Hydrodynamical research on frictional resistance of rough surface

ARCH1EF

'

Hydrodynamical Research on Frictional Resistance of

Rough Surface

By

Hideo SASAJIMA and Eiichi yosmnA.

Reprinted from

TECHNOLOGY REPORTS OF THE OSAKA UNIVERSITY Vol. 5 No. 174 Faculty of Engineering Osaka University Osaka, Japan 1956

Itchnische Eltvisizthc31

Deth

(Received July 20, 1955)

Hydrodynamical Research on Frictional Resistance of

Rough Surface

By

Hideo SASAJIM.A and Eiichi YOSHIDA

(Department of Naval Architecture)

Abstract

Roughnesses are generally classified into two types according td the

characteristic properties of their resistance coefficients. The authors clarified these characteristics theoretically by attributing them to the existence of the separation in flow or not, and gave an order of magnitude of the roughness allowance for ship's frictional resistance.

1. Introduction

Up to the present, it is likely seen that the roughness has been treated

chiefly by experimental researches. This paper will give an approach to theoretical methods on this subject.

As is well known, the roughness can be classified in two _{different types}

according to their characteristic properties, i. _{e. whether the constancy of} re-sistance coefficient appears or not for large Reynolds number. The _Nikuradse's

sand roughness or a drawn steel pipe surface belongs to the constanc_{j type, and an} asphalted surface to the non-constancy type. _{The authors attribute the difference}

between the two types to the separation in flow. The sand roughness has such

a steep slope that it forces a flow to separate. Then the resistance in larger

Reynolds number will be entirely governed by eddies. On the other hand, _{on a}

roughness with very gradual slope, which is defined " Wavy " here, the flow

always adheres to the unevenness of wall surface for any large Reynolds _number.

Therefore the resistance coefficient of such a surface may be expected to resemble to the smooth surface's. _{This property, however, has been doubtfully observed if it} might be a transition phenomenon on the way to a perfect roughness flow.

Recently the frictional resistance of ships, especially the roughness allowance,

has received particular attention in all shipbuilding countries, and we have re-ports on several experimental researches on large scale and with high _accuracy

in which the tendency of the wavy roughness is evidently observed on their

resistance curves. Although the roughness of ships involves, of course, structural

roughnesses, the most effective one is that of the painted surface. It consists of

a continuous and very gradual unevenness, which is endorsed by recent measure-ments. 1)2)3)4) So the painted surface behaves as a wavy roughness. By this fact, it will be clear why many shipbuilders have used the Froude's coefficient in the evaluation of ship's frictional resistance rather than the theoretically introduced methods : Schlichting developed the Nikuradse's sand roughness to the flat plate.

According to such considerations, the authors tried to give a theoretical

explanation of the above mentioned experimental facts from the standpoint of our modified momentum transfer theory in which the effect of viscosity was introduced. The main aims in this paper are as follows :

to classify the kind of roughnesses and to clarify the characteristic difference between them,

to indicate that the various authoritative experimental data can be

satisfactori-ly explained by the theory,

to propose a new method of prediction on the roughness allowance of ships'

frictional resistance.

2. Modified Momentum Transfer Theory5)

The original form of the momentum transfer' theory proposed by Prandtl is

- lul=11C1 ,

,(1)

R= pu'v'= pi2C2

where u' : mean velocity fluctuation in x-direction,

mean velocity fluctuation in y-direction, 1 : mixture length,

C : velocity gradient of main stream =u/y,

Reynolds stress.

These relations hold well in the whole region of boundary layer except in the neighbourhood of the wall. The equation of motion due to this idea is

= pl2C2,

where r is shearing stress. The viscosity force itC is neglected under the reason

of its smallness compared with p/22. But, in the neighbourhood of the wall it is

evident that the viscosity effect plays the main part in place of Reynolds stress, of which the latter must diminish as a result of the decay of velocity fluctuation by an obstruction of the wall. Therefore there should be a laminar sublayer and

a successive transitional layer of which the existences have been experimentally

con-firmed. They are especially, in our present purpose, expected to be the key of the

the behaviour of flow in this region.

Karman introduced an equation with respect to the energy transport and

dissipation from the Navier-Stokes' equation, i. e.

dx, L 2 \ 2

p ij

p dx2 d.x' 2 k aXk I

dru,( q2

p \-1+t-R du._ d2 q2

lizz(

Ozsii \ 2

(2)

q3/L q3/L a vq2/L2 42/22

where x2 coincides with y, and ze2, te3 are the velocity fluctuations in the

directions of x, y, z respectively, and q2=u'12-hu'22+u'32.

q3/L, vq2/L2, vq2/22 represent the orders of magnitude of the particular terms.

Karman suggested that L would be a length standard such as a diameter of the

pipe or a breadth of the channel, and he obtained the following relation

neg-lecting the first term in the right side of eq. (2).

q3/L-42/A2.

