The motion in a viscous incompressible fluid of a vortex pair toward a rigid plane wall on which slip is allowed is considered. It is shown that the centroids of vorticity do not approach the wall monotonically, and there is some rebound at a rate depending upon the viscosity and initial separation of the vortices

Saffman, P. G.

English

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BRBEF COMMUNICATIONS 
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urpose of this Brief Communications section is to present important research results of more limited scope than regular 
artic es appearing in Physics of Fluids A. Submission of material of a peripheral or cursory nature is strongly discouraged. Brief 
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abstract limited to about 100 words. 
Approach of a vortex pair to a rigid free surface in viscous fluid 
P. G. Saffman 
Applied Mathematics, California Institute of Technology, Pasadena, California 91125 
(Received 3 October 1990; accepted 26 November 1990) 
The motion in a viscous incompressible fluid of a vortex pair toward a rigid plane wall on 
which slip is allowed is considered. It is shown that the centroids of vorticity do not approach 
the wall monotonically, and there is some rebound at a rate depending upon the viscosity and 
initial separation of the vortices. 
Saffman’ considered the perpendicmar approach of a 
symmetrical vortex pair of equal and opposite two-dimen- 
sional vortices toward a plane wall in inviscid incompress- 
ible fluid. It was shown that the vortices must approach the 
wall monotonically in the absence of viscous effects. Suppose 
the wall is y = 0 and the vortices are in the first and second 
quadrants. The vorticity w(x,y,t) is antisymmetrical about 
the line of symmetry x = 0, i.e., 
w(x&t) = -w( -x,y,t) . (1) 
Let I’ andJ, both positive, be the total strength and height of 
the centroid above the wall of the vorticity in the first quad- 
rant as follows: 
r= OD 
J-s 
-“dxdy, 0 0 jj=’ m ss - ro 0 wydxdy. 
In the absence of viscosity, 
C-0 
dt- ’ 
and it was shown that 
(2) 
(3) 
(4) 
Id- -L 
z- 20 ’ J 
-v(OY)‘dy 
’ 
(5) 
where v(x,y) is they component of velocity. Thus &/dt < 0 
for all time and the vortex pair approaches the wall mono- 
tonically. 
It is known’ that this result is false in a viscous fluid with 
no slip at the wall. A boundary layer is induced, which sepa- 
rates, injecting vorticity into the fluid that causes the ap- 
proaching vortex pair to bounce back from the wall in a 
manner that depends only weakly on the viscosity. There is a 
question of whether the vortex pair rebounds in a viscous 
fluid if the surface is free, i.e., the fluid can slip but the tan- 
gential stress is zero. This describes a vortex pair incident 
vertically upward on an air-water interface. In the limit of 
zero Froude number, the surface can be modeled by a rigid 
plane. The purpose of this Brief Communication is to show 
theoretically that in this limit there is some rebound and that 
the vorticity centroid does not approach the wall monotoni- 
cally, even though no boundary layers are created, and con- 
firm the numerical evidence for rebound, e.g., Peace and 
Riley.3 (The effect of surface waves at finite Froude number 
is beyond the scope of this work. ) 
When the wall is rigid but stress free, the boundary con- 
ditions on y = 0 are 
vzo, du=O, 
JY 
where u is the x component of velocity. Slip at the wall, i.e., 
u # 0, is allowed. From symmetry, we have on x = 0, 
av u=o, -=o. 
ax (7) 
It follows from (6) and (7) that3 
w=O,. on x=0 and i=O. (8) 
Consider I,, the horizontal component of the hydrodynamic 
impulse of the vorticity in the first quadrant, 
I, =rjT= - 
SJ 
- oydxdy. (9) 
0 0 
The vorticity evolves according to the equation _ 
am a0 ati --I -u-v 
at v-+vv2w. ax ax (10) 
The fluid is supposed incompressible so that 
%+$=o. (11) 
On differentiating (9), we have, using ( 10) and ( 11) and the 
boundary conditions on II, v, and w, , 
r$+j$= [J[ -y(u$+u$)+vyV2ti]dx& 
