FREE SURFACE. X f X a. Fluid element

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1 On the Generation of Vorticity at a Free Surface Thomas Lungren 1 an Petros Koumoutsakos 1 Dept. of Aerospace Eng. & Mechanics, Univ. of Minnesota, Minneapolis, MN IFD, ETH urich, CH-809, Switz. an CTR, NASA Ames/Stanfor Univ., Moett Fiel, CA Abstract The mechanism for the generation of vorticity at a viscous free surface is escribe. This is a free surface analogue of Lighthill's strategy for etermining the vorticity ux at soli bounaries. In this metho the zero shear stress an pressure bounary conitions are transforme into a bounary integral formulation suitable for the velocity-vorticity escription of the ow. A vortex sheet along the free surface is etermine by the pressure bounary conition, while the conition of zero shear stress etermines the vorticity at the surface. In general, vorticity is generate at free surfaces whenever there is ow past regions of surface curvature. It is shown that vorticity is conserve in free surface viscous ows. Vorticity which ows out of the ui across the free surface is gaine by the vortex sheet the integral of vorticity over the entire ui region plus the integral of \surface vorticity" over the free surface remains constant. The implications of the present strategy as an algorithm for numerical calculations are iscusse. 1 Introuction Often the term free surface is use to refer loosely to any gas-liqui interface. In this paper we ene free surface ows to be iealize gas-liqui ows in which the ynamics of the gas phase is neglecte by setting the gas ensity an viscosity to zero. While we are mostly concerne with free surface ows in this paper we sometimes iscuss the connection beween these iealize ows an similar real ows with gas-liqui interfaces. In free surface ows there are many situations where vorticity enters a ow in the form of a shear layer. This occurs at regions of high surface curvature an supercially resembles separation of a bounary layer at a soli bounary corner, but in the free surface ow there is very little bounary layer vorticity upstream of the corner an the vorticity which enters the ow is entirely create at the corner. Roo (1994) has associate the ux of vorticity into the ow with the eceleration of a layer of ui near the surface. These eects are quite clearly seen in spilling breaker ows stuie by Duncan & Philomin (1994), Lin & Rockwell (1995) an Dabiri & Gharib (1997). In this paper we propose a escription of free surface viscous ows in a vortex ynamics formulation. In the vortex ynamics approach to ui ynamics the emphasis is on the vorticity el which is treate as the primary variable the velocity is expresse as a functional of the vorticity through the Biot-Savart integral. The pressure no longer appears in the formulation. However, since pressure appears in the free surface bounary conition, a suitable proceure is require to convert this to a bounary conition on the vorticity. When vortex ynamics methos are use for viscous ows with soli bounaries a similar problem arises. A vorticity bounary conition may be etermine by following Lighthill's (1963) iscussion of the problem. Lighthill note that the velocity el inuce by the vorticity in the 0

