The torque and turbulent boundary layer of rotating disks with smooth and rough surfaces, and in drag reducing polymer solutions

DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

BETHESDA, MD. 20034

THE TORQUE AND TURBULENT BOUNDARY LAYER OF ROTATING DISKS WITH SMOOTH AND ROUGH SURFACES

AND IN DRAG-REDUCING POLYMER SOLUTIONS

by

Paul S. Granville

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

TABLE OF CONTENTS Page ABSTRACT 1 ADMINISTRATIVE INFORMATION 1 INTRODUCTION 1 GENERAL ANALYSIS MOMENTUM EQUATIONS

RESISTING MOMENT OR TORQUE 6

CIRCUMFERENTIAL VELOCITY PROFILE 6

RADIAL VELOCITY PROFILE 8

SOLUTION OF MOMENTUM EQUATIONS 10

LOGARITHMIC MOMENT FORMULAS 14

GENERAL 14

SMOOTH SURFACES IN ORDINARY FLUIDS 16

ROUGH SURFACES OR DRAG-REDUCING POLYMER SOLUTIONS 17

SPECIAL LOGARITHMIC MOMENT FORMULAS 18

FULLY ROUGH SURFACES 18

ENGINEERING ROUGHNESS 19

POLYMER SOLUTIONS WITH LINEAR LOGARITHMIC CHARACTERIZATION 20

SURFACE WITH LAMINAR AND TURBULENT FLOW 21

LOCAL SKIN FRICTION 22

BOUNDARY-LAYER THICKNESS 23

ORDINARY BOUNDARY-LAYER THICKNESS 23

DISPLACEMENT THICKNESS 25

MOMENTUM THICKNESS 26

SHAPE PARAMETER 27

COMPARISONS WITH EXPERIMENTAL DATA 28

EVALUATION OF VELOCITY INTEGRALS 28

SOURCES OF EXPERIMENTAL DATA 29

VELOCITY PROFILES 30

SKEWNESS OF VELOCITY 31

LOCAL SKIN FRICTION 31

SKEWNESS OF WALL SHEAR STRESS 32

RESISTING MOMENT 32

RESISTING MOMENT FOR CASE OF MAXIMUM DRAG REDUCTION WITH POLYMER

SOLUTION 33

APPENDIX - GENERAL POWER-LAW ANALYSIS FOR SMOOTH CASE 39

REFERENCES 42

LIST OF FIGURES

Page

Figure 1 - Geometry of Flow and CoordinateSystem : 35

Figure 2 - Radial VelocityProfile Comparison for Smooth Case,

R=4.2x106

... . 36

Figure 3 - Radial Velocity Profile. Comparison for Smooth Case,

R=2x106

36

Figure 4 - Comparison of Models of Relative Skewness of

Flow 36

Figure 5 -. Variation: of Local Skin Friction Parameter a with

Reynolds Number for the Smooth Case 37

Figure 6 - Variation of Skewness of Wall Shear Stress with

Reynolds Number for the Smooth Case 37

Figure 7 - Moment Coefficient versus Reynolds Number for the

Smooth Case 38

b0,b11b2

Cf C

m

NOTATION

A S]ope of logarithmic velocity law in natural logarithms

A A defined for interactive layer

A1 Slope of logarithmic velocity law in conunon logarithms

A1 A1 2.3026 A

A Factor in torque formula, Equation (71)

A Factor in torque formula, Equation. (160)

A1 Factor in. torque..formula for fully rough;'surfaces, Equation (81)

0,a1,a2 Constants, in sees. expansion,. Equatiojt (56)

Intercept, of interactive logarithmic velocity law, Equation (159)

B1,B2 Intercepts of' logarithmic velocity law, Equations (18) and' (1.9)E

B3 Constant in quaZiOn (75)

B Derivative of B1 with respect to ln £* B Factor in torque formula, Equation (71)

B Factor in. torque formula, Equation (160)

Factor in torque formula for fully rough surfaces, Equation (79)

Constants in series expansion, Equation (55)

Coefficient of local skin friction

Torque coefficient, Equation (15)

c1,c2 Constants in Equation (70)

c3,c4 Constants in Equation (100)

c5,c6 Constants in Equation (101)

P Constant., Equation (69)

F Outer law function, Equation ('17) '

F Integral of F with respect to z/d, Equation (117)

-2 '

2.

-F , Integral of F with respect to z/, Equation (24)

Fg,F2g,Fg2 Integral of Fg, F2g, Fg2 with respect to' z/ô respectively'

FG ' ntegra1 of FG with respect to

,

g Radial velocity law, Prandtl form, Equation (37)

Integral of g2 with respect to z/6

H Shape parameter

h1,h2,h3 Ratios given in Equations (40), (41), and (42) k Roughness linear dimension

k* Roughness Reynolds number, k*

E UT ()/V

km Torque coefficient, Equation (13)

9.. Length measure

Length Reynolds number, 2 _UT

Threshold value of £* for polymer solutions

M Moment or torque

m Exponent in Equation (51)

P Factor in Equation (84)

Factors in Equations (103) and (104)

rr Normal stress in radial direction

Pq

Normal stress in circumferential direction

rz Shear stress in r-z plane

rc Shear stress in r-4 plane

P Shear stress in z- plane

q Drag reducing factor in Equation (69)

R Reynoids number, R E

-

2

R5 Reynolds number, R = wr

r Radial distance from center of disk s1,s2 Factors in Equation (111)

s3,s4 Factors in Equation (122)

s5,s6 Factors in Equation (131)

t Subscript denoting transition

U Relative velocity outside boundary layer, U =

u Relative velocity inside boundary layer for circumferential direction

u Shear velocity in radial direction, u =i/r Ip

T,r t,r r

u1 Shear velocity in circumferential direction, UT

_=l/Tw/P

Vr Velocity component in radial direction

v, Velocity component in circumferential direction V Velocity component in normal direction

w Coles wake function

z Normal distance from disk

Tangent of angle of resultant wall shear stress from

circumferential direction

Derivative of c with respect to Function given in Equation (47) Derivative of with respect to

Boundary layer thickness Displacement thickness

11 z/5

0 Momentum thickness

Factor defined in Equation (130)

X Function given in Equation (54) dX/da

p Viscosity of fluid (solvent for polymer solution)

Kinematic viscosity of fluid (solvent for polymer solution) Kinematic viscosity of solution for polymer solutions

p Density of fluid

Local skin friction parameter,

T Wall shear stress in radial direction w,r

Driving wall shear stress in circumferential direction

w,4

T Wall shear stress in circumferential direction w,

Azimuth angle

Skewness of velocity, Equation (148)

w Angular velocity

ABSTRACT

The resisting torque of disks rotating in an unbounded

fluid is analyzed on the basis of three-dimensional boundary-layer theory. Smooth and rough surfaces in ordinary fluids and in drag-reducing polymer solutions are considered. A

general logarithmic relation is derived for the torque as a

function of Reynolds number for arbitrary roughness and arbi-trary drag reduction. Special formulas are obtained for smooth surfaces, fully rough surfaces, polymer solutions with

a linear logarithmic drag-reduction characterization, and

polymer solutions with maximum drag reduction. Relations are also obtained for boundary-layer parameters such as thickness, wall shearing stress, etc. The computed results are in ex-cellent agreement with experimental data available in the literature.

