Drag coefficients of flat plates oscillating normally to their planes

rs

OKT: 1972

ARCHIEF

Lab.

y.

Scheepsbouwkunde

Technische Hogeschool

Deift

Drag Coefficients of Flat Plates Oscillating Normally

to Their Planes

_W

I it roil (iii lUll

'fue 1! Nt notable tort r I iii rr t I ir rttI liii of tire (i rag

eoefiri'nt itf osculating iktt

rlaI \tti titade liv \'Ç'iliiarii

Fronde ini 87'l [11*). ¡ it ¡tri irriti '\l'rinrrnttaI tiata obtained

with tile ose of a heav. frec-sIlirigirig trcrulir ihm, without

des-eruption of his eperiniirtt. At a i'.ii1t

ti Iii invesiiaations,

tite drag tifficient ol 1.( was ied

iii ut frit' or lie drag

of an oeiiIatiiig platt', ¡ri titi' irr'tig:itiort o! i'iicctii'ne.s of bilge kti'k. iovcstigatítN found that aiiit- of irag coulinicrits

tis uumili as i 6 WOUld iii iii't'tssir V li Stint' Ui'S tir e\jiiain their

Ilitniness, lilt

ir ir' t\i

i\iiunIutuui1 of tile

oict'ha-Itisili or nnli a large dksipatiuii of iuclg\.

Io I'rouies is-orli it

starti

tiittt iii' 1ttaiiopd to make

lulore '\t h'IISiVC tests, t Irmnilglm

t ini' Its Imiti

i'nim ri liii any

fui-titi rm'fi'rencc to tiO'n-i ¡rl

lilt' liti initie. ¡bit clii. papers

liv \lmNusttm artd KitIg .

:i.

\iionm IJ. until kt'uli'gan und (tnijicinIcr [5] c'ontrmhutuml tu knnotvlt'tlge of time problem. _t,im iImli'-Iigatioo liv i\i, 7'iinmm tin (I gLitt' iiinIr- mum tire mmninmn

tlrag tii'flìiient of ti fitti plate of itrinmite icnngtim-it liii iatiii for

a rangt' cii nmmmiituii's iii it-miiiatiiini amid Ri'\ rolL inoiii!I. 'i lii'

drag ti'1lh'iemits of ()iiilLmiinig iiniis t\nn' li-ti-i immnin'iI for

tutee tlilfcm:i'mit hint ¡laies of ¡ntlintiii' U-iit'ci mtl mii, 'i lic filate lengtir it,ns kept lm'icti at 12, i2 init'lii' juni all rnnnns su-rc mmrabi'

in a Pmimk shgimtiv \ il-r- intim tint' _{itttcs anni at a} lt'mtim of 0.3 Ii. 'liti' within td lint tinti- s:nrii'nI Itonu I),Ii5 ¡iii-in to 2,5f)

inm'lnt'». Hue conditminins i! limai -\ii'rnmiii'nmt (imiti bit' i'omtsih'm'i'ml

essentially lwo-djmìmt'rt--iimii;ii, 'iiii- ictmits .!iti,t'il in st-i-s strtmtug

depm'mmil'rrum' on the dmltmm'riiirdi's.s imst'iiiimitnii ;nmmnimhtnnle of tini'

plate. 'flic 1mg cot'fíi-ienni liti ¡cil lt-nimm limit i t tu -I ir

uscii-latimimu annimlitudes hi'twi'i'0 U. It) tinI 3 filate st ¡diii

,, \ii

obje-tive of Man tin's ituvestigatmomi ssti ti gimimi tini irrsiaht jotti the

blow oii't-itarijsmli that am-mounts mr titis reniai kahle variatiomi

of time value of V. lii ost'ilki thom ainnpiii irk'. Finali an

attempt sVtIS nradt' tri esnmiminiii' thai LiFt ti titi' 111gm' keelS p11cc'

tivcmmi.'s as a roll danmper uttrilirmiambie Lo wase generation.

Time prescrit work i-, esscimmiti liv ni i-omniinnu,ntiun of that of

M. Martin. Since lue liad stimdii'l iiilv the ttvolimmneusional case, it was of interest to inVest gatt' tito t'Ile(-t of

tinrec-dmnmen-sional hlosv conditions no tile drtng tif osciiinmtitug Platt's. This was accomphshcd liv testing plates of vai ioos aspect ratios

in a fiume Wider thnmti the ii'mmgtirs mf 11m'' plates. Aim analysis of tiri' rentons for time lniiu drnmg il arr osciilaiiiug pitmte IR

corn-parisoim is ithi that of omme ¡ni steady imnotioni is also offered.

