Study on the Non-Linear Free Surface Problem

Transcription

1 1 Study on the Non-Linear Free Surface Problem Around Bow Katsuyoshi Takekuma *, Member Summary The general equation flow near the free surface is derived by considering up to the 2nd order terms velocity components. It is given by the non-linear 2nd partial differential equation, whose mathe matical type is classified by the sign its discriminant. By solving the discriminant approximately, clear explanation the breakdown bow wave in deep, water, which is important for a full hull form, can be given. The assumptions used in the calculation are supported by the model experiments Inuid Model S According to the above test results the law conservation energy is maintained by the occurrence vorticity with lateral axis during the breakdown bow wave. Thus the mechanism the occurrence the wave breaking resistance becomes clear. 1. Introduction The investigation for applying the linearized wave making resistance theory to the design hull forms high speed ships has been advanced largely especially by the needs the rapid increase service speed container ships and car ferries, for these several years. In Nagasaki Experimental Tank, Mitsubishi Heavy Industries, Ltd., also the investigation for the im provement hull forms high speed ships employing the linearized wave making resistance theory has been also conducted as one the most important projects, and some fruitful results were obtained as shown, for instance, in the design hull forms twin screw container ships, i.e., the Kamakura Maru and three sister ships. However, through these investigations, it has been recognized that kinds ships to which the linearized wave making resistance theory can be applied are limited within fine hull forms such as a high speed liner, a container ship and a car ferry, as the assumptions used in the theory deviate considerably from actual state for full forms. The free surface phenomenon around bow for full hull forms, whose specific feature is represented by the breakdown bow wave in deep water is entirely different from the phenomena which can be explained by the linearized theory, such as the shallow water wave and the wave pattern behind a ship. So the investigations on the breakdown deep water wave around bow are inevitable. The importance this phenomenon in the performance ships, especially in the case lightly loaded condition a full ship, was pointed out by Babal),2), who experimentally showed the existence resistance component due to the breakdown bow wave, which was named the Wave Breaking Resistance by the 12 th ITTC held in 1969 at Rome. He tried to explain the phenomenon through the theoretical model hydraulic jump in shallow water, but still a pro is lacking why the shallow water theory can be applied to the deep water phenomenon. Tulin and Dagan3),4),5) presented the idealized two dimensional theory for the modeled phenomenon Nagasaki Technical Institute, Mitsubishi Heavy Industries, Ltd.

2 2 around bow, but as far as the breakdown bow wave is concerned, they refered only the Talor's11) concept instability. Witham7) and others8),9),1 ) have presented the attractive theory, i.e., the Dispersive Wave Theory which treated slowly varying wave trains, by using the non-linear 2 nd partial differential equation. But it can not always explain the breakdown bow wave in deep water. These theories, mentioned above, are applied only to the simplified case and can not give always so effective explanations as the long wave theory in the case shallow water. The purpose this paper is to show the results theoretical and experimental study in order to give the general explanation about the breakdown the bow wave in deep water surface. At first, general equation flow near the free surface was obtained by considering up to the 2 nd order terms velocity components as the non-linear 2 nd partial differential equation, whose characteristics were decided by the sign its discriminant. Next, the qualitative considerations were given by conducting some numerical calculations various shapes bow based on its approximate solution. Lastly, experiment Inuid S. 201 model was carried out, in order to compare with the results numerical calculations and to prove the validity the assumptions in the calculation. 2. The general equation flow near the free surface The general equation flow near the free surface are derived under the well-known assumptions, i.e., (1) incompressible and non-viscous fluid, (2) steady and irrotational motion, (3) equi-constant pressure free surface and (4) uniform gravity force. The rectangular co-ordinate system the equation is shown in Fig. 1. Then, (1-1) on z=h (x, y) (1-2) Fig. 1 Co-ordinate System (1-3) (1-4) (1-5) where, ( 1 ) u.v and w are velocity components longitudinal, lateral and vertical directions, respectively. ( 2 ) g is acceleration due to gravity. ( 3 ) d is depth water. ( 4 ) h(x, y) is height free surface. ( 5 ) L is length ship. ( 6 ) f(x, z) is hull surface ship. ( 7 ) U is uniform advance speed ship. In the case the linearized theory, the formula (1-2), i.e., the expression the free surface condition, is to be reduced to the well-known formula, i.e., U2+gw=0 on z= 0.

