Hydrodynamic aspects of drag reduction with additives

Lab.

_{v. Scheepsbouwkunck}

Technische Hogechooi

De

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

Bethesda, Maryland 20034

C'r4irdlin

.sbouwkundö

ische HogeschoctD44L.

DCUMENTATIE

_I:

-6

DATUM:

DO C U H C H I Al I U

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liotheek van d

HYDRODYNAMIC' ASPECTS OF DRAG REDUCTION WITH ADDITIVES

by

Paul S. Granville

Presented to Chesapeake Section of

Society of Naval Architects and Marine Engineers 19 September 1972

Washington, D. C.

(submitted for publication in MARINE TECHNOLOGY) 10 M.N. ₁₉₇₃

The Naval Ship Research and Development Center is a U. S. Navy center for laboratory

effort directed at achieving improved sea and air vehicle.. It was formed in March 1967 by merging the David Taylor Model Basin at Carderock, Maryland with the Marine Engineering Laboratory at Annapolis, Maryland.

Naval Ship Research and Development Center

Bethesda, Md. 20034

*REPORT ORIGINATOR

MAJOR NSRDC ORGANI ZATIONAL COMPONENTS

OFFIcER-IN.DIARGE CARDE ROCK SYSTEMS DEVELOPMENT DEPARTMENT SHIP PERFORMANCE DEPARTMENT 15 STRUCTURES DEPARTMENT 17 SHIP ACOUSTICS DEPARTMENT MATE RI ALS DEPARTMENT NSRDC COMMANDER TECHNICAL DIRECTOR01 OFFICER-IN.cHARGE ANNAPOLIS 04 AVIATION AND SURFACE EFFECTS DEPARTMENT COMPUTATION AND MATHEMATICS DEPARTMENT PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT CENTRAL INSTRUMENTATION DEPARTMENT NDW-NSRDC3960/46 (REV. 8-71) GPO 917-868

A A

a

a2

0.2

C.3

B1 B2 B

1,l

32

133

C CF D f f 0 k m L m P q r RD NOTATION

slope of logarithmic inner law

slope of logarithmic interactive law

slope of logarithmic resistance formula for pipe flow

slope of logarithmic formulas for maximum drag reduction for pipe flow slope of logarithmic drag formula for flat plates

slope of logarithmic formula for maximum drag reduction for flat plates slope of logarithmic resistance formula for rotating disks

slope of logarithmic formula for maximum drag reduction for rotating

disks

intercept of logarithmic inner law intercept of logarithmic outer law

Intercept of logarithmic interactive law

-intercept of logarithmi.c resistance formula for pipe flow

intercept of logarithmic formula for maximum drag reduction for pipe

f I ow

intercept of logarithmic drag formula for flat plates

intercept of logarithmic formula for maximum drag reduction for flat plates

intercept of logarithmic resistance formula for rotating disks intercept of logarithmic formula for maximum drag reduction for rotating disks

concentration of additive

drag coefficient for flat plates diameter of pipe

Fanning friction coefficient

value of f at onset of drag reduction

resisting moment coefficient for flat plates length of flat plate

characteristic length of additive.

-characteristic mass of additive type of additive

Meyer factor for drag reduction radius of rotating disk

value of at onset of drag reduction

t characteristic time of addi:tTve

u _{velocity parallel to wall In shear flows}

U

I

U y V V0 p shear velocity

velocity of flat plate or velocity outside of shear flow average velocity In pipe flow

Ooles wake factor

normal distance from wall thickness of shear layer

drag reduction characterization kinematic viscosity of solution kinematic viscosity of solvent density of fluid

wall shearing stress

ABSTRACT

The hydrodynamic aspects are presented of drag reduction

with additives. The fundamental properties of thi.s remarkable phenomenon are described. A brief history is outlined from

anomalous results in pipe flow, through strange effects in the.

Texas.. oi If ields and to the current research effo-ts. _orrela

tion by means of the velocity similarity laws of turbulent _flow

is explained for drag reduction in pipe flow arid for the boundary layers on bodies. The limits of drag reduction are also explaäned on the. basis of the interactive similarity law:. The peculiar role of viscoel.asticity 'is examined. _Naval

architectural applications .are reviewed.