N o ni end a tu re A an-ca of piatte

b h If width tif plate - w n

C distance tm'avcti-it by a pitte iii steady motion during a

cycle of tic mÇit-man Viti ip tinti t

C1 drag cocfflcient for steady motion otean dr-att coefllcicint

ti t'pth of center of plath' below water stnrfaee

e t1istncc between vortices iii a Kai-man vortex street

*) Î'Jmmmbcrs in brackets tndtciitc rcJcrences at end of paper.

OC U i-i EN T A Ti E J

-i\Iuinanmmcd R i dj a nov i e

Eolia Iiijitute of ilidraulic Reseur(h, State Uuiti'rsitv of iova, Iowa (iLs,

bIla

13

-Fnoude osnn-rnher

acceleration of gravity

length of plate

equivalent si mpic-pcnsctutimmn tength number of escilitations

distance fi-orn axis of oscittation to center of plate

Reynolds number

moment nicti nia Oil t he lystemn

nit u n nett t act i ng oit tite pl tite

rnn'>ment acting on rcrnnnmmudcr of oscittating system period of oscittation

ttme

Ctistiaceinc-nt ampli I nie uf osciltation, rE-1, viilttcit' of Kttrntittnn vortex trait

s'e ken ty of pl tit t'

weight of tmsciitatittj pamt of apparatus nass density of il rr ¡it

tintgnnttni- amplitude of oscillation

kirnernnmtic VIscosity

tic-titt uF nnitguimmr .tutplitnicte poi' cycte

ci('ctiy of umnguiar n itt)! itude per cycle ctnne

nit-) d *5)

to the presence of plate

dN -( dN

nondirnensionat atripiitucttn

5*) sutoncript F denotes values nmnitlmout plates

u tal y ¡

Let us comisider the rlirupmmtg

tiri' to a flat plate of an

itspr'ct rtmtio rigitllv att ai-lieti to a h gIn-inertia pendulum having

a period T (see Fig. 1) unii ¡merliiriimiog free oscillations. Thi'

period of tite pentluluni is assummmm'il to he constamit for small angles of t,scil lut ion.

ltg. I Sudi-ti ql l'endulmi mi Arrangement

ii

Ss'initfsti,'ehnik Dii. 9 - 1962 - Heft 45

F g L N 1' R M1, M1, It T u t'y W t-) ti t-) dN it i-) A dN b

TÏÏT

¿ht ¿oct dt,)'g 4

Orte cycle

I

¡1g. 2 _{Sketch Showiiig Loss 01 Energy per Cycle}

-As k (CIt 11(1111 Fig. 2, the energy diange (luring OOC cycle is gisu by

= -- W L (cus Ci CUS AN

where W S l'le weight of tile usci l!tlitlg lun of apilalatUs ititil L the distance from titi: cellIer tif o't'ilatioti to tile center of gravit\ of the pendulutti. here _{toil C riItrt'serlI tite a11u11} lar poitions of the plate und jtcnduluui at tite lwgilIItitig and end of une complete oscillation.

We have

A(-)

==

A

and for sittali oscillation Eq. 1 may lie ritten approxiiuLttely In tile form

dE It')

-dN 11\

where d(-),-dN ¡s the eltattue in tite lillettlar untptlitttde of i-cil-latioti in a eurnplete cycle. Ft1iiittrip tite loss iii ettt'rg\ tu tite work done b tite dumping itoitictit M1,, te obtain

cl()

₌

M11d(-) (2)

dN

Cit

in whitit tite integration is taken user a ctltllt)Iett: ivel u.S line this eneruv lts is parti\ tine to tite ditntjtitig nttunettt acting on tite plate, aoci partl dite to tuai front tite rct of the oscillating s> stern, M., Eq. (2) becomes

el ('-)

WLC df)

Md(-) +

- M dC

. (3)

dN

If r is litt- listance fritru tite axis of OiiILttiOlt lit tite t-ettttr of the plate. attd thie oloittent Ott the platt' k alnas in tite opposite setise to that of tite attgular velocity C, it cati he ex-pressed in titti form

