Vol. 44 • No. 1 February 1997
T E C H N I S C H E U N I V E R S I T E I T
S c h e e p s h y d r o m e c h a n i c a A r c h i e f M e k e l w e g 2 , 2 6 2 8 CD D e l f t T e l : 0 1 5 - 2 7 8 6 8 7 3 / F a x : 2 7 8 1 8 3 6SCHirFSfiHIlK
Autonomous Oscillations and Bifurcations of a Tug-Tanker Towby Tao Jiang
Elimination of Wave Resistance by a C a m b e r e d T w i n - H u l l at Supercritical Speed
by Xue-Nong Chen
Prediction M e t h o d for Ship Manoeuvring Motion in the Proximity of a P i e r
by Katsuro K i j i m a
Optimal Pressure Distributions for R i v e r - B a s e d A i r - C u s h i o n Vehicles
by Lawrence J. Doctors
F a r - F i e l d and N e a r - F i e l d Dispersive Waves by Francis Noblesse and Xiao-Bo Chen
Viscous Effects in Seakeeping Prediction of T w i n - H u l l Ships by Helge Rathje and Thomas E. Schellin
T h e Nature of Nonlinear Effects in Ship Wave Making by Hoyte C. Raven
Shell-Function Solutions for Three-Dimensional Nonlinear Body-Motion Problems
by J. Andrew Hamilton and Ronald W . Yeung
P u b l i s h e d b y
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N
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U -
N
E
U -
N
E
U
Das Fachbuch
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H e r a u s g e b e r P r o f . D r . - l n g . H . K e i l 3 5 0 S e i t e n , F o r m a t 1 4 , 5 x 2 1 , 5 c m , z a h i r e i c h e S k i z z e n u n d T a b e l l e n , E f a l i n , D M 9 7 , 5 0 z z g l . V e r s a n d k o s t e n , i n k l . M w S t . I S B N 3 - 8 7 7 0 0 - 0 9 1 - 6 T E I L I SCHIFFBAU - SCHIFFSMASCHINENBAU Betriebsfestigkelt schiffbaullcher Konstruktionen - BeispieleProf. Dr.-lng. H. Petershagen, Dr.-lng. W. Fricke, Dr.-lng. H. Paetzold
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Vol. 44 • No. 1 • February 1997
This issue is dedicated to Professor Dr. Som Deo Sliarma at the occasion of his 60th birthday
Tao Jiang
A u t o n o m o u s Oscillations a n d Bifurcations of a T u g - T a n k e r Tow Ship Technology Research 44 (1997), 4-12
Nonlinear horizontal oscillations and bifurcations of a tug-tanker tow operating in calm water were investigated by locally linearized stability analysis, time-domain simulation and Poincare map. The influence of towhook location, towline length, and control parameters of a PID autopilot on the dynamic behavior of the tow was systematically studied.
Keywords: tug-tanker tow, autopilot, stabihty analysis, simulation, Poincare map, oscillations, bifurca-tions
Xue-Nong Chen
E l i m i n a t i o n of Wave R e s i s t a n c e by a C a m b e r e d T w i n - H u l l at S u p e r c r i t i c a l Speed Ship Technology Research 44 (1997), 13-21
A catamaran consisting of two hulls with suitably cambered centreline can be made to have, theoret-ically, zero wave resistance in shallow water at a chosen supercritical speed. A body plan for such a 'superconductive' catamaran is proposed. Its wave resistance is numerically compared to twice the wave resistance of a straight monohull, identical to one of the component hulls except for camber, and of a conventional catamaran with two straight hulls. The wave resistance of the proposed catamaran is lowest, not only at the design speed but also in a broad interval around it.
Katsuro Kijima
P r e d i c t i o n M e t h o d for Ship M a n o e u v r i n g M o t i o n i n the P r o x i m i t y of a P i e r Ship Technology Research 44 (1997), 22-31
The hydrodynamic interaction forces between two ships and between ship and pier are predicted to evaluate the influence of these interaction forces on ship manoeuvring motions. The calculation is based on the slender-body theory. The ship manoeuvring motion in the proximity of the pier is simulated. The effect of the lateral distance between ships and pier is investigated.
Keywords: hydrodynamic interaction force, ship manoeuvring motion, slender-body theory
Lawrence J . Doctors
O p t i m a l P r e s s u r e Distributions for R i v e r - B a s e d A i r - C u s h i o n Vehicles Ship Technology Research 44 (1997), 32-36
The wave resistance of an air-cushion vehicle traveling in a channel at a constant speed is considered. The cushion is compartmented into a number of subcushions with different pressures. An optimization process is carried out. It is possible to almost completely eliminate the wave resistance at speeds up to those corresponding to a Froude number of about 0.35. At higher speeds, substantial reductions (up to 57%) at a Froude number of 0.6 are achievable.
Keywords: wave resistance, ACV, optimization, shallow water
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Francis Noblesse, Xiao-Bo Chen
F a r - F i e l d a n d N e a r - F i e l d Dispersive Waves Ship Technology Research 44 (1997), 37-43
The usual representation of free-surface effects as a Fourier superposition of elementary waves is ex-pressed as the sum of a wave component and a near-field (local) flow disturbance. This Fourier repres-entation of free-surface effects is valid for arbitrary singularity distributions and for a wide class of water waves and other dispersive waves. The wave component accounts for the contribution (dominant in the far held) of the dispersion curves, defined by the dispersion relation characterizing the class of waves under consideration, and the contribution (negligible in the far field but significant in the near field) of the region of the Fourier space, called dispersion strips, centered at the dispersion curves. The near-field component, given by a regular double Fourier integral, excludes the contribution of the dispersion strips. Keywords: Dispersive wave, free-surface effect, diffraction, radiation, Fourier analysis, far-field wave, near-field
Helge Rathje and Thomas E. Schellin
V i s c o u s Effects i n Seakeeping P r e d i c t i o n of T w i n - H u l l Ships Ship Technology Research 44 (1997), 44-52
Our method to predict seakeeping behavior of twin-hull ships includes effects of viscous damping on hulls and lift and drag on stabilizing fins. The method solves the equations of motions by a modified panel method suitable for twin-hull ships and includes linearized semi-empirical expressions utilizing experimentally determined drag and lift data. The supplemental damping on hulls und stabilizing fins is treated as quasi-steady response to harmonic exciting forces caused by free-surface waves. The sensitivity on vessel response of drag coefflcients and lift gradients for hulls and stabilizing fins is discussed. Keywords: seakeeping, viscous effect, twin-hull, ship motion, drag, lift, equivalent linearization
Hoyte C. Raven
T h e N a t u r e of Nonlinear Effects in Ship Wave M a k i n g Ship Technology Research 44 (1997), 53-61
Nonlinear effects in ship wave generation are reviewed. The magnitude and nature of nonlinearities in the free-surface boundary conditions are studied by comparing wave pattern predictions by linearised and nonlinear methods. Making a distinction between transfer effects (satisfying the free surface condi-tion at its true locacondi-tion) and refraccondi-tion effects (exact, not free-stream free-surface condicondi-tion) provides explanations for the underestimation of bow wave heights by linearised methods; for the even larger underestimation of diverging bow waves at a distance from the hull; and the phase differences in hull wave profiles.
