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Anomalous wall effects and associated drag reduction in turbulent flow
Chemical Engineering Science, 1968, Vol. 23, pp. 23 l-242.
Pergamon Press.
Printed in Great Britain.
Anomalous wall effects a&associated drag reduction in turbulent flow W. KOZICKI and C. TIU Department of Chemical Engineering, University of Ottawa, Ottawa 2, Canada (First received 30 May 1967; in revisedform 5 August 1967) Abstraet-An analysis has been conducted relating to the anomalous behaviour and associated drag reduction encountered primarily in the turbulent flow of viscoelastic non-Newtonian fluids. It is shown that these fluids conform to the same general correlation scheme formulated for the purely viscous fluids when allowance is made for an effective velocity of slip at the wall attributable primarily to an increased laminar sub-layer thickness. Effective velocities of slip in turbulent flow and laminar sub-layer thicknesses computed from the experimental data of numerous sources are presented in support of the analysis. _ INTRODUCTION
behaviour of certain polymer solutions in turbulent flow has been observed and reported repeatedly in the literature. The most striking phenomenon is the reduction in pressure drop required to maintain the average velocity of a liquid under turbulent flow conditions achieved by the addition of a small quantity of foreign material, such as high molecular weight additive. This effect was first noticed by Toms [ l] in 1948 while conducting measurements on solutions of polymethylmethacrylate in monochlorobenzene. Subsequently, numerous papers have been published reporting on the drag reduction obtained through the addition of additives. Dodge and Metzner[2] observed the lowering of friction coefficients in the turbulent flow of aqueous CMC solutions when compared with other non-Newtonian viscous fluids. Shaver and Merrill[3] reported the same phenomenon for solutions of long polymer molecules. Wells[4], Elata and Tirosh [5], and Hoyt and Fabula[6] also found separately that Newtonian fluids with minute concentrations of various additives showed remarkable drag reduction. Meter[7] measured friction factors for aqueous solutions of natrosol hydroxyethylcellulose, which possess viscoelastic properties. The friction factors were found to be significantly lower than those of purely THE ANOMALOUS
viscous fluids, and a Meter model[7] was used to correlate the data. Many tentative interpretations of the turbulent drag reduction phenomenon have been proposed, and this area of investigation still remains controversial. Astarita[8] ruled out the possibilities of the particulate and the laminar slip effects in accounting for all of the observed reduction, and proposed a scheme relying on the viscoelasticity of the fluid to explain the anomalous behaviour. Metzner and Park [9] also attributed the increased stability of the laminar flow field and low pressure drops in turbulent flow to the continuum (viscoelastic) properties of the fluid, although they concluded that the particulate effects could not be ruled out as contributory mechanisms. Meyer [lo] has proposed a formula containing two elastic fluid parameters to correlate the experimental measurements of Ernst [ 1l] and other investigators [4,5]. In this study, the analysis of drag reduction in turbulent flow of viscoelastic polymer solutions utilizes only the purely viscous properties of the fluid, consideration also being given to the anomalous behaviour in the vicinity of the conduit wall. This anomalous behaviour is attributed to two phenomena: (1) the anomalous behaviour observed under laminar flow conditions (i.e. polymer adsorption on the conduit wall, or the “separation phenomenon”), and
231
W. KOZICKI and C. TIU
(2) a laminar sub-layer thickness greater than that applicable to the purely viscous nonNewtonian fluids. An effective velocity of slip at the wall shown to be a direct result of these phenomena accounts for the observed drag reduction and the anomalies in the correlation of friction factor data. The Ryan and Johnson [ 121 stability parameter criterion has been applied to predict the critical radial position and transition point from laminar to turbulent flow, for non-Newtonian (powerlaw) fluids exhibiting an effective velocity of slip in laminar flow.
where ut represents the velocity distribution ascribable to the fluid in the absence of anomalous effects (i.e., when q(T, y) = 0 at all y Q &). At y = 0, ut = 0 and Eq. (3) becomes
sq(q y)dy=u, h
U=
,,
(4)
where u,, defined by Eq. (4), is the effective velocity of slip at the wall in turbulent flow. The composition of the function q(T, y) in the laminar sub-layer region y s Sz, shown in Fig. 1, is postulated as follows: 4 =
ANALYSIS
g(c
Y)
q=o
We consider the turbulent flow of a viscoelastic non-Newtonian fluid in a pipe. The velocity distribution is obtained by integration of the velocity gradient,? U=
“du
I
,6JQ
(1)
where y is the normal distance measured from the wall. In the present case, the velocity gradient in the vicinity of the wall is represented by du dy=
h(T,Y)
+q(T,Y)
where h(~, y) denotes the velocity gradient characteristically ascribable to a purely viscous non-Newtonian fluid and q(~, y) is a correction term necessitated by the anomalous behaviour in the wall region. We stipulate here only that q(~, y) is identically zero outside of the viscous sub-layer, i.e. at y 3 8% When the substitution given by Eq. (2) is made in Eq. (il), and an integration carried out to y greater than &, (i.e., beyond the boundary of the viscous sub-layer), one obtains ~-)z(T,Y)
U=
=
u,+~:~(GY)
d_v+~;dv4 dy dy
Y Fig. 1. Velocity profiles adjacent to a solid boundary turbulent flow.
in
where g(r, y) is the same function used in laminar flow as a correction to the velocity gradient necessitated by the anomalous behaviour [13,14,15] under laminar flow conditions. For the case in which the effective velocity of slip in laminar flow results from separation of the solute molecules, and the formation of a thin layer of pure solvent, this function is given by
tFor simplicity, we omit the over-bars commonly used to denote time-smoothed quantities which are quite obvious from the context of the discussion.
232
g = j+
7) A
;+).
(6)
Anomalous wall effects in turbulent flow
uw= Ul(S2)-4(82).
Here, f(r) represents a function of shear stress denoting the velocity gradient characteristically ascribable to the fluid in laminar flow. The boundary points used to define the regions specified in Eq. (5) denote the following, also illustrated in Fig. 1: a1 is the outer edge of the laminar anomalous layer located within the viscous sub-layer; E represents the boundary of the laminar sub-layer traditionally ascribed to those fluids referred to as purely viscous fluids [2, 9, 161; and finally, a2 is the actual sub-layer thickness in the present case involving the flow of viscoelastic fluids, which is postulated to be greater than the corresponding thickness E for the purely viscous fluids. In the subsequent calculations, it is not necessary to know the precise location of B In view of Eq. (5), the integral in Eq. (4) can be separated into u, = I;&,
Y) dy + ,;
[f(7)
-MT,
Y)I dy.
The second integral in Eq. (8) applied to an Ostwald de-Waele (power-law) fluid becomes 8s f 0
I
(7)
E.
utot+j.,“fbl dy--so& h(r, Y) dy.
1
-l&
++J($)2-.
Hence, Eq. (7) may be written as 40 =
(7) dy = _E(5~[l_(l_~~+“~]
= &@l”[
The first integral on the right is recognized as the effective velocity of slip in laminar flow. Turning to the second integral, the lower limit may be replaced by zero since f(r) = h(7, y) at y <
(10)
. .]
