**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

**Suggested Citation:**"Slamming of Flat-Bottomed Bodies Calculated with Exact Free Surface Boundary Conditions." National Research Council. 1990.

*The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics*. Washington, DC: The National Academies Press. doi: 10.17226/1604.

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Slamming of Flat-Boftomec! Bodies Calculated with Exact Free Surface Boundary Conditions S. Falch Norwegian Marine Technology Research Institute Trondheim, Norway ABSTRACT OF The impact of a flat-bottomed two-dimensional body which y falls vertically towards initially calm water is studied theoretically and numerically. The flow in the ~ compressible air layer between the body and the water ° surface is calculated by assuming that the air is i nv i sc i d and t he f 1 ow i s one-d i dens i one 1 . The ca 1 cu 1 a - A, tion of the flow in the water is based on potential theory, and a boundary- i ntegra 1 -equat i on techn i que wi th exact nom inear boundary conditions are used. The effect of var i ous deadr i seang 1 es has been studied. and the pressure acting on the bottom of the body is calculated unti 1 the body makes contact with the deformed water surf ace . L i st of symbol s b c cO cw Cp C+, C_ g h H Ho Ha K M N NA NF n2 p SO r sl SO SF So t u V VO Half width of body Ve 1 oc i ty of sound i n a i r . Velocity of sound in air at atmospheric pressure. Velocity of sound in water. Constant in nondimensional momentum equation. Family of characteristic curves. Acce 1 erat i on of gray i ty. Distance between body and water surface. Elevation of the bottom of the body at centerl ine. Value of H at start of the numerical calculation. Amp 1 i tyde o f f o rc ed heave osc i 1 1 a t i on . Number of elements on body. Two-d i Dens i one 1 mass of body. Number of e 1 events i n a i r ca 1 cu 1 at i on . Number of surface elements between x = 0 and x = b. Total number of boundary elements. y-component of the unit normal vector. Pressure i n the a i r. Atmospheric Dressure. Rad i us of cy 1 i nder . Surface of body. Cyl indrical surface of seal 1 radius. Horizontal control surface far down. The f ree surf ace of water. Vertical control surface far away. Time. Velocity of air in x-direction. Velocity of body. Ve 1 oc i ty of body at start of the numer i ca 1 ca 1 cu 1 at i on . 251 Ax At Hor i zonta 1 soace coord i Rate. x-coordinate of fluidparticle. Vert i ca 1 space coord i nate . ExDonent in the relationship between p and p. Elevation of the water surface. Deadr i se ang 1 e . Dens i ty of a i r . Density of air at atmospheric pressure. Densi ty of water. Velocity potential. St reamf unc t i on . Circular frequency of heave oscillation. Width of element along the x-axis. T i meper i ad be tween two t i mes teps . CHA PT E R 1 . I NTRODU CT'ON Section 1.1 Gener 1 a The aim of this work is to calculate the pressure on a f lat-bottomed body which is falling through air towards initially calm water and subsequently penetrates into the water. The effect of the entrapped air between the body and the water surface is specially studied. This is a particular case of the general slamming problem where an arbitrary shaped body hits the arbitrary shaped free surface of the water, or it may be the water that hits the body. The pressure and the force acting on the body due to this, will in several applications be of signif icant interest to marine designers. The classic example occurs when a ship is travelling at high speed or i n heavy sea, and the bow moves out of and reenters the water. Severe damage on shipbows has been reported as a consequence [8]. Considerable impact forces may also occur when large waves hit cross members or deckstructures of offshore oilrigs. Finally the situation must be mentioned where different kind of equip- ment such as a mini-submarine is lowered or lifted through the splash- zone. Soecia1 ly if the eauiDment is operated by a crane situated on some king of floating platform, the relative motion between the water surface and the equipment may in rough weather be considerable, caus i ng s i gn i ' i cant i moact 1 cads . The calculation of imoact loads in these situations is needed in order to determine the "ecuired strength of the structures involved. For the case of slamming of a ship bow or lowering of equipment through the solash zone, knowledge of the impact loads is also required in order to determine in what kind of weather condition cer- tain operating may be carried out. A better understanding of the whole phenomena of slamming wi 11 also rove information on how to design the shape of the structures in order to minimize the imoact loads.

To calculate the impact loads in the cases mentioned here, is very difficult. The problem has to be simplified. The purpose of such simplifications may partly be to find approximate solutions to the real problems, and partly to be a step on the road to more sof isti- cated methods. For this purpose i t seems reasonable that one way of making the simplifications is to device the problem into sub-problems. Special phenomenons can then be studied separately. The most obvious simpl if ication wi 11 be to start to study two- dimensional problems. Another simplification which seems reasonable, is to assume that the real problem may be approximated by the entry of a body into calm water with velocity equal to the relative velocity between the body and the moving free surface. Such an approximation is expected to be good if the dimensions of the body is small com- pared to the wavelength. Both the above approximat ions are made by several other authors. The bodies involved may have various different forms. Some of them lead to certain phenomenon which needs to be studied specially. In this work it has been chosen to study flat-bottomed bodies. This discussion leads to the conclusion that it is interesting to study the fo1 lowing particular case, and which may be characterized by : The calculations will be two-dimensional. That means that the resul ts wi 11 be approximately val id for a slender horizontal body . - The body wi 11 be symmetrical . - The water is initial ly calm. - The body is falling vertically under influence of gravity and the pressure force on the bottom. (The body may also have prescribed motion. ) - The body is completely rigid. - The body is almost flit-bottomed. That means it may have a small dead r i se ange 1 . For this particular case, the aim is to calculate the pressure distri- bu~ion on the surface of the body and the flow in the water and the air as a function of time, both before and after the moment when the body makes contact with the water surface. The calculation wi 11 include both the effect of entrapped air and the possibi 1 ity of large water surface elevation. This is a theoretical and numerical study. No experiments have been carried out, but the results of this work have been compared with other works, both theoretical and experimental. In the effort to achieve this aim, several simplifications, especially on the properties of water and air, will be made. The introduction, and the discussion of the justification, of these simplifications will be done in the fol lowing chapter, but they wi 11 al 1 be of such a kind that they will have more or less negligible influence on the results of the calculation. See t i on 1 . ? _Backoround Unti 1 the beginning of the nineteensixties, the theories of van Karman t1~ and Wagner t2] was used to predict hydrodynamic impact forces. These theories neglected the influence from the air, and in the case of a f lat-bottomed body, van Karman suggested that the maximum impact pressure was the acoust ic pressure of water. Then several experimental and theoretical works was done, which showed that for the case of a f lat-bottomed body the effect of the air bet- ween the body and the water surface has to be taken into account in order to predict the impact pressure more correctly. They found that the cushioning effect of the trapped air in the layer between the fal 1 ing body and the water surface reduced the impact pressure to approximately one-tenth of the acoustic pressure. The most significant of these works was experiments made by Chuang (3, 4] and Lewison and Maclean [5] and a theoretical and experimental study made by Verhagen [6]. In 1977 Koehler and Kettleborough t7] presented a theoretical study which shows in detai 1 the calculated pressure in the airgap and the surface elevation for one of the cases used in the experimental study of Lewison and Maclean. Verhagen describes that pictures taken during his experiments shows that the water elevation is at first noticeable at the edges of the body just before the plate touches the water surface. Next, the air layer breaks up into bubbles beginning at the edges and extending to the center of the body with a speed in the order of magnitude equal to the velocity of sound in air. With increasing time the bubbles drift relatively slowly outwards. Chuang [4] present some interesting sketches of the situation after impact based on photographs taken during experiments. These sketches are given in Fig. 1.1. In the theoretical work of both Verhagen and Koehler and Kettleborough a two-dimensional problem was studied. They assumed that a compressible inviscid air layer exists between the body and the water surface. As the body fal ls, the air in the layer flows one- dimensional ly outwards from the centre of the body. The water is assumed to be incompressible and Verhagen states that this is a good approximation if the velocity of the body do not exceed a few meters per second. In both works the calculation of the flow in the water is based on 1 inear free surface condition. The pressure distribution is calcu- 1 ated unt i 1 the edge of the body makes contact wi th the elevated water surface, and Koehler and Kettleborough assumes that the maximum impact pressure is reached at this time-instant. Verhagen also calcu- lates the pressure in the trapped air bubble after this time by a simplified method assuming that the pressure in the bubble and the downward velocity of the water surface within the bubble is only a f unc t i on of t i me . Koeh 1 er and Kett 1 eborough makes the ca 1 cu 1 at i on with different deadriseangles. The intention is to predict the uncer- tainty in experiments due to the fact that in an experimental set up -t is not Dossible to be sure that the bottom of the body is exactly oaral lel tO the undisturbed water surface. Some of the results Presented in the papers mentioned above wi ~ ~ us used for comparison in the present work and wi 11 then be discussed in more detai 1. As mentioned in section 1.1 it was the intention in the present work to calculate impact pressure both before and after the moment when 252

