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A model study of flow dynamics in human central airways. Part II: Secondary flow velocities
97
Respiration Physiology (1982) 49, 97-113 Elsevier Biomedical Press
A M O D E L S T U D Y O F F L O W D Y N A M I C S IN H U M A N C E N T R A L A I R W A Y S . P A R T II: S E C O N D A R Y F L O W V E L O C I T I E S
D. ISABEY* and H . K . C H A N G Biomedical Engineering Unit and Department of Physiology, McGill Univerjty, Montreal, Quebec, Canada
Abstract. Secondary velocity components perpendicular to the tube axis were measured in a 3 : 1 scale model of the human central airways. Slanted hot-wire probes were introduced axially in order to measure the secondary velocities at about 120 points for each of the 7 stations investigated. Secondary velocities in the inspiratory direction never exceeded a mean value of 18% of the mean axial velocity. Secondary velocities in the expiratory direction reached a mean value of 21.5% of the mean axial velocity. In the inspiratory direction, two unequal eddies were formed in the left main bronchus and in the right upper lobe. Moreover, maximum velocites were observed near the wall and the decay of secondary velocities in the left main bronchus was observed. The secondary flow patterns observed in the left upper and lower lobes after the second bifurcation were difficult to recognize, although they seemed to be more influenced by the second bifurcation. The complexity of the flow pattern was reinforced by viscous effects acting near the wall. In the expiratory direction, only two stations in the trachea were measured ; four uneven eddies seemed to have existed, with the ventral eddies appearing to be predominant. Overall, the secondary velocity magnitudes as well as the patterns of eddies were very dependent on the geometry of the model used. Air flow patterns Airway model Hot-wire anemometry
Secondary flow Tracheo-bronchial tree
This is Part II of a three-part study aimed at understanding the air flow patterns in the central airways. In this part, the object is to measure the secondary (tangential and radial) velocity components in various airway branches. McDonald (1960) was probably the first to demonstrate the presence of eddies in fluid flows through a branching tube. Using a visualization technique, Schroter and Sudlow (1969) showed, for inspiratory flow and different Reynolds numbers, the existence of two helical vortices resulting from secondary motion downstream of Accepted.for publication 2 March 1982 Technical assistance was afforded by Christopher H. Bracken. * Present address: U. 68 INSERM, H6pital Saint Antoine, F-75571 Paris Cedex 12, France. 0034-5687/82/0000-0000/$02.75 © Elsevier Biomedical Press
98
D. ISABEYAND H. K. CHANG
the carina. During expiratory flow, four similar vortices were observed. The first investigator who actually measured secondary velocities in curved or elliptical tubes and bifurcating systems was apparently Olson (1971); his data were extensively quoted in a recent review of pulmonary fluid mechanics by Pedley et al. (1977). To our knowledge, in the last ten years there have been no other studies of secondary motion relevant to respiratory flow. In spite of this hiatus, secondary motion is frequently invoked in the literature of respiratory physiology because it plays a direct and an indirect role in such phenomena as particle deposition and gas dispersion in airways. Experimental studies of dispersion in bent tubes by Caro (1966) have shown that secondary flows do act to promote lateral mixing of injected substances. Such mixing observed for substances of low diffusivity and in the absence of turbulence would actually be very slow without secondary flow. According to Lighthill (1972), the pattern of secondary motion observed in steady state could explain why the critical Reynolds number for transition to turbulence increases from 2000 for straight pipes to a typical 6000 for curved pipes. Applying this result to high flow in airways, some authors consider that secondary flow assists the decay in turbulence observed from large airways to small airways and hence contributes to diminished dispersion and deposition of particles. As stated by Pedley et al. (1977), the degree of mixing is notably affected by the secondary motions, particularly by the strength of these, which depend upon local velocities. To supply needed information in this field, we measured secondary velocities in the first generations of the central airways, i.e., where secondary velocities are probably greatest.
Apparatus and methods MODEL A 3 : 1 large-scale model of the human central airways specially designed for velocity measurements was used. It was constructed from acrylic plastic blocks and heattreated tubes all manually finished and polished. Airways from the trachea to the lobar bronchi were represented. The dimensions were based on the airway geometry proposed by Horsfield et al. (1971). Details of the geometry are given in Part I of this study by Chang and E1 Masry (1982). A photograph of the regular model along with the linear resistors, the special adaptors as well as the hot-wire probe assembly, is shown in fig. 1. A detailed arrangement of the linear resistors and conic restrictors used during axial velocity measurements is shown in the insert (detail a) of fig. 2; for secondary velocity measurements, special adaptors were used to replace the regular pieces as shown in the insert (detail b) of fig. 2; fig. 3 shows a sketch of the model and the positions of the stations investigated. These special adaptors were designed because the
SECONDARY VELOCITIES IN CENTRAL AIRWAYS
99
Fig. 1. Photograph of the model along with the special adaptors specifically designed for secondary velocity measurements.
secondary flow measurement technique required that the measuring probe be inserted axially from the end of a branch. These carefully machined pieces allowed normal air passage as well as distal probe insertion and were used one at a time. By pre-drilling a number of entry ports at different angles in an adaptor and rotating this piece about its own axis, one could make the probe reach the desired positions for velocity measurement. By making repeated measurements at a given station with these pieces attached at the exit and at different rotational angles, we verified that the axial velocity profiles throughout the model did not vary more than the normal variation from one experiment to the other, namely, between 5 ~ and 10~o. This was because the linear resistors placed distally to the model offered more than 95~o of the overall resistance in the model. Thus, with the aid of the special adaptors and the linear resistors, we were able to map out the secondary velocities over a given cross-section of the model with an axially-inserted hot-wire probe. Dynamic similarity between the model and the real airways was ensured with identical Reynolds numbers. Under these conditions, since the model scale was 3 : 1, the velocities in the model represented one-third the equivalent physiological flow.
100
D. ISABEY A N D H . K . C H A N G
compressed a!r .= Z
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FLOW SYSTEM
The system used to supply air in the model (see fig. 2) was essentially identical to that described by Chang and E1 Masry (1982). However, to achieve the most stable flow possible, a high-pressure stabilization tank fed by a high-pressure line (100 psi) was placed upstream of the decompression valve and the air filter. Air flow rates were monitored with a rotameter of 1% precision. A single constant flow rate of 0.6 L/sec (equivalent to a physiological flow of 0.2 L/sec) was used
S E C O N D A R Y VELOCITIES IN C E N T R A L A I R W A Y S
101
Ce
D"
B"
.2A~~o ( A' -zs C -3
Cross-section at a given station
~
R.U.
.U.
8 Li.
RM. RL
Fig. 3. Sketch of the model showing the positions of the stations of measurement.
during the experiments. As was the case in the axial flow study, the inspiratory direction of flow was achieved by connecting the air source to a second stabilization reservoir immediately upstream of the trachea to ensure a nearly flat entrance velocity profile in the trachea. For expiratory experiments, the air source was directed toward a cylindrical manifold placed below the model and attached to the five lobar bronchi. In this case, the reservoir at the top of the model was opened to room air. The flow distribution was controlled with the same system as used in the axial flow study. Linear resistors made of copper tubes packed with stainless-steel capillaries were connected to the five lobar bronchi. Since the resistance of these tubes was much greater than the overall resistance of the model, the flow was distributed in invariable proportion throughout the system.
VELOCITY MEASUREMENT
A constant-temperature hot-wire anemometer system (Disa 55M01) was used as in the axial-flow study (Chang and E1 Masry, 1982). However, slanted gold-plated probes (type 55P02) were used since they are known to be more sensitive to tangential velocity components than are straight probes (Champagne, 1967). The slanting angle, denoted by ~t in fig. 4, was always 45 °. The probe consists of two prongs sustaining a gold-plated wire 3 mm in length. The sensitive part of the wire is not gold-plated and is 1.25 mm in length and 5 /~m in diameter. The support for the probe is mounted on a micromanipulator to measure and control its longitudinal displacement. The angular position of the probe (angle ~b) is controlled using an index fixed on the support. The base of the micromanipulator is tightly screwed to the model at a distal branch (see detail b, fig. 2).
