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An experimental study of nonrheometric flows of viscoelastic fluids
183
Journal of Non-Newtonian Fluid Mechanics, 1 (1976) @ Elsevier Scientific Publishing Company, Amsterdam
183-198 - Printed
in The Netherlands
AN EXPERIMENTAL STUDY OF NONRHEOMETRIC VISCOELASTIC FLUIDS II. FLOW IN A LIQUID JET EXITING
D.D. GOULDEN
FROM A VERTICAL
TUBE
* and W.C. MacSPORRAN
Schools of Studies (Ct. Britain) (Received
FLOWS OF
December
in Chemical
Engineering,
Uniuersity
of Bradford,
Bradford,
Yorkshire
21, 1975)
Summary A laser anemometer has been used to study the region of accelerating shear flow near the exit of a vertical tube. It is in this region that the transition between steady laminar shear flow in the upstream tube and elongational flow in the downstream liquid jet takes place. Downstream velocity profiles were measured for solutions of 0.9% polyacrylamide in 85% glycerol/water and 0.9% polyacrylamide in water. Reynolds numbers (based on wall conditions in the fully developed upstream flow) ranged from 45 to 310 and Froude numbers from 0.294 to 4.11. Tubes, having sharpedged and rounded exit corners, with diameters of 1.25 cm and 1.90 cm were used Upstream velocity profiles were measured for a solution of 0.9% polyacrylamide in water. Reynolds numbers ranged from 16 to 670. Only tubes having sharp-edged exit corners were used. It was found that the transition region did not extend upstream into the tube but was confined to the downstream jet. The transition took place over a distance of about 3-5 tube diameters depending upon the value of the Froude number. The axial distance downstream from the tube exit plane at which the velocity profile first became flat increased with increasing Froude number. The magnitude of the jet velocity at this point decreased with increasing Froude number. Th.e condition of the tube exit corner was found to influence the flow in the transition region. Downstream velocity profiles obtained using tubes having rounded exit corners initially develop more slowly than, but soon catch up with and eventually overtake, the corresponding profiles obtained using tubes with sharp-edged exit corners. * Present
address:
Lever
Bros.
Ltd.,
Port Sunlight,
Wirral,
Cheshire.
184
Downstream velocity profiles obtained for the 0.9% polyacrylamide in 85% glycerol/water solution were found to develop smoothly. The transition from steady shear flow in the tube to elongational flow in the jet took place through the combined processes of acceleration of the outer layers of the jet due to radial transfer of momentum with adjacent inner layers, the process spreading steadily inwards with increasing axial distance from the tube exit plane, and acceleration of the whole due to gravity. However, the velocity profiles obtained for the 0.9% polyacrylamide in water solution did not always develop so smoothly. At a Reynolds number of 310 and Froude number of 2.06 the radial momentum transfer process was restricted to a narrow outer region of the jet until a downstream axial distance of about 2 tube diameters was reached. Thereafter, the transition to a flat profile took place smoothly.
1. ~troduc~on The experimental study reported in this paper forms part of a wider research programme designed to obtain a better understanding of nonrheometric flows of viscoelastic fluids. A previous paper [ 1 J reported experimental measurements of the developing flow both upstream and downstream from the entry plane in a re-entrant tube geometry. In this paper, the results are presented of a preliminary experimental study of flow rearrangement in the tube exit region [ 2,3]. The flow field near the exit of a vertical tube may be divided into three regions: (1) the region of steady fully developed laminar shear flow well upstream from the tube exit plane; (2) the region of elongational flow in the falling liquid jet well downstream from the tube exit plane; and, (3) the intermediate transition region of accelerating shear flow! The flow in regions (1) and (2) falls within the general classification of rheometric flows. However, the flow in region (3) is nonrheometric. The object of the present work was to examine experimentally the nonrheometric flow in the transition region. Information was sought regarding the way in which the velocity re~rangement takes place, the extent of the region both upstream and downstream from the tube exit plane, and the influence of tube exit conditions on the rearrangement. Related experimental studies are those of Gogos [4,5] for vertical extrusion of polymer melts and Allen [6,7] for horizontal extrusion of polymer solutions. Both these workers used streak photography to measure the velocity distributions. In this work, the velocity distributions were measured using a laser anemometer. 2. Apparatus A brief description of the experimental is described in detail elsewhere [ 31.
apparatus
is given below. The apparatus
2. I Laser anemometer The laser anemometer used in this study was operated in the reference mode and is illustrated in Fig. 1. As may be seen. the ontical arrangement
beam of the
185
INCIDENT BEAM
Fig. 1. Laser anemometer.
