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Velocity distribution and friction factors in flows with drag reduction


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15 SEP. 1972



and a log region where

p oix 5Y

#9-Lab. yr.


DATUM: I 9 0 . 1973



sche Hogeschoo,


o Doo,iumgEsri,E,





Michael Poreh and Yona Dimant

Technion - Israel Institute of Technology

Faculty of Civil Engineering

A simple descriptive model, based on van Driest's mixing length


ky[1 - exp(-y+/A+)] with a variable damping parameter A+, is

pro-posed to reFrasent the effect of linear macromolecules in dilute

solu-tions on the wall region in boundary layer flows. Measurements used

to support an elastic sublayer model for drag reduction are shown to be in better agreement with the proposed model. A relation between A+

and parameters of the polymer solution and the flow identified by Virk

(1971), is derived for the range where Virk's correlations are valid.

The maximum drag reduction appears to be associated with an asymptotic


The ability of minute quantities of high molecular weight polymers

to reduce the turbulent skin friction and thus to decrease the drag of

underwater bodies, 'has excited many investigations of the phenomenon of

drag reduction. Theoretical efforts to explain the mechanism of drag

re-duction, have not been very successful, probably because drag reduction

is affected by an interaction

between the molecules and the

timia-dependent, non-linear turbulent flow near the edge of the viscous subla-yer. On the other hand, experimental and semi-empirical studies have

suc-ceeded in


many features of simple drag reduction flows and

describing them in approximate phenomenological models.

The earlier descriptions of such flows employed a two-layer model

to describe the mean velocity profile (Meyer 1965, Elata et al 1965);

a viscous sublayer where






where An = 5.75, B = 5.5, u+ = u/V , y+ = yV /v and V is the shear

n _6

velocity. The term Au', which describes the upward shift of the log

profile in the conventional law of the wall representation, was

empi-rically related to the shear velocity and polymer characteristics by

the equations ' .* * * *

Au=alog0.7p7_1; V > V


. . cr * * Au = 0 ; V. < V' cr where


c is the shear velocity at the onset of drag reduction and a

is a concentration dependent parameter.

Virk and Merrill (1969) correlated measurements of the onset of drag reduction in "thin" solvents by the semi-empirical relation

(Ref14)cr =


(/R g) (4)

Where R is the polymer radius of gyration in dilute solutions, Q a

non-dimensional constant characteristic to the polymer species-splvent

combination, R is the radius of the pipe, Re the Reynolds number

based on the mean velocity and diameter, and f Fanning's friction


Integration of u+ over the area of the pipe yields an expression

for the friction coefficient f. At high Reynolds numbers and small to

moderate values of Au the contribution of the'sublayer to'the

integ-* *

ral of u+ is negligible, yielding for V> Vcr the equation

1/2 * *


) - b + a log(V /V




Where an = 4.0 and b = 0.4. Plotted on Prandtl coordinates, f-14

versus Ref2, Eq.(5) gives straight lines which intersect the

Newto-nian line (a = 0) at (Ref1/2) , where V = V

cr. This result has been


supported by numerous independent pressure-loss measurements at large

Reynolds numbers for small values of Au+. The data deviates from Eq.(5)

at large values of Au+, where Au seems to reach a maximum value

(Seyer and Metzner 1969, Whittist et al 1968) as well as near (Refia)


where a smooth transition from the Newtonian curve to the polymer

solu-tion curve is observed.

The effect of the transition region between the viscous sublayer and the log region, was first considered by Poreh and Paz (1968). The velocity in this zone was approximated by the following log law



+ + + +

U = yl

ln(Y /Y1 )4. Y1+

where y1,the


thickneseOf the viscous sublayer, was assumed to be proportional to the "thickness" yj+ in the lwo-layer model:

y +-.0.43 (7)

-and the value of y,+ was determined by the .intersection of Eqs.


(1) and (2). When Au = 0 and y,+= 11.6 Eq. (6) reduces to

u= 5.0 in 17-1.- 3.05 (8)

which had been used by von Karman to describe the buffer zone in Newtonian flows. The model has been used successfully to relate heat transfer characteristics to friction losses in dilute polymer solutions. The effect of the buffer zone on the friction coefficient was found, however, to be negligible.

Recently, Virk (1971) proposed a new 3-layer model to describe

the velocity distribution in drag reducing fluids. He termed the

transition between the viscous sublayer and the log region - elastic

sublayer and proposed to describe it by a universal logarithmic law

u= Am in y + Bm (9)

where Ant = 11.7 and B = -17.0. The "edge" of the viscous sublayer yv+ is given by the intersection of Eqs. (1) and (9). The "edge" of

the elastic sublayer, is given by the intersection of Eqs. (9) and

(2). The relation between Au+ and the thickness of the elastic layer

is given by

Au+= (Amm A )1n(Y /Y ).




