**Suggested Citation:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." 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:**"Some Numerical Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves." National Research Council. 1990.

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

Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Some Numenca] Computations about Free Surface Boundary Layer and Surface Tension Effects on Nonlinear Waves E. Campana University "La Sapienza" Roma, Italy F. Lalli and U. Bulgarelli I§tituto Nazionale per Studi ed Espenenze di Architettura Navale Roma, Italy Abstract____ This paper is a first approach to the development of a full nonlinear numerical method implementing the exact boundary conditions, taking into account viscosity and surface tension effects at the free surface. At this step, the following results have been obtained, maintaining the simplicity of the collocation method proposed by Dawson t53: 1) effective fully nonlinear numerical procedure, taking into account the effects of viscosity at the free surface on potential flow; 2) simple preliminary results for the linear gravity-capillarity problem. All the obtained numerical results have been tested with analytical solutions or experimental results. List of symbols 0xy = frame of reference fixed with the body : Ox is oriented in the opposit direction of the body velocity V, Oy is vertical upwards = body surface L = characteristic body length (chord) depth of the body center free surface unit vector normal to and S Cartesian equation of the free surface curvature of the free surface curvilinear abscissa defined along S free surface boundary layer thickness body velocity vector h S n = tax) = K = 1 = = = ~(U,O) = T = k - - v = = g = P = Rw = = = of = r = r* = P = ~ = q* = 455 = fluid velocity vector = vorticity vector = density surface tension wave number group velocity kinematic viscosity acceleration of gravity pressure wave resistance velocity potential perturbation potential simple layer density p ql IP - q *1 field point source point image source point 1. Introduction The Rankine source method succesfully adopted in recent the numerical solution of free surface flows. In the usual approach the free surface boundary conditions are linearized and applied on the undisturbed free surface. Then, using the simple layer potential function the boundary value problem is r" to an integral equation which solved numerirn11v has been years for potential can be _ _ _ _ i, . With this approach, in 1977 Dawson t53 proposed a simple and effective procedure for computing the wave resistance of ships in the steady-state case. Such procedure was based on the idea of writing the above mentioned linearized boundary conditions along the zero Froude number flow streamlines on the undisturbed free surface. In the wake of Dawson's paper several studies have then appeared in literature; some Authors deal with the linear problem, discussing about the consistency of different kinds of

linearizations t9,13], the numerical implementation of the radiation condition [10,12,14] or the numerical stability [123. On the other hand some Authors, maintaining more or less the simplicity of the collocation method, and the use of an upstream finite differences scheme t5] for the radiation condition, proposed some kinds of iterative procedures to solve the nonlinear problem [6,11,153. The present paper is a first approach to our future aim to develope a full nonlinear method, implementing the exact boundary conditions taking into account viscosity and surface tension effects at the free surface. This combined gravity-capillarity problem is interesting because can give some more informations on nonlinear waves behavior, with respect to the classical model which neglects surface tension. As a matter of fact, Longuet- Higgins r 2] proposed a model for the nonlinear transfer of energy from steep gravity waves to capillary ripples, which "ride" on the forward faces of the long waves. In this case the surface tension can be locally important and play a significant role in the generation of waves by wind. Furthermore capillary waves, taking energy from gravity waves and loosing it by viscosity,tend to delay the onset of breaking. On the other hand Marco and Ikehata, in an experimental work [8], pointed out the importance of surface tension on the shape of the wave pattern around the model, in particular at the forebody; how wave resistance coefficient depends also on- the Weber number is furthermore shown in t83. Among the free surface phenomena, the presence of a thin boundary layer can also be considered. All real fluid motions are of course rotational; even in nearly irrotational flows the relatively small amount of vorticity present in thin layers, can be crucial in determining the main flow characteristics, as it's well known. The condition of vanishing of the tangential stress at a perturbed free surface cannot be satisfied by irrotational motion. Consequently regard the incompressible flow as combination of a thin rotational free surface layer and a potential flow region. The exact amount of the vorticity jump generated at the free surface can be written as a function of its curvature and of the potential velocity t4~. So we obta~ boundary cordite effect at S; o an we a in a nonlinear dynamic ion including viscosity f course in the limiting 456 case of Re ~ on , the condition is obtained. Numerical results have been computed for the full nonlinear formulation in the presence of a free surface boundary layer, while simple linear results are presented for the gravity-capillarity inviscid problem. In order to validate the capability of the proposed model, numerical simulations of some typical test problems are described and discussed. usual Bernouilli _. Mathematical formulations____________ ____________ We consider the following mathematical formulation for a 2D steady state potential flow, due to the motion of a submerged non-lifting body in a fluid of infinite depth. The extension to the 3D case can be easily obtained. The velocity V=(U,O) of the body is directed in the negative x direction, the y axis is positive upwards and the undisturbed free surface level is given by y=O. Consequently the fluid domain ~ is bounded on the upper part by a free boundary S. and it is unbounded in the other directions. We assume that the fluid velocity, q=(u,v), can be written as: (1) qV, where (1 ') ~ = ~ In (1') the term LTx is the undisturbed flow potential and the term Stakes into account the interaction between the free surface and the body. Let be y= ~ (x) the Cartesian equation of the free boundary S. and l a curvilinear abscissa defined along S. The potential ~ (x,y) satisfies Laplace's equation inside ~ `~ , where = ~ V ~~ C ~2 is the body (2) V {(x,'~sO ~ (x'~6 t`6 n T. -I} The following boundary conditions, which include the effects of surface tension T , must be associated to equation (2) : ( ) lt orb 49 on S