Keywords: Free surface condition, wave resistance, Rankine panel method, nonlinearity, bow wave, ray theory, wavemaking
J. Andrew Hamilton, Ronald W. Yeung
S h e l l - F u n c t i o n Solutions for T h r e e - D i m e n s i o n a l Nonlinear B o d y - M o t i o n P r o b l e m s Ship Technology Research 44 (1997), 62-70
The shell-function method for time-dependent free-surface fiows efficiently computes fluid-body interac-tions by decomposing the fluid domain into an outer and inner region and recognizing certain universal functions, independent of the specific problem geometry, that need to be computed only once. The method is implemented using an isoparametric boundary-element formulation. Results are given for several linear and nonlinear body-motion problems.
Keywords: free-surface flow, CFD, panel method, time-dependent Green function, Rankine source method, wave-body interaction, higher-order boundary element
Autonomous Oscillations and Bifurcations
of a Tug-Tanker Tow
Tao Jiang, Mercator University^
1. Introduction
To salvage a large tanker disabled in the open ocean, i t is usually towed by a tug into sheltered waters. However, the operation of such a tow may be impaired by large-amplitude, low-frequency horizontal oscillations of both vessels and associated high tension-peaks in the stretched towline, even in a seemingly innocuous steady environment, frequently due to dynamically unstable equilibria.
In studying this dynamic stability problem, the low-frequency horizontal motions (surge, sway, yaw) of the vessel can be reasonably treated as decoupled from the high-frequency vertical motions (heave, pitch, roll). The local dynamics of a towed ship about its equilibrium have been frequently analyzed by classical linear stability theory. The towing system was simplified by assuming uniform tug motion, rigid towline and small towed-ship excursions, Strandhagen et
al. (1950). The global dynamics of a towed ship have been investigated by applying modern methods of nonlinear dynamics. The towing system was still simplified by assuming uniform tug motion but the towed ship was allowed to make large excursions in the horizontal plane under an elastic towline, Bernitsas and Kekridis (1986). Jiang (1996), Jiang and Sliarma (1997) proposed a more general model of the complete coupled system having six degrees of freedom, namely surge, sway and yaw motions of the tug and of the towed ship. Linear and nonlinear hydrodynamic forces associated with ship motions were modeled according to the four-quadrant maneuvering model of Sliarma (1982). The nonlinear restoring forces of an elastic towline were empirically approximated. Numerical simulations showed that the dynamic behavior of the towed ship can be qualitatively different in the unreaHstically simplified subsystem from that in the more realistic coupled system. A suitable PID autopilot on the tug can effectively stabilize the equilibrium of the entire tow, thereby reducing not only the excursions of the tug but also of the towed ship.
Based on this comprehensive mathematical model, the focus is here on nonlinear oscillations in the state space and bifurcation phenomena in the parameter space of a tow configuration. The basic restriction is still that the subject tug-tanker tow operates in calm water, implying that the tow is an autonomous system. Suitable methods of nonlinear dynamics such as locally linearized stabihty analysis, time-domain simulation and Poincaré map are applied to this autonomous system. Special interest lies on the influence of towhook location and towline length as well as of autopilot parameters on the stabihty behavior of the tow.
2. Mathematical model
To describe the arbitrary motion of a tow in the horizontal plane, an earthbound coordinate system OoOioyo is used, Fig. 1. Its origin Oo can be freely chosen. Without loss of generality, the a;o-axis is collinear with the reference course denoted by iptei- For describing the towed-ship motion as observed on the tug, a relative coordinate system Oi-Treij/rel is introduced, which parallels the earthbound coordinate system and originates at midship point O i of the tug. Following standard practice in ship maneuvering, two shipbound coordinate systems Oia;iyi and 022:21/2, centered at midship point O i of the tug and at midship point O2 of the towed ship, respectively, are used to simplify the description of external forces. The towhook location on the tug is symbolized by A i ; the towline attachment-point on the towed ship, by A2; the horizontal distance between A i and A2, by L t - The towhne direction in the horizontal plane is identified by
t/^T-'Inst, of Ship Technology, Mercator University, D-47048 Duisburg, Germany; jiang@nav.uni-duisburg.de
o„
Fig. 1: Schematic of tow configuration and coordinate systems
The velocity components of the towed ship relative to the tug (irel, ^rel) are Xre\ =
(•"2 cos V'2 " "2 sln V'2) "
{^1 COS V»! " " l siu V'l),yrei = («2 cos tp2 + ""2 sln V'2) - (^^1 COS if^i + t i l sin V'l), (1) with velocity components ( « i , ui) of the tug and («2, "2) of the towed ship resolved along ship-bound coordinates (a'l, 2/1) and {x2, ^2), respectively. The heading angles V i and ip2 are related to the corresponding yaw rates n of the tug and 1-2 of the towed ship by ^1 = lSO°/n • r i and ^2 = 18077r • r2.
The tug and the towed ship are treated each as a transversally symmetric rigid body having three dominant degrees of freedom: surge, sway and yaw. Considering a tow operating in calm water, the horizontal dynamics of the tow can then be described by three state equations
h
M j - ^ F i v i + F l i + F c i + F p i + F r i + F t i ] (2)
for the tug and additional three state equations
= M2 ^ [FlV2 + Fl2 + Fc2 + FP2 + F r 2 + Ft2]
U2(3)
for the towed ship. M i and M2 represent the generalized inertia matrix (comprising the body mass as well as the added mass in the ideal fiuid) of the tug and of the towed ship, respectively. The force-couple F i v expresses the additional body inertia and ideal fluid effects arising from the use of rotating coordinate systems as well as the Munk moment. The subscripts L, C, P and R stand for hydrodynamic l i f t , cross-flow, propeller and rudder eff'ects, respectively. According to the four-quadrant maneuvering model of Oltmann and Sharma (1984), these effects are analytically expressed as nonlinear functions of velocities, propeller rate and rudder angle. One special feature of this model is the exphcit formulation of realistic three-way hull-propeller-rudder interactions for a conventional propeller and hull-propeller-rudder behind a ship, such as supertanker or container carrier. However, for an ocean tug with two steerable Kort nozzles considered here, the description of the propeller and rudder forces has to be modified. Following Jiang (1996), the propeller and Kort nozzle are treated as one control organ, due to their non-separable hydrodynamic interactions. The force-couple F t denotes the horizontal effects of the towline tension. For a nylon rope i t can be empirically approximated as a simple nonlinear function of its elastic elongation.