laser anemometer has been considerably modified from that used in previous work [l]. The modifications were necessary to ensure that the incident, scattered, and reference beams had fixed orientations in the air, a situation which could not have been realized had the reference beam been allowed to pass through the liquid jet. Light from a continuous wave He-Ne gas laser (not shown) is split into two beams having approximately equal intensities. One, the incident beam, is redirected using two mirrors (M, and M,) and focused onto a spot in the flowing fluid using a lens (L). The other, the reference beam, is redirected using a mirror (Ms) and passed through a beam splitter (BS). The transmitted primary beam is reflected back along its path using a mirror (M4) and is further reflected from the rear face of the beam splitter (BS). It then passes through the aperture (A,) and the interference filter (IF) onto the cathode of the photomultiplier tube (PM). The secondary beam which is reflected from the front face of the beam splitter (BS) passes through the aperture (A,) and intersects the incident light beam at its focal spot in the flowing fluid. Not only does this secondary beam
166
provide convenient visual evidence of the position of the focal spot but also it serves a vital purpose which will be explained later. That portion of the light, which is scattered from the incident beam along the direction of the secondary beam, passes through the apertures (A, and A,) of the receiving optics onto the same area of the cathode of the photomultiplier tube as the primary reference beam. The apertures control the spot size in the flowing fluid from which scattered light reaches the cathode and also ensure the close alignment of the reference and scattered beams required for efficient photomixing. The interference filter is used to shut out background radiation having unwanted frequencies. The optical system described above was mounted on a 1-in.-thick vertical steel plate which was placed on the optical table described previously [ 11. The optical benches on which the transmitting and receiving optics were mounted were arranged symmetrically at equal angles to the tube/jet axis. These angles were adjustable so that, when measurements were being made in the falling liquid jet, the benches were inclined at equal angles below the horizontal as shown in Fig. 1. When measurements were being made in the tube, the benches were inclined at equal angles above the horizontal. With these arrangements the tube exit plane could be approached as closely as possible from both upstream and downstream directions. 2.2 Pump loop and test section The test fluid was pumped, using a variable speed gear pump, from a 70-gallon storage tank via a 1-in.-nominal-bore copper pipe and a flexible polythene hose to the test section. The test section consisted of a vertically mounted glass tube which discharged into a constant head receiver. The overflow from the constant head receiver passed to a 5-gal sump tank from which the fluid was returned to the storage tank, via a l-in.-nominal-bore copper pipe, by gravity flow. An electromagnetic flow meter was used for flow measurement and it was calibrated using a 50-gal weigh tank mounted on a compression load cell. The test section was mounted vertically on a l-m-long optical bench which was secured to the l-in-thick vertical steel plate. Both the glass tube and the constant head receiver were mounted on standard optical components and clamped to the optical bench. The test section could be moved relative to the fixed optical arrangement of the laser anemometer. Vertical movement of the test section was achieved by releasing the clamping screws and sliding it up or down the optical bench to the desired position before retightening the clamping screws. The vertical distance moved was read from the scale mounted on the side of the optical bench. Horizontal movement of the test section was effected using the horizontal transverse feed screws provided on the standard optical components. The horizontal distance moved was read from the vernier scale also provided. The constant head receiver formed an integral part of the test section and was moved with it. This ensured that the length and, therefore, the weight of the jet falling from the tube exit remained constant. It was felt that this precaution was necessary to avoid the risk of altering the flow conditions in the jet. Tubes having internal diameters of 1.25 cm and 1.90 cm were used in this work. Both sharp-edged and rounded exits were used.
187
3. Calculation of the velocity doppler frequencies
distributions
in the falling jet from the measured
Calculation of the velocity distributions in the upstream tube was carried out using the simple procedure outlined previously [l ] . Calcuation of the velocity distributions in the downstream falling jet requires a more complicated procedure. The complication is introduced through the longitudinal curvature of the free surface of the jet. The experimental measurements, obtained at a given vertical distance, y, consist of a set of Doppler frequencies, Afo, corresponding to a series of horizontal distances, x. The procedure whereby the measured Doppler frequency distributions, Afh (x,y), are converted into axial velocity distributions, u, (r,z), is outlined below. The relationship between the horizontal and vertical distances, and the radial and axial coordinates of the point of intersection of the incident and scattered light beams is obtained as follows. For a general position (3c,y) the configuration of the incident and scattered light beams, in relation to the free surface of the liquid jet, is as shown in Fig. 2. The angles made by the incident and scattered light beams in the air, 8,, are determined by the fixed optical components of the laser anemometer. Remembering that the secondary reference beam coincides with the scattered light beam, the vital purpose which its serves may now be stated. The angles, BRiand t!IRs,at which the incident and secondary reference
Fig. 2. Diagram of liquid jet showing the orientations of the incident and secondary reference light beams with respect to the free surface.
188
light beams are reflected from the free surface of the jet were measured as also was the refractive index of the liquid. Using these measurements all the other angles indicated may be calculated. The necessary relationships are listed below:
n=7
sin I& sin @li ’
&i =Sin-l
n=- sin G,, sin & ’ sin $,i 7
( 1
&=sin-l
,
(1)
sin#,,
( 1 -
n
,
The radii, Ri and R, of the jet at the points at which the incident and scattered light beams intersect the free surface were measured from photographs of the liquid jet. Simple trigonometric relationships may then be used to calculate the radial and axial coordinates of the point of intersection of the incident and scattered light beams. These are given by the following expressions: r=
2(x - 3ce) tan 8, - Ri(tan 0, - tan Oli) + R,(tan (tan Oli + tan e,)
z=(Y-Yd+(~-d
tan ezi -tan
se,,+
8, )
8(1 -tan
0,)
tan&-
tan f3,, + R, In general, z varies with r so that, for a given fixed vertical distance y and variable horizontal distance X, the point of intersection does not traverse an axial plane of the jet. The relationship between the measured Doppler frequency and the velocity be obtained with reference to Fig. 3 which shows the relative orientations of the wave vectors of the incident and scattered light beams, ki and k, , together with the’velocity vector, V. The Doppler frequency shift between the incident and scattered light beams is given by [8] (4) Resolution of the velocity vector into its components in the radial and axial coordinate directions gives: v . k, = V, cos 8, + U, sin Ok ,
@a)
189
(5b)
v * ki = U, cos 8li - U, sin Oli . Substitution
of eqns. (5) into eqn. (4) gives
ek - cos 8li) + u, (sin el, + sin oli) ]
AfD =~[u,(cos
(6)
.
a
The radial velocity component may be eliminated from eqn. (6) using the equation of continuity to obtain the following relationship between the Doppler frequency and the axial velocity component
AfD =f
-dr
(cos el, - cos Bli) + U, (sin 8, + sin eli)
a
.
(7)
I
Evidently, the calculation of axial velocity distributions, u, (r,z), from the measured Doppler frequency distributions, AfD (x,y), is not a simple process. However, upon making the approximations, 8, = Brs = tI1 and Ri = R, = R. eqns. (2), (3) and (7) simplify to: tan
r= (x-x0)&,
e (8)
I
z = (Y -ye)
+ R (tan 8, - tan 0,) ,
Afo = 2 F v, sin
8,
(9)
.
(10)
a
SCATTERED FREQUENCY
LIGHT t +At, \ I I’
v
‘\ INCIDENT LIGHT FREbUENCY t
3
Fig. 3. Diagram showing the orientations of the incident and scattered light *beams with respect to the velocity vector.
