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 condition is enforced and enters the ow

diusively resulting in the ux of vorticityinto the oweld In the case of a free

T Lundgren P Koumoutsakos

surface the vortex sheet remains at the surface to enforce the pressure b oundary

condition and constitutes a part of the vorticityeldoftheow The task is to

determine the strength of the vortex sheet at the free surface so as to satisfy the

b oundary conditions

For the purp ose of describing this pro cess we assume that the velo city and the

n

vorticity eld are known at time t throughout the ow eld and at the free surface

n n

and we wish to obtain the ow eld at time t t t

n

Step Given the velo city and vorticityattime t we up date the p ositions of the

vorticity carrying elements and the surface markers by solving dx dt ux t

p p

n n n n

x x tu x

p p p

We up date the vorticity eld by solving

d

r

dt

n n n

with initial condition at t t and b oundary condition x

f

n

at x x The solution to this equation whichwe denote by is still

f

incomplete It do es not satisfy the correct vorticity b oundary condition at the end

of the time step and must b e corrected in step The b oundary condition which

wehave imp osed ensures rather arbitrarily that the vorticity on the b oundary is

purely convected The correction which is needed will b e a vortical layer along the

freesurface with vorticity of order t and with thickness of order t We reason

that the additional velocity eld induced across this layer can b e neglected since its

variation is only of order t

For an incompressible ow the velo citymay b e expressed in terms of a stream

function by

u k r

and the vorticity itself is related to by

k ru r

We use the convention thatn is always outward from the uid t is the direction of

integration along the surface and k n t is a unit vector out of the page The

solution of this equation gives

where

Z

x x tlnjx x jdx

a a a

uid

and represents an irrotational ow selected to satisfy b oundary conditions It is

consistent with vortex dynamics to take this irrotational part as the ow induced

bya vortex sheet along the b oundary of the uid ie by

Z

xt x s tln jx x s jds

f f

intfc

Vorticity at a freesurface

but it must b e shown that this can b e done in sucha way as to satisfy the b oundary

conditions In this formulation the b oundary can b e either solid or free or a mix

of these but in this pap er we are sp ecically interested in free b oundaries which

separate an incompressible uid from a uid of negligible mass densityThevelo city

eld is obtained by applying giving the BiotSavart law

uxtu xtu xt

where

Z

u x x t k rln jx x jdx

a a a

uid

and

Z

u x x st k rln jx x s jds

f f

intfc

The velo city eld is also dened by these integrals for p oints outside the uid u

is continuous across the interface and u has a jump discontinuity As the p osition

vector x tends to a p oint on the interface from inside the uid whichwe will indicate

with a subscript we get

Z

s

u t PV s tn rln jx s x s jds

f f

intfc

while as the p oint is approached from the outside indicated by

Z

s

u t PV s tn rln jx s x s jds

f f

intfc

Here PV indicates the principal value of these singular integrals By subtracting

these equations it is clear that the vortex sheet strength is the jump in tangential

velo city across the interface since u t is continuous wehave

u t u t

By and the tangential comp onent of the surface velo cityis

Z

s

PV s tn rln jx s x s jds u t u t

f f

intfc

Equation is a Fredholm integral equation of the second kind the solution of

which determines the strength of the free surface vortex sheet when the right

hand side is given In the case of multiply connected domains the equation needs to

b e supplemented with m constraints for the strength of the vortex sheet where m

is the multiplicity of the domain Prager For example in the case of a free

T Lundgren P Koumoutsakos

surface extending to innity no additional constraint needs to b e imp osed as the

problem involves integration over a singly connected domain However in the case

of a bubble an additional constraintsuch as the conservation of total circulation in

the domain needs to b e imp osed in order to obtain a unique solution

The righthand side of the equation may b e determined from the quantities which

have b een up dated In particular u can b e computed via the BiotSavart integral

n

from the known vorticity eld with order t accuracy The tangential

comp onentofthevelo city of the free surface can b e computed using in the

form

n n

n

u t u t tQ u n t p

n

n n

where Q signies the righthand side of evaluated at time t The pressure

b oundary condition enters the formulation of the problem at this stage Up on

solving the strength of the vortex sheet is determined such that the pressure

b oundary condition is satised justifying the previous assertion We should add

that admits more than one solution in multiply connected domains suchas

atwodimensional bubble conguration but unique solutions may b e obtained by

using Fredholms alternative

Note that the present metho d of enforcing the pressure b oundary condition is

equivalent to previous irrotational formulations Lundgren Mansour

which employa velo city p otential

At the end of this step the p oints of the freesurface the velo city eld and the

n n n

u and The strength of the vortex sheet have b een up dated x

p

n

vorticity eld still needs to b e corrected near the free surface

Step At this step we consider generation of vorticity at the free surface Hav

ing determined the strength of the vortex sheet from Step we can compute the

normal and tangential comp onents of the velo city eld at the free surface in order

