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A Boundary Integral Approach nntive Variables for Free Surface Flows C. Casciola (I.N.S.E.A.N., Italy) R. Piva (Universita di Roma, Italy) ABSTRACT The boundary integral formulation, very efficient for free surface potential flows, has been considered in the present paper for its possible extension to ro- tational flows either inviscid or viscous. We analyze first a general formulation for unsteady Navier Stokes equations in primitive variables, which reduces to a representation for the Euler equations in the limiting case of Reynolds infinity. A first simplified model for rotational flows, ob- tained by decoupling kinematics and dynamics, re- duces the integral equations to a known kinematical form whose mathematical and numerical properties have been studied. The dynamics equations to com- plete the model are obtained for the free surface and the wake. A simple and efficient scheme for the study of the non linear evolution of the wave system and its interaction with the body wake is presented. A steady state version for the calculation of the wave resistance is else reported. A second model has been proposed for the simulation of rotational separated regions, by coupling the integral equations in velocity with an in- tegral equation for the vorticity at the body boundary. The same procedure may be extended to include the diffusion of the vorticity in the flowfield. The vortex shedding from a cylindrical body in unsteady motion is discussed, as a first application of the model. INTRODUCTION One of the most successful approaches for the analysis of free surface flow problems is given by the boundary integral equation method. In particular, if the governing equations are linear, as it is for the potential flow approximation, this approach reduces by one the space dimensions of the computational domain. Moreover, it provides a description of the boundary conditions (which are usually non linear and unsteady) more accurate than any other compu 22 tational model. As a matter of fact, the boundary integral equation method, together with some specific techniques introduced to linearize and discretize the kinematic and the dynamic boundary conditions at the free surface, leads to an extremely efficient com- putational methodology for the evaluation of the wave resistance and of the overall potential flow field, e.g. see A. However more realistic flows always contain re- gions of vorticitY different from zero. which after being generated at the body wall, usually remains confined in a narrow region close to the body itself and its wake, provided the Reynolds number is sufficiently high. The relevance of the rotational flow may be en- hanced by large separated regions about bluff bodies, by the interaction of the wake with the free surface or with other solid bodies (e.g. the propeller), or by a larger effect of the diffusion for moderate values of the Reynolds number. In all these conditions, where either confined or large vertical regions appear, the inability to intro- duce the velocity potential prevents from using the very efficient model previously mentioned. In the for- mer case (i.e. confined vertical regions) the overall picture of the flow field does not change much with respect to the potential one, so that the classical sin- gular perturbation approach, that is a boundary layer- external solution interaction model, may give suffi- ciently accurate results. In the latter case (i.e. large vertical regions) any kind of external flow field cor- rection becomes inefficient and the direct field dis- cretization of the Navier Stokes equations (or a sim- plified version of them, e.g. parabolized) seems to be the only available approach for practical applications A. Anyhow in both cases we have to deal with more complicated techniques, which require a larger com- putational effort. In particular, extending the effect of viscosity to the entire flow field, we may even spoil from a numerical point of view the wave pattern sim- ulation at the free surface, with respect to the much

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simpler potential flow model. It is reasonable at this point to try to answer the following questions: in which condition is it possible and convenient the extension of the boundary integral method to rotational flows? How large is the loss of efficiency due to the presence of non linear terms in the equations? In fact, these terms give rise to field integrals and cancelling one of the main advantages of the boundary integral formulation for potential flow (that is the only presence of boundary unknowns). Purpose of this paper is to give some prelimi- nary answers to these questions by illustrating a gen- eral boundary integral formulation for viscous flows in primitive variables (velocity components and pres- sure) which reduces to a representation for inviscid flows in the limiting case of Reynolds infinity. The theoretical analysis, developed by the authors in pre- vious papers t3, 4, 5] and briefly summarized for the reader's convenience in Section 2, is successively ap- plied to generate a set of computational models which are increasingly complicated as long as they become suitable to deal with flows presenting more relevant vertical regions. A first group of models is obtained by decoupling the kinematics from the dynamics in the integral rep- resentation which holds in the limit of zero diffusion. By doing so, we recover the purely kinematical inte- gral representation for the velocity vector, known as Poincare formula, valid also for rotational flows A. The dynamical part of the equations, still in differ- ential form, gives rise to auxiliary conditions for the free surface and for the wake. The generation of vor- ticity and its release from the body, in the classical case of sharp trailing edge, is assured by the enforce- ment of a "Kutta-type" condition, which accounts for the local viscous phenomena, as explained in details in Section 3. A further kinematical equation is required to account for the unknown position of the field dis- continuity given by the free surface or by the wake. The above assumptions provide a very simple and efficient model, able to analyze rotational un- steady flows and well equipped to treat the non linear free surface behaviour. Several computational results obtained by this method t3] are reported in Section 3. Also a steady state linearized version of the model which resembles, in terms of velocity, a classical po- tential flow model t11 largely used for the wave resis- tance calculation, is reported in Section 3, in order to show the versatility of the present approach and its capability to reproduce the most interesting positive features of the existing methods. A further step in the direction of a complete sim- ulation of the rotational flow, is attempted by includ- ing the diffusion phenomena, neglected in the previ- ous group of models, as a first order effect acting over the inviscid solution. In fact, without introducing the boundary layer equations, the viscous effects are re- covered by the boundary integral formulation for the vorticity transport equation, given in Section 2. As- suming, as in the first order boundary layer theory, that the pressure does not change normally to the body wall, we combine its value from the inviscid son lution is combined in order to obtain an integral equa- tion in the wall vorticity. This procedure, described in Section 4, allows to detect, within the limits of the approximation, the separation point along the wall, hence the position and the intensity of the issuing vortex layers for the simulation of the rotational wake region. Finally the boundary integral equations and the computational procedure for a complete model are briefly outlined in Section 5. By a complete model we mean a model in which dynamics and kinematics are fully coupled and the same viscous fundamental solutions together with the related integral represen- tations are considered in the entire flow field. More specifically the concept of interacting external and in- ternal solutions, which is typical of perturbation tech- nioues. is not adopted here. The close relationship between this model and the previous ones may be of great help to overcome the numerical difficulties mainly due to the kernel of the integral equations, which becomes highly singular for increasing values of the Reynolds numbers. The numerical results con- cerning this model are still in progress and are going to be presented in a further paper. A GENERAL FORMULATION OF FLOW PROBLEMS IN TERMS OF BOUNDARY INTEGRAL EQUATIONS The integral formulations of the Navier Stokes equations have been mainly used for studying the mathematical aspects of viscous flows. A detailed de- scription of the method is given in the book of La- dyzhenskaya t8], where the integral representation for the steady state problem is also presented. More re- cently integral formulations have received new interest for the numerical simulation of viscous flows. In par- ticular the authors investigated the flow about stream- lined bodies when no massive separation occurs A. The analysis of the boundary integral equation given in full details in a previous paper t10], is briefly summarized here for completeness. Besides, an inter gral representation for the vorticity is proposed for its relevance to the solution procedure in the case of rotational separated flows. a. The velocity representation for viscous flows We consider the case of the undisturbed fluid in uniform translation with constant velocity UOO with respect to the body frame of reference. The absolute 222

