Maximum drag reduction at high Reynolds number for a flat plate immersed in polymer solution

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Tcchnische Hogeschool

MAXIMUM DRAG REDUCTION AT HIGH REYNOLDS NUMBER FOR A FLAT PLATE IMMERSED IN POLYMER SOLUTION

by

Paul S. Granville

This document has been approved for public release and sale; its distribution is unlimited.

Ship Performance Department

ABSTRACT

A logarithmic drag formula is deduced for a flat plate moving at

high Reynolds number in a polymer solutdon for the condition:of maximum

drag reduction. This is a limiting condition which occurs when the boundary layer.is reduced to the interactive layer and the-laminar

A1

NOTATION

Slope of logarithmic' velocity law in natural logarithms

Slope of logarithmic interactive velocity law in.natual

logarithms

Slope of logaithmjc.interative velocity law in common

logarithms

FactOr inllogarithmic formula, Eq.. _17]

B1 Intercept'oflogarithmicvelocity law for condition

of no drag reduction

Intercept of logarithmic ±nträttize,veloci'ty law

Factor in.Ogarithmic formula, Eq.. [7]

Linearizi,ng factors in Eq. '[6]

CF Overall drag coefficient, Eq. [3]

C0

D.,D2 Constants in E. [3]

Drag

Reynolds number'of flat plate

Surface.area of flat plate

u Velocity component parallel to flat plate

u Shear velocity

U Yelociy of flat plate

y Normal 'distance from flat plate

v Kinematic viscosity of solution'

Density of solution

Shearing stress 'at wall

%D.R. Percent drag reduction

:MAXIMUM DRAG REDUCTION AT' HIGI-U REYNOLDS NUMBER

FOR A FLAT PLATE IMMERSED IN POLYMER SOLUTION

The development of the interactive layrer.concep*:Vfor;turbtIiet1t shear flows with drag-reducing polymer solutions provides .a method of predicting the maximum diag reduction for shear flows. A condition of

maximum drag reduction develops if the shear layer is reduced to the

laminar sublayer next to the wall and the interactive layer. A logar-ithmic law describes the interactive layer.

where

u is the velocity parallel to the wall, is the shear velOcity,

t

pis

the density of the fluid,

y is the normal distance from

wall,

v is the kinematicviscosity of the solution, and

-J

A and B are constants.

This law has the same form as that for fluids without drag reduction

= A

+-B10

I.Ar [2]

where A and B1

0 are constants.

Hence, a prediction of maximum drag reduction may be readily made.

for flat plates with boundary layers w.th uniform concéitration of polymer

solution from the results of Reference 2.

Equation 172] of Reference 2 is rewritten.in terms of A and B

in-stead of A and B1 0 as

RLC=1

_+'---

_-+

\2-

where CF is the overall.drag coefficient,. CF

drag of flat plate with area .S at velocity U and RL is the Reynolds

Number, RL

UL/)

Here B2 = 0 and then = A and Thus, there results

LCF&+

or in common logarithms

T

-+

+

- SC2 o:i)

where A1 =2.3026 A.

If CF is linearized with respect to

number of interest 4

IT

iCF

over the range of Reynolds

+

[6]

Then the usual flat-plate form is athieved for the drag coefficient for

niaximurn drag reduction

where

\ocj

g

-5c,

to2

Values of A = 11.7 and = -17.0 are given by Virk et

al.

3 derived by Giles

0ó.3oZ

is also plotted. Also plotted are the laiinar drag coefficient

L32.S

Cç

and. the turbulent Schoênherr law fOr no drag reduction CF

\oc3

RLo OZ4Z

[7]

The percentage of drag reduction, %D.R.., is also plotted in Figure 2

/0t2.(

---oD0/o

where. C, is the drag coefficient for polymer solutions. and CFO is the. drag coefficient for no drag reduction at the.same Reynolds. number. The results are most favorable.

It should be noted that the maximum drag reduction is predicted

on the.basjs of a theoretical model. In practice high shear stresses

will probably mechanically degrade the polymer molecUles and diminish

the.dra reduction so that the maxiwuin drag reduction may not be attained.

The actual friction line :lies:then between:theSthoenherr line of no

drag reduction and the line of maximum reduction. This line may be determthed by the methOd of Reference 2.

REFERENCES

Virk, P S., Mickley, H.. S., and Smith, K. A., "The Ultimate Asymptote and Mean Flow Structure. in Tbms' Phenoenon," Transactions. of ASME,

Journal of Applied Mechanics, Vol 37, Series E, No 2 (Jun 1970) Granville, P.. S., "Frictional Resistance.artd.Velocity Similarity Laws

of Drag-Reducing Dilute Polymer Solutions," Journal of Ship Research,

Vol. 12.., No.. 3 (Sep 1968).

Giles, W. B., "Similarity Laws of Friction-Reduced Flows," Journal

10

10

io6

I

Figure 1 - Drag Coefficient for Maximum Drag Reduction of Flat Plates

I J e - No Drag Recjuct1 (Schoe, Reynolds Number I I I

liii

I III

100 80 60 40 20 0 6 7 8

Log Reynolds Number log RL

U U

I

I

I

I

I

I

I

9, Drag

ledUCt

a

a

a

a

i

I

_I