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Initiation of turbulence in models of arterial stenoses
J. tliomuchmirn Vol. 12. pp. 185 196. Pergamon Press. 1979. Printed in Great Britam
INITIATION
OF TURBULENCE IN MODELS OF ARTERIAL STENOSES*
WITHAYA YONGCHAREONand DONALD F. YOUNG
Department of Engineering Science and Mechanics, Iowa State University, Ames, IA 50011, U.S.A. Abstract - The development of turbulence under both steady and pulsatile flow through models of arterial
stenoses was studied experimentally. Stenoses were represented by three severely constricted rigid-walled models with different shapes. Model geometries included a streamlined shape, a hollowed plug with blunt ends, and a thin plate orifice. Velocity fluctuations were measured with hot-film probes. Results indicate that: (a) turbulence develops at Reynolds numbers well below the critical value for flow in an unobstructed tube; (b) the critical Reynolds number varies with a dimensionless frequency parameter, first becoming less stable and then more stable as the frequency parameter is increased ; and (c) the critical Reynolds number depends on the shape of the obstruction with the orifice-type stenosis exhibiting the lowest value for the critical Reynolds
INTRODUCTION
As a stenosis develops in an artery, the nature of the flow in the vicinity of the stenosis is significantly altered. The degree of alteration is strongly influenced by the severity of the stenosis, and for moderate and severe stenoses, highly disturbed regions of flow distal to the stenosis commonly occur under physiological flow conditions. These regions of disturbed flow may remain laminar in the sense that even though the flow patterns are complex and time-dependent, they are essentially periodic and reproducible. However, it is generally found that as the stenosis increases in severity, a condition will be reached for which the flow patterns become unstable and irregular velocity fluctuations characteristic of turbulent flow ensue, at least over a part of the flow cycle. The development of turbulence is an important cdnsideration since turbulence represents an abnormal flow condition in the circulation and is related to several important clinical problems such as poststenotic dilatation (Roach, 1972), phonoangiography (Lees and Dewey, 1970), and effect on pressure losses and thus regional blood flow (Young and Tsai, 1973a, b; Young et al., 1977). Little information is available with regard to the conditions under which the flow will become turbulent when disturbed by the presence of an obstruction. A few studies involving the development of turbulence in steady flow through stenoses have been reported (Young and Tsai, 1973a; Kim and Corcoran, 1974; Azuma and Fukushima, 1976). Although there are variations in the reported values for the critical Reynolds numbers for the development of turbulence, the results do clearly show that turbulence will develop at relatively small Reynolds numbers (compared with
* Revised 23 June 1978.
the traditional
value of 2000 for pipe flow), and the
critical Reynolds number decreases as the severity of the stenosis increases. The problem becomes considerably more complex
for pulsatile flow. A few transition studies involving unsteady flows in unobstructed tubes have been reported (Sarpkaya, 1966; Sergeev, 1966; Yellin, 1966; Nerem and Seed, 1972; Hino et al., 1976), and these studies indicate that the unsteadiness can affect the transition process. Several recent studies have been concerned with the characteristics of the developed turbulence distal to a stenosis (see, e.g. Kim and Corcoran, 1974; Giddens et al., 1976; Tobin and Chang, 1976; Clark, 1976, 1977; Pitts and Dewey, 1977); however, there do not appear to be any reported systematic studies of the initial development of turbulence in the presence of a stenosis under pulsatile flow conditions although a few observations have been made (Sacks et al., 1971; Young and Tsai, 1973b). The present study was undertaken to investigate experimentally, using an in vitro model system, the conditions under which pulsatile flow through a stenosis becomes turbulent. METHODS
For a given experiment, a rigid-walled model was located in a flow system in which either a steady or pulsatile flow could be generated. Three severely constricted models (89% area reduction) were used (see Fig. 1). One model had a streamlined shape in the form of a cosine curve (model S-2), the second model was a hollowed plug with blunt ends (model P-2), and the third model was a sharp-edged orifice (model O-2). For models S-2 and P-2, the length Z0 was four times the radius R,. The radius of the unobstructed test section R, was 9.525 mm. The fluid used for all tests was a saline-glycerol
186
WITHAYA YONGCHAREON
(a)
(b)
(Cl
Fig. 1. Model geometries: (a) streamlined model (S-2), (b) blunt end plug (P-2), (c) orifice (O-2).
mixture, with different proportions of saline to glycerol used to obtain various viscosities (for more details, see Yongchareon, 1977). Steady flow was created with a constant head tank, whereas pulsatile flow was generated by a one-way valve and a pulsator which consisted of a piston-cylinder combination driven by a scotch-yoke mechanism (see Fig. 2). The one-way valve was located upstream from the pulsator and was arranged so that it blocked the flow when the piston moved against the flow direction. Since the piston oscillated harmonically, the flow waveform was approximately a half sine-wave over part of the cycle with zero flow over the remainder. Both the frequency and stroke could be varied, and the flow rate was controlled by the combination of the stroke and an orifice located at the flow outlet. The instantaneous flow rate was obtained with an in-line electromagnetic flowmeter system (Biotronex Model BL-610). The main conduit in the test facility was a brass tube and the ends of the model sections were bored in such a manner that the brass tube to model connections provided a flow conduit of constant diameter except in the region containing the obstruction. The distances
and DONALD F.YOIJNG
between the electromagnetic flowmeter and the test sections and between the test sections and the constant head tank were 1.25 and 1.85m, respectively. These distances were sufficient to assure that entrance-length effects would be negligible. The instrumentation for velocity measurements consisted of the following : cylindrical hot-film probes (Thermo-system Mode1 1270-2OW), anemometer (DISA Type 55DOl), auxillary unit (DISA Type 55D25) used as a high pass filter, linearizer (DISA Type 55DlO), RMS voltmeter (DISA Type 55D25) and two digital dc. voltmeters (DISA Type 55D30). Instantaneous flow data were obtained from oscillographs, and all data were also recorded on magnetic tape (Tandberg Series 115). The block diagram for the instrumentation used for velocity measurements is illustrated in Fig. 3. For all measurements, the sensing element of the hot-film probe was perpendicular to the tube axis. The hot-film probe could be inserted into the flow at six axial locations distal to the stenosis (Z/D = 2,4, 6, 8, 10, 12). Radial traverses could also be made. Proximal to the stenosis, point velocities were measured in a straight test section (Fig. 2).
EXPERIMENTALRESULTS
Steady ,pow The flow characteristics distal to the stenosis exhibit a strongly non-uniform spatial distribution. As the fluid flows through the constriction, a jet-like flow develops in the throat section and the main stream separates from the boundary at very low Reynolds numbers. For example, separation occurs at a Reynolds number of approximately 10 for the streamlined model S-2, for the Reynolds number Re = DU/v, with D the unobstructed tube diameter, U the mean velocity, and v the fluid kinematic viscosity. With a continued increase in Reynolds number, the separated region will extend several diameters downstream with TEMPERATURE CONTROLLER CONSTANT HEAD TANK
OVER FLOW ORIFICE VALVE;(??rai
. (VALVE
PULSATOR
FLOW .
*
I
n
STRAIGHT SECTION
Fig. 2. Schematic of experimental apparatus.
Turbulence in arterial stenoses
187
FROM HOT-FILM PROBE
I
I
AUXILIARY UNIT
I
Fig. 3. Block diagram of instrumentation
the reattachment point also moving downstream (see, e.g. Johansen, 1929 ; Young and Tsai, 1973a ; Back and Roschke, 1972). With a still further increase in Reynolds number localized velocities distal to the stenosis become time-dependent and relatively low frequency undulations can be detected with the hot-film probe. Eventually, the velocity fluctuations become irregular in appearance and the output from the probe fluctuates randomly with the signal containing high frequency components. At a given location distal to the stenosis, the intensity ratio lJ,,,/f? varies with Reynolds number as shown in Fig. 4, where U,,, is the root-mean-square value of the point velocity as obtained from the linearized output signal from the hotfilm probe, and C?is the local mean velocity at the same point. The Cl,,, value is zero for sufficiently small Reynolds numbers but increases rapidly once the disturbed flow is initiated, reaches a peak with a subsequent decrease to a relatively low level. It is noted from Fig. 4 that the intensity ratio is not strongly dependent on the radial position of the probe when
system.
near the wall, but there is dependence on radial position when comparing centerline readings with readings near the wall. Figure 4 shows that the relative intensity of the flow disturbances at a point is a strong function of the Reynolds number. Also, the relative intensity near the wall is higher than at the tube centerline. Disturbed flow near the wall was first detected at a lower Reynolds number than the Reynolds number which produces disturbances at the tube centerline. This phenomenon suggests that the flow first becomes unstable in the region of the shear layer bounding the jet emerging from the throat of the obstruction. The variation of relative intensity with Reynolds numbers at various axial positions, but the same radial position R/R,= 0.9, is shown in Fig. 5. For clarity, only smooth curves drawn in by hand are shown, and the actual data points are omitted. All curves have the same general shape, although at the near field location Z/D = 4, the curve shows a more rapid increase in the intensity after the peak value is attained. The exact
0.8‘
STEADY FLOW 0.6-
0.00 77
oiODELs-z
0.75 0.90
200
300
400
600 500 REYNOLDS NUMBER, Re
700
800
Fig. 4. Variation in relative turbulence intensity with Reynolds number at various radial positions for Z/D = 12.
