Effects of configuration on wave resistance of multiple pressure distributions

Effects of Configuration on W a v e Resistance of Multiple Pressure Distributions

Ronald W. Yeung^", Hui VJan\ S u r y a R B a n u m u r t h y \ Wan L Ham\ and J a e - M o o n Lew^

CA 94720-1740, USA. Corresponding author: r\vyeung@berkeley.edu.

2 Department o f Naval Architecture and Marine Engineering, Chungnam National University, Taejon 305-764, Korea

A b s t r a c t

The wave resistance of multiple pressure distributions (cushions) is examined. The analytical formulation is based on a linear theory involving the n o t i o n o f interference resistance as developed i n Yeung et al. (2004). Research w o r k previously carried out by Yeung and Wan (2008) and Yeung et al. (2008) is further developed to obtain the equations of wave resistance. Wave resistance of various configurations o f dual cushions and tri-cushions are analyzed using the computational code developed at the Computational Marine Mechanics Lab ( C M M L ) at the University o f California at Berkeley. The results o f the analysis show that there is a significant reduction i n the wave resistance o f the multiple pressure cushion craft compared to a single-cushion craft o f identical displacement. W i t h the aid of the expression f o r interference resistance between two cushions developed here, the wave resistance of a dual-SES ship is analyzed and presented.

K e y w o r d s

Wave resistance; powering; air-cushioned vehicles; surface-effect ship; optimal design; low emission systems; high-speed vehicles

1 I n t r o d u c t i o n

Faster speed, yet lower power consumption, has often been the design objective o f high-performance marine vehicles such as hovercrafts, Surface-Effect Ships (SES), among others. Lower power consumption also means less carbon-dioxide emission, an issue o f great environmental concern. The concept o f a m u l t i - h u l l system offers favorable possibility o f powering reduction i n steady m o t i o n . Configuration arrangement of component hulls is therefore an important design issue to address, especially at the early stages of design.

The problems o f steady f o r w a r d - m o f i o n o f multi-hulls and SES (thin hulls w i t h a pressure cushion) were analyzed i n Yeung et al. (2004), and Yeung and Wan (2008), respectively Therein, linearized theory was used to obtain the interference wave resistance, w h i c h can be either positive or negative, increasing or reducing the powering for a given speed. Results f o r a single pressure cushion are quite well k n o w n (see, e.g., Havelock (1932), Wehausen and Laitone (1960), Newman and Poole (1962) and Doctors and Sharma (1972)). The possibility of shaping the pressure f u n c t i o n w i t h i n a cushion was considered i n the interesting work o f Tuck et al. (2002). However, the effects o f combining multiple numbers o f cushions and SES's have yet to be thoroughly explored. There have been reports (Burg, 1992) o n the use of multi-cushions to successfully improve the rides and maneuverability of SES and o ü i e r cushioned crafts. Also, SES EU AS (2008) has developed a dual-SES ship w i t h t w i n cushions. Hence, developing an efficient methodology to assess the powering performance of multi-cushions is desirable. This paper addresses the m u l t i - b o d y interaction issue i n the same vein as Yeung et al. (2004) and Yeung and Wan (2008), w i t h the aim o f obtaining the necessary interference expressions f o r rapid evaluation o f the behavior o f a pressure collection.

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya P. Banumurthy Wan L. Ham and Jae-Moon Lew

W a v e r e s i s t a n c e of m u l t i - c u s h i o n s

2.1 W a v e resistance formulation of a

mono-cushion

W i t h i n the f r a m e w o r k o f l i n e a r theory, the generalized steady wave-resistance p r o b l e m i n deep water can be s u m m a r i z e d as f i n d i n g a v e l o c i t y p o t e n t i a l , (K.v.y.z), that satisfies Laplace's equation, but is subject to the combined free-surface boundary condition (Havelock, 1932):

. p f / ^ r _ ^ M , ( X ) r

ko^, (-v. y, 0)^„ (x,y, 0) = (.V. y)/pU (1) where k„ = g/U^ and U is the forward speed i n direction x (Fig. 1) and p is the water density. Here, P(x,y) is the applied (cushion) pressure, which vanishes everywhere except i n the p l a n f o r m regions 5^. I n Fig. 1, the p l a n f o r m area o f the cushion is - 1 < .v ^ 1 and - 1 ^ y ^ 1. Conditions o f decaying disturbances as z - > -oo and the absence o f upstream waves

