2005
and fixed volute, so called rotor-stator interactions, have been reported by Tsukiya et al., who used particle image velocimetry
共PIV兲 to measure the variations in the velocity field within the
blade passage of a scaled model pump as a function of the position of the passage relative to the volute cutwater 关9兴. Flow structures that existed within the blade passage were largely a function
of position relative to the cutwater.
Efforts to investigate the flow field over a range of flow rates
have increased with awareness that the flow rate through the pump
will vary during the heartbeat, even while the pump is operating at
constant rotational speed. Qualitative flow visualization by Wu et
al. within both an axial and a centrifugal pump at design and
off-design conditions showed that the flow field exhibits undesirable flow structures at both higher and lower than design flow
rates 关10兴. Manning and Miller characterized the flow within the
outlet cannula of a pump over a range of flow rates ranging from
⬃1.5 to 10.5 L / min 关11兴. Apel et al. presented measurements of
the angle of fluid at the exit of an axial pump over a range of flow
rates 共1 – 4.5 L / min兲 in order to validate computational models
through this pump 关4兴. The authors are not aware of any studies
that have characterized the flow field in a rotary pump during
pulsatile flow rate conditions.
All reported measurements of the flow through pulsatile
diaphragm-type pump are necessarily for time-varying flow, due
principally to the fact that these devices generate a pulsed flow
even when operating against constant resistance. Orime et al. presented particle streak patterns to reveal flow patterns and some
measurements of mean speed in a total artificial heart over the
cycle of heart beat by making measurements at approximately 20
times during the cycle 关12兴. Additionally, quantitative studies, using laser Doppler velocimetry 共LDV兲 关13,14兴 and PIV 关15兴 have
been used to measure characteristics of pulsatile flow through
diaphragm-type pumps.
The synchronization of measurements to the cycle of periodic
phenomenon is known as phase-averaging. This method has been
applied to the determination of mean flow and turbulent stresses
within blood pumps based on LDV measurements 关13兴. The methods are used extensively in the treatment of rotor-stator interactions on large-scale machines, and discussions on the accuracy of
methods can be found in 关16,17兴.
Copyright © 2005 by ASME
Transactions of the ASME
Fig. 1 Cut-away view of the pump showing the location of the
three measurement planes within the pump: IE-inlet elbow, BPBlade passage, CW-Cut-water.
While some of these studies of rotary blood pumps have characterized the flow field over a range of steady flow rates, none
have presented measurements of the flow field during pulsatile
conditions that would enable the evaluation of the effect of transient variations in flow rate. In the current study, particle image
velocimetry 共PIV兲 was used to characterize the flow within a centrifugal LVAD under a time-varying flow rate that is representative of the conditions when implanted. Additionally, the study
includes a prediction of the effects that specific flow features will
have on blood damage. This work addresses the effects of a timevarying flow field due to these rapid changes in flow rate through
the pump, by using PIV to make quantitative measurements of the
mean velocity and stress fields within the pump. Measurements
reported elsewhere characterize the flow field over the range of
steady flow rates at which this particular pump is expected to
operate 关18兴. In order to characterize the influence of the transient
flow rate, measurements of the internal flow field were made
while the pump operated within a mock circulatory system, which
included a model of a beating left ventricle. Additionally, comparisons of the transient flow field to the steady flow field are
presented in an effort to characterize the time-dependent effects
on the flow field.
Methods
Experimental Apparatus. Particle image velocimetry 共PIV兲 is
a technique that measures the instantaneous velocity field within
an illuminated plane of the fluid field using light scattered from
particles seeded into the fluid 关19兴. The light scattered from these
particles is collected and imaged using a camera at two separate
times so that the displacement of the particles can be measured.
PIV measures the instantaneous two-dimensional field illuminated
by the laser sheet and is therefore well suited for investigating
transient phenomena. Details on the PIV system, seed particles,
test fluid, and software 关20兴 used during these experiments have
been reported elsewhere 关18,21兴.
