Limiting conditions to similarity-law correlations for drag reduction polymer solutions

NOV.

'972ARCHIEF

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

ibliotheek van d

Onderafde1n

-choo

DOCUMEIAiE

DATUM:

Washington, D. C. 20034

-Lab. V. Scheepsbouwkunde

Technisclse I

Iu95ClOOl

LIMITING CONDITIONS TO SIMILARITY-LAW

CORRELATIONS FOR DRAG-REDUCING

0

POLYMER SOLUTIONS

by

Paul S. Granville

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

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WASHINGTON, D. C. 20034

LIMITING CONDITIONS TO SIMILARITY-LAW CORRELATIONS FOR DRAG-REDUCING

POLYMER SOLUTIONS

by

Paul S. Granvillé

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

TABLE OF CONTENTS Page ABSTRACT 1 ADMINISTRATIVE INFORMATION ...':.:...1 INTRODUCTION 1 SIMILARITY LAWS 3 VELOCITY PROFILE 3

FRICTION FACTORS OR COEFFtCIENTS 5

RANGES OF VALIDITY OF SIMILARITY-LAW CORRELATIONS 6

GENERAL 6

LINEAR LOGARITHMIC B-CORRELATION 7

ALLOWABLE REGIMES FOR tB-CORRELAT IONS 8

PIPEFLOW

10 NUMERICAL RESULTS 12 CONCLUSION 13 REFERENCES 18 LIST OF FIGURES Page

Figure 1 - Inner Similarity Law for Velocity Profile ...13

Figue 2

Similarity Law Friction Diagrath 14

Figure 3 Restricted Limits to B-Corre1ations 14

Figure 4 Subcritical Regime 15

Figure 5 Supercriticai Regime 15

Figure 6 - Limiting Values of n as Function of a arid 16

Figure 7 - Reynolds Number of Pipe Flow Corresponding to

n for Interactive Regime and to n0 for

Inception of Drag Reduction 16

Figure 8 - Limiting Values Of R as Function of

and ni

17

Figure 9 Limiting Values of for Hoyt Rheometer 17

NOTATION

A Slope of lOgarithmic law, Equation (5)

A. _{Slope of interaëtive law, Equation (3)} Constant in Equation (3)

B1 Constant in Equation (5)

B2 Factor in Euation (6)

C Concentration

of

polymer in solution

f Function of

1 _{Characteristic length for polymer} Reynolds number based on 1, 1*

_{E UTZ/V}

Value of 1* at drag-reduction inception

in _{Characteristic mass}

UT

_/m\3

Reynolds number based on m, m*

P _{Polymer species dissolved in a particular solvent}

r Radius of pipe

R _{Reynolds number for pipe flow, R = 2rV/v}

R. _{Values of R corresponding to interactive regime}

Values of R corresponding to conditibn of no drag reduction

t _{Characteristic time for polymer}

t* _{Reynolds number based on t, t E UT}

u _{Velocity in direction of flow}

Shear velocity, _UT

UTO

Value of UT at inception of drag reduction

U _{Value of u at outer edge of shear layer (centerline of pipe)}

V _{Average value of u across pipe}

y Normal distance from wall

y. _{Value of y at thickness of interactive layer}

Thickness of laminar sublayer Reynolds number, y* uy/v.

y*

_{Value of y at thickness of interact.ive layer}

Value of y at ti cknessp £ laitithar. sublayer.

Meyer drag-reduction factor; see Eqüàtiàn (7)

Thickiiess Of shear layer

Drag-reduction factor Reynolds number, Ti

o c

Critical value Of n.

Limiting value of n for interactive

regime-Ti0 Limiting value of Ti for no drag reduction

Altered von Kármn factor,

i/A

V Kinematic viscosity of solution Kinematic viscosity of solvent

p Density of solition

T Wai shearing stress

w

-t Value of T at inceptioh of drag reduction

-ABSTRACT

Similarity-law correlations for drag-reducing polymer

solutions are shown to require sufficiently thick turbulent' shear layers. Limiting conditions occurring for the shear,

layers thinned down to only a laminar sublayer and an inter-active (buffer) sublayer are examined on the basis of an

interactive logarithmic law. Subcritical and supercritical

regimes are defined on the basis of shear layer thiãkness.