188
WITHAYA YONGCHAREON and DONALD F. YOUNG
STEADY FLOW
0.8-
MODEL S-Z
0.6lb , ,? d D.4;1 z c z = P 0.2?z
0.0
..
----_______---
200
300
400
500
600
700
800
900
* 1000
REYNOLDS NUMBER, Re
Fig. 5. Variation in relative turbulence intensity with Reynolds number at various axial locations for R/R, = 0.9.
reason for this behavior is unknown. At Z/D = 12, where the disturbances were first detected, the relative intensity reaches a maximum value more rapidly than it does at the other locations. It is seen that disturbances first develop at Z/D = 12 and move upstream as the Reynolds number increases. For models P-2 and O-2, only a limited amount of steady flow data were obtained, but the general trends were the same as found for model S-2. For a given geometry, transition from laminar to turbulent flow under steady flow conditions is determined by the critical Reynolds number Re*, and in this study, the relative intensity tJ,,$~ was used as a criterion for obtaining the critical Reynolds number. For flow through complex configurations such as stenoses, it is difficult to characterize a given flow as laminar or turbulent. As suggested by Robertson and Herrick (1973, three basic flow types need to be considered (see Table 1). Luminar flows can be either steady or unsteady, but if they are unsteady the flow
Table 1 Type
characteristics can be described by well-behaved functions of time. Turbulent flow is characterized by irregular (random), three-dimensional fluctuations in the pertinent flow characteristics. There is, of course, no sudden transition from the laminar to the turbulent state but rather there will be an intermediate state in which perturbations to the main flow develop, but the perturbations are not fully irregular. These flows may be classified as disturbed flows. To further complicate matters, turbulent flows may also be transient in the sense that they may decay in space or time. Thus, the turbulence which develops distal to a stenosis will be transient for both steady flow (variation with axial distance) and unsteady flow (variation with both distance and time). For the purpose of the present study, the desired critical Reynolds number was taken to be the value which delineated the latter stage of the disturbed flow regime and the early stage of the turbulent flow regime. Values of the critical Reynolds numbers could be
Basic flow types (from Robertson and Herrick, 1975) Nature
Description
1
Laminar
Orderly flow in smooth (streamlined) layers or laminae
2
Disturbed viscous
Flow with perturbations having some regularity in time and space
3
Turbulent
Flow with irregular (random, chaotic) temporal and spatial disturbances of velocity and pressure
(a) Permanent
Stationary (in the statistical sense) turbulence
(b) Transient
Turbulence which is decaying in space or time
Turbulence
in arterial
stenoses
Fig. 7. Typical experimental oscillographs. recording of point velocity at R/R, = 0.75 in unobstructed section of test section located stream from stenosis. Instantaneous volume rate of flow as measured by electromagnetic flowmeter. (cl Hot-film recording of point velocity at R/R, = 0.75,2/D= 8 (distal to stenosis) with output fil tered to elirninate frequencies below 50 Hz. Z/D = 8 (distal to stenosis). (d) Hot-film recording of point velocity at R/R, = 0.75,
(a) Hot-film
z
Turbulence in arterial stenoses estimated from observations made of the instantaneous velocity variations on an oscilloscope screen. Observations obtained in this manner are of course subjective to some extent. It was found that random fluctuations (as opposed to the more regular undulations) appeared to occur when the intensity ratio exceeded a value of approximately 0.3. The value of 0.3 is arbitrary but was found to be a useful criterion for distinguishing the disturbed flow regime from the early stages of turbulence. Therefore, in an attempt to reduce the subjective nature of the measurement, the average of the two measured Reynolds numbers which bracketed this value of 0.3 was taken as the critical Reynolds number. The upper value corresponded to the clearly observed early stages of turbulence, and the lower value corresponded to the mild intermittent disturbances. With this criterion, the critical Reynolds number was found to be 250 + 15 for the orifice, 295 + 16 for the plug and 328 f 18 for the streamlined obstruction. The plus and minus values incorporates both the range on the bracketed values of the critical Reynolds number and the uncertainty due to measurement errors. As expected, the flow through the orifice is the least stable since the velocity profile at the orifice exit is relatively flat, which provides a high velocity gradient at the separated shear layers and hence, a highly unstable flow. It is not obvious that the boundary layer at the separated section of the streamlined obstruction is thicker than the plug-like obstruction. Nevertheless, the separation point is downstream from the throat of the streamlined obstruction, whereas the separation point is fixed at the throat (the discontinuity section) for the plug-like model. Thus, the separated shear layers for the streamlined obstruction are closer to the wall than the pluglike obstruction, and the wall may tend to damp out small disturbances that would be amplified in the separated shear layer. As the result, the streamlined obstruction provides a more stable flow than the pluglike obstruction. Flow conditions immediately upstream from the stenosis were not studied in detail, but it is apparent from the above discussion that upstream flow patterns, which vary with stenosis shape, will affect the transition process. As seen in Fig. 5, the relative turbulence intensity at Z/D = 12 is significantly greater than zero at a Reynolds number well below those required to develop turbulence at other locations. Thus, turbulence appeared to occur first at Z/D = 12. The distance from the throat of the obstruction to the axial location where turbulence was first observed is denoted as the critical length Z*. The normalized critical lengths, Z*/D. for three geometries - the orifice, the plug and the streamlined obstruction - were found to be 8, 16 and 12. respectively. Pulsatile
used in the pulsatile-flow system described in the Methods Section. Since arterial blood flow is known to be highly pulsatile with a typical ratio of peak-to-mean velocity in the range of 2-3 (Young et al., 1977), and with low velocities over part of the flow cycle, the base flow pulse shown in Fig. 6 was used for pulsatile flow studies. For all tests, the ratio T,/T was held constant at a value of approximately 0.6 which gives a corresponding constant value of 2.6 for the ratio of the peak-tomean velocity, UP/U. For pulsatile flow, the development of turbulence for a given stenosis geometry will be a function not only of the Reynolds number but also of U&l and a frequency parameter, LX,.For the present study, the frequency parameter used was Ro,,/G. Since UP/U was held constant, the critical Reynolds number depended only on the frequency parameter for a given stenosis. It is to be noted that a commonly used frequency parameter is the so-called “alpha parameter” where CL= R,m so that CL,= fi a. For the flow pulse studied, the local inertial effects which affect the transition phenomenon depend on Tl so that the parameter a, seems an appropriate index for characterizing this effect, although for a fixed ratio, TJT, the parameter o! could be used. For a given value of a,, point velocity measurements were made at several locations distal to the stenosis with R/R,= 0.75, and the value of the peak Reynolds number could be varied over a wide range. For the pulsatile-flow tests, the peak Reynolds number is based on the peak cross-sectional average velocity CT, in the unobstructed tube, i.e. Re, = U,D/v. As was the case for steady flow, three basic flow regimes could be observed : (1) laminar flow over the entire flow cycle ; (2) disturbed flow over a part of the cycle with large scale, relatively low frequency, velocity fluctuations, and (3) transient turbulence which was first initiated near the peak of the flow cycle with the disturbances damped out before the next cycle started. Typical oscillograms of the basic flow pulse and the instantaneous point velocity are shown in Fig. 7. The critical peak Reynolds number, Re,*. was again taken to be the value which delineated the latter stages of the disturbed flow regime and the early stage oft he turbulent flow regime. The problem is complicated by
flow
To investigate the effect of unsteadiness of the base flow on the development of turbulence, the same stenosis models used in the steady-flow studies were
1.91
Fig. 6. Base-flow pulsatile waveform.