{x -> oo) are also to be observed. We also note that f r o m the

dynamic fi-ee-surface boundary conditions, the linearized fluid pressiurep(x,y,z) and the longitudinal free-surface slope can be derived and are given as:

p(.x,y,z) + Pix,y) =

-gC,(.v.y) = t / 4 . „ ( . v , > ' , 0 ) - / ' , / p

(2) (3)

The velocity potential ^p{x,y,z) can be given i n terms o f derivative o f the Green function, G, as

= -^\\P{^„^)G,{x^^y^•z,0)dk^^ (4)

after p e r f o r m i n g an integration by part i n ^. The Green f u n c t i o n , G, is given by Havelock (1932): 1 , 1 , 4A-„ ,''12 sec'e ( 5 ) G(x-i,-y~j\-z,0 = —+-+-r i\ TC

\ -.—cos[A-(.v )cos9 ]cos[A^(>' - T ) )sin9 ]

0

X sin[^o (.V - ^ ) secO ] c o s [ k f , (>< - r i ) sin9 sec' 6 ]

= G,+G^

where G^ and G„. denote the terms that are symmetric (the first three) a n d a n t i - s y m m e t r i c w i t h respect to ( . v - ^ ) , respectively. In the above expression, ,-=.J(x-^)' +(y-r\f +{z-C,f and =^(x-S,y+(y-r\f+(z+Q\ The wave resistance induced by a m o v i n g cushion is given by the integral o f product o f the pressure, P, and the free-surface

(7)

^vhere Ap(k), the complex wave-making ampUtude (or the Kochin) function, is given by:

MX)=^llP(^,n)e''''^^^-^^^d^dn _{( 8 )}

In arriving at Eq. (6), we note that there was no contribution f r o m . Further, f r o m Eqs. (6), (7), and (8), we observe that

Rw^ depends quadratically on P, i.e., a reduction o f 50% i n P

will lead to a saving of 75% i n the wave resistance. So, smaller P is favored i n terms o f reducing wave resistance. However this reduction w i l l lead to an increase i n the planform area o f the cushion f o r a fixed displacement.

Fig. I Pressure funclion P(x,y) of a cushion (a=5, (i=20) in

unitized variables: x-2xIB= 'lyIL^.

A popular pressure cushion p r o f i l e o f peak P^ value that is i n f i n i t e l y d i f f e r e n t i a b l e i n the h o r i z o n t a l plane was proposed by Doctors and Sharma (1972) (shown i n Fig. 1) and is given as:

P{x, y) = -^ [tanh(a ( ? +1)) - tanh(a (x - 1 ) ) ]

4

x [ t a n h ( P ( y - H ) ) - t a n h ( P ( 7 - l ) ) ]

(9)

This hyperbolic tangent f o r m w i t h the tapering parameters a and ^, in the longitudinal and transverse directions respectively, leads to a closed f o r m expression f o r ApQ.) (see Yeung and Wan, 2008). Also, note that i t is convenient to express P^^ i n terms o f hydrostatic head, P„,l pg ( m ) , where p is the density of sea water and g is the acceleration due to gravity.

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya P. Banumurthy, Wan L. Ham and Jae-Moon Lew

2.2 Interference resistance of dual

cushions w i t h separation and

stagger

U

y i

Fig. 2 Coordinate systems: shown for two pressure cushions with separation and stagger.

In the case of two pressure cushions with separation and stagger, as defined in Fig. 2, the total wave resistance on the two pressure cushions is not only the sum o f the resistance due to pressure P^{x,y) (i.e., and pressure P^^P^^ individually, b u t also o f an interference t e r m w h i c h cannot be ignored. This term accounts f o r the effect o f pressure 1 o n pressure 2 as well as the effect o f pressure 2 on pressure 1 (^itfl^ft)' or effectively, the superposition o f the wave-interference effects o f each o f the surface distribution i n the field. Following Yeung et al. (2004), we can establish;

- K . +Rw.