102
D. ISABEY A N D H . K . C H A N G
f3
Z
W
z
Fig. 4. Angular positions (denoted by if) of the slanted hot-wire probe during velocity measurements; axes x, y, z define the inertial coordinate system; U, V, W are velocity components; ct is the slanting angle o f the probe.
During an experiment, the probe was rotated about its own axis in order to register different cooling effects due to the non-axial orientation of the local velocity vector. Note that according to Jorgensen (1971) the effective cooling velocity acting on the sensor may be expressed as U2rf = W 2 + k~V 2 + k 2 W 2
(1)
where k~ and k2 are respectively the correction factors for the yaw and pitch angles and U, V, W are the velocity components normal to, along, and binormal to the wire, respectively (see fig. 4). The hot-wire probe was calibrated in a manner similar to that used in the axial-flow study reported in the companion paper (Chang and E1 Masry, 1982). The different sensitive coefficients k~ and k 2 w e r e defined by placing the wire under appropriate angles at the maximum velocity of a known Poiseuille flow. According to Acrivlellis (1978), it is theoretically possible to obtain the three velocity components in an inertial coordinate system with a proper choice of transformation matrix and from six independent readings corresponding to six different angular positions of the probe referred by the ~b angle (fig. 4). In solving a system of equations derived from the general eq. (1), Acrivlellis obtained the following three products of components: UV =
U~ - U~ 2(1 -k~) sin 2a
(2)
UW =
U42 - U22 2(1 -k~) sin 2~
(3)
VW =
(U 2 - U~) V'r2 + U~ - U~ 2 v/2 (sin 2 a + k~ cos2 ~ - k 2)
(4)
SECONDARY VELOCITIES IN CENTRAL AIRWAYS
103
from which the sign and the modulus of the components may be easily deduced; is the probe angle with respect to the inertial coordinate system x, y, z, and U~ ... U6 are the six readings. This method is theoretically correct in laminar flow but has been shown to be questionable in turbulent flow (Bartenwerfer, 1979). Since some degree of turbulence exists even at a flow rate of 1.2 L/sec, and since we found that this led to considerable errors in the computed secondary velocity components, we were forced to make all the secondary velocity measurements at a low flow rate of 0.6 L/sec, or half of the lower flow rate used in the axial flow study. Furthermore, we modified the Acrivlellis method by: (i) inserting the probe parallel to the largest velocity component, namely the axial component as already specified, (ii) taking more than 6 readings at each point in order to eliminate the terms that introduce too large an error, and (iii) obtaining the axial flow component separately with a straight probe in the same manner as did Chang and E1 Masry (1982). For condition (ii), a computer program gave, at each point, the product of the axial flow magnitude by the secondary flow magnitude and the direction of the secondary velocity. Note that if U is the axial flow component and V and W are the secondary flow components, the two quantities above are independent of the product (V • W) which has the larger margin of error (see eq. (4)). Finally, the magnitude of secondary flow was obtained by dividing the product ( U . x / r ~ + W 2) by the axial velocity magnitude (U) that had been obtained separately. The present method was tested in measuring secondary velocities after a 180 ° bend in a pipe. The ratio of the radius of curvature to the pipe diameter was 2.3 and the Reynolds number was 1030. These parameters were chosen to be similar to those used in a previous study (Olson, 1971). The results obtained were very similar to the theoretical prediction (White, 1929) and to Olson's data as presented in Pedley et aL (1977). An estimation of error for the present method has shown that the absolute error in positioning angle was about +_ 3 ° and the relative error on secondary velocity magnitude was 10%. However, this error was increased and reached about 30~o when the magnitude of the axial velocity was small, i.e., near the wall.
Results INSP1RATORY FLOW Stations 4, 6, 7, 8, 10 whose positions are given in fig. 3, were studied during inspiratory flow. Stations 4 and 6 are located respectively at 0.9 and 3.9 diameters from the carina. These two stations were chosen in order to follow the evolution of secondary flow through the left main bronchus. Stations 7 and 8 are located respectively at 2.2 diameters and 1.7 diameters from the flow divider of the right
104
D. ISABEY AND H.K. CHANG
lung. The last three stations were chosen because secondary flow was presumably stronger in these branches than it would be in other branches. At each station secondary velocities were measured mainly along 4 diameters and at 14 different points along each diameter. Additional points near the wall along the intermediate diameters were also studied. The results obtained are presented in figs. 5a to 9a. The diameter A - A ' is in the frontal plane and the diameter C - C ' in the sagittal plane, with C always on the dorsal side. The vectors plotted on these panels represent the velocity components (normalized by the mean axial velocity) measured in the plane normal to the airway axis. Thus the true velocity at a given point of an investigated airway cross-section is the resultant vector of this plotted velocity vector and the axial velocity component. The axial velocity component is directed into the paper in inspiratory flow and toward the reader in expiratory flow. The corresponding isovelocity contours obtained from separate axial velocity measurements are presented in figs. 5b to 9b. At station 4 where the Reynolds number is 680, the isovelocity contours (fig. 5b) reveal the existence of a peak axial velocity near the inner wall of the bifurcation. This pattern is similar to the patterns observed at higher Reynolds numbers and is extensively described by Chang and E1 Masry (1982). In fig. 5a it is relatively easy to recognize two semi-circular eddies developing on each side of the diameter A-Aft The secondary velocities along this diameter are directed toward A', i.e., toward the flow divider; the two eddy streamlines along the tube wall rejoin each other at A after completing their respective semi-circles. The fluid in this region experiences centrifugal forces due to the airway curvature and tends to move toward the inner wall of the bifurcation. At the same time, there is recirculation of fluid near the walls due to the establishment of a trasverse pressure gradient across the S T A T ION
(a)
C dorsal
A
4
A
(b)
.(3'
C' dors
A' A" Fig. 5. Inspiratory flow measured at Station 4. Axial velocitiespoint into the paper. (a) Direction and magnitude of the secondary velocities; the scale gives the magnitude relative to the mean axial velocity at this cross-section. (b) Isovelocitycontours of the axial flow; numerical values indicate magnitudes normalized by the mean axial velocityat the cross-section.
SECONDARY VELOCITIES IN CENTRAL AIRWAYS STATION
(a)
105
6
(b)
A
C dorsal
C"
C dorsal
0.5
I,,I=N=N
A'
Fig. 6. Inspiratory flow measured at Station 6. See fig, 5 for legend.
pipe, balancing the centrifugal forces. The two secondary eddies are not symmetrical as they would be in an ideal bifurcation. Correspondingly, the isovelocity contours are not symmetrical about the diameter A-A'. The maximum secondary velocity along A - A ' is about 27% of the mean axial velocity while the maximum secondary velocity near the wall reaches 40% of the mean axial velocity on the ventral side and 25% of the same velocity on the dorsal side. The average secondary velocity is about 14.5% of the mean axial velocity for the data plotted in fig. 5a. Data obtained at station 6 (fig. 6a,b) for the same Reynolds number (680) differ from the data obtained at the preceding station. While the axial velocity profiles develop a bi-peak structure (see Chang and E1 Masry, 1982, and fig. 5b) and become less symmetrical with respect to A-A', the eddy (fig. 6a) appearing initially on the dorsal side becomes predominant with a maximum velocity near the wall reaching 33% of the mean axial velocity. Simultaneously the eddy on the ventral side seems rather weak. The velocity of the fluid motion in the center of the tube is only 15)/o of the mean axial velocity and is no longer parallel to the A-A" direction as it was at station 4. It is of interest to note that the stronger eddy occupies more than just the right half of the branch. Accordingly, there is not a unique stagnation point for secondary velocities but presumably a curved line of zero secondary velocities. Finally, at station 6, the overall mean secondary velocity represents 10.8% of the mean axial velocity. Data obIained at stations 7 and 8 (figs. 7a,b; 8a,b), where Reynolds numbers are respectively 460 and 560, are more difficult to describe. It is difficult to figure out the contours of eddies although one can ascertain, at least from data at station 8, that there are more than two separate eddies downstream of the second bifurcation. Interestingly, the main secondary motion along the diameter A - A ' is directed essentially toward the more immediate flow divider for both stations while the secondary motion resulting from the first bifurcation is rather weak in this region.