ADMINISTRATIVE INFORMATION

This investigation was conducted under the General Hydromechanics Research Program of the Naval Ship Systems Command. Funding was provided under Project R 023-01. Material in this report appeared informally as NSRDC Technical Note HL 133 in June 1969.

INTRODUCTION

A circular disk rotating in an unbounded fluid at rest develops a

resisting torque, or moment, which is wholly viscous in origin. A boundary layer develops on the disk surface which is three-dimensional in that there is a cross-flow velocity component, namely, the radial velocity component. The boundary-layer flow is laminar at the center of the disk and undergoes transition to a turbulent flow at some radial distance from the center.

The turbulent boundary layer on the disk behaves like other turbulent shear

flows in another important aspect: there is increased resistance due to

surface roughness and decreased resistance due to the presence of polymer

additives in the fluid.

The principal aim of this report is to develop analytical relations

between the resisting torque coefficient and the disk Reynolds numbers for the case of roughness and/or polymer additives in terms of boundary-layer

factors. The relations between magnitude and direction of local skin

of the analysis is the similarity-law correlation for ±oughness and/or

polymer additives. If the empirical factors in the similarity-law corre-. lation are known, the torque and other properties of the flow may be pre-dicted. Conversely, if the torque coefficient and Reynolds number are

measured, the similarity-law empirical factors may be deduced. . This is a

most valuable attribute since the rotating disk may then be used as an

in-struinent for experimentally obtaining similarity-law correlations for various types of irregular roughnesses and/or polymer 'additives at high

shearing, stresses. Once a similarity-law cOrrelation is known, it may be

12

used to predict the characteristics of other types of shear flows.

The development of resisting torque on the rotating diSk depends on

the development of the boundary layer on the disk, particularly the turbu-lent boundary layer at high Reynolds numbers. As originally formulated by

von

Krmn,3 the

method of integral relations is based on the integration

of the equations. of motion across the boundary layer for both the circum-ferential, and the radial directions. It is then necessary to supply relations for the circithferentia1 and rad:ia.l velocity components.

In the. classical analyses of von Krmn and Goldstein4 for smooth surfaces, the circumferential velocity relative to the rotating disk is

considered, in effect, to behave like an ordinary two-dimensional similarity-law shear flow except that the reference shear velocity is the skewed or

resultant shear velocity.' Von KrmLi used a power-law similarity law and

'Goldstein. the more geTleral logarithmic similarity law.

Von Krmn assumed

1. Granville, P.S., "The Frictional Resistance and Turbulent Boundary Layer of Rough Surfaces," Journal of Ship Research, Vol. 2, NO. 3 (Dec 1958).

A complete list of references is given on pages 42 to 44.

2'. Granville, P.S.,"The Frictional Resistance and Velocity Similarity Laws of Drag-Reducing Polymer Solutions," Journal of Ship Research, Vol 12, No. 3 (Sep 1968). .

von Krinn, T., "On Laminar and Turbulent Friction," National Advisory Committee for Aeronautics TM 1092 (Sep 1946), translated from

Zeitschrift 'far agewandte'Mathemaik und Mechanik, Vol. 1, No. 4 (Aug 1921).

Goldstein, S., "QT' tb Resistance to the Rotation of a Disc

Immersed in a Fluid," Proceedings of Cambridge Philosophical Society, Vol. 31, Pt. 2, p. 232 (Apr 1935). ' '.

that the radial velocity component was.proortional to the

relative

circum-ferential velocity componnt and the skewness of the wall shear.ing stress,

with a linear correction to provide zero velocity at the edge of the

boundary layer. Goldstein, on the other hand, assumed that the radial velocity componetit (except very close to the wall.) was proportional to the

absolute

circumferential velocity compoent, which automatically provides

zerO velocity at the edgeof the boundary layer. Comparison with

experi-mental data shows that the von Krmn model overestimates the radial

velocity component whereas the Goldstein model grossly underestimates it.

As stated before, the von Krmn and Goldstein analyses apply only to

smooth surfaces. Dorfmàn5'6 extended the Goldstein analysis to the special case of the fully rough surface.

In the approach followed here, the boundary layer on the rotating

disk is considered as a three-dimensional boundary layer The st:reanwise

velocity., component in the boundary layer is then the circumferential

velocity and the cross-flow velocity component is the: radial velocity. The similarity law for rough surfaces and for polymer solutions 'which includes the smooth case is then applied to the circumferential velocity. The

cir-cumferential wall shearing stress is taken as the reference wall shearing stress instead of the resultant wall shearing stress as used by von KArmn

and Goldstein. Certain inconsistencies are thus. avoided. The actual difference In magnitude of these two wall shearing stresses: is negligible.

The radial velocity model used is that of Prandtl which is a generalization

of the von Krmn model.

.

A general logarithmic. formula is derived for the resisting moment

of rotating disks which applies to arbitrary rough surfaces and/or

drag-reducing polymer solutiOns. The special cases considered include:

Smooth surfaces. . .

Fully rough surfaces.

Polymer solutions wjth a linear logarithmic drag-reduction

characterization.

Dorfman., L.A., "Drag of a Rotating Disk," Soviet Physics-TechMcal

Physics, Vol.. 3, No. 2, p. 353 (Feb 1958).

Dorfthan,, L.A., "Hydrodynamic Resistance and the Heat Loss of

4. Maximum drag reduction with polyier solution.

Formulas are also derived for the local skin friction, the

boundary-layer

thickness, the displacement thickness, and the niOthentüin thikes as

functions of Reynol4s number. In addition, solutions ae Obtained for the

skewness of the boundarylayer velocity and for the skewness of the wall

shearing stress. Comparisons are made with existing test data.

The appendix gives a generalization of the von

Krmn

power-law

analysis for 8mooth dIsks which provides simple formulas for limited ranges of Reynol4s number. The more àccuraté quadratic. Mager model is used here for the radial velocity profile.

GENERAL ANALYSIS

MOMENTuM EQUATIONS

The steady-state equations of motion of turbulent flow in the

neighborhood of a rotating disk4 are in cylindrical polar coordnates (see

Figure 1) r, , and z with corresponding mean velocity components vr, v, and v.. For the radial directiOn the equation is

v

v2

)

r r z 3z r . pr r rr pr p .z

and for the circumferential direction, it is

v + v r

1

r r z z r pr ar r pr

where

rr and p are the normal stress components;

rz'

p1, and p

are the shearing stress components; and

p is the density of the fluid.

The turbulentReynolds stresses are included. The corresponding equation

of continuity is

!Hrv)

+=o

(3)

'I

d

/2

-dr 2 [r3c5

S

(v)2

von

Krmn3

originally derived these equations from mothentum considerations-across flow cOntrol surfaces.

In Equations (4) and (5), Tw,r is the radial cpuiponent of the driving

wall shearing stress and T is the circumferential component of the driving wall shearing stress. The boundary conditions for a disk rotating

with angular velocityware v=wr for z = 0 and

V=Vr= 0 for z = 6.