2 ui will not in general satisfy the no-slip bounary conition. The spurious slip velocity may be viewe as a vortex sheet on the surface of the boy. In orer to enforce the no-slip bounary conition the vortex sheet is istribute iusively into the ow, transferring the vortex sheet to an equivalent thin viscous vortex layer by means of a vorticity ux. The no slip conition therefore etermines the vorticity ux, which is the strength of the spurious vortex sheet ivie by the time increment. The physical character of Lighthill's metho has le to its irect formulation an implementation by Kinney an his co-workers (1974, 1977) in the context of nite ierence schemes, an by Koumoutsakos, Leonar an Pepin (1994) to enforce the no-slip bounary conition in the context of vortex methos. Their metho has prouce benchmark quality simulations of some unsteay ows (Koumoutsakos an Leonar, 1995). For free surface ows a vortex sheet is employe in orer to ajust the irrotational part of the ow. Unlike the case of a soli wall this vortex sheet is part of the vorticity el of the ow an is use to etermine the velocity el. The strength of the vortex sheet is etermine by enforcing the bounary conitions resulting from a force balance at the free surface. This gives two conitions, which are the subject of this paper. One, a relationship between the vorticity ux an the surface acceleration the other a relationship between the vorticity at the surface an the curvature of the surface. The resulting strategy can be easily aapte to a numerical scheme an can lea to improve numerical methos for the simulation of viscous free surface ows. As a conceptually attractive by-prouct of this stuy we n that vorticity is conserve if one consiers the vortex sheet at the free surface to contain \surface vorticity". We prove in Section 4 that vorticity which uxes out of the ui, an appears to be lost, is really gaine by the vortex sheet. As an example of the signicance of this, consier the the approach of a vortex ring at a shallow angle to a free surface. It has been observe (Bernal & Kwon, 1988 Gharib, 1994) for an air-water interface that the vortex isconnects from itself as it approaches the surface an reconnects to the surface in a U-shape structure with surface imples at the vortex ens. There is a clear loss of vorticity from the water an an acceleration of the surface in the irection of motion of the ring as iscusse by Roo (1994). In interpreting this from a vortex ynamics point of view we have to istinguish between the real-ui experiment escribe an a hypothetical (or numerical) experiment with a free surface. In the free surface case the missing vorticity is foun in a vortex sheet in the surface which connects the vortex ens. We show in the appenix that vorticity is conserve for two real viscous uis separate by an interface. This means that vorticity lost from the water passes into the air an woul be expecte to be foun in a fairly thin bounary layer ragge along by the accelerating water surface, similar to the vortex sheet in the free surface case. We can imagine a limit in which as the viscosity an ensity of the gas are mae smaller an smaller this vortex layer on the gas sie of the interface contracts to avortex sheet. In Section we present the governing equations an bounary conitions aapte for a vortex ynamics formulation. In Section 3 we escribe a fractional step strategy for the enforcement of the bounary conitions at a free-surface. The conservation of the vorticity el in two-imensional free-surface ows is shown in Section 4. Conservation of vorticity in a general three-imensional context is treate in the Appenix. Mathematical Formulation In orer to introuce the vorticity generation mechanism we consier, without loss of generality, two imensional ow of a Newtonian ui with a free surface (gure 1). We consier the stresses 1

3 ^ n FREE SURFACE ^ t X f X a Figure 1: Denition Sketch s Flui element 1 in ui as negligible an when not otherwise state the ow quantities refer to ui 1..1 Governing Equations Two imensional incompressible viscous ow may be escribe by the vorticity transport equation with the Lagrangian erivative ene as! = r! + u r () where u(x t)isthevelocity,! =! ^k = ru the vorticity an enotes the kinematic viscosity. The ow el evolves by following the trajectories of the vorticity carrying ui elements x a an the free-surface points x f base on the following equation: where x p enotes x a or x f. x p = u(x p ) (3). Bounary Conitions The bounary conitions at the free-surface are etermine by a force balance calculation. For a Newtonian ui the stress tensor is expresse as T = ;pi +D : (4) where D is the symmetric part of the velocity graient tensor. The local normal an tangential components of the surface traction force are expresse as ^n T ^n an ^n T^t respectively. Balancing these two force components results in the following two bounary conitions at a free-surface.

4 1. ero Shear Stress. Assuming negligible surface tension graients, balancing the tangential forces at the free-surface results in This may be expresse ^t D ^n = 0 : (5) ^n ru ^t + ^t ru ^n = 0 : (6) For the purposes of our velocity-vorticity formulation we wish to relate this bounary conition to the vorticity el an to the velocity components at the free-surface. For a two-imensional ow, by the enition of vorticity in a local coorinate system, we have Using (6) we may rewrite (7) as! = ^n ru ^t ; ^t ru ^n : (7)! = ;^t ru ^n : (8) By some further manipulation the free-surface vorticity may be expresse in terms of the local normal an tangential components of the velocity el! ^n ^n = @s ^n = ; +u ^t where is the curvature of the surface, ene by = For steay ow, where the free-surface is stationary, u 1 ^n is zero an the rst term on the right in (11) rops out. The steay version of (11) was given by Lugt (1987) an by Longuet-Higgins (199), the unsteay form by Wu (1995). A three imensional version of (8) was erive by Lungren (1989). The sense of (11) is that vorticity evelops at the surface whenever there is relative ow along a curve interface. This conition prevents a viscous free-surface ow from being irrotational. Enforcing the vorticityelgiven by the above equation at the free-surface is equivalent to enforcing the conition of zero shear stress.. Pressure Bounary Conition. This is the conition that the jump in normal traction across the free-surface interface is balance by the surface tension. It is expresse as k^n T ^nk = ;T (1) where T is the surface tension an the vertical braces enote the jump in the quantity. Using (4), this becomes ;p 1 +^n ru ^n + p = ;T : (13) 3