An autopilot based on simple PID control is commonly installed on tugs. The measured or simulated heading is compared with the desired reference heading and the error found is used
as input to the controller. The output of the controller is fed to the rudder servo. The current rudder angle resulting from the control dynamics can be generally approximated by using a time lag T R introduced by the hydraulic steering gear. The associated rudder angle is treated as a state variable governed by
<^i = ^ ^ ^ with S* =
kp[i;i{t)
~7Pre{]+ kiiPj{t) + koMt)
(4) within the deflection range \Si\ < Suax dictated by the mechanical rudder arrangement andsubject to the available rudder-rate range \Si \ < ^^ax imposed by the steering gear. Proportional factor kp, integral factor kj, and difi'erential factor k^ are the control parameters. The heading-error integral tpj- is governed by
= -01 - Aei- (5) The trajectory of midship point O i of the tug in the horizontal plane can be evaluated by
time integrals of the tug velocity components ( i o i , yoi)
i o i = UicosV'i - uisinV»!, yoi = f i cos + u i sin V'l. (6) The trajectory of midship point O2 of the towed ship is obtained by coordinate addition
Xo2 = a^ol + Xre\, yo2 = Vol + yrel- (7)
(iCrel, yrel) are time integrals of relative velocity components, see (1).
3. Methods of investigation
For the numerical investigations performed here, the system dynamics represented by kin-ematic equations, dynamic equations and control equations are unified in canonical form as a generalized state equation:
x( t ) = f( x( 0, c ) (8)
which expresses the rate of change of state vector x, comprising 12 state variables
X = (a;rel, yrel, V^l, i^2, «1, " i , Ti, U2, «2, ^2, IpJ, Si)'^ (9) with respect to the time t as independent variable. Vector f is a nonlinear function of the
state vector x and of the time-independent parameter vector c. Vector c comprises all system parameters required for the specification of forces, including operational parameters as well as control parameters. Since the governing equation (8) does not contain time explicity, i.e. there exists no time-dependent external excitation, the dynamic system of a tow in calm water is an autonomous system.
To analyze the local dynamic behavior around an equilibrium in the autonomous system, the classical Unear stability theory can be apphed, Guckenheimer and Holmes (1986). I t comprises: (i) Identification of equilibrium x e in state space as iterative solution of the nonlinear algebraic equation
f( x E , c ) = 0. (10)
(ii) Linearization of the state equation around the equilibrium
y = A y (11)
with the Jacobian Matrix A = 9 f ( x e , c ) / 9 x and the perturbation vector y = x x e
-(iii) Evaluation of the eigenvalues of the Jacobian matrix A from
| A - I c 7 | = 0. (12)
Tug Autopilot with control equation: 6* = fcpCVi - \Kref) + ^iV/ + ^ d V i
x(0),c
x(r)
Tug - Tanker Tow with dynamic state
equation: x( 0 = f(x(/),c)
herein:
Xi2 = 8, = L
x(r +AO
Fig. 2 Schematic of numerical simulation of PID tug autopilot and tow dynamics
I is the unit matrix having the same dimension as A , and a denotes the eigenvalues. In the generic case, either all real parts are negative definite implying stable equilibrium, or at least one real part is positive definite indicating unstable equilibrium. The degenerate case of at least one real part being zero defines a center manifold in state space as well as a bifurcation point in parameter space.
The global solution of the nonlinear difi'erential equation (8) can be generally approximated by time-domain numerical simulation. Fig. 2. Due to the numerical stifi'ness of the dynamic system, the subroutine LSODA was applied to ensure efficient numerical integration. It works with controlled accuracy and variable time-step and is available via pubUc-domain library ODE-PACK.
To study the bifurcation phenomenon in a nonlinear dynamic system, the method of Poincaré map can be applied, where the continuous system is simplified by using time-discretization at a suitable Poincaré section in the state space, Thompson and Stewart (1986). Such section must fulfill the transversality condition
f - n = x - n / 0 . (13) n is the normal vector of the Poincaré section, i.e. the fiow governed by differential vector field
(8) crosses this section. In an autonomous system the stroboscopic sampling of the maximum or minimum values of one state variable satisfies the above condition and can be well implemented in numerical integration. Suppose the maximum heading angle of the tug is sampled, then the following relations hold
V^i = 0 and -i/ji < 0 (14)
which define the Poincaré section at
X = (a-'rel, yrel, V»!, V'2, «1, «1,0, «2, "2, ''2, tpj, ^ l ) ' ^ (15) with the normal vector of n = (0, 0, 0, 0, 0, 0,1,0, 0,0,0,0)'^ and satisfy the condition
Table I : Principal particulars of tug and tanker; relevant operational and control parameters
Item Dimension Tug Tanker
Hull
Length between perpendiculars [m] 60.17 325.0
Length of waterhne [m] 65.808 335.0
Beam N 16.26 53.0
Draft [m] 6.75 21.79
Block coefficient 0.539 0.829
LCB f w d of midship section x q [m] -0.544 10.35
Radius of gyration (2—axis) [m] 15.12 81.25
Propeller
Number 2 1
Diameter N 3.472 9.10
Pitch ratio 1.26 0.715
Expanded area ratio 0.58 0.682
Number of blades 4 5
Screw sense outward righthanded
Rudder Kort nozzle-rudder
Number of rudders 2 1
Rudder area
[m2]
24.65Chord length [m] 9.0
Aspect ratio [m] 1.54
Operational parameters
Unloaded towline length L t u [m] variable within (50, 1000)
Attachment point coordinate [m] variable within (-25, 0) 162.5
Attachment point coordinate yx [m] 0 0
Control parameters
Propeller rate n [min~^] 140 0
Rudder angle 5 • [ ° ] ' variable within (-40, 40) 0
Rudder speed 5 [7s] variable within (-2.5, 2.5) 0
Autopilot parameters Time lag t r [s] 2.0 Proportional factor k-p Integral factor k\ [s-1] variable within (0, 4) variable within (0, 0.3) Differential factor fco [s] variable within (0, 8)
This kind of Poincaré map is particularly suitable for application to a multi-dimensional system such as in the subject tow. The global bifurcations can be analyzed by point mapping at the Poincaré section, Kreuzer (1987).
4. Results and discussion
The ESSO OSAKA was chosen as exemplary tanker, and a typical ocean tug with two steerable Kort nozzles as exemplary tug (Table I ) .
The specific values of more than 50 parameters, which depend on the geometry of hull, propeller and rudder and which are required for the specification of hydrodynamic forces in the four-quadrant model, were identified from planar motion tests with captive models (tanker: V B D No. 1238, scale 1:65; tug: HSVA No. 2509, scale 1:16) in the Duisburg Towing Tank ( V B D ) , Jiang and Sharma (1997). The reference values (including the range examined) of design, operational, control parameters as well as autopilot parameters were carefully chosen
to simulate realistic tug operation; they are also given in Table I . Specific values of all variable parameters, as used in the individual numerical analysis, are stated in the corresponding figure captions.