106
D. ISABEY A N D H. K. C H A N G STATION
(a)
7 A
A
C dorsal
C"
Co
Oh)
.C"
dorsal
Q5 A"
A"
Fig. 7. Inspiratory flow measured at Station 7. See fig. 5 for legend. STATION
(a)
8
A
O
A
,C"
C'
dorsal
Oh)
dor,'
0.1 0.5 A'
A'
Fig. 8. Inspiratory flow measured at Station 8. See fig. 5 for legend.
This observation agrees with the axial flow results showing that the more immediate bifurcation has the stronger effect on axial flow (Chang and E1 Masry, 1982). Near the wall the effect of the strongest eddy observed at station 6 can be felt since at many exterior points, the fluid has a secondary velocity component turning counterclockwise as at station 6. Note also that near the wall, there is a large radial component directed toward the center of the tube. As at stations 4 and 6, the maximum secondary velocities are observed near the wall. They reach 2 8 ~ of the mean axial velocity at station 7 and 3 4 ~ of the mean axial velocity at station 8. The mean secondary velocity at station 7 represents 13.9~ of mean axial velocity and 17.4~ at station 8. Results obtained at station 10 (fig. 9a,b) are simpler to describe. The Reynolds number is 460. A peak of axial velocity accompanied by a high velocity gradient
S E C O N D A R Y VELOCITIES IN C E N T R A L A I R W A Y S STATION
(a)
10
A"
C"t I ventral
A"
C
1
A
107
c" ventra
(b)
-C
0.5
A
Fig. 9. Inspiratory flow measured at Station 10. See fig. 5 for legend.
is observed near the inner wall of the closest bifurcation. In fig. 9a, we observe two separated eddies but they are of unequal strengths. A maximum secondary velocity reaching 50~o of the mean axial velocity is measured near the dorsal wall. Velocities near the center of the tube lie between 16~o and 19~ of the mean axial velocity. At station 10, the mean secondary velocity represents 16.1 ~o of the mean axial velocity.
E X P I R A T O R Y FLOW
Secondary velocities in expiratory flow were measured only in the trachea because probes could not be inserted at any other branches without interfering air flow. We investigated station 3 located at 1 diameter from the carina and station 2S located at 2 diameters from the carina. The Reynolds number was 1060 at station 2S and slightly less at station 3 where the cross-section was non-circular. Even at these Reynolds numbers, we noticed that the flow regime was not completely laminar. Thus the data obtained after the AC component was filtered out were less accurate than the inspiratory flow data. Results are plotted in figs. 10a,b for station 3 and in figs. l l a , b for station 2S. The conventions for these figures are the same as those for figs. 5-9: A - A ' is still in the frontal plane and C - C ' in the sagittal plane. Secondary flow patterns (figs. 10a and 1 la) are quite complex at both stations. At station 3 the directions of secondary velocities seem to indicate the presence of 4 unequal eddies, the two upper eddies being more important. At station 2S the contours of the eddies seem different and the evolution from station 3 to this station is too complex to visualize. At station 3, the maximum secondary velocity is 4 0 ~ of the mean axial velocity and the mean secondary velocity is 21.5~o. At station 2S the maximum secondary velocity is 30~o of mean axial velocity and the mean secondary velocity 16~.
108
D. ISABEY A N D H . K . C H A N G TABLE 1 Average ratios of local secondary velocity to local axial velocity
Station
Flow direction
Percentage o f tracheal flow passing this station (~)
Reynolds Number
Average ratio of secondary velocity to axial velocity (%)*
2A 3 4 6 7 8 10
Expiratory Expiratory Inspiratory Inspiratory Inspiratory Inspiratory Inspiratory
100 100 45 45 20 25 20
1060 -680 680 460 560 460
21.4 21.5 13.1 10.0 15.2 15.7 15.0
100 n (Vs)i
* Average ratio = --N i~'~=I(~wherea)i N is the number of points measured at a station, v s is the magnitude of secondary velocity, and va is the magnitude of axial velocity.
STATION (a)
3
A
C
A
C"
dorsal
C
(b)
C"
dorsal
5 ,~
A'
Fig. 10. Expiratory flow measured at Station 3. Axial velocities point toward the reader. (a) Direction and magnitude of the secondary velocities; the scale gives the magnitude relative to the mean axial velocity at this cross-section. (b) Isovelocity contours of the axial flow; numerical values indicate magnitude normalized by the mean axial velocity at the cross-section.
SECONDARY VELOCITIES IN CENTRAL AIRWAYS STATI ON
(a)
2S
A
c
A
-
dorsal
c"
109
c
(b)
c"
dorsal
Fig. 11. Expiratory flow measured at Station 2S. See fig. 10 for legend.
SUMMARY BY STATISTICS In an attempt to estimate more precisely the strength of the secondary motion in inspiratory flow at the different stations studied, we computed point by point the percentage ratio of the secondary velocity to the local axial velocity. Table 1 gives the average value obtained at each station.
Discussion
Although the model geometry studied here is closer to the real airway geometry than are models used by previous investigators, there are still several idealized features which need to be discussed. Branches are rigid and of circular cross-section (except near a bifurcation), inside surfaces are dry and smooth. These different aspects have been extensively discussed by Chang and El Masry (1982). The use of steady-flow in this study has also been discussed by Chang and E1 Masry (1982). Based on the different criteria for quasi-steady flow available in the literature, it is shown that the present steady conditions match the physiological flow regime occurring in the major part of any quiet breathing cycle. The measurement method presents several limitations. The major difficulty is that the relative magnitude of secondary motion is, at many points, only slightly larger than the variability in measurements among experiments. This problem was overcome by taking output readings at the highest precision possible to the exclusion of measuring even small levels of turbulence intensity. With the previously mentioned modifications of the method of measurement, results obtained when the axial
110
D. ISABEY AND H.K. CHANG
velocity is very small are questionable even if the secondary velocity is large. However, data obtained downstream of a 180 ° bend (not presented) confirm the validity of the method when it is applied with the restrictions mentioned above. In addition, the secondary flow magnitudes measured in the model are quite reasonable if one compares them with the only available data in the literature. Olson's data as reported by Pedley et al. (1977) predict a maximum secondary velocity of 30~ of the mean axial velocity in the daughter branch of an ideal bifurcation. The present data give 40~o and 2 5 ~ at station 4. Similarly, we found that beyond the first bifurcation of the model even the faster fluid particles did not cover an entire helical cycle within 3 diameters. This result is close to certain data in the literature (Schroter and Sudlow, 1969). Data obtained during inspiration confirm the presence of eddies at sites of branching. These eddies develop quickly (station 4 is only at 0.9 diameter from carina) and they are of unequal strengths. For instance, in the left main bronchus it is clear that the eddy on the dorsal side becomes, at station 6, much stronger than the eddy on the ventral side. Although one must mention the geometrical asymmetry between the ventral side and the dorsal side at bifurcations (e.g., shape of carina) in attempting to explain this, it is still difficult to relate the secondary flow pattern to the geometrical particularities. Based on literature studies and on our present experiments, we can only state that secondary motion is highly sensitive to geometrical asymmetry. This observation highlights the need to use a realistic model as we did here. Another finding here is that highest secondary velocities were observed near the wall. For instance, at certain points close to the wall at station 4, the secondary velocities are 4 0 ~ higher than the secondary velocities measured near the center of the tube. Indeed, results qualitatively similar may be obtained from the data presented by Pedley et al. (1977) and also by Sobey (1976) who studied theoretically secondary flows in a tube of slowly varying ellipticity. Indeed, this kind of expansion precedes each bifurcation in our model and contributes, along with the change in direction, to the generation of secondary motion. Quite interesting is the decay in secondary velocity observed between station 4 and station 6. In the whole the secondary velocity decreases from 13.1 ','~, at station 4 to 10"/o at station 6. This decay might result from the decay in axial velocity observed at the wall, the latter effect resulting from viscous forces acting in the boundary layer to decelerate the fluid at the wall. An explanation has been suggested by Sobey (1976) who found that transverse velocities for flat inlet flow are larger than those for parabolic flow because the axial velocities near the wall are higher in the first case than in the second case. In order to understand more about these data it is instructive to compare the secondary velocity patterns with the contours of the isovelocities of axial flow. It is clear from figs. 6a and b, for instance, that the side of higher secondary velocities, i.e., the dorsal side, corresponds to the side of the higher axial velocity. To understand the reasons for this, one must mention the interesting result of Scherer (1972)
