Theoretical research on frictional resistance of a flat plate of finite breadth and a circular cylinder in axial motion

Lab.

v. Schtseptholaikunc

Technischta Hncsichcol

Delft

Theoretical Research on Frictional Resistance of a

Flat Plate of Finite Breadth and a

Circular Cylinder in Axial Motion*

By

Hideo SASAJIMA, Matao TAKAGI** and Ichiro TANAKA

Reprinted from

TECHNOLOGY REPORTS OF THE OSAKA UNIVERSITY

Vol. 9 No. 345 Faculty of Engineering

Osaka University Osaka, Japan

No. 345

(Received Dec. 20, 1958)

Theoretical Research on Frictional Resistance of a

_Flat Plate of Finite Breadth and a

Circular Cylinder in Axial Motion*

By

Hideo SASAJIMA, Matao TAKAGI** and Ichiro TANAKA

(Department of Naval Architecture)

Abstract

In order to know the three-dimensional characteristics of the frictional resistance, two typical cases are studied.

Part 1 : The effect of breadth of a flat plate on its frictional resistance.

Two complicated simultaneous equations are reduced to single equation by

adopting an elliptic coordinates system. By the ordinary 'method using

momentum relation the frictional resistance of the plate is calculated as a

function of its Reynolds number and aspect ratio. The results are

'Com-pared With experiments of Hughes.

Part 2: The effect of transverse curvature. AS a typical case with its effect, a flow along the outer surface of a circular cylinder, in its axial motion is investigated by the momentum transfer theory. Frictional

resistance is expressed as a function of the position- and radius-Reynolds

numbers and compared With Hughes' experiments.

Part 1: the Effect of Breadth on Frictional Resistance of a Flat Plate 1. Introduction and Rough Consideration

kecently Hughesl) made a series of experiments of the flat plates with finite breadth and obtained valuable results. However it appears to be very difficult to perform such kinds of experiments because they need a high degree

of accuracy. Of course he may have paid much efforts for this point, but some

ambiguity seems to remain in his conclusions, which say the effect of aspect ratio is independent of Reynolds numbers and hence the basic line of frictional resistance is considerably lower than -the Schoenherr line at large Reynolds

numbers.

:Now we consider the general features of the problem before we make a * This work is also reported in the following publications.

1 : H. SAsAgmA & M. TAKAGI, J. Soc. Nay. Arch. Japan, 101, 49, 1957 (in Japanese)

2": I. TANAKA & H. SAsmima, J. Soc. Nay. Arch. Japan, 102, 103, 1958 (in Japanese)

** the Hitachi Shipbuilding & Engineering Co., Ltd.

68 Theoretical Research on Frictional Resistance of a Flat Plate of Finite Breadth and a Circular Cylinder in Axial Motion

theoretical calculation which follows. The increase of frictional resistance of a flat plate of finite breadth compared with the infinitely wide one is probably due to the increase.of fluid quantity dragged by the plate. If it' is assumed that the boundary layer thiekness over the plate is uniform, resistance is given by the sum of the two components, the resistance as a part of the infinitely wide plate and the contribution due to the cylindrical parts around two edges. The latter may be roughly estimated from the resistance of a long wire of small radins ro in axial motion. The resistance of the wire is given by

R=27-cplr0 oic it)dy + Oyu (U

u)dy}

where y is measured from the surface of the wire. The first term being the same form as the infinitely wide plate, it represents the term corresponding to the increase of the wetted 'surface due to the thickness of the plate at edges. So it decreases the aspect ratio slightly. The second term is the 'essential one which describes the increase due to finite breadth and it is noted here that the

term is independent of 7-0. From this relation it is known that the effect of

finite breadth is more and more remarkable. as the plate breadth becomes small

compared with the thickness of the boundary layer. Therefore both for constant

Reynolds number the larger the aspect ratio is, and for constant aspect ratio the smaller the Reynolds number is, the larger the rate of resistance increase

follows. In Fig. 5 the results of this simplified calculation are also included. 2. Fundamental Equations and Velocity Distribution

For simplicity we depict in the paper only *the principle of the calculation. Details should be referred. to the original paper. The, method of calculation is

the same as the author's report2) preveously issued. First of all the approxi-mate shearing stress distribution is calculated from the equation of motion and then by the modified momentum transfer theory we get the relation between

z (zip)

Fig. 1

the shearing stress and the velocity

gradi-ent. After integrating it the velocity distribution is obtained. Substituting it into the Kartnan's momentum relation, we get the shearing stress or frictional resistance as a function of the Reynolds number and

aspect ratio.

