Similarity laws for turbulent flow of dilute solutions of drag reducing polymers

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT c:ENTER BETHESDA. MD. 20034

SIMILARITY LAWS FOR TURBULENT FLOW OF DILUTE

SOLUTIONS

ÓF

DRÁG.ÁEDUC1NG POLYMERS

by

T. T. Huang

pL.

APPROVED FOR PUBLIC RELEASE: DISTRIBUTÌOÑ LÍNLIM1TED

August 1973 _{Report 4096}

TABLE OF CONTENTS 11 Page ABSTRACT i ADMINISTRATIVE INFORMATION. . . .. t INTRODUCTION . . i

REVIEW OF VELOCITY SIMILARITY LAWS EOR ORDINARY

TURBULENT BOUNDARY LAYERS

... -.

.

4

VELOCITY SIMILARITY LAWS FOR POLYMER BOUNDARY-LAYER FLOWS 6

POLYMER TYPES AND SOLUTION PROPERTIES 9

PIJ4W%V EXPERIMENTS TO DETERMINE

... 10

SATURATED DRAG-REDUCTION LINE 13

DISCUSSION OF MEASURED

...

15

DRAGREDUCTION DOMAINS 17

COMMENT ON TIlE STRÓNGLY INTERACTIVE LAYER 18

CONCLUSION ₁₈

ACKNOWLEDGMENTS . . . 20

REFERENCES . . .

. .

.

...

29

LIST OF FIGURES..

Figure 1 - Outer Layer Velocity Profiles for OrdiÌiaÍyEÌtern1 and Internal FlOws 20

Figure 2 - Experimental Evidence of Interactive Layers in Internal Polymer Flows 21

Figure 3 - Effect of Polymer Solutions on Velocity-Defect Law, Pipe Flow 21

Figure 4 - Effect of Polymer Solutions on the Law of the Wall, Open Channel Flow 22 Figure 5 - Effect Of Polymer Solutions on Velocity-Defect Law, Open Channel Flow.

uD

22

Figure

6 - Solutions of Equation (18) in Terms of AV versus - for

2v = LV

Values

various of

Figure 7 - Effect of Pipe Entrance on Drag Reduction 24

Figure 8 - Typical Results of V/u. versus u.,./v for POLYOX WSR301 Solutions, Pipe Diameter is 3 181 Centimeters, and Temperature is 75 Degrees

Fghrenheit 25

Figure 9 Temperature Effect on Drag Redúction of POLYOX WSR-301

Figure ¡0 - Drag-Reduction Characteristics, _{versus u/z.', for POLYOX}

WSR-301 Solutions

Figure 11 - Drag-Reduction Characteristics, versus for SEPARAN

AP.30 Solutions

Figure 12 - Drag-Reduction Characteristics,

¿7

versus u../v, for MAGNIFLOC

835A Solutions

Figure 13 Drag-Reduction Characteristiôs, Ai versus ui/v, fôr Guar Gum

J2M SolutiOns .

Figure 14 - Saturated Drag-Reduction Boundary for POLYOX WSR-301,

u.e1

MAGNIFLOC 835A, and SEPARAN AP-30 at =

r'

Table i - Characteristic Values of Constants in Equation (23) . . .

...16

Page 26 26 27 27 28

P

B B b C C CT

D,

dp/dx

f

K g1 g1

= VD/i'

ti U0 U

u;

UT V VP vs

y

NOTATION

Slope of logarithmic velocity law in common logarithms for ordinary Newtonian fluid or 2.3026/K

Slope of logarithmic velocity law in common logarithms for strongly interactive layer, EquatiOn (5)

a Constant, Equation (23)

Constant of ordinary Newtonian innór logarithmic velocity law, Equation (1). Constant of modified inner logarithmic veIocit law, Equation (5)

Constant, Equation (23)

Concentration of polymer sOlutiOn in parts per millioñ V

Optimal concentration Local friction coefficient Diameter of pipe

Pressure gradient along pipe Darcy-Weibach friction fâctor

Von Krmn constant

Characteri tic length scale(s) of polymer solutions Length scale for synthetic polymers tested, 5 x

iOa cm

Length scale for guar gum J2M polymer, 137 x

lO3cm

Reynolds number

Characteristic time scale(s) of polymer solutions

Free-stream velocity Loóal mean velocity

Nondimensional mean velocity

Frictional velocity at. onset of drag reduction

Frictional velocity

Average velocity across pipe cross section Average velocity for polymer-solvent system

Average velocity for solvent alone V

Normal distance from the wall

= UT.V/l) = = ys :-yw max ¿ V+ ( V4)

T

o

Nondimensional distance from the wall Nondimensional thickness of laminar sublayer Thickness of laminar sublayer

Nondimensional thickness of strongly interactive layer Thickness of strongly interactive layer

Nondirnensional thickness of weakly interactive layer Thickness of weakly interactive layer

Constant defined in Equation (6)

Maximum value of at

ui/P =

2 x lO cm'

{ V I u

-

i'

/114 at constant ui/v, dtag-reduction function

Saturated value of

IV

Boundary layer thickness Strain rate

Eì] Steady-state intrinsic viscosity

O Momentum thickness of boundary layer ji Dynamic viscosity of fluid

Dynamic viscosity of polymer-solvent system Dynamic viscosity of solvent alone

y Kinetic viscosity of fluid = y/ (D/2) Nondimensioni y

y I

(D/2) Nondimensional yg

= '/

(D/2) Nondimensiònal y

p Mass density of the fluid

Wall shear stress

Cole's wake parameter

Subscript

ABSTRACT

Velòcity similarity laws, based oñ a fouHayer, rnean-vclocity.profìle modál áró dedúced for turbulent boundary láyers with dilute polymer solutions by theans of pipe-flow

experi-ments. Measured drag rcduction is found to have three döñiain: undersaturatéd, optimal, and oversàtürated. The drag réduction does ot increase with increasing concentratiòn in the

)vcr-saturatéd dóÌnin vheré a strong interactive layer dominates the entire liner lbgarithtnkregioti

of the bouhdary layer. Dfag reduction increases with increasing concentratioñ it the

úñdèr-saturated dômain where the four-layer profile exists ¡n the boundary layer. The bòundary be-tween the two domains gives ¿ptiiial drag réduction it is determined by the polymer type àiìd concentration and by a Reynolds nùmber based on shear velocity and boundarylayer thickness. ipe-flow experiments have been made t stû' the drag-reduction characteristics in the

undçr-saturated domain. The effects of solvent temperature, pipe diameter, polymer type and

con-centration, and WâU shear stress on the measured drag reduction have been invéstigatéd.

ADMINISTRATIVE INFORMATION

This work was authorized and fúndéd by the Naval Ship Research and Developmeñ Ceter (NSRDC)

under its Independént Explòratoy bevlöpmeht Prograrh, Task ZF6L412.00l, Work Unit i-1508-309.

INTRODUCTION

During the past 10 yr, turbulent drag redûction by dilute polymer solutions has received much atten-tioti by many investigators. These studies are stimulated both by the promise of engineering applications_and by the fundamental aspects of the problem. Most efforts thus far have been experiments wthtuThulent flow in smooth pipes similar to what was first done by Tòrns.' * Data from the pipe-flow studies may be ôlassi-fled in two main,groups: (I) gross flow measurements of pressuré drop versus flow rate, such as the worlç

1*

Toms B A. Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds

Number Proceedings First International Congress Rheology North Holland Publishing Co Amsterdam VoL 2 _pp

135-141 (1948). A conplete listing of refeiencös is given on page 29.

by Wells,2 Savins,3 Ernst,4 Elata and Tirosh,5 Fabula,6 Virk et al.,7 Hershey and Zakin,8 Van Driest,9

Paterson arid. Abernathy,10 and Huang and Santelli;1 and (2) measurements of mean-velocity profile in flows experienciñg drag reduction, e. g., data of Elata et aL,12 Ernst,4 Virk etal.,7 Goren andNorbury,13 Wells

et al.,'4 Patterson and Florez,15 Tornita,16 Seyer and Metzner,17 Tsai,18 and Wetze! and Ripken.19 A few measurements of the structure turbulence during drag redûction .'çre made by Virket al.,7 Wells;et al.,14

Seyer and Metzner,17 Rudd,20 and Chung and Graebél.21 Most of the resUlts,7'14,'17'21 aitho rather scattered, indicated a reduction. of axial turbulence intensity at a given centerline velocity, altliough the ratio of axial turbulence intensity to sheat velocity was not significantly affected by the polymer in alarge portion

of pipe core However a significant increase in fluctuation of axial velocity and a large reduction in fluctu atiön of transverse velocity relative to shear velocity near the a!l was noted by Rudd.20

2WeUs, C. S., Jr., "On the Turbulent Shear Flow of an Elasticoviscous Fluid," American Institute of Aeronautiçs and

Astronautics Preprint 64-36 (1964). . . ,

3Savins J G Drag Reduction Charactenstics of Solutions of Macromolecules in Turbulent Pipe flow Society of

Petroleum Engineers Journal, Vol. 4, p. 203 (1964). .

-4Ernst, W. E., "Investigation of Turbulent Shear Flow of Dilute AqueousCMC Sotutions," American Institute of chemical Engiuieers Journal, Vol. 12, No. 3, pp. 581-586 (1966).

5EIata, C. and J. Tirosh, "Fnctidnal Drag Reduction," Israel Journal of Technology, VoL 3, pp. 1-6 (1965).

