LAMINAR-TURBULENT TRANSITION I N DUCTS OF R E C T A N G U L A R C R O S S SECTION RICHARD W. HANKS AND HANG-CHING RUO’ Department of Chemical Engineering, Brigham Young University, Prooo, Utah
The lower critical Reynolds number for the laminar-turbulent transition has been calculated theoretically as a function of the duct aspect ratio (ratio of width to height) for the steady, isothermal flow of Newtonian fluids in straight ducts with constant, rectangular cross sections. The theoretical calculations are compared with data reported by seven independent investigators in ten systems and found to be in agreement to within experimental precision. The theoretical curve is recommended for practical calculations, and a simple “rule of thumb” for ducts of large aspect ratio is given.
HE details of the mechanism of the transition from laminar T t o turbulent flow are a t present not fully known. Various theoretical methods have been used to predict the stability of hydrodynamic flows and the transition from laminar to turbulent flow. The problems of hydrodynamic stability and the laminar-turbulent transition are different (9). The problem of hydrodynamic stability concerns the behavior of infinitesimal, time-dependent, periodic disturbances to a steady flow field. The prediction of instability does not necessarily indicate a transition to turbulence, but rather may indicate transition to a different stable form of nonturbulent flow, as in the classic example of the existence of secondary vortices predicted by Taylor (74) for the Couette viscometer. T h e problem of the laminar-turbulent transition deals with finite disturbances which result in transition to turbulent flow. Various theories have been proposed which lead to a parameter, usually related to the Reynolds number, which is used to predict the onset of turbulence. This method was reviewed by Hanks (6) and more recently by Ruo (72). Limitations and weaknesses in the various theories led Hanks (6) to utilize the Cauchy form of the continuum equations of motion to develop a generalized parameter which permits the calculation of the lower critical value of a n arbitrarily defined Reynolds number. The salient features (6) of this parameter are:
For Newtonian flow the parameter is proportional to the Reynolds number. The proportionality constant is determined by the particular definition chosen for the Reynolds number, but may be calculated theoretically from the solution of the equations of motion for laminar flow in the particular duct of interest. The parameter has the same physical significance +s the Reynolds number-that is, it represents the ratio of the magnitudes of an inertial force to a viscous force acting on an element of fluid. The numerical value of the parameter corresponding to the laminar-turbulent transition is independent of the flow channel geometry, the thermal condition of the flow field, and the rheological nature of the fluid. The third property listed above is perhaps the most useful of this generalized parameter-that is, a unique numerical value of the parameter permits one to calculate the critical value of a n arbitrarily defined Reynolds number regardless of such complicating factors as noncircular geometries, nonisothermal flows, or nowNewtonian fluid behavior (6). This latter feature has Present address, Lunday-Thagard Oil Co., Los Angeles, Calif. 556
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been adequately demonstrated (6, 7) for several different systems involving all of these complications in various combinations. The following development is a n application of this theory to steady, developed, isothermal flow of Newtonian fluids in straight ducts with constant rectangular cross sections. Theoretical Development
The generalized parameter is expressible in the form
K =
1
I
PVXW ___
I V 9 where w = VXv is the fluid vorticity and 1is the stress deviator tensor (2). In order to solve Equation 1 one must have a n algebraic expression for v which satisfies the equations of motion and appropriate boundary conditions. For the case of steady, rectilinear, laminar flow in a straight duct of constant rectangular cross section, Cornish (3)presented the following solution to the equations of motion:
where t = x / a , q = y / b , R = a / b , r = ( - d g / d z ) / ( 2 p ) , the fluid viscosity, and
I.(
is
(3) A, = ( 2 j
- 1) a2
(4)
The relation between the space coordinates, x and y, and their boundary values, a and b, is indicated in Figure 1. Cornish (3)calculated the mean flow velocity, D, as
= f
3
b2rF(R)
I t can be shown (6) for the case of rectilinear flow that Equation 1 can be written as
(7) where p is the pressure field, p is the density, and f is a body force per unit mass-e.g., gravity. Consider a physical system
a
easily be obtained. These calculations were performed with the assistance of a digital computer for a sequence of values of the aspect ratio, R = a / b . The results of these computations are presented in Figures 2 and 3. In Figure 2 the solid curve represents the theoretical values of Re, calculated as described above as a function of R. The data points represent experimental values obtained from literature sources as discussed below. The limiting value of Re, for large R corresponds to flow thrpugh infinite parallel flat plates, a limit previously (6) verified experimentally. The data for this limiting case are included here only for comparison purposes. Figure 3 is a plot of ij as a function of R. The calculations of 2 discussed above revealed that 2 occurred a t i j as given by = 0 except in the symmetrical case R = 1, Figure 3 and for which i j = = 0.637. Unfortunately, no data are available in the literature with which to compare the curve in Figure 3.
