**Suggested Citation:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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:**"A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies." 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.

A Boundary Integral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies C. M. Casciola Istituto Nazionale di Studi ed Esperienze di Architettura Naval Roma, Italy R. Piva University di Roma "La Sapienza" Roma, Italy Abstract The possibility to generate numerical models based on a boundary integral formulation for rotational free surface flows, either viscous or inviscid, is explored with the purpose to maintain some of the computa- tional efficiency proper of potential flow approaches. A simple method for the simulation of the unsteady nonlinear behavior of the free surface has been de- rived, starting from a mathematical model which de- couples the kinematical and the dynamical aspects of the flow field, without introducing the potential ap- proximation. The strict relationship with the coupled models based on the integral formulation for viscous and inviscid flows, is discussed in details to extend the present method to these more general conditions. The mathematical and numerical aspects of the re- lated integral equations are analyzed and an efficient numerical solution is proposed for the solution of the flow field generated by a moving submerged cylinder. ~ Introduction Among the several methodologies adopted for the anal- ysis of the free surface flow problems usually non- linear and unsteady a special role is played by the potential flow approximation. This actually leads to a very appealing model consisting of a linear steady equation for the field and a nonlinear unsteady bound- ary condition on the free surface. The boundary inte- gral equation method is particularly appropriate for the numerical solution of such problem for two main reasons. First, when applied to linear equations it reduces by one the space dimensions of the computa- tional domain, second it provides a description of the complicated nonlinear boundary conditions more ac- curate than any other computational approach. This technique has been recently used for the numerical simulation of the wave pattern generated by the mo- 469 tion of a submerged cylinder with both linear and non linear free surface boundary conditions to], yielding in a more efficient way the same results previously obtained by finite difference in curvilinear coodinates [51. On the other hand, the viscous flow field about a submerged body and the interaction between the viscous phenomena and the free surface wave gener- ation have been usually investigated by finite differ- ence schemes of the Navier-Stokes equations in their differential form. The proper numerical approxima- tion of the boundary conditions of the free surface 7 togheter with the need to confine the computational domain are the main difficulties connected with the use OI only becnnlque. In spite of these drawbacks, some interesting results have been presented recently for the unsteady flow about a submerged body, even in the case of breaking wave conditions A. The main purpose of the present work is to analyze the possibility to generate numerical models based on a boundary integral formulation for the simulation of rotational free surface flows either viscous or inviscid, which maintain some of the capabilities briefly dis- cussed for the potential approximation. In this case the field equations either Navier-Stokes or Euler are both non linear and unsteady thus preventing the explicit confinement of the related terms into the free surface boundary conditions, as for the potential flow. In the past few years we investigated the in- tegral equation approach for large Reynolds number flows about streamlinead bodies [3] . We like here to extend the formulation to viscous free surface flows for the applications to ship hydrodynamics. Following this idea, in section 2, the Navier-Stokes equations are written in integral form by using the fundamental solution for the unsteady Stokes operator and a representation formula for the velocity vector in the field in terms of the velocities and tractions at the boundary is obtained. The numerical solution of the