INTRODUCT ION

One .of the most exciting discoveries in fluid _{mechanics of thi.s}

century may be the observatiOn that turbulent skin friction can be

reduced by the presence of suspended particles in a flowing, fluid', whether

liquid or gaseous. _{Various types of particles are effective: .simp.Iesolids} like sand grains or asbestos fibers; macromolecules like _{those of: polymers} of complex soaps. Ft.seems that he prticIes have to be within certain.

size ranges and be long and rodl Ike in shape' to be effective.

The discovery is In-ftiguing in two major aspect: CI) on aftndamental.

basis, it focuses attention on how exactly Is .turbulent skln friction _generated

ata wall _{y.a flow, and.(2) on an englneer-tng basis, It provides a physical}

means-of reducing the enorrnqus energy losses due to turbulent sicin friction

in the. flow of fIt1ds _{i.n conduits or conversely, the movement of bodies in}

fluids. . . .

The phenomenon has received many names such as drag-'reduclng effect,.Toms'

effect, non-Newtonian effect or viscoelas-I-ic effect. _{The least objectionable k} the dreg-reducing effect by additives. In view of the contributions of Texas investigators, maybe a more suitable name Is the Texas effect. '

Credit for the discovery is difficult to assess. There were several isolated reports in the 1940's about reduced turbulent skin friction.

1*

Vanoni observed the iñreased flow of sand suspended in water in a channel.

Mysels and his associates2 noticed the increased flow of gasoHn

in aipe

when thickened with a complex soap, Napalm; and Toms3 reported the increased

pipe flow of a polymer, polymethyl methacrylate, dissolved in the organic

liquid, monochlorobenzene. There were still earlier obseatibns bn the

increasedf-low rate of silt-laden rivers. Unfortunately the reports in the

l940s seemed to lead nowhere. No subsequent investigations Were stimulated

immediately,

A new start seems to have occurred in the Texas oi lfields. Suspensions

of clay particles have long bean used as drilling muds to be pumped down

through the long drilling pipes to clear the bits among other 'functTOn. In

the 1950's, the flow of drilling mud was speeded up to build up the pressure

to help in fracturing the rock being drilled. Organic materials I Ike Guar

Gum were added to keep the clay particles from settling out. The there was th.e realization that the pumping rate was increased by. these additives

Researchers for the oil companies like Savins became interested and began to

studythe phenbménon. The earlier work of Toms was nOted. At the beginning

of 1-he 1960's, an oiL company engineer informed the Office of Naval Research of the new:phenomenOn. The Naval laboratories alsO became interested and the

whole subject of drag reduct.ionwlth additives then began the intenivë

development which continues to this day. More will be si d aboutthls ih more detailed history in this:pper.

Drag redutiOnby additives has beenstudied mostly for the urbulent

flow in pipes. It is found that the fMctfon coefficient is reduced for the'

same Reynolds number. The reduction is a function of concentratioh type Of

additive and diameter of pipe. The variation with diameter may be accOunted for if it Ts assumed in the analysis that the additive particles impart

additional parameters such as characteristic lengths, masses and/or times.

* References are listed on page 9.

There Is aisoa lower limit to how reduced the friction factor can become. This

limit of maximum drag reduction for turbulent flow is still somewhat greater than that for laminar flow. The friction coefficient for maximum drag reduction is

solely a function of Reynolds number and Is Independent of concentra-Hbn or

type of polymer. Drag eductlon has also been studied for rotating disks,flat

plates and other bodies. Here drag reduction Occurs In the exterior turbulent

boundary layer.

The hydrodynamic analysis of drag reduction may be performed on, the

basis of an extension of the well-known similarity laws of turbulent, flow for ordinary fluids. The similarity laws have proved most successful for ordinary

fluids and, have led to the universally accepted logarithmic resistance formulas -for pipe flow, rotating disks and flat plates. In a ltke manner,

logarithmic resistance formulas may be, derived for drag-reducing fluids in pipe flow, on rotating disks, and on fiat plates.