to determine the free surface vorticity and enforce the zeroshear stress b oundary

condition

Using we can compute an up dated value of the stream function on the

surface and from this compute u n s Since the surface shap e and u t

have already b een up dated wehave all the ingredients necessary to compute an

up dated value of from The next step in this pro cess is to solve the vorticity

transp ort equation for the vorticity eld using as b oundary condition For the

nal partial step we need to solve the heat equation

r

t

n

with initial condition at t t and with the b oundary condition

n

n n

x t t t

f

assuming a linear time variation of the surface vorticitybetween the two time lev

n

els The solution of this partial step is to b e added to thus yielding the

n

completely up dated vorticity eld

Vorticity at a freesurface

An analytical solution for this diusion equation can b e obtained using the

metho d of heat p otentials Friedman For a twodimensional ow the solu

tion to the ab ove equation may b e expressed in terms of doublelayer heat p otentials

as

Z Z

tt

G

xt t x x s t t s t ds dt

f

n

t

intfc

where G is the fundamental solution of the heat equation and the function s tis

determined by the solution of the following second order Fredholm integral equation

Z Z

tt

G

s t x s x s t t ds dt x st s t

f f f

n

t

intfc

Following Greengard and Strain and Koumoutsakos Leonard and Pepin

we can obtain asymptotic formulas for the ab oveintegrals Similar formu

las could help in the developmentofanumerical metho d based on the prop osed

algorithm

This up date strategy was p osed without requiring any particular numerical meth

o ds for the computational steps Wehave particular metho ds in mind however

for using this strategy for future numerical work We will use a b oundary integral

metho d similar to that used by Lundgren Mansour for the surface

computations That work was for irrotational inviscid ow Instead of the pressure

b oundary condition in the form of an unsteady Bernoulli equation was used

to access the pressure

For the vortical part of the owwe prop ose to use the p ointvortex metho d

employed by Koumoutsakos et al for viscous ow problems with solid

b oundaries In these problems the Lighthill strategy provides a vorticity ux b ound

ary condition for the second step in the vorticity up date a Neuman condition In

the prop osed freesurface strategy a Dirichlet condition is required for the second

vorticity step This mo dication can b e accomplished by using double layer heat

p otentials as suggested ab ove where single layer p otentials were used in the solid

b oundary work

Conservation of vorticity

We will showthatvorticity is conserved in twodimensional freesurface problems

vorticity whichows through the freesurface do esnt disapp ear but resides in the

vortex sheet along the surface This is shown for general threedimensional ows

in the Lundgren and Koumoutsakos

In the interior of the uid it is easy to show from Helmholtzs equation that

Z Z

d

ds dA

dt n

A S

where A is a material volume and S its surface n is outward from the region

and n is the vorticity ux in the outward direction This says that the

T Lundgren P Koumoutsakos

vorticityinA increases b ecause of viscous vorticity ux into the region there are

no vorticity sources in the interior of the uid

Everything we need to know ab out the velo city on side is contained in

We will only use the fact that b ecause the velo city on side is irrotational there

must b e a velo city p otentialu r We use ddt to mean the material deriva

tive along side and note that u u t then by some simple manipulations

du u

u ru

dt t

r u u t r t t ru

t

Then

du

u u t ru t t

dt s t

The last term in this equation is the strainrate of a surface element and maybe

expressed as

d

ds t ru t

ds dt

where ds is a material line element on side Subtracting from then gives

d p d

ds u u gy n r

dt ds dt s t

This maybewritten

d

ds ds ds

dt n s

with given by

p

u u gy

t

If weintegrate over a material segment along the interface we obtain

b b b

Z Z Z

d

ds ds ds

dt n s

a a a

From this form we see that should b e interpreted as a surfacevorticity ux Since

is a density circulation density or surfacevorticity density the last term in

whichmay b e written is the ux of surfacevorticityinto the interval at a

a b

minus the ux out at b while the rst term on the rightistheuxofvorticityinto

the interval through the surface

Vorticity at a freesurface

If the interval is extended over the entire interface by extending it to innityfor

an o cean or continuing b around to a for a closed interface like a bubble we get