In order to accomplish PIV measurements within the LVAD, a
prototype pump that allows for optical access into nearly all interior regions of the pump was designed and built, as described
elsewhere 关18,21兴. A cutaway view of the pump showing locations
of the measurement planes is shown in Fig. 1. The pump is part of
a mock circulatory loop that includes a left ventricular simulator
共to model the beating native left ventricle兲, an adjustable valve 共to
control the systemic resistance兲, and two air-filled compliance
chambers 共in which the compressibility of air models the elasticity
of blood vessels兲, and is shown schematically in Fig. 2. The loop
Journal of Biomechanical Engineering
Fig. 2 Schematic of flow loop showing pulsed ventricle simulator „PVS…, Left Ventricular Assist device under investigation
„LVAD…, arterial compliance „AC…, venous compliance „VC…, and
valve for systemic resistance „R…. Measurements of flow rate
include flow through aortic valve „AoQ… and flow through LVAD
„VADQ…. Locations of pressure measurements are aortic pressure „AoP…, ventricular „LVP…, and atrial „LAP….
is instrumented with two flow meters 共Transonic, Ithaca, NY兲 and
three pressure transducers 共Validyne, Northridge, CA兲 to monitor
the transient behavior of the ventricular simulator and LVAD under simulated physiologic conditions. The values of resistance and
compliance, as well as the pulse rate and strength of the ventricle
simulator, are adjustable so that the pump may be tested for a
range of physiologic conditions. Modeled conditions are confirmed to be physiologically relevant by comparing them to data
on the behavior and performance of the human circulatory system
关22,23兴. A cannula is connected from the apex of the left ventricle
to the inlet elbow of the pump. When making measurements in the
inlet elbow, an additional length of straight tubing is added to the
cannula before entering the inlet elbow in order to ensure the
same entrance length as was used for the constant flow rate
experiments.
During pulsatile flow experiments, a LabView® program was
used to display and record transient data, including LVAD flow
rate, ventricle flow rate, left ventricular pressure, aortic pressure,
venous pressure, working fluid temperature, rotational speed, and
the axial force on impeller. LabView was also used to control the
motor speed via a simple closed-loop feedback controller that
maintains the pump operating speed to within 20 rpm of the requested shaft speed, ⬃ ± 1%.
The synchronization of the PIV system to the heartbeat and to
the rotating shaft was accomplished by electrical triggers from
both the shaft and left ventricle simulator. Both were fed through
the digital delay generators and a logical “and” circuit was implemented in hardware to ensure that the measurement was synchronized to both the heartbeat and blade passage. For each measurement, 200 images were acquired at each phase of the heartbeat
and processed to arrive at the statistical quantities of the velocity
field that are presented here.
Mean and Turbulence Statistics. According to the Reynolds
decomposition, the instantaneous velocity ũ is though to be composed of a steady mean velocity U summed with a fluctuating
velocity u. All fluctuations, including those from turbulence and
from large-scale nonrandom phenomena, contribute to u. The
magnitude of the mean U and standard deviation u⬘ of the instantaneous velocity ũ are statistical descriptions of the flow and can
be found from a set of instantaneous velocity measurements, each
made at discrete times, by ensemble averaging 关24兴. Reynolds
normal and shear stresses are also determined from this series of
instantaneous measurements. Principal stress analysis, as outlined
by Baldwin 关25兴 was used to calculate the maximum Reynolds
stress based on measurements in the measurement reference
APRIL 2005, Vol. 127 / 255
Fig. 3 Measured physiological pressures and flow rates for a rotary LVAD over two heartbeats. Pump speed is constant at
2100 rpm.
frame.
In the case of a periodic flow, velocity variations are no longer
due exclusively to random fluctuations, but also to the periodicity
of the mean flow. In a blood pump, these periodic variations are
due to the pulsed flow rate through the pump and, in some regions, to interactions between the rotating and stationary parts of
the pump. When this is the case the instantaneous velocity can be
decomposed into components according to
ũ = U + u + ublade + upulse
共1兲
where the variations in instantaneous velocity include the variations due to blade passage ublade and to the heartbeat upulse in
addition to statistically random variations u. Because the blade
passage and pulse cycle can be considered periodic, requiring that
the nonrandom quantities of the flow field are repeatable from
cycle to cycle, they can be treated in a more exact way. The
periodic velocity fields can be represented by
ũ共␾b, ␾ p兲 = U共␾b, ␾ p兲 + u共␾b, ␾ p兲
共2兲
The instantaneous flow conditions ũ are still thought of as the
summation of a periodically oscillating mean flow U and fluctuations u. However, the velocities no longer represent time-averaged
quantities, but rather characterize the velocity measured at one
distinct phase of the periodic phenomena. All are functions of the
phase of blade passage ␾b and pulse ␾ p. U共␾b , ␾ p兲 is not a steady
mean flow, but rather a mean flow which varies periodically as a
function of the blade passage and heartbeat. While U共␾b , ␾ p兲 is
the same for a given phase of heartbeat and blade position, the
instantaneous velocity field ũ共␾b , ␾ p兲 is not necessarily the same
because this includes the contribution of the random fluctuations
u共␾b , ␾ p兲.
