1 s2.0-002980189090003 o-main

Ocean Engng, Vol. 17, No 3, pp. 201-233, 1990. 0029-8018/90 $3.00 + .00 Printed in Great Britain. Pergamon Press plc

b CDC CLb Cp Cn f (A) m' m"

PLANING AND IMPACTING PLATE FORCES AT LARGE TRIM ANGLES

PETER R. PAYNE

Payne Associates, 108 Market Court, Stevensville, MD 21666, U.S.A.

Abstract--Added mass theory has been shown to give excellent agreement with experimental measurements on planing surfaces at normal planing angles [e.g. Payne, P.R. (1982, Ocean Engng 9, 515-545; 1988, Design of High-Speed Boats, Volume 1: Planing. Fishergate. Inc., Annapolis, Maryland)] and to agree exactly with more complex conformal transformations where such a comparison is possible. But at large trim angles, it predicts non-transient pressures that are greater than the free-stream dynamic pressure and so cannot be correct. In this paper, I suggest that the reason is because, unlike a body or a wing in an infinite fluid, a planing plate only has fluid on one side--the "high pressure" side. So the fluid in contact with the plate travels more slowly as the plate trim angle (and therefore static pressure) increases. This results in lower added mass forces than Munk, M. (1924) The Aerodynamic Forces on Airship Hulls (NACA TR-184) and Jones, R. T. (1946) Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the Speed of Sound (NACA TR-835) originally calculated for wings and other bodies in an infinite fluid.

For simplicity of presentation, I have initially considered the example of a triangular (vertex forward) planing plate. This makes the integration of elemental force very simple and so the various points are made without much trouble. But the penalty is that there seem to be no experimental data for such a configur.ation; at least none that I have been able to discover. But at least the equations obtained in the limits of zero and infinite aspect ratio, small trim angles (~) and "r = 90 ° all agree with established concepts and the variation of normal force with trim angle looks like what we would expect from our knowledge of how delta wings behave in air.

I then employed the new equation to calculate the force on a rectangular planing surface at a trim angle -r, having a constant horizontal velocity u,, and a vertical impact velocity of k. This happens to have been explored experimentally by Smiley, R. F. [(1951) An Experimental Study of Water Pressure-Distributions During Landings and Planing of a Heavily Loaded Rectangular Flat Plate Model (NACA TM 2453)] up to trim angles of "r = 45 °, and so a comparison between theory and experiment is possible. The results of this comparison are encouraging, as is also a comparison with the large trim angle planing plate measurements of Shuford, C. L. [(1958) A Theoretical and Experimental Study of Planing Surfaces Including Effects of Cross Section and Plan Form (NACA Report)].

As two practical applications, I first employed the new equations to calculate the "'design pressures" needed to size the plating of a transom bow on a high-speed "Wavestrider" hull. The resulting pressures were significantly different to those obtained using semi-empirical design rules in the literature. Then I used the theory to critically review data obtained from tank tests of a SES bow section during water impact to identify how the "real world" of resilient deck plating diverged from the "model world" of extreme structural rigidity.

N O M E N C L A T U R E b e a m the cross-flow force coefficient, which in this analysis turns out to be equal to unity R cos T/~pu~b z np/qo (= Cn) the average pressure coefficient R/qo S, the normal force coefficient on wet ted area an aspect rat io correc t ion which fairs s lender wing theory into two-dimensional theory added mass pe r unit length added mass (per uni t length) at the m ax imum span or trai l ing edge

201

2( 2 P.R. P,x~qf:

P p~ qo

R S

Svv U

bl o

V

X

Y Y Z

O~

"7 z~p

p 'T

4,

static pressure ambient static pressure lp u~,, the dynamic pressure associated with the horizontal velocity ,,, the dynamic pressure associated with the absolute velocity (u7 ~, v ~7:,) at the momenl of impact force normal to the body axis body plan area viewed normal to its major axis. In planing, the nominally "wetted" area made by the intersection of the undisturbed water surface and the plate actually wetted area of a planing surface (S~ > S because of splash-up) velocity parallel to a body's axis (Fig. 1) free-stream velocity or plate horizontal velocity at impact velocity normal to a body's axis (Fig. 1) distance along the body axis (Fig. 1) body half width maximum body half width, or span draft vertical velocity dz/dl vertical acceleration dez/dt 2 angle through which a stream tube of air is deflected by a finite aspect ratio wing sin "r + ~/u~, cos 1", an "'equivalent planing angle" p - p - , vertex half angle of a triangular wing or planing plate mass density of the fluid trim angle (Fig. 1) an unknown constant which is found to be equal to C~r.

I N T R O D U C T I O N - - T H E BASIC P R O B L E M

To CALCULATE the normal force on a slender body, Munk (1924) resolved the free- stream velocity into a componen t v, at right-angles to the body ' s axis and u, parallel to the axis. The plane normal to the body is a form of Trefftz plane, but through the body instead of behind it, as in the original work of Trefftz (1921). Munk (1924) then assumed that the cross-flow in any given Trefftz plane was two-dimensional (not influenced by the cross-flow at o ther longitudinal positions) so that the elemental force dR could be expressed as

d , dR = dt (m v) dx (1)

where m ' is the two-dimensional added mass for the cross-flow in plane x. (m' = ~rp y2 for an elliptical cylinder, for example, including the limiting case of a fiat lamina.)

0 - ~ - "

uosJn l"

O ~V = U o SJl'l "L"

FIG. |. Munk's (1924) velocity components.

