Progress in the computation of the frictional resistance of ships

FIOGRESS IN THE cJAPurATION 3F TI FftIcrIONAL FFSISTANCE UF SHIPS

by

Ko E0 Schorher Dr0 g0

Tha weU4rio'mi Fraude method oí calculating the pr required to propel ship at a given speed by neans of a modal test

_{presuppoa that}

th sicti friction components of the ttai snip and model re5ietancee can

be calculated independently0 ?ci _{moat ship forms these components}

cirprisa i-.8O percent of the total, so that the

_{aclacy cf the power}

pred.iction by this method daperld9 t large extent on the acuracy of the friction data uaedQ

The exterior of ehip is an irrogulsr3.y oured Sace Thi ace'at.e det9rniiaat.ion of the tangential _{or friction forces on such a surface, either}

exp3rira9ntally or theoretically, ¡reaents great ciifficulties0

A iimlifi

cat:on is made therefore by

_{btituting for the cn'ved surfaces}

_{a plane} suxace of equal length and aree nd by assuming that the realstarc of this oquf.valent plank is ece.1 to the frictional _{resistance of the cu.reó aurfuce0} Var..ous ìstimates arid calculatona htve shown that the error thtrc<iL2.ced by

this aa umption is sr1 1, except perhops for very small mode1s o that usually no attempt is iake to ccrrac

_{for this eror0}

Th rasitane of a plank

can of aoure ha readily obtained y experiment, arid msny such experirients have been ¡nade0 However, planks

longer

_{han about fifty or sixty £et re difficult to ieke}

_{and handle,}

so that the frictional 'esis1;anoe cf hip

_{ten to twenty timen thii length}

could not be obtained by thiH direct nethod, _{Extrtipolatiori of the existthg} exp _{mental data was resorted to, but in a nce of a raticrial theory to}

g.de this extrapoLation, thoe valure wre uncertain at best0 Nevertheli3a,

Deift UniveNily of TechnoIoy Snip Hyromechanics Laboratorj

tJIirry

Mekelweg 2 - 2628 CD Deift The Netherlands

the fant that Froud& e constante, whi ch are ba8.31 on plank tests ruade. by

William )?ude in 1572, etill aie used today with only minor rvieiona, spek vieli f3r the care and jtidgnent used in thes xtrapo1ation3

Aboit the turn of the ninetentb aentuzy, effo:te to develop a ratioii

friction theory began to bear frutta The uftery ;urroundir.g thts conp1e phenomenn wa. gradually dieplled, so that today 'vo havi a fairly clear perception of ivhat is trustworthy and what still reiaain to be donc to obtain the ultimate so].ution

The mdseutal _{i.ptior in' all rat.ional aroachea to the prob].em}

is thc tptioti that a fluid in notion past a stationary 5olid bciridary c1in to the surfaot through ¿dhealve forces; in other words, if ire

stip'ulat

that the fluid has th

velccit.y U at sotre dietane Y from the well, we assume that the fluid has ro velocity at the wall and at any

distance y Ç ï hay a velocitç

. of :agnituck U o e.

conaequeics of this a anption we mar

clxe that at the dstaxrne y

froi the

al1 two contiguous 1rs a distance 4y apart. move with

7

P

Io./ìv

ic#

vo1ocit

4 J

relative to each otPr

Thialayer prcduees shear W1

stresses ith1.ci are the cause of frictional ristaoe0

ir Isaac Newton

n 1687 :ostuiated that the a.ar' str ss ()

is:Lng in tbi manner was

proportioai to the velocity gr$.ent

_{ksuring this to be trie, re}

therei'ore as a fundamental rola.ion

,L4b

(i)

/

wher?, £{ is a proportiGnaitty oon:tant callel t

cof..cient of vios'0

/

In 1383, Osborne Rsynoid shcwcd that in ordinary £luids auch as

the iJ.ow near a solid boundary apear as a. smooth glidiflg a,ction of layer over lsyer0 Equation Z)is etrietly app.lieable for t'is regirre of flow .hicb tt3

now snrlly ai1ed isou, czt lauinar flows

At higher

veloc1t12

a ¿or, of instability ocur,

id the originally mcoth flow

breiks ap into a mean motion overlaid by sna1l random to.-and-fro iotion2

of fluid partils; when this occurs Equation t ce.es to b valid

Eventually, a new ình3rently tabie f2c'& regine 5..s re-ostab1ished which

is

sadays called turbulent flow aid, as 'Will

t.ranspire later, i

by far the nore inportant regine of the tw By c&ret eçperiTerht3, Heyxioids found that the transition from 1eminax to turburnt flow occurs at a