relevant integral equation presents some difficulties at large Reynolds numbers, mainly related to the ten- dential singularity of the kernel. The limiting case of zero viscosity, that is the integral formulation for the Euler equation, is discussed in section 3 to better ob- serve the critical behavior of these equations and to device the proper way to recover simplified models for a quite general, but still sufficiently efficient, compu- tational procedure. To this purpose we like to point out that the great advantage of the potential models in the framework of the free surface Bows, is not as much connected to the introduction of the potential itself as it is to the decoupling between the kinematics (Laplace Eq.) and the dynamics (Bernoulli Eq.) of the problem. The Bernoulli equation which is non lin- ear and unsteady has to be solved at the free surface (if the pressure is not required) to give the so called dynamical boundary condition. A further kinematical equation is required to describe, in its Lagrangian or Eulerian form, the motion of the free surface in time. A brief review of the set of boundary conditions to be applied at the free surface for different approx- imations of the flow model, starting from the more general interface between two viscous fluids, is car- ried out in section 4 to understand the role of each of these conditions with regard especially to the coupling or decoupling of the field equations. More specifically, we like to understand how the free surface boundary conditions reduce for flow models still rotational, but simplified through a decoupling procedure analogous to the one for potential flows. We analyze this subject in section 5, where we derive directly from the Euler representation a decoupled model which recovers for the velocity the Poincare kinematical formula. The role of dynamics in deriving the proper boundary con- ditions is discussed in connection with this point. The proposed model resulting from the above procedure, is consistent and well equipped to treat the nonlinear free surface behavior. It has a value in itself, as the nu- merical results shown in section 6 for the submerged cylinder may indicate. Furtherly, starting from this model, for which we have studied the mathematical and numerical aspects of the solution, we intend to recover, through a backward procedure, the coupling aspects and the diffusion phenomena neglected at this moment. The direct connection between this model and the more general representations for the Euler and the Navier-Stokes equations, which has been de- scribed in the present paper, indicates the main lines to follow in the further investigations. 2 Integral formulation for vis- cous flows As mentioned before, in this section the Navier Stokes equations are recasted into an integral form. It is well known that the integral formulations for the Navier Stokes equations have been mostly used for study- ing the matematical aspects of viscous flows. A typi- cal example is the theory of hydrodynamic potentials which provides many important results for the lin- earized Navier Stokes equations. A detailed descrip- tion of the method, introduced by Odqvist, is given in [1], where the direct integral representation for the steady state problem is also presented. The aim of Ladyzhenskaja's analysis, is devoted to the existence and uniqueness of solution for the Navier Stokes equa- tions. It is on these theoretical topics that the inte- gral formulation for viscous flow has found their best application, due to the simple analysis of the proper boundary conditions. More recently, due to the developments of the bound- ary element methods, integral formulations have re- ceived new interest for the numerical simulation of viscous Bows. Although these methods have their best application in linear problems, the advantages of inte- gral formulations can be largely recovered in the nu- merical simulation of viscous flows about streamlined bodies when no massive separation occurs [3~. Actu- ally in this case the nonlinear source term related to the vorticity in the field, is confined to a small region in the boundary layer and the wake, thus allowing for an efficient discretization procedure in terms of finite elements. In this section the formulation, which has been previously described in [2~ in full details, will be briefly reviewed in order to present an extension to free sur- face problems. In order to introduce the integral for- mulation, for the sake of clarity, the Navier Stokes equations are given here: 2 +V2 +X = V.u = 0 P +uV2uVie (1) In equation (1) the term 11 = go is the poten- tial energy per unit mass related to the gravity force. The direct integral formulation for the system (1), as obtained in [2], gives the velocity of the fluid in terms of convolutions of the proper fundamental solu- tions which will be denoted in the following by u(k) and p(k) with the related traction given by tjk) _p(k)nj + R(~; + ~j ) = Uk(X ,t ) = | | (ujtj ) tjuj )) dsdt (2) It in p~OujUj )dsdt + | | Xjuji) dvdt| uju(k)dv ltO 470