The authors consider that in the turbulent velocity field 1 should be adopted for L instead of such dimensions as mentioned above. 1 can be considered to have a magnitude not so far different from A in the neighbourhood of the wall and ultimately to coincide each other at the laminar sublayer. Consequently, though the first term in the right side might not be neglected, we obtain similarly

q3//...42/22

If we put q=1C, 11=Ky1=21,

Ci=a _K2yi at Y=Yi, (3)

where k=avIK2.

This relation determines the thickness of sublayer yi. It is remarkable that

yi is combined with the vorticity directly. The proportional constant a is a function

depending only on the turbulence mechanism which should be similar at y=yi

for all kinds of the turbulent boundary

layer, and hence a is a universal constant _u,Vtt

at y=yi. K is nearly 0.4 for smooth surface. For rough surface, K will take its particular

value according to the change of mixture

length stimulated by the roughness. In

ac-cordance with K, k/v will have its own

value for each type of the roughnesses.

0

Then, the next problem is to find the

law governing the diminution of velocity

or q/1v/22.

u,/tz 71;

fluctuation in the transient region. Fig. 1 shows the measured results6) of

maxi-mum fluctuation velocities (which may be roughly three times of the

root-mean-square values) in a circular pipe. This indicates that the rapid fall of 111 is the result not of the direct contribution of the viscosity but of the restraint by the

pipe wall. For, the viscosity force must be isotropic. We will treat the

phe-nomenon in this region by the dimensional analysis. The physical quantities to be involved may be y, Yi, C, C1,1, v', p and p.

By these quantities five dimensionless ratios can be constructed, and a function-al relation exists between them.

f(v71C, ii/9v', C/Ci, 1/Y, YlYi) =0. (4)

We have the following three conditions to be considered. ( i ) Equation of motion near the wall

pCi-= pC pu'v'= pC+ plCvc

=1+ yv'

Y

Condition for the sublayer

CI a irj4,

1

or =K=

a

C v( yr

_'

Ci YvYi

A suggestion by Fig. 1 that the viscosity does not contribute explicitly

to the fall of v'.

According to conditions (i) and (ii), C/Ci and 1/y can be substituted by

other three quantities. Therefore eq. (4) becomes

f1(v71C, v/Yv', YIY1) =0, or v. /1C=Fi(v/Yv', Y1Y1)

Now, v' is indeliendent of id by the condition (iii)

.*. v'11C=F(y/y).

Although we have no effective data to determine the stridt functional form

of F, the simplest assumption will be

F=0 and F'=0

at YilY=1,

F=1 at YilY=0 .

Satisfying these conditions, we have F(Y/Y1)'=7(1 -Y1 /J) 2-Then, the Reynolds stress is

991' au )2 (3. ay

Y

Eqs. (3) and (5) are the fundamental relations to represent our modified

momentum transfer theory.

Now, the equation of motion for the turbulent region yi<yo becomes

=pp(au

ay )2 (1_ Yi )2y auOy

which can be solved in the case when r-distribution and 1 are given.

Anyway, we shall be able to put 1=Ky and z-=pk/yf in the neighbourhood of the wall. Therefore

K2y2(1 Y1 )2 ( au +v au

Yk21

v ( 7 )

Y aY aY

For the convenience of the further calculation, we will solve eq. (7) in a power series. au

kv, an(

\" ( ) ) , OY yf where a 1

a2, a3, etc. : coefficients to be determined by al only.

In the distant region from the wall, the term p8u/8y and the factor yi/y in

eq. (6) will be negligible, i. e. the equation of motion is, as in the ordinary theory,

r=11,4(

,

au \2. 8y

orau

1 J 7

(10)

ay V p

In the case of a Circular pipe of radius r,

r k2

(1--r

r

r 1

r

y y

1 y r 14-C2k

r

Therefore, the solution of eq. (10) is

8u k. y

rai

+_,

yi 5cl

+c9( Y )2 ₍₁₂₎

-ay

,y

rLy V.k rtr

-\r/jj

(8) and (12) are the solutions Of eq. (6) at the both ektreme. ends. We may

,

5)

( 6 )

(9)

with (12),

[a1/1

+1/ vk

-r

r r

-Pc2(j)2

ayy,au

k _y v Y nE=2an(Yyl )n1. Then we have the following relations.

u=-1-L[A1+2a111/1

Y log(1+ /1

-V )1 Yi 3 2a_{A. 15}2 clic@ +

y )

105 c28 ( 3 2 r + 1C.1+ 2 Yr

+" Y2

A(1.

* -an _{( Y:}

_ri,

+ailog y E Y1

n-2 n-1 \ y

1 k

...J.-- 2 v 2 [Bi + log RCf k 4 v 2-- v 1 (B2+ 2a2logRC?")], (15)

R=2 k

r [Bi

al log 4 1/21z/i+ al log

v

--'11/32+ 2a2 log 4 1/2k7p + 2a2 ,

where A1 = 1 +_{n..2 n-1}

Bi=l+ E

n=12 n-1 .02 = 1 + 2 E n=3 an an 2 8

+ 2a1( 1 +log 2+ 15 cilf+ 37Ec2K)

+a1(46

15 I- 634 cilf+ 69316 c2Klog-V2k/v),

an5

8 16 17- + c1K+ c210-2a2(1+ log4 -1/2k/v)

n-2

( 2 63 231 (16) (17)

R=2rUn.lp: Reynolds number (Um: mean velocity),

= 8v/ Um. k 1 resistance coefficient.