SECONDARY
V E L O C I T I E S IN C E N T R A L
AIRWAYS
1l l
who demonstrated for an inviscid flow that streamlines of secondary motion will be lines of constant axial velocity. According to him, this situation may be achieved experimentally for large angles of bifurcation and for Reynolds numbers approaching infinity (where viscous forces tend toward 0). Although we did not operate under such conditions, we can still consider that the pattern of secondary flows in the core flow is not profoundly dependent on viscous forces and is thus relatively insensitive to Reynolds number. This important result was demonstrated by Brundrett and Baines (1964). Consequently it is theoretically correct to look for a relation between the patterns of flow of schemes a and b presented in each figure. Using the approach proposed by Scherer, if one were to increase the Reynolds number indefinitely, at station 6 for instance, one might guess that the extrapolated flow pattern in fig. 6b could possibly approach the streamlines of secondary flow given in fig. 6a. Furthermore, one can assume that if an asymmetry existed in the isovelocity contours of axial flow, the pattern of secondary flow would likely be asymmetrical also. The same approach might be used at station 10. We conclude that the axial flow data obtained separately corroborate the secondary flow data. Secondary flow patterns obtained downstream of a second bifurcation at stations 7 and 8 are more difficult to recognize. Pedley et al. (1977) also suspected more complexity of the flow patterns beyond the second bifurcation. Secondary motion at stations 7 and 8 necessarily results from a complex combination of the motion generated by the bifurcation immediately upstream and the motion previously established at station 6. Our data show that the closer bifurcation has more influence than the farther one. It is of interest to mention that at stations 7 and 8 the Reynolds number is smaller than at station 4 or 6 and the turn of the airways is less abrupt than at station 10. Consequently, the viscous effects are presumably stronger and interact more extensively than they do at the other stations. This interaction constitutes an additional difficulty and may explain, in the absence of an adequate boundary layer theory, why we do not recognize at station 7 the simple secondary motions we might expect from inviscid fluid theory. A more complete theory has been developed by Singh (1974) for entry flow in a curved pipe. Interestingly, he finds that as the boundary layer of the retarded fluid is growing all around the tube (which results by the conservation of mass in an accelerated axial motion in the core), a radial flow converges toward the center of the cross-section. Taking into account the importance of viscous effect at stations 7 and 8, the large radial component of the secondary velocities measured near the wall at these stations can likely be attributed to this 'entry flow' effect. Somewhat related to these phenomena is the result that maximum or mean secondary velocities downstream of the second bifurcation are not drastically larger than those measured in the left mainstem bronchus. Actually, part of the gain in energy for transverse motion occurring at the second bifurcation level is probably dissipated by viscous friction at the wall. The secondary flow patterns in expiration obtained at stations 3 and 2S are also complex and the marked discrepancies between station 3 and station 2S,
112
D. ISABEY AND H.K. CHANG
separated by a distance o f one diameter, are s o m e w h a t disturbing. One must emphasize that secondary flows in expiration result f r o m the junction o f two streams c o m i n g together with different angles o f incidence, different mean axial velocities and different axial and secondary flow patterns. O u r measured axial flow patterns (figs. 10b and l l b ) show the existence o f a high velocity gradient near the wall, agreeing with Pedley et al. (1977) who have predicted a very thin b o u n d a r y layer during expiration. Accordingly, the fluid recirculation due to secondary m o t i o n in the core flow occurs in a very thin zone at the wall. U n f o r t u n a t e l y the present m e t h o d o f measurement did not allow us to investigate this region. The present findings are o f interest in studying particle deposition and gas or aerosol dispersion in the airways. The presence o f secondary eddies has been confirmed and a quantitative estimate has been provided for the first generations o f the airways. Secondary flows never exceeded 21.5'j!~, o f the mean axial flow in either the expiratory direction or the inspiratory direction. Note however, that m a x i m u m secondary velocities were observed near the wall, perhaps limiting the particle deposition. F o r reasons explained earlier in the text, we consider that the present results m a y be e x t r a p o l a t e d f o r a large range o f Reynolds numbers, at least for the measured points in the core flow. M o r e investigations o f the velocities in the b o u n d a r y layer are necessary, especially during expiratory flow.
Acknowledgements The a u t h o r s are indebted to Mr. Albert H a g e m a n n for his skillful and artistic m a n u f a c t u r i n g o f the model used in this study. This study was supported by the Medical Research Council o f Canada.
References Acrivlellis, M. (1978). An improved method for determining the flow field of multidimensional flows of any turbulence intensity. DISA INFO No. 23. Bartenwerfer, M. (1979). Remarks on hot wire anemometry using "squared signal' (letter to editor). DISA INFO No, 24. Brundrett, E. and W.D. Baines (1964). The production and diffusion of vorticity in duct flow. J. Fluid Mech. 19: 375-394. Caro, C.G. (1966). The dispersion of indicator flowing through simplified models of the circulation and its relevance to velocity profile in blood vessels. J. Physiol. (London) 185: 501-519. Champagne, F. H. (1967). Turbulence measurements with inclined hot wires. J. Fluid Mech. 28: 153-175. Chang, H.K. and O. E1 Masry (1982). A model study of flow dynamics in human central airways. Part I : Axial velocity profiles. Respir. Physiol. 49 : 75-95. Horsfield, K., G. Dart, D.E. Olson, G.F. Filley and G. Cumming (1971). Models of the human bronchial tree. J. Appl. Physiol. 31: 207-217. Jorgensen, F. E. (1971). Directional sensitivity of wire and fiber-film probes. DISA INFO No. 11. Lighthill, M.J. (1972). Physiological fluid dynamics: A survey. J. Fluid Mech. 52 (Part 3): 475-497. McDonald, D.A. (1974). Blood Flow in Arteries. 2nd ed. London, Arnold.
SECONDARY VELOCITIES IN CENTRAL AIRWAYS
113
Olson, D. E. (1971). Fluid mechanics relevant to respiration: Flow within curved or elliptical tubes and bifurcating systems. Ph.D. thesis. Imperial College, London. Pedley, T.J., R.C. Schroter and M.F. Sudlow (1977). Gas flow and mixing in the airways. In: Bioengineering Aspects of the Lung, ed. by J. B. West. New York, Marcel Dekker, pp. 163-265. Schroter, R.C. and M. F, Sudlow (1969). Flow patterns in models of the human bronchial airways. Respir. Physiol. 7: 341-355. Sherer, P.W. (1972). A model for high Reynolds number flow in a human bronchial bifurcation. J. Biomech. 5:223 229. Singh, M.P. (1974). Entry flow in a curved pipe. J. Fluid Mech. 65 (Part 3): 517-539. Sobey, 1. J. (1976). Inviscid secondary motions in a tube of slowly varying ellipticity. J. Fluid Mech. 73 (Part 4): 621-639. White, C. M. (1929). Streamline flow through curved pipes. Proc. R. Soe. London Ser. A. 103: 645-663.
