Hsiao C. Kao

Glenn Research Center, Cleveland, Ohio

A Note on Trapping Moving Vortices

NASA/TM—2000-209270

July 2000
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Hsiao C. Kao

Glenn Research Center, Cleveland, Ohio

A Note on Trapping Moving Vortices

NASA/TM—2000-209270

July 2000

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This report contains preliminary

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analysis proceeds.
NASA/TM—2000-209270 1

A Note on Trapping Moving Vortices

Hsiao C. Kao

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Summary

The topic of stationary configurations of point vortices, also known as vortex equilibrium, has received considerable attention in recent years. By observing numerical results, it is found that a “counterpart” of this system also

exists, in which moving vortices may be “trapped” by an inlet-like device to form a stationary pattern with no translational motion. After an intuitive explanation for the process, vortex trajectory maps based on numerical results

are presented. These maps exhibit two stationary points under the present conditions, which are the focal points of

vortex trajectories. A vortex upstream of these points, if within a certain offset range, will move towards these points

spontaneously and be captured there. This proposed device is also capable of trapping spinning vortex pairs and

triads. It is possible to impose a uniform stream at infinity, as long as the flow field is still dominated by the moving

vortices.

1. Introduction

In view of the fact that vortices are considered to be the “sinews and muscles” of fluid dynamics and often used

for aerodynamic performance improvements, the study of vortex dynamics has received enduring attention. New

applications and properties have been uncovered. In a review paper by Aref, Kadtke, Zawadzki, Campbell and

Eckhart (1988), the authors discussed the topic of vortex equilibrium, in which n point vortices of strength +1 and

m point vortices of strength –1 form a completely stationary system with no translational motion.

Numerical results obtained in this study suggest that a “counterpart” of this system also exists, in which moving

vortices may be “trapped” by an inlet-like device to form a stationary configuration with no translational motion. In

order to illustrate this situation, we go to figure 1. In this figure, a pair of spinning vortices moving from left to right

in a body of otherwise still fluid is being trapped by the device. The dashed line at the bottom is the ground surface.

Thus, there is an image system below the surface. If no device is present, this pair of free spinning vortices translates

parallel to the surface at a constant speed. When a device is introduced, an interaction takes place. In this case, the

free vortices establishes the flow field and the combined effect of the vortices and the device determines the vortex

paths. This assumption of no free stream velocity may be removed as long as the dominant flow field is still due to

free vortices. A brief discussion and some numerical results will be given later. This situation is somewhat similar to

Foppl’s standing vortex pair behind a circular cylinder (Saffman, 1992).

The vortex pair in figure 1 is of the same strength and same sense, spins about its centroid and moves downstream at the same time. If its initial position is within a certain offset distance from the device centerline, it is liable

to be trapped. A heavy black circle in the plot is a testimony of this state, in which two vortices rotate about an essentially stationary centroid. The pattern in figure 1 represents the vortex trajectory paths after 9800 time steps. Although it needed only 3200 time steps to reach the stationary zone, the extra 6600 steps were added to demonstrate

the immobility of the centroid. Here we use a spinning pair to bring out a more dynamic picture, even though a

single vortex is the main topic of this note.

We now give an intuitive explanation of this process with a single moving vortex. To do this, we refer to the

surface vorticity method for the potential flow by Martensen (1959). In addition to this original work, there are

many papers dealing with this method but we cite only Lewis (1981 and 1991). For calculating the potential flow

around one or more arbitrary bodies, this method places a continuous distribution of vorticity elements on the body

surfaces to separate the outer flow from the inner region of zero velocity. In the present case of zero free stream

velocity, the potential flow is due solely to the moving vortices. Thus, the strengths of these surface vorticity elements have to adjust continuously to satisfy the surface condition and are functions of time. Therefore, it is conceivable that at a particular instant one or more locations may exist, where the induced velocities form various sources

cancel one another giving rise to stationary points.
NASA/TM—2000-209270 2

2. Surface Vorticity Method

For the purpose of explanation and to make this presentation more complete, a brief description of this method

is given. A more complete account can be found elsewhere, for instance, in Lewis (1991). The complex potential

given in equation (1) represents the flow field of free vortices in the presence of bodies. The terms in this equation

comprise M surface vorticity elements of unknown strengths γm and N free vortices of given strengths Γj

