Free-streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored, vortices in an inviscid incompressible fluid. If each vortex has area A and the separation is L, there are two possible shapes if A[1/2]/L is less than a critical value 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable while the less deformed shape is stable

Baker, G. R.

Saffman, P. G.

Sheffield, J. S.

English

Caltech Authors - Main

J .  Fluid Mech. (1976), vol. 74, part 3, pp .  469-476 
Printed in  Great Britain 
469 
Structure of a linear array of hollow vortices 
of finite cross-section 
By G. R. BAKER, P. G. SAFFMAN AND J. S. SHEFFIELD 
Applied Mathematics, California Institute of Technology, Pasadena 
(Received 2 September 1975) 
Free-streamline theory is employed to construct an exact steady solution for a 
linear array of hollow, or stagnant cored, vortices in an inviscid incompressible 
fluid. If each vortex has area A and the separation is L, there are two possible 
shapes if A*/L is less than a critical value 0.38 and none if it  is larger. The stability 
of the shapes to two-dimensional, periodic and symmetric disturbances is con- 
sidered for hollow vortices. The more deformed of the two possible shapes is 
found to be unstable while the less deformed shape is stable. 
1. Introduction 
The recent observations by Brown & Roshko (1974) of organized vortex 
structures in the turbulent mixing layer have rekindled interest in the hydro- 
dynamics of arrays of parallel line vortices. Moore & Saffman (1975) argued that 
the spacing of the vortex structures was controlled by the fact that there is an 
upper limit on the line density of a linear array of vortices of finite cross-section 
in non-viscous incompressible flow. When the vortices come too close, the induced 
straining fields are too intense for the individual vortices to exist in a steady state. 
However, they restricted their analysis to uniform vortices with constant vor- 
ticity in the cores, and the critical density or spacing was determined by an 
approximate argument (which was however supported by numerical work) 
because exact analysis was too hard. 
It turns out that if the vortex cores are hollow or stagnant, so that the vorticity 
is concentrated into vortex sheets on the surfaces of the vortices, then the prob- 
lem can be solved exactly by the free-streamline theory of inviscid, incom- 
pressible, two-dimensional flow, and the purpose of this paper is to present the 
calculation as a contribution to the theory of vortices. We see no direct physical 
application of the results, but similar calculations for two-dimensional arrays 
may be of interest in the theory of uniformly rotating superfluid helium, and the 
exact results provide a means of checking the approximate argument of Moore & 
Saffman. A similar calculation was carried out (before the present work was 
done) by Hill (1975) for a single hollow vortex in a uniform straining field. 
470 G. R. Baker, P. G. Saffman and J .  8. Shefield 
+=O 
u=o  
B X 
FIGURE 1. The physical plane for a regular array of vortices with 
fore-and-aft symmetry. 
2. The physical plane 
We consider an infinite linear array of identical vortices lying on the x axis 
with centres at nL, n = 0, k 1, _+ 2, . . . . Each vortex is hollow or has a stagnant 
core. In  steady flow, constant pressure inside the cores requires that the fluid 
speed has a constant value, qo say, on the boundary of each vortex. The circula- 
tion r about each vortex is related to qo by 
= pqo, (2.1) 
where P is the perimeter of each vortex. 
At large distances, the array looks like a vortex sheet of strength 2U,, where 
urn = 4rl.L. (2.2) 
R= Um/q,, = &P/L. (2.3) 
The array is characterized by the dimensionless ratio 
We shall calculate a unique steady solution for 0 < R < 1 in which each vortex 
has fore-and-aft symmetry, i.e. is symmetrical about the x axis and the line 
parallel to the y axis through its centre. It can be shown (see appendix) that no 
solutions with this symmetry exist for R > 1 and that reflexional symmetry 
about the centre implies fore-and-aft symmetry. 