ADMINISTRATIVE INFORMATION

This investigation was funded by the Naval Ship Systems Coimnand under Subproj:ect SF 35.421.003, Task 1710, 'and by Nava.Qrdnançe Systems

Command, under TJR1O9-Ol.-03.

INTRODUCTION

The. drag-reducing effects of polymer solutiOns 'may be correlated

by the similarity laws of turbulent shear flows as developed in Reference l.

These similarity laws consider an additional length, time,' or mass to' be

introduced' into the analysis by the presence of the polymer.' The' value

of a simi1rity-law correlation lies in its universal applicability to'al.l turbulent shear flows from internal pipe flow to external boundary-layer

flows'.. The decreases in'frictional resistance or drag for one type. of

flow may be accurately predicted by data taken in ahother type of'flow.' However, as will be dethonstrated in this paper, there is an impor-tánt' limitation to the ability of the similarity-law correlation 'to aialyze drag-reduction data. If the shear flow is too' thin as in a smal'l-4iameter

pipe, the similarity-law correlation is'not applicable. The. sivation here is that the flow is' 'in a condition of maximum drag reduction due 'o the thinness of the shear layer. The laminar sublayér and interactive (buffer) sublayer remain., The purely 'turbulent portion of the shear layer' which

involves the similarity-law correlation has vanished. The'paradox is that although a ma1minn drag reduction is indicated the data are quite useless for predicting drag reduction for thicker shear layers contai'ning the usual

turbulent portion.

*References are listed on page 18.

The

original similarity-law analysis1 for drag-reducing polymer solu-tions considered the pipe-diameter limitasolu-tions on the basis of a thicker

laminar sublayer producing a thicker buffer sublayer between the laminar

sublayer and turbulent logarithmic layer. Recent experimental findings by

Tsai2 and by Tomita3 show a thicker buffer sublayer without a corresponding

thicker laminar sublayer. The velocity profile for the buffer sublayer is

correlated by a distinctive logarithmic similarity law. In fact, the buffer layer is now more properly termed the interactive sublayer since the

drag-reducing effects are ascribed to the interactions of the polymer molecules with turbulence-producing proceses occurring there. On the other hand,

Virk et al.4'5 deduce the logarithmic interactive similarity law from the

condition of maximum drag reduction. The use of the interactive logarithmic similarity law provides interesting and precise limitations which are the

subject of this paper.

The first indications of the presence of the interactive similarity

law and its accompanying maximum drag reduction may be traced to the

experiments of Hoyt and Fabula6 who observeda condition of maximum drag

reduction in tests with pipes and rotating disks. The maximum drag

reduction seemed to be independent of the type of polymer or its

concentra-tion in the soluconcentra-tion. Giles7 proposed a power-law similarity law to take

care of the condition of maximum drag reduction. Virk et al.4 later

pro-posed a logarithmic similarity law for the interactive layer to account

for maxImthn drag reduction. Actual logarithmic velocity profiles for the.

interactive layer are shown by Tsai2 and separately by Tomita.3

This report restates fbr reference the similarity laws for turbulent

shear flows of. drag-reducing polymer solutions. A simple yet very useful

aspect of the similarity law is to consider conditions at the outer edge

of the shear flow.(the centerline for fully developed pipe flows). This

leads o friction coefficients as functions of similarity-law Reynolds numbers. The interactive logarithmic velocity law then becomes a la for

maximum drag reduction, which corresponds to a limiting condition for

similarity-law correlation. The case of fixed shear layer thickness (fixed radius for pipe flow) is considered for the type of drag reduction that

limitthg regimes are identified. For pipe flow, the similarity Reynolds

number is converted to a pipe Reynolds number based on diameter and average

flow velocity. The limitations of the Hoyt9 rheometer are examined in terms

of its being a small-diameter pipe.