To study the effects of relevant parameters on the dynamic behavior of the subject tug-tanker tow, locally linearized stability analysis, time-domain simulation and Poincaré mapping were systematically performed. The numerical results are individually shown with one parameter changing within its examined range, while the other parameters keep their constant reference values. 600 ^ 400 ^ 200 •3 -200 -400
Hopf bifurcation Maximum value
Stable equilibrium
Unstable equilibrium
Minimum value
200 400 600 Towline length L^^ [m]
(a) Bifurcation diagram in phase-parameter section
800 1000
50 100 150 200 250 300 350 400
L t u M
(b) Real part of one crucial complex eigenvalue pair 5.2 15.0 ,J'4.8-|4 .6-'4.4 Tanker 130 135 140 145 150 Advance [km] (d) Trajectories for 1^^=250 m
Eigenperiod Period of limit cycle 50 100 150 200 250 300 350 400
L^u [m] (c) Period corresponding to complex eigenvalue pair in (b) „ 900-, '^SOO-•| 6oo^ ^ 500 10.0 10.5 11.0 11.5 Time [h]
(e) Towline tension for 1^^=250 m
12.0
Fig. 3: Effect of towline length on dynamic behavior of subject tug-tanker tow; reference parameter values: , i ' a i = - 5 m, kp = 2.0, kj = 0.1 s~\ fco = 4.0 s
Fig. 3 summarizes the relevant results of varying towline length. As the towhne length increases, a transition from stable equilibrium (solid line) to unstable equilibrium (dashed line), obtained by local stability analysis, occurs approximately at a length of 210 m, see the phase-parameter section spanned by relative transfer and towline length in graph (a). The bifurcation point is identified by the intersection of the real part of one complex eigenvalue pair with the stability limit defining a Hopf bifurcation, graph (b). The simulated stable equilibria (o) agree fully with those from local stability analysis, graph (a). The time-domain simulations show the qualitative change of asymptotic behavior (attractor) from a fixed point to a stable limit cycle bounded by the minimum ( A ) and maximum ( y ) values resulting from Poincaré mapping, graph (a), indicating that the Hopf bifurcation found is supercritical. Graph (c) compares the eigenperiod defined by the imaginary part of this crucial complex eigenvalue pair (dashed line) with the period of the limit cycle determined by the Poincaré map (solid line). The period difference is practically irrelevant. Amplitude and period of the limit cycle increase with the towline length. The large excursions in the horizontal plane can be illustrated by plotting the trajectories of the tug (solid line) and of the tanker (dashed line), graph (d), having the same order of magnitude as the towline length. The extremely long period of oscillation is manifested by plotting the time history of towline tension in graph (e); it is characteristic of the fish-tailing instabilities in a tow. B 100 50 0 -50 -100
Maximum value gta^le equilibrium
tJnstable equilibrium
Hopf bifurcation Minimum value *
-30 -25 -20 -15 -10 -5 0 Towhook location x^i [m]
Fig. 4: Bifurcation diagram in phase-parameter section showing effect of towhook location on dynamic behavior of subject tug-tanker tow; reference parameter values: X t u = 150 m,
kp = 2.0, ki = 0.1 s - i . Aid = 4.0 s
Towhook location can be considered as a design parameter. Fig. 4 shows its effect on the dynamic behavior in a phase-parameter section spanned by relative transfer and longitudinal coordinate of the towhook location. As the towhook location is moved towards the tug stern, a supercritical Hopf bifurcation occurs at a value of approximately -14 m, leading to the transition from stable equilibrium (solid hne) to unstable equilibrium (dashed fine), obtained by local stability analysis, as well as to a quahtative change of attractor form from a fixed point to a stable Hmit cycle bounded by the minimum ( A ) and maximum ( y ) values determined by the Poincaré map. Further investigations by using Poincaré map showed that the period and amplitude of oscillation after this bifurcation have the same order of magnitude as those in Fig. 3. A practical recommendation would be to locate the towhook near the midship point of the tug.
Turning now to the effects of the PID control parameters, Fig. 5 collects important results obtained by varying the proportional factor. As the proportional factor decreases, linearized stability analysis reveals the transition from stable equilibrium (solid line) to unstable equi-hbrium (dashed line), occurring at kp ?a 0.7, see the phase-parameter section spanned by tug heading angle and proportional factor in graph (a). The bifurcation point is a supercritical Hopf bifurcation due to the intersection of the real part of one complex eigenvalue pair with
the stabihty hmit in graph (b) and due to the quahtative change of attractor form from a fixed point to a stable limit cycle bounded by the minimum ( A ) and maximum ( v ) values resulting from the Poincaré map in graph (a). Graph (c) compares the eigenperiod defined by the imaginary part of this crucial complex eigenvalue pair (dashed line) w^ith the period of the limit cycle determined by the Poincaré map (solid line). In contrast to the extremely long period of the limit cycle in Fig. 3, the period here is quite moderate, about 1 min; this is typical of the oscillations resulting from a dynamically unstable autopilot. Although this instability, caused by undercontrolling, hardly influences the motions of the towed tanker (dashed line), the associated tug oscillations (solid line), graph (d), lead to a permanent operation of the tug rudder of large amplitude and relatively short period, graph (e), meaning a bad autopilot.
Such dynamic instabilities of an PID autopilot are also caused by overcontrolling through integral factor ki or by undercontrolling through differential factor ^ d , Fig. 6 (a) and (b), respectively. The associated bifurcations were again examined by local stability analysis as well as by Poincaré map and were found to be supercritical Hopf bifurcations in both cases.
30 20 10 -20 -30 Maximum value Hopf bifurcation Stable equilibrium Unstable equilibrium Minimum value 1 Proportional factor kp
(a) Bifurcation diagram in phase-parameter section
20-2 0 1.85 1.80-^ Stability limit 0.5 1.0 k„ 1.5 2.0 (b) Real part of one crucial
complex eigenvalue pair
35.0 35.5 36.0 36.5 Advance [km] (d) Trajectories for kp=0.5 37.0 1.0 0.0
_Period of limit cycle
EigenpCiod 0.5 1.0 k„ 1.5 2.0 30 20 ^. 10-1 M 0 ^ -10 -20 (c) Period corresponding to complex eigenvalue pair in (b)
Tug and — - tanker heading Tug rudder
3.00 3.04 3.08 3.12 Time [h]
(e) Time histories for kp=0.5
3.16
Fig. 5: Effect of proportional factor of the PID autopilot on dynamic behavior of subject tug-tanker tow; reference parameter values: L t u = 150 m, a;Ai — - 5 m, ki = 0.1 s~^,
^ O Hopf bifurcation Stable equilibrium Maximum value Unstable equilibrium Minimum value 0.1 0.2 0.3 0.4 (a) Integral factor kj [s ] 0.5
4-(JO
I
0 --4 Maximum value Hopf bifurcationUnstable equilibrium Stable equilibrium
Minimum value
0.0 0.5 1.0 1.5 2.0 2.5 (b) Differential factor kj, [s] 3.0
Fig. 6: Bifurcation diagrams in phase-parameter section showing effect of integral factor (a) and differential factor (b) of the PID autopilot on dynamic behavior of subject tug-tanker tow;
reference parameter values: L t u = 150 m, xai = - 5 m, kp = 2.0 5. Conclusion
Locally linearized stability analysis, time-domain simulation and Poincaré mapping were successfully applied to investigate nonhnear horizontal dynamics of a tug-tanker tow in calm water. Whereas the local stability analysis yielded Hopf bifurcation points in relevant parameter subspaces, defined by the intersection of the real part of one crucial complex eigenvalue pair with the stability limit, the Poincaré map revealed that these Hopf bifurcations are supercritical, identified by qualitative change of the attractor form from a fixed point to a stable limit cycle (here autonomous or self-sustained oscillation). The motions of both vessels after a Hopf bifurcation in the parameter section spanned by towhook location and towhne length are characterized by extremely long-period oscillations associated with the so-called fish-tailing instabilities of a tow proper. On the contrary, after a Hopf bifurcation in the subspace of control parameters of the PID autopilot, oscillations are only observed in the tug motions, with a moderate period corresponding to the typical dynamic instabilities of an autopilot.