Respiration Physiology (1982) 49, 97-113 Elsevier Biomedical Press
A M O D E L S T U D Y O F F L O W D Y N A M I C S IN H U M A N C E N T R A L A I R W A Y S . P A R T II: S E C O N D A R Y F L O W V E L O C I T I E S
D. ISABEY* and H . K . C H A N G Biomedical Engineering Unit and Department of Physiology, McGill Univerjty, Montreal, Quebec, Canada
Abstract. Secondary velocity components perpendicular to the tube axis were measured in a 3 : 1 scale model of the human central airways. Slanted hot-wire probes were introduced axially in order to measure the secondary velocities at about 120 points for each of the 7 stations investigated. Secondary velocities in the inspiratory direction never exceeded a mean value of 18% of the mean axial velocity. Secondary velocities in the expiratory direction reached a mean value of 21.5% of the mean axial velocity. In the inspiratory direction, two unequal eddies were formed in the left main bronchus and in the right upper lobe. Moreover, maximum velocites were observed near the wall and the decay of secondary velocities in the left main bronchus was observed. The secondary flow patterns observed in the left upper and lower lobes after the second bifurcation were difficult to recognize, although they seemed to be more influenced by the second bifurcation. The complexity of the flow pattern was reinforced by viscous effects acting near the wall. In the expiratory direction, only two stations in the trachea were measured ; four uneven eddies seemed to have existed, with the ventral eddies appearing to be predominant. Overall, the secondary velocity magnitudes as well as the patterns of eddies were very dependent on the geometry of the model used. Air flow patterns Airway model Hot-wire anemometry
Secondary flow Tracheo-bronchial tree
This is Part II of a three-part study aimed at understanding the air flow patterns in the central airways. In this part, the object is to measure the secondary (tangential and radial) velocity components in various airway branches. McDonald (1960) was probably the first to demonstrate the presence of eddies in fluid flows through a branching tube. Using a visualization technique, Schroter and Sudlow (1969) showed, for inspiratory flow and different Reynolds numbers, the existence of two helical vortices resulting from secondary motion downstream of Accepted.for publication 2 March 1982 Technical assistance was afforded by Christopher H. Bracken. * Present address: U. 68 INSERM, H6pital Saint Antoine, F-75571 Paris Cedex 12, France. 0034-5687/82/0000-0000/$02.75 © Elsevier Biomedical Press
98
D. ISABEYAND H. K. CHANG
the carina. During expiratory flow, four similar vortices were observed. The first investigator who actually measured secondary velocities in curved or elliptical tubes and bifurcating systems was apparently Olson (1971); his data were extensively quoted in a recent review of pulmonary fluid mechanics by Pedley et al. (1977). To our knowledge, in the last ten years there have been no other studies of secondary motion relevant to respiratory flow. In spite of this hiatus, secondary motion is frequently invoked in the literature of respiratory physiology because it plays a direct and an indirect role in such phenomena as particle deposition and gas dispersion in airways. Experimental studies of dispersion in bent tubes by Caro (1966) have shown that secondary flows do act to promote lateral mixing of injected substances. Such mixing observed for substances of low diffusivity and in the absence of turbulence would actually be very slow without secondary flow. According to Lighthill (1972), the pattern of secondary motion observed in steady state could explain why the critical Reynolds number for transition to turbulence increases from 2000 for straight pipes to a typical 6000 for curved pipes. Applying this result to high flow in airways, some authors consider that secondary flow assists the decay in turbulence observed from large airways to small airways and hence contributes to diminished dispersion and deposition of particles. As stated by Pedley et al. (1977), the degree of mixing is notably affected by the secondary motions, particularly by the strength of these, which depend upon local velocities. To supply needed information in this field, we measured secondary velocities in the first generations of the central airways, i.e., where secondary velocities are probably greatest.
Apparatus and methods MODEL A 3 : 1 large-scale model of the human central airways specially designed for velocity measurements was used. It was constructed from acrylic plastic blocks and heattreated tubes all manually finished and polished. Airways from the trachea to the lobar bronchi were represented. The dimensions were based on the airway geometry proposed by Horsfield et al. (1971). Details of the geometry are given in Part I of this study by Chang and E1 Masry (1982). A photograph of the regular model along with the linear resistors, the special adaptors as well as the hot-wire probe assembly, is shown in fig. 1. A detailed arrangement of the linear resistors and conic restrictors used during axial velocity measurements is shown in the insert (detail a) of fig. 2; for secondary velocity measurements, special adaptors were used to replace the regular pieces as shown in the insert (detail b) of fig. 2; fig. 3 shows a sketch of the model and the positions of the stations investigated. These special adaptors were designed because the
SECONDARY VELOCITIES IN CENTRAL AIRWAYS
99
Fig. 1. Photograph of the model along with the special adaptors specifically designed for secondary velocity measurements.
secondary flow measurement technique required that the measuring probe be inserted axially from the end of a branch. These carefully machined pieces allowed normal air passage as well as distal probe insertion and were used one at a time. By pre-drilling a number of entry ports at different angles in an adaptor and rotating this piece about its own axis, one could make the probe reach the desired positions for velocity measurement. By making repeated measurements at a given station with these pieces attached at the exit and at different rotational angles, we verified that the axial velocity profiles throughout the model did not vary more than the normal variation from one experiment to the other, namely, between 5 ~ and 10~o. This was because the linear resistors placed distally to the model offered more than 95~o of the overall resistance in the model. Thus, with the aid of the special adaptors and the linear resistors, we were able to map out the secondary velocities over a given cross-section of the model with an axially-inserted hot-wire probe. Dynamic similarity between the model and the real airways was ensured with identical Reynolds numbers. Under these conditions, since the model scale was 3 : 1, the velocities in the model represented one-third the equivalent physiological flow.
100
D. ISABEY A N D H . K . C H A N G
compressed a!r .= Z
=.,........................ ,,~'~,,,,,,.,~, ..........= ................... '~ : . . 3.way valve " =" respiratory 1"= : [ flow i l..thermomete r == " -" ' -= i ! ] r.Li
high pressure._.~ stabilizing r ~
i reservoir . ~
tank
I
"
" I f,owmeterl I
screens / . . . . . . . . . . . . . . . . . I
:
/
~"
"
II I I
i
:
airway
=-
: ,am,..
f
~
I
I
~,
"mml',,,,L. - / / /
:......_
filter
,
:
1,.,,,,,,,,,,,,,,~,,,,,.,,,,,.~
I
/
:=
i
I
.
: ",.L needle X" val ve
-
I
/
:
-= i
L..J
"1 .................... -I
" ."
"U
anemometer display unit unit
I~'-'t..
/detail
!
~
~
~%Lq.,,,;'~ ~Iv" - " ' " ~ ' .;~
~/
a
"~
/...
-
/~ ' ~ / ~
I
t
~-~,see "N~..o details a,b
~'\
X/
res,sto,
r.e.MrJcto/
~terminal "<.~t~r[2"n,?l
°'-~L~__~J
linear resist
=-'.
= ~ expiratory
hOelJe||l|JeJ|eooeo|eegoolale|||||||||||e|eoJ~ | f I OW
Fig. 2. A schematic of the flow system and details o f distal units used in axial (detail a) and secondary (detail b) velocity measurements.
FLOW SYSTEM
The system used to supply air in the model (see fig. 2) was essentially identical to that described by Chang and E1 Masry (1982). However, to achieve the most stable flow possible, a high-pressure stabilization tank fed by a high-pressure line (100 psi) was placed upstream of the decompression valve and the air filter. Air flow rates were monitored with a rotameter of 1% precision. A single constant flow rate of 0.6 L/sec (equivalent to a physiological flow of 0.2 L/sec) was used
S E C O N D A R Y VELOCITIES IN C E N T R A L A I R W A Y S
101
Ce
D"
B"
.2A~~o ( A' -zs C -3
Cross-section at a given station
~
R.U.