190
The situation where these approximate equations apply exactly arises only when the incident and scattered light beams intersect on the axis of the jet. However, if 8, is kept small and if the curvature of the free surface is small, then these simple expressions may be used. By taking 8, < 4” use of this approximate method was made possible in this work and it is believed that little loss of accuracy was incurred through its use [ 31. The method will be least accurate for the region very close to the tube exit plane in those cases where the Froude number was lowest. No attempt was made to apply the full equations in these regions because it was envisaged that elaborate numerical computation schemes involving the interpolation of unequally spaced experimental data would have been required. Moreover, to achieve a reasonable degree of accuracy using such schemes, much more extensive experimental data would be required. 4. Results and discussion Using the above apparatus, velocity profiles were measured near the exit plane of a vertical tube in which fully developed laminar flow conditions had been established. Velocity profiles measured in the jet downstream from the tube exit plane for a 0,996 polyacrylamide ip 85% glycerol/water solution are presented in Figs. 4-7. These experimeptal results reaffirm the conclusion stated previously [l]
DIMENSIONLESS Fig.
Re,
RADldL
COORDINATE
r/F&,
0.9% polyacrylamide in 85% glycerol/water. Do = 1.25 cm, VO = 71.0 = 101, Fr = 4.11, sharp-edged exit, - - - - - - fully developed tube profile.
4.
cm/s.,
191
DIMENSIONLESS
RADIAL
COORDINATE
P IR,,
Fig. 5. 0.9% polyacrylamide in 85% glycerol/water. D,-, = 1.25 cm., V, = 45.3 cm/s, Re, = 62, Fr = 1.67, sharp-edged exit, - - - - - - fully developed tube profile.
DIMENSIONLESS
0
AXIAL
COORDINATE
I 0 DIMENSIONLESS
RADIAL
‘, 1.0
0.5 COORDINATE
riR0
Fig. 6. 0.9% polyacrylamide in 85% glycerol/water. Do = 1.90 cm, Vo = 51.0 cm/s. Re, = 104, Fr = 1.40, sharp-edged exit, - - - - - - fully developed tube profile.
192 DIMENSIONLESS
DIMENSIONLESS
AXIAL
RADIAL
COORDINATE
COORDINATE
r/R,
Fig. 7. 0.9% polyacrylamide in 85% glycerol/water. Do = 1.90 cm, V. = 23.4 cm/s, Re, = 45, Fr = 0.294, o sharp-edged exit, 0 rounded exit, - - - - - - fully developed tube profile.
that the laser anemometer is an extremely powerful tool for the detailed investigation of flow fields. The transition from steady shear flow in the tube to elongational flow in the falling jet is illustrated quite graphically. The picture presented here is one of a smooth.velocity profile rearrangement which takes place through the combined actions of (a) acceleration of the outer layers due to radial momentum transfer with adjacent inner layers, the process spreading progressively inwards with increasing axial distance from the exit plane, and (b) acceleration of the whole due to gravity. Figures 4-7 cover a variety of experimental conditions for tubes having
193
sharp-edged exits, and Fig. 7 includes a comparison between sharp-edged and rounded exits, all other conditions being the same. The widest variation is in the Froude number, which may be interpreted as giving a measure of the ratio of the momentum flux (inertia force) to the gravitational force. The highest Froude number case is shown in Fig. 4 from which it might appear that the only process involved is the acceleration of the outer layers. Very little appears to be happening near the axis. However, in the absence of gravitational acceleration the momentum transfer process whereby the outer layers are accelerated would necessarily lead to a deceleration of the inner layers. That no such deceleration has occurred here indicates that the gravitational accelerational is not negligible. The reader should not be tempted to place too much importance on the comparison, on the basis of approximately equal Reynolds numbers, afforded by Figs. 4 and 6. The initial conditions in the upstream tubes, which have different diameters and therefore different elastic normal stress levels, are quite different. Some idea of the stress levels involved may be gained from the fact that for the smaller diameter tube, DO = 1.25 cm, a slight swelling of the jet was observed immediately downstream from the tube exit plane, viz: R/R, = 1.018at z/D, =
0.20. Besides the initial conditions referred to above, another factor which can affect the velocity rearrangement is the condition of the tube wall at the exit corner. The comparison between sharp-edged and rounded exit corners presented
DIMENSIONLESS
RADIAL
COORDINATE
r/I?tJ
Fig. 8. 0.9% polyacrylamide in water. Do = 1.90 cm, V, = 62.0 sharp-edged exit, - - - - - - fully developed tube profile.
cm/s,
Re,
= 316:
Fr = 2.06,
194
Fig. 7 shows that for the rounded exit the velocity profiles are initially flatter than, but soon catch up with and then overtake, the corresponding profiles for the sharp-edged exit. A similar comparison for a Newtonian 85% glycerol/water solution gave clear evidence that the flow is initially decelerated by the rounded exit [3]. Velocity profiles measured in the jet downstream from the tube exit plane for a 0.9% polyacrylamide/water solution are presented in Figs. 8 and 9. The picture presented of the flow in the transition region is largely the same as for the polyacrylamide/glycerol/water solution discussed above. However, the following two observations may be of some importance. First, Fig. 8 shows that in the higher Froude number case the velocity profiles overlap. The region of overlap extends from the tube exit plane to about 2.5 tube-diameters downstream. Velocity profiles measured at axial distances of 0.10,1.63 and 2.68 tube-diameters have been omitted from Fig. 8 for reasons of clarity. The complete set of measurements appear to indicate that the momentum transfer process is restricted to a narrow outer region of the jet until a downstream axial distance of about 2 tubein
DIMENSIONLESS
DIMENSIONLESS
RADIAL
AXIAL
COORDINATE
COORDINATE
rlRO
Fig. 9. 0.9% polyacrylamide in water. Do = 1.90 cm., V, = 33.0 sharp-edged exit, - - - - - - fully developed tube profile.
