Transitional Flow in Isosceles Triangular Ducts Richard C. Cope E . I . du Pont de Nemours and Co., Chattanooga, Tenn.
Richard W. Hanks* Department of Chemical Engineering, Brigham Young University, Protro, Utah
A study of mean velocity distributions and turbulence intensity measurements in ducts having isosceles triangular cross sections has revealed the existence of a range of mean flow Reynolds numbers for which a pronounced nonturbulent secondary flow occurs. The driving mechanism is shown to arise from the threedimensional character of the laminar flow. The secondary flow results in an approximately 6-8% increase in frictional resistance over that observed in rectilinear laminar flows. Transition from rectilinear laminar flow to a three-dimensional laminar flow with secondary circulations occurs at a critical Reynolds number predicted accurately b y the momentum stability theory of Hanks. A subsequent transition to a turbulent flow upon which is superimposed a mean secondary flow occurs at a higher Reynolds number.
I n an earlier paper (Hanks and Cope, 1970), the authors presented the results of an experimental study of the frictional resistance characteristics of Newtonian fluids experiencing the transition from laminar flow in straight-walled ducts of isosceles triangular cross section. In that paper, a theoretical analysis of the stability of the laminar flow field (Sparrow, 1962) in such ducts was performed, and a model was proposed to explain the apparently anomalous pressure drop increases observed (Cope, 1968) during transitional flow. The conclusions of the model were supported by the pressure drop measurements of Cope (1968) and the results of a flow visualization study by Hanks and Brooks (1970). However, the behavior of the flow in an equilateral triangular duct upon undergoing transition did not correspond with the predictions of the proposed model (Hanks and Cope, 1970) and it was suggested that unknown details of the turbulent flow field might be responsible for this discrepancy. The purpose of the present paper is to report the results of a detailed experimental investigation (Cope, 1970) of the mean velocity profiles and turbulence intensities which occur during the transitional flow of Kewtonian fluids in equilateral and isosceles triangular cross section ducts. The results of this investigation, coupled with the understanding which has been gained in the simpler cases of flow transition in pipes, parallel plates, and concentric annuli (Hanks, 1963, 1968; Hanks and Bonner, 1971) reveal some interesting aspects of the complex phenomena of transition in general. Review of Previous Work
Transitional Flow. The flow geometry considered herein is the isosceles triangular cross section duct. Eckert and Irvine (1956) in studying flows in such ducts suggested that macroscopic zones of laminar and turbulent flow simultaneously existed in the duct during the transition to turbulence. By this they meant the existence of two macroscopically large regions of flow extending the entire length of the duct, a concept not to be confused with the traditional laminar sublayer concept in turbulent flows or with the occurrence of turbulent bursts in a laminar stream (Lindgren, 1957; Rotta, 1956). This suggestion was based upon flow vis106 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
ualization (Eckert and Irvine, 1956) using a technique which Hanks and Brooks (1970) showed to give spurious results due to the effect of the injector. A similar phenomenon (simultaneous laminar-turbulent flow) was suggested to occur (Rothfus, et al., 1950) in concentric annuli over a limited transitional range of velocities with the turbulence being sustained in the annular region near the core. Hanks and Bonner (1971) performed theoretical calculations based upon a modified form (Hanks, 1968) of Van Driest's (1956) mixing length model of turbulent flow and showed that such an effect does in fact arise if one assumes that turbulence occurs near the core first. The theoretical results of this analysis for an annulus fail to agree quantitatively with experimental data although the qualitative behavior is as predicted. In the case of the concentric annulus the transition region is characterized by a frictional resistance curve very similar to the one shown in Figure 1for a 30" apex angle isoscelestriangular duct. Although this transition model has been suggested (Eckert and Irvine, 1956) for isosceles triangular ducts on the basis of flow visualization studies, the frictional resistance data for triangular ducts available in the literature (Carlson and Irvine, 1961; Eckert and Irvine, 1957; Nikuradse, 1930) were so scattered and inaccurate in the lower Reynolds number range that no definite conclusions could be drawn. Preliminary pressure drop data (Simisky, 1967) and later careful measurements by Cope (1968) clearly showed the existence of an extended transition region bounded by a lower and upper critical Reynolds number (Hanks and Cope, 1970). Theoretical calculations of the stability limit of the laminar flow (Hanks and Cope, 1970) based on Sparrow's (Sparrow, 1962) theoretical velocity distributions yielded quantitative agreement with the lower of the two experimentally observed critical Reynolds numbers. The data of Figure 1 are typical of the flow data obtained by Cope (1968) for a series of ducts. Analysis of Hanks and Cope (1970) revealed a condition of minimum stability occurring in the apex region (for those ducts in which the apex angle is less than 60'; otherwise the condition occurs near the base first) of the isosceles triangular ducts. Because of this, they postulated that the region of flow between the isosceles walls in the apex region became tur-
bulent first while that region near the triangular base (being the more stable) remained laminar. This model, which predicted results exactly inverse to Eckert and Irvine's apparent observations (1956), adequately accounted for the transitional pressure drop characteristics of each of the ducts studied by Cope (1968) with the exception of the equilateral duct. The stability analysis of the equilateral triangular duct predicted minimum stability points of equal magnitude symmetrically distributed in the duct (Hanks and Cope, 1970). Hence, the model proposed earlier (Hanks and Cope, 1970) would suggest that all three regions of the flow field in this case simultaneously become unstable a t a single critical Reynolds number with the consequence that no intermediate transitional region should be observed. Indeed, one would predict that the frictional resistance data should appear exactly as those for the pipe flow case. The data in Figure 2, obtained for an equilateral duct (Cope, 1968), show clearly that a transitional region does occur. One is thus left with the obvious need to determine the flow phenomenon responsible for the increased frictional resistance in the transitional range of Reynolds numbers since the