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SEAKEEPING OF HIGH SPEED SHIPS WITH TRANSOM STERN AND
THE VALIDATION WITH
UNSTEADY WAVES AROUND SHIPS
Muniyandy ELANGOVAN
by
JAPAN
Social and Environmental Engineering
Introduction 1
Mathematical Formulation 2
Numerical Method 3
Experiments for the Validation of Seakeeping 4
Interaction Effect of Incident Wave in an Unsteady Wave 5
Seakeeping of High Speed Ship with Transom Stern 6
Conclusions 7
(ii) Trimaran
(i) Monohull
(ii) Unsteady wave measurement
Outline of the Presentation
(i) Hydrodynamic forces, motions and added wave resistance
1. INTRODUCTION
Aim/Purpose
• Passenger Comfort
• Safe Transport
• Accurate estimation of wave forces for structural analysis
• Prediction of operational behavior in bad weather
• Reduce the Engine power by reducing the resistance of hull
Needed Data • Ship Motions
• Hydrodynamic Forces
•Ship performance in calm water and waves
Ship Design
(Hull)
Ship Construction
Numerical Analysis
Seakeeping Analysis
Model Test
[Cheaper, less time, prototype, no limitation]
[Expensive, more time, model size, no. data points]
Real Fluid Problem
Continuity Equation
&
Navier Stokes Equation
Viscous Fluid
Laplace Equation
&
Bernoulli’s Equation
Irrotational
Inviscid
Constant density
Ideal Fluid
Physical Problem
Incompressible
Seakeeping Analysis – METHODs
Strip Method
[Ursell,F., 1949, Korvin-Kroukovski, B.V, 1949,
Tasai, F, 1959, Watanabe, Y., 1958 ]
Unified Theory, Enhanced Unified Theory
[Newman, J.N., 1978 , Kashiwagi,M.,1995]
High Speed Strip Theory
[Takaki, 1975, Chapman R.B, 1975., Maruo, 1960,
Saito et al., 1978]
2D
Green Function Method
[Inglis, R.B., & Price W.G, 1981, Kabayashi, M,
1981, Chang, 1977, Iwashita & Okhusu, 1989]
Rankine Panel Method
[Yasukawa, H. 1990, Sclavounos & Nakos, 1990,
Gadd, 1976, Dawson, 1977]
3D
Frequency Domain
Hybrid Method:
Rankine Panel Method & Green Function Method
[Iwashita, et. al.1993]
2D 3D CFD
[Sato, et. al. 1999, Weymouth, et al.2005,
Panahi, et al. 2009, Mutsuda et al. 2007]
Green Function Method
[Beck, R.F & Liapis, 1987, Powlowski and
Bass. 1991]
Rankine Panel Method
[Maskew, B.,1991,Nakos, et. al., 1993,
Yasukawa, H, 2002.]
3D
Time Domain
Hybrid Method:
Rankine Panel Method & Green Function Method
[Lin & Yue 1980, Kataoka & Iwashita. 2004]
Two dimensional potential solvers are fast and reliable for the
prediction of ship motion. When it comes to the local forces, it fails to
capture three dimensional effect.
(i)
The Green function method treats the problem in three dimensionally.
For the linear free surface, it gives good result, but it is difficult to get
the Green function for the nonlinear boundary conditions.
(ii)
In time domain, nonlinear forces can be analyzed but the computational
value is high and treatment of radiation condition also difficult. (iii)
Hybrid method also computational time is expensive and treatment of
radiation condition is difficult. It needs more memory in computer. (iv)
Considering the above points and the flexibility for the implementation
of free surface boundary condition and radiation condition, Rankine
panel method (RPM) is selected for the seakeeping analysis. In
frequency domain, weak nonlinear boundary condition can be applied.
(v)
(vi) Due to the development of numerical solver, it is equally important to
validate the result by the experimental results. Therefore, improvement
in the measurement system is also required. Unsteady wave is treated
as a higher level of local pressure estimation, which can be used for the
comparison with numerical results. Considering the needs, unsteady
wave measurement is identified to improve the measurement system.