cm/s,
Re,
= 131,
Fr = 0.584,
195
diameters is reached. Thereafter, the transition to a flat profile takes place smoothly. Second, the velocity profiles shown in Fig. 9 at downstream axial distances of 1.10 and 1.63 tube-diameters do not have the expected change of curvature. For both of the viscoelastic fluids studied, the extent of the transition region downstream from the tube exit plane was found to depend mainly on the value of the Froude number. The distance below the tube exit plane at which a flat (or nearly flat) velocity profile was observed increased steadily with increasing Froude number. At a Froude number of 0.294 the distance was about 3.0 tubediameters and at a Froude number of 4.11 the distance was greater than 5.4 tube-diameters. The magnitude of the velocity in the flat profiles was found to vary in an opposite manner with the value of the Froude number. At a Froude number of 4.11 the flat profile had a velocity about 2.2 times the average tube velocity. The value increased steadily with decreasing Froude number until at a Froude number of 0.294 the velocity was about 4.5 times the average tube velocity. The above remarks apply in those cases where the tubes had sharp-edged exit corners. For tubes with rounded exit corners slightly longer distances were required before flat velocity profiles were achieved. The magnitude of the velocity in the flat profile was found to be considerably greater than that for the corresponding sharp-edged exit case. The extent of the transition region upstream from the tube exit plane was investigated in some detail for tubes having sharp-edged exit corners. For the 0.9% polyacrylamide/water solution the experimental conditions are summarised in Table 1. For each set of conditions, velocity profiles were measured at various axial distances extending from about 0.10 to about 4.5 diameters upstream from the tube exit plane. No deviation from the theoretical fully developed velocity profile was detected. Figure 10 shows results obtained for the lowest Reynolds number studied. A similar study using a Newtonian 85% glycerol/water solution [ 31 also indi-
TABLE
1
Upstreamexperimentalconditions(0.9% TUBE DIAMETER Do (cm)
1.90
AVERAGE
WALL
VELOCITY
7.0
SHEAR
STRESS TW
vo (cm/set
polyacrylamide/water)
)
(dyneshn2)
WALL
SHEAR
xv (1/set
REYNOLDS NUMBER
RATE
R=w )
26.7
31.8
16
1.90
20.5
5B.5
99.2
66
1.90
33.3
79.0
163.5
130
1.90
60.9
114
300
305
1.90
96.7
152
486
670
196
OL 0
t
I
1.0
0.5 DIMENSKINLESS
RADIAL
COORDINATE
r/q
Fig. 10. 0.9% polyacrylamide in water. Do = 1.90 cm., V. = 7.0 cm/s, Re, = 16, Fr = 0.026, oz/& = -0.16, *z/I.+, = -2.26, ___ fully developed tube profile. cated no deviation from the theoretical fully developed parabolic velocity profile down to a Reynolds number of 50 which was the lowest studied.
5. Conclusions The main conclusions to be drawn from this prelimin~y study of flow rearrangement near the exit of a vertical tube in which steady Iaminar flow conditions had been established are as follows. First, flow in the transition region is governed both by momentum transfer and by gravitational acceleration. Second, gravitational acceleration is responsible for the main differences observed. The value of the F’roude number appears to control the extent of the transition region. The axial distance downstream from the tube exit plane at which the velocity profile becomes flat increases with increasing Froude number. In contrast, the magnitude of the velocity at this point decreases with increasing Froude number.
197
Third, for the experimental conditions studied, the transition region did not appear to extend upstream from the tube exit plane. Fourth, the condition of the tube wall at the exit corner affects the velocity profile rearrangement in the falling jet downstream from the tube exit plane. Compared with corresponding sharp-edged values, velocity profiles obtained using tubes with rounded exit corners initially develop more slowly. However, they accelerate more rapidly and become flat at slightly greater distances below the tube exit plane. The magnitude of the velocity at this point is considerably larger than that for the corresponding sharp-edged case. Fifth, for the polyacrylamide/glycerol/water solution the velocity profile rearrangement in the transition region takes place smoothly. However, for the polyacrylamide/water solution at the higher Froude number reported, the momentum transfer process appears to be restricted initially to a narrow outer region of the jet. Perhaps this is due to delayed elastic recovery. Some unusual results are reported for this solution at the lower Froude number. Nomenclature
;.a I,
s
n r V
ur, u,? X
x0 Y Yo ;
FrO Re, Rip R, R Ro vo Afo YUJ 0,
frequency of incident and reference light beams gravitational acceleration unit wave vectors in the directions of the incident and scattered light beams, respectively refractive index of liquid radial coordinate velocity vector radial and axial velocity components, respectively horizontal distance horizontal distance corresponding to tube/jet axis vertical distance vertical distance corresponding to tube exit plane axial coordinate tube diameter p po/gDo = Froude number P VoDol~w = Reynolds number radii of jet at points at which the incident and scattered/secondary reference light beams, respectively, intersect the free surface radius of jet tube radius average velocity in tube Doppler frequency wall shear rate angle between horizontal and incident and scattered/secondary reference light beams in air angles between horizontal and incident and scattered/secondary reference light beams, respectively, in liquid angles between the horizontal and the surface normals at the points at which the incident and scattered/secondary reference light beams respectively, intersect the free surface
angles between horizontal and reflected portions of the incident and secondary reference light beams, respectively, at their points of intersection with the free surface wavelengths of laser light in air and liquid, respectively wall apparent viscosity density of liquid wall shear stress angles between the surface normals and the incident and scattered/ secondary reference light beams, respectively, in air at their points of intersection with the free surface angles between the surface normals and the incident and scattered/ secondary reference light beams, respectively, in liquid at their points of intersection with the free surface References 1 E.T. Busby and W.C. MacSporran, J. Non-Newtonian Fluid Mech., 1 (1976) 71-82. 2 D.D. Goulden and W.C. MacSporran, Velocity measurements in a falling liquid jet, Paper presented at the British Society of Rheology Autumn Conference, Univ. of East Anglia, Norwich, September, 1973. 3 D.D. Goulden, Ph. D. Dissertation, Univ. of Bradford (in preparation) 4 C.G. Gogos, Ph. D. Dissertation, Princeton Univ., 1965. 5 C.G. Gogos and B. Maxwell, Polym. Eng. Sci., 6 (1966) 353. 6 R.C. Allen, Ph. D. Dissertation, Princeton Univ., 1970. 7 W.R. Schowalter and R.C. Allen, Trans. Sot. Rheol., 19 (1975) 129. 8 R.J. Goldstein and D.K. Kreid, Trans. ASME, J. Appl. Mech., 34E (1967) 813.
