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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 330 Pressure Fluctuation on Finite Flat Plate above Wing in Sinusoidal Gust K.Nakatake, K.Ohashi, J.Ando (Kyushu University, Japan) ABSTRACT It is generally said that the amplitude of pressure fluctuation induced by a sphere varying own volume with time is proportional to the second derivative of the volume variation. Pressure fluctuation induced by a cavitating wing is a similar example. In this paper, as a modeled cavitation phenomenon, we treat the pressure fluctuation on a finite flat plate induced by a wing advancing in uniform flow with a sinusoidal gust and varying its thickness with time. The wing is represented by a simple surface panel method âSQCMâ which can treat the unsteady motion. In calculation for a finite flat plate, we use four kinds of calculation methods: the first one uses a mirror image method, the second one does the solid boundary factor method, the third one does the source distribution method and the last one does QCM. Comparing these four kinds of results for 2-D and 3-D cases, we discuss the availability of the four methods and investigate the relation between the amplitude of pressure fluctuation and the second derivative of the wing volume. 1. INTRODUCTION Pressure fluctuation on a hull surface induced by a propeller causes ship hull vibration. Pressure on the hull surface fluctuates largely because the propeller is working in the hull wake. If cavitation occurs, the amplitude of pressure fluctuation becomes larger. Many researchers, such as Huse (1968), Vorus (1974), Hoshino (1980), Wang (1981), Breslin et al. (1982), Ikehata & Funaki (1986), Kehr et al. (1996) etc. studied pressure fluctuation induced by a propeller. As a modeled cavitation phenomenon, we treat the pressure fluctuation on a finite flat plate induced by a wing advancing in uniform flow with a sinusoidal gust and varying thickness with time. In calculating pressure fluctuation, we need to model a propeller and a ship hull. There are a lot of studies on propeller characteristics and we can obtain the highly accurate results (ITTC, 1993). As to modeling of ship stern, there are a few studies using panel method, but ship stern is usually treated as an infinite flat plate, and a solid boundary factor (2.0) is used to express the effect of the flat plate (Huse, 1968). In order to represent hydrodynamically the finite flat plate above a wing, we adopt four kinds of methods: the first one is the mirror image method, the second one is the so called solid boundary factor method, the third one is the source distribution method (SDM) and the forth one is QCM (Quasi-Continuous vortex lattice Method, Lan, 1974). SDM treats the flat plate as a mere solid boundary and QCM does it as a wing. By using a simple surface panel method âSQCMâ (Ando et al., 1998), we calculate the characteristics of an unsteady wing varying its volume in a sinusoidal gust and pressure fluctuation on the flat plate above a 2-D wing and a 3-D wing in a sinusoidal gust. By comparing the obtained results, we discuss the availability of the solid boundary factor method, the mirror image method, SDM and QCM and investigate the relation between the amplitude of pressure fluctuation and the second derivative of the wing volume with respect to time. 2. CALCULATION METHODS FOR PRESSURE FLUCTUATION Let us consider the problem of pressure fluctuation acting on a finite flat plate, induced by a wing advancing in uniform flow with a sinusoidal gust. The unsteady wing is well represented by a simple surface panel method âSQCMâ and its formulations are described in the reference (Ando et al., 1998). Therefore we outline the main equations in this paper. Fig. 1 shows the schematic diagrams to represent the 2-D finite flat plate by QCM and by the mirror image method when a wing is advancing in uniform flow with a sinusoidal gust and changing its thickness with time. We take the coordinate system fixed to the wing and adopt four kinds of the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 331 calculation methods to represent the finite flat plate. The first one is the mirror image method and the second one is the solid boundary factor method, which means to multiply the pressure induced by the unsteady mono-wing by the factor 2.0. The third one is the source distribution method (SDM) and the fourth one is QCM (Lan, 1974). Here we describe the formulations for SDM and QCM, since other methods are included partly in these formulations. Fig. 1 Coordinate System and Schematic Diagram for Wing