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velocity is expressed as u = uOO + u, where u is a per turbation velocity. We also introduce a perturbation stagnation pressure P + ~ pU2-pOO-~ pU2 with so that at infinity P = 0. Introducing these quantities the governing equations become V u=0 (1) ,uV2u - VP _ p flu = X (2) where the right-hand side term X contains the nonlin- F ear terms and the gravity force X = -pu x (+ pg Notice that with the above expression of the v2E equations, the non linear terms, which give rise to a field integral, have to be accounted for only in the regions where the vorticity is not negligible. The integral representation of the velocity as a solution of the system (1, 2) is given by t4] Uk(X*,t*) = ~ ~ (Ujt; )-tjU; )) dSdt -To Jan PVaUjUj dSdt (3y + ~ ~ Xjujk)dVdt-~ pusujk)dVdt~ where the stress vector t is modified to include the dynamic pressure. As shown by (3) the field veloc ity is given in terms of surface as well as volume in tegrals. The first surface integral gives the effect of the boundary values of the velocity and the modified traction. The second surface integral gives the effect of the momentum flux due to the boundary motion. In free surface flows, this is a non linear contribution, because the boundary normal velocity component to is strictly dependent on the fluid velocity field. This source of non linearity is localized on the free surface as in the case of potential flow. The first volume in tegral, related to the term X, accounts for the body lotion become t5) force and for the vorticity effects in the fluid. This source of non linearity is within the field equations, and in particular is connected to the rotational flow region, which may be more or less confined, depending on the flow field. Finally the second volume integral gives the effect of the initial conditions. The funda mental solutions uji) and tjk) are given by t4] u(k) = [jkF- `' ~(4) p(k) = _ 49G bit* - t) tj ) = -p( )ni + ~ ( ~ + ~) ni where for the two-dimensional case G = -lnr 2,r 1 -'2/4 (t -t) 47rp 1 (4~(t, t) ) and E1 is the exponential integral. b. The inviscid flow as limiting case (6) We consider now the limiting case of the previ- ous theoretical formulation as the Reynolds number goes to infinity, i.e. as ~ goes to zero. In fact the pa- rameter L,t could be more appropriate as we can see from the expressions of the two functions F and E ap- pearing in the fundamental solutions. We can easily see that their distributional limit is given by F = - -H(t*-t)~(x*-x) E = - -H(t*-t) G(x*-x) and the equation V2E = F reduces to V G = b(x*-x) (7) Therefore the expressions of the fundamental so u~k) = _-(t*-t) (0Cii ~z`02` Ba jade) t(k) = LOG bit*-t)nj (9) Combining them with (3) we obtain the repre- sentation valid for rotational inviscid flows, which in vector notation reads 223

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( ' ) Man ( ) + lo Jan p [(n V)VG-nV2G] dSdt -Jordan va [(u V)VG-uV2G] dSdt (10) + lo In (V* x V*G x X) dVdt -In(V* XV*GxuO)dv where the vector identity (b ~ V)VG-bV2G = V* x V*G x b (11) has been used in the expressions of the volume inte- grals, and uO is the initial velocity. In order to keep the vector notation, when study- ing a two dimensional flow, we consider a cylindrical body in a three-dimensional space. In particular we assume a local set of orthonormal coordinates defined at each point of the boundary by the unit tangent vector ~ (anti-clockwise), the unit normal vector n (external to the fluid domain) in the cross section and k = ~ x n. The integral representation (10) corresponds to the differential model for inviscid flows given by the Euler equations. c. Analysis of the boundary integral equation The integral representation (3) for viscous flow gives, for x* going to the boundary, an integral equa- tion which is a constraint between the values assumed by the velocity and the traction at the boundary. In the limit a factor c (= 2 for smooth boundaries) ap- pears at the left-hand side to account for the jump properties of the double layer kernel t(k). If either the velocity or the traction is assigned as boundary con- dition, we obtain an integral equation of first kind for the unknown traction or an integral equation of sec- ond kind for the unknown velocity respectively. Usu- ally for free surface flows about submerged bodies we have a mixed-type boundary condition, that is the traction is assigned at the free surface and the velocity at the body wall. This exactly resembles the poten- tial flow formulation that for the same physical case requires Neumann and Dirichlet boundary conditions for the body and the free surface, respectively. At increasing values of the Reynolds number the kernel of the integral equation tends to become sin- gular, in any of the described cases, as shown by the expressions (4) and (5) of the fundamental solution u(k) and t(k). Actually, the functions F and E appear- ing in these expressions, become sharper and sharper as the kinematical viscosity goes to zero. The main difficulty in solving directly the integral equations for large Reynolds number flows is essentially related to the crucial behaviour of these functions rather than to the presence of the volume integrals. A deeper insight on the properties of these equa- tions for large Reynolds numbers, is provided by the analysis of the limiting case of zero diffusion. It ap- pears from (8) that the kernel u(k) has a hypersingu- lar behaviour when the collocation point approaches the boundary. It follows an interesting comparison with the viscous case, showing the computational dif- ficulties to be expected asymptotically for increasing values of the Reynolds number. In particular u(k) is composed of two terms which are both singular. The first one is the well known hy- persingular term which appears in the double layer representation of the velocity potential for the Neu- mann problem t11~. The second one is a Dirac delta function on the boundary itself. By combining the two terms together with a few vectorial identities and the Stokes theorem for a closed surface t11,12], the second surface integral may be re-expressed in the form (VP x n) x VGdS (~12) an which corresponds to a vortex layer of density By = (VP x n) with P = lot* Pp aft. Moreover the kernel does not show now the same singularity as in the original form. By similar manipulation through known vecto- rial identities the third surface integral may be re-set in the form V Xi ~ 2va(V X u)GdSdt * 0 an (13) of* ~ v* x I J VatU X VG)dSdt 0 an which is not presenting particularly attractive features with respect to the original form. The volume integral containing the non linear convective term, through an integration by parts may be written as rt* ~ / / V* x V*G x XdVdt = so an (14) V* x ~ G x (V x X)dV - V* x ~ Gn x XdS with X = Jo* Xdt. By the same procedure, the volume integral con- taining the initial velocity term gives -~ (V* x V*G x us) dV = -V* x ~ G`~odV + V* x ~ Gn x UodS n an (15) Notice that in this form the initial term integral is not extended to the entire flow field, but it gives a 224