Journal of Non-Newtonian Fluid Mechanics, 1 (1976) @ Elsevier Scientific Publishing Company, Amsterdam
183-198 - Printed
in The Netherlands
AN EXPERIMENTAL STUDY OF NONRHEOMETRIC VISCOELASTIC FLUIDS II. FLOW IN A LIQUID JET EXITING
D.D. GOULDEN
FROM A VERTICAL
TUBE
* and W.C. MacSPORRAN
Schools of Studies (Ct. Britain) (Received
FLOWS OF
December
in Chemical
Engineering,
Uniuersity
of Bradford,
Bradford,
Yorkshire
21, 1975)
Summary A laser anemometer has been used to study the region of accelerating shear flow near the exit of a vertical tube. It is in this region that the transition between steady laminar shear flow in the upstream tube and elongational flow in the downstream liquid jet takes place. Downstream velocity profiles were measured for solutions of 0.9% polyacrylamide in 85% glycerol/water and 0.9% polyacrylamide in water. Reynolds numbers (based on wall conditions in the fully developed upstream flow) ranged from 45 to 310 and Froude numbers from 0.294 to 4.11. Tubes, having sharpedged and rounded exit corners, with diameters of 1.25 cm and 1.90 cm were used Upstream velocity profiles were measured for a solution of 0.9% polyacrylamide in water. Reynolds numbers ranged from 16 to 670. Only tubes having sharp-edged exit corners were used. It was found that the transition region did not extend upstream into the tube but was confined to the downstream jet. The transition took place over a distance of about 3-5 tube diameters depending upon the value of the Froude number. The axial distance downstream from the tube exit plane at which the velocity profile first became flat increased with increasing Froude number. The magnitude of the jet velocity at this point decreased with increasing Froude number. Th.e condition of the tube exit corner was found to influence the flow in the transition region. Downstream velocity profiles obtained using tubes having rounded exit corners initially develop more slowly than, but soon catch up with and eventually overtake, the corresponding profiles obtained using tubes with sharp-edged exit corners. * Present
address:
Lever
Bros.
Ltd.,
Port Sunlight,
Wirral,
Cheshire.
184
Downstream velocity profiles obtained for the 0.9% polyacrylamide in 85% glycerol/water solution were found to develop smoothly. The transition from steady shear flow in the tube to elongational flow in the jet took place through the combined processes of acceleration of the outer layers of the jet due to radial transfer of momentum with adjacent inner layers, the process spreading steadily inwards with increasing axial distance from the tube exit plane, and acceleration of the whole due to gravity. However, the velocity profiles obtained for the 0.9% polyacrylamide in water solution did not always develop so smoothly. At a Reynolds number of 310 and Froude number of 2.06 the radial momentum transfer process was restricted to a narrow outer region of the jet until a downstream axial distance of about 2 tube diameters was reached. Thereafter, the transition to a flat profile took place smoothly.
1. ~troduc~on The experimental study reported in this paper forms part of a wider research programme designed to obtain a better understanding of nonrheometric flows of viscoelastic fluids. A previous paper [ 1 J reported experimental measurements of the developing flow both upstream and downstream from the entry plane in a re-entrant tube geometry. In this paper, the results are presented of a preliminary experimental study of flow rearrangement in the tube exit region [ 2,3]. The flow field near the exit of a vertical tube may be divided into three regions: (1) the region of steady fully developed laminar shear flow well upstream from the tube exit plane; (2) the region of elongational flow in the falling liquid jet well downstream from the tube exit plane; and, (3) the intermediate transition region of accelerating shear flow! The flow in regions (1) and (2) falls within the general classification of rheometric flows. However, the flow in region (3) is nonrheometric. The object of the present work was to examine experimentally the nonrheometric flow in the transition region. Information was sought regarding the way in which the velocity re~rangement takes place, the extent of the region both upstream and downstream from the tube exit plane, and the influence of tube exit conditions on the rearrangement. Related experimental studies are those of Gogos [4,5] for vertical extrusion of polymer melts and Allen [6,7] for horizontal extrusion of polymer solutions. Both these workers used streak photography to measure the velocity distributions. In this work, the velocity distributions were measured using a laser anemometer. 2. Apparatus A brief description of the experimental is described in detail elsewhere [ 31.
apparatus
is given below. The apparatus
2. I Laser anemometer The laser anemometer used in this study was operated in the reference mode and is illustrated in Fig. 1. As may be seen. the ontical arrangement
beam of the
185
INCIDENT BEAM
Fig. 1. Laser anemometer.
