J M a r Sci Teclinol (2014) 1 9 : 4 1 5 ^ 2 4 D O I 10.1007/S00773-014-0257-2
O R I G I N A L A R T I C L E
Analytical methods to predict the surf-riding threshold
and the wave-blocking threshold in astern seas
Atsuo M a k i • Naoya U m e d a • M a r t i n Renilson • Tetsushi Ueta
Received: 29 M a y 2013/Accepted: 4 February 2014/Published online: 27 February 2014 © J A S N A O E 2014
Abstract F o r the safe design and operation o f high-speed craft, i t is important to predict the behavior o f these c r a f t i n f o l l o w i n g and astern quartering seas as i t is w e l l k n o w n that serious problems can occur when a vessel is forced b y the waves to travel at wave speed. The s u i f - r i d i n g threshold is the lower speed h m i t above w h i c h the vessel w i l l be f o r c e d to travel at wave speed (usually on the face o f the wave) w h i c h is generally accepted to be a prerequisite f o r broaching. W h i l e a vessel travelling below this speed w i l l experience significant changes i n its longitudinal speed i n the wave, i t w i l l not be f o r c e d to travel at wave speed. For high-speed craft, the wave-blocking threshold also becomes important. This is the upper speed, below w h i c h the vessel w i l l also be f o r c e d to travel at wave speed (usually on the back o f the wave) and is related to the possibility o f b o w diving. B y the
A . M a k i
Graduate School o f Maritime Sciences, Kobe University, 5-1-1 Fukae-Minami, Higashinada, Kobe 658-0022, Japan Present Address:
A. M a k i ( H )
Naval System Research Centre, Technical Research and Development Institute, Ministry o f Defence, 2-2-1 Nakameguro, Meguro, Tokyo 153-8630, Japan e-mail: atsuo_maki@yahoo.co.jp
N . Umeda
Department of Naval Architecture and Ocean Engineering, Graduate School o f Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0971, Japan M . Renilson
Higher Colleges o f Technology, Abu Dhabi, U A E T. Ueta
Centre f o r Advanced Information Technology, Tokushima University, Minami-Josanjima, Tokushima 770-8506, Japan
application o f a p o l y n o m i a l approximation to the wave-induced surge force, including the nonlinear surge equation, an analytical f o r m u l a to predict both the s u i f - r i d i n g and the wave-blocking thresholds is proposed. Comparative results o f the surf-riding threshold and wave-blocking threshold predicted u t i l i z i n g the proposed f o r m u l a and the thresholds predicted using numerical bifurcation analysis indicate f a i r l y good agreement. I n addition, previously proposed analytical f o r m u l a e are examined. I t is concluded that predictions o f these thresholds obtained using the analytical formulae based on a continuous piecewise linear approximation and M e l n i k o v ' s method agree w e l l when used to predict these thresholds w i t h predictions obtained f r o m numerical b i f u r -cation analysis and those obtained experimentally using free-running models. As a result, i t is considered that these t w o calculation methods could be recommended f o r the early design stage tool f o r avoiding broaching and b o w d i v i n g .
K e y w o r d s S u i f - r i d i n g • W a v e - b l o c k i n g • A n a l y t i c a l f o r m u l a • Nonlinear dynamics
1 Introduction
W h e n a vessel runs i n f o l l o w i n g and/or stern quartering seas, i t is i n danger o f broaching [ 1 , 2 ] or b o w d i v i n g [ 3 - 7 ] , w h i c h generally precede a capsizing event. Broaching is a phenomenon i n w h i c h a vessel cannot maintain the desired course despite engaging m a x i m u m steering. A l t h o u g h several experimental [ 8 ] , numerical and analytical studies [9] have shown that this phenomenon occurs even at l o w Froude nuinbers, f o r example < 0 . 3 , this k i n d o f broaching is often avoidable by u t i l i z i n g o p t i m a l rudder control [ 1 0 ] . I n general, broaching i n the high-Froude region is consid-ered to be more dangerous.