contact occurs. It is then clear from fig. 1.1 that the method used for calculating the flow in the water must necessarily be based on exactly non-linear boundary condition. Such a method has therefore been used, and this is the main deviation from earlier works. It has however not been possible within the l imit of the present research project to ful ly develop a method which calculates the pressure on the body during al l phases of the impact. The computer programme which has been made, calculates the flow in the air and the water prior to the moment when contact i s made between the body and the water sur- ~ace . CHAPTER 2. OVERALL DESCRIPTION OF THE PROBLEM AND THE SOLUTION The problem to be solved is the one described in section 1.1. Fig. 2.1 shows the situation at time t = 0 which is the starting point of the numerical calculation. y - , H a V A right-handed, earth-f ixed coordinate system is used with the x-axis horizontally, the y-axis vertically and positive upwards, and the ori- gin at the intersection between the undisturbed water surface and the centerl ine of the body. b is the halfwidth of the body and ~ is the deadrise angle. The elevation of the keelpoint (that is the intersec- tion coins between the centerline and the bottom of the body) is designated H(t), and the velocity of the body V(t). V is defined to be pos i t i ve cowards . , he numer Cal calculation starts (t = 0) when the body has reached the Dosition H = Ho, and the velocity is V = VO. Because the effect of the viscosity in the water is confined to thin boundary layers, the water may be regarded as inviscid. This means gnat -~e 'low in the water will be irrotational and it may be charac- terized by a velocity Dotentia1 ¢(x,y,t). Ante- the moment when the bottom of the body has gassed the water sur- face and continue to move downwards through the water, it is possible that vorticity will be shed from the edge of the body. This effect, however, will not be studied in this work. It will probably have negl igible inf luence on the pressure on the bottom of the body. On the other hand, if a study of the effect of a secondary impact as shown in Fig. 1.1, as well as an exact calculation of the pressure along the sidewal l of the body are to be carried out, it may be necessary to take this vortex shedding into account. Whether the water might be regarded as incompressible or not depends on whether the duration of the impact is large compared to the time it takes for a pressure shock wave in the water to travel a distance equal to the hal f width of the body. If the width of the impact pressure-time history curve is called At, the half-width of the body is b and the velocity of sound in water is cw, then the condition for the water to be regarded as incompressible becomes: Experiments and earl ier works have shown that this condition holds for the range of velocity which has been studied, so the water will be t rea ted as a n i ncompress i b l e f l u i d . 253 Finally, it is assumed that the effect of surface tension can be neg l ected . For the same reason as for the water, it is assumed that the air may be treated as inviscid. However the compressibility of the air will not be neglected. This is necessary because the calculation shows that the f low in the air beco- mes superson i c . Verhagen [6] assumes the flow in the air to be isentropic (=reversible ad i abet i c ) wh i ch means that: = (~)r , y= 1.4 0 0 On the other hand Koehler and Kettleborough !7~ assumes that y is closer to 1.0, which is the value for an isothermal process, and he argues that the temperature wi l l be kept constant because of al l the waterspray in the gap between the body and the water surface. In this work both values are used. For the ideal ized case which is studied here, and also for the experiments used to compare with, the heat-transfer from the air in the gap to the surroundings will be negligible during the very short time of the pressure build up, which means that the flow in the air will be isentroic. This assumption wi l l be used through the paper if not otherwise stated. On the other hand in a more realistic slamming situation where the water is not initially calm, and there is wind and waves, the situation will be different. Because of the spray in the air, the contact-surface bet- ween the air and the water will be very large, which means that the heat transfer wi l l be much faster. It is possible that this causes the flow in the air to be isothermal. When the body approaches the water surface, the pressure in the air between the body and the water starts to bui ld up at the same time as the air starts to flow outwards from the centerline towards the edges OF the body. The pressure rise causes the water surface to deform, and this situation is illustrated in Fig. 2.2. - Y ~ 1 Lo |h(x,t) ~ (x t) F i g . 2 . 2 t is known that the pressure wi l l not exceed the atmospheric Dressure significantly until H << b. This means that the flow in the air may be regarded as one-dimensional, that is the flow is paral lel l to the x-axis and al l quantities are a function of x and t only. This assumption has also been made by other authors [6], [7]. The elevation of the water surface is designated :~(X,t), and the distance between the body and the water surface is h(x,t). The rela- tionship between these two is h(x,t) = H(t) + ~ x - ((x,t) (2.2) The one-dimensional flow in the air is characterized by the horizontal ve l oc i ty u ( x, t ), t he pressure p ( x, t ) and t he dens i ty p ( x, t ) . As mentioned above the pressure bui ld up in the gap under the body causes the water surface to deform. Or in other words, before the body makes contact the flow in the water and particular the free sur- face elevation depends on the pressure in the air above the water sur- face. On the other hand the calculation of the pressure in the air