two-regime model is apparently not correct. The data of Figure 1 are replotted in Figure 3 as the more conventional logarithmic: plot of f us. Re which masks the intermediate transitional range (as defined above) so clearly revealed for the same data in Figure 1. Figure 3 demonstrates that the data in the transitional region more nearly approximate a laminar than a turbulent curve up to a Reynolds number of about 1650. The theoretical lower critical transition Reynolds number for the data in Figure 1 is Re = 1180 (Hanks and Cope, 1970). This value represents the highest value of the Reynolds number a t which a rectilinear laminar flow may be sustained in the 30' triangular duct according to the stability theory used. Turbulent Flow. A number of investigations of turbulent flow in ducts having noncircular cross sections have shown t h a t in such ducts a secondary mean flow is characteristically observed t o be superimposed on the mean primary flow. This secondary flow consists of circulating cell pairs in the cross section plane of the duct. Helmholtz's theorems (Lamb, 1932) on circulation require that these cells occur in pairs with opposite senses of rotation. Recently, a number of attempts have been made t o measure secondary velocities in turbulent flow in noncircular ducts, to make quantative predictions of secondary flow patterns, and to discover the driving forces responsible for secondary flows. Prandtl (1927) suggested that the secondary currents in a straight noncircular cross section channel move toward the corners along the bisector of a corner angle and then outward along the walls. His conclusion is substantiated by mean velocity contour maps (Brundrett and Baines, 1964; Hoagland, 1960; Nikuradse, 1930). Hoagland (1960), studying fully developed turbulent flow in rectangular ducts, concluded that a force of significant magnitude to drive a secondary flow will result whenever a gradient of wall shear stress exists. He observed a secondary flow pattern consistent with that predicted by Prandtl. Xalaika (1962) showed that earlier theoretical analyses of turbulent flow did not account adequately for the secondary flow which superimposes additional energy dissipation over the general flow pattern. For example, the contribution to the frictional resistance made by the secondary flow velocities in square ducts causes the predicted value of the friction factor to be 12% lower than the experimental value (Hartnett, et al., 1962). These latter investigators concluded that "the neglect of secondary flows in the analysis accounts for the difference between the analytical and experi-
c ' " " ' ' ' " ' ' ' ' ' ' l
f*Rt
"i B
12
-
Transifion 4 T u r b u l r n l
Laminar-
I
I
I
8
6
I
I
IO
I
I
12
I
I
14
I
16
1
1
18
1
1
20
1
A=
-
1
22
Re/100
Figure 2. Experimental data for f.Re vs. Re in equilateral triangular duct (Cope, 1968)
3.0 0
.
""0"
2.0 200 f
1.5
1.0
1000
1500 Re
2000
)O
Figure 3. Data of Figure 1 replotted as f vs. Re on log-log coordinates
mental results." Leutheusser (1963), observing that the secondary flow effect in a rectangular channel distorted the isovels (lines of constant axial velocity) toward the corner regions, concluded that a t extremely high Reynolds numbers the strength of these (secondary) currents should become zero since the distribution of the wall shear stress approaches uniformity. This statement suggests that a more marked effect on the main flow by the secondary flow might be observed at Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
107
low and moderate Reynolds numbers than a t extremely high Reynolds numbers. Brundrett and Baines (1964) observed that the maximum secondary velocity in rectangular ducts with aspect ratios of 1 to 1 and 3 to 1 were approximately 1-1.50/0 of the center line velocity. Tracy (1965) observed that for rectangular ducts having width to height ratios greater than 5 the secondary motions are suppressed to the corner regions and gradually vanish as this ratio becomes large. Liggett, et al. (1965), compared calculated and experimental values of the secondary velocities and obtained good agreement. However, their method does not provide information concerning how the primary and secondary flows interact to form the established pattern. I n all previous experimental studies only fully developed turbulent flows were considered because it has been shown that secondary flow cannot exist in a fully developed (uniform) laminar flow (Einstein and Li, 1958; Maslan, 1958). Development of Proposed Model of Transition
The observed slight increases in the frictional resistance reflect an increase in the axial pressure gradient. This pressure gradient in turbulent flow may be seen to arise from a number of sources by considering the time-averaged momentum equation (we use standard Cartesian indicia1notation)
where w p = ~ ~ is the~ vorticity ~ ofv the flow, ~ p ,is the ~pressure, F is the external body force field, and p j k is the viscous stress tensor. The bracketed terms may be combined to yield __ the divergence of the "Reynolds stress" tensor p ( u j ' ~ k ' ), j which represents the turbulent momentum flux. For a steady state flow in which the external body forces are neglected, one may thus calculate the turbulent pressure gradient ( P = P 1 / 2 p i 7 i 7 ) as
+
Terms b through d vanish for a rectilinear laminar flow and the axial pressure gradient serves only to balance the viscous tractive forces as represented by term a. This condition corresponds to the region labeled laminar in Figure 1. For a laminar flow around corners, in an entrance or in helical coils, terms b and c give rise to an increase in the pressure drop as a result of the additional energy required to maintain a mean secondary motion. The order of magnitude of such an increase is in agreement with that observed in the "transition" region increase in the f . R e product shown in Figure 1. At higher values of the Reynolds number, the transition to a fully turbulent motion occurs with the result that term d causes a large increase in the frictional resistance for a proportionately small increase in the Reynolds number. This state is identified as the turbulent region in Figure 1. It is suggested that the observed increase in the frictional resistance curve for isosceles triangular ducts, as shown by the data in Figures 1 and 2, reflects the successive contribution of each of the terms in eq 2 as a function of increasing Reynolds number. That is, the laminar portion of Figure 1 is due to the action of term a in eq 2, with the upper boundary of this 108 Ind. Eng. Chern. Fundam., Vol. l l , No. l , 1972