certain value of the nondimi onal ratio

VL

í

/4

whe:e, V is a characteristt veJ.00it' of th, l'mw

i.. i

a charaoteriati lh

p Ia the density of

t11f

,fr

is th

vis coity of th

meciu

iS the nematic visosi f the methi

ail expssed in con3ictent pbyLaJ. mits

The resistance of a flat. plate In lcn±nar flow wia caì.ulated by Blasiuz in 1905 on th3 beds of rational miachan.cs ard _{a bypcthess} introdued by Prandtl in l90L

_{Frathtl hd asrd that the effect of}

coeity on the f.ow of a fìz. aver

. solid bouncary s

'find to a

very narrow layer cice to the h ndry in other trds he aated that the

flow inside of this so-3ILUed

_{bc-unthry lrr was governed by- the geral}

Navier.-3tokes equations for v.acoue fluld, while outside of

_{this lirer}

F

f

_(%JLJ

the flow 's governed by the Euler equaticns for non-.viscoue f].iz.d By this division of the ilor nto two re4cns and the assuiaption that

the boundary layer lra3 thin, the NaViEir-tOke3 equations could be

simplfied0 In articular, thooe equations becaie solvable for the two dimensione]. caso of the r.ecous flow along a flat plate0 As

mentioned above, these cloultiona cere carried through by Blasis who obtained the foiowing fo-ula a end ros uit

there

I

is the frtctionai resistance in poundz

S

i

the wetted aurfaoe in sqaaro feet

C

i

coefficient of mean peciflc friction

R is the Reynolds NwIer defined above

This equation,

ile entirely valid

has only minor practical vale for the Nava]. Architect, because expo imsnit.a show that lvx £1nr

along a flat plate breaks down ir the Rernold Number range 1X),O)O /i vJ2-Ç

to 3OO 000; that te, for a pie on,y two iaet long, the f low ,cea es to be entirely iinar at a apeed of tbout one foot per second0 To

make matters worse, the tr'.nsition f om puro lEnLthar to pure trbLent flaw is not stiden or sharply 1efined Thus, 1f no specie]. precautions

are taken, it 1s found by experirnt that stable tirbulent flow s2crig a flat platu is not reaohed ntU he tynoics Number 1,000,000 8

ceeded1be.een this lower li.t of the tn'hulent regirìe

and the

uqer ,ïznit of the 1amina' regine, the i10

i med ax foUs no

defirtho la

Ezperimentere who must ork in th! re.ngo atterpt therefore to render the flow turbulent Lr .iens o turbulence atimulating devicea such a wires 3P nand trIp3 attached to the bo' of the model, or struts, roda cito0 tced ithead of he model0

Obviously, the diffieultie presented by mizod flow are a2.so iaLniirized bj using large rather than vaU modeli0

In contrat to th viscous £low orob].em of the fiat plates the

corresponding turbulent f 1iw problem proved conaidsr&],y more dlfticult

to solvo This arose frcn tho faot already ¡iritioed that turbulence

is a rand3m tc»andfro motion superimposed ori a mean flow which does

not lend itoelf reai].y to rathnatical uxpreesion

tcept by state'

tical nethcds

In spite of thi

difficulty a rational theory was developed between 1920 and 1932

Fily t1iruh the wx'k of Lndig

Frandtl arid Theodore von arm'an _{Thia thet.ry, while y no means}

cozplete and contair!ng iiy simpliricatioï has produced pressions

Mch interpret experimental facts _{oeedingly well 55 will be sean} later0

The Íouidation of this therry aga..n the ass umption thet th entire friction process is confined to the botidary layer0 At the edge of this layer

pred to be thin

the velocity is

qtal to the veiocty of the

undisturbed flowwhile inside of the layer the near. velocity dcreues

from U

to zero at the wan

'Y

To th oiw attention, let O-

in Figura I be s stationary wafl oonciding with the .taxin of an x.y coorciinato ayotezt; tne .floc i asetned to te

two dimensional, uniform and ittesdy to th

_{left of O and outide cf the}

boundary layer0 _{The thicknese of this layer (groatly exaggerated in the} figuro) is denoted by

_J

and i*i premied to be the dist&nce from th waLl where the velocity is ninety-nine p*rcent of the d.sturbed velocity

