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On the Generation of Vorticity at a Free-Surface

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On the Generation of Vorticity at a Free-Surface Center for TurbulenceResearch Annual Research Briefs On the generation of vorticity at a freesurface By T Lundgren AND P Koumoutsakos Motivations and ob jectives In free surface ows there are many situations where vorticityenters a owin the form of a shear layer This o ccurs at regions of high surface curvature and sup ercially resembles separation of a b oundary layer at a solid b oundary corner but in the free surface ow there is very little b oundary layer vorticity upstream of the corner and the vorticity whichenters the owisentirely created at the corner Ro o d has asso ciated the ux of vorticityinto the ow with the deceleration ofalayer of uid near the surface These eects are quite clearly seen in spilling breaker ows studied by Duncan Philomin Lin Ro ckwell and Dabiri Gharib In this pap er we prop ose a description of free surface viscous ows in a vortex dynamics formulation In the vortex dynamics approach to uid dynamics the emphasis is on the vorticityvector which is treated as the primary variable the velo city is expressed as a functional of the vorticity through the BiotSavart integral In free surface viscous ows the surface app ears as a source or sink of vorticityand a suitable pro cedure is required to handle this as a vorticity b oundary condition As a conceptually attractivebypro duct of this study we nd that vorticityis conserved if one considers the vortex sheet at the free surface to contain surface vorticity Vorticity which uxes out of the uid and app ears to b e lost is really gained by the vortex sheet As an example of the signicance of this consider the the approachofavortex ring at a shallow angle to a free surface It has b een observed Bernal Kwon Gharib that the vortex disconnects from itself as it approaches the surface and reconnects to the surface in a Ushap ed structure with surface dimples at the vortex ends There is a clear loss of vorticityfromtheuid and an acceleration of the surface in the direction of motion of the ring as discussed by Ro o d Since vorticity is conserved the missing vorticity has b een attened out into a vortex sheet which connects the vortex ends In a real waterair interface the connection is in a thin vortex layer in the air The completion of vortex lines along the surface allows one to maintain the physical picture of closed vortex tub es When vortex dynamics metho ds are used for viscous ows with solid b oundaries avorticity b oundary condition may b e determined by following Lighthills discussion of the problem Lighthill noted that the velo city eld induced by the vorticity in the uid will not in general satisfy the noslip b oundary condition This spurious slip velo citymaybeviewed as a vortex sheet on the surface of the b o dy Permanent address Dept of Aerospace Engr Mechanics Univ of Minnesota Minneap olis MN T Lundgren P Koumoutsakos n t Free Surface X 2 f FLUID ELEMENT S X 1 + a Figure Denition sketch In order to enforce the noslip b oundary condition the vortex sheet is distributed diusively into the ow transferring the vortex sheet to an equivalent thin viscous vortex layer by means of a vorticity ux The vorticity ux is the strength of the spurious vortex sheet divided by the time increment For free surface ows a vortex sheet is employed in order to adjust the irrotational part of the ow Unlike the case of a solid wall this vortex sheet is part of the vorticity eld of the ow and is used in order to determine the velo city eld The strength of the vortex sheet is determined by enforcing the b oundary conditions resulting from a force balance at the free surface The physical character of Lighthills metho d has led to its direct formulation and implementation by Kinney and his coworkers in the context of nite dierence schemes and by Koumoutsakos Leonard and Pepin in or der to enforce the noslip b oundary condition in the context of vortex metho ds Their metho d has pro duced b enchmark qualitysimulations of some unsteady ows Koumoutsakos and Leonard The present strategy can b e easily adapted to sucha numerical scheme and can lead to improved numerical metho ds for the simulation of viscous free surface ows Accomplishments In order to intro duce the vorticity generation mechanism we consider without loss of generalitytwodimensional ow of a Newtonian uid with a free surface Fig We consider the stresses in uid as negligible and when not otherwise stated the ow quantities refer to uid Mathematical formulation Twodimensional incompressible viscous owmay b e describ ed bythevorticity transp ort equation d r dt Vorticity at a freesurface with the Lagrangian