The limit R = 0 corresponds to an array of point vortices or a single vortex in 
unbounded fluid, according as the limit is reached by P -+ 0 or L -+ to. R = 1 
gives avortex sheet in which each vortex is pulled out longitudinally and squeezed 
sideways to lie along a length L of the x axis. Notice that in the limit R = 1 
and the limit P = 0 the area A of each vortex is zero. 
The deformation of the cores is conveniently measured by P/A+, which has the 
minimum value of 2nt for a circle and becomes large with the eccentricity. We are 
interested in how the deformation depends upon the spacing for vortices of given 
size and strength. The area A is a more basic measure of the size than the peri- 
i 7 r b  
+= 
47 1 
D E , A  B 
,k  B 
meter and A*/L is a dimensionless quantity that specifies the relative spacing of 
the array. The procedure is to calculate P/A4 and A+/L as functions of R,  and 
by eliminating R obtain the deformation in terms of the spacing. 
The physical plane is shown in figure 1. Because of the symmetry it suffices 
to  calculate the flow inside the contour ABCDE. Either the direction or magni- 
tude of the velocity is known on the contour, and themethods of free-streamline 
theory can therefore be applied by mapping the potential plane into the hodo- 
graph plane. 
3. The mappings 
We introduce the complex variable z = x + i y ,  the complex potential 
w = q5 + i$, the complex velocity 
u - i v  = dw/dz = q e-ie, (3.1) 
(3.2) 
and the hodograph variable 
Q = log (qo/q) + iI9 = T + i0, say. 
The potential and hodograph planes are shown in figure 2. B is a stagnation 
point because of the symmetry. The Schwarz-Christoffel transformations 
ir ir w = - l o g [ ( ~ + 1 ) ~ - ( ~ - l ) ~ ] - - l o g 2 ,  2n 4n (3.3) 
Q = - log [{(b - 1) (6 + l)}a - {(b + 1) (6 - 1)}4] + 4 log (6 - b) + log 2 (3.4) 
transform the interiors of the contours into the upper half of the 6 = E + ir plane, 
with E -+ = - 1, C +  < = 1, B -+ E = b, and A +  5 = 00, where = -00, D -+ 
b = (1 + R4)/2R2. (3.5) 
The physical plane follows from integrating 
The quadrant of the vortex surface from D to C is mapped into the part of 
472 C. R. Baker, P. G. Saffman and J .  8. Shefield 
I I 
0 0.38 
Ai/L 
FIGURE 3. Perimeter length as a function of inverse distance between the cores. Variables 
are normalized with A).  The seven dots on the bottom half of the curve are #he values for 
R = 0.1 (0.1) 0.7. 
the real 5 axis from = - 1 to 5 = 1. Making the substitution .$ = - cos 2h, we 
find for the parametric equation z = Zfh) = X ( h )  +iY(h)  of the vortex with 
centre a t  the origin 
1L - - \I T1* ) U l l l  1 = - \I - 1 r - J  i511111 - 
27r 27r 
where 0 < h < 2n gives the complete perimeter. 
The vortex is obviously circular as R -+ 0, and flattens to the slit 
-+L < x < +L as R +  I. 
The perimeter P is 2RL. The area A is found by numerical integration, which 
gives AIL2 as a function of R. Figure 3 shows a plot of PIA3 against At/L. Note 
the maximum value of A*/L for a value R = R, + 0.805. 
Linear array of hallow vortices of Jinite cross-section 473 
4. Discussion 
For a given value of A*/L, there are either two or no possible steady states. If 
hollow or stagnant-cored vortices of given size are placed in an array such that 
their separation is too small, there is no possible steady state and the vortices 
presumably disintegrate. For the vortex of largest area for given L, the length of 
the major axis is 0.71L and that of the minor axis is 0-25L. 