and Flat Plate Firstly, the induced velocity due to the bound vortex on the camber surface and the shed vortex at time tL is expressed as follows. (1) where, NÎ³: number of divisions of camber surface L: number of shed vortices In Eq. (1), and are the induced velocities due to the bound vortex and the shed vortex with unit strength, respectively, and Î(tL) is a circulation around the wing at time tL, and Î³v(Î¾v, tL) is the strength of the bound vortex on each panel. Next the induced velocity due to the source panel on the wing surface is expressed as (2) where, NÏ: number of divisions of one side of the wing In Eq. (2), expresses the induced velocity due to a line source with unit strength, and Ïj(tL) does the source strength on each panel at time tL. SDM uses the similar equation with strength Ïp to represent the finite plate. Adding these velocity components to the relative inflow velocity we have for a mono-wing (3) and the boundary condition to be satisfied on the wing and camber surfaces is the solid boundary condition, (4) where, unit normal vector at each panel and the unsteady Kutta condition (Ando et al., 1998). Next QCM treats the finite plate as a thin wing and gives the following expression for the induced velocity (5) where, the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 332 lp: length of plate : number of divisions of plate. In Eq. (5), and are the induced velocities due to the bound vortex on the flat plate and the shed vortex with unit strength, respectively, and Îp(tL) is a circulation around the flat plate and is the strength of bound vortex on each panel on the flat plate. As to divisions of the finite flat plate, we divide it into three parts i.e. the front part, the wing part and the aft part, and adopt cosine divisions for each part. Summing up all velocity components, we have total velocity in case of presence of the finite plate as (6) The boundary conditions on the flat plate are also the solid boundary condition and the unsteady Kutta condition. The unknown strengths are Î³, Ï and Î³p or Ïp, but we solve the boundary conditions for the wing and those for the flat plate, iteratively. In this case, we need usually several times for enough convergence. Lastly, the unsteady pressure p(t) is expressed in the wing-fixed coordinate system as follows. (7) where, In Eq. (7), Ï, p0, and are water density, the ambient pressure and the velocity potential in the wing-fixed coordinate system, respectively. can be obtained analytically and is evaluated by two-points upstream difference scheme. We define the unsteady pressure coefficient Cp as (8) and denote Cp for the mono-wing case by 3. RESULTS OF 2-D PROBLEM Let us consider the case where a wing with NACA0012 section is set under a finite flat plate in uniform flow VI with a sinusoidal gust of vertical velocity We take the vertical distance d=0.5c, (c: chord length), the angle of attack Î±=0Â°, the reduced frequency k(=Ïc/2VI)=1.0, the amplitude Ï 0=0.1VI and the time increment ât=Ïc/72VI. Ï expresses the circular frequency of the gust and the finite flat plate has length lp of 5c or 10c and the center of the plate is set so as to coincide with the midchord of the wing. In this case, we calculate the pressure fluctuation on the flat plate induced by the wing in uniform flow with a sinusoidal gust by the four methods. Firstly we calculate it by QCM and SDM representing the flat plate and then compare these results with those by the mirror image method and the solid boundary factor method. Number of panel divisions of the wing surface is 60 along the perimeter and number of discrete vortices on the camber is 29 and those of the flat plate are 49 (29 for wing part, 10 for fore and aft parts) for QCM and number of source panels is 10 per chord length c for SDM. 3.1 Pressure fluctuation due to wing in a gust Fig. 2 shows the instantaneous pressure distribution (time step 280) calculated by QCM and SDM on the flat plate of two length 5c and 10c and Fig. 3 shows vortex distribution Î³p and source distribution Ïp at the same time step. From Fig. 2 we find that QCM gives nearly the same pressure distribution for lp=5c, 10c, while SDM does different distributions for two kinds of lp though Ïp is nearly same. Î³p distribution of QCM shows a little different distributions between 5c and 10c, but Î³p satisfies the unsteady Kutta condition in addition to the solid boundary condition and shows a little rise near the leading edge as a usual thin wing. Fig. 4 shows a comparison of pressure distributions obtained for lp=5c at two time steps 245, 280 by QCM and SDM and Fig. 5 shows