2
Figure 1. Geometry of rectangular duct showing location of recta nguIa r Ca rtesia ii coordinate system used
Comparison with Experiment
consisting of a Newtonian fluid flowing in fully developed, steady, laminar flo\v in a horizontal duct of rectangular cross section for which Equation 7 becomes
By introducing the d:imensionless variables defined above, together with a dimensionless velocity field u = V / O , one can express Equation 8 in the form
where
A literature search was conducted for data obtained in rectangular systems which spanned the laminar-tyrbulent transition Reynolds number. From these data, values of Re, were obtained for comparison with the present theoretical calculations. From the published review papers which tabulate lower critical Reynolds numbers for different noncircular ducts it is apparent that copsiderable confusion exists as to the values to be used. For example, Rothfus et al. ( 7 7 ) quote the critical Reynolds numbers measured by Davies and White (4)and Cornish (3) as 2960 and 2360, respectively. Eckert and Irvine (5) list 2800 and 2800, respectively. Similar discrepancies exist for other cases. Therefore, it was deemed necessary to reanalyze the existing data. T h e theoretical solutions of the equations of motion for various straight ducts can be written in the form fRe = X
(laminar)
(14)
Empirical correlations for turbulent flow when written in the same way yield f R e = cRem, rn Equations 2, 5 , and 12 show that Z(6,7 , R) is independent of R e and depends only on the geometry of the flow field. The lower critical value of the Reynolds number occurs when K = 404, a unique constant ( 6 ) . The problem of predicting Re, has now been reduced to that of determining the extremal values of Z, frorn xvhich Re, is easily computed 4s
>1
(turbulent)
(15)
Clearly, the product f R e is a sensitive measure of the laminar or turbulent nature of the flow. Hence, i f f R e is plotted on linear coordinates as ordinate with R e as abscissa, all laminar flow data should cluster about the line given by Equation 14.
m0-
-
-
where we have set K = 404 in Equation 9 and represents the extremal value of Z evaluated a t 5 = $ and 7 = ij. According to the theory (6), ancl 7 are the coordinates of the spatial location within the cross section of the flow field which is least stable to a disturbance--that is. a disturbance introduced a t the point having coordinates ,$ and i j should result in transition to turbulence. Because of the infinite wries involved in Equation 2 the analytical calculation of i j , and 2, while straightforward, is tedious. A simpler, but effective, method \\as therefore used. Z was calculated for a mrsh of points 5 and 7 . From these calculated values of Z a hie-dimensional contour plot was made (72) which permitted graphical evaluation of 2, and f . By successively refining the mesh size sufficient accuracy could
z
E,
E,
1500,L
I
A
2
I
I
3 4 ASPECT RATIO, R
I
I
5
I
6
'
Figure 2. Critical Reynolds number, Re, defined Equation 13 as a function of duct aspect ratio, R
OD
by
Experimental points derived from fRe data published in references cited
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ASPECT R A T I O , R Figure 3. Spatial coordinates of theoretical location of turbulence inception as a function of duct aspect ratio, R ij occurs along line = 0. For R = 1 limiting value is i j = 0 . 6 3 7 . metric case only, = 0 . 6 3 7 also
As soon as departure from laminar flow is experienced, the data should begin to rise sharply with increasing R e as indicated by Equation 15. The intersection of a smooth curve drawn through the nonlaminar data with the line f R e = h then determines Re,. As a n illustration of this process, the data of Allen and Grundberg (7) are plotted as described in Figure 4. Using this technique, one estimates Re, = 2300 from their data rather than the value 2400 listed in the literature (5, 77). All data points plotted in Figure 2 for comparison with the theoretical curve were obtained from the reanalysis of the published data of the authors cited using this technique. The data of Whan (75) and Davies and White (4) for ducts of large aspect ratio (essentially infinite parallel plates) were compared previously ( 6 ) with theoretical calculations for infinite parallel plates and found to be in excellent agreement therewith. These data are included here to substantiate the limiting behavior of the present theory. The asymptotic limit of a rectangular duct is represented by infinite parallel flat plates. I t is apparent from Figure 2 that for aspect ratios in excess of about 6 to 10 the behavior of the system is within 2 to 370 of that of infinite parallel plates. This is well within the limits of experimental precision of the published data, and therefore distinctions with respect to R above about R N 10 are fruitless. Most of the data reported in the literature are in good agreement with the theoretical curve in Figure 2. The two values a t R = 3.52 listed by Nikuradse (70) and Schiller (73) are so greatly a t variance as to cast doubt on the validity of the data. Certainly the theoretical value of Re, falls within the range indicated by the data. Mors precise measurements might clear u p this problem. Hoivever, the 1:alues of Cornish ( 3 ) and Allen and Grundberg (7) for aspect ratios bracketing these questionable points agree well with the calculated values. 560
