The drag and side force of base ventilated surface piercing struts

-Working Paper No. 380-8R1 December 1991 TEcHNIScHE UUBU

Lab

Archiof Mekeiweg 2, 22B CD Dc Tel.: 015.788873.FalCOlS-78

THE DRAG AND SIDE FORCE OF

BASE VENTILATED SURFACE PIERCING STRUTS

By Peter R. Payne

Payne Associates 300 Park Drive

Severna Park, Maryland 21146 USA Office 410-647--4943 Fax 410-647-0954

.---J I

ABSTRACT

The drag of a basevented surface piercing strut and the side force developed by any

slender surface piercing strut when yawed are both investigated with the aid of "added

mass" theory. The equations developed are very simple and seem to agree well with the

limited amount of experimental data with which they are compared. They would certainly seem to be adequate for all normal engineering purposes.

INTRODUCTION

Shen and Wermter (1979) remark that "Because.... the flow about a surface piercing str-at is so complex, a reliable mathematical theory is not yet available for predicting the side

force characteristics and the inception of ventilation on strut.s."

This is probably still

correct, so far as ventilation prediction of the sides is concerned, but the difficulty with

sideforce calculations is at least partly due to using theoretical concepts developed in

aeronautics, which do not fit well into a free surface problem.

The simplest version of the theory developed in this paper gives the side force

coefficient of a surface piercing strut as

c

side force

Y - dynamic pressure x side area

F

-sin2acosaa

puS

immersed side area.

where a is the yaw angle of the strut, and S is the

This equation is approximately correct when the flow is attached to both sides of

the yawed strut, as in Figures 1 and 2, taken from Kaplan (1953) which show the flowover the lee or suction side. The level of the water attachment line is clearly falling as the water

moves aft over the chord, and below that line the water is still attached to the strut's

surface. But after the "stall" occurs, as in Figure 3, the flow is completely detached from

the suction face, and the side force is half the value given by the equation above.

This particular strut (Kaplan's "model 1" has ,a "circular arc" section, as shown in

Figure 4. Up to 60% of its ten inch chord the section is described by a 24.71 inch radius; behind that by one of 11.45 inches. The total thickness is 1.5 inches. Figures 5 and 6 show it above and below the surface at zero yaw, and Figure 7 presents Kaplan's drag data for these tests. The drag coefficients for the three different drafts are clearly different, due to a combination of wave making drag and spray drag. Figure 8 shows the highest speed data

points of Figure 7 plotted against immersion, after changing the drag coefficient to

drag

CDC _{= dynamic pressure} x (chord)2

It seems clear that there is a residual resistance at zero draft- usually called "spray drag" even though some of it may be due to the strut termination at the other end

_{- and}

this corresponds to

c

spray drag

0 112

DS - dynamic pressure x (thickness)2

-or about half of Hoerner's (1965) value of CDS = 0.24 (Hoerner's

pp. 10-13). Also significant is the observation that the location of the strut's maximum

thickness (at 40% versus 60%) does not seem to have much effect on this value.

2

Model #1 is the only strut which Kaplan yawed and measured the resulting side

force. At zero yaw, its drag coefficient (oli immersed side area) presumably falls somewhat be i ow the value of about C = .013 as its skin friction falls with increasing Reynold's

number.

This paper is primarily concerned with base vented struts such as Kaplan's strut #4

in Figure 4.

At high speeds, when the base is vented, its drag is no more than that of

model #1. However, the theory of a base vented strut does not seem to have been put on a firm footing. Hence the investigation which is reported here.

Figures 9 and 10 show model #4 fully vented at 23 and 35 feet second.

The

"pocket" of air behind the base obviously extends much further behind the strut at the

higher speed.

- - - -r --

_5_-__

- -, ,.-. t_ L_

-FIGURE 1.

Model #1 with end plate, 0° rake -

14° yaw; i chord submergence;

V = 9.96 ft/sec.

