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An experimental study of pulsatile flow in a curved tube
J. B~kmLS 0 Pcqamon
Cal-9290/79/1cal-0793
Vol. II. pp. 793405. PRII Ltd. 1979. Printed in Great Bntain
xmml
AN EXPERIMENTAL STUDY OF PLJLSATILE FLOW IN A CURVED TUBE* K. B.
CHAWRAX,~
T.
L. YE.ARW(X)I)~and D. W. WIETIVG
Tulane University Schools of Medicine and Engineering. New Orleans, LA 70118. U.S.A. Abstract - Experimental measurements of axial velocity
profiles at various cross-sections in a curved tube due to a pulsatile flow of uniform entry are reported. The curved tube with the radius of 1.5 cm and radius-radius of curvature ratio of 0.1 was mounted in a mock-circulatory system which simulates physiological pulsatile flow for a blood-analog fluid. The experiments were carried out for a time-avera_ped Reynolds number of 1019, a Dean number of 322 and a Womersley parameter of 21.89. Flow visualization was used to qualitatively investigate the nature of the flow, complementing the quantitative measuremenfs made by hot-film anemometry. The fluid in the curved tube accelerated and moved downstream during the forward flow duration (systole) in the pulsatile flow cycle. However, during the remainder of the cycle (d&stole), recirculation occurred, with fluid particles near the outer wall of the curved tube continuing their motion downstream, and fluid particles near the inner wall moving in a retrograde fashion back upstream. Immediately past entry into the curved tube, the maximum axial velocity was observed near the outer wall of the tube. The maximum velocity shifted towards the inner wall at the apex of the curve and shifted back towards the outer wall in the far downstream region. The velocity profiles for the pulsatile flow are compared with the corresponding results for steady flow at each cross-section for a Dean number of 569. The implications of the results with respect to the hemodynamic theories of atherogenesis are briefly discussed.
INTRODUCTION
(1973) for the range of Dean number to include the entire laminar flow regime. Yao and Berger (1975) and
Considerable attention has been devoted to the theoretical and experimental study of steady and unsteady flows in curved tubes. An experimental study of the velocity profiles due to a pulsatile flow of uniform entry in a curved tube is the subject of this paper. In addition to its relevance to the analysis of pulsatile entry Row in curved tubes, an important motivation behind this study is the fact that the phenomenon of atherosclerosis is known to selectively occur at arterial sites of curvature, bifurcation and branching (Texon, 1960; Caro et ul., 1971). Typical curvature sites in the human circulatory system with frequent incidence of atherosclerotic plaque formation are the aortic arch, and the cerebral and coronary arteries. Even though the exact mechanism of atherogenesis is not fully understood, several hemodynamic theories for the genesis of this phenomenon have been suggested, as briefly discussed by Chandran et al. (1974). A comprehensive review of these theories can also be found in a paper by Bergel er a/. (1976), as well as Car0 (1977). Dean (1927,1928)described theeffectsofcentrifugal force in the production of secondary flow in fully developed steady viscous flow in tubes with small curvature. Numerical analyses of steady viscous fully developed flow in curved tubes have been reported by McConologue and Srivatsava (1968) and Greenspan l Received 24 August 1978. t Present address: Hemodynamics
Laboratory, Division of Materials Engineering, The University of Iowa, Iowa City, IA 52242, U.S.A.
Singh (1974) reported viscous
flow
on theoretical
analyses of steady
of uniform entry to discuss the flow development in the entrance length region of curved tubes. Agrawal et al. (1978) measured the velocity profiles in the entrance region of curved tubes and pointed out that the theoretical analyses of Yao and Singh are inaccurate, due possibly to the assumptions used in their solution procedure. Fully developed oscillatory viscous flow (sinusoidal with no mean flow) in tubes of small curvature has been the subject of works by Lyne (1971 J and Zalosh and Nelson (1973). While the effect of centrifugal force in steady flow in curved tubes was to shift the maximum axial velocity towards the outer wall of the curve, Lyne (1971) showed that in the case of oscillatory flow, the maximum axial velocity is shifted towards the inner wall of the curve. Chandran er ol. (1974) reported on the analysis of oscillatory viscous fully developed flow in elastic tubes of small curvature as a simulation of flow in curved arteries. They showed that in the cross-section of the bend, the secondary flow changed from a quadri-helical to a bi-helical pattern during one cycle, with the net eh%ctof shifting the maximum axial velocity towards the inner wall of the curve. More recently, Smith (1975) presented a theoretical analysis of unsteady viscous flow (sinusoidal with a non-zero mean) in curved tubes. It is worthwhile pointing out here that, except for the work of Chandran et al. (1974). all the other works cited above dealt with rigid curved tubes. Peronneau et trl. (1974) reported preliminary irl vitro studies of pulsatile flow (steady flow with sinusoidal oscillations), as well as irr cico studies in dogs.
793
K.
794
B. CHA.VDRAN, T. L. YEARWOODand D. W. WIETI~G
Fig. 1. Schematic of the mock-circulatory system.
Plane A
\
Plane8 \
PlaneC
PlaneD I
PimE /
Fig. 2. Schematic of the curved tube used in the experiments indicating the planes in which velocity measurements were obtained.
The nature of pulsatile flow in the entrance region of curved tubes (such as that occurring in the aortic arch) has not been well understood. Specifically, for the corresponding aortic arch in humans, the average radius is about 1.5 cm and the radius of curvature is about 4.5 cm with a radius-radius of curvature ratio t: of 0.33. Moreover, the flow is further complicated by an asymmetric entry condition, due to the presence of valves, as well as an irregular geometry with tapering and branchings at the bend. To delineate the effect of curvature alone in a puisatile flow with uniform entry, this paper reports an experimental analysis in a curved tube of regular geometry and a radius-radius of l
Vitamek, Inc., P.O. Bos 25359, Houston, TX 77005,
U.S.A.
curvature ratio of 0.1. Qualitative studies were conducted through a flow visualization technique and quantitative measurements of the velocity profiles at various cross-sections in the curve were made by hotfilm anemometry. A lower value of c was chosen for the curved tube in this study rather than the approximate value mentioned for the aortic arch (c = 0.33), for the following reason. Even though we only measured the axial velocity component in this study, we are planning on using a three-dimensional probe to study the radial and tangential velocity components subsequently. It was felt that for smaller values of r7 the radial and tangential velocity components wiIl not be of the same order of magnitude as the axial velocity component and thus the results will be relatively easier to interpret. Better understanding of the nature of the secondary flow in this region will be valuable in interpreting the results in tubes with larger curvatures in the future. The experiments with three-dimensional probes both in the curved tube and in a model of the aortic arch are, at present, being performed in our laboratory. EXPERIMENTAL
METHODS
The experiments were, conducted by using a mockcirculatory system previously described by Wieting (1968, 1969) and fabricated by Vitamek, Inc.* The basic components of this system are described in the schematic diagram of Fig. 1. Briefly, this system provides a pulsatile flow for a blood-analog fluid using
Fig. 3, Flow visualization
photo_eraph
during
systole for the flow straightener.
Fig. 1. Flow visualization
photograph
during
diastole
795
for the How straightener.
Fig. 8. Flow visualization photograph in the curved tube in systole. Fig. 9. Flow visualization photograph in the curved tube in diastole.
796
An experimental study of pulsatile flow in a curved tube rates from zero to 199 beats per mm, mean flow rates from zero to 10 I. per min, a systolic duration from zero to 300 msec, and good approximations of physiological wave forms in the aorta, left ventricle and left atrium. A curved tube test section as illustrated in Fig. 2 was machined out of two blocks of plexiglas? and cemented together. The radius of the tube a is 1.5 cm and the radius of curvature R is lScm, with the resulting radius-radius of curvature ratio ofO.1. To remove the effect of any flow disturbance created by the aortic valve as the flow comes out of the aortic valve chamber, a 6-in. flow straightener was mounted in between the aortic valve and the curved tube. This length was chosen on the basis of previous studies by Wieting (1969) in which it was determined that the disturbances caused by a human homograft aortic valve and most prosthetic aortic valves were dissipated by a point 6 in. downstream (distal) to the valve. Typical flow visualization pictures in the flow straightener during systole and diastole are shown in Figs. 3 and 4. respectively. All the studies reported in this paper have been conducted with the flow straightener in the system and a Gott Butterfly aortic valve.* The Gott valve was chosen because it has-no measurable back flow, and the disturbances caused by the valve can be minimized through appropriate use of the flow straightener. The mock-circulatory system is primed with a blood analog fluid composed of 529, glycerol in water, which at 37°C has a viscosity of 0.04 poise. The system is tuned to provide a mean flow rate of 5.1 I./min at a rate of 72 beats per min and a systolic duration of 270 msec. Simulated blood pressure in the ventricular chamber is measured with a Statham P23Ht pressure transducer and the time dependent flow rate is measured by means of a Statham Model M4001 electromagnetic flow meter: and an In Co Metric Model K26 mm cannulation-type electromagnetic flow probe: placed immediately downstream from the curved tube. The flow visualization technique used in the qualitative analysis has been previously described by Wieting (1969). Briefly, the technique employs neutrally buoyant white spherical plastic beads suspended in the blood-analog fluid and illuminated with a specially designed slit light. Flow visualization and. photography are performed perpendicular to the plane of the slit light beam in the center plane of the curved tube. Still photographs are taken throughout the cycle using a Nikon FZSB 35 mm camera equipped with a motor Prosthetic heart valve donated to Dr. D. W. Wieting by Vincent Gott, M.D., Dept. of Surgery, Johns-Hopkins University, Baltimore, MD 21204. t Statham Inst.-Physiological Div., 2230 Statham Blvd., l
Oxnard, CA 93030. : In Vioo Metric Systems, P.O. Box 217, Redwood Valley, CA 95470. 8 Red-Lake, 2991 Corvin Drive, Santa Clara, CA 95051. 1’Thermo-Systems, Inc., P.O. Box 3394, St. Paul, MN 55165.
