A Time Domain Numerical Simulation Method for Nonlinear Ship Motions Shukui LIU1, a, Apostolos PAPANIKOLAOU2, b, Wenyang DUAN3, c 1Address of first author: Ship Design Laboratory, National Technical University of Athens, Athens, Greece 2Address of second author: Ship Design Laboratory, National Technical University of Athens, Athens, Greece 3Address of third author: College of Shipbuilding Engineering, Harbin Engineering University, Harbin, Heilongjiang, P.R.China a
b
c
[email protected],
[email protected],
[email protected]
Keywords: Time domain simulation, nonlinear ship motions, Green function
Abstract A three-dimensional nonlinear time domain simulation method for solving the problem of a ship undergoing large-amplitude motions originally developed at Harbin Engineering University is introduced in this paper. In this method, the body boundary condition is satisfied on the instantaneous wetted surface of the moving ship, while the free-surface boundary condition is linearized. The transient free-surface Green function, which includes a Rankine part and a free-surface memory part, is employed to formulate the relevant boundary-value problem and its numerical solution is accomplished by the solution of integral equations for the strength of sources and the mixed source-dipole distribution on the wetted body surface. The main advantage of this method lies in its capability to investigate large-amplitude motions of bodies with forward speed, that is, various nonlinear effects of interest to ship’s safety. Some of the preliminary results obtained by this approach, referring to the radiation and diffraction problems, at zero and non-zero speeds, are herein presented and compared with those from the frequency domain code of NTUA and with other published theoretical and experimental results and show a fully satisfactory agreement. Introduction The accurate prediction of the seakeeping behavior of ships in heavy seas is of great importance to ship design and operation. Since the early 50ties, many hydrodynamicists addressed the problem of a surface ship sailing in waves by approximate analytical and simplified or more advanced numerical methods in the framework of linear ship motion theories. The strip theory was the first one which delivered accurate enough results for practical applications to ship motions’ prediction and enjoy a wide application even today. Grim [1] and Korvin-Kroukovsky [2] did the pioneering work on strip theory, whereas Gerritsma & Beukeman [3] and Salvesen et al. [4] further improved this theory for practical applications in the 60ties. But due to some inherent limitations of strip theory (linearity of responses, quasi 2D approach, slender body, low speed and high frequency assumptions), its application is confined to a certain extent. With the rapid advance of computer technology in the 70ties, various 3D approaches [5, 6, 7, 8] were successfully developed for zero speed problems. For the nonzero speed 3D problem, Chang [9] was the first to present a numerical solution on the basis of the Green function method, while Inglis & Price [10] and Guevel [11] later followed further improving relevant theory for practical applications. Although results of these theories appear closer to relevant experimental data, some intricate numerical problems related to the significantly more complex corresponding Green function and the treatment of the singularity at the intersection of the body boundary and the free-surface hindered their wide application. Iwashita & Ohkusu [12] appear to have developed a satisfactory numerical solution to the 3D, nonzero speed problem based on the Green function method. Another approach is the so-called Rankine source method, which use a distribution of simple Rankine sources over the body surface as well as on a carefully-chosen part of the free surface. Many researchers, including Bertram [13] and Nakos & Sclavounos [14], applied this method to practical utility. But in all above methods, the body boundary condition is still satisfied on the mean body position, thus they are linear with respect to the motion responses. An alternative