The coefficients V k/v, an, c1 and c2 are obtained so as to agree with the

experiments of pipe flows.

/7i7=6.82 when K=0.4,

a2= 0.366, a2=0.300, a3=0.238, a4=0.184

a5= 0.135, a6= 0.093, a7= 0.056, etc.

c1=8.4, c2= 3.8.

For other kinds of flows, e. g. a flow along a flat plate, a flow with pressure

gradient, if r and I are given in funtions of y/8, the solution will be similarly

obtained with the unaltered K and V klv values for the same roughness.

s.)

3. _{Hydrodynamical Treatment of Rough Surface}

I. Equivalent wail' surface

In the case when we concern with a rough surface, it is the first problem

where the base plane must be placed in order to measure the heights of

pro-tuberances, the thickness of sublayeE, etc. This is called here the equivalent

wall surface.

We have, however, no reasonable method* to estimate this surface, except to select an appropriate value to the experimental results.

After all, the authors put the distance between this surface and the top of

protuberance to be asKs (Ks : _{height of protuberance), and assume that the}

position of this surface has no dependency on Reynolds number.

H. Distribution of shearing stress

By what mechanism the shearing stress in the near region of protuberances

is distributed is the second problem. In the case when a flow separates the

shearing stress will rapidly decrease from the vicinity of the top of protuberance,

as shown by t-r' _{in Fig. 2, and will become independent of Reynolds number.}

With respect to its _{behaviour we have, however, been informed of no}

experi-Fig. 2.

Sheciring Stress in Protuberance

in Fluid

asKi. TopofProtuberance

Ks

* Schlichting showed a criterion such that the distance between this surface and the wall

surface was given by the following ratio.

total volume of protuberances

area of wall surface

This criterion will become irrational in the extreme case when the wall surface is

roughened by very slender protuberances, such as needles.

'/

Shearing Stress Distribution when no Separation exists

mental report measured directly. Anyway, the shearing stress in fluid adding to

the head resistance is equal to that for the smooth surface.

If the flow has no separation, the rough and smooth surfaces have the same shearing stress distribution. Namely, the wavy roughness has the same distri-bution with that of the smooth surface.

0

III. Variation of mixture length

It may be an appropriate consideration

that the mixture length will be influenced by protuberances or unevenness on the wall

re-gardless of the existence of separation, because the scale of local velocity fluctuation created

by the roughness will be an order of the

natural turbulence. But since this effect will

,8KT Y2

or fiH be limited in the neighbourhood of the wall

Fig. 3. only, we may, for the sake of simplicity, put

as follows. (see Fig. 3) 1=K'y=pKy for Cly.. 13.1C,, or 0_- y1:31/,

l=pligKs

or pifgH for

gics<y. AVG or i3H<y- pe9H, 1=Ky for Y> Y2= /*Ks or y>Y2---- P511,

where

p=K/K,

K': proportional constant of the increased mixture length,

H: wave height of unevenness of the wavy surface,

g : a proportional constant.

Sand Roughness

The stimulation for the turbulence is in this case caused by the eddies

flowing out from the backside of protuberances. Hence, p will have a certain value for any Kg as long as it is not too large. j3, cannot be determined without

selecting a compatible value for the experimental data. We will indicate in the

following chapter how to choose these p and 3. Wavy Roughness

The increment of mixture length, as easily considerd, should be a function of HI2 (A: wave length of the wall unevenness). Pig. 4 shows the calculated

results of p for HIA (see chapters 4 and 5), where a general tendency can be

observed in spite of the insufficiency of the spots. H/2 for ordinary paints is in

As for there is no method to define this coefficient as in the case of the

sand roughness and we ought to select it to be suitable to the experimental data.

1.8

1.6

1.2

1.010

/5

20

30

40 50 60

Fig. 4.

When the roughness is not small enough

compared with the thickness of the boundary

layer, this case does not belong to our present subject, the problem must be turned

to that of the form effect which will have

little influence on the resistance. Hence it is

required to prepare a certain limitation to this

respect. The simplest assumption is to put

19H<T8, where

r

is a constant and a the thick- 0 .Y2

ness of the boundary layer. It is compatible Fig. 5.

with Fromm's experiment that r-=-:0.05 using ,9=3.

The above consideration is for the roughness of regular. waves, while the actual surfaces consist of sundry elements whose slopes, as seen in Fig. 8, decrease

generally corresponding to the increase of their wave lengths. For these surfaces

the envelope of each mixture length (see Fig. 5) may be replaced for them.