797
drive and a Nikkor 50 mm 1: 1.4 lens using a range of shutter speeds between l/8 and 116Osec. High-speed movies are taken with a Redlake Hycam 16 mm camera4 equipped with a Switar 25 mm 1: 1.4 lens at 100 frames per sec. After the flow pattern studies were completed, small diameter holes were drilled in the central plane of the curved tube at five symmetric stations, A-E, as illustrated in Fig. 2. These holes allowed the insertion of the hot-film velocity probe across the entire diameter of the tube. The velocity profiles at the crosssections A-E were measured at different times in a cycle using a TSI” Model 1056 constant-temperature anemometer. The hot-film velocity probe was calibrated in a steady flow system at 37°C under, Poiseuille flow, and the calibration was linearized using a TSI Model 1052 41hOrder Polynomial Linearizer. A TSI Model 1210-20 W hot-film probe is suitable for these studies. The hot-film sensor was mounted on support pins 12.7 mm away from the end of a cyclindrical shaft 3.2 mm in diameter. The probe shaft was in the plane of the cross-section of the curved tube and the hot-film sensor and its supports projected from the end of the shaft. The projected area of the probe shaft was less than 104; of the area of cross-section of the tube when the probe was inserted to the maximum radial depth for measurement. During forward or reverse flows, the probe shaft was not interfering with the fluid flow over the probe itself. The probe was inserted through the hole and positioned so that the hot-film probe was perpendicular to the axial flow direction. Even though the axial velocity component is the predominant component of flow going over the probe, the probe actually measures the vector sum of the axial and radial flow components in the central plane of the curved tube and will introduce some error in the measured values. After the study reported in this paper, we have employed a three-dimensional probe by Thermo-Systems, Inc. to determine the three velocity components in the curved tube. Some of the preliminary measurements with the three-dimensional probe in steady flow show that for the curved tube under consideration (with a radius-radius of curvature ratio ofO.l), the radial velocity component was, at the most, 10yOof the axial velocity. We conservatively estimate that our axial velocity measurements are accurate to within 15% of the measured value due to the factors discussed above. DATAREDUCTION The flow visualization and hot-film anemometry techniques were used in a complementary fashion in order to obtain the final results. Analysis of the highspeed movie data showed that streamlined flow existed during the forward flow phase (systole) of the cycle. However, during the period in which theaortic valve in the system was closed (diastole), a reversed flow was observed along the inner wall of the curve, while along the outer wall of the curve, a forward flow of small
798
K. 8. CHAMXAN,T. L. YEAFSQOLIand D. W. WETIXG
magnitude was still observed. Thus, the flow visualization technique afforded us a qualitative analysis of the flow. An attempt to quantify the magnitudes of the velocity by a frame-by-frame analysis of the high-speed movies, however, yielded very inaccurate results. The results were quantitatively inaccurate for two reasons. First, particles tended to appear and disappear, moving in and out of the plane of illumination due to the obvious three-dimensional character of the flow. Second, the time lapse between each frame was at least 10msec and thus severely limited the precision to which instantaneous velocities could be estimated by carefully measuring particle displacements from frame-to-frame. Hot-film anemometry was then used to provide a quantitative basis for the qualitative data obtained by the flow visualization technique. It should be noted that the hot-film anemometer technique is insensitive to the direction of flow, since a flow in the reverse direction also produces a d.c. voltage signal output similar to that of the forward flow. Hence, the flow visualization technique was valuable in interpreting the data from the hot-film
anemometer, since the signal output from the anemometer appears to be rectified if flow reversal occurs, The first derivative of that signal will contain an appropriate discontinuity corresponding to .the inflection point of the output signal. Ideally, the inflection point for the velocities would occur at a zero voltage reading. However, under the unsteady flow conditions, a zero flow measurement is not likely to be obtained, due to the local disturbances of flow in the vicinity of the probe during flow reversal. Figure 5(a) shows the voltage-time signal output of the anemometer probe for a typical flow cycle at a point very near the inside wall of the curved tube. It is known that reversed flow occurs during that portion of the cycle indicated by the positive sign, An inspection of the first differential of the bridge output curve in Fig. 5(b) shows a dramatic discontinuity at the closing of the valve when the fluid acceleration is most negative. This curve is obtained by a first order finite difference approximation of the voltage-time signal output curve. It is surmised that flow reversal has occurred at this point in the cycle, at approximately 330 msec. In addition, another flow reversal must take place when forward flow is re:established at thecommencement of the next cycle. Inspection of Fig. 5(a) again indicates that a rapid rise in fluid acceleration has occurred just
-
Fig. 5. Anemometer signal output (a), the first derivative of the signal (b) and the velocity-time curve for a point near the inner wall (c) of the curved tube.
(bl
Fig. 6. Anemometer signal output (a). the first derivative of the signal (b) and the velocity-time curve for a point near the outer wall of the curved tube.
799
An experimental study of puisatile Row in a curved tube
Table 1. t1
[J
[I
rs
(6
r-
fa
(9
fl0
9 10
1.6 2.0 2.0 1.8 2.2 2.1 1.8 2.1 2.0 2.1
4.5 4.4 4.4 4.8 4.7 4.6 4.0 4.8 4.4 5.0
4.9 5.2 5.0 -7 ::; 5.0 5.2 5.0 5.2 5.2
1.5 1.2 1.6 2.0 2.0 1.8 2.4 2.0 2.0 2.6
3.3 2.5 2.6 3.0 3.2 2.0 2.6 2.8 3.1 2.2
2.2 1.6 1.8 1.6 2.0 2.4 0.8 2.2 2.4 1.9
2.2 1.9 1.4 1.2 1.2 2.0 2.2 2.3 2.8 2.2
1.9 2.0 1.8 1.0 1.2 1.9 1.6 1.6 1.6 2.0
1.9 1.4 1.2 0.8 1.4 1.g 1.6 1.2 1.2 1.8
1.4 1.6 1.2 0.9 0.8 1.4 1.8 1.6 1.2 1.2
Average S.D.
1.97 0.18
4.56 0.28
5.11 0.12
1.91 0.41
2.73 0.43
1.89 0.48
1.94 0.52
1.66 0.34
1.43 0.35
1.31 0.3 1
V,,
25.40
58.80
65.89
24.63
35.20
34.37
‘5.01
21.40
18.44
16.89
Cycle
2 4 5 6
8
Plane A, 0.60 cm from the inner wail. Velocity magnification factor = 12.894cm/set per V d.c. t, = ri = r3 = rr = rs = t6 r7 = ts =
time of initial rise of pressure in flow straightener, at I = 0.00 msec. time of peak in pressure (systole), approximately 67 msec. 100 msec later. inflection point of pressure curve as valve closes, approximately 297 msec iowest drop in pressure, approximately 345 msec. time of peak residual rise in pressure (diastole), approximately 427 msec. 100 msec later, approximately 527 msec. 100 msec later, approximately 627 msec. 727 msec. c9 = 100 msec later, approximately r10- 70 msec later, approximately 797 msec. after a mild dip at the end of the cycle (t = 830 msec,
approximately), as indicated by the arrow. This mild dip is very characteristic when reversed flow dominates the latter portion of the cycle, and it is not seen at other points in the tube where forward flow persists during the entire cycle, as seen in Figs. 6fa)and (b). The fluid velocity curve shown in Fig. 5(c) is then constructed using the calibration constants. In Figs. 6(a) and (b), the signal output and its first derivative are shown for a representative point near the outer wall of the curved tube, and a similar analysis is used in constructing the velocity profile seen in Fig. 6(c). Such an analysis for the direction of flow correlated very well with our qualitative analysis of the high-speed movie. Table 1 illustrates the anemometry data for a single point in the curved tube, one of ten such points in each of the five planes shown in Fig. 2. At each point, the d.c. voltage output of the anemometer for 10 successive cycles was recorded on a Dynograph* strip chart recorder, Model 5048, simultaneously with the pressure in the flow straightener by way of a fluid-filled tap located 3 in. downstream from the Gott valve. From each of the 10 cycles, 10 standardized time points were read from the strip chart recording of the voltage curve in accordance with specific characteristics of the pressure curve. These standard time points are shown in Fig. 7(b) on the pressure curve. Thus, the pressure curve served as a timing standard by which the velocities
at each location
about
the tube could
he
* Beckman Instruments, Inc., Offner-Division, 3900 River Road, Schiller Park, IL 60176.
