Analysis and modification of theory for impact of sea-planes on water

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REPORT No. 810

ANALYSIS AND MODIFICATION OF THEORY FOR IMPACT OF SEAPLANES ON WATER

T\' \VILBUR L. MAYO

SUMMARY

An ana1,,.is of arailable theory on seaplane impact and a propwed in'dification thereto are presented.

In jreïìs

methods the rer-al! rn.oînent?Lm of the float and virtua1 mass

kas been as iincd to remain constant durinq the impact hnt

the present i 'alysis shows t1at this assumption is riguroosly correct off!, "hen the resnitant Velocity ùf the float iS nori,aI to the keti. The proposed rnothflcatwn chidly iflPOlI'eS eon-sideration ol the fact that forward velocity of the seaplane float causes inoìn 'finn to be pacsrti into tite hy(1ro(i/Jnunic downv'ash (an action t

at i

the entiic cons(Ierat1on in the ease (f the planingfloai ) and considera/ion of the fact that, foi an unjiact

with trim, t/ rate of penetration is determind not only by the velocity con o)nent normal fo the keel but alNo by the l'elocity component arallel to Ike keel, n'h ich ten(/Ñ lo iediice the pen et rat ion.

The ana1is of previous treatnients includes (t (1i,(U.'k,OTt °.f each of t'e important c(),ür?but()ns to f/U? SOf?il;on of the

impact problem. The development of the concept of flow in transverse planes, the momentum equations, the aspe et-ratio corrections, the effect of the generated ware on the rirtual mass,

the distribution of surface pressure, and the conditions for

maximum. impact force are discussed in detail. Impact treatments based on flow in longitudinal planes, as for bodies of very high aspect ratio, have been omitted since they seemed to

be of no 'interest for the problem of the typical float.

The momentum passed to the downwash is evaluated as the product of the momentum of the flow in the transverse plane at

the step by the rate at which such planes slide off the step.

Simple eqnations are given that permit the use of planing data to evaluate empirically the momentum of the flow in the trans-verse plane at the step. On the basis of such study, rnodfiea-tian of the gneral equations of the previous theory is supple-men ted by nwilflcation of the formula for the mosupple-mentum of the flow in the plane element. This improvement can be made because the flow in tite plane is independent of the flight-path angle.

Experimental data for planing, oblique impact, and vertical drop are usd to show that the accuracy of the proposed theory is good. Wagner's theory, which has been /he most popular theory 1LJ. to the present, is compared with the neu theory and with recent (la fa for oblique impacts. The data show that the loads calculated by Wagner's equation are excessive, particularly for high trims. Use im this equation of the proposed formula

7773 10--48

for the momentum of the flow in- the planes reduces the calcu-lated force but the values are still excessive.

INTRODUCTION

A number of theoretical papvrs on the impact of scaplaties

Oil vati'r W(F(' published I)etwe1I I )29 nil I 938 Lit the

f)FOpOSaIS pr(S('flte(I vere not uviierallv accepted. B(('ause a it n(1('q1at(' theory oit -Ite}i to J)nse revision of dvsign

1'Pq1lirPflì('fls vas iIeP(It'(i, 1mfl analysis of previously 1)l11)lis11(d

work was iituk'itakeii at tite Langley i\I eniorial Avronaut icnl

I_-! i o )it t ()1.

Th(' it rìalvsrs vas coiici'iiiv1 chiefly vit-1i the tren t meut s tilitt took ])1Oprr (ogluz:l11cc of tue- low aspect ratio of the

S(1J)1O1I( flout. 'FIt('S t-r(!fltflj('IltS W'P1C (OIflIfloflIV E);lsC(1 Oit t lie íLSSlIfllptiOfl tluit the ovt'r-aII moment-um of the

sea-J)lflhu(' n nl ilV(l1O(1VI1flfli( viit liai I1ULSS 1efllfliflS coiit iìt ( II1i11 I he iJn1)a(t. r1I1(. C1ri(f (Iift('re1(e I)et\V('('fl tin' t i'ti

t-flents was in

I he (let cmi ination of the magnitude of the

virtual mass.

In the present paper a critical survey of the 1)I'('vioUS treatments and a proposed theory are presented. Twon-siSteflci('S Iii t lie pr'vious I ileory, which apjear to inva I dat p

it when a component of motion parallel to the keel Exists,

are shown.

The proposed theory, based essentially on

accepted physical concepts, provides a logically consist-eat 101(1 ilfliíi((i

treatment, which is applicable through the

entire range of oblique impact-, including the end point of

the planing float. Planing data and dat-a for oblique anti vertical impacts m.a(le in the Langley impact basin are

analyzed to show tue general applicability of the proposed met 110(15.

A similar analysis and modification was first prepared in

1941 hut was given only limited

circulation. Further

development followed as more modern test results, particu-larly those fiorn th Langley impact basin, becamo available.

Tite present report has been prepared in order to give the

theory in a form consi(lerably shorter and better confirmed by experiment than the earlier version.

SYMBOLS

ß angle of (leali rise

angle between flight path and plane of water surface r angle of hull keel with respect to plane of water

REPORT NO. 81 0NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

I resal ant' velocity VV verti al velocity

1' vcrtial velocity at instant of contact

i

,, v(lo(it.y normal to keel

velocity normal to keel at instant of contact

i horizon tal velocity

in mass of sea pinne

ni niass of seaplane corrected for eccentric impact

nì, virtual mass associa.te(l with hydrodynamie flow

i)eneath float

x half-width of flat-plate equivalent of bottom i leie't,h of wetted area

y depth of immersion, normal to plane of water surface

t time

u eli ective l)OttOfli. slope (dy/dx)

F

resiil ant hydrodynam.ic. force

flj max mum hvdrodyuimic force normal to plane of

Wmaz

w 1er surface (livided by weight of seaplane

TF tota weight of seaplane

g aece ration (lue to gravity

p rn.a density of fluid

Wliere, units are not specified any consistent system of units may be used.

IEVIEW OF PREVIOUS LITERATURE

VON KÁRMÁN

Basic theo "y.The ear1iet impact theory, and apparently

(hat u poil w lì ich all subseqiì ci it work has been based, was a(lvanced by von Kîrmiuiì (reference 1).

Von KIri un considers a wedge-shape body d rapped

vertically ini o a horizontal water surface; the tota.!

ittoitien-turn of the is-edge and tite virtual mass of (lie flow are

as-SUflle(l to renain constant. With tite assumption of two-dimensional flow in planes normal to the keel line, von Kármiín considers that at each instant the virtual mass is equal to the mass of water contained in a semicylimler of

length equal to the length of the wedge and of tliaineter equal to the width of the wedge at the plane of the vater surface. The following sketch is a cross-sectional view of tite wedge and the wo ter mass:

Tite momentum equation of reference i is written

m V,0 = m V, + p ir.r21 V,

Tuis equation leads to the following equation for force:

F=

V,02Prf I cot fi

(i4w)

In the case of the flat bottom the forceequation (2) yields an infinite foi-ce. Voti Kiírintíri considers the effect of cotti-pressiltili ty of t he water in red tiring these 1011(15 for tite flat bottom. In reference 1, for a vertical velocity of about 6 feet per second, a pressure of tite order of 425 pounds per square inch is calcula ted and tite elasticity of the structure is con eluded to be an important consideration in sticht a case.