region being determined by stability considerations (Hanks, 1969; Hanks and Cope, 1970). We propose that the increase in f .Re in the transition region portion of the curve in Figure 1 arises from an increased energy dissipation due to the superposition of a three-dimensional secondary motion, described by terms b and c in eq 2, on top of the laminar motion due t o term a. The upper bound of this region would also be determined by stability considerations (Hanks, 1969) applied to the three-dimensional flow field. From a detailed knowledge of this field quantitative calculations could be made in a manner similar to that employed for the simpler concentric annulus case (Hanks and Bonner, 1971). Finally, the steeply rising turbulent portion of the curve in Figure 1 is due primarily to the action of term d in eq 2 with a smaller contribution from term a arising principally near the duct walls. This latter flow field should have a two-dimensional axially invariant secondary flow pattern similar to that observed by Brundrett and Baines (1964) for rectangular ducts. This means that once transition to turbulence has occurred, terms b and c of eq 2 add vectorially to zero. This proposal, that the additional energy dissipation resulting from the action of mean secondary velocities provides the increase in axial pressure loss reflected by the f . R e product in the transition region, differs from all previous analyses of secondary flow in uniform straight channels. Previous investigators have postulated that in the absence of turbulent flow no driving forces exist for the secondary flow; a conclusion which is valid if the flow is only two-dimensional. The essential feature of the proposed model is the three-dimensionality of the laminar flow which drives the secondary circulation cells. If one assumes that a mean secondary flow is superimposed upon the primary flow in the transition region, then the effect of this secondary motion upon the stability of the entire flow field can in principal be predicted by stability calculations. For this purpose a suitable representation for the secondary flow pattern must be known or assumed. An approximate analysis may be conducted for the equilateral duct by assuming the secondary flow pattern shown in Figure 4. This assumed secondary flow pattern was developed (Cope, 1970) by using the results of Brundrett and Baines (1964) obtained in a square duct as a model. By assuming the secondary velocities to be approximately 1.5% of the local axial velocity (calculated by Sparrow's (1962) method) for laminar flow a series of secondary velocity vectors was drawn. These vectors were then adjusted so as to cause finite difference approximations to the derivatives to fall on smooth curves and satisfy the requirements of continuity. The details of this estimation process are found elsewhere (Cope, 1970). The stability calculation is then made using the stability parameter K defined as (Hanks, 1969)
(3) This parameter is calculated numerically using the assumed velocity profile shown in Figure 4 and a map of K values throughout the cross section is prepared. According to the theory (Hanks, 1969) upon which K is based, whenever K < 404 the flow is stable. The results of this approximate calculation, carried out a t one axial position (Cope, 1970), showed a marked decrease in the magnitude of K everywhere in the cross section when the assumed secondary flow field was included compared with the values K would have if no secondary flow pattern were assumed. In fact, regions of the flow which had K > 404 without secondary flow became areas of K < 404 when secondary flow were assumed. Also, the loca-
tions of maximum K , which correspond to points of minimum stability, shifted position toward the corners when the secondary flow was assumed. This result seems qualitatively reasonable. Since this entire calculation was only approximate and appeared t o be quite sensitive t o the magnitudes of the secondary velocity gradients no attempt was made t o compute quantitative results. Rather, the qualitative confirmation of the stabilizing effect of the secondary flows was accepted as a sufficient result of the calculation. Another interesting result of this approxiniate calculation is the observation that the secondary velocities appear t o contribute a greater increase t o the shear stress terms than they do t o the secondary vorticity terms. If the above model of the transition range, in which a mean secondary flow is superimposed upon the primary laminar stream creating a three-dimensional laminar flow, is correct, one may expect the following observable phenomena t o occur. (1) The frictional resistance curve of f . R e us. Re should increase slightly over the laminar value as a result of the additional energy dissipation caused by the mean secondary motion in the transition range. (2) There should be no turbulence intensity in the flow stream for at least the first part of the transition range of Reynolds numbers. (3) The flow stream should be stabilized due to the presence of the secondary motion and transition t o turbulence should be significantly delayed beyond the upper limit of stability of uniform laminar flow. (4) Velocity profiles measured on planes parallel to and successively approaching the triangular base should demonstrate a convection of the maximum velocity away from the altitude of the triangle and toward the corner regions. ( 5 ) Velocity profiles measured along given curves in the cross section but at different axial positions should not be similar. A n inquiry of the above conditions was made by measuring the mean velocity profiles and turbulence intensity at various transverse positions and a t two axial locations for a series of Reynolds numbers below, above, and in the transition region (as defined by the pressure loss data) for two different triangular ducts. Because it is extremely difficult to measure both the magnitude and direction of the mean secondary, velocities accurately (Hoaglaiid, 1960; Patterson, 1966; Tracy, 1965) a n alternate approach was used to infer their presence. Alean velocity profiles were measured with a pitot tube and the behavior of the positions of maximum velocity and the shapes of the isovels were used to infer the presence of secondary flows. Experimental Work
The recirculating flow loop used consisted of a 55-gpni centrifugal pump, a filter section, a turbine flow meter, the test section, and a stainless steel reservoir containing a temperature control system. Details and specifications are given by Cope (1970). Aqueous solutions of polyglycol a t 25OC were used as the test fluids. These solutions are Xewtoiiian and had viscosities ranging from 22 to 60 cP depending upon the concentration. A constant temperature DISA Model 55.401 hot film anemometer and Thermo-Systems hot film parabolic probe were used to measure the turbulence intensities. The pitot tube was coiistructed by nesting two hypodermic needle tubes of differing size in a piece of brass tubing rrhich formed the bent tube portion. The needles were 0.020 and 0.035 in. o.d., respectively. The tip of this probe was made approximately hemispherical by shaping it n ith emery cloth
Table 1. Isosceles Triangular Duct Dimensions Overall length apex angle, deg
12-ft base, in.
Height, in.