The momentum equations are changed to the usual form of boundary-layer momentum equations by considering the velocity relative to the

rotating disk u in the circumferential direction

- u = v - rw

and, the relative circumferential wall shearing stress T is

w, -t

-

T.' w, 6

J

v v dz r 0

(*)]- r2w26

= - r2 (1 (5) )2 d (-)=.. r Y (8) A three-dimensional, boundary-layer flow exists next to the rotating disk with thickness 5. Integration of the equations of motion across the

boundary layer and incorporation of the equations of contthuity result in the von Krm.n momentum

equations4

for a rotating disk; in the radial direction, this is given by

-

(r ,'

v2dz)_

J

v dz = - r

T

(4)

0

In the circumferential direction, it is given by

Then the boundary conditions become u = 0 for z = 0 and u = U = rw for z = 6,

and

or

- Ir

dr

RESISTING MOMENT OR TORQ!JE

The resisting moment or torque is given by

dM = 2rrr dr) r

or for the whole disk area on one side

r

r 2

M=.27r1

T r dr

W,

-ó

From the circt.mferentia1 moment equation, E4uation(9),

= 2p rw

_{() (}

I

0

-

.-

. -

,

'3

A dimensionless moment coefficient km is. defined by von Karman- for

both

sides of the disk as

-1

__1

-

52.

t I

prw

J0

A frequently used alternate moment coefficient C is defined as

6

C 2 k

Ii' In

CIRCUMFERENTIAL VELOCITY PROFILE

The circmiferentià1. flow relative to the disk u, 1.e, the relative

streamwisè flow, is assumed to obey .sirnilr-iZ> laws like that of pipe flow

iv

U

4 IZ

7 ViF7

U (13) (14)

(10)

(12)

and of boundary-layer flow on flat plates'2 .there is

_{no strewise}

pressure gradient for the disk flow here. In analyses of three-dimensional turbulent boundary layers, the streainwise flow in the boundary layer is considered as a two-dimensional flow.

The inner similarity law fOr all cases including rough surfaces and

polymer solutions is given by

ru

z

Lv

£*

(16)

where 9 _(UT 2./v) and 9.. _{s a characteristic length scale of roughness} or polymer. The outer similarity law is

--

FI-UT LIS

where uT =

T/P

is the tangential shear velocity and v E 1.1/p is the kinematic viscosity of the fluid.

The two similarity laws are considered to overlap so that it can easily be shown1 that within the region of Overlap, the inner law becomes

A in {* ,.00]

I ,

and the outer law becomes

UT =

- A ln*+ B2

B1 is a constant for smooth surfaces in ordinary fluids but, for

rough surfaces and for polymer solutions, it varies with 2..

_andother

pertinent factors which are constant fora given ituation. Equations (18)

_ai'4

(19) produce

u 6 a = A in + B1[ _,...] T , U T,q) (17) (i8)

Also

where

F2 d (i-) ,a constant (24)

RADIAL VELOCITY PROFILE

The radial velocity profile vr.represents the cross-flow velocity

component of three-dimensional boundary layers. Prandt17 suggests a cross-flow velocity profile of form

U

(25)

where g[1] 0 and ct the tangent ofthe angle of the resultant. wall shearing stress from the circumferential direction.

Von Krinn had

already implicitly used this form for the disk with

7. Prandtl, L., "On Boundary Layers in Three-DimensiOnal Flow, B I G S 84 (Aug 1946), also British M A P Report and Translation 64

(May 1946).

a

A inJ+ B1[9* ,.J +B

(21)

The outer law encOipasses almost all the boundary láyér.1 Adcord-ingly, for considerations Of meitin changes, the Outer law is asSüièd to hold all the way to the wall, 1 e , 0 < z/6 < 1 Then

g[t]=

i-f

as the simplest relation that satisfies boundary conditions. For cross flows in general, Mager8 finds a good fit to experimental data by

2 rzl

(

z\

[J=

_'-T)

Of course a g[z/cS] may be selected for rotating disks that conforms exactly to the experimental findings. Other forms of cross-flow velocity profiles are the triangular relations of Johnston9 and the polynomial relations of Eichelbrennér.10

Goldstein4 assumes somewhat arbitrarily a radial velocity profile of the form

v

r

U

U\u

everywhere except close to the disk; for the remaining distance close to the disk wall, he assumes

V

r u

IT- =

iT

Mager, A., "Generalization of Boundary Layer Momentum-Integral Equations to Three-Dimensional Flows Including Those of Rotating Systems," National Advisory Committee for Aeronautics Technical Report 1067 (1952).

Johnston, J.P., "On the Three-Dimensional Turbulent Boundary

Layer Generated by Secondary Flow," Transactions of American Society of

Mechanical Engineers, Journal of Basic Engineering, Vol. 82, Series D,

No. 1 (Mar 1960.).

Eichelbrenner, E.A., "La Couche Limite Tridimensionelle en Regime Turbulent d'un Fluide Compressible," NATO Advisory Group for Aerospace

SOLUTION OF MOMENTUM EQUATIONS

From the relation fOr boundary-layer thickness (5, Equation (21), and

since by definition - U

rw

(30) 2 2 a- a and

the momentum equations, (8) and (9), become for the radial direction p

and for the circumferential direction

The Reynolds number R is defined as

2 a2

d[R3/2

e T T w,r = w, p - p i

A clue to th& form of th solution to the citumferentia1thomentwn

Equation, (33), namely R[aJ, is indicated as fôl1ows If E4uation (33) is

rewritten, as -a-B1 B2 A 10

() d(f

13/2

dR

d?jda.

)

d(f)

a

an inegration by parts from R = 0 to R = R results in cY-B-B 1

R=5ea3S

()(d(

and and in genéràl are constants.

0

r

1 10

I

IR3'2 e A a2

J

.R3'2

L

0

0

1 2

d(f)

= a2 g2/

d()=HFg

(1 2h2 h3 h1 a (38) (39) Sc:

/V \

iv \

LI)

\U / \U /

For the Prandtl form of the radial velocity profile, from Equations (22)

and (25),

= a g

ft]

(1 . F (37)

the integrals of the velocity profiles required for Equations (32) and (33)

become where h1 = Fg2/Fg (40) h2 = Fg2/g2 ₍₄₁₎ h3 = F2g2/g2 (42)

[1 =

[1

d ( (43) (

K)

d

(.)]da

(36)

The momentum equations, (32.) and (33), become for the radial direction.

g2_[a-B--B

a-B -B

c2Re

_'

_a(1

+)]=+(e

()

and for the circumferential direction

where

r

a-B-B

e

A

(i

-2 A

R=ca e

where

[a]

is to be determinedas well as

Prior to the solution of

[a],

[aI, and a[a], it is expedient to

obtain some intermediate relationships.

The factor B1 is constant for

smooth surfaces but B1[2* ,"] for rough surfaces and for

disks in

drag-reducing polymer solutiOn. Since, by definition., 9

_u

9../v

i'7

R1"/a,

then

dB1[2*]

- B'

(

-_

-

l\2Rdc7

where E

dB1/d[ln 9}.