In deriving the integral representation formula (2) from the Navier Stokes equations in differential form, the term X =u x ~ (` = V x u) has been assumed as a known forcing term; furthermore the traction vector t(k) is modified in the form i I (~::i + {1:rj) (3) where: p=P+tu2+0 (4) is the Bernoulli group. For the sake of clarity it is to be remarked the difference between the traction vector tj involved in the boundary conditions and the modified traction vector tj (tj = tj + 2u2 + II). As shown by (2) it appears that the solution u is given in terms of surface as well as volume integrals. The first surface integral gives the effect of the bound- ary values of the velocity and of the modified traction. Nonlinear terms are present in this integral through the modified tractions tj. The second surface integral accounts for the mo- tion of the boundary, which moves with velocity ZJa (normal velocity component of a geometric point be- longing to the boundary), and gives the effect of the momentum flux due to the boundary motion. This integral gives a nonlinear contribution in free surface flows for the dependence of the boundary ve- locity u~ on the fluid motion. The source of this non- linearity are always confined to the boundary as the ones acting in potential flows. Different is the case of the first volume integral which is related to the term X and accounts for the rotational effects in the fluid. This source of nonlin- earity, which directly comes from the field equation, is confined to the rotational flow region. In the gen- eral case it would require a large computational effort and it would penalize the computational procedure in comparison with other computational approaches. However for the flow about a submerged streamlined body, either attached or without a massive separa- tion, this term can be efficiently treated by finite ele- ments tecniques retaining in this way a great deal of the computational efficiency of the boundary element method. In order to complete the description of (2) the last volume integral carries the information on the initial conditions. The integral representation gives a boundary inte- gral equation for the collocation point, at which the velocity is evaluated, approaching the boundary. To account for the jump properties of the double layer kernel t(k) a factor c (= 2' for smooth boundaries) appears in the limit at the left hand side . The ob- tained integral equation is a constraint between the 471 values assumed by the velocity and the traction at the boundary. If we know one of them, for a given configuration of the domain representation (2) gives a Fredholm equation for the remaining one (either a first kind for the unknown traction or a second kind for the unknown velocity). The fundamental solutions u(k) and p(k) satisfy the equations: V . up) = 0 `5' TV U(k) _ Vp(k) _ p_ = §*e(k) which must to be considered in the sense of distribu- tion theory with fixx*) and bitt*) Dirac delta functions in space and time respectively. The explicit solution of (5) for the free space problem in two di- · menslons gives ?1( ) = biiiF t7D id (6) pot) = _ adG bit* - t) where: E = G = 2 Intr) (7) 1 * er2/4V(t*t) F = petit t) At*ty 47rp ~ (4L'(e. 2~) V2E = F where El is the exponential integral. Looking at the fundamental solutions it appears that all the terms in the integral representation that contain a derivative with respect to ok can be recasted in the form of a gradient flak = a3*) by taking the derivative with respect to ok out of the integrals. These terms give an irrotational contribution to the velocity field, while the rotational effect is related to the function F. which is not reducible to a gradient form. The function F is the fundamental solution for the heat transfer equation and it becomes sharper and sharper as the kinematical viscosity ~ goes to zero. The function E has the same behaviour as it ap- pears from the last equation in system (7~. The main difficulty in solving these equations for large Reynolds number flows is related to the crucial behaviour of the functions F and E, rather than to the evaluation of the volume integrals. In order to gain a better insight on the proper- ties of representation (2) in the case of large Reynolds numbers, we analyze the limiting case of Re infin-

