rodyna - Caltech Authors

Eighth Symposium

NAVAL RODYNA

HYDRODYNAMICS I N THE OCEAN ENVIRONMENT

Office of Naval Research Department of the Navy

UNSTEADY, FREE SURFACE FLOWS; SOLUTIONS EMPLOYING THE LAGRANGIAN DESCRIPTION OF THE MOTION

Christopher Brennen, Arthur K. Whitney

CaZifornia Institute of TechnoZogy Pasadena, CaZifornia

ABSTRACT Numerical techniques for the solution of unsteady f r e e surface flows a r e briefly reviewed and consideration i s given to the feasibility of methods involving pararne t r i c planes where the position and shape of the f r e e surface a r e known in advance. A method for inviscid flows which u s e s the Lagrangian description of the motion is developed. This exploits the flexibility in the choice of Lagrangian reference coordinates and i s readily adapted to include terms due to inhomogeneity of the fluid. Numerical results a r e compared in two cases of irrotational flow of a homogeneous fluid f o r which Lagrangian l i n e a r i z e d solutions can be constructed. Some examples of wave run-up on a beach and a shelf a r e then computed.

I.

INTRODUCTION

T h e r e a r e many instances of unsteady flows in which analytic solutions, even approximate ones, a r e not available. This i s p a r ticularly t r u e of f r e e surface flows when, for example, non-linear waves o r even slightly complicated boundaries a r e involved. Though analytical methods a r e progressing, especially through the use of variational principles (Whitham [1965]) and, in some c a s e s , the non-linear shallow water wave equations yield important results ( C a r r i e r and Greenspan [ 19581) t h e r e is s t i l l a need f o r numerical. methods. Indeed, numerical "experiments can be used to complement actual experiments.

"

Until very recently numerical solutions in two dimensions

Brennen and W h i t n e y

invariably s e e m e d to employ t h e Eulerian description of the motion though the Lagrangian concept has been used f o r s o m e t i m e in the much s i m p l e r one-dimensional c a s e (e.g. , Heitner [ 19691 , Brode [ 19691 ) and to m a k e s m a l l t i m e expansions (Pohle [ 19521 ) . P e r h a p s the b e s t known of t h e s e E u l e r i a n methods i s the Marker-and-Cell technique (MAC) begun by F r o m m and Harlow [ 19631 and f u r t h e r refined by Welch, et al. [ 19661 , H i r t [ 19681 , Amsden and Harlow [ 19701 , Chan, S t r e e t and Strelkoff [ 19691 and o t h e r s . The m o s t difficult p r o b l e m a r i s e s in attempting t o reconcile the initially unknown shape and position of a f r e e s u r f a c e with a finite difference s c h e m e ahd the n e c e s s i t y of determining d e r i v a t i v e s a t that s u r f a c e . In the s a m e way, few solutions exist with curved o r i r r e g u l a r solid boundaries. In steady flows, mapping techniques have been e m ployed to t r a n s f o r m t h e f r e e s u r f a c e to a known position (e. g. , Brennen [ 19691 ). It would t h e r e f o r e s e e m useful t o examine the use of p a r a m e t r i c planes f o r unsteady flows. The Lagrangian d e s c r i p t i o n in i t s m o s t g e n e r a l f o r m ( L a m b [ 19321) involves s u c h a plane and by suitable choice of the r e f e r e n c e c o o r d i n a t e s , t h e f r e e s u r f a c e can be reduced t o a known and fixed s t r a i g h t l i n e . However a d i s c u s s i o n of other p a r a m e t r i c planes and mapping techniques is included in Section 3 .

--

The m a j o r p a r t of this paper i s devoted to the development of a n u m e r i c a l method f o r the solution of the Lagrangian equations of motion in which full u s e i s made of the flexibility allowed in the choice of r e f e r e n c e coordinates. F o r the m o m e n t , we have r e s t r i c t e d ourselves to c a s e s of inviscid flow. Very r e c e n t l y , H i r t , Cook and Butler [ 19701 published details of a method which employs a Lagrangian tagging s p a c e but i s otherwise s i m i l a r to the MAC technique. This i s f u r t h e r d i s c u s s e d in Section 4B.