Comparison with tests.Von Ktírmtn divides tite force

equation (2) by tite ai-ca of the float in the plane of tite water

surface to ohtaimt an average pressure and compares titis

average pressai-e a.t the instant of contact (that is, for very small values of x) witit the experimental pressures reported in reference 2. Von Ktírmnumn observes that titis equation agrees approximately with tite pressure data if tite vertical

velocity is assumed to be of tite order of the sinking speed before flare-off.

Later tests sitow that tite flare-off during landing reduces tite, vertical velocity at contact to a fraction of the sinking speed of tite airplane before f lare-ofF. Furtltermore, local

pressures may be considerably greater titan the average

pressure. Data from tests of a modern flying boat, which included measured vertical velocities, showed pressit res much greater titan those computed by von Krmtín's formula

PABST

Introduction of F,,, ìflr,

and aspect-ratio

correction.-PtiIist (ref.ei'i'uve 3) COflSi(iSr5 tite velocity of pencli'at.ion tc be iepreseitted l)V tite velocity iiornìtì,l to the keel ratliet' than by the vertical velocity. Tite tvo velocities nì.n.y 1)e fluiti: (iiffeìeiìt \Vlli'rì flue trilli IS not zelo; for exafl).i)le, ill tue limit-ing case of a l)111C plaitlimit-ing motion, the resultant velocity may have a large coni.poiunt rtorni.aI to tILe keel but the vertical component of velocity is equal to zero. Tile full implicaI ions of titis change will be developed in subsequent discussion.

Pabst also points out in reference 3 that, if tite paint ai

impact is not directly below the center of gravity,

flic

eccentricity of the impact is taken into consideration by

multiplying the mass of the seaplane m by the factor in which i is the radius of gyrrttion of the airplane about tite

center of gravity and r is tite distance from tIte center of pressure to tite center of gravity. Titis reduction of tite

mass intro(iuces two problems. First, instead of the decrease in velocity according to impact theory previously developed, V,, may actually increase during the impact duc to change in

trim and tite peak force may occur at an increased rather

than a i-ed uced V. Second, tite reduced mass of tite seaplane

will vary iluring th impart as a result of tite variation of tite eccentricity of tite impact due to shift of tite center of I)reSSUre.

By applying titese modifications and introducing ait

empirically determined aspect-ratio factor in tite equation

originally given by voit Kirmmn, Pabst obtained tite following equation for tite impact force:

,

/

32\

irpxl /

x\

itt \\liRIL t1( iitai tiìaS ?flW iS MItUI to

2 when i

is greater ti an 2x and the factor I

-

_{is an empiririil}

asI)vct-rat io corretion (dcterm.iiwd by vibasI)vct-ration of submerged ihites).

Iii a 1at r publicatioti (rcfereitce 4) Pabst presel1ts the

foh1owiiu 1uatìori, which is aj)parelltly _{an empirical} re-vision of iii. first equation (equation (3)):

-O I14-V 2p120 786e _{cot ß}

F=

fl (4) in, of

in which l).7S6e°'4

_{replaces tite term ir(i_}

x2) equation (3 _{The virtual mass in equation (4) is}

irp i2x2 lx 2

/l2T?

(i

0.850 12+4x2) inst('i1(1 of 1rpxl(1 x 2

\

i as in equatin (3).

A formub 'for maximum _{pressirrt' that is tire saine as that}

of von 1rIri iit1ii except for tite repla((rrlent of V by I ,, is

presente(l Ir- Pabst in refenrice 3.

Ela sticity. I'abst's consid eration nf t lie fi at -bott ont fi uit (ail angle of dead lise of 0°) is conccrn((1 Wi t h t lie effect of elasticity of the seaplane iii (i.eternuhlirìg tue liydrodyriarrìic force. Pabsi presents equat ions for various spring-arititiìrtss

combinations but obtains a solution only for tite case of a rigid inassless float c.onneet((l to a rigid seaplane by a Iltass-less spring.

As has been noted, when tite effect of elasticity is neglected, a theoretical curve of impact force against angle of dead ilse approaches infinity as the angle of dead rise approaches zero.

In order to talco into account elasticity at low angles of dead rise without making a complete analysis, Pabst

pro-poses that, in tite low-angle range, tite curve be changed into its tangent passing through tite point calculated for _{OE0 angle} of (letl(l IIS('.

Comparison

_{with tests.A point representing}

_force

recorded on Bottomley's float of _{20X° angle of dea& rise} (reference 5), which apparently was corrected to 100 by one of Pabst's equations in order to correspond to Pabst's

data for 9)° angle of dead rise, was found to lie between the rigid-body curve and tite tangent line drawn to _approximate

the effects of elasticity of tite structure at small _{angles of}

dead rise. Although titis agreement is good, further study

indicates that Pabst's equation can agree with Bottomley's data only within a very limited velocityrange.

Pabst presents flight data in a plot of vertical force on tire float against the longitudinal position of tire hydrodynamic center of pressure (reference 4). _{1f tite loaded area is} as-stimed to })C Proportional to tite distance of tite center of pressure from tite step, the slope of a line from the step

loca-tion on the abscissa of titis plot to ami experimental point

represents art average pt'eSSlIle for tite impact represemited by tire l)OiIlt. _{For a particular seaplane. on tite basis that.}

tite average pressate is tlettrminetl chiefly by tite flight parameters and tliitt the flight parameters causing highest

pressures are apl)roOchì((i for various positions of tIte center of pressure (variation of the center-of-pressure position duc chiefly to seaway), tire maxinlumn points for positions of the center of pressure ranging from the step to the center of the Ilorit tend to lie along the same sloping line.

rçie experimental program condu cted by Pabst cot isisted of successive tests of flat and V-bottom floats installed on the sante seaplane. _{Limit ]ines of the type discussed in tire} J)1CV1OUS J)a1agrapit were drawn for botEt sets of data. Pabst Sl1OWC(I that the ratio of tite slopes of these experimental

limit lines was 0.8 and observed that the theoretical ratio

of tite slopes of these lines was 0.7 (reference 4).

Pabst's correlation of flight data indicates that tire theory gives values of tire proper order; how-ever, the accuracy of the correlation is questionable l)ecaUse of the many ripproxi-natioris involved in (hawing and interpreting tite limit litres,

computimig tite flat-bottom force, _{and computing the}

V-bottom force. Asi(lC frorri these uncertainties, direct proof

of tire theory requires evaluation of tire absolute slope of

these lines rather titan their relative slopes.

WAGNER

Virtual ntass,\Vi t h tire object of approxini ateiy account-ing fur tite Wa VC generated by the float, Wagner (references G arai '7) _{1)1tS'irt5 a solution winch gives a value of tite} virtual mass different from liait of von Ktrnumn and Pabst. As has b)Cefl mrote(l, Von Kírinruii 1111(1 Pabst defined tite ii'tuai mass (for two-(limnelisional flow) as tue mass of tite r()iltne of ¡luid contaitted in a semicylinder of diameter

equal to t ut Wi(l tir of the float in tite plaire of the water

surface. _{TIriA virtual mass corresponds to one-half that} given by convent ional hydrodynamic tireory for submerged motion of ellipses with width equal to the width of tite float

in the plane of the water surface and therefore constitutes

a first approximation for the case of an object passing

through tite water surface.

Wagner considered that, since tite water displaced by an immersing float rises along the sides of tire float, the widtit

of tire wetted surface and the virtual _{mass of tire flow are} greater tiran those based on the float width in the plane of tite undisturbed water surface.