ity. An integral formulation related to the differential model for inviscid flows (that is the Euler equations) is obtained in the next section. 3 The limiting case for inviscid flows The Euler equations for an incompressible fluid are obtained from the Navier Stokes equations for the Reynolds number infinity. Consequently one of the boundary conditions for viscous flow should be re- laxed, due to the lowering of the order of the partial differential equation. For this limiting case a bound- ary integral formulation is directly derived from the one valid for Navier Stokes equations, rather than from the differential model. Some introductory con- siderations on the proper boundary conditions to be applied is presented while the general discussion on the free surface boundary conditions for both viscous and inviscid models is given in the next section. As a first step we obtain the limiting expressions for the fundamental solutions. As it appears from (6) the pressure fundamental solution p(k) doesn't de- pend on the kinematic viscosity a, so that it retains the same expression as for the viscous case. This is not surprising since the pressure fundamental solution is related to the compressibility effects in the fluid, which in both cases is assumed of constant density. Looking at the limiting behavior of the function F in (7) the following relationship holds: Limo in f (X) F (X, X* ; t, t* )dV = (8) -lI(t* - t~f~x*) from (8) it directly follows the distributional limit for the function F as: F = H(t*telexx*) (9) The limiting value for the function E (note that V2G = 6(xx*) is given by V2E = H(t*telexx*) (10) Therefore we have: E = H(t*t)G (11) Finally the limiting expressions for the velocity fun- damental solutions are 472 ~ ~ ( j aziazi a~jozE ) with the corresponding traction vector given by: t(k) = _~(tt No (13) The direct combination of (12) and of (13) into the integral representation (2) readily gives the limiting formulation which is given below in vector notations with the explicit expressions of the surface integral while the remaining volume integrals are denoted by Iv u = |`n(t*) (u n) VGds + (14) o l2(t) [(n V)VGnV2G] dsdt+ Ito lift) Ha [(U V) VGuV2G] dsdt + Iv It clearly appears from (14) that the kernels in the second and third integrals have an hypersyngular behaviour when the collocation point approaches the boundary. The formulation (14) does not present a direct computational interest, but it has great interest for the comparison it provides with the viscous case showing in particular the computational difficulties to be expected for large Reynolds number. In this case in fact the fundamental solutions do not present a real hypersingular behavior, but they are very difficult to be numerically evaluated in an accurate way for their close relationship with the hypersingular ones. Moreover, the following analitycal manipulations to set eq. (14) in a better form from the compu- tational point of view will give some usefull sugges- tions for an accurate treatment of the viscous equa- tions. Following the procedure suggested for potential cows by Hsiao and Nedelec, eq. (14) can be rewritten through some known vector identities, in the form u = {n(~*) (u n) VGds+ (~15) /~0 i;n(~) (VP x n) x VGdsdt +V* x ~ ~ 21J~ (V x u) Gdsdt to en(~) V* X ~ J. L,o (U X VG`) dsdt + Iv to no) where V* = en if* . In the present form the hypersin- gular kernel doesn't appears any more and the con- tribution of the modified pressure P is closely related to that of a vortex layer of density by = VP x n. Fur- ther manipulations may be performed on (15) in or- der to show its close relationship with an uncoupled

model based on the Poincare kinematic formula as it is shown in the next section. At the present stage it is of some interest to notice that eq. (15) could be used in the numerical simulation of nonlinear free surface flows with no difficulty other than the computation of the volume integrals. It is also worthwhile to notice that, for a two dimensional case, the velocity u at any point in the field is given in terms of two scalar valued functions at the boundary, instead of the two vector valued functions appearing in (2~. This fact is strictly related to the lower order attained by the differential model for the limiting case. As a result only one inte- gral constraint is expected among the two scalars as the collocation point approaches the boundary. Ac- tually the boundary integral equation is given by the normal component of the vector formula (153. The other component gives, even on the boundary, a rep- resentation for the tangential velocity component in terms of given normal component and the modified pressure. In the next section the proper boundary condi- tions for the free surface will be reviewed for both the viscous and the inviscid model. The kinematical description of the interface will also be presented. 