It is shown that a symmetrical vortex pair consisting of equal and opposite vortices approaching a plane wall at right angles must approach the monotonically in the absence of viscous effects. An approximate calculation is carried out for uniform vortices in which the vortices are assumed to be deformed into ellipses whose axis ratio is determined by the local rate of strain according to the results of Moore & Saffman (1971)

Saffman, P. G.

English

Caltech Authors - Main

J .  Fluid Mech. (1979), vol. 92, part 3, p p .  497-503 
Printed in Qreat Britain 
497 
The approach of a vortex pair to a plane surface 
in inviscid fluid 
By P. G .  SAFFMAN 
Department of Applied Mathematics, California Insitute of Technology, 
Pasadena 
(Received 18 September 1978) 
It is shown that a symmetrical vortex pair consisting of equal and opposite vortices 
approaching a plane wall at right angles must approach the wall monotonically in 
the absence of viscous effects. An approximate calculation is carried out for uniform 
vortices in which the vortices are assumed to be deformed into ellipses whose axis 
ratio is determined by the local rate of strain according to the results of Moore & 
Saffman (1971). 
1. Introduction 
The behaviour of a vort'ex pair in a ground effect is of some interest for aerodynamic 
and environmental reasons. The simplest theoretical treatment models the vortex 
pair by two rectilinear line vortices of equal and opposite circulation at  the same height 
above an infinite plane. Each vortex moves in the velocity field induced by the other 
and the images in the plane. The trajectories are easily computed (see Lamb 1932, 
p. 223) and are found to be the curves 
where x = 0 is the plane of symmetry, y = 0 is the wall, and 2x0 is the initial separation 
of the vortices when infinitely far from the plane. According to this elementary 
solution the vortices approach the wall monotonically as they separate from one 
another, the final asymptotic distance from the wall being x,,. 
Dee & Nicholas (1968) reported on flight test observations of trailing vortices pro- 
duced by an aircraft flying close to the ground. They found moderate agreement 
between the observed trajectories and those predicted by the elementary theory when 
allowance is made for additional horizontal displacement produced by a cross-wind. 
Harvey & Perry (1971) noted, however, that the theory and the observations differed 
in one noteworthy respect. In  many of the tests, the vorticesrose again after having ap- 
proached the ground, i.e. they appeared to bounce. Harvey & Perry carried out some 
wind-tunnel experiments on a single vortex shed by a half-span wing over a moving 
floor and concluded that the cause of the vortex bounce is separation of the boundary 
layer on the ground underneath the vortex. They argued that boundary-layer separa- 
tion produces a secondary vortex whose development slows down the horizontal 
motion of the primary vortex and makes it rise. It was reported that in some configura- 
tions the horizontal motion of the primary vortex could be reversed. 
Recently and in ignorance of the earlier work, Barker & Crow (1977) observed 
rebounding for a vortex pair, generated in water, approaching either the free surface 
or a horizontal rigid plane immersed in the water. They claimed that the phenomenon 
Xg(X2f y2) = x2y2, (1.1) 
0022-1 120/79/4206-6060 $02.00 @ 1979 Cambridge University Press 
17 FLM 92 
498 P. G. Saflman 
can be attributed to the effect of finite core size, which is of course neglected in the 
elementary theory. However, they gave no evidence to support their contention 
apart from reference to unpublished numerical simulations and an unsubstantiated 
statement that core deformation would lead to unsteadiness that could explain the 
effect. 
We shall present below simple arguments to show that in the framework of an 
inviscid theory, the rebounding cannot be explained by finite core size and that the 
velocity of the vortices towards the wall cannot change sign. We shall also calculate 
the paths and shapes of the vortices by means of an extension of an approximation 
introduced by Moore & Saffman (1975) which was used quite successfully by them 
and others (e.g. Bilanin, Teske & Williamson 1977) in similar contexts. 