= F1 r Ct1 U Ar C _(i-)

2

where C> is it drag coefficient. A is the area of litt _Pial. is the nia density tif thi

utili attd C is

litt ittstitntittteouS angular Velocity of tite plate. hence Eq. (3) Itcuontes

et 1J4 tl p A rit _W1k)11 -

+

Îl1-df) ei) (IN (5) tiC (I)

_{- dN}

- ,' - ó.ta*-

-witere C11 is ti mutai valtit: of Ct sit-it titat

Ct _(tt

j

(, f)' dC =

_t'

f2 iif

t-l11 t-t11

Let US ¿ti-sttttìC tite hitrtttotiit' tilotitltt

°-tt

= C1 sui -T \!c titen obtain t'lt _T

(t dtJ =

t-)!) O T 2-rt

= -

--CO5 -

tit = --

-T j 3Tt

w hit-It, substituted into (5). vicids

lite ucigitt t,f tite plate is ver stttail in comparison w jilt tho lutuitt tI tite etitjrt tiseiii;tlina s\ lttti. Sn that we can (-otisitler hit tite tie of a lt'1 Itiatt- lis ott affect the magnitittie if hi htt tetti)

_{ti lit. (i. t'}

iii-li matt liteti Ite approxittiated liv (tIC, tiA)1-. w litri- litt- sititi-ript I' denotes values without tite plate. Sithistitutittg titis 'ilut its Ep. (7) and putting

ti(-) tif) _{( tl(-)} tIN tIN dN)1. ve obtain

=

I (i A o et f) _I li

3r' WL

\\ LC

3\L

I-1 A (6) (8)

I

Schi(Tstechnik 13cl. 9 - 1962 11It 45 ₁₄

-1-or these plates 3 \\ L 16 t ri1 -rd is a constant, equal to 755. litt- troitit:m niav aiso ht ¿tnal,i-I dimensionally. 'Tite

itlittt drtie utteflicittit tait iii: c'qtrt'- --d as a function of the qtmtttiitiec 1t. y, g, h. I, ti

..

ntttl T. wltere y is time kinematic ktttsit - litt _{act-r-hI'rtttion} of grai ity, L the length _{of tise}

pin te, J timi tfcjtth tif the center of tite plate below lite water

51m nl itCC. attti x0 r (r) Thìe t theorem then yields the fune-t lofune-tfune-tai relafune-tion

L11-f

(ri'

h

h' - lt

L tI x1

wlttre R is tut Reynolds number xb/vT and F the Froude

nitnhlter x/i' gd

lit lit k i - \pression C1> is shttttvti to depend upon tise Reynolds number. luis parameter WitS tini treated in the present work. Keulegt i [Sj has slioitti tha t there is no correltttion between tIme eoelftcients am-md tite Ri-i tiohls ntttttlter. This result appears to lie reasonable for a sltarp-edgcd hut plate.

;M)ilrilttlS 811(1 Irm'e(lurc

Titi' apitaratus ueti itt tise investigation, showti in Fig. 1, «-tttisisted of ii imigh-ittertia pitiditiutni, a water tank 2.5 feet

'i i J e,ti record t r, u t rai n gtu ni, a nil a pois t er. TI t pend nl u ru

was mita it 01) of a crositar tt ppttnl t'ti on steel kmtife edges.

'l'ont vet in-ui mtu'miibem- atìtl a hloricotitti otte were attached to

t ii t- _{rossi itt r to lot-ms u recta mi ni e. il itse itieti-imers had etti al}

cctitOts ntade up of 2 i-ittt-h imeI 1mmigits. 'i'iie oscillation was

Jtrmtiittceti Liv tistmnnttml dispitmictitemit of two 31 X I O-ittcli steel

routìds Susltemtti(il fromim tim> (tUiS of tite crossbar.

fM1d(-).

(7)

i lra. lair. 1april al lout

'ilges. Iras

5115-tile (ciller o! tite lier -r le o iii i n'il i i

gli. lo

the itotlwn of titis strut, parallel to

tIti- itolar liti

initiait

to the xis of tite tank. were faicrti-i titi' \ut jails li''-t platc,

i'ach i-itiiii thick. 'fin J)iLL1'.

willi ala'it ratios ranaitig

frotti i Iv 1)_l.. were te,ti-il. l'ivi' tif dviii 1titi tIti' slittil' tinca,

16.0 squale ittihi's. anil titi. oilier live liad antis from 7.0 to 30.31 square incites. '1 li' acune-Irle rittirartit stits of titi' list

plates are given

ill laidi'

I and tite dìaractctisties (if the

oscillating s\stein itt labte 2.