4L r'---

'-z. 4'

:

..:

;-::

kS

FIGURE 2.

Model #1 with end plate, 0° rake -

18° yaw; 1 chord submergence;

3

-

-

-FIGURE &

Model #1 with end plate, 0° rake -

22° yaw; i chord submergence;

V = 7.50 ft/sec.

FIGURE 4. Sections of the two struts considered in this paper.

FIGURE 5. Model #1, 0° yaw - 0° rake; i chord submergence; V = 35.04 ft/sec.

MODEL .1 CIRCULAR ARC SECTION

MODEL .4 OGIVAl. SECTION

uJ 0.040 0.030 o u, o 0.020 o z o I.. u-uJ o o 0.010 o 0.000 O 0 fr

FIGURE 6. Model #1, 0' yaw - 0° rake; i chord submergence; V = 35.04 ft/s.

KAPLAN'S MODEL 1

FROUDE NUMBER ON CHORO

IIIIERSION IN CHORDS OPEN SYMBOLS ARE FOR MAXIMUM THICKNESS o 2.0 AT 60%. THE SOLID POINTS ARE FOR THE

o 1.0 STRUT REVERSED TO PUT THE MAXIMUM

G 0.5 ThICKNESS AT 40%.

FIGURE 7. Drag coefficient on immersed side area for Kaplan's model #1 at zero yaw.

o o O _G O ç o u o a g . G 8 .0 G KAPLAN ¡9 -I0 20 30 40 50 6.0 70

ri

«J3HYflOS OIOH

6

VENTILATION DRAG

Payne (1966) suggested that transom drag could be explained by a transom suction

coefficient CDs, if we postulate that the water surface behind the transom falls below the

undisturbed surface by an amount (5) proportional to this suction like a manometer. Thus if the transom depth below the undisturbed surface is z, the static pressure at the heel is

pg(z-5)

=

_{pgzpu CDS} (1)

so the surface depression is

5-ii

_-

29CDS

or

--where F =

From the first of equations (2) we see that CDS can be estimated very simply by

putting a scale on a vessel's transom and measuring the average water depression at various speeds. The transom is ttclearll when

C

F_f>1

or F _2/CDs

The hydrostatic drag component due to this effect (Payne, 1988) is

=

1ou _CDS bz[1 4gz] C2 DS or

=C F

Dfrontal DS

when the transom is dear [F > _2/CDS]

.pgbz2 1

Dfrontal

bz F

or just half the value given by berner (1965) (his pp. 10-14, Eq. 35)

The first step in our study is to determine the value of the stern (or trailing edge)

suction coefficient from experimental data. Payne (1988) quotes values ranging from .08 to

0.125 for conventional transom stern ships and 0.17 to 0.22 for blunt forms like barges.

But no data has been analyzed for much deeper, narrower forms like surface piercing struts with blunt trailing edges.

(3) }

Analysis of Kaplan's (1953) strut drag experiments

In the absence of spray drag the total strut drag is, from equation (4)

D

_{= PVC2CDO + .pgbz2 pgb}

(z_5)2 CD = CDO +

(b/c)

(b/c) I_.l2

F L zJ

[i]2

=

(b/cJ

[CD CDO]

CDFi_Ji_

F (b/ c) [CD CDØ] (7)

CDS=_.[1_J1_ F

(b/c)

[CD_CDO]] (8)

If there was a spray drag component, this would add a term

IbasedonC

Dspray

Dspray CZ _Dspray

+pb2

to equation (6), so that the apparent value of the constant term CDO

would vary with depth of immersion.

The first step was to plot the experimental drag coefficient against Froude number based on immersion, and Figure 11 shows that this seems to collapse the data quite well, as

we would expect from equation (6).