11. LACRANGIAN EQUATIONS O F MOTION The g e n e r a l inviscid dynamical equations of motion in Lagrangian f o r m a r e ( L a m b [ 19321 ):

where X,Y.Z a r e t h e C a r t:esian coordinates of a fluid p a r t i c l e a t . . t i m e t , F , G , H a r e t h e components of extraneous f o r c e acting upon it, P is the p r e s s u r e , p the density and a , b , c a r e any t h r e e quantities which s e r v e t o identify the p a r t i c l e and which v a r y continuously f r o m one p a r t i c l e to the next. F o r e a s e of r e f e r e n c e (X, Y , 2) a r e t e r m e d E u l e r i a n coordinates, ( a , b , c ) Lagrangian coordinates. Suffices a , b , c , t denote differentiation.

Lagrangian S o Z u t i o n s of Unsteady Free S u r f a c e Flows

If X,, Yo, Z, i s the position of a p a r t i c l e at s o m e r e f e r e n c e t i m e t o (when the density i s p,) then the equation of continuity i s simply

Frequently it i s convenient to define a , b , c a s identical to X,, Yo, ZO, thus reducing the R.H. S. of ( 2 ) t o p,; however it will be s e e n in the following sections that flexibility in the definition of a , b , c I s of considerable value when designing numerical methods of solution. If the extraneous f o r c e s , F , G , H , have a p o t e n t i a l S2 and P , if not uniform, i s a function only of P then, eliminating S2 + +/p f r o m (1):

w h e r e , f o r convenience, the velocities Xt, Y,, Z a r e denoted by a r e related to t h e Eulerian U, V, W. The quantities vorticity components , 5 , , 5 2, G3 by

r,, re,r3

(Thus, of c o u r s e , vorticity changes with t i m e a r e due solely t o changes in the coefficients of the L.H. S. of (4) which, in t u r n , r e p r e s e n t s stretching and twisting of the vortex l i n e , ) Given the vorticity distribution c(X, Y, Z) a t some initial t i m e , to, r ( a , b , c) (which is independent of time) may be obtained through Eqs. (4) and used in the final f o r m of the dynamical equations of motion, namely Eqs. (3) integrated with r e s p e c t to t i m e .

Brennen and Whitney

F o r incompressible, planar flow the equations reduce t o XaYb- YaXb = F ( a , b )

Continuity:

(or differentiated w. r . t .

Motion:

UbXa

(5)

t):

- UaXb + VbYQ- VaYb = - r ( a , b ) .

By introducing the vectors Z = X conveniently combine to:

+ iY

and W = U

-

(7) iV, ( 6 ) and (7)

Other types of flow have also been investigated. F o r example, in the c a s e ~f a heterogeneous, o r non-dispersive stratified liquid in which p is a function of ( a , b ) , Eq. (8) becomes:

The integral t e r m therefore manufactures vorticity. The methods developed for a homogeneous fluid in Sections 4A t o D a r e modified in Section 4E to include such effects. 111. OTHER PARAMETRIC PLANES It may be of interest to d i g r e s s a t this point to consider other parametric planes ( a , b ) , which a r e not necessarily Lagrangian. That i s to s a y the restrictions X+(a,b,t) = U, Y+(a,b,t) = V a r e abandoned s 6 that U, V a r e no lohger either ~ u l L r i a no r Lagrangian velocities. Provided J = 8 ( ~ , Y ) / B ( a , b # ) 0 , o r m, the equation f o r incompressible and irrotational planar flow remains

To incorporate one of the advantages of the Lagrangian s y s t e m , it is required that the f r e e surface be fixed and known, say on a line of constant b. Then the kinematic and dynamic f r e e surface conditions a r e respectively

Lagrangian S o Z u t i o n s o f Unsteady Free S u r f a c e FZows

Now a useful choice concerning the ( a , b ) plane would be to require the mapping from (X, Y) to be conformal. Then, of c o u r s e , (10) simply reduces to the Cauchy-Riemann conditions U, = Vb, U b =V, s o that W = U iV i s an analytic function of c = a + ib o r of Z.