5 Using the continuity equation, expresse in local coorinates, we get ^n ru ^n = ;^t ru ^t (14) = (15) Therefore = p 1 = p + ^t ; u ^n : + u ^n where p is the constant pressure on the zero ensity sie of the interface. Since pressure oes not occur in the vorticity equation, the pressure conition must be put in a form which accesses the primary variables. From the momentum equation at the free-surface we obtain (17) ^t u 1 = ; + ^n r! ; g^j ^t (18) where g is the gravitational constant ^j is upwar. For our purposes this equation may be put in a more tractable form by further manipulation. First we observe that an ^t u 1 = u 1 ^t + u 1 ^t (19) u 1 ^t = u 1 ^n ^n ^t : (0) Then using the fact that the free-surface is a material surface we obtain the kinematic ientity ^n ^t = ^t ru ^n (1) Using this ientity we n 1 ; u 1 ^t : () u 1 ^t = u 1 1 ; u 1 ^t u1 ^n ; + ^n r! ; g^j ^t : (3) We emphasize that the material erivative here is taken following a ui particle on sie 1 of the interface. 4

6 With p 1 substitute from (17) this formula may be regare as equivalent to the pressure bounary conition. Except for the ux term all the terms on the right-han-sie of the equation are quantities ene on the surface an erivatives of these along the surface. We prefer to think of the role of the vorticity ux in this equation as a term which moies the surface acceleration, rather than consier that the equation etermines the ux. Using a strategy analogous to Lighthill's for a soli wall, we propose a fractional step algorithm that enforces the pressure bounary conition in a vorticity-velocity framework. This strategy allows us to gain insight into the evelopment an generation of vorticity at a viscous free surface an can be use as a builing tool for a numerical metho. 3 A Fractional Step Algorithm In orer to show that the free-surface bounary conitions are satise in a velocity-vorticity formulation we consier the evolution of the ow el uring a single time step. In a manner similar to Lighthill's approach for a soli bounary, a vortex sheet is employe to enforce the bounary conitions. The vortex sheet becomes part of the vorticity el of the ow. The ierence between the soli wall an the free surface is the role of the surface vortex sheet in ajusting the velocity el of the ow. In the case of the soli wall the vortex sheet is eliminate from the bounary (so that the no-slip bounary conition is enforce) an enters the ow iusively, resulting in the ux of vorticity into the owel. In the case of a free surface the vortex sheet remains at the surface to enforce the pressure bounary conition an constitutes a part of the vorticity el of the ow. The task is to etermine the strength of the vortex sheet at the free surface so as to satisfy the bounary conitions. For the purpose of escibing this process we assume that the velocity an the vorticity el are known at time t n throughout the ow el an at the free surface an we wish to obtain the ow el at time t n+1 ( t n + t). Step 1. Given the velocity an vorticity at time t n we upate the positions of the vorticity carrying elements an the surface markers by solving x p = = u(x p t) We upate the vorticity el by solving x n+1 p = x n p + tu n (x n p) : (4)! = r! (5) with initial conition! =! n at t = t n an bounary conition! =! n (x f ) at x = x f. The solution to this equation, which we enote by! n+1=, is still incomplete. It oes not satisfy the correct vorticity bounary conition at the en of the time step an must be correcte in step. The bounary conition which we have impose ensures (rather arbitrarily) that the vorticity on the bounary is purely convecte. The correction which is neee will be a vortical layer along the free-surface with vorticity of orer t an with thickness of orer (t) 1=. We reason that the aitional velocity el inuce across this layer can be neglecte, since its variation is only of orer (t) 3=. For an incompressible ow the velocity may be expresse in terms of a stream function by u = ;^k r (6) 5