4 Free surface boundary condi- tions for couplet! models In both the integral formulations presented in the pre- vious sections the kinematical and dynamical parts of the flow problem are strictly coupled. The proper boundary conditions for the free surface problem in the coupled models is introduced here in some details in order to show the differences with the correspond- ing treatment in decoupled models (splitting of kine- matics and dynamics) to be introduced in the next section. The two possible kinematical descriptions of the free surface (Lagrangian and Eulerian) are also intro- duced and discussed at the end of this section. The boundary conditions which directly follow from the in- tegral formulations are treated first. Let us consider the general case of an interface between two different fluids. Two different cases are considered. In the first one both fluids are assumed to be viscous while in the second one they are considered both inviscid. For a given position of the interface the boundary condi- tions are given by the dynamical balance of the free surface and by the no slip property of the viscous flu- ids. Neglecting surface tension effect, the boundary conditions are u = uu t = to (17) where the subscript u stays for the upper fluid and the values of both members of (16) and (17) are un- known. From the integral representation (2) it follows that the boundary conditions are appropriate for the interface problem. Collocating the integral represen- tation for the lower fluid at a point belonging to the free surface, one integral constraint between u and t is enforced. Analogously the other integral equation is obtained for u';, and t". In this way at each point of the interface we have four vector unknowns and four independent vector equations that is two bound- ary integral equations and two boundary conditions. the case of no motion of the fluid acting upon the interface the problem is more simple and only one boundary condition, the dynamical one, has to be im- posed, in the form, for instance, of a known pressure distribution: t = pan (18) In this case we have one unknown (the fluid ve- locity at the free surface) and one integral equation where the value specified by (18) for the traction vec- tor is introduced. A similar procedure leads to the boundary condi- tions to be used for the formulation (15) for the second case. Due to the inviscid character of the fluids it is expected to get a discontinuity in the tangential ve- locity component across the interface while only con- ditions on the normal velocity component and on the pressure may be used In = Sun P = Pu (19) (20) We have now four scalar unknowns and four scalar equations: the two boundary conditions and the two integral equations relating the normal velocity com- ponent to the pressure. As in the viscous case, for no motion of the fluid acting upon the interface, a given pressure distribution is assigned. In order to complete the formulation, the kine- matical description of the interface has to be briefly recalled. As it is well known, for the geometrical anal- ysis of a moving surface two possible decription are available. The Lagrangian one, gives the position of the geometrical points on the surface as a function of a Lagr~ngian parameter I. Denoting by Xf this function, once the velocity of the point labeled by ~ were known, the surface geometry may be determined (16) solving the initial value problem: 473

dXf - = us x(~, o) (21) = x0(~) (ED: where Do is the set of values of the Lagrangian pa- rameter ~ and Uf is assumed to coincide with the lo- cal fluid velocity. Some problems are expected in the case of the interface between two inviscid fluids due to the discontinuity of the velocity across the interface. Some intermediate value for the tangential component should then be assumed to move the free surface. In the Eulerian description the free surface config- uration is defined by the implicit function f (x, t) = 0. In the restrictive hypotesis in which a single valued function is assumed for the free surface configuration, the geometric description is given in the more usual form z = If (x, t) . In this case the only normal compo- nent of the surface velocity (~a) is defined. By equat- ing its value to the normal fluid velocity the proper equation for the kinematic evolution of the free sur- face is attained. The two geometrical formulations are equivalent. In fact the Lagrangian description is slightly more general than the Eulerian one, in which some reg- ularity assumptions on the function f must be ac- cepted. Moreover, whenever an Eulerian description exists, the two formulations are equivalent. Only in the computations some difference appears. Actually the Eulerian description doesn't modify significantly the dimensions of the boundary elements, while the Lagrangian one concentrates the elements near the crest of the waves where sharper velocity gradients ap- pears and more computational accuracy is required. For these reasons the Lagrangian description is pre- ferred in the computational model for the nonlinear free surface problem discussed in section 6. From the previous analysis it clearly appears the simple use of the boundary conditions for free surface problems when using coupled formulations in integral form. Coupled models allow to impose