416 J Mar Sci Technol (2014) 1 9 : 4 1 5 ^ 2 4
Since one o f the prerequisites o f broaching is the phe-nomenon o f surf riding, the estimation o f the surf-riding tlKeshold is important to assess a vessel's safety i n f o l l o w i n g and/or stem quartering seas. However, at higher Froude numbers the surf-riding phenomenon disappears and, i n contrast, the ship overtakes the wave. The transition region at which this occurs is caUed 'the wave-blocking thi'eshold'. Figure 1 shows a schematic view o f the surf-riding and wave-blocking thi-esholds. I n this figure, the abscissa represents the wave steepness, while the ordinate represents the nominal Froude number. A s shown in this figure, both nominal Froude numbers f o r the surf-riding and wave-blocking thi-esholds coincide at the wave celerity when wave height tends to zero. The transition is considered to be the threshold o f b o w d i v i n g , i.e., the lower l i m i t o f its occurrence. Therefore, this threshold c o u l d provide a m e a n i n g f u l insight into a vessel's behavior when running at higher Froude numbers. B o w d i v i n g occurs when a vessel's bow is immersed into the back of the wave o w i n g to the vessel's f o r w a r d speed and amplitude o f the wave relative to the b o w .
I f either o f these phenomena occur, broaching or b o w d i v i n g , passengers could be i n j u r e d at the best, or at the worst, the ship c o u l d capsize f r o m the resuhing y a w and extreme induced r o l l m o t i o n . W i t h regard to broaching, i t is one o f the three m a j o r capsizing scenarios identified i n the new generation o f intact stability criteria w h i c h w i l l be added to the 2008 Intact Stability Code (IS code) b y the Internarional M a r i t i m e Organizadon ( I M O ) [ I I ] .
The second generation intact stability criteria to be estab-lished at I M O consist of: vulnerability criteria; and perfor-mance-based criteria. I f a ship fails to pass the vulnerability
Surf-riding threshold Wave-blocking threshold
Oscillatory motion with ship overtaking w a v e
Oscillatory motion with w a v e overtaking ship - I 1 1 1 1 i — •
V\/ave Steepness
Fig. 1 Schematic view o f the surf-riding threshold and wave-blocking threshold
criteria, its safety is to be assessed against the performance-based criteria, utilizing numerical simulation or its equivalent. Thus, i t is important that any vulnerability criterion is easy to apply, ensures a conservative safety level, and is not based upon empirical data applicable to only a l i m i t e d vessel type. A vulnerability criterion f o r suif riding in regular f o l l o w i n g seas can be used i n place o f that f o r capsizing due to broaching because, as noted, surf riding is the prerequisite to these phenomena and travelling i n f o l l o w i n g seas is the most sus-ceptible heading f o r surf riding. T o estimate the surf-iiding threshold or wave-blocking threshold i n regular f o l l o w i n g seas, an analytical solution is obviously most suitable because it is easy to apply, and does not rely on empirical data which are only applicable to a l i m i t e d vessel type.
I n this paper, the authors develop a generalized f o r m u l a f o r predicting the s u i f - r i d i n g and wave-blocking thresholds by making use o f M e l n i k o v ' s method [ 1 2 ] . Based upon previous research [13], the authors propose an analytical formulae f o r estimating the surf-riding and wave-blocking thresholds by m a k i n g use o f the continuous piecewise linear ( C P L ) approximation.
Finally, a new approach f o r predicting the surf-riding and wave-blocking thresholds, based upon third order polyno-m i a l approxipolyno-mation f o r the wave-induced surge f o r c e is proposed. Predictions using both approaches are validated using results f r o m free-running model experiments.
2 Reducing the nonlinear surge equation
2.1 Basic autonomous surge equation
The co-ordinate system used i n the f o r m u l a e presented i n this paper is illustrated i n F i g . 2. A n inertia co-ordinate system o - (^(.with the o r i g i n at a wave trough has the ^ axis pointing toward the wave direction. The ship fixed co-ordinate system, G - xz, w i t h the origin at the center o f gravity o f the ship has the x-axis p o i n t i n g toward the bow f r o m the stern and zaxis d o w n w a r d . Here, the ship's l o n -gitudinal velocity, ii, is defined as it = + c where c indicates the wave celerity.
Initially, the generalized f o n n o f the approximate polynomial nonlinear surge equation needs to be obtained. This approach f o r the analysis has been used previously by the authors [13], and is thus only briefly summarized here.