Likewise

From the partia,l solution of the circumferential equatiOn, (36

a-B-B

R

a2

e A

[.1

Hence let

a-B1-a-B-B r

e

21

(2A+ B)

da

(2AB)

L

a

12

h1'

1.

R'2

dR

a!] - 2

2

da

(45)

(48)

(49)

(46)

[a]

(47)

Hence Also, in general, where d do [e(m a' = and ' -do da dB1[.*] _B do =

(2A+B)

2m 2A + B e [1

/

2h A 1a

Ii

+ a 2 2 2 2h h

r

B' (2A +

B)

, 2A + B 2g a a (1 - +

_--i

_I]

_-

L

(a.

₊ a

_a2/L

2 a _2a

fF22

)[l 2A + A (a-. +

fj

2 (50) (51) where m is a constant.

By using R[a], given in Equation (47), and performing the indicated

operations, the circumferential momentum equation, (45), is reduced to

(2A+B)[3.

h1 ]

+A.-+.j(3A.B')

+ 5 La 2 a

a(i4)

a

=

[

2AB

The radial momentum equation, (44), is reduced to

(52) (53) (54) 5Fg (1 a-B. -B A

L'

B 2

and

Let

and

Substitution of the telãtions for and into the reduced momentum

equatiOns, (52) and (53), and equating coefficients in like powers of 1/a

produces b (2A + B) = - dà / b1 b2

Ii+- +-.--+

a a2

+.

I)

14 (55) (56) b0 = 5 (57)

* (2A + B)

h1 /

3A+4B

=*(2A4 B) (h1 +

and so on.

LOGARITHMIC 4)MENT FORMULAS

GENERAL

With the insertion of the relation for 6 from Equation (21), the relation for the moment áoefficient, Equation (13), becomes

1

A'

k

ae

m R

U)'lU)

0

Using the relations for the radial velocity profile and the appropriate

solution, R[a], gives km[a]; km is then related to Reynolds number R im-plicitly through parameter a. However, a may be eliminated and km related to R in logarithmic formulas to follow.

By inserting the velocity profile integral, Equation (39), and the expression for R, Equation (47), the overall force coefficient km in Equation (13) is related to the local force coefficient a as

and

1

r

(58)

By reiteration, the a's within the brackets are replaced by km'S such that

41Tg/

h

(1L

(59)

k=

' a

More explicitly with given by the 'series, Equation (55), and b0 by Equation (57) + h1) 4irj E

l(b1

b1(b1 + h1) - b2 + + (60) (61) - a or 2 4irl (b1 + h1) [1 + a2 b1(b1 '-h1) - b2 a = in a 2 4

a =j1 -

(b1+h1)

-mL

or by binomial expansion 2 m (62) (63) (b1+h1) h1+3h12)

v- viç

-

j-

(4b2-b12+2b1 _4ii a =

v:i k[

Inserting the velocity profile integral, Equations (39) and (21), into the moment equation, (13), yields

a-B1-B2.

4i1

A

k=ce

Pg

Substituting for by EquatiOn (56) and taking the natural logarithm results in

ln km ln 4ir - in R

(66)

Eliminating the

d's

through Equations (62) to (64) gives, in general, (1 h1 a + ln a - ln a +

0

A B

+B) -

:.. -

_:-A

2

vV

16 .[--- b12.+ 2 b1h1 +

3h12).+

a1

I/+ ..

(67) SMOOTH SURFACES IN ORDINARY FLUIDS.

Throughout the report, these are termed the smooth case. Here B = 0, and the logarithmic moment law becomes in common

logarithms . . (64) (65) - Al log Fg

,/Y)

D +

S..

(68). Where A1 2.3026 A and 1 = A in R ₊

,/log

RI/ic +

1/.(B10

+ B2) -A a a

ROUGH SURFACES QR DRAG-REDUCING POLYMER. SOLUTIONS

Ignoring the negligible effects of B in the coefficients such as b1 and b2 before uiç in Equation (67) (see Equation (57)) provides a logarithmic moment law for rough surfaces and for disks in drag-reducing polymer. so

lutions ... A

AI

iF

4r

cD

(69) and B c2b)

_[v

(B10 + B2) =

A1/log (I5

)+.c1D]

(7)

g

Equation (71) is the same general form for the logarithmic moment jaw as

obtained by Goldstein.4

1

AlogRJ/ç+B+

1

where B10 = B1 for smooth surfaces in ordinary fluids.

Bi]

1,0' (74)

If is linearized with respect to 1/

+ ₍₇₀₎

and terms of order higher than are neglected,then

I

- A log Rvç

+ B

SPECIAL LOGARITHMIC MOMENT FORMULAS

FULLY ROUGH SURFACES

and

18 1

For fully rough surfaces

B1B3_Aln*=B3Alflk*

(75)

where B3 is a constant. Here 9= k, the roughness length scale. Then B = -A.

Since by definitiOn

k* = k

a r (76)

= B3 + A1 log a - A1 log R + A1

iogf

(77)

From Equation (63) 1 (2A + B) (78) 4'rr

loga

log -ç-- log 2.3026

+ ..

Then from Equation (67)

lpg-+

B1

(79)

=l

1 (80) where log

_{.j. +}

B A

V4

1 (81) 1

c_5A/11

-4ir (,50A h1

B1

ENGINEERING ROUGHNESS

For surfaces with engineering roughness (Colebrook-Whité roughness)1

/ k* B = B

- Am

(1+-1 1,0 \ p where B2-B10.

p= e

A (84) Then

From Equation C76) and Equation (64),

1 1

+-(L

R-)=

+1/R

f+

p p a r

p p 4rj

m m Likewise 1 \+B =

Substituting into Equation (67) produces

h\

h1

-.i)

(82)

(83)

20

2

For r/k, + , Equation (88) reduces to the mQmnt formula for smooth

sur-faces; for RIf+

, Equation (88) reduces to the moment formula for fully

rough surfaces.

POLYMER SOLTIONS WITH LINEAR LOGARlTHMIC

CHARACTERIZATION*

It has been found experimentall,y that for spme polymer

solutions2

B1

is a linear 1garth4c function of 9. Then

B10 - A1 log

+ q(log £* - log ) 2* >

B1 B.1 - A1 10

O

where B10 is a constant,

is the kinematic viscosity of the solutioii,

v

is thekinematicyiscosity of thesolvent,

is the drag-reduction threshold va]ué of 9, and B = dB1/.3d (log ) = q/2.3..

Since from Equation (76) and Equation (64),

R

r

VvcL1

_m;'

r

(2A + B')

After the present report was written, Poreh and Miloh extended the

Goldstein analytical model to this case ("Rotation of a Disk in Dilute Polymer Solutions," Journal of Hydronautics, Vol. 5, No. 2, Apr 1971).

-A

B

a

a]

then

B1 - B10 = - A1 log + q [log

Ryç_. log,--+ log

Physically, the flow in the boundary layer is laminar starting from the center of the disk at r = 0 and undergoes transition to turbulent flow

ãt.r =.r specified by a Reynolds number of transition Rt.