_{In order to evaluate}

tite increased width, Wagner assumed that tire particles at tire top of the upflow or generated wave move vertically

upward and, with differences in elevation (lire to slope of tire generated wave neglected, these particles

_{niose in}

accordance with the following equation for tire velocity

distribution in the plane of a flat plate in immersed motion (reference 6):

T7=V/1-2

(5)

where

V _{velocity of particle passing through plane of plate in} getting around tire plate

V velocity of free stream relative to plate

x distance from _{particle in plane of plate to center of}

plate (.r>r)

Considering % as the float vid th at the top of the wave and x as t lie dist anee from a pa it ide in the wave to t lie plane of symu ictry of tue float, Wagner integrated equation (5) to determine x as a fniirtion of the depth of immersion.

For the case of triangulur cross section, solution for X

yielded a value equal to ir!2 times the 1ìa1f-vidt1ì of the float in the plane of the water surface. Wagner defined tue

virtual mass as the mass of a semicvlinder with diameter

equal to 2r. Thus, for the case of the triangular V-bottom, (lie virtual mass specified by Wagner is (ir/2)2 times the

virtual mass specified by von Kíírinmn and Pabst. The igiiifìcance of this difference is evident from the fact that it

results in a 2.47:1 increase of the force calculated for a

specific (Ini ft 1111(1 VelOci t Y.

The assumptions that the particles in the wave move

verti-cally according to equation (5)

and that the float width

at the top of the wave determines tue virtual mass seem very arbitrari. The most direct evidence that the method is ina(lequnte lies in the fact that a later and more exact

solution for t e case of triangular cross section gives values that are Sum nuit jolly different. Although t lie met lind is ina(lequate it remains important l)ecause a l)ett er solui t i(Jfl for the case f the float bottom with transverse curvature

has not been )rovided.

Pressure d stribution.---Continuing along the saine lines,

Wagner (reliienee G) stated that the niaxinium

exists at the top of

the calculated wave and equals the

dynamic pre ure based on a velocity equa

I to t lie rate

of increase of the flou t }i:ilf-vidtli at this point. Alt liough t his assunipt ion may he lacking i n rigor, improvement is not a t t empt ed in the presen t J)U pci.

1f the momentum of the float and the virtual mass is

assumed constant, the equation for maximum piesSure is

172 1

]ar

(

+)2

where

r'.2X maximum pressure

Ve, initial rate of effective penetration,

V,+(T'V,)r+VwT,,,

(approx. equal to V,)

V wind velocity

V wave velocity

r, angle of wave slope

Wagner presented the following equation for determining

the pressures, other than the maximum, on the bottom at a

given instant: pV02 _{j i 2 u}

L/i_

m, 2

2(-,-1

(7) w hiere

p pressure at a point x<x (curve to be faired into

solu-tion of equasolu-tion (6) for x=x)

x luoii.ontal distance from point for which presure is

cal-cuiluteil to center of Íloat

(6)

The present paper gives ari Uil prove(l theoretical method for deterininiiig I lie instantaneous velocities subsequent to

the initial conI act velocities which may lead to large

cor-rections to equations (G) und (7) for large penetrations. \\ugner out i il1('(l equatiOn (7)

froni a solution of the

equa tai u u

ò,

1(ò

p òt

2òx

where

, velocity potential iii plane of flat plate

(V xrp2)

Equation (8) is incomplete because a term representing

the dynamic head of the free stream is lacking. The force increase obtained when equation (S) is corrected is, for con-ventional angles of dead rise, of the order of 10 percent.

r1hie validity of the assumed velocity potential is some-what uncertain. Comparison of solutions of equations (6) und (7) with experiment has shown that the principal

fac-t ors are aj)pIoxinìufac-te(l hufac-t fac-the experimenfac-tal (lufac-ta have lucen

complicatu'(l liv seaway and otliei factors which have not

peiniitted adequate evaluat loti.

Force equations. Except for the folli iwing differences, Wagner's 1(11(1 ((luiatiOli is the SQ!ii(' us that a(lvancNl by von Eí rni(ii

(u) The new method of obtaining the virtuial mass is

incnra)ru

The velocity used iii the iiionient urn equation is

(Itfilleul ill flerently.

The equal I ùii is written as a n integral for a non-ree t u uigula r pl'ssuule a rea.

Reduction of the sealdune mass for the case of

ee((ntric impact is contempla teil.

The foice equa t ion given by Wagner (references 6 and 7) is:

F=

irpV2 ' x dl ₍₉₎

/

1)

m,\3j

u

Wagnei limited this equation to angles of ilinil iise less than -30°. Without giving theoretical justification Wagner ree-omniended that tite equation be multiplied by the following factor if the angle of (leali rise is less thait 10°:

i-0.i5log

(10)

where

angle of slope of huill bottom at edge of impact area,

radians

1f the float is a fluted prism and is at zero trim, u is

deter-mined by x and is in(lepen(hellt of i. For a particular coni-hination of ii 011(1 X, use of an iulerease(l value of i results in an increase in impact area bu t a decrease in velocity been use the virtual mass is greater. By part ial differentiation of t lie force equation wit h respect to I the force for a part icuular value of X (and u) can he shown to be a maximum whìeuì lis

such that

(8)

ANALYSIS AND MODIFICATION OF THEORY FOfl L1PACT OF SEAPLANES ON WATER

\Vngner deny ti this relationship and substitut NI it ill ('qUt-turn (9) to ob am the followi 1g equation

O.3lTe 2mr

p 0

J ?P.r2'

\vIlert3

maximum volite of force that can occur for a

par-tictiltir value of J regar(lleSs of the impact length The procedure for using equation (12) is to determine, first

m, i

tite volite of for whicli

= when the length of the

im-pact area is equal to tite length of the forebody of the

sea-plane. For smaller values of x the critical length is greater

than the lengi h of the float, and therefore equation (12) is

not valid. F r larger values of x the critical lengths will be

less titan the ''ngth of the float and the statistical variation

of the length t i the impact a ea in seaway is assumed to

per-mit tite critic 1 lengt li and iìiaximum force for each of the larger values f x to be attained during the life of the float.

Wagner state tha.t equation (12) is to be solved b means of trial-and-vt 'or substitution of these largei' val tas of ,i in combination i :ith corresponding values of ii; thus tite value of x is (lctel'm ned for which a maximum value of tite iìiaxi-mum forces w 'thin the range of the formula exists.

ii u in equi lion (12) is ('Oflsidercd to be a ('OnStant as for the case of a t tiangular pìisniatic float, it is obvious that t lie equation will rive Inaxinium ioree wheii a rnininiijin valute of

x is substitut d. By subst il itt ing tite minimum val w' of ,uu to which this quation was limited and sul)stittlting tIte volite of u for a tri tirgular prisma i Ic ¡inuit, Wagner ot1tai tit'd the following equi tiou for the maximum force on such a flout t:

(i L,Qr17 2

_L))

¡3

where

Fmax maxi muni hydrodynamic force

'maz length of tite forebody of the float

It should be remembered that equations (12) and (13) are based on a relationship between x and i which was so

(lcrived tito t tite force calculated for a l)alticUlar value of j is greater titan it would have bcvn if i had been either greater or smaller. The maximum force (luring tut' impact time

history of a triangular prismatic float, however, will occur at

a value of .r less than that to which equation (13) must be

limited, that is, at a value of x foi which tite force would be

greater if the float were longer. Solution of titis profilent (reference 8) itas shown t hat equation (13) shot 11(1 be derived from tite relationship

m i

ut 5

rather than from

niw

i

In

Tite effect of such correction is to i itereast' the ('ahulutted

niaxiinmtm fom'('e foi'

the triangular prismatic float by

24 percent.