Identification of solver and marine industry needs
Scope of Present Research
Presently, unsteady waves are measured by Okhusu’s method
[Okhusu, Kyusu University, Japan] to estimate the added wave
resistance. This formulation does not include the interaction term
between the incident wave and the steady wave. The interaction term is
derived, and effect is studied by applying to the modified Wigley hull.
(i)
To treat the transom stern condition, until now there is no proper
boundary condition to treat the transom stern by panel method. From
the experimental observations, a new boundary condition is derived to
treat the transom effect mathematically in numerical method.
(ii)
Transom boundary condition is applied to monohull and numerically
predicted hydrodynamic qualities have been compared with
experimental data for validation. This application has been extended for
trimaran as well.
(iii)
2. MATHEMATICAL FORMULATION
Governing Equation
Laplace Equation
2.1 Definition of Problem
Fig.1 Coordinate system n
(i) Body boundary condition
(ii) Free surface [KC & DC] (iv) Control Surface
(v) Bottom Surface
(iii) Radiation condition
2.2 Velocity Potential
Fig.2 Double body flow Fig.3 Uniform flow
where
Unsteady Velocity Potential
Total Velocity Potential
, ,
(i) Double Body Flow Potential
(ii) Steady Velocity Potential
Free Surface
Hull Surface
Free Surface
Hull Surface
2.3 Boundary Value Problem
(iii) Steady Velocity Potential
[Baba, E., 1976]
DBF
NK
Free Surface
Hull Surface
(i) Unsteady Velocity Potential [Yasukawa, H, 1990]
[Timman & Newman, 1962]
2.3 Boundary Value Problem (Cont.)
DBF
NK (ii) Unsteady Velocity Potential
where ,
,
(i) Steady wave
(ii) Steady pressure
(iii) Steady forces and moment
2.4 Hydrodynamic Parameters
(iv) Unsteady Wave
(v) Unsteady Pressure
2.4 Hydrodynamic Parameters (Cont.)
where
where
,
,
2.4 Hydrodynamic Parameters (Cont.)
(vi) Added mass and Damping Coefficient
(vii) Wave Exciting Forces and Moments
2.4 Hydrodynamic Parameters (Cont.)
(viii) Ship motion
where
,
3. NUMERICAL METHOD
3.1 Computation of Velocity Potential
(i) Direct Method
(i) Indirect Method
where ,
Q = (0, 0, -d)
L = 2.0 m
d = L / 10 = 0.2 x
y
z
Fig.4 Free surface with point source
3.2 Point Source Problem
[ Bessho, M., 1977] (i) Analytical Formulation
where
where
Green Function
Wave Term
3.2 Point Source Problem (Cont.)
,
[Bertram, V, 1990] (ii) RPM – Panel Shift Method
Velocity Potential
Free Surface
Integral Equation
3.2 Point Source Problem (Cont.)
[Scalvounos & Nakos, 1990] (ii) RPM – Spline Interpolation Method
Velocity Potential on the free surface
Radiation condition
Integral Equation
3.2 Point Source Problem (Cont.)
Fig. 5 RPM-PSH: Wave pattern at Fn = 0.2, Ke L =5.0, t = 0.447
3.3 Numerical Results
RPM-PSM RPM-SIM
Fig. 6 Wave pattern at Fn = 0.2, Ke L =5, t = 0.447
RPM-PSM RPM-SIM
3.3 Numerical Results (Cont.)
Fig.7 RPM-SIM: Wave pattern at Fn = 0.2, Ke L =30, t = 1.095
3.3 Numerical Results (Cont.)
RPM-PSM RPM-SIM
Fig. 8 Wave pattern at Fn = 0.2, Ke L =30, t = 1.095
RPM-PSM RPM-SIM
3.3 Numerical Results (Cont.)
3.4 Integral Equation for Ship
Velocity Potential
Integral Equation
where
,
4. EXPERIMENTS FOR THE VALIDATION OF SEAKEEPING
Unsteady waves are generated by both the diffraction of incident
wave and the motions of the ship due to the incident waves
1
Unsteady Wave Measurement
Okhusu proposed a method for measuring ship generated unsteady
waves and the evaluating the wave amplitude function which can be
used for estimation of added wave resistance.
