Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics (1994) / Chapter Skim
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Session 1- Wavy/Free Surface Flow: Panel Methods 1
Session 1- Wavy/Free Surface Flow: Panel Methods 1
Pages 1-42
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From page 3... ...
first introduced the mixed Euler-Lagrange method for solving two-dimensional fully nonlinear water wave problems. This time stepping procedure requires two major tasks at each time step: the linear field equation is solved in an Eulerian frame; then the fully nonlinear boundary conditions are used to track individual Lagrangian points on the free surface to update their position and potential values.
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From page 4... ...
. The successful implementation of an EulerLagrange method to solve fully nonlinear water wave problems requires a fast and accurate method to solve the mixed boundary value problem that results at each time step and a stable time stepping scheme.
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From page 5... ...
On the free surface, the kinematic condition is used to time step the free surface elevation and the dynamic condition is used to update the potential. Many different approaches are possible to time march the free surface boundary conditions.
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In the material node approach no extra derivatives need to be evaluated and this probably explains why this is the approach most often used. With material nodes one must always be concerned that the nodes do not penetrate the body surface between time steps since they are unconstrained.
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From page 7... ...
NUMERICAL TECHNIQUES In the usual manner, the integrals may be discretized to form a system of linear equations to be solved at each time step. In the desingularized method, the source distribution is outside the fluid domain so that the source points never coincide with (20)
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From page 8... ...
The free surface elevation and the potential are then updated by a time stepping procedure. In the free surface updating, if material nodes are used, the spatial derivatives of the free surface elevation, Vet, are not required.
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From page 9... ...
Material nodes were used and the time step size was At~i; =.05 . A fourth-order Runga-Kutta-Fehlberg time stepping scheme is used.
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1 1~ / \ 0 2 4 ~ 8 10 Figure 2: Comparison of fused horizontal nodes versus material nodes. 0.2 0.1 n~ 0.0 T~dimensional / \ _ I .
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Calculations have shown little variation in the numerical results with node spacing schemes. Figure 6 shows the typical wave profiles generated by a heaving rectangle in infinite water depth.
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0 - ~ I '1 1 : 0.0 0.5 1.0 1.5 2.0 ~ . m4H/g Figure 8: Added mass coefficient vs frequency for a heaving two-dimensional rectangle in infinite water depth, B = 2.0, H = 1.0?
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From page 13... ...
, - _ w v \_/ 0 . 2- ~Time = 48.0 \~_/ ~ ~Time = 36.0 \ I_ _ ~Time = 24.0 ~~ 0-0- 1 1 1 1 0.0 5.0 10.0 15.0 20.0 25.0 AH Time = 60.0 ~Time = 12.0 Figure 10: Comparison of the wave profiles using 2 and 60 planes of symmetry at the end of different periods for a heaving ~reedimensional cylinder in infinite water depth,R= l.O,H= l.O,a/H=O.l, m~ = ~l3 13
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From page 14... ...
5 0.0 2.0 4.0 6.0 8.0 10.0 x/R Figure 12: Comparison of instantaneous wave profiles for a heaving cylinder in finite water depth, h/H= 2.0, R = 1.0, H = .5, a/H = .5, symmetry plane number = 60 cylinder moves upward, however, the bottom is closer to the free surface and a phase difference appears between the curve of the force and the motion as a result of the wave damping.
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From page 15... ...
O -2.0~ ' ~ 1.~ ~I I ~ ~3 . 0 ~ 0.0 10.0 20.0 30.0 t~g/Hup~'er Figure 14: Time history of the heaving amplitude and vertical force on a double cylinder in infinite water depth, RUpper = 1.0, HUpper = 1.0, Rbo~tom = 1.5, Hbo~om = 0 5, a/Hupper = 0~50 Wigley Hull Starting from Rest Results for the free surface waves and wave resistance of a Wigley hull starting from rest have been computed using the desingularized approach.
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From page 16... ...
Figure 15: Wave contours for the Wigley Hull, Froude number = .25, coarse and, NF = 1920, NB = 369, to= 14 Figure 16: Wave contours for the Wigley Hull, Froude number = .25, Dme and, NF = 247S, NB = 510, to= 14 1 6 Figure 17: Shaded rendering of the waves generated by the Wigley Hull, Froude number = .25, fine ~id, t - = 14 16
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An expanded scale of the wave profile along the hull for the last time step is given in figure 19. Also plotted in figure 19 is the wave profile along Wigley hull as experimentally measured by the University of Tokyo (see Noblesse and McCarthy 1983 or Noblesse et al.
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From page 18... ...
~X ._1 ~ ,l,~' ~/ - _ _ / / :, P Bt ~ -- EXP W__ ~ TOTAL -P~ ~~ - 1 pVt V~ 1 1 1 1. 1 1 2.0 4.0 6.0 8.0 10.0 12.0 t~ 14.0 16.0 18.0 Figure 21: Wave resistance components for Wigley Hull, Froude number = .25, fine grid 18
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From page 19... ...
The time step size was i\t~=.07 for both calculations. CONCLUSIONS The desingularized method is a viable alternative for solving fully nonlinear water wave problems.
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Mi (1986) , "Rankine Source Methods for Numerical Solution of the Steady Wave Resistance Problem," Proceedings 16th Symposium on Naval Hydrodynamics, University of California, Berkeley, pp.
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From page 21... ...