(11)
where an expansion using the binomial theorem has been effected. If terms beyond the first in the series are neglected, the result corresponds to assuming f(r) equal to a constant, f(r& in the integral on the left side of Eq. (1 l), an approximation usually justified on the grounds that S, is sufiiciently small. Popovich and Hummel[ 171, utilizing a photographic technique, recently observed a noticeable curvature in photographs depicting the shape of the velocity profile in the wall region in turbulent flow of Newtonian fluids. It therefore appears advisable to ascertain that subsequent terms in the series of Eq. (11) are inconsequential in the evaluation of the integral. We now introduce the dimensionless variables
U-9
u+=_!!_ u*
(12)
y+ = -YU*P 7)W
(13)
in Eq. (8), utilizing the result of Eq. (1 l), to obtain + _ uf = [&+ _ ( 2)03-5nN2fl@j2+2 uw
(9) In other words, the first two terms on the right side of Eq. (8) represent the velocity at y = S, if the laminar sub-layer were to extend this distance from the wall. The third term in Eq. (8) is the velocity at y = a2 traditionally ascribed to the purely viscous non-Newtonian fluid in turbulent flow. Substitution of Eqs. (9) into Eq. (8) yields the following simple relationship for the effective velocity of slip in turbulent flow
-
+
ut+(&+)
. . . ]
(14)
where +=
(1+3n)[Re(f)1-n’2]1’n’
(15)
Expressions for the dimensionless velocity utf for purely viscous non-Newtonian fluids in turbulent flow have been presented. Krantz and Wasan[ 181 gave the following form for the velocity profile of power-law fluids in the 233
W. KOZICKI
and C. TIU
region near the wall which includes the laminar sub-layer and buffer zone,
wall, k. Thus, the shape of the velocity profile in the turbulent core will be the same as predicted by Eq. (20) above except that it will be shifted linearly upward in a plot of u+ versus y+ 4 += 1-L+ -!&n(y+)2]y+ 2y [ (for a particular n, Re and corresponding f) by + ~d+(y+)~+u~+(y+)~ 0 G y+ c yC+ (16) the amount b. Ernst[ 1 l] also observed experimentally that the shape of the velocity profile where for the turbulent core in the flow of viscoelastic 3.03 iiuids formed by the addition of small amounts (17) of polymer additives to water (such that the u4+=5?GY flow behaviour index n was substantially unity) . were the same as for the pure solvent except u5+ = .qg (18) that they were shifted upwards. A plot of his data is shown in Fig. 5. and If the shift in the velocity profile from that 0.6y,+ = 2.42 lny,++Z(n,Re) - 1.09 (19) predicted by Eq. (20) is designated by AR, then the effective velocity of slip is given by where the function I(n, Re) has been defined by 40 -= AB. Bogue and Metzner [ 163. (21) u* The expression developed by Krantz and Friction factor-Reynolds number correlations Wasan was constructed to give a continuous and smooth transition to the logarithmic velocity of experimental data are also amenable to distribution expression developed by Bogue and evaluation of the effective velocity of slip in Metzner[ 161 for the turbulent core at y+ = ye+ turbulent flow. The Dodge and Metzner[2] given by Eq. (19). The following is the universal expression relating the friction factor and velocity profile suggested by Bogue and Metzner Reynolds number of purely viscous pseudofor the turbulent core, plastic fluids is ut+= 2*42Iny++I(n,Re)
+c(y/&f) Y+ > yc+
[Re(f)1-n”2] (2-v
where the functions Z(n, Re) and c(y/R,P,f) have been tabulated by these authors. Thus, if effective velocity of slip data for both laminar and turbulent flow is available, Eq. (14) in conjunction with Eqs. (16,20) enable one to determine, by a trial and error procedure, the viscous sub-layer thickness SZ+. The method used for calculation of the anomalous layer thickness has been indicated elsewhere [ 151. Evaluation of the effective turbulentflow
velocity
(22)
For viscoelastic fluids exhibiting slip effects, we utilize the same expression with the consequence that the relevant friction factor and Reynolds number expressions are now given by Eqs. (19,20),
2Til
2u*2 ((u)-z&)2
f=,((u)-um)2=
(23)
and Re =
of slip in
Equations (3) and (4) indicate that deviations from the universal velocity profile proposed by Bogue and Metzner [ 161 may be expected by virtue of an effective velocity of slip at the
-f$.
&“‘((u) - 40)2-nb 8’2’~‘K’
(24)
where n’ = d log TJd log
234
Q(u)--u,t) D
(25)
Anomalous wall effects in turbulent flow
and K’=T,/
[
mediate radial position postulated not to exceed 808, the critical value 2,. Subsequently, Hanks [21] generalized this criterion to the following form in rectilinear flow
1. (26)
8((u)--u,,) D
”
The latter physical parameters of the fluid are determined from laminar flow measurements. Equation (22) may be expanded, utilizing Eq. (23), and rewritten in the following form which yields the effective velocity of slip in turbulent flow,
1 [Vu. ul which can be applied to channels of arbitrary cross-section, with ff, = 404. For power-law fluids exhibiting anomalous behaviour at the wall in laminar flow, it is found that Z,,,,, occurs at A, given by
&=(-&r”“+“[$.$+ l]N(“+l) (32)
LJ
-slog
-
{Rf?(f’)‘-““2}
+O$
1
(27)
where
(31)
K = ZP]l-_
and also that
3n+l
and
( >
X --y
Re, = Dn’(U)2--nP @l’-1KI
*
(29)
It should be noted that the mantissa of the logarithmic term in Eq. (27), Re’(f’)1-““2, is equal to Re(f) 1+‘12, since the velocity terms cancel each other. Hence, as indicated by Eq. (27), the effective velocity of slip in turbulent flow can be readily evaluated from the deviation between the measured friction factory and the corresponding value yielded by the unmodified Dodge-Metzner correlation. Prediction of the laminar-turbulent transition point The point of transition from laminar to turbulent flow of a fluid exhibiting anomalous behaviour in laminar flow can be expected to be different from one which exhibits no anomalous effects. Ryan and Johnson[ 121 introduced a stability parameter Z for flow in pipes, defined as Z_-~u _-
rw (
-- h dr >
(30)
which is zero at both the centreline and the wall, and reaches a maximum at some inter-
&l’n =
808.
(33)
Hence, the following expression for the critical value of the quantity f’ = 27,/p(~)~ is yielded for these fluids by the stability parameter criterion, (34) where (3n + 1)2 dn)
=
n
1 (
n+2
(fl+2m+1) >
(35)
and B(n,$$
= [(&$S+
l](fl+2)l(n+1!(36)
The function q(n) has been tabulated by Longwe11[22]. The corresponding expressions for the critical values of the friction factor, f, and Reynolds number, Re, denoting the point of transition from laminar flow, are readily deduced from the above results, as shown by the following relationships: 235
W. KOZICKI
Re = c
o”‘[(u> ---QI~-~‘P=- 16 @I’-1K’ fc’
and C. TIU
(38)
The critical position ratio A, marking the point in the flow field where the propagation of disturbances is initiated, when 2 is at its maximum value, is shown plotted in Fig. 2, with U,,/(U) as a parameter. Figures 3 and 4 are similar plots of the critical friction factor&. It must be emphasised that the above relationships determined by consideration of the viscous characteristics of fluids should be regarded only as tentative in the case of fluids exhibiting appreciable viscoelasticity. Hopefully, any serious limitations to be uncovered will serve to stimulate additional research and investigation in the area. DISCUSSION
lo’
9
I6
n
OF RESULTS
The present analysis has been applied to a wide cross-section of experimental data available in a variety of different forms and for fluids of widely differing flow characteristics. Any inconsistencies or anomalies attributable to the assumed mechanism were thus more amenable
Fig. 3. Critical friction factor for power-law fluid with a positive laminar effective velocity of slip.
to evaluation. In the absence of inconsistencies, the availability of data over a wide range of conditions provides a basis for comparison and
n Fii. 2. Plot of critical radial position X, for a power-law fluid with an effective velocity of slip.