the boundary conditions directly in terms of boundary velocity and tractions. The integral form allows for a simple use of the boundary conditions also in the discrete form, with no additional difficulty as in the numerical mod- els based on the differential equations. In the next section a decoupled model is intro- duced as a semplified version of the inviscid one. In this case the definition of the boundary conditions is not as direct as for the coupled models. A general kinematical repre- sentation and the dynamical boundary condition In this section a purely kinematical integral represen- tation will be recovered as a semplified version of the integral formulation for inviscid flows. It follows a direct similarity between the two representations, in the sense that boundary integral equations with sim- ilar properties are attained in the two cases. The procedure which leads from the inviscid inte- gral formulation (15) to the present one is based on the idea of a back substitution of the Euler equations in differential form into the inviscid integral repre- sentation. In this way the dynamical variables are dropped out thus reducing (15) to a purely kinemat- ical formula. In order to apply in a simple form the above procedure a quite drastic approximation is per- formed by considering the fluid domain fI to be fixed in time. A more rigorous approach should reach the same conclusion through fairly more complex calcu- lations without leading to a deeper understanding of the analogy between (15) and the present model. Un- der this assumption the surface integrals containing the boundary velocity L,a no longer appear and the representation (15) may reduce to a much simpler ex- pression. Actually, by combining the Euler equations and the vorticity transport equation `~9~ + VP + X = 0 (22) aB: + V x X = 0 (23) the following representation is attained from ea.(151 by a simple integration in time ~ ~ _ , u = ~ (u · n) VGds (~24~) isntuxn~x VGds x VGdv n Representation (24) is the well known Poincarei formula when the velocity field is assumed solenoidal. This is a purely kinematical identity which can be used to obtain boundary integral equations for the kinematics of the flow. When the collocation point approaches the boundary, two equations follow from eq. (24) by performing the tangential and normal . ~ . projections. 474

cu. + Isn upon* dS con + |6n US 45n' dS = _ l; | u"Gds+( (25) = ~ * | U?GdS + 17 (26) where It stays for the contribution of the volume inte- grals. Equation (25) is a second kind Fredholm equa- tion for the unknown us or a first kind equation with Cauchy kernel for the unknown a". Analogously for eq. (26~. The properties of the Eredholm equation are well known from the potential theory while matching properties can be found to hold for integro differen- tial one (see (~7~. As expected the equations (25) and (26) are not independent, in the meaning that, what- ever is the unknown, for a given data the solution of the first one is also a solution of the second one. When the boundary is a solid wall the value of an is assigned and us is the unknown. Different is the case of a free boundary, where an is the unknown. In this case to determine the boundary data u, the dynamical part of the model should be used. The procedure is strictly similar to that used for potential flows where the values of the potential ~ are evaluated by means of the Bernoulli's equation. In the present case, due to the vorticity in the field, which has been retained, no potential function exists. The Euler equations must be used to obtain the relation between the tangential velocity component us and the known pressure distri- bution. In order to write the required equation we shall fix a point on the free surface, labeled by the Lagrangian variable I. Due to the boundary motion, the unit tangent vector A, at the boundary point ~ will change in time. By projecting the Euler equation on this tangent vector we obtain DUr [3~ u · Dt Bt = _o (2r In order to semplify eq. (27) the second term on the left hand side is written as: u bitt = UTAH Ott +unn~ Butt (28) Noting that ~ · ~ = 0 the final expression is DUE = Anne . B~ _ 1 an _ 90?] (29) Bt 2 B~ B~ where the dynamic boundary condition P = COnSt has been used. This nonlinear evolution equation for the tangential velocity component gives the free boundary condition for the two boundary integral equation (25) and (26~. It is worthwhile to add few more comments on the relationship between the present decoupled model and the coupled inviscid one. The normal component eq. (15) is very similar to (26~. In the case of the coupled inviscid formulation the role of us is played by the tangential derivative of the dynamic pressure. A part from the integration in time the