(10)

where ^ i r ^ and are each given by Havelock's equations (1932), or Eqs. ( 6 ) , ( 7 ) , and ( 8 ) here. To c o m p u t e the interference resistance, consider the t w o local frames o f reference, Oix^y^z^ and 02X2y2Z2 P'g- 2- Using Eqs. (3), (4), we can write the expression o f the interference resistances ^ • O i - f / i (pressure 2 acting on the wave slope at cushion 2 generated by cushion 1) and ^ n ' ^ ^ ^ (pressure 1 acting o n the wave slope at cushion 1 generated by cushion 2), i n a manner s i m ü a r to Eq. (6), as:

T h e n c o m b i n i n g Eqs. (11) and (12), and m a k i n g use o f .Vj = * i - St, y j = ^1 - sp, and Z2 = ^i(Fig. 2), we can show that the summed interference resistance on the two pressure cushions can be written as:

R - _{J | r f v i r f > . , P}

, ( > ^ l . > ' l) j J/ ' 2 e 2 . 1 l 2 )

^'^ ^ (13) X V , (•'^1 - s t - ? 2 ; >'i - s p ^ i 2; 0, o)fi/5 2'''i 2

O f interest is that only G„, survives i n this summation. O n substituting the expression f o r G„^^, Eq. (13), is written as:

ƒƒ ''^2'^2^2e2.Tl2) ƒ sec'e cos[Ao(.Vi-st^2)sece]

xcos[Ao(>'i-sp--ri2)sinO seo^6](ie

Equation (14) is still unwieldy. However, i f both cushions are symmetric about their o w n A'-axes, i.e. port and starboard symmetry, the result simplifies greatly i n a manner similar to the h u l l - t o - h u l l interference problem o f Yeung et al (2004). Under this assumption, i n Eq. (8) can be written as:

^p(X) = ^ J j. / ^ ^ P g . T l ) . ' V 4 TCpf/ s

xcos(ko}Jl^ -h]), f o r P(x,y)=:PCx-y)

and the interference resistance is given by:

(15)

00 dX cos(A-QspXVx^ -1)

W>.'-i

; miApAp^ )cos(A-o>.st) +5(Ap Ap )sin(Ao>'St)}

(16)

Here, 9? and 3 denote real and imaginary parts, respectively and

A is the complex conjugate o f the Kochin function. Similarly,

i f the pressure cushions are symmetric about their ^-axes, i.e. fore-aft symmetry, then Eq. (8) can be written as:

"Pi "n • Gtj.vj.,j ( x 2 - t « K i ; y 2 + s p ^ i i ; z 2 . 0 ) ^ ^ ^ ^ = ƒ{ ACvi,>',>;2.vCvi.>'i)rfv,<^, fy2 = - l J J' ^ - l 4 ' l/ l ( v„ > ' , ) | j r f 4 2 < ^ l 2 / ' 2 e 2 . T l 2 ) X G,, (A;i-st-42;>'i-«p^i2;^i.o) (11) (12)

xcosCAflX^), for P(^x,y) = Pi-x,y)

and the interference resistance is given by,

R,. -o-pr.2p^>-cos(AostX)

x{3?(/lp^ Ap^)cosikgX-Jx^-lsp)

+:S(Ap^ \ )sin(AoXVx^-lsp)}

(17)

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya R Banumurthy Wan L. Ham and Jae-Moon Lew

The pressure distribution used here is given by Eq. (9) on a rectangular planform (Fig. 1) has both port-starboard and fore-aft symmetry. Thus, in this case, the Kochin function is purely real and is given as:

(19) - h i )

and the interference resistance simplifies to:

11

-lsp)cos(ArflStX.)

(20)

Equations (16), (18), and (20) show explicitly how the stagger and separation behveen the pressure cushions can influence the total wave resistance. These new expressions can be computed concurrently w i t h the mono-cushion resistances, 7?„y,7= 1,2 given b y E q . (6).

2.3 Arbitrary number of pressure

c u s h i o n s

The expression f o r the interference resistance benveen two cushions can be used to analyze a family o f multiple pressure cushions. Suppose there are n cushions, labeled 1 to n , then the total wave resistance o f the system is given by:

1=1 1=1 j=i+i

(21)

I n particular f o r the tri-cushion case (Fig. 3), the total wave resistance o f the tri-cushion is given by:

+l^p,^Pi +l^p^<^p^ (22)

sp,'2

L p i

2.4 W a v e resistance of a d u a l - S E S

Fig. 4 Configuration of a dual-SES system.