The phase-averaged statistics are calculated according to the
same equations, but all of the instantaneous measurements
ũ共␾b , ␾ p兲 are made at the same phase of both the blade passage or
the heartbeat. Equations 共1兲 and 共2兲 may be related by realizing
that ublade is the rms of U共␾b , ␾ p=const兲 for ␾b = 0 to 2␲ over a full
cycle of blade phases. Similarly, upulse represents the magnitude of
variations of mean velocity during one pulse cycle rms
兵U共␾b=const , ␾ p兲 for ␾ p = 0 to 2␲其. Additional reference on phaseaveraging techniques using PIV data may be found in Zhang et al.
关17兴, Sinha et al. 关26兴, Wernert and Favier 关16兴, and Wernet 关24兴.
Blade-to-Blade Variations. The velocity within the blade passage of a centrifugal pump is not uniform. Generally, the velocity
at the exit of the blade passage is highest near the suction side 共the
blade face away from the direction of rotation兲 of the blade and
lowest near the pressure side 共toward the direction of rotation兲 of
the blade 关27兴. Because of this some periodicity is expected as the
result of interactions between the rotating and stationary parts of
the pump, even at a steady flow rate 关28兴. According to Eq. 共1兲,
256 / Vol. 127, APRIL 2005
the magnitude of the velocity fluctuations caused by the passing of
the blade is quantified by comparing measurements in which the
data acquisition is not synchronized to the position of the impeller,
“free-running mode,” and measurement for which the acquisition
is synchronized with rotational position to insure phase-averaged
measurements at one specific phase of the blade passing event,
“passing-blade mode.” For the purposes of investigating blade-toblade variations, the flow rate through the pump is held constant
to eliminate further variations due to flow rate. Statistical treatment of free-running measurements 共no consideration of blade
phase兲 includes influences of the both random variations u and
variations due to blade passage. Because the blade passage is
periodic, fluctuations due to blade passage can be treated as functions of the phase of blade passage, rather than as a random phenomenon, as was shown in Eqs. 共1兲 and 共2兲.
Measurements for 20 phases of blade position are reported for
several constant flow rates. The variations due to the passing blade
are removed because all images are acquired at one particular
phase of the blade passage. The rms of the fluctuating velocity
u⬘共␾b兲 is a measure of the magnitude of u共␾b兲. The mean of the
instantaneous fields is U共␾b兲, where ␾b is the phase of the impeller.
Selection of Operating Condition. A single set of operating
conditions for testing pulsatile flow was established based on the
measurements of the global pump performance over a range of
physiologic conditions. The time variations of ventricular and aortic pressures and flow rates through the LVAD and aortic valve are
shown in Fig. 3. The ventricle simulator was beating at
60 beats/ min with a systolic duration of 0.40 s and a diastolic
duration of 0.60 s. The strength of the ventricle’s contraction was
set so that the left ventricular pressure varied from
10 to 90 mmHg, which is sufficiently high that the aortic valve
opens with each pulse. The systemic resistance remained constant,
and the LVAD rotational speed was adjusted to achieve a
6 – 7 L / min combined flow rate through the LVAD and the native
aorta. This occurred at a speed of 2100 rpm. Under these conditions, the flow rate through the LVAD varies from approximately
1 L / min during diastole to 11 L / min during systole for a mean
LVAD flow of 6 L / min. The combined flow rate of the heart and
the assist pump is about 7 L / min at a mean aortic pressure of
95 mmHg and atrial pressure of 10 mmHg 共Fig. 3兲. The rotational
speed remained at 2100 rpm and all other variables remained the
same for all of the experiments presented herein.