to the formulation of the problem in the frequency domain is to work in the time domain, enabling the address of large amplitude ship motion problems. The original work on a time domain approach in this area might be credited to Finkelstein [15] and Cummins [16]. Later Beck & Liapis [17] and Korsmeyer [18] investigated the linearized radiation problem at zero speed while King et al. [19] the linearized diffraction problem with forward speed. Beck & Magee [20] and Ferrant [21] presented convincing results on submerged bodies undergoing large-amplitude motions, while Lin W M & Yue D., Lin et al., Shin et al. [22, 23, 24] showed the applicability of the method to large amplitude ship motions. In this paper, we will introduce a three-dimensional numerical time domain method for solving the problem of a ship undergoing large-amplitude motions by employing a transient free-surface Green function, which has been under
development since the early 1990s at Harbin Engineering University [25, 26, 27] and is being further developed by the first author in the framework of his PhD study at NTUA[28]. This work is in line with earlier work of others [e.g. 22], but with particular focus eventually on ship’s safety in heavy seaways for improved ship design and operation. In this method, the earth-fixed Cartesian coordinate system is employed. The body boundary condition is satisfied on the instantaneous wetted surface of the moving body, while the free-surface boundary condition is linearized. The transient free-surface Green function, which includes a Rankine part and a free-surface memory part, is employed to formulate the relevant boundary-value problem. The numerical solution is thus accomplished by solving an integral equation for the velocity potential or source strength distribution on the wetted body surface. Once the velocity potential and tangential velocities are obtained, the hydrodynamic loads can be calculated by integrating the hydrodynamic pressure resulting from the Bernoulli equation on the wetted surface. The main advantage of this method lies in its capability to investigate large-amplitude motions of bodies with forward speed, that is, various nonlinear effects of interest to ship’s safety. In this paper some preliminary results obtained by this method, referring to the radiation and diffraction problems, at zero and non-zero ship forward speed, are presented and compared with the frequency domain 3D code of NTUA [29] and with other published theoretical and experimental results. Problem Formulation Basic Assumptions Consider a general 3D body floating on the free surface and undergoing arbitrary 6-DOF motions. An earth-fixed Cartesian coordinate system is chosen with the x-y plane coincident with the undisturbed free surface and z-axis pointing upwards. The fluid is assumed to be homogeneous, incompressible, inviscid and its motion irrotational. The water depth is infinite. The flow field can be described by a velocity potential:
Φ T ( p , t ) = Φ I ( p , t ) + Φ ( p, t ) where
(1)
Φ I is the incident wave potential, Φ = Φ T − Φ I is the disturbed flow potential, t is time, and p (x, y, z) is a
point in the flow field. In the fluid domain Ω ( t ) , which is bounded by the free surface S f and the control surface
Sc at far field, Φ ( p, t ) satisfies Laplace’s equation:
∇ 2 Φ ( p, t ) = 0
( t ) , the body surface Sb ( t )
(2)
Where p is an arbitrary point in
Ω ( t ) and t is time.
The body boundary condition is applied on the instantaneous body boundary Sb ( t ) :
∂Φ ∂Φ I =V ⋅n − ∂n ∂n
(on Sb ( t ) , t>0 )
(3)
n is the unit normal vector pointing out of the fluid domain Ω ( t ) , and V is the instantaneous velocity of the body surface. The linearized condition is imposed on the free surface S f
∂ 2Φ ∂Φ +g =0 2 ∂t ∂z
(on S f
(t ) :
( t ) , t>0)
(4)
where g is the gravitational acceleration. The initial conditions at t=0 are:
Φ=
∂Φ =0 ∂t
(on S f
( 0) ).
(5)
For finite time, the conditions on the control surface at infinite are:
Φ,
∂Φ , ∇Φ → 0, (on Sc , t>0). ∂t
Related Integral Equations The time-domain Green function which satisfies the linearized free-surface condition [30] is:
(6)
1 1 ɶ + G ( p, t; q,τ ) G ( p, t; q,τ ) = − rpq r ' pq ɶ where G ( p, t; q,τ ) is the free-surface memory part : ∞ Gɶ ( p, t; q,τ ) = 2 ∫ gkek ( z +ζ ) J 0 ( kR ) sin gk ( t − τ ) dk
(7)
0
where
p = p ( x, y, z ) and q = q (ξ ,η , ζ ) are the field and source points respectively, R =
( x − ξ ) + ( y −η ) 2
2
rpq = R 2 + ( z − ζ ) ， rpq' = R 2 + ( z + ζ ) ，and J 0 is the zero-order Bessel function. 2
2
p is in the fluid domain Ω ( t ) , we apply Green’s identity to Φ ( q,τ ) and Gɶ ( p, t; q,τ ) to obtain a boundary integral equation for Φ . After some transformation, we get the integral equation for the disturbed potential Φ ( p, t ) as following: Suppose that
2πΦ ( p , t ) +
∂ Φ (q, t ) ∫∫ ∂ nq Sb ( t )
1 1 − rpq r ' pq
1 1 − − r rpq ' pq
∂Φ ds = ∂ nq q
∂Φ t ∂Φ ∂Gɶ 1 ∂Gɶ = ∫ dτ ∫∫ Gɶ −Φ dsq + ∫ Gɶ −Φ V dl (8) N q 0 ∂nq g wl (τ ) ∂τ ∂τ Sb (τ ) ∂nq where p ∈ Sb ( t ) . This integral equation corresponds to a mixed distribution of dipoles and sources over the body surface and its solution leads to the velocity potential on the body surface. An alternative integral equation on the basis of a sources model and facilitating the calculation of the velocity potential derivates is given next:
Φ ( p, t ) =
1 4π
1 1 ds + − rpq rpq' q
∫∫ σ ( q, t )
Sb ( t )
ɶ ( p, t ; q,τ ) ds + 1 ɶ ( p, t ; q,τ ) V v dl , , d τ σ q τ G σ q τ G (9) ( ) ( ) q N n q ∫0 S∫∫τ ∫wl (τ ) g ( ) b where σ ( q, t ) is the source strength and p ∈ Ωb ( t ) . Notice that vn = ( N ⋅ n ) VN and apply the body boundary condition for p so we obtain the equation for σ :
+
1 4π
t
∂Φ ∂Φ I 1 1 = vn ( p , t ) − = σ ( p, t ) + ∂n p ∂n p 2 4π +
1 4π
∫∫ σ ( q, t )
Sb ( t )
∂ 1 1 − dsq + ∂n p rpq rpq'
∂Gɶ 1 ∂Gɶ d τ σ q , t ds − σ q , τ V v dl ( ) ( ) q N n q ∫0 S∫∫τ ∂n p g ∫wl (τ ) ∂n p b( ) t
(10) The idea is to solve the source strength with Eq. (10) and then evaluate the velocity potential by using Eq. (9). Or if we use Eq.[8] directly, then we need to calculate the second differentiation of the Green function, which is less accurate and more complicated. Hydrodynamic Forces The unsteady pressure is given by Bernoulli’s equation:
∂Φ 1 p = −ρ + ∇Φ ⋅ ∇Φ ∂t 2 Integrating (10) over the instantaneous wetted surface
Sb ( t ) we can obtain the hydrodynamic forces acting on the body:
∂Φ 1 + ∇Φ ⋅∇Φ nds F = − ρ ∫∫ ∂t 2 sb ( t )
(11)
Once the forces are obtained we can apply the harmonic analysis as below: ∞
Fij (t ) = ∑ ( Aij (n) cos nωt + Bij (n) sin nωt )
(12)
n =0
in which
Aij (1) and Bij (1) are corresponding to the added mass and damping coefficient.
Numerical Solutions – Discussion of Results To solve Eq.(8) and Eq.(10), a panel method[31] is employed. The wetted body surface is divided into MN quadrilateral panels and the waterline intersection is divided into
NW segments. The velocity potential and its derivates on each panel
and segment are assumed to be constant while the equations are satisfied on the geometrical center. Take Eq.(8) as an example, the discritized form is:
∑ {C Φ MN
j =1
ij
M nj
+ Dij Φ iM } = EEi
i = 1, 2,..., MN
(13)
the influence coefficient matrixes and are:
1 1 Cij = − ∫∫ − ' ds , r r ∆s j
∂ 1 2π + ∫∫ ∂n − r ' ds j ∆s j Dij = ∂ 1 − 1 ds ∆∫∫s ∂n j r r ' j
i= j i≠ j
MN 1 Nw m m m m ɶ ɶ EEi = ∆t ∑ ∑ Φ nj ∫∫ G j ds − Φ j ∫∫ Gnj ds + ∑ Gɶ km ∫ ΦτmkVNkm dlk − Φ mk ∫ GɶτmkVNkm dlk g k =1 ∆l m = 0 j =1 ∆s j ∆s j ∆lk k where m is the index for time τ and M ∆t = t , i, j and k are the indices for field point p , source point q and waterline ɶ m means the normal velocity ∂Φ on j th panel at time τ . segment ∆l . For instance, G nj ∂n Since EEi is a convolution integral, the solution must be done with a time-marching technique thus burden the M −1