The wave resistance o f a SES was given by Yeung and Wan (2008) as a sum o f the wave resistance o f the two demi-hulls which contain the cushion, the wave resistance o f the cushion, the i n t e r f e r e n c e resistance o f the d e m i - h u l l s a n d the interaction wave resistance of the cushion and the demi-hulls. Now, w i t h the availability o f cushion-to-cushion interference expression, we can compute the wave resistance o f a dual-SES seen i n Fig. 4. The expression for the wave resistance o f a dual-SES can be written as:

R]- -RsES^ +PsES2

+ ^ / / , « / / 3 +RH^**H, + % 2 « / / 4 +RHi**H,

+ ^^//,«P2 + ^ / f 2 « . P j +-'^//3«P, +PHti^P, (23)

+ R

The wave resistance o f the SES's and the cross hull-to-cushion i n t e r f e r e n c e resistance {RHt<->P,-RH2<r^p^-RH,<->P,, a n d « i j ) can be computed as i n Yeung and Wan (2008). The h u l l - t o - h u l l i n t e r a c t i o n between the h u l l s i n the SES's ( % , « / / 3 >-^//,«//4 'PH,<^II,> and Rff^^t^H,) can be computed using the expression given i n Yeung et al. (2004). Finally, the cushion-to-cushion interference resistance Rp^ is computed using Eq. (20).

The equation for the hull-to-cushion wave resistance, requhed to compute the resistance o f the dual-SES, developed i n Yeung and Wan (2008) (Eqs. (24) and (25)) appears very complex. However, i t can be simplified by using the Kochin f u n c t i o n developed f o r a cushion w i t h port-starboard symmetry (Eq. (15)). The interference resistance equation between the h u l l and the cushion can be written as:

,,2ldXcos(kaspX-Jx^ - 1 )

Fig. 3 Configuration of a tri-cushion craft. ^{Ap Ai) sin(AoStX) -I- S^Ap 4 ) cos(i^oStX)}

(24)

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya R Banumurthy, Wan L. Ham and Jae-Moon Lew

where, is the Kochin f u n c t i o n o f the h u l l (see Yeung et al. (2004) or Eq. (7) o f Yeung and Wan (2008)) and sp and st are the separation and stagger bet^veen the cushion and the h u l l .

3 R e s u l t s a n d d i s c u s s i o n

3.1 Dual-cushion s y s t e m s

First, a dual cushion configuration of pressure cushions (Fig. 2) is studied as a function o f varying speed, separation, and stagger. For Bp I Lp = 0.5 , we compare the performance o f the dual cushions, each o f peak pressure P^^, against a m o n o cushion o f the same displacement and p l a n f o r m . The m o n o -cushion resistance (shown i n Fig. 12), therefore, has a pressure o f 2 P „ , applied over the same "footprint" as the individual cushion i n the dual-cushion system. We note that the wave resistance o f each o f the demi-cushions is 25 % o f the mono-cushion resistance, since the wave resistance varies quachafically with the nominal pressure in the cushions. Thus, the effect o f speed, sp, and st o n the interference wave resistance is directly related to the total resistance curves since the non-dimensional total resistance (Rj IRf^ o f the dual-c u s h i o n system w o u l d be the i n t e r f e r e n dual-c e resistandual-ce ( ^ f l « B , / « o ) plus 0.5.

Rr

^0 1 K

F i g . 5 R^IRt, vs. F„ and sp/B^ for dual cushions in parallel ( s t= 0 ) .

Figure 5 shows the t o t a l wave resistance relative to R^ f o r d u a l cushions i n a p a r a l l e l c o n f i g u r a t i o n . S i g n i f i c a n t interference drag occurs w h e n s p / B p is about u n i t y and F„ is b e l o w the f i r s t resistance h o l l o w o f the m o n o c u s h i o n ( F i g . 12). N o t e that f o r large F„ or sp, the d u a l -c u s h i o n resistan-ce approa-ches the expe-cted value o f 5 0 % o f that o f the m o n o - c u s h i o n . Also, the wave resistance r a t i o , Rj^ /RQ , never exceeds 1, i.e., the wave resistance o f the dual cushion i n parallel c o n f i g u r a t i o n is always smaller t h a n that o f the m o n o - c u s h i o n .