the corresponding Î³p and Ïp distributions. From these Figures, we notice that QCM results give more stable and reasonable pressure distributions than SDM results even if Ïp shows smoother distribution. This seems to be due to the stable flow produced by the unsteady Kutta condition. Fig. 6 shows a comparison of amplitude of pressure fluctuation âCp between QCM and SDM. We find QCM and SDM give fairly different distributions of âCp except the central part. Especially SDM gives larger âCp than QCM in the fore part and does lower âCp after the central part than QCM, and then gives large âCp near the trailing edge. From these results, we think that QCM gives more stable and realistic pressure fluctuation for 2-D problem than SDM. Fig. 7 and Fig. 8 show the contributions of the unsteady component and the velocity component to the total âCp in case of QCM and SDM, respectively. We must notice that these curves are not additive since each pressure fluctuation has phase difference. the authoritative version for attribution. From these Figures, we understand that QCM gives more reasonable behavior near the leading edge of the flat plate than

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 4 Pressure Distribution on Flat Plate Fig. 2 Pressure Distribution on Flat Plate Fig. 6 Amplitude of Pressure Fluctuation on Flat Plate on flat Plate PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST Fig. 5 Singularity Distribution on Flat Plate Fig. 3 Singularity Distribution on Flat Plate Fig. 7 Component of Amplitude of Pressure Fluctuation 333

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. on flat Plate Fig. 10 Amplitude of Pressure Fluctuation on Flat Plate Fig. 12 Comparison of Pressure Distribution of 2-D Wing Fig. 8 Component of Amplitude of Pressure Fluctuation on flat Plate PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST Fig. 13 Time History of Lift Coefficient Fig. 9 Pressure Distribution on Flat Plate Fig. 11 Component of Amplitude of Pressure Fluctuation 334

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Fluctuation Pressure Fluctuation Fig. 14 Comparison of Solid Boundary Factor Fig. 16 Effects of Frequency kt on Amplitude of Pressure Fig. 18 Effects of Phase Difference Ï on Amplitude of PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST Pressure Fluctuation Fig. 15 Variation of Wing Section Fig. 17 Effects of th on Amplitude of Pressure Fluctuation Fig. 19 Effects of Phase Difference Ï on Amplitude of 335

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 336 SDM. We think that the Kutta condition stabilizes the flow over the flat plate. Next we compare QCM results with those of the mirror image method and the solid boundary factor method. Fig. 9 shows a comparison of pressure distributions at time step 280 on the flat plate obtained by the three methods. The mirror image method gives nearly the same results as but different distribution from QCM result near the aft part of the plate. Fig. 10 shows a comparison of amplitude âCp and Fig. 11 does components of âCp 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, Ït the circular frequency of thickness variation and Ï 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(=Ïtc/2VI) instead of Ït, we show in Fig. 16 the relations between the amplitude of pressure fluctuation âCp 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 âCp changes almost linearly with kt for kt=1.0~3.0 and âCp does quite differently for kt<0.05. For small kt, the limit of âCp will be the value of the quasi-steady case. Next we show some results of âCp 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, Ï=0 and th=0.01t0, 0.05t0, 0.1t0, we show the relation of âCp and th in Fig. 17 and get the linear relation between âCp and th. By setting the phase difference Ï as âÏ, âÏ/2, âÏ/4, Ï/2, Ï between thickness variation and the gust, we show the relation between âCp and Ï in Fig. 18. The phase lead of thickness variation (Ï=âÏ/4) gives a maximum value of âCp, because maximum lift and maximum thickness work together for maximum âCp. Fig. 19 shows the effects of phase difference on âCp. 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, Ï 0=0.1VI, and the same conditions for the gust and thickness variation tw (t) and phase difference Ï. 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 âCp 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 âCp 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.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Wing and Flat Plate Fig. 24 Pressure Distribution on Flat Plate Fig. 22 Pressure Distribution on Flat Plate Fig. 20 Coordinate System and Schematic Diagram for PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST Fig. 23 Pressure Distribution on Flat Plate Fig. 21 Pressure Distribution on Flat Plate Fig. 25 Amplitude of Pressure Fluctuation on Flat Plate 337