The most effective olymei found to date is. polyethylene oxide ç.Poiyox) particularly in Its high molecular weights of well over a million,. Polyox as a drag-reducing agent has been intensively studied and has a wide

litera-ture. To complicate matters Polyox has two peculiar properties, namely, a

variable viscoelasticity and a propensity for mechanical degradation1 The

variable viscoelasticity which also shows.up in laminar flow has led to great

confusion with the drg-reducing property in turbulent flow.' En fact, some

investigators associate the two. The mechanical degradatron and its associated loss of drag-reduction effectiveness has sharply limited the engineering

applications of Fôlyox, especially fOr continual use at high shea,r rates.

The naval architectural applications of drag reduction by additives

include model and full-scale tests. Additives have 'been used to simulate a high Reynolds number in model towing tests. In prctIcal application,

concentrated solutions are injected from slots Tests on British mine-sweeper HMS HIGHBURTON with slot, injection proved successful.

There are many aspects to drag reduction by additives besides the

purely hydrodynamic aspects considered in this paper. Reference should

5 6 7 8

HYDDYNAM1C NATURE. OF DRAG-REDUCNG, PHENOMENON WITH ADDITIVES

To undersan.d i-he basic features of drag reduction, consider the

friction diagram (Figure I) for fully-developed pipe flow where the Fanning

frittion coefficient f is. plotted against Reynolds number RD. The reference velocity is 1he average velocity V through the pipe nd the reference length. is the pipe dIameter 0. The wall shearing stress which is a meayre of

the friction, loss is non'dimens.ionali.zed as

I

w

I 2

There are two choices for k'inemati:c viscosity In defining Reynolds number,

namely, that of thësolUtion or the solvent v0, v always being greater

than Or > The drfference in density p for solution and solvent

is negligible. Since engineers are more interested in the ress drag

reduction, the Reynolds number is now defined With solvent kinematic viscosity or

Ro

VD

0

There are then different friction lines for laminar flow for the

solvent and for the polymer sblution due to the diffrence in kinematic viscosity. Likewise for turbjlnt flow.

Figure I Shows drag reducfTbn for a polymer solution

of i-he same

concentration and type of polymer but different diameters o pipe. Note

that the onset

of

drag reduction from the turbulent line of the polymer Solution varies with pipe diameter. Note also thai-the smaller diameter pipe provides drag reduction at a smal ler Reynolds number and gives greater

drag reduction at the same Reynolds number.

Note the limiting line of maximum drag reduction which, Is' still higher than the laminar line. The line of maximum drag reductiOn is Independent

of

diameter, concentration and type of polymer. For very srnlI pipes, drag reduction may start before -I-he norm& occurrence of transition from laminar to turbulent flow. There is, then, no onset of drag reduction from a

The effect of Just varying concentration is more complicated. In

general, at low concentrations increasing the concentration increases the drag reduction. There Is however an optimum value of concentration where

an increase In concentration reduces the drag reduction.

Similar friction diagrams may be shown for rotating disks and for flat

plates. _{For -otating disks, the friction coefficient is given In terms of}

a resisting moment and the Reynolds number by the circumferential velocity

and disk radius. For flat plates the drag coefficient is given by the

total drag over the area and the Reynolds number by the plate velocity and

plate length.

h{ ISTORY

Increased

flow

rates in the turbulent regime due to suspended particles began to be noticed by different Investigators in the 1940's. Vanoni I

in 1946 observed that water with suspended sand flowed more rapidly in an open channel. l:t was noted that the effect is dependent on concentration and size distribution of the suspended sand.

Mysels and associates found a reduced pressure drop for the pipe flow of gasoline thickened with Napalm, a complex soap, during investigations

in World War II. A patent9 was secured in 1947 but a complete account

of the experiments did not appear until years later,2''°

Toms3 studied the flow of a polymer, polymethyl methacrylate, dissolved

in an organic liquid, monochlorobenzene, in various sized pipes. A reduced

pressure drop was found for the solution in the turbulent regime and not In

the laminar regime. Also, the reduction is a function of pipe diameter

and polymer concentration at the same Reynolds number,

Unfortunately, these early isolated observations did not stimulate any

further Investigations. The full implications of the phenomenon were yet

to be understood. MentTon should be made of still earlier isolated

observations of increased flow of silt-liden rivers; the earliest reported

being in l883. An independent line of development began by accident in the

Texas ol Ifields. In the 1950's an ollwell drilling operation called

fracturing began to be developed. It involved the increased flow of dri Hing ñiuds to help fracture the rock being drilled. Organic additives like Guar, Gum were used to stabilize the mud particles suspended in water. It was

happily found that the additives were reducing the tremendous pumping power

required in fracturing opertions.11''2

Savins13''4 an investigator for the Socony-Mobil Oil Co. In Texas

began to realize the full implications of the phenomenon at the start of the

95g The work of Mysels and of Toms. was rediscovered.