W i i ds z z i z z m m

m

M

m j

j

N

=+ = − ( ) + − ( )j

= =

Φ Ψ ∑ ∑ γ Γ

2 2 π π

1

1 1

log log , ( )

where z = x + iy represents a point in the flow field, zm = xm +iym refers to the midpoint in the surface element sm,

whose vorticity strength per unit length is γm, and dsm is the length of this element (fig. 2). The symbol zj = xj + iyj

refers to the location of Γj. The circulation γm or Γj is defined to be positive when clockwise in accordance with

Lewis’ convention. The total number of surface vorticity elements is M, which may be the sum of several bodies.

The derivative of W gives the velocity components u and v in the x- and y-direction,

uxy ds y y

xx yy

y y

xx yy

m m

m

M

m

m m

j

j

N

j

j j

(,) = ( ) −

( ) − + − ( )

+ −

= = ( ) − + − ( ) ∑ ∑ γ

2 2 π π

2

1

2 2

1

2 2

Γ

vxy

ds x x

xx yy

x x

xx yy

m m

m

M

m

m m

j

j

N

j

j j

(,) = − −

( ) − + − ( ) − −

= = ( ) − + − ( ) ∑ ∑ γ

2 2 π π 1

2 2

1

2 2

Γ

In these two equations as well as in equation (1), variables are given in dimensionless quantities. The characteristic

length is L, the length of the body, and the characteristic velocity is Γo/L, where Γo is the representative strength of a

free vortex. If we replace x and y in equation (2), which is a field point, by xn and yn, the coordinates of the surface

element sn (fig. 2), and impose the Dirichlet condition that the internal tangent velocity component is zero, equation (2) become equations (3), which can be used to determine the unknowns γm in equation (1).

1

2

1

4

1

2

1

2

1

1 1 π

γ γ

π π

m mbnm n n

n m mn

M

jvn j

j

N

ds K s s ds

R , , , , Ksz n M

,

( ) + − 



 



 = − ( ) =

=≠ =

∑ ∑Γ L

Kss yy xx

xx yy

bnm nm n nm n

nm nm

, cos sin ( ) = ( ) ( ) − − − ( )

( ) − + − ( )

θ θ

2 2 3

and

Kss

yy xx

xx yy

vn j

nj n nj n

nj nj

, cos sin ( ) = ( ) − − − ( )

( ) − + − ( )

θ θ

2 2

Here Kb( ) and Kv( ) are two influence coefficients representing the tangential velocity components at sn induced by

other surface vorticity elements and by free vortices respectively. The symbol Rn is the radius of curvature at sn and

θn the tangential angle (fig. 2). The second term in the first equation accounts for the self-induced velocity of γn at

element sn.
NASA/TM—2000-209270 3

Equations (3) are a linear system of M equations for M unknowns, which can be solved by any standard method.

The Gaussian elimination method was used for every example shown here.

With the quantities γm determined, one can use equations (2) to calculate the convective velocities for vortex Γk

by changing z to zk, where zk is the position of Γk at the instant t. The evolution of zk is then given by

dx

dt

u dy

dt

v k

k

k = = k , .

For a system of N vortices, there are 2N such equations, whose solutions give the vortex paths. At each time step one

has to update γm, which requires solutions to equations (3). A second-order method is used here for calculating the

time development from t to t + ∆t.

xt t xt ut ut t t

yt t yt vt vt t t

k k kk

k k kk

( ) + = + −− [ ]

( ) + = + −− [ ]

∆ ∆∆

∆ ∆∆

() () ( )

( )

() () ( )

1

2

3

4

1

2

3

If the vortices are sufficiently far from the body, it is adequate to calculate vortex paths based on the above procedure. However, if they approach sufficiently close to the surface, accuracy may deteriorate, and precautions

should be taken to remedy the situation. Lewis (1991) suggested a few such remedies, which will be followed

closely in this note. Some details will be given later.