2.00
30 60
1.07 1.73
1.50
-+-+---$-+-+-
+-•
7
Figure 4. Approximate secondary flow field assumed for theoretical stability calculation in equilateral triangular duct
and exaniining it under a microscope. The pitot tube was calibrated taking into account the “Barker effect” (Barker, 1922) which causes a n increase in the measured static presure drop due to the viscous forces a t low Reynolds numbers (based upon the pitot tube radius). The fluctuating signal from the DISA anemometer was fed to a Tektronix 564 storage oscilloscope which allowed the trace of the velocity us. time plot to be recorded and photographed to provide permanent records of the observed events. Average velocity measurements were determined with a Cox Model An’ 24 turbine flow meter calibrated at the system viscosity. Fluid density measurements were made using a calibrated pycnometer and an hlGW Lauda constant temperature circulator Type D40 which maintained a given temperature to within +O.0loC as determined by an XBS secondary standard thermometer. The viscosity measurements were made in the same bath using a calibrated Ubbelohde viscometer. A Pace-Wanco Model P7D pressure transducer was used to measure the pressure differential from the pitot tube. The signal from the transducer was conditioned by a Pace hlodel CD-10 carrier demodulator and then displayed on a HewlettPackard Model 3430A digital voltmeter. The details of the two isosceles triangular ducts used are given in Table I. Experimental Results
In order to determine if the flow in the equilateral duct in transition was three-dimensional in nature, mean velocity profiles were measured at tlvo different axial positions: L / D = 123 and L / D = 93. The profiles were first recorded for the longer length in order t o ensure that no obstructions in the path of the flow stream ahead of the pitot tube or hot film anemometer existed t o introduce spurious disturbances. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
109
,
+x2
Figure 5.
I I
Coordinate system used for equilateral triangle
-
I
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I
-
-x3 0
0.00
0
0.10
-
5c
r
I
I
I
I
I
I
I -
a 0.10
LID = 123
0-
A 0.20
0
4 0.1
02
0.3
0.4
0.5
0.6
O.?
4 2
Figure 7. Velocity profiles (ft/sec) measured in the equilateral duct for Re = 1700, I / D = 123
5 -
3 -
2 -
I
I
I
0
0.1
0.2
I 0.3,
I
I
I
0.4
0.5
0.6
O.?
X2
Figure 6. Velocity profiles (ft/sec) measured in the equilateral duct for Re = 1 100, I / D = 123
Then the necessary holes were cut in the duct a t the shorter L / D value and a second series of measurements was made. The static pressure wall taps were inspected with a 4X magnifying glass to be certain that they were identical. The pitot tube reference coordinate system used is defined in Figure 5 and typical mean velocity profiles measured a t L I D = 123 are shown in Figures 6, 7 , and 8. The solid linea in these figures represent the mean velocity profiles for the case of rectilinear laminar flow in the equilateral duct as given by Knudsen and Katz (Knudsen and Katz, 1958). Dashed lines are used to indicate a faired curve through the experimental data in the turbulent range because a t present no theoretical means exist to calculate transitional velocity profiles that would include the effect of secondary velocities. Figure 9 is a summary plot a t one particular location for all four Reynolds numbers showing the steady shift of the point of maximum velocity away from the altitude toward the corner. Figure 10 shows faired isovel curves derived from the experimental measurements on the left side in comparison 110 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
L-A 0I
0
0.1
0.2
0.3
-%
0.4
0.S
0.6
0.7
Figure 8. Velocity profiles (ft/sec) measured in the equilateral duct Re = 2400,I/D = 123
with theoretical laminar isovels computed for the same Reynolds number on the right side. Figure 11 contains photographs of oscilloscopetracings for (a) an open pipe, (b) the 30' apex angle isosceles triangular duct a t several positions and one fixed Reynolds number, and (c) the 60' equilateral triangular duct a t the point of maximum velocity for a series of Reynolds numbers. These pictures
I
I
I
SI-
I
I /
I
I
0.5
0.6
t
+
4
*3
7
6
u, 6
5
4
3
2
0
0.1
0.2
0.3 0.4 -X2
0.7
Figure 9. Summary plot of profiles (ft/sec) for equilateral duct showing shift of location of maximum velocity toward the corners
illustrate the qualitative features of the flows in the different ducts. Figures 12 and 13 are photographs of oscilloscope tracings of anemometer outputs a t the position of maximum stability parameter K ( z ) in a pipe and in the equilateral triangular duct, respectively. At each location, the measured turbulence intensity is indicated as 707’.Figure 14 shows the set of tracings for the equilateral triangular duct in relation to the frictional resistance curve illustrating the existence of an extended region of increased pressure drop with no turbulence intensity observable. Figure 15 shows a plot of turbulence intensity measured a t two positions as a function of Reynolds number in comparison with measured values of the f . R e product for the equilateral triangular duct. The probe positions correspond to the location of maximum velocity, labeled U,, and the position of maximum theoretical K ( x ) , labeled Kmx. These data correspond to the traces shown in the previous figures and quantitatively define a region of Reynolds number in which a definite pressure drop increase occurs but for which
%T=O. Figure 16 shows the same comparison as shown in Figure 15 for a 30’ apex angle isosceles triangular duct. The two probe positions shown here are the two positions of relative maxima of the stability parameter K ( z ) as calculated from the earlier work (Hanks and Cope, 1970). The position of the smaller of the two maxima is labeled KmlnimSx. These data clearly show that the two-macroscopic zone model of earlier work [Eckert and Irvine, 1956,1957;Hanks and Cope, 19701 is incorrect. Figure 17 contains a summary of mean velocity profiles measured a t a fixed position in the equilateral triangular duct for a series of Reynolds numbers and a t both L I D positions. Solid curves are calculated theoretical rectilinear laminar profiles. Dashed curves represent the faired data a t L I D = 123 while the dotted curves represent the same thing a t L I D = 93. No solid curve is shown for the highest Reynolds number
Figure 10. Velocity isovel contours for equilateral duct, Re = 1700, = 123
LID