6Fabula, A. G., "The Tôms Phenomenon in the Turbulent Flow of Very' Dilute Polymer Solutions," Proceedings Fourth International Congress of Rhológy, lnterscience Publications, New York, Part 3, pp. 455-479 (1965).

7virk P S et al The Toms Phenomenon Turbulent Pipe Flow of Dilute Polymer Solutions Journal of Fluid Mechanics, Vol. 30, Part 2, pp. 305-328 (l967).

8Hershey H C and J L Zakin A Study of Turbulent Drag Reduction of Solutions of High Polymers in Orgamc

Solvents," Cheniistry Engiñeers Science, Vol. 22, p. 1847 (1967). V

V

9Van Driest, E. R, "Turbulent Drag Reduction of Polymeric Solutions," Journal of Hydronautics, Vol. 4, No. 3, pp. 120-126 (1970).

10Paterson, R W. and F. H. Abernathy, "Turbulent Flow Drag Reduction and Degradation with Dilute Polymer Solutions," Journal of Fluid Mechanics, Vol. 43, Part 4, pp. 689-7 10 (1970).

11Huang, T. T. and N. Santelli, "Drag Reduction and Degradation of Dilute Polymer Solutions in Turbulent Pipe Flows," NSRDC Report 3677 (Aug 1971).

'2Elata C et al Turbulent Shear flow of Polymer Solutions, Israel Journal of Technology VoL 4 No 1 pp

87-95 (1966). .,

13Goren,. Y. and J. F. Norbury, "Turbulent Flow of Dilute Aqueous Polymer Solutions," Journal of Basic Eng., Trans-actions of American Society of Mechanical Engineers, Paper 67-WA/EF-3, Vol. 89, p. 814 (1967). V

'4Wells, C. S. et al., "Turbulence Maasurements in Pipe Flow of a Drag-Reducing Non-Nôwtoñian Eluid,"VArnerican Institute of Aeronautics and Astronatitics Journal, VoL 6, No. 2, pp. 250-257 (1968).

V

15Patterson G K. and G L Forez Velocity Profiles during Drag Reduction Chapterin Viscous Drag Reduction Edited by C. S. Wells, Plenum Press, New York pp. 231-250 (1969).

16lomita, Y., "Pipe Flows of Dilute Polymer Solution, Parts I and II," Bulietin of the Japanese Sòciety of Mechanical Engineers, Vol. 13, No. 61, pp. 926-942 (1970).

'7Sêyer, F. A. and A. B., Metzner, "Turbulent Phenomenon in Diag-Reducing Systems," American Institute of Chemical Engineers Journal, VoL 15, No. 3, pp. 426-434 (1969).

1 8Tsai, F., "The Turbulent Boundary Layer in the Flow of Dilute Solutions of Linear Macromolecules," Ph.D. Thesis Umversity of Minnesota (1968).

'9Wetzel, J. M. and J. F. Ripken, "Shear and Diffusion in a Large Boundary Layer Injected with Polymer Solution;" University of Minnesota, St. Anthony Falls Hydraulic Laboratory Project Report 114 (Feb 1970).

20Rudd, M. J., "Velocity Measurements Made with a Laser Dopplermeter on the Turbulent Pipe Flow of a Dilute Pol'ther Solution," Journal, of Fluid Mechanics, VoL 51, Part 4, pp. 673-685 (1972).

Velocity profiles during.drag reduction were measured by Wetze! arid kipken1 in flow over à large channel floor. l'heir ròsults ¡ndicatèd that the same velocity similarity laws héld foi both iñte(nal arid ex-ternal boundary-layer flows during drag reduction. Total flat-plate drag reduction by ejectiOns of polymer solutions was reported by Wu and Tulin.22

Although the mechanism responsible for drag reduction is still not undertood, the recent velocity pro-file measurements in internal flows by Tsai,1 8 and by Seyer arid Metzner1 confirm the interactive layers velócity model proposed by Van Driest,9 Virk et al.,23 and Virk.24

The measured velocity profiles also show evidence of the interactive layei velocity profile in an external boundary layer during drag reduction.19 In drag-reducing flow, the mean-velocity profile in both the iriterìial and external turbulent boundary layers can be divided into

A viscous sub layer,

A strongly interactive layer, characterized by a smaller Von Krmn constant,

A weakly interactive làyer, characterized by a parallel. upward shift of velocity profile in a sernilogarithrnic plot, and

An outer Wake region.

On the basis of the four-layer model, the drag redüction at a given boundary layer thickness and wall shear stress may be classified accordingto, three distinct domains.

Oversaturated: in this case the drag reduction reaches. its maximum possible value and cannot be increased by increasmg còñcentration, and the entire linear logarthmic region of the boundary layer is domi mated by the strongly interactive layer.

Undersaturated: here the dragreduction effectiveness increases' with increasing concentration, and all four layers are present in the boundary layers.

Optimal: this is the boundary between Items (1) and (2).

The drag reduction in the oversaturated domain is independent of concentratiofl and can be derived by assuming thät the strongly interactivè layer dominates the entiré linear logarthniic region of the boundary layer. However, the amount of drag reduction in the uñdersaturated domain is a function of polymer type, concentration, wall shear, solvent temperature, and boundary-layer thickness. Since the three drag-redùction domams exhibit different characteristics, it is of fundamental importance that the particular domain be clearly dèfined in each flow' tuàtioñ.

Many potential applications foi polymer drag reductionsuch as in fire hoses, pipelines, and laige ex-terrial boundary layer flowsare in the undersaturated or optimal domain. In these cäs'ès the boundary layer is usually thick, and the wall shear stress is rather high. There are piesently very few experiments in th

22Wu, J. and M. P. Tulin, "Piag Re4ucti'on by Ejecting Additive Solutions irito Pure-Water Boundary Layer," American Society of Mechanical Engineers Gas Turbine and Fluid Engineering Conference San Francisco Calif Paper

72-FE-12 (1972).

23Vu P. S.et al., "The. tJltimate Asymptote and Mean Flow Sfructure in Toms Phenomenon," American Society of Mechanical Engineers Journal of Applied Mechañics, VoL 37, pp. 488-493 (1970).

24Vffk, P. S., "An Elastic Súblayer Model for Drag ReductiOn by Dilute Solutions of Linear Macromolecules," Journal of

Fluid Mechanics Vol 45 Part 3 pp 417-440 (1970)

range of practical interest. In all cases, special caution must be paid to polymer experiments because the measured drag reduction can vary from batch to batch and can be affected by the methods used for mixing and transferring the polymer solutions It is necessary to use strictly standardized procedures when takmg measûrements. in carefully coñtrolled experiments.

The primary objective of this study is to characterize dilute polymer solutions within the framework of the fourlayer, velocity-similarity laws for turbulent. boundary layers. The drag-reduction domains will be

defined The particular validity of these laws in the undersaturated domain at high shear stress and with thick boundary layers will bè emphasized.

REVIEW OF VELOCITY SIMILARITY LAWS FOR ORDINARY TURBULENT BOUNDARY LAYERS

For ordinary Newtonian fluids, it is well established in the literaturô2528 that thç velocity distri-bution within a two-dimensional turbulent boundary layer with zero pressure gradient has the òllowiiig empirically based similarity properties: a viscous sublayer extends from the wall to a small distance frOm the wall a law of the wall applies from the edge of the viscous sublayer to about 15 percent of the boundary layer thickness, and a velocity-defect làw applies to the outer region of a boundary layer with smooth and

rough wälls.

:För. a thick boundary layer, it is usually assumed that the viscous sublayer (ü7u7 = U7 y/y) applies

from the wall to u7 y/y = 10.8. Thé law of the wall then dominates fron that point to approximtel'y y/5 0.1.5 without introducing significant error by neglecting the "buffer" zone. According to experimental

data and dimensiOnal analysis, the law of the wall assumes the following functional relation

2.3026

fu7y\

fuTy\

log (-J +B=.A log (i-FR

u.

K

\.P

/

\v /

where ii is the mean velocity at distance y from the wail,

u,1. is the friction velocity (u7 = where r,is the wall shear stress and p is the mass density

of the fluid)

(1)

Sch1ibhting, H. "Boundary-Layer Theory," Sixth.Edition,McGraw-Hill BookCornpany, New York (1968). 26Hinze, J. O., "Turbulence," McGraw-Hil Book Company,, New York (1959).

27"Proceedings Computation 'of Turbulent Boundary-Layers l98 AFÖSR-IFPStañfórd Conference," Edited by S.J.

Kline et al., Stanford University (1969). ,

K is the Von Krmn constant,

y is the kinematic viscosity,

& is the boundary-layer thickness, and the common logarithm to the base 10 is used.

The law of the wall holds for both internal pipe flows and external or flat-plate boundary layers close to the wall. The "best universal" value for K is 0.41 and for B is 5.