The atiutuiar tllsp!at'etnt'itt I tolti lic vertical wris ltlt'Lliiil'd by reading lit' nla\t!luutil (Iis!iiLii'uuti'iii if

a pointer agai it

a scale

aliltrati'd in detani'i. 'lite pulii tul oscillation was

olitainr'il li means of a aItlionu1 rioT ii-r unii a strain gage. The l)oittlrr used in niu-itsiuring

lic aiiglrs ut siing of tite

pendutunt w as 4.6 feet long. It a as ntade fiotti a liar of small

diameter attached at the luit tolti of tue lItighI A 37 circular

arc of I .6-foot radius, marked oli in divisions of 0.23 on

a piece of plywood, served u.s a scale.

in all tests tite angular ilispiacetiieuit aus lirouglit to atiiitit

20° by littntl through a iuotnliu-r of sss ings li-fore tite pendu-lurn wa released. Univ the ltiaxiiutiultt angles at the 'ud of each swing Ivere read Tests for evcr late w ere repi-ati'd three or four titiies in ordr't' to avoid errons nl tiucastireitti'ttt. These ditta were then piottu'd agluitist tile utuitiliu'r nf Oscilla-tions ti) iluttuin tile declittiog-tingle curses for cult plate. Tite value of df') dN was dit'11 girt-ti by the siope ul lic envelope

CUrVe of u dualin in g-angle e arre. 'l'alite 1

Characteristics uf l'est Plates

Plate f 2h A

inc-tics mi-Itt's itides 21,

1 12,127 11.923 7.51 lIt.4 2 . 12125 1 2 :13 12125 3 12,123 t,27 lair illS 4 12123 175 2122 11.1)3 5 12125 2,511 lieS t 6 8000 21mO 16.111 4.171 7 tuRI 2.11 8 5,611 2.113 tui pi 2.1111 ti 4.911 :127 is.io 1.711 10 4.00 4,090 19.00 liJO l'utile 2

Charactcristtcs of Oaci11atíu System

Weight nf oppatatus lOii.lilJpounds

Pc-riad i t'system in air w ittint plate _190seconds

Oistniuc' fr-nm centcu' uf osr'itl;it

ii In

enti- '- of gravit'tif usci t I,'tiii system 1797 inctics

DisItiitcn rrorn center ofoscillation to

cc'tuterttrie of plate 2t1,59 inches

1('ii I I(251111

During the expeniullerut tut' infini-ni'' of tite oscillating

system oit Ihe daiuipiiig of an oscillating platt' -was taken

separately.

l)ata for

lie damtitig of hic oscillating systitu without a test plate, dite to tile virtual strut in natur.

luir-(lumpitig of the apparatus, annI knifc-eulg' friction. arr lililtit

in Fig. :l. fri Figs. 4a aliti 4h lint' plutted (.' against N from tue

nicasurt-nuents with tite various test plates. Cotiuparison of

20 'n ii /5 /0 lb 5 do dN 2 0

'4

8-1g. 4a Derlining-Angte Curves for System with I'iates

5

o

2 4

6 8 /0 /2 /4 /6 /8

9, in deqre0s

ltg. 1h uugitlar Damping pur ¡full t mur 5stem willi

l'la tes

me-s

Jig, 5a Angular Damping per Roll for System with

Plates

i)

2 4

_{6 8}

_{/0 /2 /4 /6 ¡8} 19, ¿ut du'grees

- 15 -

Setutlfstedunik titi. 9 - 1902 - Heft 45

\s

ti raie 'n rife

A ri/e

G / 'fc V Pmo/e No. /

5ne. No.2_No.3

No. 4 No.5 o P/ale No. 6 .:fs1

&.