It is realized that the actual values of left ventricle pressure,
peripheral resistance, and compliance are highly variable from
patient to patient and will change within any single patient. While
the nominal conditions used here are completely realistic, they
correspond to a ventricle with a significant pulse pressure when on
ventricular assist. Nonetheless, for the purposes of prescribing
Transactions of the ASME
some realistic conditions to the pump in order to investigate the
internal fluid dynamics, this is an ideal test case for measurements
of pulsed flow because the variations in flow rate through the
pump 共from 1 to 11 L / min during each heart beat with a mean of
6 L / min兲 are extreme for this situation. This represents an upper
limit of the variations in flow rate and can, therefore, be used to
bound the effect of transient flow rate.
A reference phase, ␾ = 0 is assigned to a measurement near the
end of diastole, 0.15 seconds before the contraction of the ventricle and corresponding increase in LVAD flow rate begin. Phaseaveraged measurements were made at 20 phases 共every 0.05 s兲 of
the heartbeat, and are illustrated by symbols in Fig. 3. The selected phases are chosen to show portions of the pulse during
which transient effects are most prominent.
Unsteady versus Quasi-Steady Flow. One method to characterize the transient effects within a flow is to compare the instantaneous or phase-averaged velocity and shear field to that of
steady flow at the same flow rate. If the frequency of oscillation is
significantly low, then the fluid behaves quasi-steadily. This is to
say that the instantaneous velocity and shear fields are essentially
identical to those of steady flow at the instantaneous flow rate.
However, at a certain frequency of oscillation, the high acceleration forces become significant and the instantaneous fluid field
differs from the steady case.
Although theoretical estimates of this critical frequency are
available for simple geometries, the application of this theory to
the heart pump is complicated by both the geometry and the fact
that turbulent momentum transfer dominates over molecular viscosity in most regions of the pump.
Because of these complications, a direct comparison of the
phase-averaged and steady-state flow fields is used here. Within
each configuration, measurements of the instantaneous flow field
for pulsatile flow are compared with steady flow measurements at
the same or similar flow rates to determine the validity of the
quasi-steady assumption.
Fig. 4 Phase averaged speed in the inlet elbow at 8 phases
with phase, ␾, and flow rate, Q, indicated for each.
Results
Time-Varying Flow Rate. Contour plots of mean velocity
within the inlet elbow are shown in Fig. 4 at phases 0 to 16␲ / 10.
Four of them 共␾ = 0 , 2␲ / 10, 4␲ / 10, 6␲ / 10兲 cover the sudden increase in flow rate during the beginning of systole, and four of
them 共␾ = ␲ , 12␲ / 10, 14␲ / 10, 16␲ / 10兲 cover the transition from
systole to diastole. The strong contraction of the ventricle begins
just after 2␲ / 10 共flow rate, Q, ⬃1 L / min兲 and changes to
11 L / min in 0.2 s. Similarly, between phases of ␲ and 14␲ / 10,
the flow rate drops from 10.5 to 1 L / min during 0.2 s. There are
obvious recirculation patterns within the inlet elbow at the low
flow condition. These recirculation regions form at the beginning
of diastole and continue until the heart contracts
共13␲ / 10– 3␲ / 10兲.
Profiles of mean velocity and rms as a function of z position are
shown in Fig. 5 to illustrate the time evolution during the acceleration phase. The position of the profile is y = −0.022 m, which is
very near the inlet of the elbow. The same charts show the velocity and rms profiles for steady flow at 11 L / min as a dashed line,
for comparison to the pulsed flow cases. The profile of speed
quickly reestablishes itself during the contraction after the sudden
acceleration of flow. As compared to the steady case, the shape of
the velocity profile is significantly flattened at 4␲ / 10– 5␲ / 10 and
slightly so at 6␲ / 10, but is essentially identical to the steady flow
case by 7␲ / 10. The velocity profile is completely established between 0.15 s 共3␲ / 10兲 and 0.2 s 共4␲ / 10兲 after the sudden acceleration begins and is never very different from the steady flow
profile.
The rms of velocity, however, does not exhibit as rapid a response to the sudden change in flow rate. The rms of velocity does
not approach the steady-state profile until approximately 9␲ / 10.
Journal of Biomechanical Engineering
At 7␲ / 10, the flow rate is high 共⬎8 L / min兲, but levels of rms are
everywhere lower than 0.05 m / s. This is about one-third of the
steady-state values reported at 9 L / min.