7 an the vorticity itself is relate to by! ^k ru = ;r : (7) We use the convention that ^n is always outwar from the ui, ^t is the irection of integration along the surface, an ^k = ^n ^t is a unit vector out of the page. The solution of this equation gives =! + (8) where!(x) 1 =; ui!(x a t)lnjx ; x a jx a (9) an represents an irrotational ow selecte to satisfy bounary conitions. It is consistent with vortex ynamics to take this irrotational part as the ow inuce by a vortex sheet along the bounary of the ui, i.e. by 1 (x t)=; (x f (s 0 ) t)lnjx ; x f (s 0 )js 0 (30) but it must be shown that this can be one in such a way as to satisfy the bounary conitions. In this formulation the bounary can be either soli or free or a mix of these, but in this paper we are specically intereste in free bounaries which separate an incompressible ui from a ui of negligible mass ensity. The velocity el is obtaine by applying (6), giving the Biot-Savart law u(x t)=u! (x t)+u (x t) (31) where an u! (x) = 1 u (x) = 1 ui!(x a t) ^k rln jx ; x a jx a (3) (x f (s) t) ^k rln jx ; x f (s)js : (33) The velocity el is also ene by these integrals for points outsie the ui u! is continuous across the interface, u has ajumpiscontinuity. As the position vector x tens to a point on the interface from insie the ui, which we will inicate with a subscript \1", we get (u ^t)1 = (s) ; ; P:V: 1 while as the point is approache from the outsie, inicate by \", (u ^t) =+ (s) ; P:V: 1 (s 0 t) ^n r ln jx f (s) ; x f (s 0 )js 0 (34) (s 0 t) ^n r ln jx f (s) ; x f (s 0 )js 0 : (35) 6

8 Here P.V. inicates the principal value of these singular integrals. By subtracting these equations it is clear that the vortex sheet strength is the jump in tangential velocity across the interface since u! ^t is continuous we have = u ^t ; u1 ^t : (36) By (31) an (34) the tangential component of the surface velocity is (s) ; ; P:V: 1 (s 0 t) ^n rln jx f (s) ; x f (s 0 )js 0 = u 1 ^t ; (u! ^t)1 : (37) Equation(37) is a Freholm integral equation of the secon kin the solution of which etermines the strength () of the free surface vortex sheet when the right han sie is given. In the case of multiply connecte omains the equation nees to be supplemente with m constraints for the strength of the vortex sheet, where m+1 is the multiplicity of the omain (Prager, 198). For example in the case of a free surface extening to innity no aitional constraint nees to be impose as the problem involves integration over a singly connecte omain. However, in the case of a bubble, an aitional constraint, such as the conservation of total circulation in the omain, nees to be impose in orer to obtain a unique solution. The right han sie of the equation may be etermine from the quantities which have been upate. In particular u! can be compute via the Biot-Savart integral (3) from the known vorticity el! n+1= with orer t accuracy. The tangential component of the velocity of the free surface can be compute using (3) in the form (u 1 ^t) n+1 = (u 1 ^t) n + tq n (u 1 p 1) (38) where Q n signies the right han sie of (3) evaluate at time t n. The pressure bounary conition enters the formulation of the problem at this stage. Upon solving (37) the strength of the vortex sheet is etermine such that the pressure bounary conition is satise, justifying the previous assertion. Note that the present metho of enforcing the pressure bounary conition is equivalent to previous irrotational formulations (Lungren & Mansour, 1988, 1991) which employ a velocity potential. At the en of this step the points of the free-surface, the velocity el an the strength of the vortex sheet have been upate (x p n+1, u n+1 an n+1 ). The vorticity el (! n+1= ) still nees to be correcte near the free surface. Step. At this step we consier generation of vorticity at the free surface. Having etermine the strength of the vortex sheet from Step 1 we can compute the normal an tangential components of the velocity el at the free surface in orer to etermine the free surface vorticity an enforce the zero-shear stress bounary conition. Using (8,9,30) we can compute an upate value of the stream function on the surface an from this compute u 1 ^n =@s. Since the surface shape an u 1 ^t have alreay been upate we have all the ingreients necessary to compute an upate value of! 1 from (7). The next step in this process is to solve thevorticity transport equation for the vorticity el using! 1 as bounary conition. For the nal partial step we nee to solve the = r! (39) 7