F i g . 2 Co-ordinate systems
J Mai- Sci Teclinol (2014) 19:415-424 4 1 7
The equation representing nonlinear surge m o t i o n i n this paper is described as f o l l o w s :
{m + m,)lc + [R{u) - T{ii, n)] - X,, = 0 (2.1) I n this equation, a dot denotes differendation w i t h respect to time /, where: R: the ship resistance, T: the pro-peller thrust, in: the ship mass, in^: the added mass i n the
X direcdon, ii: the instantaneous ship velocity i n the X direction, and n: the propeller rate. Although additional
terms such as thrust variation due to wave particle velocities also exist, they are not essential to obtain sufficient agree-ment w i t h the results f r o m model experiagree-ments. Therefore, these terms are ignored to s i m p l i f y the expression.
Assuming that the h u l l f o r m is longitudinally symmet-ric, the F r o u d e - K r y l o v force is represented as a first order approximation as f o l l o w s :
X , , « / s i n ( f c ^ G ) (2.2) where wave number, k, is defined as 2n/A, and 1 is the
wavelength. Here, the phase o f the sinusoidal f u n c t i o n representing X„. is ignored (See Appendix). The resistance curve, R{ii), and the thrust coefficient curve, Kj{J), can be approximated by «.-th polynomials as f o l l o w s : ^ ( " ) ~ = ' 0 + + ' " 2 " ' + • • ' i = 0
n
KT{J) «Yu
+ K\J + K2J^ + • (2.3) (2.4) i = 0where each and ;c,- are chosen on the basis o f a polyno-m i a l fit o f the resistance and the thrust coefficient curves, obtained f r o m the tank tests or f r o m numerical calcula-tions. Note that: / = M(1 - Wp)lnD.
Then T{ii; n) becomes:
T,{u-n) = {\-tp)pn''D'KT{J)
Y-
- 2 £ , / - 4 (2.5)where t^, and Wp are the thrust deduction and wake fraction, respectively. These are taken at their still-water values, as normal. Here, D and p are the propeller diameter and water density, respectively. Substituting these equations into E q . 2.1 yields: n + in.)^c + Y Y A • 4 C ' ' + / s i n ^ f G '•=1 j = i V J = Te{c,,;n)-R{c,,) where c,- is: Wp) PK; (2.6) (2.7)
Here, c„, is the wave celerity. This equation represents the approximate generalized expression o f surge.
3 Prediction of the surf-riding and wave-blocking thresholds
3.1 B r i e f review o f existing work
M e l n i k o v ' s method [12] is a p o w e r f u l analytical approach w h i c h makes i t possible to test f o r the existence o f ho-moclinic or heteroclinic connections o f an invariant man-i f o l d to a saddle. I n surge motman-ion man-i n f o l l o w man-i n g seas, the heteroclinic b i f u r c a t i o n point is equivalent to the surfriding and waveblocksurfriding thresholds. A n application of M e l -n i k o v ' s method to E q . 2.6 has already bee-n c a -n i e d out by the authors and the generalized result, presented by M a k i et al. [ 1 3 ] , can be shown to be as f o l l o w s :
Te{c„;n)-R{c,,] f /=i j=i V 2 y V 2 y where i f k y
n
(m+in^y^ (3.1)Here, F represents the Gamma f u n c t i o n . I t can be assumed that R(u) can be fitted w i t h cubic p o l y n o m i a l and that KT can be fitted w i t h quadratic polynomial. Therefore, substitute n = 3, and put K3 = 0, then the f o l l o w i n g con-dition is obtained: Teic R{C,,) _ 4 ( c i + 2 C 2 C . , + 3 C 3 C ^ , ) f
+
%^fk{m + mx) 2 ( C 2 + 3 C 3 C , , ) 32C3V7 k{m + m,,) 2j:[k{i (3.2)Furthermore, M a k i et al, [13] developed an expression f o r predicting the surf-riding threshold by u t i l i z i n g the C P L approximation. I n the C P L approximation approach, the sinusoidal f u n c t i o n representing the F r o u d e - K r y l o v force is approximated by three piecewise linear f u n c -tions, and a continuity condition is imposed at the boundary o f each linear f u n c t i o n . I n each region, surge equation is approximated by linear equation. Then, each f u n c t i o n can be solved analytically, and the continuity condition f o r each hnear solution leads to the simple f o r m f o r m u l a calculating the heteroclinic b i f u r c a t i o n point, that is the s u r f - r i d i n g threshold. This f o r m u l a can be solved w i t h N e w t o n method. This result is b r i e f l y summarized below.