laser anemometer has been considerably modified from that used in previous work [l]. The modifications were necessary to ensure that the incident, scattered, and reference beams had fixed orientations in the air, a situation which could not have been realized had the reference beam been allowed to pass through the liquid jet. Light from a continuous wave He-Ne gas laser (not shown) is split into two beams having approximately equal intensities. One, the incident beam, is redirected using two mirrors (M, and M,) and focused onto a spot in the flowing fluid using a lens (L). The other, the reference beam, is redirected using a mirror (Ms) and passed through a beam splitter (BS). The transmitted primary beam is reflected back along its path using a mirror (M4) and is further reflected from the rear face of the beam splitter (BS). It then passes through the aperture (A,) and the interference filter (IF) onto the cathode of the photomultiplier tube (PM). The secondary beam which is reflected from the front face of the beam splitter (BS) passes through the aperture (A,) and intersects the incident light beam at its focal spot in the flowing fluid. Not only does this secondary beam
166
provide convenient visual evidence of the position of the focal spot but also it serves a vital purpose which will be explained later. That portion of the light, which is scattered from the incident beam along the direction of the secondary beam, passes through the apertures (A, and A,) of the receiving optics onto the same area of the cathode of the photomultiplier tube as the primary reference beam. The apertures control the spot size in the flowing fluid from which scattered light reaches the cathode and also ensure the close alignment of the reference and scattered beams required for efficient photomixing. The interference filter is used to shut out background radiation having unwanted frequencies. The optical system described above was mounted on a 1-in.-thick vertical steel plate which was placed on the optical table described previously [ 11. The optical benches on which the transmitting and receiving optics were mounted were arranged symmetrically at equal angles to the tube/jet axis. These angles were adjustable so that, when measurements were being made in the falling liquid jet, the benches were inclined at equal angles below the horizontal as shown in Fig. 1. When measurements were being made in the tube, the benches were inclined at equal angles above the horizontal. With these arrangements the tube exit plane could be approached as closely as possible from both upstream and downstream directions. 2.2 Pump loop and test section The test fluid was pumped, using a variable speed gear pump, from a 70-gallon storage tank via a 1-in.-nominal-bore copper pipe and a flexible polythene hose to the test section. The test section consisted of a vertically mounted glass tube which discharged into a constant head receiver. The overflow from the constant head receiver passed to a 5-gal sump tank from which the fluid was returned to the storage tank, via a l-in.-nominal-bore copper pipe, by gravity flow. An electromagnetic flow meter was used for flow measurement and it was calibrated using a 50-gal weigh tank mounted on a compression load cell. The test section was mounted vertically on a l-m-long optical bench which was secured to the l-in-thick vertical steel plate. Both the glass tube and the constant head receiver were mounted on standard optical components and clamped to the optical bench. The test section could be moved relative to the fixed optical arrangement of the laser anemometer. Vertical movement of the test section was achieved by releasing the clamping screws and sliding it up or down the optical bench to the desired position before retightening the clamping screws. The vertical distance moved was read from the scale mounted on the side of the optical bench. Horizontal movement of the test section was effected using the horizontal transverse feed screws provided on the standard optical components. The horizontal distance moved was read from the vernier scale also provided. The constant head receiver formed an integral part of the test section and was moved with it. This ensured that the length and, therefore, the weight of the jet falling from the tube exit remained constant. It was felt that this precaution was necessary to avoid the risk of altering the flow conditions in the jet. Tubes having internal diameters of 1.25 cm and 1.90 cm were used in this work. Both sharp-edged and rounded exits were used.
187
3. Calculation of the velocity doppler frequencies
distributions
in the falling jet from the measured
Calculation of the velocity distributions in the upstream tube was carried out using the simple procedure outlined previously [l ] . Calcuation of the velocity distributions in the downstream falling jet requires a more complicated procedure. The complication is introduced through the longitudinal curvature of the free surface of the jet. The experimental measurements, obtained at a given vertical distance, y, consist of a set of Doppler frequencies, Afo, corresponding to a series of horizontal distances, x. The procedure whereby the measured Doppler frequency distributions, Afh (x,y), are converted into axial velocity distributions, u, (r,z), is outlined below. The relationship between the horizontal and vertical distances, and the radial and axial coordinates of the point of intersection of the incident and scattered light beams is obtained as follows. For a general position (3c,y) the configuration of the incident and scattered light beams, in relation to the free surface of the liquid jet, is as shown in Fig. 2. The angles made by the incident and scattered light beams in the air, 8,, are determined by the fixed optical components of the laser anemometer. Remembering that the secondary reference beam coincides with the scattered light beam, the vital purpose which its serves may now be stated. The angles, BRiand t!IRs,at which the incident and secondary reference
Fig. 2. Diagram of liquid jet showing the orientations of the incident and secondary reference light beams with respect to the free surface.
188
light beams are reflected from the free surface of the jet were measured as also was the refractive index of the liquid. Using these measurements all the other angles indicated may be calculated. The necessary relationships are listed below:
n=7
sin I& sin @li ’
&i =Sin-l
n=- sin G,, sin & ’ sin $,i 7
( 1
&=sin-l
,
(1)
sin#,,
( 1 -
n
,
The radii, Ri and R, of the jet at the points at which the incident and scattered light beams intersect the free surface were measured from photographs of the liquid jet. Simple trigonometric relationships may then be used to calculate the radial and axial coordinates of the point of intersection of the incident and scattered light beams. These are given by the following expressions: r=
2(x - 3ce) tan 8, - Ri(tan 0, - tan Oli) + R,(tan (tan Oli + tan e,)
z=(Y-Yd+(~-d
tan ezi -tan
se,,+
8, )
8(1 -tan
0,)
tan&-
tan f3,, + R, In general, z varies with r so that, for a given fixed vertical distance y and variable horizontal distance X, the point of intersection does not traverse an axial plane of the jet. The relationship between the measured Doppler frequency and the velocity be obtained with reference to Fig. 3 which shows the relative orientations of the wave vectors of the incident and scattered light beams, ki and k, , together with the’velocity vector, V. The Doppler frequency shift between the incident and scattered light beams is given by [8] (4) Resolution of the velocity vector into its components in the radial and axial coordinate directions gives: v . k, = V, cos 8, + U, sin Ok ,
@a)
189
(5b)
v * ki = U, cos 8li - U, sin Oli . Substitution
of eqns. (5) into eqn. (4) gives
ek - cos 8li) + u, (sin el, + sin oli) ]
AfD =~[u,(cos
(6)
.
a
The radial velocity component may be eliminated from eqn. (6) using the equation of continuity to obtain the following relationship between the Doppler frequency and the axial velocity component
AfD =f
-dr
(cos el, - cos Bli) + U, (sin 8, + sin eli)
a
.
(7)
I
Evidently, the calculation of axial velocity distributions, u, (r,z), from the measured Doppler frequency distributions, AfD (x,y), is not a simple process. However, upon making the approximations, 8, = Brs = tI1 and Ri = R, = R. eqns. (2), (3) and (7) simplify to: tan
r= (x-x0)&,
e (8)
I
z = (Y -ye)
+ R (tan 8, - tan 0,) ,
Afo = 2 F v, sin
8,
(9)
.
(10)
a
SCATTERED FREQUENCY
LIGHT t +At, \ I I’
v
‘\ INCIDENT LIGHT FREbUENCY t
3
Fig. 3. Diagram showing the orientations of the incident and scattered light *beams with respect to the velocity vector.