"e

'-'-,

X P/ole No, 7 A P/ale No, 8 - ' P/ale No. 9 G P/ale No, /0

____________

5 /0 /5 20 25 30 N

Fig. 4h iiecttuuiog-Angte Curves for System with Plates ctlnsu's s liii liIo'.0 ni F'ig.,'i shows that the ulaniping

b ilititmit hmm' tust itlaht's is 'tosiuhi'rahlv smaller titan dial with

t tu' te--t ItlLt[i's. luit titi- fmnìi'r most hue takt'n julo consideration

iteeause it inlluetiecs titi- int,il result for C11.

o /00 /50 200 250, 300 ₃₅₀

N

Fig, 3 Dec1iningAngle Curve Our System without Plate

5 /0 /5 20 25 30 35 40 N 20 /5 /0 7 -i 3 2 20 /5

/0

ut, 'Q 5 Q) 3

9 8 C,,6 5 4 3 2 o

From 11cc.3, (Iuta for (d9 1l"i)1. ieijiiin-d hi uliputiria C1

for the rauc 9

1 tu () ¡ ari ultuilu-il. From _{lig. -I_m}

arid 11m In- lnpc of time Laiiccrmit oas immeaimrd nr (\irv ch-cn-i nie i-e-mili. ulmtaiimmml fur c-cm li 1mhiic are ¡iIi hil iii I _{na ¿mInI}

5h, fron) 1111011 ii is iIli lo rmad mlIi-m-tl die ahmes of sm-i us 9. Frommi

li-i

m-iirmç-s¿hill m-lmiciiinmi ( ans COmputed -t interval-. mml I mliiem. 'I h iIU--. if C1, m(-nus X,,/I) ari ncn in Taule timil 1I,IIml iii

II. (I

..\mm mmriIlIi L))

this plut I1h data olmi;miiil liii

III pi

II liii silhhIimIIm

\lhjd) arc iiiiiIar iii ca-ii-ial hnJ,. lii im-.tmic \ Ii 1cm nilniiv gives al ic-s for C1 ol lii- ml Il i- i ) t ¡ a 1m-tri em-iI 2 ai I 1.

AS X,,1) decreases to i-ru. iIi mciiiic of Ci iiii-rea and difference lmclwccii timm- alcm,--c I C Jur hie simile mimic

of x,/h fui iiifferent ¡mia[- inm-rm-ams. ¡mir \, Ii i. tlmc drag

coefficient br plate iN0. ¡ I) is C, .2 tui1 for ¡iicttc \ic. I

is C11 - ¡ ¡.5. Dccremisiima ,, Ii [nidi i ralciil\ iin-riai-s the

value of C,. In geiiercil. Iuma,- ir. tIce miiut sii il-_lug feattimm uf these tic rvms is the rciitarkahlc o ay in o him-li C11 arics

with ,t,,/b.

'Thiee mi-pills ¿Ire Iii (-ssemutLml ;iccim-miuiiuiL ri liii hic rmriuI;s of Kculccciii and Carimt-iiim-r 5I fur cm ii\mil plaie iii Liti 01il

lating (lull. The gm-mi-cal -liape nl lii mitime-. unii tIlt valuc of C1 _{are in gond clgrm-u-inm-i11 o iii \lcuiiiucs 1m-_tilts IhJ for time}

case of ¿in oscillaiicicc plate o lili icmlmiiitc Nimect ratlu. i. e..

under lv ii-dimensional Ilmuw _{moiimlitiiilis .il)1 ç,} tIn- sumullcst value for ( is betm cmii 2 ¿mcccl 3. i list resu Its arc greLitcr diari

the value ol 1.6 nieasmircd by I-tootle.

Fig. 7 i a plot of Ct1 uguicNl I;2 li fur litiirciil \ahlies of

x,/h. In

i im- same figure ilme curve of C1, against aspect ratio

90/ 0.02

Schlffstcchmrmik 13d 9 - toc;2 - lIeft 43

lid liii- 11cm c..m,-m, (mf simiils llm,,m . Thus o as done to

corn-¡icmmi- valu-s uf C1, ;muml nr lic; al1ic geonictrieal shape of pl ii - ni il-t- imamlv no i ci im-Im -cid r il ow tond it ions. fly

emmici-¡m.mriiicc Iii- d-lic mc- of C11 ¡cc -teamls how with one mml the curves

('ir mmmi-h-amlv Ilari - say \, Im ¡ . 017 can see that lIle dIng

uI ti,imimis iii ullsim-amiv timo ¿ir,- ¿iliocit 3 to 5 times _{as great as}

li,m-m- (mir shad V llmmw. I u c'oii-tiI the shapes of iIi _{curves in}

tilt; issu casis aie t-r similar.