Toms' data were

rep lotted in the form of friction factors versus Reynolds humber. Here the onset effect and the diameter effect are obvious. The reduced friction due

to paper fiber suspensions'5and even for dust suspended in air'6 were noted.

17

18..

Sayins also noted thai- Shaver and Merrill and Dodge and Metzner in

their studies of the pronounced non-Newtonian effects of the turbulent flow of pseudoplastic solutions had encountered the drag-reduction phenomenon for

someof the polymers. Savins also applied the term drag reduction and made

the distinction between drag-reducing effects and non-Newtonian effects. One

of the characteristics of the drag-reducing effect is that the viscosity of the solution, though higher than that of the solvent, is still Newtonian

under shear.

Another Texas engineer, Horace Crawford of the Western Co., informed

the Office of Naval Research and the Naval laboratories about this phenomenon

in 1961. Soon afterwards, Pabula19 of the Pasadena Annex of the Naval Ordnance Test Station discovered the drag-reducing properties of water solutions of Polyox, the most effective of the polymers to date. Very

dilute solutions, even as low as one-half per million, give noticeable drag reduction. Now followed an ever-increasing and widening research, in this phenomenon both here and abroad which still continues.

The correlation of data from pipe tests, particularly the diameter

effect proved puzzling. Meyer,2° another investigator from Texas., solved this by correlating the data on the basis of similarity of the wall

hering

stress. A linear logarithmic relation was found, Elata etal2' Introduced

the idea of a relaxation time as a correlation parameter. Granvlile22

pro-posed then characteristic lengths, time and/or masses for the analysis, and

23

by applying the overlapping concept of Millikan to the similarity laws found that correlation follows from the wall shearing stress. Granvi lie22

also extended the similarity method to flat-plate boundary layers and obtained

logarithmic formulas for the drag of flat plates in polymer solutions.

Granvi I 1e24 further extended the method to the more complex boundary layers on rotating disks.

The limiting condition of maximum drag reduction was observed experimentally

25 . 26

by Hoyt and Fabula, A power-law analysis followed by GI les, which

. . . 27

inherently has a limited range of Reynolds number. Virk et al fitted a logarithmic formula for the maximum drag reduction and deduced the iiteractive velocity similarity law. Granville derived the maximum drag reduction for

28 . . 24

flat plates as well as for rotating disks from the interactive velocity similarity law.

Since practical applications involve injections of concentrated polymer

solutions either from slots29 or continuously from a surface,3° various

studies and analyses have been made for such flows,

Studies on ship models culminated in the recent successful test on the

31

British minesweeper ElMS Highburton.

ANALYSIS AND CORRELATION

Overaf'P Considerations

The first step in the analysis of resistance data for drag-reducing

effect by additives is to introduce the idea that characteristic lengths, masses, and/or times are being imparted by the additive particles. This

immediately explains the diameter effect in pipe flow, the radius effect for rotating disks, and the length effect for flat plates since another

dimensionless ratio besides Reynolds number arises in the analysis. The

characteritVic mass, or a àharacteristic time representing the additive

particle. The characteristic length Z may be substituted by a characteristic

mass m or char-acteristic timet by

rotating diks

and flat plates

where

The nondimensional resistance relationships are then for pipe flow

f

(m)l/3

=-Ff

p

C is the concentrafidn of the additive,

P represent the Tnd of additive, V

k is the moment coefficient defined in Reference 24,

m

w is the angular velocity of the. disk,

r is the radius of the disk,

CF is the drag coefficient of the flat plate defihed in Rf 22,

U is the velocity of the flat plate,

V

L is the length of the flat plate,

m or t may be used instead of 2. by use of Equation (3) in forming the nondimensiOnal ratiO.

