Elimination of wave resistance by a cambered twin hull at supercritical speed

Elimination of Wave Resistance by a Cambered Twin-Hull

at Supercritical Speed

X u e - N o n g C h e n , Universitat Stuttgart^

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

Clien & Sharma (1996a,b) found t h a t the wave resistance of a single-hull ship at supercrit-ical speed in a rectangular channel of suitable w i d t h can be made t o vanish t o t a l l y w i t h i n the framework of a linear shallow water wave approximation and, f u r t h e r m o r e , even in more ac-curate nonlinear models, namely, Kadomtsev-Petviashvili ( K P ) and Boussinesq models. The mechanism is t h a t the bow wave, after reflection f r o m the channel sidewall, hits the a f t e r b o d y and counteracts the stern wave so t h a t the resultant wave in the ship wake disappears. B y analogy to electrical conductors, the term "shallow channel superconductivity" was proposed for this phenomenon.

T h e favorable effect of destructive interference between bow and stern waves in open deep water on wave resistance has been well-noticed and investigated. I n principle, ship waves can be completely eHminated i n some cases, e.g. at certain speeds, w i t h i n linear theory (Tuck 1991,

Tuck and Tulin 1992, Tulin and Oshri 1994). The favorable wave interference between t w i n hulls at supercritical speed in shallow water is more significant and less sensitive t o speed t h a n in deep water. T h i s has been reported, on the basis of linear theory and model experiments, e.g. by Eggers (1955) and Heuser (1973), and by Chen and Sharma (1994a) d u r i n g the validation of a nonlinear theory by model experiments. While investigating a ship moving in a channel off the centerline, they found f o r a Series 60 ship model, i n both numerical c o m p u t a t i o n and model experiments, up t o 30% wave resistance reduction in the supercritical speed range at the best off-center track compared t o centerline motion. The reason is t h a t wave dispersion is much weaker in shallow water t h a n in deep water, especially in the supercritical speed range. As distinguished f r o m Kelvin's wave pattern i n deep water, or even in shallow water at subcritical speed, the ship wave pattern at supercritical speed is similar to shock waves i n supersonic flow. The bow and stern waves appear as a wave crest and t r o u g h extending t o i n f i n i t y along oblique characteristic lines w i t h nearly permanent wave forms. I n a special case a suitable ship forebody can generate a permanent oblique solitary wave on each side (Mei 1976). I f the wave-crest meets the wave-trough, they can largely cancel each other. I n this sense, superconductive ships are more feasible i n shallow water.

Busemann (1935) proposed already t o cancel the aerodynamic wave drag at supersonic speeds by means of a biplane configuration in which the two foils f o r m a kind of Venturi tube. The same basic idea underlies the work by Chen and Sharma (1996a). The present problem, however, is more complicated because of 3-D effects introduced by cross flow under the t w o keels. While the wave resistance of a ship in a narrow channel can be made t o vanish completely, i t seems t h a t the wave resistance reduction by o p t i m a l wave interference between the straight t w i n hulls of a conventional catamaran i n shallow b u t otherwise unrestricted water cannot exceed about 50%. However, complete resistance elimination is theoretically possible! Cancellation of far-field waves can be accomplished by a catamaran in which each of the t w o hulls has a cambered centreline.

The theoretical basis used here f o r treating 3-D flow effects in a catamaran hull on shallow water is the concept of blockage of the transverse flow (induced by the other hull and by the camber) around a hull section in shallow water. The blockage is quantified by an asymptotic potential difference between the t w o sides (Newman 1969). T h i s concept was exploited also by (Chen & Sharma 1994a) t o t r e a t the asymmetric flow about a slender single-huU ship moving asymmetrically i n a channel. Generalization t o locally varying yaw angle allows the treatment

'Math. Inst. A, Univ. Stuttgart, Pfaffenwaldring 7, D-70569 Stuttgart, Germany

of a cambered hull and, hence, the determination of camberlines such t h a t waves are created on one (the inner) side of a catamaran hull only! To obtain a superconducting catamaran, such a cambered hull and its m i r r o r image (the other catamaran hull) have t o be configured so as to eliminate waves behind the catamaran. This is achieved by applying a two-soliton solution as in Chen & Sharma (1996a) t o f i t the afterbody t o the forebody w i t h respect t o b o t h cross-sectional area and camber. I t means t h a t at the designed supercritical speed the bow wave generated only on the inner side by each of the two forebodies hits the other a f t e r b o d y and cancels its stern waves on b o t h sides. Consequently, the catamaran has theoretically zero wave resistance, and this state depends no longer on channel w i d t h .