References
BERNITSAS, M.M.; KEKRIDIS, N.S. (1986), Nonlinear stability analysis of ship towed by elastic rope, J. Ship Res. 30/2
GUCKENHEIMER, J.; HOLMES, P. (1986), Nonlinear oscillations, dynamical systems, and bifurca-tions of vector fields, Appl. Math. Sciences 42, Springer
JIANG, T. (1996), On the dynamic instabilities in a towing process and the possibilities of stabilization, Jahrbuch Schiffbautechnische Gesellschaft 90 (in German)
JIANG, T.; SHARMA, S.D. (1997), Dynamic stabilization of a tug-tanker tow by applying active control on the tug, 11. Ship Control System Symp., Southampton
KREUZER, E. (1987), Numerische Untersuchung nichtlinearer dynamischer Systeme, Springer
OLTMANN, P.; SHARMA, S.D. (1984), Simulation of combined engine and rudder maneuvers using an improved model of hull-propeller-rudder interactions, 15. Symp. Naval Hydrodyn., Hamburg
SHARMA, S.D. (1982), Kriifte am. Unter- und Überwasserschiff, 18th training course Institut fiir Schiff-bau, Hamburg
THOMPSON, J.M.T.; STEWART, H.B. (1986), Nonlinear dynamics and chaos, John Wiley & Sons STRANDHAGEN, A.G.; SCHOENHERR, K.E.; KOBAYASHI, P.M. (1950), The dynamic stability on course of towed ships. Trans. SNAME 58
Elimination of Wave Resistance by a Cambered T w i n - H u l l
at Supercritical Speed
Xue-Nong Chen, Universitat Stuttgart^ 1. Introduction
Chen & Sharma (1996a,b) found that the wave resistance of a single-hull ship at supercrit-ical speed in a rectangular channel of suitable width can be made to vanish totally within the framework of a linear shallow water wave approximation and, furthermore, even in more ac-curate nonhnear models, namely, Kadomtsev-Petviashvili (KP) and Boussinesq models. The mechanism is that the bow wave, after reflection from the channel sidewall, hits the afterbody and counteracts the stern wave so that the resultant wave in the ship wake disappears. By analogy to electrical conductors, the term "shallow channel superconductivity" was proposed for this phenomenon.
The favorable eff'ect of destructive interference between bow and stern waves in open deep water on wave resistance has been well-noticed and investigated. In principle, ship waves can be completely eliminated in some cases, e.g. at certain speeds, within hnear theory (Tuck 1991, Tuck and Tulin 1992, Tulin and Oshri 1994). The favorable wave interference between twin hulls at supercritical speed in shallow water is more signiflcant and less sensitive to speed than in deep water. This has been reported, on the basis of hnear theory and model experiments, e.g. by Eggers (1955) and Heuser (1973), and by Chen and Sharma (1994a) during the validation of a nonlinear theory by model experiments. While investigating a ship moving in a channel off the centerhne, they found for a Series 60 ship model, in both numerical computation and model experiments, up to 30% wave resistance reduction in the supercritical speed range at the best off-center track compared to centerhne motion. The reason is that wave dispersion is much weaker in shallow water than in deep water, especially in the supercritical speed range. As distinguished from Kelvin's wave pattern in deep water, or even in shallow water at subcritical speed, the ship wave pattern at supercritical speed is similar to shock waves in supersonic flow. The bow and stern waves appear as a wave crest and trough extending to infinity along oblique characteristic lines with nearly permanent wave forms. In a special case a suitable ship forebody can generate a permanent oblique solitary wave on each side (Mei 1976). If the wave-crest meets the wave-trough, they can largely cancel each other. In this sense, superconductive ships are more feasible in shallow water.
Busemann (1935) proposed already to cancel the aerodynamic wave drag at supersonic speeds by means of a biplane configuration in which the two foils form 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 fiow under the two keels. While the wave resistance of a ship in a narrow channel can be made to vanish completely, i t seems that the wave resistance reduction by optimal wave interference between the straight twin hulls of a conventional catamaran in shallow but 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 two hulls has a cambered centreline.
The theoretical basis used here for treating 3-D flow effects in a catamaran huh on shallow water is the concept of blockage of the transverse flow (induced by the other huh and by the camber) around a hull section in shallow water. The blockage is quantified by an asymptotic potential difference between the two sides (Newman 1969). This concept was exploited also by (Chen & Sharma 1994a) to treat the asymmetric flow about a slender single-hull ship moving asymmetrically in a channel. Generalization to locally varying yaw angle allows the treatment
of a cambered hull and, hence, the determination of camberlines such that waves are created on one (the inner) side of a catamaran hull only! To obtain a superconducting catamaran, such a cambered hull and its mirror image (the other catamaran hull) have to be configured so as to ehminate waves behind the catamaran. This is achieved by applying a two-soliton solution as in Chen & Sharma (1996a) to fit the afterbody to the forebody with respect to both cross-sectional area and camber. I t means that at the designed supercritical speed the bow wave generated only on the inner side by each of the two forebodies hits the other afterbody and cancels its stern waves on both sides. Consequently, the catamaran has theoretically zero wave resistance, and this state depends no longer on channel width.
2. A Simple T h e o r y for a C a t a m a r a n with C a m b e r e d Hulls 2.1. Formulation of the mathematical model
Consider a catamaran with two hulls of length /*, beam 6* and draft c/* moving along the centerline of a shallow channel of water depth h* and width w* at speed U*. Each hull can be curved as defined by a local yaw angle tp{x), Fig. 1. The two hulls are mirror-symmetric and arranged symmetrically about their common centerplane, which coincides with the channel centerplane. Ideal flow is assumed. The ship is free to t r i m and sink. The same symbols are used to denote dimensional variables and their nondimensional forms, the former being marked with asterisks. Unless otherwise stated, all variables are nondimensionalized by reference to water depth h*, gravity acceleration g* and water density p*. For example, / = l*/h* is nondimensional ship length. So = Sg/h*"^ is nondimensional midship section area, s = s*/h* is nondimensional separation between the mid-points of the midship sections of the two hulls, etc. The nondimensional speed U = U*/y/g*h* is usually called the depth Froude number
Fnh-z
y = -0 y = +0
symmetric centerline channel bottom \ i i i i i \ i i i i i ' i Fig. 1: Scheme of the problem and the near-field cross-flow
Due to symmetry, only the port half is considered. A Cartesian coordinate system Oxyz moving at the same speed as the ship is used with origin O located in the center-point of the port hull, plane Oxy on the quiet free surface, z positive upward, and x positive forward. The flow is then governed by the Laplace equation in the fluid domain and boundary conditions on the shiphull surface, free surface, channel bottom and sidewalls.