because no theory is available for turbulent profile calculations. Figure 18 shows photographs of oscilloscope tracings for the probe being located a t approximately the point in the equilateral duct cross section (indicated by the cross in the diagram) where the maximum deviations were observed between the measured velocities and the theoretical laminar curves. These traces were obtained a t a series of Reynolds numbers and the measured turbulence intensities are also recorded. Figure 19 shows a series of photographs of typical turbulence “bursts” a t Re = 1610 in the equilateral triangular duct a t the position of maximum velocity. In this picture the time scale of the recording oscilloscope was speeded up by a factor of 10 in order to display the details of the probe output. Discussion of Results
The data shown in Figure 6 are a series of mean velocity profiles along planes parallel to the duct base a t different altitudes for the case L I D = 123 and Re = 1100 which corresponds to the flat horizontal region of the frictional data in Figure 2. From the close agreement between the experimental data and the solid theoretical curves calculated by Sparrow’s method (Sparrow, 1962), we conclude that under this set of conditions the flow corresponds to fully developed, rectilinear laminar flow. For this same Reynolds number there was observed a slight ( -2,5y0a t the center line) indication that the flow JTas not completely developed a t L I D = 93, but the difference was so slight that no measurable effect on the pressure losses could be found. When the Reynolds number was increased to 1700, however, a very marked change in the situation occurred as shown by the data in Figure 7. These data correspond to those in Figure 6 in all details except that Re = 1700. It is most apparent from these data that the flow field is far from being a fully developed rectilinear flow (solid theoretical curves). lnd. Eng. Chem. Fundam., Vol. 1 1 , No. 1, 1972
11 1
-Re POS~TION' 960
.00
Q
0.70
1515
0.50
1515
0.30
1515
0.16 'in. from
Q
(aI
tbi
10)
IC1
2200
1515
base
1230
Q
1340
1490 1660 2580
B
1015
t-
t+
Figure 1 1. Oscilloscope tracings for qualitative comparison of transition in (a1 1.5-in. diameter DiDe; (b) isoceles triangle; (c) equilateral triangular duct
30" apex angle
Re % _ _T 1500
0
1700
0
1900
0.29
2100
0.85
ec./di".
t-
datK
tive deviations a t X z = 0 (the duct altitude) to 2&25% in the vicinity of X 2 = -0.35 and X 3 = -0.30. In particular, for the profiles in the lower half of the duct, the location of maximum velocity is seen to be shifted away from the center line of the duct toward the corner region. Since this type of velocity profile behavior is characteristic of the action of mean secondary motions in the plane of the cross section (Bruudrett and Baines, 1964; Hoagland, 1960; Nikuradse, 1930), we interpret the data of Figure 7 as indicating the existence of a mean secondary motion in the plane of the cross section. Since, a t the Reynolds number corresponding to these velocity data, a frictional resistance increase of 7.5% over the rectilinear laminar value is observed, it seems reasonable to conclude that this increase is caused by the additional energy dissipation associated with the mean secondary motion. That it is indeed due to this cause and not to turbulent energy dissipation will be shown below. 112 Ind. Eng. Chem. Fundam, Vol.
11, No. 1, 1 9 7 2
position in 1.5411. pipe. %T is the per cent turbulence
From Figure 2, it appears that Re = 2400 corresponds to turbulent flow since a very marked increase in thef.Re product is observed which is Characteristic of this condition (Hanks, 1968). For this reason, no theoretical curves can be computed for comparison with the Re = 2400 velocity profile data presented in Figure 8. Even so, the shapes of the faired data curves (dashed curves) still clearly reflect the transposed maximum characteristic of mean secondary flows. In this case the secondary mean motions are superimposed upon a turbulent basic flow as opposed to being superimposed upon a laminar basic flow as in Figure 7. Furthermore, it is evident that these secondary motions penetrate more deeply into the corner regions. The degree of distortion of the isovels toward the corners is clearly shown in Figure 9 where a series of velocity profiles a t Xa = -0.30 and a series of Reynolds numbers is shown. A curve is drawn through the points of maximum velocity to show how this point progressively shifts toward the corner as the Reynolds number is steadily increased. This distortion
_ Re _ %T 1100
0
1200
0
1300
0
1500
0
1600
.81
1700
1.74
-R e_%T 1800 2.61
2000
3.56
5 0 mvldiv.
L
t+
0.5
t-+
st:cldiv.
Figure 13. Oscilloscope tracings for hot film probe located at Kmsxposition in equilateral triangular duct. %T i s per cent turbulence intensity I
20
l4
1 I
3 $41
20
r-m
2
18 I
16 14
8
10
12
14
16
18
20
22
mllm Figure 14. Oscilloscope trocings for equilateral duct with position. Tracings correlated with hot film located at K,, f.Re vs. Re data
is revealed in another way by the series of experimental isovels on the left side of Figure 10. The isovels on the right side are what would exist if the flow were fully developed rectilinear laminar motion. The gross distortions caused by the secondary currents are plainly evident from Figure 10. From these mean velocity profiles it seems quite evident that secondary motions exist. The question of real interest is their origin and driving force. For this purpose hot film anemometer measurements were made. In order to become familiar with the characteristic response of the hot film
Figure 15. Comparison of turbulence intensities at two locations in equilateral triangular duct as a function of Reynolds number with f.Re data for same duct
anemometer and probe, preliminary measurements of the instantaneous velocity as a function of time were made in several ducts and are shown in Figure 11. While no qnantitative measures of turbulence intensity are available for these particular photographs, they contain some very useful and significant qualitative information. They were made to test the validity of the earlier suggested model (Eckert and Irvine, 1956; Hanks and Cope, 1970) that in the transition range of Reynolds numbers, macroscopic zones of both laminar and turbulent flow coexisted simultaneously. Thus, preliminary velocity traces were made in the 30’ duct a t four positions measured from the base of the duct toward the apex along the altitude for a Reynolds number from which the frictional Ind. Eng. Chem. Fundam.,Vol. 11, No. 1, 1972
113
I
I
I
I
I
4
I
I -0.30
I
4
J -I
5 1500 1700 2400
0
A 0
LID.
