Frictional drag reduction

ARCHIEF

B, C -

constants

D

-

pipe diameter

-I

- friction coefficient g - gravitational acceleration

k

-

von Karman's universal constant

R piperadius

r

-

radial distance from pipe axis

Re - Reynold number

- Reynolds number of laminar sub-layer

1,,, - maximum velocity u local mean velocity

INTRODUCTION

For some time now it has been established that

re-duction of the friction óoefficient in turbulent flow can

be obtained by adding foreign matejials to a fluid. This has been observed,, while using woodfibers, by Daily and Bugliarello [1] and small spherical pOly. sterene particles suspended in water, by Elata and Ippen [2]. The same holds. for dust particles in air, as observed by Sproulle [3]; or for solutions of lông polymer molecules, as reported by Shaver and Merrill [4], Ripken and Pilch [5] et al.

In order to explain. the observed drag reduction, variOus theories have been proposed, by Shaver and Merrill [4], Vanoni and Brooks [6], Savin [7], and others. Most authors issume that the non-Newtonian behaviour, which characterizes many solutions, is res-ponsible for the decrease in friction coefficient in turbulent flow. Lately it has been proven by Hoyt and Fabula [8] that even Newtonian fluids.at minute concentrations of some additives -(as low as 5 ppm) show remarkable drag reduction.

ISRAEL JOURNAL OF TECHNOLOGY. VOL. 3 No. 1. 1965. pp. 1-6.

Proc. VII Israel Ann. Conf. AVIATION and ASTRONAUTICS, February 1965

Frictional Drag Reduction

C. ELATA AND J. T[ROSH***

Techn ionIsrael Institute of Technology, Haifa

Received, December 15, 1964

- ABSTRACT

This paper deals. with changes in the turbulent boundary layer resulting in frictional drag reduction. Experiments are described showing a remarkable decrease uifriction coefficientsfordilute solutions of separan AP 30 and guargum.Itis shownfrom these data, and those available in literature for all kinds of additives, that drag reduction effects nearly always coincide with adecreasein the universal

constant k.This contradicts the generally accepted Reynolds similarityprinciple;whichis theconven-tional assumption on which the analysisofthe turbulent boundary layeris usuallybased. Furthermore, it appears that an additional entity must exist which influences the mechanism of turbulent flow.

- 'NOTATION U0 Uj - :-V V Yo V p

velocity at outer edge of laminar sub-layer

- reference velocity

- .,/7shear velocity

mean cross-sectional velocity

- distance from solid wall

- thicknessoflaminar sub-layer

- dynamic viscosity - kinematic viscosity

- density

- wallshearstress

The various theories proposed are, by necessity phenomenological, just as those for turbulent flow of a "simple" fluid. No attempt will be made 'here to

choose between these theoHes Or to suggest a

differeflt-mechanism. , . '

The purpose of this paper is to. investigate changes in the flow characteristics which. lead to.drag -redu-tion, which are still far from being understood. Some definite conclusions are made on which any future, phenomenological theory will have to 'be base4.

The analysis is based on experiments, conducted

recently at the Hydraulics Laboratory 4t the Technion,

as well as on'dàta published by others. The experi

ments show reduction of frictional resistance in pipes of Op .to 55%. with extremely dilute solutions.

THE TURBULENT BOUNDARY LAYER

The turbulent boundary layer can be analyzed with the aid of dimensional considerations, as shown by Townsend 9]. The basic assumption which is usually made, 'is that close to the boundary the flow is purely

This research was carried out Under contractNo. 62558-4093 for the Office of Naval Research.

** Senior Lecturer, Facultyof CivilEngineering. Research Engineer, FacuLty ofCivilEngineering.

Office of Naval ResearO

-'

inerican Embassy

London

Lab.

_{v. Scheepsbouwkunde}

Technische

_Hogeschool

Deift

2 C. ELATA AND 3. TIROSH (Israel 3. Techn. ylacous,%while farther away flow is inertial, i.e.,

Inde-pendent of viscosity. This second assumption Is the Reynolds similarity principle, assumed valid for flow

at high Reynolds numbers.