Consider now a fluid particle ¿i. distance y from th walL _{If its 7e].00ity} in Sectton D..E is u, it must Imve a _{nailer velocity (} 4&

in Section E$, because the retai'din

effect of thi wall is

_tie0

It follo, that the slopes of the velocity prot'iies

are becomi.ng steeper with dounstreain distance; this nrroning of the velocity

_{profile in the}

diraotion muet be acoompanied by an increaee ìn their

_{iidth in the !}

dtroction so as to preserve continty of floti- _{Thus, by elementary} reasoning we may conclude that _J

_{is a ructioii of x}

To connect the velocity cistribution with the &iear 8treSs at the

ic1l (ihat is, the frictional drag

the plate): vo1 Kar!nan app1.ed the momentum theorem to the bondar layer

This theora t4te5 that. the

change of momenti

in unite time is equrl to the applied fcte

Thw,

coridoring a strip of unit width nonl to the plane of the paper, we

have

r

()

'where, the integral at the left :epreeGute th frictional drag per unit

width of the plate betwe3n stations O .nd z;

the integral at the

r.ght represents the moiientun loes in the boundary layer betwerA

these stations;

'io i the raU shear stree;

(,Ñ44 dj

_is

_{he fluid Ia3n passtng thrcugh oe e1iient of width}

dy in urii time;

Dfferentiating Equation ¿ with reapet to , we get

ç

(ø

f(u4J

This

prescn ahows that if the

_{ìriation of' the velocity 24 i a}

knci function o _{y, the shear strass le obtained me a f unct±on of the}

bourdaxy layar thickness

I

If we also knor or are abJ.e to deduce he

vaatou of

as a function of z, the friction force is fo.md

Prandt].

and von 1rman reasoned thht tne twn functions u .e g(y) an 5 Cf Cx)

cannot b entirely Independent and r.ìat 'be cortected through the Ethea?

stress and the characteristIc properties of the me±Lunh _{Thus, whn the} velocity distribution was assì.ned to be given _{by an exponential function}

of the type and .1 C)

Cf

(3

Id

(17f

(Li'

(apr.)

it aen*d reas lable that th proportioaìity fecto

? _{'thou1d be a}

function of C. and )1 oniy. that is, that. wu should hava

L)(V(3

)))

Dy .ana1ing this roiation by the theoy Í d nn1oL13

en equation for r

s obtained !dllCh h

the fn

E

ihore, C ie a constant.

Thaerting Equations 6 and 3 ii.to the onantun oquatin J we get

I..-. _:

---;-¿j

¡n. 2 11.

.

* L . ) ,

(7)

itter, 1j and K2 are constactj to ba deterid.nd by- expriinent

_{To detertre}

these ontant&, Prandtl anà ro _{Karman con.jetirod that th rnechauism}

lt zhould be possible to obtan them as coil ag the sponent (n) front

veLi'distribution aiicì çresuredrop

asurimnts in circular p3.pes

Using va1ue found by Blasi

fr

an ans].rsis of many pipe friction experiments, the final equationa were

r

_:e37

E1.

OO4

R

whore, R le the Reynolds Nmber

ELticn 12 when conparc uith the results of experiments on fLit plates im for to hold on.iy mithin a limitad range of Reynolds Nunbr;

outaic(e

f this range, part

ularly &t the biher Reyncld Mmbe:r, the

dev1atiori w.s marked.

_{In rarcspect, this scaied not sur}

isng since

tho simple exponential lay for ve1ocItr distribution wc after ali. only a

convriìent mathematical sxpreision, .&nd waii not founzied on pbyslcal Iactc0

Horever, the results were en aging enough to pursu3 this line of

rseoring firther

ro obtain a rational solution oe other phyicel fact wa obviously

neeasary0 This wa given by an eben3ion of Ecation 3 deued by

Reynolds in h.s previously mntioned voi'k Reynolds had postulated that

in turu1ent flow the major por-Uors of the pressure 1c _{in pipes (or the} friction drag of plates) cazas troei the mornentis exchange of relatively ].arge c1uiips of fluid partioi.ea moir.g back and forth acre the rean .f1c and only a minor portion can from visous shear0

_{In matheu*tial iagige}

this wa expreaeed br the ganaral c'uatton

r

M _(p3)

/

7wu

3y _{untw cons1.derations Reynolds than founi that, if M}

end $ zre

the velocity fluctuations pa.tUel to and ¿cres the n*an 1ow re3peot1ve1y

the _{eoond member of Equetion 13 1t proportiona]. to the product} c

)

that e, the thear etres equation becoiee

wh.era

the b'r indicates the trtpora1 mn vri1'ie of the ro&ct (A.I'