compared and contrasted. The pressure signal was repeatable with negligible cycle-cycle variation to serve as the timing standard for the velocity measurements. The average voltage and the standard deviation was calculated over the 10 cycles for each time point as shown in Table 1, and the subsequent velocities were obtained by multiplication of the voltage by the calibration constant. For these experiments, the calibration constant was 12.9 cm/set per V d.c. As evident from Table 1, the standard deviation was relatively low compared to the measured voltage during the systolic duration of the pulsatile flow cycle. However, relatively large variations in the voltage signals from the mean were observed during the diastolic duration, even though the voltage signals at various cycles were qualitatively similar. This variation during diastole is attributed to local fluid disturbances around the probe, such as the random dissipation and decay of transient eddies and low frequency vortex shedding from the probe shaft expected at very low axial velocities. The standard deviation in velocity measurements are indicated in our measurements for one of the planes (in Fig.’ 10). RESULTS AND DISCUSSIONS
The flow parameters in the experimental set up are given below: Density of the fluid, p = 1.13 g/cm’; Viscosity, p = 0.04 poise; Frequency of the pulsatile flow, f1.20 set-’ ;
72 min-’
=
800
K. B. CHASXAS, T. L.
YEARWOOD
1
and
D. W. WIETIX
I
0
,400
I
,800
sec.
(0)
1
1
0
,400
I
,800
sec.
(bi
Fig. 7. Typical flow rate curve obtained from the electromagnetic flow meter (a) and a typical pressure signal from the flow straightener (b).
I,2
Womersley
parameter,
(>
IZ= u f
= 21.9.
The maximum flow rate QmXXand the average flow Q.,in a cycle were obtained from the electromagnetic flow meter. A flow rate vs time curve obtained from the flow meter along with a typical pressure signal from the flow straightener just up stream of the curved tube are shown in Figs. 7(a) and (b). The cross-sectional area A of the curved tube is 7.07cm2. Then the cross-sectional average axial velocity w, the Reynolds’ number (Re = 2p@u/p) and the Dean number D( = Re(a/R)“‘) are computed and the values are shown below: rate
Q(l/min)
w(cm/sec)
Re
D
18.1
42.4
3600
1140
Pulsatile flow Maximum during a cycle Time-averaged value in a cycle
5.1
12.0
1020
320
Steady flow
9
21.2
1800
570
Typical still photographs from the flow visualization study during systole and diastole are shown in Figs. 8 and 9, respectively. A careful analysis of the photographs as well as of the high-speed movies showed that during systole, fluid in the curved tube accelerated downstream. During diastole, however, recirculation occurred, with fluid particles near the outer wall of the curved tube continuing their motion downstream, and fluid particles near the inner wall moving in a retrograde fashion back upstream. This motion is not caused by back flow, since the Gott prosthetic aortic valve has no appreciable back flow (Fig. 7a). The magnitude of velocity at the inlet of the curved tube was not directly measured so as to avoid drilling holes in the flow straightener and thus introducing another source of possible flow disturbance before the fluid enters the curved tube. However, velocity measurements were made in another tube with the same geometry as the flow straightener at the same axial distance away from the valve and the velocity distributions at early and peak systole (0 msec and 67 msec in Fig. 7b) are shown in Fig. 10. The velocity distribution was observed to be symmetric with re-
An experimental study of pulsatile i3ow in a curved tube
Fig. 10. Velocity profile at the inlet of the curved tube: (a) early systole (0 msec): (b) peak systole (67 msec).
spect to the center of the tube, reasonably flat at the point of entry into the curved tube, with a variation of less than 10% between maximum and minimum magnitudes during peak systole. As is evident from Fig. 4, flow reversal occurred along the cross-section of the flow straightener during diastole and the quantification of the velocity profiles was not attempted in this section during diastole. PLANE
I
The velocity profiles obtained at Planes A, C and D during systole (125 msec in Fig. 7b) and during diastole (650 msec in Fig. 7b) are shown in Fig. 11. In Plane A. during systole, the maximum axial velocity (W) is observed near thl outer end of the curved tube. However, in Planes C and D, the maximum velocity is observed at the inner end with the skewing of the velocity more pronounced in Plane C. During diastole, a coherent reversed flow is observed along the inner radius ofthe curve with a slight forward flow along the outer radius. The magnitude of the reversed flow is seen to be a maximum in Plane C. The velocity profiles at Planes A, B, C, D and E during various times in the cycle are shown in Figs. 12-16, respectively. In these figures, velocity profiles of measurements due to steady flow in the curved tube at the same planes are also superposed in dotted lines for a direct comparison between steady and pulsatile flows. In the case of steady flow, the mock-circulation unit was replaced by a steady flow system where the blood-analog fluid was pumped by a centrifugal pump. The steady flow velocity profiles shown in the figures are for a flow rate of 9 l/min (with a corresponding Dean’s number of 569) compared to 5.1 limin (and 322) for the pulsatile flow. It is interesting lo note that the steady flow profiles and the pulsatile flow profiles
A
20
PLANE
1:
PLANE
C
PLANE
D
0 10
5
0
5
801
10
20
t
60
2ot
Fig. 11. Cross-sectional axial velocity profiles during systole (a) and diastole (b) in Planes A, C and D. The ordinates denote axial velocity in cm/set. In all the figure-s, left side is the outer wall and the right side is the inner wall of the tube.
802
K. 9.
CHANTIRAN,
T. L.
and D. W. WIETISG
YEARWooD
PLANE PLANE
C
100
A
t
b
2 50
-
VI
---_--______---N\
-30
-
-20
\OUTER
WALLV
t
Fig. 12. Cross-sectional axial velocity profiles at Plane A during different times in a cycle: (a) early systole (0 msec); (b) peak systole (67 msec); (c) mid-diastole (627 msec); (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 3.40 from the entrance.
PLANE
Fig. 14. Cross-sectional axial velocity profiles at Plane C during different times in a cycle: (a) early systole (0 msec): (b) peak systole (67 msec); (c) mid-diastole (627 msec): (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance - IO.20 from the entrance.
B
90 L
PLANE
0
1oc 80 -
9c EC
70 ti : 50 -
; 6C
____-----___
P-
$ 5c =
4c
c 0
30
ii >
20
: z Q OUTER -30
-
-40
-
WALL
Fig. 13. Cross-sectional axial velocity profiles at Plane B during different times in a cycle: (a) early systole (0 msec); (b) peak systole (67 msec); (c) mid-diastole (627 msec); (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 6.8a from the entrance.
\
WALL
0 -10
C
INNER
10
d
YgzJ$-g
-20 -30
C
Fig. 15. Cross-sectional axial velocity profiles at Plane D during different times in a cycle: (a) early systole (0 msec); (b) peak systol (67 msec); (c) mid-diastole (627 msec); (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 13.60 from the entrance.
An experimental study of pulsatile flow in a curved tube PLANE
E
Fig. 16. Cross-sectional axial velocity profiles at Plane E dcrinp different times in a cycle: (a) early systok (0 msec); (b) peak systole (67 msec); (c) mid-diastole (627 mscc): (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 17.04 from the entrance.