3
Unsteady wave patterns physically show the pressure distributions
over the free surface that can be considered as local physical value.
This unsteady wave is superior to pressure measurement on the hull
surface from the point of view of the cost and the convenience.
Therefore it is valuable to utilize the unsteady waves in order to
validate the numerical computation methods more precisely.
2
Unsteady wave measurement analysis is made considering the
number of wave probes and the unsteady wave second order term
4
4.1 Introduction
Fig.9 Layout of ship motion, forces and moment measurement system
(a) Motion free measurement setup (b) Forced motion measurement setup
Fig. 10 Diagram - Unsteady wave measurement arrangement
Wave Probes
4.2 Unsteady wave measurement
2nd order wave
1st order wave Steady wave
Total wave (i) Steady wave, (ii) Diffraction wave, (iii) Radiation wave
(3 unknowns)
(5 unknowns)
4.2 Unsteady wave measurement (Cont.)
(i) Total wave (up to 1st order)
(ii) Total wave (up to 2nd order)
4.2 Unsteady wave measurement (Cont.)
where
Modified Wigley hull(blunt) mathematical expression
Fig. 11 Plan view of the modified Wigley hull
Fig. 12 Wave probe dependency study
at y/(B/2)=1.4; Fn=0.2, l/L=0.5, c=p
4.3 Experimental Results
Fig. 13 Diffraction wave - 2nd order term
at y/(B/2)=1.4; Fn=0.2, l/L=0.5, c=p
4.3 Experimental Results (Cont.)
5. INTERACTION EFFECT OF INCIDENT WAVE
IN AN UNSTEADY WAVE
Okhusu proposed unsteady wave measuring method, which does not
include the interaction term of the incident wave and steady wave.
1
Interaction effect of Incident wave in an Unsteady wave
Here, interaction term formulation is carried out to see the influence
of incident wave and steady wave in the unsteady wave analysis by
computed DBF and measured NK wave.
2
Hydrodynamic forces, moment, ship motions and waves are
numerically computed and compared with an experimental result to
validate the Rankine panel computational code.
3
5.1 Introduction
5.2 Influence of Steady and Incident wave in an Unsteady wave
(ii) Unsteady wave elevation
where
,
(iii) Unsteady wave elevation (diffraction)
where ,
(i) Measured unsteady wave
5.3 Treatment of influence term in Numerical Method
(i) Double body flow formulation
(ii) Neumann Kelvin formulation
then,
then, ,
Unsteady wave elevation
Unsteady wave elevation
Interaction Term
DBF
[Double body flow formulation]
Fig. 14 Perspective view of modified Wigley hull
5.4 Treatment of influence term in DBF
Fig. 15 Interaction between Double body flow and Incident wave
at y/(B/2)=1.4; Fn=0.2, l/L=0.5, c=p
5.4 Treatment of influence term in DBF (Cont.)
Interaction Term
[Neumann Kelvin formulation]