_CT1 )

This equation supp1.ed th rnaing link _{in the development of the}

turìclent Ifloir thry

_{?randti uti1ied it In the fo].1oiIzg}

_ma.nner

_First,

hs au'med that the first abar of

_{quat.ion Lt was negligible conpired to} the !teC3d meiber in the oiter higWj turth.lent part of tint °bounthiy layer,

while vry close to the 'al1

_{iero ths prsnca o! the iia]J. tended to}

danp ox the turbulent fluctu _{tho aeconcl tern vas ligible coipared to}

t:e first0

This is

_{usfl,' wpreed br the tateent tha.}

_{a ianar ub}

layer uxierilea the turbulent bouidazy _{rsrg a tact which hai ben -erified}

cpinta11r0 In

cccrdanc _{th this aesumption, ths ahrar atxes}

equation for turbulent flow bseaie 5iinp

r

_-

9(7 ¡ _.

()

J3 7,4

)/

(i r)

tier, the expressîon

under the rd5 cal _{.ndicat the mean square} va1a of the velocity f1uctwAt.ions _and

_{Is the oca11ed}

cor-xation coeftiient0

Tc reduce Eque.tion ]. _{fwther, Prpndtl aseuined t1t the root mean} e quare of the longitr1inal velocity fluctuation,

j?

_{, was proportionú}

to the slope of the velocity profile of the rasan ilw mu1tplie by

a

charicterististic length

oh he called the 0lrizirAg length"; he f iwther

aasuned that there exi3ted a definite correlation between the longitudine].

ars.! tranev'e'ee fluctuations0 The p1aus1b.1ity of this reasoning is illustrates in 1gure 2, and farther d1scured in the next paragraph0

Pru

R

i

L

I

In &gure Z let A. _{be a portion of the mean velo ci}

_{profil izi the}

turhlont part of the bounda

ieter; let

be the average velocity in the

fluid Ltjer (l

a distance y frorn th walJ and,41and vJ be the turbulent

veloci4y fluctuations in this ngion0 Assume now that in the turbulcnt r4xing p:rocess a maze of fluid particles le transported from Layer (1) to a neighboring Layer (2) a distance _{farther fr3m t}

_{waU and that this}

aas retains its îdentitq until .t reaches Layer (2) but then

_{mixes v±th the}

surrouxing fluid suddenly and orplee1y; obviously, the _{transport velocity}

partcle3 when merging with the flzid in Layer (2) ie ori the a'7rag8

prportinal to

Fuz'therïnore to preserve continuity of

flow, for each flass of prtici.s transported from Ler (i) to Layer (2)

ar eqtal mase rnu5t be transported in the reverso directions it follows that

I

scie corrsiation must «xit oetween the taneport velocity ( 7) arid the velocity inci'eaea or rea

mixîn; process led ?rar1t1. to set

so that Fuation l

became

.1

,E"

=

) This concept cf the turbulent

2..

z

.q

or, ccnticti;7 C

_-

,i /C

) irto a single factor, it rsductxi to

r

_-

_C")

This general eqition iM indepondet of the bouriary

_{shape so that it}

iray be prased to be valid for a circular pi.pe.

_{It is esily p'oved that}

the sIesr stress in a pipe vaise linearly with

_{distance from the wall; in}

other wcrds

if (a) denotis the pipe radius,

_{y the distance frcm the wail,}

-1j

Inserting this relation ini Equation 16, it beccmes

r

/

i 6/a

4Á E3

J

L/r' )i1

ci

where, C Is a constant of integr&ttior0

Von iarnian

_{this sa e equation by a different line of}

_aoning Experirtta on rectangular cond.ts with smeoth and rough walls had

dexnon-strited that when th wsfl 3hear stresr Z' ¿ich is proportional to

_prese

drop ir a pipeJ 'ias held constant, the thape of the re1oity distr1butior. curre in the highly turbulent entrsl règion was Independent of discharge, roughness coefticient, etc0

_{In artcular, the Lollouing "velocity defect"}

t'.)