When this envelope is regarded as a mixture length of an equivaleht regular waves, K is nearly the same with that of the smallest element and y2 is much

prolonged.

IV. Laminar sublayer and vorticity distribution

The condition to determine the thickess of sublayer yi for the smooth surface

is, as mentioned in the previous chapter,

814/8y=k/A, kooll K2. (19)

X FROMM

Now, we will suppose a rough and a smooth surface with an equal resistance

ro on their walls.

i) Sand Roughness

In Fig. 6, A0AA'A" is &u/j' for the smooth surface. In this case the mixture

length varies as OMN, and r/p does as 0'0"PQ. At first, by the effect of p

au/ay decreases as AoBB"A", except in the laminar sublayer of which the

thick-ness is contracted to yi/p from yi according to eq. (19). Next, by superposing

the effect of r/,u, au/ay is reduced to 0"B'B"A" where B', the intersection of r/p

and k/p2y2, gives the ultimate thickness of the sublayer. Consequently we can consider as follows.

The area A00"B'B"B= velocity loss due to the variation of shearing stress

distribution.

The area BB"A"A =velocity loss due to the variation of mixture lengh.

Y1 Y1 y2

Fig. 7.

o y/p y,

I otsKs 19Ks

Thickness ofSublayer

Fig. 6.

ii) Wavy Roughness

In this case, there exists only the

effect of the increase of mixture length which behaves as eurve OMN in Fig. 7. Corresponding to this, 8u/ay is

con-tracted to A0BB"A" from A0AA" for the smooth surface, and the thickness of

sublayer to from yi as in the above

case. Then,

V. Limit of wavy surface

We have already defined the wavy roughness by the absence of separation. Now we will consider the limiting slope where the separation will not appear in

the flow.

Fromm9) measured Cf of four kinds of wavy surface IIA, JIB, IIC, and IID according to the wave height in order. By his experiments, IIA was represented by separate curves per each breadth of channel, and they became constant for higher Reynolds number. This fact shows that IIA belongs to the roughness of

a solid drawn pipe. On the other hand, IIC and IID varied parallellikely with

Reynolds number but not with the breadth. IIB had an intermediate character,

H/A.

2

+ WALL 6ALJOE RECORDS 9 RED-OXIDE PAINT

SAL-PA/NT clearof Seams

0 in wayoffairing

*RED-OXIDE PAINT X AL-PAINT

A NORFOLK HOT PLASTIC 0 MARE ISLAND HOT PLASTIC

y FLAME SPRAYED HOT PLASTIC 71 COLD PLASTIC a ANTI-CORROSIVE Fig. 8.

...iminl

111.11111111

_{7 EIMMilinglIER1}

z

1 MISINUIREMII

omm.

WMP. 15.\

4i C)

MINIM El ITIUM.

1---

WARIER

glinnaMs,

---1 .:

...

Iffidilhailignikkg

111110.11111,1

Biwa=

..i.

,....

Eli

,.._ Z

,

Q.. c,

z

lulls"

alibi

ct 14.1 per P . inch - '1

IIIN . 1

2 ACI.P.

, FREQUENCY= ,,

..

UNDULATION

ncx

3 2 / 0-2 8 6 5 4 O-j 8 6 5 4 3 2

1. e. dependent on both Reynolds number and the breadth. Hence it seems that there existed a weak separation.

After all, in his experiments, the limit of wavy roughness was between IIB

and IIC. The limiting slope is given by H/2=1/10 1/15, or H/),=-;1/12, the mean value, may be a criterion for it.

Fig. 8 shows the results measured on the "Lucy Ashton "4) and the values1)2)

obtained by the authors in the same process with the former. Even though there

remain some questions in comparing these directly with the limit above mentioned,

they indicate an order of actual surface. " Norfolk " hot plastic has H/2 of 1/15, i. e. this would be almost a critical material which could be used as economical

ship paint. "Mare Island" and "Flame sprayed" hot plastic and Anti-Corrosive " have H/A of 1/20, 1/40 and 1/50 respectively. Others all have not more than

1/50, and are quite safe from the separation.

Next we will add a curious phenomenon pointed out by Hopf,10) who measured

the resistance of a surface with wire netting pasted by paraffin. In the initial

stage of the test, Cf varied along with that of a smooth surface. When the

paraffin was washed away by the stream after several experiments, Cf was

shifted to that of the type belonging to a solid drawn pipe and crossed with the

initial one. A similar phenomenon was observed between a new and a several years

old aqueduct. It contradicts to our knowledge that the steeper the slope of

protuberances the lesser. the resistance. But it can be explained as follows. In

the case when the limit is slightly exceeded, the separation covers only a part of

the backside of protuberances, and the eddy resistance is not so large to overcome

the fall of the resistance due to the diminution of mixture length from the initial stage.

