The vertical impact of a wedge on a fluid

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TECHNICAL NOTE

THE VERTICAL IMPACT

_{OF A WEDGE ON A FLUID}

PETER R. PAYNE Payne, Inc., 1933 Lincoln Drive,

Annapolis, MA 21401, LISA.

AbstractVon Kárusán first proposed_{a rnethød of calculating the force}

on a vertically

ins-pacting wedge in _{929. Starting with Wagner}

(193t, 1932), quite a number of authors hase suggested 'improvements' to tise theory. One purpose of this_{present note is to suggest that} many of the proposed changes

are actually incorrect and that the original

theory is superior to most of its successors. The other

purpose is io suggest a way in which the_{rigor of the Von} }(ármdn theory can be improved arid_{how the maximum pressure}

away from the keel can be calculated. Where comparison is

possible, both the keel pressure and the maximum pressare predictions agree well with

nteasarcntents by Chuang (1966).

NOTATION

Trins angle of a planing plate,

or a time parameter in the appendix. rp _{Total spray sheet area/submerged}

wedge area.

t

l)earjrise angle (Fig. I).

i

_{Local surface elevation or the}

ratio MIA!.

y

Mass density of the fluid. h _{Width or beam of a planing}

surface or wedge.

2v

Actual wetted width of a svedge.

F _{Vertical force acting on the wedge.}

virtual ntass of prism

[(Ça)

- _{virtual usais of equivalent flat plate'}

- fl/n.

Nominal (still water intersection)_{wetted lengtls.} I,, _{Actual wetted length.}

For a flat La!

I. - I

splash-up distance,

planing plate.

M.

Distance belween still water intersection and the stagnation line. es' _{Virtual water moss per unit}

length.

M

_{Wedge struclural ntass per unit length length.} m,

Water muss in one spray sheet, per unit length.

P

_'-'i-fl/it.

R _{Force normal to one side of the}

wedge or on a planing surface, R, _{The stagnation line contribution to R.}

u,

Free stream velocity of a planing plate. k

Absolute velocity of the stagnation lìue,

A velocity. .v

Horizontal distance away from the wedge vertex._«-e/e. 2y _{Nominal (still water Intersection)}

wetted width of u wedge. r

Immersion distance of thesvedge apes. ca _{R/fl yu,'b/ (for a planing surface).}

421

Ocean Engng, Vol.8, No, 4._{pp 'tt- 41f', 19s1.}

0029-80l8/8t/04042l-l6 S02,/0 Printed is Great Britain.

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422

INTRODUCTION

TitE ADVENT

of seaplanes led to an nterest in the calculation of the forces on their bottoms

during landing. Von Kármsiit (1929) provided the first theoretical solution by suggesting

that the flow around a two-dimensional wedge

(Fig. I) could be approximated by the flow

PETER R.PAsse

FIG.t. An impacting wedge.

around an expanding flat plate of half-width y =

zftan Çi. The virtual mass ni'

1/2 p it ri

associated with the expanding plate would

therefore increase as z2 and assuming

conserva-tion of momentum he wrote

Mî0

(M + m')

2tx

i C

il =

sin-

-n

X

(li

M being the mass per unit length of the

wedge structure.

From this it is easy to show that vertical

hydrodynamic force on the wedge will be

F= _-f-1(±)±

_tanL

₂

If reliable experimental data for

vertically impacting wedges had been

available at that

time, Equation (2) would have been

found to be quite accurate and much

subsequent

labor saved. But lacking such a comparison, each new worker who took up

the subject

seemed to want to add an additional

refinement.

Wagner (1931, 1932) was the first to add such complexity in papers which

made no

reference to Von Kármán's prior work. In addition to using the virtual mass

concept to

calculate the force on the wedge, he went on to use

the expanding plate analogy to calculate

the local water surface elevation, obtaining

(2)

(3)

where

tif

The vertical impaci of a ssedge on a fluid

'i-z

Fa.. 2. Wagners water nue hypothesis.

where e = ir/2 i, the actual wetted width, according to Wagner, as shown in Fig. 2.