-

-

In this way, John [ 19531 has constructed some special, exact I), has the particular analytic solutions. The kinematic condition, (I solution W(a,t) = Zt(a,tl on.the f r e e surface, which implies W(c,t) = by analytic continuation, If, in addition,

.=

where K i s r e a l on the f r e e s u r f a c e , then the dynamic condition thereon i s also satisfied. John discusses s e v e r a l examples f o r various choices of the function K. The potential of such methods may not have been fully realized either analytically o r numerically. In the l a t t e r c a s e , however, the conformality of the (X,Y) to (a,b) mapping i s not necessarily a great advantage, whereas a fixed and known f r e e surface position most certainly is. The digression ends h e r e and the following sections develop a Lagrangian numerical method f r o m the equations of Section 2. IV,

A NUMERICAL METHOD EMPLOYING LAGRANGIAN COORDINATES

A method f o r the numerical solution of incompressible, planar flows i s now described. It attempts to take full advantage of the flexibility in the choice of Lagrangian coordinates. A.

Time Variant P a r t

The method uses an implicit scheme with central differencing over t i m e , t . Thus z P ( a , b ) i s determined a t a s e r i e s of stations in t i m e , distinguished by the intfger , p. Knowledge of velocity values, enables z P + ' ( ab) , t o be found from z V t f l , at a midway station p + Z through the numerical approximation

zp+'= zPt

TZ!+"

(error order

7

3

Zltt)

(14)

Brennen and Whitney

z!,

where 7 i s the t i m e interval. Acceleration v a l u e s , , needed in the f r e e s u r f a c e condition (Section 46)a r e approximated by (z!"~ ~ : " ~ ) /(re r r o r o r d e r r4Z + + t t ) . Thus the main p a r t of the solution involves finding P ; ' ' ~ knowing zP,z : - ' ~ and t h e i r previous values

-

.

The f i r s t t i m e s t e p (from p = 0 to p = 1) r e q u i r e s a little special qttention. C l e a r l y z o ( a ,b) is chosen t o f i t the required initial conditions. But f u r t h e r information is r e q u i r e d on a f r e e s u r f a c e which will enable the accelerations in that condition to be found ( s e e Section 4 C ) . B.

Spatial Solution

A method of the present type i s r e s t r i c t e d t o a finite body of fluid, S. However, S, could be p a r t of a l a r g e r o r infinite m a s s of fluid if an "outer" approximate solution of sufficient a c c u r a c y was available to provide the n e c e s s a r y matching boundary conditions a t the interface. The region, S , need not be fixed in t i m e . It would indeed be d e s i r a b l e , f o r example, t o "follow" a bore.

In a g r e a t number of c a s e s of widely different physical geo m e t r y including all the examples of Section 6 , it i s convenient to choose S t o be rectangular in the ( a , b ) plane. This rectangle (ABCD, F i g . i ) i s then divided into a s e t of elemental rectangles. The motion of e a c h of t h e s e cells of fluid i s t o be followed by d e t e r mining the Z values a t all the nodes.

Ib

Fig. i

NODE NUMBERING IN FREE SURFACE CONDITION :

Rectangular Lagrangian Space, . The Numbering Conventions Used 122

S , Showing the

L a g r a n g i a n S o Z u t i o n s o f ~ n s t e a dPpee ~ S u r f a c e FZows ,,. , Making the assumption of straight sides the actual a r e a of a cell in the physical plane Is