9 with initial conition! =0att = t n,anwith the bounary conition!(x f ) = (! n+1 1 ;! n 1 )(t ; t n )=t (40) assuming a linear time variation of the surface vorticity between the two time levels. The solution of this partial step is to be ae to! n+1= thus yieling the completely upate vorticity el! n+1. An analytical solution for this iusion equation can be obtaine using the metho of heat potentials (Frieman, 1964). For a two-imensional ow the solution to the above equation may be expresse in terms of ouble-layer heat potentials as!(x t+ t) = 0 (x ; x f(s 0 ) t; t 0 )(s 0 t 0 ) s 0 0 (41) where G is the funamental solution of the heat equation an the function (s t) is etermine by the solution of the following secon orer Freholm integral equation : ; 1 (s t) + t+t t (s 0 (x f(s) ; x f (s 0 ) t; t 0 ) s 0 0 =!(x f (s) t) : (4) Following Greengar an Strain (1990) an Koumoutsakos, Leonar an Pepin (1994) we can obtain asymptotic formulas for the above integrals. Similar formulas coul help in the evelopment of a numerical metho base on the propose algorithm. This upate strategy was pose without requiring any particular numerical methos for the computational steps. We have particular methos in min, however, for using this strategy for future numerical work. We will use a bounary integral metho similar to that use by Lungren & Mansour (1988, 1991) for the surface computations. That work was for irrotational invisci ow. Instea of the pressure bounary conition in the form of (3), an unsteay Bernoulli equation was use to access the pressure. For the vortical part of the ow we propose to use the point vortex metho employe by Koumoutsakos et al (1994, 1995) for viscous ow problems with soli bounaries. In these problems the Lighthill strategy provies a vorticity ux bounary conition for the secon step in the vorticity upate, a Neuman conition. In the propose free-surface strategy a Dirichlet conition is require for the secon vorticity step. This moication can be accomplishe by using ouble layer heat potentials (as suggeste above) where single layer potentials were use in the soli bounary work. 4 Conservation of vorticity We will show that vorticity is conserve in two-imensional free-surface problems vorticity which ows through the free-surface oesn't isappear but resies in the vortex sheet along the surface. (This is shown for general three-imensional ows in the Appenix.) In the interior of the ui it is easy to show from Helmholtz's equation that A1!A = s 8

10 where A 1 is a material \volume" an S 1 its \surface", n is outwar from the region an ;@!=@n is the vorticity ux in the outwar irection. This says that the vorticity in A 1 increases because of viscous vorticity ux into the region there are no vorticity sources in the interior of the ui. Everything we nee to know about the velocity on sie is containe in (31,3,33). We will only use the fact that, because the velocity on sie is irrotational, there must be a velocity potential(u = r). We use = to mean the material erivative along sie 1, an note that u ; u 1 = ^t, then by some simple manipulations + u 1 ru (44) Then + 1 u u ; ^t r^t ; ^t ru1 : (45) ^t + 1 u u ; 1 ; ^t ru1 ^t : (46) The last term in this equation is the strain-rate of a surface element an may be expresse as 1 ^t ru 1 ^t = s (47) s where s is a material line element on sie 1. Subtracting (18) from (46) then gives + s + 1 u u 1 ; + p 1 + gy ; ^n r! : (48) This may be written with s = ; s ; u u 1 p ; + gy5 1 ; : (50) If we integrate (49) over a material segment along the interface we obtain b a s = ; s ; b s : From this form we see that shoul be interprete as a surface-vorticity ux. Since is a ensity (circulation ensity or surface-vorticity ensity) the last term in (51), which may be written a ; b, is the ux of surface-vorticity into the interval at a minus the ux out at b, while the rst term on the right is the ux of vorticity into the interval through the surface. 9