kernels are identical, so that the numerical methods for the two equations are expected to be essentially the same. The main difference between the two formulations is due to the fact that from (15) it follows only one integral equa- tion, while in the kinematical decoupled model any of the two equations (25) and (26) may be selected. This feature of the kinematical model may be used to avoid the solution of the Cauchy type equation which is more difficult from the numerical point of view. 6 The numerical simulation for a submerged body Some numerical results obtained by the formulation introduced in the previous section are presented for the unsteady flow about a submerged body. The flow field is assumed to be irrotational and the nonlinear free surface configuration is followed in time by the Lagrangian description. The evolution of the wave pattern generated from rest by a moving cylinder is simulated and the steady state configuration is reached in the case in which no breaking wave occurs. The numerical procedure adopted makes use of the dynamical equation (29) end of the kinematical equa- tion (21) to evalute us on the free surface and the free surface configuration respectively. The following notations are used: [FIB and [Q' denote the body boundary and the free surface respectively. [LOO de- note the fictitious boundary used to cut the computa- tional domain at a finite distance from the submerged body. When treating the flow in a channel, the fi- nite depth bottom is denoted by [Qb while ~QOO is given by an inflow part (Bali) and an outflow part (AGO). As a boundary condition to be used in the boundary integral equation, the normal inflow veloc- ity, time dependent in general, is be assigned on [Eli. The zero normal velocity condition is assigned on the bottom boundary for both the channel flow or the infi- nite depth case. At the outflow boundary the normal velocity component is also given or a suitable radi- ation condition is applied. On the free surface the tangential velocity component is evaluated by the dy- namic equation (29). The reference frame is assumed connected with the body, which is moving at a costant depth beneath the indisturbed free surface. The dy- namic equation is used in the inertial frame connected to the indisturbed fluid and the proper value of us is evaluated by changing the reference frame to that 475

fixed to the body. As shown in the previous section any of the two boundary integral equation either (26) or (25) can be used to determine the unknown velocity compo- nent. The most efficient choice, from the computa- tional point of view, is the use of the second kind EYed- holm equation, in order to avoid the variational tec- niques required for an accurate solution of the Cauchy type equation. To this purpose a mixed approach is used here by assuming the normal component (26) to hold on the free surface and the tangential com- ponent (25) on the other boundaries. The equation which follows from this mixed formulation does not present a Cauchy type singularity and can be easily solved by numerical methods. In the case of multi- connected regions, some attention must be devoted to the eingenvalue related to the circulation around the body which in the present numerical study will be assumed to be zero. In the computational procedure standard finite el- ements tecniques are used to discretize the equation. In more details, piecewise linear shape functions are used to describe the geometry while piecewise con- stant functions are used for the unknown. In order to assure the solution uniqueness the condition of zero circulation around the body is imposed. The com- putational procedure is splitted in two parts. The boundary unknowns are evaluated first by solving the boundary integral equation. With the velocity dis- tribution on the free- surface completely known the configuration of the domain and the values of us are updated. The second order accurate Adams Bash- forth scheme is used to perform this calculation. No redistribution procedure is used for the free surface panels which tends to concentrate near the crest of the waves. Due to the mean motion, at each time step a new panel is added at the inflow side and one is canceled at the outflow, where a Sommerfeld radi- ation condition is used. By the described numerical tecnique the simula- tion of the transient wave system due to the motion of a circular cylinder has been performed. The ob- tained results are compared with those presented by Haussling and Coleman [5] and with those of Ligget and Liu [6] for the potential flow equation. Haussling and Coleman use a finite difference approach in gen- eralized curvilinear coordinates and Ligget and Liu a boundary element approach. Both use an Eulerian description