depends on the width of the gap h(x,t) which partly depends on the p = Oo at x = + b elevation of the water surface ((x,t). This may be illustrated like this: h(x,t) - p(x,t) - ((x,t) - h(x,t) (2.3) I_ The first arrow will be treated in chapter 3 and the second arrow in chapter 4. The third arrow is given by eq. (2.2), and the arrow from o to h indicates that h depends on the pressure through the motion and the force on the body. Of course, to get a solution to our problem, al l calculations have to be made simultaneously, which in the numerical procedures means that all the calculations will be carried out in each timestep. How this is done, is described in chapter 5. CHAPTER 3. THE AIR REGION The assumptions and simplifications made in chapter 1 and 2 which regards the air region, may be summarised as: The body is two-dimensional, symmetrical, rigid and flat-bottomed. It f a l l s ver t i ca l l y under i n f l uence of gray i ty and pressure or has a prescribed vertical motion. The flow in the air is one-dimensional, inviscid, compressible and isentropic. The situation is shown in Fig. 3.1 where the coordinate system and the symbols are the same as introduced in chapter 2. Y ~ b (3.4) However this boundary condition can only be used as long as the flow is subsonic. Afted the flow has become supersonic, the problem will be completely determined in the region we are interested in without any boundary condition at x = + b. The initial condition will be determined in the same way as in [6] and [71. The initial height, Ho' of the falling body is assumed to be sufficiently above the free surface so that the air is not behaving compressible and the pressure distribution within the air layer is essential ly atmospheric so that no initial effect wi l l be felt by the water. At "his height the water free surface will be undisturbed and for the case ~ = 0, h(x,O) will be a constant. Since p(x,O) also is assumed to be constant, an initial velocity distribution may be derived from the continuity equation (3.2) . Since the problem is symmetrical, u will be zero at the centerline (x = 0), and the result is: u(x,0) = ~ h = - ° x (3.5) ~t=0 0 and hence if the velocity is assumed constant: at(x~o) = h2 at X ~ = O2 X o Using this initial velocity distribution and holding p(x,0) constant = pO, an initial air pressure distribution can be obtained from the equation of motion (3.1). With the boundary con- dition p = pO at x = b, the result is: 1 e 1 D(X,O) po ~ po O2 (b2 - X2) (3.6a) ~ , H o i ~ U(X,t) p(X,t) and hence from eq. (3.3): ,, ,~ Q(X't) For this f now the eacation of motion becomes an au 1 so U = _ _ at ax p ax We eccet-,~n o' continuity: aX(UP~) + at(pk) = 0 ~he relationship between oressure and density is: = (or 0 0 In this chapter h(x,t) is assumed to be known, which means that there are three equations for the three unknowns u, p, p. By the use of equation (3.3) p may be eliminated from eq. (3.1), which means that eq. (3.1) and (3.2) contains only the two unknowns u(x,t) and p(x,t). These unknowns are to be determined in the region x c[-b,b], t ~ [O. tkl, where tk is the time where the body has reached a point where this calculation is no longer valid. The equation system is hyperbol ic, and in order to solve the problem, initial-conditions at t = 0 is needed for both the unknowns. As long as the flow is sub- sonic, a boundary condition along x = ~ b for one of the unknowns is a l so needed. At the edge of the body the air will flow as a free jet (see [6] and '7l ), which means that the following boundary condition should be used: 254 (3.1) 3.2 H (3 3) b << 1 p(x,o) = p (Pig )r o (3.6b) The derivation of the initial condition was based on the assumption that '`e oressure distribution is essential ly atmospheric -hat: v2 0 2 2 b << 0 H ° o which gives the fol lowing restriction on the choice of Ho H V _ >> 0 D i'p O O -h55 must hold at the same time as Th i s means which earl ier has been introduced as a restriction which is necessary in order to use a one-dimensional model. Initial-condition for the case ~ ~ O may be derived in a similar way and it can be shown that the deviation from the above results is small if H << 0. b The problem described here has been solved using the method of charac- teristics with specified time intervals, similar to the method used by Verhagen [6]. The method together with a numerical scheme called CHAR-S is described in ref . [ 17 ] .

CHAPTER 4. THE WATER REGION Section 4.1 Introduction The assumptions and simplifications made in chapter 1 and 2 which regards the water region, may be summarized as: The problem is two-dimensional and symmetrical and the water is initial ly at rest. The water is assumed to be inviscid and incompressible which means that the flow may be represented by a velocity potential. The surface tension is neglected. Two typical situations which will be treated, are illustrated in Fig. ~ . 1. Y ~ ~ _ _ _ ^~ p(x,t) Flg. 4.1 a) ¢~_ ~ ~ ~ (x,t) b) ~ ~ 'he coordinate system and the notation is the same as introduced in chapter 2. In this chapter it is assumed that the pressure distribu- tion over the water surface and/or the motion of the body is known, and the f low in the water is examined. There are several methods which may be ar,Dlied to obtain this flow. No detailed evaluation of these methods has been made in this work. It was however a demand that the method should not have any other restrictions than those men- t i oned i n chanter 1 and 2 . The method chosen is a boundary-integral-equation technique with exact nom inear boundary condition based on Faltinsen [13] . This method is described in section 4.2. Running of a computer program derived from the method, shows that problems may occur in some situations. The problems are connected to the intersection point between the body and the free surface. This will be explained in more detail in section 4.4. Some effort has been made to overcome this problems. A modified ver- sion of the method used in section 4.2 has been developed. This is described in section 4.3. The modifications have however not been very successful in solving the intersection-point problem. In section 4.4 some numerical examples are presented and compared by the work of other authors. ect_n 4.2 A boundarv-inteural-equation technique Let the water be infinite in extent and be of infinite depth. Since we have assumed irrotational flow in an incompressible fluid, there wi l l exist a velocity potential ~ which satisfies the Laplace equation in the fluid: 2 2 0 Ox ay 'he pressure p in the air over the water surface may differ from the atmospheric pressure pO, and since the surface tension is neglected, the dynamic free surface condition can be written as: at + 2[(ax) + (ay) ] + go + Pw = 0 (4~2) where g is the acceleration of gravity. The kinematic free surface condition is: ~ _ ~ ~ ~ - O at ay + ax ax ~ on y = ((x,t) ( 4.3 ) If there is a body penetrating the water surface, the boundary con- dition on the wetted body surface will be: (4.4) where V as before is the vertical velocity of the body, 2n is the y- component of the unit normal vector to the body, which is assumed to be positive into the fluid, and is the derivative along this normal an vector . The i n i t i a l cond i t i on are set to be: ~ = 0 on the free surface $ = 0 (4.5) The boundary value problem defined by eq. (4.1-4) has to be solved for each timesteo. This will be done by applying Green's second identity to t he ve l oc i ty potent i a l ~ and the f unct i on ~ def i ned by: ~ ~ (x,t) ¢(x,y) = in ~x-x1)2 + (y_y1)2 ~ 1k ~ x where (x1, ye) is a point in the fluid-domain. We can then or i te: S'(~ Ids ° where S' = S + SO + SS + SF + S1 YE ~ ~XU`r F ~ ~ S n Fig. 4.3 S: (4~6) l ,~'(X1rY:' S is the wetted body surface, SO is a vertical control surface at x = _=, SB is a horizontal control surface far down in the fluid, SF is the free surface and S1 is a cylindrical surface of small radius and with axis through (x1,yl) perpendicular to the x-y plane. Since the problem is transient with the fluid initially at rest, the contribution to (4.6) from So, and SB are both zero. This means that eq. (4.6) can be written as a (x,y x y ) ds x, an (x ,y ) = J | (x,y) ¢(x,y;x ,y ) ~ ( 1 1 s+sF an(X,y; 1 1 an(x,y) (4~7) For *x* > bF(t), where bF is large compared to the half width of the body b, we can, since the problem is symmetrical, approximate ~ to the velocity potential of a dipole with singularity in origo and axis along the y-axis: 255 ¢(x.y) ~ 2 2 (4.8) x +y where A at th i s stage i s unknown. tF(t) is assumed to be so large that no wave has reached the points x = +bF(t), which means that the free surface outside those points