The fact that there is negligible scatter due to

immersion depth justifies the earlier assumption that spray drag is small. The largest likely spred in CDO as between one half and two chords immersion is

CDo = .0024

which would make C .07, a figure much lower than the 0.24

Ds pr ay

quoted by Hoerner (1965)(pp. 10-13) and only 60% of the value which we deduced for

Kaplan's model #1. Presumably this is because the maximum thickness of Kaplan's model

#4 was at the trailing edge, so the local surface angle to the flow is much smaller than

model #1 whose maximum thickness was at about onehalf the chord back from the

leading edge.

(6)

(b/c) r

_{Ii!1 =C}

r2

(b/c)

CD

-. -.- ' .. ---,',----.-.. __f"_. -8 O. 10 0.08 I.J Q Q 0.05 E z o z 0.04 Q u. u-3 Q 0.02 0.00 01 02 - 05 2 5 IO .20

SOUARC OF THE FROUDE NUMSER ON I19IERS ION DEPTH

FIGURE 11. Total drag coefficient as a function of Froude number.

Now, by virtue of equation (6) the high speed data is plotted against 1/Fi in

Figure 12 and extrapolated to give CDO = .0125.

KAPLM1 (1953? DATA FOR MSOEL 4

0.050 < 0.040 Q u, Q 0.030 z o I-_z 0.020 Q I-u. "j 8 ° 0.010 0.000

KAPLAÌ4C9S3I DATA FOR MOOL 4

KAPLANS o C o oO a_E rb G _Ó o C o -O . I14IERSION IN CHORDS

02

.01

00.5 . KAPLAN2 0.00 0.10 0.20 0.30 . 0.40 0.50

INVERSE OF THE SQUARE OF THE FROUDE NUMSER ON IP?IERSION DEPTH

FIGURE 12. Determining the total drag coefficient CDO at infinite Fronde number.

O o

9

Next wè substract the skin frictiondrag component to determine the pressure drag component. Kaplan used turbulence generators set back one inch from the leading edge, measuring .05 inches in diameter and .03125 inches high, spaced at 0.3 inches.

berner

(1965, pp. 5-8) gives CD = 0.8 for such excretions and this gives a drag increment of 0.71

pounds per unit chord immersion depth at 35 feet/second. Because the first inch or so of the boundary layer is almost certainly laminar, the drag increment of the turbulators was

ignored and the skin friction was calculated by assuming that the boundary layer was

turbulent over the entire chord, using llama's (1954) relationship.

= 3.46 logia R - 5.6 (9)

where the Reynolds number R =

and c = chord

ZIO = speed

y = kinematic viscosity of the water

It was initially reasoned that if there was a net pressure drag on the section after the skin friction had been subtracted, the average local velocity over the foil's surface

would be

close

to the free stream value, and so

there was no need to use a

"supervelocity" term for the velocity.

The result of this calculation is shown in Figure 13, where it is apparent that the

pressure drag coefficient so calculated is about CD? .005

Theoretically, if there was no boundary layer thickness growth, CDP should be zero.

Now equations (4) to (6) would tell us the total strut drag if we knew the values of the

suction coefficient CDs, since CDO is now known. But when we compare theory with

experiment, in Figure 14, we see that CDS is clearly not a constant. So equation (8) was employed to derive CDS from the data, with the result shown in Figure 15.

An attempt to correlate with the aspect ratio (wetted depth/thickness) of that

part of the trailing edge which was still wet was quite unsuccessful, leading to the puzzling

tentative conclusion that CDS was a unique function of immersion Froude number. Very

roughly

K r-ans

DS F

where the data in Figure 16 supports Kt r ans 0.4.

w 0.030 z o I-z w Q w 0.010 V) w 0.000 C-0.6 0.10 .c 0.08 w w Q u) Q 0.05 z o z ! 0.04 o u. u-w 8 00.02 C-O 7 0.8 1.0

KAP 1953) DATA FOR HODEL .4

0.4 0.3 0.2 0.1 O Q. 0.00 01 02 05 2 5 10 20

SQUARE OF THE FROUDE NUMBER ON IttIERSION DEPTh IPtIERSION IN CHORDS

0 2.0

0 1.0

o o.s

50

FIGURE 14. The experimental data comared with equation (6) taking CDO = .0125.