V

Also v for the solution may be used instead of v for the solvent or

V 0 V V

carrier in forming the Reynolds number.

A cOnvenieht measure Of characteristic length 2. (or m ort by

Equation (3)) may be made from the onset of drag reduction in pipe flow at

8

(4)

(5)

f end CR ) for pipe diameter D by

o

Do

rTo

CR)

0

Do

Similarity Laws of Turbulent Shear Flows

The frictional resistance due to turbulent flow of ordinary fluids defied rational analysis until the development in the early part of this century of the velocity similarity laws by Prandtl and by von Krma'n working

independently on different aspects. The analysis resulted in the well-known logarithmic resistance laws for pipe flow, flat plates, and rotating disks.

A key ingredient in correlating velocity profiles is the wall shearing stress Since the velocity at the wall is zero, the next obvious para-meter is the derivative of tangential velocity u with normal distance y at

du

the wall or () which is equal to - where

is the coefficient of viscosity.

dyw

Frendtl primari ly developed the inner velocity law and von Karm'n the outer velocity law. In a simplifying analysis Millikan23 later showed that the

logarithmic form of the velocity similarity laws resulted from the over-lapping of the inner and outer laws within the shear layer.

To account for the drag-reducing effects of additives, Granvi I 1e22 extended the Frandtl-von Krmn-Mi II ikan approach by introducing the idea of characteristic lengths, masses, and/or times for the additives into the simi larity-lew analysis and obtained for the inner law

uy

uL

u

-f[

t

- -

,,

C, F]

T where

u = is the shear velocity.

From equation (3), a characteristic mass m or time t may be substituted

forL.

The concept of characteristic length already exists in analyses of

rough surfaces32 and this type of analysis of drag-reducing effects may be called the negative roughness analogy.

The outer law remains unaffected

-y

-6

where

6 is the thickness of the shear layer,

and U is the value of u at the outer edge of the shear layer.

In the region where the inner and outer laws overlap, logarithmic relations result22 and 1-he relation for the inner law is

uy

ut

u

- -

A In + B + B[ T

T

uL

where A and B1 are constants, B, as a function of -i-- and C, epitomizes

the drag-reducing qualities of additive P. For overlapped part of the outer law

- - A In + B2

T

where B2 is a constant for the particular flow situation. The whole outer

law may be rewritten as

U

-Alnf

B2(I-I/2w[*J)

(12)

t

33

where w[6J is the wake function of Coles. Coles gave a table of values

for w. There is a fitted equation by Hinze34

w = I - cos (w-) (13)

and a fitted polynomial by Moses35

= 3

()2

- 2

(f)3

(14)

10

(9)

See Figures 2 and 3 for plots of the inner and outer laws.

Equating Equation (10) and (II) producesa relation' 'for skin friction coefficient.

U Utô

See Figure 4 for a plot of - against

--T

The B

L-1-

, C, Pj is the relation for drag reduction

and once known may be used to determine the drag rédution in pipe flow, forflat plates and for rotating disks.

A special case of B is the Meyer2° relatioh wh,ich holds near the onset of drag 'reduction and when written with characteristic 1eng+h2 becomes

u u

= q [log --- log (16)

u

where

(_I)

_V,O

is the onset value of

; '

''

V and

q

=f[C, P1

. ' .

A condition of. maximum drag reduction exists when the shear layer thins to the interactive line shown in Figure 2.. Virk et a12? proposed a

logarithmic relation, ,. . . . U V U

-

UTY -.

= A ln+ B

(18) T U

_pU

u T V W = A lnL_ + B1 4 B

+ BE-- , C, F].

(15)

which may also be written UTY

= A In

_uy

V

uy

(L)

V UTY

-to show that the interactive line passes through A and B are constants. (See Figure 2).

Thicker shear layers where the Bcorrelaiion holds maybe said o be in

a conditiOn of interediate drag reduction. ..

A study of the limiting conditions of the SB-correlation is given in

Reference 36.