will coincide with the x-axis. Hence the contribution to (4.7) from the part of SF with x > bF is: J {~(x~y) an(x,y; 1 _ ¢(x,y;x1,yl) fix y']d~y=onAI(x1,y1) (4~9) and the contribution from the part of SF with x ~ -bF is: |Fl¢(X,Y) an(x y; 1 _ ¢(x,y;x1,yl) fix y,idxly=o.AJ(x1'y1) (4.10) An analytical expression for I(x1,yl) and J(x1, Y1) are given in ref. r13]. When solving the integral equation (4.7) at each timestep, a. will be known on the body by the use of the boundary condition on the body (4.4), and ~ will be known on the free surface. At the first timestep ~ will be known on the free surface from the initial condition (4.5), which also determines the position of the free surface at this time. When the problem has been solved for one timestep, the free surface conditions are used to find ~ on the free surface and the position of the free surface for the next timestep. This is done by the following timestepping procedure: If we follow a fluid particle on the free surface, we can write: DO = a. a. a. a. a. Dt at + ax ax + ay ay Using the dynamic free surface condition (4.2) this becomes: Dt = ~ 9: + 2[(a.)2 + (alp) 2] _ P-PO (4.11) By the use of the kinematic free surface condition (4.3) we can f urther or i te: and r ~ _ Ot ~ By Ox F a. Ot ~ ax where OF is the x-coordinate of the fluid particle. A fourth order Runge-Kut~a method is used to perform the sidestepping. No investiga- t~ons have been done in this work to f ind out whether any other methods would have been useful. (4.13) When solving the integral equation (4.7), the wetted body surface and that part of the free surface lying between x = -bF and x = bF are divided into straight-line elements (see Fig. 4.4). , . . · ~ ~ F \ \ i, ~ ( X i ' Y i ) Fig . 4 . 4 The elements are symmetrical ly arranged. The total number of elements are 2NF, whi le the number of elements on the body is 2K. (For the situation illustrated in Fig. 4.'a, K will be equal to zero.) The midpoint of element number i are denoted (x.,y. ), and the surface of this element will be called Sj. It is assumed that both ~ and a`P are constant over each element. For element no. i these constants wi 11 be denoted ~ = ¢(x.,y. ) and a`P respectively. Due to the sym- metry of the problem it is clear that Hi = ~ i and a. = a. where ~ jj is defined by: If we now adultly the integral eq. (4.7) NF times, putting (x1,yl) equal _ _ to (xj,yj ), i = 1,2,...,NF, the result is: 256 -27r¢(xj,y<) = ~ And J anin'/(x-xj) + (y-yj) ds(x,y) an S IS lniX~Xi ) + (y-yj ) ds(x,y) + A I(xi,yj) + A J(Xj,yj) (4.14) For I x I > bF, ~ is assumed to be given by eq. (4.8). We now assume that am is determined by this expression also on the element NF: al A = anNF x 2 NF Hence: NF anon If this is inserted into eq. (4.14), and all the known parts are gathered on the right-hand side of the equation, we will have for 2rrd,j ~ ~ is | anini(x-x,. ) + (y-y; ) ds j-k+1 ,lnj J ; in (:x Xj ) + (y-yj ) ds (4. 15) - r;(xj,yj) + J(Xj'Yj)lXNF anNF (4~16) -~+! is is anlniX Xj ) +(y-yi ) ds . ~ _ ~ (4~12) j--1 an: S IS ~ni/(x Xj ) +(Y-Y; ) ds -or i = !c+1, k+2,...NF, we will have similar equations but with 27r~j on the ozone- s de of the eoual sign. Reese ecuat offs may be written in matrix form as A · x = b where ~ he ;~:<nowns are: (4.17) 1 ¢2' ~K' an k+1'--- anNF) (4.18) A is an NF x NF matrix with elements: S j +S j ~ x ~ Y ) ~y-y i ) ds ( x, y ) (4. 19) Aj j = - | ln~i ) +(Y~Y; ) ds(x,y) - ~ jNF[I (Xj,yj )+J(xj,yj )]X2 i=1, . .,NF j=k+1, . ., NF i j21 and S JS ln ~x-xj ) +(y-yj ) ds(x,y) i=1, . .,NF j-k+i j s JS an`X.Y)lniX-Xi) +(y-yj) ds(x,y) - 2nd ] (4.20)

djj 1 if i = O if i '+ j When the algebraic equation system (4.17) has been solved, the flow in the f luid wi l l be determined for the actual time instant we are dealing with. In order to step the solution forward, the substantive derivatives given by eq. (4.11-13) will be applied to the fluid par- ticle at the midpoint of each element on the free surface. We then need to know as and a. for the midpoints, which can be calculated when the equation system (4.17) has been solved. The Sidestepping wi l l give new values for the velocity potentidal ~ on the free surface, and new positions of the fluid particles where these ¢-values are valid. Knowing these new positions for the fluid particles, new elements have to be arranged. The midpoints of these new elements will not always coincide with the positions of the fluid particles, and in that case the value of ~ at the new midpoints will be deter- mined by interpolation. How the new elements are arranged and how the interpolation are carried out, are explained in more detai l in Appendix A. After the fluid motion has been determined, the Bernoulli equation may be used to obtain the force on the wetted surface of the body. However, another procedure is used. This is derived by Faltinsen [13] and he finds that the force on the wetted surface of the body is: F = Pw dt | finds + | pwGyndS (4~21) Section 4.3 A Modif fed Method As mentioned in section 4.1, some problems may occur at the intersec- tion between the body surface and the water free surface. In an attempt to overcome the problems some modifications to the pre- viously described method has been developed. The modifications have been based on the fol lowing considerations: In the method in section 4.2 new position of the free surface and new values of ~ on the free surface are calculated by timestepping of fluid Canticles at the mid- points of the old elements. It is then necessary to create new free surface elements and calculate the value of ~ at the midpoints of these. One way of doing this is described in Appendix A. However if the curvature of the free surface is large, the position of the new elements are strongly dependent on the detai ls of this new-element- creation-orocedure. so avoid this it would have been better to use the endpoints of the elements as fluidparticles which shall be given new position and new ¢-value by the timestepping procedure. This would have determined the new free-surface elements directly. Another consideration which has been made, was inspired by Lin et.al. [14~. They used the Vinje and Brevig [15] approach based on Cauchy's theorem and made one modif ication to this method at the intersection point. In these methods the velocity potential ~ and the stream func- tion ~ varies linearly over each boundary element. ~ is known on the body by the use of the body boundary condition, and ~ is known on the water free surface in the same way as in section 4.2. Vinje and Bre- vig chooses ~ to be the known function at the intersection point bet- ween the body and the free surface. The modification made by Lin et.al. [14] was that they assumed that both ~ and ~ was known at the intersection point, and reduced the algebraic equation system with one equation. Lin et.al. [14] found that this modifications improved the method and that the problems at the intersection point was reduced. 'I a similar modification could be made to the method in section 4.2, it would be worth trying. ,he above considerations has lead to the following modifications in the method in section 4.2: The wetted surface of the body and that part of the free surface which lies between x = -bF and x = bF are divided into 2NF elements as before. Now the endpoints of element no. ~ will be (xj,yj) and (xj+1,yi+l). The velocity potential varies l inearly over each element. The value of ~ at the endpoint is now denoted ¢; = ¢(Xj,yj ). However 3¢ is still assumed to be a constant over each element. This corresponds to ~ varying l inearly Over the elements. a<P is this constant value for element no. i. We now need to introduce for each element two functions w1(x,y) and W2(x,y). They are defined to vary linearly over the elements, w1 decreasing from 1 to 0, and w2 increasing from O to 1 when moving along the element from the symmetry! ine and outwards. 'I we amply the integral equation (4.7), putting (x1,yl) equal to ( x i, y j ) the resul t wi l l now become: NF / 2 (x; Eli ) j21 {¢i | w1 an ln'/(x-xj ) +(y-yj ) dS(x,y) + ~j+1 J W2 an 1ni(X-Xj ~ +(Y-Y; ) dS(x,y) as | lnix-xj)+(y-yj)2 dS(x,y)} ~ Sj+S j + A[I(xj,yj) + J(xj,y;)] (4.22) By applying the same expression for A as before (4.15) and by gathering the known terms on the right hand side, (4.22) may for i=1,2,...K be rewritten as: q i ¢1S IS 1 aniniX xj ) +(y-y; ) dS K r a / i-2 i S IS 1 an ln~/(x-xj )2+(y_y )2 dS / + J W2 ln1/(X-Xj )2+(y_y; )2 dS] Sj ;+S_ ( j-1 ) ~ - ~ ~ l a n j S J S i/ ( x x i ~ + ( y-y j ) dS - ~ (X;,y; ) + J(Xj'Yi )]XNF anNF NF /- = ~ '1' [ I - at anin1/(x-xj ) +(Y-Yj ) dS , ,] ~ /- S J ~ W2 ln/(x-x )2+(y y )2 dS] j-1 -(j-1) qNF+lS :+S W2 anin)/;X~Xi ) +(Y-Yj ) dS NF -NF ~ an J [nix-x; ) +(y_y; )2 dS (4.23) For i = k+', NF, the term 2nq)i will be on the other side. These equations may be written in matrix form: A x = b we th unknowns: X = (q~l'¢-''--4K' ank+1 anNF and the elements of the coefficient matrix A will now be: 257 (4.24)