at y/(B/2)=1.4; Fn=0.2
NK
Fig. 16 Measured steady wave
5.5 Treatment of influence term in NK
Fig. 17 Interaction effect between Kelvin wave and Incident wave
at y/(B/2)=1.4; Fn=0.2, c=p
l/L=0.5 l/L=0.6
l/L=0.7
NF = 140 x 38 = 5230
NH = 70 x 20 = 1400
Fig. 18 Perspective view of modified Wigley hull
5.6 Computational Domain
Fig. 19 Computational grid
Fig. 20 Steady Kelvin wave pattern of modified Wigley model (blunt) at Fn=0.2
5.7 Numerical and Experimental Result (Cont.)
Fig. 21 Added mass and damping coefficient due to
forced heave motion at Fn=0.2
5.7 Numerical and Experimental Result
Fig. 22 Added mass and damping coefficient due to
forced pitch motion at Fn=0.2
5.7 Numerical and Experimental Result (Cont.)
Fig. 23 Wave exciting forces and moments at Fn=0.2, c=p
5.7 Numerical and Experimental Result (Cont.)
Fig. 24 Ship motions at Fn=0.2, c=p.
5.7 Numerical and Experimental Result (Cont.)
Fig. 25 Heave radiation wave at Fn=0.2, KL = 30
5.7 Numerical and Experimental Result (Cont.)
Fig. 26 Diffraction wave at Fn=0.2, l/L=0.5, c=p
5.7 Numerical and Experimental Result (Cont.)
Fig. 27 Heave radiation wave at y/(B/2)=1.4 for Fn=0.2, KL = 30, 35
Fig. 28 Diffraction wave at y/(B/2)=1.4 for Fn=0.2, l/L=0.5 , 0.7, c=p
5.7 Numerical and Experimental Result (Cont.)
[forced motion, | x3 |=0.01m, | x5|=tan^(0.01/0.42), |x7|=0.01 m]
Fig. 29 Total unsteady wave at y/(B/2)=1.4 for Fn=0.2, c=p
5.8 Influence of Amplitude of Incident wave
l/L=0.7
l/L=0.9
[forced motion, | x3 |=0.01m, | x5|=tan^(0.01/0.42), |x7|=0.01 m]
5.8 Influence of Amplitude of Incident wave (Cont.)
Fig. 30 Total unsteady wave at y/(B/2)=1.4 for Fn=0.2, c=p
l/L=1.1
l/L=1.4
[Maruo, H, 1960]
[Okhusu, M, 1977]
(i) Added Wave Resistance
(i) Kochin Function
5.9 Estimation of Added wave Resistance
where
where ,
at y/(B/2)=1.4; Fn=0.2, l/L=0.5, c=p
Fig. 31 Kochin function computed with diffraction wave
5.9 Estimation of Added wave Resistance (Cont.)
5.9 Estimation of Added wave Resistance (Cont.)
Fig. 32 Added wave Resistance at Fn=0.2, c=p
Steady wave resistance (average) = 0.4382 kgf, L = 2.5 m
5.9 Estimation of Added wave Resistance (Cont.)
Fig. 33 Added wave Resistance at Fn=0.2, c=p
Steady wave resistance (average) = 0.4382 kgf, L = 2.5 m
Derived interaction effect of incident wave and steady wave in the unsteady
wave have been applied for unsteady wave analysis in various ways
1). Influence of 2nd order term in the wave. - not dominant
2). Interaction effect between double body flow and incident wave. – small
3). Interaction effect between Kelvin's wave and incident wave – Remarkable
1
Steady wave, diffraction wave, radiation wave and unsteady pattern which are
numerically computed are compared with experimental data, and it is matching
well with experiment. Only in the amplitude, there is a small difference.
2
The experiments were carried out for a modified Wigley model and the obtained
results of hydrodynamic forces, ship motions, unsteady wave fields and added
wave resistance were used for the validation of the present RPMs. Through the
comparisons, it was confirmed that the present RPM code is effective for the
seakeeping estimations.
3
5.10 Concluding Remarks
6. SEAKEEPING OF HIGH SPEED SHIP WITH TRANSOM STERN
TRANSOM STERN BOUNDARY CONDITION
HIGH SPEED - MONOHULL
To treat the transom stern condition, until now there is no
proper boundary condition to treat the transom stern by panel
method. A new boundary condition is derived from the
experimental observation to treat the transom effect
mathematically in numerical method.