LOGARITHMIC RISTANCE FORMULAS

The most hotable consequences of the. velodit.y simi larity laws: for

ordinary fluids are logarithmic resistance formulas for the turbulent

regime. There are Prandtl's universal law of friction for srncx*h pipes,37

the Karman-Schoenher formulas for smooth flat pla±es,37 and the Goldstein

fOrmul.a for smooth rotating disks.37

By ap:lyings.imi lari$y laws for drag-reducing additivs, Grahvi lIe obtained enera:l.: logarithmic resistance formüIa for flat plates.2 and

24

rotating disks. Formulas were also obtained for flat plates and. rotating

disks for the special cases of the Meyer linear logarithmic characterization

- . . 24,28

and for the condition of maximum drag reduction. These formulas are

now listed: FIp. Flow General 12 '9) (20)

Linear log (Meyer)

+ _log

(F

_{) + - I b} ( - 1 o

.1

voj

2.

Ordinary fluid

-a, log

' (Prandtl)37

0

Maximum drag reduttion

a log (r-)

(Virk et al)27

NumerUcal values = 4.0, = 19.0, u2. q [log --- - log

(1._)]

1 3 = -0.4 (Fraridtl)37 =

-324

(Virk et aLY27 (21) (.22) (2-3)

Flat Plate 22 General 22 Linear log = log (CF -42 V 37 Ordinary fluid log (CF

Maximum drag reduc-t-1on28

log (CF Numerical values 4. 13 11.33

log (CF.L)±

log J2 () -

log ( ) (25) V 0 . (Karman-Schoeriherr) (26,) (27) (Schoenherr)37 -32.44 (Granvi I Ie)28 14 24

Rotating Disks 24 General 24 Linear log 37 Ordinary fluid q)

log k) +

2

-

L1

log (k ) +

-'k3

m v 3 0 24 Maximum drag reduction

log 3 24. Numerical values = 3.46

a

= 11.33

2

log (Jkm _) + 15 47r = -2.18 -32.44 u 2 4ir T

log5- (-.--)

(28) (29) (Goldstein) (30)

VISCOELASTIC EFFECTS

A complicating factor In the Investigation of drag

additives i the presence of Viscoelasticity in the. fluid, particuiarly

with POlyox. The viscoelastic effects interfere with interpreting

drag-reduction effects and result In other confusions. An additional compl

ica-tlon with Polyox sOlutiors Is the var1ab1lity Of the viscoelasticity which

seems to depend on the method of preparation. Gently stirring or aging the solution seems to eliminate the viscoelasticlty,38'39 which Is ascribed to

the entanglements of the Polyox molecules. On the other hand, dilute solu-tiofls of Guar Gum seem to lack viscoelasticity. A simple test of visco-elasticity may be made through the Weissenberg effect wherein, the fluid

climbs a spinningrod.

A manifèstãtion Of elasticity In fluid flow Is the increase :fl normal

tress or pressure in directions normal to the streamline. A simple

analcgy is to Imagine the streamlines like curved rubber bands under tension.

Then there is a tendency to straighten out and a consequent force at right

angles.

The effeetsof fluid elasticity may be listed as follows:

40,41,42

I Pitot tube errors

38,39,43,44

2, Pressure drag reduction in laminar regime 45,46

3 Inhibition of cavitation Inception

42,47,48 . .

Faster spreading of jets 39,48 Fully rough flows

Wãl I attachment (Coanda effect)39'48

38 Vortex stretching 49 Orifice flows Pipe contractions50 . Vortex shedding51

I. The normal stress difference due to elasticity affects both the

static pressure reading and the stagnation pressure so that the dynamic pressure or velocity reading is less.

2. Spheres dropped iii a tank attain a higher terminal velocity even through the Reynold number is still in the lamipar range.. The elasticity of the fluid causes a narrower wake and a subsequent smaller pressure drag.

Cavitation on head forms is delayed due to changs In .pressure

distributiOn arising from the normal stress difference. . .

Jets spread faster and have the small scale eddying suppressed

by. theelastic rOperties. . .

Flows in the fully rough regime show drag reduction arising from. the. effect of elasticity on the local separation behind the roughness elements.

Without elasticity uch as with Guar Gum, there Is no drag reduction in

the fully rough regime. . .

Jets of elastic fluids show a greater attachment to a.rOtat.ing

cylinder. .

7 Vortex stretching is thade more difficUlt by the elastici.tyôf...

the fluid. . . .