Aj = 2~6j + | w1 anini(x-xj) +(y-yj) dS(x,y) ~ -. ., = 1 V F , j=1 A jj - 2n5; ~ | wl aninix-x )2~(y_yj)2 dS ~ 2~ ASj - = 0 ~-2 3 k j 1 anj bF / + J w a ln1/(x-x )2+(y_y )2 dS (4.25) , -1 - ( j-1 ) A; = _ J lnix-xj ) +(y-yj ) dS i= , NF ~ ~3 ink ~ 2, . . .NF ~ IS jNF rI (Xj All )+J(xj tyi )]XNF The elements of the vector b is: NF /- i j=~+1 j [ S IS 1 an (x Xj ) (y yj ) dS i=l, NF ~ ~ /- - r W1 aan lnl//(x-xj)2+(y-yj)2 dS - 27r~jj] / J W2 an ln'/(x-xj )2+(y_yj )2 dS / J lni(x-x )2+(y_y )2 dS 5 5 1 1 B = ¢, - J w' an ~nix-xj) +(y-yj) dS - 2?TAj ] i= ~ ~c (4.26) NF / -2 i ~ t~ w1 an lni(x-xj )2+(y_y )2 dS ~ .+_ Sj 1+S 2 an ln/(/x-xj ) +(y_y; )2 dS - 27r,i ] / NF4~- J W2 an lni(x-xj )2+(y_yj )2 dS NF -NF + ~ an | lnix-xj ) +(y-yj ) dS (4.27) In order to step the solution forward, given by eq. (4.11-13) will now be applied to the fluidparticles at the endpoint for each element on the free surface. To carry out this, a`P and a~p have to be known at the endpoints. However with the model used here, a. and atP will generally be singular at the endpoints. This means that 3$ and a. have to be calculated at the midpoints of each element, and then the value at the endpoints are found by inter- polat ion. Another modification to the method in section 4.2 is also made. Since the f luid is incompressible, we wi l l have that: J an dS = 0 ~ 8 5 x (4.28) 258 If the diDole aDproximation, eq. (4.8), are used far away from the body, it turns out that the contribution from SB and Sy' is zero. The contribution from SF outside x = +bF may also be calculated by the use of the dipole approximation in this area. Still assuming that a. is constant over each element eq. (4.28) becomes: ( 4.29 ) where ASj is the length of element no. j. Using the same expression for A as before (see eq. (4.15) ), this can be rewritten as: 1 a s + = - a sj 4.30 j_~+l anj ~Sj + anNF ( NF bF ) j-1 j ( ) witn the unknown terms on the left side. This eauation may be used instead of one of the equations in the equation system (4.24). [f equation no. i is interchanged, the new coeff icients are: A. . = 0 ~ ,] j=1, . . . ,k Aj j = ~Sj + (jNF b j=k+1, . . . NF i j - 1 a n j j (4.31) The introduction of the eq. (4.30) corresponds to the use of ~ as known at the intersection point in the Vinje-3revig-method. This may be explained as fol lows. lf we look at that part of the free surface SF having x > 0, it is true that: Si an sJ aS ~x. *1 (4.32) which means that St aPds will be known if ~1, which is ~ at the inter- sect,on point, is given. The method described in this section is used in a computer program. This has been run for the case of a cylinder oscillating vertically in the free surface. The method has been tried with and without interchanging one of the equation in the equation system (4.24) with the equation (4.30). The only way to avoid instability in the solu- tion is to interchange equation (4.30) with equation number K+1. This means that the equation formed by setting (x1,yl) equal to the inter- sec'ion point (xk+1,yk+l) in the integral equation (4.7), is not used. In this ~orm the modified method in this section has been compared w,th the method in section 4.2 and with the numerical results obtained by other authors. This is presented in section 4.4. The numerical results shows that the modified method do not handle the ~ntersection Doint oroblem any better than the method in section 4.2. If the modified method is used in the case k=O, which means that there is no body penetrating the free surface, it is not possible to avoid instability in the solution whether the equation (4.30) is used or not . Instability here means that a<P on the free surface shows large n oscillations for each element when moving along the surface, see Fig. 4.5. This is not unexpected since in this case the eq. (4.7) will be a Fredholm integral equation of 1. kind over the entire boundary. Sectior d.4 Numerical results and discussion The computer orogrammes developed form the two methods described in section 4.2 and 4.3 are compared with the work of other authors. For Drevity the two programmes are called SLM and SLM-Mod. respectively. Two d i f f eren t cases a re ca l cu l ated . In the f irst case both programmes are used to calculate the force on a

AL an ~- Fi 9 . 4 . 5 cylinder with forced vertical oscillation (position of center: y(t) = Ha sin at). The results are compared with results obtained by Faltinsen using a programme based on the same method descried in sec- tion 4.2 and Dresented in [133, and also with results presented by Vinje and 8revig [15]. Fig. 4.6 shows the results for the three values of the amplitude: Ha/r = 0.1, 0.3 and 0.6 (r is radius of the cy 1 i nder ), ~r/g = 1. 0. The good conformity between the results for Ha/r = 0.1 indicates that there are no bugs in the programmed, but the deviation in the results for Ha/r = 0.3 and 0.6 is difficult to explain. The reason why the case Ha/r = 0.6 is not calculated for higher time values, is that the calculation breaks down shortly after because of problems with the element lying closest to the intersection point. In Fig. 4.7 the elements on the cylinder and the free surface are shown for the two last timesteps before the programme SLM breaks down. As seen from the force curve in Fig. 4.6 the programme SLM-Mod. tends to break down even before. The second case which are used to test the programmed, is one which are presented by Doctors [16]. He uses a triangular pressure-element acting on the free surface of the water. The pressure-element as a function of position x and time t is defined by Fig. 4.8. The free surface elevation which is the result of the pressure-element, is calculated by a method which is based on potential theory and with 1 i near f ree surf ace cond i t i on. P-Po pa / P-Po pa \ At~x = 1 _ \ _x/Ax o / , \ _ that -1 0 1 0 1 2 Fig. 4.8 1 A In fig. 4.9 results presented in [16] are compared with results obtained by the programme SLM. Because SLM is based on exactly free surface conditions, the value of the parameter Ax has to be chosen. Since the intention is to compare with the linear case, the parameter Ax has to be chosen in such a way that the slope of the free surface is small. From the figure it is seen that the maximum value of oW9(tPo is approximately equal to one. So the chaise has been made: pw96x/po - 100 or ax = 1000 m. The agreement in the results are good, and since this case are very similar to the slamming case for the time-period before the body makes contact with the water, the programme SLM seems to be well suited for calculating f lat-bottomed slamming. As mentioned before the programme SLM-Mod. is not able to calculate this case proper 1y because a. on the free surface oscillates when moving along the surface. FORCE ON CYL I NOER I N HEAVE MOT I ON u, ~ _ o - u, o o o u, to i a? Heave ampLetudo: HA/r - O ~ FaLt~n.on /13/ V~nJo / t5/ ' SLM SLM-Mod o _ ~ . . . . I . · · · I ~ I r I I '0 0 / 0 2 0 3 0 ~' 0 5 0 OMEGA ~ Fig. 4.6 a FORCE ON CYL I NOER I N HEAVE HOT I ON In . o u, o _ if Heave amplitude HA/r = 0 3 I? FaLLen6en /43/ Venue / 15/ ' SLM ~ SLM Hod T ~ ~ ~ , , I 1 0 2 0 OMEGA Fig. 4.6 b 259