Z Z

d

ds ds

dt n

intfc intfc

Now letting A in b e the entire uid we get

Z Z

d

ds dA

dt n

uid intfc

Adding and gives

Z Z

d d

dA ds

dt dt

uid intfc

It is in this sense that vorticity is conserved

We b egan this approach as an attempt to obtain an evolution equation for

whichwould eliminate solving an integral equation to up date Equation

or might app ear to playsuch a role but the o ccurrence of the velo city

p otential in the equation makes it unuseable for this purp ose Since could b e

expressed byanintegration over the surface involving the time derivativeof

would involve a surface integral of d dt therefore an integral equation for d dt

would result defeating the purp ose

A similar result can b e shown for the conservation of vorticity in threedimensional

ows Lundgren and Koumoutsakos

Ped ley problem

A problem solved byPedley as part of a study on the stability of swirling

torroidal bubbles gives an example which illustrates some concepts discussed here

One can describ e the ow as a p otential vortex of circulation swirling around

a bubble cavity of radius R The ow is induced bya vortex sheet of strength

R at the bubble interface At some initial time one turns on the viscosity

and vorticity b egins to leak from the vortex sheet into the uid The circulation at

innity remains constant therefore the strength of the vortex sheet must decrease

with time

We p ose this problem in the form describ ed in Section Since the ow is axially

symmetric the vorticity satises

t r r r

The vorticity b oundary condition is

V R

T Lundgren P Koumoutsakos

where V u t is the tangential comp onent of the velo city at the interface with

the tangent convention used earlier V is negative for p ositive swirl and R is the

constant radius of curvature of the surface The pressure b oundary condition

is

V

t r

The velo city inside the bubble is zero so u t The strength of the vortex sheet

is therefore V a p ositive quantity The sense of the problem is that since

is required to b e nonzero a layer of p ositivevorticitymust develop in the uid

The resulting ux of vorticity out of the interface causes to decrease with time

Equations and may b e combined into a single b oundary condition

t R r

Therefore the problem is to solve with this b oundary condition and with

initial conditions for all rRand R for r R This last condition

prevents the trivial solution

For large tR Pedley gives an approximate solution

r

exp

R R

This satises exactly but has a relative error of order in the b oundary

condition For small another approximate similarity solution is

p

x

p

expx Erfc

R

where x r RR This solution satises the b oundary condition exactly but

neglects the last term in requiring that b e small enough that the vortical

layer is thin compared to the radius of the bubble

Further details of the solution are unimp ortant here This problem illustrates

b oth conservation of vorticity and generation of vorticity when there is ow along

a curved freesurface

Conclusion

In this pap er wehavepresented a strategy for solving free surface viscous ow

problems in a vortex dynamics formulation This strategy centers on determining

suitable b oundary conditions for the vorticity in analogy with Lighthills strategy

for solid b oundary ows The two free surface b oundary conditions play distinct

roles in determining free surface viscous ows Wehave shown that the pressure

b oundary condition determines the strength of a vortex sheet at the free surface

which determines the irrotational part of the ow The pressure force mo dies the

surface velo city from which the vortex sheet strength is found by solving an integral

Vorticity at a freesurface

equation The zero shear stress b oundary condition on the other hand determines

the value of the vorticity at the surface providing a Dirichlet condition for the

vorticity equation

Wehaveshown that vorticity is conserved for b oth two and threedimensional

free surface ows the vortex sheet b eing considered part of the vorticity eld It

follows that vorticity whichmight app ear to b e lost by ux across the free surface

now resides in the vortex sheet It was shown in the app endix that vorticityis

conserved for two viscous uids in contact across an interface It is physically clear

that in the limit as the density and viscosity of one of the uids tend to zero the

vorticity in that uid would b e conned to a thin surface layer Vorticitywould

then b e conserved in the remaining uid plus a contribution in the surface layer

Therefore the conclusions wedraw for free surface ows are physically reasonable

for real uids

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