192
WITHAYA YONGCHAREONand DONALD F. YOUNG
the presence of the unsteady base flow which by itself will give relatively large values of the intensity ratio U,,,/o even for the laminar regime. In turbulent pulsatile flow, the intensity ratio contains contributions from both the unsteady base flow and any random fluctuating velocity characterizing turbulence. If the base flow waveform in the turbulent flow region distal to the stenosis was the same as that in the proximal laminar flow, the contribution to U,,,.ii due to the base unsteady velocity could be obtained from measurements obtained in the unobstructed section upstream from the obstruction. Thus, the difference between V,,Jii for the distal turbulent and proximal laminar flows would give an indication of the degree of turbulence. Unfortunately, in the region close to an obstruction, the base velocity waveform is different from that in the unobstructed tube, and therefore, the difference between proximal and distal intensity ratios is not a good index of turbulence. The contribution of the base flow can be reduced by using a high pass filter which ideally allows only frequencies higher than the cut-off frequency to pass. The difficulty, however, is the loss of low frequencies of the disturbed flow. After considerable experimentation, it was found that the most reliable index for determining the critical Reynolds number was the value of the difference between Urm$Up measured by a hot-film probe located at the distal location of interest, and Urms/UP measured proximal to the stenosis, with the low frequency components eliminated at both locations with a highpass filter. The cut-off frequency was chosen in such a way that the II,,,, after filtering out the low frequency components, was approximately l”/;, of the peak velocity of the proximal laminar waveform. In these experiments, the cut-off frequency was set at 50 Hz. A typical plot of the intensity difference (described
MODEL
100
n
above) vs the peak Reynolds number, Re,, is shown in Fig. 8. Even though some low frequency components of turbulence were eliminated, the general trend of curves is the same as that found in the steady flow (Fig. 5). In Fig. 8, the arrow indicates the Reynolds number at which the initial stages of the turbulent waveform was observed. At a lower Reynolds number, the value of the intensity difference is due in part to the change in the proximal and distal base flow waveforms. From a visual inspection of all the hot-film signals as observed on an oscilloscope screen, it was determined that turbulence developed when the intensity difference exceeded a value of approximately 0.03. By using the same procedure as was followed for steady flow, the average of the two measured Reynolds numbers which bracketed this value of 0.03 was taken as the critical Reynolds number. Again, the upper value corresponded to clearly observed early state of turbulence, and the lower value to mild, intermittent disturbances. With this criterion, the critical Reynolds numbers Rep* were obtained for the three model stenoses at various values of the frequency parameter. These values, along with the normalized critical lengths Z*/D, are tabulated in Table 2. A plot of the critical Reynolds number vs the alpha parameter, r,, for the orifice, the plug and the streamlined obstruction is given in Fig. 9, and the effect of the alpha parameter on the critical length is shown in Fig. 10. From Fig. 9, it is observed that the critical Reynolds number first decreases with the alpha parameter, reaches a minimum value, and then increases to a value which is higher than the steady flow (a, = 0) value. The general trend of the effect of the alpha parameter on the critical Reynolds number is the same for all three models. However, the obstruction shape influences the critical Reynolds number, and its effect is the same as
S-2
A.,
:’
300
500
..,_..A--
700 PEAK
REYNOLDS
_,~~__.._...~.-o---
900 NUMBER,
1100
1360
‘15(
Rep
Fig. 8. Variation in turbulence intensity difference with peak Reynolds number at various axial locations for probe located at R/R, = 0.75with z, = 13.3.
Table 2. Critical
Reynolds
numbers
and critical
Alpha parameter, 0 Model number O-2
P-2 s-2
193
in arterial stenoses
Turbulence
lengths
a,
6.2
23.3
13.3
-. Re;
Re:
Z*/D
250 + 15 295 + 16 328 t 18
8 16 12
Rez
Z*/D
8 16 12
121 f 27 195 + 18 273 + 26
In the present study, attention has been focused on the effect of stenosis shape and the frequency para-
ALPHA
I
OO
I
3.9
7.7
PARAMETER,, I
/ 11.6
15.5
1
I
I
I
5
10
15
20
FREQUENCY
Fig. 9. Effect of frequency
parameter
2 2 4
DlSCUSSlON AND SUMMARY
frethickness fi 2v w wrth o being the oscillation quency. Some experimental data obtained in the ascending and descending aortas of large dogs (o! > 6)
I
407 5 60 462 + 47 528 rt 38
tended to support the contention that the critical Reynolds number increases with the alpha parameter. From Fig. 10, it can be observed that the critical lengths at an alpha parameter of 6.2 are the same as at the steady flow value (c(, = 0). However, for higher values, the critical length decreases as the alpha parameter increases. This phenomenon may be due to the fact that as the alpha parameter increases the available time for turbulence to diffuse is decreased. This result is similar to observations by Kirkeeide and Young (1977) in which it was reported that the position of the maximum wall vibration, which should be closely correlated to a point of maximum turbulence intensity downstream from the obstruction, was found at a shorter distance downstream from the obstruction for pulsatile flow than for steady flow at the same Reynolds number. The effect of the geometry still has an important role on the critical length as in the steady flow case : in general, the orifice appears to yield the shortest critical length.
the steady flow case; i.e. the critical Reynolds number for the orifice is the smallest over the entire experimental range of alpha parameters. The effect of the alpha parameter on the critical Reynolds number can be partly explained by considering the well-known velocity profiles of an oscillating flow through a straight tube. For alpha parameters between 0 and 1, the velocity profile is approximately the same as the parabolic profile for a steady flow, i.e. the flow is quasi-steady. For alpha parameters between 1 and 10, the velocity profile changes dramatically with the alpha parameter and has inflection points which would tend to promote the instability of the flow. For high values of alpha parameters, the velocity profile becomes “plug-like” with viscous effects confined to a thin layer near the wall. Nerem and Seed (1972) have suggested that the critical Reynolds number increases with the alpha parameter for relatively large values of the alpha parameter. The theoretical justification for this type of relationship is based on the idea that transition is governed by the Reynolds number U6/v where 6 is the Stokes-layer in
600
4 8 6
335 + 55 429 f 40 431 f 42
Z*/D
Rez
Z’/D
and stenosis
PARAMETER,
geometry
119.4
II 25
at
on critical
peak Reynolds
number.
WITHAYA YONGCHAREON and DONALD F. YOUNG
194
0
0
I 5
I 10 FREQUENCY
Fig. 10. Effect of frequency
parameter
1 15
meter on the criticai Reynolds number. The results, as summarized in Table 2 and Fig. 9, show that both the shape and the frequency parameter do have an effect. Since this study was restricted to one stenosis size, corresponding to a severe stenosis, it is of interest to speculate on the effect of stenosis severity. As noted in the introduction, there have been a few reported values for critical Reynolds numbers for steady-flow through constrictions. These results are plotted in Fig. 11 and clearly show the strong influence of the stenosis size. The values are scattered over a relatively wide range. Probably the two principal factors contributing to the scatter are: (1) the stenosis shapes used in the various studies were not the same, and (2) the criteria and methods employed for the detection of turbulence were different for each of the studies. The values cited by Kim and Corcoran (1974) are higher than the others, but in their study the flow was referred to as fully turbulent when the spectrum of turbulence contained frequencies higher than 100 Hz and relative intensities larger than 0.1. Thus, their values would be expected to be higher than those characterizing the early stages of turbulence. The solid line in Fig. 11 is based on data obtained by Sacks et al. (197 1) from animal experiments in which circular orifice plates were implanted in the descending aortas of dogs. In these studies, the critical Reynolds number was defined as the minimum Reynolds number at which a murmur (as detected with a catheter microphone) could be discerned. Although the experiments are for pulsatile flow, the form of equation for the critical Reynolds number Re* = 2384 (d/D)‘, which correlated their data is based on steady-flow
P-2
0-K0DEL
S-2
o-MODEL
O-2
I 20
PARAMETER,
and stenosis geometry observed.
