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Twenty-First Symposium on NAVAL HYDRODYNAMICS
In the case of variable grid spacings, the values of and are determined as follows.
(18a)
(18b)
where,
(19)
The schemes above are compared for the period of ten times of wave period (T) of the case of T=1.5sec, and the results are shown for the vertical variation of the density-function in Fig.1. It is obviously noted that the higher the order of derivative of the differencing error is, the less the density-function is diffused in the vertical direction. The smaller time increment (DT) is also very effective for suppressing the numerical diffusion. As shown in Fig.2 the discontinuity of the density-function disappears and a very sharp interface is obtained when the time increment is reduced to the half value of the original one for the third order upwind scheme. Therefore, the third order upwind scheme is employed hereafter.
The simulated waves for three cases of wave period, T=0.9, 1.2 and 1.5sec, are presented in Figs.3 and 4, which indicate respectively the wave profile with a sine curve for each case and the time variation of wave height with measurements. It is demonstrated that at least ten waves are generated with sufficient degree of accuracy and the magnitude of the error due to the numerical modelling is of the same order with that of experiments. Therefore it may be safe to say that the density-function method employed here can have sufficient accuracy for simulating waves without special treatments for suppressing the numerical diffusion when it uses sufficiently fine grid spacing and small time increment.
FREE SURFACE SHOCK WAVE ABOUT A WEDGE MODEL
The nonlinear features of ship waves in the near field had been noticed in the 1970s and the detailed structure and mechanism of nonlinear bow waves are experimentally investigated by Miyata et al.[1][2][3]. It is elucidated that the nonlinear bow wave had a lot of common properties with supersonic shock waves and the nonlinear bow waves involving these properties are called free surface shock wave (FSSW). The typical properties of FSSW are (1) steepness of the wave slope, (2) discontinuity of velocities satisfying the shock wave condition, (3) free surface turbulence on and behind the wave front, (4) systematic change of the wave-front-angle depending on the Froude number (Fd) and the ship configuration and (5) dissipation of wave energy into momentum loss far behind the ship. Also the FSSW is limited in the thin layer near the free surface.
Although the above properties are recognized by experiments, the details of the FSSW structure had to be investigated by numerical simulations. However, due to the property of (1), wave breaking occurs at the wave-crest point, which makes the numerical simulation of FSSW significantly difficult. The finite-difference method mentioned in 2.1 with density-function method is applied to this problem and a wedge model of which half entrance angle is 20deg and draft is 0.1m is chosen for the simulation [20]. The simulations are carried out at three Froude numbers based on the draft, 0.8, 1.1 and 1.4, with normal grid spacing. The case of Fd=1.4 is also simulated with fine grid spacing of which minimum spacing is 1/4 of that for the former case. See the Ref.[20] for other details for computational conditions.
The systematic change of the wave-front-angle depending on the Froude number is realized showing good agreement with the experimental results and other properties of FSSW including 3D wave breaking phenomenon are recognized in the simulations by the finite-difference method. In this paper only the result of the case of Fd=1.4 with fine grid spacing is shown. The time-sequential overviews of the wave at Fd=1.4 are presented in Fig.5 The uniform flow is accelerated until dimensional time (T) reaches 1.77sec. Spilling breaker appears at T=1.564sec before the flow acceleration is ceased and the wave crest overturns at around T=1.932sec. The plunging wave front breaks at T=2.024sec and the wave again develops. The breaking wave front is laterally extended after T=2.208sec and the above process of breaking is periodically repeated. The secondary wave also shows breaking features in the vicinity of the body surface, however the accuracy is supposed to be inferior to the foremost wave due to the influence of the momentum deficient motions of the foremost wave.
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