236
Anomalous wall effects in turbulent flow
ia2 , , , ,
(
,
,
I
,
Fig. 4. Critical friction factor for power-law fluid with a negative laminar effective velocity of slip.
for observation of patterns or trends which may exist. Ernst [ 1l] conducted velocity measurements (using a probe 0405 in. thick) close to the wall in the turbulent flow of a 0.05 per cent aqueous CMC solution, whose physical properties are not substantially different from water, in two different diameter pipes and at two different Reynolds numbers for each pipe. A plot of his experimental data is shown in Fig. 5 together with the computed laminar sub-layer thickness determined for each set of data, situated on the laminar sub-layer curve in the figure. This figure serves to given an indication of the reasonableness of the assumed mechanism as well as the reasonableness of the computed values for the laminar sub-layer thickness when compared with the experimental points. We see that the interpretation of the lowering of frictional losses due to an increased laminar sub-layer thickness attributable to the presence of long-chain polymer molecules in the high-shear field is supported by the point velocity measurements of Ernst. Figure 6 shows a plot of the laminar sub-
IO -
0
D= 1427”
,
Ra = 960 x10’
-
al
D=OeY
,
Rr.
-
8
D=l42?
,
Re=4.59.105
-
.
D=O&
,
Re=2.11
-
689.10’
1.10’
Y+ Fig. 5. Plot of Ernst’s experimental velocity profile data showing extended laminar sub-layer.
237
W. KOZICKI
layer thickness S, plotted as a function of the shear stress at the wall rW, computed from the average velocity-pressure drop data of Metzner and Park[9] for the turbulent flow of O-3 per cent J-100 solutions in three different diameter pipes. The smooth curves drawn through the
and C. TIU
The lowest curve in the figure which has been included for comparison purposes gives the approximate location of the laminar sub-layer applicable to the purely viscous fluids, taken as given by y+ = E+G 5.0. A similar curve showing the thickness of the laminar anomalous layer is not included in this figure since it falls well below the latter curve. Figure 7 shows the effective velocity of slip in turbulent flow computed from the data of Metzner and Park. The effective velocity of slip attributable to anomalous effects under laminar flow conditions, computed from the laminar flow data, is also shown. It is seen that the separation phenomenon occurring near the
=I-----
SLIP
40-
H 2 I
i
I
oe
16
2.4 20-
t
w
,
lb/f?*
Fig. 6. Plot of laminar sub-layer thicknesses computed from data of Metzner and Park. LAUINAR
experimental points were extrapolated to intersect the critical radial position versus shear stress curves denoting the locii of points on the laminar-turbulent transition curves characteristic of the fluids exhibiting anomalous behaviour in laminar flow. These curves depicting the points of transition from laminar flow characterized by an effective velocity of slip which is a function of the shear stress at the wall were computed utilizing the relationships presented. The point of intersection of the laminar sub-layer curve for turbulent flow with the critical radial position curve based on laminar flow considerations is taken to indicate the point of transition from laminar to turbulent flow at the given conditions.
SLIP
1
0
tw *
lb/f?
Fig. 7. Effective velocity of slip including laminar slip evaluated from data of Metzner and Park.
wall can account for only a small fraction of the total effective velocity at the wall and that most of the effective velocity of slip in turbulent flow and the associated drag reduction is attributable to the increased thickness of the laminar sublayer. Figure 8 presents a S, versus rro plot of the experimental data measured by Shaver and Merrill[3] for some CMC solutions in which
238
Anomalous wall effects in turbulent flow
0
@?#I%
WC-TO
OEQRAOED
8
04lK
cm-To
DEQRAOED
0
o!!ou
CMC-708
Fig. 8. Laminar sub-layer thicknesses from data of Shaver and Merrill.
the location of the axis of the tube is also shown, to gain a proper perspective of the physical situation. The uppermost point on each curve represents the computed critical radial position and corresponding critical shear stress, determined for the system assuming no anomalous behaviour in laminar flow, implied by Shaver and Merrill[ 31. It is seen that the computed laminar-turbulent transition points are consistent with the extrapolated curves of the laminar sub-layer thickness. It is worthy to note here that in the case of some systems, in the transition from laminar to turbulent flow, the actual transition is not as abrupt as with simple Newtonian systems but more gradual resulting in velocity profiles and calculated friction factors not substantially different from those applicable to laminar flow over a considerable range of r,,,; hence, the reference to turbulence suppression. In view of the size of the sub-layer thickness at the wall, one might also
conceive the possibility of a cylindrical laminar flow regime of indeterminate thickness about the axis of the pipe, so that over some range of rW beginning at the critical value the actual region of turbulent flow is the annular region confined by the two laminar zones. The velocity profile measurements of Shaver and Merrill131 and of Eissenberg and Bogue[23] lend some support to this possibility. Meyer[ lo] has correlated the data of Elata and Tirosh[S] in terms of a critical friction velocity at which no noticeable displacement in the velocity profiles of dilute viscoelastic non-Newtonian fluids from those for water was observed. A re-examination of the same problem in the light of the present discussion suggests that a constant critical shear stress valid for all fluids is questionable, although it may hold to a good approximation. Figure 9 shows the laminar sub-layer thicknesses computed from these data as a function of T,, together with the
239
W. KOZICKI
e 0 #.I\
and C. TIU
concentration, expressed as parts per million by weight, for polyethylene oxide and other natural products in distilled water calculated from the data reported by Hoyt[24]. In most cases, the dimensionless thickness is observed to increase with the concentration of the additive; 0
MEYER’S
EXTRAPOLATION
I
,
,
CONSTANT
R*’
4
VELOCITY 14,000
oNA (SALMON 1
m 6UM
+s
TRAGACANTH
TAMARIND
Fig. 9. Laminar sub-layer thicknesses obtained from data used by Meyer.
GUM
CHATTI
0 TAPIOCA
of Sz with TV corresponding to Meyer’s extrapolation of the data. The dashed curves represent the authors’ extrapolation of the same data to the points denoting the computed critical radial positions and shear stresses marking the points of transition from laminar to turbulent flow. Figures 10 and 11 show the variation of the dimensionless laminar sub-layer thickness with
OEXTRINE
variation
TOBACCO
4-
CONCENTRATION,
10
-0-O
+z
-o-o-o-o-c-~
m!
CONSTANT
“ELCJCITY
Re * 14.000
6 ‘
--------
0
LmTlLLEO
20
VIRUS
ppn
Fig. 11. Dimensionless sub-layer thickness for various natural product solutions used by Hoyt as a function of concentration.
-l-----l o/o
MOSAIC
RNA
WATER
40
0
60
CONCENTRATION.
100
pprn
Fig. 10. Dimensionless sub-layer thickness for Hoyt’s polyethylene oxide solution.