\Vagner (reference 6) worked hypothetical examnples in which his equations gave values of t hie proper order, but lie

(1 I i miot. coimehat t' t lie eq tint jolis ivi u lì lillY (xperiinvrlt ai it t a.

Improved solution for the flow beneath an immersing

triangular prism.By means of an iteration piocess, the

details of Witi('il are not given, \Vagner (reference 7) made a rather exact solution for the flow I)eneathì an immersing tri-angular prismil vithì aim angle of (li'n(l rise of 18°. Titis

solui-tion involved only one flow pattern,

silice siniilar but

enlarged flow pattermis were assurne(I to exist at ahi times. Titis assumption is reasonable since. tite previous defiiiit ions of the virtual mass for the ease of the immersing triangular prism 118(1 been on that basis.

Rather titan make separate calculations for other aught's of dead rise, Vsagner estimated tite effect of angle of dead

rise by Writing tite equation of ari arbitrary curve that

paes t h rougit tite calculated point for a u angle of dead

rise of 180 a ¡ah t hat asymptotically aitproaches zero force at an angle of (1(0(1 mise of 900 and infinite force at ari angle of (leali rise nf 0°. Tite force equation written by Wmigiier

was only for constant-velocity inimersion, anti gave for

vertical penetraI ion

Fr(_1)

PVV2yl (15)

Titis eqitutt Ori was used by Svdnw to evaluate tite virtual maass, wit iii was titen applied itt formulas for

variable-velocity impact. rrltis work is discussed in a subsequent

section o!' tite prescrit paper.

MEWES

\1('S'eS WI'()te Wagner's equations in terms of acceleration rather titan force (reference O).

Tite equation for curved bottoms as written by Mewes was limited to mass loadings high enough to cause peak force to occur at maximum float width. The form coefli-cient u, whicit Wagner determined by an integration process for each float form, was approximated by Mewes in ari ex-pression that contains the angle of dead rise at the keel, tite. angle of (lead rise at the chine, and tlìe distance between the side of tite float and the point of intersection of tangent hin's

to time bottom surface at tite keei and the chine. He also

presented an approximation to Wagner's correction factor

for fluite angle of tlead rise (equation (10)).

Mewes' eqnat ion for Fn for' the triangular prisitiat Ic

float is riot obtained.directly from Wagner's but it may be obtained by incorporating th(' cotiection of equation (13),

which was previously (liScusSetI.

\leves gives an equation foi' tite flat-bottom float, which

apparently was obtained by substitution of values for tite

st ritcturtil elasticity in Pabst's equation.

- TAUB

Taub's st mdv (reference 10) (liti not introduce new

tite-oretica.h concepts. Rather, it

used Wagner's tlteory to

predict the effect of designi trends on design loads.

SCIIMIEI)l'N

Schmied e I'S analysis (rl feience

il) assumed tua t

t h e

impact. flI'('il at ilitreiciit. stages of the impact process con-sists of similar ('llil)Ses and that. tue flow action is of the type

specified by Wagner. In this way he obtained an equation

that, becau c of its assumed relu I iorìship between x and 1, took into at omit growth of virtual mass due to both growth of w id tu an growth of length of the wetted area. Al though this anaIysi was eniefnllv (leVeloped, the simplifying as-suml)t ions i itli regard to hull shape and manner of contact with seawa prevent this case from being of great practical interest.

W EINIG

Weinig used previous theory to indicate the effect of

deformatioi of the hull cross section on the impact force

(refercnce 12). The types of deflection he considered arc

too simple o represent tite deformations of an actual sea-plane hull lut the magnit ides calculated intlitate that. the

structural lasticity of tite bottom proper of a V-hot tom

seaplane wjl I have little effect on theresu1taiìt hydrodvnamic force.

S YI)O W

Source 01 virtual mass. The moment ouI equation used

in previous ,theory gives, in general t cims, tite following value for t li e hydrodynainic force:

F=(rn

dTÌ dm,

=

_dï

+

_di

where

i '? velocity component substituted in momentum equation

For the constant-velocity case solved by Wagner in

obtain-ing equation (15) tile first term of equation (16) is 0. By equating the remaining term to equation (15) and

integrat-ing, Sydow (reference 8) obtained a valuo for rn, that lie used to effect a variable-velocity solution.

The value of ìn, obtained by Sydow is

r/'ir

\2

2

-'flW t\ß) P1Y

(li)

For the widely used angle of dead rise of 22° this value of

in is 1.56 times the virtual mass given l)y von Krmnán and 0.56 times tue virtual mass given by Wagner in his general soluitin. The theoretical basis of equation (17) is more

sound than that of the previous definitions; tile importance

of the tiirrerences is shown by the fact that for equivalent instantaneous conditions these ratios of virtual mass can

be regarded as force ratios.

Force equations.By using hie new formula for virtual

nass, in von Kuinin's equation, Sydow ohtancd the follow-ing force equation:

Vr02 irpi _{/ ir}

FCy_l)Y

(16)

(18)

Eq nation ( 1 8) aiutI previous inpact 'q uations are I)ase(l on

tite assumption of a weightless Inass. Sydow (lerivedl a second equation for a (Irol) test, which included the

momen-turn due to the weight acting for time time of the impact.

This equation for a drop test ivill not 12e (11SCIISSC(l li(reill

because it is not rca(iily colIil)aIal)le with the previous e(!uations atol l)ecause the previous equations are fiore representative of seaplane impact, for which the wing lift

approximal ely balances the weigh t.

In his colisidelation of elasticity, Sy(io%v iiicluded

equa-tions for tite float both with weight and without weight.

This treatment (livi(lc(1 the total mass imito a hull iimas and all upper mass spring-connected to cacTi other. Equations were ivritten both for ivide floats and for floats sufficiently Ïìarlow to enlise maximum force on the sprung mass to occur

after the chines are imlnersed. The equations fou these cuses will not be discussed because they are not necessary for an evaluation of the basic hv(lrodynamic theory.

Comparison with experimentAn experimental check is

obtai uid for veut iraI dvop of litt1! masses (angles of dead

rise of 0°, 10°, 20°, and 30°) spring-connected to an upper

mass. The drop data are corrected for the effect of gravity by t lieoret icully derived factors and compared with corn-putted values fou tite case of wing lift. The spring constant is motjificd to fit tite data. Titis modification is assumed to represent. tite (fr((t of time elasticity of the bottom proper.

Because of this moditira.ion and other approxiniations the

exact accuracy of the theory is ¡iot established. The

hydro-dynamic theory used by Sydow, however, appeals to give

approximately correct results for vertical drop of a

tri-titiguilar i1ism at zero trim into smooth water.

K R E PS

Kreps (reference 13) used Wagner's treatment of time

virtual mass.. The finite-keel

factor recommended by

Kreps differs, however, from that advanced by Wagner. The aspect-ratio factor determined . experimentally by Pabst was In(orporated in Kreps definition of the virtual mass .

Oe idea was advanced by Kreps that lIad not been

in-ClU(le'dl 111 the previous theory. This idea was the inclusion of a force term representing the resistance of the instantane-ous flow pattern in addition to tue force previinstantane-ously associated with the rate of change of the flow pattern. The new term made use of the familiar flat-plate air-drag coefficient. of I .2S

In addition to time fact that this coefficient is of doubtful validity, the theoretical soundness of the entime term is

questiouiahie. This subject vihi l)e discussed further in a subsequent section of this report.