Planing and impacting plate forces 203

Performing the differentiation in Equation (1)

dR dv dm' - - = m ' - - . ( 2 ) dx dt + v dt

Munk (1924) was concerned only with steady-state forces, so he ignored the transient dv/dt term. And for the remaining term, he wrote

dR dm' dm' dx dm' dx d x - V ~ = V ~ - d t = u ° s i n ~ dx dt ' (3)

On the assumption that the axial flow velocity was u = Uo cos "r, he wrote

dx dt - Uo cos "r (4)

and thus obtained

dR 2dm' 1 2dm' . - Uo ~ sin ~ cos r = ~ Uo ~ - sin 2"r (5)

which can be integrated directly to give

1 2 , R = ~ Uo mx sin 2~ (6)

where m" is the value of the added mass (per unit x) at the maximum span (Jones, 1946).

So far, we have covered familiar ground; and for small values of "r, Equation (6) is known to give excellent agreement with experiment (Payne, 1988).

The average loading Ap = /5 - t% on the body will be

R 1 2 , Uomx . A p S 2 S sm2"r (7)

where S is the area of the body projected normal to the velocity vector v. To put values into this, assume that the body is a fiat plate wing, of span 2Y, so that

m" = "troY 2 (8)

and

1 'trY 2 Ap = ~ pu~ ~ - sin 2-r.

If, for example, the planform is triangular (y = x tan 0),

y2 X 2tan 20 - - tan 0

S X 2 tan 0

and

A p = qo rr tan 0 sin 2-r

for a fully immersed triangular plate.

(9)

(lO)

204 P . R . PAVNL

r X - ,

S c h e m e 1.

For a planing plate, the added mass approaches half the value given by Equation (8) as the angle "r ~ 0, but has an additional "cavity term" (Payne, 1981) for finite angles. Let us ignore this additional term for the moment and write

~r tan 0 Ap = q,, - - - f sin 2"r for a triangular planing plate (11)

then for -r = 45 ° , sin 2 = 1, and

0 = 0 ° 10 ° 20 ° 30 ° 40 °

A p _ 0 0.277 0.572 0.907 1.318. qo

It was the anomaly of Ap > qo that presumably led to the idea that added mass theory is only valid for slender surfaces and bodies at small trim angles.

The effect o f axial velocity

The axial velocity u will actually diminish in regions of high static pressure and increase in low pressure regions. A body of revolution has both high pressure (upwind) and low pressure in its lee, so the average axial velocity is not much different, perhaps, to Uo cos -r. But the planing plate only has fluid in its "high pressure" side.

At the leading edge (of a triangular planing plate, for example) the axial velocity will clearly be u = Uo cos -~. But as a normal force develops when moving aft, the local average static pressure will increase to

dR 1 1 1 A/~ = dx 2y = 2 pu~, cos: -r - 2

by Bernoulli. So, from Equation (3)

uv dm' 1 1 ___ = 2 ~ IX2

2y dx 2 9 u ° c ° s ~ ' r - 2 9

and

vu dm' U 2 "}- . . . . . . U 2 C O S 2 T = 0

yp dx

p U 2 (12)

(12a)


SO

and

dx v dm' A ( v dm' l 2 u = I t = 2yp dx + ~1\2y9 dx ] + uI c°s2 "r. (13)

dx Substituting for d t and v in Equation (3) gives

dR 2dm' . / 1 dm' 1 dm' . - Uo ~ sin "r ~ c o s 2 'r + sin 2 "r sin "r.

2py dx 2py dx

For the triangular plate, where conventional slender plate thoery gives

m r ,/1- = ~ p tan 2 0 x 2 d m t dx - ~rp tan 2 0 x = 7rpy tan 0

(14)

1 dm' -:r tan 0.

2py dx 2

Therefore,

i R 2 ' 'IT = Uomx sin • cos 2 T + tan e sin 2 T - ~ tan 0 sin -~

/ 1 2 , . ) i 7r 'rr = ~ Uo mx sin 2"r 1 + ~(tan 0) 2 tan 2 "r - ~ tan 0 tan ~" (15)

-- (Munk's result) × (Bernoulli correction).

The Bernoulli correction is plotted in Fig. 2 for various vertex angles 0. Applying this to Equation (11)

qo 2tan0s in2" r 1 + tan0 tan 2 ~ - ~ t a n 0 t a n ' r . (16)

Figure 3 compares this modified theory with Munk's, for slender triangular plates, and shows that it predicts substantially reduced pressures above about T = 20-40 °. For a trim angle of 45 °, Fig. 4 shows that the average pressure coefficient (or force coefficient) reaches a maximum of 1/2 at the limit of 0 = 90 °, instead of the obviously incorrect value of infinity given by the Munk theory.

The cavity and cross-flow complications

So far, we have treated the problem in a very simple way. In fact, there are additional normal force components due to the added mass of the cavity and to the cross-flow, which gives additional normal force components, so that we then have

dR dm' dx - uv ~ + Cocqo2y sin 2 T (17)

206 P . R . P A v ~

0.8

o o , \ " , 4 , \ \

0

0.2

0 20 ° 40 ° 60 ° 80 ° TRIM ANGLE ~-

F=G. 2. The Bernoulli correction to Munk ' s (1924) equation, as a function of trim angle, for triangular planing plates; see Equation (15).

CR = ZX5 qo

0 .6

MUNK'S (1924) THEORY ( ~ fen e sin 2 r )

PRESENT THEORY, EQ. (16)

L INEAR APPROXIMATION

0.4

0 .2 S

0 = 15 °

" \

\ \

O 20 ° 4 0 ° 60 ° IO ° TRIM ANGLE "t"

Fl6. 3. Average added mass pressure component as a function of trim angle for triangular planing plates.


MUNK (1924) c.= ~ -~ ton e q.