The implication of these calculations is that  the explanation of rebounding given 
by Harvey & Perry is the correct one, a t  least for rigid walls. However, the phenomenon 
was also observed by Barker & Crow when the vortices approached a free surface. 
Since the boundary layer a t  a free surface is much weaker than that a t  a rigid 
boundary (and is in a sense continually separating) it is most unlikely that secondary 
vortices would be produced in this case as required by Harvey & Perry’s explana- 
tion. Unless surface contamination in Barker & Crow’s experiment had the effect 
of making the free surface behave like a rigid boundary with respect to slip, their 
observations that the nature of the boundary had little effect upon the gross 
properties of thevortex motion would suggest an inviscid explanation. Since an inviscid 
explanation dose not exist, the observations on rebounding a t  a free surface present 
an unsolved problem. 
2. Theoretical considerations 
From symmetry, it is sufficient to consider the vorticity o in the positive quandrant 
bounded by the wall (y = 0 )  and the axis of symmetry (x = 0).  The motion is two- 
dimensional with velocity components (u, v). The total vorticity in the quadrant is 
given by 
The circulation r is conserved when viscosity is neglected. Vjscous effects would pro- 
duce vorticity a t  the wall and cause diffusion of vorticity across the axis of symmetry. 
The components ( Iz ,  I,) of hydrodynamic impulse of the vorticity are 
The co-ordinates ( E , j j )  of the vorticity centroid are 
E = -r,/r, g = Z/r. (2.3) 
It follows from Euler equations for incompressible inviscid fluid that 
Integrating by parts, using the kinematic boundary conditions 
u = O  on x = O ,  v = O  on y = O ,  (2.5) 
Approach of a vortex pair to a surface 499 
which follow from symmetry, and using (2.3) to replace I, by y, we obtain 
Similarly, 
Here vo= v(0, y ) ,  uo = u(2,O). (2.8) 
Supposc that the vortex pair is initially at  a large distance from the wall x = 0. 
Then (2.6) predicts that ij decreases at  a rate depending upon the velocity field on the 
axis of symmetry. As the pair approaches the wall, the vortices begin to separate 
under the action of the image vorticity in the wall. This is also shown by (2.7), according 
to which Z increases when uo is not negligibly small on the wall. Eventually, the 
vortices will be so far apart that v,, is negligible. The centroids of the vortices will then 
have reached an asymptotic distance from the wall and continue parallel to it. 
Since the right-hand side of (2.6) is always negative, inviscid dynamics predicts 
that the vorticity centroids approach the wall monotonically, however large the 
vortices or their deformation may be, and asymptote to a line parallel to the wall. 
We cannot assert that the final motion of each vortex is steady. It is possible that 
the final state is a periodic or aperiodic oscillation. However, if the final motion is 
steady, then the time reversibility of the Euler equations ensures that the final dis- 
tance from the wall is equal to one-half the initial separation. 
We should mention the possibility that the rebounding of the vortices is apparent 
and not real. Hooker (1936) claimed that the observed increasing spacing ratios of 
K&rmSn vortex sheets is due to a divergence between the centroid of vorticity and the 
centre of rotation of a vortex as it spreads. The centre of rotation is the instantaneous 
stagnation point of the flow as measured by a given observer, depends of coarse on 
the observer’s speed, and would be indentified as the natural centre of the vortex by 
common flow-visualization techniques. Thus deformation of a vortex as it approached 
the wall might lead to an apparent movement of the vortex away from the wall. 
However, we shall estimate this effect in our approximate calculation to be described 
in the next section and show that it does not occur. 
3. An approximate calculation for uniform vortices 
The deformation of the vortices and its effect on their trajectories can be estimated 
simply if we assume that the vortices are elliptical with axis ratio and orientation 
given by the theory of Moore & Saffman (1971) for steady uniform vortices in uniform 