~ (em :~)+ T ~ on S 2~se .~) (5) ~ ~ O - (6) km | V i| bounded ~x'_oo on ~o3 Here K(l) is the curvature of S. considered negative when the centre of curvature lies on the fluid side of the free surface: en (7) K=.~ Ax (4~ ,2 )3~ x From now on, in will present, an conditi _ of ~ whenever the d to have finite depth. We are going to c kinds of approximation previously described, boundary conditions obtained from.(3) and for computational precisely four models every model we extra boundary on must be imposed at the botton omain is supposed Therefore, in this case, the linear model describing the motion of a submerged body is constituted by (2), (5), (6')~9) and (8) or (3'). It is worth to notice that, for this formulation, theorems for solution existence and uniqueness have been given in r7] 2.2 The linear formulation with T/O Without neglecting surface tension, conditions (9) and (8) can be written as: (10) ~YXxi t,= ~ ', _ T rr (11) A= _ u ~ 4, 1 ~ x pit L~ Hence, in this case s the linear model is given by (2), (5), (6), (10) and (11) or (3'). Onsider different s for the problem in which the given on S. In the nonlinear case, with T=0, (4), are suitable condition (9) becomes: purposes. More are delineated. z.3 The nonlinear formulation with T=0 z.1 The linear formulation with T=0 (12) ~ ~ ~ §~' = 0 on S while the nonlinear Bernouilli equation give: Assuming T=0 in (4) and neglecting the nonlinear terms both in (3) and in(4), we get the linearized free - ~ / UP &2 ~ surface conditions: (13) ~ ~ ~ ~ ~ ~4J on S (3~) U ~ =-7, on y=0 (8, ~ 5 _ U ~ on y=0 and the well known Neumann-Kelvi n condition: (9) U y +~=0 ~ xx g where, for the car __. we have (see 2.6): (6') km Ivy ~ IT x ~ oo di t i on At infinity, Therefore the nonlinear model, neglecting surface tension, is given by equations (2), (5), (6), (12) and (13) or (3) 2.4 The nonlinear formulation with TWO Assuming TWO, the unified free surface condition is obtained by eliminating ~ between (3) and (4~: (14) ~ ~ ~ ~ ~ = T t~ So the nonlinear model with surface tension is given by equations (2), (4), is), (6), (14). It is worth to observe that conditions (9) , (10) , (12) can be easily deduced from (14~. In the sequel we confine ourselves to consider the models given in 2.3 on S 457