Hopf's experiment gives an important suggestion that the limit of wavy

sur-face does not depend on Reynolds number. Suppose two surface A and B, where

A is

slightly steeper, and let the limit become smaller with the increase of

Reynolds number. For small Reynolds number both remain to be wavy and Cf

of A is higher. With a growth of Reynolds number, A is the first to separate

and crosses B, then B separates and crosses again. We have no informations with

respect to such an absurd property between similar surfaces.

On the contrary, if the limit increases with Reynolds number, a surface which has the separation at first will become purely wavy at a large Reynolds number, and so Cf must begin to fall. Such a fact does not exist also.

Consequently, a surface which was once wavy remains to be so forever, i. e. the limit -is not influenced by Reynolds number. The property of wavy .surface

separation type.

4. Rough Circular Pipe I. Principle of calculation in general

The authors have already clarified that the flow on the rough surface had the

velocity losses of two types, i. e.

1) the one due to the variation of shearing stress, and

ii) the other due to the variation of mixture length.

Here, we will denote them by Jul, and Ju, respectively, aud ki.Yi (velocity

at y=y1) by U1. (See Fig. 6)

A) Velocity Loss due to the Variation of Shearing Stress Distribution

For the convenience of calculation, the shearing stress distribution between

the top of protuberance ancl the equivalent wall surface is assumed to be a

straight line of which the- intersection with the baseline is given by sceaKs.

i ) For

a) when crK8

Jul 1an

PYz an ( yl Vi-1]

(71 p L nt2 n

+alog

_VI

,2n-1 \ PYi J

1 ( yr, sagics \2 1

2 \ Yi Yi ₎ asKs

Where. yz is the abscissa of intersecting point of the shearing stress curve and

the vorticity curve for the rough surface, (in this region k / y2 curve can be

replaced by the vorticity curve without an appreciable error) and can be obtained by the following relation,

.Y1 Cz e 1 .Y1

Pyi Co 1

1e PasK,

of which Ci is the vorticity for the rough surface at y=yz and Co=k/y.

b) when asKg< Yi/

dal 1 _{(1+ e)} asKg ₍₂₂₎

Ui 2 Y1

ii) For e<O,

a) when yi/P, dui

2 n-1

[1+

an 4- al log PYi p

-

ii=2 n-1 .1 Pyl 1 Yi 2ea8Ks 2p .3),

1e asKs \ Yi

-(2.0) (21) PY n-'1] (23)

b) when asKs< Yil

dui 1 1 cr.K.

111

2 1e

"

B) Velocity Loss due to the Variation of Mix:lure Length

This loss exists only when ,31C5>yi or [31-1>yi.

4U2

[1

ai+ E

_{+al log} Y9

p , -

E

n, an(

n-1

(25)

n

Yi

nJ.

where y2=p9lf5 or pRH. (26)

In the result, the total velocity loss du is represented by

du dui du2

Ui_Ui

These velocity losses, which appear only in the neighbourhood of the wall as above mentioned, can be regarded as the loss in U. (mean velocity), or the loss

of Reynolds number.

If we put Cf and C'f to be the resistance coefficients at Reynolds numbers

R for the smooth surface and R' for the rough surface, both having an equal re-sistance,

C1R2=C1'R'2,

.*. log Cf,

2 log(1

z114) , (28) Cf

where LIR= R R' . ₍₂₉₎

On the other hand,

JR du du 11 (LIU' du,

R (1

.

U1 U. 1- U1 Um

Urn

131+ al log 41/ 2

-1/V

v+ al log Y1

(B2+2ez1 log 4V

2 1/ k/v+2a2 log r).

(31)

Thus, the resistance coefficient for the rough pipe can be' calculated by eqs. (15), (16) and (20) (31).

The constancy of Cf for very large Reynolds number can be simply shown

as follows.

In eq. (20) for large Reynolds number, i. e. for yz)yi, y2)yi,

Yi Yi

..

yl-÷EasKe..

Yz EasKs

Jul 1 rill_

+ al log Pcas+ al log Ks 1

'

P L

2 n 1

_{Yi J}

(24)

(27)

4/42 ( 1 an

--).

)E1ai+

E

_{+al logpti ± ai log Kg}

U1 P n-2 n-1 Yi

l

.

"

__b. 1+ z an 1

n-1

+alllogea, 4 logp

(1-1)

+

(1-1)

log0 +al log K18

P P yl

n-2 p

du

or a 4 ai logK8 (32)

Yi where a is a constant not containing IC1.

If we use the shear velocity 14, (32) becomes

du

a' + 5.75 log10 , d : a constant. UT

The velocity distribution for the smooth surface is =b + 5.75 log10 , b: a constant.

Y1

Therefore, the velocity distribution for the rough surface is

u du

=b' +5.75 logio , b' : a constant,

Ur UT If,

which contains no viscosity term and is governed by if, only.

Namely, the constancy of resistance coefficient for large Reynolds number is proved.

