V O L U M E 67, N U M B E R 19 P H Y S I C A L R E V I E W L E T T E R S 4 N O V E M B E R 1991

Particle Transport by Angular Momentum on Three-Dimensional Standing Surface Waves

Makoto Umeki Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan

(Received 13 March 1991)

The transport of fluid particles on three-dimensional standing surface waves is studied by the inviscid irrotational theory. When two internally resonant modes are excited simultaneously, a second-order vortical drift velocity emerges on the surface proportional to the angular momentum. Particles can move periodically, quasiperiodically, or chaotically over a wide distance depending on the size of the container. The presented theory is applied to the smallest system having a transport effect which has been dis­covered in capillary-wave experiments.

PACS numbers: 47.35.+i, 05.60,+w, 47.20.Tg

The study of mixing and transport in fluids from a ki­nematic point of view confirmed a close relation to the theory of dynamical systems and chaos [1], Analyses were mainly applied to temporally periodic or steady flows. Among many hydrodynamic systems, tendril whorl [2], blinking vortex [3], and ABC flows [4] are prototypes exhibiting Lagrangian turbulence. Recently, Ramshan-kar, Berlin, and Gollub [5] provided a new example of Lagrangian turbulence: the transport of particles on a fluid surface by standing waves. They observed that not only spatiotemporal chaos but also almost stationary and weakly modulated wave states produced a significant drift of particles characterized by a normal or fractional Brownian motion. It is well known that a two-dimen­sional nonlinear traveling (gravity [6] or capillary [7]) wave can produce a horizontal drift of particles; however, standing waves have been very little investigated. The ex­istence of a boundary layer [6,8] may produce such streaming; however, its magnitude is very small. There are no theories on transport by three-dimensional waves. The purpose of the study reported in this paper is to pro­vide an explanatory mechanism for this horizontal drift phenomenon in terms of the theory of inviscid irrotational fluid. It shows that, even when the surface wave is oscil­lating exactly periodically with no spatial modulation, particle transport can occur due to the angular momen­tum.

Recent theoretical and experimental studies [8-14] of low-dimensional mode competitions in parametrically ex­cited surface waves have made it clear that there are two kinds of mixed states, i.e., one without an angular mo­mentum M and the other with it. When two internally resonant modes are excited, the surface displacement may be expressed as

2

77(f,x,>>)=6tfi2 nsinit + 0, V,Oc , y ) + Oie2) , (1)

where a\ =*K\ tanh(Kjd), K\ denotes a wavelength, d the depth of the fluid, t the time nondimensionalized by the fundamental angular frequency ft)/, and y/j the eigenfunc-tion. The latter state, denoted by Mb, arises from the breaking of the symmetry (0i,02)—* (0i-r-7r,02) or (0j, 02 + /r). The periodic mode-competition state, in which

the amplitude r, and the phase 0, may be described by the slow time variable T, arises via a Hopf bifurcation of the Mb state which appears on one side of the codimension-2 bifurcation point in the space of two pa­rameters, i.e., forcing amplitude and frequency. A hor­izontal drift has been found to exist when the Mb or mode-competition state is excited.

We consider, in a simple case having this drift velocity, that two degenerate modes (ra , /0 = l and (« ,m) = 2, m > n >: 0 integers, in a square cylinder of length / and depth d, are resonantly excited. The linear eigenfunction of mode im,n) is represented by y/mnix,y) = [ ( 2 — 8mo) x(2 — 8no)]l/2co$(m7rx)cos(nxy), where the coordinates x,j> £ [0,1] are made dimensionless by /. The second-order modes are denoted symbolically as ( l l ) = ( m ± m , n±n), (22) = in ±n,m ± m ) , and (12) = (m ±n,m ±n) [15]. A velocity field on a fluid surface is given by

ivx,vy) = ( a x , dy)(l>(t, x,yyz)\z = nu,x,y), (2)

where

<p==s^</>i(t)y/i(x,y)cosh[Kj(z+d)]/cosh(Kid) (3)

is a velocity potential. Expanding (2) up to the second order leads to

vx,y(T,t,x,y) = eo2v\Xyy(T,t,x,y)

+ e2a2lv2x,y(^,x,y) + V$XJ(T,t,x,y)] .