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3
0
~ I;
I
For this special sym-
I
I I I I
I
I
I
"y
22 21 20
19
Figure 4. Flow data of Allen and Grundberg ( I ) for a rectangular duct with R = 3.92 Illustrating technique used to determine Re, from published fRe data. For this case, Re, = 2300
The value of Re, derived from Schiller's (73) data for the square duct agrees well with that obtained from the data of Lea and Tadros ( 8 )for a square duct, and both agree well with the theoretical value. The data of Lea and Tadros ( 8 ) for the rectangular duct of aspect ratio R = 2.04 were too scanty to be definitive. Unfortunately, these data were reported only in the form of a small crowded graph with four different sets of data superimposed thereon. The data in question did not permit a very precise determination of Re,. The highest laminar point had a Reynolds number slightly less than 1900, while the next point (which \yas clearly nonlaminar) had a Reynolds number of 2230. Thus, somewhere between these
values the transition occurs. The best graphical estimate one can make from this set of data appears to be Re, ‘v 1960, xrhich is subject to considerable error because of the paucity of data. The theoretical value of 1900 is included within the uncertainty of the data. Rothfus et al. ( 7 7 ) proposed empirically that the product Rec(zl,,,/D) is a constant equal to 4200. The ratio t ~ ~ is easily shown to be a function of R independent of Re. Consequently, the curve of lie, us. R can simply be converted into a curve of Re,(u,,,,,lo) us. R. Table I contains the theoretical values of Kec(umaT,’D),a s suggested by those authors, for the points tabulated in their paper. This parameter is clearly not a constant as suggested by Rothfus t t a / . ( 7 7 ) . Obviously, if the experimental va1ue.s of Re,, which agree well with the theoretical curve of Re, us. R, were to be multiplied by the / U , same nonconstant behavior would theoretical ratio Z : ~ , ~ ~ the be observed. The existence of the minimum in the product Rec(LNm:,x!D) as a function of R is indicated even in the tabulation of Kothfus et al. ( T 7 ) , although not to quite the degree shown by the theoretical results tabulated in Table I. However, as a n approximation? their parameter does succeed in removing much of the. variation of Re, with R. Conclusions
-4theoretical curve of the lower critical Reynolds number (defined in terms of hydraulic diameter) hai been calculated as a function of the aspect ratio, a / b , for the steady isothermal rectilinear flow of Newtonian fluids in straight ducts of constant rectangular cross section,. Data reported in the literature by seven independent investigators obtained \\ ith ten different systems were compared \$ith the theoretical calculations of this study and found to be in good agreement The present study is thought to represent the first systematic theoretical calcula-
Table 1.
Theoretical Values of Re,(v,,,/V) Aspect Ratios Aspect Ratio, R hle, 1I n d o 2.10 2060 1. o 1.97 1900 2.04 1.97 1060 2.36 1.86 2085 2.92 1.79 231 5 3.92 1.50 2800 >10
for Selected ReAvmax16) 4326 3743 3861 3878 4144 4200
~
tion of the lower critical Keynolds number for this geometry, and is believed to be more reliable than available empirical correlations. Further experimentation, particularly \vith reference to the spatial location of the laminar-turbulent transition within the flow field, is desirable. The theoretical curve presented in Figure 2 is recommended ~ for / use 6 in determining the lotver critical Reynolds number for the laminar-turbulent transition in rectangular ducts. As a practical rule of thumb, when the duct aspect ratio exceeds about IO, the critical value of R e is approximately 2800 if defined in terms of the hydraulic diameter. Critical values of any Keynolds number defined in terms of an arbitrary characteristic length, L. may be obtained from Figure 2 by multiplying Re, from that graph by the ratio L/’D,,. Acknowledgment
The authors thank the Brigham Young Universiry Computer Center for making its facilities available for the calculations. literature Cited (1) Allen. J.: Grundbcrg. N. D.. Phil. M q . 23, 490 ( 1 937). ( 2 ) Bird, R. B., Stewart. I V . E.. Lightfoot, E. N..“Transport Phenomena.” \Viley. New York. 1962. (3) Cornish. R. J., Proc. Roy. Soc. (London) 120A, 691 11928). (4) Davies, S. J.. \Vhite, C. hi..Ibid.. 119A, 92 (1928). (5) E>ckert, E. R. G.. Irvine. T. F.. Jr.. Proceedings of Fifth Midwest Conference on Fluid Mechanics, p. 122. University of Michigan Press, ;inn Arbor, Mich.. 1957. (6) Hanks. R. \V., A . I . Ch. E. ,J. 9 (1). 45 (1963); “Generalized Criterion for the Laminar-Turbulent Transition in the Flow of Fluids,” Oak Ridge Gaseous Diffusion Plant. Union Carbide Nuclear Co.. Oak Ridge. Tenn.. U.S. Atomic Enerpv Cornsn. Rent. K-1531 ih’ov. 19. 1962). 17) Hanks. R. \