The local mean velocity u o the flow close to 'a smooth boundary, can in general be expressed by

=

p) (1)

where y denotes the distance from the wall; ti,, the boundary shear stress, viscosity and p densIty. I follows from dimensional analysis that

u /u' = U /U'(yu' /v) (2)

In which U'

J7 is the so.called shear velogity,

and v the kinematic viscosity.

ClOse to the boudary floW Is essentially laminar (though not necessary steady) and

'u/u' =yu'/v

, ' (3)

This is the equation for th ó-callcd'ianhlflar sub-layer.

At some distance fromthe boundary flOW' becomes inertial, the direct óffect of viscous stressc&:becomes

negligible and thà local veincity u can be expressed by wall distance 'y' and 'some'refSrence velocity u1

öxlsting at some distance Yi from the wall. Then

u/u'

u/u'(y/yj;uL/t4')' (4),

where u1 rePzesnts an' arbItrarY, translatiofl velocity

uty1 which can beassied any yalue withOutvlolatlng

the similarity principle.

At some distance from thS Wall both Eqs (2) and' (4)

must be valid. By 'equating Eqs. (2) and (4) and the velocity derivatives with y' from' both equations, It

can be ashy

scan that

u/u'

(1'/k)lfly/Y1+Ui/U'

-

₍₅₎

(t/kflyudlv+Ui/U'_(1flc)inYIt4'/V

This is the equation for the sO-called inner turbulent boundary layer. Here k "von Knrman'i Universal Constant" Is a constant, representative for thS inertial

flow.

It Is convenient to,choOse as reference velocity u1 in Eq. (5), the velocity u0 at the Intersection point Ye

of Eqs. (3) and (5). Eq. (5) becOmes then

u/u'

(1/k)lnyu'Jvl+ R0 - (1i/k)ln R0 (6)

where

R0=y0u'/v=u0fu'.

Point Yo is situated in a region where in fact neither Eq. (3) nor Eq. (5) hold as was shown experimentally. Irinay [10] Indicated that a transition layer of dis-tinct characteristics exists between the laminar sub-layer and the inner turbulent boundary sub-layer.

Farther away from the boundary Eq. (6) does not hold. Different equations were suggested for the outer

boundary layer; Hama [11] assumed a parabolic

equation of the form

'(Urn

u)/u"

C(i -

y/R)2 (7)

Here Urn represents maximum velocity at distance R;

C is a constant. Eq. (7) Is compatible with Eq. (4), as expected.

To snmn1ize, the turbulent boundary layer can be divided Into four regions:

The laminar Sub-layer Eq. (3)

'The transition layer

The ináer turbuleni boundary layer Eqs. (5) and (6) 'The outer turbulent boundary layer Eq. (7).

The friCtion coefficient f can be found by late-. grating the velocity profile over the boundary layer. 'For pipe flow an expression for I can be derived by integrating Eq. (6) from r =Oto r

=.R -

Yo' where r

is the radial distance from the pipe axis. This

cx-presslon is a approximate one, whereby the discharge in the outer ring of thickness Yo Is' neglected, since

y /R 41 It 'Is

furthCrmore based on the

approxima-tiOn that Eq. (6) holds near the pipe axis. Although thá

actual' velocity profile deviates from the logarithmic

law in thá neighbourhOod of the pipe axis, the relative wCight of the discharge is small in the same area.

Introducing the above-mentioned approximations one can write

1 = (1 /J8 k)In Re%Jf + B( R0, k) (8)

where / 8(u' /V)3 as defined by Darcy-Welssbach;

Re VD/v,; D Is the pipe diameter and' V the mean cross-sectional velocity.'

This expression, proven valid for pipe flow of

simple fluids, holds as well for flow with additive as suggested by available data.

While the values of k can be found from the velocity

measurements, it is suggeSted by Townsend [3] that the most accurate method of' measuring k is obtained from frictional resistance data represented ,by,Eq.(8).

Vol.3,1965) C. ELATA AND J. TIROSH 3 Experimental evidence of "pure" fluids indicates that

both k and R0 are essentially independent of Reynolds

number, although a comparison by Hinze [12] shows that a large scattering of data exists.

As shown by Lumley [13], Eq. (8) implies that

changes in the friction coefficient, at the same Reynolds

number, can become evident only through changes in the values of k and/or R0.

It is useful to analyze the meaning of such possible

changes.