O.

/

KAPLAN 12

0.00 0.10 0.20 0.30 0.40 9.50

INVERSE 0F THE SQUARE 0F THE FROUDE NUMBER ON Et-?IERSION DEPTH

FIGURE 13. Determining the pressure drag coefficient CDP (skin friction subtracted) at infinite Fronde number.

1.0 0.8 i-z Q 0.8 'I. Ui o t) z o I-U 0.4 Ui U, 0.2 0.0 a I.0 0.8 t.-z Ui C.)

:

0.6 It. U,_0.4 W U, 0.2 0.0

DERIVED FROM KAPLAN (1953) MODEL .4 DATA

FROUDE NUMBER BASED ON IIERSION SUBMERGENCE IN CHORDS

o 2.0

a 1.0

o 0.5

ENTERPRISE OF MIAMI DATA. REE ENTERPRISE REPORT FOR MARCH 7th )989

I Not. th.t propeller reco proxlmit>' and tho compt.tc transom shape

make thi. s much more complett problem)

FIGURE 15. The base suction coefficient as deduced from equation (8) for the Kaplan data, together with some observations made on board the 60foot Wavestrider "Enterprise

of Miami"

DERIVED FROM KAPLAN (1953) MODEL 4 DATA

11

FIGURE 16. Very roughly, the suction coefficient CDS appears to vary inversely as the

immersion Froude number, using this preliminary analysis.

s o a o 00 I . a . o o a a KAPLAN9 O G 00 0 s g - o SUBMERGENCE CHORDS 02.0 o 1.0 o 0.5

.

4/F1

/

00 lo 20 30

INVERSE OF THE FROUDE NUMBER BASED ON I??iERSION

12

Not surprisingly, in view of how we obtained equation (10) a comparison between Kaplan's drag data and equation (6) then' gives reasonable agreement, as Figure 17 shows. In performing this calculation, it was assumed that CDS could not become arbitrarily large

as F -' 0, but would be limited to the value given by berner's (1965) fully submerged

base drag coefficient (pp. 3-21 Eq. 39)

0.135

{CFB]T where 40 30 z o 20 o I0 o cl C ri '-df

For Kaplan's strut this gives limiting CDS values of the order of 0.4 - 0.5.

The agreemént between theoryU and experiment in Figure 17 is perhaps acceptable for normal engineering calculations, but clearly skirts around some of the fundamentals of the problem. It was eventually realized that the separated flow of the strut's trailing edge

was the same as the flow off the transom of a conventional planing boat, and that this flow

gave a suction force on the sides of the strut, just as it does on the bottom of the boat.

Payne (1988) called this effect "dynamic suction".

KAPLAN (1Q53) IOEL .4

FIGURE 17.

Comparison between "theory" and experiment when equation (10) is

substituted into equation (6).

o o e o KAPLAN 13 10 20 30 40

SPEED IN FEET PER SECOND

SUBRGENCE IN CHORDS

02.0 Q LO

0 0.5

DYNAMIC SUCTION

Payne (1988)(pp. 50-61) focused attention on a phenomenon that has been known

both theoretically (Sedov, Mamo, Squire and Cumberbatch, for example) and

experimentally (the inevitable low speed squatting of planing models in the towing tank)

for decades, without the two aspects of the problem being brought together and recognized as different manifestations of the same phenomenon.