Fig. 6 Rj. IRfj vs. F„ and st/i;> for dual cushions in tandem (sp=0).

The corresponding results o f having the dual cushions i n tandem ivith st being varied are shown i n Figs. 6 and 7. I n Fig. 7, Xg is the m a x i m u m (transverse) wavelength o f the Kelvin wave system. The oscillatory patterns i n the interference resistance ( F i g . 6) are m o r e c o m p l e x i n the t a n d e m configuration. A t lower F„, the transverse waves generated by the cushions interact and leads to a series o f "humps and hollows". This is clearly seen i n Fig. 7 where the humps and hollows are systematically one-unit value o f st / XQ apart. Note that i n Fig. 7, st is bounded by/.p and KLp, w i t h K i n this study selected to be 11. The equation o f bounding contour f o r st is given by: st / XQ x F „ ^ = K / 2jt • Besides that, a valley o f l o w total drag occurs f o r a combination o f F^ and st / Lp. This valley extends to larger values o f st / Lp. Also, the oscillatory patterns disappear at high speeds except at small st, since the cushions generate primarily divergent waves at high F^, (Fig. 6 ) .

Fig. 7 Rj. /Tij vs. F„ and st/?.^ for dual cushions in tandem ( s p= 0 ) .

Next, the effects o f varying b o t h stagger and separation are shown in Figs. 8 and 9 for F„= 0.4, which is a point of m i n i m u m -resistance ratio i n Fig. 5, and f o r a higher Froude number, F „ = l . These p l o t s p a r a l l e l the s o - c a l l e d W e i n b l u m configurations o f di-huUs. The behavior at the two speeds is drastically different, but Rj- /Roof 20% is achievable f o r a wide range o f sp - st combinations. A t lower speed, Rj-1 RQ displays more oscillations w h i c h can be attributed to the interaction of the transverse waves. Also, at sp = 0 and st = 0, the demicushions overlap and is equivalent to the m o n o -cushion. Hence, the wave resistance ratio goes to 1.

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya R Banumurthy Wan L. Ham and Jae-Moon Lew

Figures 10 a n d 11 i l l u s t r a t e s a possible d u a l c u s h i o n c o n f i g u r a t i o n w i t h m i n i m u m resistance at the t w o F „ ' s . The dark l i n e i n Figs. 10 and 11 marks the p o s i t i o n o f cushion 2 w i t h respect to c u s h i o n 1 at the o r i g i n f o r a c o n f i g u r a t i o n w i t h f i j - /RQ K 0.2. Note that i f s t / L p < l a n d

sp/Bp<], t h e d e m i - c u s h i o n s o v e r l a p a n d hence the

configurations are o f n o practical significance. Similar fi-gures can be generated f o r a given speed and used to position two hovercrafts traveUng together i n a manner where the f l e e t e x p e r i e n c e s m i n i m u m wave r e s i s t a n c e . T h i s configuration arrangement does not line up w i t h the Kelvin-Wave cusp fines o f 19.48° emanating f r o m the lead cushion's fore and aft corners.

I n the above study, i t is seen that the wave resistance o f a dual cushion system never exceeds that of the mono-cushion i n any o f the cases. Thus, the interference effects o f the demi-cushions i n the dual cushion system o f t e n results i n c o n f i g u r a t i o n s w i t h a s i g n i f i c a n t r e d u c t i o n i n wave r e s i s t a n c e c o m p a r e d t o m o n o - c u s h i o n o f e q u a l displacement.

Table 1 Particulars of the individual cushions i n a tri-cushion system and a comparative mono-cushion (CK=5, /J=20). Dimension Single Cushion _{(Individual Cushion)} Tri-cushion

Lp (m) 50 50 Bp dm) 25 25 P„ (seawater) 1.0 0.3333 P „ ( K P a ) 10.066 3.3553 3.5

i

^

I

Fig. 1 0 Contour plot Rj-IIif, vs. s t / i p and sp/B^ for a dual cushion at F„ = 0 . 4 . The dotted lines represent the Kelvin-wave cusp lines generated at the starboard bow and stern corners of the fore-cushion.