A change in R0 does not contradict any of the

assumptions utilized in the development of the equa-tions for the velocity profile. In Figure 1, velocity profiles are compared at constant boundary shear. A decrease in thickness of the laminar sub-layer corres-ponds to a decrease in R0 and as can be seen, to a

decrease in discharge. This is equivalent to an increase

of the friction coefficient, as occurs, in fact, in the

flow over a rough boundary.

LAMINAR FLOW

2 SMOOTH TURBULENT FLOW

3.ROUGH TURBULENT FLOW 4. TURBULENT FLOW WITH ADDITIVES

U

Figure 1

Schematic comparison of various velocity profiles at same wall shear stress

A decrease in friction coefficient, attributed to an increase in thickness of the laminar sub-layer can therefore easily be envisaged. A decrease of f may

also be attributed to a decrease of the universal constant k. In this case, however, Reynolds principle of

similar-ity, which implies k = const., will be violated. in the following it will be seen that drag reduction phenomena caused by additives are nearly always connected with a decrease in the value of k; such a decrease must be the result of changes in structure of the turbulent boundary layer.

DESCRIPTION OF EXPERIMENTS AND EQUIPMENT

Some preliminary experiments were conducted in a rotating cylinder system (Plate 1). A cylinder, free to

rotate in a cylindrical sleeve, was connected to a D.0 motor-torque meter combination manufactured by

Kempf and Remmers. The cylinders are part of the

viscosity measuring system of an Eprecht Rheomat-15.

With this combination rotational speeds of up to

5000 RPM could be reached.

Plate 1

Rotating cylinder apparatus

The system was filled with water, and with solutions

of separan AP3O* at concentrations of 10, 20 and 30 ppm. Torque readings were compared for dif-ferent concentrations at various speeds of rotations.

For the more relevant measurements a circulation system was used, consisting of a test section

and a

return loop (Figure 2).

The test section is made up of four parallel circular

smooth perspex pipes with diameters of 12, 22, 35 and

50 mm. The length of these pipes was selected to be

* A floccutating agent processed by Dow Chemical Co.

200 Dlong enough to measure pressure drops down-stream from the initial length of flow establishment. Each pipe has ten pressure tap connections, divided equally along the pipe with a distance of 20D bet-ween them. The pressure taps consist of four cir-cumferential holes diametrically opposed and

en-closed in a ring to obtain more accurate average

pressure readings. The pressure gradient can be read

from two differential manometers designed to measure

pressure drops between 0.005-1.0kg/cm2 over the test section length.

The return loop is constructed of 4

pipe. Two closedimpeller pumps can provide 80 m3 /h against a combined head of 50m water.

For the rate of flow measurements two venturirneters

are incorporated into the system covering the possible range of flow rates from 0.5-80 m3 /h. A separate volumetric flow rate measuring system is provided to calibrate the venturimeters for each fluid.

A series of pressure drop and discharge measure-ments were made along the 50mm pipe and some complementary measurements along the 22mm pipe. Guar gum* was used as additive, at concentrations of

50, 100, 200 and 400 ppm. This high molecular weight

polymer was found not to degradate, i.e., it maintains its properties, and was therefore found suitable for

consistent experiments.

The viscosity of the solutions was measured. with the

Rheomat-15 Eprecht viscometer.

RESULTS AND DISCUSSION

The results from the preliminary experiments with the rotaüng cylinders are plotted in Figure 3. Torque reduction of 32% could be achieved at concentrations of 30 ppm. Reynolds number based on gap width at

* Processed by Stein and Hall under the trade name 'Jaguar'.

Figure 2

The circulation system

maximum speed

of rotation was approximately

Re = 6000.

The results from the pipe experiments were plotted in the conventional friction coefficient-Reynolds

number diagram, (Figure 4). Separate lines represent data for runs with different concentrations. The re markable decrease in friction coefficient is again evident from this diagram. Reynolds numbers were

based on actual viscosities. These measured viscosities

0504

prRsp,, eipts

REYNOLDS NUMBER

Figure 4

Frictional resistance in smooth pipes for guar-gum solutions (log-log)

ft. P.M

Figure 3

Torque reduction from measurements with

rotating cylinder system

3C 20 201 lM PPM

.--b...