190
The situation where these approximate equations apply exactly arises only when the incident and scattered light beams intersect on the axis of the jet. However, if 8, is kept small and if the curvature of the free surface is small, then these simple expressions may be used. By taking 8, < 4” use of this approximate method was made possible in this work and it is believed that little loss of accuracy was incurred through its use [ 31. The method will be least accurate for the region very close to the tube exit plane in those cases where the Froude number was lowest. No attempt was made to apply the full equations in these regions because it was envisaged that elaborate numerical computation schemes involving the interpolation of unequally spaced experimental data would have been required. Moreover, to achieve a reasonable degree of accuracy using such schemes, much more extensive experimental data would be required. 4. Results and discussion Using the above apparatus, velocity profiles were measured near the exit plane of a vertical tube in which fully developed laminar flow conditions had been established. Velocity profiles measured in the jet downstream from the tube exit plane for a 0,996 polyacrylamide ip 85% glycerol/water solution are presented in Figs. 4-7. These experimeptal results reaffirm the conclusion stated previously [l]
DIMENSIONLESS Fig.
Re,
RADldL
COORDINATE
r/F&,
0.9% polyacrylamide in 85% glycerol/water. Do = 1.25 cm, VO = 71.0 = 101, Fr = 4.11, sharp-edged exit, - - - - - - fully developed tube profile.
4.
cm/s.,
191
DIMENSIONLESS
RADIAL
COORDINATE
P IR,,
Fig. 5. 0.9% polyacrylamide in 85% glycerol/water. D,-, = 1.25 cm., V, = 45.3 cm/s, Re, = 62, Fr = 1.67, sharp-edged exit, - - - - - - fully developed tube profile.
DIMENSIONLESS
0
AXIAL
COORDINATE
I 0 DIMENSIONLESS
RADIAL
‘, 1.0
0.5 COORDINATE
riR0
Fig. 6. 0.9% polyacrylamide in 85% glycerol/water. Do = 1.90 cm, Vo = 51.0 cm/s. Re, = 104, Fr = 1.40, sharp-edged exit, - - - - - - fully developed tube profile.
192 DIMENSIONLESS
DIMENSIONLESS
AXIAL
RADIAL
COORDINATE
COORDINATE
r/R,
Fig. 7. 0.9% polyacrylamide in 85% glycerol/water. Do = 1.90 cm, V. = 23.4 cm/s, Re, = 45, Fr = 0.294, o sharp-edged exit, 0 rounded exit, - - - - - - fully developed tube profile.
that the laser anemometer is an extremely powerful tool for the detailed investigation of flow fields. The transition from steady shear flow in the tube to elongational flow in the falling jet is illustrated quite graphically. The picture presented here is one of a smooth.velocity profile rearrangement which takes place through the combined actions of (a) acceleration of the outer layers due to radial momentum transfer with adjacent inner layers, the process spreading progressively inwards with increasing axial distance from the exit plane, and (b) acceleration of the whole due to gravity. Figures 4-7 cover a variety of experimental conditions for tubes having
193
sharp-edged exits, and Fig. 7 includes a comparison between sharp-edged and rounded exits, all other conditions being the same. The widest variation is in the Froude number, which may be interpreted as giving a measure of the ratio of the momentum flux (inertia force) to the gravitational force. The highest Froude number case is shown in Fig. 4 from which it might appear that the only process involved is the acceleration of the outer layers. Very little appears to be happening near the axis. However, in the absence of gravitational acceleration the momentum transfer process whereby the outer layers are accelerated would necessarily lead to a deceleration of the inner layers. That no such deceleration has occurred here indicates that the gravitational accelerational is not negligible. The reader should not be tempted to place too much importance on the comparison, on the basis of approximately equal Reynolds numbers, afforded by Figs. 4 and 6. The initial conditions in the upstream tubes, which have different diameters and therefore different elastic normal stress levels, are quite different. Some idea of the stress levels involved may be gained from the fact that for the smaller diameter tube, DO = 1.25 cm, a slight swelling of the jet was observed immediately downstream from the tube exit plane, viz: R/R, = 1.018at z/D, =
0.20. Besides the initial conditions referred to above, another factor which can affect the velocity rearrangement is the condition of the tube wall at the exit corner. The comparison between sharp-edged and rounded exit corners presented
DIMENSIONLESS
RADIAL
COORDINATE
r/I?tJ
Fig. 8. 0.9% polyacrylamide in water. Do = 1.90 cm, V, = 62.0 sharp-edged exit, - - - - - - fully developed tube profile.
cm/s,
Re,
= 316:
Fr = 2.06,
194
Fig. 7 shows that for the rounded exit the velocity profiles are initially flatter than, but soon catch up with and then overtake, the corresponding profiles for the sharp-edged exit. A similar comparison for a Newtonian 85% glycerol/water solution gave clear evidence that the flow is initially decelerated by the rounded exit [3]. Velocity profiles measured in the jet downstream from the tube exit plane for a 0.9% polyacrylamide/water solution are presented in Figs. 8 and 9. The picture presented of the flow in the transition region is largely the same as for the polyacrylamide/glycerol/water solution discussed above. However, the following two observations may be of some importance. First, Fig. 8 shows that in the higher Froude number case the velocity profiles overlap. The region of overlap extends from the tube exit plane to about 2.5 tube-diameters downstream. Velocity profiles measured at axial distances of 0.10,1.63 and 2.68 tube-diameters have been omitted from Fig. 8 for reasons of clarity. The complete set of measurements appear to indicate that the momentum transfer process is restricted to a narrow outer region of the jet until a downstream axial distance of about 2 tubein
DIMENSIONLESS
DIMENSIONLESS
RADIAL
AXIAL
COORDINATE
COORDINATE
rlRO
Fig. 9. 0.9% polyacrylamide in water. Do = 1.90 cm., V, = 33.0 sharp-edged exit, - - - - - - fully developed tube profile.