O-MODEL
I 25
3
at
on axial location
at which turbulence
was first
observations of Johansen (1929). Specific values for the alpha parameter in the descending aorta of the dog were not given in the paper by Sacks et al. (1971), but a reasonable estimate for this parameter is approximately 6. It is to be noted from Fig. 9 that for this 4
-
I-
0 0 D O
,022 0.1
ZACK5 ET AL. (1971) YOUNG KIM & AZUM PRESENT STUDY
0.2
0.4
0.6
0.8
1.0
d/D
Fig. 11. Some experimental steady-flow data illustrating effect of stenosis severity on the critical Reynolds number.
195
Turbulence in arterial stenoses value of Z, the critical Reynolds number, would be about the same as the steady-flow value. The data shown in Fig. 11 suggest that the critical Reynolds number decreases with the square of the diameter ratio, d/D, or directly with the area ratio, Al/A,, where A, is the minimum area in the stenoses and A, is the area of the unobstructed vessel. The results of the present study cannot be directly applied to stenotic flow in arteries since stenosis geometries do not generally coincide with the model configurations, and arterial flow waveforms do not have the simplified shape utilized in the model experiments. However, stenosis geometries can be approximately related to the model geometries by broadly classifying the various shapes into three types as described in Table 3. Adjustments in the critical Reynolds number to account for shape can be approximated with the equation
where values for the shape factor, K, are given in Table 3. These values are based on the steady-flow data given in Table 2 and indicate that the critical Reynolds number may increase by 20-307; as the stenosis shape varies from a type 1 to a type 2 or 3. The effect of eccentricity of the narrowed lumen on critical Reynolds number is unknown, although there is some evidence (Seeley and Young, 1976) to suggest that eccentricity may not be very important, at least for severe stenoses. With regard to the effect of arterial waveforms on the development of turbulence it is difficult to characterize the pulsatile flow variations in the various arteries since the exact nature of the waveform varies from artery to artery. However, the fine detail of the flow waveform is not thought to be highly significant, and a typical arterial waveform can be approximated with the flow pulse used in the present study (Fig. 6) with the proper selection of rr/t. A reasonable value for this ratio is l/3 so that a, = $a, where a is the commonly used alpha parameter based on the heart rate. AS an illustration of how the results of the present study could be used to obtain an estimate of the critical Reynolds number for a given arterial stenosis we assume that the following data are specified : stenosis
type = 2; percent stenosis = 80, unobstructed lumen radius = 3 mm ; heart rate = 60 beats/min ; blood visblood density = 1050 kg/m3, cosity 0.004 N-s/m’, ~~/7 = l/3. With these data a = 3.8 and a, = 6.6. From Fig. 9 the critical Reynolds number for this a, for the orifice (89”/, stenosis) is 130. We assume that for an 80% stenosis the Reynolds number is increased by the ratio of (Al/A,) for an 80”/ stenosis to (Al/A,) for an 89% stenosis. Since this ratio is 1.8 the estimated critical Reynolds number for the orifice is 230. Since the stenosis of interest is a type 2, for which the shape factor is 1.2, the critical Reynolds number is expected to be 280. This Reynolds number corresponds to a peak velocity of 0.18 m/s. Peak velocities in excess of this value are commonly found in human arteries. and therefore, it is likely that for the stenosis described, turbulence would be generated. The ability to predict whether or not turbulence is present for a given stenosis should prove to be useful, particularly in conjunction with clinical techniques utilizing bruits as a means for detecting stenosis, and in gaining a better understanding of turbulence-induced effects, such as post-stenotic dilatation. It should be emphasized that the extension of the results of the present study to stenoses with different area ratios is speculative and additional studies are required to more precisely determine the effect of area ratio on the critical Reynolds number. In summary : The critical Reynolds number for the development of turbulence in pulsatile flow through a stenotic obstruction depends on numerous factors which include stenosis shape and size, and the nature of the base flow waveform. Turbulence will develop at Reynolds numbers well below the critical value for an unobstructed tube. For the severe stenoses used in the present study : (a) The critical Reynolds number was reduced as the stenosis shape became more abrupt. (b) The critical Reynolds number varied with frequency parameter with the Row first becoming less stable and then more stable as the frequency parameter was increased (c) The axial location (critical length) at which turbulence was first observed was a function of both stenosis shape and frequency parameter,
Table 3. Basic stenosis types _
~___ Type
Classification
Description
Abrupt, localized constriction of short length
Shape factor, K _____ 1.0
1
Orifice
2
Phlg
Relatively long, narrowed lumen with the narrowed lumen having an approximately constant diameter
1.2
3
Streamlined
Change in lumen diameter occurs gradually with no abrupt changes in cross section
1.3
196
WITHAY&
YONGCHAREON and
with the shortest length observed for the orifice. The critical length tended to decrease as the
frequency parameter increased. As the Reynolds number was increased beyond the critical value, the location of the most intense turbulent fluctuations moved upstream. 4. Although not specifically covered in the present investigation, past studies suggest that the critical Reynolds number decreases with the area ratio AI/A, and, as a first approximation, decreases in direct proportion to this ratio. Acknowledgements
- This work
DONALD
F. YOUNG
Roach, M. R. (1972) Post-stenotic dilation in arteries. In Cardiovascular Fluid Dynamics (Edited by Bergel. D. H.), Vol. 2. Academic Press, New York. Robertson, J. M. and Herrick, J. F. (1975) Turbulence in blood flow. Dept. Theoretical and Applied Mechanics Report No. 401, University of Illinois. Sacks, A. H., Tickner, E. G. and MacDonald, I. B. (1971) Criteria for the onset of vascular murmurs. Circulation Res. 29, 249-256. Sarpkaya, T. (1966) Experimental determination of the critical Reynolds number for pulsating Poiseuille flow. J. bas. Engng 88, 589-598. Seeley, B. D. and Young, D. F. (1976) Effect of geometry on pressure losses across models of arterial stenoses. J. Biomechanics 9, 439448.
was supported by the Engineering Research Institute, Iowa State University, through funds made available by the United States Public Health Service Grant HL 11717 from the National Heart, Lung and Blood Institute.
Tobin, R. J. and Chang, I. D. (1976) Wall pressure spectral scaling downstream of stenoses in steady tube flow. J.
REFERENCES
Yellin, E. L. (1966) Laminar-turbulent transition process in pulsatile flow. Circulation Res. 29, 791-804. Yongchareon, W. (1977) Development of turbulence in pulsatile, nonuniform flow. PhD Thesis, Iowa State University. Young, D. F. and Tsai, F. Y. (1973a) Flow characteristics in models of arterial stenoses - I : Steady flow. J. Biomechanics
Sergeev, S. I. (1966) Fluid oscillations in pipes at moderate Reynolds numbers. Fluid Dynamics (Mekh. Zh.) 1, 121-122.
Biomechanics 9, 633-640.
Azuma, T. and Fukushima, T. (1976) Flow patterns in stenotic blood vessel models. Biorheology 13, 337-355. Back, L. H. and Roschke, E. J. (1972) Shear-layer flow regimes and wave instabilities and reattachment lengths downstream of an abrupt circular channel expansion. J. appl. Mech. 39,677-681. Clark, C. (1976) Turbulent velocity measurements in a model of aortic stenosis. J. Biomechanics 9, 677-687. Clark, C. (1977) Turbulent wall pressure measurements in a model of aortic stenosis. .I. Biomechanics 10, 461-472. Giddens, D. P., Mabon, R. F. and Cassanova, R. A. (1976) Measurements of disordered flow distal to subtotal vascular stenosis in the thoracic aortas of canines. Circulation Res. 39, 112-119.
Hino, M., Sawamoto, M. and Takasu, S. (1976) Experiments on transition to turbulence in an oscillatory pipe flow. J. Fluid Mech. 75, 193-207.
Johansen, F. C. (1929) Flow through pipe orifices at low Reynolds numbers. Proc. R. Sot. Land. 126,231-245. Kim, B. M. and Corcoran, W. H. (1974) Experimental measurements of turbulence spectra distal to stenosis. J. Biomechanics 7, 335-342.
Kirkeeide, R. L. and Young, D. F. (1977) Wall vibrations induced by flow through simulated stenoses in models and arteries. J. Biomechanics 10, 431-441. Lees, R. S. and Dewey, C. F. (1970) Phonoangiography ; A new noninvasive diagnostic method for studying arterial disease. Proc. natn. Acad. Sci. 67, 935-942. Nerem, R. M. and Seed, W. A. (1972) An in oivo study of aortic flow disturbances. Cardiouascutar Res. 6, l-14. Pitts, W. H. III and Dewey, C. F., Jr. (1977) Spectral and temporal characteristics of post-stenotic turbulent wall pressure fluctuation. ASME Paper 77-BIO-8, presented at the ASME Joint Applied Mechanics, Fluids Engineering and Bioengineering Conference, New Haven, Conn.