2. A S i m p l e T h e o r y for a C a t a m a r a n w i t h C a m b e r e d H u l l s 2.1. F o r m u l a t i o n of t h e m a t h e m a t i c a l m o d e l

Consider a catamaran w i t h two hulls of length /*, beam 6* and d r a f t m o v i n g along the centerhne of a shallow channel of water depth h* and w i d t h w* at speed U*. Each hull can be curved as defined by a local yaw angle ip{^)y Fig- 1- The t w o hulls are mirror-symmetric and arranged symmetrically about their common centerplane, which coincides w i t h the channel centerplane. Ideal flow is assumed. The ship is free t o t r i m and sink. T h e same symbols are used t o denote dimensional variables and their nondimensional f o r m s , the f o r m e r being marked w i t h asterisks. Unless otherwise stated, all variables are nondimensionalized by reference to water depth h*, g r a v i t y acceleration g* and water density p*. For example, I = l*/h* is nondimensional ship length. So = S*/h*'^ is nondimensional midship section area, s = s*/h* is nondimensional separation between the mid-points of the midship sections of the t w o hulls, etc. The nondimensional speed U = U*/\/g*h* is usually called the depth Froude number

Fnh-Due to symmetry, only the port half is considered. A Cartesian coordinate system Oxyz moving at the same speed as the ship is used w i t h origin O located i n the center-point of the port huU, plane Oxy on the quiet free surface, z positive upward, and x positive f o r w a r d . The flow is then governed by the Laplace equation in the fluid domain and boundary conditions on the shiphuU surface, free surface, channel b o t t o m and sidewalls.

T h e problem w i l l be solved by matched asymptotic expansions, Chen and Sharma (1994a,1995). T h e flow region is divided i n t o near field and f a r field. I n each field multiple-scale expansions are applied, and the mathematical model is closed by asymptotic matching. The model is simplified f o r b o t h fields to obtain analytic solutions and t o design a catamaran explicitly. Therefore, the far-field equation is j u s t repeated w i t h a few explanations, and the crucial matching condition is rederived by simple reasoning.

In the far field, one can apply shallow water wave theory to derive various model equations. For mathematic simplicity, the K P equation is taken here, although i t is not the best i n the supercritical range, Chen and Sharma (1996b). Its stationary f o r m is

y z

\ I I I I I F i g . 1: Scheme of the problem and the near-field cross-flow

(1 - U'^)^^^ + ^yy + SU(p^(p^^ + —-if, _xxxx = 0; ₍₁₎

ip is the depth-averaged velocity potential.

In the near field around a single hull of the catamaran, one can apply shallow-water slender-body theory. F i g . 1. T w o key results of the slender-slender-body theory are employed: (i) the asymptotic transverse velocity corresponds to the time rate of change of the hull cross-section at a location fixed to the ground, superimposed on the mean (over depth) transverse velocity due t o asym-metry, e.g. Chen and Sharma (1995)] (ii) the mean transverse velocity yields an asymptotic potential difference between the two sides of a body floating in shallow water, Newman (1969). In mathematical terms,

V n \ y = ± 0 = ^ \ V r ^ + %, (2)

^ip = 2VnC{x). (3) Av? = V'ly=+o - ^\y=-o is the potential j u m p , y = ± 0 or 0* the p o r t / s t a r b o a r d side of the

p o r t component hull, Vn the local normal velocity, F„ its mean value, Vr the local tangential velocity, S{1) the lengthwise cross-sectional area d i s t r i b u t i o n , and C[x) the blockage coefficient for the cross section at x. I n terms of far field variables, the normal and tangential velocities can be expressed and approximated by neglecting disturbance velocities ipx, (py, which are small compared t o the mean velocity U, as