during systole are relatively flat in Planes A and B. However, in Plane C, the velocity profiles for both cases have pronounced asymmetry, with the steady flow velocity magnitudes being higher towards the outer wall and the pulsatile flow velocity magnitudes being higher towards the inner wall. Lyne (1971) was the first to discuss the difference in secondary flows between steady and oscillatory flows through a theoretical analysis. He suggested that in the case of oscillatory flow, the viscous effects are confined to a region close to the wall and the inviscid core region tends to move the maximum axial velocity towards the inner wall. This effect was corroborated by Chandran et al. (1974) in analyzing oscillatory flow through curved elastic tubes. The present study confirms that a difference in the secondary flow is found between steady and pulsatile flows where the pulsatile flow is a combination of several modes of oscillatory flow superposed with a steady flow component. It is also interesting to note that in Plane D, the pulsatile flow profiles during systole become less skewed relative to those in Plane C. By Plane E, the skewing of the profile was shifted towards the outer wall. This phenomenon could be due to the fact that an inviscid core region such as that referred to by Lyne (1971) is not present in the downstream region of the tube. A recent work of part&&r concern to this problem is that by Singh et 01. (1978) in which they employed a boundary layer analysis to investigate the nature of pulsatile Row in the entrance to the aorta. As an expansion upon the steady entry flow in a curved tube of Singh (1974), they analytically examined the consequences of a pulsatile pressure gradient (sinusoidal with a non-zero mean) imposed upon flow in the entry region of a curved, rigid circular tube, with r. of the
803
same order of magnitude as the present experimental study. They showed that the fluid velocity and wall shear will initially be higher near the inner wall of the curved tube owing to the external flow and geometric factors. However, as the flow develops, they predict that the effect of curvature will be to increase the secondary flow of the boundary layer, drawing fluid azimuthally from the outer wall to the inner wall. As the fluid is drawn towards the inner wall, the boundary layer there would become thicker, while at the outer wall, the boundary layer would become thinner. If this is true, the wall shear and fluid velocity at the outer wall will become greater than at the inner wall as the flow develops further downstream. Furthermore, at this location downstream, their analysis. predicts the gradual development of a reversed flow near the inner wall as a result of the deceleration of the fluid at the close of the systolic phase. The quantitative precision of their analysis by comparison with our results are quite poor, possibly because they assume that axial flow comes to rest everywhere in the tube by the end of each cycle. Nevertheless, our results can be seen to be qualitatively in agreement with their analysis. The shifting of the velocity profiles between Planes C and E may very well be the result of the secondary flows as indicated by their analysis. Furthermore, their analysis of flow reversal along the inner wall as a result of fluid deceleration is qualitatively supported by our results for flow profiles during diastole. The results of Singh et al. (1978) suggest that an inviscid core of fluid would persist throughout our curved tube past Plane E. The problem ,of entry length in the tube under pulsatile conditions cannot be solved by our present results, however, without knowledge of the secondary flows. It ‘is not known if coupling of the inner and outer boundary layers, therefore, could take place by Plane E under the present experimental set-up. The velocity profiles reported by Peronneau er a/. (1974) agree qualitatively with the results reported in this work. In their in uioo experimentation in the aortic arch, they show reversed flow along the inner radius of the curve during peak reverse flows. They also show similar profiles in their in vitro studies. However, no direction quantitative comparison can be made between their results and ours because of the difference in tube geometry. Moreover, they used a sinusoidal flow superposed on a steady flow, whereas our enperimental set-up replicated a typical pulsatile flow similar to that produced in man by the left ventricle. Agrawal et al. (1978) have reported on the flow development in the entry region for steady viscous flow of uniform entry into curved tubes for Dean’s number ranging from 139 to 565. Their results show that immediately after entry into the curved tube, the maximum axial velocity is measured towards the inner wall and as the flow devefops, the maximum axial velocity progressively shifts towards the outer wall in the downstream region. This effect of skewing towards the inner wall in the upstream region was more pronounced at lower Dean’s numbers (at about 180) at
804
K. B. CHASDRAN,
T.
L.
axial distances of 2-12 times the radius of the tube. At higher Dean’s numbers (at about 565), the velocity profiles in the upstream region were observed to be relatively flat before skewing towards the outer wall in the downstream region. Our results for steady flow (Dean number = 569) show a similar trend, with the profiles being relatively flat in Planes A and B (axial distances of 3.4 and 6.8 times the radius of the tube) before skewing. towards the outer wall in the downstream region (Planes C, D and E). However, in the case of pulsatile flows, the axial velocity distribution shows a trend opposite to that for steady flows, possibly due to the reasons discussed by Lyne (1971) and Chandran et ol. (1974). In the pulsatile flow situation with a time averaged Dean number of 322, the maximum axial velocity magnitude was observed towards the outer wall in Plane A (axial distance = 3.40), followed by a progressive shift of the maximum axial velocity towards the inner wall of the curved tube in Planes B and C (axial distances of 6.8~ and 10.2~. respectively). It is also interesting to note that in Planes D and E (axial distances of 13.6~ and 17a, respectively), the maximum axial velocity is seen to be shifting towards the outer wall once again. During diastole, however, a reversed flow was observed towards the inner wall and a slight forward flow towards the outer wall in all the planes. The maximum values of reversed flow velocity magnitudes were observed in Plane C at the top of the curve. Even though an attempt was not made to compute the wall shear stress distribution in a cross-section from these velocity measurements, it is evident from Figs. 12-16 that the higher magnitudes of wall shear are, in general, observed at the inner wall of the bend, due to pulsatile flow in a curved tube. Moreover, a large variation in the magnitude and direction of the shear stress is also observed in this region compared to the outer wall of the bend, due to the flow reversal during a cycle measured towards the inner wall. Several investigators have proposed that the wall shear stress distribution dueto pulsatile blood flow plays a vital role in atherogenesis. Fry (1968) has demonstrated that abnormally high shear stresses damage the endothelial lining and suggested that these are precisely the sites for lipid deposition. On the other hand, Caro et al. (1971) postulated a shear dependent mass transfer mechanism for atherogenesis and suggested that lipid deposition occurs where the wall shear stresses are relatively low. They correlated their postulation with observed incidence of atherosclerosis in the inner end of the curve such as in the aortic arch. Their arguments are based on steady flow qualitative study in a model of the aortic arch where low wall shear stresses are expected at the inner end of the curvature. In contrast, however, the present study on pulsatile flow in curved tube shows that higher magnitudes of wall shear stress are observed towards the inner wail of the bend. Moreover, the magnitudes of the wail shear near the inner wall fluctuate between a maximum in one direction to the other during a cycle,
YEARWOOD
and D. W. WIETISG
due to the flow reversal observed in diastole. As suggested more recently by Caro (1977), the effect of such large fluctuations in wall shear due to pulsatile flows around a curve, as well as in bifurcation and branching sites on the endothelial lining, need to be investigated to improve our understanding of the mechanical effects of blood flow in atherogenesis. We chose a curved tube with a radius-radius of curvature ratio of 0.1 with an aim to understand the nature of the secondary Row in a model of the aortic arch. The study in the aortic arch will be further complicated by the increase in curvature, irregular cross-section of the lumen, tapering and branching of arteries at the top of the curve. Effects of each of these complications need to be lineated before a better understanding on the fluid dynamic effects on atherogenesis can be obtained. At present, work is proceeding in our laboratory with a three-dimensional hot-film probe to understand the nature of the development of the secondary flows both in the curved tube and in the model of the aortic arch. The flow visualization technique used in the present study has a disadvantage in that it does not illustrate the nature of the secondary flow perpendicular to the plane of illumination. Nevertheless, this technique was useful in interpreting the results from the hot-film anemometry measurements.
Acknowledgement
-The support of this work by a grant from National Institutes of Health (NHLBI-HL18156-01-02) is gratefully acknowledged.
REFERENCES Agrawal, Y. C., Talbot, L. and Gong, K. (1978) Laser anemometer study of flow development in curved circular pipes. J. Fluid Mech. 85, 497-518.
Bergel, D. H., Nerem, R. M. and Schwartz, C. J. (1976) Fluid dynamic aspects of arterial disease. Atherosclerosis 23, 253-261. Caro, C. G.. Fitzgerald, J. M. and Schroter, R. C. (1971) Atheroma and arterial wall shear observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. R. Sot. (B) 177, 109-159. Care, C. G. (1977) Mechanical factors in atherogenesis. Cardiovascular Flow Dynamics and Measurements (Edited by Hwang, N. H. C. and Normann, N. A.). Chapter 13, p. 473. University Park Press, Baltimore. Chandran, K. B., Swanson, W. M., Ghista, D. N. and Vayo, H. W. (1974) Oscillatory flow in thin-walled curved elastic tubes. Ann biomed. Etqng 2, 392-412. Dean, W. R. (1927) Note on the motion of fluid in a curved pipe. Phil. Mug. (S) 7, 4, 20, 208-223. Dean. W. R. t 1928) The steamline motion of fluid in a curved pi&. Phil. ‘Mug: (S) 7, 5, 30, 673-693. Fry, D. L. (1968) Acute vascular endothelial changes associated with increased blood velocity gradients. Circulation Res. 22, 165-197. Greenspan, D. (1973) Secondary flow in a curved tube. /. fluid Mech. 57, 167- 176. Lyne, W. H. (1971) Unsteady viscous flow in a curved pipe. J. Fluid Mech. 45, 13-31. McConalogue, D. J. and Srivastava, R. S. (1968) Motion of a fluid in a curved oiw. Proc. R. Sot. (Al 307. 37-53. Peronneau, P., Hi&& J., Xhaard, I$ ‘Deliuche, P. and Philippo, J. (1974)The effectsof curvature and stenosis on
An experimental study of pulsatile flow in a curved tube pulsatile Row in uiro and in ritro. Cardiocascular Applied Ultrasound Vol. I, pp. 203-215. North Holland, Amsterdam. Singh, M. P. (1974) Entry flow in a curved pipe. _I.Fluid Meth. 65, 517-539. Singh, M. P., Sinha. P. C. and Aggarwal, M. (1978) Flow in the entrance of the aorta. J. Fluid Mech. 87, 97-120. Smith, F. T. (1975) Pulsatile flow in curved pipe. J. Fluid Mech. 71, 15-42. Texon, M. (1960) The hemodynamic concept of atherosclerosis. Bull. N.Y. Acad. &fed. 36, 268.
805
Wieting, D. W. (1968) A method for analyzing the dynamic flow characteristics of prosthetic heart valves. ctS,M15 Paper No. 6%WA/BHF-3. Wieting. D. W. (1969) Dynamic Row characteristics of heart valves. Ph.D. Dissertation. University of Texas. Austin. Texas. Yao, L. S. and Berger. S. A. (1975) Entry flow in a curved pipe. J. Ffuid Mech. 67, 177-196. Zalosh, R. G. and Nelson, W. G. (1973) Pulsating flow in curved tube. J. Fluid ,Uech. 59, 693-705.