+, +A


Thus, when Au+ becomes small the elastic sublayer deminishes. Except

+ +

for small values of R (R = RV /v) and the large values of Au +, the

contributions of both the viscous and elastic layers to the integral

of u+ are small (see table 3, Virk (1971) ). In these cases, the

det-ails of the sublayer are insignificant and Virk's model gives the same friction coefficient as the model of Meyer and Elata. Virk termed this

case - the polymeric regime,. (Note that the last term in Virk's

fric-tion factor relafric-tion for this regime, Eq. (12) in Virk (1971), is


tical to Au in Eq. (10).



In the other extreme case where the elastic sublayer becomes large and the contribution of the log region to the integral of u+ is negligible, the friction coefficient is described by a universal

law obtained by integration of (9),

l/f = 19.0 log(Ref ) - 32.4 (11)

Equation (11), termed the maximum drag reduction asymptote, descri-bes reasonably well the maximum values of drag reduction obtained in many investigations at small values of R.

A very similar, but slightly more complicated 3-layer model, has

been offered independently by Tomita (1970).

Virk's analysis of data in the polymeric regime has yielded an

additional contribution. He has correlated semi-empirically the depen-dence of the slope of the straight lines in Prandtl's coordinates, which are described by Eq.(5), to identifiable polymeric parameters.

Defining a fractional slope increment A in Pram:Wits coordinate

sys-tem, which is proportional to a in Eq.(2), A =


US =

a/,/g7- (12)

p s


Sp is the slope with polymers and S = An = 4.0 is the Newtonian

slope, Virk showed that

1h 3h.

A = a/iTT = K(J(C/M) N (13)



is Avogadro's number 6.02 x 1023, C concentration as a weight

fraction, M molecular weight, N number Of backbone chain links and K a characteristic constant of the species-solvent combination. The para-meter A appears as well in an expression which Virk derived

theoreti-cally for the turbulent strain energy of the macromolecules.

Virk's correlations describe a large volume of the data in the polymeric regime and in the maximum drag reduction regime. It should be noted, however, that the correlations proposed for the two regimes are not related. The equations proposed for the polymeric regime are

unaffected by the details of the elastic sublayer, whereas Eqs, (9)

and (11) proposed for the maximum drag reduction regime, are indepen-dent of the polymer propoerties; Thus, it appears to the authors that

the correlations do not prove the exittance of an elastic sublayer,


in character from the corresponding layer in a Newtonian fluid. We

shall show that this transitional zone in dilute polymer solutions

is similar to the conventional buffer sone in a Newtonian fluid by

deriving the entire velocity profile in the wall region for both

cases using Van Driest's mixing length model.

ky [1 - exp(-1,+/A+)] (14)

letting A+ be a function of the polymer-solvent properties and the

shear. The model which gives a continuous velocity distribution can



applied to other boundary layer flows and to problems of

heat transfer and diffusion.


In analogy to the damping of harmonic oscillations near a wall, van Driest (1958) proposed that the turbulent mixing length near a

wall be described by Eq. (14) where A+= 26 is a dimensionless

univer-sal constant for smooth boundaries and k =


= 0.4. There is some

doubt whether A and

Bn are truly Reynolds number independent. Coles

(1954) for instance, suggests that An slightly increases at low Reynolds numbers. Accordingly,the shear stress in a turbulent pipe flow, given

by T=p(v+(2Idu/dyl)du/dy, can be described by the equation:

T1-= (1


- exp(-y+/A+)fide/dy+



where T+= T/Tw and T = pV .Equation (15) may also be written as



-1 + 4k2y+2[1 - exp(- y+/A+)]2.Tt


In order to find the mean velocity profile, van Driest Used the

cons.-tant'shear approximation, namely

T = Tv,

or T+ = 1. Denoting the

velo-city obtained in this manner by


one can write that

duo 2


1 + + 4k y4.2 [1 - exp(-y+/20")]2

Integration of Eq. (17) gives for large values of y the log law

uo = k-1 in y++ B (Eq.2) where the value of B is


Very close to the wall, where exp(-y+/A+) =1, the solution of Eq.

(17) is uo+ = y+ (Eq.1). A comparison with measurements in Newtonian

fluids (van Driest 1958) shows that the velocity profile obtained from Eq.(17) is in good agreement with measurements in the sublayer,

buffer zone and the log region in zero pressure gradient boundary layers and pipe flows. A deviation of the data from the log law is

observed in the outer region of the flows.