So, said Wagner, we should consider the plate has having a width of it

i' instead of 2)'.

The present wriler fails to see why Vagner made such an arbitrary choice, or why

subsequent svorkers followed it so unquestioningly. The virtual mass concept is based on

relationships between the ss'edgc and the niais body of the fluid; not on localized

perturba-tions of the fluid close to the body. In fact, as will be shown later, Von Kármán's expanding

lamina theory gives excellent agreement with experiment on the assumption that its width

is 23. Wagner's theory therefore gives results which are (5/2)2 too large in most cases.

Wagner's "correction" to Von Kárnsán's theory was adopted by nearly all subsequent

workers who therefore had to devise further corrections to offset the errors it caused.

Pabst (1931) addrd an "aspect ratio" correction and Kreps (1943) added a "fluid-dynamic

resistance" force K p r2 which seems to the present writer to be rather like "adding in the

date", since the analysis leading to Equation (2) is already intended to give this force. We

are here dealing with a potential floss problem in which the concept of steady State

"resist-alice" has no meaning.

Excellent reviews of the gros ing body of analytical conjectures have been provided by

Mayo (1943) and by Hisplinglioff and Doherty (1950) the last being particularly noteworthy

for its combination of careful analysiswidexperimentation, Bisplinghoffcontputed the virtual

mass for an expanding wedge (instead ol' using the expanding flat plate approximation)

and achieved the same results as Scdov (1933) and Taylor (1930) of whose work he was

apparently unaware. He went front this to the computation of the water surface rise for an

expanding prism, which gives a lower splashed-up wetted width than Wagner's analysis;

ita mely

Vit

which tends to Wagner's result of ie/2 as 5 u 0. Bisplinghoff employed Equation (4) to

detine the size of his expanding prism, following the example of every worker since Wagner.

As a result, his theory predicted forces svhiclt were larger than the ones he measured

experi-mentally.

therefore,

±

M

M +m'

M-

pitztcot2t

(4) C

ittan

y

2KÇS

(3)

or, quite closely,

18' 360 450 540

f((3)

0.9422 0.8903 0.8012 0.7566 0.7164

p 90

Withf()

t, Equation (7) is identical with Equation (2).

lt is perhaps worth noting here than Von Kárrnán's result can be obtained in quite a

different way. Payne (l980b) has shown that the stagnation line force on a flat planing plate

is given by (1/b -s 0).

R,

p u,,2 ¡ it tan T. (8)

Near the stagnation line on one side of a wedge, we may regard the plate surface, of

immersed length ¡ = z/sin

as moving horizontally at a speed u, = dy/di = dz/dt. cot II

424 PETER R. PAYNE

lt is the purpose of this paper to suggest that most of the corrections applied to Von

Kármán's theory are inappropriate

THEORY

Let in' refer to the virtuat mass of water associated with the immersed (shaded) portion

of the wedge in Fig. I and s' refer to the downward velocity dz/dt of the wedge.

The force on the wedge is given by

d , di?:'

, th

F = - (in r) = e2

- + in

. ₍₅₎

dt

dz

dt

From Taylor (1930), the virtual mass of an immersed wedge s

in' = 4 p

lt y2f([±)

₄

_{p a z cot2 f(),}

₍₆₎

therefore,

din'

dz

= p it z cot2 pf(J3).

Making this substitution in (5)

F=E'( [(5)2

(7)

tane p

where, from Payne (1980a) for

The veritcat impact of a wedge on a asid

425

and a trins angle

_{r = 3. And since F = (2R,/b)}

cos p Equation (8) reduces to the

non-transient part of Equation (2).

lt can be argued that

_{Equation (7) is incomplete}

because we have only employed

_the

virtual mass associated with

_{the immersing wedge. There}

is an additional mass to be

con-sidered; that of the water

_{in the spray sheets, which}

is clearly not part of the mass in'. The

spray sheet mass (in, say) is being_{pushed normal to the wedge}

sufrisce at a velocity i/sin

_3,

so from Equation (5) there is

_{a normal force on each side equal}

to

±

dai

2

R= -,-

--sin p dt

sin

The corresponding vertical

force contribution will be

LF,

2 Rcos Ç

=_

[(±)i d

+

,

If the spray sheet

_{area is}

M,

ipzs p

2tanp'

dm,

_pipz

dz

tan'

ig being an as yet unknown ratio.