Cal-9290/79/1cal-0793
Vol. II. pp. 793405. PRII Ltd. 1979. Printed in Great Bntain
xmml
AN EXPERIMENTAL STUDY OF PLJLSATILE FLOW IN A CURVED TUBE* K. B.
CHAWRAX,~
T.
L. YE.ARW(X)I)~and D. W. WIETIVG
Tulane University Schools of Medicine and Engineering. New Orleans, LA 70118. U.S.A. Abstract - Experimental measurements of axial velocity
profiles at various cross-sections in a curved tube due to a pulsatile flow of uniform entry are reported. The curved tube with the radius of 1.5 cm and radius-radius of curvature ratio of 0.1 was mounted in a mock-circulatory system which simulates physiological pulsatile flow for a blood-analog fluid. The experiments were carried out for a time-avera_ped Reynolds number of 1019, a Dean number of 322 and a Womersley parameter of 21.89. Flow visualization was used to qualitatively investigate the nature of the flow, complementing the quantitative measuremenfs made by hot-film anemometry. The fluid in the curved tube accelerated and moved downstream during the forward flow duration (systole) in the pulsatile flow cycle. However, during the remainder of the cycle (d&stole), recirculation occurred, with fluid particles near the outer wall of the curved tube continuing their motion downstream, and fluid particles near the inner wall moving in a retrograde fashion back upstream. Immediately past entry into the curved tube, the maximum axial velocity was observed near the outer wall of the tube. The maximum velocity shifted towards the inner wall at the apex of the curve and shifted back towards the outer wall in the far downstream region. The velocity profiles for the pulsatile flow are compared with the corresponding results for steady flow at each cross-section for a Dean number of 569. The implications of the results with respect to the hemodynamic theories of atherogenesis are briefly discussed.
INTRODUCTION
(1973) for the range of Dean number to include the entire laminar flow regime. Yao and Berger (1975) and
Considerable attention has been devoted to the theoretical and experimental study of steady and unsteady flows in curved tubes. An experimental study of the velocity profiles due to a pulsatile flow of uniform entry in a curved tube is the subject of this paper. In addition to its relevance to the analysis of pulsatile entry Row in curved tubes, an important motivation behind this study is the fact that the phenomenon of atherosclerosis is known to selectively occur at arterial sites of curvature, bifurcation and branching (Texon, 1960; Caro et ul., 1971). Typical curvature sites in the human circulatory system with frequent incidence of atherosclerotic plaque formation are the aortic arch, and the cerebral and coronary arteries. Even though the exact mechanism of atherogenesis is not fully understood, several hemodynamic theories for the genesis of this phenomenon have been suggested, as briefly discussed by Chandran et al. (1974). A comprehensive review of these theories can also be found in a paper by Bergel er a/. (1976), as well as Car0 (1977). Dean (1927,1928)described theeffectsofcentrifugal force in the production of secondary flow in fully developed steady viscous flow in tubes with small curvature. Numerical analyses of steady viscous fully developed flow in curved tubes have been reported by McConologue and Srivatsava (1968) and Greenspan l Received 24 August 1978. t Present address: Hemodynamics
Laboratory, Division of Materials Engineering, The University of Iowa, Iowa City, IA 52242, U.S.A.
Singh (1974) reported viscous
flow
on theoretical
analyses of steady
of uniform entry to discuss the flow development in the entrance length region of curved tubes. Agrawal et al. (1978) measured the velocity profiles in the entrance region of curved tubes and pointed out that the theoretical analyses of Yao and Singh are inaccurate, due possibly to the assumptions used in their solution procedure. Fully developed oscillatory viscous flow (sinusoidal with no mean flow) in tubes of small curvature has been the subject of works by Lyne (1971 J and Zalosh and Nelson (1973). While the effect of centrifugal force in steady flow in curved tubes was to shift the maximum axial velocity towards the outer wall of the curve, Lyne (1971) showed that in the case of oscillatory flow, the maximum axial velocity is shifted towards the inner wall of the curve. Chandran er ol. (1974) reported on the analysis of oscillatory viscous fully developed flow in elastic tubes of small curvature as a simulation of flow in curved arteries. They showed that in the cross-section of the bend, the secondary flow changed from a quadri-helical to a bi-helical pattern during one cycle, with the net eh%ctof shifting the maximum axial velocity towards the inner wall of the curve. More recently, Smith (1975) presented a theoretical analysis of unsteady viscous flow (sinusoidal with a non-zero mean) in curved tubes. It is worthwhile pointing out here that, except for the work of Chandran et al. (1974). all the other works cited above dealt with rigid curved tubes. Peronneau et trl. (1974) reported preliminary irl vitro studies of pulsatile flow (steady flow with sinusoidal oscillations), as well as irr cico studies in dogs.
793
K.
794
B. CHA.VDRAN, T. L. YEARWOODand D. W. WIETI~G
Fig. 1. Schematic of the mock-circulatory system.
Plane A
\
Plane8 \
PlaneC
PlaneD I
PimE /
Fig. 2. Schematic of the curved tube used in the experiments indicating the planes in which velocity measurements were obtained.
The nature of pulsatile flow in the entrance region of curved tubes (such as that occurring in the aortic arch) has not been well understood. Specifically, for the corresponding aortic arch in humans, the average radius is about 1.5 cm and the radius of curvature is about 4.5 cm with a radius-radius of curvature ratio t: of 0.33. Moreover, the flow is further complicated by an asymmetric entry condition, due to the presence of valves, as well as an irregular geometry with tapering and branchings at the bend. To delineate the effect of curvature alone in a puisatile flow with uniform entry, this paper reports an experimental analysis in a curved tube of regular geometry and a radius-radius of l
Vitamek, Inc., P.O. Bos 25359, Houston, TX 77005,
U.S.A.
curvature ratio of 0.1. Qualitative studies were conducted through a flow visualization technique and quantitative measurements of the velocity profiles at various cross-sections in the curve were made by hotfilm anemometry. A lower value of c was chosen for the curved tube in this study rather than the approximate value mentioned for the aortic arch (c = 0.33), for the following reason. Even though we only measured the axial velocity component in this study, we are planning on using a three-dimensional probe to study the radial and tangential velocity components subsequently. It was felt that for smaller values of r7 the radial and tangential velocity components wiIl not be of the same order of magnitude as the axial velocity component and thus the results will be relatively easier to interpret. Better understanding of the nature of the secondary flow in this region will be valuable in interpreting the results in tubes with larger curvatures in the future. The experiments with three-dimensional probes both in the curved tube and in a model of the aortic arch are, at present, being performed in our laboratory. EXPERIMENTAL
METHODS
The experiments were, conducted by using a mockcirculatory system previously described by Wieting (1968, 1969) and fabricated by Vitamek, Inc.* The basic components of this system are described in the schematic diagram of Fig. 1. Briefly, this system provides a pulsatile flow for a blood-analog fluid using
Fig. 3, Flow visualization
photo_eraph
during
systole for the flow straightener.
Fig. 1. Flow visualization
photograph
during
diastole
795
for the How straightener.
Fig. 8. Flow visualization photograph in the curved tube in systole. Fig. 9. Flow visualization photograph in the curved tube in diastole.
796
An experimental study of pulsatile flow in a curved tube rates from zero to 199 beats per mm, mean flow rates from zero to 10 I. per min, a systolic duration from zero to 300 msec, and good approximations of physiological wave forms in the aorta, left ventricle and left atrium. A curved tube test section as illustrated in Fig. 2 was machined out of two blocks of plexiglas? and cemented together. The radius of the tube a is 1.5 cm and the radius of curvature R is lScm, with the resulting radius-radius of curvature ratio ofO.1. To remove the effect of any flow disturbance created by the aortic valve as the flow comes out of the aortic valve chamber, a 6-in. flow straightener was mounted in between the aortic valve and the curved tube. This length was chosen on the basis of previous studies by Wieting (1969) in which it was determined that the disturbances caused by a human homograft aortic valve and most prosthetic aortic valves were dissipated by a point 6 in. downstream (distal) to the valve. Typical flow visualization pictures in the flow straightener during systole and diastole are shown in Figs. 3 and 4. respectively. All the studies reported in this paper have been conducted with the flow straightener in the system and a Gott Butterfly aortic valve.* The Gott valve was chosen because it has-no measurable back flow, and the disturbances caused by the valve can be minimized through appropriate use of the flow straightener. The mock-circulatory system is primed with a blood analog fluid composed of 529, glycerol in water, which at 37°C has a viscosity of 0.04 poise. The system is tuned to provide a mean flow rate of 5.1 I./min at a rate of 72 beats per min and a systolic duration of 270 msec. Simulated blood pressure in the ventricular chamber is measured with a Statham P23Ht pressure transducer and the time dependent flow rate is measured by means of a Statham Model M4001 electromagnetic flow meter: and an In Co Metric Model K26 mm cannulation-type electromagnetic flow probe: placed immediately downstream from the curved tube. The flow visualization technique used in the qualitative analysis has been previously described by Wieting (1969). Briefly, the technique employs neutrally buoyant white spherical plastic beads suspended in the blood-analog fluid and illuminated with a specially designed slit light. Flow visualization and. photography are performed perpendicular to the plane of the slit light beam in the center plane of the curved tube. Still photographs are taken throughout the cycle using a Nikon FZSB 35 mm camera equipped with a motor Prosthetic heart valve donated to Dr. D. W. Wieting by Vincent Gott, M.D., Dept. of Surgery, Johns-Hopkins University, Baltimore, MD 21204. t Statham Inst.-Physiological Div., 2230 Statham Blvd., l
Oxnard, CA 93030. : In Vioo Metric Systems, P.O. Box 217, Redwood Valley, CA 95470. 8 Red-Lake, 2991 Corvin Drive, Santa Clara, CA 95051. 1’Thermo-Systems, Inc., P.O. Box 3394, St. Paul, MN 55165.