418 J Mar Sci Teclinol (2014) 19:415-424
Using n — 3 and K3 = 0 i n Eq. 2.6, tlie f o l l o w i n g expression is obtained:
(m + m.yic -f-Ai(c„, ; H ) ^ G + A 2 ( C , „ ) < ^ G
+^3^1
+fsm(k^G) = Te{c,y, n)-R{c,,) (3.3) I n Eq. 3.3 the terms defined by: Ai(c„.; n), A2(c,„), A 3 , r(c„„ n), /?(c,„) are as f o l l o w s : ' Ai(c„, ; H.) = 3r3cl, + 2(/-2 - 12)^,,. -t- ' ' i - T I « A2(cuO = 3 r 3 C „ , -f- (;-2 - T 2 ) • A 3 = r 3 (3.4) re(c„,, n) = T24 -H Ti;;c,„ - f T Q H ^ ../?(c,„) = r 3 c f „ - F ; - 2 c 5 - | - / - i c , „
This is identical to the equation introduced by Spyrou [14].
Taking the quadratic regression o f the damping terms i n Eq. 3.3, Spyrou obtained the f o l l o w i n g analytical f o r m u l a to estimate the surf-riding threshold:
4y2 (3.5)
- a = 2 e ' ' " ' ' [ c K C 0 s l / T - a s m X n ] (3.9) Here: = R e [ c 3 ] , c, = I m [ c 3 ] , = ReU^] and
A/ — Ini[23], where: a, = / i ( / 0 / ( m . + m.v) (3.10) a 2 = / / ( m + m,.) (3.11) y.3 = [Te{c, n)-R{c)]/{m + m,) (3.12) In 2X,{cl + c]) (3.13)
/ l l = CRI, + CIXR + C / i 2 , 4 2 = CRXR - CiXi + CRX2
(3.14) 1 , X(y.2 4 4 a 2 _ - K l ± ^Ja\ + 16^2/^ 2 ^ _ - a i ± i / a j - 160:2/2 C3,4 = ± where: 4 4 a 2 ( A i - f ^ 4 , 3 ) (As - i 4 ) (3.15) (3.16) (3.17) A l (c,„ ; /O + '42(c,„.) ^ + ^ 4 3 ^ ^ ^ / = ] / = i i=i ;=1 (3.6)
On the other hand, M a k i et al. [13] approximated this damping term by the linear regression:
m =
Ai(c,„ ; n) 1^ <^^^/<^c+A2(c„..) ^^d^a4-A3 CcdÜG (3.7)
The above expression, however, leads to the f o l l o w i n g :
(m + m,)U + m t c +fsm{k^G) = T{c,,, n) - /?(c,„) (3.8)
This is the expression f o r a nonlinear pendulum equa-tion, which cannot be solved analytically.
T o overcome this d i f f i c u l t y , the sinusoidal term is approximated using the C P L f u n c t i o n . Then, the f o l l o w i n g f o r m u l a w h i c h can be used to estimate the surf-riding threshold is obtained:
3.2 The wave-blocking threshold
The method to predict the wave-blocking threshold is almost identical to that used to predict the surf-riding threshold. However, the difference is that the traj ectory on the upper phase plane is employed in the formulation f o r the wave-blocking threshold, whereas f o r the surf-riding threshold it is on the lower plane. The detailed explanation o f the methodology is omitted, however, the final resuhs are ihustrated below.
The generalized condition o f the wave-blocking threshold obtained by applying M e l n i k o v ' s method is represented as: / ^ ( c , „ ; ƒ
i=i j=i
/r
J + 2 ( 3 . I f Assuming n — 3 and KJ the f o l l o w i n g is obtained:0 as explained above, then
J Mar Sci Technol (2014) 19:415-424 419 r , ( c „ , ; » ) - /?(c,,) _ 4 ( c i + 2c2C,, + Scjc^,) ƒ
+
ny^fk{m + in,) 2(C2 + 3C3C„,) 3 2 c 3 \ ^ (3.19) On the other hand, applying the C P L approximation method the f o l l o w i n g bifurcation condidon is obtained:-a = 2e''"''[cji c o s A / T - c / s i n i y i ] where: «1 = i 3 ( « ) / ( ' " + '«-ï) «2 =f/{i"+m.,) c/.i = [ r , ( c , n) - / ? ( c ) ] / ( m + m,-) T = — I n 0!A//12 + /l2 2 A / ( c | + c 2 ) (3.20) (3.21) (3.22) (3.23) (3.24) Ax = C R I , + CIXR + C1X2, M = CRXR - CiX, + CRX2 (3.25) 1 , ^ « 3 4 4(X2 •1,2 C3,4 - K i ± V K | + 1 6 « 2 / i , 2 ' -a.\ ± i / a j - 1 6 a 2 / A 4 4(X2 ( A 3 - A 4 ) (3.26) (3.27) (3.28)
w i t h the f o l l o w i n g defined as: CR — R e [ c 3 ] , c / — I m [ c 3 ] ,
-XR = Re[;.3]and Xi = I m [ l 3 ] . Equadons 3.19 and 3.20 are
solved numerically using Newton's method.