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contribution only in the rotational region and along the boundary. It is worth to notice that the representation (10) of the velocity vector, for x* approaching the bound- ary, gives only one integral equation, when considering its normal projection. This fact is strictly related to the lower order of the corresponding differential equa- tion for the inviscid case. Hence, only one integral constraint is expected among the two scalar quanti- ties (u n) and P. as the collocation point approaches the boundary. Namely, we obtain a Fredholm integral equation of second kind if P is assigned and a first kind Cauchy-thype integral equation if the boundary condition prescribes the normal velocity component. Also in this case, for the problem of the free sur- face flow about a submerged body we have a mixed- type boundary condition and the resulting system of discretized equations has to be carefully analyzed in order to determine the mathematical properties of the corresponding integral operator A. d. The integral representation for vorticity In some of the models we are going to discuss in the next sections, we consider also the integral rep- resentation for vorticity in regions close to the body. Let us write the vorticity transport equation in the form \7xx (16) where the nonlinear terms are taken as a source at the right-hand side of the equation. The integral repro sensation for the solution of (14) is given, in the case of a fixed fluid domain, by ((x*,t*) - J-o /onL,(~8n-F3~)dSdt /o /n (V x x) FdVdt-/n (oFdV (17) where F is the fundamental solution of the diffusion equation given by (6) for the two-dimensional case. We notice that this representation is not independent from the representation (3) for velocity. Actually it can be obtained by performing the curl of (3) and by accounting also for the expression of the stress vector t in terms of its normal and tangential components t15] t = -pn+,u~xn (18) which is valid for a traslating rigid body. For the collocation point x* approaching the boundary we obtain an integral equation which is a constraint between the values of ~ and ~ at the bound ary. No simple boundary conditions are available for ~ or ~ and the latter is usually approximated by using the intensity of the vortex layer at the body boundary and by assuming a reasonable model for the diffusion of vorticity t13, 14~. For ~ assigned as known bound- ary condition, a second kind integral equation for ~ is obtained. The kernel &oF still presents some computa- tional difficulties for large Reynolds numbers. THE DECOUPLING OF KINEMATICS AND DY- NAMICS: A MODEL FOR ROTATIONAL INVIS- CID FLOWS Let us consider the representation of velocity (10) combined with the new expressions (12) and (13) of the surface integrals, for the analysis of inviscid flow fields. The representation contains in its terms both the kinematical and the dynamical aspect of the phys- ical model, as clearly shown by the two unknowns, namely the normal velocity and the stagnation pres- sure. A coupled integral formulation like (10) allows for a straightforward enforcement of the boundary condition in terms of the boundary velocity and pres- sure. In addition their discretization does not imply any further difficulty as in the numerical models based on differential equations. Therefore a coupled integral formulation would be ideal for the simulation of free surface flow fields, where a particular attention has to be paid for a simple and accurate application of the boundary conditions. However, the presence of some computational difficulties (mainly related to the calculation of the surface integral (13) and the vol- ume integral (14~) suggested to still follow the classical path of decoupling the kinematics from the dynamics, as successfully experienced by all the potential flow models. a. The Poincare formula for kinematics A purely kinematical integral representation is obtained by eliminating the dynamical variables through a back substitution of the Euler equation and of the vorticity transport equation (in their differential form) into the integral representation (10) for inviscid flows. For the sake of simplicity, we apply first the above procedure to a fluid domain fixed in time, so that the integrals which account for the free surface motion are dropped out. We will see later that their inclusion, is not going to modify the final result, al- though it complicates significantly the overall pro- cedure. After introducing the new integral expres- sions (12), (14), (15) and combining with the vorticity transport equation for inviscid flow integrated in time 225