The problem will be solved by matched asymptotic expansions, Chen and Sharma (1994a,1995). The flow region is divided into near field and far field. In each field multiple-scale expansions are applied, and the mathematical model is closed by asymptotic matching. The model is simplified for both fields to obtain analytic solutions and to design a catamaran explicitly. Therefore, the far-field equation is just repeated with 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 it is not the best in the supercritical range, Chen and Sharma (1996b). Its stationary form is
1/2
(1 - U'^)(px:v + <^yy + ^U(p^(fi^^ + -—^:oxxx = 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. Fig. 1. Two 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 to 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,
1 ^ O
Vn\y=±0 = T ^ V r ^ + Vn, (2)
Aip = 2VnC{x). (3) A(p = <^|y=+o — 'p\y=-o is the potential jump, y = ± 0 or 0^ the port/starboard side of the
port component hull, Vn the local normal velocity, Vn its mean value, Vr the local tangential velocity, S{1) the lengthwise cross-sectional area distribution, and C{x) the blockage coefficient for the cross section at a;. In terms of far field variables, the normal and tangential velocities can be expressed and approximated by neglecting disturbance velocities ip^i f y , which are small compared to the mean velocity U, as
Vn = ^ c o s V » - ( t t ^ - f^) sin ~ T T ^ c o s V ' + f / s i n V», (4) oy ox oy
Vr = - ( ^ - U)cosil) - —sinij^ ^ Ucosip. (5) ox oy
Substituting (4), (5) and (3) into (2), one obtains an approximate matching condition: dip 1 dS Aip
Q^\y=±o - -d[ ^. = ± 0 = T:^U— + - UUml^ix). (6) 2 C ( x ) c o s V ' ( . t ) If the local yaw angle ip{x) is smaU, cosV» ~ 1 and dS/dl ^ dS/dx result in
This will serve as a simplified boundary condition at the ship location for the K P equation (1). See Chen and Sharma (1994a,1995) for a more accurate boundary condition involving the disturbance velocity and local free-surface elevation. The model stih has to be closed by imposing the boundary condition ipy = 0 on the channel sidewaü and centerplane, and the K u t t a condition at the ship stern x = -1/2:
'Px\y = 0+ = ^x\y = 0- • (8)
2.2. Superconductive catamaran solution
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 two steps. First, non-existence of outward waves requires that the transverse velocity must vanish on the outer side:
The potential along the outer side of the port hull, in the far-field point of view, is then constant. As there is no discontinuity at the bow, this constant is equal to the potential on the inner side at the bow:
By using (7) and (10), the no-outward-wave condition (9) yields a prescription for the camberline:
(p{x,Q~) must be determined by the inner side condition. Substituting (11) into (7) for the inner side y = 0~ simply provides the boundary condition on the inner side:
dip _ dS
d^^'=-'-^d^ y=-0 = U— . (12)
So far i t has been ensured that a hull curved according to (11) will 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 that for a straight hull of same cross-sectional area. Consequently, all the waves are generated on the inner side. Normally, they would extend into the wake behind the catamaran, with or without reflection from the inner side of the other huU. The next step, therefore, must ensure that the waves are confined to the area between the two hulls. As the reduced problem (1) with (12) is exactly the same as that of a straight hull with twice the displacement, all that is needed is merely an application of the original idea of shallow water superconductivity in Chen and Sharma (1996a). Second, non-existence of wake waves requires that the stern waves be canceled by the bow wave of the opposite hull. The cross-sectional area distribution should be so selected that 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. it will be perfectly absorbed. The bow wave can be an obhque solitary wave. Thus the two oblique solitary waves stemming from the two forebodies constitute a configuration of an oblique interaction of two identical sohtons in the area between the two hulls, and the two afterbodies are then determined by the incident wave. This two-soliton solution of K P Equation (1) can be used to derive the ship shape, Chen and Sharma (1996a). The major expressions needed to specify the catamaran are:
the potential on the inner side, which is used in (11):
Wk (:os\\(2kxi) + J~A2e^\){2kx) . , , \/^AÜ
^(3-',0 ) = — — : . — r 7 : 7 H : with k = - ^ ^ 7 7 - , A2
3 cosh(2A;.Ti)-I-VA2Cosh(2fc.T) 2U '
U ^ - l - AU [/2 - 1 - 4 A C / ' (13) the cross-sectional area distribution, which is obtained inversely from (12) as </?y|y_o- is known, being half the value given by (A 8) in the Appendix of Chen and Sharma (1996a):
8 - 1 - Af/sinh(2fc.Ti) 3 cosh(2A;.Ti) + y/M cos\\{2kx) and the mean separation between the two hulls:
(14)
s = 4.Ti/v/f/2 - 1 - AC/. (15) A and xi are two parameters, free to be chosen for designing various hulls; theoretically xi can be arbitrary, and 0 < A < (C/^ — l ) / ( 4 { / ) . The theoretical ship length / is infinite, but Sa{x) vanishes exponentially at infinity so that it can be safely truncated. Since the disturbance velo-city at the stern on the inner side also vanishes at x = - 0 0 , the K u t t a condition (8) is satisfied. Actually, to construct the catamaran, the two above steps are reversed, i.e. one starts with the two-soliton interaction solution between the two hulls and then derives the cross-sectional area distribution and the local yaw angle.
3. Proposed Design of a Superconductive C a t a m a r a n
In this section a possible body plan design is proposed. In 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 coefhcient; (ii) the velocity-potential jump 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 I* = 60 m, draft d* = 2.5 m, and design speed U* = 42.84 km/h (corresponding to a depth Froude number Fnh — 1-7). The cross-sectional area distribution, which automatically determines the displacement volume, the camberline and the mean hull separation will now be derived from the superconductive solution, choosing reasonable values A = 0.20, xi = 2.5 for the two free parameters. Other details of huh geometry such as beam, body plan, and wetted surface area will be decided by practical considerations. For better overview, the final results are given here in advance: mean huh separation s* = 20.08 m, beam 6* = 4.992 m, midship section area 5* = 11.224 m^, displacement volume V* = 332.4 m^, and wetted surface area 5* = 371.55 m^, leading to hullform coefficients CB = V*/{l*dlb:) = 0.444, CM = St/{d*X) = 0.8994 and Cs = S*^/VVH^ = 2.631. The derivation now follows. Since Sa{x) is an even function, 5'a(-//2) = 5'a(//2), the following truncation yields a finite ship length /:
S{x) = Sa{x)-Sa{l/2), (16) with S{-l/2) = S{l/2) = 0. Since Sa{l/2).lin, if 1/2 - xi is large enough, the practical
cross-sectional area distribution S{x) is close to the theoretical one Sa{x). The displacement of each component hull is
[•1/2
V* = h*^ / S(x).dx = 332A (17)
J-l/2
The slenderness ratio V/l^ of this hullform is very close to that of a projected catamaran in Heuser (1973).