2.5

2.0

1.5

/ I.O /

/ PRESENT THEORY EQUATION 06)

O 20* 40* 60* 80* PLATE VERTEX HALF ANGLE O

FIG. 4. Average added mass pressure at • = 45 °, as a function of plate vertex angle.

where C o c is the cross-flow coefficient, normally assumed to be the value determined by Bobyleff (1881) = 0.88 for a flat plate with a free-streamline separated wake in an infinite fluid. Also we have, for the added mass

where

m' ~_ 2 pyZ (1 + sin "Of(A)

is an empirical "aspect ratio" correction obtained by Payne (1981). The local static pressure will now be governed by

d R 1 1 2 1 2 - uv din ' AI~ - ~ ~y - d~22 pUo - 2 pu 2y dx + C o c qo sin2 I" (18)

where the value of + was cos 2 "r before, but is now initially unknown. Re-arranging Equation (18) gives the quadratic relationship

v dm' u 2 + - - ~ - u + UZo ( C o c sin 2 "r - + ) = 0 . ( 1 9 )

PY

2118 P . R . PAYNE

If we use +qo = qo cos 2 T as the upstream dynamic head, Equat ion (19) would give u = 0 at ~ = tan ' (1/X/Q)c-) or "r = 46.38 ° for Co~. = 0.88. This is obviously unrealistic. If + = 1, u > 0 when "r = 90 ° . The only value which gives u --~ 0 as -r --~ 90 ° is 6 -- CDC. But then, in the limit "r --~ O, u/uo --~ X/G~.. So the only value of Co<, which gives sensible results at both angular limits is CDC = 1.0 (Fig. 5).* In what follows, we shall retain Co<, instead of replacing it with unity in order to preserve generality.

The solution to Equat ion (19) is, therefore

v d m ' ~/(2~y d ~ ' ) 2 u = - - + + sin 2 ~') (20) u,2CDc (1 - 2py dx

Making the substitution in Equat ion (17)

-dx = 2yCDcq°s in2"r+vd~ ' 2py + bloCD( . 2 ( 1 - s i n 2 . r ) - 2pyV . (21)

For the tr iangular planing plate,

m' "rr ( 2 s i n [ t = ~ p t a n e 0 x 2 l + ~ t a n 0 / f ( A )

dm' ( 2 sin "r \ ~lx = rrp tan 2 0x 1 + 3 ian O) f ( z ) (22)

U

U o C0$ T

0.8~-

0.6

0.4

0.2

I i

i

I

2 0 * 4 0 o

TRIM ANGLE T °

\ 6 0 * 8 0 *

U FlG. 5. The ra t io -- for a flat plate . CDC = 1. The circles are cos "r, so u =u,, cos 2 "r in this example .

/'/o COS I"

• dm ' E q u a t i o n (19) gives u = u,, cos , , if ~ = 0, so the di f ference is due to the added mass t e rm in this example .

* Odd ly enough , Bol lay (1937, 1939) conc luded CDC = 2.0 for a wing (and hence CDc- = 1.0 for a p lan ing surface) on en t i re ly d i f ferent theore t i ca l g rounds .


and

v dm' ~r [ 2 sin 'r \ 2py dx - u o s i n ' r ~ t a n 0 11 + ~ ) f ( A ) .

Since the square brackets of Equation (21) do not contain any terms in x, direct integration gives

rr ( 2 s i n ' r ) R = CocSqosin2"r + U2o s in ' r~pY 2 1 + 3 tan-0 f (A ) ×

+ CDc(1 - sin2,r) ~ . . ~ \ 2 O y / 2fly t . ld(,

and

2 sin ,r \ CR = Coc sin 2 "r + 7r sin r tan 0 1 + ~ tanOn 0) f (A ) x

~/\~fpy + Coc (1 - sin 2 r) 2py " (23)

Equation (23) is plotted in Fig. 6 for various vertex angles. This looks very reasonable and like what we see with delta wings, except for the latter's dip in normal force between r -~ 30-60 ° because of the bursting of the air vortex structure above the wing and the concomitant loss of upper surface suction. See, for example, Fig. 7 on p. 18-5 of Hoerner and Borst (1975).

~"t"

CR I " 0 1 ~ 7 " ~ ~ C D C

o.,I o I / I

0 " 6 ~ ~ T /

0.4

I / / / / I ./ I

20 ° 40 ~ 60* 80 ° TRIM ANGLE IN DEGREES

FIG. 6. Normal force coefficient on a triangular planing plate, according to Equation (23).

21(i P. R, P.',',~t:

Note that in the limit T ~ 0

10CR tan 0 A ---> = ( 2 4 )

7r 0~ \ l + tan 2 0 \/16 + A 2

which is cor rec t in the l imits A ~ 0 and A --> ~. A l so for infinite aspect ra t io (no cross-flow or cavi ty) , Equa t ion (16) becomes

[,/ I2 ] CI~ = :2 sin 2-r 1 + tan T - 2 tan "r . (25)

As shown by Fig. 7, this agrees with small angle theo ry (CR = ~r'r). R e m a r k a b l y , ins tead of going off to some infinity as "r - ~ 90 °, the solut ion "stal ls" l ike a real wing, at a be l ievab le C~.,,,,., and "r ....... and then the force coefficients d rop off to zero at -r = 90 ° .

Rectangular flat plate with vertical velocity The a d d e d mass at a loca t ion x beh ind the still wa te r surface in tersec t ion is

a p p r o x i m a t e l y

m ('2)2( ) = ~ p 1 + ~/~ sin "r f(A) (26)

7r- (

r

o. i l

0.3 C L ~ ~

02. ~

0.1

O 20 ° 40 ° 60 ° 80 ° TRIM ANGLE IN DEGREES

Fro. 7. Predictions of the theory for the added mass normal and lift coefficients at infinite aspect ratio. (No cavity or cross-flow terms.)


where

SO

,/ 2 f ( A ) = l/ 1 +

d m t 'IT dx - 6 pb sin "rf(A)

v = Uo sin r + 2 cos r = `/Uo

where `/ = sin -r + - - cos r Uo

v dm' rr . 2py ~ - - 6 sin r `/Uo.