Profiles of mean and rms of velocity are also shown in this
same section during the deceleration of flow at the end of systole
in Fig. 5. Between 13␲ / 10 and 15␲ / 10, the flow along the outer
curvature of the elbow reverses direction even though the pump
flow rate is positive, creating a recirculation region 共Fig. 5兲. This
recirculation region persists throughout diastole, is not present
during steady flow conditions at 1 L / min and is the most pronounced transient effect within the pump.
Contour plots of the mean velocity are shown at the same
phases of the heartbeat within the blade passage in Fig. 6. The
flow within the blade passage at all flow rates is nearly identical to
the steady flow case at the same flow rate. This is demonstrated in
Fig. 7, which shows the mean velocity, rms, MRSS, and viscous
shear at one phase in the heartbeat. The selected phase of 13␲ / 10
is immediately following the sudden deceleration in the flow, but
the similarity between phase-averaged and steady flow cases is
evident at all phases of the heartbeat.
Contour plots of the mean velocity in the cutwater are in Fig. 8.
The flowfield within the cutwater region responds very quickly to
the sudden flow acceleration at the beginning of systole. Profiles
共at x = 0.02 m兲 of the mean speed and MRSS are shown for all
measured phases in Fig. 9. Mean velocity, rms, MRSS, and viscous shear stress are nearly identical to the steady-state measurements at approximately the same flow rate, other than the fact that
the size of the recirculation zone for the phase-averaged case is
slightly larger than for steady case. At each phase, the steady
profiles at the nearest steady flow rate are also shown as a dashed
APRIL 2005, Vol. 127 / 257
Fig. 5 Time evolution of mean speed and rms of fluctuating velocity near the inlet of inlet elbow during the rapid acceleration of
flow and through systole and then through deceleration and diastole compared to steady flow at 11 L / min
line. The profiles of mean velocity show that the size of the recirculation region changes between phases 5␲ / 10 and 7␲ / 10, but is
reestablished at 7␲ / 10. The recirculation region in the pulsatile
case is more pronounced than for the steady flow case. The recirculation region continues during the deceleration of flow to
12␲ / 10, at which time the flow rate is 9 L / min. At 13␲ / 10, the
flow rate has dropped to 4 L / min and the recirculation region
does not exist, as it the case for steady flow.
Blade-to-Blade Variations. A comparison of measurements of
steady flow that are not synchronized 共top兲 and synchronized
共middle and bottom兲 to the phase of the blade passage is shown in
Fig. 10. The top images show mean speed U and rms u⬘ = rms共u
+ ublade兲. The middle and lower images show mean velocity U共␾b兲
258 / Vol. 127, APRIL 2005
and rms u⬘共␾b兲 at two phases of the blade passage acquired by
synchronizing the acquisition to a specific phase of the blade position. The first is at a reference phase ␾b of zero degrees, when
the impeller blade is just passing the cutwater tongue and the
second when the cutwater is centered between two blades, ␾b
= 180. Passing blade measurements were made at 20 phases of the
blade passage and for three different flow rates, 3, 6, and 9 L / min.
Measurements here are shown only for two phases because they
represent the extremes of the cyclical variations, which are generally small.
Blade-to-blade variations of the mean velocity are evident
within the 2 mm outside of the impeller outer diameter 共gray line
in Fig. 10兲. Immediately downstream of the blade, the mean veTransactions of the ASME
Fig. 7 Phase-averaged flowfield at 13␲ / 10 in the blade passage compared to steady-state measurement at approximately
the same flow rate, 4.5 L / min
Fig. 6 Mean Speed in the blade passage
locity is 2.3 m / s, whereas it is 2.7 m / s for the majority of the
blade passage. To an observer fixed relative to the housing and
near the impeller exit, the velocity would vary as a function of
blade passage. It can also be seen that there is very little difference
between the three cases except within the regions near the outer
diameter of the impeller. The mean velocity and rms of fluctuating
velocity fields are nearly the same for the 0 and 180 deg blade
phases. This can be written as Ui,j共␾b = 180兲 ⬇ Ui,j共␾b = 0兲. Additionally, they look nearly identical to the circumferentially averaged case shown at the top of the figure. This indicates that the
effect of the blade passage is minimal, i.e., ublade is small. The
limited effect of blade-to-blade passage was also seen for flow
rates of 3 and 9 L / min. The separation at the volute tongue and
resulting separation region present at 9 L / min is not affected by
the blade passage.