calculation. After mapping the quadrilateral elements into standard unit squares the integration of the memory-related terms may be obtained by one-point or four-point Gauss quadrature depending on panels’ characters. As to the submerged case, the waterline integral vanish automatically. For the floating case, the waterline integral has not been included yet. 1. Radiation problem of a floating hemisphere The first set of numerical results obtained include added-mass and damping coefficients for a floating hemisphere studied by Hulme [32]. At this stage, we use the linearized body boundary condition. Starting from rest, the hemisphere is undergoing a sinusoidal motion z = A sin ωt in deep water. Results by the present method, generated by use of 256 panels and 40 time steps per period, are compared with those of the frequency domain code NEWDRIFT [8,30] (with 100 panels) and of Hulme’s semi-analytical method. Fig 1. and Fig 2. show a comparison of the added-mass and damping coefficient in heave, noting that A33 was divided by 2πρ R / 3 and B33 by 2πρω R / 3 non-dimensionalization. The agreement between all methods over the entire frequency range is very good, except for the irregular frequencies problem appearing in the results of the frequency domain code NEWDRIFT that was herein not treated. 3
3
B33
A33
LIU Hulme's
0.8
LIU Hulme's
0.3
0.2 0.6
New Drift 0.1
New Drift
0.4
0.0 0
2
4
kR
6
8
0
Fig. 1 Comparison on added-mass
2
4
6
kR
8
Fig. 2 Comparison on damping coefficient
2. A submerged sphere undergoing large-amplitude motion Ferrant [21] presented results of the heaving problem of a submerged sphere, with a mean depth of submergence equal to the diameter ( z0 / R = 2.0 , Fig. 3). In order to compare with Ferrant’s results, the same number of discrete panels, which is 200 on the whole surface, is used by NEWDRIFT and the present method (Fig. 4). Fig. 5 and Fig. 6 show us the comparison of the added-mass and damping coefficient in heave, noting that results are divided by and
4πρ R 3 / 3 for A33,
2πρω R 3 / 3 for B33 for non-dimensionlization. SL
A
x
S b(t)
Fig. 3
Z = Z 0 + A cos ωt
Fig. 4 Mesh of the Sphere 0.225
0.7
0.6
Ferrant Nonli Ferratn Linear LIU Nonli
B33
A33
Ferrant Nonli Ferrant Linear LIU Nonli
0.150
New Drift
0.5
New Drift
0.075 0.4
0.000 0.3 0.0
0.8
1.6
Fig. 5 Comparison on added-mass
2.4
kR
3.2
0.0
0.8
1.6
2.4
kR
3.2
Fig. 6 Comparison on damping coefficient
The agreement among the various methods is again good and a clear nonlinear effect on the added mass and damping values is noted. 3. Diffraction problem of a submerged sphere with forward speed
A submerged sphere advancing in regular deep-water waves at constant forward speed has been analyzed by Wu G X [33] with a linearized potential method. A distribution of sources over the surface of the sphere is expanded into a series of Legendre functions, by extension of a method used earlier by Farell [34] in analyzing the wave resistance on a submerged ellipsoid. It is noted that Wu’s calculation has been done in frequency domain, while calculation in this paper in time domain. For comparing the force amplitude results, an orthogonalizing analysis of the diffraction forces is performed and the results are given under the following form:
F ( t ) = ( Freal cos ωt + Fimag sin ωt ) / Aρ gπ r 3 k
Freal and Fimag are the so-called non-dimensional real part and imaginary parts of the wave exciting forces. All the results below were obtained with a 100-panel meshing of the demi-hull and 52 time steps’ calculation per period. The abscissas in Fig 7 and Fig 8 are the reduced wave number kR. 0.0
F1
F3
0.0
-0.4
-0.4
W uF1r W uF1i LiuF1R LiuF1I NDF1r NDF1i
-0.8
-1.2
0.0
0.4
0.8
kR
W uF3r W uF3i LiuF3r LiuF3i NDF3r NDF3i
-0.8
-1.2
1.2
Fig. 7 Comparison on wave load F1
0.0
0.4
0.8
kR
1.2
Fig. 8 Comparison on wave load F3
4. Resistance of a submerged ellipsoid An ellipsoid with its major axis parallel to the free surface and moving with forward speed has been studied by Farell [34] and Chen X B [35]. In this paper we present results in comparison to those of Chen XB for an ellipsoid with the lengths of major and small axes 2a=2.3 and 2b=0.4 respectively, focal distance is c=1.132475 and different submergences and speeds. The cosine-panel method is employed in the calculation to approximate the ellipsoid at the ends more precise. A perspective view of the demi-ellipsoid with a mesh of 288 panels is shown in Figs. 9 and 10. The Froude number and resistance coefficients depicted in Figs 11 to 14 are defined respectively as:
Fn = 2 gc , Cw = −1000 F1 / (πρ gc3 ) .