6,395-410.
Young, D. F. and Tsai, F. Y. (1973b) Flow characteristics in models of arterial stenoses - II: Unsteady flow. J. Biomechanics 6; 547-559. Young, D. F., Cholvin, N. R., Kirkeeide, R. L. and Roth, A. C. (1977) Hemodynamics of arterial stenoses at elevated flow rates. Circulation Res. 41, 99-107. NOMENCLATURE d D K R
minimum diameter of stenosis unobstructed tube diameter shape factor radial coordinate from tube centerline unobstructed tube radius R, Re Reynolds number. DU/v critical Reynolds number, DU/v Re* peak Reynolds number, DUdv Re, critical Reynolds number, DU,/v Re: t time cl time-averaged cross-sectional mean velocity ii time-averaged point velocity time-averaged peak point velocity 0, peak cross-sectional mean velocity u&J u 11116 root-mean-square value of point velocity instantaneous cross-sectional mean velocity U(t) Z axial coordinate Z* critical axial length Lx alpha parameter, R,&& modified alpha parameter, R,J2/5,v a, 6 Stokes-layer thickness Y kinematic viscosity of fluid r period of flow waveform pulse width of half sine-wave 51 w oscillation frequency.
INITIATION
OF TURBULENCE IN MODELS OF ARTERIAL STENOSES*
WITHAYA YONGCHAREONand DONALD F. YOUNG
Department of Engineering Science and Mechanics, Iowa State University, Ames, IA 50011, U.S.A. Abstract - The development of turbulence under both steady and pulsatile flow through models of arterial
stenoses was studied experimentally. Stenoses were represented by three severely constricted rigid-walled models with different shapes. Model geometries included a streamlined shape, a hollowed plug with blunt ends, and a thin plate orifice. Velocity fluctuations were measured with hot-film probes. Results indicate that: (a) turbulence develops at Reynolds numbers well below the critical value for flow in an unobstructed tube; (b) the critical Reynolds number varies with a dimensionless frequency parameter, first becoming less stable and then more stable as the frequency parameter is increased ; and (c) the critical Reynolds number depends on the shape of the obstruction with the orifice-type stenosis exhibiting the lowest value for the critical Reynolds
INTRODUCTION
As a stenosis develops in an artery, the nature of the flow in the vicinity of the stenosis is significantly altered. The degree of alteration is strongly influenced by the severity of the stenosis, and for moderate and severe stenoses, highly disturbed regions of flow distal to the stenosis commonly occur under physiological flow conditions. These regions of disturbed flow may remain laminar in the sense that even though the flow patterns are complex and time-dependent, they are essentially periodic and reproducible. However, it is generally found that as the stenosis increases in severity, a condition will be reached for which the flow patterns become unstable and irregular velocity fluctuations characteristic of turbulent flow ensue, at least over a part of the flow cycle. The development of turbulence is an important cdnsideration since turbulence represents an abnormal flow condition in the circulation and is related to several important clinical problems such as poststenotic dilatation (Roach, 1972), phonoangiography (Lees and Dewey, 1970), and effect on pressure losses and thus regional blood flow (Young and Tsai, 1973a, b; Young et al., 1977). Little information is available with regard to the conditions under which the flow will become turbulent when disturbed by the presence of an obstruction. A few studies involving the development of turbulence in steady flow through stenoses have been reported (Young and Tsai, 1973a; Kim and Corcoran, 1974; Azuma and Fukushima, 1976). Although there are variations in the reported values for the critical Reynolds numbers for the development of turbulence, the results do clearly show that turbulence will develop at relatively small Reynolds numbers (compared with
* Revised 23 June 1978.
the traditional
value of 2000 for pipe flow), and the
critical Reynolds number decreases as the severity of the stenosis increases. The problem becomes considerably more complex
for pulsatile flow. A few transition studies involving unsteady flows in unobstructed tubes have been reported (Sarpkaya, 1966; Sergeev, 1966; Yellin, 1966; Nerem and Seed, 1972; Hino et al., 1976), and these studies indicate that the unsteadiness can affect the transition process. Several recent studies have been concerned with the characteristics of the developed turbulence distal to a stenosis (see, e.g. Kim and Corcoran, 1974; Giddens et al., 1976; Tobin and Chang, 1976; Clark, 1976, 1977; Pitts and Dewey, 1977); however, there do not appear to be any reported systematic studies of the initial development of turbulence in the presence of a stenosis under pulsatile flow conditions although a few observations have been made (Sacks et al., 1971; Young and Tsai, 1973b). The present study was undertaken to investigate experimentally, using an in vitro model system, the conditions under which pulsatile flow through a stenosis becomes turbulent. METHODS
For a given experiment, a rigid-walled model was located in a flow system in which either a steady or pulsatile flow could be generated. Three severely constricted models (89% area reduction) were used (see Fig. 1). One model had a streamlined shape in the form of a cosine curve (model S-2), the second model was a hollowed plug with blunt ends (model P-2), and the third model was a sharp-edged orifice (model O-2). For models S-2 and P-2, the length Z0 was four times the radius R,. The radius of the unobstructed test section R, was 9.525 mm. The fluid used for all tests was a saline-glycerol
186
WITHAYA YONGCHAREON
(a)
(b)
(Cl
Fig. 1. Model geometries: (a) streamlined model (S-2), (b) blunt end plug (P-2), (c) orifice (O-2).
mixture, with different proportions of saline to glycerol used to obtain various viscosities (for more details, see Yongchareon, 1977). Steady flow was created with a constant head tank, whereas pulsatile flow was generated by a one-way valve and a pulsator which consisted of a piston-cylinder combination driven by a scotch-yoke mechanism (see Fig. 2). The one-way valve was located upstream from the pulsator and was arranged so that it blocked the flow when the piston moved against the flow direction. Since the piston oscillated harmonically, the flow waveform was approximately a half sine-wave over part of the cycle with zero flow over the remainder. Both the frequency and stroke could be varied, and the flow rate was controlled by the combination of the stroke and an orifice located at the flow outlet. The instantaneous flow rate was obtained with an in-line electromagnetic flowmeter system (Biotronex Model BL-610). The main conduit in the test facility was a brass tube and the ends of the model sections were bored in such a manner that the brass tube to model connections provided a flow conduit of constant diameter except in the region containing the obstruction. The distances
and DONALD F.YOIJNG
between the electromagnetic flowmeter and the test sections and between the test sections and the constant head tank were 1.25 and 1.85m, respectively. These distances were sufficient to assure that entrance-length effects would be negligible. The instrumentation for velocity measurements consisted of the following : cylindrical hot-film probes (Thermo-system Mode1 1270-2OW), anemometer (DISA Type 55DOl), auxillary unit (DISA Type 55D25) used as a high pass filter, linearizer (DISA Type 55DlO), RMS voltmeter (DISA Type 55D25) and two digital dc. voltmeters (DISA Type 55D30). Instantaneous flow data were obtained from oscillographs, and all data were also recorded on magnetic tape (Tandberg Series 115). The block diagram for the instrumentation used for velocity measurements is illustrated in Fig. 3. For all measurements, the sensing element of the hot-film probe was perpendicular to the tube axis. The hot-film probe could be inserted into the flow at six axial locations distal to the stenosis (Z/D = 2,4, 6, 8, 10, 12). Radial traverses could also be made. Proximal to the stenosis, point velocities were measured in a straight test section (Fig. 2).
EXPERIMENTALRESULTS
Steady ,pow The flow characteristics distal to the stenosis exhibit a strongly non-uniform spatial distribution. As the fluid flows through the constriction, a jet-like flow develops in the throat section and the main stream separates from the boundary at very low Reynolds numbers. For example, separation occurs at a Reynolds number of approximately 10 for the streamlined model S-2, for the Reynolds number Re = DU/v, with D the unobstructed tube diameter, U the mean velocity, and v the fluid kinematic viscosity. With a continued increase in Reynolds number, the separated region will extend several diameters downstream with TEMPERATURE CONTROLLER CONSTANT HEAD TANK
OVER FLOW ORIFICE VALVE;(??rai
. (VALVE
PULSATOR
FLOW .