The theory does not specify any detailed geometry of the cross sections. I t can be designed by practical considerations. Here, for simplicity, mathematical hull lines described by exponential functions are used:
y{x, z) = ±- y / f , , , (1 - e x p { - ƒ (.T)[. + d{x)])), (18) l-QX^{-f{x)d[x)}
y = ±\b{x) is the waterline,
z = -d{x) = -do {I - 0.9exp[-1.2(,x- + 6 ) ] } { l - 0.99exp[6(.T - 6)]} (19) the keel-hne, and f{x) a parameter function chosen to be f{x) = 20 sech(0.3.T). Integrating
(18) over -d{x) < z <0 yields
Since S{x) is known from (16), b{x) can be determined from the above equation as
h( \ - ^ l \ 1 - e x p { - / ( a - - ) r / ( x ) } .^n '^•'^ - ' ' ^ • ' ^ d ( , x - ) + j J ^ [ e x p { - / ( , x - ) c f ( . T ) } - 1 ] •
The hull camber is determined by theory as moderated by practical requirements of real flow. For the given clearance c = I - do = 0.5, the theoretical yaw angle according to (11) together with (13) is too large, the maximum value being about 18°. To obtain a more suitable camber,
the fohowing considerations are introduced: (i) Any 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 than the theoretically predicted value, (ii) The theoretical model based on potential theory with 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 jump across the hull. Since the potential jump is determined by the outer field solution, it cannot account for this effect. To keep the term A.ip[x)/C{x) in (7) realistically, especially in the afterbody, either the blockage coefhcient should be increased or the potential jump decreased artificially. The former artifice, tried in Chen and Sharma (1994a), improves the prediction of side force and yaw moment on a ship at drift angle. Therefore the theoretical potential jump in (11) is reduced by dividing i t by the function
t a n ^ ( . ) = - 1 ^ + ^ ( / / 2 , 0 - ) - ^ ( x , 0 - ) 1
' 2dx^ h{x) 2UC{x) (22)
where fd{x) = 2 . 5 + (6 - a;)/12. Since dym/dx = ta.nip{x), integrating (22) once yields the camberline ym (x). AdditionaUy, a suitably sized small skeg should be fitted under the keel in the afterbody to increase the blockage coefficient to approximate
C{x) = 0.95 exp(-0.18a;). (23) The actual design may have to be done by experiment rather than by theory. Fig. 2 shows the difference made by the skeg (solid line) and the theoretical blockage coefficient (dotted hne) for a rectangular cross section of beam b{x) and draft dg, Taylor (1973):
C{x) = h { x ) { l / c - l ) + ^[l (24)
c = 1 - do is the clearance under the keel. Fig. 2 also shows the final lengthwise distributions of cross-sectional area, blockage coefficient and local yaw angle for a component hull of the proposed design. Fig. 3 the body plan, centered about the camberhne, and a top view of the cambered twinhuü.
o.osr
Fig. 2: Distribution of cross-section area (left); blockage coefficients according to (23) (solid line) and (24) (dotted hne) (middle); local yaw angle (right)
DWL
4 6 forward direction
Fig. 3: Body plan centered about the camberhne and top view of the cambered twin-hull
4. Numerical Experiments
The computer program SHALLOWTANK, 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 with model measurements of Heuser (1973) was obtained. This program is now extended to handle the case of cambered hulls. The original program, already involving asymmetric flow, is in principle valid for the present pur-pose. Only the parts related to the yaw angle, which was constant over the entire ship length before, had to be modified to allow yaw angles ip being a function of x.
For the sake of record, numerical details underlying the computed results shown in this paper are listed below. The grid size is Iship = 20, Ax = 0.10033 (Aa;* = l*/{2Iship) = 1-5 m), A y = 0.028078 (Ay* = 1.25503 m), smahness parameter e = 0.11186, see definitions in Chen and Sharma (1995). The time steps finally selected after a few trials to overcome numerical instability are AT = 0.005 for Fnh < 1-3, AT = 0.004 for 1.3 < Fnh < 1-8, AT = 0.003 for 1.8 < Fnh < 2.3, and AT = 0.0025 for Fnh > 2.3. The huh separation used is s* = 22.59 m since numerical experiments at design speed Fnh = 1-7 revealed that its optimum value for the truncated huh S{x) is somewhat different from the theoretical design value s* = 20.08 m for the infinite huh Sa{x). A straight monohull, identical to each component huh of the proposed design except for camber, is used as a reference for comparing wave resistance and ascertaining the favorable effect of wave interference. The catamaran calculations hold for a channel width 150.62 m, which is large enough to avoid sidewah effects and only half of which is modeled w
in the computer by virtue of symmetry. The monohuU calculations using the same grid without taking advantage of symmetry hold for a symmetric channel of half the width, except for two points Fnh = 1.2 and 1.3 where extra calculations in the full-width channel become necessary to avoid sidewall effects. The calculated range extends longitudinally about 10 ship lengths upstream and about 20 ship lengths downstream.
Fig. 4: Calculated specific wave resistance (left); estimated specific total resistance (right) for the monohull (dots), the conventional catamaran (c) and the cambered twin-hull (s) Fig. 4 shows the calculated specific wave resistance and the estimated specific total resistance of the cambered twin-huh, the same twin-hull without camber (conventional catamaran), and the uncambered component huh (monohuh). Specific resistance is resistance/weight. Total 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 multiplied by a form factor 1 + k. By reference to comparable hullforms, the form factor is roughly estimated to be 1.4 for the cambered twin-hull, 1.32 for the conventional catamaran, and 1.24 for the monohull, Jiang
et al. (1995). The two catamarans have almost the same wave resistance in the sub- and near-critical speed ranges, but i t is higher than that of the monohull due to unfavorable wave interference. The 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 in the supercritical speed range. A t least in the interval 1.5 < Fnh < 2.6, the wave resistance of the conventional catamaran is already significantly lower than that of the monohull, and the cambered twin-huh is even better. A similar trend persists for the total resistance despite larger assumed form factors for the catamarans. A t the design speed Fnh = 1-7 the cambered twin-huh reduces the specific resistance with respect to the monohull by about twice as much as the conventional catamaran. This is not surprising since the camber supresses the outward 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 that the lower waves are generated by the cambered twin-huU on the outer side and in the wake than by other two ships. The inner-side waves trapped between the two hulls are about twice as high for the cambered twin-hull as for the conventional catamaran.
Fig. 5. Calculated wave patterns at Fnh = 1-7: (a) monohuh, (b) conventional catamaran, (c) cambered twin-huh
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 twin hulls is much lower than that of its uncambered component monohull and also than that of a conventional uncambered catamaran. This 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 to other traffic and less ecological damage to the river banks. Cambered hulls in a catamaran seem quite natural since the presence of a neighboring hull destroys the symmetry of flow around a straight hull and one might hope to partly restore the symmetry by camber. However, the mechanism exploited in this paper is not the restoration of symmetry but the complete cancellation of stern waves by diverting all bow waves toward the inside. The underlying inviscid potential theory could not do justice to unavoidable viscous effects in a real fluid. So I do not claim to have found the overall optimum. Verification by model experiment and further optimization by trial and error based on flow observation are indispensable.