3.3 N e w analytical f o r m u l a f o r predicting the surf-r i d i n g and wave-blocking thsurf-resholds
The approach given i n the previous secdon was obtained using a C P L approximation f o r the sinusoidal t e r m i n E q . 3.3. A n alternative is to make use o f a p o l y n o m i a l approximation f o r the same term. I n E q . 3.8, the non-di-mensionalization is as f o l l o w s : f k
-t
(3.29) m + nix w h i c h yields: d^y n^y . + s m y = -dt^ d iq
where each coefficient is defined as: /?(«) \Jfk{m -V nix) m + iih [ r , ( c . , , n)-R{c,,)]k in + nix ( 3 . 3 i :
Here, the sinusoidal f u n c t i o n is approximated by a p o l y n o m i a l o f a t h i r d order as foUows:
s i n y pa - / 0 ' ( ) ' - (3-32) Finally Eq. 3.30 can be transformed as f o l l o w s :
y + Py - /0'(y - y i ) (y + y i ) = - (3.33)
It is w o r t h noting that the periodicity o f wave-induced force does not disappear using this approximation. H o w -ever, w i t h careful scrutiny i t can be seen that the approx-i m a t approx-i o n f o r one wave approx-is suffapprox-icapprox-ient sapprox-ince the heteroclapprox-inapprox-ic orbit j o i n i n g t w o saddles can be considered w i t h i n one wave. Assuming aj < 02 < « 3 , an analytical factorization, such as Cardano's technique, yields:
« ' ( } ' - y i ) ( y + y 1) - f ^ = A((3' - «1) (}' - «2) (y - « 3 ) (3.34) Then Eq. 3.33 becomes:
y - I - Py - i-i{y - fli)(y - « 2 ) ( y - 0 3 ) = 0 Putting:
y - a\
(73 - ai
the f o l l o w i n g equation is obtained:
X +
jix +
/ l ï ( l -x) {x
- a) = 0 where: ^ «2- fli
fl3 -ai fl = /i(cZ3 - fli (3.35) (3.36) (3.37) (3.38)Here note that 0 < f l < I . As pointed out b y M a k i et al. [ 1 5 ] , the state Eq. 3.37 is identical i n f o r m to the F H N ( F i t z H u g h - N a g u m o ) equation except f o r some o f the coefficients. N o w the f o l l o w i n g solution is assumed [ 1 5 ] :
x = cx{l-x) (3.39)
Here x can be calculated as:x = c^l-x){l-2x) (3.40)
so that substituting Eq. 3.40 into E q . 3.37 yields:(3.30) j,U,-2c') + (ë' + pë-ficl 0 (3.41)
420 J M a r Sci Teclinol (2014) 1 9 : 4 1 5 ^ 2 4
I f the above equation is satisfied f o r A ' ^ 6 (0, 1), then i t is necessary to sadsfy the f o U o w i n g condidons:
+ Pc - fia = 0
E h m i n a t i n g c f r o m these equations yields:
0
(3.42)
(3.43)
This equation represents the condition o f the surf-riding or the wave-blocking threshold that is to be satisfied. I n this equation, the upper sign corresponds to the wave-blocking threshold, w h i l e the lower sign corresponds to the suif-r i d i n g thsuif-reshold.