111 vaLuo _{in this equation except J re known o det.ermlnable by eperinent0}

By such experiraenta, t wa forxt that the ¡nixing length A? varied ver nesrly according to the relation

,

_Á7

whs, k i

a coefficient hring approxiraate1y the value O.!O Iiwerting this relation in EquatIon L6b, the final equation is

:

(3

law was found to hold

, ,4t

where, ,U

ta the

odrnu velocity at the center of the onduit

,Li.

is the velocity

_{distance y from the wU}

a is the radius or halt breadth of the conduit0

This xperinental fact suggested that the shape of th velocity profila wa conditioned wholly by loc1 factors in othei worde, that the local velocity ai

the stress pattem at different distances frnii the wail

were

closely connected with the devatF of the mean velocd.ty profiie Von Kazm8n _{now conjectured tha the individual ±1.ow patteria in arr one}

eecton &croa the boundary 1yer might be dynamically similar0 _?asuance

of this thought to ita logica. conclusion lad to the condition that

scoesiive derivatives of the

_{locity profile must be related by the follcwin} ecuation

cL/d.

d

4<

-.--.---.-.--.-- p

It will be noted that eath of these ratios

he the dimen3ioI7 of a length;

it aeeirnd reaGonable to use the "mixing length" (I), defined by- Pramiitl,

as the charao'!,erietio length of the system, so that Equat. 19 could be

written

9

A _J

/)LA

(z.)

₄₄

/

É

1 '

the relation between the Ehear stresa end the velocity distribution became

7

ina].1r wten the sheer stra: in the boundz.x7 layer of the straight plank

was asurci', as a first approxinwtion, to bo cqual to the a1l shear etret3s

r,,

Equat.1.on 21

-s

which, when integrated with prcper values for the constaritt of thtcgation

reduces to Equation 170

It wtU he noticed that according to Equation 17., the velocity

--beriw minim infinity at the vail, that ir, tr yO

ThiM difficulty is svcided by st.tçulating that tìi'a logarithmic velocity law holds oriij to the

edge of the laminar sublayer0 The formula ny also be expEcted to break dour3

at the outer edge of the boundary layer since the shear stress nut graduaUy

redc to zero as the boundary layer

rges into the undisturbed riuid, carefully considering these Íact ¿lee mp!-Foerstr Ibdrvmechsz3isce

Poblone des Sc}tfsantriebc, 1932, pp, 66-67 von Kamen tranatorrned

Equation 17 into the foU ovir.ig one

U 5

í ()

*

& c

.

where

_ì,(J

Jis a function which is independent o± the Reyno].da Ni.ber

j.

C2 is the valua of

h()whsnY

¡a

ia

;vii,

This equation, when introduced into the mo'tentuin integral, Equation % ¡nd when the integration was extended from zero to c) _{(rather than from thu} By

reor

the Laminar s ublayar to following equation

vth ares ami are nnerice1 onttafltB

kf te 80.ie tr8fl8forTatiOfl this bscorrLeS'

tias sutitu,ed for the value ¡k U

for e.znplioit.y

,

j

Differentiating t.io expreion 1r. the bracket ami splitting vari*bJ.o, we get

LI)

_{_jc}

_-

_LrL

I

a

This equations when integrated, resulta in

i

or, y th'cspping the second and third terrr, whieh are

_{aU coipud to the fir,}

this redize to

C

cl as stricU

re4uixad), led to the

df

*1

/0

-- J

_¿J

f'

-

_z

z

* z

c

)7

(24)

cÑ

(zr)

(z

(7T)

,

_s,)

Mow it wLU b _{reriebered that by definition v had}

j

*

(u)

we may aleo define & "local coefficient of frictiot" by the relation

/

introducing these three expressions in Equation 27 end mbirIng constants,

we get

-

_{(Ri; Cf'}

_{J *}

_B

where, A

iT

k and B (2k)

Equation (29) &s relation betw9en ths local coefficient cf friction at dis tance x from the 1eadin dge of the plate and tie corr&poricithg

ieynolds Number; by "local. coefi4icient of friction

is rart the

coeffloi3nt of friction of the jagt foot of a plate having a length of

(x) feet

Obviously, for practIcal calulatioris we need to know th meezl

coefficient of friction rather than the local ooefticient0 The latter was derived by the criter fc quatio 29 In the following ranner0

For an exponential forniuz of the type of Eqation (3)

or (12), it

is easily shown that the coneotion betieen the local _{aid the mean frLction} coefficient is given by

F

Cf,

₎

where, _{is the exponent in the EcponentIa1 fcrmula}

Now when Equation 29 _{plotted on do'ible iogar.thmic eoordintM paper,}

we get a shaU _{curve concave zpard; it may thei'e.tore be considered} equivalent to an exponential £onula uith slowly _{changing ecponnts}

Corßiderin (ni) as variable, _{we niiy thus introdueu Equation 3(L) in Eqution 29} The resulting expression may be further

_{reduced by expanding jirn}

leah to

m-!!5

(9i)

Froiu experimental data wa know that in the Jeyno1d8 Nuiaber rrnge of

greatest practical interest,

xr.e1y, R

lO to R

lO- the value of ¡n lies between 0.10 and 02O; t.he value of B is uiknc'wn

Hoior, if B

i

mss.md to have the order of rignitixie l4 we an e3timnte the relative nagnitudes of the terrz on the right hand side of Eq:ation 3].