and 2.2. 2.5 The effect of viscosity at the free surface Let us consider the model described in 2.3, in which we will introduce, in the hypothesis of large Reynolds numbers, the modifying effect of viscosity. The introduction of a boundary layer at the free surface can be considered as a first step of a zonal splitting of the global problem into several interacting ones: - the irrotational subdomain; - the boundary layer at the free surface; - the boundary layer about the body km 1q| = U - the wake; 1 a= Ho Furthermore the new dynamic boundary condition improves some convergence properties of the numerical procedure t153. To deduce the above mentioned condition we start from the consideration that the irrotational motion does not fulfil the condition of zero tangential stresses at the free surface (if the free boundary has a non zero curvature). Consequently, we introduce into the model given in 2.3 a partition of the flow field into an inviscid region, free from vorticity, and a thin viscous layer at the free boundary. Across the second region a vorticity jump, connected with the curvature of S. is considered, as well as the consequent velocity jump, due to the viscous component inside the boundary layer. In fact, although a rigid boundary is the commonest source of vorticit,~, in the case of a free boundary the vanishing of the tangential stresses generates vorticit~ and consequently a viscous boundary layer. Across this layer the above mentioned finite jumps of the velocity and of the vorticity are generated. To include the effects of the vorticity at the free surface, we start from the steady Navier-Stokes equation: (~5) [(at V)] = ~ V ~ ~ ~ ~q !21) ~< ~ ^¢ where p = p + 9~5 , p is the dynamic pressure, p Is the atmospheric pressure and ~ is the fluid density. By using the following identities: (16') (q V)9 = V(2q 9) 9 _ (16'') ~q = ~ VX and, for y= ~q, (x): (171 V (42 q q §~) 9 ~ ~ Since the integral on the left hand side of (17) is path independent, we can integrate both sides of (17) along S. which in the steady flow is a streamline. Requiring that: we obtain: ~ , ~ ~ 9t -2- 9 9 -A ~vxz since qx4~de =0 . The kernel of the curvilinear integral can be trasformed as follows: ( 1 9 ) (Van; )~t = - d! _ ~ where n is the unit vector normal to S. oriented outwards. By substituting (19) into (18), we can Basils ~et: (20) 7~= Us_ ~ q q + ~ ~ ~; ~e' 2, &, ~ ~ ~ _ ~ ~ The normal derivative of ~ , if the viscous layer is sufficiently- thin, can be expressed as the ratio between the vorticity jump across the boundary layer and its thickness`: Con- we shall use Helmholtz cle.on~position and a boundary layer approximation to evaluate the above mentioned v-orticit:- Inch velocity jumps, ill terms of the velocity- potential `>raciient and of the geometrical curvature of the flee surface. The 458

expression of ^t can be readly evaluated [43 as: (21') at = 2 ~ |vI| Moreover an extimation oscillatory boundary layer can be given r4~ by: (22) ~ = (U!V) Furthermore we observe velocity distribution in domain can be decomposed as: ~ vT qua where q is required to satisfy: or V'9`r=0 Vx9~~t of the thickness that the the fluid In particular the velocity at the free boundary can be thought as the sum of [V\' and the jump ^|9~| of the rotational component across the boundary layer. This jump can be estimated r 43: (23) ~15~1- ~ ^: - hence, we have: (23') Iql = IVII+AI6Vrl KIWI + +` a: - IvII +~s K 1~74 ~ B>' introducing (21),(_2),(23~) in (20), neglecting terms of order ~ we get: (24) 9(~)= U _ ~4 ]' (~+ IS)+ + ~ ~ 5 K(4) {~.~ And hence: The unified free surface boundary condition is obtained from (3), `24): (25) j~4~ (~+4&K)~: 4~ ~ ~ ~ me (SKI (' _ ~ K) - o Ve observe that the usual in~iscid nonlinear dynamical condition can lye obtained from (25) in the limiting case of ~ ~ O (i.e. Re-. oo). 2.6 An integrodifferential formulation for the gravity-capillarity problem Let's consider now the linear problem with surface tension described in 2.2. We are going to develope some mathematical calculations in order to obtain an integrodifferential formula_ tion, in which the third order derive_ tive present in (10), that create some difficulties for numerical implements_ tion, does not appear. Moreover, we intend to explore the behavior of the solution. Solving (11) with respect to it follows: (26) ~xx~ 95 ~ = iT ~x= ~ that is: (27) qtx)= ~ X LAI ~ ~ Je OFFS) d'] ~ o ~ 6-dX[A2- ~ J~'~)di] o where An and A are arbitrary constants, and ~ = IT Since ~ is bounded for l~1--. no, we find: ^1- ~ i ~ ,rC')d; A2 = - ~ J ~ Id) Hi Cam CO (28) 4(X)- _ ~ [I JO )~)di + ~ -~xJ ~, rt;,4~] At this point, three different formulations involving respectivel>- ~ , by and ~~, will be obtained Substituting the Expression of P (if) and integrating by parts we ~et: 459