Although the logarithmic moment formulas are derived for complete turbulent flow over the disk, a correction may be made for the presence of

(2A - log 9,* + S...] _(.91) 0 Finally where (92) 47r 1 r A log R 2 - 1-c2 D q ±og £

its

and ' 4'rr (93) + .1-c2 D q A1 log + (94)

B2 = B

+

c2D

q log - 5(2 3026) q log

The onset of drag reductiOn = 1, then from Equation (90),

is given by 9,* =

£*.

If 9, is chosen so

the onset of drag reduction

occUrs

when

(95.)

(2Al+)/]

SURFACE WITH LAMINAR AND TURBULENT FLOW

For laminar flow6

.1.935

km (96)

laminar flow in the central part of the disk. The correction is required only for an intermediate range of Reynolds numbers.

The moment of the area up to transition is subtracted for the

turbu-lent flow and added on for the laminar flow. Then the corrected k for m the disk with partial laminar flow becomes simply

/R\512r

km = (km)turb -(--)

_{I(1(m,t)turb -}

(kmt)iam (97)

Obviously the correction becomes smaller for increasing R. _R = 2.9 x 10

has been observed for smooth surfaces)1

LOCAL SKIN FRICTION

The components of the local skin friction or wall shearing stress

and

are expressed in terms of parameter a by Equations (30) and

(31); a may be obtained as function of Reynolds number R as follows:

Substitution of ci and from Equations (56) and (55) in the relation

for R, Equation (47), produces a-31-B2

A /

a+b

a,+ab +b

R=ab ae

(1+

1

l,

11

2...j

(98)

00

a

a2

1

To obtain a more compact although slightly more approximate relation, the logarithm is taken and terms of order 1/a and higher are dropped so

that

A(a1+b1)

a

= A1 log R -

A1 log a b

₀₀

+ B +

B2 - A1 log a -

+ ..

(99)

1 a

11. Gregory, N. et al., "On the Stability of Three-Dimensional

Boundary Layers with Application to the Flow Due to a Rotating Disk,"

Philosophical Transactions of Royal Society, London (A)

248, p. 155 (1955);

also in "Boundary. Layer Effects in Aerodynamics," Symposium at National

Physical Laboratory, Great Britain (1955); published by Philosophical Library,

New York (1957).

where A1 - 1 + c4 A1 + C6 A(a1 + b1) and by and nondiinensionally by or

B1 +B2 -

Ajiog

a0b0 + c3 + 2.3026 (a1+b1) - 1 + c4 A1 + c6 A(a1+b1)

For the smooth case (ordinary fluids), p1 and p2 are constants. Also for a coefficient of local skin friction C where C

E T

f f w,

2

BOUNDARY-LAYER THICKNESSES

ORDINARY BOUNDARY-LAYER THICKNESS

From Equation (21), the ordinary boundary-layer thickness is given

a-B1-B2 va A

= - e

rw (p1 log R + p2) 2 pU (103). (106) (107) If log c3 + c4 a (100) and 1 C5 + C6 a (101) then a = 'l log R + p2. (102)

or where and Also 0- 131 A

To obtain

Was

a function of Reynolds number R, the coimnon logarithm of Equation (107) produces

log log o log R

+

s2= p

p2 ,+ C3

+ C3

-24 (108) (109) (110) (112) (113) (114)

The log o àndo are convert:éd to functions of R through Equations (100) arid

(102).

Then

log 5VF

_{[il (*}

+

c4)

DISPLACEMENT THICKNESS

From the usual definition of displacement thickness, 6

6 ES

(1 -dz (115) (116) where

FE

Fd()

(117) or From Equation (107),

and from Equation (108),

log

6*

r

log

6*/=

log

To obtain 6*Vas a function of R., the logarithm is taken so that

a- B1 - B2

A

a-B1-B2

A

The elimination of a by means of Equation (102) produces

\ -2 B1 jlog R + log F / .1 (118) (119) (120) (121) U) S3 (122)

where and Also 1

42

=s R

_{r 3} Then MOMENTUM THICKNESS

From the usual definition of momentum thicknêss,

GE

4 A .2 1 2 a From Equation (107), A

(4)

26 (124) (125) (126) log s3 og F +--. (p - B1 - B2) (123) or ü = - 01 (129) where

Then as before,

where

and

Finally,

SHAPE PARAMETER

The usual definition of shape parameter

H is

or

log 55 = log

- c.3

I-

S 1W 6

°iVi

S5 R

= 1

-

c4)

-S4

Oy.-=s3R

-s5

H=

cS*

H

=-Then from Equations

(125)

and

(134),

i

-

-J- R64

S3

1 B2

_/1

Ai2(,A,I

C4)

(130)

(131)

(132)

(135)

(136)

(137)

COMPARISONS WITH EXPERIMENTAL DATA

EVALUATION OF VELOCITY INTEGRALS

Numerical values are required for the various velocity integrals designated in the formulas derived in this report. These may be obtained

directly from experimental data or be evaluated indirectly from analytical

models as follows:

For the circumferential velocity, Equation (17),

F=-Aln+B2

(l_.)

(138) Here r E z/5 and w is the Colès'2 wake function which is given a polynomial

13

fit by Moses as

f=3i12-2n3

(139)

For the radial velocity, Equation (25), the Mager relation is used: 2

g = (1-n).

The various integrals then become

g

4(A

+fB2)

29 2 2 1 Fg = -(-A2 A B2 + -j-- B2 ) -'2 137 73 Fg

--A -

B2 28

Coles, D., "The Law of the Wake in the Turbulent Boundary Layer,"

Journal of Fluid Mechanics, Vol. 1, Pt. 2 (Jul 1956).

Moses, H.L., "The Behavior of Turbulent Boundary Layers in

Ad-verse Pressure Gradients," Massachusetts Institute of Technology Gas Turbine Laboratory Report 73 (Jan 1964).

Then

SOURCES OF EXPERIMENTAL DATA

An account of early résistance experiments with rotating disks is

given by Dryden et al)4 Data fOr higher Reynolds numbers and for smooth and rough surfaces are presented by Theodorsen and Regier)5 Resistance

measurement with polymer solutions are reported by Hoyt and Fabula,16 by

F. = A + 2 B2 85 2 274 29 2 h - A + .A B2 + - B Fg

3(--A +4.B2)

.137 73 60 A

28B2

(145) (1.46) (147)

Dryden, J.L. et al., "Hydrodynamics," National Research Council Bulletin 84 (1932); reprinted by Dover Publications, New York, p. 352 (1956).

Theodorsen, T. and A. Regier, "Experiments on Drag of Revolving

Disks, Cylinders and Streamline Rods at High Speeds," Natiohal Advisory

Committee for Aeronautics Technical Report 793 (1944).

Hoyt, J.W.. and kG. Fabula, "The Effect of Additives on Fluid Friction," 5th Symposium on Naval Hydrodynamics, Office of Naval Research

ACR-ll2, U.S. Government Printing Office,. Washington,, D.C; :(Sep 1964).