2Q00 3000 SEPARAN 1P 000 30 C 5000 1000

I

- V l

PPM anPPM N 100 PPM BUARGUM GUARRUN

-£

GUARGUM GUARGUM :

-C. ELATA AND J. TIROSH [Israel J. Techn.

2 5 IC 5 QJ( 0.02 0.015 0.01 0.008 z 0 I.-0 w 0

I-are represented in Figure 5. The rheogram for. guar gum solutions in water over an extended shear rate (Figure 6) was taken from Hoyt and Fabula [8].

Wall shear stresses in the pipe experiments varied from 0.007 grJcm2 to 2gr/cm2, which coincides with

the range of shear stresses in Figure 6.

IA 0 .12 TIMPRATU9I, 54.iO'C GUARGUM CONCEN1RAIION Figure 5

Relative viscosities of guar

PPM gum solutions

Another presentation of the friction data was made in Figure 7. Values of von Karman's constant were calculated from the slopes in the _{1 /..Jf} log Re ./f diagram and are given in Table I. As can be seen, values vary from k 0.40, till k = 0.09.

TABLET

COMPARISON BETWEEN ORAC1 REt)UTION AND k .VAI.UES

FROM FIGURE 7 AT Re 5 x 105.

From an examination of data presented in literature

it was found that reduction Of friction coefficient nearly always coincides with a decrease in k values.

The.e findings, representing data from experiments with different kinds of additives, are compared in Table II.

The data of wood fiber suspensions do not follow

this trend' however.

From the fact thatkis not a "universal" constant .as is established here, it may be concluded that the

Reynolds similarity principle is violated, as explained

before. This indicates that away from the boundary additional forces besides pressure and inertial forces should be taken into accOunt in the turbulent

boun-dary layer.

it is not clear if the additives have a local and direct

effect on the decrease in k values, representing less

effective momentum transfer cause by changes' in

turbulent ttuctuEe. It. seems more likely to assume

thattlie additives influence cOnditions at the boundary

which then indiE5ctly cause the observed changes in

structure.

It is shown above that drag reduction for fluids

with different additives coincides with. similar change in flow characteristics. It seems reasonable to assUme that the drag reduction in all 'those cases is associated with similar changes in the flow mechanism. A general

phenomenological explanation, covering the effect of the different kinds of additives should therefore, be looked for. It .is questionable to explain frictional drag reduction as resulting, from non..Newtonian be-haviour of the. .solution, since :a similar effect was found for Newtonian fluids. The same holds for the hypothesis that elastic effects may be the cause of

drag reduction as proposed by Metzner and Park [14]

o

14

COMCENTR r -WPPI.l:IGO.' -3

4

Iv

p'-

_£000

'

.d'

. U-"-

yl

TEMPERATURE It ' +WA1CR o so PPM GUAPGUM V IX PPM RUARGISI ' '

/

90.13 , -

/

k0.I7 I - 0.40 -. i_-22IoqLT-DJ +_*V Concentration of - guar gum In ppm % Drag reduction 0 0.40 50 23 0.23 100 32 0.17 200, 42 0.13 400 .55 0.9

Vol. 3, 19651 C. ELATA AND J. TIROSH 5

- 00 5 _0' 5 !O S _o 5 _o' S S i°

WALL SHEAR RATE, SEC' Figure 6

Rheogram for guar gum sOlutions in water from Hoyt and Fabüla [8]

2 5 2

Figure 7

Frictional resistance in smooth pipes for guar gum solutions (semi.log)

200 200 100 1 I, 12 I0 9 a 7 5

since such a theory apparently does not hold for

particle suspensions nor for dilute solutions of some polymers. Again, the drag reduction phenomenon

found with sand suspensions was based on the

buoyant weight of the sand, but suspensions of neut-rally buoyant particles show a similar though less

effective result.

TABLE II

The most intriguing case is. undoubtedly the drag reduction found with dilute Newtonian. solutions as presented here. As far as could be established such solutions do not exhibit elastic properties at the rein-vant minute concentrations. To describe the pheno-menon some additional parameter should be found

however. This be seen from dimensional analysis,

since in order to express a functional relationship as presented by the experimental data in Figures 4 and 7

an additional parameter is needed. This parameter must

of course be based on an additional entity not yet considered in the analysis. It is not clear if this entity

which should be connected to the molecular properties

of the solution or to the particle properties of the sus-pension can also appear in the form of an additional continuum property.