cm/s,
Re,
= 131,
Fr = 0.584,
195
diameters is reached. Thereafter, the transition to a flat profile takes place smoothly. Second, the velocity profiles shown in Fig. 9 at downstream axial distances of 1.10 and 1.63 tube-diameters do not have the expected change of curvature. For both of the viscoelastic fluids studied, the extent of the transition region downstream from the tube exit plane was found to depend mainly on the value of the Froude number. The distance below the tube exit plane at which a flat (or nearly flat) velocity profile was observed increased steadily with increasing Froude number. At a Froude number of 0.294 the distance was about 3.0 tubediameters and at a Froude number of 4.11 the distance was greater than 5.4 tube-diameters. The magnitude of the velocity in the flat profiles was found to vary in an opposite manner with the value of the Froude number. At a Froude number of 4.11 the flat profile had a velocity about 2.2 times the average tube velocity. The value increased steadily with decreasing Froude number until at a Froude number of 0.294 the velocity was about 4.5 times the average tube velocity. The above remarks apply in those cases where the tubes had sharp-edged exit corners. For tubes with rounded exit corners slightly longer distances were required before flat velocity profiles were achieved. The magnitude of the velocity in the flat profile was found to be considerably greater than that for the corresponding sharp-edged exit case. The extent of the transition region upstream from the tube exit plane was investigated in some detail for tubes having sharp-edged exit corners. For the 0.9% polyacrylamide/water solution the experimental conditions are summarised in Table 1. For each set of conditions, velocity profiles were measured at various axial distances extending from about 0.10 to about 4.5 diameters upstream from the tube exit plane. No deviation from the theoretical fully developed velocity profile was detected. Figure 10 shows results obtained for the lowest Reynolds number studied. A similar study using a Newtonian 85% glycerol/water solution [ 31 also indi-
TABLE
1
Upstreamexperimentalconditions(0.9% TUBE DIAMETER Do (cm)
1.90
AVERAGE
WALL
VELOCITY
7.0
SHEAR
STRESS TW
vo (cm/set
polyacrylamide/water)
)
(dyneshn2)
WALL
SHEAR
xv (1/set
REYNOLDS NUMBER
RATE
R=w )
26.7
31.8
16
1.90
20.5
5B.5
99.2
66
1.90
33.3
79.0
163.5
130
1.90
60.9
114
300
305
1.90
96.7
152
486
670
196
OL 0
t
I
1.0
0.5 DIMENSKINLESS
RADIAL
COORDINATE
r/q
Fig. 10. 0.9% polyacrylamide in water. Do = 1.90 cm., V. = 7.0 cm/s, Re, = 16, Fr = 0.026, oz/& = -0.16, *z/I.+, = -2.26, ___ fully developed tube profile. cated no deviation from the theoretical fully developed parabolic velocity profile down to a Reynolds number of 50 which was the lowest studied.
5. Conclusions The main conclusions to be drawn from this prelimin~y study of flow rearrangement near the exit of a vertical tube in which steady Iaminar flow conditions had been established are as follows. First, flow in the transition region is governed both by momentum transfer and by gravitational acceleration. Second, gravitational acceleration is responsible for the main differences observed. The value of the F’roude number appears to control the extent of the transition region. The axial distance downstream from the tube exit plane at which the velocity profile becomes flat increases with increasing Froude number. In contrast, the magnitude of the velocity at this point decreases with increasing Froude number.
197
Third, for the experimental conditions studied, the transition region did not appear to extend upstream from the tube exit plane. Fourth, the condition of the tube wall at the exit corner affects the velocity profile rearrangement in the falling jet downstream from the tube exit plane. Compared with corresponding sharp-edged values, velocity profiles obtained using tubes with rounded exit corners initially develop more slowly. However, they accelerate more rapidly and become flat at slightly greater distances below the tube exit plane. The magnitude of the velocity at this point is considerably larger than that for the corresponding sharp-edged case. Fifth, for the polyacrylamide/glycerol/water solution the velocity profile rearrangement in the transition region takes place smoothly. However, for the polyacrylamide/water solution at the higher Froude number reported, the momentum transfer process appears to be restricted initially to a narrow outer region of the jet. Perhaps this is due to delayed elastic recovery. Some unusual results are reported for this solution at the lower Froude number. Nomenclature
;.a I,
s
n r V
ur, u,? X
x0 Y Yo ;
FrO Re, Rip R, R Ro vo Afo YUJ 0,
frequency of incident and reference light beams gravitational acceleration unit wave vectors in the directions of the incident and scattered light beams, respectively refractive index of liquid radial coordinate velocity vector radial and axial velocity components, respectively horizontal distance horizontal distance corresponding to tube/jet axis vertical distance vertical distance corresponding to tube exit plane axial coordinate tube diameter p po/gDo = Froude number P VoDol~w = Reynolds number radii of jet at points at which the incident and scattered/secondary reference light beams, respectively, intersect the free surface radius of jet tube radius average velocity in tube Doppler frequency wall shear rate angle between horizontal and incident and scattered/secondary reference light beams in air angles between horizontal and incident and scattered/secondary reference light beams, respectively, in liquid angles between the horizontal and the surface normals at the points at which the incident and scattered/secondary reference light beams respectively, intersect the free surface
angles between horizontal and reflected portions of the incident and secondary reference light beams, respectively, at their points of intersection with the free surface wavelengths of laser light in air and liquid, respectively wall apparent viscosity density of liquid wall shear stress angles between the surface normals and the incident and scattered/ secondary reference light beams, respectively, in air at their points of intersection with the free surface angles between the surface normals and the incident and scattered/ secondary reference light beams, respectively, in liquid at their points of intersection with the free surface References 1 E.T. Busby and W.C. MacSporran, J. Non-Newtonian Fluid Mech., 1 (1976) 71-82. 2 D.D. Goulden and W.C. MacSporran, Velocity measurements in a falling liquid jet, Paper presented at the British Society of Rheology Autumn Conference, Univ. of East Anglia, Norwich, September, 1973. 3 D.D. Goulden, Ph. D. Dissertation, Univ. of Bradford (in preparation) 4 C.G. Gogos, Ph. D. Dissertation, Princeton Univ., 1965. 5 C.G. Gogos and B. Maxwell, Polym. Eng. Sci., 6 (1966) 353. 6 R.C. Allen, Ph. D. Dissertation, Princeton Univ., 1970. 7 W.R. Schowalter and R.C. Allen, Trans. Sot. Rheol., 19 (1975) 129. 8 R.J. Goldstein and D.K. Kreid, Trans. ASME, J. Appl. Mech., 34E (1967) 813.