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5^-S^o = If X X (19) 2 ur + ion UT[,n.dS = (24) the integral representation (10) becomes ut *,t* = VP x n x VGdS ) fan ( ) +J&.n (u n) VGdS-V* x ~ G`dV (20) +V* x / An x (us-X) dS time Combining now the Euler equation integrated in (u - us) ~ X = -VP (21) we obtain a kinematical representation for the velocity vector u(x*,t*) - -[ (u x n) x VGdS+ + },.,ntu n)VGdS-V* x ~ G`dV (22) where on the right-hand side both u and I, as function of time, are given for t-t*. The velocity representa- tion (22) is the well known Poincare formula usually written as [61 U(:l:*) = V* t/n(V · u)GdV-/ (u n)GdSlJ -V* X t/n(V X u)GdV + / (u x n)GdS) (23) which is a velocity integral representation satisfying only the kinematical equations V x u = s. and V u = Q In the present case Q is identically equal to zero. Let use underline the fact that the splitting be- tween kinematics and dynamics, inherent to the clas- sical velocity potential formulation, is here recovered by recombining the Euler and the vorticity transport equations with the original coupled formulation. In order to verify the equivalence with the Poincare for- mula also in the case of a moving boundary, as is the case for the free surface, it is more convenient to op- erate in a reverse way, that is to differentiate in time equation (22), accounting for the variation in time of part of the boundary an. A brief note on these cal- culation is reported in Appendix A. Finally we deduce the boundary integral equa- tions which follow from (22) by performing the tan- gential and the normal projections for the collocation point 2:* approaching the boundary (assumed smooth) 226 ; unGdS+I~T 2 an + [n un en* dS- (~25~) + 2, * ,/; urGdS + Itn where I`' and I`n give the contribution of the volume integrals in the two projections, respectively. Equation (24) is a second kind Fredholm equa- tion for the unknown or or a first kind integral equa- tion with a Cauchy type integral (the kernel a8G is singulars for the unknown an. The opposite is valid for equation (25~. The two integral equations are com- pletely equivalent in the sense that if you solve the first one, the solution will satisfy also the second one. The choice may depend on the assigned boundary condi- tion and on the preference about the numerical tech- nique to be used. b. The dynamics of the free surface For a solid wall the value of un is assigned and u, is the unknown. Instead, for a free boundary an is the unknown, and the dynamical part of the model should provide the boundary value or. The procedure parallels exactly the one used for potential flows where ~ is the unknown and the values of the potential ~ are evaluated by means of the Bernoulli equation. In the case of rotational flows, the Euler equations must be used to relate the tangential velocity component to the pressure distribution assigned as boundary condition. Notice that the coupled representation (10) contains in it the pressure and no additional dynamic condition would be required. We write the Euler equation for a point of the free surface, labeled by the Lagrangian variable I. The tangent projection of the Euler equation on the free surface at point ~ gives Du ,__! ~P_93'7 (~26) where ~ has the usual meaning of free surface elevation and If is equal to zero for the boundary condition of assigned constant pressure. The unit tangent vector at point ~ changes in time for the boundary motion. Therefore Du D(u ~ Dr _ . ~ = -u Dt Dt Dt and, for r. DD' identically equal to zero, we finally obtain

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DOT DT Brl _ = unn -- 9- The non linear evolution equation (27) for the tangential velocity component gives at each time the required boundary value Or for each of the two equa- tions (24) and (25~. We finally need to complete the formulation by means of a Lagrangian description in time of the fluid interface. Denoting by Xf((,t) the position of the geometrical point, labeled by I, of the free surface the new interface geometry is determined by solving the initial value problem ``' = Of X'(6,O) = Xo(~) ~ ~ D' (28) where DO is the set of values of the Lagrangian pa- rameter ~ and U' is assumed to coincide with the local fluid velocity u. c. The dynamics of the wake The kinematical representation (22) expresses the field velocity as function of its boundary value and of the field vorticity both at the present time t*. Hence, we have to determine the distribution of vor- ticity by adding a dynamic equation. For instance, the vorticity transport equation for incompressible invis- cid flows in 2-D provides the very simple result of con- stant vorticity along the motion. We consider in this . . . . . . . . . . . section the physical case of very large Reynolds num- ber flows about streamlined bodies with sharp trailing edge, which do not experience any boundary layer sep- aration [16~. We may simulate these conditions by the zero diffusion model with a vertical wake downstream of the body. These wakes (or free vortex sheets) are given by surfaces of discontinuity characterized by the fact that both pressure and normal fluid velocity are continuous across them, while the tangential compo- nents of velocity may admit a jump, that is a concen- trated vorticity Ok = [u] x n _ [trek (29) where ~ ~ is the symbol for the jump across the discon- tinuity surface. The volume integral It appearing in (22), if the field vorticity is only concentrated on the wake, may be expressed as It =-V* x / [Ur]Gds (30) 0. where on is the wake surface. Similarly to what has been done for the free sur- face, we introduce a dynamic equation to study the evolution of furl, which is given by t17~. DW (Jluri) = 0 (31) where w is the velocity of a point ~ of the wake, DO is r271 the material derivative along the wake motion and J = |~| is the Jacobian of the trasformation xu, = xw(6,t) which gives at each time the position of the point ~ belonging to the wake. Equation (31) is equivalent to state that JO Judd = const. (32) along the motion, which has the physical meaning of conservation of concentrated vorticity for a portion (~! < ~ < 62) of the wake. The initial value ATE of the vortex layer intensity at the trailing edge is taken to be the limit ATE = km I.UT(X+) + Ur(X_~l (33) I+-TO I_ _TE where :z:+ are points on the upper and lower side of the body and uric+) the corresponding tangential compo nents of the velocity. In a sudden start the value of ATE decreases in time asymptotically to zero when the body reaches the steady state. The equation (31) plus the initial condition (33) is called a Kutta-type con dition, because at steady state t satisfies the classical Kutta condition of zero vorticity at the trailing edge. To describe the evolution of the geometrical con figuration of the wake we adopt a Lagrangian model completely analogous to (28) where w = (u+ +u_~/2. Ago = w (28) d. Comparison with potential flow models We describe now the solution procedure for the model consisting of the integral equations (24) or (25) plus the dynamics contribution for the free surface (eq. 27) and for the wake (eqs. (31) or (32~) to which the evolution equation (28) has to be added. A com- plete theoretical analysis for the solution of the two equations (24) and (25) has been recently performed [10] for the case of flow past bodies, that is for as- signed normal velocity component on the boundary. The same would be for assigned tangential component all over the boundary. The contemporary presence of free surface and body complicates the analysis, be- cause we have a mixed-type boundary condition, that is un assigned at the body wall and a' at the free surface. Let us recall first the main findings of the previous analysis for the case of uniform boundary condition. We consider here only the second kind Fredholm integral equation (i.e. the tangential component (24) for the unknown ur or the normal component (25) 227