+
7
93
ALL
5 4
-..__. E X P E R IM E N T A L
6
>
(
0.1
0.2
0.3
-
0.4
0.5
0. 6
0.7
x2
Figure 117. Composite curves for velot:ity profiles (ft/sec) as a furiction of l / D and Re a t Xa = -0.30 in. equilateral triangul'ar duct
Id i v.
duct with probe a t Xs = -0.35 in., Xa = -0.30 in. This observed velocity profiles and theoretical laminar profiles.
plot of the friction factor us. Reynolds number for this duct which was shown in Figure 3. Part c of Figure 11 shows that the frequency of turbulent bursts passing the probe in a given time interval increases as the Reynolds number is increased from a laminar value (Re = 1230) through the three-transitional values shown to a tnrbulent (as indicated by the frictional resistance curve of Figure 2) value of Re = 1815. It is particularly evident that a t Re = 1490 no turbulence is
above, strongly suggest that the increase in the f . R e is primarily due to the action of the mean secondarj and not to turbulence. The remaining feature of t nomenon which must he explained is the mechanism 1: of which this nonturbulent secondary motion is driven. It has been amply and clearly demonstrated in fully developed turbulent flow (Brundrett and Baines, 1964) that secondary mean motions arise from and are driven by differences in the turbulence intensities in the transverse directions. This mechanism was explained by Brundrett and Baines (1964) by considering the mean vorticity equation for steady flow I
t-
-
Figure 19. Oscilloscope tracing of typical turbulence "burst" for Row in equilateral triangular duct for Re 1610. Time scale is expanded by 10 X over other pictures shown herein
present. From these preliminary ob I that the original model of a "tw angular cross section is essentially ~ i i t i o i i ~aiiu c ~ uittli m e observed increase in the frictional resistance data must be ascribed to a different phenomenon. Quantitative measurements of the percentage turbulence intensity for the pipe flow case measured a t the radial position r/R = l/'& corresponding to the position of maximum K(r) (Hanks, 1963) are shown in Figure 12. The traces for Reynolds numbers of 1100, 1200, and 1300 are not shown because they are identical with the trace for Re = 1500. These data are included for comparison with similar data for the triangles. A feahure of particular interest to note here is the existence of intermittent bursts of turbulence which occur with increasing frequency as the transition to turbulence at Re = 2100 occurs. This behavior is identical with that observed by others (Lindgren, 1957; Rotta, 1956). This type of behavior is characteristic of a transition to turbulent flow and is accompanied by a sudden steep rise in the product f .Re in relation to the laminar value. Figure 13 illustrates the instantaneous velocity and percentage turbulence intensity observed in the equilateral duct for a range of Reynolds numbers with the probe located at the K , position. The first measurable turbulence intensity occurs a t a Reynolds number uf 1600. The data of Figure 14 clearly show that this value of the Reynolds number is well into the transition region. Therefore, it is clear that the mechanism responsible for the upward drift of the f , R e data above the rectilinear value of f.Re = 13.3 results in a measurable increase in the f . R e product of the flow stream before any turbulence intensity is detectable. This conclusion is quantitatively substantiated by the data shown in Figure 15 for the equilateral duct and the same conclusion can be drawn for the 30" duct from the summary in Figure 16. In an effort to be certain that these conclusions were sound, we made many measurements in the range between the lower critical Reynolds number and the Reynolds number of lowest detectable turbulence intensity with the sensitivity of the anemometer set as high as it could be set and failed to observe any indication of turbulence. These data, together with the mean velocity data discussed
In one- and two-dimensional flows. term h. which tnev physically interpreted as representing the increase in vortcity due to the stretching of vortex lines, vanishes exactly. Thus,,since 1 ~ : ~ ~ ~:~~ JZz~~.:~ Germs a ana a1 comwinen represem simply brit: uiiiusiou and convection of mean vorticity (Aris, 1962), only term c can represent a production of vorticity and hence provide a driving force for the secondary cellular motion. In the present situation, however, we have demonstrated the presence of a secondary cellular flow under conditions for which term c of eq 4 is zero. Therefore, term b must be uouvanishing in order to provide the source of vorticity required to sustain the secondary cells. However, the nonvanishing of term b in a laminar flow (as the flow must be if term c vanishes) is a necessary and sufficientcondition that the flow be three.. . , (Aris, . IYoLJ. -, ,. . . aimensional 11 tuis couclusion is CorrecTI. mean velocity profiles measured at different 8bxial locations should differ from each other measurably. Sm:h measurements will thus refute or verify the validity of this I:iroposition. ,"I., rl*m..llo+*n+n +Le+ The data presented in Figure 17 cIesLLJ uylllylluyluyc the flow is three-dimensional. The data for Re = 1500 (square data points) are of special interest. It is clear that both experimental curves (dashed, L / D = 123; dotted, L / D = 93) differ markedly from the theoretical rectilinear laminar curve (solid curve) and simnltaneously show the shift in maximum . . . . ... . . . . . velocity toward the corner which is characteristic of secondary flow. Furthermore, for Re = 1500, the per cent turbulence intensity is zero. The differencebetween the two experimental curves (which exceeds all reasonable estimates of experimental error and is, therefore, significant) is obvious and thus clearly demonstrates the three-dimensionality of the flow. Since term c in eq 4 can provide the source term for the additional vorticity associated with the mean secondary cells, the existence of nonzero values of term b is not required to provide the driving mechanism in turbulent flows and so term h can vanish. Thus, one might expect to find a trade-off occurring between terms h and e as the Reynolds number increases. In this trade-off one would expect term b to vanish gradually in favor of term e. The data of Figure 17 indicate that this trade-off actually appears to take place. In the top set of curves (Re = 1100) the separation between L/D = 123 and L/D = 93 is clear and T = 0 (T represents per cent turbulence intensity). The same is true for the second set of data (Re = 1500) which we have already discussed above. However, the third set of data (Re = 1700), for which T 21 2% a t the K,, position, show the profiles at the two axial locations to be very close to each other in both shape and degree of deviation from the rectilinear laminar curve (solid line). Finally, the curves for Re = 2400 (for which T = 4%) a t the two positions are identical everywhere except very near the corners. It appears that by this value of Re, the effect of term b has nearly vanished and the flow is essentially two-dimen-
.