In the outer layer, the velocity defect derived from observation follows the universal form of

U0 - ii

_2.3026

log ()

2

[i

+

(

Y

____

___

-

+-UT K K &1J

/

= 9.6 fi -

L

,flow over flat plate,y

0.i5 6

UT

\

6/

where U0 is the free-stream, outer flow velocity, and Z is Cole's29 wake parameter, which is a constant for equilibrium flow, and the last bracket of this equation is Hinze's26 approximate wake function. Laws (1) and (2) overlap near the wall (y!& <0.15); so that to a good approximation, we obtain

2.3026

/u&\

2&2

log (I+B+

-u1 K K

A large collection of outer layer profiles is shown in Figure 1. External boundary-layer data with zero pressure gradient are taken from Kebanoff and Diehl,30 Freeman,3' and Schultz-Grunow.32 Internal pipe-flow data taken from Nikuradse,33 Laufer,34 Seyer and Metzner,17 and Wetzel and Ripken.19 Equation (2) with K = 0.41, and 2 = 0.5, fits the data of the external boundary-layer flow very well, and

the same equation with the same K but different 7, 0.20, also fits the data of the internal flow well. The

29Coles, D., "The Law of the Wake in the Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 1, PP. 191-226 (1956).

30Klebanoff, P. S. and Z. W. Diehi, "Some Features of Artificially Thickened Fully Developed Turbulent Boundary Layers with Zero-Pressure Gradient," National Advisory Committee for Aeronautics Report 1110 (1952).

31Freeman, H. B., "For Measurements on a 1/40-Scale Model of the U.S. Airship AKRON," NationalAdvisory Com-mittee for Aeronautics Report 432 (1932).

32Schultz-Grunôw, F., "Neues Widerstandsgestetz fur glatte Platten," Luftfahrtforschung, Vol. 17, No. 239 (1940); also National Advisory Committee for Aeronautics Technical Memorandum 986 (1941).

33Nikuradse, J., "(esetezmassigkeiten der turbulenten Stromung in glatten Rohren," VDI-Forschungsh, 356 (1932). 34Laufer, J., "The Structure of Turbulence in Fully Developed Pipe Flow," National Bureau of Standards Report 1974 (1952); also, National Advisory Committee for Aeronautics Report 1174 (1954)..

5

or

Uoü.

only difference between the internal and external boundary-layer profile is the value chosen for the wake parameter in Equation (2). The empirical formulas of Hama,35 Equation (2a), are also shown in Figure 1

and appear to fit the data best at the outer edge of the boundary layer y/8 0.6.

The rest of this paper will discuss how the ordinary velocity similarity laws are affected by the polymer solutions and how these laws can be modified to characterize drag reduction. As in the development of ordinary similarity laws, the modified laws for polymer solutions are established according to the

experi-mental evidence rather than theoretical analysis.

VELOCITY SIMILARITY LAWS FOR POLYMER BOUNDARY-LAYER FLOWS

The mean velocity profiles of internal flows measured by Seyer and Metzner17 on l-in, smooth pipe, Tsai18 on a 6- by 15-in, rectangular duct, and Wetzel and Ripken19 on 4-in, rough pipe are plotted in

Figure 2 in the form of the law of the wall, and in Figure 3 in the form of the velocity-defect law. Similarly, the data of the external boundary layer obtained by Wetzel and Ripken'9 in an open channel are shown in Figures 4 and 5. This channel flow is not two-dimensional and the floor is hydraulically rough. On the basis of these data, the nondimensional velocity profile for a turbulent boundary layer with drag-reducing polymer solutions can be divided into four layers expressed as follows

Viscous Sublayer (0

_{y <yg )}

U UT Y

U+ E - =

(4)

UT P

Strongly Interactive Layer (vg Y _YS)

UAlogy+B

(5)

where and are determined by the best fit of data in the strongly interactive layer shown in Figure 2, i.e., A' = 30, and

= 20.2.

Weakly Interactive Layer y A &)

u = A logy + B + L7

(6)

where X = 0.15 for flow over flat plate, and A i for pipe flow,

35Hama, F.R., "Boundary-Layer Characteristics for Smooth and Rough Surfaces," Transactions of Society of Naval Architects and Marine Engineers, Vol. 62, Pp. 333 (1954).

or

4. Outer Wake Región

Uoii

i\

_r

Alog IJ + I l+cosir

u.

\6/

_KL

U0ii

_/

9.6 (i

_-

_-r--)

,flowoverflatplate,y0.l56

U.

_\

_/

where all quantities are as previously defined, and p and z.' are the mass density and kinematic viscosity of the solvent. The two constants A and B are taken from data for Newtoniän fluids, i.e., A = 5.62 and B = 5. A three-layer model for pipe flow with polymer solutions was first proposed by Van Driest9 and later by Virk et al.23 and Virk.24 In their models the outer wake region was neglected.

At the intersections of the layers, the velocity must be continuous, añd the thickness of the viscoUs sublayer is assumed not to be affected by polymer solution Thus evaluation of Equations (1), (4), and

(5) at y = y yields

Aiogy +y =Älogy +B

or

B=-2A)logy

Evaluation of Equations (5) and (6) at y y and use of Equation (8) yield

= (At' A) log ys

, or, ---- = 10

Ys

1-A

Yg Yg

For flow without drag reduction, y = ,

A A,= B, and

0.

As shown in Figures 2 and 4 the law of the wall has two layers: a weakly interactive layer, specified by Equation (6) and characterized by a shift of ', and a strongly interactive layer, specified by

Equation (5) and characterized by a smaller Von Karman constant. The strongly interactive layer is absent for the ordinary Newtonian flUids. The value of is 'positive fôr drag-reducing polymer flows in contrast with flows with roughness whiôh shows negative It also can be seen (internal flOw) in Figure 3 and (external flow) in Figure 5 that the velocity-defèct law in the form Of EquaÚon (7) is valid for polymer solutions and roughness for the entire range of y

y

6. Since the large-scale mixing processes are con-trolled mainly by inertia rather than by viscosity, the velocity defect depends only upon wall shear and is independent of how the wall shear arises. Thus, (U0

-

uT)/u.1. versus y/5 is universal for _' where y

(:.)]

y.y6

₍₇₎

can be obtained from Equation (9). It is to be noted from Figures 3 and 5 that the velocity defect assumes

a different slope, i.e., (U - ii)/u = -

log (y!6) + constant, forYg i'

The local frictional coefficient c7, = r/(p UO2 /2) can be calculated by evaluating Equations (6) and

(7)aty

y, giving

r

/

iu6\

1/-

=B+tB+ - Ii+cos(ir

-Il

+Alog

(-I

(10)

KL

/

whereK 0.41,

= 20, and

6 = D/2 for fully developed pipe flowD is the diameter.

For external flows the value of 2 = 0 5, and the boundary layer thickness increases with distance along a

streamline To compute c'., 6, and the momentum thickness O the equation for boundary layer momentum

is used with the velocity profiles specified in Equations (4) through (7). The integration of the two-dimensional equation for boundary layer momentum gives

dO

(u

dx

if the iiorñ-ial Reynolds stress and streamwise variation of direct stress are neglected.

The computation procedures are well developed for ordinary twodimensiona1 turbulent boundary layers with zero pressure gradient, e.g., Landweber36 arid Coles.37 These methods have been extended by Granv'111e38 and by McCarthy39 to apply to the computation of flat-plate, frictional-drag reductioll with

polymer additives Both require functional information about ¿' however, both neglect the presence of

the strongly interactive layer. A series of controlled pipe-flow experiments have been performed to in-vestigate the characteristics of and, consequently, the thickness of the strongly interactivê layer (Equation (9)). These experiments supply the necessary information .requirçd to characterize completely the modified sund.ty laws for boundary-layer flOws of polymer solutions.

36Landweber L The Fnctional Resistance of Flat Plates m Zero Pressure Gradient Transactions of Society of Naval Architects and. Marihe EÌiiieers, Vol. 61, pp. 5-32 (1953).

37Coles, D, "The Problem of the Tûrbulent Boundary Layer," Z. Angew. Math., Physics 5, pp. 181-203 (1954). 38Granville, P S Fnctional Resistance and Velocity Sinulanty Laws of Drag Reducing Dilute Polymer Solutions Journàl of Ship Researth, Vol. 12, No. 13 (1968).

39McCarthy, J. H., "Flat-Plate Frictional-Drag. Reduction with Polymer Injection," Journal of Ship Resçarch, Vol. 15, No. 4 (1971).

9

POLYMER TYPES AND SOLUTION PROPERTIES

Four commercially available polymer typcs were used in this study. Three synthetic polymers used

were:

POLYOX WSR-301, blend 8259 W, a polyethylene oxide polymer of Union Carbide Co.,

SEPARAN AP-30, a polyacrylarnide copolymer of the Dow Chemical Co., and

MAGNIFLOC 835A, an anionic charged polyacrylamide polymer of American Cynamid Co. The one natural polymer investigated was guar gum J2M of the Western Co.

The viscoSity of the polymer sOlutions was measured directly at 25 C by a Well-Brookfjejd_Cone-Plate

MicrO Viscometer.1' _{In this apparatus the relationship letween shear stress and strain rate at varióus}

con-centratiOns was measured by a torque-spring meter at various cone angular cone speeds. The intrinsic vis-cosity of a polymer solution is defined as

i)

PS (i')

[tJlim

-C

c-øO where is the polymer solution viscosity,

is the solvent Viscosity at the same temperature,

e is polymer concentration in grams per deciliter (g/dl), and is strain rate ¡n 1/sec.