Number suffices r e f e r to the four v e r t i c e s , numbered anticlockwise; other node numbering conventions a r e shown in Fig. 1, If this a r e a i s to remain m a l t e r e d after proceeding in time from station p to p + 1 through Eq, (14) then

where the t e r m s on the L. 1-1. S. , second line a r e numerical corrections required to p r e s e r v e continuLty m o r e exactly and prevent accumulation of e r r o r over a l a r g e number of time steps. The num e r i c a l value of the L. H.S. a t some point In the iterative solution is termed the continuity residual, R,. Assuming linear variation in velocity along each side of the cell, evaluating the circulation around 1234 and setting this equal to the known, initial circulation, I?, yields (in the c a s e of a homogeneous fluid):

Slight hesitation i s required h e r e since, f o r validity, the Z and W values in this equation should relate to the s a m e station in time. But by choosing to a ply it a t the midway stations and substituting zP+@=Z' t (r/Z)Ztpf'the T t e r m s a r e f und to c ncel and (17) porsists when the values r e f e r r e d to a r e ZB and W R, la the c i r c u lation residual. The modification of (17) in the case of a heterogeneous fluid is delayed until section 4E. Combining ( i 6) and (17) produces the cell equation:

-

2 q

Permanent Cell Circulation T e r m

Brennen and W h i t n e y

-

(We + W,,

-

W,

-

W p ,

- z,))

if required

= 0 = RI -t iR, = R, the cell residual.

(18)

The higher o r d e r correction, included for completeness, allows the shaoe of the cell sides and the variations in velocitv alone them to be of &bic form. Without it the neglected t e r m s a r e bf o r d z r ZaWbbbr ZabWab, etc., with it they a r e of o r d e r etc. Values r e f e r r e d to a r e ZP and ,vP

+gawbbbbb,

.

Though this derivation of the cell equation is instructive, it can be obtained m o r e directly (except f o r the continuity correction) by integration of (8) over the a r e a of the cell in the (a,b) plane (using Taylor expansions about the center of the cell). Z

P

The cell equations m u s t now be solved f o r being known, in o r d e r to proceed in time.

wp*=

(U

-

iV),

In a recently published p a p e r , Hirt, Cook and Butler [ 19701 take a r a t h e r different approach in which the ( a , b ) pl&e is employed m e r e l y a s a tagging space. The equations a r e written in essentially Eulerian t e r m s , no derivatives with respect to a , b appearing. The numerical method (LINC) i s s i m i l a r to that of the MAC technique ( F r o m m and Harlow [ 19631 , Welch, [ 19661 Chan, Street and Strelkoff [ 19691 , etc.) and involves solving for the p r e s s u r e a t the center of a cell a s well a s for the vertex velocities. Advantages of the method described in the present paper are: the p r e s s u r e has been eliminated (though this may be disadvantageous in compressible flows); no special treatment i s required for cells adjacent t o bounda r i e s ; inhomogeneous density t e r m s a r e relatively easily included. However, since the LINC s y s t e m i s based on the Eulerian equations of motion, the inclusion of viscous t e r m s is m o r e easily accomplished than in the present method where such an attempt leads t o horrendous difficulties.

s.

C.

.

Boundary Conditions

To complete the specifications, a condition upon Wp + m i s required a t each of the boundar odes. This usually takes the f o r m of an expression connecting UPYl'and vpfl F o r example, solid boundaries, whether fixed o r moving in t i m e , may be prescribed by a function, F ( X , Y , t) = 0. Then the required relation is

.

Lagrangian S o Z u t i o n s o f Unsteady Free S u r f a c e Flows

Dynamic f r e e surface conditions a r e simply constructed f r o m Eqs. (1). If, for example, the only extraneous force is that due to gravity, g, in the negative Y direction, the condition on a f r e e surface such as AB, Fig. 1 , is XttXa t (ytt f g)ya

T 8 --(' aa

: -

XaaYa (X, + Y2o 3/2 ) aa a

where T is the surface tension if this is required. Unlike the field Eqs. (8) o r (18) these boundary conditions may not be homogeneous in a l l the variables. In a given problem only the boundary conditions a r e altered by different choices of typic 1 length, h (perhaps an initial water depth), and typical time, in the above example. Then, using the s a m e l e t t e r s for say the dimensionless variables$ g and ~ / pin Eq. (20) would be r e placed by 1 and S = ~ / p g h The numerical f o r m of that condition used a t a f r e e surface node such as 0 (Fig. 1) is:

.