.U.
8 Li.
RM. RL
Fig. 3. Sketch of the model showing the positions of the stations of measurement.
during the experiments. As was the case in the axial flow study, the inspiratory direction of flow was achieved by connecting the air source to a second stabilization reservoir immediately upstream of the trachea to ensure a nearly flat entrance velocity profile in the trachea. For expiratory experiments, the air source was directed toward a cylindrical manifold placed below the model and attached to the five lobar bronchi. In this case, the reservoir at the top of the model was opened to room air. The flow distribution was controlled with the same system as used in the axial flow study. Linear resistors made of copper tubes packed with stainless-steel capillaries were connected to the five lobar bronchi. Since the resistance of these tubes was much greater than the overall resistance of the model, the flow was distributed in invariable proportion throughout the system.
VELOCITY MEASUREMENT
A constant-temperature hot-wire anemometer system (Disa 55M01) was used as in the axial-flow study (Chang and E1 Masry, 1982). However, slanted gold-plated probes (type 55P02) were used since they are known to be more sensitive to tangential velocity components than are straight probes (Champagne, 1967). The slanting angle, denoted by ~t in fig. 4, was always 45 °. The probe consists of two prongs sustaining a gold-plated wire 3 mm in length. The sensitive part of the wire is not gold-plated and is 1.25 mm in length and 5 /~m in diameter. The support for the probe is mounted on a micromanipulator to measure and control its longitudinal displacement. The angular position of the probe (angle ~b) is controlled using an index fixed on the support. The base of the micromanipulator is tightly screwed to the model at a distal branch (see detail b, fig. 2).
102
D. ISABEY A N D H . K . C H A N G
f3
Z
W
z
Fig. 4. Angular positions (denoted by if) of the slanted hot-wire probe during velocity measurements; axes x, y, z define the inertial coordinate system; U, V, W are velocity components; ct is the slanting angle o f the probe.
During an experiment, the probe was rotated about its own axis in order to register different cooling effects due to the non-axial orientation of the local velocity vector. Note that according to Jorgensen (1971) the effective cooling velocity acting on the sensor may be expressed as U2rf = W 2 + k~V 2 + k 2 W 2
(1)
where k~ and k2 are respectively the correction factors for the yaw and pitch angles and U, V, W are the velocity components normal to, along, and binormal to the wire, respectively (see fig. 4). The hot-wire probe was calibrated in a manner similar to that used in the axial-flow study reported in the companion paper (Chang and E1 Masry, 1982). The different sensitive coefficients k~ and k 2 w e r e defined by placing the wire under appropriate angles at the maximum velocity of a known Poiseuille flow. According to Acrivlellis (1978), it is theoretically possible to obtain the three velocity components in an inertial coordinate system with a proper choice of transformation matrix and from six independent readings corresponding to six different angular positions of the probe referred by the ~b angle (fig. 4). In solving a system of equations derived from the general eq. (1), Acrivlellis obtained the following three products of components: UV =
U~ - U~ 2(1 -k~) sin 2a
(2)
UW =
U42 - U22 2(1 -k~) sin 2~
(3)
VW =
(U 2 - U~) V'r2 + U~ - U~ 2 v/2 (sin 2 a + k~ cos2 ~ - k 2)
(4)
SECONDARY VELOCITIES IN CENTRAL AIRWAYS
103
from which the sign and the modulus of the components may be easily deduced; is the probe angle with respect to the inertial coordinate system x, y, z, and U~ ... U6 are the six readings. This method is theoretically correct in laminar flow but has been shown to be questionable in turbulent flow (Bartenwerfer, 1979). Since some degree of turbulence exists even at a flow rate of 1.2 L/sec, and since we found that this led to considerable errors in the computed secondary velocity components, we were forced to make all the secondary velocity measurements at a low flow rate of 0.6 L/sec, or half of the lower flow rate used in the axial flow study. Furthermore, we modified the Acrivlellis method by: (i) inserting the probe parallel to the largest velocity component, namely the axial component as already specified, (ii) taking more than 6 readings at each point in order to eliminate the terms that introduce too large an error, and (iii) obtaining the axial flow component separately with a straight probe in the same manner as did Chang and E1 Masry (1982). For condition (ii), a computer program gave, at each point, the product of the axial flow magnitude by the secondary flow magnitude and the direction of the secondary velocity. Note that if U is the axial flow component and V and W are the secondary flow components, the two quantities above are independent of the product (V • W) which has the larger margin of error (see eq. (4)). Finally, the magnitude of secondary flow was obtained by dividing the product ( U . x / r ~ + W 2) by the axial velocity magnitude (U) that had been obtained separately. The present method was tested in measuring secondary velocities after a 180 ° bend in a pipe. The ratio of the radius of curvature to the pipe diameter was 2.3 and the Reynolds number was 1030. These parameters were chosen to be similar to those used in a previous study (Olson, 1971). The results obtained were very similar to the theoretical prediction (White, 1929) and to Olson's data as presented in Pedley et aL (1977). An estimation of error for the present method has shown that the absolute error in positioning angle was about +_ 3 ° and the relative error on secondary velocity magnitude was 10%. However, this error was increased and reached about 30~o when the magnitude of the axial velocity was small, i.e., near the wall.
Results INSP1RATORY FLOW Stations 4, 6, 7, 8, 10 whose positions are given in fig. 3, were studied during inspiratory flow. Stations 4 and 6 are located respectively at 0.9 and 3.9 diameters from the carina. These two stations were chosen in order to follow the evolution of secondary flow through the left main bronchus. Stations 7 and 8 are located respectively at 2.2 diameters and 1.7 diameters from the flow divider of the right
104
D. ISABEY AND H.K. CHANG
lung. The last three stations were chosen because secondary flow was presumably stronger in these branches than it would be in other branches. At each station secondary velocities were measured mainly along 4 diameters and at 14 different points along each diameter. Additional points near the wall along the intermediate diameters were also studied. The results obtained are presented in figs. 5a to 9a. The diameter A - A ' is in the frontal plane and the diameter C - C ' in the sagittal plane, with C always on the dorsal side. The vectors plotted on these panels represent the velocity components (normalized by the mean axial velocity) measured in the plane normal to the airway axis. Thus the true velocity at a given point of an investigated airway cross-section is the resultant vector of this plotted velocity vector and the axial velocity component. The axial velocity component is directed into the paper in inspiratory flow and toward the reader in expiratory flow. The corresponding isovelocity contours obtained from separate axial velocity measurements are presented in figs. 5b to 9b. At station 4 where the Reynolds number is 680, the isovelocity contours (fig. 5b) reveal the existence of a peak axial velocity near the inner wall of the bifurcation. This pattern is similar to the patterns observed at higher Reynolds numbers and is extensively described by Chang and E1 Masry (1982). In fig. 5a it is relatively easy to recognize two semi-circular eddies developing on each side of the diameter A-Aft The secondary velocities along this diameter are directed toward A', i.e., toward the flow divider; the two eddy streamlines along the tube wall rejoin each other at A after completing their respective semi-circles. The fluid in this region experiences centrifugal forces due to the airway curvature and tends to move toward the inner wall of the bifurcation. At the same time, there is recirculation of fluid near the walls due to the establishment of a trasverse pressure gradient across the S T A T ION
(a)
C dorsal
A
4
A
(b)
.(3'
C' dors
A' A" Fig. 5. Inspiratory flow measured at Station 4. Axial velocitiespoint into the paper. (a) Direction and magnitude of the secondary velocities; the scale gives the magnitude relative to the mean axial velocity at this cross-section. (b) Isovelocitycontours of the axial flow; numerical values indicate magnitudes normalized by the mean axial velocityat the cross-section.