The measured intrinsic viscosity is 17 dl/g for the present POLYOX WSR-301 solution,U and it is 12 dug for the guar gum J2M solutions. Both solutions behaved like Newtonian fluids for concentrations less than 500 ppm. Within this range the polymer viscosity for POLYOX and guar guam can be approximated

by

[]c+0.4 []2 c2

(12)

Significant non-Newtonian behavior or shear thinning wäs noticed for MAGNIFLOC 835A and SEPARAN

AP 30 solutions, although it appeared that both solutions may have had a Newtonian range at sufficiently high shear rates. The viscosity measured varied appreciably With the angle of cone and Volume of solution used. The intrinsic viscosity for MAGNIFLOC 835A was about 40 dl/g. The intrinsic Viscosity for SEPARAN AP-30 solutions varied between 100 and 200 dl/g; this large variation was due to its highly non-Newtonian behavior and sensitivity to cone angle used. The weight-average molecular weights quoted by the manufacturers were

5 x 106 for POLYOX WSR-301, (2-3) x 106 for SEPARAN AP-30,

16 x 106 for MAGNIFLOC 835A, and (0.5-2) x 106 for guar gum J2M.

PIPE-FLOW EXPERIMENTS TO DETERMINE &

Since is the basic hydrodynamic characteristic of diluted drag-reducing polymers in turbulent

boundary layers, the experimental technique used to obtain information about this factor is of fundamental importance. Theoretical prediction of is not possible at present. Thus a simple pipe-flow experiment.

has been selected to determine zi The concept of using to characterize polymer drag reduction was first proposed by Meyer,40 which was analogous to that used by Hama35 for roughness.

By dimensional analysis the for dilute polymer solutions depends upon the following groups of variables; see also Granville38

/ g 2

(Ti

T

Ti

= g ₍

, - ,

, e, polymer type, temperature,

P P

roughness, mechanical, chemical, thermal, and other

degradation)

where is characteristic length scale(s) of the polymer solution, and t. is characteristic time scale(s) of the

solution. The relevant £ and t are not specified in this study, except in a relatively arbitrary way; their

determination is a subject for further study. Relaxation time and radius of gyration have been proposed

for t and

In pipe flow one may define a readily measured mean-flow pipe velocity V as

1__

±.

Q

-

irD

-

-d (1

-U

T

where Q is the volume flow rate, and = y/(D/2). Using the velocity profiles specified in Equations (4) through (7) with 2 = 0.20, the integration of Equation (14) yields

40Meyer, W. A., "A Correlation of the Friction Characteristics for Turbulent Flow of Dilute Non-Newtonian Fluids in Pipes," American Journal of Chemical Engineering Journal, Vol. 12, No. 3, PP. 522-525 (1966).

VP

- = A log

u7 2V u7D r- i

2Q I

1 1 ¡

+ - I - - - --

-K

_L2

2

\

2

A parameter may be further defined

D

¿V17+Alog

_L D5 2.3026 VP _Vs U7 U7 at constant u1Jv

where the subscripts p and s represent, respectively, the polymer-solvent system and solvent alone.

Accord-mg to Equations (15) and (16), V may be written as

(ZA)

11

IA_\

2 i

I

(ZA)

\ 2.3026/

2.3026

I

2Z

f

Uryg

f

i

-

--(

2.3026\e

4

/

p

\

3

For flow without drag reduction, Z = A, i = 0, and y = , so that Equation (15) reduces to

u7D ₃

A log

+B

-2 p 2 +

__1_ ('e

K L2 2 +

- ?-

-cos ir

_-

- __

-cos rs +

(17)

/ A

\

2Z

/

\

u7Yg I

-(16) I J+

\2.3026/

2.3026 I g

- -)

/

-y

\ £

-COSirg + ir 2 _ir2 - sin ir I I

-77.2 cos ir + ir sin (15) vs UT

where the difference between the viscous sublayer thickness of the solvent and the polymer-solvent system

is assumed to be negligibly small. If one uses D = D5 = D, and uyg Iv = 10.8, then the difference be-tween the measured V and is

V

2

XA

t cs

iB

2

1BKL\

cg 2 _ir2

L)]

cos ir E

(E

2

-i

-

1

- Eg

(18) +

sin7rE

_sinirE

ir ir

A A

where Eg = lO.8/(uD/2v), and = 10 from Equation (9).

The solution of Equation (18) is shown in Figure 6, which provides the correction term for the desired from the measured V. The difference between the desired ¿I and the measured V decreases with increasing uD/2 i'. Thus, for a given range of u, the V measured in the larger diameter pipe is a close approximation to í.. It also should be noted that the first term of the right hand side of Equation (18) is much larger than the second term. As an example, if one plans to measure a value of to 20 at

1000 dyn/cm2, a pipe diameter of 1.7 cm will give an error equal to or less than 5 percent (Figure 6). A flow facility that would give minimum mechanical and chemical degradation was designed to

measure for drag-reducing polymer solutions at high wall shear stress. Four 10-m-long test pipes-0.385-,

1.918-, 3.181-, and 5.080-cm IDwere used. The inside surfaces of the pipes were carefully polished. A large holding tank containing 800 liters was connected to the test pipes, which were oriented vertically. The flow rate through each pipe was regulated by varying the air pressure in the holding tank. Testing was done by forcing the polymer solutions through the instrumented length of test pipe into a weighing barrel. The solution was then discarded to avoid mechanical degradation. The controlling ball values were located at the downstream ends of the pipe to eliminate solution degradation. Abrupt inlets were first used to encourage transition from laminar to turbulent flow soon after the inlet. However, data in Figures 7a and

7b show that abrupt inlets cause significant degradation, especially in the 0.385-cm pipe; see Figure 7a. Thus, bell-mouth pipe entrances were installed on the four pipes tested. The wall shear stress could be computed from the pressure drop along the pipe. To allow flow to become fully established, the first pressure tap was located 120 diam from the pipe entrance. The pressure drops along intervals of three

lengths of the pipes were measured with Dynasco differential pressure gages. Signals from the gages were averaged electronically for 5 sec and were then displayed by digital _{voltmeters The pressure drops}

presented m the figures are the average values of the measurements of 2 or 3 successive length mtervals, and the discrepancies between the readings are within 2 percent. This indicates that fully developed 'turbulent flow was established, and that no serious degradation occurred along the pipes. High quality control was maintained by carefully mixing and trathfèrring the tested polymer solutions. Except for guar gum, the dry pOlymer powders were dissolved in distilled water at a ratio of 2000 ppm, i day before each experiment, and then were diluted by well wateì to the desired concentration before each test. The guar gum polymer was mixed directly into the well water at the desired conceñtration 10 min before the test because the

so-lution was found to degrade after I day of storage. The _{same batch of polymer was used for each polymer}

type. The experimental setup and the standard procedures were designed to minimize th mechanical and

chemical degradation of the solutions.

Typical gross flow, drag-reduction data of V/u7 versus u./v are plotted in Figure 8; shear stress has

been calculated from the formula r, =(D/4) (dp/d.x)where dp/dx is the measured pressure gradient along the pipe, and V is the measured average velocity defined in Equatioii (14). The reason for not uilñg a 'mdre complete nondimensional parameter u. _{Li/v as shówn in Equation (13) is that the appropriate}

character-istics length scale L for the polymer solutions is not known a priori. _{Nevertheless, a suitable L may be}

inserted in the figure without difficulty whenever it is found. The subsequent data shown in Figures 9 through 13 are the dragreduction characteristics of the polymer solutions investigated, plotted as versus

u7/v. The quantity

V

_{was Obtained froin"Euätion (16), making use of data such as those shown ¡ii}

Figure 8. Then.'wasobtaired.froïn the méastired

_{V, using Équation (18). The values of('}

V)/i

for most data are lss than 5 'percent '(Figure 6). Data' for which

_{- V)/' is larger}

than 10' percent are not included. 'Ïn reducing data, the Von Krrñn-Ptändtl resiStance equation forthe

smooth pipe is sed for the solvent rather than Equation (15), i.e., ' '

u7D

= 5.657 log - + 0.292

(19)

SATURATED DRAGREDUCTION LINE

As shown in Figure 7 and Figures 10 through 13, the drag reduction for the four polymersolutions

increases with concentration to a saturation limit above which further increases of polymer cOncentration

produce no 'further reductions. This phenomenon was' first reported by Fábular6 and Hoyt and Fabula,41

4'Hoyt, J. W. and A.G. Fabula, "'theEffect of Additives on'Flüid Friction," Fifth Symposium of Naval Hydrodynamics, Edited by J. K. Lunde' and S. W. Doroff, Offl of Naval Research, Department of the 'Navy, ACR-12, pp. 947-974

and later by Virk et aL"2 and Virk.24 We assume that for the case of saturated drag reduction, the

strongly interactive layer dominates the entire pipe, the weakly interactive layer is absent, and the outer wake region is negligible. Then, performing the integration in Equation (14) gives

fv\

log

(±.)

= 2v 2 2.3026

3Olog (-1 48.8

(20) i,

The Darcy-Weibach friction factor can then be written as

/3

1

B I

+lo

/

_{i \4.6052 2}

g32)

- - log (R

-= 10.6 log (Rv7) - 22

(21)

where R = Dy/v is the Reynolds number f = 8 r/(p V2) = 8u2/V2, and the subscript i represents the

values for saturated drag reduction. Equation (20) agrees well with the data shown in Figure 7a and other data collected by Huang and Santelli11 with 2 and taken from the velocity profile shown in Figure 2, i.e., A = 30, and = 20.2. Thus, the assumption is valid that the strongly interactive layer dominates the entire pipe for saturated drag reduction in pipe flow. Similarly, at the saturated drag-reduction con-dition, the in Equations (8), (16), and (20) becomes

Du

(

V)

= (

- A) log

(A - A)

₊ 4.6052 s y

/Dur\

= 24.25 log I- 1

48.9 V where y = 10.8, and y = D/2. 3 + log 2) (22)

DISCUSSION OF MEASURED

The functional relationships between the measured _{and its nondimensional group of variables in} Equation (13) are discussed in the following paragraphs, and useful formulas for Ai are presented. The in-fluence of roughness and mechanical and chemical degradation are not included.