Our familiar notions of ideal fluid flow break down when water flows out from under the transom of a boat. Assume that the transom heel is at a depth z below the undisturbed

water surface and that the boat is traveling fast enough for the water to separate cleanly from the transom heel. The total head pressure of the water flowing directly under the

boat is, relative to axes moving with the boat

P=p+pgz+pu

(12)

Equating this to the conditions just at the transom heel, Bernoulli's equation gives

us

p+pgz+pu20=p+pu2

r]2

Li

This speeding up of the flow amounts to a kind of ticirculationhi around the hull (opposite in sign to the circulation around a wing) which in the case of a twodimensional planing plate gives an average downward acting pressure of

'Pav =

zpg or LC

'»

Pay p 1L F 13 (13) (14)

Sedov gets k =

+

6.9 and Maruo, Squire and Cumberbatch all get similar values. Payne (1988) purports to show that this value adequately explains the observed low speed squatting of a seaknife model during tow tank testing, although more recent

work has raised some doubts about that analysis.

The same phenomenon should apply to the flow separating from the trailing edge of

a ventilated strut. The incremental suction pressure due to it, if the velocity increment across the chord stayed at the equation (13) value, would be pgz. That is to say, the strut

surface would not experience the hydrostatic pressure due to depth when at speed. From equation (13) the velocity ratio due to this effect at a depth h is

H2

UO

The average of this down to a depth h is

Í-12

=i+

I)hoJ 2

14

The strut length associated with this average is, from equation (2)

- 2g

DS

r]2

CDS

=1

+.

av

Below that, where there is water in contact with the trailing edge, the velocity ratio will be constant at

rui2

Lj

=l+Cs

Thus the overall average velocity ratio is

2

-C

avh1!] +

[1[1+cDS]

= F

! [i +

+

[1

+ C]

-.-C2

= 1 + CDs_F

_-P

=1±

'z

1F2< 2

LzÇ

1F2> 2

LzC

Determination of the suction coefficient CDS from experimental data

We may write the total drag for F as

C2

D = .pucz CDP + 2Cf 1 + 1.2

+ CDS

-+ pgbz2 pgb(z_5)2

Dividing throughout by pucz

CD = CDP + 2Cf [ + 1.2 + CDS F + (b/ c) (bi c) 1 2 F (18) (20)

I

(17)

I

(19)

Substituting equation (2) for ¿5/z

CD = CD? + 2Cf [1+1.2 ] + 2C1 CDS - Cf

Fy

+

(b/2c)

(b/c)

[ 1 F

CDS+

_1]

lCD_2

[1+1

.

2J

DP]

or

C_---C +

F DS

_[4+2Cf]

which may be solved for CDS. When F >

CDP + 2Cf

[i +

+

(b/c)

= C

FJ

F

As Figure 18 shows, this gives an entirely reasonable variation of CDS with the

aspect ratio of the submerged portion of the trailing edge. The values at low aspect ratios

are somewhat lower than the values established earlier by Payne (1966).

This is to be

expected because this early analysis did not identify the increase in skin friction due to "dynamic suction" but lumped all the drag in the suction coefficient term.

If CDS _{= CDSO + KDSA}

as in Figure 18, where CDSO = 0.1

KDS = .031

then the submerged aspect ratio (A) is CDS

A=[1_] =[i_F__]

CDS and _{CDS = CDS0 + KDs}

[1_F]

[1 +

F]

CDS = CDSO + KDS C

+K

DSO DS b K

DS Zv2

2 5.L'z

:'7

15

- .21 _.t _1 -;r'. -'-t.. r

0.0

o

spray drag D

= _{pub2 CDS}

strut drag coefficient

_{[F <2/CDS]}

CD = CDP + 2C1 [ +

i.4

+ CDS - F

+ CDS F

[F _2/CDS]

CD=CDP+2Cf

[1i.2+_L1

C

.(b/c)

FJ

F

DATA FROM KAPL.AN'S MODEL .4

Ic

=.05

L Dspray (26) Ce,- 0.1 + .031(M) 00 . . o -' - KAPLANI6 4 6 8 10 12 14

ASPECT RATIO OF PiTTED PORTION OF TRAILING EDGE

IttIERSION INCHORDS

0 2.0

D 1.0

0.5

FIGURE _{18. The suction coefficient CDS plotted as a function of the wetted aspect ratio of}

the trailing edge.