240
Anomalous wall effects in turbulent flow
f’
function defined by Eq. (28) body force vector stability parameter defined by Eq. (3 1) fluid consistency index for Ostwald deWaele fluid, lb,-secn/ft2 K’ function defined by Eq. (26) 12 flow behaviour index for Ostwald deWaele fluid, dimensionless n’ function defined by Eq. (25) P pressure, lbt/ft2 R radius of the pipe, ft r radial distance measured from the centreline, ft Re Reynolds number defined by Eq. (24); subscript c denotes critical value function defined by Eq. (29) local point velocity, ft/sec velocity vector average velocity, ft/sec total slip velocity, ftlsec laminar slip velocity, ftlsec laminar velocity distribution defined by Eq. (9) 4 turbulent velocity distribution defined by Eq. (9) u+ dimensionless velocity defined by Eq. (12) u* V/(7,/p), friction velocity, ftlsec + function defined by Eq. (17) u4 + function defined by Eq. ( 18) u5 radial distance measured from the Y wall, ft dimensionless distance defined by Y+ Eq. (13) + dimensionless thickness defined by YC Eq. (19) z stability parameter defined by Eq. (30)
however, a maximum thickness may be attained at some concentration beyond which the thickness becomes substantially independent of the concentration of the additive. For example, the maximum points occur at 20 ppm and 100 ppm for polyethylene oxide and tapioca dextrine solutions, respectively. In contrast, mosaic virus and RNA do not show any noticeable drag reduction.
f I? K
CONCLUSIONS
The conclusions drawn from this study are: (1) There is considerable evidence that drag reduction observed with viscoelastic nonNewtonian fluids can be attributed mainly to an increased laminar sub-layer thickness at the wall. This increased thickness may be a manifestation of eddy suppression, due to the mechanical resistance associated with preferred orientations and alignment of the polymer molecules in high shear fields. Stabilizing influences associated with shear-stress induced normal stresses may also play a prominent role in suppression of eddies near the wall. (2) For conclusive proof, additional velocity profile measurements in the wall region, perhaps by the recently described photographic technique [ 171, are required. (3) Future investigations should also take into consideration the molecular characteristics (e.g., molecular weight and chain-length distributions) of the polymer additives for a deeper insight and greater understanding of the problem. This information might also enable one to set up a general correlation scheme for use in prediction of the effective velocity of slip in turbulent flow which is needed in addition to the friction factor data. Acknowledgments-The authors wish to gratefully acknowledge the financial assistance received from the National Research Council of Canada. The second author was also the recipient of a National Research Council Studentship.
Greek laminar anomalous layer thickness, ft laminar sub-layer thickness of the viscoelastic fluid, ft 62’ dimensionless laminar sub-layer thickness of the viscoelastic fluid E laminar sub-layer thickness of the purely viscous fluid, ft A r/R, dimensionless radius 2
NOTATION D
f
diameter of the pipe, ft friction factor defined by Eq. (23); subscript c denotes critical value 241
C.E.S.-D
W. KOZICKI X,
p q 7, b
and C. TIU
critical radius defined by Eq. (32) density, lb&k3 non-Newtonian viscosity, lbf- sec/ft2 non-Newtonian viscosity at the wall, lb,- sec/ft2 solvent viscosity, lbt- sec/ft2
7 TV 4 cp( n) O(n)
shear stress, lb,/fP shear stress at the wall, lb#t2 function defined by Eq. ( 15) function defined by Eq. (3 5) function defined by Eq. (36)
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1949. DODGE D. W. and METZNER A. B., A. 1. Ch. E. Jll959 5 189. t:; SHAVER R. G. and MERRILL E. W., A. I. Ch. E. Jll959 5 181. 141 WELLS C. S., Am. Inst. Aeronautics Astronautics J. 1965 3 1800. 151 ELATA C. and TIROSH J.,Proc. 7th Israel Am. Co& Aviation anddstronautics; IsraslJ. Tech., 1965 5 1. (61 HOYT J. W. and FABULA A. G., 5th Symp. on Naval Hydrodynamics, Bergen, Norway 1964. METER D. M.,A. 1. Ch. E. Jll964 10 881,887. ii; ASTARITA G., Ind. Engng Chem. Fundls 1965 4 354. 191 METZNER A. B. and PARK G., J. Fluid Mech. 1964 20 291; PARK G., M. Ch. E. Thesis, University of Delaware 1963. [lo] MEYER W. A., A. I. Ch. E. Jll%6 12 522. [11] ERNSTW.D.,A.I.Ch.E.J/196612581. [12] RYANN.W.andJOHNSONM.M.,A.I.Ch. E.JIl9595433. [13] OLDROYD J. G., J. ColloidSci. 19494 333. [ 141 WILKINSON, W. L., Non-Newtonian Fluids. Pergamon Press 1960. [ 151 KOZICKI W., HSU C. J. and TIU C., Chem. Engng Sci. 1967 22 487. [16] BOGUE D. C. and METZNER A. B., Ind. Engng Chem. Fundls 1963 2 143. [ 171 POPOVICH A. T. and HUMMEL R. L., Chem. Engng Sci. 1967 22 2 1. [18] KRANTZ W. B. and WASAN D. T., Paper presented at the Fifty-Ninth National Meeting, Amer. Inst. Chem. Engrs., 1 Columbus, Ohio, May 1966. 1191 OLDROYD J. G., Proc. Int. Cong. on Rheol. Vol. 2, p. 130, Scheveningen, Holland 1948. PO1 KOZICKI W., CHOU C. H. and TIU C., Chem. Engng Sci. 1966 21665. PII HANKS R. W.,A. 1. Ch. E. Jll963 9 1. LONGWELL P. A., Mechanics of Fluid Flow. McGraw-Hill 1966. ::;; EISSENBERG D. M. and BOGUE D. C., A. 1. Ch. E. Jll964 10 723. 1241 HOYT J. W., Paper presented at the Symposium on Rheology, ASME Applied Mech. and Fluids Engng Conf., Washington, June 1965. R&um&Une analyse a Cd faite sur le comportement anormal et la r&u&ion de traide qui Iui est associ6, rencontres premierement dam un courant turbulent de fluides vixqueuxdlastiques nonnewtoniens. 11est d6montr6 que ces fluides se conferment au m8me plan &n&al de correlation form& pour les fluides purement visquenx, quand on prevoit une v&cite effective de glissement le long des parois, attribuable tout d’abord B l’epaisseur accrue de la sous-couche lamellaire. Les v&tcit& effectives de glissement dam un courant turbulent et dans des 6paisseurs de sous-couches lamellaires compt6es a partir des donnees exp&imentales de sources nombreuses sont p&e&es pour soutenir I’analyse. Z-f-Es wurde eine Analyse des abnormalen Verhaltens turd der damit verbundenen Abnahme des Widerstandes durchgefilhrt, den man in erster Linie bei der Turbulenzstr~mung viskoelastischer, nicht newtonischer Flilssigkeiten begegnet. Es wird bewiesen, dass diese Flilssigkeiteo die gleichen allgemeinen Beziehungen aufweisen, die tiir die rein viskosen Flbssigkeiten formuliert wurden, wenn die effektive Slipgeschwindigkeit an der Wand berilcksichtigt wird, die in erster Linie der grosseren laminaren Substratendicke zuzuschreiben ist. Werte fiir effektive Slipgeschwindigkeit in Turbulenzstriimung und Laminarsubstratendicke, errechnet aufgrund experimenteller Messungen von zahlreichen Quellen, werden zur Erh&tung der Analyse angegeben.