(say)

(27)

(28)

(29)

(30)

using the splash-up equation of Savitsky and Neidinger (1954). Because of the vertical velocity ~r, the variation of added mass with time is

~ - = 1 + 0 . 3 ~ ~ > 1

= l . 6 - 0 . 3 e b (b < 1 )

where

Once again, the square bracket in Equation (21) does not contain terms in x, so that direct integration gives

R=CDcSwqo`/a+u2o` /2 9 1 + 3b sin~ f(A)

`/sin T + CDC (1 -- `/2) _ 6 `/sin "r (31)

_ R Sw ~ b ( 4 2 . ) CR qoS - s Coc,/2 + `/~ e 1 + ~ ~ sin -r f(A)

× ~ ` / s i n "r + CDC (1 -- `/2) _ 6 ` / s i n "r (32)

which for planing (2 = 0) simplifies to

) CR = CDC sin 2 "r + ~ ~ sin "r 1 + ~ ~ sin -r f (A)

x sin 2 T + CDC (1 -- sin 2 ~) - ~ sin 2 "r (33)

+

+

II r

i

+

i i

H-

+

I J

11

f I

+ +

~ i

~ I i

II

11

II

÷

-I

H ÷


and

A R I - sin "r dt 2 p

[[ 2} 1+ ~ ~ s m ' r + l + ~ s i n v

(38)

A comparison with Smiley's experiments

Smiley (1951) dropped a heavy, one-foot wide plate into the water when travelling at a speed Uo. He read acceleration data from accelerometers whose accuracy was only --+ 0.2 g, linear to about 100 Hz. He integrated this to obtain velocity and again to obtain draft, the latter also being measured independently with a resistance bridge device. As Table 1 shows, the accuracy of the accelerometer was not particularly good by 1987 standards. His overall accuracy estimates were as follows:*

Horizontal velocity: Initial values for landings (ft./sec) . . . . . . Time histories for planing runs (ft./sec)

Initial vertical velocity (fl./sec) . . . . . . . . . Draft (ft.) . . . . . . . . . . . . . . Model weight (lb.) . . . . . . . . . . Vertical acceleration (g) . . . . . . . . . . Time (sec) . . . . . . . . . . . . . .

+ 0.5 -+1

-+ 0.2 -+ 0.03

-+2 --- 0.2

-+ 0.005.

In Fig. 8, we have compared Equations (32) and (38) with Smiley's data (by integrating the equation of motion) indicating the accuracy limits in the usual way. It seems clear that predicted acceleration velocity and immersion fall well within the experimental range, particularly as the initial impact velocity can vary by - 0.2 feet/sec.

Figures 9-13 contain additional comparisons between theory and those experiments in which more than two or three data points were obtained. An analysis of this comparison shows that theory is always within Smiley's data accuracy tolerance and that the errors are randomly distributed with respect to both impact speed and immersion depth Froude number Uo/X/~. So on this evidence, the theory seems to be correct.

The average pressures associated with these calculations are given in Fig. 14 and as can be seen, pressure not only exceeds the dynamic head ½ pU2o but also exceeds that absolute value ½ p(U2o + Zo) by a significant margin. These pressures are based on the actually wetted area.

* Smiley did not give a tolerance on trim angle setting, which is not as easy to control as one might think.

214 P. R. P.x'~Ni

TABI,E 1. SMII_EY'S I)RAFI READIN(iS

Run

R u n No. 1

Run No. 2

Run No. 3

t (scc)

0.007 0.(117 0.029 0.071

O.0O8 0 .017 O.O27 0.054 0.072 0 .086 0.091 0. 138

(}.0l(I 0 .02 l 0.030 0.056 0.073 0.085 0 .089 0 .123

Error

I).33 0.06 0.0

-0 .067

{).33 {).33 0.125 0.20 0.12 0.(155 0.1t) 0.19

0.0 0.17 0.11 0.13 0.11 0 .10 0.095 (I.042

= 60, w = 1,176 lb. Error = bridge circuit rcading-accclcrometer inlegjation

bridgc circuit reading

IC

¢.D z 0.8

z _o

~ 0.6 J

~_ 0.4

z_

~ 0 2

~A

SMILEY'S[ 0 IMMERSION DATA ~ [ ] VELOCITY (1951) k • ACCELERATION

Y

=

/ /

4~

/ / / > : /

z o

z

4 3

o 2 ~

uJ

0 .05 O. I 0.15 0.2 TIME iN SECONDS

Fro, 8. A c c u r a c y analys is o f Smi ley 's run 14, using his formal prec is ion data: 1 = 15 °, w = 1,176 lb. 4.1 f t . /sec, u,, = 39.3 -+ 0.5 f t . /sec.

SM

ILE

Y'S

/-

0

IMM

ER

SIO

N

DA

TA

t

O V

ELO

CIT

Y

(~J5

1)

• /~

CC

ELE

RA

TIO

N

1.2

~~

-o

/ "-

o °

z 1.

0 --

I0

a_

o - i

ixl

! u_

~

/ z

< q

• w

I k

/ >

0.6

--6

j LL

_z

g ~

0.2

2

0 --

0

.05

0.

1 O

. 15

02

TIM

E

IN

SE

CO

ND

S

Fno.

9.

Sm

iley

's

run

15

; "r

=

15 °

, w

= 1,

176

lb.,

~,

, =

7.7

ft./

sec,

u,

= 43

.5

ft./

sec.

SM

ILE

Y'S

/'

- 0

IMM

ER

SIO

N

DA

TA

t

O V

ELO

CIT

Y

(195

1)

• A

CC

ELE

RA

TIO

N

1.4-

1.2

~.r

~

-~

/ /

/ I.O

/ ¢

/ / 0.

8 I ,

f ! '

b r-

//

0.2

~

O

'~0

.0

5 o.

I o.