Vn = ^cosé - - U)smé ^cosé+ Usm'ib, (4) dy ox ay

Vr = - ( ^ - [ / ) c o s V - 5 ^ s i n V « C/cos^^. (5)

OX oy

Substituting (4), (5) and (3) into (2), one obtains an approximate matching condition:

If the local yaw angle 'ip{x) is small, cos V' ~ 1 and dS/dl ~ dS/dx result in

This w i l l serve as a simplified boundary condition at the ship location f o r the K P equation (1). See Chen and Sharma (1994a,1995) f o r a more accurate boundary condition involving the disturbance velocity and local free-surface elevation. The model still has t o be closed by imposing the boundary condition ^py = Qon the channel sidewall and centerplane, and the K u t t a condition at the ship stern x = -1/2:

^Ay=Qi = ^x\y=o- • (8)

2.2. S u p e r c o n d u c t i v e c a t a m a r a n s o l u t i o n

A "superconductive catamaran" would radiate no waves outwards and backwards and, hence, would have zero wave resistance at its design speed. This is achieved by a logical combination of t w o steps. F i r s t , non-existence of o u t w a r d waves requires t h a t the transverse velocity must vanish on the outer side:

The potential along the outer side of the p o r t hull, i n the far-field point of view, is then constant. As there is no discontinuity at the bow, this constant is equal t o the p o t e n t i a l on the inner side a t the bow:

^{x,0+) = const. = y ) ( / / 2 , 0 - ) . (10)

By using (7) and (10), tlie no-outward-wave condition (9) yields a prescription f o r the camberline:

t a n ^ f o ; ) - - i ^ + y - ( V 2 . 0 - ) - y > ( a . , 0 - )

!fi{x,0~) must be determined by the inner side condition. S u b s t i t u t i n g (11) i n t o (7) f o r the inner side y = 0~ simply provides the boundary condition on the inner side:

So f a r i t has been ensured t h a t a hull curved according to (11) w i l l generate no waves on its outer side, on which the transverse velocity is zero. Meanwhile, i t pushes all displacement flow toward the inner side, on which the transverse velocity becomes twice as much as t h a t f o r a straight hull o f same cross-sectional area. Consequently, all the waves are generated on the inner side. Normally, they would extend into the wake b ë h i n d the catamaran, w i t h or w i t h o u t reflection f r o m the inner side o f the other hull. T h e next step, therefore, must ensure t h a t the waves are confined t o the area between the t w o hulls. As the reduced problem (1) w i t h (12) is exactly the same as t h a t of a straight hull w i t h twice the displacement, a l l t h a t is needed is merely an application o f the original idea of shallow water superconductivity in Chtn and Sharma (1996a). Second, non-existence o f wake waves requires t h a t the stern waves be canceled by the bow wave of the opposite hull. T h e cross-sectional area d i s t r i b u t i o n should be so selected t h a t the bow wave generated by the inner side of each component hull and incident upon the inner side of the other hull will not be reflected, i.e. i t w i l l be perfectly absorbed. T h e bow wave can be an oblique solitary wave. Thus the t w o oblique solitary waves s t e m m i n g f r o m the t w o forebodies constitute a configuration of an oblique interaction of two identical solitons i n the area between the t w o hulls, and the two afterbodies are then determined by the incident wave. This two-soliton solution of K P Equation (1) can be used t o derive the ship shape, Chen and Sharma (1996a). T h e m a j o r expressions needed to specify the catamaran are:

- the potential on the inner side, which is used in (11):

, 8Uk cosh(2A;.ri) + ^/A^expi2kx) . ^ , y/MU , - I - AU ¥'(•^•.0 ) = ^ „ „ „ u / o , . „ . N , / ^ _ . . , u / o . , . ^ w i t h k = — — , A2 =

3 cosh(2A;.ri)-Fv/Alcosh(2A:.x') 2U ' U^-1-4AU' (13)

the cross-sectional area d i s t r i b u t i o n , which is obtained inversely f r o m (12) as (py\y-Q- is known, being half the value given by ( A 8) in the A p p e n d i x of Chen and Sharma (1996a):