(4)

Here r is an Oie2) slow time variable, v\Xyy iv2X,y) denotes the first- (second-) order velocity, and the param­eter a=a/l denotes the ratio of the wavelength to the size of the container. The drift and oscillatory velocities are denoted by v$x,y and v\x,y,v°x,y, where the time average of the latter vanishes with respect to the period of the fast surface oscillation. The velocity is expressed as

(v\x,v{y)= (dx,dy)((/>]y/i+foVi) , (5)

0 / T / ( T ) C O S U + 0 / ( T ) ] , for i - l , 2 , (6)

(v?x,v$y)*~M(dy,-dx)v(x,y), (7)

¥ ( J C , J 0 — X c / > s in[ ( /w±/ i )^x]s in[ ( /w±/ i )^v] (8) iG<12>

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V O L U M E 67, N U M B E R 19 P H Y S I C A L R E V I E W L E T T E R S 4 N O V E M B E R 1991

(seeRef. [15]),

£ c lri2sin(2f + 20ity,-(x,jO + Z c/r22 sin ( 2 / + 202>V/(*>>>)

/ e < 11 > / G < 2 2 >

+ 2 Cir\r2sin(2t + 0\ + 02)Vi(x,y) . / e ( i 2 >

(9)

The linear approximation reduces to the first-order veloci­ty v\Xyy, and the second-order velocities v2x?y arise from nonlinearity of the kinematic and dynamic surface boundary conditions. For the details of the coefficients c7-and others, the reader may refer to a forthcoming paper [16]. The actual angular momentum associated with the surface displacement was shown to be proportional to M sssr\r2sin(0\ — #2)* which we call simply an angular mo­mentum [14]. The drift velocity is divergence-free and vortical due to the surface deformation, and the vorticity

n is given by n ^dxUy — dyvx = — M A t d j ) , where ¥ is a stream function for the drift and A=d /dx2 + d2/ by1. It may be of interest that there is a rotational motion of the fluid particles although the flow itself is ir-rotational. Since the order of the oscillatory flow is lower than that of the drift flow, the familiar perturbation analysis in chaos theory is of little use where the time-dependent perturbation has a smaller magnitude than the unperturbed flow, which is usually independent of time.

k0.54

0.8

0.6

0.4

0.2

0.0

1 ' ' ' ' 1 '-i-^rr~

-J I I L_

0.2 0.4 0.6 0.8

0.52

FIG. 1. Poincare plots of particle trajectories on the surface starting at three different points, (0.1,0.1), (0.3,0.3), and (0.5,0.5). (a) Case ^oSBS0.63, doubly periodic motion; (b) case AQ^Q.60, triply periodic motion; (c) enlargement of the trajectory starting at (0.5,0.5); (d) case AQ^O.SI, random motion.

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When two internally resonant modes 1 and 2 begin to compete on a slow time scale, the fluid flow can be quasi-periodic or chaotic with a short period of oscillation and a long (or infinite) period of competition. Trajectories of fluid particles in each case were examined by numerical integration of Oc,j>) = (vXyvy) with the velocity field (4). The temporal evolution of the slowly varying amplitudes (pi,qi)—ri(sinOi,cosOi), / = 1,2, was described by the two-mode Faraday-resonance system [14],

dx (/>/,?/) = d d

dq{ ' 9/7/ H-a(pi,qi) , (10)

for 1=1 ,2 , where the Hamiltonian H is given by

H- 1 Z Wp? + qft+A*ip?-q?)+ {Mp? + q?)2] 1-1

+ jC(pt + qt)(pi + ql)+{D(p}q2-p2qi)\

(11)

with the normalized amplitude Ao, frequency offsets /?i =0.55, / ? 2 = — 0.45, and damping coefficient a = 0 . 1 . We may assume that the motion of particles does not de­pend directly on the method of excitation. As nonlinear coefficients, the values in the deep-water approximation tanh(K-irf) = 1, calculated by the average Lagrangian method, were used: ,41 =0.25 , C = 0 . 3 5 , Z) = - 0 . 7 5 . Stationary, periodic, and chaotic states are realized, re­spectively, by v4o=0.63, 0.60, and 0.57.

Numerical Poincare plots of trajectories of representa­tive particles for the two-dimensional mode ( m , « ) = ( l , 0) and 6=0 .1 are shown in Fig. 1. The coefficients in (8) and (9) are chosen as C2o=co2==^/2/8, c\\ = ( 3 — V2)/14, and cf\ =0 .5 . For the drift velocity field, four vertices are unstable saddle points, and the center of the square is neutrally stable. When the Mb state is excited, the motion of the fluid is periodic and the motion of the parti­cles is quasiperiodic, i.e., a slow drift superimposed on the fast orbital motion. When periodic mode competition occurs, the wave oscillation is doubly periodic, and the motion of particles is triply periodic. When the mode competition becomes chaotic, particles also move ran­domly.

Some properties of the flow dependent on the parame­ter e should be emphasized for exactly periodic flows. First, since the drift velocity is proportional to 62, so is the frequency of the drift of the particles. Second, as e becomes larger, the magnitudes of the drift and oscilla­tion velocities become closer. Third, the divergence of the flow (4) is not zero but periodic both temporally and spa­tially. After expansion of the velocity (7) at the center (x,y) =(0.5 ,0 .5) and some manipulation, the frequency of the drift motion fd near the center can be obtained as McDK€2a2. As long as e remains small, the ratio of fd to the fundamental frequency normalized to 1 is small and no phase locking is observed. Figure 2 shows the Poin­

care plots for 6=0 .8 and ^4o==0.63. Although this value may be too large to assume the validity of truncation up to Oie2), an interesting phenomenon is found: "spread­ing" of particles, in which tori of particle trajectories disappear. A particle starting at (0.5,0.6) spreads out­ward rotationally and passes through apparently stochas­tic and regular (resonancelike) regions. This spreading is observed numerically up to 6—0.3; however, it is unclear whether spreading disappears completely at nonzero e or becomes invisibly weak as e approaches zero.