II. Circular pipe of sand roughness

As is well known, C7-curve of this type of roughness has a rise-up at a certain

Reynolds number and then approaches to a constant value. For the theoretical

Fig. 9. 1105100C, NO. r /K, 0(3 p cx,/p 6 p (162/00q1 . / 2 3 60.0 , 0.10 .20 .25 040 .60 .70 250 3.00 2.80 0.1 /.50 /.20 I. /5 05q5 .562 a 6 ,

764 COLEBROOK& WHITE'sNIKURADSE s experiment- .560.565

A B C

BLASIUS'ra-mula.

NIKURADSE's . ifor Smooth Pipe

AUTHOUR's . ... I a

j...-%.,... ,.. ...

B.,:,..,

1 ....,.... A . 40 50 6° fog Udiv 0.7 0.6 0.5 a4 04

treatment, we will consider here such a surface with small spheres of diameter K. distributed compactly.

It is evident by the argument in the previous chapter that the sand roughness

has both of the velocity losses. Fig. 9 shows the results of our calculation and

of the experiments by Nikuradsen) and Colebrook & White.12) (log 100 C1). in the

annexed table indicates the value of log 100 G1 when Cf becomes constant for

large Reynolds number. Fig. 10 is the dimensionless representation of Fig. 9.

1

A

5.75 lo 11

7

06

/

id . NO. r/Ks cis P I,

t

1

600 0/6

a .20 0.40 :60

/.50

1.20 010

a

b b # NIKURADSE

IPT4

ri

Ea II

13 76 COLRickasEARfOoorsKngi 001-11W

A///t

2:pe

AUTHOR J

imam

_{_______}

08

12 16 18

20

22

(09 ur1(4/1, Fig. 10.

The ranges of the parameters are

(4=0.15-0.25, 8/a.= 2.5-3.0,

p=1.15-1.5,

which were selected by the following consideration.

As the model surface consists of compactly distributed spheres of diameter

K., only the upper half parts of them are effective. By using Schlichting's idea

S 2

( Ka \

K, Ife 1 1

-3 7r \ 2 )

\

2 ) 2 5- Ifs 6

Therefore, as will be a value not so different from 0.2. a, 'governs the Rynolds number from which Cf rises up rapidly, and the above range is adequate to this point.

e gives an influence on the value of (log 100 Cr) and on the behaviour of

/0

°L0

loglOOCi _{NO /31-/../r p} , / 2 3 3:100 . 1, 1.2 /.4 /6

11111111..-

B A 8 C BLASIUS' Formula N/KURADSE'f ,4 U TH 0 UR 's Ffoi: Smooth Pipe

1Awilitti

...

_ix ligke,_-91111411411111111111111

11111612Vis

---1

11111111.114111

C

.gm

approach to this Value. Anyway, it must not be zero, and about 0.1 is the best.

For E._ 0.01, Cf considerably decreases and approaches to a constant value after it arrived at a certain maximum.

p and 0/a. affect the slope of the rise-up and the value of (log 100

8/a8 is the leading. factor on the former.

By such considerations, the values of these parameters will be necessarily

restricted in the ranges previously given,

III. Circular pipe of wavy roughness

The rough surface of this type has the velocity loss 4142 due to the change' of

Mixture length only. We showed in Fig. 11 the calculated results together with

50 6.0 _foc'Ud/'

Fig. 11.

Fromm's9) and others' experimental data, where p was assumed to be 1.0-1.7

according to the slope of surface unevenness. The full lines correspond to H /r

=1/100 except that for p=1.7, and the arrows at both ends to H/r=1/50 _and

1/200. The non-marked broken lines are various experiments reproduced from Hopf's1.0) treatise. IIB, IIC, and IID belong to Fromm's of which particulars _are

given in the Table 1. 1113 is shown for the comparison, though it is not purely.

wavy. Since his apparatuses were narrow channels, we must pay such an attention 0. 0.6 0.5 0. 123 a2

Table 1 Therefore, 4z42 1 )[1.

al+ E

n-2 n-1

-4. ' 1 )czi log 4R 4u1 1 R U. 1 Cf log Cf, -9.2 log p

that, when the re§ults are converted into that of circular pipes, the half breadth of

channel is used for HI?' and the hydraulic

radius for Reynolds number.

The non-dependency -on r can be observed

even for IID, hence 01-1-0.054r where 0 is

put 3.0 according to the calculation for the sand roughness. Then the curves for H/r= 1/50 represent approximately the maximum resistance of a definite slope.

In the previous chapter we have deduced that the limiting slope allowable for the

wavy surface lies between IIB and IIC. Thus the curve for p. 71.7H/r= 1/50

representats Cf of the roughest wavy surface.

We can prove the parallellike run of the resistance coefficient for the wavy surface to that for the smooth in log-log plotting.

When Reynolds number is largely increased, Y2)Y1, an

+al log po +a1 log r 1

IIC IID HlIZM 0.66 0.44 0.27 HP, 1/10 1/15 1/25 Hit- 0.044 0.029 0.018 P . 1.85 1.55 1.40 U. _{B1+ al log 4 1/2k7v + al log} a/log .