Before proceeding to solve equations (2) to (4) for a given geometry, it is essential to make a comparison

between the exact and numerical predictions. Although it is more convenient to consider a single point vortex in a

uniform stream past a circular cylinder, which has exact solutions for any given vortex position, this is not a good

comparison, because the vortex motion is partially driven by the free stream. A more stringent test for the present

investigation is to consider the trajectories of a vortex pair near an orifice in an infinite body of otherwise still fluid,

which was studied by Sheffield (1977). This case is akin to the present investigation, because the presence of free

vortices creates the velocity field, and the combined effect of vortices and the orifice determines the vortex paths.

Sheffield considered three cases and had graphical representations for two. His figures contain, however, no

numerical information, which makes comparison virtually impossible. For this reason, we recomputed the trajectories in one of his cases by solving numerically the equations given in his paper, which are shown as two dotted lines

in figure 3 for two different initial positions, corresponding to those in his figure 2. Although these curves are

numerically generated, the governing equations are based on the Schwarz-Christoffel transformation and are exact.

By contrast, two solid-line curves in figure 3 obtained by the surface vorticity method are totally numerical. The

close agreement in figure 3 suggests that the present method is capable of predicting vortex trajectories in the presence of two bodies (two parts of the orifice). A few particulars are perhaps pertinent. The smallest interval between

elements is 0.03L, with L being the half width of the orifice, and the total number of vorticity elements is 144 without recourse to the symmetry property. (Only the lower half is plotted in fig. 3.) A slightly better agreement can still

be expected, if surface elements are placed more densely near the sharp corners.

3. Geometries of Proposed Device

In order to find locations of stationary points for moving vortices, we begin with the geometry. Two such configurations are shown in figures 4. The device in figure 4(a) is labeled as model A, which is symmetrical both longitudinally and laterally and consists of essentially two forward sections of an NACA-0012 airfoil. The elements at the

nose region are picked directly from the coordinates for the NACA-0012 and those in the midsection are obtained by

curve fitting to ensure a convex shape. Since the arrangement of surface vorticity elements may affect the appearance of stationary points, coordinates for one such distribution with 17 elements in the forward quadrant are given in

table 1. Other distributions with more surface elements have also been used in computations and will be discussed

later.
NASA/TM—2000-209270 4

To show that the appearance of stationary points is not an exclusive property of model A, model B is introduced

in figure 4(b). This is a simpler geometry than model A, consisting of two parallel surfaces and two semicircular

caps. Geometric quantities such as curvatures and tangent lines are known exactly, except at junctions between a

circular arc and a straight line.

4. Computational Procedure

Models in figure 4 all lie in the upper half plane with a dashed line at y = 0 to denote the ground surface. The

existence of the ground surface is equivalent to imposing an image system in the lower half plane. Most calculations

in this investigation were performed in the upper half plane with the image system included in the influence coefficients in equations (3). See, for example, Lewis for details. A few sample cases were, however, obtained in the entire space to verify the correctness of the half-plane method.

Information needed for computation is as follows. The vortex strength is of unity and the sense is negative

(counter-clockwise). It moves from left to right. The time step for a regular run is 0.00375, but it may be reduced to

less than 0.000375 if necessary. The computational procedures for two or more vortices are essentially the same as

those for a single vortex.

Under the conditions that the vortex is not near the body and the ratio of the local thickness to the length of

local element is greater than 0.64 (an empirical quantity by Lewis), accurate numerical solutions can be expected.

This ratio is amply satisfied in the present investigation. However, if a vortex moves close to the surface, inaccurate

solutions begin to appear. Thus, steps have to be taken to alleviate this problem. Fortunately, stationary points

present in the present study are not very near the surface. Nonetheless, we still gave heed to Lewis’ precautions. For

instance, we made sure that the ratio between the distance from the vortex to the surface and the length of the local

element was much greater than 2.0, and the net circulation around the body perimeter due to the externally located

vortex was zero in accordance with Kelvin’s theorem. If the latter condition was not satisfied, an adjustment was

made either according to Lewis’ treatment (Chap. 8) or by distributing the unbalanced circulation evenly among all

surface elements.