steady straining fields. We apply this theory by assuming (cf. Moore & Saffman 1975) 
that the appropriate value of the uniform straining field is the rate of strain at  the 
centre of the vortex due to the other vortex and the images in the plane. The configur- 
ation of the vortex pair and its image is then as shown in figure 1. The centre of the 
vortex in the first quadrant is at  (X, g). The semi-major and semi-minor axes have 
length a and b, respectively and the major axis is inclined at angle 8 to the x axis. 
We require the velocity and rates of strain induced a t  (Z, g) by the other three 
vortices. Consider a uniform elliptical vortex, of strength K ,  bounded by 
X2/a2 + Y2/b2 = 1. (3.1) 
17-2 
500 P. G .  Saflman 
.@ 
L 
r 
y@- 
L 
X 
- r  
FIGURE 1. Configuration of the vortex pair and its image in the well y = 0.  
An elementary calculation shows that the complex potential w(2) (2 = X + i Y )  
outside the ellipse is 
N = - i ~ / 2 7 ~ ( c +  +e-25), (3.2) 
where Z = ccoshc, c2 = a2-b2, 5 = g+ir. (3.3) 
The complex velocity a t  the point ( X ,  Y )  relative to  the axes of the ellipse is 
i K  
U - i V  = _. - = --ee-C. 
Denote the rate-of-strain tensor eij = &(aui/axj + auj/axi) a t  (X, Y )  by 
(3.4) 
(3.5) 
Then (3.6) 
Relative to axes Oxy which make an angle q5 with those of the ellipse, the velocity 
components (u, v )  and corresponding components (a ,  p)  of the rate-of-strain tensor 
are given by 
A - iB = (d2w/dz2) = ( i K / n c 2 ) e - c  cosech 5. 
u - iv = ( - iK/r ic)e-Ee@, a - ip = (iK/7TC2)e-c ezi$ cosech 5. (3.7) 
Approach of a vortex pair to a surface 501 
Quadrant Strength X Y e 
2 - r  25 cos 0 25 sin 0 e 
4 -r  2g sin 0 - 29 cos 0 e-n 
3 r 2(52 + 32)t cos (e - tan-lJ/Z) - 2(52 + y2) t  sin (8 - tan-lg/z) - e 
TABLE 1. Parameters for the calculation of the velocity and rate-of-strain at the centre of the 
vortex in the first quadrant. 
The values of K ,  X ,  Y and Q for the contribution a t  ( E ,  I) by the vortices in the 
second, third and fourth quadrants are listed in table 1. 
Substitution of these values into (3.7) and summing over the three vortices gives 
the velocity (4, V )  and rate of strain (01, p )  a t  the centre of the vortex. The motion and 
shape of the vortices are determined by the equations 
dZ/dt = ZL, d j j ld t  = V, (3.8) 
E ( E  - 1) nab 
P n  = - ( E 2 + / 7 2 ) k  r 8 = 3 tan-l =+- a 4' (€2+ I) (€+ 1 )  (3.9) 
Here E = a / b  is the axis ratio of the ellipse. Equation (3.8) assumes that the vortex 
moves with the velocity at it5 centre. Equation (3.9) is the result of Moore & Saffman 
(1971) for the shape of a uniform vortex in a uniform straining field. The size of the 
vortex is conserved during the motion; hence 
ab = R2, (3.1) 
where nR2 is hhe area of the vortex. Note that (3.9) has no solution if 
(nab/F)(E2+P2)* > 0.15. (3.11) 
When ij = co, we have the case of a single vortex pair. Then 8 = in-, the major axes 
If this occurs, the vortex is expected to disintegrate. 
of the pair are parallel, and the equation for E reduces to 
-- (E 1)2 - I- [ 1 + $  (4]-". 
€2+ 1 
(3.12) 
The solution of this equation giving 6 as a function of R/Z is shown in figure 2, together 
with the values of b / 3  and - 4nZV/F. Note that solutions exist for all R/Z. Further, 
b / Z  -f 1 as R/Z --f co, i.e. the vortices become long and elongated and approach one 
another more closely as the distance between their centres tends to zero. The speed of 
the pair is less than the velocity of two point vortices a t  their centres. This behaviour 
is qualitatively similar to that of the exact solution for two hollow vortices (Pockling- 
ton 1894). Notice that the condition (3.11) is never violated, so this simple theory 
predicts that a vortex pair never disintegrates however close the vortices are. Of 
course, the theory may not be accurate for values of a/b  much larger than one. 
We can estimate the position of the centre of rotation by finding the point a t  which 
the velocity produced by the vortex in the first quadrant is equal and opposite to 
the velocity of its centre. Inside the ellipse, the stream function in Cartesian co- 
ordinates of the velocity relative to the centre is 
7,k = (r/2nabc2) [X2(b2 - ab) - Y 2  (a2 - ab)].  (3.13) 