Equation (IO) then becomes

2 PZ

_[()2

+ 4 z

2J tan2 p

which is the same form

_{as (7) but with 2ip replacing irf(p).}

So the complete equation for

_the

vertical force becomes

F =

'.LL

_{[af(p) + 2g] [(±)2 +}

4 z

EJ.

tan' p

A rough idea of the

_{spray sheet area ratio g can be}

obtained from the work of Bispling.

hoff and Doherty (1950) since

_{this spray sheet area}

must be equa! to the immersed area of

the half wedge, minus the area of the water elevation.

Bisplinghoff and Doherty's

solution for the water surface elevation is

fl2KPcl+PP(p+l)p(p+I)(p+2)

_a

3.'

l0.

42

P(P+l)(P+2)(p+3)

216e

(9)

(IO)

(4)

VE

K=

r

- +

2

EI

r(1cos

\

itf

z

=(

tanit)

2

v'luan t

(I

+

F(I

+ ht)

i'

P +

Ici

Integrating (13) with respect to x to find the area of the pited up water gives

$(ii

z)dx +(c y)-

=cf Oi

-

z)dx +(c z/tan)2_

cz(Í 1) +

2Kc2

{ (. I)

-P(P+1)(1

_\

P(P+I)(P+2)Il

30 '

J

-

210 _P(P+1)(P+2)(P+3) (1

1512

U7

/

From continuity the area of the spray sheet will be the difference between this

and the

submerged area zt/2 tan 13. For computation convenience, we express the ratio (t

ci) of

elevated water area to the submerged area as

(ittan13

xtan13(l)+ntan13

{ (l)+... }

2K13

')

(ittanít

i\2

₊

istan 13

+

P(P+ I)

+

P(P+ 1)(P+2)

\2K13

i

K

13

210

(15) 1512

This is plotted in Fig. 3. lt is interesting to note that as 13 - O the spray sheet area

tends

to zero and the surface deformation is "all wave". The reverse is true of 13 -s- it/2. This s

the same trend as Sedov (1933) and Pierson and Leshnover (1950) obtained for flat p!sflhiig

plates and Pierson (1950) obtained for vertically impacting wedges.

6

N

O 10 20'

tOaO

_50' ß degrees 101o. 3.

Spray sheci and ctevaiecj wuier trois oturing wedge impaci.

Unfortunately, although Bisptinghoff

_{and Doherty's solution for the water stirface}

elevation is better than Wagner's (which

corresponds to zero spray area) it is far removed

from reality, as Fig. 4 shows. We

_{cannot therefore expect good agreement using}

ci front (IS)

in Equation (12).

O Bisplinghoff et al e,per,mentoI points

5F

Nwogner(_f_)

Risplinghoft ei. of

IO 20 30 40 Deadrise orgie l, degrees Fin. 4. _{Comparison beiween experirneni and theory of ihr}

raiio of acisat wetiesi width.

COMPARISON WITH EXPERIMENT

In Figs 5 to 9, we compare Equations

(2), (7) and (12) (using the equation of motion

from the Appendix) with the

experiments of Bisplinghoff and Doherty (1950). For all but

3 = 20 the experimental data

drops away from the theory after a certain time interval.

Roughly, this corresponds to where the wedge becomes totally immersed, but

_Bisplinghoff

does not supply enough model

dimensions for titis to be checked precisely.

g

g 60' no The veriicat impaci of a wedge on a fluid

427

426 l'Ei-PR R. PAYNE

II

dueto series truncation

N

iiytan

2K)3

2K3

where=x/c

1.4 y .2 O 0E 02

3 \Í

(14)

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428 00 200 0005 Equation 7 with Wagner splash up z-069 In.