1
, I
~~~~~~~~~~1 ~~1
1
1
~
~
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yIIayy
!nd. Ens. them. Fundmrn., Vo~ I . l l , N o . l , 1 9 7 2 115
sional again with the secondary motion being driven by the turbulence intensity differences as explained above. The above experimental results indicate the existence of the hierarchy in the stable forms of motion which can occur as a function of increasing Reynolds number for flow in noncircular conduits. The first level of this hierarchy appears to be a rectilinear, laminar, one-dimensional flow. The second level appears t o be a stable, three-dimensional, nonturbulent transitional flow containing secondary flow cells in the plane of the cross section. Finally, the third level is a fully developed, turbulent flow consisting of two-dimensional mean secondary cells superimposed upon a one-dimensional mean axial flow, This sequence of flows is quite analogous to the sequence observed in the Taylor vortex phenomenon in a rotatioiial Couette viscometer. Figure 18 is included to illustrate that even a t the point in the equilateral cross section where the maximum deviation from the rectilinear laminar profiles was observed (indicated by cross in the diagram), there is no measurable turbulence intensity until Re = 1500. At this Reynolds number, the measured profiles have already exhibited the pronounced bulging effect attributed to a mean secondary current. At this point we take occasion to make a rather interesting observation. If one recalls the fact that the data plots shown in Figures 7 and 10 represent only half of the duct, the other half being given by a mirror image reflection about Xz= 0, then it becomes evident that the velocity profiles shown possess multiple points of inflection. Similar character caii be inferred for cross plots a t constant X s for the same data. In classical two-dimensional hydrodynamic stability theory (Lin, 1955), it is well known that the existence of an inflection point implies instability. Furthermore, Betchov and Criminale (1964) concluded that mean turbulent profiles were unstable when they possessed points of inflection. However, the present data clearly show that the three-dimensional secondary flow which occurs in the transition region of these ducts is stable. X o disturbances grow t o cause transition to turbulence aiid the motion caii be observed over long periods of time and reproduced a t will by approach from either higher or lower Reynolds numbers. Therefore, it appears that the existence of the mean secondary circulation, F\ hile introducing inflections into the mean velocity distributions, exerts a stabilizing influence on them which counterbalances this normally destabilizing influence. This result suggests strongly that inferences drawn from simple one-dimensional flows (and theoretical analyses of the same) involving only two-dimensional disturbances may provide misleading guides for more complex threedimensional phenomena. A similar conclusion follows from the work of Spielberg and Timan (1960), mho concluded that a n essential difference exists betrreen two- and three-dimensional stability characteristics of classical Poiseuille pipe flow. Therefore, it appears that further study involving complex three-dimensional floir s is required before a complete understanding of stability of laminar flow and transition therefrom is to be had. In order to examine the nature of the turbulent bursts observed in the equilateral duct as the transition to turbulence is approached, photographs were made of a number of oscilloscope tracings using a time scale ten times shorter than that used for the pictures shown in previous figures. Several typical bursts are shown in Figure 19. These bursts are characterized by a short time duration and a high intensity as contrasted with the longer time duration and lower intenqity of the turbulent bursts evident in the pipe flow case shown in Figure 116 Ind. Eng. Chern. Fundarn., Vol. 11, No. 1 , 1972
12. Just prior to the transition to turbulence the duration of the turbulent bursts in the pipe flow case averages approximately 0.45 sec per burst as compared with only 0.15 sec for those in the 30 and 60" isosceles triangular ducts under similar flow conditions suggesting that the turbulent bursts are being damped more strongly in the triangular ducts than in the pipe. This conclusion is consistent with the approximate stability calculations based on the assumed mean secondary flow field in the triangular ducts (Cope, 1970) and bhe observation above concerning the stabilized inflected profile flow. That is, the presence of the secondary flow appears to have a definite stabilizing effect on the primary flow with the result that strong turbulent bursts are damped out faster for the flow conditions in the triangular cross sections than are similar bursts in the pipe flow where no secondary flows exist. Conclusions
From the above observat'ions one can draw t'he following conclusions. 1. In the transition region of triangular ducts velocity profiles measured on planes parallel t o arid near the base of a n equilateral duct show the position of the maximum velocity to be displaced a m y from the altit'ude toward the corner region. This distortion is interpreted as indicating the presence of a secondary circulation superimposed upon the primary flow. 2. In the transition region of such flows no measurable turbulence intensity occurs unt'il a Reynolds number is reached just prior to t'he abrupt break in the f . R e us. Re curve, indicating the onset of turbulent flow. Thus, the additional energy dissipation occasioned by the mean secondary circulations mentioned above appears t o be responsible for the slight increases observed in the frictional resist'ance data in the tmiisition region. 3. Bursts of turbulence in the isosceles ducts are damped approximately three t'imes faster than are similar bursts in pipe flow under similar dynamic conditions. This indicates that the observed mean secondary motion exerts a stabilizing effect on the primary laminar flow aiid is responsible for delaying the transition to turbulence beyond the stability limit of a rectilinear laminar flow. 4. The existence of inflection points in the mean axial velocity distributions does not necessarily imply instability as suggested by Betchov and Criminale (1964). The superposit'ion of transverse secondary motions which cause the inflections to develop appears to provide a stabilization of the flow simultaneously. This phenomenon requires further investigation before it is fully understood. 5. The driving force for the mean secondary motion encountered in the transition region arises from t'he threedimensionalit,y of the flow. As transition progresses this mechanism is gradually replaced by a turbulent mechanism which has been described by previous investigators. 6. The transition from a developed rectilinear laminar flow to a nonturbulent, three-dimensional, secondary flow occurs a t Reynolds numbers which are in quantitative agreement wit'h those calculated theoretically (see Hanks and Cope (1970) for experimental verification) for the stability limit of the rectilinear laminar flow. literature Cited
Ark, R., "Vectors, Tensors, and the Basic Equations of Fluid Mechanics," p 115, Prentice-Hall> Englewood Cliffs, K . J., 1463
Barker; A I . , Proc. Roy. SOC.,Ser. A 101, 435 (1922). Betchov, R., Criminale, W.0.)Phys. Fluids 7, 1920 (1964). Brundrett, R., Baines, W. D., J . Fluid Mech. 19, 373 (1964). Carlson, L. W.. Irvine, T. F.. Jr., Trans. A S M E , J . Heat Transfer 83, 441 (1961).