FORCE ON CYL I NDER I N HEAVE P1OT I ON . o - u' o' o o' up, to 10.0 y Heave sop ~ LtUde: HA/r c 0.6 6: ~ F=~neon \ / ~ V~nJ. /,S/ \ / ' SLM ~ ~ SLM - ad Fig. 4 .6 c 7 CHAPTER 5. CALCULATION OF FLAT-BOTTOMED SLAMMING Sect ion 5.1 G-n_ral As ment toned at the end of chapter 3 and ~ the programme CHAR-S which was based on the method described in ref. [17] and the programme SLM which was developed from section 4.2, will be used as subprogram- mes in the overal l calculation of the f lat-bottomed slamming problem. For each t imestep CHAR-S is used to calculate the pressure acting on ah" water surface between x = -b and x - b. Outside these points the reassure is Set equal to the atmospheric pressure. The value of the pressure at the midpoint of each free surface element are then used in eq~'atir~n (~.11) in order to step the solution in the fluid forward. In the programme CHAR-S it is necessary to use much smaller timesteps than wi l l be used in the overal l programme. When the solut ion in the water region has been obtained at time tj, then the pressure in the airgap at tj will be calculated. In this cancellation h(x,t) will be needed, and it is found by the use of 'inPar interpolation between h(x,tj_1) and h(x,tj). Oh. calcu'at~d pressure distribution in the airgap and the above statement that the Dressure outside the gap is set equal to the atmo~nheric pressure together results in a pressure distribution acting on the water surface which is not smooth at the edge of the body . The pressure distribution before and after the flow in the air has reached supersonic speed are i l lustrated in f ig. 5. I. In a real case however the pressure distribution will probably be smoothed out over a distance in the order of magnitude equal to the width of the airgap at the edge (see [6l ). In the present numerical r.~1r:~l;~, ion, only the value of the pressure at the midpoint of each free surface element is used as input to the calculation of the flow in the f luid. This means that the pressure distribution are smoothed out over a distance equal to the size of the surface elements. P ~ .~£tcr the f low has reached Fig. 4.7 Boundary elements for circular cylinder in forced ~ supersonic speed heave motion: y(t) = []asin(.`,t), lla/r ~ 0.6, .','r/q - 1.0 ,,~ ~ == ~ on SURFACE ELEVATION DUE rc PRESSURE El£MLNT I I b ~ x Up o o to in o to _ - 10. t. -b Fig. 5. l sec tick! 5 ? t)·mensiona1 analys_s The Dressure, a, under the body and the surface elevation, A, will be calculated as a function of x and t. The parameters involved in the prob l em are: ~a'fwidth of the body Her . se ang l e "ass of hotly Pos i t i on 0 f t he body when t he numerical calcu'at ion starts Velocity at this Point VO t)-ns. ty o' water Pw Aemosr,her ic pressure ,Oo Density of air at atmospheric pressure pa Ad i aba t i c constant Y Acceleration of gravity 9 260 b [m] ~ [ ] M [kg/m] He [m] [m/s ] [kg/m' 3 [Da] [kg/m'] [ _ ] [m/s 2 3

One oQss;b~- choice is then to write that: p/Do and fib are a 'unct ion of: x tV VO V M H P _ _0 _ 0 ~ YPo/Po ~ O b2 ' b , pa , v, r ~ _ _ Here Vo//ypO/po is again the Mach number, and VO/~; has the same form as a Froude number. Instead of M/oWb2 it would have been possible to use Mg/pob, but since the motion of the body will be domi- nated by i ts Mass and the pressure force on the body rather than the gravi ty force, the former wi l l be used. In addition to these parameters which describes the physical problem, the computer programme also contains parameters which def ines how the free surface initia11y is divided into elements, as well as the i nterva l At ( t ) between the t imesteps . In the calculations which have been made with the programme, free surface elements of eoual length have been used. The number of ele- ments between x = 0 and x = b is Cal led NA, and NP is as before the total number of elements along the positive x-axis. This means that the end of the last element is at the position: x = bF = NF NA In the CHAR-S suburogramme, which calculates the pressure in the air, the number of elements along the x-axis between x = 0 and x/b ~ 1.0 is N - 2 NA. Sect ion 5.3 Numerical results and_iscussion Three cases which has been investigated in the l itterature wi l l be used for comparison. The f irst one is one of the cases which was measured experimental ly by Lewison and Maclean [53 . The same case was also calculated numerical ly by Koehler and Kettleborough [7] . The model used by Lewison and Maclean was completely flat-bottomed, but Koehler and Kettleborough made calculations for the same body but with three different deadrise angles (e = 0. '9 = 0.25 and ~ = 0.5 deg. ). The two next cases was investigated both experimental ly and numeri- ca, ly by Verhagen [fi1 . The model is the same for both cases but two different drop heights are studied. Data for al l the three cases are given in table 5.1. ___ Case _ . A I 8 Lc , . - b [ml ._ _ 1.5 0.2 0.2 [kg/m] 4014 ~ 0 20.0 20.0 _ _ Table 5.1 Drophe i gh t [m] 1.52 0.04 0.40 [5], 17] [6] [6 In the present work the starting point of the numerical calculation is selected so that Ho/b = 0.2 in case A and C, and Ho/b = 0.1 in case C, and then VO is calculated by assuming that only gravity forces are acting prior to this moment. For case ~ Ho was selected to be the same as used by Koehler and Kettleborough. The values of Ho and VO for the three cases are given in table 5.2. Case Ho VO [m] 1m/s ] I A 0.3 -4.88 8 O .02 -O .626 C 0.04 2.66 Table 5.2 261 'n order to compare with the results presented by Koehler and Kettleborough [71, case A in table 5.1 has been calculated with three different deadrise-angles. In [7] y = 1.0 was used, but there is no specific information about the number of elements. However the f ;gures in [7] indicates that NA = 5. These values together with NF = 20 has therefore been used in this comparison. The results are presented in Fig. 5.2-8. Fig. 5.2 compares the pressure-time history at the centerline. The curves from [7] are translated along the t-axis so that the body has the position H = 0.0 at the same time as in the present calculation. In order to see the effect of the chaise of y, case A with ~ = 0.0, NA = 5, NF = 20 has been calculated also with y = 1.4. In Fig. 5.10 the pressure at the centerl ine is compared for the two values of the ad i abatconstant . In section 5.1 it was mentioned that the discontinous pressure at the edge of the body wi l l, in the present numerical model, be smoothed out over a distance equal to the size of the surface elements. To see the effect of this, case A with ~ = 0.0, y = 1.4 has also been calculated with NA = 20, NF = 80. The results are given in Fig. 5.11-12. The inf luence on the pressure and the surface elevation at the centerl ine is seal l, but the shape of the free surface and the pressure close to the edge of the body is very different. NA = 5 is probably far to less to handle the steep pressure-curvature at the edge of the body, and this value of NA has only been used in the present work to make the comparison with the results presented by Koehler and Kettleborough [7]. It may even be that NA = 20 is to smal l but further investigation of this has not been done. As mentioned before, case A has also been investigated experimentally. In !5] time-history curves at x/b = 0.0, 0.5 and 1.0 are prestend. If those curves sha1 l be compared with the present calculation, it is not clear where the t-axis shal l start since the position of the body is not 9 i ven . The same prob l em a l so occurs when pressure- t i me h i story for case B and C are to be compared with the curves presented in [63. Koehler and Kettleborough assumes that the maximum impact pressure is reached a. She moment of contact between the body and the water sur- face. This would of course solve the problem of locating the t-axis since the time of maximum pressure is given in the curves presented in [5] and [6] . However the maximum pressure wi l l probably not be reached at the moment of contact. The argument for this is that at the timeinstant when contact is made, results from the present numeri- cal method shows that the downward velocity of the water surface has not reached the same magnitude as the velocity of the body. Which means that after the time of contact the airbubble under the body, wi l l be compressed for some period of time. Verhagen [6] uses a simplified method to calculate the pressure in the airbubble after contact, but it is not indicated on the curves at what time the contact is made. Because of this problems, any direct comparison between the pressure- time history curves presented in [5] and [6] and those calculated by the present method has not been plotted. 'he results of the present calculation of case B and C are instead given i Fig. 5.13-16. The pressure on the centerl ine at the moment of contact as presented in Fig. 5.12, 1`, 16 is significantly smaller than the maximum pressure found in [5] and [6]. Koehler and Kettleborough explains this deviation for case A by assuming that the larger maximum pressure obtained in experiments occurs because the flat bottom of the body may not be completely paral lel to the water surface. This explanations may wel l be true, and the very large influence of small variations in the deadrise- ans e, which has been calcualated both by Koehler and Kettleborough r7l and in the oresent work, support this. However the deviation between the pressure calculated at contact by the Dresent method and the maximum pressure obtained in experiments