5. Results and Discussion

5.1. Models A and B with a Single Moving Vortex

Based on the surface element distribution given in table 1, a series of computations for a moving vortex was

made, whose results were plotted in figure 5 in the form of a trajectory map. The dotted lines (short dashed lines)

represent vortex paths and the long dashed line at the bottom refers to the ground surface. A vortex in the space

above the topmost dotted line can slide over the upper body smoothly with some distortion near the surface. Below

this line there are two distinct regions, where vortex paths converge to two stationary points. A vortex anywhere in

the upper region will converge to the upper stationary point and be trapped there. Likewise, a vortex in the lower

region will be trapped by the lower stationary point.

The convergence process to the stationary point is somewhat analogous to the convergence of an infinite series.

As a vortex comes close to the stationary station, the translational velocity approaches zero (10–8~10–12) in the limit

and each translational distance is smaller than the one before it.

The fact that a moving vortex in various upstream positions converges spontaneously to the stationary point is

somewhat unexpected. One may, therefore, assume that this stationary point is also an equilibrium point. To put it to

a test, we displaced a vortex to various arbitrary locations around the stationary point and computed their paths. The

finding that these paths all converged to the stationary point in the cases tested is an indication that this assumption

is probably correct.

In the computations for figure 5, no values of bound circulation or the Kutta conditions were imposed. As a

result, the calculated circulations around two bodies were not zero and were, in fact, a part of the solution. By specifying different values of bound circulations, it is possible to obtain different solutions. These solutions were found to

contain no stationary point. Therefore, it seems that for a set of surface elements distributed around an inlet-like

device such as model A, there is only one solution that accommodates stationary points.
NASA/TM—2000-209270 5

Lewis’ corrective measures stated previously were all adopted here. To satisfy Kelvin’s theorem of zero net

circulation in a closed loop exterior to a vortex for model A in figure 5, the unbalanced circulation for one case was

made to be zero by adjusting the vorticity values on two surface elements nearest to the final position of the stationary vortex. However, in all subsequent computations the unbalanced circulation was distributed equally among surface elements. This latter procedure is deemed to be more reasonable, since we deal with moving vortices especially

in the case of spinning. Note that imposing Kelvin’s theorem for zero circulation is not the same as making the

bound circulation to be zero. (See Lewis, 1991 for details.)

The numerical experiments conducted so far seem to indicate that the appearance of stationary points depends

not only on the geometry of the device but also on the arrangement of surface elements. For instance, one can

reconfigure the surface element distribution by decreasing or increasing the number of elements or re-arranging

elements. Moreover, even though the geometry is symmetric, the element distribution does not have to be. We

examined all these cases and found that the locations of stationary points were not fixed. A different distribution

would result in a different set of stationary points. Although the new locations are in general only slightly different

from those in figure 5, the connecting vortex paths can be substantially dissimilar. In one of these cases with 82 elements per body, there was only one stationary point situated right in front of the lower body somewhat like a stagnation point. For this reason, it is felt that the existence of stationary points is not an exclusive property of the

geometry. This raises the question whether stationary points can be easily observed in an experiment by simply installing a device like model A or B.

To demonstrate that geometries other than model A can also accommodate stationary points, a trajectory map

for model B with 94 elements per body is plotted in figure 6. This figure is seen to be rather similar to figure 5. As in

the previous case, we also re-configured the element distribution and reached the similar conclusion of shifting stationary points. Furthermore, in one case with 126 elements per body, no stationary point was found. This notwithstanding, we believe that there is sufficient evidence to suggest that some sets of vorticity elements distributed

around model B are capable of producing stationary points in the flow field.

5.2 Vortex in a Uniform Free Stream Velocity

The trajectory maps in figures 5 and 6 were based on the assumption that the body of fluid is motionless at infinity. To remove this restriction and to see the effect of including the free stream velocity, we repeated the necessary computations with the free stream velocity added. Using the free stream velocity U∞ as the reference velocity,

we can add a term in equation (1) to become

W i i ds z z i zz z m m

m

M

m j

j

N

=+ = − ( ) + − ( )j +

= =

Φ Ψ ∑ ∑ γ Γ

2 2 π π

5

1 1

log log , ( )

where z in the last term represents the free stream velocity.

