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Fig. 10 shows a comparison of amplitude ∆Cp and Fig. 11 does components of ∆Cp composed of the flat plate, the wing and the interaction components. From these Figures, we understand that the difference between QCM and other two methods is caused by the flat plate component due to the finite length effect. Next we investigate the effect of the flat plate on the lift of 2-D wing and check also the solid boundary factor 2.0. We show the pressure distributions on the wing surface in Fig. 12, and the lift coefficient CL in Fig. 13, and the solid boundary factor Sb at the center of the flat plate in Fig. 14. We understand that the finite flat plate of CL gives similar effect on the wing characteristics to the mirror image, because QCM results are almost same as those of the mirror image method As to the solid boundary factor, the value of the mirror image method converges to 2.0 with increase of the distance d, while the value of QCM shows a different tendency and seems to converge to 1.0. Accordingly, we think that the solid boundary factor 2.0 is not always applicable to the finite flat plate. 3.2 Pressure fluctuation due to thickness-varying wing in a gust Let us consider the case where a thickness-varying wing is in uniform flow with a sinusoidal gust. We assume that the wing thickness tw(t) varies with time by the following expression, (9) In Eq. (9), t0 is the original wing thickness, th the amplitude of thickness variation, ωt the circular frequency of thickness variation and ψ the phase difference from the sinusoidal gust. Then the second derivative of wing volume with respect to time becomes as (10) In Eq. (10), c1 is a constant known from the expression of NACA wing section. Then the amplitude of of the volume variation becomes At first, we consider the case where the wing is in uniform flow and is varying only upper surface. (see Fig. 15) Introducing kt(=ωtc/2VI) instead of ωt, we show in Fig. 16 the relations between the amplitude of pressure fluctuation ∆Cp at the center of the flat plate and kt, when kt changes from 0.5 to 3.0 and from 0.001 to 0.05. We find by three methods that ∆Cp changes almost linearly with kt for kt=1.0~3.0 and ∆Cp does quite differently for kt<0.05. For small kt, the limit of ∆Cp will be the value of the quasi-steady case. Next we show some results of ∆Cp when the wing is in a gust and is changing the thickness with phase difference from the sinusoidal gust. In cases of kt=1.0, ψ=0 and th=0.01t0, 0.05t0, 0.1t0, we show the relation of ∆Cp and th in Fig. 17 and get the linear relation between ∆Cp and th. By setting the phase difference ψ as −π, −π/2, −π/4, π/2, π between thickness variation and the gust, we show the relation between ∆Cp and ψ in Fig. 18. The phase lead of thickness variation (ψ=−π/4) gives a maximum value of ∆Cp, because maximum lift and maximum thickness work together for maximum ∆Cp. Fig. 19 shows the effects of phase difference on ∆Cp. Therefore we understand that occurrence of cavitation with phase difference may affect the amplitude of the pressure fluctuation on the hull. 4. RESULTS OF 3-D PROBLEM Corresponding to the 2-D problem, we perform calculations for the 3-D problem. Let us consider a case where a 3-D wing with a span s=3c and NACA0012 section is set in uniform flow with a sinusoidal gust and a finite flat plate with a breadth 2s and a length 5c is set at a vertical distance d=0.5c above the wing. (see Fig. 20). We take lp=5c, 6c, k=1.0, υ0=0.1VI, and the same conditions for the gust and thickness variation tw (t) and phase difference ψ. Fig. 21 shows instantaneous pressure distributions obtained by QCM and SDM on the midspan line (y=0.0) on the flat plate at time step 148 and Fig. 22 does the pressure distributions on the midchord line (x=0.5c) at the same time step. These distributions seem to be plausible around the fore and wing parts. Though SDM gives some edge effect to Cp, but degree of the effect is not large compared with 2-D case. Figs. 23 and 24 show a comparison of Cp obtained by the four methods in both x, y directions. It is interesting to notice that the four methods give similar distributions except the aft part and SDM can give reasonable Cp above the wing. We think that this may be due to the weaker edge effect compared with 2-D flat plate. We show a comparison of the amplitude distribution of the pressure fluctuation ∆Cp in Fig. 25 and Fig. 26. Around the wing part, all methods give nearly the same values, but do different tendencies in the aft part as the 2-D case. Fig. 27 shows the relations between ∆Cp and kt in case of the thickness-varying wing. We find again linearity between them in the range of kt=1.0~3.0. Then we show in Fig. 28 a comparison of the solid boundary factors at the center of the plate obtained by QCM and the mirror image method and finally in Fig. 29 a comparison of the phase difference effects the authoritative version for attribution.

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