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 338 on âCp by the four methods. From these results we understand that the solid boundary factor method is not always applicable to the finite flat plate and SDM is also applicable to estimation of pressure fluctuation on the flat plate near the wing. Since these results are obtained only by numerical calculations, we must evaluate them by corresponding experiments. These calculation methods are easily extended to the case of a pitching or heaving wing in uniform flow. Fig. 26 Amplitude of Pressure Fluctuation on Flat Fig. 27 Effects of Frequency kt on Amplitude of Pressure Fluctuation Fig. 28 Comparison of Solid Boundary Factor Fig. 29 Effects of Phase Difference Ï on Amplitude of Pressure Fluctuation 5. CONCLUSION We applied QCM, SDM, the mirror image method and the solid boundary factor method to the problem of amplitude of pressure fluctuation induced on the finite flat plate set above a wing in uniform flow with a sinusoidal gust. From the obtained results, we conclude as follows. â¢ The mirror image method, the solid boundary factor method, QCM give nearly the same amplitude of pressure fluctuation on a finite flat plate in the upstream and upper regions of the wing in 2-D and 3-D problem. SDM is similar only in 3-D problem. â¢ Only QCM seems to give reasonable amplitude distribution on a finite flat plate in lengthwise direction. â¢ The amplitude of pressure fluctuation varies linearly with the second derivative of wing volume with respect to time in a certain range of circular frequency of gust. ACKNOWLEDGEMENT The authors would like to deeply thank Mrs. Yasuko Yamasaki for her typing of this manuscript. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 339 REFERENCES J.Ando, S.Maita, K.Nakatake, âA New Surface Panel Method to Predict Steady and Unsteady Characteristics of Marine Propeller,â Proc. of 22nd ONR Symposium on Naval Hydrodynamics, Washington, 1998. J.P.Breslin, R.J.Van Houten, J.E.Kerwin and C.-A.Johnsson, âTheoretical and Experimental Propeller -Induced Hull Pressures Arising from Intermittent Blade Cavitation Loading and Thickness,â Trans SNAME., Vol. 90, pp. 111â151, 1982. T.Hoshino, âPressure Fluctuation Induced by a Spherical Bubble Moving with Varying Radius,â Trans. of the West-Japan Society of Naval Architects, Vol. 58, pp. 221â234, 1979. T.Hoshino, âEstimation of Unsteady Cavitation on Propeller Blades as a Base for Predicting Propeller Induced Pressure Fluctuations,â Journal of the Society of Naval Architects of Japan, Vol. 148, pp. 33â44, 1980. E.Huse, âThe Magnitude and Distribution of Propeller-Induced Surface Forces on a Single-Screw Ship Model, â Norwegian Ship Model Experiment Tank Publication, No. 100, 1968. M.Ikehata & H.Funaki, âAnalytical Characteristics of Oscillating Pressure Distribution above a Propeller,â Journal of the Society of Naval Architects of Japan, Vol. 159, pp. 71â81, 1986. 20th ITTC Propulsor Committee Report, 1993 Y.-Z.Kehr, C.-Y.Hsin and Y.-C.Sun, âCalculations of Pressure Fluctuations on the Ship Hull Induced by Intermittently Cavitating Propeller,â Proc. of 21st ONR Symposium on Naval Hydrodynamics, pp. 882â897, 1996. C.E.Lan, âA Quasi-Vortex-Lattice Method in Thin Wing Theory,â Journal of Aircraft, Vol. 11, No. 9, pp. 518â527, 1974. W.S.Vorus, âA Method for Analyzing the Propeller-Induced Vibratory Forces Acting on the Surface of a Ship Stern,â Trans. SNAME, Vol. 82, pp. 186â 210, 1974. G.Wang, âThe Influence of Solid Boundaries and Free Surface on Propeller Induced Pressure Fluctuations,â Norwegian Maritime Research, No. 2, pp. 34â46, 1981. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 340 DISCUSSION T.Hoshino Mitsubishi Heavy Industries, Ltd. Japan The authors should be congratulated for their efforts to make clear the solid boundary effect of a finite flat plate on the pressure fluctuation induced by a wing. I have the following questions. (1) Fig. 6 shows that there is very large difference of amplitude of pressure fluctuation between QCM and SDM, especially in the aft part of the plat. This difference would be due to the consideration of trailing vortices shedding from the trailing edge of the plat in QCM. Does this difference become small, if the aft part of the flat plat was lengthened? (2) In the calculation of the pressure fluctuation induced by a propeller, the