Fig. 8 Rj- IR^ vs. !t/Lp and sp/B^ for a dual cushion at Fn=0.4.

2S.

S 1.6

r---O.S

Fig. 9 RflR^ vs. gilLp and sp/B^ for a dual cushion at F n= 1 . 0 .

•

/

3.2 Tri-cushion crafts

The tri-cushion configuration used i n this section is described in Fig. 3. The equation to compute the wave resistance is given by Eq. (22) and the principal particulars of the i n d i v i d u -al cushions i n the tri-cushion system and a mono-cushion, having equal displacement as the tri-cushion system, are presented i n Table 1.

Fig. 11 Contour plot vs. Rj /R„AlLp and sp/B^ for a dual cushion at F„ = 1 . 0 . The doUed lines represent the Kelvin-wave cusp lines generated at the starboard bow and stern corners of the fore-cushion.

The tri-cushion system as shown i n Fig. 3 has certain restrictions imposed on the configiu-ation. The outrigger cushions (cushion 2 and 3) are always parallel, i.e., st between cushion 2 and 3 = 0. The m a i n cushion (cushion 1) can have a stagger relative to the outriggers but always lies exactly between t h e m , i.e..

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya R Banumurthy Wan L. Ham and Jae-Moon Lew

separation between the outriggers is sp and separation between either o f the outriggers and the main cushion is sp/2.

6 0 0 ? — , . . .o«. . 'j" . 0 , 8 . , . , , , 1,5 I Tri-Oishion I

Speed(m/s)

Fig. 12 Comparison of wave resistance of a mono-cushion, two configurations of dual cushions and a tri-cushion. A prehminary study is carried out to compare the wave resistances of a tri-cushion configuration, mono-cushion (both described in Table 1) and two dual-cushion configurations, all of them having the same displacement. The tri-cushion is configured to havesp/Bp = 2.5 and st Z i p = 1.5 and the t\vo dual-cushions are configured to have sp/ Bp = 2.5 and st=0 (catamaran-hke)

andsp/Bp = 1.25andst/Z,p = 1.5 (staggered catamaran-like).

The results of the study (Fig. 12) show that the tri-cushion offers the greatest savings in terms of wave resistance compared to the other configurations. This study is followed by a detailed analysis on the effects o f speed, sp and st on the wave resistance o f the tri-cushion and is compared to that o f the mono-cushion, RQ (Fig. 12). We note that the wave resistance o f the m o n o -cushion is 9 times that of the individual -cushions i n tri--cushion. Thus, the total wave resistance ratio can be written as:

^0 PQ ^ ^0 PQ

3 RQ

wheteRi„tf = 2Rp^^p^ +Rp^.^p^. Thus, the interference effects

can be deduced directly from the total-resistance plots since the non-dimensional total resistance {Rj. /RQ) w o u l d be the interference resistance plus 1/3.

Fig. 13 Rj-/R^ vs. F„ and sp/B^, for a tri-cushion configuration with sl/Lp=i.S.

First, the effects of varying separation and speed, f o r a fixed St (st / Lp = 1 . 5 ) , o n the wave resistance o f a tri-cushion are analyzed and the results are presented i n Fig. 13. Figure 13 shows a large valley o f l o w resistance {Rp / RQ « 0 . 1 5 ) at F„ = 0 . 7 f o r the e n t i r e range o f sp. A t l o w speeds the oscillatory pattern i n the wave resistance is attributed to the interaction o f the transverse waves generated by the m a i n cushion and the outriggers. Note that unlike the case o f dual cushions, the oscillations do not disappear at large values o f St. A t higher speeds, the osciUatory pattern disappears b u t t h e i n t e r f e r e n c e e f f e c t s are s t i l l o b s e r v e d f o r 1.5 < s p / i J p ^ 2.4, since Rj- /RQ is around 0.5 and does n o t go to 0.33.

Fig. 14 Contour plot of Rj-IRf^ vs. F„ and st/ip for a tri-cushion with sp/Sj,=2.5.