N o w solving Eq. 3.43 using the iterative method, the time domain trajectory can be obtained as:
q\t)=x=l/[\ + tx^{-ët + cl)' (3.44)
where c > 0 and d e (—oo, oo) is an arbitrary constant determined by the i n i d a l condidon. Equadon 3.44 is alternatively represented as:
I - I - tanh ct - d (3.45)
4 Validation of the analytical formulae against n u m e r i c a l bifurcation analysis
T o validate the proposed formulae, comparative calcula-tion between the formulae and numerical b i f u r c a t i o n
Table 1 Principal particulars of the O N R tumblehome vessel
Items Values
Length 154.0 m
Breadth 18.8 m
Draught 14.5 m
Block coefficient 0.535
analysis was carried out f o r the O N R ( O f f i c e of N a v a l Research) tumblehome vessel [16], The principal charac-teristics and the body plan o f this vessel are shown i n Table 1 and Fig. 3, respectively.
Figure 4 shows the wave-blocking threshold using the piecewise linear approximation method, i.e., Eq. 3.20 w h i c h is compared w i t h that obtained f r o m the numerical b i f u r c a t i o n using the C P L approximated wave-induced surge force. Here, the numerical b i f u r c a t i o n analysis is based on the method utihzed by M a k i et al. [ 1 7 ] . I n this method, the conditions o f e q u i l i b r i u m points, their eigen
• Numerical bifurcation analysis for CPL system
Analytical fonnula (CPL approximation)
0.045 0.05
Fig. 4 Comparison o f the wave-blocking threshold predicted using the Numerical Solution, and that predicted using the Analytical Solution w i t h Piecewise Linear approximation, f o r X/L = 1.0
Sinusoidal function Approximated function
Fig. 3 Body plan o f the ONR tumblehome vessel
Fig. 5 Linear, quadratic and cubic approximation o f the sinusoidal function
J Mar Sci Teclinol (2014) 1 9 : 4 1 5 ^ 2 4 421
value, their eigen vector and the connection o f the t w o trajectories f r o m the t w o e q u i l i b r i u m points, (the so-called stable and unstable invariant manifolds), are numerically solved using the N e w t o n method. The trajectories are obtained using the R u n g e - K u t t a method. I n this figure, the abscissa is the wave steepness, w h i l e the ordinate indicates the nominal Froude number, defined as the ship velocity i n calm water w i t h the same propeller revolutions. Since there is no discernible difference between the two, i t can be concluded that the proposed f o r m u l a is consistent w i t h the numerical b i f u r c a d o n analysis f o r predicting the wave-blocking threshold.
• Numerical bifurcation analysis for polynomial system
Analytical formula (polynomial appro,\imation)
0.4-I 0.35- 0.25- 0.2-o 0.3 ip • • • [j* 1.2 ll l l 1.4 1.6 AIL
Fig. 6 Comparison o f the surf-riding threshold predicted using the Numerical Solution, and that predicted using the Analytical Solution w i t h the polynomial approximation, f o r HI?. = 0.05
• Numerical bifurcation analysis for polynomial system
Analytical formula (polynomial appro,\imation)
0.52
0.02 0.025 0.03 0.035 0.04 0.045 0.05 H IA
Fig. 7 Comparison o f the wave-blocking threshold predicted using the Numerical Solution, and that predicted using the Analytical Solution with the polynomial approximation, f o r XIL = 1.0
T o validate the results o f the p o l y n o m i a l approximadon method the appropriate C P L curves are determined keeping the zero crossing points the same as those o f the original f u n c d o n between (~3n/2 and 3n/2). As a result, the sinu-soidal f u n c t i o n can be represented as f o l l o w s :
s i n y ; Ti){y+n) (3.46)
The approximadon result is shown i n F i g . 5.
Figures 6 and 7 show the surf-riding and wave-block-i n g thresholds, respectwave-block-ively, that are obtawave-block-ined by solvwave-block-ing 3.43. The numerically obtained thresholds f o r the approximated p o l y n o m i a l surge equation are also plotted. I n F i g . 6 the abscissa is the wavelength to ship length rado, w h i l e i n Fig. 7 the abscissa is the wave steepness. Since there is no discernible difference between the results f r o m the t w o methods f o r both thresholds, i t can be concluded that the proposed f o r m u l a 3.43 is consistent w i t h the numerical b i f u r c a t i o n analysis f o r predicting these thresholds.