(29) ~q(x)= -I_:Q We STY 'by, _ x Q J e ~ 1J~ d) ~ so From (29), deriving with respect to x and applying the kinematical condition ( 3 ' ) : ( 3 0 ) ~ ~ _ ~ ~ ~ 5~ [Q~XJ4,~l Ad x JO The formulation involving y fol lows directly from (28), de Diving with respect to x and `'sing again condi t ion ( 3 ' ): - ~ _ _ ~ ~ dxJ -d) g' e J e ~l~ ~~] _ ~ Finally, to wi th y (28): ), get the last express ~ on we integrate by parts (32) ~(x)~~ g~`{x 2-g,[ 1 Bit,' ~ ~ r ~ ~' ~ and the unified condition is obtained as above: ~ 3 3 ) y _ _ p IJ [~d ~ J e ~' ~ 3) AX Ix ^' ~ The relations ~ 3~)), ~ 31 ) or can be used instead of ~ 10 I, to the calculati. in of the third ~eriNiRti\'e yxxy G! yyyy in u me r i c a l i m p l a m e n t a t i 0 !~ . We notice that: f'-~n~ f l) ) w (I?) when In o~- `~f the So-' f ~-ee wave f~ 1 1 '1; t, y a I- i ~ b l e s cons tent s: f ~ ~ ) i~Voi d t? r ~ e r the ... ~ obtai n 1 ma? () , t ha t i ~ ~ ~ on . den- to stlldv some properties ution' we co~si<1e~- the ~ Ample problem, given by ( 2 i, ( 10 ), means of the -epa~-at ion off He can ~;~-ite-, up a~-bit~-~- t 34 ) (p - Me { ~ ~ ~ where k is the separation constant. Substituting (34) in (10) we get the dispersion relation: (35) ~ _ MU ~ ~ 9~=0 r T u i t h: 4~' = ]°]J (A _ ~ ) C ~ (A ~ ~ ~ ) Hence, if respectively: be= ~ tic ~ (pure capillary waves ) Since we consider the steady case, the phase velocity is zero, while the group velocity is given by: g or T ~ O we obtain (pure gravity waves) (36) ~ = ( _ _ ~ ) U Uk which can be written' (37) ~ = ~ (I ~ ) 4kT - Us The positive sign must be gravity waves, and the negat capillary waves: it is recognize that the positive for 460 chosen for ive one fol- easy to .= group velocity is the farmers, and

egative for the tatters. So it follows capillary opposite erections L1,2J, as it is well known from experimental observation. n that gravity waves and ripples propagate in d 2.7 A simplified formulation for the linear gravity-capillarity problem Now we reconsider the linear formulation described in 2.2. Let us assume: ~ c Y= ~ k! (38) ~ c where the superscripts G and C indicate the terms connected respectively to the gravity and capillary waves; in fact, as it has been shown in the preceding section, in the capillarity-gravity problem two wave sistems are present: the capillary way-e preceding the body, and the gravity one following it. So we can write: ~(7x+:X) tY+tY U({X ~ AX ~ ~ ~ (7 F? )- p <~=0 km ~ = km iV9t = 0 ~ ~ _ oo ~ ~ o() = km |~| = 0 x_ +0o Since the can apply the effects, solving (39) U )4 _ Sty (40) Mix + ~ ~ ~ p ~xX = 0 problem is linear, we superposition of the two splitted problems: (41) ~ 1 ~{ 1 = 0 X ~ _= (42) ~ SAC _ SAC (43) ~ My +~: _ T Arc _ o (44) ear l~cl=0 X .~= These formulations can be strongly simplified assuming: (45) ~ - - C by) t~ Tc= _ (Gil that is, the free surface is sinusoidal. Of course, hypothesis (45) can be a good approximation only for the far field: the solution available with this assumption can so be considered asymptotical. We have: ( 4 6 ) gym + (~+ T ( Ad) ) ~4 _ (47) Myth (~+ T LACE) c and, considering (35), we get finally: (48) 74 = _ Yx tJk~4 ( 49 ) i,t,`+ ~ ye = ~ C TIC (50) ~ = x v-he (51) ~ sky =0 OX y In order to solve problem (48), (49) condition (41) must be considered, as well as condition (44) for problem (50), (51). 3. Numerical procedure and results The numerical solution of the mathematical models given in 2.3, 2.S and 2.7 is reached by means of a simple layer formulation: 461