References
CHEN, X.-N.; SHARMA, S.D. (1994), Nonlinear tiieory of asymmetric motion of a slender siiip in a shallow channel, 20. Symp. on Naval Hydrodyn, Washington, pp.386-407
CHEN, X.-N.; SHARMA, S.D. (1995), A slender ship moving at a near-critical speed in a shallow channel, J. Fluid Mech. 291, pp.263-285
CHEN, X.-N.; SHARMA, S.D. (1996a), Zero wave resistance for ships moving in shallow channels at supercritical speeds, J. Fluid Mech.
CHEN, X.-N.; SHARMA, S.D. (1996b), On ships at supercritical speeds, 21. Symp. Naval Hydrodyn., Trondheim, pp.147-158
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Prediction M e t h o d for Ship Manoeuvring M o t i o n
in the Proximity of a Pier
K a t s u r o K i j i m a , Kyushu University^
1. Introduction
Recently, several long bridges that connected between islands were built above main sea routes in Japan. Large piers in the sea routes are needed to support such large bridges. These restrict the course of ships. When ships are sailing in the proximity of each other, the interaction forces between ships influence the manoeuvring motion. Furthermore, the interaction between ship and bank or bridge pier should be taken into account for safe navigation. Yeung and
Tan (1980) and Kijima (1984,1987,1994) proposed prediction methods for the hydrodynamic interaction forces between ships and between ship and bank wall for this problem. Here, a prediction method based on the slender-body theory for hydrodynamic interaction forces between ships and between ship and pier having circular or oval cross section is proposed.
2. Formulation
The speed of two ships running straight in the proximity of pier is denoted by Ui (i = 1,2). The coordinate systems fixed on earth and on each ship are represented by o - xy and Oj- - a;,?/,-(i = 1,2) respectively, Fig. 1. Spi and Su are lateral and longitudinal distance between ship i and the pier and Spi2 and Stu are lateral and longitudinal distance between the midships of ship 1 and ship 2.
F, .v,t
cm
Ship ; Ship 1 7 ^ . 0C
2> Ship 2 c Ship 23
c Ship 1 Ship 1 Ca,sc 1 Case 2Fig. 1: Coordinate system Fig. 2: Geometry of the piers
Assuming small Froude number, the free surface can be regarded as a rigid wah. Then double-body models of the two ships can be considered. The velocity potential (f>{x,y, z;t) expresses the disturbance generated by the motion of the ships. </> should satisfy:
V^ct>{x,y,z;t) = 0; dcf>] dn d(t> dz_ d(j) [dni = 0; = 0; B, (1) (2) (3) (4) (5) <j)^0 at yjx] + yf + zf oo.
Bi and C are surface of ship i and the pier respectively, fi is the water depth, {nj;)i the a;
'Dept. of Naval Arch, and Marine Systems Eng., Faculty of Engineering, Fukuoka, Japan
component of inward unit normal on Bi. To simplify the problem, assumptions are made using the slenderness parameter e:
1. Ship breadth B and draft d are relatively small compared to ship length L;
L = 0{1), B^O{e), d = 0.{s) (6) 2. The order of the water depth h and Spi2, Spi, Sp2 that represent the lateral distance
between the ships and between the ship and the pier are:
h = 0{e), Spu = 0 { l ) , SpuSp2 = 0 { l ) . (7) Under these assumptions, the problem can be treated as two-dimensional separately in the
inner and outer region.
3. Inner Solution
The following conditions should be satisfied in the inner region: d^i
id Nil Uiit){n,)i (8)
Lateral distance between the ship and the pier and between the ships, Spi {i = 1,2) and Spu, will be considered in the outer region assuming that they have same order of ship length L:
xi = 0 { l ) , yi = z i = 0 { e ) .
The velocity potential $ i in the inner region should satisfy following conditions:
dyf
+
0; 0.(9)
(10)
(11) Tixi is the cross section of the ship i at Xi, Ni the inward unit normal on the cross section K j , . .
The velocity potential $j- in the inner region can be replaced by the velocity potential repres-enting the two-dimensional problem of a ship cross section between parallel walls represrepres-enting the bottom and its mirror image above the water surface. Then $ i can be expressed as
$,-(2/„ z,; Xi; t ) = Ui{t)é^\y,, z,) + V*[xi, m Y ' { y i , Zi) + fi{xi, t ) .
(2),
(12) and are unit velocity potentials for longitudinal and lateral motion, V* is the cross-flow velocity at Ej,,., and fi is a term being constant in each cross-section plane, which is necessary to match the inner to the outer solution.The outer hmit of and can be written as
Ah
lim ^^^\yi,z,) = -yi±Ci{xi). |yd»^
Si{xi) is area of the cross section of ship i at Xi, and S[{xi) = coefficient Ci{xi) is estimated by Taylor's (1973) formula.
Finahy, the outer limit of the velocity potential $ i is written as u,m{x,)
(13) (14)
dSi{xi)/dxi. The blockage
hm ^i[yi,Zi\Xi\t) = -•
4. Outer Solution
The following conditions should be satisfied in the outer region:
xi = yi=0{l), zi=0{e). (16) The Taylor expansion of the velocity potential (/) fot z = 0, neglecting higher-order terms, is
0
4>i = Mx,y,0\t)+ —(l)i{x,y,z;t)
z=0
1 9 ^ ^
z' +
z=0
~ (t)oi{x,y;t) + (t)ii(x,y]t) • z + (t)2i{x,y\t) • z^.
The leading-order term (fy^i satisfies the two-dimensional Laplace equation d(t>oi , d(t)oi
+
0.(17)
(18) dx"^ dy"^
Hereafter (pi is substituted by </>ot.
The velocity potential is represented by distributing sources and vortices along the body axis:
2
<t>i[^,y\t) = "t\\ I Oj{sj,t)G^p{x,y;i,r,)ds,+ [ jj{s„t)G^;'\x,y;^,rj)dsA .
j=i ^ K-'^j •'^i'^j ) (19)
aj{sj,t) and yjisjjt) are the source and vortex strengths respectively. Lj and Wj denote the integration along ship j and vortex wake shed behind the ship j, respectively. The Green function G^.^\x, y\x, h) and G^^\x, y; x, h) are defined as
G r ' ( . T , y ; e , 7 ? ) = \n^{x-0' + {y-vr + Hl''\x,y;^,r^), G\''\x,y;^,r,) = Un'' (^^^ + Hi^\x,y;^,ri).
(20) (21)
H^"^ and iï|^^ are harmonic functions arising from existence of the pier. They are determined to satisfy the conditions
dH.
= 0, dH.
dn 0 (22)
J c
By expanding 4>i for yi and translating the coordinate system, the inner limit of ^ ; is obtained: lim (f>i{^,y;t) 1 . 2 / a,{sj,t)G'-^\x,,0;^i,Q)ds,+ [ j,{s,,t)G'-;'\x„0;^,,Q)ds, + \ I (r,{sj,t)^^{xi,0;^i,0)dsj+ [ 7j{sj,t)^^^{x,,0;i,,0)d.s, } yi I JLJ uyi JLjWj oyi dG\ ds: ^TT JLi-uj, ^ Z, Jx, Z, (23) dHi^ dyi Xi,0;^i,0)dsi dHi^'^^