11 If the interval is extene over the entire interface, by extening it to innity for an \ocean", or continuing b aroun to a for a close interface, like abubble, we get s = s: Now letting A 1 in (43) be the entire ui we get ui!a s : Aing (53) an (5) gives ui!a + s =0: (54) It is in this sense that vorticity is conserve. We began this approach as an attempt to obtain an evolution equation for which woul eliminate solving an integral equation, (37), to upate. Equation (48) or (49) might appear to play such a role, but the occurance of the velocity potential in the equation makes it unuseable for this purpose. Since coul be expresse by an integration over the surface involving, the time erivative of woul involve a surface integral of = therefore an integral equation for = woul result, efeating the purpose. 4.1 Peley's problem: Vorticity outsie a swirling cylinrical bubble A problem solve by Peley (1968) as part of a stuy on the stability of swirling torroial bubbles gives an example which illustrates some concepts iscusse here. One can escribe the ow as a potential vortex of circulation ; swirling aroun a bubble cavity of raius R. The ow is inuce by a vortex sheet of strength 0 = ;=R at the bubble interface. At some initial time one turns on the viscosity an vorticity begins to leak from the vortex sheet into the ui. The circulation at innity remains constant therefore the strength of the vortex sheet must ecrease with time. We pose this problem in the form escribe in Section. Since the ow is axially symmetric the vorticity satises The vorticity bounary conition @t + : (55)! 1 = ;V 1 =R (56) where V 1 = u 1 ^t is the tangential component of the velocity at the interface (with the tangent convention use earlier V 1 is negative for positive swirl), an R is the constant raius of curvature of the surface. The pressure bounary conition (3) : (57)

12 The velocity insie the bubble is zero so u ^t = 0. The strength of the vortex sheet is therefore = ;V 1, a positive quantity. The sense of the problem is that since! 1 is require to be non-zero a layer of positive vorticity must evelop in the ui. The resulting ux of vorticity out of the interface causes to ecrease with time. Equations (56) an (57) may be combine into a single bounary 1 : (58) Therefore the problem is to solve (55) with this bounary conition an with initial conitions! =0forall r>ran! = 0 =R for r = R. This last conition prevents the trivial solution. For large ( t=r ) Peley gives an approximate solution! = 0 R exp ; r : (59) 4R This satises (55) exactly, but has a relative error of orer ;1 in the bounary conition. For small another approximate similarity solution is! = 0 R exp(x +4) Erfc x p +p where x =(r ; R)=R. This solution satises the bounary conition exactly but neglects the last term in (55), requiring that be small enough that the vortical layer is thin compare to the raius of the bubble. Further etails of the solution are unimportant here. This problem illustrates both conservation of vorticity an generation of vorticity when there is ow along a curve free-surface. 5 Conclusion In this paper we have presente a strategy for solving free surface viscous ow problems in a vortex ynamics formulation. This strategy centers on etermining suitable bounary conitions for the vorticity in analogy with Lighthill's strategy for soli bounary ows. The two free surface bounary conitions play istinct roles in etermining free surface viscous ows. We have shown that the pressure bounary conition etermines the strength of a vortex sheet at the free surface, which etermines the irrotational part of the ow. The pressure force moies the surface velocity, from which the vortex sheet strength is foun by solving an integral equation. The zero shear stress bounary conition, on the other han, etermines the value of the vorticity at the surface, proviing a Dirichlet conition for the vorticity equation. We have shown that vorticity is conserve for both two an three imensional free surface ows, the vortex sheet being consiere part of the vorticity el. It follows that vorticity which might appear to be lost by ux across the free surface now resies in the vortex sheet. It was shown in the appenix that vorticity is conserve for two viscous uis in contact across an interface. It is physically clear that in the limit as the ensity an viscosity of one of the uis becomes small, the vorticity transmitte to that ui woul be conne to a thin surface layer, a vortex sheet in the limit. Kelvin has shown (Lamb, 1945, art.145) that vortex lines cannot begin nor en in the interior of a ui. This is also true for two viscous uis in contact, since the proof only requires 11 (60)