We take an elliptic coordinates system

H. SASAJIMA, M. TAKAGI and I. TANAKA 69

x=ccosl*osv,

y=csinhEsinv,

(1)

z=2.

Under such coordinates, it is proved that v and rz is negligible as is shown later and we are allowed to consider the case along z-axis only. Equation of motion along z is

u Ow Ow 1

Or .1

( 1 \

cA

+w oz

pc.1 O pc O AP.

and the Karman's relation is

y2,-a to6pw(Ww)c2,12d= crosinv, ( 3 ) where A=1/ sin2v+ sinh2

r=rtz (henceforth),

es= corresponding to the boundary layer thickness..

Neglecting the left hand side of (2), we get

r=rosin72/2 . ( 4 )

In laminar sublayer,

(p/ cA) x (Ow/Of) .

(5)

Therefore using (4) we get

w / w*= a , ( 6 )

where w*= ro/ p, a= R*Esinv, R*= u*c / v.

Thickness of laminar sublayer is given by the relation

si2(r/p)9.si=k

(7)

where k is a universal constant, s is the distance along v=constant and si is

the thickness_ of sublayer. If we put a suffix one for all the values correspond-ing to s=si, we can rewrite (7) as follows.

al=21/k/v Ti f/l+r,2 / +n2+111(7-1+1/1+ ri2) ( 8)

where r=e/sinv.

In turbulent region, we apply the mixture length theory in terms of s, i. e.,

dw pK2s2(

2' 0 4

(9)

as

Here mixture length is assumed. to be Ks.

Therefore wliv*=- (2/K) t/1+7-2/{11/1+r2+1n(r-V1-Fr2)}dr+tri

4

j/1-1- (1)/

(1.)2

:+111

(f+ 1/1+ (f)2 )1da-

(10)

70 Theoretical Research on Frictional Resistance of a Flat Plate of Finite

Breadth and a Circular Cylinder in Axial Motion

where

r=a/8, 0=Roin2v.

Or approximately

w/w* = criEl + (4.6) -11g (alai)]. (11)

Velocity distributions are shown

in Fig. 2,

(10) being full and

(11) dotted. In what follows we use (11)

for the sake of

too

convenience.

3. Frictional Resistance

4.4,ye

.Fig. 2 Velocity-distribution with parameter P

Substituting (11) into (3)

and neglecting the variation of (xi along z, which is found to be small, we get

a14[Ki]+ a18[K2]/(Rcsin272) 2 RL. (12)

at the aft end L of the plate.

Here RL=-WL/v, R,=-Wc/v,

[K1]=C1 (0.0900 +0.0535 1gC1+ 0.01027 1g21) (-0.0428+0.0708 lg.])

[K2]=C3(0.01825 + 0.01695 1gC1+ 0.00592 1g2 C1+ 0.000881 1g3Ci+ 0.0000540 1g4)

(-0.0294-0.00500 1gC1+ 0.00421 1g2C1+ 0.000911 1g3Cid-0.0000659 1g4).

It is also assumed that the turbulent flow occurs from the leading edge of the

plate. ai is a function of Ci and 72, and the relation between these is obtained from (8) and the following relation

Ti/a12(1+ 41.6 1gC/)/Rcsin272, (13)

which comes from (11).

From (12) we know ci as a function of RL numerically. Then we can calculate the following quantities.

Thickness of boundary layer,

Es= C7a72(1+ lei)/Rasin272.

Local frictional resistance coefficient,

1 2

=

D/

KL

H. SASAjIMA, M. TAKAGI and I. TANAKA 71

1

where [Li] = (0 . 1229 + 0 . 04731gC/) O. 0428 + 0 . 06141g2C1)]/(1.+

₄₆

[L2] = (Ô 0207 + 0 . 00974 1gc,+0.001142 1g2)

(-0.0294+0.00549 1gC1+0.00280 1g2),

D,, =paw (W w)ck ,

1 resistance per unit breadth. (16)

si tsm2o

Total rseistance coefficient,

Cf= clidX = Cf'shivdv.

c o

(17)

c wiz

Rate of increase of total resistance of the plate with finite breadth over the

infinitely wide one,

r IT."(

-cCf' 1 )sinaxhi, (18)

JO

where Cfce means c, of a infinitely wide plate.

4. Calculated Results and Comparison with Experiments

Results are shown in Figs. 3, 4, and 5. From the upper figure of Fig. 3 it

5

Fig. 3 Boundary layer thickness and the increase of frictional resistance 1gCl), 4,A= SO i

norm

_

mil

,

mi

Elm'

50° 70°

II

IMES

IIIIIIIL-"

,

i...--nit&

:

/Co,

I le. Ce

21

mjm..-7

11211.1" 111171111111112

.