Another way to interpret the measurements is by observing the
levels of rms for the circumferentially averaged and the phaseaveraged cases. Variations due to the blade passage contribute to
the rms for the free-running measurements, but not to the rms of
the phase-averaged measurements. The fact that the levels of rms
of fluctuating velocity are very nearly the same, except immediately adjacent to the impeller, for the free-running and phaseaveraged measurements, demonstrates that the blade passing has
very little effect on the velocity field in this region. This can be
written as u + ublade ⬇ u共␾b兲, therefore, ublade ⬇ 0. Within 2 – 3 mm
Journal of Biomechanical Engineering
outside the impeller diameter, levels are slightly 共⬃0.05 m / s兲
higher for the free-running case than for the frozen-blade cases.
This is because the variation of mean velocity as the blade passes
contributes to the levels of rms only in the free-running case. The
magnitude of ublade is less than 0.1 m / s immediately adjacent to
the blade and approximately zero further from the blade. This is
⬍25% of the levels of rms caused by random fluctuations.
Error Analysis. Statistical quantities estimated based on phaseaveraging assume that the flow is periodic and also constant during the acquisition period for every phase. Ideally, the magnitude
of fluctuations and statistical quantities is affected by only random
turbulent fluctuations of the fluid velocity. Realistically, two additional factors, cycle-to-cycle variations and finite duration of
phase window contribute to the statistical describing the flow.
In these experiments, the first of these factors corresponds to
variations in the transient flow rate caused by the beating ventricle
from one heartbeat to the next. The range of instantaneous flow
rates at any given phase due to the cycle-to-cycle variation is a
maximum of ±0.5 L / min with a standard deviation of 0.3 L / min
共3% of the maximum flow rate and 50% of the lowest flow rate兲.
During pulsatile flow, the rms of phase-averaged velocity measurements at some locations 共inlet elbow centerline at 7␲ / 10
shown in Fig. 4兲 is as low as 3% of the maximum mean velocity,
indicating that the combined effects of the cycle-to-cycle variations have an rms of less than 3% of the maximum measured
values.
Similarly, even if the flow rate is exactly periodic, the mean
flowfield may not be exactly periodic. This is particularly evident
APRIL 2005, Vol. 127 / 259
waiting for the first blade passage during the correct phase of the
heartbeat, the effective phase-averaging window size is 0.0057 s
共blade-passing period at 2110 rpm兲, or 2 deg of the pulse cycle.
During the sudden flow deceleration at the end of systole, the
flow rate changes approximately 0.4 L / min during a 0.0057 s
time period. If measurements are distributed randomly within this
phase window, the rms of the variation of flow rate is
⬃0.1 L / min, which is about 1% of the maximum flow rate. Measurements presented for pulsatile flow within the blade passage
are most affected by this at phases of 2␲ / 10 and 12␲ / 10, during
the rapid acceleration and deceleration of the flow. A comparison
of the flowfield at 12␲ / 10, shown in Fig. 11, reveals that the
levels of rms are slightly 共⬃0.1 m / s or 3% of the mean velocity兲
higher in this case than for the steady flow case. This is in contrast
to the case shown at 13␲ / 10, as was shown in Fig. 7, at which
phase the levels of all statistics are identical to the corresponding
steady flow case.
Discussion and Conclusions
Fig. 8 Phase averaged mean velocity field in the cut-water at 8
phases of pulsatile flow
in the inlet elbow during pulsed flow. Two adjacent counterrotating vortices form during diastole near the center of the inlet elbow. The exact position of these vortices is not exactly the same
during each heartbeat, so a fixed point within the elbow may be at
a different position within a vortex during each hearbeat. The
inclusion of these cycle-to-cycle variations in the statistical treatment of velocity leads to artificially high values of rms and Reynolds stresses. In this case, statistics, such as rms and MRSS, do
not characterize small-scale random variations of the flow, but
rather are artifacts of cycle-to-cycle variations in the flow.
The second effect is caused by acquiring data over a finite
duration of phase and statistically treating this as though it were
all acquired at the identical phase. This is often done in phaseaveraged measurements to increase the amount of data obtained.