= 2 A2

Anifilokhiev and Ferguson,17 and by Smaliman and Wade.'8 Velocity surveys

11 . 19 20

have been performed by Gregory et al., by Stain, by Gadd, by Chain and

21 . 22

Head, and by Erian.

VELOCITY PROFILES

For the

smooth case,

Stain19 experimentally obtained for Equation (19) values of 2.3026 A = 5.6, B1 = 5.6, B2 = 0. These values will be used for numerically evaluating the formulas of the report.

Radial velocity profiles from the Mager model compare well with ex-perimental data for

smooth surfaces

shown in Figures 2 and 3. Results from an eddy-viscosity model by Cooper23 are also compared in Figure 3.

Amfilokhiev, W.B. and A.M. Ferguson, "The Variation of Friction Drag with Surface Roughness in Dilute Polymer Solutions," University of

Glasgow, Department of Naval Architecture Experiment Tank Report 8 (Aug

1968).

Smailman, J.R. and J.H.T. Wade, "The Influence of Hydrodynamic Drag of High Molecular Weight Compounds," (Canada) C.A.S.I. Transactions, Vol. 2, No. 1, p. 37 (Mar 1969).

Stain, W.C., "The Three-Dimensional Turbulent Boundary Layer on a Rotating Disk," Mississippi State University Aerophysics Department, Research Report 35 (Aug 1961).

Gadd, G.E., "The Effect on the Turbulent Boundary Layer of Adding

Guar Gum to the Water in Which a Disk Rotates," National Physical Laboratory (England), Ship TM 42 (Nov 1963).

Cham, T-S. and M.R. Head, "Turbulent Boundary-Layer Flow on a

Rotating Disk," Journal of Fluid Mechanics, Vol. 37,. Pt. 1, p. 129 (Jun

1969).

Erian, F.F., "The Turbulent Flow due to a Rotating Disk,"

Clark-son College of Technology (Potsdam, N.Y.) Department of Mechanical Eng-ineering (Apr 1970).

Cooper, P., "Turbulent Boundary Layer on a Rotating Disk

Calcu-lated with an Effective Viscosity," AIAA Journal, Vol. 9, No. 2, p. 255

(Feb 1971).

SKEWNESS OF VELOCITY

As shown in Figure 1,, the skëwñess of boundary-layer flOw is given

by where

tan - ₍₁₄₈₎

At the wall, z = 0, tan =

a.

The relative skewness is given by For the Prandtl radial-velocity model,

tan t4

a.

Hence the von Ka'rma'n linear model is

tan

- 1 -

--a

-

-and the.Mager quadratic model S

tanq_ (

\2

-\-1

Up to the wall, the Goldstein model is

and at the wall, it is

t$fl

= 1 a

tn

U U (149)

The comparison 'in Figure 4 shows that the Mager model lies between the other two. _{Since the Mager model is close to the experimental data, it} is considered that the flow skewness is overestimated by the von Karman model aiid underestimated by the Goldstein model.

LOCAL SKIN FRICTION

The variation of local skin friction parameter _{with Reynolds number}

is given by EquatiOn (102). Over the usual range of . (20 to 28), it is

c6 = -0.0018. The local skin friction for the

smooth

case, Equation (102), becomes

a = 4.96 log R - 5.74 (154)

This compares very well with the test data shown in Figure 5. The test

data for local skin friction were obtained from the velocity profiles by

the Clauser procedure.

SKEWNESS OF WALL SHEAR STRESS

The skewness of the wall shear stress a. was given by Equation (56); to order , it is

a!

a

. _2. (i

_L

a a

For

smooth

surfaces with A1 = 5.6, B1 = 5.6, B2 = 0, Equation (57) gives a0 = 3.845 and a1 = 1.738. A close fit obtained graphically over the usual

ranges of values of a produces

a 0.0107 (156)

a

Substitution of Reynolds number R for a in Equation (102) yields for the

smooth case

4.395

= 4.96 log R 574

0.0107 (157)

Figure 6 shows a favorable comparison with test data and other pre-dictions. The test data were obtained visually from streak lines that emanated from artificial roughnesses in a sublimation process.21

RESISTING MOMENT

The resisting moment coefficient as a function of Reynolds number was given by Equation (71) as

- A log

R%/ç +

B

By a graphical determination, i.e., c1 = 0.124 and i.e. c2 = -0.00379.

32

For the smooth case with A1 _{5.6, B1 = 5.6, and B2 = 0:}

- 3.456 log Ri/ic - .176 (158)

Figure 7 i-ndicates a very agreeable correlation with more recent test data which were unavailable at the time of the Goldstein _correlation in 1935. There is still need for accurate data at high _{ReynOlds numbers.}

RESISTING MOMENT FOR CASE OF MAXIMUM DRAG REDUCTION WITH POLYMER SOLUTION

24 The development of the interactive layer _{concept by Virk et al.}

for turbulent shear flows with drag-reducing _{polymer solutions provides a}

method of predicting the maximum drag reduction for _{rotating djsks. Such a}

condition of. maximum drag reduction develops if the boundary layer is reduced just to the laminar sublayer next to the wall and the interactive

layer. _{A logarithmic law describes the interactive layer; for rotating}

disks, it may be written in the circumferential _{direction as}

T,

V (.159)

where

where A and B are constants.

_Thi

_{equatOn has the}

Equation _{(18) fOr the smooth case, B1 = constant.}

Hence the torque formula for the smooth case if A is substituted for A and B for B or

- A log

R.i/+

same form as

Equation (71), holds

(160)

24. Virk, P.S. et al., "The Ultimate Asymptote and _{Mean Flow Structure}

in Toms' Phenomenon," Transactions American Society of Mechanical _Engineers, Journal of Applied Mechanics, Vol. 37, Series E, _{No. 2 (Jun 1970).}

and B (l-c2D) + - log

g/c2+

C1DI (162) A1 = 2.3026 A

Since there is no overlapping, B2=0. From Equations (69), (146), and (147), D = 3.026 A2/rr. From Equations (140), (144), and (143) Fg = 11/18 A, F2=2A2 and -2 = 1/5. Then

vc

A-1 and A1V 47r 3.026 -2

_Ac

ii 2 (161) (163) B-1

c --.

log (-

v

2) 3.026 2 = (1 3.026 2

C2)L

(164)

With the values of A=1l.7 and B= -17.0 given by Virk et al.,24 the resisting

moment for the case of maximum drag reduction with polymer solutions

be-comes

1

= 11.33 log R5vc - 32.44

Here R E wr2/v where is the kinematic viscosity of the solution.