A similar, although not identical, behaviour holds for a single hollow vortex 
of area A and circulation I' in a uniform irrotational deformation with strain 
rate 8. Hill (1975) has shown that there are either one, two or no steady states 
according as e A / r  < 0.03, 0.03 < eA/F < 0-1 or 0-1 < eA/I'. Following Moore & 
Saffman (1975), we can estimate the critical value of A*/L for an array from the 
result for a single vortex by putting E = ar/6L2 in the critical value for the 
single vortex. This gives an estimate of 0.43 for the critical value of At/L.  The 
exact value is 0.38, so that the approximate argument of Moore & Saffman (1975) 
appears to be reasonable. The exact value of P/A* for the critical vortex is 4.2 for 
the array and 4.5 for the single vortex. 
The existence of two possible configurations of the array suggests that at least 
one of them is unstable, and this should be the most deformed. We shall now 
verify this idea, by investigating the linear stability to infinitesimal perturba- 
tions of an array of hollow vortices, and demonstrate the existence of a class of 
disturbances to which the array is unstable for R > R, and stable for R < R,. 
5. Stability of an array of hollow vortices 
We shall restrict attention here to infinitesimal periodic disturbances with 
reflexional symmetry about the centre of each vortex, which leave the centres 
undisplaced, because our interest lies in the stability to variations of shape. 
Stability of the array to disturbances which alter the positions of the vortices, 
i.e. of the type considered by Lamb (1932,s 166) for point vortices, is a matter 
for further study. (The effect of finite core size might have a bearing on the fact 
that Brown & Roshko (1974) did not appear to find the Lamb-type instability.) 
It is sufficient to consider the strip -QL < x < $L, y > 0 and to use as in- 
dependent co-ordinates the undisturbed velocity potential and stream function. 
The strip is 0 < + -= $r, -GO < @ < 0. A deformation of the boundary is de- 
scribed by the curve 
The disturbance to the velocity potential is denoted by @($, @, 1 ) .  Then 
= s(+,t), o < + < ir. (5.1) 
a w / a p  + a w . / a p  = 0, (5.2) 
@ - t o  as $-+--co. (5.3) 
For a hollow vortex, the pressure must be constant on its boundary; this gives a 
dynamic boundary condition 
474 
on 0 < cj5 < *I?, 4 = 0. In  addition there is a kinematic condition 
G. R. Baker, P. G. Saffman and J .  S. Shefield 
a@ ias as 
a+ - pb2 at + @ 
-- -- (5.5: 
satisfied on the undisturbed vortex. 
normal modes of the form 
The symmetry requires that the disturbance has period +I' in q5. We look for 
CD = c Q) 47rncj5 +wt)exp($Inl+), 
= m ancos(--l, 4nnq5 +wt), 
- m  
6 
b - cos (4nq5/r) - (5.7) 
where w is to  be found. Inserting (5.6) and (5.7) into the boundary conditions and 
carrying out some straightforward algebraic manipulations, we obtain the 
recursion relation 
a,,, + (1:' sinhp u -t n)2 
for - co < n < co, where b = coshfl, p = -log 2R2 and a = wI'/dnq$. The eigen- 
values u are determined by the requirement that a,+O as n+ & co. If (+ is 
complex, the motion is unstable. 
In  the limit = co, R = 0, the eigenvalues are obviously 
(+=n-tl+nl*, n = + 1 , + _ 2  ,.... (5.9) 
It is easy to  verify directly that these are the natural frequencies of a single 
hollow vortex. For p large but not infinite, the eigenvalues can be expanded as 
power series in e-8, and i t  is found that I+ remains real provided that the regular 
perturbation scheme remains valid. 
For smaller p, numerical means need to be employed, and the method of 
Laplace (Jeffreys & Jeffreys 1950, p. 486) is convenient. For given fl, wk assume 
a value of u and calculate al/ao and a-Ja0 as functions of (+ such that a, -+ 0 
as 1.1 -+ +a. Substitution into the recursion relation (5.8) for n = 0 gives an 
equation determining u. The details are as follows. For n positive, define 
(5.10) 
It can be shown that a2 = O(n-2) for large n when a, decays as n + co. From 
the recursion relation, 
(5.11) 
The asymptotic behaviour of a,' gives a starting value from which i%+(v,fl) can 
be calculated numerically. We proceed similarly for the a_, (n > 0), defining 
(5.12) 
Linear array of hollow vortices of finite cross-section 475 
and calculating a;(r ,  p). Because of the symmetry of the recursion relation 
a u , P )  = a$( -u,P). (5.13) 
The recursion relation for n = 0 gives 
f(a2,P) = ao+(cr,P)-tag(cr,P) = 2*eBcoshp-2. (5.14) 
Since the left-hand side can be found numerically as a function of r, the roots of 
(5.14) are obtained in a straightforward manner as functions of p. 