We have already seen that the effect of drag reducing additives

is to change the value of B in the log law. It is therefore natural

to examine the possibility of describing the velocity distribution in

such flows by the integral of Eq.(17) with values of A+ larger than 26. We have also seen that the contribution of the velocities in the

vis-cous sublayer and the buffer zone to the calculation of the friction

or drag coefficient in the polymeric regime is small. Thus the proposed

model would be useful only if it can describe the velocity distribution near and in the maximum drag reduction regime. Now, the maximum drag

reduction regime corresponds to large values of Al- and small values of


and one sees from Eq.(17) its asymptotic solution for small values

+ +

of R /A is given by u+= y+. Since we do not expect the velocity at any

point in the pipe to exceed the velocity given by the parabolic

distri-bution in a laminar flaw,

+ +

u= y (1 - y+/2R+), (18)

one has to disqualify this solution. The reason for the failure of this

solution is of course the assumption T = T1,7whichis valid only close to

the wall. We shall show later that although the error introduced by this

assumption in Newtonian flows is small, it is large for small values of

e/A./.. In view of this difficulty, we shall modify van Driest's solution by taking into account the variation of the shear stress in the pipe as well as the different character of the flow near the center of the pipe.

The proposed model for drag reducing flows in pipes assumes that the velocity distribution is composed of two parts


-u = -ul



The first part, describing the law of the wall, is given by the solution


dui+ dy+



7 y

/R )

*A computer program for the calculations of u+ and f is available on

request from the authors.


1 + 1/1+ 4k2y+ (1 - eicp(-1,+/A+)] (1 - y+/R+)

It is easy to see that the limit of Eq.(20) for small values of 111-/A+




= 1 - y /R-I-



which describes the parabolic velocity distribution (18). This result

implies that


which is zero near the wall, has to vanish identically

for small values of R+/A+. In other words, the deviation from the law

of the wall has to decrease as the region where the damping is effect_ve

increases. This condition is satisfied by the following equation

pro-posed for ul+.


- cos(Tril-/e)][1 - expf-.2e/er (22)

U2 = 2K

where 11 = 0.67 is a universal constant for pipe flows. The value of II

has been determined so that the Newtonian friction factor at Re = 5-105 would satisfy Eq.(5) with a = 0, an = 4.0 and bn = 0.4. Note that for

large values of le/A+, which is always the case if A+ = 26, the

exponen-tial term in Eq.(22) vanishes and u2+ becomes identical to Coles' Wake Function.

Undoubtedly, many other schemes can be used to describe the

dev-iation of the velocity profile near the center of the pipe from ul+and

its dependence on A+. As we shall see later the relative contribution

of u2+ is very small and thus any consistent model which complies with


boundary conditions would be

satisfactory. The choice of Coles'


Function is justified mainly for convenience in future applications of

the model to boundary layer flows.


A clear distinction between the new model and the



approximation used by van Driest, is the dependence of the velocity

profile on R.f. Both u1+ and u2+ are functions of 12+ and it is riot

possible to describe u+ as a function of y+ and A+ alone. We have

plo-+ +

tted in Fig.1 numerically computed distributions of ul, u and u0+ for

+ *

A = 26 and A = 300 . We see.that the various velocity


curves for R+= 10.000 are practically the same at small, valuesof

The maximum difference between u+ and ul is about 5% for

A = 26 at the center of the pipe and only half of it for A= 300. This indicates of course that the contribution of the u2 is rela-tively small. Another interesting observation is that the

differen-ces at this value of R+, between u+ and uo+ can be hardly noticed. They are better distiguished in Fig.2 where velocity defects u+


u+max- u1 andand u+max - uo+ are plotted: Note that at center of the

+ +

pipe du /dy = 0 whereas duo+/dy+ # 0.

We have also shown in Fig.1 the distribution of u+ for R+= 1000

and R+= 100. We see that the differences between the velocity profiles

for R+= 10.000 and R+=1000 are small. Practically the same profile is

also obtained in the Newtonian case. for R+= 100; however, the velocity

distribution for


100 and A+= 300 does not coincide any more with

the other profiles which have larger values of le/A+. The velocity

distributions according to the various models for


100 are plotted

separately in Fig.3. We see from this figure that the difference

bet-+ +

ween u and uo for A+= 300, is large. Note that the velocity u+ near the wall merges with the parabolic equation u+= y+(1 - y+/2R+) whereas

uo is tangent to the u+= y+ curve and goes Above the parabolic profile.

We have also plotted in this figure Virk's ultimate profile (Eq. 9).