Time equations proposed by Kreps were use(l by him to interpret drop-test (iuta for floats having angles of (l('ad rise

up to 30°. This analysis was not a direct comparison of

theoretical all(i experimental force values; instead, StU(lV ol

the total velocity dissipated during the impact part of the

immersion was involved. Approximate agreement. of thr theory with drop-test data was indicated.

Fightrath

Piane cf ter

5tríOCe J

Sloe view

PROPOSE!) THEORY

Increase )f flow momentum in fixed planes.For a long

narrow pris attic float in vertical drop at zero trim, the how will occur P inni ruy in traììsverst' planes normal to flic water surface. W lien t lie float in vert ital drop is considered to

have a trim angle with respect to the water surface, since

11 aid part ici s will he accelera t ed normal to the platitig, the transverse laites must be considered normal to flic keel

ra t lier tha n to tite water surface. The flow in a piatte for

n ptìrtntila

depth and velocity of the iflunersing

cross

section is

ubstantially the same as for zero trim.

The difference il i the depth of tlìe keel at (lifferent points along the lengt Ti sets up longitudinal

pressure gradients and

thereby ciii ages the cross-plane-flow and end effects, l)ut the effect o these longitudinal variations and of finite keel

length will, as in i)reviolls treatnients, be approximated by the aspect-I tija factor.

For obliqi e impact, tue motion of tite float can he analyzed in terms of I te component motions l)arlllleI and perpendicular to tite keel. 1f the float is prismatic and the nose projects

1)eyond tito water surface, the niotion parallel to the keel

(loes not ca ISO any change of the float cross sec tioti in tlìt transverse f ow planes, winch are regarded as fixed in space.

This stítcn cnt applies to the flow planes as long as they remain (hilcutly beneath the float; when the step passes

through tli flow f)iane, the intersected float cross sect ion instantly va nishes and flic plane becomes part of tite wake of tite float. Figure 1 shows tite inclined flott t, the velocity cornponeri s, and a normal hOW 1)10flC.

1go of

generated vvave

End view

FIGURE 1.Prismatic float at positive trim.

For the commonly assumed frietionless fluid, motion of

the prismatic float l)orPendicular to tite stationary flow planes will not aflect tite flO%v within tite plane in any way. Titus

tite flow in the plane will be determined by the manner of growth of tite intersected float form without regard to the fact that cross-plano velocity of the float causes different

cross sections of tite float to be in contact with tite piatte at different times. Tite flow process within a particular flow piatte will i)rOPerly 1)egin when tite keel at the water surface reaches that flow plane. Beginning at titis instant tite keel line of the float will penetrate tite plane at velocity V until

the plane slides off tite rear of tite float.

The pressures

registered in tite plaiie at any iiìstartt should, for tite eOri

Si(lere(1 float iii an ideal fluid,

he tite same as if similar

(nhtìrgcnl('nt (if tite float se(tioll vithin tite pltìne occurred,

ill Uil ('qIIi\ltieltt l', Yel'ti('tÌl penetration. For vertical

penetration the flow J)iafle is iii contact vitit only one cross section of tite float but for an obmlique impact lite flow plane

is progressively in contact with all sections.

Iii tito precç(ling (lisclission, if a float of uniform cross

section liiL(l not been assumed I Ito mot ion parallel to t lie

keel woititi have been seen to cause an increase in tite inter-se(t('(l float cross section in tite flow planes beneath tite float.

If i it longitudinal curvature is riot too great, tite flow will stili occur primarily in plamics nomini to a I)ase line parallel to tite keel at flic step. The primary effects of the combined notions normal 1111(1 ¡)alall('i to tite keel will be approximately

1l)resIttcd if tite reactions in fixed flow planes normal to

tite keel at tite step are computed on tite basis of the absolute ill <'t'e'.150 of tite ifltelS('Cte(l Iloat cross section in each plane as (ietelflhiIte(1 by the combined iriotioiìs of the float.

The sitittmtttioii of tite reactions of flic indiVi(lUal flow

planes being acted upon by tite float must equal the total

¡ate of citange of tite momentum of the fluid. Since is defined as tite moment urn of tite flow directly beneath the float (that flow' wit i'ht a fleets tite acceleration derivative),

the mate of change of the momentum of tite flow directly

beneath the float is (ìri,TÇ). In order to obtain tite total rate of duuitge of tite moment um of lite fluid, however, tite

rate at winch moment titi is imparted to tite downwasii in

connicI.ion with flow platas sliding off tite Step must be added.

Where in, is tite virtual mass of the plane at the step as included in tite determination of m, tite rate of momentum passage to the (lownwash is equal to tite momentum of this

plane multiplied by tite number of planes sliding off tite step per unit time. Titus tite complete equation is

(10)

j;; _{bydrodynani ic force in tim e V-(hirect ion}

ills virtual mass of flow plancat step (per unit distaitce in

keel direction)

VP velocity of the float parallel to keel (rate at witich flow planes sTille along keel and off step)

In a subsequent section of tite present report, tite same force equation is derived by integrating the force along tite float.

_{The solution is made for fixe(1 trim; if the local}

penetration velocity and acceleration are treated as variables

along tite keel, itwever, variable-trim solutions

can be

obtained.

The variation along the keel of the instantaneous reaction of tite individual flow planes determines tite instantaneous lengthwise (histriliut ion of the load. Further, tite pressure

distribution in tite transverse planes gives the pressure

dis-tribution over tite bottom area. Previous impact theory

concerning tite pressure distribution in a flow plane is (iirectly

applicable to tite proposed theory if the velocity and

RC-celeration of tlte float cross section are defined by the theory proposed herein.

Comparison with

_{previous theory.I'revious impact}

thcory \VÌS as('(ì 011 tlic ÌS1Iflj)tiOfl I hat th( inoment.unì of

th(' S('8 phi tì tlfl(I virtual in as t'vinai ris coust an t (I tiring the impact. _{t (lì an flS1I11)t iOn 1'('(IUireS hint tue secon(1 terni} of equltio11 I 9) be O, 1)IIt _{t bis t 'rn is O only \vlìen T', is thy} rcsiiltant vcOcity (that is, V=O).

For the J) isniatic Jiont wi t lì l)O\V above tue vater surface (at posit i v t ifl).), t lÌ( V(lOcity (OfllpOflC'!lt of 11w float pituI1I to t lie keel (IO('S not causo iiange of the momentuni

in _{t lic llOim al flow planes a ud therefore is witlìoii t force}

(ffeCt. _{Titis COfl(litiOfl nieans that the effect on th'} instan-taneous force of the rate of momentum J)assagc to the

(lown-vash (1 LI( _{to a velocity corn1)onc'nt paraiJel to the keel is}

1)alance(I by the effect of this velocity component in causing (owing to its vertical component) Icss increase of tuìe virtual mass. For this condition that the velocity COfl1I)OIleflt

J)ra11e] to tite keel does not affect the instantaneous force for a given draft an(1 velocity, tut' previous theory wolII(l he justifie(l iO i 'glectilig tl1( V(lO(ily parall(I to the keel if the force eqllati(n IÌa(l beeiì written in terms of the instantaneous velocity 8tH (lraft. 'Plie error of the previous theory for

this ease lip in t la' fact that the total m.omentuni. of the

seaplane an vhf iial mass -as tssiimtl to he (onstan t (see

equation (i )) an(1 tlu Illofllcntum left behind itì the down-vashì 'oras thrvhy neghecte(1.