Note that the roots occur in pairs, k u. The roots are known for large ,8, so 
the procedure is to follow the roots numerically as ,8 decreases. The smallest 
positive root crl(p), say, turns out to be the one of interest. As p decreases, crl 
decreases from 1 - 1/2* at ,8 = co to zero at /3 = 0.434. This value can be found 
analytically as the recursion relation can be solved in closed form (using genera- 
ting functions) when cr = 0. For ,8 less than 0.434, equation (5.14) is found to have 
roots with a2 < 0, demonstrating that there is an exchange of stabilities. It can 
be shown that the other roots remain real. 
The critical value of p at which the array becomes unstable to disturbances of 
the type considered here gives the same value of R, 0.805, as that a t  which At/L 
is a maximum, thereby demonstrating that, when there are two possible con- 
figurations, the more deformed is unstable to disturbances for which the less 
deformed is stable (cf. Moore & Saffman 1971). 
This work was supported by the U.S. Army Research Office, Durham, under 
contract DAHC 04-75-C-0009. 
Appendix 
Consider a member of the linear array of hollow or stagnant vortices with the 
geometry as shown in figure 4. We assume periodicity of the array and reflexional 
symmetry only. 
Using the hodograph variable defined by (3.2) we then require that 7 satisfies 
Laplace's equation in the strip ABCDEF and the following boundary conditions: 
T =  0 along CD, 
7 has period *I' in q5, 
rN- logR as $+--co. 
Moreover, we want 7 to have the correct behaviour at  the stagnation points 
B and E .  Noting that dwldx N (w - w,)* at a stagnation point, we can separate out 
the singular behaviour of r at such points by using functions which behave 
locally as required. Including terms ensuring correct asymptotic behaviour and 
without violating (A2), we have 
-+log 
476 G .  R .  Baker, P .  G .  Saffman and J .  S. Shefield 
FIGURE 4.Physical and potential planes for an array with reflexional symmetry but not 
necessarily fore-and-aft symmetry. $, is the value of the stream function at  the stagnation 
points B and E. 
where H satisfies Laplace's equation and (A2), and is bounded on the strip. 
Clearly 
a0 4nn4 H = exp(4nn$/r) +B,cos- 
a=O 
Now for r to satisfy (A 1) we require A,  = 0, and hence there is fore-and-aft 
symmetry. Further, 
4nn9 
0 = -log 2R - &log 
so that the B, are all uniquely determined. Note that for the correct asymptotic 
behaviour we require Bo = 0, and so 
Writing b = 1 + 2 sinh2 (2n$,,/I'), we find that (A 4) implies 
b + (b2-  l)* = l /R2. 
Since b > 1, then R c 1. 
REFERENCES 
BROWN, G. L. & ROSHKO, A. 1974 On density,&%@s and large structure in turbulent 
HILL, F. M. 1975 Ph.D. thesis, Imperial College, London. 
JEFFREYS, H. & JEFFREYS, B. S. 1950 Methods of Mathematical Physics. Cambridge 
LAMB, H. 1932 Hydkodynamics. Cambridge University Press. 
MOORE, D. W. & SAFFMAN, P. G. 1971 Structure of a line vortex in an imposed strain. 
In Aircraft Wake Turbulence and its Detection (ed. Olsen, Goldberg & Rogers), p. 339. 
Plenum. 
MOORE, D. W. & SAFFMAN, P. G. 1975 The density of organized vortices in a turbulent 
mixing layer. J .  Fluid Mech. 69, 465. 
mixing layers. J .  Fluid Mech. 64, 775. 