SIMILARITY LAWS

VELOCITY PROFILE

The two similarity laws,' the inner law and the outer law, provide the linkage between the wall shearing stress and the velocity profile

u [y].

For drag-reducing polymer solutions, the inner law is stated (see

Figure 1) as:

U

ft *

_Ly

'I

1*,C,PJ

where u is the mean velocity in the direction of flow,

UT iS

the shear velocity, _UT

is the wall shearing stress, p is the density of solution,

E u1y/v,

y is the normal distance from the wall,

'' is the kinematic viscosity of the sOlution,

1*

_{E UTZ/V}

1 is the characteristic length due to polymer,

C is the concentration of polymer in the solution, and P is the polymer species dissolved in a particular solvent.

As shown in Reference 1, a characteristic time t or characteristic mass m may also be considered in place of 1. Instead of l, there results

t or m* where

* Ur/m\l/3

m

=(-V

The expression, Equation (1), takes different forms for different

layers inside the shear layer.

(1)

or

For the lc7rinar üblcryr Of thickness

U _a a

(2)

UT

The interactive, layer ofthickness (y.* - has a logarithmic

relation

=Iny'

-UT a Y

= A ln +

T

to indicate that the interactive logarithmic line passes

through

L Here

is the intersection of the laminar profile, Equation (2), and the

logarithmic profile for no drag reduction, Equation (5), with AB = 0. is a constant but y* is a function of .1*,

y.*[l].

The slope A may be considered as the reciprocal of a modified von Krmn constant

K,

A

1/K.

The outer lw, is unaffected by the polymer solution and has the forñ

U-u

UT

where U is the maximum velocity at the outer edge of the shear layer (at

centerline for fully developed pipe' .lçw) and is the thickness of the shear layer (5 = r = radius for pipes).

As shown n Reference 1, the overlapping of the inner and outer

laws results in the logarithmic law which is the similarity law providing

correlation for drag-reducing polymer solutioiis

=.A lay'

+B1

+ _{B[I',,C, P1}

or

U-u

Aln-+B2

UT &

yy* y?(1']

[y] L&J (3) (4)

where A is the slope of the logarithmic law, A =

1/K, K

= von Krmn constant;

B1 is a constant (for flows without drag reduction);

B2 -is a factor which is -constant for a particular type of shear layer (B2 is close to zero for pipe flows); and

= f[l*, C, P] represents the similarity-law correlation.

A particular type of siB-function is the Meyer8 empirical relation = a ln u /u for u u which occurs at lower values of u (u

T

T,0

T

T,0

- T

T,0

is the drag-reduction inception value of uT) The- Meyer relation is

generalized in Reference 1 to

1* U 01

sB=a1n7,

1*1

T'

0

where a = f[C, P], a 0. For convenience, a value of l is assumed to

produce 10* = 1, so that 1 ₌

_V/UTO

Then

B=a1nl*

(8)

From tests with Polyox solutions (WSR-30l), Tsai2 obtained A = 12.7 and from Polyox solutions (Alkox-C) Tomita3 obtained A = 10. From data for maximum drag reduction of polyier solutions flowing in

small pipes, Virk et a1. deduced A = 11.7. The interactive logarithmic velocity profiles with these various values of A and a y* = 11.6 are plotted*in Figure 1 where a value of A = 2.5 is also used.

FRICTION FACTORS OR COEFFICIENTS

The results at y = ó (where u = U) are an important but implicit

consequence of the similarity laws. A friction factor or coefficient

U/uT = (T/pU2Y'1/2 becomes a function of a Reynolds number n

See Figure 2.

*The power-law of Giles7 obtained from maximum drag reduction for

- / - . 396\

Guar Gum is -also plotted fcr the interactive region _\U/UT

= 535y*

)

5

For tKe interactive regime., r

U

-Aln

UT

Here the shear layer ends in the interactive laye. This is the. condition of maximum drag reduction.