797
drive and a Nikkor 50 mm 1: 1.4 lens using a range of shutter speeds between l/8 and 116Osec. High-speed movies are taken with a Redlake Hycam 16 mm camera4 equipped with a Switar 25 mm 1: 1.4 lens at 100 frames per sec. After the flow pattern studies were completed, small diameter holes were drilled in the central plane of the curved tube at five symmetric stations, A-E, as illustrated in Fig. 2. These holes allowed the insertion of the hot-film velocity probe across the entire diameter of the tube. The velocity profiles at the crosssections A-E were measured at different times in a cycle using a TSI” Model 1056 constant-temperature anemometer. The hot-film velocity probe was calibrated in a steady flow system at 37°C under, Poiseuille flow, and the calibration was linearized using a TSI Model 1052 41hOrder Polynomial Linearizer. A TSI Model 1210-20 W hot-film probe is suitable for these studies. The hot-film sensor was mounted on support pins 12.7 mm away from the end of a cyclindrical shaft 3.2 mm in diameter. The probe shaft was in the plane of the cross-section of the curved tube and the hot-film sensor and its supports projected from the end of the shaft. The projected area of the probe shaft was less than 104; of the area of cross-section of the tube when the probe was inserted to the maximum radial depth for measurement. During forward or reverse flows, the probe shaft was not interfering with the fluid flow over the probe itself. The probe was inserted through the hole and positioned so that the hot-film probe was perpendicular to the axial flow direction. Even though the axial velocity component is the predominant component of flow going over the probe, the probe actually measures the vector sum of the axial and radial flow components in the central plane of the curved tube and will introduce some error in the measured values. After the study reported in this paper, we have employed a three-dimensional probe by Thermo-Systems, Inc. to determine the three velocity components in the curved tube. Some of the preliminary measurements with the three-dimensional probe in steady flow show that for the curved tube under consideration (with a radius-radius of curvature ratio ofO.l), the radial velocity component was, at the most, 10yOof the axial velocity. We conservatively estimate that our axial velocity measurements are accurate to within 15% of the measured value due to the factors discussed above. DATAREDUCTION The flow visualization and hot-film anemometry techniques were used in a complementary fashion in order to obtain the final results. Analysis of the highspeed movie data showed that streamlined flow existed during the forward flow phase (systole) of the cycle. However, during the period in which theaortic valve in the system was closed (diastole), a reversed flow was observed along the inner wall of the curve, while along the outer wall of the curve, a forward flow of small
798
K. 8. CHAMXAN,T. L. YEAFSQOLIand D. W. WETIXG
magnitude was still observed. Thus, the flow visualization technique afforded us a qualitative analysis of the flow. An attempt to quantify the magnitudes of the velocity by a frame-by-frame analysis of the high-speed movies, however, yielded very inaccurate results. The results were quantitatively inaccurate for two reasons. First, particles tended to appear and disappear, moving in and out of the plane of illumination due to the obvious three-dimensional character of the flow. Second, the time lapse between each frame was at least 10msec and thus severely limited the precision to which instantaneous velocities could be estimated by carefully measuring particle displacements from frame-to-frame. Hot-film anemometry was then used to provide a quantitative basis for the qualitative data obtained by the flow visualization technique. It should be noted that the hot-film anemometer technique is insensitive to the direction of flow, since a flow in the reverse direction also produces a d.c. voltage signal output similar to that of the forward flow. Hence, the flow visualization technique was valuable in interpreting the data from the hot-film
anemometer, since the signal output from the anemometer appears to be rectified if flow reversal occurs, The first derivative of that signal will contain an appropriate discontinuity corresponding to .the inflection point of the output signal. Ideally, the inflection point for the velocities would occur at a zero voltage reading. However, under the unsteady flow conditions, a zero flow measurement is not likely to be obtained, due to the local disturbances of flow in the vicinity of the probe during flow reversal. Figure 5(a) shows the voltage-time signal output of the anemometer probe for a typical flow cycle at a point very near the inside wall of the curved tube. It is known that reversed flow occurs during that portion of the cycle indicated by the positive sign, An inspection of the first differential of the bridge output curve in Fig. 5(b) shows a dramatic discontinuity at the closing of the valve when the fluid acceleration is most negative. This curve is obtained by a first order finite difference approximation of the voltage-time signal output curve. It is surmised that flow reversal has occurred at this point in the cycle, at approximately 330 msec. In addition, another flow reversal must take place when forward flow is re:established at thecommencement of the next cycle. Inspection of Fig. 5(a) again indicates that a rapid rise in fluid acceleration has occurred just
-
Fig. 5. Anemometer signal output (a), the first derivative of the signal (b) and the velocity-time curve for a point near the inner wall (c) of the curved tube.
(bl
Fig. 6. Anemometer signal output (a). the first derivative of the signal (b) and the velocity-time curve for a point near the outer wall of the curved tube.
799
An experimental study of puisatile Row in a curved tube
Table 1. t1
[J
[I
rs
(6
r-
fa
(9
fl0
9 10
1.6 2.0 2.0 1.8 2.2 2.1 1.8 2.1 2.0 2.1
4.5 4.4 4.4 4.8 4.7 4.6 4.0 4.8 4.4 5.0
4.9 5.2 5.0 -7 ::; 5.0 5.2 5.0 5.2 5.2
1.5 1.2 1.6 2.0 2.0 1.8 2.4 2.0 2.0 2.6
3.3 2.5 2.6 3.0 3.2 2.0 2.6 2.8 3.1 2.2
2.2 1.6 1.8 1.6 2.0 2.4 0.8 2.2 2.4 1.9
2.2 1.9 1.4 1.2 1.2 2.0 2.2 2.3 2.8 2.2
1.9 2.0 1.8 1.0 1.2 1.9 1.6 1.6 1.6 2.0
1.9 1.4 1.2 0.8 1.4 1.g 1.6 1.2 1.2 1.8
1.4 1.6 1.2 0.9 0.8 1.4 1.8 1.6 1.2 1.2
Average S.D.
1.97 0.18
4.56 0.28
5.11 0.12
1.91 0.41
2.73 0.43
1.89 0.48
1.94 0.52
1.66 0.34
1.43 0.35
1.31 0.3 1
V,,
25.40
58.80
65.89
24.63
35.20
34.37
‘5.01
21.40
18.44
16.89
Cycle
2 4 5 6
8
Plane A, 0.60 cm from the inner wail. Velocity magnification factor = 12.894cm/set per V d.c. t, = ri = r3 = rr = rs = t6 r7 = ts =
time of initial rise of pressure in flow straightener, at I = 0.00 msec. time of peak in pressure (systole), approximately 67 msec. 100 msec later. inflection point of pressure curve as valve closes, approximately 297 msec iowest drop in pressure, approximately 345 msec. time of peak residual rise in pressure (diastole), approximately 427 msec. 100 msec later, approximately 527 msec. 100 msec later, approximately 627 msec. 727 msec. c9 = 100 msec later, approximately r10- 70 msec later, approximately 797 msec. after a mild dip at the end of the cycle (t = 830 msec,
approximately), as indicated by the arrow. This mild dip is very characteristic when reversed flow dominates the latter portion of the cycle, and it is not seen at other points in the tube where forward flow persists during the entire cycle, as seen in Figs. 6fa)and (b). The fluid velocity curve shown in Fig. 5(c) is then constructed using the calibration constants. In Figs. 6(a) and (b), the signal output and its first derivative are shown for a representative point near the outer wall of the curved tube, and a similar analysis is used in constructing the velocity profile seen in Fig. 6(c). Such an analysis for the direction of flow correlated very well with our qualitative analysis of the high-speed movie. Table 1 illustrates the anemometry data for a single point in the curved tube, one of ten such points in each of the five planes shown in Fig. 2. At each point, the d.c. voltage output of the anemometer for 10 successive cycles was recorded on a Dynograph* strip chart recorder, Model 5048, simultaneously with the pressure in the flow straightener by way of a fluid-filled tap located 3 in. downstream from the Gott valve. From each of the 10 cycles, 10 standardized time points were read from the strip chart recording of the voltage curve in accordance with specific characteristics of the pressure curve. These standard time points are shown in Fig. 7(b) on the pressure curve. Thus, the pressure curve served as a timing standard by which the velocities
at each location
about
the tube could
he
* Beckman Instruments, Inc., Offner-Division, 3900 River Road, Schiller Park, IL 60176.
compared and contrasted. The pressure signal was repeatable with negligible cycle-cycle variation to serve as the timing standard for the velocity measurements. The average voltage and the standard deviation was calculated over the 10 cycles for each time point as shown in Table 1, and the subsequent velocities were obtained by multiplication of the voltage by the calibration constant. For these experiments, the calibration constant was 12.9 cm/set per V d.c. As evident from Table 1, the standard deviation was relatively low compared to the measured voltage during the systolic duration of the pulsatile flow cycle. However, relatively large variations in the voltage signals from the mean were observed during the diastolic duration, even though the voltage signals at various cycles were qualitatively similar. This variation during diastole is attributed to local fluid disturbances around the probe, such as the random dissipation and decay of transient eddies and low frequency vortex shedding from the probe shaft expected at very low axial velocities. The standard deviation in velocity measurements are indicated in our measurements for one of the planes (in Fig.’ 10). RESULTS AND DISCUSSIONS
The flow parameters in the experimental set up are given below: Density of the fluid, p = 1.13 g/cm’; Viscosity, p = 0.04 poise; Frequency of the pulsatile flow, f1.20 set-’ ;
72 min-’
=
800
K. B. CHASXAS, T. L.
YEARWOOD
1
and
D. W. WIETIX
I
0
,400
I
,800
sec.