University Press. 


Structure of a linear array of hollow vortices of finite cross-section

J .  Fluid Mech. (1976), vol. 74, part 3, pp .  469-476 
Printed in  Great Britain 
469 
Structure of a linear array of hollow vortices 
of finite cross-section 
By G. R. BAKER, P. G. SAFFMAN AND J. S. SHEFFIELD 
Applied Mathematics, California Institute of Technology, Pasadena 
(Received 2 September 1975) 
Free-streamline theory is employed to construct an exact steady solution for a 
linear array of hollow, or stagnant cored, vortices in an inviscid incompressible 
fluid. If each vortex has area A and the separation is L, there are two possible 
shapes if A*/L is less than a critical value 0.38 and none if it  is larger. The stability 
of the shapes to two-dimensional, periodic and symmetric disturbances is con- 
sidered for hollow vortices. The more deformed of the two possible shapes is 
found to be unstable while the less deformed shape is stable. 
1. Introduction 
The recent observations by Brown & Roshko (1974) of organized vortex 
structures in the turbulent mixing layer have rekindled interest in the hydro- 
dynamics of arrays of parallel line vortices. Moore & Saffman (1975) argued that 
the spacing of the vortex structures was controlled by the fact that there is an 
upper limit on the line density of a linear array of vortices of finite cross-section 
in non-viscous incompressible flow. When the vortices come too close, the induced 
straining fields are too intense for the individual vortices to exist in a steady state. 
However, they restricted their analysis to uniform vortices with constant vor- 
ticity in the cores, and the critical density or spacing was determined by an 
approximate argument (which was however supported by numerical work) 
because exact analysis was too hard. 
It turns out that if the vortex cores are hollow or stagnant, so that the vorticity 
is concentrated into vortex sheets on the surfaces of the vortices, then the prob- 
lem can be solved exactly by the free-streamline theory of inviscid, incom- 
pressible, two-dimensional flow, and the purpose of this paper is to present the 
calculation as a contribution to the theory of vortices. We see no direct physical 
application of the results, but similar calculations for two-dimensional arrays 
may be of interest in the theory of uniformly rotating superfluid helium, and the 
exact results provide a means of checking the approximate argument of Moore & 
Saffman. A similar calculation was carried out (before the present work was 
done) by Hill (1975) for a single hollow vortex in a uniform straining field. 
470 G. R. Baker, P. G. Saffman and J .  8. Shefield 
+=O 
u=o  
B X 
FIGURE 1. The physical plane for a regular array of vortices with 
fore-and-aft symmetry. 
2. The physical plane 
We consider an infinite linear array of identical vortices lying on the x axis 
with centres at nL, n = 0, k 1, _+ 2, . . . . Each vortex is hollow or has a stagnant 
core. In  steady flow, constant pressure inside the cores requires that the fluid 
speed has a constant value, qo say, on the boundary of each vortex. The circula- 
tion r about each vortex is related to qo by 
= pqo, (2.1) 
where P is the perimeter of each vortex. 
At large distances, the array looks like a vortex sheet of strength 2U,, where 
urn = 4rl.L. (2.2) 
R= Um/q,, = &P/L. (2.3) 
The array is characterized by the dimensionless ratio 
We shall calculate a unique steady solution for 0 < R < 1 in which each vortex 
has fore-and-aft symmetry, i.e. is symmetrical about the x axis and the line 
parallel to the y axis through its centre. It can be shown (see appendix) that no 
solutions with this symmetry exist for R > 1 and that reflexional symmetry 
about the centre implies fore-and-aft symmetry. 
The limit R = 0 corresponds to an array of point vortices or a single vortex in 
unbounded fluid, according as the limit is reached by P -+ 0 or L -+ to. R = 1 
gives avortex sheet in which each vortex is pulled out longitudinally and squeezed 
sideways to lie along a length L of the x axis. Notice that in the limit R = 1 
and the limit P = 0 the area A of each vortex is zero. 