*
I
n
STRAIGHT SECTION
Fig. 2. Schematic of experimental apparatus.
Turbulence in arterial stenoses
187
FROM HOT-FILM PROBE
I
I
AUXILIARY UNIT
I
Fig. 3. Block diagram of instrumentation
the reattachment point also moving downstream (see, e.g. Johansen, 1929 ; Young and Tsai, 1973a ; Back and Roschke, 1972). With a still further increase in Reynolds number localized velocities distal to the stenosis become time-dependent and relatively low frequency undulations can be detected with the hot-film probe. Eventually, the velocity fluctuations become irregular in appearance and the output from the probe fluctuates randomly with the signal containing high frequency components. At a given location distal to the stenosis, the intensity ratio lJ,,,/f? varies with Reynolds number as shown in Fig. 4, where U,,, is the root-mean-square value of the point velocity as obtained from the linearized output signal from the hotfilm probe, and C?is the local mean velocity at the same point. The Cl,,, value is zero for sufficiently small Reynolds numbers but increases rapidly once the disturbed flow is initiated, reaches a peak with a subsequent decrease to a relatively low level. It is noted from Fig. 4 that the intensity ratio is not strongly dependent on the radial position of the probe when
system.
near the wall, but there is dependence on radial position when comparing centerline readings with readings near the wall. Figure 4 shows that the relative intensity of the flow disturbances at a point is a strong function of the Reynolds number. Also, the relative intensity near the wall is higher than at the tube centerline. Disturbed flow near the wall was first detected at a lower Reynolds number than the Reynolds number which produces disturbances at the tube centerline. This phenomenon suggests that the flow first becomes unstable in the region of the shear layer bounding the jet emerging from the throat of the obstruction. The variation of relative intensity with Reynolds numbers at various axial positions, but the same radial position R/R,= 0.9, is shown in Fig. 5. For clarity, only smooth curves drawn in by hand are shown, and the actual data points are omitted. All curves have the same general shape, although at the near field location Z/D = 4, the curve shows a more rapid increase in the intensity after the peak value is attained. The exact
0.8‘
STEADY FLOW 0.6-
0.00 77
oiODELs-z
0.75 0.90
200
300
400
600 500 REYNOLDS NUMBER, Re
700
800
Fig. 4. Variation in relative turbulence intensity with Reynolds number at various radial positions for Z/D = 12.
188
WITHAYA YONGCHAREON and DONALD F. YOUNG
STEADY FLOW
0.8-
MODEL S-Z
0.6lb , ,? d D.4;1 z c z = P 0.2?z
0.0
..
----_______---
200
300
400
500
600
700
800
900
* 1000
REYNOLDS NUMBER, Re
Fig. 5. Variation in relative turbulence intensity with Reynolds number at various axial locations for R/R, = 0.9.
reason for this behavior is unknown. At Z/D = 12, where the disturbances were first detected, the relative intensity reaches a maximum value more rapidly than it does at the other locations. It is seen that disturbances first develop at Z/D = 12 and move upstream as the Reynolds number increases. For models P-2 and O-2, only a limited amount of steady flow data were obtained, but the general trends were the same as found for model S-2. For a given geometry, transition from laminar to turbulent flow under steady flow conditions is determined by the critical Reynolds number Re*, and in this study, the relative intensity tJ,,$~ was used as a criterion for obtaining the critical Reynolds number. For flow through complex configurations such as stenoses, it is difficult to characterize a given flow as laminar or turbulent. As suggested by Robertson and Herrick (1973, three basic flow types need to be considered (see Table 1). Luminar flows can be either steady or unsteady, but if they are unsteady the flow
Table 1 Type
characteristics can be described by well-behaved functions of time. Turbulent flow is characterized by irregular (random), three-dimensional fluctuations in the pertinent flow characteristics. There is, of course, no sudden transition from the laminar to the turbulent state but rather there will be an intermediate state in which perturbations to the main flow develop, but the perturbations are not fully irregular. These flows may be classified as disturbed flows. To further complicate matters, turbulent flows may also be transient in the sense that they may decay in space or time. Thus, the turbulence which develops distal to a stenosis will be transient for both steady flow (variation with axial distance) and unsteady flow (variation with both distance and time). For the purpose of the present study, the desired critical Reynolds number was taken to be the value which delineated the latter stage of the disturbed flow regime and the early stage of the turbulent flow regime. Values of the critical Reynolds numbers could be
Basic flow types (from Robertson and Herrick, 1975) Nature
Description
1
Laminar
Orderly flow in smooth (streamlined) layers or laminae
2
Disturbed viscous
Flow with perturbations having some regularity in time and space
3
Turbulent
Flow with irregular (random, chaotic) temporal and spatial disturbances of velocity and pressure
(a) Permanent
Stationary (in the statistical sense) turbulence
(b) Transient
Turbulence which is decaying in space or time
Turbulence
in arterial
stenoses
Fig. 7. Typical experimental oscillographs. recording of point velocity at R/R, = 0.75 in unobstructed section of test section located stream from stenosis. Instantaneous volume rate of flow as measured by electromagnetic flowmeter. (cl Hot-film recording of point velocity at R/R, = 0.75,2/D= 8 (distal to stenosis) with output fil tered to elirninate frequencies below 50 Hz. Z/D = 8 (distal to stenosis). (d) Hot-film recording of point velocity at R/R, = 0.75,
(a) Hot-film
z
Turbulence in arterial stenoses estimated from observations made of the instantaneous velocity variations on an oscilloscope screen. Observations obtained in this manner are of course subjective to some extent. It was found that random fluctuations (as opposed to the more regular undulations) appeared to occur when the intensity ratio exceeded a value of approximately 0.3. The value of 0.3 is arbitrary but was found to be a useful criterion for distinguishing the disturbed flow regime from the early stages of turbulence. Therefore, in an attempt to reduce the subjective nature of the measurement, the average of the two measured Reynolds numbers which bracketed this value of 0.3 was taken as the critical Reynolds number. The upper value corresponded to the clearly observed early stages of turbulence, and the lower value corresponded to the mild intermittent disturbances. With this criterion, the critical Reynolds number was found to be 250 + 15 for the orifice, 295 + 16 for the plug and 328 f 18 for the streamlined obstruction. The plus and minus values incorporates both the range on the bracketed values of the critical Reynolds number and the uncertainty due to measurement errors. As expected, the flow through the orifice is the least stable since the velocity profile at the orifice exit is relatively flat, which provides a high velocity gradient at the separated shear layers and hence, a highly unstable flow. It is not obvious that the boundary layer at the separated section of the streamlined obstruction is thicker than the plug-like obstruction. Nevertheless, the separation point is downstream from the throat of the streamlined obstruction, whereas the separation point is fixed at the throat (the discontinuity section) for the plug-like model. Thus, the separated shear layers for the streamlined obstruction are closer to the wall than the pluglike obstruction, and the wall may tend to damp out small disturbances that would be amplified in the separated shear layer. As the result, the streamlined obstruction provides a more stable flow than the pluglike obstruction. Flow conditions immediately upstream from the stenosis were not studied in detail, but it is apparent from the above discussion that upstream flow patterns, which vary with stenosis shape, will affect the transition process. As seen in Fig. 5, the relative turbulence intensity at Z/D = 12 is significantly greater than zero at a Reynolds number well below those required to develop turbulence at other locations. Thus, turbulence appeared to occur first at Z/D = 12. The distance from the throat of the obstruction to the axial location where turbulence was first observed is denoted as the critical length Z*. The normalized critical lengths, Z*/D. for three geometries - the orifice, the plug and the streamlined obstruction - were found to be 8, 16 and 12. respectively. Pulsatile
used in the pulsatile-flow system described in the Methods Section. Since arterial blood flow is known to be highly pulsatile with a typical ratio of peak-to-mean velocity in the range of 2-3 (Young et al., 1977), and with low velocities over part of the flow cycle, the base flow pulse shown in Fig. 6 was used for pulsatile flow studies. For all tests, the ratio T,/T was held constant at a value of approximately 0.6 which gives a corresponding constant value of 2.6 for the ratio of the peak-tomean velocity, UP/U. For pulsatile flow, the development of turbulence for a given stenosis geometry will be a function not only of the Reynolds number but also of U&l and a frequency parameter, LX,.For the present study, the frequency parameter used was Ro,,/G. Since UP/U was held constant, the critical Reynolds number depended only on the frequency parameter for a given stenosis. It is to be noted that a commonly used frequency parameter is the so-called “alpha parameter” where CL= R,m so that CL,= fi a. For the flow pulse studied, the local inertial effects which affect the transition phenomenon depend on Tl so that the parameter a, seems an appropriate index for characterizing this effect, although for a fixed ratio, TJT, the parameter o! could be used. For a given value of a,, point velocity measurements were made at several locations distal to the stenosis with R/R,= 0.75, and the value of the peak Reynolds number could be varied over a wide range. For the pulsatile-flow tests, the peak Reynolds number is based on the peak cross-sectional average velocity CT, in the unobstructed tube, i.e. Re, = U,D/v. As was the case for steady flow, three basic flow regimes could be observed : (1) laminar flow over the entire flow cycle ; (2) disturbed flow over a part of the cycle with large scale, relatively low frequency, velocity fluctuations, and (3) transient turbulence which was first initiated near the peak of the flow cycle with the disturbances damped out before the next cycle started. Typical oscillograms of the basic flow pulse and the instantaneous point velocity are shown in Fig. 7. The critical peak Reynolds number, Re,*. was again taken to be the value which delineated the latter stages of the disturbed flow regime and the early stage oft he turbulent flow regime. The problem is complicated by
flow
To investigate the effect of unsteadiness of the base flow on the development of turbulence, the same stenosis models used in the steady-flow studies were
1.91
Fig. 6. Base-flow pulsatile waveform.