(XI - X J ~ ( U : + ' ~ -

u:-"\

+ (YI - Y3)

p

p+1/2

(Vo

P-1/2

- vo

+

7)

where P i s a s s e s s e d a t each node as

and the accelerations have been replaced b~ She expressions given in Section 4A. Again, Eq. (21) r e l a t e s uo+' to v:+'l2 since all other quantities a r e known.

If the liquid s t a r t s f r o m r e s t a t t = 0 (as in the examples of Section 6 ) then difficulties a t the singular point t = 0 can be avoided by choosing to ? p l y the condition at t = r / 4 r a t h e r than t = 0. Using Ztt = 2Zt /r and Z = ZO + (r/4)2;I2 a t that station the special boundary condition becomes

Brennen and W h i t n e y

in the c a s e of z e r o surface tension. D.

Method of Solution

*

It remains to discuss how the equations may be solved to find P+1/2 a t every node. Due to the non-linear t e r m s in (18) and some boundary conditions a s well as to the fact that a good estimate of wP+'l2can be made f r o m values at previous time stations, a simple iterative o r relaxation scheme was employed. Such a method involves visiting each cell in turn and adjusting the W values a t its vertices in s u c h a way that repetition of the p r o c e s s reduces the cell residuals, R, to negligible proportions, But, on a r r i v a l a t a particular cell, t h e r e a r e an infinite number of ways in which its four vertex values can be altered in o r d e r to dissipate the single cell residual. However, experience demonstrated that a procedure based 4 ) was superior in convergence on the following changes (A W,,2, and stability to any of the others tested:

Here w i s a n overrelaxation factor and A i s the a r e a of the cell, which i s unchanged with time and given by the expression (15). T h e s e incremental changes have a simple and meaningful physical interpretation. As can be s e e n from Fig. 2, they a r e a combination of two changes, one representing pure stretching and the other pure rotation, which dissipate respectively the continuity and circulation components of the residual. Having visited each and every c e l l , the boundary conditio s were then imposed. Where these w e r e given in the f o r m A.U p+l&+ B . v ~ * ' / ~ +C = 0 = R,, A,B,C being constants and RBthe residual, the following changes were made, the choice being based upon experience:

The whole p r o c e s s was then repeated to convergence,

E. Inhomogeneous Fluid In a non-dispersive, inhomogeneous fluid, p(a,b) , which i s inodependent of t i m e , will be prescribed t h r ~ u g hthe initial choice of Z (arb). Indeed in many c a s e s it will be convenient to choose Z O in

Lagrangian S o Z u t i o n s of U n s t e a d y Free S u r f a c e F l o w s

Fig. 2(a). The Cell in the Reference Plane (a,b)

INCREMENTAL VELOCITY CHANGES WHICH DISPERSE THE CONTINUITY RESIDUAL, R, (PURE STRETCHING)

INCREMENTAL VELOCITY CHANGES WHICH DISPERSE THE CIRCULATION RESIDUAL, R, (PURE ROTATION)

Fig, 2(b), The Cell in the Physical Plane (X,T)

such a way that p is some simple analytic function of (a,b), This i s particularly desirable because by substituting for p , pa, p b in Eq. ( 9 ) , this can then be integrated over a cell a r e a ( a s in Section 4B) to produce a convenient additional t e r m , 8p*L/2on the L. H.S. of the will depend upon that cell Eq. (18). Since the expression for '8" P choice of p(a,b) an example will illustrate this. If p i s to be constant along the f r e e surface, A B , Fig, 1, and along the bed, CD, it may be possible to choose Z 0 such that p is a linear function of b , say p = pco (1 + yb) where y = p A B / p C D - 1 and b = 1 on AB, Then,

Brennen and Whitney

where p = y ~ b / ( i+ yb34), b34 being the b value on s i d e 34 of the cell and Ab the difference a c r o s s each and every cell. The f i r s t t e r m i s of o r d e r p, the second o r d e r p2. The boundary conditions a r e usually identical to the homogeneous case.