Effect of Temperature and Thermal Degradation: the effect of temperature on drag reduction is embedded in the kinematic viscosity of the solvent for POLYOX solutions from 40 to 75 F (Figure 10), for SEPARAN solutions from 40 to 100 F (Figure 11), and for guai gum solutions from 70 to 100 F (Figure 13). The data in Figure 9 show that high temperatures of 85 and 100 F _{cause significant thermal} degradation of i7 for POLYOX solutions.

Effect of Pipe Diameter or Boundary-Layer Thickness: the measured for POLYOX solutions at small concentration of < 33 ppm is independent of pipe diameter if the diameter is larger than 1.918 cm (Figure 10); however, A _{depends upon pipe diameter if the diameter is less than 0.385 cm; see Figure 7a} in which drag reduction is saturated for cases with a bell-mouth entrance. The measured for SEPARAN

and MAGNIFLOC solutions at < 50 ppm is independent of the pipe diameters tested (Figures 11 and 12).

This also is true for guai gum solutions of < 500 ppm as shown in Figure 13. Since the measured iiis independent of pipe diameter in the undersaturated drag-reduction domain, the measured values of are

expected to be universal for both internal and external boundary layers.

Effect of Polymer Concentration: in the undersaturated drag-reduction domain, the measured

is proportional to '/at a given value of ui/v for the four polymers tested (Figures 10 through 13). This

is also in agreement with other guai gum data summarized by Virk24 and Poreh and Mioh.42

Effect of ui/v and Polymer Types: the measured _{increases with increasing ui/v, provided} u7/v exceeds the onset value for the drag-reduction effect; levels off at a maximum value; and, finally, drops off for all polymers tested. At all the concentrations tested, the curves of for the three synthetic

polymers attain their maximum values at the same ui/v = 2 x 10 cm. However, the value of u./v for

the onset of drag reduction appears to decrease with increasing concentration; see Figure 7b. For the range of concentrations investigated, the value of drops from its maximum value as

UT/v (ui/v) at ma [log

(uTl)]2

(0.5 2.5;

= 5x i0

cm;

u. L

c5O

ppm) (23) max [108

42Poreh, M. and T. Mioh, "Rotation of a Disk in Dilute Polymer Solutions," Journal of Hydronautics, VoL 5, No. 2, Figure 4, p. 64 (1971).

15

where is the maximum value of Aif at ui/P=

2 x l0 (cm'), and o, b, and s in parts per

million are constant, depending only upon the polymer types. The length scale is defined as L = (1/

(u7/i.') at maximum

Aif)

= 5 x iO cm for the three synthetic polymers. The values of a, b, and s are ob-tained from the best fit of data and are shown in the formula of Figures 10 through 12 and Table 1. The empirical formula of Equation (23) is valid ¡n the range of experimental data, i.e., 0.5

u.

Li/u 2.5.

Equation (23) is significantly different from the basic results of Virk.24

TABLE i - CHARACTERISTIC VALUES OF CONSTANTS IN EQUATION (23)

*

UT L1

-

b2

= 10 p

which depends upon concentration (Figures 7 and 8) and is consistent with the results of Paterson and Abernathy10 but is contradictory to the Virk results.24

The measured

if

for guar gum solutions exhibits different characteristics:

¿if

increases linearly

with log [(u/v)/(u/v) onset' from the onset of drag reduction (u1./v = 7.3 x

102 cm) to

UT/P

= 4 x i & cm, and then drops off (Figure 13). The dragreduction at various concentrations has a common onset point atUT/P= 7.3 x 102 cm1. So, in the case of guar gum solutions, a length scale is

defined corresponding to the onset point of drag reduction, i.e., L2 =

(l/(U/v)

_onset = 1.37 x i0 cm.

According to data shown in Figure 13, the measured

Aif

for the guai gum solution can be approximated

by

Il

5.5; £2 = 1.37 x l0 cm; c

500 PPm)(24)

if=

l.11og

(U7 L2

/

u L2

i) _f

\

p

A similar result, i.e.,

Aif

,_.c log (u7 L 2/v), was first observed by Meyer,40 which is valid only for a poor

drag reducer. a2

_(cs)

Polymer types a b s POLYOX WSR-301 5.7 25 0 SEPARAN AP.30 3.0 16 2.5 MAGNIFLOC 835A 4.0 19 1.5

Once ,A7is known, the thickness of the strongly interactive layer can be calcUlated from Equation (9). As shown in Figures 10 through I 3, the POLYOX solutiOns tested are superior for drag reduction, compared to the other three polymers at corresponding concentrations. MAGNIFLOC solutiôhs are slightly better drag reducers than the SEPARAN sOlutions. The guai gum solutions are. the least effective of all. No conclusive evidence exists to correlate the length scales of the polymer solutions in the high shear field with any rheological length scales of polymer solutions such as radius of gyration or uncoiled

length.

5. Effect of u2 t/P: a similar proce4ure can also be made to obtain t from from a plot of

versus u72!'. For the polymer solutions tested, the data do not correlate as well as using versus_UT/v.

Since we have found that

¿7

depends upon u g _where g _{is a constant independent of}

concen-tration and u7,it can be seen thatt. = £ 1/u7. The use of time scale t.is not as fruitful because t would

have to vary with ut..

DRAG-REDUCTION DOMAINS

As shown in Figure 7ä, the drag reduction may change from the under to the over-saturated domains when the. solution. concentration is increased with u. Dlv kept constant. Similarly, the drag reduction may shift from the over to the under-saturated domain when u,. Div is increased, and the concentration is

main-tained at a constant vaine. For a given ù Div, an opt mal concentratioñ may then be defined as that valüe of c at which the maximum drag reduction occurs. Oversaturation exists when c> _cm, and under

saturation exists when c> . At the saturated drag reduction ( V) is given by Equation (22). Since

in the presentexperiments the ratio (&- V)/< 0.05, it may be approxirnatedby

Thus the. optimal concentratIon for the three synthetic polymers can be obtained by substituting Equation (23) into Equation (22), i.e.,

(Du,.\

_r

24.3 log (

_\2v/

I

.41.6+b Ilog

L V

Cm=

with g ₌

_{x l0 cm. Figure 14 shows}

_{versusDu,./2v at U7 £1/v = I for the three synthetic}

polymers -tested. A few experimental data points, e.g., from Figure 7a, are also presented. Good agreement is noteth The lines shown in Figure 14 represent the optimal drag-reduction domains for each polymer. The oversaturted drag-reduction characteristics are independent of concentration, and as shown before, te strongly interactive layer dominates the entire boundary layer. The undersaturated drag-reduction domain is tò.the right óf the line, and this is where most of the results of the present study apply. lt is

interesting to note that at given values of the concentratiOn and u,./v, the drag reduction may be

17

oversaturated in a small pipe and be undersaturated for a large pipe. Thus, for a given flow situation, it is important to determine the drag-reduction domain before the similarity laws deduced here can be properly

applied.

The optimal concentration for guar gum solutions may also be estimated, i.e., 24.3 log Cm = 1.1 log

uD

- 41.6

₂ UT g2 V

COMMENT ON THE STRONGLY INTERACTIVE LAYER

The strongly interactive layer shown in Figure 2 represents a tentative and approximate meanvelocity

profile in the buffer zone of a turbulent boundary layer with drag reduction since the data are taken from two experiments only. When the strongly interactive layer extends to the pipe axis, the approximate velocity profile does give accurate saturate drag.reduction line (Equation 21) which is commonly observed. As shown in Figure 2, the strongly interactive layer is between the laminar sublayer and the common logarithmic pro-file. Thus, the strongly interactive layer may be the by-product of the velocity intermittency between the laminar flow and the common wall turbulence. Because most of the data reported in the work are taken when the strongly interactive layer is small compared with the pipe radius (Figure 6), the slight uncertainty of the strongly interactive layer thickness and profile has little effect on the similarity laws discussed in this paper. However, we believe that the strongly interactive layer can only be satisfactorily determined after more accurate velocity profile data and more reliable turbulence measurements in this critical zone become

available.

CONCLUSION

Drag reductions caused by dilute POLYOX WSR.30l, SEPARAN AP-30, MAGNIFLOC 835A, and guar gum J2M solutions was measured in four different smooth pipes, having ID's of 0.385, 1.918, 3.181,

and 5.08-cm, over a range of high wall shear stresses. Certain conclusions can be drawn for internal flow of homogeneous polymer solutions. Nevertheless, the results in the form of velocity similarity laws are ex-pected to be valid for external boundary layers since universal forms of the drag-reduction function have been deduced.