1.49 21h13 4 47 S56 741 8.93 11h41 11-lic 12.29 i 407 imL45 17.115 111.34 20. 02 0227 22.10 21.29 21313m) ('laie No t (1.07 4_50 3.83 3-49 3-mo 2.90 2.03 2h18 2.56 2.50 2.44 2.57 2.32 2.213 2.21 2.10 2.29 2.06 005

9/

020.3 0.507/

L 3

7 /0

20 30 5070 /00 1/23

Fig. 7 %crsus 1,21) for Various %aiues of

- 16 -a pd/io

-l'lì/e

--.---_-I iVi Apmct 1/23 /9.40

/1

I.O

-

--. r/tn)

//

-_/ , /,

/0

--licite No. 2 ('tIll e No. 2 Plate No.4 Plate No.5 0.93 mm-20 mm.74 9.9(1 0,93 9,110 0_37 10.90

I .ilii 3-lis i 49 (i 30 1.06 6-ils 0.74 7_85

2.79

lis

2.23 4_70 1.94 5.35 1.12 5 _7(; 3.72 2.1)8 4.05 2.13 4.45 1.411 50m; 4.03 3:15 3.62 2.611 4.83 1.80 4_23 5_50 t . mils 4.46 _{3.31 3.19 3.87 2.23 4.10} f51 2.91 5.21 2.116 3_72 3.41 2.60 3.90 7.44 2_mil 5.93 2.92 4.25 3.22 2.98 3_95 8.57 ml rim 2.81 4.78 3.17 3_35 3.40 9.30 2.04 7-44 2.72 5_31 3.07 3.72 22h 10.24 2.38 11.23 2.67 9_88 2.99 4.11 3.16 11.111 2.50 11.03 2 mIS 6.30 2.119 4.46 3.04 12.09 2.43 9.117 2-Sci 11.90 2.81 4.04 2.91 1202 2.27 10.41 2.511 7.42 273 5.20 2.02 13.92 2.20 11.14 2.49 7.96 2,65 5.56 2.73 14.08 2.22 11.110 2.42 8.49 2.57 5.90 2.63 12 .01 2-ici 12.04 2.39 9.02 2.51 17.32 2.29 16.74 2.12 12.40 236 9.56 2_43 6.70 2.54 05 Ic)

/5 20 23 30 35 40

4.5

'0

cm5 rO 65 /.0

/5

80 85

90 9.5 X,

ltg. 6

t

v.crmii,, x,/b Various \pcCt Ratios

ii

I-' -u _X,' X X X t_2 i) C1m i) b b b ¿0 IO 8

-5

c_p 4

.3

'2'

I)aiìipiiig 1pI('(itaI,i),Iut iii

iiteidv 1Iov

It 1)1)5 lO'Cll 5l(t',Vll 11V

1\iIlrtin iI lilt

(Ill t'.u(ITJlting 118)1

plate causes a pair (If \'(O'tjCCs III 11h01 alOi (rull' at tile edges

and IUOVC laterally UOdV during '01h 11)111 ('(('1) ((1 tile tI('il-latoso. At the VerY hcai,lIliIIg ((1 tIIC 1)1(1101)1 t!))' hI0.Ili)11'I uf tile

tWO vOrtit'cs is sVfllfllCtl'it'ai. if hIC 0)311101) II) ((III' (1il'(('ti(Il( fer. sists bug enough, tile flow ll)'l.OflWs Il(lstllihil 1(30! II 5l(r'('' 'lu))

of VOit ('CO ISshed :Iiterllate!\ frnuìi tile tI(I) 'ligI's (If tile plate Ifl tile iI,aflner of the Karola)) (unIx 1(011. 'I le (xlcllt Il) 0111(11

vortices develop depends uf(o)) the p)'riOd uf tile (l5)'!l!atil(li. It appears that till.: ugh aiues (Ii lu' drag cIo'Illl'iI'rIt ('UI)

be ihttI'iIllllId to the eflerav ¿'xpcudcd il) 'on ex fui'tuatiuu. ill

steady 3111(3 ion a pair ((f (urli ('(:5 (If till' Kal'Illan tx'ail is shed as the plate moves u distance e j\ III

e

i + UIU

where e Is the space l)'Iweel) u pair of ortIes in a row of tue Karman vortex trail, u is till' \el(Il'Its uf a \'urt'x in the ti'aii, and L ill' velocity o. tilt' ¡'(lute rI'lati'e to huid at rest at iii-unity 171. iflce, a('('un(iing (I tile il)eIO,Ili'CIl)('litS uf KallIlall

and lIniIa'h [8], we hase «2h 5.5 und uiU 0.2, Eq. (9) yields, in dimensiolisless fOrIli,