may also be partly explained by assuming that the maximum pressure will not be reached until a short time-period after the calculated contact. The argument for this was stated above. PRESSURE-T I ME H I STORY CURVE ~ 1 o ~_ ~o r~ Pr~ont thooru ~ KoahLer /~/ o- I I.,,,I, I i I 0.110 0.115 0.180 0.185 0.190 0.195 0.200 t.VO/b Fig. 5.2 Case A, ,=1.0, NA=5 SLIRFACE ELEVAT I ON AND POS I T I ON OF BODY ~ - l THETA - O .000 dog b - 1.500 ~ n - ~o 1' . ooo kg/ - . HO ~ O .300 n ~n- VO '~ -~.880 m/e - o- NA ~ 5 . NF - 20 tVO/b - 0.115000 0 - O. 185000 0_ 1 O. 191200 8- o. o'_ _~ttt~ o_ I ' I ' I ' 1 ' I ' I ' 1 ' 1 ' 1 ' 1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 t.4 1.6 t.8 2.0 x~b Fig. 5.3 Case A, ~ = 1. 0 PRESSLIRE D I STR I BUT I ON ., P~ ~ THETA - 0.000 dog b - 1.500 ~ A° M - 4014.000 koJa .~ 11 HO - 0.500 ~ N I ~ VO '' ~ .~0 ~. N I i tVO/b - 0.115000 1 o 1 o. 185000 1 A 1 o. 189000 I / ~ I 0.190000 . I / \ ~ 0.190400 j/ \l O 190800 o 11 \\ ~A\\ o_ ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 0.0 0.2 O.t 0.6 0.8 1.0 1.2 1.' 1.6 1.8 2.0 x/b Fig. S . 4 Case A, . . SURFACE EL EVAT I ON AND POS I T I ON OF BODY - 0 N ,~ 0. ~ o - O- ~ o O o U) O o - - THETA - 0.250 dog b - 1.500 ~ M - 4014.000 ko/a HO s 0.500 ~ VO = - .880 m/e NA - 5 NF - 20 t.VO/b - 0.180000 O. 190000 O. 195000 ~n oo.- ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 ' 1 0.0 0.2 0.4 0.6 0.8 1.0 ~ .2 1.' 1.6 ~ .8 2.0 x/b Fiq. 5.5 Case A, ,=1.0 262

- ~ PRESSURE D I STR I BUT I ON . ~ THETA ~ 0.250 deg o~ b - 1.500 ~ M - 4014 .000 kg/m HO ~ 0.300 a o_ ll VO- ;~.880~/e I l (VO/b - O. 190000 I lt 1 0.195800 ' ~ O 195000 o_ · 0 0 ~ ~ _', 0.0 0.2 0.4 0.6 0.8 1.0 l.2 1.' 1.6 f.8 2.0 x/b Fig. 5.6 Case A, '=l.O SURFACE ELEVAT I ON AND POS I T I ON OF BODY N 0: 8 o O ~ I 1 1 1 ' 1 ' 1 - 1 ' 1 ' 1 t0 0 0.2 0.4 0.6 o.e 1.0 1.2 1.4r 1.6 1.8 2.0 THETA - 0.500 deg b - 1.500 ~ M - 40 ~ S.OOO kg/ - HO ~ 0.500 ~ VO ~ -~.880 - /. NA - 5 NF - 20 tVO/b - 0.180000 O. 190000 O. f91200 x/b Fig. 5.7 Case A, >=1.0 PRESSURE D I STR I BUT I ON 0 ~n ~ _ - u)~ THETA - O .500 deg b - ~ .500 n M - 1014 .000 ko/. HO = 0 .500 ~ vo ~ -''rr.880 111/8 NA - 5 NF - 20 tVO/b - 0.180000 O. 190000 O. 192000 O. 193000 O. 195600 O. 191200 - _' o I . I . I . I · I . I . I . I · I . I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1 .`rr 1.6 1.8 2.0 x/b Fig. 5.8 Casr~ A, ~r=1.0 MESSURE-T I ME H I STORY CURVE 0 o r o ID 263 GAMMA GAM1A - 1.4r .` - 1.0 O_ , , , , I , , , , I 8. ,5 0.180 0.185 0.190 0.195 tVO/b Fig. 5.10 Case A, NA-5

r SURFACE ELEVAT I ON AND POS I T I ON OF BODY 0 o- ~ 0 - o- o o o o u' o o g o THETA - O .000 deg b - l .500 m M - 40 /~.000 kg/m HO = 0 .300 m VO = -~.880 m/e NA 8 20 NF ~ 80 tVO/b - O. /15000 O. /83000 O. /88200 f; o o- I . I , I . I , I , I , I . I , I . I 0.0 0.2 0.' 0.6 0.8 /.0 1.2 1.' 1.6 1.8 2.0 x/b Fig. 5.11 Case A, .=1.4 PRESSURE O I STR I BUT I ON SURFACE ELEVAT I ON AND POS I T I ON OF BODY I 8 o ; THETA ~ O .000 deo b - O .200 m n - 20 .000 kg/m HO = 0 .020 m VO = -O .626 m/c NA - 20 NF - 80 (VO/b ~ O .0150 0.0840 0.0882 ~_ o _ · ~ · I · I · I · I - 0.0 0.2 O.' 0.6 0.8 t.0 l 1.2 l.4 1.6 1.8 2.0 x/b Fig. 5.13 Case 8, ,=1.4 PRESSVRE D I STR I BUT I ON THETA - 0.000 deg b - /.500 ~ rl ~ 4014.000 kg~m HO '5 0 . ~iOO ~ VO ~ -~.880 e/e _ :./5 NA - 20 NF - 80 (VO/b - O. /15000 O. /83000 O. /85000 O. /86000 O. /81000 O. /81600 O. 188200 N _ _ ~ _ ·~ ~ 066000000~ ~ i - . ~ V _. o~ ' 1 ' 1 ' 1 ' 1 ' I ' 1 ' I ' 1 ' I 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 /.6 1.8 2.0 x/b Fig. 5.12 Case A, !=1.4 264 Fig. 5.14 Case B. `~51.4 THETA - O .000 deg b - 0.200 ~ M - 20.000 ko/a HO 5 0.020 ~ VO - -0.626 - /. NA - 20 NF - 80 t-VO/b 0.0800 0.0810 0.0860 O .0e82 0 0 12 0 14 0 16 0 18 1 10 1 12 1 11 1 16 1 18 2 1 U.0 0.2 0.4 0.6 0.8 1.0 1.2 1.' 1.6 1.8 2.0 x/b