SECONDARY VELOCITIES IN CENTRAL AIRWAYS STATION
(a)
105
6
(b)
A
C dorsal
C"
C dorsal
0.5
I,,I=N=N
A'
Fig. 6. Inspiratory flow measured at Station 6. See fig, 5 for legend.
pipe, balancing the centrifugal forces. The two secondary eddies are not symmetrical as they would be in an ideal bifurcation. Correspondingly, the isovelocity contours are not symmetrical about the diameter A-A'. The maximum secondary velocity along A - A ' is about 27% of the mean axial velocity while the maximum secondary velocity near the wall reaches 40% of the mean axial velocity on the ventral side and 25% of the same velocity on the dorsal side. The average secondary velocity is about 14.5% of the mean axial velocity for the data plotted in fig. 5a. Data obtained at station 6 (fig. 6a,b) for the same Reynolds number (680) differ from the data obtained at the preceding station. While the axial velocity profiles develop a bi-peak structure (see Chang and E1 Masry, 1982, and fig. 5b) and become less symmetrical with respect to A-A', the eddy (fig. 6a) appearing initially on the dorsal side becomes predominant with a maximum velocity near the wall reaching 33% of the mean axial velocity. Simultaneously the eddy on the ventral side seems rather weak. The velocity of the fluid motion in the center of the tube is only 15)/o of the mean axial velocity and is no longer parallel to the A-A" direction as it was at station 4. It is of interest to note that the stronger eddy occupies more than just the right half of the branch. Accordingly, there is not a unique stagnation point for secondary velocities but presumably a curved line of zero secondary velocities. Finally, at station 6, the overall mean secondary velocity represents 10.8% of the mean axial velocity. Data obIained at stations 7 and 8 (figs. 7a,b; 8a,b), where Reynolds numbers are respectively 460 and 560, are more difficult to describe. It is difficult to figure out the contours of eddies although one can ascertain, at least from data at station 8, that there are more than two separate eddies downstream of the second bifurcation. Interestingly, the main secondary motion along the diameter A - A ' is directed essentially toward the more immediate flow divider for both stations while the secondary motion resulting from the first bifurcation is rather weak in this region.
106
D. ISABEY A N D H. K. C H A N G STATION
(a)
7 A
A
C dorsal
C"
Co
Oh)
.C"
dorsal
Q5 A"
A"
Fig. 7. Inspiratory flow measured at Station 7. See fig. 5 for legend. STATION
(a)
8
A
O
A
,C"
C'
dorsal
Oh)
dor,'
0.1 0.5 A'
A'
Fig. 8. Inspiratory flow measured at Station 8. See fig. 5 for legend.
This observation agrees with the axial flow results showing that the more immediate bifurcation has the stronger effect on axial flow (Chang and E1 Masry, 1982). Near the wall the effect of the strongest eddy observed at station 6 can be felt since at many exterior points, the fluid has a secondary velocity component turning counterclockwise as at station 6. Note also that near the wall, there is a large radial component directed toward the center of the tube. As at stations 4 and 6, the maximum secondary velocities are observed near the wall. They reach 2 8 ~ of the mean axial velocity at station 7 and 3 4 ~ of the mean axial velocity at station 8. The mean secondary velocity at station 7 represents 13.9~ of mean axial velocity and 17.4~ at station 8. Results obtained at station 10 (fig. 9a,b) are simpler to describe. The Reynolds number is 460. A peak of axial velocity accompanied by a high velocity gradient
S E C O N D A R Y VELOCITIES IN C E N T R A L A I R W A Y S STATION
(a)
10
A"
C"t I ventral
A"
C
1
A
107
c" ventra
(b)
-C
0.5
A
Fig. 9. Inspiratory flow measured at Station 10. See fig. 5 for legend.
is observed near the inner wall of the closest bifurcation. In fig. 9a, we observe two separated eddies but they are of unequal strengths. A maximum secondary velocity reaching 50~o of the mean axial velocity is measured near the dorsal wall. Velocities near the center of the tube lie between 16~o and 19~ of the mean axial velocity. At station 10, the mean secondary velocity represents 16.1 ~o of the mean axial velocity.
E X P I R A T O R Y FLOW
Secondary velocities in expiratory flow were measured only in the trachea because probes could not be inserted at any other branches without interfering air flow. We investigated station 3 located at 1 diameter from the carina and station 2S located at 2 diameters from the carina. The Reynolds number was 1060 at station 2S and slightly less at station 3 where the cross-section was non-circular. Even at these Reynolds numbers, we noticed that the flow regime was not completely laminar. Thus the data obtained after the AC component was filtered out were less accurate than the inspiratory flow data. Results are plotted in figs. 10a,b for station 3 and in figs. l l a , b for station 2S. The conventions for these figures are the same as those for figs. 5-9: A - A ' is still in the frontal plane and C - C ' in the sagittal plane. Secondary flow patterns (figs. 10a and 1 la) are quite complex at both stations. At station 3 the directions of secondary velocities seem to indicate the presence of 4 unequal eddies, the two upper eddies being more important. At station 2S the contours of the eddies seem different and the evolution from station 3 to this station is too complex to visualize. At station 3, the maximum secondary velocity is 4 0 ~ of the mean axial velocity and the mean secondary velocity is 21.5~o. At station 2S the maximum secondary velocity is 30~o of mean axial velocity and the mean secondary velocity 16~.
108
D. ISABEY A N D H . K . C H A N G TABLE 1 Average ratios of local secondary velocity to local axial velocity
Station
Flow direction
Percentage o f tracheal flow passing this station (~)
Reynolds Number
Average ratio of secondary velocity to axial velocity (%)*
2A 3 4 6 7 8 10
Expiratory Expiratory Inspiratory Inspiratory Inspiratory Inspiratory Inspiratory
100 100 45 45 20 25 20
1060 -680 680 460 560 460
21.4 21.5 13.1 10.0 15.2 15.7 15.0
100 n (Vs)i
* Average ratio = --N i~'~=I(~wherea)i N is the number of points measured at a station, v s is the magnitude of secondary velocity, and va is the magnitude of axial velocity.
STATION (a)
3
A
C
A
C"
dorsal
C
(b)
C"
dorsal
5 ,~
A'
Fig. 10. Expiratory flow measured at Station 3. Axial velocities point toward the reader. (a) Direction and magnitude of the secondary velocities; the scale gives the magnitude relative to the mean axial velocity at this cross-section. (b) Isovelocity contours of the axial flow; numerical values indicate magnitude normalized by the mean axial velocity at the cross-section.
SECONDARY VELOCITIES IN CENTRAL AIRWAYS STATI ON
(a)
2S
A
c
A
-
dorsal
c"
109
c
(b)
c"
dorsal
Fig. 11. Expiratory flow measured at Station 2S. See fig. 10 for legend.
SUMMARY BY STATISTICS In an attempt to estimate more precisely the strength of the secondary motion in inspiratory flow at the different stations studied, we computed point by point the percentage ratio of the secondary velocity to the local axial velocity. Table 1 gives the average value obtained at each station.