\VitIi r('g: rd to l)('Ictration, the error in

thì(' previous theory is tu; t L' _{sin r was not subtracted fioni i ¡u}

tion V(.O(itY Ihnt exists Foi t

aloi'.

_{ctiiallv, I his iieg1rt ('(I} teiiïi niay ci use the Ia i to be el.irn.hi ri a t :1 tulLe i hat

T' is of inre magnitude in a dovnwiì il

direction.

Con-sideration oi this terni leuds to small pelai rations and sinai! forces for l(w flight-path ¿Ingles. hut the previous tli('ory

(for the nOi mal lange of trims, essentially a salut ion for

vertical (i mp) always gives large ¡)('llet rations and forces. This theory gives no values of the maximum penel rai ion;

it was explained t hat lIeLri('ete(i buoyancy forces finally stopped t lie peiit'ttat ion of tut' float.

If the float is not prismatic, the instantaneous-force equa-t ion developed by equa-the previous equa-theory is correcequa-t only when h ',, is the resultant velocity. Titis restriction is (hile to the Inc.t that the velocity component parallel to the keel causes

increase of the flow pattern in the flow pianes beneath the

float. As previously discussed, the momentum passed to

the downwash is deterniinod by time float cross section at the

step without regard to such iìonsimilarity of the forward

cross sections as may i)ave substantial effect on the virtual mass.

Since tile previous impact theory did not disclose the in-adequacies discussed, tile proposed theory olens a new field for study und a(ivancement.

For the planing float the first term of equii tion (19) is (J and the hiydiodynamic force is equal to the last term. When methods of tue previous impact theory are used to (iefilw the

flow in tut' plane at the step, this last term provides a

theoretical V-bottom planing formula that is more advanced

than previous planing theory based on an approximate flat

plate and two-dimensional flow in loilgi tll(1 mal planes. Also,

tins last term permits tIne use of ¡diuning thata to evaluate

experimentally tine momenitum of the flow in the normal

planes for a specific cross section, afl(l tile complete equation provides means whereby the results of such studies can i)e iiSc(h for calculating transient (impact) and oscillatory

mo-tions of till' float. Examples of such use of experimental

planing data will be shown later in this text.

A later section of tilis report will correlate both tine pri'-viotis I hieorv und time pi'oposi'ii theory with experinlen i al (huta for oblique and vertical impact in order to shiow tine

inadequacy of tite previous theory and prove tue merit of

the proposed I hueory.

Derivation of force equations.A prismatic float nit

posi-tive triai is hiowii in figure 1._\. flow plane is shown iii tine side View. The depthi of penetration of tue float imito this plane is represented by z in this figure. According to pre-ceding interpretations the momentum of the flow in tut' plan e

clement (fixed in space) can be represented, in the case of triangular float cross section and similar flow at different

(lepthlS of immersion, by

tlz

dt viiere

z pend rai on in the PIlule

ds thickness (If piane

K thìeoietical

coefficient; different values specified by

ill bce nit treat uiienits, varies according to angle of

(leali liSC

'l'tue force irainsuuui i ted by this plane to the font can be obIta meni from the following eq nat ion

d17

dz

_I[2(1+0z(dz)2l(l.

₍₂₁₎

where

F

force (normal to keel) registered by individual flow plane Tue total force can be obtained by summing tile forces in the individual flow pianes. Thus,

(I'

Fnj slit

, jç[z.i

+2z()}1s

Whieii s tait r is substituted for z afl(l, for fixed-trini impact, V,, is substituted for -, the following equation is obtained

FJ'ThK(s2

tn2r

+2s tan

rV2)dS

d V

Ky2 V2

3 sin r cos2 r+Sifl T CO

(20)

(22

Time virtual mass is a fictitious mass that, if it is

con-sidered to ¡nove at velocity V,, has the aggregate momentum of all the flow particles in all the flow pianes directly beneath thle float. Tiii rnnìSs Can be ohtaifle(l by integrating eqii-tion (20) over the wetted length; the first term of equaeqii-tion (:

ANALYSIS AND MODIFICATION OF

however, reprisents ÌHW out1, by inspection,

Ky

¡H W

= 3 sin r cos2

1f the moin(ntilrn of the uloat and virtual mass is to remain

constant as specified by the previous impact theory, the drn

bist

term of equation

(22) must equal V

This

equality hohls, however, only when V is the resultant

velocity. The following algebraic equivalent of equation (22) nhay be written

K

F =

+

y2ü

+KY2T9V1 ₍₂₄₎

' 3 sin r C0s2 r sin rcos2r cos2 r where

!/ velocity i wmal to watei' surface (V1)

V9 velocity I float parallel to keel

(- '

sii r+ V, cos r)

The respertiv terms of equation (24) represent m, Vn a n (1 I ie rate at which momentum is hupa rl ed to

the dosvnwasl rì connection with tue planing action.

Tite inoiiwi tuin of the flow in Ihit' flow ¡)llll1( at. the Step (equation (20) ) is Ky2T7/cos2 r. The velociiyV is a measure

of the rate ai which these planes slide .olt I lie stoJ). Tue l)ro(lIlct of t li 'e terms is ('qului to the last leim of equation (24), which is ii-i agreemerìt \vitlt the t hieoiy for the flll)11'll-tinn passed 10 the (iownwasli.

1f ¡ìi, and the flow in the plane at the step nie lletermine(l, equation (24) can he obtained tluouErhl direct substitution of these quauttit ¡es in equation (19). The solution presented was chosen in order to explain the theory better.

Momentum of the flow in the plane.As indicated in a

previous sert ion of this report, the most advanced treatment of the flow- in a plane being penetrated by a nontriangular prism (Wagner) is of questionable accuracy. For the case

of a float that is both nonprismatic and of nonuniform cross section, the in crease of the float cross section in the Stationary planes niay correspond to an equivalent vertical Penetration

durini which the float changes shape. Since the previous

theory lias riot treated this case, the effect of the previous

changes of flow pattern oui the flow for a given cross section is questionable. Although this effect is iiot. believed to be important for conventional proI)1(ms (as will be indicated

in a comparison of theory with experiment), importance

would limit the proposed theory for oblique impact to floats of uniform cross section along the keel; curvature along the keel for Ibis case of uniform cross section would alter only

the rate of penetration of the given cross section into the

flow planes passing beneath the float and could be considered. Although the proposed theory offers new ways to handle

the more complicated cases, the present report will be re-st ticted to obtaining solutions for a triangular prism and

comparing these solution s Wi tI experimental (Jata.

(23)

TIIEOHY FOlt IMPACT OF SEAPLANES ON WATER 9 Analysis of the previous theory indicated that the virtual

mass used by Sydow (equation (17)) represents the best

theoretical solution of the flow in normo.l planes for the case

of the triangular prism. Since equation (17) was derived from equation (15), this definition of the virtual mass is for two-dimensional flow. Probuhly the best way to correct for enti lOSs, consi(lering relative simplicity and probable ac-curacy for conventional n agIes of dead rise and trim, is by use of Pabst's cinpiricil I aspect-ra t io factor

i

2A

For bodies of low aspect ratio with the two-dimensional flow considered to occur in transverse planes, the effective aspect

ratio to be used in this factor is time length to mean beam

ratio. Based on the area beneath the plane of the water

surface for a triangular prism at l)OSitive t.tim, this ratio is tan ß

tan

r

The empirical factor was (letelmlned as an over-all effect,

but the saune total force will be obtained if it is applied to

I lie planes hid i vid uallv. Introduction of the factor and the

effective aspect rutio into equation (17) gives

r/r

\

(

tiiflr

12 jT7

1) pdt1

_{2 tan /3} vhiere

w,, vi ut ual muss of the (low in the plane

The aspect-mat io factor (expression (25)) was obtained in

submerged vibration tests and is somewhat questionable.