Cope, R. C., M.S. Thesis, Brigham Young University, Provo, Utah, 1968. Cope, l., 1960. Knud;;;, J. G., Katz, D. L., “Fluid Dvriamics and Heat Transfer, p 103, bIcGraw-Hill, New York, N. Y., 1958. Lamb, H . , “Hydrodynamics,” 6th ed, Chapter 7 , Dover Publications, New York, S. Y., 1932. Leutheusser, II. J., Proc. Amer. SOC.C i d Eng., J . Hydraul. Diu. 89 ( S o . HY3), 1 (May 1963).
Liggett, J. A., Chiu, C. L., Rliao, L. S., Proc. Amer. SOC.Civil Eng., J . Hydraul Diu. 91 (No. HY6), 99 (Nov 1965). Lin, C. C., “The Theory of Hydrodynamic Stabilitv,” “ , Cambridge ” University Press, New Yoik, N. k.,1955. Lindgren, E. R., Arch. Fysilc 12, 1 (1957). Xialaika, J., Hydr. Diu.J . (HY6) 88, 1 (Nov 1962). hlaslan, S.H., Quart. Appl. Math. 16, 173 (1958). Nikuradse, J., Zng.-Archiu. 1, 306 (1930). Patterson, G. K., Ph.D. Thesis, Universitv of Missouri. Rolla. N o . , 1966. Prandtl, L., Proc. Intern. Congr. Appl. Mech., 2nd Congr., Zurich 62 (1927). ltothfus, It. R., Yionrad, C. C., Senecal, T’. E., Ind. Eng. Chem. 42, 2511 (1930). Rotta, J., Ing. Archiv. 24, 258 (1956). Simisky, P. L., h1.S. Thesis, Brigham Young University, Provo, Utah, 1967. Sparrow, E. AI., A.I.Ch.E. J . 8 , 599 (1962). Spielberg, K., Timan, H., Trans. ASME, Ser. E., J . Appl. Mech. 27(3), 381 (1960). Tracy, H. J., Proc. Amer. SOC.Civil Eng., J . H y d r . Diu. 91 (No. HY6), 9 (Nov 1965). Van Driest, E. R., J . Aeronaut. Sei. 23, 1007 (1956). RECEIVED for review March 31, 1971 ACCEPTED October 4, 1971 AIChE Meeting, Chicago, Ill., Nov-Dec 1970. This work was supported by XSF Grants GK-1922 and GK-15893.
Stability of an Exothermic Reaction inside a Catalytic Slab with External Transport limitation James C. M. Lee, Lakshminarasimha Padmanabhan,” and Leon Lapidus Department of Chemical Engineering, Princeton Lniversity, Princeton, N . J. 08540
The stability of an exothermic reaction A + B occurring inside a catalytic slab with a significant external heat transfer limitation i s investigated. By averaging the distributed state variables over the total catalyst volume, useful stability criteria may be obtained in a straightforward manner. It i s shown that the low-temperature steady state may be unstable.
w h i l e efficient techniques are available to treat the stability of systems whose dynamical behavior is represented by @.D.E.’s,there is no systematic means to determine stability in systems described by P.D.E.’s. A typical example of such a system is that of a porous catalyst particle with int’ernal chemical reaction. During the past few years extensive research has been carried out in this direct’ionand some significant results have in fact been obtained. Typical works have been those of Gavalas (1968) usiilg topological methods, Luss, et al. (1967, 1968), and Kuo and Amuiidson (1967) on nonself adjoint’ eigenvalue problems, and Berger and Lapidus (1968) and h’ishimura and Matsubara (1969) using Liapunov sufficieiicy conditions. These results, although obt’ained by the use of rigorous mathematical techniques, are quite conservative and many important stability problems of the reacting catalyst remain unsolved. The main difficulty stems from the occurrence of a n irreducible set of P.D.E.’s and complex eigenvalue problems in function space. Giiless some new technique of attacking bifurcatioii and eigenvalue problems in
function space are developed, it seems that no significant progress is imminent along these lines. In the meanwhile, or even as the only feasible approach, it may be fruitful to turn toward approximation techniques. Since the stability of a lumped system is more easily analyzed, approximation of a P.D.E. (distributed) system by a corresponding O.D.E. (lumped) system may provide a n efficient means to obtain the information regarding complex stability. Hlavacek, et al. (1969a, 1969b), used a heuristic method to carry out such a n approximation; Luss and Lee (1972) recently used an averaging technique also. I n the present work we use this same averaging procedure t o investigate the stability of a n exothermic reaction A + B occurring inside a catalyst slab with a significant external heat transfer resistance. Stability Analysis
The system under study is an infinite catalyst slab of thickness 1. @neside is exposed to the reactant in the fluid phase and Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
117