(52, T(r, 1~[~+Q' ~ ~ 1 ~ ~ in which r = jp_ r' = tp_~ r" = lp_ with p field point, c~ , quay ~ , q ~ ~ source points, being. ~' the image of ~ with respect to the undi_ sturbed free surface. The surfacers and a local portion of S are discretized, by means of the classical collocation method E53, with segments tangent to the boundaries. 3.1 The nonlinear gravity problem In the numerical procedure the free surface S is followed, step by step, updating its discretization and, of course, the influence matrices. The iterative scheme consists of two cycles. An 'internal' one, in which the nonlinear problem given by (2), (5) and (12) is solved with an iterative procedure; when the solution of this system satisfies the required accuracy, the free surface is updated in the 'external' cycle by means of (13). At the first step, to initialize the procedure, the potential flow and the free surface configuration are computed with Dawson method. The computat double iteration summarized as in t tonal steps, for the scheme, can be he following: - solve the linear problem (2) (I), (6), (a), (I) to initialize the numerical procedure; `~' - ext~em)na1 loop (index m): 7< ~ are known on each bounder element i; 3 - internal loop (index j) : solve the nonlinear system (j+4 > I ~ = 0 At, 2(j) (+1) Ajar) II ~ ~ {a ~ - o t~ Hi ~ Hi in which the quadratic terms are considered explicitly;, and the condition II is imposed on the previous appproximation (m) of the free surface configuration when ma|- All/ l^~| is lower then a pref~.xe;] tollerance go to step 4 ; 4 - external loop (index m) : calcu_ ration of the new approximation of S. with III ~ - - ~ U l' ) when make |~-~` /~^ l is less then a required tolerance, the numerical procedure is considered to be concluded. Near the downstream bou if the damping zone suggeste present, an oscillatory be been observed, though for al grid points convergence reached; t evident as as it coul notice the more rapid_ for wave he obvious sit quantity, applicatior The s been used formulatior the effect surface; t instead conside~-abl of the met involving ~ We read in (12) or i the four p scheme urn p to ndary, also d in (53 is havior has l the other has been his phenomenon becomes more the Froude number increases, d be expected. Moreover we t convergence is reached 1 v for wave rag i stance than rather average norm i r~ 1 the app geo 1 o c be exp be 462 ight; this behavior ce the former is an can be useful for r ame nume I ical strategy has also for the nonlinear (24), (25), which includes s of viscosity at the free he use of such formulation of (12), (13) extends y the range of applicability hod 15 , though the terms iscosity are rather ~mall. cork that the term ~~` n (25) is discretized with Dints finite differences ~ - r posed by Dawson; the ownstream damping zone, in which a two Dints operator is used, is fixed equal a quarter of the wave length. To validate the present method, numerical procedure has been lied to a submerged hydrofoil, whose metrical characteristics find ation in the fluid dynamic field can found in fig.1; several erimental results for this case can found in r33. In all the examples the free boundary has been discretized by- 250 panels, and the boundary of the body by 40 elements. The number of panels per wave length has been chosen equal to 40, and about 80 free surface elements hare beeen L?lacecl upstream the boa: leading edge. In the first two examples we compute the wave pattern corresponding to two different Froude numbers (.4406 and .~`l7), with the Froude number defined as If/ ~ (L is the chord lent-). The ;~av-e pattern, computed is- the nonlinear- p~-ocedu~-e described before, by means of (is), is compared with the experimental data