502 P. G .  Saffmn 
1.2 
6 
5 
$ 4  
3 
2 
1 
I .o 
0.8 9 
x 
0.6 5 
9 
L' 
0.4 1'; 
. 
t= d 
I 
0.2 
0 
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 
RJX 
FIGURE 2. Properties of a vortex pair as functions of R I Z .  Each vortex is appoximated by an 
ellipse of major axis a and minor axis b with centres 25 apart. R2 = ab. x' is the displacement of 
the apparent centre of rotation relative to the centre. 
0 '  i I I I I I 
1 2 3 4 5 6 
K 
FIQURE 3. Approximate calculation of vortex pxir in a ground effect for R / x ,  = I .  -, path 
of centre of vortex; ---, path of apparent centre of rotation. Initial, intermediate and final 
shapes are shown. 0, position of apparent centre of rotation. 
It follows easily that the co-ordinates of the centre of rotation relative to the centre 
and the x, y axes are 
x' = (r/r) (a  + b )  [Ti sin 8 cos 6(a - b )  - @(a cos2 8 + b sin2 8)] ,
y' = (r/I')(a+b)[Ti(asin28+b cos28)-~sin8cos8(a-b)]. } (3.14) 
Approach of a vortex pair to a surface 503 
It  is assumed in this calculation that the centre of rotation is inside the vortex, which 
turns out to be the case. The values of x'/x are also shown in figure 2. It will be noted 
that for distorted vortices, E > 2 say, the centre ofrotationisdisplacedfrom the centroid 
of vorticity by a significant amount, which might make the vortices appear to be 
further apart than they actually are. 
Results for a vortex pair approaching a wall are shown in figure 3 for R/xo = 1.0. 
These were obtained by numerical integration of (3.8) using a multistep Adams- 
Moulton integration scheme with time step 5xi/2I'. A modified Euler scheme was 
used to specify the initial values for the multistep procedure. The values of Ti and G 
were found by solving (3.9) and (3.7) by Newton's method with the values given by 
table 1.  The integration was completely straightforward. Similar results are found 
for other values of R/xo. For smaller R/xo, the deformation of the vortices is reduced 
and the trajectories of the centroid and apparent centre of rotation are closer. The 
converse holds for larger R/xo. For values of R/xo up to 2.0, for which the axis ratio 
approximates 4, the trajectories of the centroids are indistinguishable from those of 
point vortices. The path of the apparent centre of rotation always approached the 
wall monotonically. The calculated paths are symmetrical about the line x = y and 
the approximation does not allow for the existence of oscillations which might occur 
in reality. In any event, it now seems clear that the explanation of the observed 
rebounding must be elsewhere than the inviscid dynamics of finite vortices. 
The work was supported by the U. S. Army Research Office, Durhen (DAAG 
29-78-C-001 I )  and the Department of Energy (EY-76-S-03-0767). 
R E F E R E N C E S  
BARKER, S. J. & CROW, S. C. 1977 The motion of two-dimensional vortex pairs in a ground effect. 
BILANIN, A. J., TESKE, M. E. & WILLIAMSON, G. G. 1977 Vortex interactions and decay in 
DEE, I?. W. & NICHOLAS, 0. P. 1968 Flight measurements of wing tip vortex motion near the 
HARVEY, J .  K. & PERRY, F. J. 1971 E'lowfield produced by trailing vortices in the vicinity of 
HOOKER, S. G. 1936 On the action of viscosity in increasing the spacing ratio of a vortex sheet. 
LAMB, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press. 
MOORE, D. W. & SAFFMAN, P. G. 1971 Structure of a line vortex in an imposed strain. In  Aircraft 
Wake Turbulence (ed. J. H. Olsen, A. Goldburg & M. Rogers), pp. 339-354. Plenum. 
MOORE, D. W. & SAFFMAN, P. G. 1975 The density of organised vortices in a turbulent mixing 
layer. J .  Fluid Mech. 69, 465-473. 
POCKLINGTON, H. C. 1894 The configuration of a pair of equal and opposite hollow straight 
vortices of finite cross section mwing steadily through fluid. Proc. Cumb. Phil. SOC. 8, 
J .  Fluid Mech. 82, 659-671. 
aircraft wakes. A.I .A.A.  J .  15, 250-260. 
ground. R.A.E. Tech. Rep. no. 68007. 
the ground. A.I .A.A.  J .  9, 1659-1660. 
Proc. Roy. SOC. A 154, 67-89. 
179- 187. 


The approach of a vortex pair to a plane surface in inviscid fluid

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The approach of a vortex pair to a plane surface in inviscid fluid

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