Choosing the free stream to be unity, the vortex strength to be –1.5, and the remaining conditions to be the same

as for figure 6, we carried out the calculations and plotted the results in figure 7. The similarity between figures 6

and 7 is obvious with stationary points approximately at the same locations. The passageway between the space of

two bodies in figure 7 is, however, considerably larger. Therefore, the chance for a vortex being trapped is reduced.

Note that this is a rather intense vortex. As its strength decreases, the regions, in which vortex paths converge to the

stationary points, will contract and eventually disappear altogether.

5.3 Model A with Two or Three Moving Vortices

In the above we presented trajectory maps for two different models, which have the capability of trapping a

single moving vortex. It would be of interest to see whether this capability remains effective as two or more vortices

are introduced. The answer is affirmative as indicated by the sample cases in figures 8 to 10. No trajectory maps are,

however, given, partly because of the difficulty of laying out a map with multiple vortices. The surface element

distribution adopted for these computations was somewhat different from that used for figure 5. The present
NASA/TM—2000-209270 6

distribution had 90 elements per body instead of 66 elements per body. The purpose is simply to show that both

distributions accommodate stationary points.

The vortices shown in these figures are of equal strength and of the same sense. Therefore, they rotate about the

centroid at a constant speed and move nearly parallel to the ground surface from left to right initially, since the presence of the device is only scantly felt. This is the reason that the trajectories far from the device are equally coiled.

For a group of two or more vortices placed uniformly around the circumference of a circle of radius a and in a free

space of still fluid, they will undergo circular motions at a constant angular velocity Ω given by the equation

(Saffman, 1992, Chap. 7)

Ω Γ = − ( ) n

a

1

4

6 2 π

, ()

where n is the number of vortices and Γ is the strength.

We first study the fate of a pair of spinning vortices moving from left to right. Four sets of these trajectories

were depicted in figures 8 and 9, two in each figure. As in earlier cases for a single vortex, each set was computed

separately. The purpose of figure 8 is to show that the stationary points for a pair lie approximately at the same

places as in figure 5, and the purpose of figure 9 is to demonstrate that two pairs of vortices can converge to the

same stationary point much like a single vortex. In figure 8 the vortex strength of these two pairs is the same but the

separation distances are different. Thus, the upper pair spins slower than the lower pair. We regard the two dark

disks to be the stationary stations, because at these stations the forward motion of the centroid ceases, but the vortices continue to rotate. If the centroid wobbles a little and the spinning radius changes a little, the vortex paths after a

long period of time cover the entire area, which then becomes black. (The vortex strength and the time step chosen

for these two cases were unity and 0.001875 respectively.)

We now consider the case of three like-signed vortices of the equal strength with the angular velocity given by

equation (6). This is a triad of spinning vortices much like a pair of spinning vortices. The uniform coiling of

upstream vortex trajectories in figure 10 attests to this assertion. Two sets of initial conditions for these triads were

selected to lay claim that they will be captured like a single vortex. This is shown in figure 10. Finally, it is noted

parenthetically that similar trajectory patterns for spinning vortices, though not shown, have been obtained for

model B.

6. Concluding Remarks

It is seen in the above that stationary points can occur in two types of inlet-like devices. The geometry of the

device appears to play a more important part in determining the existence of stationary points, but the role of the

element distribution pattern cannot be neglected. It is not, however, clear at this stage the relationship between these

two or how to separate one from the other. Further study is needed.

Since both models can accommodate stationary points, it is reasonable to assume that there are other configurations that have the similar property.

Although the space between two bodies and the height of the lower body from the ground surface were fixed at

70 and 50 percent body length in the present study, different values of these lengths have been examined. As far as

the space between two bodies is concerned, it can be as large as 1.5 times body length and still accommodate stationary points.

The trajectory maps in figures 5 to 7 point to the fact that there are two zones, one for each body. Thus, there is

a likelihood that three or more bodies can be stacked together to produce three or more zones with one stationary

point for each body.

The preceding evidence suggests that two or three spinning vortices can have similar characteristics of a single

vortex. Therefore, it is possible that some of the present findings are valid for a vortex with a finite core, since a

group of spinning vortices bears some resemblance to a vortex with a finite core.
NASA/TM—2000-209270 7

References

Aref, H., Kadtke, J.B., Zawadzki, I., Campbell, L.J. and Eckhardt, B., 1988: “Point vortex dynamics: recent results

and open problems,” Fluid Dynamics Res. 3, pp. 63–74.