obtained in [3] and with the linear numerical solution computed by Dawson's method [5]. As it's shown in fig.2 and 3 the full nonlinear formulation gives a numerical solution which fits the experimental data better than the linear one. We notice that in this range of Froude number the inviscid nonlinear formulation gives almost the same results of (25), the only difference being a small damping in the wave amplitude for the viscous formulation, as it can be expected. This behavior is shown in fig.4 (Fr=.704), in which the typical steep wave shape can be recognized; the damping appears to be significant rather far from the body: at a distance about 10*L a 4 per cent attenuation is present. In fig.5 the wave resistance as a function of the Prude number is plotted (for this figure, Fr=U/Vg~h): also in this case the nonlinear method behaves better then the linear one, with respect to the experimental data. We remark that the inviscid formulation gives practically the same results for the wave resistance, but its range of applicability is narrower with respect to (25), in fact for Fr>.57 the numerical procedure diverges E ~ while the use of (25) allows one to obtains results up to Fr=.65. 3.2 The linear gravity-capillarity problem We solve numerical!: both the splitted problems (2), (5), t48), (49), (41) and (2), (5), (50), (51), (44), by 3 means of D;~lvson r S method, noticing that for the capillarit:- problem the second derivative must be discr*tized with . downstream finite differences operator, in order to satisfy the radiation condition. In fig.6 the wave pattern obtained with this procedure is shown: the dashed liners indicate the .splitte'-l solutions, while the continuous line indicate the superposition of them. In this case the body is a circular curling der (diameter d = (?.01 m), with ~ depth of the center- equal to the diameter. Surface tension value is .0/4 N/m; for the discretization, 2~, panels per capillary wale length hare been chosen, with ~ total ntlml~~~ `,f I)) panels used. the to with Fi~.7 shows the behavior of wave lengths accuracy with respect the discretizatron, in com~-ison the analytical Values. lineally, in fig.X the ~r;-'lues gravity- anal capillary- bale lengths functions of the velocity- I! plotters, in comparison with f ~ s trot the dispersion relation (35); in this case, 30 panels per wave length have been used. Conclusions Both the proposed numerical models seem to be promising and suitable to have more investigations. At the present the introduction of viscosity effects is rather effective in order to get convergence with highly nonlinear boundary conditions. The model with surface tension can be a very deep tool for describing nonlinear energy transfer phenomena; the results obtained with the simplified linear model are preliminary steps for studying a radiation condition suitable also for the nonlinear model. Acknowledgements We wish to acknowledge Prof. P. Bassanini, Dept. of Mathematics, Univ. 'La Sapienza' of Rome, for the helpful suggestions and encouragements, and Eng. T. Coppola, who implemented the computational codes. References - 9 463 1 Lamb, H., Hydrodynamics, 6th ea., Cambridge University Press, 1945. Longuet-Higgins , M. S. , "The Generation of Capillary Waves by Steep Gravity Waves", Journal of Fluid Mechanics, vol. 16, 1963. Salvesen, N., "On Second Order Ware Theory for Submerged T~-o-Dimensio_ nal Bodies",6th Symposium on Naval Hydro., Washington, 196G. 4 Batchelor, G. A., An Introduction to Fluid Dynamics, Cambrid~e,1S67. 5 Damson, C. W., "A Practical Compu_ ter Method for Solving Ship-Wave Problems" ,2nd Int. Conf.Numerical Ship Hydro., Berkeley, 1977. Daube, O., Dulieu, A., "A Numerical approach of the Nonlinear Wave Resistance Problem". 3rd Tnt If Numerical Do, C., Uniform Constant ,~}~en~ati Primiceri Maroo, H. discussic Flow ~.rc posium or '9,86. Jensen, G., Mi, 'Rhine Sourc Numerical Ship Hydro., Paris,1981. Guevel, P., "Waves on a Flow in a Channel of Depth", Research Notes in cs, 'S. Vol I, Fasano & 0 Ed., London 1983. , Il~ehata, M., "Some ns on the Free Surface und the Bow", 16th Sym_ Natal Pedro., Berkeley, 7. X. ~ Soding, H., ~ Methods for Solutions of the Steady

Wave Resistance posium on Naval 1986. 10 Jensen, Radiati State of St n.1., 1 1 Rong, H. rical ~ Ship-Wa ?~/. 2 Sclavour "Stabil thods f Forward Naval H Raven, Theme ~ Naval H 14 Nakatake J., Kat Hydrod, Hydrofc 5 Campana, F.,Bul~ Free Su means ISRP, c Problem", 16th Sym_ Hydro., Be~-keley , P. S., on Condi Ship Wa~- ip Res 987. ,Liang,X ethod fo ~ve Prob "On the Sumerical tion in the Steads-- e Problem", Journal ea~ch, Vol. 31, ., Wang,H.,"A Nume_ r Solving Nonlinear rem", ITTO, Kobe, os, P. D., Nakoc ity Analysis of E or Free-Surface ~ Speed", 17th Syn ydro., The Hague, H. C., "Variatic y Dawson", 17th Syn ydro., The Hague, , K., Kawagoe, T. aoka, K., "Calcu] namic Forces Ac ifs", SMSSH, Varnz E., Lalli, F. arelli, U.,"Fully rface Flow Comput of Moving Panels hangai, 1989. y , D. E., anel Me_ ~lows with rposium on 1988. ns on a IpOsium on 1988. , Andou, ation of tiny on , 1988. Pitolli, Nonlinear ation by method" , , f REE SURFACE ~R-~ ~ ~ .l4e .175 .te' .152 .1's .t48 .tt~ 1 ~ ~ ~ o .l .t ·' ·4 ~ ~ ·' ~ ~o ~1\ ~ ~ 1 h=~.S ~ 1 - Fig.l: Cross section of the hydrofoil [33; quantities expressed in ft. 464