15

0.2

TIM

E

IN

SE

CO

ND

S

FIG

. 10

. S

mil

ey's

ru

n

18;

-r =

30

° ,

w

= 1,

176

lb.,

~,

, =

6.8

ft./

sec,

u,

, =

39.1

ft

./se

c.

o z 0 ,,o,

69

t~

_z

E _=.

§.

t~

5"

bO

~h

1.4

1.2

z z 1.

0 O

0.8

# I-

0.6

z_

z o 0.

4

SMIL

E¥'S

['-

O I

MM

ERSI

ON

D

ATA

t D

VEL

OC

!TY

(195

1)

• AC

CEL

ERAT

ION

0.2

J /

a..

"%,

/ /

/

f /

,/

fl

o co

_z

8

8 ,,-d

>

6 j

05

o

I O

15

o z

TIM

E

IN

SE

CO

ND

S

FI6.

11

. S

mil

ey's

ru

n 19

;'r

= 30

°,

w

= 1,

176

lb.,

2,

=

9.1

ft./

sec,

u,

, =

23.5

ft

./se

c.

z z O d g z <[

m

_z

SMIL

EyIS

/"

0

iMM

ERSI

ON

D

ATA

t D

VEL

OC

ITY

(195

1)

• AC

CEL

ERAT

ION

1.4

t.2

LO

/ /

/ /

/ 08

/

0.6

l// /

/~

0.4

/ --

i I •

/

0,

2/

~/

0

/

o111

"

\

.05

0.1

0 15

02

Ti

ME

IN

SEC

ON

DS

FiG

. 12

. S

mil

ey's

ru

n 21

: •

= 45

°,

w

= 1,

176

lb.,

5.

=

6.L~

ft./

sec,

u.

=

39.3

ft

./se

c.

z o co

~o ~

O

F. t.u

_z

r.3

S >

6 d o_

4 2

IJ

z 7R

SM

ILE

Y'S

/"

0 IM

MER

SIO

N

DAT

A LL

. 0

VELO

CIT

Y (1

951}

•

ACC

ELER

ATIO

N

1.2

m (.9

z

I.O

.J

0.8

o z g_

~ 0.

4

/ 0

0

ss

~

/ _J

0.2

'~

0 .0

5 0.

1 0.

15

0.2

TIM

E

IN

SEC

ON

DS

FIG

. 13

. S

mil

ey's

ru

n 2

2; "

r =

45 °,

w

= 1,

176

lb.,

~0

=

9.3

ft./

sec,

u,

=

25.0

ft.

/sec

.

o z 0

.~10

~ I--

z 8

- >-

q

6 j LU

ii z ~ 1.

5 tlJ

E

.~ I.c

CL

laJ 0.

5

T =I

S °

E 4

NO

° 4

22

1.14

tl~

"

21

1.03

1 "~

E

19

1.15

0 ~,

~

18

1.03

0

15

L.O

SL

0 0.

5 o.

I

0.15

0.

2 E

LA

PS

ED

T

IME

IN

S

EC

ON

DS

FIG

. 14

. A

vera

ge

pres

sure

ti

me

hist

orie

s fo

r th

e ca

lcul

atio

ns

in

1 U

2 2

I 2

Fig

s 9-

13.

Qo

=

~ p

( ,,

+

z,,)

, qo

=

~

pu,,

.

Pre

ssu

res

= to

tal

forc

e t

"--.I

2h~ P .R . PAV~.

A comparison with Shuford's planing plate measurements

Shuford (1958) tested flat planing plates up to trim angles of 34 ° and his results are compared with both the classical Munk calculation and the theory of this paper in Figs 15-19. It is unfortunate that no data are available for trim angles larger than 34 ° because the two theoretical approaches are only just beginning to diverge significantly at this angle, thanks to the dominant effect of cross-flow.

It is also unfortunate that Shuford did not measure the draft of his models , so that we have to infer it by using the splash-up equation of Savitsky and Neidinger (1954). With that caveat, the present theory appears to give a low estimate of the force below T = 30 ° and a slight over-estimate at 34 ° for l,Jb > 2.0. Figure 19 puts this in the perspective of the total angular range from 0-90 ° .

BRASS 0 WITH WINDSCREEN BRASS [ ] NO WINDSCREEN BRASS <~ NO WINDSCREEN -NO SPRAY SCREEN SHUFORD

PLASTIC ~, (1958) • WEINSTEIN 8 KAPRYAN

MUNK (1924)

CLB THEORY

0.7

o.o S/ . e j

0.5

0.4 /

0.5

0.2

OA

0 2 4 6 8 I0 WETTED LENGTH / BEAM

Fro. 15. Comparison between theory and experiment. Flat plate at -r = 12 °, C , = 18.2, C /~( = l .

BRAS

S ~

WI,~

.WIN

DSC

REE

N

"~

BRAS

S C

N

O W

IND

SCR

EEN

B

RA

SS

~

NO

W

IND

SC

REEN

-NO

SPR

AY S

CR

EE

N

SH

UFO

RD

P

LAS

TtC

~

U

958)

•

WEI

NST

E]N

8

KAPR

YAN

1.6

CLB

M

UN

K TH

EOI

11

92

4)'

~

1.4

1.2

Z~

T.

EO

I o

//

0.6

/

0.2 /

0 2

4 6

8 I0

W

ETT

ED

LE

NG

TH /

BE

AM

FIG

. 16

. C

om

par

iso

n

bet

wee

n

theo

ry a

nd

ex

per

imen

t.

Fla

t

pla

te

at r

=

18 ° ,

C

v =

18

.2,

Co

c =

1.0.

CLB

3.C

2.5 1.5-

,

1.0~

0.5 / 0

2

// /

MU

NK

TH

EO

RY

'~

(192

4) /

'PR

ES

EN

T TH

EO

RY

4 6

B

I0

WE

TTE

D

LEN

GT

H/B

EA

M

FIG

. 17

. C

om

par

iso

n

bet

wee

n

theo

ry

and

ex

per

imen

t.