The length of this phase duration is known as the phase-averaging
window size and is typically anywhere from 0.2 to 10 deg in published experiments. The effects of this on statistical treatment of
phase-averaged measurements are discussed in detail in Zhang et
al. 关17兴.
For experiments within the inlet elbow and cutwater, image
acquisition was synchronized to only the heartbeat and images
were acquired at a precise phase of the heartbeat to within less
than 10−10 s 共the jitter of the laser兲. In this case, only cycle-tocycle variations contributed to derived statistics. For measurements in the blade passage, the measurement is synchronized to
both the heartbeat and passing blade. A finite-sized window of
pulse phase is used to ensure that both these events occur during
every cycle of the heartbeat in order to keep the data acquisition
rate sufficiently high 共1 Hz兲 and to keep the laser running. By
260 / Vol. 127, APRIL 2005
With the exception of the inlet elbow at low flow rates and
cutwater recirculation region at high flow rates, the pulsatility of
the flow has a minimal effect on the flow field. In all regions that
were investigated, the flow field during and immediately after
sudden accelerations and decelerations was compared to a steady
flow case at the same or a similar flow rate. Little difference was
detected in either the mean flow or turbulent statistics, indicating
that the quasi-steady assumption is valid in this pump.
Differences between the steady and unsteady cases are most
pronounced in the inlet region and the cutwater recirculation zone.
This can be expected because these are both regions with large
boundary layers, relatively low momentum 共due to slow speeds兲
and low effective viscosity 共due to low turbulence intensities兲. At
a mean velocity of 1 m / s, fluid will travel ⬃300 mm during the
time between 3␲ / 10 and 9␲ / 10. This is slightly greater than the
length of the inlet cannula. If the fluctuations in flow that contribute to rms in the inlet elbow 共discussed elsewhere 关18兴兲 are the
result of entrance effects from the ventricle into the inlet cannula,
it follows that the fluctuations are not apparent in the inlet elbow
until enough time has elapsed during systole that fluid has traveled
from the entrance of the inlet cannula into the inlet elbow. This
could explain the phase delay of the rms profile as compared to
profiles of mean speed.
Although measured velocity and shear fields were slightly different from the steady cases during the sudden accelerations and
decelerations at the beginning and end of systole, they are essentially identical to the steady flow case during the high flow and
low flow conditions that constitute the majority of the pulse cycle.
This indicates that unsteady terms are not significant in the flow
field, so that the quasi-steady assumption is valid at this pulse rate.
It would be expected that unsteady terms become more significant
at increased pulse rate and decreasingly important at slower pulse
rates. Measurements and computations of the steady flow field at
the corresponding flow rate are a good representation of the flow
field at any phase of the heartbeat.
The current study addresses the flow within a magnetically suspended VAD using a prototype with an impeller suspended on a
shaft. The potential significance of this is that the shaft-mounted
impeller is rigidly suspended, whereas the position of a magnetically suspended impeller is a function of the forces exerted on the
impeller. During the pulse cycle, variations in the flow rate will
affect the fluid forces exerted on the impeller and therefore the
exact position of the impeller as controlled by the magnetic suspension. The precise effect that this variation on geometry would
have on the time-varying flow field is not known. In the magnetically suspended prototypes of this pump variations in impeller
position are never more than 0.10 mm over the range of flow rates
presented here, which is rather small compared to a blade height
of 2 mm and most characteristic dimensions of the pump.
Transactions of the ASME
Fig. 9 Profiles of mean velocity and MRSS at x = 0.02 m within the cut-water as a function of pulse phase „solid line… compared to
contours for steady flow cases „dashed line…
Measured viscous and Reynolds shear stress levels within this
pump during time-varying flow are similar to those of steady flow
conditions at the corresponding flow rate. Over the range of flow
rates investigated here, and, therefore, the range of measured
shear stresses, pump-induced hemolysis is not expected to be significant 关18兴. Stress levels at some flow rates do, however, exceed
those reported to activate platelets and therefore have the potential
to initiate thrombus formation 关18兴. The pulsatile nature of the
flow with a native heart may prove to be beneficial because it
increases blood washout by changing the flow patterns within the
Journal of Biomechanical Engineering
pump during every heartbeat. As an example, the recirculation
regions that exist in the diffuser at high flow rates is not present at
low flow rates. Fluid streamlines are directed toward the inner
wall of the diffuser and create wall shear levels that act to wash
this region during diastole. Similarly, the recirculation region that
exists in the blade passage at low flow rates is not present for high
flow rates and levels of viscous shear on the pressure side of the
blade are in the range of 2 Pa, which is reportedly large enough to
discourage thrombus formation 关29,30兴.