Namely, the differende between log Cf and' log Cf' becomes constant.

5. Flat Plate of Wavy Roughness

The assumptions under which our theory' was developed to the flat plate are as follows.

i ) The velocity distribution for the smooth flat plate is similar with that for the sniooth circular pipe, where, to simplify the calculation, 1=Ky and r = const.

throughout the radius of pipe are adopted.

ii) The velocity loss exists only near the wall surface.

where

11 ' Y2

>

Velocity for the smooth surface

uo= {A+ ai log

E

an ( 721

Yi 721

n-2 n 1 \

72

Velocity loss due to the roughness

= [4]-=

IA

al+ al log

Velocity for the wavy surface u = uo du.

Velocity at the outside of the boundary layer

Resistance on the one side of the plate

5

D= P(U u)udy = pkU

Jo vi[G][ .53

where

721

A 1+ E

_{n.2 n-1}

(33)

[E]= (11A-24+ (a1a2) (A al)

E

an log

721 2 n-3 (n 1) 2) 721

--[4][ai+ 721 ₂

ai a2 + E

_{n.3 (i2-1) (n-2)} 1+_{agii log} 721

an

A

The unknown quantities are 72 and (3, which are determined by eq. (36) and

'Carman's momentum equation, i. e.

k a

ro _{op (U u)udy.}

To solve this equation, we assume 722/721--const, which means y2 or H varies

proportionally to The variation of yi is so small at x= 100m. is equal to

1.35 times of the value at x=lm.--- that its effect can be practically neglected.

If it is necessary, the correction is easily done by dividing the length of the plate.

The solution is as follows.

Ux

(k \2[R]

+ const., v 721 (38)

Tr_

1 an 2

ala2

+ L 3 +AI

E

..3

a2 2 2anJ. a9+ E n.3 (n 1) (n-2)

/

1 2 an

E

71=3 n (n-1)2 4 (13----3nt:4 n(n 1) 1 U [G] [G] A+ a1 log na 721

E

721,-1_[4] n-2 n ha. n.=-2 n 1 \ 729

[R]=[R,][I1] [41+ [41[432,

[R]=a1242+4a1A+6(4+24(A-2a1) log 1 + a;.

(1og-93

Vi

2a.

4_ A2( a2) +a1Al2a1+

3 n-3 (n-1) (-2) 5

2a.

ai212ai+ a2 + an, aiez2 12+ _(n-1)21 ...]lo

n=.3

[h]=---2cii (A 2(4)+24 log 1

721

(ai 2a2)

+2a11 2 + al+ E

_{n=3(n-1) (n}

an

2 a/a2721( log ---_)+)2+ 3 aiagh( lo )3 1 1 Uyi k [6], v x v [G] k [R] 71i Thickness of the boundary layer

U8 _{_} k [G] 8 [G]

72/ x k [R]

e) Displacement thickness of the boundary layer

Uai 1 8

(Uu)dy

k [N] V o

n

) 111°g vi 1

[h]=

The integral constant in eq. (38) is determined bY the condition

at the

-transition point, which depends upon the career in the laminar part. But, for

the present purpose, we put zero under the assumption that the flow is already turbulent at the leading edge of the plate.

Then' the following results are obtained.

a) Coefficient of the total friction

v [E]

Cf =

_-.

2

pU2k

[R][G] (39)

2

Coefficient of the local friction

To

_2 1.0 1

C'f - ₁ k _[G]2 (40)

2p U2

Thickness of the laminar sublayer

81 . [N]

x k [R]

[N]----A +ai log 1

A+ al E

(n-11=11.111

11111 1111kii.iAM

0 /P6i/-L'..! 10" 3.10' 0 'JO' /0' a. loy/CeCi. _{/261U/. - 3 10'} I 07' 0 6

rWil4 MINIM

5

11111111111WMFAIM

_{p- I 7} J /0 10" 10" 3./0"

0111

pm/L 0 I 3.0_310' 10 lay L/L 10 0 5C110ENHERR' x pRANDTL-SCHLJCIIT1N0 A SCI-I0ENHERRI-00004 WIRAOA

.0 NORFOLX HOT PLASTIC' D HARE ISLAND HOT PLASTIC E FLAME SPRAYED MOT PLASTIC

45' pHA - 10-4 3-10' 10" .7-10" /0" 3-/O" p /.4 "RI& - 10" , 3.60' 10'_qtr. p-1.2 pH& -Fig. 12. 07 _{'41%441411%.4,64} - J /0 '6

'%-111

f) Momentum thickness of the boundary layer

U z5L D k [E] 751 1

Cf (44)

v pvU v 721[G] x 2

If we put [4]=0 in

the above relations, we obtain the solutions for the

smooth surface, which coincide sufficiently with the Prandtl-Schlichting's result.

n = (SC,L lCia TING) 4

111t4141111111111

110 1 I I I 1111111 IMis4".1411th.111 0 2 0 04 02 Fig. 13. logULA, 10.' (SCHLICHTINfil 9 0 log CILA., 13 7--- I. pH/L-111.11h.'..414S11..41444441" /;:47 1.3 t L2

Figs. 12 and 13 indicate the behavior of the resistance coefficients for the

wavy roughened plates corresponding to the wave slope and height. The two parameters p and H/L in the figures represent the shape (slope) and the size of roughness respectively.