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for the unknown an). This formulation exactly corre- sponds to a Neumann internal problem for potential flow with a simple layer representation. A complete equivalence with regard to existence and uniqueness of the solution and compatibility conditions, holds. Namely, let us consider for instance equation (24), the compatibility condition for the right-hand side An* | '9r. ./an UnGdS + Itr] dS* = 0 (34) is identically satisfied, giving the existence of the solu- tion, for any assigned normal velocity at the boundary and for any distribution of vorticity in the domain Q (notice that It gives the velocity induced by the field vorticity whose circulation is identically zero be- ing the vorticity only external to an). Moreover, it is known from the potential theory that the solution is not unique and it may be expressed in the form Ur = Ur + OKUT (35) where uP is a particular solution and up is the eigen- solution, that satisfies the homogeneous equation for the Neumann problem. The solution for the homoge- neous problem is given by the simple layer density for the related Robin potential, for which the property J;n us dS ~ O holds. By imposing the further condition of conserva- tion of total vorticity, we find the circulation ~ around the body, as far as we know by (32) the vorticity is- sued at each time on the wake. Finally from the value of I' we can find the constant c' appearing in (35) and determine uniquely the solution or. As we said before, we do not need to consider here the singular integral equations of first kind which may require more sophisticated numerical techniques t12~. In fact, a mixed approach is used, that is we assume the normal component (25) to hold on the free surface and the tangential component (24) on the other boundaries. Altough a deeper theoretical analysis would be required to study the mathematical properties of the corresponding integral operator, we experience a nice behaviour of the discretized equa- tion from a numerical point of view. The above illustrated solution procedure is sim- ple enough to be comparable, for both theoretical and numerical aspects, to the classical one for potential flow. By the present procedure we may not only re- produce the results obtained by potential flow mod- els, but also obtain an immediate extension to rota- tional inviscid flows, with vorticity confined in wakes or blubs. Moreover with regard to the free surface, the present model provides a very efficient unsteady tech- nique to study the non linear evolution of the waves. Few sample numerical results are reported. The transient wave system due to the motion of a sub- merged circular cylinder, previously discussed in t3] is here reported as a test case for the accuracy of the model. The numerical results shown in Fig. 1, com- pare favourably with other classical results 18,19. ,' ~ ~ ~ ~ ~ . . . .... Figure 1: Free surface configurations after a sudden start of a submerged cylinder. Submergence h = 2, Fr = .566, D = 1, t = 4.5,6,7.5,9. solution, ~ Ref. 18, ~ Ref. 19 - present A more interesting case is given by the unsteady motion of a slightly submerged lifting airfoil. After a sudden start of the airfoil the vortex layer shedding from the sharp trailing edge, interacts with the free surface giving rise to a wave pattern which feels the influence of the wake vorticity. The airfoil at three different values of the angle of attack is shown in Fig. 2. 228

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a" =- ~ ~(38) Figure 2: Free surface and wake configurations af- ter a sudden start of a lifting airfoil. Submergence h-.75 * chord, Fr = 1 (with respect to the chord). Angle of attack 0°, 10°, 20°. t = 3 e. A steady linearized free surface condition To complete the comparison with potential flow models, we briefly discuss now the application to steady linearized boundary conditions at the free sur- face, frequently used for an efficient calculation of the wave resistance. Equation (27) which describes the dynamics of the free surface, for steady state and a zeroth order linearization reduces to UOO ~ r = - 9,90 (36) considering that the unit tangent vector r in this ap- proximation is constantly aligned with the undisturbed velocity UOO- The fluid interface motion given by the Lagrangian description (28) is here conveniently ex- pressed through its linerized Eulerian form UOO Bt7 = u" (37) Combining (36) and (37) implies the well known Neumann-Kelvin condition usually written in terms of the velocity potential. In- troducing (38) into the right hand side integral of the equation (24) and integrating by parts we obtain Uoo ~ / BurGdS = _Uoo ~ at/ Ur3GdS 9 Br* an' B~ 9 Br* an' B~ (39) with ~Q = aQb + aQf where b and f stay for body and free surface respectively. Combining with (39), the integral equation (24) becomes 2 r + Jan u, fin* dS + 9 Or. fan us ~, dS = ~ DIG dS + I (40) This integral equation gives, for assigned normal ve- locity at the body boundary oaf, the tangential ve- locity component on the entire boundary ~Q. Once u, is known, we easily obtain the free surface eleva- tion ?7 by integrating equation (36~. The tangential derivative of the free surface integral in (40) is dis- cretized by the upwind finite difference scheme used in the potential flow model proposed by Dowson A. The steady linearized version of the present model exactly reproduces the numerical results pb- tained by the above mentioned potential model. For instance, the wave pattern generated by a submerged lifting airfoil is shown in Fig. 3, for a zero contribu- tion of the term I`T. However, the present tecnique may include some rotational effects relevant to steady state conditions. ODD ~D ~4 Figure 3: Wave elevation in a steady problem for a NACA0012 airfoil. Submergence h = 1 * chord, Fr = .5 (with respect to chord). Angle of attack 10°. Trailing edge position x = 4.5, y = 229