For the iegular drag-reductiQn regime or 4B-correlat-ioh regithe,

-A1nii+B1+B2+iB(1,C,PJ,

!?*1i[l°J

(10) Here the shear layer consists of the laminar sublayer, the interactive

layer, and a purely turbulent portion. For the Meyer Or ithear logarithmic

AB-correlation, Equation (7),

+B2+alnl,

RANGES OF VALIDITY OF SIMILARITY-LAW CORRELATIONS

GENERAL

Tie regular similaritylaw correlation given by Equatioh (10) is

applicable only if the hear layer is thicker than the interactive, layer or

> The limiting condition n1[l*] for the similarity-law or

correlatiOn occurs at the intersection of U/UT fOr both regimes as seen in

Figure 2. For this limiting condition

U-..

-A1h1-BA!n1+B1+B2+B

(12) UT (9) or

B=(A-A)inn1-B-B1-B2

-(13)

which is plotted in Figure 3. Here = 11.7 and B2 0. Wells10 proposed. the following enp1.rica1 criterion as. a limiting con4it-iQn

AB=O.O331

(14)

As seen in Figure 3, agreement is only in the neighborho.od of = 14,000.

LINEAR LOGARITHMIC LB-CORRELATION

For the case of constant thickness shear layer (constant-diameter pipe), n is by definition + B + B2 111711=

A-A-a

B

= 7

(15) (17)

as a

point

of

inception

of drag

reduction n0

for

any value of .

The intersection of Equation (16) for constant 6/i with Equation (9), the interactive line, or the line of maximum drag reduction represents

the limiting value of r, ri, for iiB-corre1ations as a function of and 6/i or

a

B

1n-A-A-a

For constant ct, in is then a linear function of in 5/i.

7

(18) Then for the linéar logarithmic B-correiation, Equation (U) becomes

!(A+a)In11+B

+B2-a14,

71>111 (16) Therefore in a friction-factor diagram like Figure 2, U/UT against fl plots as a straight line for constant /l.

The intersection of Equation (16) for constant 6/i with the line of

LXB = 0 represents a point of inception of drag reduction. Equating

ALLOWABLE REGIMES FOR LB-CORRELATIONS

It is evident from Figure 2 that various allowable regimes for correlations may be delineated depending on the position of drag-reductIon

inceptionfl0 _{These are discussed below and summarized in Table 1.}

Critical Point

The intersection of the interactive line itself and the AB 0 line

provides a critical value of or n

From Equation (12) with AB = 0,

and from Equation (17) /l

fj\

0,c (20)

where (S/i) is the critical value of 6/Z. Consider straight lines fr

various value of a in Figure 2. emanating from for constant in

accordance with Equation (16). It is evident that if the slopes A + a are less that for the interactive regime or (A + 0.) < A., there can be

no additional intersection other than at . On the other hand, for

slopes A + a eqial to or greaterthan A., wh{ch is physically impossib1e,

the friction line matches that for the interactive line and no AB-correlation

is possible. Hence for /Z ₌

Oa<(X-A)

No restriction in similarity-law correlation for -i>ii

(=A)a<oo

Complete restriction for

Subcritical Regime /l <

In Figure 4-, consider lines of slope A +. 0. Oipared to A, the slope of the interactive line It is evident that no intersection is possible

IT)??0.

-.

A-A

B+B1+82

TABLE 1

Allowable Regimes for iB-Correlation

9

Regime

Range ofa

Range of

for

AB-correlation

Critical

&

f6\

=

Oa<(-A)

( - A) a <co

No range

Subcritical

_7<V11

o

fO\

Oa<(A)

- A)

a <co

No range

Supercritical

S

Oa<(-A)

for LA + a) . A and that there is intersection fOr (A + a) < Hence for

S/l < (6/i),.:

Oa<.(A-A) No restriction in sinUlarity-law correlation for

and restriction for

(A-A)a<oo Complete restriction for

Supercritical Regime 6/i. >

In Figure 5, consider lines of slope A + a compared to , the slope

of the inte±'a.tive line. It is evident there is no intersection and no restriction for (A

+cL)

A and that there is intersection and restrictiOn

for (A +

_)>

A. Hence for 6/i >

Ii):

Oa<(-A) No restriction in similaritylaw correlation for

(-ia<oo Restriction fot

and no restriction for

Figure 6 shows limiting value

of n

as a function of a and 6/i. Two

examples are in4icated: (1) 100 ii 940 allowable range for 6/i = 100.

and a = 18 and

(2)

35 allowable range for 6/i = 3 and = 4.