It is remarkable that Reynolds number at the maximum curvature of Cf

changes not with p but with H/L. This property is useful for the prediction of

the frictional resistance of actual ship from the model test. We will now describe it by some examples.

Let the model be a flat plate coated by the same paint with an actual ship,

and its length sufficient to avoid the laminar effect. Testing speed is required to

cover the range of the maximum curvature, so it must be much higher than ordinary

0,

0/L-7..7c7q7c74771V4)

test especially when the paint is smoother one. We are able to evaluate the

resistance of ship by adopting the actual L when p is obtained from the model

test.

Dr. Toddl)2) reports in his paper some of data for the painted surfaces of

21 ft. long, which fit exactly for our present purpose.

" Norfolk " hot plastic has its maximum curvature at R=2.5x 107, hence p=1.7

and RH/L=1.5 x 10-5. When this paint is used on a ship of 525 ft. long, C1 is

obtained by shifting the model result along UfiH/v=cOnst.-curve in _{an amount} of log 525/21. The roughness allowance is 4c1=0.0004 at R=109. _Strictly

speak-ing, a correction for the variation of 722/121 is necessary, _{in this case log 525/21}

+log 1.23 _{we are, however, on the safe side if we neglect it. The values for}

other paints are shown in Table 2. ilmean has the meaning of the height of

equivalent regular waves as explained in the chapter 3 assuming =3. 4C1

is for the same ship at R=109.

Table 2

Plotting H.a. in Fig. 8, they seem small beyond our expectation. Therefore it may be said that the dominant cause of the roughness effect is on the steeper

elements regardless of their amplitude, and that more gradual elementsare not more than to hasten the commencement of the effect z1C1 in the above table contains no allowance for the structural roughnesses such as butts and seams of plating, rivet

heads, ruggedness of welding lines etc. For the ships of welded construction,

however, they would give too small effect to disturb the above estimation. _Above

all, a moderate value of %IC1 for recent ships may be the magnitude of an order of 0.0001, which is only a quarter of the customarily used.

6. Conclusion

The roughness are classified into two essentially different types according

to the existence of the separation or not, of which the non-separation type is

defined "Wavy ".

the modified momentum transfer theory is used for the calculation. This

involves the viscosity effects and gives the thickness of the laminar sublayer

Norfolk

hot plastic Mare Islandhot Plastic Flame sprayedhot plastic A ntiorrosive R at mak.

curvature 2.5)<10T 2.5)<107 4)<107 5><10T ?

Hmecya 0.O013' . 0.0013' _{0.0008" 0.0005"} .

P 1.7 1.5 1.2 1.2

which is important to solve the problems of rough surface.

The sand roughness is explained by the change of shearing stress and the

increase of mixture lengh, both of which bring the losses of velocity near the

protuberances.

The local disturbance produced on the wavy surface is connected only to

the augmentation of mixture length by which the loss of velocity is brought in the near region of the wall surface.

The characteristic of Cs-curve for the sand roughness is theoretically

obtained.

1) A surface which was once wavy keeps its property even when Reynolds number becomes large. The Characteristic of the wavy surface must be

dis-tinguiShed from the transitional property of a surface of separation type.

The steepness limit of the wavy surface is estimated from the Fromm's experiment. A surface with practical paint has the steepness less than- this limit and belongs to the wavy.

Cs of the wavy roughened flat plate is calculated for -various shape- and

size-parameters. The results agree well with the recognized properties of the ship's

friction. A method of prediction of the ship's frictional resistance is indicated

with some examples. dcr = 0.0001 may be considered as a good estimation for

ordinary ship paints.

The roughness effect of the paint seems to be determined by an element of the smallest in height but the steepest in slope.

As for a flat plate of sand roughness, Schlichting's work based on the

Nikuradse's experiments seams to be reliable. If our method were developed to

this subject, a similar , result would be obtained. A fouled ship's surface belongs to

this type.

References

Todd ; S.N.A.M.E,. 1951. Todd ; S.N.A.M.E., 1954.

Con, Lackenby, Walker ; T.I.N.A., 1953.

Denny ; T.I.N.A., 1951.

Sasajima, Tanaka ; Kansai Zosen Kyokaishi 80, 1955 (in Japanese).

Fage ; Phil. Mag., (7), 21, 1936.

Karman ; Jour. of Aeron. Sci., Feb., 1934.

Schlichting ; Ingen. Archiv., 7, Heft 1, 1936.

Fromm ; Z.A.M.M., Heft 5. 1923.

Hopf ; Z.A.M.M., Heft, 5, 1923.

Nikuradse ; Forsch. Arb., Heft 289, 1929.