Fla

t p

late

at T

=

30 ° ,

C

v =

18.2

, C

o(-

=

1.

"0

E _=..

.-t

(/Q bO

CLB

~.0

2.5

2.0 ,5

0.5

0 8

MU

NK

(i

92

4) // /2

,/

TH

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Y

4 6

8 I0

WE

TT

ED

L

EN

GT

H/B

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M

Fro

. 18

. C

omp

aris

on

bet

wee

n

theo

ry a

nd

exp

erim

ent.

F

lat

pla

te

at "

r =

34 ° ,

Cv

= 18

.2,

Ct)

c =

1.

O S

HU

FOR

D (1

9,58

) ME

AS

UR

EM

EN

TS

CLB

12

,1

" N

/ ,

,.o

/~

\

//~

I ',

\_~MUNKTHEORY

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PRES

ENT

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~~

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.2 0

20*

40 °

60 °

80 °

TR

IM A

NG

LE

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FIG

. 19

. C

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and

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2.0,

('

~,

= 1.

hJ

v


A practical application

The equations presented replicate Smiley's data to within the accuracy of his measurements, and so may be employed to calculate forces on impacting surfaces in other situations. It so happens that two of my recent designs, Figs 20-23, have bows which to a first approximation may be regarded as rectangular flat plates at trim angles of 45°! While there is an obvious urgent need to determine realistic "design pressures" for the plating of these bows, I have not been able to find anything at all helpful in the literature concerned with "design pressures" to help me with this.

To simplify the calculation, I considered only vertical motion (no pitch) with the plate impacting on a calm water surface with a vertical velocity of 10 ft./sec. Taking the weight associated with the bow to be one-fifth the total displacement, or 10,000 lb. acting on the bow, Fig. 24 shows the submerged motion of the bow at various speeds, and Fig. 25 the peak average pressure (normal force divided by area actually wetted, including the splash-up area) as a function of speed and bow plate angle. Notice that impact pressure varies as V n, where 1 < n < 2.

Figure 26 gives the variation of average pressure with bow immersion. Damping contributes a substantial hysteresis loop.

In Figs 27 and 28, we can see the effect of varying the weight carried by the plate. Comparing Fig. 27 with Fig. 26, we see that peak acceleration occurs much later in the impact than maximum average pressure; an observation first made, I believe, by Heller and Jasper (1961), " . . . the maximum effective pressure for the entire boat does not occur when accelerations are greatest". Not surprisingly, the greater the load, the larger the plate's penetration into the water. But in Fig. 28, we see that the pressures are not much affected by the load carried, only the vertical acceleration. And then most remarkable of all, the peak vertical acceleration is not much affected by trim angle.

A comparison with some SES bow panel impact tests

During the 3K SES program, a full-size bow panel was fabricated with the same scantlings as those intended for the prototype vehicle.

The model weighed 740 lb. (after run No. 20) and had a beam of (about) 44 in. Gersten (1975) describes the test set-up; Band (1977) gives some of the results for

~o = 18.76 ft./sec

:?o (= Uo) = 70 ft./sec

for • = 0.75 °, 2 °, 5 ° and 10 °. Here we have considered 2 °, 5 ° and 10 ° because of the obvious air entrainment and water smoothness problems at the lowest angle.

Considering the problems of "ringing" and local dynamic deflection locally, the agreement between the theory (of this paper) assuming that the flow separated cleanly at the knuckle and acceleration measurements (Figs 29-30) is probably acceptable. Failure to reach the "r = 2 ° peak acceleration of 59 g is possibly due to air entrainment, but it is more probably due to a soft accelerometer mount (< 2,000 Hz) or an indication that rigid body response cannot be expected of such a model when the excitation force peaks in 1 msec. Note that in all the cases, the integral of the experimental data (with respect to time) is roughly the same as for the calculated curve. Thus, the calculations must be giving realistic velocities and displacements.

IJ

IJ

l

I I

f

Fro

. 20

. L

ines

of

the

24-f

oot

Wav

cstr

ider

pro

toty

pe.

F U

Sta

-5

Look

ing

forw

ard

Sta

-3

Look

ing

aft

f ~l~

l~lb

~

ll~

. ~"~

," ~

VI[

$T

RIO

£R

> 2 r~

FIG. 21. The 24-foot Wavestrider in action on the Chesapeake Bay.

UmYEI'rRID£R

!

FroG. 22. Side and plan elevations of the 60-foot Wavestrider.

223

FI(,. 2 3 The 60-foot Waves t r ide r . Enterprise.

o~ I [ I

O6

0.5

0.1

• 0.5 0.1 0.15 0.2

TIME IN SECONDS

0.4

z

~ 0.3

0.2

TS

\ 0.25

FIG. 24. I m m e r s i o n - t i m e his tory for a ver t ica l impac t at 10 ft./sec, v = 45 °, A = l(J,000 lb., b e a m = 20 ft. F igures on the curves give speed in knots .

224


FIG. 25.

2 v 150 I

I 140 I

i / J ! r = 4 5 °

i ,// t2C I

/ /

IO0 / J'7100 I / /

I /

r / ~ ~ / / r = ~,O* .~, 8c , ' / - ~,~ -z60 I

, . 1 I ~ 4c

, y

o 20 4o 60 80 ioo SPEED IN KNOTS

The effect of speed on maximum average pressure during the impact of a fiat plate. A = 10,000 lb., beam = 20 ft., V2/50 and V2/100 are for V in knots (see Danahy, 1968).

50

40

~z 3o nn .J

~j2o

o_ IO

f J 1 1

I I

.05 0.1 0.15 0 . 2 IMMERSION IN FEET

r""~ -[.-= 4 5©

T = 1 5 °

Fro. 26. Pressure vs immersion during a 10 ft./sec vertical impact at 50 knots. A = 10,000 lb., beam = 20 ft.