A.

Accuracy

If the cell equation, (18), i s used without the higher o r d e r spatial correction, an indication of the e r r o r s due to neglected higher order spatial derivatives can be obtained by assessing the value of that correction and inferring i t s effect upon the final values of W. Unfortunately, the m e s h distribution and m e s h s i z e required f o r a solution of given accuracy will not be known a p r i o r i and can only be arrived a t either by t r i a l and e r r o r o r by using some technique of rezoning. The l a t t e r method in which cells a r e subdivided where and when thp, violence of the motion demands i t , can be difficult to program satisfactorily ahd has not been attempted thus fat. E r r o r s due to higher o r d e r temporal derivatives a r e most easily r e Jated by ensurin that, f o r each cell, both T W ~ - W Z , -and~ T ~ W ~ - W ~ ~ /a r~e cZo m ~ f -o rZt a b ~l y l e s s than unity. A workable rule of thumb can be devised in which a suitable T for a particular time step i s determined f r o m the W and Z values of the preceding step.

Lagrangian S o l u t i o n s o f U n s t e a d y Free S u r f a c e Flows

B.

Stability of Cell Relaxation

Suppose the central m e m b e r , cell A , of the group of cells shown in Fig. 3 contained a residual RA which was then dissipated according to the relations (23;. Transfer functions, DAB,DAC,etc., will describe the residual changes, ARBSetc. , in the surrounding cells where

Fig. 3,

2-Plane

F o r example

where AA i s the a r e a of cell A. F o r convergence of the relaxation method it is clearly necessary that the w for each cell be chosen so that all w (D a r e significantly l e s s than unity. It i s instructive to inspect the case in which all the cells a r e roughly geometrically similar in the Z plane. Then

I

Brennen and W h i t n e y

where d l , d 2 a r e the lengths of the cell diagonals. F o r square c e l l s , y , = y O= 0 and €he situation i s stable. However difficulties may a r i s e when the cells a r e very skewed o r elongated and it i s in such situations, in general, that c a r e has to be taken with the relaxation technique. C,

Observation on the Cell Equation

One feature of the basic cell equation, (18), itself demands attention. Note that without the higher o r d e r spatial correction, the residuals, R in a l l of the cells (of Fig. 3) remain unaltered when the W o r Z values at alternating points (say the odd numbered points of Fig. 3) a r e changed by the s a m e amount, Such alternating " e r r o r s " must be suppressed. Some damping is provided by the higher o r d e r spatial correction since i t i s not insensitive to these changes. But experience showed this t o be insufficient unless all the boundary conditions also inhibited such alternating "e rrors." Solid boundaries usually provide adquate damping, F o r instance, in Fig. 4(a) fluctuations in U on BC, DA and in V on BC a r e obviously barred, But the f r e e surface provides little o r no such suppression and a s will be seen in the next section this can lead to difficulties. It is of interest to note that some of the solutions of Hirt, Cook and Butler [ 19701 exhibit the s a m e kind of alternating errors.

In the MAC technique, neglected higher o r d e r derivatives of the diffusion type and with negative coefficients (a "numerical" viscosity) can lead to a numerical instability if not counteracted by the introduction of sufficient r e d viscosity. In the present method, a s with that of Hirt, Cook and Butler [ 19701 , the convection t e r m s which cause that problem a r e not present. The higher o r d e r spatial correction does contain t e r m s of diffusion o r d e r , but it cannot be directly correlated with a viscosity since viscous t e r m s a r e of a different form (i. e.