8 kVm-l-AUsmh{2kx,)

"^^^ 3 c o s h ( 2 f c . T i ) + ^ c o s h ( 2 A ; , T ) ' ^ ' and the mean separation between the t w o hulls:

s = A x x / V U ^ - 1 - AC/. (15)

A and xi are two parameters, free t o be chosen for designing various hulls; theoretically Xi can be a r b i t r a r y , and 0 < A < (C/^ - 1)/(4C/). T h e theoretical ship length / is i n f i n i t e , but Sa{x) vanishes exponentially at i n f i n i t y so t h a t i t can be safely truncated. Since the disturbance velo-city at the stern on the inner side also vanishes at x = - o o , the K u t t a condition (8) is satisfied. A c t u a l l y , t o construct the catamaran, the two above steps are reversed, i.e. one starts w i t h the two-soliton interaction solution between the t w o hulls and then derives the cross-sectional area d i s t r i b u t i o n and the local yaw angle.

3. Pi'oposed D e s i g n of a S u p e r c o n d u c t i v e C a t a m a r a n

In this section a possible body plan design is proposed. I n order to avoid excessive camber two practical modifications have been made: (i) a skeg (vertical fin) is installed under the keel in the afterbody to increase the blockage coefficient; (ii) the velocity-potential j u m p between the two sides of each hull is reduced artificially by a factor, depending on longitudinal position, to roughly simulate the unavoidable viscous damping effects.

Let the given quantities be water depth h* = 5 m, ship length /* = 60 m , d r a f t d* — 2.5 m , and design speed U* — 42.84 km/h (corresponding t o a depth Froude number Fnh = 1-7) • T h e cross-sectional area d i s t r i b u t i o n , which automatically determines the displacement volume, the camberline and the mean hull separation will now be derived f r o m the superconductive solution, choosing reasonable values A = 0.20, xi — 2.5 f o r the two free parameters. Other details of hull geometry such as beam, body plan, and wetted surface area w i l l be decided by practical considerations. For better overview, the final results are given here in advance: mean hull separation s* = 20.08 m , beam 6* = 4.992 m , midship section a r e a 5 * = 11.224 m^, displacement volume V* = 332.4 m^, and wetted surface area = 371.55 m^, leading t o hullform coefficients CB = V*/{l*d*X) = 0.444, CM = S*Jid*bl) = 0.8994 and Cs = 5 * / ^ / F ^ = 2.631. The derivation now follows. Since Sa{x) is an even f u n c t i o n , Sa{—l/2) = Sa{l/2), the following truncation yields a finite ship length /:

S{x) = Sa{x)-Sa{l/2), (16) w i t h S{-l/2) — S{l/2) = 0. Since Sa{l/2).lltl, i f 1/2 - Xi is large enough, the practical

cross-sectional area d i s t r i b u t i o n S{x) is close to the theoretical one Sa{x). T h e displacement of each component hull is

f l / 2

V* = h*^ / S{x).dx = 332.4 m^ (17)

J-112

T h e slenderness r a t i o V/l^ of this hullform is very close t o t h a t o f a projected catamaran in Heuser (1973).

T h e theory does not specify any detailed geometry o f the cross sections. I t can be designed by practical considerations. Here, f o r simplicity, mathematical hull lines described by exponential functions are used:

»<^' ^' = ^ - e x p { - / M 4 ^ - ) ) - + '

z = -d{x) = -do { 1 - 0.9 exp[-1.2(a; + 6 ) ] } { 1 - 0.99 exp[6(.T - 6 ) ] } (19) the keel-line, and f { x ) a parameter f u n c t i o n chosen to be f { x ) — 20 sech(0.3.^). Integrating

(18) over -d{x) < z <0 yields

Since S{x) is known f r o m (16), b[x) can be determined f r o m the above equation as

Uf ^ CV ^ 1 - e x p { - f { x ) d { x ) }

'•W = 5 W , ( , , , + _ ^ [ e x p ( - / W J M } - l ) - ' ' ' '