Virk's profile is quite close to 12+ but it also gives at one region

slightly larger velocities than in a laminar pipe flow.

Measured velocity distribution are compared with the calculated

profiles of u+ in Figs. 4 - 7. The values of A+ were chosen arbitrarily

(The data is taken from Virk (1971), Fig.3, using the same symbols to

denote the various entries.) The agreement with the data is very good. In particular the velocity profiles in the maximum drag reduction

re-gime, Figs. 6 and 7, describe the measurements much better than the



The dependence of the friction factor f-1/2 on RefV2 as a function

of A, has been obtained numerically and plotted in Fig.8. One sees that at large values of Ref 112 the variation of f-112 for constant

val-ues of A+ is described by a logarithmic law. The Newtonian case A+= 26


coincides with the line describing the equation f = 4.0 log Ref 2_Q,4 Integration of the theoretical limit of Eq.(19) for small values of

R+/A+ gives the laminar friction law.

f = 16/Re (23)

Several data points appearing in Fig. 1 of Virk (1971), near

Ref1/2= 200(e= 70), are quite close to Eq.(23). However, the available data at larger values of RefV2 indicate that the values of A+ obtained so far in dilute polymer solutions are bound by

201= 350.

At the polymeric regime, as defined by Virk, an approximate

rela-tion between e'and the polymeric properties can be found using Virk's correlations. At large values of Ref, where the friction factor cur-ves for different values of A+ are described by parallel lines, Au+ is uniquely related to A+. From Fig.8 it was found that at this range

+ +



= 40

log(A-/An + 4) - 28

At small values of Ref° the relation between Au+ and A+ depends on the values of Ref, however, if Au+(A+) is measured along straight lines originating at Ref1/2> 1000 and having slopes which do not exceed the slopes recorded in actual measurements, the deviation from Eq.(24) is

less than 5%. The re from Eq.(3) and for V > tion of Au+ (24)

lation between A+ and the shear stress can now be obtained

* *

. This equation is composed of two expressions; for V <


Vcr It is suggested that a better description of the varia-is obtained by the single equation

* *

Au+= (a/4) log(1 + (V /V )41.


* *

Equation (25) deviates from Eq.(3a) at V > 2V by less than 3% and

* * cr *

is practically zero for V< V

r/2. The values of Vcr according to Eq. c

(25) should be determined by the


of the straight line

(3a) with the Newtonian profile, which is exactly the procedure used



by Virk. It follows from Eqs.(24 and (25) that

+ + *

5(1 + (V "*


kiatisovii 4

dr (26)

where a is related to the polymer properties by Eq.(13).

We have used Eqs.(26) and (12) to calculate the variation of

C1/2 versus Ref1/2

for solutions of the polymers AP-30 and Guar Gum.

(Estimated values of the critical shear and molecular properties are

given by Whitistt et al (1968) And Virk (1971),table 5). The


curves for the three solutions,and curves for constant values of A+

are compared with the ,measurements of Whitistt et al (1968) in Figs:


9 - 12. At small and moderate values of V* /Vcr the agreement between

the data and the theoretical calculations (solid lines) seems to be

satisfactory even for small values of Ref. The agreement is


sur-prising as it merely reflects the adequacy of Virk's correlations and the slight improvement due to the use of the continuous equation (25)

rather than equations (3a) and (3b). The phenomenon of maximum drag

reduction, however, appears now in a different light. One

sees that

* *

when V /Vcr becomes large, the data deviates from Eq.(26) and seem to be correlated with curves of constant A+.The measurements in the

concen-trated polyox solutions and the smaller pipe-diameters seem to be bound

by the curve A+ = 350, which is close to Virk's maximum drag reduction

asymptote in the range RefV2 < 1000. However, the deviation from the

lines which are calculated using Virk's

polymeric regime correlations,

and the approach to the maximum value of eldo not occur only near the maximum drag reduction asymptote. It appears that for each solution, there exists a maximum value of A+ (or au+) approximately independent of the pipe diameter. Only when R+ is small the curves coincide in


limited region with Virk's maximum drag reduction asymptote

(11).This evidence is not manifested in Virk's model which predicts drag

reduc-tion values


the order of 90% for very large shear rates. It is also

interesting to note that the measurements of drag reduction with alum-inium distearate in an organic solvent shown in Fig. 12

(McMillan et

al, 1971) exceed the maximum drag reduction curve and appear to reach



In the absence of a theoretical model for drag reduction mechanism

there is no way at present to determine whether the asymptotic value of

A is determined by properties of the particular polymers used,

expe-rimental limitations, a dependence of drag reduction on the existance of a minimum level of turbulence necessary to deform the macromolecules in solution, degradation or other causes.