242
Pergamon Press.
Printed in Great Britain.
Anomalous wall effects a&associated drag reduction in turbulent flow W. KOZICKI and C. TIU Department of Chemical Engineering, University of Ottawa, Ottawa 2, Canada (First received 30 May 1967; in revisedform 5 August 1967) Abstraet-An analysis has been conducted relating to the anomalous behaviour and associated drag reduction encountered primarily in the turbulent flow of viscoelastic non-Newtonian fluids. It is shown that these fluids conform to the same general correlation scheme formulated for the purely viscous fluids when allowance is made for an effective velocity of slip at the wall attributable primarily to an increased laminar sub-layer thickness. Effective velocities of slip in turbulent flow and laminar sub-layer thicknesses computed from the experimental data of numerous sources are presented in support of the analysis. _ INTRODUCTION
behaviour of certain polymer solutions in turbulent flow has been observed and reported repeatedly in the literature. The most striking phenomenon is the reduction in pressure drop required to maintain the average velocity of a liquid under turbulent flow conditions achieved by the addition of a small quantity of foreign material, such as high molecular weight additive. This effect was first noticed by Toms [ l] in 1948 while conducting measurements on solutions of polymethylmethacrylate in monochlorobenzene. Subsequently, numerous papers have been published reporting on the drag reduction obtained through the addition of additives. Dodge and Metzner[2] observed the lowering of friction coefficients in the turbulent flow of aqueous CMC solutions when compared with other non-Newtonian viscous fluids. Shaver and Merrill[3] reported the same phenomenon for solutions of long polymer molecules. Wells[4], Elata and Tirosh [5], and Hoyt and Fabula[6] also found separately that Newtonian fluids with minute concentrations of various additives showed remarkable drag reduction. Meter[7] measured friction factors for aqueous solutions of natrosol hydroxyethylcellulose, which possess viscoelastic properties. The friction factors were found to be significantly lower than those of purely THE ANOMALOUS
viscous fluids, and a Meter model[7] was used to correlate the data. Many tentative interpretations of the turbulent drag reduction phenomenon have been proposed, and this area of investigation still remains controversial. Astarita[8] ruled out the possibilities of the particulate and the laminar slip effects in accounting for all of the observed reduction, and proposed a scheme relying on the viscoelasticity of the fluid to explain the anomalous behaviour. Metzner and Park [9] also attributed the increased stability of the laminar flow field and low pressure drops in turbulent flow to the continuum (viscoelastic) properties of the fluid, although they concluded that the particulate effects could not be ruled out as contributory mechanisms. Meyer [lo] has proposed a formula containing two elastic fluid parameters to correlate the experimental measurements of Ernst [ 1l] and other investigators [4,5]. In this study, the analysis of drag reduction in turbulent flow of viscoelastic polymer solutions utilizes only the purely viscous properties of the fluid, consideration also being given to the anomalous behaviour in the vicinity of the conduit wall. This anomalous behaviour is attributed to two phenomena: (1) the anomalous behaviour observed under laminar flow conditions (i.e. polymer adsorption on the conduit wall, or the “separation phenomenon”), and
231
W. KOZICKI and C. TIU
(2) a laminar sub-layer thickness greater than that applicable to the purely viscous nonNewtonian fluids. An effective velocity of slip at the wall shown to be a direct result of these phenomena accounts for the observed drag reduction and the anomalies in the correlation of friction factor data. The Ryan and Johnson [ 121 stability parameter criterion has been applied to predict the critical radial position and transition point from laminar to turbulent flow, for non-Newtonian (powerlaw) fluids exhibiting an effective velocity of slip in laminar flow.
where ut represents the velocity distribution ascribable to the fluid in the absence of anomalous effects (i.e., when q(T, y) = 0 at all y Q &). At y = 0, ut = 0 and Eq. (3) becomes
sq(q y)dy=u, h
U=
,,
(4)
where u,, defined by Eq. (4), is the effective velocity of slip at the wall in turbulent flow. The composition of the function q(T, y) in the laminar sub-layer region y s Sz, shown in Fig. 1, is postulated as follows: 4 =
ANALYSIS
g(c
Y)
q=o
We consider the turbulent flow of a viscoelastic non-Newtonian fluid in a pipe. The velocity distribution is obtained by integration of the velocity gradient,? U=
“du
I
,6JQ
(1)
where y is the normal distance measured from the wall. In the present case, the velocity gradient in the vicinity of the wall is represented by du dy=
h(T,Y)
+q(T,Y)
where h(~, y) denotes the velocity gradient characteristically ascribable to a purely viscous non-Newtonian fluid and q(~, y) is a correction term necessitated by the anomalous behaviour in the wall region. We stipulate here only that q(~, y) is identically zero outside of the viscous sub-layer, i.e. at y 3 8% When the substitution given by Eq. (2) is made in Eq. (il), and an integration carried out to y greater than &, (i.e., beyond the boundary of the viscous sub-layer), one obtains ~-)z(T,Y)
U=
=
u,+~:~(GY)
d_v+~;dv4 dy dy
Y Fig. 1. Velocity profiles adjacent to a solid boundary turbulent flow.
in
where g(r, y) is the same function used in laminar flow as a correction to the velocity gradient necessitated by the anomalous behaviour [13,14,15] under laminar flow conditions. For the case in which the effective velocity of slip in laminar flow results from separation of the solute molecules, and the formation of a thin layer of pure solvent, this function is given by
tFor simplicity, we omit the over-bars commonly used to denote time-smoothed quantities which are quite obvious from the context of the discussion.
232
g = j+
7) A
;+).
(6)
Anomalous wall effects in turbulent flow
uw= Ul(S2)-4(82).
Here, f(r) represents a function of shear stress denoting the velocity gradient characteristically ascribable to the fluid in laminar flow. The boundary points used to define the regions specified in Eq. (5) denote the following, also illustrated in Fig. 1: a1 is the outer edge of the laminar anomalous layer located within the viscous sub-layer; E represents the boundary of the laminar sub-layer traditionally ascribed to those fluids referred to as purely viscous fluids [2, 9, 161; and finally, a2 is the actual sub-layer thickness in the present case involving the flow of viscoelastic fluids, which is postulated to be greater than the corresponding thickness E for the purely viscous fluids. In the subsequent calculations, it is not necessary to know the precise location of B In view of Eq. (5), the integral in Eq. (4) can be separated into u, = I;&,
Y) dy + ,;
[f(7)
-MT,
Y)I dy.
The second integral in Eq. (8) applied to an Ostwald de-Waele (power-law) fluid becomes 8s f 0
I
(7)
E.
utot+j.,“fbl dy--so& h(r, Y) dy.
1
-l&
++J($)2-.
Hence, Eq. (7) may be written as 40 =
(7) dy = _E(5~[l_(l_~~+“~]
= &@l”[
The first integral on the right is recognized as the effective velocity of slip in laminar flow. Turning to the second integral, the lower limit may be replaced by zero since f(r) = h(7, y) at y <
(10)
. .]