N e x t , the results f o r the case o f varying speed and stagger w i t h sp f i x e d at 2.55p are presented i n F i g . 14. T h e oscillatory pattern displayed i n this study is similar to the case o f a dual cushion system i n tandem. The smaller region shows a regular series o f h u m p s and hollows, w h i c h disappear w i t h increasing speed. Significant r e d u c t i o n i n the wave resistance is observed f o r a large range o f st at different Fj^ and the total resistance of the tri-cushion system reaches a m a x i m u m o f 62% of the mono-cushion resistance! Also, note that the large valley i n the resistance starts at St / Z.p= 0.5, w h i c h suggests that the outriggers travel i n the wake o f the m a i n cushion and the interference o f the wave systems o f the cushions leads to a drastic r e d u c t i o n i n the total resistance.

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hui Wan, Surya R Banunnurthy, Wan L. Ham and Jae-Moon Lew

Fig. 16 vs. sp/Bj, and s t / i p for a tri-cushion at F„ = 1.0.

The effects o f varying sp and st at F„ = 0.377, which is the location o f a hoUow i n the wave resistance (see Fig. 12) and at a higher F „ = 1.0 are presented i n Figs. 15 and 16. The lower bound o f sp and St is set to 1, to avoid overlap o f the cushions. At lower speeds the wave resistance displays a regular oscillatory pattern which corresponds to the interference o f transverse waves. Note that /RQ is less than 0.5 f o r the entire range of sp and st and hence even the worst configuration o f the t r i -cushion, i n terms o f wave resistance at F„ = 0.377, has a resistance o f 50% o f that o f the mono-cushion. For the case o f F„ = 1.0, a large valley (Rj. / RQ < 0.15) is seen f o r the entire range o f sp and st varying between 1,5 Lp and 3 Lp (Fig. 16). Note that at F „ = 1, there is a small range o f sp and st values where Rp is as small as 10%/?QI

< 0.6

0.1 0.2 0.3 0.4 0.5 0.6 0.7

stAo

Fig. 17 Contour plot of /ij-Ziio vs. st/?.o and A for a tri-cushion at F „ = 1.0.

The effects o f varying the stagger as well as the fractional displacement carried by the center cushion i n the tri-cushion system on the total resistance are presented i n Fig. 17. Here, A is defined as the ratio o f the displacement carried b y the m a i n cushion to the total displacement o f the tri-cushion craft, i.e., A = A j / A p . Thus, when A = 1.0, the tri-cushion craft is actually a mono-cushion craft and Rp /RQ is 1 whUe when A = 0, the tri-cushion craft becomes a dual-cushion (the outriggers carry all the displacement). Note that A is varied by changing P„, i n each o f the three cushions such that the total displacement remains constant. Figure 17 shows a m i n i m u m value o f Rp /RQOCCUVS around the region o f A = 0.33, i.e., the

m a i n cushion and the each o f the outriggers carry equal displacement. This justifies the initial choice o f taking P„, to be equal i n the three cushions, for this F „ . We also note that the total resistance ratio shows an oscillatory pattern f o r a given A as the stagger changes.

3.3 A d u a l - S E S s y s t e m

Yeung and Wan (2008) studied the effects o f varying the fractional displacement carried by the cushion o f a SES o n the wave resistance o f the SES. Here, w i t h the aid o f Eq.(23), the effects o f v a r y i n g the separation and stagger o f the component SES's on the wave resistance o f a dual-SES c r a f t (see Fig. 4) is analyzed. The wave resistance o f the dual-SES is compared w i t h that o f a single SES and a catamaran o f equal displacement. Each component o f the SES (a "demi¬ SES") consists o f Wigley huUs o f length 50 m , beam 2 m and draft 2.25 m and the cushion has a length o f 40 m , breadth 10.5 m and a n o m i n a l pressure head (P„,) o f 0.7810 m . The cushion is assumed to carry 60% o f the total displacement o f 1000 m ' of the dual-SES.

(5P-WVB

Vt (nVs)

Fig. 18 Total Resistance, R,-, vs. (sp-li')/B and speed, U, for a dual-SES with st = 0 compared to wave resistance of a SES and a catamaran of equal displacement.