The deformation of the cores is conveniently measured by P/A+, which has the 
minimum value of 2nt for a circle and becomes large with the eccentricity. We are 
interested in how the deformation depends upon the spacing for vortices of given 
size and strength. The area A is a more basic measure of the size than the peri- 
i 7 r b  
+= 
47 1 
D E , A  B 
,k  B 
meter and A*/L is a dimensionless quantity that specifies the relative spacing of 
the array. The procedure is to calculate P/A4 and A+/L as functions of R,  and 
by eliminating R obtain the deformation in terms of the spacing. 
The physical plane is shown in figure 1. Because of the symmetry it suffices 
to  calculate the flow inside the contour ABCDE. Either the direction or magni- 
tude of the velocity is known on the contour, and themethods of free-streamline 
theory can therefore be applied by mapping the potential plane into the hodo- 
graph plane. 
3. The mappings 
We introduce the complex variable z = x + i y ,  the complex potential 
w = q5 + i$, the complex velocity 
u - i v  = dw/dz = q e-ie, (3.1) 
(3.2) 
and the hodograph variable 
Q = log (qo/q) + iI9 = T + i0, say. 
The potential and hodograph planes are shown in figure 2. B is a stagnation 
point because of the symmetry. The Schwarz-Christoffel transformations 
ir ir 
w = - l o g [ ( ~ + 1 ) ~ - ( ~ - l ) ~ ] - - l o g 2 ,  2n 4n (3.3) 
Q = - log [{(b - 1) (6 + l)}a - {(b + 1) (6 - 1)}4] + 4 log (6 - b) + log 2 (3.4) 
transform the interiors of the contours into the upper half of the 6 = E + ir plane, 
with E -+ = - 1, C +  < = 1, B -+ E = b, and A +  5 = 00, where = -00, D -+ 
b = (1 + R4)/2R2. (3.5) 
The physical plane follows from integrating 
The quadrant of the vortex surface from D to C is mapped into the part of 
472 C. R. Baker, P. G. Saffman and J .  8. Shefield 
I I 
0 0.38 
Ai/L 
FIGURE 3. Perimeter length as a function of inverse distance between the cores. Variables 
are normalized with A).  The seven dots on the bottom half of the curve are #he values for 
R = 0.1 (0.1) 0.7. 
the real 5 axis from = - 1 to 5 = 1. Making the substitution .$ = - cos 2h, we 
find for the parametric equation z = Zfh) = X ( h )  +iY(h)  of the vortex with 
centre a t  the origin 
1L - - \I T1* ) U l l l  1 = - \I - 1 r - J  i511111 - 
27r 27r 
where 0 < h < 2n gives the complete perimeter. 
The vortex is obviously circular as R -+ 0, and flattens to the slit 
-+L < x < +L as R +  I. 
The perimeter P is 2RL. The area A is found by numerical integration, which 
gives AIL2 as a function of R. Figure 3 shows a plot of PIA3 against At/L. Note 
the maximum value of A*/L for a value R = R, + 0.805. 
Linear array of hallow vortices of Jinite cross-section 473 
4. Discussion 
For a given value of A*/L, there are either two or no possible steady states. If 
hollow or stagnant-cored vortices of given size are placed in an array such that 
their separation is too small, there is no possible steady state and the vortices 
presumably disintegrate. For the vortex of largest area for given L, the length of 
the major axis is 0.71L and that of the minor axis is 0-25L. 
A similar, although not identical, behaviour holds for a single hollow vortex 
of area A and circulation I' in a uniform irrotational deformation with strain 
rate 8. Hill (1975) has shown that there are either one, two or no steady states 
according as e A / r  < 0.03, 0.03 < eA/F < 0-1 or 0-1 < eA/I'. Following Moore & 
Saffman (1975), we can estimate the critical value of A*/L for an array from the 
result for a single vortex by putting E = ar/6L2 in the critical value for the 
single vortex. This gives an estimate of 0.43 for the critical value of At/L.  The 
exact value is 0.38, so that the approximate argument of Moore & Saffman (1975) 
appears to be reasonable. The exact value of P/A* for the critical vortex is 4.2 for 
the array and 4.5 for the single vortex. 