, like

1vVXyr dt). Also, the higher o r d e r spatial 2

correction has a beneficial r a t h e r than a destabilizing effect. The F r e e Surface By including previously neglected derivatives, the n u r n e r i c d f r e e surface condition (without surface tension) is found to correspond m o r e precisely to:

Lagrangian S o Z u t i o n s o f Unsteady F r e e S u r f a c e F l a w s

where Aa i s the a difference a c r o s s a cell and the second w d third t e r m s constitute truncation e r r o r s . Inspect this in the light of a linearized standing wave solution (see Section 6A), i.e., cos Y =b

+ M cos A

sin t cos k a e k b

where the variables a r e non-dimensionalized a s in Section 4C and k i s the non-dimensional wave number in the a , b plane. Then, the second and third t e r m s of Eq. (26) will be insignificant provided

respectively.

Or, in t e r m s of a wavelength, X = 2rr/k:

since A a AX, the X difference between points on the f r e e surface. The f i r s t condition states the inevitable; namely, that the solution will be hopelessly inaccurate for (a,b) plane wavelengths comparable with the mesh-length Aa. Given that the f i r s t condition holds then the second says that r -0. -

FIG. 4 ( a )

FIG.

4(c)

A

; SHELF X

FIG. 4 ( b )

FIG. 4 ( d )

L a g r a n g i a n S o Z u t i o n s o f U n s t e a d y F r e e S u r f a c e Flows

I n the examples to follow ( s e e Figs. 4(a) t o (d)) s a t i s f a c t o r y n u m e r i c a l solutions could b e obtained by ignoring a l l but one of t h e singularities The exception was the s h o r e l i n e , point A, Fig. 4(c). If i s the angle between t h e tangent t o t h e f r e e s u r f a c e at A and the horizontal then c o r r e l a t i n g the two boundary conditions yields :

.

( z ~ +=) -~ eia/(cot

cos a

+ s i n a)

(28)

Thus the sign of d e t e r m i n e s the direction of the acceleration up o r down the beach. If t h e fluid s t a r t s f r o m r e s t at t = 0 , P = 0, then ( 2tt)t,o = 0 and s u c c e s s i v e differentiation of the b a s i c equation (8) and the boundary conditions yields (for i r r o t a t i o n a l motion):

z t t t t , zatt,zbttfz t t t t t

= 0 o r co, unless a = 2 4 IT

(29) IT

Z t t l t t t ,Zatttt ,Z b t t t t = 0 o r m, unless a = 4 o r 77 These relations suggest a behavior which is logarithmically singular i n t i m e a t t = 0 unless a = r/2n, n integer. Roseau [ 19581 found s i m i l a r l o g a r i t h m i c singularities in periodic solutions f o r the g e n e r a l c a s e which excluded a = n / n and another s e t of p a r t i c u l a r angles ( s e e a l s o Lewy [ 19461). But a s y s t e m a t i c analysis of t h e s i n g u l a r behavior (especially f o r t # 0; has not a s yet b e e n completed. R a t h e r , since the relations (29) no longer n e c e s s a r i l y hold if the condition of irrotationality n e a r that point is relaxed, the p r o b l e m was circurnvented numerically by replacing the circulation condition on the s i n g l e c e l l in t h a t c o r n e r by the condition (28) a t the point A and the t i m e t = 0 was avoided by applying (28) at t = 7/4 just as was done with the general f r e e s u r f a c e condition (Section 4C). Note that s t r o n g singularities could b e introduced by unsuitable mapping to t h e ( a ,b) plane.

VI. SOME RESULTS INCLUDING COMPARISONS WITH LINEAR SOLUTIONS A.

Lagrangian Linearized Solutions

L i n e a r i z e d solutions to the Lagrangian equations a r e obtained by substituting X = a + Y = B + q into t h e equations of continuity and motion and neglecting all multiples of derivatives of 5 and q. F o r incompressible and irrotational p l a n a r flow the Cauchy-Riemann conditions f a = - qb, = qa r e s u l t s o t h a t f + i q , and t h e r e f o r e Z - c (where c = a + ib) is a n analytic function of c. In the absence

c,

cb

Brennen and W h i t n e y

of s u r f a c e tension the f r e e s u r f a c e condition reduces to ( g = 1 i n the dimensionless v a r i a b l e s )

ttt .t gq, = 0

(30)

only when the additional assumption t h a t q t t