5 Validation of several analytical methods against free-running model experiments
T o validate experimentally the proposed methods, predic-tions computed by all the methods were compared w i t h results obtained f r o m a free-running model experiment caiTied out i n the seakeeping and maneuvring basin o f N R I F E (National Research Institution o f Fishing E n g i -neering) w i t h a scale model o f the O N R tumblehome vessel. I n the experiment, the autopilot heading, xc, was set to - 5 ° f r o m the wave direcdon, because ii has been shown that the e f f e c t o f a small deviation i n the heading on the
Wave direction
Auto pilot heading
Fig. 8 Definition o f the heading angle and the autopilot heading w i t h respect to the wave direction
422 J M a r Sci Teclinol (2014) 1 9 : 4 1 5 ^ 2 4
Fig. 9 T i m e series o f the ^ Pitcli (degrees) Yaw (degrees) oscillatory motion with
Fn = 0.45 and ic = - 5 . 0 ° f o r XIL = 0.8 WX = 0.05 F i g . 10 T i m e series o f surf riding w i t h Fn = 0.4 and yc = - 5 . 0 ° for XIL = 0.8, HI X = 0.05 Pitch (degrees) -2 -4H -6 10 15 20 25 30 t ( s ) 90 60 30 0 -304 -60 -90-* Yaw (degrees) 10 15 20 25 30 t ( s ) 40 30 20H 10¬ 0--10-^ -20 -30 -40 Roll (degrees) 10 20 30 l ( s ) Rudder (degrees) 15 20 25 30 t ( s ) 0.6¬ 0.55¬ 0.5¬ 0.45-• O 2 L L
1
C:-> 0.15¬ 0.1;
: : - { '^•y&t- 1 \ % 0 - 1 i i ( 1 « 1 P %•
•
- { j . A . A i A n ' — Q 0 1 1 I 1 8 l l l l 1 2 1 l l l l 4 1 l l l l 6 1 l l l l 8 2Oscillatory motion (EXP.) Surf-riiding (EXP.)
Numerical bifurcation analysis Melnikov's m e t h o d
C P L apploximation method Polynomial m e t h o d
XIL
Fig. 11 Comparison o f the predicted surf-riding and wave-blocking threshold f o r the three methods w i t h the experimental results as a function o f wavelength to ship length ratio, w i t h a wave steepness o f 0.05
surf-riding threshold is negligibly smaU [17] and a heading o f 0° could cause a collision w i t h the tank w a l l at the beginning o f the run. The definitions o f heading angle, and auto p i l o t heading xc. are given i n F i g . 8. I n i t i a l l y the model d r i f t e d near the wave maker and then the propellers
and the autopilot controls were activated. The propeller revolutions were set to the specified n o m i n a l Froude number during the model runs. A propordonal autopilot w i t h a rudder gain o f 1.0 and a dme constant o f 0.163 s i n model scale was used.
J Mar Sci Teclinol (2014) 19:415-424 423
Fig. 12 Comparison o f the predicted surf-riding and wave-blocking threshold f o r several methods and experimental results, as a function o f wave steepness, at a wavelength to ship length ratio o f 0.8
• Oscillatory motion (EXP.) O Surf-riding (EXP.)
A Numerical bifurcation analysis Melnikov's method
C P L apploximation method Polynomial m e t h o d
0.03 0.035 0.04 0.045 0.05 0.055 0.06 HIX
Figure 9 shows the time series as an example o f oscil-latory motions f o r the upper nominal Froude number region where the model overtakes the wave. This can be i m p l i e d f r o m the time history o f the pitch m o t i o n . A l t h o u g h a long time record could not be obtained because o f the high speed and the l i m i t a t i o n of the tank size, the p i t c h up-crossing tendency after about 22 s suggests that the model was overtaking the wave.
I n contrast. F i g . 10 shows the time series f o r a slightiy lower nominal Froude number, still higher than that cor-responding to the wave speed, and illustrates a case o f wave b l o c k i n g . The convergence behavior o f the p i t c h m o t i o n suggests that the model was captured o n a wave and f o r c e d to run at the wave celerity. These t w o examples indicate that the wave-blocking threshold exists between a Froude number o f 0.4 and 0.45. However, the wave-b l o c k i n g threshold may wave-be affected wave-by the i n i t i a l conditions [ 1 8 ] . T o exclude the dependence upon the i n i t i a l condi-tions, model runs w i t h various i n i t i a l conditions are rec-ommended as a task f o r the future.