The results of the present study are consistent with the interactive layer, mean-velocityprofile

ob-served during drag reduction and measured by Seyer and Metzner19 and Tsai18 about internal flow and by Wetzel and Ripken14 about external flow. The four layers consist of

A viscous sublayer,

A strongly interactive layer, characterized by a small Von Ka'rrnn constant,

A wealdy interactive layer, characterized by a parallel upward shift by of the semilogarithniically (26)

r

u7

¿i

b [loe

_j

0.5

4. An outer wake region where the effect of polymer is fúnctióñally unimportant. The drag reduction can accordingly be divided into three domains:

1. Oversatúrated, inwhich the strongly interactive layér dominates the entire boundary layer 'and fOr which drag reduction is independent of concéntration. I

2. Undersaturated, in which all four layers are present and for which the Ai increases with increasing concentration, and

3. Optimal, defined as the boundary between domains (1) and (2).

Drag reduction in terms of a friction factor for domains (1) and (3) can be specified by assuming 'that the strongly interactive layer dominates the entire boundary layer. The derived line is in good

agree-ment with data.

Drag reduction in terms of

A7

in the undersaturated domain was investigated by a series of smooth pipe-flow experiments. The main conclusions are

increases with the square root of concentration fOr the four polymer solutions tested. Ai for the three synthetic polymers tested can 'be approximated by

1.1 s/Flog

V

19

i,

where a and b are conStants depending on polymer type, c and s are concentrations 'in parts per million, and ,

g _{the length scale is taken as the value of l/(u7/v) at maximum which is 5 x i0 cm}

for the three synthetic polymers tested. . '

3. The Ai for the natural polymer or .guar gum solutions can be approximated by

5.5; c 500 ppm

where g2 is taken to be the value of' l/(u7/v) at the onset of drag reduction, which is found to be

g

2 = 1.37 x iO3 cm.

To reasonable aèc'uracy, the effect of temperaturé on the functional form of

7

can be ábsorbed entirely in the kinematic viscosity of the solvent for most of the temperature range tested. This is not true for POLYOX at, 85 F because significant thermal degradation appears to' occur. It has beén found that

ari abrupt pipe entrance causes significant mechanical degradation of the solùtion and Should not be used. The pipe diameter is only important in determinin.g the proper dragreduction domain. The value

of is independent of pipe diameter in the undersaturated drag-reduction domain.

2.5; c 50 ppm.

The author is indebted to Mr. J.H. McCarthy for many valuable and stimulating discussions during the course of this work. The author would also like to thank Messrs. N. Santelli and G.S. Belt for their assistance during the experiment and Dr. Wetzel of the University of Minnesota, who furnished the tabu. lated data shown in Figures 4 and 5.

ACKNOWLEDGMENTS

y, 6

Figure 1 _{- Outer Layer Velocity Profiles for Ordinary External and Internal Flows}

O £ FLATPLATE R5 4.8 o 10' 1.5x 106 1.0 z 10 2.7 iO 3.8 z 10' 7.1

1'

I -KLEBANOFF KLEBANOFF FREEMAN31 SCHULTZ-GRUN0W I - 0.4 I SCHULTZ.GRUNOW32 SCHULTZ.GRUNOW32 & & fl I DIEHL3° 0.5) DIEHL30..._... I 0

.

A O 0 0 0

2!

(K-0.4fl-0j UT

III

Uil

èO.15) PIPE FLOW

L

j

VO

R..

t, 3.2x 10'NIKURADSE 5.. 105 LAUFER 1.60 1O5SEYER 3z 105 WETZEL 6*io WETZEL J 9.2 z i0 NIKURADSE33 & METZNER'7 & RIPKIN1° & RIPKIN19

J.

I u.. -5. z log ..1

-t-02 0.4 0.6 0.8 10 12 11 10 9 8 U D 6 5 4 3 2 o o

D 40 35 30 35 16 .10 6 ii 30 r 10 SOLVENT: WATER u 30 log y - 20.2 WETZELAND RIPKEN19 4" ROUGH PIPE POLYOX WSR'301 PPM R,, 9.4 6x105 o 6x10 2

3'

4 - YUç log y Dg -;;-o 0.01. 002 0.04 0.06 0.99 01 y,6 21

SEYER AND METZÑER'7

1 SMOOTH PIPE HYDROLYZED POLYÄCRYI.AMIDE (ET-597) PPM

R,,

-ORDINARY FLUID l'SAI'8 6" r 15" RECTANGULAR DUCT POLYOX WSR.301 PPM 5

Figure 2 - Experimental Evidence of Interactive Layers in Internal Polymer Flows

02 04 06 08 1.0

Figure 3 - 'Effect of Polymer Solutions on Velocity-Defect Law, Pipe Flow.

-n - !9 POLYMER 1000-SEVER 2.0, O ..- SEVER O - WETZEL -WETEL - WETZEL

()

& & & & METZNER1 C METZNER1 & METZNER' RIPKEN'9 RIPKEN'9 RIPKEN'9 -- iO4 £314 r

.01.6 r 10

WATER 6 r 106 8 r 10 6x O _{,o WATER}

I!i

I II

iI4

111111

UIIII. II

iÌi WI!W iiuiuiii

.&62Ig(!)+%0[1+r(w')]

H

- '

:.uuIuI

o 35 40 A o

o

50 -o 4 1000 '134x104 4 1000 314x104 O 16x106

-lo 10 0 5 6 B 102 30 28 26 24 6 4 2

f

u 30IÖgy -20.2

.iï

Q + 5.0 LW E WAT AT ER ER C (POLVOX WSR 301) 223ppm l23ppm

Figure 4 - Effect o Polymer Sölutions on the Law of the Wall, Open Chañnel Flow

iL O PURE WATER A PURE WATER OPURE WATER - 106 C(POLYOX WSR301) Bppn L 24 ppn, 223 l23ppn,

o

2- 6 8 001 2 4 6 8 0.1 2 8 8 1:ò y11 o PUR

o

PUR 8 10 2 8 10

45

40 2I 30 20 30 18 , 18 14 12 10

uD

Figure 6 - Solutions of Equation (18) m Terms of versus

urD

for vanous Values of

-2v - - . 23 0% 10 0 -102 1

70 00 50 30 20 10 o 70 00 50 40 20 10 Io 6 8 10 -t

(an)

1iguxe la Pipe Diameter ¡s 0.385 Centimeter, and Temperature is 75 Degrees

Fahrenheit-0

103 . fi

UT

)1)

Figure lb - Pipe Diameter is 1.918 Centimeters, and Temperature is 75 Degrees Fahrènheit

Figure 7 - Effect of Pipe Entrance on Drag Reduction

4 s i CONCENTRATION . i ABRUPT ENTRANCE. 0 PIPE .. PIPE

. H

BELLMOUTH ENTRANCE ("a) - (UTD) -5

Saturated Drag Reduction Line

10 .A

£

20 0 50

0

c(pp,n( ix

o

a

(-r) + )Smoottt Pipe)

44!t

_I

-. - 5.657 log i OPEN POINTS SÓLID POINTS . .- I ABRUPT BELLMÖUTH PIPE i ENTRANCE PIPE.ENTRANCE -(opi"!

(Saturated Dreg Reductmn

(!).30Io8.8

Lu.e)

.ir -r-.

20

--

IEP

_{uiul..--4:; -t1}

-2.6

5.657 log * 0.292 (SMOOTH PIPE)

4 6

40 10 30 28 20 > i', im

I

15 10 5 o 102 o 30 log 9 -48.8

- 5.57 log () + 0.282 (Smooth P(pe)

10 5 20 1.0 o I 33 25

Figure 8 - Typicäl Results of V/ui versus ui/v for POLYOX WSR-301 Solütions, Pipe Diameter is 3.181 Centimeters, and Temperature is 75 Degrees Fahrenheit

PIPE ID. -3.181 cm TEMPERATURE o p ooF 70F 75F 85F 100_F 4 8 8 103 u?

- (1)

Figure 9 - Temperature Effect on Drag Reduction of POLYOX WSR-30 1 at 20 Parts per Million

8 iO3

(cnr1(

35 30 35 10 s 0

FIgure 10 Drag-Reduction Characteristics, versus urli', for POLYOX WSR.301 So1utions

35 30 35 20 t5 102 10 5 o 102 2 4 6 8 10 2 4 i' n nr cm') OPENPOIN1S $01.10 POINT HAI.FF1.LED POINT V

PIPE ID., TEMPERATURE

3.191 cm 70°F 1.918 ctr 7° 1080 mi' ADDITIONAl. 20 PPM g - -i,.5x10-cm

735[IngL

r _)J a d

.