(0)
1
1
0
,400
I
,800
sec.
(bi
Fig. 7. Typical flow rate curve obtained from the electromagnetic flow meter (a) and a typical pressure signal from the flow straightener (b).
I,2
Womersley
parameter,
(>
IZ= u f
= 21.9.
The maximum flow rate QmXXand the average flow Q.,in a cycle were obtained from the electromagnetic flow meter. A flow rate vs time curve obtained from the flow meter along with a typical pressure signal from the flow straightener just up stream of the curved tube are shown in Figs. 7(a) and (b). The cross-sectional area A of the curved tube is 7.07cm2. Then the cross-sectional average axial velocity w, the Reynolds’ number (Re = 2p@u/p) and the Dean number D( = Re(a/R)“‘) are computed and the values are shown below: rate
Q(l/min)
w(cm/sec)
Re
D
18.1
42.4
3600
1140
Pulsatile flow Maximum during a cycle Time-averaged value in a cycle
5.1
12.0
1020
320
Steady flow
9
21.2
1800
570
Typical still photographs from the flow visualization study during systole and diastole are shown in Figs. 8 and 9, respectively. A careful analysis of the photographs as well as of the high-speed movies showed that during systole, fluid in the curved tube accelerated downstream. During diastole, however, recirculation occurred, with fluid particles near the outer wall of the curved tube continuing their motion downstream, and fluid particles near the inner wall moving in a retrograde fashion back upstream. This motion is not caused by back flow, since the Gott prosthetic aortic valve has no appreciable back flow (Fig. 7a). The magnitude of velocity at the inlet of the curved tube was not directly measured so as to avoid drilling holes in the flow straightener and thus introducing another source of possible flow disturbance before the fluid enters the curved tube. However, velocity measurements were made in another tube with the same geometry as the flow straightener at the same axial distance away from the valve and the velocity distributions at early and peak systole (0 msec and 67 msec in Fig. 7b) are shown in Fig. 10. The velocity distribution was observed to be symmetric with re-
An experimental study of pulsatile i3ow in a curved tube
Fig. 10. Velocity profile at the inlet of the curved tube: (a) early systole (0 msec): (b) peak systole (67 msec).
spect to the center of the tube, reasonably flat at the point of entry into the curved tube, with a variation of less than 10% between maximum and minimum magnitudes during peak systole. As is evident from Fig. 4, flow reversal occurred along the cross-section of the flow straightener during diastole and the quantification of the velocity profiles was not attempted in this section during diastole. PLANE
I
The velocity profiles obtained at Planes A, C and D during systole (125 msec in Fig. 7b) and during diastole (650 msec in Fig. 7b) are shown in Fig. 11. In Plane A. during systole, the maximum axial velocity (W) is observed near thl outer end of the curved tube. However, in Planes C and D, the maximum velocity is observed at the inner end with the skewing of the velocity more pronounced in Plane C. During diastole, a coherent reversed flow is observed along the inner radius ofthe curve with a slight forward flow along the outer radius. The magnitude of the reversed flow is seen to be a maximum in Plane C. The velocity profiles at Planes A, B, C, D and E during various times in the cycle are shown in Figs. 12-16, respectively. In these figures, velocity profiles of measurements due to steady flow in the curved tube at the same planes are also superposed in dotted lines for a direct comparison between steady and pulsatile flows. In the case of steady flow, the mock-circulation unit was replaced by a steady flow system where the blood-analog fluid was pumped by a centrifugal pump. The steady flow velocity profiles shown in the figures are for a flow rate of 9 l/min (with a corresponding Dean’s number of 569) compared to 5.1 limin (and 322) for the pulsatile flow. It is interesting lo note that the steady flow profiles and the pulsatile flow profiles
A
20
PLANE
1:
PLANE
C
PLANE
D
0 10
5
0
5
801
10
20
t
60
2ot
Fig. 11. Cross-sectional axial velocity profiles during systole (a) and diastole (b) in Planes A, C and D. The ordinates denote axial velocity in cm/set. In all the figure-s, left side is the outer wall and the right side is the inner wall of the tube.
802
K. 9.
CHANTIRAN,
T. L.
and D. W. WIETISG
YEARWooD
PLANE PLANE
C
100
A
t
b
2 50
-
VI
---_--______---N\
-30
-
-20
\OUTER
WALLV
t
Fig. 12. Cross-sectional axial velocity profiles at Plane A during different times in a cycle: (a) early systole (0 msec); (b) peak systole (67 msec); (c) mid-diastole (627 msec); (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 3.40 from the entrance.
PLANE
Fig. 14. Cross-sectional axial velocity profiles at Plane C during different times in a cycle: (a) early systole (0 msec): (b) peak systole (67 msec); (c) mid-diastole (627 msec): (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance - IO.20 from the entrance.
B
90 L
PLANE
0
1oc 80 -
9c EC
70 ti : 50 -
; 6C
____-----___
P-
$ 5c =
4c
c 0
30
ii >
20
: z Q OUTER -30
-
-40
-
WALL
Fig. 13. Cross-sectional axial velocity profiles at Plane B during different times in a cycle: (a) early systole (0 msec); (b) peak systole (67 msec); (c) mid-diastole (627 msec); (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 6.8a from the entrance.
\
WALL
0 -10
C
INNER
10
d
YgzJ$-g
-20 -30
C
Fig. 15. Cross-sectional axial velocity profiles at Plane D during different times in a cycle: (a) early systole (0 msec); (b) peak systol (67 msec); (c) mid-diastole (627 msec); (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 13.60 from the entrance.
An experimental study of pulsatile flow in a curved tube PLANE
E
Fig. 16. Cross-sectional axial velocity profiles at Plane E dcrinp different times in a cycle: (a) early systok (0 msec); (b) peak systole (67 msec); (c) mid-diastole (627 mscc): (d) late diastole (780 msec). Dotted line shows the steady flow axial velocity profile. Axial distance = 17.04 from the entrance.