pressure fluctuation is calculated on the plate fixed in space, not moving with propeller. On the other hand, the pressure fluctuation on the plat moving with the wing is calculated in the present paper. In this case, the pressure fluctuation due to the thickness effect can't be considered and only the loading effect, is considered. I think that this would be the reason why there is large difference between QCM and SDM. AUTHOR'S REPLY Thank you for your raising questions. (1) We think that this difference is due to both the shedding vortex (QCM) and the unstable flow (SDM) near the trailing edge. In case of 2-D problem, lengthening of only aft plate does not improve the difference near the leading edge of the plate. Lengthening of plates forward and afterward improves the difference in the fore and upper parts of the plate (See Fig. 2), but still it can not improve that in the aft part of the plate. In case of 3-D problem, however, the effects of shedding vortex and the unstable flow seem to be small for the fore and upper parts of the plate (see Fig. 21). (2) In this calculation, only the loading effect is considered in the sense that the relative position of the flat plate to the wing is unchanged. We think that large difference between QCM and SDM is due to the unstable flow (SDM) at the trailing edge. Especially this effect is sensitive in the 2-D problem. DISCUSSION K.Kim Naval Surface Warfare Center, USA Cavitation-induced hull pressure has been a continuing subject for many researchers in this field. The authors tried different numerical schemes to predict induced pressure on a flat plate above a pulsating wing in a sinusoidal gust. I have some questions. (1) It is not clear where the boundary condition expressed in Equation (4) was applied; on camber surface, on the wing surface or on both camber and the wing surface? (2) It appears that Fig. 9 (pressure distribution at time step 280) and Fig. 10 (amplitude of pressure fluctuation) are identical. I am wondering if Fig. 10 should be replace by Cp figure. (3) For Fig. 7, the authors stated that total Cp was not the sum of the amplitude of the components due to the phase angle difference. In Fig. 11, however, total Cp appears to be sum of the components. Was phase angle considered here or not? (4) For 3-D case, the authors applied unsteady Kutta condition at downstream edge of the flat plate. How did the authors treat the side edges in terms of boundary condition? (5) In Figs. 16 and 27, Cp is expected to be sensitive to the time-step size for different reduced frequencies. Did the authors use different time step size in these figures? (6) Judging from the large discrepancies in predicted pressure distribution by QCM and SDM, it appears that the validity of the SDM is questionable. I would like to suggest that the authors revisit the formulation and/or numerical schemes to identify possible causes of the discrepancy. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PRESSURE FLUCTUATION ON FINITE FLAT PLATE ABOVE WING IN SINUSOIDAL GUST 341 AUTHOR'S REPLY We thank the discusser for his minute discussions and reply as follows: (1) The boundary condition (4) was applied to on both the camber and the wing surfaces. (2) We considered always the phase difference in our calculation. We show the pressure fluctuations of components with time at x/c=3.0 in Fig. A-1 in case of Fig. 7, and that in Fig. A-2 in case of Fig. 11. From these Figures, we understand that in case of Fig. 7 the phase difference between unsteady and velocity components is so large that two components cancel to each other, on the other hand, in case of Fig. 11, the phase differences among three components are small, then the total amplitude is nearly equal to sum of three components. (3) We did not apply any boundary condition at the side edges, because its effect seems to be small. (4) We tested several time-step sizes in the calculation and confirmed the used one is sufficient for the given range of reduced frequency. Therefore we used the same time-step size. (5) We think that the validity of SDM is not fine behind the unsteady wing especially in the 2-D case. The main cause is that SDM can not satisfy the Kutta condition at the trailing edge of the flat plate. This condition assures the smooth flow downwards at the trailing edge and we think that the flow field near and downstream the trailing edge is not expressed by SDM in principle. Fig. A-1 Component of Pressure Fluctuation or Flat Plate at x/c=3.0 the authoritative version for attribution. Fig. A-2 Component of Pressure Fluctuation or Flat Plate at x/c=3.0