PIPE FLOW

The restrictions in similarity-law correlatiOns have been obtained

interm

of parameter for given 6/i and a. For fully develope4 flow irn circular pipes, the restrictions are more useful in term$ of Reynolds

number R 2rV/v where V is the average velocity across the pipe and r is the radius (here 6 = r). y definition

Un UT

_R

(21)

and

=

2i

Ur

Now we obtain V/ut as a function of

In general, as given in Reference 1, for circular pipes

=2f

.dy*

i y*y*

(22)

In the case of n. for the interactive regime, the shear layer is reduced to the laminar sublayer and the interactive layer. By using the

appropriate velocities, Equations (2) and (3), there results

V

3

1 I

(AY2 Y3

(23) The corresponding R, R., is 2 *3 R1 = in - (3 + 2)n + 4y 2y2 -3 (24)

which is plotted in Figure 7. Here A = 11.7, YL* = 11.6.

In the case of ri0 at drag-reduction inception, the shear layer is the usual one without drag reduction. The result is

/A *2

.3

V

3 1 111tYL L

=AIn0-A+B1 +B2+--(2Ay-y2)---

(25)

and

R0 = 2Afl0 hi - (3A - 2B1 - 2B2)170 + 4Ay -

2y2

-

iAy2

.Zy3

(26)

which is plotted in Figure 7. Here A = 2.5, B1 = = 11.6.

NUMERICAL RESULTS

For POlyOx WSR 301, White11 suggests

a=23jt for a<20

(27)

where C concentration in parts per million. Also u

0 0.08 ft/sec.

Thenl=/ü

=l.6X103in.

T,0

The variation of viscosity with concentration for

POlyox WSR

301

from data obtained by van Driest12 is fitted by

V0

V

= 2.04 X 103C10 (28)

where' is the kinematic viscosity of water.

For Guar Gum, Reference 1 quotes Elata data which in.the notation

of this paper is = 0.0113C for C < 800 ppm. Also in Reference 1,

1 = 4.0

x l0 in. and

-t _{1 = 5.25 X i0''}

p0

FrOm Equation (19), ri = 11.6.. Hence for Polyox WSR 301,

cS

r

= 0.0188 in. and for Guar Gum, ô

r.

0.0046 in.

Also a critical

C c

c

c

value Of

= A -

A = 9.2 whic1. corresponds to C = 16 ppm for Polyox WSR 301 and C 810 ppm for uar Gum.

Plots of ri. against R1 and against R0 are shown in Figure 7. The plot of c against ri for constant 6/i in Figure 6 is converted to one of against. R1 for constant r/l jn the case of pipe flow in Figure 8. All

'these results assume transition to turbulent flow has Occurred.

As an example, the. rheOmete of HOyt9 may be examined in accordance with the principles stated in this paper. The radius of 0.0215 in. gives

a r/i = 13.25 for Polyox WSR :301 and r/l = 53.6 for Guar Gum. Since these values.of r/l are greater than (r/l)c = 11.6, the rheoineter is operating in the supercritical regime. Hence there is no limitation to measuring

flows with. c less than 9.2 which corresponds to C = 16 ppm for Polyox WSR 301 and to C = 810 ppm for Guar Gum.