It has been shown that the effect of linear macromolecules in dilute solutions on the flow in the wall region, can be described by van Driest's mixing length model with a variable damping parameter A+. If the Reynolds number of the flow R large, the constant shear

approximation used by van Driest can be used. When R4711+ is not large, it is necessary to take into consideration the variation of the shear

stress with the distance from the wall. The velocity distribution in

the outer region is modified in this case using Coles' Wake Function multiplied by a factor. The factor decreases as the damping action of the molecules increases. Although the model does not explain the dam-ping mechanism it suggests a similarity between flows with and without

polymers which is not present in the elastic sublayer model. The model

does not explain the nature of the maximum drag reduction asymptote either, however, it is pointed out that the maximum drag reduction

curves for a given polymer might be associated with a maximum value of

the damping parameter A+. REFERENCES

Coles, D. J. of Appl..Math. & Physics. (ZAmP) Vol.5 No.3, 1954. Elata, C., Lehrer, J. & Kahanovitz. 1966, Israel J. Tech. 4, 87.

McMillan, M.L., Hershey, H.C. & Baxter, R.A. 1971. "Drag reduction"

Chem. Eng. Prog. Symposium Series, 111, 67, 27.

Meyer, W.A. 1966, A.I.Ch.E.J. 12., 522.

Poreh, M. & Paz, U. 1968, Inter. J. Heat Mass Transfer, 11, 805. Tomita, Y. 1970, Bull. J.S.M.E. 13, 935.

van Driest, E.R. 1965, J. Aero. Sci. 23, 1007.

Virk, P.S. & Merrill, E.M. 1969, Viscous Drag Reduction (Ed. C. S. Wells)

Plenum Press.

Virk, P.S. 1971, J. Fluid Mech. 45, 417.

Whitistt, N.F., Harrington, L.J. & Crawford, H.R. 1968. Clearing


I 1 1 1 1 I I I 1 1 1 1 1 1 1 1 I 1 1


1 1 1 1 1 1 1 1 10 102 103 104 Fig.1

Velocity distributions according to the various models(A+=








0.5 0.6




Fig.2 Defect velocities in the various models (124-= 10000, 11+= 26).

60 50 Y.(1-


1 I 1 1 1 1 11 R.2100





ell 300 11+ 226


113 + +


Fig.5 Velocity distributions for R+= 1890.





60 50 40


30 20 10









-DATA 103 Fig.6

Velocity distributions for 114-=-600


Velocity distributions for





4 0 2 0 10 ^ 16 Re 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1

/ 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1/2A+ = 350


= 19.0 logio(Re. f1/2) 32.4



225 -2C0 175


V2_ 1 f

4.0 logio (Re


) 0.4




90 -so






II, Li.'










Friction factor curves as a function of e

70_ 60 ..11


106 3 0 1 f1"2


40 35 30 1 25 20 15 10 5 10 . 40 35 30 25 1 20 15 10 5 02 law I I I


I 1 I 11- ITT


0.18" ID 0.416" ID

Eq.( 26)



T 7I,

1 ill .../...LO3g0...1 T326705 I ..V


/ 7

''...;."'' ...4'...''''' 2252(5) ...**- .... _....".

...'"--7 ,...;.,2,






...'s 1152°5

//7 </'






7 ...a...;4..-

...:..,...- /is_



.... - ...3.,....- _-.-:,- ---- 18


. ---,..-- ---..;,..-- ,.,..-:...- 50



.../' .../.../ 26 ..- ....---",



..---- -%




,---10 W.P.P.M. 1 1 1 1 11111 1 1 1 11111 1 1 1 1 1111 MM.


Measurements of friction coefficients



50 W.P.P.M.

0.1e ID

1.624 ID



1 1 1 11111 1 TflTTfl TA-30 1j3661 _ 200 -125 100 --"*" 90 50 26 1 I I 11111 1 1 1 1 1111 105 103 104 Relerf"

Fig. 10

Measurements of friction coefficients



I 1 11 1 1 1 11

10 102 103



35 15 10 5 -

AP-30100 W.RP.M


0.41e ID


ID 6.0.


T I T 00' , 1 1111111







275 -_,....-"..-.°




..:::....:...,-;....217050 ...;',...--** --- 152°5










-_, --- ...100



...;...%...- 670°

ii //



/* -'----


I /

/ ---/

---., --":..--- ..,..-..../ 26 ...- ...::...-1 1111111 ma 1 1



los 3 10


Fig.11 Measurements of friction coefficients

Fig.12 measurements of friction coefficients

10 102 io3




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