(11)
where an expansion using the binomial theorem has been effected. If terms beyond the first in the series are neglected, the result corresponds to assuming f(r) equal to a constant, f(r& in the integral on the left side of Eq. (1 l), an approximation usually justified on the grounds that S, is sufiiciently small. Popovich and Hummel[ 171, utilizing a photographic technique, recently observed a noticeable curvature in photographs depicting the shape of the velocity profile in the wall region in turbulent flow of Newtonian fluids. It therefore appears advisable to ascertain that subsequent terms in the series of Eq. (11) are inconsequential in the evaluation of the integral. We now introduce the dimensionless variables
U-9
u+=_!!_ u*
(12)
y+ = -YU*P 7)W
(13)
in Eq. (8), utilizing the result of Eq. (1 l), to obtain + _ uf = [&+ _ ( 2)03-5nN2fl@j2+2 uw
(9) In other words, the first two terms on the right side of Eq. (8) represent the velocity at y = S, if the laminar sub-layer were to extend this distance from the wall. The third term in Eq. (8) is the velocity at y = a2 traditionally ascribed to the purely viscous non-Newtonian fluid in turbulent flow. Substitution of Eqs. (9) into Eq. (8) yields the following simple relationship for the effective velocity of slip in turbulent flow
-
+
ut+(&+)
. . . ]
(14)
where +=
(1+3n)[Re(f)1-n’2]1’n’
(15)
Expressions for the dimensionless velocity utf for purely viscous non-Newtonian fluids in turbulent flow have been presented. Krantz and Wasan[ 181 gave the following form for the velocity profile of power-law fluids in the 233
W. KOZICKI
and C. TIU
region near the wall which includes the laminar sub-layer and buffer zone,
wall, k. Thus, the shape of the velocity profile in the turbulent core will be the same as predicted by Eq. (20) above except that it will be shifted linearly upward in a plot of u+ versus y+ 4 += 1-L+ -!&n(y+)2]y+ 2y [ (for a particular n, Re and corresponding f) by + ~d+(y+)~+u~+(y+)~ 0 G y+ c yC+ (16) the amount b. Ernst[ 1 l] also observed experimentally that the shape of the velocity profile where for the turbulent core in the flow of viscoelastic 3.03 iiuids formed by the addition of small amounts (17) of polymer additives to water (such that the u4+=5?GY flow behaviour index n was substantially unity) . were the same as for the pure solvent except u5+ = .qg (18) that they were shifted upwards. A plot of his data is shown in Fig. 5. and If the shift in the velocity profile from that 0.6y,+ = 2.42 lny,++Z(n,Re) - 1.09 (19) predicted by Eq. (20) is designated by AR, then the effective velocity of slip is given by where the function I(n, Re) has been defined by 40 -= AB. Bogue and Metzner [ 163. (21) u* The expression developed by Krantz and Friction factor-Reynolds number correlations Wasan was constructed to give a continuous and smooth transition to the logarithmic velocity of experimental data are also amenable to distribution expression developed by Bogue and evaluation of the effective velocity of slip in Metzner[ 161 for the turbulent core at y+ = ye+ turbulent flow. The Dodge and Metzner[2] given by Eq. (19). The following is the universal expression relating the friction factor and velocity profile suggested by Bogue and Metzner Reynolds number of purely viscous pseudofor the turbulent core, plastic fluids is ut+= 2*42Iny++I(n,Re)
+c(y/&f) Y+ > yc+
[Re(f)1-n”2] (2-v
where the functions Z(n, Re) and c(y/R,P,f) have been tabulated by these authors. Thus, if effective velocity of slip data for both laminar and turbulent flow is available, Eq. (14) in conjunction with Eqs. (16,20) enable one to determine, by a trial and error procedure, the viscous sub-layer thickness SZ+. The method used for calculation of the anomalous layer thickness has been indicated elsewhere [ 151. Evaluation of the effective turbulentflow
velocity
(22)
For viscoelastic fluids exhibiting slip effects, we utilize the same expression with the consequence that the relevant friction factor and Reynolds number expressions are now given by Eqs. (19,20),
2Til
2u*2 ((u)-z&)2
f=,((u)-um)2=
(23)
and Re =
of slip in
Equations (3) and (4) indicate that deviations from the universal velocity profile proposed by Bogue and Metzner [ 161 may be expected by virtue of an effective velocity of slip at the
-f$.
&“‘((u) - 40)2-nb 8’2’~‘K’
(24)
where n’ = d log TJd log
234
Q(u)--u,t) D
(25)
Anomalous wall effects in turbulent flow
and K’=T,/
[
mediate radial position postulated not to exceed 808, the critical value 2,. Subsequently, Hanks [21] generalized this criterion to the following form in rectilinear flow
1. (26)
8((u)--u,,) D
”
The latter physical parameters of the fluid are determined from laminar flow measurements. Equation (22) may be expanded, utilizing Eq. (23), and rewritten in the following form which yields the effective velocity of slip in turbulent flow,
1 [Vu. ul which can be applied to channels of arbitrary cross-section, with ff, = 404. For power-law fluids exhibiting anomalous behaviour at the wall in laminar flow, it is found that Z,,,,, occurs at A, given by
&=(-&r”“+“[$.$+ l]N(“+l) (32)
LJ
-slog
-
{Rf?(f’)‘-““2}
+O$
1
(27)
where
(31)
K = ZP]l-_
and also that
3n+l
and
( >
X --y
Re, = Dn’(U)2--nP @l’-1KI
*
(29)
It should be noted that the mantissa of the logarithmic term in Eq. (27), Re’(f’)1-““2, is equal to Re(f) 1+‘12, since the velocity terms cancel each other. Hence, as indicated by Eq. (27), the effective velocity of slip in turbulent flow can be readily evaluated from the deviation between the measured friction factory and the corresponding value yielded by the unmodified Dodge-Metzner correlation. Prediction of the laminar-turbulent transition point The point of transition from laminar to turbulent flow of a fluid exhibiting anomalous behaviour in laminar flow can be expected to be different from one which exhibits no anomalous effects. Ryan and Johnson[ 121 introduced a stability parameter Z for flow in pipes, defined as Z_-~u _-
rw (
-- h dr >
(30)
which is zero at both the centreline and the wall, and reaches a maximum at some inter-
&l’n =
808.
(33)
Hence, the following expression for the critical value of the quantity f’ = 27,/p(~)~ is yielded for these fluids by the stability parameter criterion, (34) where (3n + 1)2 dn)
=
n
1 (
n+2
(fl+2m+1) >
(35)
and B(n,$$
= [(&$S+
l](fl+2)l(n+1!(36)
The function q(n) has been tabulated by Longwe11[22]. The corresponding expressions for the critical values of the friction factor, f, and Reynolds number, Re, denoting the point of transition from laminar flow, are readily deduced from the above results, as shown by the following relationships: 235
W. KOZICKI
Re = c
o”‘[(u> ---QI~-~‘P=- 16 @I’-1K’ fc’
and C. TIU
(38)
The critical position ratio A, marking the point in the flow field where the propagation of disturbances is initiated, when 2 is at its maximum value, is shown plotted in Fig. 2, with U,,/(U) as a parameter. Figures 3 and 4 are similar plots of the critical friction factor&. It must be emphasised that the above relationships determined by consideration of the viscous characteristics of fluids should be regarded only as tentative in the case of fluids exhibiting appreciable viscoelasticity. Hopefully, any serious limitations to be uncovered will serve to stimulate additional research and investigation in the area. DISCUSSION
lo’
9
I6
n
OF RESULTS
The present analysis has been applied to a wide cross-section of experimental data available in a variety of different forms and for fluids of widely differing flow characteristics. Any inconsistencies or anomalies attributable to the assumed mechanism were thus more amenable
Fig. 3. Critical friction factor for power-law fluid with a positive laminar effective velocity of slip.
to evaluation. In the absence of inconsistencies, the availability of data over a wide range of conditions provides a basis for comparison and
n Fii. 2. Plot of critical radial position X, for a power-law fluid with an effective velocity of slip.