The apparatus ope±ates at a fixed speed of .V = 0.49 in.Isec which.

will give a variable Reynolds number since the viscosity depends on the

concentration even for a fixed temperature. Consider the Reynolds number for water at room temperature R = 1.4 X. o4 for the results shown in

Figure 9. It is seen from the flatness of the curve that a 9.5 (C -

l.ppth)

represents the upper limit of a for the iB-characterization of Polyox WSR 301. There is, however., no limit iti Figure 9 in a for Guar Gum, outside of a 20

as the physical limit.

CONCLUS ION

Similarity-law correlations for drag-reducing polyners should not

be madeif the shear layer is too thin, e.g.., as that, in a capillary tube.

Exact limitations may be deduced from the intersection of the logarithmic

interactive line and the similarity-law logarithmic line. Subcritical and supercritical regimes may be. defined on the basis of shear-layer thickness

The Hoyt rheometer should be used with caution for similarity-law

correlations. U? 50 10 INTERACTIVE _{-.. 11.6 + A n} 11.6 TSAI -12.'7 VIRK ET AL 11.7 TOMITA. 10 GILES-5.35V

1

/

k/

4

/

/7

TURBULENT: f2.5In ? +5.5+B OR 11.6+2.5 In

Figure 1 '- Inner Similarity Law for Velocity Profile

.13.

Go Go 20 10

--__.;:;;).t.

-..-1110 11 B 0 60 -50 40 8 30 20 10 0

Figure 3 - Restricted Limits to B-Coi'ielations

10 io2 'a3

'1

u,.S

Figure ₂ Similarity Law Friction Diagram

io2 2 6 B 2

AB 0 40

U.,. 30

UT 60 50 20 10 0 10 15

Figure 5 - Supercritical Regime

A

lid:'

I S 20 - -

-A%fl7)12

AS 10

AB0

-TuRBUt.' --4lI -50 (EXAMPLE) _'lo.c -1 10 102

Figure 4 - SUbcritical Regime

40 U UT

10

,7 _{SIMILARITY REYNOLDS NUMBER}

Figure 6 - Limiting Values of n

as Function of a and tS/l

10 io2

- --

__1

EXAMPLE-ALLOWABLE RANGE 9 _d

_____

III

FOROIB 1OO'794O

I'j4,i

ISUPERCRITICAL

I

I

_Lt?

:

UI_

r116

r1SflNTERAC1IVE

____________

LIMIT)

liii

__________ 6 F (fl4IERA RANGE TIVE LIMIT) -6

-

3 1

,////4////

(INTERAcTIVE

______

LIMIT)

/,'f"

EXAMPLE-ALLOWABLE VhW'

F-I

4

SUBCRITICAL

riii

FORa4:a5

liii

1111

liii

liii

.

II.!

PIPE REYNOLDS NUMBER

FIgure 7 - Reynolds Number of Pipe

Flow Corresponding to n

for Interactive

Regime and to

for Inception of Drag Reduction

20 18 16 14 12 10 9.2 8 6 4 2 0

14 12 0 UI 18 16 0 PIPEREYNOLDS NUMBER. R

-Figure 8 - Limiting Values of R as Function of

and ni

,,, 53.6(GUAR GUM)

tc

-I ID

.,',,

.i EXAMPLE-ALLOWABLE RANGE FOR GUAR GUMa 15

'

.4 X i02 46 X -13.25 (POLYOX WSR 301) F

,

.

III

-

I_._______

-I

-I 10

PIPE REYNOLDS NUMBER. R -V

Figure 9 - Limiting Values of R for Hoyt Rheoineter (r = 0.0214") 17

uhF1

I

I

[dill

I!!

I

tIII__

I

'I!- I

liii

11.6 _CRITICAL

''JJjj

ii

r

-Lio

p

N1 5INTERACTIVE UMT

liii

Ill.

us,

7/

P,,lIlI

77//P'- _/". -SUBCRITICAL

liii

III

20 18 16 14 12

REFERENCES

Granville, P. S., "The Frictional. Resistance aM Velocity Similarity Laws of Drag-Reducing Polymer Solutions," NSRDC Report 2502

(Sep 1967); .alsp J. Ship Res., Vol. 1.2, NO. 3 (Sep 1968).