Ix'")

-PIPE ID. TEMPERATURE

1.918cm 40°F

::

::

3181cm 00°F

8.080cm 80°F

ADDITIONAL s Pci

flhi!iIt II

V

t;?;!:

5080 cm TEMPERATURE

::

80°F ._. - ______ V V 2.5 1.0 V . PIPE.I.D. V TEMPERATURE V V I 1 I I I (U1] V 2, - 5" 10 cm -OPEN POINTS 3.181 cm 75°F HALF.FILLEO POINTS 1.918cm V 75°F

I)

SOLID POINTS 0.385 cm 75°FV _.. e ADDITIONAL 20 PPM - -V 3.181 cm 1.818cm 50°F b V -V

-

1.918 cm V 40°F V -0- - 3.181 cm 100°F 50 () 1.918cm 100°F V 20 V V 2 4

35 10 G 35 10 5 o PIPE I O. TEMPERATURE OPSN POINTS 3.181 an 75F SOLID POINTS 1.818 an 7VF - 4,/:g -19[hIE (u?t)]2

I

5x 10. cnr 5.0

Figure 12 - Drag-Reductiôn Characteristics, versus urli.', for MAGNIFLOC 835A SolUtions

9 10

U7

- (cnr1)

Figure 13 - Drag-Reduction Characteristics,

ff

versus u.,.jv, for Guar Gum J2M Solutions

27

8 iO

PIPE ID. TEMPERATURE

OPEN POINTS 3.181 cm 1O,F

HALF.FILLED POINTS 1.918cm 7SF SOLID POINTS 0.385 an 1SF

4

3.111 an 85F 3.181cm 100F -- L - -1.1 1.37 V'Io n 1Q 1h cm) c (ppm) 500

-

-Is.. -300 ' -u 100 102 4 10 U7 )cm1) 2 4 8 1O 20 n' 1 15

200 2 i OVER.SAÏURATEDDRAG-REbUCTION 1243Io 2

DOMAIN IS To THE LEFT OF THE

-OPTIMAL DRAG REDUCTION ) LINE _{/em =} - 41

_{J + 2.5}

2\

/

PIPE I.D. = 1.918cm

/

DU Iog 4i.2 (24.3 2 + PIPE I.D. = 0.385 cm 1.5 -

:

/

4 PIPEI.D.3.18ic

----f----

-

--

----/,;?b

PIPE I D = 1 918 cm

/

ì"j1'

j

PIPE ID. 0.385 cm

2IogT..4t2)2

m ₅₇

--/

's

:--

-

--102 1 0 i 5 Du 2v

Figure 14 - Saturated Drag-Reduction Boundaiy fc,r POLYOX WSR-301, MAGNIFLOC 835A,

U1

and SEPARAN AP-30 at -

-V 100 80 60 40 E E 20 1000' 800 600 400 10 8 6

REFERENCES

I. Toms, B. A., "Some Observations on the Flow of Linear Polymer Solutions Through Straight Tubes at Large Reynolds Numbers," Proceedings First International Congress Rheology, North Holland Publishing Co., Amsterdam, Vol. 2, pp. 135-141 (1948).

Wells, C. S., Jr., "On the Turbulent Shear Flow of an Elasticoviscous Fluid," American Institutes of Aeronautics and Astronautics Preprint 64-36 (1964).

Savins, J. G., "Drag-Reduction Characteristics of Solutions of Macromolecules in Turbulent Pipe Flow," Soàiety of Petroleum Engineers Journal, Vol. 4, p. 203 (1964).

Ernst, W E., "Investigation of Turbulent Shear Flow of Dilute Aqueous CMC Solutions," American Institute of Chemical Engineers Journal, Vol. 12, No. 3,_{pp. 581-586 (1966).}

Elata, C. and J. Tirosh, "Frictional Drag Reduction," Israel Journal of Technology, Vol. 3,

pp.

1-6 (191-65).

Fabula, A. G., "The Toms Phenomenon in the Turbulent Flow of Very Dilute Polymer Solutions," Proceedings Fourth International Congress Rheology, Interscience Publications, New York, Part 3,

_pp

455-479 (1965).

Virk, P. S. et al., "The Toms Phenomenon: Turbulent Pipe Flow of Dilute Polymer Solutions," Journal of Fluid Mechanics, Vol. 30, Part 2, _{pp. 305-328 (1967).}

Hershey, H. C. and J.

L

_{Zakin, "A Study of Turbulent Drag Reduction of Solutions of High} Polymers in Organic Solvents," Chemistry Engineers Science, Vol. 22, p. 1847 (1967).

Van Driest, E. R., "Turbulent Drag Reduction of Polymeric Solutions," Journal of Hydronautics, Vol. 4, No. 3, _{pp. 120-126 (1970).}

Paterson, R. W. and F. H. Abernathy, "Turbùlent Flow Drag RedUction and Degradation with Dilute Polymer SOlutions," Journal of Fluid Mechanics, _{VoL 43, Part 4, pp. 689-710 (1970).}

Huang, T. T. and N. Santelli, "Drag Reduction and Degradation Of Dilute Polymer Solútions in Turbulent Pipe Flows," NSRDC Report 3677 (Aug 1971).

Elata, C. et al., "Turbulent Shear Flow of Polymer Solutions," Israel Journal of Technology, Vol.. 4, No. 1, pp. 87-95 (1966).

l3 Goren, Y. and J. F. Norbury, "Turbulent Flow of Dilute Aqueous Polymer Solutions," Journal of Basic Eng., Transactions of American Society of Mechanical Engineers, Paper 67-WAIEF-3, Vol. 89, P. 814 (1967).

Wells, C. S. et al., "Turbulence Measurements in Pipe Flow of a Drag-Reducing Non-Newtonian Flüid," American Institute of Aeronautics and Astronautics Journal, Völ. 6, No. 2, pp. 2C)-257 (1968).

Patterson, G. K. and G. L. Florez, "Velocity Profiles during Drag Reduction," Chapter in "Viscous Drag Reduction," Edited by C. S. Wells, Plenum Press, New York, pp. 231-250 (1969).

Tornita, Y., "Pipe Flows of Dilute Polymer Solution, Paits I and II," Bulletin of the Japanese,. Society of Mechanical Engineers, Vol. 13, No. 61, pp. 926-942 (1970).

Seyer, F. A. and A. B. Metzner, "Turbulent Phenomenon in Drag-Reducing Systems," American Institute of Chemical Engineèrs Journal, Vol. 15, No. 3 pp. 426-434 (1969).

Tsai, F., "The Turbulent Boundary Layer in the Flow of Dilüte Solutions of Linear Macromolecules," Ph.D. Thesis, University of Minnesota (1968).

Wetzel, J. M. and J. F. Ripken, "Shear and Diffusion in a Large Boundary Layer Injécted with Polymer Solution," University of Minnesota, St. Anthony Falls Hydraulic Laboratory Project Report .114. (Feb 1970).

Rüdd, M. J., "Velocity Measurements Made with a Laser Dopplermeter on the Turbulent Pipe Flow of a Dilute Polymer Solution," Journal of Fluid Mechanics, Vol. 51, Part 4, pp. 67 3-685 (1972).

Chung, J. S. and W. P. Graebel, "Laser Anemometer Measurements of Turbulence in Non-Newtonian Pipe Flows," The Physics of Fluids, Vol. 15, No. 4, pp. 546-554 (1972).

Wu, J; and M. P. Tulih, "Drag Redùctión by Ejecting Additive Solutions into Pure-Water Boundary Layçr," American Society of Mechanical Engineers Gas Turbine and Fluid Engineering Conference,. San Francisco, Calif., Paper 72-FE-12 (l972) .

-Virk, P. S. et al., "The Ultimate Asymptote and Mean Flow Structure in Toms Phenomenon," American Society of Mechanical Engineers Journal of Appliçd Mechanics, Vol. 37, pp. 488-493 (1970).

Virk, P. S., "An Elastic Sublayer Model for Drag Rdüction by Dilute Solutions of Linear Macromolecules," Journal of Fluid Mechanics, Vol. -45, Part 3, pp. 417-440 (1970).

Schlichting, H.,. "Boundary-Layer Theory," Sixth Edition, MòGraw..Hill Book Company, New York (1968).

Hinze, J. O., "Turbulence," McGraw-Hill BoOk Company, New York (1959).

"PrOceedings Computation of Turbulent Boundary-Làyers 1968 AFOSR-IFPStanford Conference," Edited by S. J. Kline et al., Stanford University (1969).

Milhikan, C., "A Critical Discussion of Turbulent Flows ¡n Channels and Circular Tubes," Proceedings of the Fifth International Congress of Applied Mechanics, Cambridge, Mass., pp. 386-392 (1938).

Coles, P., "The Law of the Wake in, the Turbulent Boundary Layer," Journal of Fluid Mechanics,

Vol. 1, pp. 191-226 (1956).

Klebanoff, P. S. and Z. W. Diehi, "Some Features of Artificially Thiôkened Fully Developed Turbulent Boundary Layers with Zero-Pressure Gradient," National Advisory Committee for Aeronautics Report 1110 (1952).

Freeman, H. B., "Force Measurements on a 1/40-Scale Model of the U.S. Airship AKRON," National Advisory Committee for Aeronautics Report 432 (1932).

Schultz-Grunow, F "Neues Widerstandsgestetz fur glatte Platten," Luftfahrtforschung, Vol. 17, No. 239 (1940); also National Advisory Committee for Aeronautics Technical Memorandum 986 (1941).

Nikuradse, J., "Gesetezmassigkeiten der turbulenten Stromung in glatten Rohren," VDI-Forschungsh, 356 (1932).

34; Laufer, J., "The Structure of Turbulence ¡n Fúlly Developed Pipe Flow," Natiönäl Bureau of Standards Report 1974 (1952); also, National Advisory Committee for Aeronautics Report 1174 (1954).

Hama, F. R., "Boundary-Layer Characteristics for Smooth and Rough Surfaces," Transactions of Society of Naval Architects and Marine Engineers, Vol. 62, pp. 333 (1954).

¡..andweber, L., "The Frictional Resistance of Flat Plates in Zero Pressure Gradient," Transactions of Society of Naval Architects and Marine Engineers, Vol. 61, pp. 5-32 (1953).

Coles, D., "The Problem of the Turbulent Boundary Layer," Z. Ailgew. Math., Physics 5,

pp. 181-203 (1954).

Granvi}ie, P. S., "Frictional-Resistance and Velocity Similarity Laws of Drag-Redúcing Dilute Polymer Solutions," Jäurnal of Ship Research, Vol. 12, No. 13 (1968).