Elastic effects decrease the discharge coefficient in. flow through an orifice. Nor-elastic .Guãr Gurr has same discharge coefficient as ordinary

fluids. .

.

Pressure loss.across pipe fittings like valves. and other sudden.

cOntractions are unaffeOted by GuarGum, a non-elastic additive. ..

NAVAL ARCH ITECThRAL APPLICATIONS

Initial applications of drag-reducing polymers in naval architecture

were made to shipmodeltes-tirg. An inexplicable problem.éncountered In some

towing tanks is the occésional reduced resistance found while towing ship

models. Hoyt52 ascribes th eira+Tc behavior to drag-reducing polymers, polysaccharides, secreted by algae growing in the basin water.

Another exis+Ihg problem In measuring the resistance of surfaceshIps

with towed ñiodelsis the inability to satisfy the Reynolds number, asWel.l as the Froude number. TO overcome this, Emerson53 simulated é full-scale Reynolds number in terms of a lower friction coefficient by towing a model in

a Polyox-filled basin. Large reductions in model resistance.resulted1

Tot-hi used a basin filled with Guar Gum solution to simulate'a full-scale Reynolds number in testing ship models in a restricted section of the .St.

Practical applications in reducing drag in a body.moving -'in water:

involve some method of Injecting additives into the

exferna!bounda.ry-layer The simplest method is to inject a highly concentrated solution

through a slot. The polymer solution diffuses throuh: the, existing boundary layer and gradually becomes more. dilute downstream of the slot. -There is,

in a sense, .a concentration boundary layer developing :ir-jdet existing

-momentum boundary layer. The concentration boundary .layer has i,t orig:ifl at the slot and has a concentration profile which varies, downstream. The. thickness of the concentration boundary, layer increases downstream until it

equals the thickness Of the momentum boundary layer. The concentration

profile-keeps 11-s same'shape in theunited boundary layers. Since the polymer solution is only effective in its drag redUction close-to the wall, the

effective concentration 'Is the. wall concentration. The.wal I concentration

affects the rate'of growth of the momeniurn boundary layer..' This Interactt!on.

is allowed for in an analysis by Granville55 for the case of the united boundary layers.. An approximate analytical' solution for-thi.s case Is also

given by MoCarthy.'56 An ahalysis using Lagrangian cpnsidertiQflS is given

by POreh and Hsu57 for the whole diffusion process starting from the slot,

Experimental evidence .of 'the' efficacy of slot injection on a body 58.

59.

of revolution Is reported-by Vogel and Pattersqn. Dove .found drag reductioh by injecting polymer sOlution from afrigte model., ,

A'bold'fUll.-Scale experiment In producing drag-reduction by slot

-Injection was accompIished on 'the British minesweeperHMS HIGHBURTPN.31.

The experiment was carried outunder'the adverse conditions of arough hu,I!,

ba&weather, and an-Insufficiency of design data for slot Injection.

Nevertheless, '. substantial drag reduction was achiev8d. The.HIGHBURTON ..

experiment confirmed-the technical feasibility of drag reduction by additives on hips. -Iheeconomlcal feaslbillty'fOr commercial shipping isanothér

matter since It depends mostly on therelative price'of additive andfuel.

There' Is a 'li've!y. discUssion in Reference .31 about 1-be possibilitie,s f the,

application of dragreductionbY additives to ships. The.whole subject !S,

still 'i.n'the- process of development and It Is premature +0 arrive at fast

conclusions.

REFERENCES

I. .Vanonl, V.A.,"Transporl-ation of Suspended Sed!ment by Water," Trans. of Am. Soc. of Civil Engineers, Vol. III, p. 67 (1946)'. Agoston, G.A., Harte, W.H., Hottel, H.C., Klemm, W.A., Mysels, K.J.,

Pomeroy, H.H'., and Thompson, J.M., "Flow of Gasoline Thickened by Napalm," Industrial and Engineering Chemistry, Vol. 46,-No. 5, pp. 1017-1019 (May 1954).

Toms, B.A., "Some Observations on the Flow of Linear Polymer So!utions

Through Straight Tubes at Large Reynolds Numbers," Proceedings of

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Figure 1 - Typlial Friction Diagram for Pipe Flow

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