First, the effect o f varying speed and separation on the total resistance o f the dual-SES is analyzed and the results are presented i n Fig. 18. The separation, sp, is defined as the centerline-to-centerline distance between the demi-SES and stagger, st, is m i d s h i p - t o - m i d s h i p distance between t h e m . Separation, sp, is allowed to vary between Wind (F-l-56, where

Wis total w i d t h o f the SES ( = 14.5 m ) and B is beam o f the

component hulls in the demi-SES. Figure 18 shows an expected decrease i n the wave resistance o f the dual-SES as they move apart. Also, note that a significant reduction i n the wave resistance o f the dual-SES when compared to a catamaran is observed at very high speeds. A t the h m n p speed, the dual-SES displays a reduction i n the wave resistance when compared to a SES o f the same displacement.

The results o f varying the stagger and speed o n the wave resistance o f the dual-SES i n tandem is shown i n Fig. 19, i n w h i c h we observe that there is a pocket i n the range o f 1 . 2 ^ s t / Z . S l . 6 having a significant drop i n the wave resistance (nearly half when compared to configurations w i t h st / 1 > 1.8).

Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ronald W. Yeung, Hul Wan, Surya R Banumurthy, Wan L. Ham and Jae-Moon Lew

However, the resistance at the h u m p speed f o r almost the entire range o f st is o f the same order as the resistance o f the single SES.

x i o '

I

Fig. 19 Rp vs. st/L and U for a dual-SES with sp = 0.

Fig. 20 Total Resistance, Rj-, vs. (sp - WyB and st/i for a dual-SES at t/ = 12.28 m/s.

Next, the wave resistance o f the dual-SES is computed at speed f / = 12.28 m/s ( h u m p speed f o r parallel configuration of the dual-SES) f o r varying values of sp and st. The results i n Fig. 20 show the existence o f a range o f sp and st values where the wave resistance is smaller than the single SES at the h u m p speed, which is around 600 k N .

significant reduction i n wave resistance compared to a mono-cushion. The interference effects were presented as functions o f speed, the separafion (sp) and the stagger (st) between the cushions. I n the tri-cushion case studied, i t was shown that for a range sp and st, the wave resistance of the tri-cushion can be as low as 10 % as that o f the mono-cushion.

The framework f o r computing the wave resistance o f a dual-SES was developed. The wave resistance o f the dual-dual-SES was f o u n d to be significantly less than a catamaran o f same displacement and was also f o u n d to be less than the wave resistance o f a single SES at the h u m p speed f o r parallel and tandem configurafions.

It is important to note that the final optimal spacing of m u l t i -cushions and SES's i n terms o f m i n i m u m wave resistance is not necessarily the best design solution considering the possible importance o f other requirements on the vessel, such as stability, maneuverability, etc. The tools developed are meant to provide the designer a fast and efficient m e t h o d to evaluate possible c o n f i g u r a t i o n s i n the early design stages. M o r e r e f i n e d p r e d i c t i o n s t a k i n g i n t o a c c o u n t o f viscous effects a n d nonlinearities, similar to Ciortan et al. (2007), can be utilized in the detailed evaluation stage.

5 A c k n o w l e d g m e n t s

The research has been supported by the O f f i c e o f Naval Research Grant No. N000140310211, under Award N o . ORG 29 via the Florida Atiantic University Consortium. We are grateful to M . Dhanak, L. J. Doctors, and K. Cooper f o r their comments. The support by the Korea Science and Engineering Foundation (MEST Grant ROl-2008-000-20852-0) f o r J - M . Lew's visit to U C Berkeley is also acknowledged.

6 R e f e r e n c e s

4 C o n c l u s i o n s

The mathematical f o r m u l a t i o n o f the wave resistance f o r a multiple number o f pressure cushions, taking i n t o account the interference resistance among the cushions, is developed in this work. The expression f o r the interference resistance is presented i n terms o f the Kochin f u n c t i o n . The expression for the interference resistance between a h u l l and a cushion, first presented by Yeung and Wan (2008), was simplified w i t h the pressure cushions having port-starboard symmetry

The effect of configuration of a dual cushion system and a t r i -cushion system was studied and the results were compared to a mono-cushion of equal displacement. Both systems showed

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Effects of Configuration on Wave Resistance of Multiple Pressure Distributions Ranald W. Yeung, Hui Wan. Suiya R Banumurthy, Wan L. Ham and Jae-Moon Lew

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