Lewis, R.I., 1981: “Surface vorticity modelling of separated flows from two-dimensional bluff bodies of arbitrary

shape,” J. Mech. Engng Sci., 23, pp. 1–12.

Lewis, R.I., 1991: Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems, Cambridge University Press, New York, N.Y.

Martensen, E., 1959: “Berechnug der druckverteilung an pitterprofilen in ebener potentialstromung mit einer

Fredholmschen integralgleichungen,” Arch. Rat. Mech. Anal., 3, pp. 235–270.

Saffman, P.G., 1992: Vortex Dynamics. Cambridge University Press, New York, N.Y.

Sheffield, J.S., 1977: “Trajectories of an ideal vortex pair near an orifice,” Phy. Fluids, 20, pp. 543–545.

TABLE 1.—COORDINATES FOR

MODEL A ON THE FORWARD

QUADRANT

x y

0.0020000

.0050000

.0080000

.0120000

.0180000

.0360000

.0700000

.1000000

.1300000

.1600000

.2050000

.2500 

.3000

.3500

.4000

.4500

.5000

0.0078146

.0122131

.0153151

.0185770

.0224718

.0308125

.0408627

.0468276

.0511933

.0544181

.0576546

.0598

.0619

.06327

.0641

.06467

.06483
NASA/TM—2000-209270 8

Figure 1.—A pair of spinning vortices trapped by an 

 inlet-like device as it moves from left to right. 

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

Vortex initial positions

Ground surface

Figure 2.—Surface vorticity elements on a 

 two-dimensional body.

Sn n

Sm

m dSm

Figure 3.—Lower-half trajectories of an ideal vortex

 pair approaching an orifice.

Sheffield's exact solution

Present numerical results

Figure 4.—Geometry of an inlet-like device. 

 (a) Model A. (b) Model B.

Ground surface

Ground surface

(a)

(b)
NASA/TM—2000-209270 9

Figure 5.—Vortex trajectory map with model A for a

 single vortex moving from left to right.

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

* Stationary points

*

*

Figure 6.—Vortex trajectory map with model B for a

 single vortex moving from left to right.

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

* Stationary points

*

*

Figure 7.—Vortex trajectory map with model B for a

 single vortex moving from left to right in a uniform

 stream flowing from left to right. Vortex strength

 = –1.5.

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

* Stationary points

*

*

Figure 8.—A pair of spinning vortices trapped by 

 model A as it moves from left to right from two 

 different initial positions.

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

Vortex initial positions
NASA/TM—2000-209270 10

Figure 9.—A pair of spinning vortices trapped by 

 model A as it moves from left to right from two 

 different initial positions.

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

Vortex initial positions

Figure 10.—A triad of spinning vortices trapped by 

 model A as it moves from left to right from two 

 different initial positions.

2.5

2.0

1.5

1.0

0.5

0.0

–0.5

0.0 1.0 2.0

x

–1.0

y

Vortex initial positions
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A03

A Note on Trapping Moving Vortices

Hsiao C. Kao

Vortex; Vortex-body interaction; Spinning pairs or triads; Ground surface

Unclassified -Unlimited

Subject Category: 02 Distribution: Nonstandard

Responsible person, Hsiao C. Kao, organization code 5860, (216) 433–5866.

The topic of stationary configurations of point vortices, also known as vortex equilibrium, has received considerable

attention in recent years. By observing numerical results, it is found that a “counterpart” of this system also exists, in

which moving vortices may be “trapped” by an inlet-like device to form a stationary pattern with no translational motion.

After an intuitive explanation for the process, vortex trajectory maps based on numerical results are presented. These

maps exhibit two stationary points under the present conditions, which are the focal points of vortex trajectories. A vortex

upstream of these points, if within a certain offset range, will move towards these points spontaneously and be captured

there. This proposed device is also capable of trapping spinning vortex pairs and triads. It is possible to impose a uniform

stream at infinity, as long as the flow field is still dominated by the moving vortices.