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A VISCOUS-INVISCID INTERACTION MODEL FOR SEPARATED FLOWS The model described in the previous section is appropriate for the simulation of inviscid rotational flows with either a field vorticity whose initial posi- tion and intensity are assigned or a vortex sheet is- suing from a sharp trailing edge whose intensity is determined by a Kutta-type conditions. Therefore, the vorticity generated at the body wall may be ac- counted for, always in the limit of Reynolds number going to infinity, but it is restricted to flows where no separation occurs. Actually, the diffusion phenom- ena were neglected everywhere in that model if we exclude, in a sense, the unsteady generation of the wake by the Kutta-type condition which provides the limiting behaviour of a viscous fluid. A further step in the direction of a complete simulation of rotational flows (i.e. with separated regions) is attempted by including an internal viscous solution which is going to modify the external inviscid solution. In fact, we do not introduce the boundary layer approximation, that would be inappropriate to the present aim, but we try to recover the viscosity effects in a more con- sisting way through a boundary integral formulation for the full vorticity transport equation, which is valid in principle in the entire flow field. However, as in the first order boundary layer theory, we introduce sev- eral drastic assumptions to simplify as much as pos- sible the numerical solution of the boundary integral equation resulting from the representation (17) for x* appoaching the boundary (assumed smooth) LIP ~ {u2N = __ _ B ~B~ 2 MU' (43) It follows the boundary condition (42) for the nor- mal derivative of vorticity. The two volume integrals of the non linear and the initial terms, appearing in eq. (41), are confined to very narrow layers (in the limit of Re ~ oo that we consider here) where the vorticity is different from zero. Consistingly with this approximation, the fundamental solution F is taken constant across these layers. Hence, we may compute the integral across the layer of the non linear term V X X = V X Huron-2lnS~) ~ ~ (V x Den = k / [fir (ur`) -~ funny] dn In particular, for a layer close to the body boundary, whose thickness is i, taking into account that us"-0 at the body, ~-- 0 at the outer edge of the layer and that within the layer ~ =-~ we have: ~ (V x Den = _ ~.721 k (44) Instead, for a wake of thickness 264 with a jump furl --~ and a tangential velocity defined as wr u, +u, 2 2~(2*~t*) = lo |dnV (fun -F37~) dSdt | (V x Dan =-kin 2'1 =-k~9r(WrY) (45) I* (41) By the same reasoning, we obtain respectively + lo In (V x X) FdVdt - | ~oFdV This is a Fredholm integral equation of the second kind if ~ is assigned as boundary condition. From the differential Navier Stokes equation on the body boundary we obtain the general local identity which gives in the tw~dimensional case, i.e. for I= ok, Vp x n =-p(V x <3 x n = -wok (42) In the framework of a first order theory we as sume that the pressure doesn't change normally to the body wall within the viscous layer close to the body itself, and its value is given by the inviscid rotational model of Section 3. As a matter of fact, from that model we find us, and the pressure follows from the Euler equation at the wall / ~Odn = -urology (46) o /_`S^Odn = -~0 (47) By combining the approximate values of the vol- ume integrals (44), (45), (46), (47), with the integral equation (41), we have a relatively simple model to determine the value of the wall vorticity. As a further crucial feature of the model, we place the separation point at the wall position where S. is changing its sign. A vortex layer is issuing from this separation point and its intensity is determined by a Kutta-type condition completely analogous to the one introduced in Section 3 for the wake at the sharp trailing edge of a streamlined body t20;. 230

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( \ / Figure 4: Wake evolution for flow past a cylin- der. Separation point at fixed positions ax =+ 108°. t= 1,2,3,4 The numerical procedure alternates the solution of equations (24,25) for the external inviscid flow, with the solution of eq. (41) for the internal viscous layer. They are connected through the condition (42) which in a first order approximation relates the external pressure gradients with the generation of vorticity at the wall t21~. The simplifying hypotesis for the cal- culation of the volume integrals may be released to obtain better approximations. The model has been applied to a cylindrical body. We didn't consider here the presence of the free sur- face to focuse our attention on the generation scheme for the separated regions. In fact the model of gen- eration still requires a deeper understanding and it seams resonable to select a test case for which a large experience is available. Several computational mod- els, using a Kutta-type condition for the calculation of separated flows in the framework of vortex methods have been recently presented 23,24,25. First we discuss some numerical results with two fixed separation points in symmetric position on the cylinder boundary (c' =+ 108° from the front stagna- tion point). A symmetric rear separated region, like the one shown in Fig. 4, is obtained if no pertura- bation is introduced in the solution procedure. By retarding of one time step the lower side of the cylin- der with respect to the other, we introduce a large oscillation in the two vortex layers. By advancing in time, they assume a configuration (see the sequence in Fig. 5) which resembles the initial displacement of the vortices in the classical Karman street. For a more stable behaviour of the vortex layer we adopted a de-singularization technique t22] in order to elimi- nate the singularity of the kernel for x ~ x*. 231

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,~/, l.''> /( Figure 5: Wake evolution for flow past a cylinder with an initial perturbation. Separation points at fixed po- sitions c' =+ 108°. t = 3, 5, 7, 8 Finally the complete procedure including the in- teraction with the internal viscous solution, has been applied to the case of Re = 104. At each time step the Kutta-type condition is applied in a new position corresponding to the zero value of vorticity which is determined from the solution of the integral equation (41~. The sequence in Fig. 6 shows the motion of the separation points towards the rear part of the cylin- der, starting from the initial position of 90°, which 232

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\ - - \ - \ ~7 corresponds to zero tangential derivative of pressure (i.e. zero vorticity for no volume integrals in the first inviscid solution). The wiggles appearing in the wake configurations in the present case, ace given to the motion of the separation point (i. e. the source of the vortex shedding) and not to a vortex layer instability. The convergence to a steady state solution is uncer / me/ ) '¢, / Figure 6: Wake evolution for flow past a cylinder. Separation point at variable positions determined by the solution of the vorticity equation. t = 1, 2, 3 fain in the sense that a steady state is almost reached, but is not mantained, as if it were an unstable solu- tion. We don't have at the moment sufficient data to understand if this is just a numerical instability or it simulates the ineherent physical instability present at those values of the Reynolds number. 233