SURFACE ELEVAT I ON AND POS I T I ON OF BODY o. ~ 0 - o o o o- o 8- o o fir. '0.0 0.2 0.' 0.6 o.e ,.o 1.2 l.' Fig. 5.15 Case C, >=1.4 PRESSURE D I STR I BUT I ON _ 1 THETA - 0.000 deg b - 0.200 ~ M - 20.000 ko/- HO ~ 0.040 ~ VO ~ -2.658 -/e NA - 20 NF- 80 tV0/b - 0 .1820 0.1910 O. 1992 ~ A_ `'NL: _' i of I I ' I ' I ' I ' I ' I ' I ' I ' I ' I 0.0 0.2 0.t 0.6 0.8 t.0 1.2 I.S 1.6 f.8 2.0 x/b Fig. 5.16 Case C, ,=1.4 THETA - O .000 deg b - 0 .200 n n - 20.000 kg/m HO ~ O .040 a. V0 = -2.658 a/e NA - 20 NF- 80 (V0/b - 0. /9f0 0. /950 0.1910 0.998, 0. 1992 265 CHAPTER 6. CONCLUSIONS The discussion at the end of the previous chapter shows that it is necessary to calculate the pressure on the bottom of the body also after the time of contact between the body and the elevated water sur- face. Without such information it is not possible to compare the pressure bui ld-up calculated in the time-period before contact by the present numerical method with experimental results. As regarding the calculation of the flow in the water, this work has shown that the method described in section 4.2 may be used in the pre- sent situation without instability problems, despite the fact that the integral equation which is used is a Fredholm integral equation of 1. kind along the entire boundary. The comparison with Doctors which was made in section 4.4 also indicates that the surface elevation due to a pressure distribution acting on the water surface is calculated quite correct l y. For the purpose of calculating the flow in the air, three methods have been used. They suoDort each other, but no independent comparison wi th other resu, ts has been made. The resul ts of the present work shows that the inf luence of a seal l dead r i seang 1 e i s very l a rge . The present work also shows that the smoothening of the calculated discontinuous pressure distribution has a large effect on the shape of the water surface and the pressure close to the edge of the body. REFERENCES t1] van Karman, T.H., "The impact on Seaplane Floats during Landing", NACA TN 321, 1929. [2] Wagner, H., "Uber Stoss und Glei tvorgange an der Oberf lache van F 1 uss i gke i ten", Ze i tschr . f . Angewandte Matheeat i k und Mechanik, Band 12, Heft 4, 1932, pp. 193-215. [3] Chuang, S. L., "Experimental Investigation of Rigid Flat-Bottom Body Slamming", OTNSROC Report 2041, 1965. [4] Chuang, S.L., "Experiments on Flat-Bottom Slamming", Journal of Shin Research, Vol. 10, No. 1, 1966, pp. 10-17. rS~ Lewison, G. and Maclean, W.M., "On the Cushioning of Water Impact by Entrapped Ai r", Journal of Ship Research, Vol . 12, No. 2, 1968, pp. 116-130. [6! Verhagen, J.H.G., "The Impact of a Flat Plate on a Water Surface", Journal of Ship Research, Vol. 11, No. 4, 1967, rho. 211-223. r7l Koeh'e", B.R. and Kettleborough, C.F., "Hydrodynamic Impact of a Fa, l ing Body upon a Viscous Incompressible Fluid", Jour- ~al of Ship Research, Vol. 21, No. 3, 1977, pp. 165-181. ~82 va~ammoto, Y., Iida, K., Fukasawa, T., Murakami, T., Arai, M. and Ando, A., "Structural Damage Analysis of a Fast Ship Due to Bow Flare Slamming", Int. Shipbuilding Progress, Vol. 32, No. 369, 1985, pp. 124-136. 293 Abbot, M. B., "An Introduct ion to The Method of Characteris- t i cs ", Amer i can E l sev i er, New York, 1966 . rho! Ames, W.F., "Numerical Methods for Partial Differential Eauatior;s", 2. edition, Academic Press, 1977.

[11] Landau, L.D Press, 1959. , and Lifshitz, E.M., "Fluid Mechanics", Pergamon [12! Lister, M., "Mathematical Methods for Digital computers", edited by Ralston, A. and Wilf, H.S., John Wiley & Sons, Inc., 1960, pp. 165-180. 'i3! Faltinsen, O.M., "Numerical Solutions of Transient Nonl inear Free-surface Motion outside or inside Moving Bodies, Proc, Second Int. conf. Num. Ship Hydro., Berkeley, Sept. 1977, Dp. 347-357. f gal . . r;5l Some of the endpoints cannot be found in this way. (See Fig. A.2. ) s Pi Ps Q 6 A, W.M., Newman, J.N. and Yue, D.K., "Nonlinear Forced Motions of Floating Bodies", Proc. 15th Symp. Naval Hydro- dynamics, Hamburg, Sept. 1984, Session I, pp. 1-15. Vinje, T. and Brevig, P., "Nonl inear Two-Dimensional Ship Motions", Proc. 3rd Int. Conf. Num. Ship Hydro., Paris, June 1981, pp. 257-266 (See also "Nonl inear Two-Dimensional Ship Motions", SIS Report No. 18, NHL, Trondheim, 1980). x=bF Y ~ ~ _ x Fig. A.2 The endpoint P in Fig. A.2 is the intersection point between l 1 3 . Endpoint P s the intersection point between l and The y-coordinate of endpoint P is: r'6l, Doctors, L.H., "Solutions of Two-Dimensional Slamming by Means of Fini te Pressure Elements", Proc. 3rd Int. Conf . Num. Shio Hydro., Paris, June 1981, pp. 559-577. Y(~36 ) = 2 y(Q ) - y(Ps ) 217; Falch, S., "A Numerical Study of Slamming of Two-dimensional and the x-coordinate is: Bodies", Dr.ing.thesis, Division of Marine Hydrodynamics, The Norwegian Institute of Technology, 1986. x(P ) = Xb Append i x A This appendix describes how new elements on the free surface are generated and how the velocity potential ~ at the midpoints of the new elements is calculated in the programme SLM which is based on section 4.2. Suppose that the fluid-particles at the midpoints of the old elements on the free surface are given new positions by the use of the time- stepping procedure described in section 4.2. These new positions are designated Qj (i = 1,...4) in Fig. A.1. ~< ~ Fig. A. 1 12 Qua For the new elements which will be generated, the position of the -ight endpoint of an element is common with the position of the left endpoint of the next element. the position of the endpoitn P lying between Q and Q in Fig. A.1 is f ound as to l l ows: is the straight line through Q which is parallel with the line between Q and Q . 3 1 is the straight line through Q which is parallel with the line between Q and Q . s _s 2 ~ 't (new) = ~ ~ (¢ - ~ ) ' l is the normal at the midpoint on the line between Q and Q . 2 ~ 2 S IS P is the intersection between l and l ~ a P is the intersection between l and l and f inal ly P is the midpoint between P and P . 1 2 _.d l . ~ 3 The y-coordinat of endpoint P is: 3 y(P3) = 2 Y(Qs) ~ Y(P ) and the x-coordinate of endpoint P is found by the condition that P shal l be on the body surface. The midpoints of the new elements will not coincide completely with the points Q. which is the position the f luidparticles was given by the sidestepping procedure. The time-steping also gives new values of at the points Q. The value of ~ at the midpoints of the new ele- ments are then calculated by l inear interpolation along the free sur- face. An examples of how this is done in detail is illustrated in Fig. A.3. and ~ are the values of the velocity potential at Q and Q . us \ 51 \ P F i g . A . 3 - S 3 :~( n ew ) Q2 P is a newly calculated endpoint, and we want to f ind ~ (new) at the new midpoint Q (new). s is the distance PQ, s the distance : · ~ 2 PQ and s is the distance PQ (new). The following expression for 2 3 2 the new velocity Dotential is then used: 266

DISCUSSION _ by J. Matusiak My question concerns the importance of air cushioning loads. It is obvious that in a case of flat plate inspecting water surface it has a very pronounced effect. However an increase of deadrise angle might significantly increase an "escape of air" and thus decrease its influence on slamming loads. Did you conduct a numerical experiment in which you calculated loads (both pressures and total forces) for the air cushioning being included and disregarded. If so what is practical limit value of deadrise angle for which air does not cushion the slamming loads? Author's Reply This is of course a most interesting and important question. First of all, since I only carry on the calculation until the moment when the body makes contact with the water-surface, I do not find the maximum pressure. Secondly, I did not make any comparison with slamming loads calculated without the effect of air cushion, so I don't know the limit value of the deadrise angle. But if you look at Figs 5.5- 5.8 in the paper, you can see that a change in the deadrise angle of half a degree gives a dramatic change in the result. So my guess is that the effect of air cushion will be small for a deadrise angle in the order of one, two or three degrees. 267