McCarthy, J. H., "Flat-Plate Frictional-Drag Reduction with Polymer Injection," Journal of Ship Research, Vol. 15, No. 4 (1971).

Meyer, W. A:, "A Correlation of the Friction Characteristics for Turbulent Flow of Dilute Non; Newtoniin Fluids in Pipes," Anierican Journal of Chemical Engineering Journal, VoI. 12, No. _{, Pp.}

522-525 (1966).

Hoyt, J. W. and A. G FabUla, "Thé Effect of Additives on FlUid Frìctiòn," Fifth Symposium of Naval Hydrodynamics,, Edited y J. K. Lunde and S. W. Doroff, Office ofNaval Research, Department of the Navy, ACR-12,-pp. 947-974 (1964).

42: Píh, M. and T. Miloh, "Rotation of a Disk in Dilute Polymer SolutiOns," Journal of

Copies _Copies

AAMRDC 3 NAVUSEACEN, Pasadena

1 Dr. J. W. Hoyt

3 CHONR _i Dr. J. G. Waugh

2 ONR 438 (Fluid Dyn Br)

i

br. W. D. White

1 ONR 472 (Chem Br)

1 NAVWPNSCEN

2 NRL, Surf Chem Br

1 Dr. R. C. Little 6 NOL

i Dr. R. E. Wilson

1. US Naval Academy 1 Dr. A. E. Seigel

Prof Bruce Johnson, Eng Dept 1 Dr. A. May

1 N. Tetervin

1 NAVPGSCHOL 1 Dr. V. C. Dawson

Prof T. Sarpkaya, Dept Mech Eng

4 NPTLAB NUSC

i

USN ROTC & NAVADMINU, MIT 1 P. E Gibson i R. H. Nadolink

i

NAVWARCOL

NLONLAB NUSC

15 NAVSHIPSYSCOM

2 SHIPS 2052 4 NAVSEC

1 SHIPS 031 _{1 SEC 61Ó1E01}

i SHIPS 034 1 SEC 6110-01 1 SHIPS 0341 1 SEC 6136 1 SHIPSO3412 1 SEC6114D i SHIPS 0342 1 SHIPS 037 12 DDC 1 SHIPS 037C

1 SHIPS 0372 1 BUSTOS, Hydraulic Lab

1 PMS 381

i PMS393 i Lib of Congress

3 PMS 395 Sci & Tech Div

NAVAIRSYSCOM

_i

MARAD

1 AIR 5301 Aero & Hydro Br

3 NAVORDSYSCOM

1 ÓRb 035, Weap Dyn Div

1 ORO 05431, Hydro Pro Gng

1 NELC

3 NAVUSEACEN, San Diego

1 Dr. A. G. Fabula 1 Dr. T. Lang

33

INITIAL DISTRIBUTION

1 NASA, Fluid Phys Av (RRF)

Ira R. Schwartz

1 Arizoña State U

Dept Chem Eng Prof N. S. Berman

i Univ of Bridgeport

Dept Mech Eng

i Cal Tech, Hydrodyn Lab Prof A. J. Acosta

i Colorado State Univ

Dept of Civ Eng Prof J. P. Tullis

i Univ of Delaware Dept of Chem Eng

Prof A. B. Metzner

i Univ of Florida

Sch of Eng

Prof E. R. Londgren

i Univ of Illinois

Dept Chem Eng

Prof T. J. Hanratty

2 State Univ of Iowa

Iowa Inst Hydraulic Res

i Prof L. Lándweber

i Prof J. F. Kennedy

i Iowa State Univ Dept Chem Eng

Prof W. H. Abraham

i Kansas State Univ Dept Chem Eng

Prof L. T. Fan

i Univ of Mass

Dept of CivI Eng

Prof C. E. Carver, Jr.

INITIAL DISTRIBUTION (Continued)

Copies Copies

i Brown Univ 4 MIT

Dept of Mech 2 Dept Nay Arch

Prof R. I. Tanner 2 Dept Chem Eng

i Univ of Calif, Berkeley 2 Univ of Mich

Dept of Chem i Dept Nay Arch

Prof M. C. Williams 1 Dept Eng Mech, W. P. Graebel

i Univ of Calif. San Diego 3 Univ of Minnesota

Prof A. Ellis p St. Anthony Falls Hydraulic Lab

2 Univ of Missouri

Dept Chem Eng

i

Ohio State Univ

Dept Chem Eng Prof H. C. Hersey

i Univ of Oklahoma, Res Inst Prof E. F. Blick

5 Penn State Univ, ARL

i Dr. J. L. Lumley

i Dr. B. Parkin

i R. E. Henderson

i

Dr. F. W. Boggs

i

Purdue Univ

Sch of Eng, Prof A. T. McDonald

2 Univ of Rhode Island

i Dept Mech Eng, Prof F. M. White

i Dept Ocean Eng, Prof T. Kowalaski

2 Stanford Univ

i Dept Civ Eng, Prof E. Y. Wu

i

Dept Mech Eng, Prof S. J. Kline

2 Stevens Inst, Davidson Lab

i J. Breslin

i

Univ of West Virginia

Prof W. Squire

i Univ of Wisconsin Dept Chem Eng

INITIAL DISTRIBUTION (Continued)

Copies CENTER DISTRIBUTION

35

i WPI, Alden Res Lab L. C. Neale Copies 1 Code 01 Dr. A. Powell i SNAME _{1 012 Dr. R. C. Allen} 1 Aerojet-General 1 11 W. M. Ellsworth Hydrodyn Dept 1 115 R. J. Johnston

i Esso Math & Sys Inc 1 15 Dr. W. E. Cummins

Dr. Richard Bernicker

1 1502 G. R. Stuntz, Jr. 1 General Electric Co

R&D Center 1 1508 CDR F. J. Moran, Jr.

W. B. Giles _{1 152 R. Wermter}

4 Hydronautics, Inc 1 154 W. B. Morgan

1 V. Johnson 1 Dr. Jin Wu

1 1541 P. Granville

1 M. P. Tuline _{1 1542} Dr. B. Yim

2 LTV Research Center 1 1544 R. Cumming

Dr. C. W. Wells

3 1552

i North American Rockwell

Ocean Sys Dr. E. R. van Driest 1 1 1 J. H. McCarthy T. T. Huang N. Santelli 1 1556 D. Cièslowski 1 Oceanics, Inc A. Lehman 1 156 J. Hadler

1 Union Carbide Corp

1 1802.2 Dr. F. Frenkiel Dr. F. W. Stone _{1 1802.3 Dr. H. Lugt} 1 1805 Dr. E. CuthiU 1 1843 J. Schot 1 19 Dr. M. M. Sevik 1 194 G. J. Franz 1 2762 E. M. Petrisko 1 2833 M. Greenberg

UNCLASSIFIED

orturit). no,r,..sua.wtI - - --

t

- DOCUMENT CONTROL DATA - R & D

S..'curity c-laseilicationoflitio, bodyufabst,ael and indini annOtaticn niuI be entered when- the overall report is

claseilied)-I. ORIGINATINGACTlVITY (Cotporate author) -

-Naval Ship Research and Development Center

-Bethesda, Maryland 20034 20. REPORT-SECURITY CLASSIFICATION UNCLASSIFIED 2b. GROUP 3. REPORT TITLE

-SIMILARITY LAWS FOR ÏURBULEÑT FLOW OF DILUTE SOLUTIONS OF DRAG-REDUCING POLYMERS

4. DESCRIPTIVE NOTES (Typeofrepereand inclusive dates)

-5. AU THOR(S) (Flr,et name, middle initial, last name)

-T.T.Huang

-5. REPORT DATE - -

-AUgust 1973

la. TOTAL NO. OF PAGES 7b. ÑO. OF REFS

42

-Sa. CONTRACT OR GRANT NO. -

-b. PROJECT NO. _{Task ZF61.412.00I}

- Work Unit 1-1508-309

C.

-d. - - . .

-Sa. ORIGIÑATORS REPORT ÑUMBERIS)

4096

-9bOTHER REPORT NO(S) (Any other numbers that may .e assigned this report)

lo:oIsTRIeuTIoN STATEMENT - -

-APPROVED FOR PUBUC RELEASE: DISTRIBUTION UNLIMITED

Il. SUPPLEMENTARY NOTES

--

-IS. SPONSORING MILITARY ACTIVITY

-Naval Ship Research and Development Center

13. ABSTRACT - - -

-Velocity sithilarity laws, based on a four-layer, mean-velocity-profile model are deducêd for turbulent boundary layers with dilúte polymer solutions by means of pipe-flow

experi-meñts. Measured drag reduction is found to have three domains: undersaturated, optimal, and

oversaturated. The drag reduction does not increase with increasing conceñtration in the over-saturated domain where a strong interactive -liyer dominates the entire linear logarithmic region of the b °undary layer, Drag redUctiOn increases with increasing concentration in the undersaturated domain where the four-layer profile exists ¡n the boundary layer. The boundary between the two -domains gives optimal drag reduction; it is determined by the polymer type and concentration and by a Reynolds number based on shear velocity and boundary-layer thickness. Pipe-flow experimeñts have been made to study the

drag-- - reduction charaôteristics iii the undersaturated domain. The effects of-solvent

temperature, pipe diamôter, polymer -type and concentratioñ, and wall shear stress on the measured drag reduction have been investigated.