Fig. 9 Mesh of the Ellipsoid
Fig. 10 Mesh at the end
0.5
Cw
Cw
2.0
0.4 1.5
Liu C hen
1.0
0.3
Liu Chen
0.2 0.5
0.1
0.0
0.0 0
60
120 T im eStep 180
Fig.11 Fr=0.45 d=0.252c
0
60
120 TimeStep 180
Fig.12 Fr=0.45 d=0.5c Cw
Cw
2.4
0.8
Liu C hen
1.8
0.6 1.2
Liu C hen
0.4 0.6
0.2
0.0
0.0 0
60
120 Time
Step 180
Fig.13 d=0.252c Fr=0.35
0
60
120 Tim eStep 180
Fig.14 d=0.252c Fr=0.6
5. Diffraction problem of a Wigley-hull The experiment data of a Wigley-hull form with a mid-ship section coefficient equal to 2/3 and length-breadth ratio of 10 were published by J.M.J. Journée [36]. These data refer to hydrodynamic coefficients for heave and pitch, vertical motions, wave loads and added resistance in head waves. In order to validate the code, some calculations were carried out herein on the wave loads acting on this Wigley hull. A perspective view of the Wigley hull discretized with 250 panels is shown below. Present results are compared with those resulting from the frequency domain code NEWDRIFT. The dimensionless results are defined as below, in accordance to Journée:
F1 =
F 10 F 20 F 30 F 50 F2 = F3 = F5 = kς a ρ g ∇ , kς a ρ g∇ , C33ς a , kς a C55
Fig.15 Panels used to represent the Wigley hull
0.6
F3
F1
0.6 NDF1r NDF1i LiuF1r LiuF1i
0.4
N D F3r N D F3i LiuF3r LiuF3i
0.4
0.2 0.2
0.0 0.0
0.5
1.0
1.5
λ/ L
-0.2 2.0
Fig.16 Fn=0 β=π wave load F1
0.5
1.0
1.5
λ/ L
2.0
1.5
λ/ L
2.0
1.5
λ/ L
2.0
Fig.17 Fn=0 β=π wave load F3 0.8
F1
F5
0.6
NDF5r NDF5i LiuF5r LiuF5i
0.4
F1ND F1LIU F1test
0.6
0.4
0.2 0.2
0.0
0.5
1.0
1.5
λ/ L
0.0
2.0
0.5
1.0
Fig.19 Fn=0.2 β=π Wave load F1
F5
Fig.18 Fn=0 β=π wave load F5
F3
0.6
0.6
F3ND F3LIU F3test
0.4
F5ND F5LIU F5test
0.4
0.2
0.2
0.0 0.5
1.0
Fig.20 Fn=0.2 β=π Wave load F3
1.5
λ/ L
0.0 2.0
0.5
1.0
Fig.21 Fn=0.2 β=π Wave load F5
Conclusions A 3D time-domain approach to predict the ship motions, originally developed at Harbin Engineering University is further developed by the first of the authors to simulate large amplitude ship motions in heavy seas. In this method the linearized free-surface condition is employed but the body boundary condition is applied on the exact wetted surface of the moving body. In the present paper, a validation of the code, for different bodies at zero or nonzero speeds, is systematically investigated. The results, including basic hydrodynamic coefficients, resistance, wave exciting forces and moments, are compared with published numerical or experimental data and the frequency domain code of NTUA NEWDRIFT. Good agreement is noted throughout these comparisons, which shows a promising future of the present approach. Yet the present code is in a preliminary stage and some more work is necessary to address the entire large amplitude ship motions problem. Particularly, a satisfactory numerical approach to the waterline integral term in the integral equation for the velocity potential needs to be developed and the equations of ship motions formulated and solved for practical applications. Acknowledgement