13 continuity of the velocity el. The vortex ynamics formulation suggests that vortex lines on't en at free surfaces (where the velocity is iscontinuous) but can take an abrupt ben an continue in the surface to complete a close circuit. This is a physically reasonable result in light of the limit process escribe, but can't be prove irectly in terms of vortex tubes, since a tube in the free surface woul have innite vorticity in zero cross section. 6 Appenix. Vorticity conservation in three-imensional ows 6.1 Real Viscous Fluis Separate by an Interface In three-imensional ows there is a simple kinematic result which was shown by Truesell (1954). For smooth vorticity els, which ten to zero fast enough at innity vorticity is conserve in the form ui!v =0: (61) This means, for instance, that the average vorticity inavortex ring is zero vorticity on one part of the ring is cancelle by vorticity in the opposite irection on another part of the ring. Equation(61) can be prove by using the ientity! = r(!r) (6) (which requires only r! = 0) where r is the position vector. Integrating this over a nite volume V an using the ivergence theorem gives!v = ^n!rs : (63) V S Equation(61) follows upon letting V be the whole space. It is easy to see that (61) is still true if there is a stationary soli boy inclusion in a viscous ui since! ^n =0at no-slip bounaries. Consier now a case where two real uis with ierent nite viscosities are separate by an interface. The vorticity is iscontinuous so (63) is only vali for volumes on either sie of the interface. Since the velocity iscontinuous in viscous uis it follows that! ^n is continuous across the interface an therefore, by cancellation of the two surface integrals, we see that (61) is still true in this case. However, if one of the uis is non-viscous, as for free surface problems, one can no longer assume continuity ofthevelocity an (61) cannot be prove. 6. Free Surface Flows For three imensional free surface ows we will prove that!v + S 0 (64) ui 1

14 where the vortex sheet strength is ene by =^n (u ; u 1 ) (65) this being the circulation per unit length aroun a surface element in the plane of the velocity jump. Only the Biot-Savart law is require for the proof the uis on't nee to satisfy the Navier-Stokes equations. Consier the omain to consist of a vortical incompressible ui on sie 1 separate from an irrotational ui on sie by an interface which extens to innity. The bounary of the vortical ui is the interface plus a surface S 1 which will be taken to innity. The irrotational part is boune by the interface an a istant surface S. Since! = ru we have, by Gauss's theorem,!v = ^n u 1 S + ^n u S : (66) ui S1 Since vorticity is zero on sie we can also write 0=; ^n u S + S ^n u S (67) where the normal on the interface is irecte towar sie. Aing these equations, an ening a temporary quantity, we n!v + S = ^n us : (68) ui S1[S The velocity el inuce by the vorticity el of the ui an the vortex sheet is given by the three-imensional Biot-Savart law u = ; rg(r r 0 )! 0 V 0 ; rg(r r 0 ) 0 S 0 (69) ui where 1 g(r r 0 )=; 4jr ; r 0 j (70) assuming that the velocity is zero at innity. compact, say zero outsie of some region, then u = 1 4 If the vorticity an vortex sheet are suciently r r 3 (71) asymptotically for large r. Now consier this asymptotic result in the integral on the right-hansie of (68). If the istant surface is a sphere centere at the origin the integral is zero since then r is in the same irection as ^n. But this result is inepenent of the shape of the istant surface. Since vorticity is zero in the istant region one can apply Gauss's theorem in the region between two istant surfaces, obtaining ^n u S = ^n u S : (7) Surf 1 Surf 13

15 Since the integral is zero for a sphere it is zero in general. We have thus prove that = 0. Therefore (64) is true. The physical connection with the case of two viscous uis is clear. If one consiers the limit as the viscosity of the secon ui becomes small, the vorticity el in that ui will reuce to a thin vortical layer along the interface an (61) will reuce to (64). We can show the more limite result that = = 0 using the metho employe for twoimensional free-surface ows in Section 4. This is analytically intensive but very instructive. Using the enition (65) the material erivative offollowing the ui on sie 1 of the free-surface is =^n (u ; u 1 )+ ^n (u ; u 1 ) : (73) Using the kinematic equation r= = r ru for a ui element normal to the surface, one can show an therefore using this an (u ; u 1 )=;^n we n ^n = ^n ru 1 ; ^n ru 1 ^n ^n (74) = ^n (u ; u 1 ) ; ^n ^n ru 1 : (75) From the Navier-Stokes equation for the ui on sie 1 one can easily obtain ^n u 1 = ;^n r p1 + gy ; r! 1 ^n + ^n r! 1 : (76) There is no Navier-Stokes equation on sie since the ensity is zero, but an irrotational velocity el is ene there by the Biot-Savart integral. Following an analysis similar to the twoimensional case, we have + u 1 ru (77) an therefore ^n u +(u 1 ; u ) ru + r u u + u u (78) +(u 1 ; u ) ru 1 ; (u ; u 1 ) r(u ; u 1 ) (79) = + u u +(u ; u 1 ) ru 1 ^n +(u ; u 1 ) r(u ; u 1 ) ^n : (80) The last term may be written (u ; u 1 ) r(u ; u 1 ) ^n = ;^n r (u ; u 1 ) (u ; u 1 ) 14 ; (u ; u 1 )! 1 ^n (81)