Using equation (25) for CDs, equations (20) and (22) are used to calculate the drag of Kaplan's #4 model in the following form.

16 1.0 0.8 I.. z 'J 0.6 t.-tu 8 z o 0.4 0.2

J .-;J__ - - . -

-..

.-.. .

--

j

-17

where in equation (27)

CDP is the pressure drag coefficient

1.2 b/c gives the increase in velocity due to its thickness of the

section (Hoerner [1965], pp. 6-6, Eq [7])

[CDS

- F

CS/4]

in equation (19)

gives the increase in velocity due to dynamic suction, as

The last term in equation (27) is the hydrostatic drag due to base suction.

In equation (28) the term 1/Fi gives the increase in velocity over the strut's surface, as in

equation (19) and the term (b/c)/F is the hydrostatic drag of the cleared trailing edge.

As shown in Figure 19, these equations give a much better fit to the data than

equation (6). It

is also much more satisfactory to have the base suction çoefficient

expressed as a function of wetted aspect ratio than merely an empirical function of

immersion Froude number. So we are now in a position to investigate the pressure forces.

DATA FROM KAPLANS MODEL .4 40 30 (n z o Q. 20 o io C0p .0011 Cd.p .05 SUBMERGENCE IN CHORDS o 2.0 3 1.0 o 0.5

THEORY OF THIS REPORT

FIGURE 19. Equations (27) and (28) compared with the experimental data of Kaplan

(1953).

o

40 30

o 10 20

'-

.r - '- ., ...- .- - -J

18

CALCULATION OF THE PRESSURE FRCE ON A WEDGESECTIONED STRUT

From Taylor (1930) and Sedov (1950) the added mass of a twodimensional vertical lamina swaying in the free surface is

m'

2D = (per unit length) (29)

where z is its depth in the water. Note that the corresponding value

for a twodimensional lamina moving broadside in an infinite fluid is

m2D [zJ2

so the ratio

swaying, added mass Z2 8

0.8106

heaving added mass

_!pz2

By analogy with added mass planing theory, the added mass due to one side of the swaying lamina will be half the value given by equation (12)(Lamb, 1921). Also, we will expect, by

the same analogy, that a finite aspect ratio swaying plate will have a lower added mass

than a twodimensional one, so that the value associated with one side may be written as

m' =f(A)pz2

(30)

wheref(A) has yet to be determined.

Using the method of Munk

(1924) given in Payne (1988) p. 41, and assuming that the wedge section, of half angle O

has its chord inclined to the flow at a yaw angle a, then the normal forces on its two

surfaces are

F = m'ud sin(O± a) cos(O

a) =

sin 2(0± a)

=plLo2.f(A)z2sin2(0± a)

(31)

or _CF

=

= 3

[]f(A)

sin 2(0± a) (32)

The side force component, at right angles to the flow direction, is

= .

[]f(A) [sin

2(0 + a)cos(O + a)sin2(O a)cos(O a)]

(33)

and the pressure drag coefficient is

when a = O

CD? =

[]f(A)

sin2û sinO

Since CDP does not vary with z/c, in practice we conclude that

f(A)-2

z

so that m' = !Fpcz

and _{CDP =} sin2O sinO

(per side)

A wedge having b/c = 0.15 would therefore have CDP = .0071

significantly larger than the figure of .001 used for

Kaplan's

parabolic section in Figure 19, as would be expected, because the pressure drag of a

parabolic section is theoretically zero in the absence of boundary layer thickness.

Figures 20 and 21 give the side force and drag coefficients calculated from equations (33) and (34) for three different thickness. When O = 0.

C =--sin2acosa

Y ir

CDP=sin2asin a

and these relationships may be adequate for many purposes. COMPARISON WITH EXPERIMENTAL DATA

Figures 22 and 23 compare equations (33) and (34) with Kaplan's data for his model

#1, which has a sharp trailing edge.