The comparison in Figure 8 shows excellent agreement between Equation (165) and some test data of Hoyt and

Fabula)6

34

SKEWNESS OF FLOW

V

RELATIVE ANGLE, tan p = 0 = ABSOLUTE ANGLE, tan 0 = V

V

atz=O,tan0=O,tancx

arc

SKEWNES$ OF WALL SHEAR STRESS,

-w,q

TYPICAL VELOCITY PROFILES

- = RELATIVE CIRCUMFERENTIAL VELOCITY

Figure 1 -. Geometry of Flow and Coordinate System

N z

_1-

r

RADIAL DISTANCE AZIMUTH ANGLE

(f) z NORMAL DISTANCE FROM DISK

o v

r

= RAbIAL VELOCITY

0

uJ v CIRCUMFERENTIAL VELOCITY = NORMAL VELOCITY -J >-LJ I-= ANGULAR VELOCITY

M = RESISTING MOMENT OR. TORQUE = RADIAL LL SHEAR STRESS

-CIRCUMFERENTIAL WALL SHEAR STRESS

1.0 0 0.12 0 0.08 = 0.06 0 _, 0.04 C 0.02 0

NORMAL DISTANCE z INCHES)

Figure 2 - Radial Velocity Profile Comparison for Smooth Case, R = 4.2 x 106 0 36 R I 4.2 x DATA-GADD p REDICTED I io6 SMOOTH I (REF. (MAGER I CASE 20) MODEL) I

xx

XTEST 0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.: 0 I TEST PREDICTED PREDICTED 1 R DATA -(MAGER I 2 x CHAM COOPER 1 io6 SMOOTH & HEAD MODEL) (EDDY I (REF. EQ. (27) I CASE VISCOSITY) 21)

r

0

RU

-("r4!lIIU

/1p4P 4 5 6 7 8 9 10 11 12 RELATIVE NORMAL DISTANCE.

Figure 3 - Radial Velocity Profile Comparison for Smooth Case, R = 2 x 106

01 02 03 04 05 06 07 08 09 1.0

RELATIVE POSITION IN BOUNDARY LAYER+

Figure 4 - Comparison of Models of Relative Skewness of Flow

0.8 0.6 I.. U, U, 1 0.4 w uJ = 0.2 0.12 >1 _0.10 0.08 0.06 0.04 0.02 0

30 0.4 0.3 0.2 0.1 28 26 2 2 S 02 04 06 08 5 w2 log R =

1og--Figure 5 - Variation of Local Skin Friction Patameter a

with Reynolds Number for the Smooth Case 4----

_{-EQ. (102)}

CHAM HEAD TEST_{(A1 5.6,}

TEST DATA(REF. (-REF. 4) DATA (REF. B1 = 5.6, 19) 21) 2 = 0) X, STAIN ---GOLDSTEIN GOLDSTEIN (EF. 4) HEAD

VON K.ARJIAN (REF 3)

02 04 06 08 ₆ ub

logR

Figure 6 - Variation of Skewness of Wall Shear Stress with

6 18 17 16 15 14 13 12 0 0.4

7

.-04 06 38

00

00 A 0

00

00 I I 0 8 0.2 0.4 0.6

0 THE000RSEN & REGIER - 24 DISK - AIR & FREON

X HOYT & FABULA - 18 DISK - WATER (REF. 16)

C SNALLMAN & WADE - 10 DISK WATER (REF. 18)

log RV

Figure 7 - Moment Coefficient versus Reynolds Number for the Smooth Case

06 08 ₆ 02 04

2

06 08

log R

=

1og-Figure 8 Maximum Drag Reduction with Polymer Additives

0.8 2 (REF. 15)

6

4

0

TEST DATA - HOYT

0 3 INCH X 18 INCH D 36 INCH

& FABULA (REF.

DIAMETER DIAMETER DIAMETER 16) = 11.33 log R km - 32.44 EQ. (165), 02 4 E 3 2

APPENDIX

GENERAL POWER-LAW ANALYSIS FOR SMOOTH CASE

It is of interest to repeat the von Krm analysis for the smooth case with two principal improvements: (1) the power is left arbitrary for

the similarity law and (2) the quadratic Mager model is used for the radial velocity profile..

Although von Krmn assumed that the similarity law held for the resultant of the circumferentjal and radial wall shear stresses, it is preferable to make the simpler assumption that the similarity law holds

only for the circumferential wall shear stress. Consequently

____ - C [nJ

(uTz)

(Al)

where C[n] is a constant for a particular n and n is an arbitrary power.

Then

n

u (z U

-For the radial velocity profile, the Mager model

v

r

u/

z

becomes

vn (l-.)2

Substitution of the velocity profiles, Equations (A2) and (A3), into momentum equations, Equations (8) and (9), produces

2 2 1 dr - f2 r2 w2 - rc (A4) and 2

d4

w T = r2 w,4 p (AS)

where

and

Let

t =

Yr

where I and t are constants.

Then

T

p and with Equation (Ag),

1 4 6 4 1. .2n+1 2n+2 + 2n+3 2n+4 + .2n+5 2 2n n+1) (2n+1) 1 2 1 1 2 'l - n+1 n+2 + n+3 2n+1 + 2n+2 2n+3 2 a

22

1+ -T

w a

f,

'I(3+t)

r

f

=

-From Equations (AlO)

and

(All),

2.

2n

2n

2(1-nt)

n+l n+l n+l .

ñl

ñ+l

w y r

40

- (3+t)f1 + (4+t)f3

From the sithi1aiity law, Equation (Al), the circwnfërêntial wall

shearthg stress is . . ' (A9) (Al 2) (Al 4) and 2 a £3

Y(4t)

rl+t T (All) p .(Ai0) 2. 2 2n 2n p = c_

(rw)'

-l.

(Al 3)

Substitute Equation (A14) into E.4iation (All) and equate powers so that

Then Equation (Al2) becomes

(l+3n)f2

(4 + 8n)f1 + (5+lln)f3

Also from Equations (All) and (A14)

l+n

_.2

2n

- r l+3n 1l+3n 1+3n

fv\jj

L(5th)f3i

Then Equation (A9) becomes

r

-11+n 2 2n

- I

13n

Jl+3n l+3n _R l+3n [(5+lln)f3J

Also the torque coefficient in Equation (13) becomes

k =4irfc-

'l m 3 .r 2n R1 +3n where 1 +n C1 = 4ir f3c [C n)f3ai l+3n l+3n (A15) (Al9) (A20)

REFERENCES

Graiwille, P.S., "The Frictjonal ReSistance and Turbulent

Boundary Layer of Rough SurfaceS," Journal of Ship Research, Vol. 2, No. 3

(Dec i958)

Granvillê, PS., "The Frictional Resistance and Velocity

$imilarty Laws of Drag-Reducing Polyi

Solutions," Journal of Ship

Research, Vol.

12,

No. 3 (Sep 1968).

,

.,

.,

von Karman, ,T., "On Laminar and Turbulent

Fiictibn," National

Advisory Conmitteê for Aeroiaitjcs 't'M 1092 (Sep 1946); translated from

Zeitschrift ftr angewandte Mathematik und Meehanik, Vol. 1, No.

4 (Aug

1921).

Goldstein, S., "On the Resistance to the RotatiOn of a Disc

Immersed in a Fluj.d," Proceedings of Cambridge Philosoplical Society,

Vol. 31, Pt. 2, p;'232 (Apr1935).

Dorfman, L.A., "Drag of a Rotating Ràugh Disk," Soviet

Physics-Technical Physics, VOl..

3,' No. 2, p.. 353 (Feb 1958).