An additional conclusion might be drawn from the results obtained here. As can be seen clearly from Figure 7, the effect of the guar gum at all concentra-tions appears only at Re'..jf 9 x or from

Fi-Based on viscosity of water Based on effective viscosity

Based on apparent viscosity at the pipe wall.

gure 4, at Re6 x 1O. The thickness Yo of the

laminar sub-layer decreases with increasing Reynolds number. If analogous to the influence of protrusions in a rough pipe the influence of the long molecules appear when the ratio of molecule length to sub-layer

thickness reaches a certain value, then for given

molecule length, Yo should have a critical value. It can easily be seen that Yo may be expressed as

follows:

y0/D =

Ro.[8/Re,ff

where R0 is a constant, usually taken as 11.6. It then follows that

From some of the experiments at various pipe

sizes, there is an indication that this is in fact the case, although data are not presented here, since they were not yet complete.

P.EFERENCm.

DAILY, 3. W. AND BIJOUABELLO, G., 1961, BasIc data for

dilute fiber suspensions in unifonn flow with shear, Tappi,

ELATA,C. AND Ipprpj, A. T., 1960; The dynamics of cpen

channel flow with suspension_s of neutrally buoyant

particles, M. 1. T. Hydrodynamics Lab., TechnicaiReport

No 45.

SPRoULL, W. T., 1961, ViscosIty of dusty gases, Nature,

190, 976.

Ssuvan, 0., n Msiuuu., E. W., 1959, Turbulent flow

of pseudoplastic polymer eolutlons in straight cylindrical tubes, A. I. Ch. E. J., 5, 181-188.

RIPERN, 3. F. AND Pn.cn, M., 1963, Studies of the

reduc-tion of pipe fricreduc-tion with the non-Newtonian additive

CMC, St. Anthony Falls Hydraulic Laboratory, Technical Paper No. 42, Series B.

VAN0NI VITO A. n Buooan, N. H., 1957, Laboratory

studies of the roughness and suspended load of alluvial

streams, Sedimentation Lab., California Institute of Terii-nology, Report No. E-68.

SAVINS, 3. 0., 1964, Drag reduction characteristics of

solutions of macromolecules in turbulent pipe flow; Soc. Pet. D*g. Jour., 203-214.

Hoyr, 3. W. Fntn.A, A. G., 1964, The effect of

additives on fluid friction, Fifth Symposium on Naval

Hydrodynamics, Bergen, Norway.

TOWNSEND, A. A., 1956, The structure of turbulent shear flow, 1st ed., Cambridge University Press.

fluay, S., 1960, Accelerations and mean trajectories In

turbulent channel flow, Trans. ASME, 961-972.

HAMA, F. R., 1954, BOundary layer characteristics for

smooth and rough surface, Trans. SNAME, 62, 333-358. HINzn, 3. C., 1961, Turbulent pipe flow, Mdcardque de Ia

turbulwzce, Coil. Intern. du Centre de Ia Rechercbe

Scientifique, 129-165.

Lwny, 3. L., 1964, The reduction of skin friction drag,

Fifth Symposium on Naval Hydrodynamics, Bergen,Norway.

METZNER, A. B. w PARK, M. 0., 1964, Turbulent flow,

characteristics of viscoelastic fluids, 3. Fluid Mecls., 20,

291-303.

cOMPAIUSON BETWEE DRAO RnDUCIION

DIFFERENT ADD1TIVFS.

n k

VALU FOR

Average

concent-Author Material ration

0 '0 Drag reduction k Reynoldsnumber Vanoni sand 0.361 and (0.09mm) 13.5 0.30 5.8 x 105(1) Brooks sand 0.808 28 0.22 5.8x105 (0.15mm) Elata polysterene 15 and particles 5 0.30 2x 105(2) Ippen (0.12mm) 25 10 0.25 2x105 Ripken CMC 0.1 and 28 0.34 5 x 105() Pilch 0.25 64 0.19 5x103 Shaver alginate 0.3 6 0.35 5 x 105(3) and vitanex 0.52 44 0.20 5x105 Merrill CMC 70 0.35 56 0.13 5 x 105