1.25 e 7.0. (10)

2h 2b

For the same flSCLllI \'('iu('it\ ill 051'Ill1lIOI\' 11101 0)1. u (air of

vortices is shed as the ¡(18)11' IlluVeS a IhsIILIlcC 2x11. henne

an oscillating plate sheds %u1 tItIS at tilt: saille rate as 1111e ill steads' motion when 11 h ''' 7(3. Rut. ¿u'connhug to I"ig. 6.

cor-responding to tills (lIllIC of \)( h 0 110 ve Cn - 3.Ó for a liai

plate of ilímite 10-flIt

10)111. )\h('r('10-, for stl'ads ill(Itil)l1.

C1 2.0. 'This mli ('all's that fu' )'llr'l'g'( ol a ylInlex u4',,('rIll)'(i by an oscillating piale is 11110(11 1.8 111111'S greater than flat (If

one in a vortex trail. h' lli'therlIlorl'. wino ,x5h is less 1118111 7f)

for tile same mean velocity, the frequency of vortex generation

l'or tilO larger sali ' i till' lIli I Illensional frequency

para-t Ill' lirag )'((('ípara-til'i('llpara-t para-t) h,b'cll'llses and apara-t para-tile SOOSC para-time the (ilhI('l'('ll)'t( il('lWcl'lI lithO I'n'Ifu'i'llts for plates Will)

)hf-lire111 asi)t'('t i'LIti(H l(l'('(OIU's s(llaill'r, In general there is U n'ud boards lilt: ('llllst,((It (aille of 2ff for infinite

nondinien-50)1(111 Ulllillitu(ll'. 1)111 tui5 is approached very' slowly

8111(1 lILLO 1)01 attained ill tile tests.

(Eingegangen uno 16. Oktober 1961)

,\FkIO)\\ h'h811n'lIt

'his ll1'o'tigatiOlI ((Ils tlt'ttOI 1111 i at the Iowa Instittite of

i h (Il',Inhh' ill'H'al'('ll LIlI)! (00- 5tlì1P,11'lb'li by tile Office of f\avai Ul'»',lll'll, nOdI')' ('I)(Iti'Ijl't '(Il)))' ((ii hi). The author wjsji1's to l'\Ill'l '-" i,is ¿lillll'l'I'iaIlllll III lh'u[l''.'_ll' Loi_us Lanliweber fui'

iIi". 011ilflIlI('l' 1)111 _{eill.'ulII'Ilgelnl'IIt} 81)111 for his ('i'itieal review

(If till' )llallusl'i'Iflt_

Ii u Il I i eIa ohy

[1J t-' 1' 0 U d e , W., "Oi'i Resistance in Rolting of Ships," Naval Science 1874.

[2j M e N o w n . J. S., "Drag in Unsteady Flow," Proc, Ninth

trltn)'flatjoflal Congreso of Applied Mechanics, Brussels,

11)77.

[3] M c N o w n and K e u I e g ilri , "Vortex Formation and

Resislance in Periodic Motion," Proceedings, ASCE,

.JLli)Lln1'V 1959.

i'] A Ii t o n , I',. "FIlmai (In,

o.f a Vortex at ttie Edge of a

l'late." RACA. Tech, i\tcm, 15933. (,Ausbildung cines Wirbets

13)1 dir Kante einer l'tatto") Göttingen, Dissertation,

Inge-nicor-Archiv, VIII. X, pp. 411-427, 1939.

f5J K e u t e g il n . G, 12., and C il r p e n t e r , L, l'i.. "Forces

on. Cylinders lInrI UtOt' _{ill on Oscillating Ftutd," Journ.} of 11es, of Nati. Nur, Stand., 11es, Paper 2857, 1958,

[63 M n r t i Ii. M.. "flott J)aiiiping DIle to Bilge Keels," Ph. D. dissci'tation, State University of Iowa, June 1959.