Equation 7 spray neQiected) Expanding lamino theory

Equation 12)w,th Thea- 8 retical spray sheet moss)

0.00 0015 0020 Time, sec

Fio. 5. Bisplingholi's IO' deadrise drop test. Fon. 6. Bisplinghoff's 20' deadnise drop tes). PETER R. PAYNE 50 or -30

E -20

-io

/,

-Equotion 7

-

Espanditg lamina theory Equation 2

Fon. 7. Bispllnghoff'S 30' deodrise drop test.

0005 OOiO 0015 Time, sec 0.020 05 .0 05 20

Tite vertical impact of a cscdge on a fluid

Equation 7

/1 Expanding

arrisa theory -- Equation 12 001 0.02 003 004 005 006 Time, sec

Fon. 8. Bisplingholì's 40 deadrise drop tesi.

I

- --

Equation 7 Expanding lamina thnory

Equation 2

002 006 006 Time, sec Fici. 9. Bisyliaghoff's 50' deadnise drop tesi.

000 01

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t'lo. Il.

of the still 0.5

Fia. 12.

0.ad,b5i OnSIC P, decree'

Fia. li.

Fio. 12. A comparison between Ap pV, and the experiments of Chuang for

:

Fia. 13. A comparison between Ap jpV,' and the experiments of Chuang or

3 45 6

0rn hei5ht, in.

Fia. 14.

The vertical impact of a i'edge on a fluid

7.5

-

Wngner

---

\.---Present Theory (cnnstnnt velocity) FIa. 15.

Fia. (4. A comparison between Sp lpV,' and the experiments of Chunng for

_6.

Fio. (5. A comparison between Ap f _{Ivy,' and the experiments of Chaang for}

3,

expanding lamina approximation is closest to reality, At large deadrise angles,theespanding

lamina over-estimates the virtual mass cotitribution by roughly the same amount

_{as the}

quantity of water in the spray sheets. lt

would be intellectually satisfying to and

a way of

calculating the additional spray sheet

_{mass, as this ittight lead to some increase in precision.}

But lacking this, the theory is still

adequate for engineering purposes.

lt follows that tIte various "improvements"

_{which have been proposed are either}

unnecessary because they have little effect, or harmful (like Wagner's splash-up factor)

because they degrade the theory's accuracy.

We have also seen that Von Kárnidn's equation for the maximum pressure

_{at the keel}

is essentially correct and agrees with the

_{measurements of Chuang (1966) for low deadrise}

angles. Equation (16) gives a slightly better fit to the data, and this difference [due to

_1(p)]

is probably important at the higher deadrise angles for which no data is available.

TIse nsaximum pressure

_{ph",2 also agrees well with that part of Chuang's}

data where

the model acceleration was reasonably low during the impact. Since he did

not measure

and report force, acceleration and velocity

_{during impact, and silice his scatter is rather}

substantial, a meaningful comparison

_{with his low deadrise data would require}

quite

involved calculations beyond the scope of this paper.

REFERENCES

BisPLIrJaHoFp, R. L. and DonEcry, C. S. 1950.

A two-dimensional study of the impact of wedges on a water surface. Contract No. NOa(s)-992t, Department_{of Acto Engineering, M.I.T.}

CIIUAN0, S-L. 1966. Slanxniing of rigid

wedge-shaped bodies with various deadrise angles. David Taylor Model Basin Report 2268.

KREPS, R. L. 1943. Experimental investigation

of Impact in landing. NACA TM 1046. Msvo, W. L.

943. Analysa and modification of theory for inspaci of seaplanes on water. NACA TM 1(8)8.

4.5 6 Drop heighti in.

7.5 433 432 PETER R. PAYNE boo 400 200 loo Wuner

--'2 50 9 50 20 ₂₀ 20 Io a a a .5

fi

_a

_-

Present then,y (constant veEcityl E no. . .