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CONCLUDING REMARKS AND PERSPECTIVES In the present paper we tried to answer several questions about the possibility and the convenience to extend boundary integral methods (usually very efficient for potential flows) to rotational free surface flows, either Inviscid or viscous. Starting from an integral formulation in primi- tive variables for unsteady viscous flows, we deduced a set of simplified models strictly connected, the one to the other, through their relevant mathematical struc- ture. Actually the basic integral equations are very similar, so the experience may be transferred from simpler to increasingly complicated models. The analysis of the limiting case for viscosity go- ing to zero, has been of great help to understand the behaviour of the integral equation for tendentially sin- gular kernels. The decoupling of kinematics and dy- namics has been another crucial feature to reduce the integral representation to a form (the Poincare for- mula) easy to be treated from a computational point of view. A further point to be stressed is the coupling of the original equations in primitive varaibles with the integral equation for vorticity, which leads to a more convenient approach for viscous flow solutions. The theoretical analysis suggested a first model for the study of free surface flows with regions of as- signed vorticity or vortex layers generated by the body motion. The resulting numerical procedure, efficient as the potential one, allows for an easy evaluation of several rotational effects. A second more refined model, suitable for the investigation of separated flows with large rotational regions, was provided by the interaction with a first order viscous solution. The model implies a numeri- cal procedure, which still requires a further analysis. However the numerical results are very promising in spite of the simplicity of the model. The latter may be considered as a first step to- wards the fully coupled and fully viscous model which is the straightforward approach to study the genera- tion of vorticity and the creation, through separation, of large vertical regions. The integral representation (3) for the velocity vector and the scalar version of the (41) for the vorticity (2D case) lead to boundary in- tegral equations in the same unknowns if we account for the relations (18) and (42) valid at the body wall. A mixed procedure which solves alternatively the nor- mal component of the first integral equation for the dynamic pressure and the second integral equation for the vorticity, is now in progress. REFERENCES 1. Dowson C.W., PA practical computing method for solving ship wave problem", 2nd Int. Confer- ence Numerical Ship Hydrodynamics, Berkeley, 1977. 2. Miyata H., Sato T., and Baba N., "Difference Solution of a Viscous Flow with Free Surface Wave about an Advancing Ship", J. Comput. Phys., no. 72, 1987, p. 393. 3. Casciola C.M., and Piva R., PA Boundary In- tegral Formulation for Free Surface Viscous and Inviscid Flows about Submerged Bodies", Proceedings of the 5th International Conference on Numerical Ship Hydrodynamics, Hiroshima (Japan), Sept. 25-29,1989. 4. Piva R., and Morino L., "Vector Green's Func- tion method for Unsteady Navier Stokes Equa- tions", Meccanica, vol. 22, 1987, pp. 7~85. 5. Piva R., Graziani G., and Morino L., UBoun- dary Integral Equation Method for Unsteady Viscous and Inviscid Flows", Advanced Bound- ary Element Methods, Cruse T.A. (ed.), Spring- er Verlag, New York, USA, 1987. 6. Brard R., Vortex theories for bodies moving in water", Proceedings of 9th Symp. on Naval Hydrodynamics, Brard R., and Castera A. (eds.), U.S. Gov. Printing Office, 1972, pp. 1187- 1284. 7. Piva R., UThe Boundary Integral Equation Me- thod for Viscous and Inviscid Flows", Proce- edings ISCFD, Oshima K. fed.), Nagoya, Japan, 1989. 8 . L adyzhenskaj a, O. . A ., The Mathemat ic al Theory of Viscous Incompressible Flows, Gordon & Breach, New York, 1963. 9. Casciola C.M., Lancia M.R., and Piva R., "A General Approach to Unsteady Flows in Aero- dynamics: Classical Results and Perspectives", ISBEM 89, East Hartford, USA, 1989, Springer Verlag, Berlin. 10. Bassanini P., Casciola C.M., Lancia M.R., and Piva R., "A boundary integral formulation for the kinetic field in aerodynamics. Part I: Math- ematical analysis. Part II: Applications to Un- steady 2-D Flows", submitted to European J. Mech., B/Fluids, 1990. 11. Nedelec J.C., Approximation des equation integrates en mecanique et en physique", Lec- ture Notes, Centre de Mathematiques Applique- es, Ecole Polytechnique, Palaiseaux, France, 1977. 234

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12. Hsiao G.C., "On boundary integral equations of the first kind", China-US Seminar on Boundary Integral Equations and Boundary Element Met- hods in Physics and Engineering. Jan. 1988. Xi'an, People's Republic of China. 13. Schmall R.A., and Kinney, R.B., Numerical Study of Unsteady Viscous Flow Past a Lifting Plate", AIAA Journal, Vol. 12, 1974, pp. 156 1673. 14. Wu J.C., "Numerical Boundary Conditions for Viscous Flow Problems", AIAA Journal, Vol. 14, 1976, pp. 1040-1049. 15. Berker R., Integration des equations du Move- ment d'un Fluide Visquex Incompressible", Encyclopedia of Physics, Flugge S. (ed.), Vol. VIII/2, 1963. 16. Morino L., ~Helmoltz Decomposition Revisited: Vorticity Generation and Trailing Edge Condi- tion", Computational Mechanics I. 17. Friedrichs K.O., Special Topics in Fluid Dyn- amics, Gordon & Breach, London, 1966. 18. Liu, P.L-F., and Ligget J.A., Applications of the Boundary Element Method to Problems of Water Waves", Developments in Boundary Element Methods - 2, Banerjee P.K., and Shaw R.P., Applied Science publishers, London, 1982. 19. Haussling, H.J., and Coleman R.M., ~Finite- Difference Computations Using Boundary Fit- ted Coordinate Systems for Free Surface Po- tential Flows Generated by Submerged Bodies", Proceedings of the 2nd Inter. Conference on Numerical Shin Hydrodynamics Wehausen J.V.. and Salvesen N. (eds.), 1977. 20. Sears, W.R., "Unsteady Motion to Airfoils anbd Boundary Layer Separations, AIAA Journal, Vol. 14, No. 2. 21. Lighthill, M.J., "Introduction. Boundary Layer Theory, Laminar Boundary Layers, L. Rosen- head (ed.), Oxford at the Clarendon Press, 1963, pp. 4~59. 22. Krasny, R., Computation of vortex sheet roll- up in the Trefftz plane", Journal of Fluid Mechanics, Vol. 184, 1987, pp. 123-156. - 23. Katz, J., "A discrete vortex method for the non steady separated flow over an airfoils, Journal of Fluid Mechanics, Vol. 102, 1981, pp. 315-328. 24. Kiya M., and Arie M., "A contribution to an in- viscid vortex-shedding model for an inclined flat plate in uniform flown, Journal of Fluid Mechanics, Vol. 82, Part 2, 1977, pp. 223-240. 25. Sarpkaya T., An inviscid model of two-dimensi- onal vortex shedding for transient and asymp- totically steady separated flow over an inclined plate", Journal of Fluid Mechanics, Vol. 68, Part 1, 1975, pp. 109-128. APPENDIX A We remove here the assumption of fixed fluid domain, introduced in section 3a to simplify the calculations for the equivalence between the kinematical represen- tation stemming from (10) and the Poincare formula (23~. Hence, we have to consider in (10) also the sur- face integral accounting for the motion of the free sur- face, neglected in section 3a. An easy way to prove this equivalence is to derive with respect to time the Poincare formula. In particular the first surface inte- gral in (23) perfectly coincides with the first in (10) therefore we focuse our attention on the second sur- face integral -V* x ~ (u x n)G dS =-V* x k / urGdS an(t) an(t) (1A) where now ~Q is a function of time, described by the parametric equation x = x(t,,t) with ~ ~ D`. The time derivative of this integral, expressed in terms of the Lagrangian parameter ~ is -/. urG dS = d t an(t) dt /De Or [Xt6,t),t] GtX((,t),x*~1 J(: to do, = iD dt~urJ)G+urJ dtVGd~ (2Ay where-is a derivative for a given point ~ and J is the Jacobian of the transformation. We introduce in t2A~ the following identities Dw where n f d Dwur dJ dt(Ur]) = Dt J + or dt DWU dr dJ = Jo Dt · r+ U d t) + Or d t (3A) Is the total derivative following the free surface motion. Using the definition of J J l~x 235