226 P . R . PA'JNI-

0.5

0.4

0.3 F- W W U-

Z

~ o.z

Q

0.1

MAXIMUM IMMERSION - - I M M E R S I O N AT PEAK ACCELERATION

O 5,000 IO,OOO 15,OO0 20,000 WEIGHT tN POUNDS

r=45 o

T =15 °

FIG. 27. I m m e r s i o n for p e a k acce le ra t ion and m a x i m u m immers ion as a funct ion of load on a flat p la te impac t ing ver t ica l ly at 10 ft./sec. V = 50 knots , b e a m = 20 ft.

In Fig. 31, we compare the measured stagnation line velocity with calculations. At r = 2 °, the measurements fall substantially below the C,,/f = 1 curve, and the same trend is clear for T = 5 ° and 10 °. Assuming no errors in data reduction (the original data and analysis have not been located), this probably means that the model is "whipping" (pitching-up) during the impact.

The comparison between measured and calculated pressures is given in Figs 32-34. Once again, the original data are not available, and E .G.U. Band has remarked of the average pressures that they were ' % . . some kind of average calculated by Rohr". Particularly for 1 --- 5 °, they imply twice the acceleration measured and calculated in Fig. 29.

The maximum pressures were calculated from

p =½ova2

where Vs is the speed of the stagnation relative to stationary water axes. This speed can be calculated as follows:

Uot

Scheme 2.


70

60 ca

z

_~ 5o

r~ U.I _1

o ° 4 0 . 1 PRESSURE ,,~ p E , . ~ . ~ , ~ _ ~ . ~ K I T = 4 5 °

~ 30 z

-- pE/~K ~ : 15 °

z_ ,,, /

~.) I( - l i , l F= 15°

n.. .T=45 o (3_

0 5,000 I0,000 15,000 20,000

WEIGHT IN POUNDS

FIG. 28. P e a k a v e r a g e pressure and p e a k acce l era t ion as a funct ion of load on a flat plate impact ing vert ica l ly at 10 f t . / sec . V = 50 knots , b e a m = 20 ft.

In time t, the stagnation line has moved a horizontal distance:

z ew S = Uot + - - -

tan r t

and therefore

ds z ~w V__,Sn=dt=uo+ tan-r t "

So, if Vs were the absolute speed of the stagnation line

C e m . x - - ~ 2pUO

Band (1977) obtains a slightly larger value for the absolute velocity Vs because the stagnation line is on the plate, above the water, at a height

,)sin, 1 /

--.:~ P R. P,~YNI

t o

0 ZO

Z

Z 0 F. -

¢ 10 b.I .._1 1/.I 0 o <~

RUN 29~ T = I O °

0

©

f ©

3 0

0

. 0 0 2 . 0 0 4 . 006 . 0 0 8

ELAPSED TIME IN SECONDS .01

'4-0 RUN 3 5 , ~ ' = 5 °

3O m

",.9 Z

0 2 0 I.- ,¢(

iJ.i ._1 t,i o o <~ I 0

_ t

/ /

:3

0 .002 . 0 0 4 . 0 0 6 .OOS .01

E L A P S E D T IME IN SECONDS

FIG. 29. C o m p a r i s o n be tween theory and m e a s u r e d acce le ra t ion for r = 5 ° and 10 °.

above the undisturbed water surface. So, the stagnation line has a vertical velocity'

and

( ~ ~/2 ~2(~ )2 ~s = U o + t a n , r ~ ] + - 1

Cem~x = (1 + "~ e.,


RUN 40,T=2"

5, 4 o c~

I I i

b 2 0 i

I0.

_B

I i

1 J .002 . 0 ~ .006 .008

ELAPSED TIME IN SECONDS .OI

Fro. 30. Comparison between theory and measured acceleration for "r = 2 °.

0 ,u I 00

_---J

800 k-

6 0 0

o 400

_J

z 20C o_

Q T ' - 2 ° - - T FROM SAVITSKY E T A L

O r = 5" O / ~ T =10" . . . . . ~ = 1.0

[ ] r-

C [ ]

O

[] [ ]

~ - . T -_ IO o

T= 2 °

' ~ r = 5 °

io 15 z o ~ DISTANCE FROM TRAILING EDQE IN INCHES

FIG. 31. Comparison between calculated and measured stagnation line velocity relative to the trailing edge.

230 P. R, PA'¢NE

z

g

z

z_

0.

.5c k ! / i o apM~

[] APAv G - - J~,W FROM SAVITSKY

I . . . . . . . . . __~. I . ET AL 300 . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . "-- ~ - : ,.0

2OC ~ j - ~ .................. O~ . . . . . . . .

\ \ \ APMA x

\ \ i 0 = 15C \ ~ . . . . . . . . . . . . . . . . . . . . .

\ I

" , . ]

10c

5 I0 15 2 0 WETTED LENGTH IN INCHES

FIG. 32. Maximum and average pressures for "r = 10 °.

Reference to Fig. 31 shows that the measured stagnation line velocity was, for T = 2 °

at ~w = 2 4.8 8.0 in.

measured Vs = 760 730 400 ft./sec

sothat ½9V~ = 3930 3627 1089 lb./sq, in.

But the measured Apmax = 250 500 500 lb./sq, in. if one draws a line through the data points in Fig. 34. Possibly, the pressure transducer diameter was too large to record the peak pressure ½pV~ or possibly could not respond to the nsec duration of its presence.

This comparison shows that it is just as important to have a good theory to check an experiment with, as is the converse. In the present case, the theory checks well with


z

_z

900

800

700

600

400

\ 0 Ap MAX

J~ FROM SAVITSKY x, El" AL

,ew i-- E = t.O

-~"'-~. O!

0

0

!_1

~,<~ %.