1
Transom stern Boundary Condition
6.1 Introduction
(i) Steady Problem
Steady Velocity Potential
[Hughes & Bertram, V, 1995]
Fig. 34 Transom steady wave
6.2 Transom stern boundary condition
(ii) Unsteady Problem
Monohull in the Experimental Lab.
Unsteady Velocity Potential
Fig. 35 Snap shot of the transom stern in the motion measurement test
Total wave – smoothly separating from the dry transom
(i) Diffraction Velocity Potential
Diffraction wave plus incident wave is equal to zero
6.3 Formulation of Transom stern condition
(ii) Radiation Velocity Potential
At Transom: Unsteady wave elevation = Vertical unsteady displacement
Only z comp.
where
6.3 Formulation of Transom stern condition (Cont.)
Fig. 36 Treatment of Transom stern boundary condition
6.4 Treatment of Transom stern condition (Cont.)
Monohull
Fig. 37 Plan view of the monohull
6.5 Hull Data
NH = 1480 (74 x 20)
NF = 3888 (162 x 24)
NFA = 297 (99 x 3)
Fig. 39 Computational Grids
Fig. 38 Perspective view of Monohull
6.5 Hull Data (Cont.)
Fig. 40 Steady wave at Fn = 0.5 Fig. 41 Measured sinkage and trim
6.6 Numerical and Experimental Result
Fig. 42 Added mass and damping coefficient due to forced heave motion at Fn=0.5
6.6 Numerical and Experimental Result (Cont.)
Fig. 43 Added mass and damping coefficient due to forced pitch motion at Fn=0.5
6.6 Numerical and Experimental Result (Cont.)
Fig. 44 Wave exciting forces and moments at Fn=0.5, c=p
6.6 Numerical and Experimental Result (Cont.)
Fig. 45 Ship motions at Fn=0.5, c=p.
6.6 Numerical and Experimental Result (Cont.)
Fig. 46 Wave pressure on the hull at Fn=0.5, l/L=1.1, c=p
6.6 Numerical and Experimental Result (Cont.)
Fig. 47 Total unsteady pressure at Fn=0.5, l/L=1.1, c=p
6.6 Numerical and Experimental Result (Cont.)
Fig. 48 Wave pressure at Fn=0.5, l/L=1.1, c=p
6.6 Numerical and Experimental Result (Cont.)
Fig. 49 Total unsteady pressure at Fn=0.5, l/L=1.1, c=p
6.6 Numerical and Experimental Result (Cont.)
at Fn=0.5, KL=30, x3 = 0.02 m
Fig. 50 Comparison of measured and computed wave pattern
at Fn=0.5, l/L=0.7, c=p, H/l=1/20
Fig. 51 Comparison of measured and computed wave at y/(B/2) = 1.52
at Fn=0.5, KL=30,
x3 = 0.02 m
at Fn=0.5, l/L=0.7,
c=p
Fig. 52 Added wave resistance (Fn=0.5, c=p) with transom and sinkage & trim effect
6.6 Numerical and Experimental Result (Cont.)
HIGH SPEED - TRIMARAN
INCLUDE TRANSOM STERN BOUNDARY CONDITION
Trimaran
Fig. 53 Plan view of Trimaran
6.7 Hull Data
Fig. 55 Computational Grids
Fig. 54 Perspective view of Trimaran
6.7 Hull Data (Cont.)
Fig. 56 Steady Kelvin wave pattern at Fn = 0.5
6.8 Numerical and Experimental Results
6.8 Numerical and Experimental Results (Cont.)
Fig. 58 Steady pressure on the hull at Fn = 0.5 Fig. 57 Steady wave view at Fn = 0.5