Tsai, F. Y. E., "The Turbulent Boundary Layer in the Flow of

Dilute' Solutions of Linear Molecules," Ph.D. Thesis, University of Minnesota, University Microfilms 69-11, 466 (1968).

Tornita, Y., "PipeFlows of Dilute Aqueous Polymer Solution:.

Part I, Experimental Study of Pipe FrictionCoefficieflt; Part II,

Correla-tion. of th Frictional Characteristics," Bulletin of the JSME, Vol. 13,

No. 61., p. 926 (Jul 197b).

Virk, P. S., Mickley, H. S. 'and $mith, K. A., "The Ultiniate

Asymptote -and Mean Flow Structure in Toms Phenomenon," Trans. ASME, J. Appi. Mech., Vol. 37, Series E, No. 2 (Jun 1970).

Virk, P. S.., "An. Elastic Sub layer Model for Drag Reduction by Dilute Solutions of Linear Macromolecules," J. Fluid Mech., Vol. 45,

Part 3 (Feb 1971).

Hoyt, J. W. and Fabula, A. G., "The Effect of Additives on Fluid Friction," Fifth Symposium on Naval Hydrodynamics (Sep 1964), Office of

Naval Research'ACR-1.l2, U. S. Government Printing Office, Washington, D. C.

.7. Cues, W. B., "Similarity Laws of Friction-Reduced Flows,

J. Hydronautics, Vol. 2., No. 1 (Jan 1968).

Meyer, W. A., "A Correlation of the Frictional Characteristics foi Turbulent Flow of D1ute Viscoelastic NonNewtonian Fluids in Pipes.,"

AIChE Journal, Vol 12., No.. 3 (May 1966). ,

Hoyt, J. W., "A Turbulent-Flow Rheometer," in "Symposium on Rheology," A.W. Morris and J. T. S. Wang, editors, Am. Sac. Mech. Eng.,

New York (1965).

Wells, C. S., "Use of Pipe Flow Corre]ations to Pre4ict

Turbu-lent Skin Friction for Drag-Reducing Fluids," J. Hydronautics," Vol. 4

No. 1 (Jan 1970).

White,.F. M., "An Analysis of Flat-Plate Drag with Polymer

Additives," J. Hydronautics, Vol. 2, No. 4 (Oct 1968).

Van Driest, E. R., "Turbulent Drag Reduction of Polymeric

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Seeu,gIyclaaslfication of title. body of abatrart and indexing annotation rnIvt be entered when the overall report is ctasaifS.d) I. ORIGINATING ACTIVITY (COrPOt.t àithOr) -. - .

-Naval Ship Research and Development Center Washington, D. C. 20034

2a.REPORT SECURITYCSj1FI'ATION

UNCLASSIFIED

2b. GeuP - . .

-3. REPORT TITLE - . -- .

-LIMITING CONDITIONS TO SIMILARITY-LAW CORRELATIONS FOR DRAG-REDUCING POLYMER

SOLUTIONS . -. . .

4. DESCRIPTIVE NOTES (Type of report and inclusive dates)

--5. AUTHOR(S) (Ftrt name middle initial, la.tn.the)

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-Paul S. Granville

6. REPORT DATE - - -

--August 1971

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IS. ABSTRACT -

-Similarity-law correlations for drag-reducing polymer solutions are shown to require sufficiently thick turbulent shear layers. Limiting conditions occurring for the shear layers thinned down to only a laminar sublayer andan interactive (buffer) sublayer are examined on the.basis of an interactive logarithmic law. Subcritical and supercritical regimes are defined on the basis of -shear layer

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UN(TASTPIFfl ecurLty Ci.iisthcation UNCLASSIFIED. Security Classification

KEY WONDI

LNI A

L.INK B LINP C

WOLE WY WOLE WY MOLt WY

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Polymer solutions, drag-reducing effects Similarity law correlations

Ranges of validity Allowable regimes Pipe flow