The existence of two possible configurations of the array suggests that at least 
one of them is unstable, and this should be the most deformed. We shall now 
verify this idea, by investigating the linear stability to infinitesimal perturba- 
tions of an array of hollow vortices, and demonstrate the existence of a class of 
disturbances to which the array is unstable for R > R, and stable for R < R,. 
5. Stability of an array of hollow vortices 
We shall restrict attention here to infinitesimal periodic disturbances with 
reflexional symmetry about the centre of each vortex, which leave the centres 
undisplaced, because our interest lies in the stability to variations of shape. 
Stability of the array to disturbances which alter the positions of the vortices, 
i.e. of the type considered by Lamb (1932,s 166) for point vortices, is a matter 
for further study. (The effect of finite core size might have a bearing on the fact 
that Brown & Roshko (1974) did not appear to find the Lamb-type instability.) 
It is sufficient to consider the strip -QL < x < $L, y > 0 and to use as in- 
dependent co-ordinates the undisturbed velocity potential and stream function. 
The strip is 0 < + -= $r, -GO < @ < 0. A deformation of the boundary is de- 
scribed by the curve 
The disturbance to the velocity potential is denoted by @($, @, 1 ) .  Then 
= s(+,t), o < + < ir. (5.1) 
a w / a p  + a w . / a p  = 0, (5.2) 
@ - t o  as $-+--co. (5.3) 
For a hollow vortex, the pressure must be constant on its boundary; this gives a 
dynamic boundary condition 
474 
on 0 < cj5 < *I?, 4 = 0. In  addition there is a kinematic condition 
G. R. Baker, P. G. Saffman and J .  S. Shefield 
a@ ias as 
a+ - pb2 at + @ 
-- -- (5.5: 
satisfied on the undisturbed vortex. 
normal modes of the form 
The symmetry requires that the disturbance has period +I' in q5. We look for 
CD = c Q) 47rncj5 +wt)exp($Inl+), 
= m ancos(--l, 4nnq5 +wt), 
- m  
6 
b - cos (4nq5/r) - (5.7) 
where w is to  be found. Inserting (5.6) and (5.7) into the boundary conditions and 
carrying out some straightforward algebraic manipulations, we obtain the 
recursion relation 
a,,, + (1:' sinhp 
u -t n)2 
for - co < n < co, where b = coshfl, p = -log 2R2 and a = wI'/dnq$. The eigen- 
values u are determined by the requirement that a,+O as n+ & co. If (+ is 
complex, the motion is unstable. 
In  the limit = co, R = 0, the eigenvalues are obviously 
(+=n-tl+nl*, n = + 1 , + _ 2  ,.... (5.9) 
It is easy to  verify directly that these are the natural frequencies of a single 
hollow vortex. For p large but not infinite, the eigenvalues can be expanded as 
power series in e-8, and i t  is found that I+ remains real provided that the regular 
perturbation scheme remains valid. 
For smaller p, numerical means need to be employed, and the method of 
Laplace (Jeffreys & Jeffreys 1950, p. 486) is convenient. For given fl, wk assume 
a value of u and calculate al/ao and a-Ja0 as functions of (+ such that a, -+ 0 
as 1.1 -+ +a. Substitution into the recursion relation (5.8) for n = 0 gives an 
equation determining u. The details are as follows. For n positive, define 
(5.10) 
It can be shown that a2 = O(n-2) for large n when a, decays as n + co. From 
the recursion relation, 
(5.11) 
The asymptotic behaviour of a,' gives a starting value from which i%+(v,fl) can 
be calculated numerically. We proceed similarly for the a_, (n > 0), defining 
(5.12) 
Linear array of hollow vortices of finite cross-section 475 
and calculating a;(r ,  p). Because of the symmetry of the recursion relation 
a u , P )  = a$( -u,P). (5.13) 
The recursion relation for n = 0 gives 
f(a2,P) = ao+(cr,P)-tag(cr,P) = 2*eBcoshp-2. (5.14) 
Since the left-hand side can be found numerically as a function of r, the roots of 
(5.14) are obtained in a straightforward manner as functions of p. 