= 
JJ( 
-&(YUw) -5 (YV@> 
+ VW + vyV2w 
> 
dx dy 
JJ( 
a% ah ZZ vw+vy- a2 + vy- ar’ > dxdy, (12) 
where the double integrals are over the first quadrant. It is 
assumed that w vanishes exponentially at infinity, so that 
there are no contributions from boundary integrals at infin- 
ity. 
Consider now, separately, the three terms in the last line 
of (12). First, 
984 Phys. Fluids A 3 (5), May 1991 0899-8213/91/050984-02$02.00 Q 1991 American Institute of Physics 984 
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s IVW dx dy 
dl d. a 1 = --u*--((uv) ---uz 
ax 2 ay ax 2 
dxdy 
vi dy , (13) 
where v0 = ~(0, y). 
Second, 
a2w 
Yv ax2 -dxdy= --Y 
since vorticity must diffuse out of the first quadrant and 
there cannot be a local minimum so that do/dx > 0 on x = 0, 
Y>O. 
Third 
*y au? J fidxdy=vj-&=(y$j-$]dxdy=O 
(15) 
on integrating with respect toy. Thus 
dy<O, (16) 
and the vertical component of the first moment of the vorti- 
city distribution decreases monotonically. 
However, it does not follow from ( 16) that djVdt < 0, 
because 
dF’ -ati m aLTo 
dt= --y 0 dxx,o s I 
dy - Y 
I I 0 ay y=o 
dx<O, 
(17) 
since the integrands are positive. Hence dJ/dt is the sum of -_ 
terms of opposite sign, and there is no reason why the height 
of the centroid above the wall should decrease monotonical- 
ly. 
Indeed, we expect that J will asymptotically increase. 
To see this, note that in a way similar to that by which (16) 
was derived, we can show that 
dx>O. (18) 
Hence the vorticity moves asymptotically away from the 
centerlinex = 0, and the right-hand side of ( 16) will tend to 
zero. It follows that as t+ 00, 
rj?- const (19) 
and the continuing decrease in F due to vorticity diffusion 
through the wall will lead to an eventual increase inJ and an 
apparent rebound from the wall in a way which will depend 
upon the value of the viscosity and the initial distribution of 
vorticity. Note that the inviscid flow field satisfies the 
boundary conditions (6) and (7), so that no boundary lay- 
ers appear. 
An estimate of the trajectory after rebound is obtained 
by making the rough approximations 
ry=K, dr vr dZ l- -= ---2’ dt Y dt=-i$ 
(20) 
5 
4 i 
0’ I I 8 
0 1 2 3 4 5 
X 
FIG. 1. Rebounding trajectories given by asymptotic formulas of Eq. (21); 
-, Re = 50, - - -, Re = 100. Peace and Riley: 0, Re = 50, X, Re = 100. 
Ohring and Lugt: + , Re = 50, Fr =i 0.356. 
where K - r\$Co, and PO, -To are the initial values of I and 2. 
These give 
, z--i,+=1og 
8av 
The curves given by Eq. (21) are plotted in Fig. l’for t > 0, 
corresponding to rebound with the minimum distance from 
the wall assumed to occur at J = X = I,, for two Reynolds 
numbers Re = IO/y = 50 and 100. Distances have been 
scaled so that 2, = 1. Also shown on the curve are numerical 
data given by Peace and Riley3 for the same Reynolds 
numbers taken from Fig. 4 of their paper. The agreement is 
remarkably good. Results from a numerical study4 allowing 
for surface deformation at a Reynolds number of 50 and a 
Froude number of 0.356 are also plotted. Note that the nu- 
merical data plot the point of vorticity maximum, which 
does not necessarily coincide with the vorticity centroid, and 
is plotted for the vortices moving away from the wall. 
ACKNOWLEDGMENTS 
I wish to thank Dr. A. Leonard for bringing this topic to 
my attention. 
The research was supported by U.S. Department of En- 
ergy Grant No, DE-FG03-89ER25073. 
‘P. G. Saffman, J. Fluid Mech. 92,497 ( 1979). 
“J. K. Harvey and F. J. Perry, AIAA J. 15,50 (1971). 
‘A. J. Peace and N. Riley, J. Fluid Mech. 129,409 ( 1983). 
“S. Ohring and H. J. Lugt, David Taylor Research Center Report No. 
89/013, 1989. 
985 Phys. Fluids A, Vol. 3, No. 5, May 1991 Brief Communications 985 
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Approach of a vortex pair to a rigid free surface in viscous fluid

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Approach of a vortex pair to a rigid free surface in viscous fluid

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