The first author is grateful to Prof. Apostolos D. Papanikolaou and Prof. Wenyang Duan for their guidance and supervisions. The present study is supported by the ASIA LINK programme (ASI/B7-301/98/679-044 (072433)) and the financial support is greatly appreciated. References [1] Grim, O.: Berechnung der durch Schwingungen eines Schiffskörpers erzeugten hydrodynamischen Kräfte, STG-Jahrbuch(1953), p. 277-299. [2] Korvin-Kroukovski, B.V.: Investigation of ship motions in regular waves. Trans. SNAME, 63(1955), p. 386-485. [3] Gerritsma, J. & W. Beukelman: Analysis of the Modified Strip Theory for the calculation of ship motions and wave bending moments. Int’l. Shipbldg. Prog., 14(156)(1967), p. 319-37. [4] Salvesen, N., E. O. Tuck & O. Faltinsen: Ship motions and sea loads. Trans. SNAME (1970), 78, p. 250-87. [5] Faltinsen, O.M. and Michelsen, F.C.: Motions of large structures in waves at zero Froude number. Proceedings of the Symposium on the Dynamics of Marine Vehicles and Structures inWaves, 1975, p. 3-18. [6] Chang, M-S and Pien, P.C.: Velocity potential of submerged bodies near a free surface application to wave-excited forces and motions. Proceedings, 11th Symp. Naval Hydrodyn., London(1976). [7] Garrison, C.J.: Hydrodynamic loading of large offshore structures: Three-dimensional source distribution methods. In: Numerical methods in offshore engineering, edited by Zienkiewicz, O.C. et al (1978), Ch. 3, p. 87–140. [8] A. Papanikolaou: On Integral-Equation-Methods for the Evaluation of Motions and Loads of Arbitrary Bodies in Waves, Journal Ingenieur - Archiv., Vol. 55(1985), p.17-29. [9] Chang, M-S.: Computation of 3-D ship motions with forward speed. Proceedings, 2nd International Conference on Numerical Ship Hydrodynamics, Univ. California, Berkeley, (1977), p. 124-135. [10] Inglis, R.B. and Price, W.G.: A 3-D ship motion theory comparison between theoretical prediction and experimental data of the hydrodynamic coefficients with forward speed. Trans. Roy. Inst. Nav. Arch., Vol.124(1981). [11] Guevel, P. and Bougis, J.: Ship motions with forward speed in infinite depth. Int’l. Shipbldg. Prog., No.29(1982). [12] Iwashita, H. & M. Ohkusu: The Green function method for ship motions at forward speed. Schifftechnik, 39(1992), p. 3-21. [13] Bertram, V.: A Rankine source approach to forward speed diffraction problems. Proc. 5th Intl. Workshop on Water Waves and Floating Bodies, Manchester (1990). [14] Nakos, D.E. & Sclavounos, P.D.: Ship motions by a three-dimensional Rankine panel method. Proc. 18th Symp. Naval Hydro.(1990), Ann Arbor, Michigan. [15] Finkelstein,A.:The initial value problem for transient water waves. Comm. Pure App. Maths.,10(1957), p.511-22 . [16] Cummins, W.E.: The impulsive response function and ship motions. Schiffstechnik, 9 (1962), p. 124-135. [17] Beck, R.F. & Liapis, S.J.: Transient motions of floating bodies at zero forward speed. J. S. R., Vol.31, No.3(1987). [18] Korsmeyer, F.T.: The first and second order transient free-surface wave radiation problems. PhD Thesis, Dept. Ocean Eng., MIT, Cambridge, Massachusetts, (1988). [19] King, B.K., Beck, R.F. and Magee, A.R.: Seakeeping calculations with forward speed using time domain analysis. Proc. 17th Symp. Naval Hydro., Hague, Netherlands, (1988). [20] Beck, R.F., Magee, A.R.: Time domain analysis for predicting ship motions. Proc. IUTAM Symp., Dynamics of Marine Vehicles and Structures in Waves, London, (1990).
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