Dorfman, L.A., "Hydrodynaic Resjstance and the Heat Loss of

Rotatiig Solids," Oliver

Boyd, 'London, (1963).

Prandtl, L, "On. Boundary Layers in ThreeQiitensiOna1 Flow,"

B.I.GS. 84 (Aug 1946); a.so Briti1:i M.A.P. Report

Translation 64 (May

1946).

Mager,A., "Generalization of BoundaryLayer Momentum-Integral

Equations to Three-Dimensional Flows Including Those of Rotating

Systems,"

National Advisory Committee for Aeronautics Technical Report

1067 (1952).

Johnston, J.P., "Oti the Three-Dimensional Turbulent

Boundary

Layer Generated by Secondary Flow," Transactions

of American Society of

Mechanical Engineers, Jurnal of Basic Engineering, Vol. 82, Series D.,

No. 1

(Mar

1960).

Eichelbrenner, E.A., "La Couche Limite Tridimensionellé en

Rgime

Turbülênt d'un Fludé Compressible," NATO Advisory Group 'for Aerospace

Research AGARDograph

97

(May 1965).

.11. Gregory, N. Ot àl., "On the Stability of Three-Dimensional Boundary Layers with Application to the Flow Due to a. Rotating Di'k," Philosophical Transactions of Royal Society, London (A) 248, P. 155 (1955); also in "Bouidary Layer ffect in Aerodynamics.," Symposium at National Physical. Laboratory, Great Britain (1955);. published by Philosohicãl Library, New York (1957).

12. Coles, P., "The Law of the Wake in the Turbulent Boundary Layer," Journal. of Fluid Mechanics, Vol. 1, Pt. 2(Jul 1956).

13.. Moses, H.L.., "The Behavior of Turbulent Boundary Layers in Adverse Pressure Gradients," Massachusetts Institute of Technology Gas Turbine Laboratory Report 73 (Jan 1964).

Dryden, J.L. et al., '!Hydtodyiamics," National Research Council Bulletin 84 (1932).; reprinted by Dover Publications, New York, p. 352 (1956).

Theodorsen, T. and A. Regier, "Experiments on Drag of Revolving Disks, Cylinders and Streamline Rods at High Speeds," National Advisory

Committee for Aeronautics Technical Report 793 (1944).

Hoyt, J.W. and A.G. Fabula, "The Effect of Additives on Fluid

Friction," 5th Symposium on Naval Hydrodynamics, Office of Naval Research

ACR-112, U.S.. Government Printing Office, Washington, D.C. (Sep 1964).

Aflfilokhiev, W.B. and A.M. Ferguson, "The Variation of Friction

Drag with Surface Roughness in Dilute Polymer Solutions," University of

Glasgow, Department of Naval Architecture Experiment Tank Report 8 (Aug

1968).

Smaliman., J.R. and J.H.T. Wade, "The Influence of Hydrodynamic Drag of High Molecular Weight Compounds," (Canada) C.A.S.I. Transactions, Vol. 2, No. 1, p. 37 (Mar 1969).

Stain, W.C., "The. Three-Dimensional Turbulent Boundary Layer on a Rotating Disk," Mississippi. State University Aerophysics Department Research Report 35 (Aug 1961).

Gadd, G.E., "The Effect on the Turbulent Boundary Layer of Adding Guar Gum to the Water in Which a Disk Rotates," National Physical

Cham, T-S. and M.R. Head, "Turbulent Boundary-Layer Flow on a Rotating Disk," Journal of Fluid Mechanics, Vol. 37, Pt. 1, p. 129 (Ji.m

1969).

Erian, F.F., "The Turbulent Flow due to a Rotating Disk," Clarkson College of Technology (Potsdam, N.Y.) Department of Mechanical

Engineering (Apr 1970).

Cooper, P., "Turbulent Boundary Layer on a Rotating Disk Calculated with an Effective Viscosity," AIAA Journal, Vol. 9, No. 2, p. 255 (Feb 1971).

Virk, P.S. et al., "The Ultimate Asymptote and Mean Flow

Structures in Toms' Phenomenon," Transactions American Society of Mechanical Engineers, Journal of Applied Mechanics, Vol. 37, Series E, No. 2 (Jun

1970).

Copies 1 ONR 472 2 DIR, NRL 1. Dr. R.C. Little 9 NAVSHIPSYSCOM 2 SHIPS. 2052 l SHIPS 031 1 SHIPS 0341 1 SHIPS 034lC 1 SHIPS 03412 1 SHIPS 03421 2 SHIPS 0372 3 NAVORDSYSCOM 1 NORD 035B 1 NORD 054131 2 NAVAIRSYSCOM

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4 NUSC, Newport

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CO DIR, NELC (San Diego)

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CE TER DISTRIBUTION

Code

1

1500

1

1540

25

1541

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1552

1

1556

UNCLASSIFIED .Sccuritv Classification

FORM

_1A74

_{(PAGE 1)}

DOCUMENTCONTROLDATA R&D

S.'crity ilosi1ation ol title, body olabotract ad on hit nn,uot he entered when the uvcrali report Is clnssjfied, ORIGINA TING ACTIVITY (Corporate author)

Naval Ship Research and Development Center

Bethesda, Maryland 20034

-20. REPORT SECURITY CLASSIFICATION

- UNCLASSIFIED

2b GROUP

.-3. REPORT TITLE -

-THE TORQUE AND TURBULENT BOUNDARY LAYER OF ROTATING DISKS WITH SMOOTH AND ROUGH SURFACES, AND IN DRAG-REDUCING POLYMER SOLUTIONS

4. DESCRIPTIVE NOTES (Tjpé ol report and inclusive dates)

5. AU THORISI (First name, middle initial, last ñae) -

-Paul S. Granville

5. REPORT DATE April 1972

70. TOTAL NO. OF PAGES

54

7b. NO. OFREF5

24

-80. CONTRACT OR GRANT NO.

-b. PROJECT NO.

R023-Ol

-d.

90, ORIGINATORS REPORT NUMBERISI -- -

-3711

Sb. OTHERREPORT MOISt (Any other numbers that may be assigned this report)

tO. O!STRIBLjTION STATEMENT

Approyed for public re-lease: Distribution unlimited

II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Naval Ship Systems Command

The resisting torque of disks rotating in an unbounded fluid is analyzed on the basis of three-dimensional boundary-layer theory.

- Smooth and rough surfaces in ordinary fluids and in drag-reducing

pol.ymer solutions are considered. A general logarithmic relation is derived for the torque as a function of Reynolds number for arbitrary roughness and arbitrary drag reduction. Special formulas are obtained

for smooth surfaces, fully rough -surfaces, polymer solutions with a

linear logarithmic drag-reduction characterization, and polymer so-lutions with maxinum drag reduction. Relations are also obtained for boundary-layer parameters such as thickness, wall shearing stress, etc. The computed results are in excellent agreement with experimental

IINC LASS [F liii)

SecuritiCLassi1ication

(PAGE 2) SëcHty Classification

LINK. A LINK K LINK C

KEY WORDS

- ROLE WI ROLE WI RO I.E WT

Rotating disks; rough urfaces; drag reduction; polyther solutions