3 Study on the Non-Linear Free Surface Problem Around Bow 3 By the combination equations (1-1)-(1-4), general formula free surface motion is expressed in the following non-linear 2nd partial differential equation, whose mathematical type is decided by the sign its discriminant. (1-6) where F(x, y ; a, v) is a function excluding any terms iv. The discriminant derived from formula (1-6) is shown as follows. (1-7) The mathematical type formula (1-6) is classified the hyperbolic, parabolic and elliptic ones according to the sign its discriminant. It seems impossible in practice to obtain analytical or numerical solutions equation (1-6), since it is one the most difficult equations to solve as entirely different types solutions coexist in an equation, which is named Tricomi's equation, like the trans-sonic phenomenon in aerodynamics, at which sub-sonic, sonic and super sonic flow coexist in flow field. But it seemed to be possible to give some qualitative explanation by the use the characteristics discriminant, in the same manner as for the trans-sonic phenomenon. For example, it may give the explanation mathematically to the Taylor's" concept instability for the breakdown wave in deep water, that equation (1-7) has a unique point, when the acceleration vertical fluid motion is nearly equal to that gravity. Exact calculation the discriminant (1-7) is impossible, however, since the velocity components are obtained as a set solution equation (1-6). An attempt is made, therefore, to calculate the velocity components by the approximate treatment, mentioned in the following section. 3. Approximate treatment the equation Approximate equations can be derived by the perturbation method, using the parameter s= U2/gL assuming that the velocity components due to wave motion are small enough in the order compared with those due to the presence body without free surface. ( 1 ) First order approximation the equation is shown as the following linearized formula, whose quantitative solution can be easily obtained, for instance, by Hess-Smith's method under the assumption rigid wall free surface. (2-1) (2-2) (2-3) where, uo, vo and wo are the velocity components 1st order solution. ( 2 ) 2nd order approximation the equation is shown in the following non-linear 2nd partial differential equation. (2-4)

4 日本 造船 学 会 論 文 集 4 第 132 号 (2-5) where ul, vi and wi are the velocity components is a function excluding It is comparatively discriminant, any terms 2nd order solution, wi. easy, however, to give a qualitative which is composed and F(x, y; uo, yo; ul, v1) explanation velocity components by the use the following due to first order approximation only.. (2-6) 4. The distinct explanation on the boundary However, range numerical calculation the physical meanings the discriminant, especially for the phenomenon where different types fluid motion coexist, can not be given. the qualitative by the breakdown Examples characteristics the specific phenomenon around bow, which is represented deep water wave as shown in Fig. 2, may be predicted as shown in the following examples by analogy with the similar by numerical phenomena calculation, to the breakdown shallow water wave, and to the shock wave in the trans-sonic flow. Example (1) Transfer the front line the breakdown bow wave due to the change load condition. Fig. tion 3 shows for a qualitatively breaking stance, Fig. Loaded 2 Condition Examples Light Bow Wave Loaded Full Condition Hull hull to the observed as shown front line from the forward Deep a result full numerical form. front in line in Fig. shoulder 2. bow load wave for in- The part as calcula- corresponds experiments, breakdown bow, It location wave entrance condition transfers to the becomes lighter. Example (2) Transfer Forms front line the breakdown bow wave due to the change advance speed. As shown in Fig. 4, a result numerical calculation tively phenomena breakdown as for Inuid Model corresponds the well S. 201 quantitato the observed that the angle front line the bow wave tends to be sharper, advance speed model becomes higher. In the above examples, defined as the discriminant Fig. 3 Calculated Front Lines Full Hull Forms matically, the front line is being zero mathe- and as the boundary between blue

5 Study on the Non-Linear Free Surface Problem Around Bow 5 water and white spray experimentally. An important assumption used in the above-mentioned calculation is the vertical derivatives velocity components at free surface are given by the following equations. (3-1) where u*, ƒë* are the velocity components at z=0, obtained by the numerical calculations under the assumption rigid wall free surface. TF is the draft fore end bow. L is the length ship. n is the parameter which corresponds to the vertical Fig. 4 Comparison Measured and Calculated Front Lines derivatives velocity distributions and in the above mentioned calculations n=3.0 was assumed in order to coincide the front line given by the calculation with the experiments. The adequacy the above-mentioned assumption was examined experimentally, as described in the next section. 5. Measurements velocity components Measurements the velocity components near the free surface around bow Inuid model S. 201 were carried out in the smaller tank Nagasaki Experimental Tank in order to prove the following items by the use 5 holes Pitot tube. The schematic illustration the instruments is (1 ) inuid Model (2) 5 holes Pitot tuts Fig. 5 Schematic Illustrations the Instrument shown in Fig. 5. Item (1) Are the velocity components due to the wave motion small enough in the order compared with those due to the presence body without free surface? Item (2) Are the vertical derivatives velocity components at free surface, used in the numerical calculations, adequate? Item (3) What is the factor which links the different types fluid motions, corresponding to the heat transfer in the case shock wave? The particulars and lines Inuid model S. 201 used in the tests are shown in Table 1 and Fig. 6. Dynamic and static pressures at prescribed positions whose schematic arrangements are shown in Fig. 7 were measured at a advance speed U/ ãgl =0.2 by the use the instruments shown in Fig. 5, and velocity components were calculated. The vertical distributions velocity components obtained by the above mentioned experiments are hown in Fig. 8.