236
Anomalous wall effects in turbulent flow
ia2 , , , ,
(
,
,
I
,
Fig. 4. Critical friction factor for power-law fluid with a negative laminar effective velocity of slip.
for observation of patterns or trends which may exist. Ernst [ 1l] conducted velocity measurements (using a probe 0405 in. thick) close to the wall in the turbulent flow of a 0.05 per cent aqueous CMC solution, whose physical properties are not substantially different from water, in two different diameter pipes and at two different Reynolds numbers for each pipe. A plot of his experimental data is shown in Fig. 5 together with the computed laminar sub-layer thickness determined for each set of data, situated on the laminar sub-layer curve in the figure. This figure serves to given an indication of the reasonableness of the assumed mechanism as well as the reasonableness of the computed values for the laminar sub-layer thickness when compared with the experimental points. We see that the interpretation of the lowering of frictional losses due to an increased laminar sub-layer thickness attributable to the presence of long-chain polymer molecules in the high-shear field is supported by the point velocity measurements of Ernst. Figure 6 shows a plot of the laminar sub-
IO -
0
D= 1427”
,
Ra = 960 x10’
-
al
D=OeY
,
Rr.
-
8
D=l42?
,
Re=4.59.105
-
.
D=O&
,
Re=2.11
-
689.10’
1.10’
Y+ Fig. 5. Plot of Ernst’s experimental velocity profile data showing extended laminar sub-layer.
237
W. KOZICKI
layer thickness S, plotted as a function of the shear stress at the wall rW, computed from the average velocity-pressure drop data of Metzner and Park[9] for the turbulent flow of O-3 per cent J-100 solutions in three different diameter pipes. The smooth curves drawn through the
and C. TIU
The lowest curve in the figure which has been included for comparison purposes gives the approximate location of the laminar sub-layer applicable to the purely viscous fluids, taken as given by y+ = E+G 5.0. A similar curve showing the thickness of the laminar anomalous layer is not included in this figure since it falls well below the latter curve. Figure 7 shows the effective velocity of slip in turbulent flow computed from the data of Metzner and Park. The effective velocity of slip attributable to anomalous effects under laminar flow conditions, computed from the laminar flow data, is also shown. It is seen that the separation phenomenon occurring near the
=I-----
SLIP
40-
H 2 I
i
I
oe
16
2.4 20-
t
w
,
lb/f?*
Fig. 6. Plot of laminar sub-layer thicknesses computed from data of Metzner and Park. LAUINAR
experimental points were extrapolated to intersect the critical radial position versus shear stress curves denoting the locii of points on the laminar-turbulent transition curves characteristic of the fluids exhibiting anomalous behaviour in laminar flow. These curves depicting the points of transition from laminar flow characterized by an effective velocity of slip which is a function of the shear stress at the wall were computed utilizing the relationships presented. The point of intersection of the laminar sub-layer curve for turbulent flow with the critical radial position curve based on laminar flow considerations is taken to indicate the point of transition from laminar to turbulent flow at the given conditions.
SLIP
1
0
tw *
lb/f?
Fig. 7. Effective velocity of slip including laminar slip evaluated from data of Metzner and Park.
wall can account for only a small fraction of the total effective velocity at the wall and that most of the effective velocity of slip in turbulent flow and the associated drag reduction is attributable to the increased thickness of the laminar sublayer. Figure 8 presents a S, versus rro plot of the experimental data measured by Shaver and Merrill[3] for some CMC solutions in which
238
Anomalous wall effects in turbulent flow
0
@?#I%
WC-TO
OEQRAOED
8
04lK
cm-To
DEQRAOED
0
o!!ou
CMC-708
Fig. 8. Laminar sub-layer thicknesses from data of Shaver and Merrill.
the location of the axis of the tube is also shown, to gain a proper perspective of the physical situation. The uppermost point on each curve represents the computed critical radial position and corresponding critical shear stress, determined for the system assuming no anomalous behaviour in laminar flow, implied by Shaver and Merrill[ 31. It is seen that the computed laminar-turbulent transition points are consistent with the extrapolated curves of the laminar sub-layer thickness. It is worthy to note here that in the case of some systems, in the transition from laminar to turbulent flow, the actual transition is not as abrupt as with simple Newtonian systems but more gradual resulting in velocity profiles and calculated friction factors not substantially different from those applicable to laminar flow over a considerable range of r,,,; hence, the reference to turbulence suppression. In view of the size of the sub-layer thickness at the wall, one might also
conceive the possibility of a cylindrical laminar flow regime of indeterminate thickness about the axis of the pipe, so that over some range of rW beginning at the critical value the actual region of turbulent flow is the annular region confined by the two laminar zones. The velocity profile measurements of Shaver and Merrill131 and of Eissenberg and Bogue[23] lend some support to this possibility. Meyer[ lo] has correlated the data of Elata and Tirosh[S] in terms of a critical friction velocity at which no noticeable displacement in the velocity profiles of dilute viscoelastic non-Newtonian fluids from those for water was observed. A re-examination of the same problem in the light of the present discussion suggests that a constant critical shear stress valid for all fluids is questionable, although it may hold to a good approximation. Figure 9 shows the laminar sub-layer thicknesses computed from these data as a function of T,, together with the
239
W. KOZICKI
e 0 #.I\
and C. TIU
concentration, expressed as parts per million by weight, for polyethylene oxide and other natural products in distilled water calculated from the data reported by Hoyt[24]. In most cases, the dimensionless thickness is observed to increase with the concentration of the additive; 0
MEYER’S
EXTRAPOLATION
I
,
,
CONSTANT
R*’
4
VELOCITY 14,000
oNA (SALMON 1
m 6UM
+s
TRAGACANTH
TAMARIND
Fig. 9. Laminar sub-layer thicknesses obtained from data used by Meyer.
GUM
CHATTI
0 TAPIOCA
of Sz with TV corresponding to Meyer’s extrapolation of the data. The dashed curves represent the authors’ extrapolation of the same data to the points denoting the computed critical radial positions and shear stresses marking the points of transition from laminar to turbulent flow. Figures 10 and 11 show the variation of the dimensionless laminar sub-layer thickness with
OEXTRINE
variation
TOBACCO
4-
CONCENTRATION,
10
-0-O
+z
-o-o-o-o-c-~
m!
CONSTANT
“ELCJCITY
Re * 14.000
6 ‘
--------
0
LmTlLLEO
20
VIRUS
ppn
Fig. 11. Dimensionless sub-layer thickness for various natural product solutions used by Hoyt as a function of concentration.
-l-----l o/o
MOSAIC
RNA
WATER
40
0
60
CONCENTRATION.
100
pprn
Fig. 10. Dimensionless sub-layer thickness for Hoyt’s polyethylene oxide solution.