Fig. 59 Added mass and damping coefficient due to forced heave motion at Fn=0.5
6.8 Numerical and Experimental Results (Cont.)
Fig. 60 Added mass and damping coefficient due to forced pitch motion at Fn=0.5
6.8 Numerical and Experimental Results (Cont.)
Fig. 61 Wave exciting forces and moments at Fn=0.5, c=p
6.8 Numerical and Experimental Results (Cont.)
at Fn=0.5, KL=30, x3 = 0.02 m
Fig. 62 Heave Radiation wave
6.8 Numerical and Experimental Results (Cont.)
Fig. 63 Diffraction wave
at Fn=0.5, l/L=0.7, c=p, H/l=1/20
6.8 Numerical and Experimental Results (Cont.)
Fig. 65 Computational Grids
Fig. 64 Perspective view of Monohull
6.9 Hull Data
(Sinkage and Trim)
Fig. 66 Ship motion at Fn=0.5, c=p.
6.10 Numerical and Experimental Results
Fig. 67 Wave pressure on the hull at Fn=0.5, l/L=1.1, c=p
(with TSC, sinkage and trim effect) (with TSC, without sinkage and trim effect)
6.10 Numerical and Experimental Results (Cont.)
(with TSC, sinkage and trim effect) (with TSC, without sinkage and trim effect)
Fig. 68 Total unsteady pressure at Fn=0.5, l/L=1.1, c=p
6.10 Numerical and Experimental Results (Cont.)
Numerically computed results of hydrodynamic forces, ship motions,
unsteady wave fields and added wave resistance were compared with
experiment and good agreement is observed.
2
6.11 Concluding Remarks
Transom stern boundary condition is derived based on the experimental
observation and applied for high speed monohull 1
In the Kochin function calculation for the added wave resistance, the
amplitude of H1 is less than 10% of H2 for the conventional ships advancing
at low forward speed. It is easily noticed that the amplitude of H1 is larger
than that can be seen in the conventional ships. It is suggested that the
contribution of H1 is valuable for the added wave resistance estimation.
4 The accuracy of the seakeeping estimations was fairly improved by taking
account of the effect of the sinkage and trim incorporated with the present
transom stern condition
3
Transom condition is applied for the trimaran and predicted hydrodynamic
forces are showing good agreement with experimental result
Sinkage and trim also considered for numerical calculations of wave
pressure and unsteady pressure and compared with experimental results.
Influence of main hull is observed in the outriggers in pressure plot.
5
6
7. CONCLUSIONS
CONCLUSIONS
The interaction effect between the incident wave and double-body flow in
steady flow is not remarkable. The consideration of its effect scarcely
affects the analyzed diffraction wave.
1.
In the analysis of diffraction wave, the effect between the incident wave
and the steady Kelvin wave is remarkable, and the effect can be also
confirmed in the Kochin function. This affects the estimation of the added
wave resistance about 4% in magnitude.
2.
The experiments were carried out for a modified Wigley model and the
obtained results of hydrodynamic forces, ship motions, unsteady wave
fields and added wave resistance were used for the validation of the
present RPMs. Through the comparisons, it was confirmed that the
present RPMs are effective for the seakeeping estimations.
3.
CONCLUSIONS (Cont.)
A flow model was proposed to satisfy the phenomena denoted in point 4.,
and a corresponding boundary condition was derived.
5.
It was confirmed that the Rankine panel method with transom stern
condition well explains the experimental results. Additionally, the
accuracy of the seakeeping estimations was fairly improved by taking
account of the effect of the sinkage and trim incorporated with the
present transom stern condition.
6.
The proposed method is applied to the trimaran and the results are
compared with experiments. The effect of sinkage and trim is taken into
account with the transom stern condition. The comparisons of obtained
results with experiments show good agreements for the trimaran.
Interaction effect between the main hull and outriggers are observed
around stern part only in the unsteady pressure distributions.
7.
From the high speed monohull analysis, It was confirmed from the
present experiments (H/l=1/50) that the transom stern was completely
dry even when the ship is freely oscillating in waves provided that ship
advances at high speed. All the waves that consist of the incident wave,
steady wave, radiation waves and diffraction wave flow away smoothly
from the bottom part of the dry transom stern.
4.
Thank You