For the given clearance c = 1 - do - 0.5, the theoretical yaw angle according to (11) together w i t h (13) is too large, the m a x i m u m value being about 18°. To o b t a i n a more suitable camber,

the following considerations are introduced: (i) A n y yaw angle will cause more vortex shedding, increasing viscous resistance and counteracting the reduction of wave resistance. So, in reality the best local yaw angle should be smaller t h a n the theoretically predicted value, (ii) T h e theoretical model based on potential theory w i t h the K u t t a condition at the stern does not hold well for strong vortex shedding beginning well ahead of the stern. Generally speaking, this viscous effect will weaken the potential j u m p across the hull. Since the potential j u m p is determined by the outer field solution, i t cannot account for this effect. To keep the t e r m A(/3(a;)/C(.'c) in (7) realistically, especially in the afterbody, either the blockage coefficient should be increased or the potential j u m p decreased artificially. The former artifice, tried in Chen and Sharma (1994a), improves the prediction of side force and yaw moment on a ship at d r i f t angle. Therefore the theoretical potential j u m p in (11) is reduced by dividing i t by the f u n c t i o n

t a n ^ M - - i ^ + y ( V 2 , 0 - ) - y ( x , 0 - ) 1

(22)

where fd{x) = 2 . 5 + (6 - x)/l2. Since dy^/dx

= tanV'(a;).

C{x) = 0.95 exp(-0.18a;). (23)

The actual design may have to be done by experiment rather than by theory. F i g . 2 shows the difference made by the skeg (solid line) and the theoretical blockage coefficient (dotted line) for a rectangular cross section of beam b{x) and d r a f t do, Taylor (1973):

C{x) = \b{x)[llc- 1) + ^ [ 1 - ln(4c)] + | - c 2 + '^c\ (24)

c = 1 - do is the clearance under the keel. F i g . 2 also shows the final lengthwise distributions of cross-sectional area, blockage coefficient and local yaw angle f o r a component hull of the proposed design, F i g . 3 the body plan, centered about the camberline, and a t o p view of the cambered twinhuU.

Q.05f

Fig. 2: D i s t r i b u t i o n o f cross-section area (left); blockage coefficients according t o (23) (sohd line) and (24) (dotted line) (middle); local yaw angle (right)

DWL 1 - 6 -4 -2 - 1 2 4 6 fonvard Uireclion

Fig. 3: B o d y plan centered about the camberline and t o p view of the cambered t w i n - h u l l

4. N u m e r i c a l E x p e r i m e n t s

The computer program S H A L L O W T A N K , Chen and Sharma (1994a, 1995), has been val-idated by several model experiments, including towing near a tank wall, which is essentially equivalent to a catamaran configuration. A direct application to a projected catamaran was reported by Jiang et al. (1995), in which good agreement w i t h model measurements of Heuser (1973) was obtained. T h i s program is now extended t o handle the case of cambered hulls. The original program, already involving asymmetric flow, is in principle valid f o r the present pur-pose. Only the parts related to the yaw angle, which was constant over the entire ship length before, had t o be modified to allow yaw angles V' being a f u n c t i o n of x.

For the sake of record, numerical details underlying the computed results shown in this paper are listed below. The grid size is I.hip = 20, Aa; = 0.10033 (Aa;* = l*/{2Iship) = 1-5 m ) . A y = 0.028078 ( A y * = 1.25503 m ) , smallness parameter e = 0,11186, see definitions in Chen and Sharma (1995). T h e t i m e steps finally selected after a few trials t o overcome numerical instability are A r = 0.005 f o r Fnh < 1-3, A r = 0.004 f o r 1.3 < Fnh < 1.8, A r = 0.003 f o r

1.8 < Fnh < 2.3, and A r = 0.0025 f o r Fnh > 2.3. T h e hull separation used is s* = 22.59 m since numerical experiments at design speed Fnh = 1-7 revealed t h a t its o p t i m u m value f o r the truncated hull S{x) is somewhat different f r o m the theoretical design value s* = 20.08 m f o r the infinite hull Sa{x). A straight monohull, identical to each component hull of the proposed design except f o r camber, is used as a reference f o r comparing wave resistance and ascertaining the favorable effect of wave interference. The catamaran calculations hold f o r a channel w i d t h w* = 150.62 m , which is large enough to avoid sidewall effects and only half of which is modeled in the computer by virtue of symmetry. The monohull calculations using the same grid w i t h o u t taking advantage of s y m m e t r y hold f o r a symmetric channel of half the w i d t h , except f o r t w o points Fnh = 1.2 and 1.3 where extra calculations in the f u l l - w i d t h channel become necessary t o avoid sidewall effects. T h e calculated range extends longitudinally about 10 ship lengths upstream and about 20 ship lengths downstream.