derivative dened as d u r dt t where uxt is the velo city k ru the vorticity and denotes the kinematic viscosity The ow eld evolves by following the tra jectories of the vorticity carrying uid elements x and the freesurface p oints x based on the a f following equation dx p ux p dt where x denotes x or x p a f Boundary conditions The boundary conditions at the freesurface are determined by a force balance calculation For a Newtonian uid the stress tensor is expressed as T pI D where D is the symmetric part of the velo city gradient tensor The lo cal normal and tangential comp onents of the surface traction force are expressed asn T n and n T t resp ectively Balancing these two force comp onents results in the following two b oundary conditions at a freesurface Zero Shear Stress Assuming negligible surface tension gradients balancing the tangential forces at the freesurface results in t D n This may b e expressed n ru t t ru n For the purp oses of our velo cityvorticity formulation we wish to relate this b ound ary condition to the vorticity eld and to the velo city comp onents at the freesurface For a twodimensional ow by the denition of vorticity in a lo cal co ordinate system wehave n ru t t ru n Using wemay rewrite as t ru n By some further manipulation the freesurface vorticitymay b e expressed in terms of the lo cal normal and tangential comp onents of the velo cityeld u n s T Lundgren P Koumoutsakos u n n u s s u n u t s where is the curvature of the surface dened by t n sFor steady ow where the freesurface is stationary u n is zero and the rst term on the rightin drops out The steady version of was given by Lugt and by Longuet Higgins the unsteady form byWu A threedimensional version of was derived by Lundgren The sense of is that vorticity develops at the surface whenever there is relative ow along a curved interface This condition prevents a viscous freesurface ow from b eing irrotational Enforcing the vorticity eld given by the ab ove equation at the freesurface is equivalent to enforcing the condition of zero shear stress Pressure Boundary Condition This is the condition that the jump in normal traction across the freesurface interface is balanced by the surface tension It is expressed as kn T n k T where T is the surface tension and the vertical braces denote the jump in the quantity Using Eq this b ecomes p n ru n p T Using the continuity equation expressed in lo cal co ordinates weget n ru n t ru t u t t u s s u t u n s Therefore u t p p T u n s where p is the constant pressure on the zero density side of the interface Since pressure do es not o ccur in the vorticity equation the pressure condition must b e put in a form which accesses the primary variables From the momentum equation at the freesurface we obtain p du n r g j t t dt s where g is the gravitational constant j is upward Vorticity at a freesurface For our purp oses this equation may b e put in a more tractable form by further manipulation First we observe that du du t dt t u dt dt dt and dt dt u u n n dt dt Then using the fact that the freesurface is a material surface we obtain the kine matic identity dt n t ru n dt u n u t s Using this identitywe nd u n p du t u n u t u n n r g j t dt s s We emphasize that the material derivative here is taken following a uid particle on side of the interface With p substituted from this formula may b e regarded as equivalent to the pressure b oundary condition Except for the ux term all the terms on the right handside of the equation are quantities dened on the surface and derivatives of these along the surface We prefer to think of the role of the vorticity ux in this equation as a term which mo dies the surface acceleration rather than consider that the equation determines the ux Using a strategy analogous to Lighthills for a solid wall we prop ose a fractional step algorithm that enforces the pressure b oundary condition in a vorticityvelo city framework This strategy allows us to gain insightinto the development and gen eration of vorticity at a viscous free surface and can b e used as a building to ol for anumerical metho d A fractional step algorithm In order to show that the freesurface b oundary conditions are satised in a velo cityvorticity formulation we consider the evolution of the ow eld during a single time step In a manner similar to Lighthills approach for a solid b oundary avortex sheet is employed to enforce the b oundary conditions The vortex sheet b ecomes part of the vorticity eld of the ow The dierence between the solid wall and the free surface is the role of the surface vortex sheet in adjusting the velo city eld of the ow In the case of the solid wal l the vortex sheet is eliminated from the b oundary so that the noslip b oundary
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