Because Kaplan used a twelve inch diameter

endplate on the bottom of his strut for these side force measurements (to simulate a foil?

an approximate correction for this was made using Figure 1-10 (p. 33) of Payne (1988

which uses the work of Bryson (1953) and Summers (1953) to obtain the relationship.

Force with endplate

_-

₊ 7/4h

Force without endplate

e

where y is the submerged strut length

h is half the endplate span

The measured side force agrees well with equation (33) (and therefore [38]) until the flow separates entirely from the lee side at around 20° yaw angle. In the stalled condition

the lift force is very roughly half the value given by equations (33) and (38) as would be

expected, since one side of the strut is still deflecting the flow, but the endplate is no

longer effective. 19 (35) (39) } (38)

20 0.50 0.40 I-z 0.30 u-u. uJ o C.) Ui e o u. 0.20 UI o U, 0.10 0.30 0.20 C.) U. u-Ui o C., o Ui 1 V, 0.10 a. 0.00 o ..d:)ç-.t.j;t .erM:....U. 8 = 00 50 100

YAW ANGLE IN DEGREES

FIGURE 20. Theoretical side force coefficient on a wedgesectioned strut for three wedge halfangles O.

FIGURE 21. Theoretical pressure drag coefficient on a wedgesectioned strut for three

wedge halfangles O.

10 20 30

YAW ANGLE IN DEGREES

30 20

1.0 0.8 z lU 0.6 u. I-u. o Q lU Q o 0.4 u. o u, 0.2 0.0 o 0.40 0.30 z u. Q u. 0.20 o Q 0.10 0.00 o 10 20

YAW ANGLE IN DEGREES

THEORY FOR A 10 FT/SEC. O EXPERIMENT FOR A CIRCULAR ARC

TRI ANGULAR 7.5 FT/SEC. D SECTION WITH A SHARP TRAILING

SECTION EDGE. (KAPLANS MODEL .1)

FIGURE 22. A comparison between theory and Kaplan's side force measurements with a sharp trailing edge sectioned strut.

D D D D D D o D 0

00

D o KAPLAN 14 / / / / D 8 .

_---::

₀ , ! -o .032 + (2/u) ln2 KAPLANIS sina 10 20 30

YAW ANGLE IN DEGREES

THEORY FOR A 0 FT/SEC. O EXPERIMENT FOR CIRCULAR ARC

TR I ANGULAR 7.5 FT/SEC. D SECTION WITH A SHARP TRAILING

SECTION EDGE. (KAPLANS MODEL si)

EQUATION (38)

FIGURE 23. A comparison between theory and Kaplan's drag measurements with a sharp trailing edge sectioned strut.

z____._.

22

The agreement with the drag data n Figure 23 is much poorer, as expected, because

a triangular sectioned strut has a higher drag than one whose sections is made up of

circular arcs, with a sharp trailing edge. However, equation (38) gives the rate of drag

increase surprisingly well up to where the lee side separates, indicating that our theoretical model gives realistic results.

Figures 24 through 26 compare the simplest version of the theory - equations (38)

and (39) - with the data of lolling et al (1975). In this case, the strut of the TAPi foil

system was parabolic in section, with its maximum thickness at the blunt trailing edge.

The data are quite scattered and have a marked zero offset, so the mean zero offset of each

data set was subtracted to produce the comparisons between theory and experiment in

Figures 24-26. Except for two high angle data points in Figure 25 (where we are seeing additional side force developed by the foil), it would seem that the theory agrees with

experiment to within the apparent experimental accuracy.

No strut drag comparison can be made with the TAPi data because the strut drag

is presumably small compared with that of the foil which it supports. CONCLUSIONS

The side force developed by a yawed surface piercing strut, base vented or not, is adequately given by the coefficient

C

sin2acosa-a

Y ir ir

when the flow is attached to both sides and by half that value when one side is ventilated. When there is a foil on the bottom of the strut, the attached flow

side force is increased by the amount given by equation (39). When one side of the strut is ventilated the "endp1ate't effect is zero.