Note that the roots occur in pairs, k u. The roots are known for large ,8, so 
the procedure is to follow the roots numerically as ,8 decreases. The smallest 
positive root crl(p), say, turns out to be the one of interest. As p decreases, crl 
decreases from 1 - 1/2* at ,8 = co to zero at /3 = 0.434. This value can be found 
analytically as the recursion relation can be solved in closed form (using genera- 
ting functions) when cr = 0. For ,8 less than 0.434, equation (5.14) is found to have 
roots with a2 < 0, demonstrating that there is an exchange of stabilities. It can 
be shown that the other roots remain real. 
The critical value of p at which the array becomes unstable to disturbances of 
the type considered here gives the same value of R, 0.805, as that a t  which At/L 
is a maximum, thereby demonstrating that, when there are two possible con- 
figurations, the more deformed is unstable to disturbances for which the less 
deformed is stable (cf. Moore & Saffman 1971). 
This work was supported by the U.S. Army Research Office, Durham, under 
contract DAHC 04-75-C-0009. 
Appendix 
Consider a member of the linear array of hollow or stagnant vortices with the 
geometry as shown in figure 4. We assume periodicity of the array and reflexional 
symmetry only. 
Using the hodograph variable defined by (3.2) we then require that 7 satisfies 
Laplace's equation in the strip ABCDEF and the following boundary conditions: 
T =  0 along CD, 
7 has period *I' in q5, 
rN- logR as $+--co. 
Moreover, we want 7 to have the correct behaviour at  the stagnation points 
B and E .  Noting that dwldx N (w - w,)* at a stagnation point, we can separate out 
the singular behaviour of r at such points by using functions which behave 
locally as required. Including terms ensuring correct asymptotic behaviour and 
without violating (A2), we have 
-+log 
476 G .  R .  Baker, P .  G .  Saffman and J .  S. Shefield 
FIGURE 4.Physical and potential planes for an array with reflexional symmetry but not 
necessarily fore-and-aft symmetry. $, is the value of the stream function at  the stagnation 
points B and E. 
where H satisfies Laplace's equation and (A2), and is bounded on the strip. 
Clearly 
a0 4nn4 H = exp(4nn$/r) +B,cos- 
a=O 
Now for r to satisfy (A 1) we require A,  = 0, and hence there is fore-and-aft 
symmetry. Further, 
4nn9 
0 = -log 2R - &log 
so that the B, are all uniquely determined. Note that for the correct asymptotic 
behaviour we require Bo = 0, and so 
Writing b = 1 + 2 sinh2 (2n$,,/I'), we find that (A 4) implies 
b + (b2-  l)* = l /R2. 
Since b > 1, then R c 1. 
REFERENCES 
BROWN, G. L. & ROSHKO, A. 1974 On density,&%@s and large structure in turbulent 
HILL, F. M. 1975 Ph.D. thesis, Imperial College, London. 
JEFFREYS, H. & JEFFREYS, B. S. 1950 Methods of Mathematical Physics. Cambridge 
LAMB, H. 1932 Hydkodynamics. Cambridge University Press. 
MOORE, D. W. & SAFFMAN, P. G. 1971 Structure of a line vortex in an imposed strain. 
In Aircraft Wake Turbulence and its Detection (ed. Olsen, Goldberg & Rogers), p. 339. 
Plenum. 
MOORE, D. W. & SAFFMAN, P. G. 1975 The density of organized vortices in a turbulent 
mixing layer. J .  Fluid Mech. 69, 465. 
mixing layers. J .  Fluid Mech. 64, 775. 
University Press. 


Structure of a linear array of hollow vortices of finite cross-section

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