[7] Prnnnttt. L ,, lind T jetions, O., Applied Hydro- and

Aerodynamics, McGraw-Hill Book Co., New York, 1936,

p. 136.

[81 K a r man, T. y.. annt Rub nc h . H., "Über den

Mechzinis-issus dos Flüssigkc'its- und Lui (widerstandes," Physikatische

Zeitschrift, January 1912.

- 17 -

Sn'hiIttectnsik 133cl. _{9 - 1962 - Heft 4} (1.47 (1.113 10.40 31 213 0.4)3 0.111 9.10 6.10 ((,:33 0.011 9.70 5.8:3 (1.23) 0.6(3 9.44 5.97 0.2:1 0.47 9.3(2 6.54

( oit'hisiotì

3 1 1(1 4.85 1.23 4.79 8,59 4.110 31.3(1 5.04 0.70 4.73

4 il'il 4,25 1.32 4:10 1:12 4.35 1.14 4.319 0.93 4.17 I il(' iligll Vai)I)'s 1)1 io' Ti)('all drag coeftiei('nt Ct, for sillufi

5 3.83 2.3(2 4.06 1.115 :1.76 1.4:3 3.75 1.16 3.319 _{lllllllli,Il('llSil(l))li 1Ii)If)lit(I)i('' \(l'i( 11h' lilie to tuo' periodic for'}

2.79 3.513 2.42 3.61) LS? 1.7) 3.44 1.41) 3.57

)ll8)til(l( 11f starti)lg ('0rtiI'('- a)l(l tut: partially formed wake. 7 8 3.72 3.35 3.1)3 2.92 3.')) 2.21) 2.))) 3.14 2.931 .1 .99 3.17 3.00 1.13 1.8)1 3.08

2.83 I(I)' 11h" 511(111' ('lIllO' ((I x 'II. lIn' li ('81g coefficients increase will)

9 4.18 3.03 3. I)) 2.3(1, 2.31) 2.5(1 2.83 2.09 2.70 ('('t I alio fur nsjo'cl (tills greatl'r tiuos one. Decreasing till:

10 11 4, 5.34 2.218 2.87 4.03 4,45 2.02 2.73 3.2)) 3)13 2.7:1 2.61 2.315 :1.15 2.70 2.55 2.22 2.312 2.76 2.49 ostn'et

'LItio 1:elow t (t iller'as's tile drag coefficient. In fact,

it is clan' froiii gclnnl'tni'.I cu10.idl'rations (neglecting effect

12 13 SII (3.04 2,74 2.71 4,85 5.25 2.315 2.00 3.95 4.29 2.4 1) :3.4:) 2.4(1 2 50 2.43 2.733 :1,02 2.41

2.34 of i(o,I(l)ianil's) 10)31 tile 111w )'I}nditinfls arc identical, and

14 (1 51 2.63 5.335 2.53 4.00 2.43 4.0(1 2,27 3.26 2.43 111111' 11)11 (110g C'lIefIit'i('lltS OtO tile same, whetller a plate is

15 31.3)6 2.57 (3.05 2.50 4.9:) 2.:))) 4.27 :3.2 3 3.49 2.31) _{((t'iI'llt('d ci'ti1'aiiv 01' 11111 i,,ulltaii( , Therefore, ¡Ile C0V(' of}

16 7.43 2.51 6.47 2.47 5.27 2.23 4.57 2.23 3.72 2.33

(T se HIS llsi)('Ct rutio ((Il ilIg-log should he symmetrical

17 7.1(0 2.39 0.336 2,42 5.11(3 2.2)) 4.84 2.23 5.95 2.30

18 8.36 2.41 7.27 2.37 5.92 2.23 5.12 2.20 4.18 2.26 ulout an aspect ratio oh one.

X X le JIl)'i'O))self ill ill'. I'I''I' IO till ¡11111 it appears reasonable

CII CI) () CII CII _{1111)1 tIn' ii'ag ('Oli iI)'iellt ('(((lili IIi-Il illI'l'('Iise. hhiis III' laVI' a}

b t) b b b

If((;,iit;)ti\l' i(l((i('i''tlllllfillO ((f till' llI('('I(I))lislll (If till) rl'l((:ii'i1(l(le

l'I ;t e I' (I.e i Io t: I'll te suniatilIlI of tile drag of 0)1 o.st'IlIatiIIg plate.