Discussion
Although the model geometry studied here is closer to the real airway geometry than are models used by previous investigators, there are still several idealized features which need to be discussed. Branches are rigid and of circular cross-section (except near a bifurcation), inside surfaces are dry and smooth. These different aspects have been extensively discussed by Chang and El Masry (1982). The use of steady-flow in this study has also been discussed by Chang and E1 Masry (1982). Based on the different criteria for quasi-steady flow available in the literature, it is shown that the present steady conditions match the physiological flow regime occurring in the major part of any quiet breathing cycle. The measurement method presents several limitations. The major difficulty is that the relative magnitude of secondary motion is, at many points, only slightly larger than the variability in measurements among experiments. This problem was overcome by taking output readings at the highest precision possible to the exclusion of measuring even small levels of turbulence intensity. With the previously mentioned modifications of the method of measurement, results obtained when the axial
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D. ISABEY AND H.K. CHANG
velocity is very small are questionable even if the secondary velocity is large. However, data obtained downstream of a 180 ° bend (not presented) confirm the validity of the method when it is applied with the restrictions mentioned above. In addition, the secondary flow magnitudes measured in the model are quite reasonable if one compares them with the only available data in the literature. Olson's data as reported by Pedley et al. (1977) predict a maximum secondary velocity of 30~ of the mean axial velocity in the daughter branch of an ideal bifurcation. The present data give 40~o and 2 5 ~ at station 4. Similarly, we found that beyond the first bifurcation of the model even the faster fluid particles did not cover an entire helical cycle within 3 diameters. This result is close to certain data in the literature (Schroter and Sudlow, 1969). Data obtained during inspiration confirm the presence of eddies at sites of branching. These eddies develop quickly (station 4 is only at 0.9 diameter from carina) and they are of unequal strengths. For instance, in the left main bronchus it is clear that the eddy on the dorsal side becomes, at station 6, much stronger than the eddy on the ventral side. Although one must mention the geometrical asymmetry between the ventral side and the dorsal side at bifurcations (e.g., shape of carina) in attempting to explain this, it is still difficult to relate the secondary flow pattern to the geometrical particularities. Based on literature studies and on our present experiments, we can only state that secondary motion is highly sensitive to geometrical asymmetry. This observation highlights the need to use a realistic model as we did here. Another finding here is that highest secondary velocities were observed near the wall. For instance, at certain points close to the wall at station 4, the secondary velocities are 4 0 ~ higher than the secondary velocities measured near the center of the tube. Indeed, results qualitatively similar may be obtained from the data presented by Pedley et al. (1977) and also by Sobey (1976) who studied theoretically secondary flows in a tube of slowly varying ellipticity. Indeed, this kind of expansion precedes each bifurcation in our model and contributes, along with the change in direction, to the generation of secondary motion. Quite interesting is the decay in secondary velocity observed between station 4 and station 6. In the whole the secondary velocity decreases from 13.1 ','~, at station 4 to 10"/o at station 6. This decay might result from the decay in axial velocity observed at the wall, the latter effect resulting from viscous forces acting in the boundary layer to decelerate the fluid at the wall. An explanation has been suggested by Sobey (1976) who found that transverse velocities for flat inlet flow are larger than those for parabolic flow because the axial velocities near the wall are higher in the first case than in the second case. In order to understand more about these data it is instructive to compare the secondary velocity patterns with the contours of the isovelocities of axial flow. It is clear from figs. 6a and b, for instance, that the side of higher secondary velocities, i.e., the dorsal side, corresponds to the side of the higher axial velocity. To understand the reasons for this, one must mention the interesting result of Scherer (1972)
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V E L O C I T I E S IN C E N T R A L
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1l l
who demonstrated for an inviscid flow that streamlines of secondary motion will be lines of constant axial velocity. According to him, this situation may be achieved experimentally for large angles of bifurcation and for Reynolds numbers approaching infinity (where viscous forces tend toward 0). Although we did not operate under such conditions, we can still consider that the pattern of secondary flows in the core flow is not profoundly dependent on viscous forces and is thus relatively insensitive to Reynolds number. This important result was demonstrated by Brundrett and Baines (1964). Consequently it is theoretically correct to look for a relation between the patterns of flow of schemes a and b presented in each figure. Using the approach proposed by Scherer, if one were to increase the Reynolds number indefinitely, at station 6 for instance, one might guess that the extrapolated flow pattern in fig. 6b could possibly approach the streamlines of secondary flow given in fig. 6a. Furthermore, one can assume that if an asymmetry existed in the isovelocity contours of axial flow, the pattern of secondary flow would likely be asymmetrical also. The same approach might be used at station 10. We conclude that the axial flow data obtained separately corroborate the secondary flow data. Secondary flow patterns obtained downstream of a second bifurcation at stations 7 and 8 are more difficult to recognize. Pedley et al. (1977) also suspected more complexity of the flow patterns beyond the second bifurcation. Secondary motion at stations 7 and 8 necessarily results from a complex combination of the motion generated by the bifurcation immediately upstream and the motion previously established at station 6. Our data show that the closer bifurcation has more influence than the farther one. It is of interest to mention that at stations 7 and 8 the Reynolds number is smaller than at station 4 or 6 and the turn of the airways is less abrupt than at station 10. Consequently, the viscous effects are presumably stronger and interact more extensively than they do at the other stations. This interaction constitutes an additional difficulty and may explain, in the absence of an adequate boundary layer theory, why we do not recognize at station 7 the simple secondary motions we might expect from inviscid fluid theory. A more complete theory has been developed by Singh (1974) for entry flow in a curved pipe. Interestingly, he finds that as the boundary layer of the retarded fluid is growing all around the tube (which results by the conservation of mass in an accelerated axial motion in the core), a radial flow converges toward the center of the cross-section. Taking into account the importance of viscous effect at stations 7 and 8, the large radial component of the secondary velocities measured near the wall at these stations can likely be attributed to this 'entry flow' effect. Somewhat related to these phenomena is the result that maximum or mean secondary velocities downstream of the second bifurcation are not drastically larger than those measured in the left mainstem bronchus. Actually, part of the gain in energy for transverse motion occurring at the second bifurcation level is probably dissipated by viscous friction at the wall. The secondary flow patterns in expiration obtained at stations 3 and 2S are also complex and the marked discrepancies between station 3 and station 2S,
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separated by a distance o f one diameter, are s o m e w h a t disturbing. One must emphasize that secondary flows in expiration result f r o m the junction o f two streams c o m i n g together with different angles o f incidence, different mean axial velocities and different axial and secondary flow patterns. O u r measured axial flow patterns (figs. 10b and l l b ) show the existence o f a high velocity gradient near the wall, agreeing with Pedley et al. (1977) who have predicted a very thin b o u n d a r y layer during expiration. Accordingly, the fluid recirculation due to secondary m o t i o n in the core flow occurs in a very thin zone at the wall. U n f o r t u n a t e l y the present m e t h o d o f measurement did not allow us to investigate this region. The present findings are o f interest in studying particle deposition and gas or aerosol dispersion in the airways. The presence o f secondary eddies has been confirmed and a quantitative estimate has been provided for the first generations o f the airways. Secondary flows never exceeded 21.5'j!~, o f the mean axial flow in either the expiratory direction or the inspiratory direction. Note however, that m a x i m u m secondary velocities were observed near the wall, perhaps limiting the particle deposition. F o r reasons explained earlier in the text, we consider that the present results m a y be e x t r a p o l a t e d f o r a large range o f Reynolds numbers, at least for the measured points in the core flow. M o r e investigations o f the velocities in the b o u n d a r y layer are necessary, especially during expiratory flow.
Acknowledgements The a u t h o r s are indebted to Mr. Albert H a g e m a n n for his skillful and artistic m a n u f a c t u r i n g o f the model used in this study. This study was supported by the Medical Research Council o f Canada.
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Olson, D. E. (1971). Fluid mechanics relevant to respiration: Flow within curved or elliptical tubes and bifurcating systems. Ph.D. thesis. Imperial College, London. Pedley, T.J., R.C. Schroter and M.F. Sudlow (1977). Gas flow and mixing in the airways. In: Bioengineering Aspects of the Lung, ed. by J. B. West. New York, Marcel Dekker, pp. 163-265. Schroter, R.C. and M. F, Sudlow (1969). Flow patterns in models of the human bronchial airways. Respir. Physiol. 7: 341-355. Sherer, P.W. (1972). A model for high Reynolds number flow in a human bronchial bifurcation. J. Biomech. 5:223 229. Singh, M.P. (1974). Entry flow in a curved pipe. J. Fluid Mech. 65 (Part 3): 517-539. Sobey, 1. J. (1976). Inviscid secondary motions in a tube of slowly varying ellipticity. J. Fluid Mech. 73 (Part 4): 621-639. White, C. M. (1929). Streamline flow through curved pipes. Proc. R. Soe. London Ser. A. 103: 645-663.