16 by using the vector ientity u ru = ru = ; u (r u). The secon term on the right of(80) requires more manipulation to get it into the esire form. First write (u ; u 1 ) ru 1 ^n = (^n ru 1 ^n) : (8) Then by expaning the yaic ru 1 in a local coorinate system with orthogonal base vectors ^t1, ^t ^n = ^t1 ^t we n (u ; u 1 ) ru 1 ^n = ; r S u 1 + r S u 1 ; r S u 1 ^n ^n +(u ; u 1 )! 1 ^n : (83) The symbol r S is the surface component ofr r S = r;^n ^n r. Substituting (81)an (83) into (80) gives ^n u + u u Then substituting (76) an (84) into (75) we get ; ; r S u 1 + r S u 1 ; r S u 1 ^n ^n : + r S u 1 = r S u 1 ; ^n r+r! 1 ^n ; ^n r! 1 ;f ru 1 ^n +^n ru 1 g ^n (85) (84) + u u ; + p 1 + gy (86) is essentially the same quantity as in (50). The brackete term at the en of (85) is zero because shear stress is zero at the free-surface (see (6)) an S r S u 1 = (S)=, where S is a material surface element on sie 1 of the free-surface, so one may write Now we use 1 S (S) = r Su 1 ; ^n r+r! 1 ^n ; ^n r! 1 : (87) r S u 1 = r S ( u 1 ) ; (r S ) u 1 (88) = r S ( u 1 ) ;! 1 ^n u 1 (89) an write, nally, 1 S (S) =r S ( u 1 ) ; ^n r+r! 1 ^n ;f^n! 1 u 1 + ^n r! 1 g : (90) (This equation may be erive from Wu (1995, eq.61a) upon interpreting his \interface velocity" as u 1 an calculating the acceleration on sie in terms of the material erivative on sie 1 as we have one here.) The rst three terms on the right can be reuce to line integrals when integrate over a surface patch, an therefore are surface \ivergence" terms, an the brackete term is the ux 15

17 of vorticity into this patch through the surface (see (94) below for this interpretation of vorticity ux) after integrating over a patch we get S = ^m u 1 s ; ^ts + ru 1 ^ts ; (^n! 1 u 1 + ^n r! 1 )S (91) S1 C1 C1 C1 where S 1 is a material surface on the free-surface, C 1 its bounary curve, ^t is a tangent vector on C 1 in the irection of integration, an ^m = ^t ^n is the normal to C1 in the surface. In orer to interpret the vorticity ux in (90) we note that Helmholtz's vorticity equation in the interior of the ui,! S1 =! ru + r! (9) has a vortex stretching term on the right-han-sie which may be written! ru = r(! u) (93) since the ivergence of the vorticity is zero. Therefore, using the ivergence theorem V!V = S (^n! u + ^n r!)s : (94) Thus we ientify r! as a iusive ux tensor an!u as a stretching ux tensor. (These are uxes relative to the motion of the ui. In general there is also a convective ux tensor, ;u!.) The surface vorticity ux is interprete from (91) in a similar way. The ux of surface vorticity in the irection ^m on the surface is ^m u 1 ; ^t + ru1 ^t with ^t = ^n ^m. This can also be written as ^m otte into a ux tensor. If the material volume V intersects the free-surface in the surface S 1 of (91) then the ux of vorticity into the volume V across S 1 is equal to the ux of vorticity out of the vortex sheet across this part of the surface. If we exten the volume to innity, so it covers the entire ui, the line integrals go to zero an we obtain ui!v + Therefore total vorticity is conserve, as was to be shown. S =0: (95) 16

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