As previously discussed, t hie (let.ailed derivation of time two-dimensional solution (equation (17)) was not given and how far time iteration process was carried and what approxima-t ions were made are noapproxima-t known. A large amounapproxima-t of phmiriiuìg

d n t a therefore was analyzed to determine t he adequ acv

of e(lufltion (2G). For this analysis equation (19) vas used

in

vbich, for planing, the first term is

O, 17= V cos r,

V,,= V sin r,

and F cos

r=

TV. The following equation results:

IV=m,V2 sinr costm r (27)

From equation (27)

TI' 1,'2'un reos2 r

Substitution of experimental values of IV, V, and r

equation (2S) gave experimental values of in3. Substitution in equation (2G) of experimental values of /3, r, anti the value of z at the step y/cosr gave corresponding theoretical values of in,: Comparison of these values in(hieated that a factor of 0.82 should he inserted in equation (26) Io give the follow-ing equation:

m=0.82(_1)

(29)

(25)

(2G)

B&'caue of the aspect-rat io foctor, equation (29) does not

hold for si tall angles of tienti rise. rrl)( _{study iiid icated}

tiitit tin' eq nation will be satisfactory for angles of (len(l rise l)etweefl U and 30° l)lit l)egiTLS to be unduly excessive lot.

angles sitia lier than 100. A variation of eot2ß wit h angle

of (lead rise may he better than the variation of (-

i)

with angle of (lead rise, but in absence of clear requirement

for change the variation used by Wagner in equation (15)

was retained. As previously discussed, this variation

ap-1)eat'd to result from arbitrary miring through a force solu-tion for 1S angle of dead rise, infinite force at 90° angle of

dead rise, and zero force at 00 angle of (lead rise.

The

cot2ß variation used prior to that time could lia-e been

faired by Wagner through these points; thus it seems that

\\agner probably had sorne reason for changing the variation

Equation (22) was derived on the basis that rn=Kz2 (is. Titis equai ion will be corrected for aspect ratio and for variation cl angle of dead rise if K is replaced by equation

(29) divid4 1 by z2 ds, which results in

ir

tan r \

,2

F=0.S2

i) (1_2

_{tan 13) iIi r}

CORI 'ELATiON OF THEORY WITH EXPERIMENT

TI' (IV

For I lie I)laflIn

float P= -

- =0, i ,= T , sin r,

COST (lt

and equati ni (30) takes tli' foiiii

IV _0.82iryT'2 sin

(i

tan r) (ir

1)2 (31)

from wItiel it can be determined that

i W

1'

₍₃₂₎

tan T

0.S2ir sin

tau 13,

Figure 2 shows comparisons of this equation wit i

experi-mental (hit a (reference 14) for a Vbottom float.

(100(1

dTÇ,

os r 3 COS + l

(30)

.20

y=

agreement exists over hie runge of the experimental (lota

Slitily of t he ('xperinìelit uT d tita ilali(ate(l that its scattei S1li)WS uiiaiiily the ilitteduracies III the measurements of (liafi and tlitit agr'ernent between measured aiitl tIteoretical dat would geuierally be very good if tite values of di-alt were takei front faited curves of (li-aft against velocity.

rFle 5(11(1-line curves in ligure 3, -liie1ì rel)rescnt applica tien of tlìe proposed theory to obli(1ue impacts, also show

good agreement of the theory with

experiment. Thes

curves and dat a vere included in reference I 5.

As discussed in tite preceding auialysis, reference I 3 con tains a proposal that a force term representing the resistauct of the steady-flow pattern be added to the force in tlìe pine

duc to iate of change of the flow pattern. Although refer ence 13 suggests that the familiar flat-plate an-drag coeffl

dent of i .28 be used, perhaps a better approximation cai

be obtained by use of tite coefficients (leveloped by Bobylef

(reference 1G) for a stream impinging on a bent lamina For the more-or-less mean angle of dead rise of 22° thi

coefficient is 0.79; the variation with tingle of dea(1 lise small for conventional ranges of angle of (lead rise.

1f a steady-flow term based on tite coefficient of 0.79 i incltided in flit' force equation, a coefficient of 0.75 instea( of 0S2 must be ihìtIO(IIICC(l into equation (29) so that tl results of t lie final solution \vilI be subjected to rnininiuri (-hange. The equal ions given iii reference i 5, vltielì givt tite soliil-liiit' CUIVeS of fii,uire 3, were based on such nÌo(hifì cation of ('qilatioht (22) 1111(1 i--hat'sent, within i or 2 percent soltit ions of eqilat ion (22). I ii nidei' to (heck the eqilat ion.

of ttft'i-ence I 5, 1 , si ii r in ti te ii elinit ion of should I )P ei lin

ihhiitC(l on the l)asis that for seaplane impact it is negligiltl as conul)are(l with I ' cos T.

SUIr('(luleflt

theoretical study lias

indicated that tu steady-flow terni represents the force in the l)laflC when tb

chines are immersed but that when tite chines are no

immersed titis force is included iii the term representin increase of tite virtual mass of the flow in tite plane am

should itot be added as suggeste(l in reference 13. For titi.

reason the steady-flow term was not included in eqttatioi

(1 9) an(l subsequent equati()lìs.

The effect of tite empirical (-orrection factor 0.82 to Sydow.

virtual mass is to increase the calculated draft for planin 0 percent and decrease tite calculated maximum force fo a severe impact 6 percent Without this correction fai

agreement thus would exist between theoretical and expert mental results.

The long-short-dash curves in figure 3 were derived ot tite basis of Wagner's formula (for two-dimensional how reduced according to Pabst 's aspect-ratio factor. Evet

with this reduction the calculated forces are excessive

partièularly for the steeper flight paths (relative to th1

wave surface) that represent severe impacts and for th

higher trims (relative to tite wave surface) at which moderi seaplanes operate in open seaway in order to obtain maxiniun

wing lift and to red tice the landing speed. C- r fan r 2)O82-1)ff doto ('-efer-ence /4

-2 Leper/mental (i (deg)

r

5 9 2f00 3.__0

7_A

I _IVI _5 Ib /0/b e 0 4 a O

/0-?

ó ¿

lo

REL'OIIT NO. 810NATIONAL ADVISORY COMMITTEE FOil AIRONAUTICS

O .04 .08 /6 .20 .24 .28 32

- Experimental dr-of lì'

.3

.1

ANALYSIS ANT) MODIFICATION OF THEOIfY FOR IMPACT OF SEAPLANES ON \vA'rEIf

03

o /2

Theoretio/ cur- ves

New method, nev coo ff2 dents

- Woqr'er, Pabst opect-ratic reduction - - Previous method new coefYi,eni's

The long-dash curves in figure 3 ai-e based on tue imrnen-turn equation as used by Wagner 1111(1 others, luit with the definition of the virtual mass of the flow in the normal planes

replaced by tile values used in the equations of the theory

proposed herein. Comparison of these curves with the curves of the proposed theory shows that the method of the previous theory still results in forces that are too large. Differences between the curves, paiticulaily at higher trims, justify use of the newer equation.