: drag of fore/first foil D2 : drag of aft/second foil En : Froude number (= V// L : lift Lo : lift of fore/first foil or 2 : lift of aft/second foil Loo Rtu SF Sc Sw V : lift of mono foil in unbounded flow . wavema.king resistance : still water surface : surface of circular cylinder : hydrofoil surface : uniform flow velocity at infinity c : hydrofoil chord length f : submergence of leading edge(fore foil)
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From page 22... ...
However, when we try to solve the wave resistance problem in the high speed range by Rankine source method, it is doubtful whether we can use LSA to represent the basic flow in the same way as the low speed range t2]
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Next, we give some uddihon~1 explanations to the case of ~ point doublet to express the circular cylinder. Denoting the strength of ~ point doublet with the ax~ in theme direction, the disturbed velocity potential )
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, (b) show comparisons of source distributions on still water surface due to the circular cylinder obtained by LSA and HSA at Fn = 0.85,10.0.
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We think that HSA gives more correct wave profiles at high speed than LSA compared with the analytical method. Surface Source Distribution Next, we show some results of the case to use the surface source distribution for the circular cylinder.
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From page 26... ...
We notice that the lift and the drag by LSA have a tendency to converge to each constant value and on the contrary by HSA, lift approaches the value in the unbounded flow and drag does zero in high speed range (Fit ~ 5.0~. In Fig.18 wave profiles are shown in the relatively high speed range (Fr,, ~ 1.0)
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From page 27... ...
In Fi&s.25~(c) we show the wave profiles and the gage patterns of the tandem Ail system coming HSA with LSA.
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From page 28... ...
We proposed a, new Rankine source method named HSA which uses the inverse image above the still water surface in the high speed range. We carried out the numerical calculations for a, circular cylinder and 2-D thin hydrofoil in the high speed range a,nd confirm that both forces a,nd wave profiles agree very well to the analytical ones.
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From page 29... ...
where ¢) wj is the disturbed velocity potentials to express the basic disturbance to use the double model or the inverse image and i\~'uj expresses the effect of the free surface on the each foil surface, respectively.
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From page 30... ...
Using HSA, since there is no analytical solution for hyclrofoils, we define the wave profiles in the limit of high speed analytically from the linearized free sr~rfa,ce condition, V71.~-(¢sy~y=o = 0 (A-17) It is shown as 71 = V J(¢sy~y=od~ + const (A-18)
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From page 31... ...
Wave profiles due to point doublet > F LSA HSA 1.0 ~ , -~.C, a' 1 ~Fn=1 0.0 , , 10.0 Fig.3(b) Source distribution on still water surface due to circular cylinder USA HSA - Analytic ,~ -10.0 0 V > -1.C Fn=1 0.0 10.0 1, <1!
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From page 32... ...
Cp A NACAOO 1 2 a =5 2-D Panel method -~- SQCM Fig.9 Comparison of 2-D panel metllod and SQCM 32 LSA 0 HSA 1.0 Analytic ll Fn=0.85 Fig O(a) N`~7ave profiles due to circular cylinder , oo · ~ O O · O /~: / '; / '~ 0 LSA · HSA Analytic ~QS' ~ ~ 1.0 ~ Fn 2.0 L 5.0 ~ Fn 10.0 Fig 7 \`Vave making resistance coe~cient Cw of circu'ar cylinder - Wave Flow /' - Basic Fiow 3=n=1 0 0 (c)
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From page 33... ...
'' 10 '''''O ~° LSA · HSA -0.5 (I$IACA0005 , a = 5 ° , f/c=0.956) t~L ~ ~ ~ ~ ~ ~ I I ~ ~ ~ t~ -~ I I I I ~ I I ~ L r,,, Fig.10 Wave profiles due to hydrofoil (NACAO0057 h/c = 0.956' ~ = 5°)
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Comparison of wave profiles between HSA and LSA 1.0 ~ ~ -~_ 0.5 / .
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Fig.16(a) Wave profiles versus stagger Al s/c=l.20 in \ /~\' 20.0 .
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1 5.0 10.0 15.~.0 ~,1,,,1,,,,,,,,,1 11 ~Q 1 5.0 20.0 , , O.t ~\ I I I I I I I I 1 1 1 1 1 1 1 1 1 1 ,\ 1 5.0 10.0 15.0 20.0 \ r Fn=10.00 HSA ~ LSA (NACA001 2, a =5 ,f/c=1 .O,h/c=O.O,s/c=1 0.0) Fig.18 Wave profiles versus Fn 36
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Comparison of wave profiles and patterns bet~veen LSA and HSA (Coarse n~esh)
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it=-0.01 - HSAFn=4.0 Fig.22(b) Comparison of wave profiles arid patterns between LSA arid HSA (Coarse mesh)
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From page 39... ...
_ - HSA Fn= 2.0 I I 1 141 1 1 -' 1 , I Fig.25(b) CompalisoI1 of wave profiles and patters 19etweeI~ LSA arid HSA (tandem foil, s/c = 6 0)
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From page 40... ...
Comparison of CL (aft foil) I i ' 1 ' 1 ' 1 ' 1 CD Aft foil s/c: 2.5 6.0 1 0.0 f/c_ O h/~-0 O HSA ~ ~ ~ ;''1' " .
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From page 41... ...
We agree with the discusser in thinking that both results should agree with each other on an ideal numerical calculation. However, we have tried to use finer mesh on still water surface using 2-D circular cylinder problem and have also checked them using donhle Precision (REALMS)
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From page 42... ...
The point is that the speed range should be respected in which both results agree, and when they begin to diverge numerically, we should choose the HSA or the LSA. We wish to consider the surface ship prob lem in the near future.
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