II

216:161M

11112511M...._

---Ilia

-- - -- -

---_

In-

-5. 10° 2C' So' - 70*

l.0

0.5

72 Theoretical Research on Frictional Resistanc

Breadth and a Circular Cylinder

Cf 0 040 0 035 ae3o 0.0215 Fig. 10% 40 106 0 51? 143

4 Experimental results of various authors

10 20 30 40 5' VB

Fig. 5 Calculated iesults of the increase of resistance

e of a Flat Plate of Finite

in Axial Motion

is known that aw/av is negligi-bly small over a large part of the breadth except the region near edges, where it is of the same order to Ow/fl. If the cause of the Reynolds stress is supposed to be due to the

velo-city gradient of main stream, the

ratio of Ow/O, and aw/ae may represent the ratio of the corre-sponding shearing stresses, i. e.

rz and rtz. Therefore the figure

serves to judge whether is

negligible or not as compared

with rtz. Comparing the upper

figure with the lower, in which (C//C1... 1) denoting the distribution of resistance

ele-ment along breadth is plotted on the basis of 72, it i known

that the contribution of aw/av near edges reaches only a few per cent of the total resistance. Hence, as was said before, it is proved that the assumpeion r,,z

=0 is

fairly satisfied in the elliptic coordinates.

' In Fig. 4 experimental

re-sults of many authors are shown

as well as the mean lines drawn by Hughes (curved)

and the

author (straight). Fig. 5 shows

the calculated r. The qualita-tive characteristics of the

calcu-lated curves seem to agree with the experiments but the magni-tudes are much smaller. This

may be due to the assumptions

and simplifications involved and

H. SASAJIMA, M. TAKAGI and I. TANAKA 13 must be studied in future. The important feature is that r is influenced by

Reynolds numbers, i. e. r becomes small with the increase of Reynolds numbers.

This is not the same as Hughes' conclusion that r is independent of Reynold'

numbers. His basic line may rather approach to the Schoenherr line at large

Reynolds numbers, as was pointed out by Wieghardt.2)

Part 2. Effect of Transverse Curvature L Introduction

In order to investigate the effect of transverse curvature on frictional- re-sistance, we consider the turbulent boundary layer along the outside surface of a circular cylinder in axial motion. Researches made previously in this field appear to be very 'few. So far as the author knows, there are two theoretical

studies, Landweber's4) and Eckert's5) and one experimental research of Hughes.1)

Besides it is said that experiments were made in U. S. A.,(1) but we have had no chance to see them. Of these, Landweber's calculation was made by the use of the velocity distribution of 1/7th power law and the relation of

1/4th power law according to Ellsworths") paper. Eckert's calculation is the same as Landweber except that it is extended to compressible flows.

The method of calculation using the power law of velocity distribution and KarnAn's momentum integral is proved to be useful to solve the flow along the flat plate, irrespective of their simplicities. If we take analogy from the fiat

plate flow, it is well imagined that the method of power formula may be a convenient way to the case of our present attention. However in the present problem, momentum increases with the square of the distance from the centre of the cylinder. So 8 does not grow up so thick as the flat plate and the velocity distribution becomes fuller. Therefore ro-,-.8 relation, momentum integral and so the frictional resistance will also be different from the case of

the power formula. In fact the comparison of the calculated results of this way with the experimental value does not agree as cited in the Hughes' paper.1) In this report the calculation is made in the same manner as Part 1, considering the effect of radius r0.

2. Method of Calculation

Boundary layer equation along x is

Ou Ou 1 0

u +v

y,,, r = pr r (rr), (19)

Ox

74 Theoretical Research on Frictional Resistance of a Flat Plate of Finite Breadth and a Circular Cylinder in Axial Motion

left hand side,

r=roro/r.

In laminar sublayer, r=,u0u/Or and r=ro.

Hence u/u*=-u*y/v, (y=rr0). (20)

Thickness of sublayer is given by ro/p=1)1012, where ,,'7jT6.82. In turbulent region

r--pP(Ou/Or)2 (21)

where 1 is mixture length. For simplicity we put 1=Ky (K=0.4) in (21) and using the above r we get the volocity distribution,

U 1

r

A _{in?, +1,14(1 / 1-}

1)]

u* KLai

7

6(1/1±1371+1)

Ala1=4.13, 13=8/ro, 72Y/Ô,

where 0(0720(1 is assumed.