It is well known that when a surface piercing strut is a rudder, or when, as is the

case with most hydrofoils, the struts used to support the foils develop sideforce in a turn,

the sudden ventilation of the lee side (and the concomitant loss of sideforce) causes

considerable difficulty with lateral control. For this reason, most high speed race boats use

rudders whose crosssection is wedgeshaped, so that ventilation cannot occur until the

rudder angle is greater than half the wedge angle. (Equation (32) and Figures 20 and 21.) Added mass theory can be employed to calculate the pressure drag of a surface

piercing strut. The results obtained in this paper for a triangular sectioned strut appear

reasonable, but have not been checked against experiment. The theory gives zero pressure

drag at zero yaw for a section (like Kaplan's parabolic strut #4) which has zero surface

angle to the flow at the trailing edge. It gives a drag increase due to yaw of

CD = sin2a sine

42

ca

which is seen in Figure 23 to agree with one experiment up to the

angle at which the leeside flow separates.

The velocity over the surface of a base vented strut is significantly greater than

conventional hydrodynamic theory suggests, due to the fact that the trailing edge is at ambient static presure, rather than the hydrostatic pressure appropriate to its depth below the free surface. Thus its drag is greater than expected, because of the increased skin

z 0.50 0.40 0.30 I-bi o Q bi o b. 0.20 bi Q Q, 0.10

I

z bi Q 0.30 b. b. bi 8 bi C.) o b. 0.20 bi o Cil 0.00 0.50 0.40 0.10 0.00 o o

TAP-i STRUT. HOLLINO ET AL (1975) IPVIERSION - 0.459 STRUT CHORDS

2 4 6

. ..:

8 IO

23

YAW ANGLE IN DECREES

FIGURE 25. Comparison between the simple theory of equations (38) and (39) with the

experimental data of Hoffing et al (1975) for an immersion of 0.918 strut chords.

t/c= 0.18.

0 80 O 70 (2/w) KNOTS KNOTS tn(2D) x ENDPLATE In(2ß) FACTOR (1/wI g D D O D 4 SPEED 070 80. 50 40 IN KNOTS

-jj41

--2 4 6 B IO

VAN ANGLE IN DEGREES

FIGURE 24. Comparison between the simple theory of equations (38) and (39) with the

experimental data of Hoffing et al (1975) for an immersion of 0.459 strut chords.

t/c= 0.18.

Ui Q o 0.50 0.40 0.30 u-- 0.20 UI o u, 0.10

FIGURE 26. Comparison between the simple theory of equations (38) and (39) with the

experimental data of Holing et al (1975) for an immersion of 1.873 strut chords.

t/c = 0.18.

REFERENCES

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Fluid Mechanics, 4, Pt. 5, pp. 466-478.

Hama, F.R. 1954. SNAME Transactions, pp. 333-358:

berner, S.F. 1965. Fluid Dynamic Drag, S.F. berner, Midland Park, New Jersey.

lolling, H., E. Baker and E. Hood. 1975. Experimental Evaluation of the Performance of

the TAPi Supercavitating Hydrofoil Model at 80 Knots, DTNSRDC Report

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Shiffstechnik, Vol. 14, pp. 57-64.

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'10

4 6

C.

25

Munk, M. 1924. The Aerodynamic Forces on Airship Hulls, NACA TR 184.

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Payne, P.R. 1988. Design of High Speed Boats, Volume 1: Planing. Fishergate, Inc., 2521 Riva Road, Annapolis, Maryland, 21401.

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Translated and edited by C.K. Chu, H. Cohen and B. Seckler. John Wiley & Sons, Inc., New York, New York. (1965).

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Summers, R.S. 1953.

On Determining

the Apparent Additional _M?.ss of a

WingBodyVertical Tail Cross Section, Journal of the Aeronautical Sciences,

Vol. 20, No. 12, pp. 856-857. (December).

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