192
WITHAYA YONGCHAREONand DONALD F. YOUNG
the presence of the unsteady base flow which by itself will give relatively large values of the intensity ratio U,,,/o even for the laminar regime. In turbulent pulsatile flow, the intensity ratio contains contributions from both the unsteady base flow and any random fluctuating velocity characterizing turbulence. If the base flow waveform in the turbulent flow region distal to the stenosis was the same as that in the proximal laminar flow, the contribution to U,,,.ii due to the base unsteady velocity could be obtained from measurements obtained in the unobstructed section upstream from the obstruction. Thus, the difference between V,,Jii for the distal turbulent and proximal laminar flows would give an indication of the degree of turbulence. Unfortunately, in the region close to an obstruction, the base velocity waveform is different from that in the unobstructed tube, and therefore, the difference between proximal and distal intensity ratios is not a good index of turbulence. The contribution of the base flow can be reduced by using a high pass filter which ideally allows only frequencies higher than the cut-off frequency to pass. The difficulty, however, is the loss of low frequencies of the disturbed flow. After considerable experimentation, it was found that the most reliable index for determining the critical Reynolds number was the value of the difference between Urm$Up measured by a hot-film probe located at the distal location of interest, and Urms/UP measured proximal to the stenosis, with the low frequency components eliminated at both locations with a highpass filter. The cut-off frequency was chosen in such a way that the II,,,, after filtering out the low frequency components, was approximately l”/;, of the peak velocity of the proximal laminar waveform. In these experiments, the cut-off frequency was set at 50 Hz. A typical plot of the intensity difference (described
MODEL
100
n
above) vs the peak Reynolds number, Re,, is shown in Fig. 8. Even though some low frequency components of turbulence were eliminated, the general trend of curves is the same as that found in the steady flow (Fig. 5). In Fig. 8, the arrow indicates the Reynolds number at which the initial stages of the turbulent waveform was observed. At a lower Reynolds number, the value of the intensity difference is due in part to the change in the proximal and distal base flow waveforms. From a visual inspection of all the hot-film signals as observed on an oscilloscope screen, it was determined that turbulence developed when the intensity difference exceeded a value of approximately 0.03. By using the same procedure as was followed for steady flow, the average of the two measured Reynolds numbers which bracketed this value of 0.03 was taken as the critical Reynolds number. Again, the upper value corresponded to clearly observed early state of turbulence, and the lower value to mild, intermittent disturbances. With this criterion, the critical Reynolds numbers Rep* were obtained for the three model stenoses at various values of the frequency parameter. These values, along with the normalized critical lengths Z*/D, are tabulated in Table 2. A plot of the critical Reynolds number vs the alpha parameter, r,, for the orifice, the plug and the streamlined obstruction is given in Fig. 9, and the effect of the alpha parameter on the critical length is shown in Fig. 10. From Fig. 9, it is observed that the critical Reynolds number first decreases with the alpha parameter, reaches a minimum value, and then increases to a value which is higher than the steady flow (a, = 0) value. The general trend of the effect of the alpha parameter on the critical Reynolds number is the same for all three models. However, the obstruction shape influences the critical Reynolds number, and its effect is the same as
S-2
A.,
:’
300
500
..,_..A--
700 PEAK
REYNOLDS
_,~~__.._...~.-o---
900 NUMBER,
1100
1360
‘15(
Rep
Fig. 8. Variation in turbulence intensity difference with peak Reynolds number at various axial locations for probe located at R/R, = 0.75with z, = 13.3.
Table 2. Critical
Reynolds
numbers
and critical
Alpha parameter, 0 Model number O-2
P-2 s-2
193
in arterial stenoses
Turbulence
lengths
a,
6.2
23.3
13.3
-. Re;
Re:
Z*/D
250 + 15 295 + 16 328 t 18
8 16 12
Rez
Z*/D
8 16 12
121 f 27 195 + 18 273 + 26
In the present study, attention has been focused on the effect of stenosis shape and the frequency para-
ALPHA
I
OO
I
3.9
7.7
PARAMETER,, I
/ 11.6
15.5
1
I
I
I
5
10
15
20
FREQUENCY
Fig. 9. Effect of frequency
parameter
2 2 4
DlSCUSSlON AND SUMMARY
frethickness fi 2v w wrth o being the oscillation quency. Some experimental data obtained in the ascending and descending aortas of large dogs (o! > 6)
I
407 5 60 462 + 47 528 rt 38
tended to support the contention that the critical Reynolds number increases with the alpha parameter. From Fig. 10, it can be observed that the critical lengths at an alpha parameter of 6.2 are the same as at the steady flow value (c(, = 0). However, for higher values, the critical length decreases as the alpha parameter increases. This phenomenon may be due to the fact that as the alpha parameter increases the available time for turbulence to diffuse is decreased. This result is similar to observations by Kirkeeide and Young (1977) in which it was reported that the position of the maximum wall vibration, which should be closely correlated to a point of maximum turbulence intensity downstream from the obstruction, was found at a shorter distance downstream from the obstruction for pulsatile flow than for steady flow at the same Reynolds number. The effect of the geometry still has an important role on the critical length as in the steady flow case : in general, the orifice appears to yield the shortest critical length.
the steady flow case; i.e. the critical Reynolds number for the orifice is the smallest over the entire experimental range of alpha parameters. The effect of the alpha parameter on the critical Reynolds number can be partly explained by considering the well-known velocity profiles of an oscillating flow through a straight tube. For alpha parameters between 0 and 1, the velocity profile is approximately the same as the parabolic profile for a steady flow, i.e. the flow is quasi-steady. For alpha parameters between 1 and 10, the velocity profile changes dramatically with the alpha parameter and has inflection points which would tend to promote the instability of the flow. For high values of alpha parameters, the velocity profile becomes “plug-like” with viscous effects confined to a thin layer near the wall. Nerem and Seed (1972) have suggested that the critical Reynolds number increases with the alpha parameter for relatively large values of the alpha parameter. The theoretical justification for this type of relationship is based on the idea that transition is governed by the Reynolds number U6/v where 6 is the Stokes-layer in
600
4 8 6
335 + 55 429 f 40 431 f 42
Z*/D
Rez
Z’/D
and stenosis
PARAMETER,
geometry
119.4
II 25
at
on critical
peak Reynolds
number.