for the free surface. Although these ap- proaches are completely different the three solutions show a good agreement (fig. 13. It could be noted that the present solution shows clearly the effects of non- linearity given by the adopted Lagrangian description of the free surface. In fact, with the present method a steady state solution couldn't be reached due to the steeping of the wave crest, which cause the solution to break before the last instant showed by Ligget and Liu. In the case, shown in fig. 3, although the Froude number is higher than in the previous one, a steady state solution has been attained due to the greater submergence of the body. Fig. 4 shows the time his- tory of the forces acting on the cylinder. Although the free surface configuration appears to reach the steady state, both lift and drag are oscillating around the presumed steady value. In fig. 8 a simulation with a smaller channel depth shows the evolution towards the breaking of the wave, which is attained by an over- turnig of the front face of the first crest behind the body. 7 Concluding remarks A computational method for unsteady free surface flows has been derived from a mathematical model which decouples the kinematical and the dynamical aspect of the flow field. The method, valid in gen- eral for rotational flows, has been shown to have the same computational efficiency of the potential based approaches. The strict relationship with the coupled models based on the integral formulation for viscous and inviscid flows, have been discussed. While the numerical solution of the Fredholm integral equation gives very satisfactory results, the numerical tecnique for an efficient as well as accurate resolution of the Cauchy type equation is to be completed. This is the crucial point for the numerical implementation of the limiting coupled formulation for inviscid flows. The viscous model, should be treated in similar way by re- casting the equation in terms of vortex layers as shown for the inviscid case. References t1] Ladyzhenskaja, O.A., (1986~: The Mathematical Theory of Viscous Incompressible Flows, Gordon & Breach, New York, NY, USA. t2] Piva, R. and Morino, L., (19873: "Vector Green' Function Method for Unsteady Navier Stokes Equations", Meccanica, Vol. 22, pp. 76-85. t34 Piva, R., Graziani, G., and Morino, L., (1987~: Boundary Integral Equation Method for Un- steady Viscous and Inviscid Plows", Ed.: T.A. Cruse, Advanced Boundary Element Methods, Springer Verlag, New York, NY, USA. [4] Miyata H., Sato T., Baba N. "Difference Solu- tion of a Viscous Flow with Free Surface Wave about an Advancing Ship", J. Comput. Phys., 72 (1987), 393 476

[5] Haussling, H.J., and Cole- man, R.M., (1977) :~ Finite-Difference Computa- tions Using Boundary Fitted Coordinate Sys- tems for Free-Surface Potential FlowsGenerated by Submerged Bodies~, Eds.: J.V. Wehausen and N. Salvesen, The Procedings of the Second Inter- national Conference on Numerical Ship Hydro- dynamics. [6] Liu, P.L-F. and Ligget J.A., (1982~: "Applica- tions of the Boundary Element Method to Prob- lem~ of Water Waves", Eds.: P.K. Banerjee and R.P. Shaw, Developments in Boundary Element Methods - 2, Applied Science Publishers, Lon- don. [7] Casciola, C.M., Lancia, M.R., and Piva, R., (1989~: "A General Approach to Unsteady Flows in Aerodynamics: Classical Results and Perspec- tives~, ISBEM 89, East Hartford Usa. [8] Nedelec, J.C., (1977~: " Integral Equations with Non Integrable Kernels, Integral Equations and Operator Theory, Vol. 5, pp. 562-572. [9] Hsiao, G.C., (1986~: " On the Stability of Integral Equations of the First Kind with Logaritmic Ker- nels", Archive for Rational Mechanics and Analy- sis, Vol. 94,2 pp.179-192, Springer Verlag, Berlin. t = 6. t = 7.5 · · . ~0a . t = 9 Fig. 1. Free - surface configurations after the start of a submerge cylinder. Submergence h = 2. Fr = .566 D = 1 _ present solution ° Ligget and Liu · Haussling and Coleman 477

0.150 0.100- C. 050- , , . . . I.Io 1.90 9.00 9.10 t.20 t.30 Fig. 2. Steeping of the wave for the same case as in Fig. 1. t = 10. o t= 15 t30 t = 45 . ~ 0 t=60 Fig. 4. Configurations of the computational domain for the some case as in Fig. 3. Fig. 3. Free surface configuration vs time Fr = .95 D = 1 Submergence h = 3 Dt = .25 t= 75 0 t = 80 a o t = 85 o t = 90 478

/ 1 1 ,~\ - - ~ Fig. 7. Pressure distribution on the cylinder. Same case as in Fig. 5. t = 85 l.tO- Ill~ 8'll. _' I! ~ ~~- t. 000 5 11 15 28 25 30 15 40 10 20 ]' 41 51 11 T' Fig. 5. Wave elevation. Fig. 6. Time History of the forces on the cylinder. Fr = .95 Same case as in Fig. 5 (forces adimentionalized by Submergence h = 3. pu2D) Dt = .25 t = 85 ~ _ _ o Fig. 8. Wave overturning. D = 1 Fr = .95 Submergence h = 3 Channel depth ~ = 4 t= 14.25 Dt = .25 479