3'

.1, 50'

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PABST, W. 1931. Landing impact of seaplanes. NACA TM 624. Vol. 22, No. t. Zeitschriftfils Elugtechnik unter Motorluftschillahrt Verlag von R. Ooldenbourg, Mnchen und Rerlin.

PAYNE, P. R. 1980a. The floreta1 force on a planing surface. J. Hydronaut, to be published.

PAYNE, P. R. l980b. The normal force on a flat planing plate, Including low lengtit to beam ratios. Ocean Engng. 8, 25 l-257.

PIERSON, J. D. 1950. The penetration of a fluid surface by a wedge. Stevens Institute of Technology Report No. 381.

PIERSON, J. D. and LES1tNOvER, S. 1950. A study of the flow, pressures, and leads pertaining to prismatic vee-planing surfaces. Steseos Institute of Technology Report No. 382.

SEDOV, L. 1933. Outline of the theory of impact in the landing of a seaplane. Technil,a Vurdahmmoga Flota, No. lO, pp. 120-124.

SMIt.EY, R. F. 1951. An experinsenlal study of water-pressure distributions during landings and planing of a heavily loaded rectangular flat plate model. NACA TM 2453.

TAYt0R, J. L. 1930. Some hydrodynamnical inertia coefficients. Phil, Mag. 9 (55). VON KARMAN, T. 1929. The impact of seaplane floats dating landing. NACA TN No. 321. WAGNER, H. 1931. Landing of seaplanes. NACA TN No. 622.

WAGNER. H. 1932. The phenomena of impact and planing on water. NACA Translation 1366, National

Advisory Committee for Aeronautics, Washington, D.C. ZAMM,

Vertical aedge impact equation of flint Ion

F

t' n :1(8)

+

(per unit length)

tan R

tanfi

= (dynamic reaction) + (hydrostatic force).

Thus the equation of motion IR

[M.

pxz'f(8]2 pBf31

pg z' =gAl,.

tan' P tan'P tan p

Let so that l.et Tlteu M, tan' P (f1')

(y +z')J +2z(±)' + 2:Rz. =4?.

n

/

t (fIl setr').

APPENDIX

d'z

¡dz \'

y nf(P) M, tan P

(y + Z)

+ 2z

+

= 2 tan 13

1114 (),/(1

a)

for [z

2(fl

=

): - 2s/

4-2 ian"

For the general case, Equation (6) must be solved numerically, but the cases of small hydrostatic foECe

(z g 2 (dz/dr)'( and small dynamic force (z g 2(dz/dr)') can be solved ir, the phase plane if the KIlS. ts

neglected:

(AS)

for

I

_/dz',.

So, in both cases, _{1/4 Iv-j sa untque function of z'/y} tris also possible, for small time

parameters e to truncate a series solution for_{Equation (6) to obtain}

so that

)sotiz., =v,)

lien

(Alt

_{j. F-dz}

y 2(th)']

(dz\

_{n' (dz}

drJo L y dr dz

¡dz\

4e' ¡dz dr

dr/,L

y o

- 12e' ¡dz

dr'

_-

o

However, r os generally too large for this to be useful.

Energy nuit,, irin liVe assume

z z,,, sin f), z,,, sin 0

J

(h., ros (ii - liz,, cou I)

2 - hi' z,, sin h),

_-

_{' z,, sin 0,}

13

M,i.,'

- j' Pdz

_j'

Fz,,cosfldO

j F2 dz

i

2 tan' 13, Z dz

p Rf(9) (3' _I

-

2 tan' 13 sin' O ros O do Pttf(P) (1' z.,' p nf(Ç3) I ')' d

_I

_{cos' OsinO dO}

tan'ß

J p ,tf(fl) (3' z,,' 4 tan' R (AtO) (A9)

434 _{The vertical tmpact of a wedge}_{on a huid}

435 PETER R. PAYNE