240
Anomalous wall effects in turbulent flow
f’
function defined by Eq. (28) body force vector stability parameter defined by Eq. (3 1) fluid consistency index for Ostwald deWaele fluid, lb,-secn/ft2 K’ function defined by Eq. (26) 12 flow behaviour index for Ostwald deWaele fluid, dimensionless n’ function defined by Eq. (25) P pressure, lbt/ft2 R radius of the pipe, ft r radial distance measured from the centreline, ft Re Reynolds number defined by Eq. (24); subscript c denotes critical value function defined by Eq. (29) local point velocity, ft/sec velocity vector average velocity, ft/sec total slip velocity, ftlsec laminar slip velocity, ftlsec laminar velocity distribution defined by Eq. (9) 4 turbulent velocity distribution defined by Eq. (9) u+ dimensionless velocity defined by Eq. (12) u* V/(7,/p), friction velocity, ftlsec + function defined by Eq. (17) u4 + function defined by Eq. ( 18) u5 radial distance measured from the Y wall, ft dimensionless distance defined by Y+ Eq. (13) + dimensionless thickness defined by YC Eq. (19) z stability parameter defined by Eq. (30)
however, a maximum thickness may be attained at some concentration beyond which the thickness becomes substantially independent of the concentration of the additive. For example, the maximum points occur at 20 ppm and 100 ppm for polyethylene oxide and tapioca dextrine solutions, respectively. In contrast, mosaic virus and RNA do not show any noticeable drag reduction.
f I? K
CONCLUSIONS
The conclusions drawn from this study are: (1) There is considerable evidence that drag reduction observed with viscoelastic nonNewtonian fluids can be attributed mainly to an increased laminar sub-layer thickness at the wall. This increased thickness may be a manifestation of eddy suppression, due to the mechanical resistance associated with preferred orientations and alignment of the polymer molecules in high shear fields. Stabilizing influences associated with shear-stress induced normal stresses may also play a prominent role in suppression of eddies near the wall. (2) For conclusive proof, additional velocity profile measurements in the wall region, perhaps by the recently described photographic technique [ 171, are required. (3) Future investigations should also take into consideration the molecular characteristics (e.g., molecular weight and chain-length distributions) of the polymer additives for a deeper insight and greater understanding of the problem. This information might also enable one to set up a general correlation scheme for use in prediction of the effective velocity of slip in turbulent flow which is needed in addition to the friction factor data. Acknowledgments-The authors wish to gratefully acknowledge the financial assistance received from the National Research Council of Canada. The second author was also the recipient of a National Research Council Studentship.
Greek laminar anomalous layer thickness, ft laminar sub-layer thickness of the viscoelastic fluid, ft 62’ dimensionless laminar sub-layer thickness of the viscoelastic fluid E laminar sub-layer thickness of the purely viscous fluid, ft A r/R, dimensionless radius 2
NOTATION D
f
diameter of the pipe, ft friction factor defined by Eq. (23); subscript c denotes critical value 241
C.E.S.-D
W. KOZICKI X,
p q 7, b
and C. TIU
critical radius defined by Eq. (32) density, lb&k3 non-Newtonian viscosity, lbf- sec/ft2 non-Newtonian viscosity at the wall, lb,- sec/ft2 solvent viscosity, lbt- sec/ft2
7 TV 4 cp( n) O(n)
shear stress, lb,/fP shear stress at the wall, lb#t2 function defined by Eq. ( 15) function defined by Eq. (3 5) function defined by Eq. (36)
REFERENCES
111TOMS B. A., Proc. Int. Congr. Rheol. Part 2 p. 135. North-Holland
1949. DODGE D. W. and METZNER A. B., A. 1. Ch. E. Jll959 5 189. t:; SHAVER R. G. and MERRILL E. W., A. I. Ch. E. Jll959 5 181. 141 WELLS C. S., Am. Inst. Aeronautics Astronautics J. 1965 3 1800. 151 ELATA C. and TIROSH J.,Proc. 7th Israel Am. Co& Aviation anddstronautics; IsraslJ. Tech., 1965 5 1. (61 HOYT J. W. and FABULA A. G., 5th Symp. on Naval Hydrodynamics, Bergen, Norway 1964. METER D. M.,A. 1. Ch. E. Jll964 10 881,887. ii; ASTARITA G., Ind. Engng Chem. Fundls 1965 4 354. 191 METZNER A. B. and PARK G., J. Fluid Mech. 1964 20 291; PARK G., M. Ch. E. Thesis, University of Delaware 1963. [lo] MEYER W. A., A. I. Ch. E. Jll%6 12 522. [11] ERNSTW.D.,A.I.Ch.E.J/196612581. [12] RYANN.W.andJOHNSONM.M.,A.I.Ch. E.JIl9595433. [13] OLDROYD J. G., J. ColloidSci. 19494 333. [ 141 WILKINSON, W. L., Non-Newtonian Fluids. Pergamon Press 1960. [ 151 KOZICKI W., HSU C. J. and TIU C., Chem. Engng Sci. 1967 22 487. [16] BOGUE D. C. and METZNER A. B., Ind. Engng Chem. Fundls 1963 2 143. [ 171 POPOVICH A. T. and HUMMEL R. L., Chem. Engng Sci. 1967 22 2 1. [18] KRANTZ W. B. and WASAN D. T., Paper presented at the Fifty-Ninth National Meeting, Amer. Inst. Chem. Engrs., 1 Columbus, Ohio, May 1966. 1191 OLDROYD J. G., Proc. Int. Cong. on Rheol. Vol. 2, p. 130, Scheveningen, Holland 1948. PO1 KOZICKI W., CHOU C. H. and TIU C., Chem. Engng Sci. 1966 21665. PII HANKS R. W.,A. 1. Ch. E. Jll963 9 1. LONGWELL P. A., Mechanics of Fluid Flow. McGraw-Hill 1966. ::;; EISSENBERG D. M. and BOGUE D. C., A. 1. Ch. E. Jll964 10 723. 1241 HOYT J. W., Paper presented at the Symposium on Rheology, ASME Applied Mech. and Fluids Engng Conf., Washington, June 1965. R&um&Une analyse a Cd faite sur le comportement anormal et la r&u&ion de traide qui Iui est associ6, rencontres premierement dam un courant turbulent de fluides vixqueuxdlastiques nonnewtoniens. 11est d6montr6 que ces fluides se conferment au m8me plan &n&al de correlation form& pour les fluides purement visquenx, quand on prevoit une v&cite effective de glissement le long des parois, attribuable tout d’abord B l’epaisseur accrue de la sous-couche lamellaire. Les v&tcit& effectives de glissement dam un courant turbulent et dans des 6paisseurs de sous-couches lamellaires compt6es a partir des donnees exp&imentales de sources nombreuses sont p&e&es pour soutenir I’analyse. Z-f-Es wurde eine Analyse des abnormalen Verhaltens turd der damit verbundenen Abnahme des Widerstandes durchgefilhrt, den man in erster Linie bei der Turbulenzstr~mung viskoelastischer, nicht newtonischer Flilssigkeiten begegnet. Es wird bewiesen, dass diese Flilssigkeiteo die gleichen allgemeinen Beziehungen aufweisen, die tiir die rein viskosen Flbssigkeiten formuliert wurden, wenn die effektive Slipgeschwindigkeit an der Wand berilcksichtigt wird, die in erster Linie der grosseren laminaren Substratendicke zuzuschreiben ist. Werte fiir effektive Slipgeschwindigkeit in Turbulenzstriimung und Laminarsubstratendicke, errechnet aufgrund experimenteller Messungen von zahlreichen Quellen, werden zur Erh&tung der Analyse angegeben.
242