during systole are relatively flat in Planes A and B. However, in Plane C, the velocity profiles for both cases have pronounced asymmetry, with the steady flow velocity magnitudes being higher towards the outer wall and the pulsatile flow velocity magnitudes being higher towards the inner wall. Lyne (1971) was the first to discuss the difference in secondary flows between steady and oscillatory flows through a theoretical analysis. He suggested that in the case of oscillatory flow, the viscous effects are confined to a region close to the wall and the inviscid core region tends to move the maximum axial velocity towards the inner wall. This effect was corroborated by Chandran et al. (1974) in analyzing oscillatory flow through curved elastic tubes. The present study confirms that a difference in the secondary flow is found between steady and pulsatile flows where the pulsatile flow is a combination of several modes of oscillatory flow superposed with a steady flow component. It is also interesting to note that in Plane D, the pulsatile flow profiles during systole become less skewed relative to those in Plane C. By Plane E, the skewing of the profile was shifted towards the outer wall. This phenomenon could be due to the fact that an inviscid core region such as that referred to by Lyne (1971) is not present in the downstream region of the tube. A recent work of part&&r concern to this problem is that by Singh et 01. (1978) in which they employed a boundary layer analysis to investigate the nature of pulsatile Row in the entrance to the aorta. As an expansion upon the steady entry flow in a curved tube of Singh (1974), they analytically examined the consequences of a pulsatile pressure gradient (sinusoidal with a non-zero mean) imposed upon flow in the entry region of a curved, rigid circular tube, with r. of the
803
same order of magnitude as the present experimental study. They showed that the fluid velocity and wall shear will initially be higher near the inner wall of the curved tube owing to the external flow and geometric factors. However, as the flow develops, they predict that the effect of curvature will be to increase the secondary flow of the boundary layer, drawing fluid azimuthally from the outer wall to the inner wall. As the fluid is drawn towards the inner wall, the boundary layer there would become thicker, while at the outer wall, the boundary layer would become thinner. If this is true, the wall shear and fluid velocity at the outer wall will become greater than at the inner wall as the flow develops further downstream. Furthermore, at this location downstream, their analysis. predicts the gradual development of a reversed flow near the inner wall as a result of the deceleration of the fluid at the close of the systolic phase. The quantitative precision of their analysis by comparison with our results are quite poor, possibly because they assume that axial flow comes to rest everywhere in the tube by the end of each cycle. Nevertheless, our results can be seen to be qualitatively in agreement with their analysis. The shifting of the velocity profiles between Planes C and E may very well be the result of the secondary flows as indicated by their analysis. Furthermore, their analysis of flow reversal along the inner wall as a result of fluid deceleration is qualitatively supported by our results for flow profiles during diastole. The results of Singh et al. (1978) suggest that an inviscid core of fluid would persist throughout our curved tube past Plane E. The problem ,of entry length in the tube under pulsatile conditions cannot be solved by our present results, however, without knowledge of the secondary flows. It ‘is not known if coupling of the inner and outer boundary layers, therefore, could take place by Plane E under the present experimental set-up. The velocity profiles reported by Peronneau er a/. (1974) agree qualitatively with the results reported in this work. In their in uioo experimentation in the aortic arch, they show reversed flow along the inner radius of the curve during peak reverse flows. They also show similar profiles in their in vitro studies. However, no direction quantitative comparison can be made between their results and ours because of the difference in tube geometry. Moreover, they used a sinusoidal flow superposed on a steady flow, whereas our enperimental set-up replicated a typical pulsatile flow similar to that produced in man by the left ventricle. Agrawal et al. (1978) have reported on the flow development in the entry region for steady viscous flow of uniform entry into curved tubes for Dean’s number ranging from 139 to 565. Their results show that immediately after entry into the curved tube, the maximum axial velocity is measured towards the inner wall and as the flow devefops, the maximum axial velocity progressively shifts towards the outer wall in the downstream region. This effect of skewing towards the inner wall in the upstream region was more pronounced at lower Dean’s numbers (at about 180) at
804
K. B. CHASDRAN,
T.
L.
axial distances of 2-12 times the radius of the tube. At higher Dean’s numbers (at about 565), the velocity profiles in the upstream region were observed to be relatively flat before skewing towards the outer wall in the downstream region. Our results for steady flow (Dean number = 569) show a similar trend, with the profiles being relatively flat in Planes A and B (axial distances of 3.4 and 6.8 times the radius of the tube) before skewing. towards the outer wall in the downstream region (Planes C, D and E). However, in the case of pulsatile flows, the axial velocity distribution shows a trend opposite to that for steady flows, possibly due to the reasons discussed by Lyne (1971) and Chandran et ol. (1974). In the pulsatile flow situation with a time averaged Dean number of 322, the maximum axial velocity magnitude was observed towards the outer wall in Plane A (axial distance = 3.40), followed by a progressive shift of the maximum axial velocity towards the inner wall of the curved tube in Planes B and C (axial distances of 6.8~ and 10.2~. respectively). It is also interesting to note that in Planes D and E (axial distances of 13.6~ and 17a, respectively), the maximum axial velocity is seen to be shifting towards the outer wall once again. During diastole, however, a reversed flow was observed towards the inner wall and a slight forward flow towards the outer wall in all the planes. The maximum values of reversed flow velocity magnitudes were observed in Plane C at the top of the curve. Even though an attempt was not made to compute the wall shear stress distribution in a cross-section from these velocity measurements, it is evident from Figs. 12-16 that the higher magnitudes of wall shear are, in general, observed at the inner wall of the bend, due to pulsatile flow in a curved tube. Moreover, a large variation in the magnitude and direction of the shear stress is also observed in this region compared to the outer wall of the bend, due to the flow reversal during a cycle measured towards the inner wall. Several investigators have proposed that the wall shear stress distribution dueto pulsatile blood flow plays a vital role in atherogenesis. Fry (1968) has demonstrated that abnormally high shear stresses damage the endothelial lining and suggested that these are precisely the sites for lipid deposition. On the other hand, Caro et al. (1971) postulated a shear dependent mass transfer mechanism for atherogenesis and suggested that lipid deposition occurs where the wall shear stresses are relatively low. They correlated their postulation with observed incidence of atherosclerosis in the inner end of the curve such as in the aortic arch. Their arguments are based on steady flow qualitative study in a model of the aortic arch where low wall shear stresses are expected at the inner end of the curvature. In contrast, however, the present study on pulsatile flow in curved tube shows that higher magnitudes of wall shear stress are observed towards the inner wail of the bend. Moreover, the magnitudes of the wail shear near the inner wall fluctuate between a maximum in one direction to the other during a cycle,
YEARWOOD
and D. W. WIETISG
due to the flow reversal observed in diastole. As suggested more recently by Caro (1977), the effect of such large fluctuations in wall shear due to pulsatile flows around a curve, as well as in bifurcation and branching sites on the endothelial lining, need to be investigated to improve our understanding of the mechanical effects of blood flow in atherogenesis. We chose a curved tube with a radius-radius of curvature ratio of 0.1 with an aim to understand the nature of the secondary Row in a model of the aortic arch. The study in the aortic arch will be further complicated by the increase in curvature, irregular cross-section of the lumen, tapering and branching of arteries at the top of the curve. Effects of each of these complications need to be lineated before a better understanding on the fluid dynamic effects on atherogenesis can be obtained. At present, work is proceeding in our laboratory with a three-dimensional hot-film probe to understand the nature of the development of the secondary flows both in the curved tube and in the model of the aortic arch. The flow visualization technique used in the present study has a disadvantage in that it does not illustrate the nature of the secondary flow perpendicular to the plane of illumination. Nevertheless, this technique was useful in interpreting the results from the hot-film anemometry measurements.
Acknowledgement
-The support of this work by a grant from National Institutes of Health (NHLBI-HL18156-01-02) is gratefully acknowledged.
REFERENCES Agrawal, Y. C., Talbot, L. and Gong, K. (1978) Laser anemometer study of flow development in curved circular pipes. J. Fluid Mech. 85, 497-518.
Bergel, D. H., Nerem, R. M. and Schwartz, C. J. (1976) Fluid dynamic aspects of arterial disease. Atherosclerosis 23, 253-261. Caro, C. G.. Fitzgerald, J. M. and Schroter, R. C. (1971) Atheroma and arterial wall shear observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. R. Sot. (B) 177, 109-159. Care, C. G. (1977) Mechanical factors in atherogenesis. Cardiovascular Flow Dynamics and Measurements (Edited by Hwang, N. H. C. and Normann, N. A.). Chapter 13, p. 473. University Park Press, Baltimore. Chandran, K. B., Swanson, W. M., Ghista, D. N. and Vayo, H. W. (1974) Oscillatory flow in thin-walled curved elastic tubes. Ann biomed. Etqng 2, 392-412. Dean, W. R. (1927) Note on the motion of fluid in a curved pipe. Phil. Mug. (S) 7, 4, 20, 208-223. Dean. W. R. t 1928) The steamline motion of fluid in a curved pi&. Phil. ‘Mug: (S) 7, 5, 30, 673-693. Fry, D. L. (1968) Acute vascular endothelial changes associated with increased blood velocity gradients. Circulation Res. 22, 165-197. Greenspan, D. (1973) Secondary flow in a curved tube. /. fluid Mech. 57, 167- 176. Lyne, W. H. (1971) Unsteady viscous flow in a curved pipe. J. Fluid Mech. 45, 13-31. McConalogue, D. J. and Srivastava, R. S. (1968) Motion of a fluid in a curved oiw. Proc. R. Sot. (Al 307. 37-53. Peronneau, P., Hi&& J., Xhaard, I$ ‘Deliuche, P. and Philippo, J. (1974)The effectsof curvature and stenosis on
An experimental study of pulsatile flow in a curved tube pulsatile Row in uiro and in ritro. Cardiocascular Applied Ultrasound Vol. I, pp. 203-215. North Holland, Amsterdam. Singh, M. P. (1974) Entry flow in a curved pipe. _I.Fluid Meth. 65, 517-539. Singh, M. P., Sinha. P. C. and Aggarwal, M. (1978) Flow in the entrance of the aorta. J. Fluid Mech. 87, 97-120. Smith, F. T. (1975) Pulsatile flow in curved pipe. J. Fluid Mech. 71, 15-42. Texon, M. (1960) The hemodynamic concept of atherosclerosis. Bull. N.Y. Acad. &fed. 36, 268.
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Wieting, D. W. (1968) A method for analyzing the dynamic flow characteristics of prosthetic heart valves. ctS,M15 Paper No. 6%WA/BHF-3. Wieting. D. W. (1969) Dynamic Row characteristics of heart valves. Ph.D. Dissertation. University of Texas. Austin. Texas. Yao, L. S. and Berger. S. A. (1975) Entry flow in a curved pipe. J. Ffuid Mech. 67, 177-196. Zalosh, R. G. and Nelson, W. G. (1973) Pulsating flow in curved tube. J. Fluid ,Uech. 59, 693-705.