6 6 Table 1 Principal Particulars Fig. 7 Schematic Arrangment Measured Points Considering the above mentioned results, it may be recognized that the assumptions which either mentioned in item (1) (2) or used in the approximate treatment general equation and numerical calculations for front lines are comparatively close to actual state. For reference, the comparison measured and calculated velocity components is shown in Fig. 9. The values the acceleration are calculated by the use measured velocity components near the free surface and are shown in Fig. 10. It is considered to be the experimental pro Taylor's11) concept instability for the breakdown deep water wave that the acceleration due to the vertical fluid motion is nearly equal to that due to gravity just behind the front line as shown in Fig. 10. Law conservation energy must be also satisfied at boundary range where different types flow coexist. For instance, it is well-known that heat transfer plays an important role for the conservation energy, in the case aerodynamics. However, in the case breakdown bow wave there is no theoretical explanation, except a few suggestion, obtained by the flow observation, like a necklace vorticity model. The author tried to find an important factor for the conservation energy, based on the result above mentioned experiment. The general equation for conservation energy incompressible fluid is given as follows.

7 Study on the Non-Linear Free Surface Problem Around Bow 7 Fig. 8 Vertical Distribution Velocity Components Fig. 9 Comparison Measured and Calculated Velocity Components (4-1) where, v is the coefficient Kinematic viscosity. In order to compare the order each term the above mentioned equation, the calculation was carried out by the use the measured velocity components, and the results are shown in Fig. 11. According to the above examination, it is shown that the equation (4-1) can be reduced to the following formula, as the values 4 th and 5 th terms the equation (4-1) are negligible comparing with others. Therefore, (4-2) Referring to the results above mentioned examination as shown in Fig. 11, the following informations may be derived.

8 8 Near the front line the breakdown bow wave, law conservation energy is satisfied by the occurrence vorticity with lateral axis, to which some part energy due to wave motion transfers. Thus, the vorticity in the present case corresponds to the heat transfer in the case aerodynamics. The resistance a ship due to this vorticity with lateral axis is considered to be the wave breaking resistance, whose component was discovered by Babal),2) in the wake behind a ship, and the theoretical explanation the mechanism its occurrence was tried by Baba2), Tulin and Dagan3),4),5). 6. Conclusion Fig. 10 Acceleration due to the Vertical Fluid Motion Through the theoretical and experimental investigations, the following conclusions are obtained. (1) The general equation fluid near the free surface is newly introduced by considering the 2 nd order terms the velocity components. It is given by the non-linear 2nd partial differential equation whose type is decided by the sign its discriminant. (2) The qualitative explanation the breakdown Fig. 11 Energy Distribution bow wave in deep water can be given by solving the discriminant the equation approximately through the assumptions that velocity components due to wave motion are small enough in the order compared with those due to the presence body without free surface by the use assumed value the vertical derivatives velocity components at free surface. 3) Taylor's11) concept instability for the breakdown wave in deep water was proved experimentally. The breakdown wave occurs at the place where the acceleration due to the vertical fluid motion becomes nearly equal to that due to gravity.

9 Study on the Non-Linear Free Surface Problem Around Bow 9 (4) In the case the breakdown bow wave in deep water, law conservation energy is satisfied by the occurrence vorticity with lateral axis, to which some part energy due to wave motion transfers. This gives the clear explanation for the mechanism the occurrence the wave breaking resistance. Acknowledgements The author expresses his gratitude to Dr. K. Taniguchi, the director and manager Nagasaki Technical Institute, MHI, to Dr. K. Watanabe, the assistant manager the same and to Mr. K. Tamura, the manager Resistance and Propulsion Research Laboratory for their instruction and encouragement. He thanks also the member Nagasaki Experimental Tank for their assistance. References 1) Baba. E.: Study on separation ship resistance components, Journal the Society Naval Architects Japan, Vol. 125 (June, 1969). 2) Baba. E.: A new component viscous resistance ships, Journal the Society Naval Architects Japan, Vol. 125 (June, 1969). 3) Dagan, G. & Tulin, M.P.: Bow waves before blunt ships, Hydronautics Inc. Tech. Rep. (1969). 4) Dagan, G. & Tulin, M.P.: The free surface bow drag a two dimensional blunt body, Hydronautics Inc. Tech. Rep. (1970). 5) Dagan, G. & Tulin, M.P.: Two dimensional free surface gravity flow past blunt bodies, J. Fluid Mech., 1972, Vol. 51, Part 3. 6) Wu, T.Y. : A singular perturbation theory for non-linear free surface flow problems, I.S.P. (March, 1967). 7) Witham, G.B. : A general approach to linear and non-linear dispersive waves using Lagrangean, J. Fluid Mech., 1965, Vol. 22, Part 2. 8) Lighthill, M.J.: Some special cases treated by the Witham theory, (M.J. Lighthill discussion meeting). 9) Howe, M.S.: Non-linear theory open channel steady flow past a solid surface finite wavegroup shape., J. Fluid Mech., 1967, Vol. 30, Part 3. 10) Hogstraaten, H.W.: Dispersion non-linear shallow water waves, J. Engineering, Mathematics, 1968, Vol. 11, No ) Taylor, G.I.: The instability liquid surface when accelerated in a direction perpendicular to their plane, Proc. Roy. Soc., 1950, Part 1.