APRIL 2005, Vol. 127 / 261
Fig. 10 Cut-water configuration for free-running and 2 phases
of passing blade arrangements at steady flow rate of 6 L / min,
2100 rpm
Implications for Pump Design. The fact that the internal fluid
field varies during each heartbeat cannot be ignored when designing or testing LVAD devices. One proposed design strategy is as
follows. Instead of considering the nominal operating condition as
the time-averaged value of flow, two separate operating conditions
should be designed for. These correspond to the high-flow 共systole兲 and low-flow 共diastole兲 conditions. During operation with a
beating ventricle, the pump spends approximately 80% of the time
operating at either of these operating conditions, whereas it spends
less than 10% of its time operating at the flow rate corresponding
to the time-averaged mean flow. Certain requirements to insure
biocompatibility will apply to either one or both of these flow
rates. In some ways, this is a more rigorous requirement for the
flow within these pumps because it places requirements at more
than one operating point. In other ways, it actually relaxes the
design requirements because some specifications must be met at
one, but not both, of the two flow rates.
Levels of viscous and Reynolds shear stress must be kept below
acceptable levels and exposure times for both design flow rates
and, preferably, all flow rates between their design flow rates. The
requirements of blood washing to clear areas of stagnant flow and
potentially wash solid surfaces with minimal levels of viscous
shear are not required at all times. The periodic 共once per heartbeat兲 clearing of these structures and washing of walls may be
sufficient to prevent thrombosis in the pumps. Every area within
the pump does not have to have ideal flow patterns at the same
phase of the heartbeat, so long as the fluid is washed at least once
per cycle. Examples in this pump are the washing of the cutwater
recirculation region during the low-flow case and washing of the
recirculation in the blade passage during the high-flow condition.
While either of these flow rates would be unattractive operating
points for steady flow, they complement one another during pulsatile flow through the pump.
It is realized that the values of maximum and minimum flow
rates will vary based on physiology, so it is not obvious that there
are two clear choices for the values of flow rates that should be
selected. Nonetheless, the pump must perform in a variety of situations, almost none of which are for the steady flow at 6 L / min.
The quasi-steady assumption can greatly simplify the design
process. Computational modeling and experimental testing at two
steady flow rates is significantly simpler than modeling and measuring the true transient behavior within the pump. This treatment
can only be justified based on measurements, such at those presented herein, that demonstrate that the quasi-steady assumption is
valid for both the low-flow and high-flow conditions in this pump.
Acknowledgments
Financial support for this work has been provided by the Utah
Artificial Heart Institute and the Department of Health and Human
Services, National Institute of Health, National Heart, Lung, and
Blood Institute Grant No. R01 HL64378-01. MedQuest® Products, Inc., supplied substantial assistance with the fabrication of
the prototype pump. The authors would also like to thank F.
Scarano and M. L. Reithmuller for use of their PIV algorithm.
Detailed comments and suggestions from the two anonymous reviewers were very helpful in preparing the final version of the
manuscript.
Nomenclature
␦ ⫽ boundary layer displacement thickness
␯
␯␶
␻
MRSS
ũ
ṽ
U
u
u⬘
ublade
upulse
Fig. 11 Comparison of flow at 12␲ / 10 in the blade passage to
steady flow at the same flow rate, 6 L / min
262 / Vol. 127, APRIL 2005
⌽blade
⌽pulse
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
kinematic fluid viscosity
effective turbulent viscosity
angular frequency of oscillation
major Reynolds shear stress
instantaneous velocity in the x-coordinate direction
instantaneous velocity in the v-coordinate direction
time averaged-mean velocity
fluctuating component of velocity
rms fluctuating velocity, u
fluctuating component of velocity caused by blade
passage
⫽ fluctuating component of velocity caused by heart
pulse
⫽ phase of blade passage
⫽ phase of heart pulse
Transactions of the ASME
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