Tile offsets of tise float with wInch the experimental data

of figure 3 verc obtained are given iI! reference 15. Tite float was a illo(iel of a four-engine flying boat except that the afterbody and chine flare were removed- Agreement of the

data with solutions of tile proposed theory for the case of a prismatic float indicates that pulled-up how has little effect on tile resultant force.

Although, as previously indicated, the proposed theory

can be used for consideration of zero trim if tise bow shape is not too blunt, solutions for this case are complicated and ai-e limited by the fact that they must he made for arbitrary bow shape. Solution for zero trim therefore is not given in

the present paper. In figure 3, however, tile experimental accelerations for zero trim can be seen to he 10 lo 20 percent less lutIn tise experimental accelerations for 3° trim.

The experimental data in ligure 4 represent vertical drop

(,=9O0) of the float, with which the data of figure 3 were

oi)tauile(1, whit the afteibody added. The (irops were ma(le at. 3° tri in, and the resultant (vertical) velocity therefore -as

not quite normal to time keel. Since soluit ion of equation (30) is much easier when the velocity is normal to the. keel, the theoretical curve in figure 4 was obtained by solving equation (30) for a flight-paths angle of 87°. Tile difference between 87° and 90° flight-path angle is not important. Agreement with experiment is shown.

2/2 20 o /0 20

i':

lui I

i..

FIOrE E 4.Varint ion of load-factor eoeflìei,'rit for vert leal drop. Test polnt from unj,ublishal

data. ß ut step, V°; r, .6 .5 Eperfmer7fa/ 1 (deg) doto O ¿ 4 6 8 /0 /2 /4 /6 /8

fIiç'ht-poth 0-79/e, , deg

FIGIRE_{3.Variation of 1riaj-factor cofficirnt with flight-path angir for a float having an angle of d' I rise of 22O. (li 1-line curves horn reference 15.)}

'Fije fore oing correlations are for the cotalit ions of uixed

trini aiìtl inooth val ('I'. Tut' tltt'oretical CIII'VeS will give l)Proxima e represeiital ion of free-to-trim inipact if used on ti te i asi ol the trim when the step cont acts the uvat ('r. Tite case of tite seaplane landing into a swell will le approxiniated i f the tri ni and the flight -pat ii angle are defined relative to

the inclined surface of tite swell rather titan relative to the

horizontal.

Good agreement of ti te theory wit h experiment is lud irat ed bitt if the velocity is small enough the theory will lie ittade-quaLe. One reason for titis inadequacy is that the dynamic force will he reduced SO muni that lite buoyancy force, which

has been assumed negligible, will become important. A

second reason is that tite effect of gravity on the flow pattern,

particularly with regard to lite generated wave, may be

substantial.

Pertiitent data that affect tite unportance of the gravity

forces are tot given in figi es i to 4, which a re based o i lbe

dynamic ferces alone. Tite range of the data, howevr. is

sufficient I tietermine that the gravity effect is negli ii)le

iii the land ng impact of ili seaplanes that have been flown. Some detti pertaining io titis sliJ)Ject Were intliitl&ii in

refereine I

..

For very low velocities tite virtual mass might be ex-pected lo tpproaeh tite value Specifi('(I by von Kt1rmtn. Such traiìition would reduce lite virtual mass used in lite

computat lens of the i)Icscut paper about 20 percent. 'l'ue

manner of titis reduction is of interest only in connection

viiii slow planing, for which it can be readily deterrttiit((1 by empiric ii means.

CONCLUSIONS

The analysis of previous impact theory, modifications of

titis theory, ami comparisøli with experiment lead Ii) t lR

followiiìg conclusions:

i. The assinnption of previous treatments of imj)ttet

theory that tite total momentum of tite float atti! virtual mass is constant is applicable when the resultant velocity of tite float is normal to the keel but is not applicable for

tite usutti oblique impact, in which the velocity component parallel to 11w keel causes momentum to be lost to the (loWil-wash behind the float.

2. Comparison of previous theory with recent, more

accurate data for oblique impact shows that titis theory

greatly overestimates the impact force. Use of newer

co-efficieiits in the previous theory only slightly reduces tite

forces. Disagreernetit wiLli tite data is larger at thin liig1ie trims tutu lower Il iglit-path angles.

3. A moulifi('d t Iteory has been developed that tales intu

account tite loss of moment tun to tue downwashi.

(oo

agreeniemit lias been obta med with data for vertical drop oblique iflil)act, and pinning.

LANGLEY \ÍEMORIAL AEIuoN.tTTwL LAluoltAToity,

NATIONA L ADVISORY COTMITTEE FOR AERONAUTICS,

LANGLEY FuI':LD, VA., August20, 1945.

REFERENCES

I . you Ttírni1it, Tb. : Tite Iiiipact on Seapliurie Floats during Latidin NACA TN No. 321, 1929.

Thompson, 1'. L. : Water-1'reisure T)iìtrihution on a Seaptari

Float. NACA Ilep. No. 290, 1928.

I'a1)it \\'i!lielin: Theory of the Lauding Impact of Saplane

NA('.\ TM No ..580, 1930.

Pahit, Wilhelm: Landing litipact of Seaptanes. NA(A Ti No. 624. 1931.

I Bottomley G. II. : The Impact of a Model Seaplane Float on Water I:igliteitith Series. lt. & I. No. 553, I3riti.ih A. C. A., 1919.

a. \Vagiior, IIerIx'rt: Landing of Staplarics. I\ACA I'M No. 622

1931.

7. \Vagti(r Ilnrixrt : tIx'r Stoíi- iiid Cleitvorgfiiige an der Oberu1icli

Voti F'Iûigkeiteit. Z. f. a. .\I. M., lId. 12, lieft 4, Aug. 1932

PP 19:3-215.

S. Sydov, .J. : Ílr (1(11 EinfIiiii veti 1'ederiiiig itiirl I'ieluìig auf dei

I au cleit »i. .j a i i rl) . 1 o: I i r d ti il id i eu Luft fait it for.ch u i i g

11. Ohh'iibourg (\I mich). Pr).I 329-1 338. (Available as Britisr

Air Ministry Tranislation No. Sb!.)

9. Mewes. E.: l)ie Stosskrhfti' ait Seel!ugzeugen bei Starts un

Lanidiungeni. Vereinigung für Luft fahrtforschung .Jalirh. 1933 Ber! in. It. Olden bourg (\ Inn i cli), 1935.

10. Taub, Josef: Load Assumpt ions for the Landing Impact of Sea

planes. N ACA TM No. 643, 1931.

Il. Seh mieden, ('.: P ber den Landestoss vor! Fi ugzeugschwinnmer,

Iiìg.-Arcliiv., Bd. X, Ili'ft 1, Feb. 1939, pp. i-13.

W'einig, F'.: Inipnet of a Vee-Tvpi Seaplane on Water with Refer

ence to Elasticity. NACA 'l'\I No. 810, 1936.

Kreps. lt. L.: Experimental Investigation of Impact in Landin on Water. NACA TM No. 1046, 1943.

Parkinson. John B.: A Complete Tank Test of a Model of t

Flying-Boat hull-N. A. C. A. Model No. 11-A. NACA L

No. 470, 1933.

\Iavo, \Vilbiir L.: Theoretical and Experimental Dynamic Load for a Prismatic Float having an Angle of Dead Rise of 22h°

NACA liB No. L5F15, 1945.

Lamb, horace: Hydrodvuantics. Sixth ed., Cambridge L'ni

Press, 1932.

U GOVO4NMENT PIPI1IN(ì FFIC '41