0.06

0.04

0.02

Fig. 4: Calculated specific wave resistance ( l e f t ) ; estimated specific t o t a l resistance (right) f o r the monohull (dots), the conventional catamaran (c) and the cambered t w i n - h u l l (s)

F i g . 4 shows the calculated specific wave resistance and the estimated specific t o t a l resistance of the cambered t w i n - h u l l , the same t w i n - h u l l w i t h o u t camber (conventional catamaran), and the uncambered component hull (monohull). Specific resistance is resistance/weight. T o t a l resistance is the sum of the wave resistance computed by S H A L L O W T A N K and the viscous resistance estimated using the I T T C 1957 correlation line multipHed by a f o r m f a c t o r 1 + k. By reference t o comparable hullforms, the f o r m f a c t o r is roughly estimated to be 1.4 f o r the cambered t w i n - h u l l , 1.32 f o r the conventional catamaran, and 1.24 f o r the monohull, Jiang et al. (1995). The t w o catamarans have almost the same wave resistance in the sub- and near-critical speed ranges, b u t i t is higher than t h a t of the monohull due to unfavorable wave interference. T h e monohull has only half the displacement and is, therefore, equivalent to an equal-displacement catamaran of infinite hull separation. The advantage of a catamaran shows

up i n tlie supercritical speed range. A t least i n the interval 1.5 < Fnh < 2.6, the wave resistance of the conventional catamaran is already significantly lower than t h a t of the monohull, and the cambered twin-hull is even better. A similar trend persists f o r the t o t a l resistance despite larger assumed f o r m factors f o r the catamarans. A t the design speed Fnh — ^7 the cambered t w i n -hull reduces the specific resistance w i t h respect t o the mono-hull by about twice as much as the conventional catamaran. T h i s is not surprising since the camber supresses the o u t w a r d waves and doubles the inward waves, thereby also doubling wave interference. Numerical calculation of the wave patterns created at Fnh = 1.7 (Fig. 5) showed t h a t the lower waves are generated by the cambered t w i n - h u l l on the outer side and in the wake than by other t w o ships. The inner-side waves trapped between the t w o hulls are about twice as high for the cambered t w i n - h u l l as for the conventional catamaran.

Fig. 5. Calculated wave patterns at Fnh = 1-7: (a) monohull, (b) conventional catamaran, (c) cambered t w i n - h u l l

5. C o n c l u d i n g R e m a r k s

The computed specific resistance of the proposed superconductive catamaran comprising cambered t w i n hulls is much lower than t h a t of its uncambered component monohull and also than t h a t of a conventional uncambered catamaran. T h i s reduction is seen over a broad interval in the supercritical speed range about the design point. I t is brought about by almost eliminating the outer-side waves and confining the inner-side waves to the area between the two component hulls. Besides the desirable economy of propulsion, this feature creates less hazard t o other t r a f f i c and less ecological damage t o the river banks. Cambered hulls in a catamaran seem quite natural since the presence of a neighboring hull destroys the s y m m e t r y of flow around a straight hull and one might hope t o p a r t l y restore the symmetry by camber. However, the mechanism exploited in this paper is not the restoration of s y m m e t r y b u t the complete cancellation of stern waves by diverting all bow waves t o w a r d the inside. T h e underlying inviscid potential theory could not do justice to unavoidable viscous elfects in a real fluid. So I do n o t claim t o have f o u n d the overall o p t i m u m . Verification by model experiment and f u r t h e r o p t i m i z a t i o n by t r i a l and error based on flow observation are indispensable.

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