It will be

four&1 that within the Reynolds Nther range atate1, the second terni 'aries only from 1O26 to l0l9 and the third term varie, from 0,73 to ..'083 It fo]ìow that Equation 31 e vez nearly equal to

£v

--

M

-ir

T)

Nf

where, A andM are ccnetanta tr be luated by axperimerc

Shortly before von arman published the theoretical wrk discuned in preceeding pageai the writer had undertaken an ¿niüyis of all the wtant xperinental data with f'riction planes and carried out

ne,

experlm9nta at low Reynolds Niber0 AU of these available test data Were recputed to a uniform Laeis and plotted in the forn of C e.gainrt

Reynold3 Thimber R0 _{Taking into 'or1Fidarat1o1 the various onditiona of}

the indiiidual tests, e nean cur't

wae drawn to represent the frictional resistance of an average te)mically smooth plane surfaces _Vaious

C;

matheatica]. expresio were tried to obtain an eqzation for this n curvos among than 'ras von Karran a Eq:atton (29) tranforned to Eqi.tion 32 as

explained ì1xìe0

This eqtion vas found to be en excellent fit o'er the

whole _{Reynolds Number range cowered by experimente when the cone tants had}

the following numerical values It

A

converting the natural logan bhn to the base _{ten logarithm, the f..na1 frïctiori}

f orniulì for smooth straight p1ne surfaces thus b9

-

k

(1C)

r-and the relation between the n and the loca], friction coefficiente

becane

C277

Ç.

O 2fl

(33)

(34)

Equation (33) t. used since 19h8 by the _{model basins on the AnertccLn} Continent as standard formula for the calculation of

_{the tuikulent friction}

of hlp niote1a0 The sai formula le used ror the calculatlx of tk _ship friction with the addition of &, conatiuit value O0C()oL to allow lcr the

roughne e t the averga ship bot tcmç this roughness _{allowance was obtained} by compering the res1atance couuted frcm tl'rust resurernents c a number

C:j:

1Jhile etxcellent practical re3ult are obtained b thc3e foraiulsa,

.t

ay be peoted that still bet.ter results iiU be obteined rhon the roughness

aflcvrant:e be broken dowu for ar.uus ship aurfaces

altar more ath bettr

relt from Ethip thrust rieasurement9 become availab].e

For complAteness of this paper, .t my be mentioned that Prandt]. and

Schlichting carried out a simt:Lar anaLysis as that by von Kanîian and the

wrter; t,he reaiOEta oi' thi8 analysis becante a'rilahle about the a.ne time

and cculd be axpressed by the empiricc3. formula

Thì f orriula agrees with Equation 33 within about three percent over the whole range of Ryno].ds Number which s of practical irteret The atructur o1 th18 formula is of course simpler than that of Equation 33, which is en advantaCe in some cases, but it 1ack the theoretical justific&tion which eupport Equation 33

In 19LjC) ßch2lz-Grunow measured the drag of a plate set flush intc the

bottom of a ipecially constructed wind tunnl Tiieee teata gave soni)at

lower vaues than thoee computed by Equation 33 and 3S, but suba tantiated

the reasoning that led to the derivation of those formulae0 Schzrunows

anayeis, which followed th sane pattani, yielded a vezy cozrplx equations on account of tbi complexity, he prop3Ld the following interpolation fortiula

0e 417

-I

on1.iaion it rcay be sa.d that., while perha _{we do not have the}

u1tirna'e f _{we do have r.oi f orrnite for smooth Qurfaces ìih}

_re

based on rational theorl and which

_{coare weil r1th}

_{ristIng experimontal}

r8sulti

Cx.rab1e f ormui ae ior moderately _{ro ugh surfaces 3uCh 18}

£ouvd on slipi are not so well. tablìehed0 However, for praotica'L purpoaes

the diract approach by

_{way of tust eesureumrts on ships}

_{apse to}