- o - - ~- cM o - o - c~ - o o · - o o o" IS on - ~ I --or r I 4. on S 00 lo _ _ o O_ ~ I" \ a/ \ ,~1 o . _ o ~ At, hair , ~ A, At,, ~ ah. ~ Am' An' q I'll 1 /I O- I ~ ~ 1 1 1 1 1 1 1 1 1 O-7S 1.3S 1.95 2.5S 3.15 3.75 4.3S TIC A< At /--';~' "a ~ i'= .~' ~ _~ ARC ~ ~ ARC \',_. X I L ~ ~ ~ 1 ~ _ 4 95 5. SS 6.1 5 Fige 2: U/Vg*L S . 4406 - Fir' <) ~ a' ~~ 5 .7 f>~ '\' \\~ - - ~ ~ ~1 7 ~r Al ~c ~r 4 r I 6 04) XtL 1 - 7 00 6.Ot) 9 t)0 10.00 11 ·00 12-00 Fig.3: U/V~ = .617 Waves generated by t!,e submerged hydrofoil, ~che arrows indicate the posi~cion of the trailing cdge. Dawson method full nonlinear method ****~** experimental [33 465

o - - ~~ - Q _ o C: o +_ 1 U. ID o o U. C' o ~~ _ o ) O <a _ ~' ,4 O 0 - do CY C) I_ o O -- ~ ' `1 1 1 1 1 1 1 1 1 0.32 0.40 0.46 ~ . S6 0.64 ~ 72 A A: A A \ v l so s 00 , \l v X/L ~ , . ~ ., t s .00 ~ 7 . SO 20 .00 2~ . SO 25 .00 7 SO 10.00 Fig.4: U/V~ = .704 / /^ / ~ / - / full nonlinear method full nonlinear with viscous correction / / ~ ~ w __ _ U/ - 466 Figs: Wave resistance versus U/V~ experimental [3] ooooooo Dawson method 446"~` full nonlinear method

0 ~ - ~o - - ~ · ~ ~ - o o - - ~ to o - -0~80 -0.64 -t) 48 -O 32 -0.46 -O.OC 0.46 0~32 ,,< 1t'1 ~ ~ (~N . . ~ ~ 1- 0.48 0.64 0.~0 Fig. 6: Gravity-capillarity steady free surface flow past a submerged circular cyi inder: - gravity and capillary waves superposition o, - _ :~ 0 _ ~ At_ ~ _ ~ m_- _ 1 ~ - ~ 00 ~ 6 00 r ~ I I I I r- 32.00 &8 . OC 64 Fig. 7: Wave lengths versus N = number of panels U = .234 m/s per wavelength; L = .01 m h = ,01 m Fox- ~ ru , c~ . _ ~ c' ._ 1 me ~ ~ 1 O- 1 (~) Fig .8 ~ Dispe2~sion relation ( 35 ) L = .01 m - h = .01 m 467 analytical curve oooooooo nwnerical values

DISCUSSION by K. Nakatake I appreciate your results about capillary waves in front of the hydrofoil. I would like to know the way of decay of that kind of wave. And how large is the effect of surface tension on the wave behind the hydrofoil? Author's Reply I thank Prof. Nakatake for his kind appreciating comment about our work. I remark that our computations about gravity-capillary waves have been performed by means of a simplified linear model which, for its nature, cannot give any information about attenuation. In the linear model, in fact, the capillary wave amplitude remain constant, as well as for the gravity wave. Anyway, since the attenuation increases with the curvature of the free surface, as it has been pointed out dealing with viscosity effects at the free boundary, the damping of capillary ripples must be much stronger with respect to the gravity waves. For the 2nd part of the question, the effect of surface tension on the wave behind the body is easily evaluated. In fact, if the pure gravity wave length is equal to 2nF2 , in the presence of surface tension the wave length beCome(1~71-4/77QF2)nF2. 468