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taking the time derivative of &~ ~ = TJ d ax Bw d dr dJ dt(~) Be, = dt(rJ) = Jdt +Tdt its scalar product times u divided by J dr 1 dJ dw unit + u,Jdt = u. d and combining these results with the expression (3A) divided by J. leads to .~d.~.(u~]) = dot ~ + us ~ (~4A) The free surface velocity w has the normal com- ponent w" = a", while for the tangential component we may assume wr = 2ur. The relationship between the material derivative and D" is Du DWU Bu D! = Dt + (or -Wr)o which combined with(4A)gives ~7,d`(ur]) = Dt ·7 + ~r(2Un) (5A) where the last term results after some tediuos manip- ulation by UT BU BW {1 1 2 + -U di9 = ~ ~ ( 2 Un) Now from the Euler equation Du _ ~ ( p ~ 9z) (6A) from(5A)plus(6A)combined with(2A)we obtain / ()[-~ (P+9z-2u2)G]dS+ +Jr [Uris G+uTu"0G]dS (7A) Integrating by parts the first integral and factorizing -, leads to IOn(t) [[Jr (P +9Z-2Un + BUT) + Urgent] dS which corresponds exactly to the term / (PnjUk-IJ U U(k)) dS (9A? including the second and the third surface integrals of equation (10). Actually by introducing l~j = Pnj-L,oUj and using the vector notations ~I&n`~) ( ) (1OA) IlxVG= (~Ton Hoper) with Tin = _ _ u2 = P + us _ u" + go JET = -on or by comparing we may verify the exact correspondence between(lOA)and (1A) In conclusion we see that the term(iOA)which in the Euler equation is integrated in time, even in this case of moving boundary, perfectly coincides with the time derivative of the term(iA,of the Poincare formula. 236

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DISCUSSION Gerard Van Oortmerssen Marin Research Institute Netherlands, The Netherlands The authors have given a rather fundamental analysis of free-surface flows and showed results of boundary integral computations for some basic cases involving simple geometries. Could you please elaborate on the perspective for applying these methods for more realistic cases of practical relevance. AUTHORS' REPLY I'd like to thank Dr. Van Oortmerssen for his question which gives me the opportunity to discuss the perspectives of our work. We have presented in this paper several models for the analysis of Cortical flows at different levels of complexity. In particular, the simplest one, for inviscid attached flows about bodies with a sharp trailing edge, may be directly used for applications in ship hydrodynamics. For instance, slightly submerged hydrofoils have been studied accounting properly for the nonlinear effects due both to the free surface and to the wake. The extension to the three-dimensional case has been completed from the theoretical point of view (see ref. 10) while its numerical application is still in progress. On the other hand, more complex models for flows about bluff bodies still require a large amount of basic work, mainly about the vortex shedding modelling, so that the application to real problems is far ahead and presently the investigation is confined to two-dimensional test cases. In the framework of this application we have proposed two possible approaches. In the first one we considered the integral representation for the complete Navier-Stokes equations. A flow field simulation by this model would require a very large computational effort and only the simple case of two vortices is now under investigation. The second simplified approach is essentially based on the coupling of an external solution, obtained by the Poincare identity, with an internal one for the detection of the separation point locations. Even for this model, a deep investigation is required to better understand its capability to represent the physical phenomenon and its range of applicability. DISCUSSION Philippe G. Genoux Bassin d'Essais des Carines, France Is your solution able to simulate alternated vortices (Strouhal effects) in the case of a flow past a cylinder? AUTHORS' REPLY In the present approach, the vorticity field is modeled by vortex sheets issuing from the separation points on the cylinder. We find that, after perturbing the system, the symmetric solution no longer exists and is replaced by a flow which shows the developments of alternate vortices. The solution is obtained with fixed (prescribed) separation points. Moreover, we found that under these conditions the flow doesn't evolve towards a periodic solution. Actually the strength of the vortex layer, still oscillating, decreases in time. This behavior may be explained by the fact that we keep the separation points fixed. In order to develop a procedure with moving separation points, we have introduced the interaction between the external and the internal solution. Presently we have an increasing oscillatory motion of the separation points; therefore, a comparison of the Strouhal number seems to be premature. DISCUSSION J.M.R. Graham Imperial College of London, United Kingdom Have the authors compared their prediction of the separation points on the circular cylinder in impulsively started flow with other published results? For example, the analysis of the onset of separation (Van Dommelen & Shen) and other similar work. AUTHORS' REPLY As I said in the previous answers, the model we propose for vortex shedding after a bluff body is still under investigation and we obtained only some preliminary results. We don't consider however, the model suitable for studying the onset of separation at its very initial stage. Let me stress again the point that our purpose here is to devise a simplified model able to analyze recirculating or separated regions without solving the complete Navier Stokes equations. Presently, we are still in the stage to reproduce the physical phenomenon and to understand the essential features to be included in the model. From this point of view, we are very interested to consider the suggestions in the paper you mention. 237

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