\ 0 ~PMAx

" O

iO0 ~ "" ~, ~, E3

~ - ~ .~-vG

0 5 I0 15 ZO WETTED LENGTH IN INCHES

Flo. 33. Maximum and average pressures for • = 5 °.

\

Smiley's experiments with an essentially rigid model and then alerts us to problems and inconsistencies in similar tests of a flexible model.

CONCLUSIONS

The Munk/Jones approach to calculating the added mass normal force on a body has been modified to account for the fluid being on only one side (the bottom) of a planing plate and the fact that it must be moving significantly slower than freestream when the plate is developing high static pressures at large trim angles. The equations resulting from this modification are well-behaved at the unlikely limits of infinite aspect ratio and 90 ° trim angle, as well as for more practically reasonable values.

The difference between the new equations and classical theory is not significant for trim angles less than about 20 °. Even for "r = 45 °, the total calculated force on a planing plate is reduced by only 20% if the length-to-beam ratio is large. The effect of the modification on added mass force is large, but the total force is dominated by the cross-

_3_ P . R . PAv~,l

6,000

• 5,00C

0 4~00( - - - -

z

z_ 5,00C - - --

a . 2 , 0 0 ¢

1,000

. . . . . . . [- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- - - - - - - L . F

o i !o o o o i

5 I0 15 20 WETTED LENaTH IN INCHES

FIG. 34. M a x i m u m a n d a v e r a g e p r e s su re s for -r : 2 °.

flow force component at these large trim angles. At smaller length-to-beam ratios, where the cross-flow component of lift is less important, the new theory predicts a markedly smaller lift than the Munk formulation.

The theory explains the impacting plate drop tests conducted by Smiley to within his stated experimental error limits. It is less satisfactory in explaining Shuford's planing plate data at high length/beam ratios. This discrepancy may possibly be due to the fact that his nominal wetted length has to be inferred, using a relationship of unknown validity.

Applying the equations developed to a practical "real boat" problem, we find that few of the semi-empirical rules--rules admittedly meant to apply to conventional boat lines--for estimating "design pressure" seem to be correct for this rather unusual problem. The pressures developed are not related to the weight associated with the surface nor do they vary as V 2.

Finally, we have used theory to critically review data obtained from tests with a "real world" SES bow section, built like the real ship, and so necessarily somewhat resilient. The comparison seems to indicate that the accelerometer is ringing on its mount and too softly mounted to follow the maximum acceleration at "r = 2 °. The model seems to be "whipping" (bow up) during the impact, so that the stagnation line velocities are lower than they would be for a rigid model. Also, a combination of inadequate frequency response and/or too large a diameter results in the pressure gauges reading peak pressures (at "r = 2 °) which are an order of magnitude less than we would expect for a rigid model, and almost an order of magnitude less than they should be, based on the observed stagnation line velocity. It would have been difficult to detect these problems without the theoretical comparison.


Acknowledgements--I am greatly indebted to John D. Pierson and E.G.U. ("Bill") Band for many helpful suggestions and some significant corrections during the course of this investigation.

R E F E R E N C E S

BAND, E.G.U. 1977. Contributing to design criteria report slam and wave-induced analysis report: comparison of impact/planing methods. In Structural Design Criteria Report. Rohr Marine, Inc., Chula Vista, California (CDRL No. E02V).

BOBVLEFF, D. 1881. Journal of the Russian Physio-Chemical Society, xiii. In Lamb's Hydrodynamics, 1945, 6th Edition, p. 104, Dover Publications.

BOLLAY, W. 1937. A theory for rectangular wings of small aspect ratio. Presented at the Fluid Mechanics Session, Fifth Annual Meeting. I.Ae.S. (January).

BOLLAV, W. 1939. A contribution to the theory of planing surfaces. Proceedings of the Fifth International Congress for Applied Mechanics, pp. 474-477. John Wiley, New York.

DANAHY, P.J. 1968. Adequate strength for small high-speed vessels. Mar. Tcchnol. (January), 63-71. GERSXEN, A. 1975. Impact experiments with a full scale wet deck panel of a two thousand ton SES. Naval

Ship Research and Development Center, Bethesda, Maryland. SPD-P-534-01 (August). HELLER, S.R. and JASPER, H. 1961. On the structural design of planing craft. Trans. R. lnstn, nay. Archit.

103, 49-65. HOERNER, S.F. and BORST, H.V. 1975. Fluid Dynamic Lift: Practical Information on Aerodynamic and

Hydrodynamic Lift. Liselotta A. Hoerner, Brick Town, New Jersey. JONES, R.T. 1946. Properties of low-aspect-ratio pointed wings at speeds below and above the speed of

sound. Report NACA TR-835. MUNK, M. 1924. The aerodynamic forces on airship hulls. Report NACA TR-184. PAYNE, P.R. 1981. The virtual mass of a rectangular flat plate of finite aspect ratio. Ocean Engng 8, 541-545. PAYNE, P.R. 1982. Contributions to the virtual mass theory of hydrodynamic planing. Ocean Engng 9,

515-545. PAYNE, P.R. 1988. Design of High-Speed Boats, Volume 1: Planing. Fishergate, Inc., Annapolis, Maryland. SAVZTSKY, D. and NEIDINGER, J.W. 1954. Wetted area and center of pressure of planing surfaces at very low

speed coefficients. Sherman M. Fairchild Publications Fund Paper No. FF-11. Institute of the Aeronautical Sciences, New York.

SHUFORO, C.L. 1958. A theoretical and experimental study of planing surfaces including effects of cross section and plan form. NACA Report 1355.

SMILEY, R.F. 1951. An experimental study of water pressure-distributions during landings and planing of a heavily loaded rectangular flat plate model. Report NACA TM 2453 (September).

TREFFTZ, E. 1921. Prandtlsche traglfachen- und propeller-theorien. Z. angew, math. Mech. 1, 206.

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