4 -6 a

the boundary layer, v=1,

u* KLai

U 1 A

40((11//11++30±11)) 1123)

(22)

In Fig. 6 the velocity distributions are shown. Harman's integral is 1 0 8 r oh] =

ru(Uu)dy.

o (24) Or rewriting it,

(11-/u )2= aaR {Rsh+ Rs2/2/Rro} . (25)

z

where Rz= Ux/v, 140=Ur0/v 12,3=U0/v,

01!2}u

--u

(1- -u-)4

(26)

2[ 2

( 4(V1+13 1) \

1 lt

(27) 3 KO n

[3(/ 1+3 +1) TE K

By solving (25) with I, and /2 expressed as (27), we get the relation of Rz-,-,u*/U

Fig. 6 Velocity distriiniiion Substituting u* \ 2

1 c9.(1/

(20) and (22) 4 into (26) we get )}

( 4 (-1/ 1+8 1)

1 +0 _1) +

/=/T)

/ _Ki9 u* 1 U

20/

ow 1+0 +1)

1+ 0 1) )

1 (.

.12=(1)2

II KO ( 3u*

, 1+19

r 13 At the edge of

H. SASAJIMA, M. TAKAGI and I. TANAKA 75

or Re-72, with parameter Rro. Therefore we can calculate relation from

Cf= rodx/1 pU2x= (Rah+ Ra2.121Rro) /5i , (28)

In this paper integration was made numerically with the conditions that the turbulent boundary layer occurs from the leading edge of the cylinder.

However the above equation can be also solved by series expansion in 1/r0. This goes as follows. If we put

A _{1 14*}

Co 1 1/1+13

C2= vki

/4KRr0,

K Tri

(29)

1+ R +1 , /

2 -1-

-then we can expand the right hand side of (25) in power series of 2 or e and integrate it easily. The results are

1 4/2,0 5 (n2+2 4 K2 3 n.-Itk

+2[n2+2 +i (-4 +2+2n+n2)AnwelS2"

_{n r=}

n

for large Rz, where

An,,.= (-1)r-1-112(n-1)(n 2) (fl + 1) / nr+1

This gives a relation between Rz and 0). While we can express Cf by to from

(28).

c1_ 5 4 R K2 a E n2+2 4 / 1 (31) 1--

"

...A co n

`n )2 /

.

From (30), and .(31) we know Cf as a function of R.

For example, the first approximation is expressed as follows.

R.=

(V

klv /K) e

ekR]/

(02,

[R]=1 4c0+ 60)2+ 2D(1-20)+20-0),

C1=2K2co2[G]/[R], [G]= 1-2(0+ 22(1co) .

3. Results and Application

The calculated results are shown in Figs. 7 and 8. Comparison of the cal-culation with Hughes' experiments shows fair agreement. It is rather interesting

to see that the mixture length theory using such simple r and 1 is valid for

such case. In Fig. 8 m=C f /C fp is plotted on the R/ R70 base, where Cis, means

Cf of the flat plate.

Now consider the application to ship resistance. We replace a ship by a

two-2n)

76 _{Theoretical Research on Frictional Resistance of a Flat Plate of Finite}

Breadth and a Circular Cylinder in Axial Motion

dimensional cylindrical body of the same length *and of the same midship section.

We assume that the bilge parts obey the law of resistance of a circular cylinder

4

0

1000 2000 z-A, 3000

Fig. 7 Comparison of calculated results with experiment

ago

6.0 6.5

70

7-5 "g

Fig. 8 Increase of resistance over flat plate

171. 9

'If

t

L°3 /0"°11111

Al111011PPI

: --R, i 0 LAN;wee" Rx- I 0

AMPP'-Se . 1

he!.

$4,904 etfyo_ r 109 It, 00 \--uullift

Iliks%th.,..11M11111

' k:

-.. ---..,_

1...-

-

0 611r615 2. 1.8 1.6 1-1.2 I.0 9 8 7 6

H. SASAJIMA, M. TAKAGI and I. TANAKA 77

obtained above and the rest does of the flat plate. In geometrical similarity, the increase of frictional resistance due to bilge parts is larger in models than

ships because m increases with the decrease of I?,0. So we only need to consider

the case of models. After numerical calculation we know th t even in a small model the effect of transverse curvature is so small that we need no

considera-tion. (For a model of L=1.2m,. RL=9 x105 and R,-0=1.4x104, the increase is

2%). Therefore it is concluded that the effect of transverse curvature must be

considered only for special cases of very fine forms.

References G. Hughes, TINA, 94, 287 (1952) & 96> 314 (1954) H. Sasajima, Technol. Repts. Osaka. Univ., 5, 371 (1955) K. Wieghardt, Schaff und Hafen, H2, 72 (1955)

W. M. Ellsworth, T.M.B. Repts., 726, (1950)

H. U. Eckert, j. Aero. Sci., 19, 23 (1952)