Figures 11 and 12 show a comparison o f the predicted surf-riding and wave-blocking thresholds using the pro-posed three formulae w i t h the experimental results. I n F i g . 11 the abscissa is the wavelength to ship length ratio, w h i l e the abscissa is the wave steepness i n F i g . 12. A l l the predicted thresholds agree w e l l w i t h the results f r o m the experiments, where the threshold is between the runs showing oscillatory motion, and those showing surf r i d i n g or wave blocking (sohd circles and h o l l o w squares i n Figs. 11 and 12, respectively).
Compared w i t h the results obtained f r o m the numerical b i f u r c a t i o n analyses, the C P L method consistently gives an overestimation o f the nominal Froude nuinber f o r the s u i f -r i d i n g th-reshold and an unde-restimation o f the n o m i n a l Froude number f o r the wave-blocking threshold. This is because the C P L approximation underestimates the wave-induced surge force [13]. Further, as has been shown pre-viously [ 1 3 ] , a shght underestimate o f the n o m i n a l Froude
number f o r the predicted s u r f - r i d i n g threshold c o u l d be caused by the d i f f r a c t i o n effect i n the wave-induced surge force.
On the other hand, the results obtained using the polyno-m i a l approxipolyno-mation polyno-method predict the correct trends f o r the thresholds, but give unsatisfactory results i n a quantitative sense. However, f o r the condition shown here, this method yields results on the conservative side o f safety. Thus, this method is considered to be acceptable f r o m a practical point o f v i e w f o r operational guidance i n the f o l l o w i n g seas.
Further, as is observed i n the case o f the prediction o f the surf-riding threshold [13], the threshold predicted by the f o r m u l a based on the M e l n i k o v ' s method shows f a i r l y good agreement w i t h that obtained by the numerical b i f u r c a t i o n analysis and experiments f o r the wave-blocking case.
Summarizing the above results, it is concluded that the formulae based on M e l n i k o v ' s method, the method based o n the C P L approximation method and that based on the p o l y n o m i a l approximation method provide s u f f i c i e n d y accurate predictions, and hence these t w o calculation methods could be used f o r the vulnerability criteria i n c l u -ded i n new generation intact ship stability code (IS code).
6 Concluding r e m a r k s
The main conclusions f r o m this w o r k are summarized as f o l l o w s :
1. B y applying t w o analytical approaches, i.e., M e l n i -k o v ' s method and the C P L approximation method, analytical predictions f o r estimating the s u r f - r i d i n g and wave-blocking thresholds have been obtained. 2. B y approximating the wave-induced force using a
polynomial f u n c t i o n , analytical f o r m u l a e to estimate the surf-riding and wave-blocking threshold have been demonstrated.
424 J M a r Sci Technol (2014) 1 9 : 4 1 5 ^ 2 4
3. As a result of the comparison w i t h the free-running model experiments, i t is concluded that the approach based on M e l n i k o v ' s method, that based on the C P L approximation method and that based on the polyno-m i a l approxipolyno-mation polyno-method provide sufficient accu-racy f o r this particular case. This is required f o r the vulnerability criteria i n the new generation intact stability code.
Acknowledgments The work was supported by the US O f f i c e o f Naval Research Contract No. 0014-06-1-0646 under the administra-tion o f Dr. Patrick Purtell. This research was partly supported by a Grant-in-Aid for Scientific Research of the Japan Society f o r Pro-motion o f Science (No. 24360355). The authors express their sincere gratitude to the above organization. Further, the authors are grateful to John Kecsmar f r o m A d Hoc Marine Designs L t d . for his compre-hensive review f o r preparing his paper as an expert i n small craft technology and native English speaker.
Appendix. Proof that the phase of the sinusoidal function can be neglected in the surge equation
The s i m p l i f i e d surge equation is as f o l l o w s :
( m -1- m . . , ) ^ G + / ? ( " ) ^ G - ! - ƒ s i n / : ( ^ G - ^ p )
= T{c,,., n) - R{c,.) ( A l ) where represents the phase o f wave-induced surge force.
W h e n
^ ' G = ^ G - è p ( A 2 )
the f o l l o w i n g can be obtained:
{m + / » , ) ^ ' G + Pin)i'G + ƒ s i n ^ 4 = r ( c „ . , n) - /?(c,„) ( A 3 )
Comparison o f A l and A 3 demonstrates that the phase,
^P, does not affect the surf-riding threshold.
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