WITHAYA YONGCHAREON and DONALD F. YOUNG
194
0
0
I 5
I 10 FREQUENCY
Fig. 10. Effect of frequency
parameter
1 15
meter on the criticai Reynolds number. The results, as summarized in Table 2 and Fig. 9, show that both the shape and the frequency parameter do have an effect. Since this study was restricted to one stenosis size, corresponding to a severe stenosis, it is of interest to speculate on the effect of stenosis severity. As noted in the introduction, there have been a few reported values for critical Reynolds numbers for steady-flow through constrictions. These results are plotted in Fig. 11 and clearly show the strong influence of the stenosis size. The values are scattered over a relatively wide range. Probably the two principal factors contributing to the scatter are: (1) the stenosis shapes used in the various studies were not the same, and (2) the criteria and methods employed for the detection of turbulence were different for each of the studies. The values cited by Kim and Corcoran (1974) are higher than the others, but in their study the flow was referred to as fully turbulent when the spectrum of turbulence contained frequencies higher than 100 Hz and relative intensities larger than 0.1. Thus, their values would be expected to be higher than those characterizing the early stages of turbulence. The solid line in Fig. 11 is based on data obtained by Sacks et al. (197 1) from animal experiments in which circular orifice plates were implanted in the descending aortas of dogs. In these studies, the critical Reynolds number was defined as the minimum Reynolds number at which a murmur (as detected with a catheter microphone) could be discerned. Although the experiments are for pulsatile flow, the form of equation for the critical Reynolds number Re* = 2384 (d/D)‘, which correlated their data is based on steady-flow
P-2
0-K0DEL
S-2
o-MODEL
O-2
I 20
PARAMETER,
and stenosis geometry observed.
O-MODEL
I 25
3
at
on axial location
at which turbulence
was first
observations of Johansen (1929). Specific values for the alpha parameter in the descending aorta of the dog were not given in the paper by Sacks et al. (1971), but a reasonable estimate for this parameter is approximately 6. It is to be noted from Fig. 9 that for this 4
-
I-
0 0 D O
,022 0.1
ZACK5 ET AL. (1971) YOUNG KIM & AZUM PRESENT STUDY
0.2
0.4
0.6
0.8
1.0
d/D
Fig. 11. Some experimental steady-flow data illustrating effect of stenosis severity on the critical Reynolds number.
195
Turbulence in arterial stenoses value of Z, the critical Reynolds number, would be about the same as the steady-flow value. The data shown in Fig. 11 suggest that the critical Reynolds number decreases with the square of the diameter ratio, d/D, or directly with the area ratio, Al/A,, where A, is the minimum area in the stenoses and A, is the area of the unobstructed vessel. The results of the present study cannot be directly applied to stenotic flow in arteries since stenosis geometries do not generally coincide with the model configurations, and arterial flow waveforms do not have the simplified shape utilized in the model experiments. However, stenosis geometries can be approximately related to the model geometries by broadly classifying the various shapes into three types as described in Table 3. Adjustments in the critical Reynolds number to account for shape can be approximated with the equation
where values for the shape factor, K, are given in Table 3. These values are based on the steady-flow data given in Table 2 and indicate that the critical Reynolds number may increase by 20-307; as the stenosis shape varies from a type 1 to a type 2 or 3. The effect of eccentricity of the narrowed lumen on critical Reynolds number is unknown, although there is some evidence (Seeley and Young, 1976) to suggest that eccentricity may not be very important, at least for severe stenoses. With regard to the effect of arterial waveforms on the development of turbulence it is difficult to characterize the pulsatile flow variations in the various arteries since the exact nature of the waveform varies from artery to artery. However, the fine detail of the flow waveform is not thought to be highly significant, and a typical arterial waveform can be approximated with the flow pulse used in the present study (Fig. 6) with the proper selection of rr/t. A reasonable value for this ratio is l/3 so that a, = $a, where a is the commonly used alpha parameter based on the heart rate. AS an illustration of how the results of the present study could be used to obtain an estimate of the critical Reynolds number for a given arterial stenosis we assume that the following data are specified : stenosis
type = 2; percent stenosis = 80, unobstructed lumen radius = 3 mm ; heart rate = 60 beats/min ; blood visblood density = 1050 kg/m3, cosity 0.004 N-s/m’, ~~/7 = l/3. With these data a = 3.8 and a, = 6.6. From Fig. 9 the critical Reynolds number for this a, for the orifice (89”/, stenosis) is 130. We assume that for an 80% stenosis the Reynolds number is increased by the ratio of (Al/A,) for an 80”/ stenosis to (Al/A,) for an 89% stenosis. Since this ratio is 1.8 the estimated critical Reynolds number for the orifice is 230. Since the stenosis of interest is a type 2, for which the shape factor is 1.2, the critical Reynolds number is expected to be 280. This Reynolds number corresponds to a peak velocity of 0.18 m/s. Peak velocities in excess of this value are commonly found in human arteries. and therefore, it is likely that for the stenosis described, turbulence would be generated. The ability to predict whether or not turbulence is present for a given stenosis should prove to be useful, particularly in conjunction with clinical techniques utilizing bruits as a means for detecting stenosis, and in gaining a better understanding of turbulence-induced effects, such as post-stenotic dilatation. It should be emphasized that the extension of the results of the present study to stenoses with different area ratios is speculative and additional studies are required to more precisely determine the effect of area ratio on the critical Reynolds number. In summary : The critical Reynolds number for the development of turbulence in pulsatile flow through a stenotic obstruction depends on numerous factors which include stenosis shape and size, and the nature of the base flow waveform. Turbulence will develop at Reynolds numbers well below the critical value for an unobstructed tube. For the severe stenoses used in the present study : (a) The critical Reynolds number was reduced as the stenosis shape became more abrupt. (b) The critical Reynolds number varied with frequency parameter with the Row first becoming less stable and then more stable as the frequency parameter was increased (c) The axial location (critical length) at which turbulence was first observed was a function of both stenosis shape and frequency parameter,
Table 3. Basic stenosis types _
~___ Type
Classification
Description
Abrupt, localized constriction of short length
Shape factor, K _____ 1.0
1
Orifice
2
Phlg
Relatively long, narrowed lumen with the narrowed lumen having an approximately constant diameter
1.2
3
Streamlined
Change in lumen diameter occurs gradually with no abrupt changes in cross section
1.3
196
WITHAY&
YONGCHAREON and
with the shortest length observed for the orifice. The critical length tended to decrease as the
frequency parameter increased. As the Reynolds number was increased beyond the critical value, the location of the most intense turbulent fluctuations moved upstream. 4. Although not specifically covered in the present investigation, past studies suggest that the critical Reynolds number decreases with the area ratio AI/A, and, as a first approximation, decreases in direct proportion to this ratio. Acknowledgements
- This work
DONALD
F. YOUNG
Roach, M. R. (1972) Post-stenotic dilation in arteries. In Cardiovascular Fluid Dynamics (Edited by Bergel. D. H.), Vol. 2. Academic Press, New York. Robertson, J. M. and Herrick, J. F. (1975) Turbulence in blood flow. Dept. Theoretical and Applied Mechanics Report No. 401, University of Illinois. Sacks, A. H., Tickner, E. G. and MacDonald, I. B. (1971) Criteria for the onset of vascular murmurs. Circulation Res. 29, 249-256. Sarpkaya, T. (1966) Experimental determination of the critical Reynolds number for pulsating Poiseuille flow. J. bas. Engng 88, 589-598. Seeley, B. D. and Young, D. F. (1976) Effect of geometry on pressure losses across models of arterial stenoses. J. Biomechanics 9, 439448.
was supported by the Engineering Research Institute, Iowa State University, through funds made available by the United States Public Health Service Grant HL 11717 from the National Heart, Lung and Blood Institute.
Tobin, R. J. and Chang, I. D. (1976) Wall pressure spectral scaling downstream of stenoses in steady tube flow. J.
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Young, D. F. and Tsai, F. Y. (1973b) Flow characteristics in models of arterial stenoses - II: Unsteady flow. J. Biomechanics 6; 547-559. Young, D. F., Cholvin, N. R., Kirkeeide, R. L. and Roth, A. C. (1977) Hemodynamics of arterial stenoses at elevated flow rates. Circulation Res. 41, 99-107. NOMENCLATURE d D K R
minimum diameter of stenosis unobstructed tube diameter shape factor radial coordinate from tube centerline unobstructed tube radius R, Re Reynolds number. DU/v critical Reynolds number, DU/v Re* peak Reynolds number, DUdv Re, critical Reynolds number, DU,/v Re: t time cl time-averaged cross-sectional mean velocity ii time-averaged point velocity time-averaged peak point velocity 0, peak cross-sectional mean velocity u&J u 11116 root-mean-square value of point velocity instantaneous cross-sectional mean velocity U(t) Z axial coordinate Z* critical axial length Lx alpha parameter, R,&& modified alpha parameter, R,J2/5,v a, 6 Stokes-layer thickness Y kinematic viscosity of fluid r period of flow waveform pulse width of half sine-wave 51 w oscillation frequency.