Figure 3 shows Poincare sections of exactly periodic flows for the three-dimensional modes ( m , « ) = ( 3 , 2 ) [Fig. 3(a)] and (6,1) [Fig. 3(b)]. Parameters are chosen as r i = r 2 = l, 0\ = 0 , 0 2 = T T / 2 . Case (a) stands for m— n and (b) for the two-dimensional limit m^>n. Chaotic dy­namics is found in both cases. The routes to chaos are breaking of large-scale tori by subharmonic resonance in Fig. 3(a) and transverse intersections of heteroclinic or­bits surrounding small-scale cells leading to itineracy be­tween local attractors in Fig. 3(b). The Poincare map is conservative in the sense that the square is mapped onto itself, but locally dissipative or expanding by the same mechanism as the spreading in Fig. 2.

In conclusion, particle transport on three-dimensional standing surface waves is studied. The second-order drift velocity is derived for the internally resonant modes (m,n) and (n,m) in a square container. Numerical in­tegrations for two-dimensional modes explore not only ro­tational motion but also spreading of particles for large e. Three-dimensional flows are shown to produce chaotic dynamics, and two different routes to chaos are pointed out.

The present analysis can be extended to other situa­tions. For nearly resonant modes in a circular container

0.0 0.2

- I 1 1 1 I I L _

0.4 0.6 0.8

FIG. 2. Poincare plots of particle trajectories starting at (0.5,0.6) for 6 = 0 . 8 and , 4 o = 0 . 6 3 . Tori disappear and particles spread outward.

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9x9 regular lattice, (a) (m,rt)=(3,2), 6=0.2; (b) (myn) = (6,1), 6=0.25. The iteration number of the plot is 1000.

[9], although the angular momentum of each mode asso­ciated with its completely degenerate components tends to vanish due to linear damping [14], there exists a second-order drift velocity related to a modes coupling with both of the two dominant modes, and it is propor­

tional to M. Extension of the present analysis to the spatiotemporal

chaotic state, such as the experiments of capillary waves by Ramshankar, Berlin, and Gollub [5], by introducing the spatial and temporal modulation is also possible but remains to be done. In their experiment, the ratio of the wavelength to the size of the container was very small, of order 0.03. However, the drift with a correlation for an almost stationary state may be understood by the rela­tively low-dimensional mixed state with angular momen­tum.

This work was supported by a Grant-in-Aid (02952-085) for Scientific Research from the Ministry of Educa­tion, Science and Culture in Japan.

[1] J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport (Cambridge Univ. Press, Cam-brdige, England, 1989).

[2] D. V. Khakhar, H. Rising, and J. M. Ottino, J. Fluid Mech. 172,419 (1986).

[3] H. Aref, J. Fluid Mech. 143, 1 (1984). [4] T. Dombre, U. Frisch, J. M. Greene, H. Henon, A. Mehr,

and A. M. Soward, J. Fluid Mech. 167, 363 (1986). [5] R. Ramshankar, D. Berlin, and J. P. Gollub, Phys. Fluids

A 2, 1955 (1990). [6] M. S. Longuet-Higgins, Philos. Trans. Roy. Soc. London

A 245, 535 (1953). [7] S. J. Hogan, J. Fluid Mech. 143, 243 (1984). [8] S. Douady, J. Fluid Mech. 221, 383 (1990). [9] S. Ciliberto and J. P. Gollub, Phys. Rev. Lett. 52, 922

(1984); J. Fluid Mech. 158, 381 (1985). [10] F. Simonelli and J. P. Gollub, J. Fluid Mech. 199, 471

(1989). [11] E. Meron and I. Procaccia, Phys. Rev. Lett. 56, 1323

(1986); Phys. Rev. A 34, 3221 (1986). [12] J. W. Miles, J. Fluid Mech. 146, 285 (1984). [13] Z. C. Feng and P. R. Sethna, J. Fluid Mech. 199, 495

(1989). [14] M. Umeki, J. Fluid Mech. 277, 161 (1991); M. Umeki

and T. Kambe, J. Phys. Soc. Jpn. 58, 140 (1989); M. Umeki, Ph.D. thesis, University of Tokyo, 1991 (unpub­lished).

[15] Four combinations [or summation in (8)] of ( ± , ± ) are taken except for (0,0). When n =0, only one combination must be chosen.

[16] M. Umeki (to be published).

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