The Motion of a Point Vortex in Multiply Connected Polygonal Domains
TThe Motion of a Point Vortex in MultiplyConnected Polygonal Domains
El Mostafa Kalmoun, Mohamed M S Nasser, Khalifa A. Hazaa
Department of Mathematics, Statistics and Physics, Qatar University,P.O. Box 2713, Doha, [email protected]; [email protected]; [email protected]
Abstract
We study the motion of a single point vortex in simply and multiply con-nected polygonal domains. In case of multiply connected domains, the polyg-onal obstacles can be viewed as the cross-sections of 3D polygonal cylinders.First, we utilize conformal mappings to transfer the polygonal domains ontocircular domains. Then, we employ the SchottkyKlein prime function to com-pute the Hamiltonian governing the point vortex motion in circular domains.We compare between the topological structures of the contour lines of theHamiltonian in symmetric and asymmetric domains. Special attention is paidto the interaction of point vortex trajectories with the polygonal obstacles. Inthis context, we discuss the effect of symmetry breaking, and obstacle locationand shape on the behavior of vortex motion.
Keywords . Point vortex motion, conformal mapping, Schottky-Kleinprime function, polygonal domains
Since the seminal work by Helmholtz [13], 2D point vortex motion has sparked in-tense research due to its important applications in fluid dynamics and geophysics [1,3, 22, 24]. A point vortex is a model of potential flow in which vorticity of the sur-rounding flow is present only at a single point. In addition to the various physicalapplications, the 2D point vortex model, despite its simplicity, provides a mathemat-ics playground in the words of Aref [1], that links a variety of mathematical areassuch as dynamical systems, ordinary and partial differential equations, Hamiltoniandynamics, theory of polynomials, elliptic functions, to name a few.The dynamics of point vortices is governed by a Hamiltonian system, which canbe derived from the Euler equations describing the velocity field and pressure of a2D incompressible and inviscid flow. If the fluid domain has no boundaries, likethe whole complex plane for instance, the Hamiltonian of the governing dynamicalsystem is written in terms of the complex Green function [24]. The presence ofsolid boundaries might break symmetries that would otherwise exist and thus theHamiltonian system is constrained by the fluid non-penetration condition expressedby zero normal velocity on all solid boundaries. For simple geometries containingspecial symmetries like the upper-half plane, a channel between two infinite wallsor the exterior of a circular cylinder, an explicit formula of the Hamiltonian of1 a r X i v : . [ phy s i c s . f l u - dyn ] F e b a point vortex can be obtained by the classical method of images in which solidboundaries are treated as streamlines of a flow field. This is usually combinedwith conformal mapping ideas as can be seen in [22, 24] for a variety of examples.Similarly, in bounded and simply connected domains, basic results concerning singlevortex motion are well established [9]. In this case the generalized Hamiltonian isknown as the Kirchhoff-Routh path function [15, 23].As many fluid domains are not simply connected (e.g. in aerodynamics), agrowing interest has centered around studying N -point vortex dynamics in multiplyconnected domains. Lin [17] had early proved the existence and uniqueness ofthe Kirchhoff-Routh path function in multiply connected domains but no explicitformula was given. Recently, Johnson & McDonald [14] studied the motion ofone point vortex in doubly and triply connected domains exterior to either oneor two circular cylinders using elliptic functions, provided that zero circulationsare imposed around the circles. Under the same boundary conditions, Crowdy& Marshall [5] presented an analytical formula for the Hamiltonian in multiplyconnected circular domains by applying methods of complex function theory. Inparticular, their analysis makes use of a special transcendental function known asthe Schottky-Klein prime function . The formula can be employed for a generalmultiply connected domain as long as the conformal mapping onto the canonicalcircular domain is known. Indeed, this formula was already applied to study pointvortex motion around an arbitrary finite number of circular cylinders [6] and throughgaps in walls [7]. When the multiply connected domain is symmetric with respectto the real axis, Sakajo [25] showed that the two point vortices’ motion is reduced tothat of a single point vortex in a multiply connected semicircle. Thus the Crowdy& Marshall analytic formula was again utilized to plot the trajectories of two pointvortices for several circular domains.In this paper, we study the motion of a single point vortex in simply and multiplyconnected polygonal domains. In case of multiply connected dmains, the polygo-nal obstacles can be viewed as the cross-sections of polygonal cylinders in 3D. Ourmethod will first make use of conformal mappings to transfer the polygonal domainonto the circular domain where the Crowdy & Marshall analytic formula can be em-ployed. The objective of our study is to compare between the topological structuresof the contour lines of the Hamiltonian in circular and polygonal domains. We aimalso to emphasize how these trajectories interact with the polygonal obstacles. Inthis context, we discuss the effect of symmetry breaking, and obstacle location andshape on the behavior of vortex motion.
We present in this section a numerical method for computing the Hamiltonian ofa point vortex motion in multiply connected polygonal domains. The method isbased on using conformal mappings to map polygonal domains onto circular ones,and then appealing to the analytic formula presented in [5] for the Hamiltonian inmultiply connected circular domains.Let G denote the multiply connected domain exterior to m non-overlappingpolygons Γ , . . . , Γ m (clockwise orientated) and interior to a polygon Γ m +1 (in thecounterclockwise orientation). Such a domain will be called a polygonal domain.We assume that no corner of these polygons is a cusp. Let us suppose that theinterior of G is occupied with incompressible fluid. The flow is assumed to undergoirrotational motion except for a single point vortex. Therefore, the circulationsaround the polygonal obstacles Γ , . . . , Γ m are zero, and the point vortex has agiven circulation χ . It is known that there exists a conformal mapping ζ = f ( z ) from the polygonaldomain G onto a multiply connected circular domain D interior to the unit circleand exterior to m non-overlapping circles [16]. We denote the inner m circles by C k , k = 1 , , . . . , m and the unit circle by C m +1 . By Carath´eodory’s theorem [16,p. 111], the mapping function f can be extended to G = G ∪ ∂G in such a way thatthe polygon Γ k is mapped onto the circle C k , for k = 1 , , . . . , m + 1. Note that ∂G denotes the boundary of the domain G , which consists of m + 1 piecewise smoothJordan curves. For the uniqueness of the conformal mapping, we fix a point α in G ,and we assume that f ( α ) = 0 and f (cid:48) ( α ) > G . Thus, computing the conformal mapping f requiresalso computing the centers and radii in the circular domain. In this paper, theconformal mapping f will be computed using the MATLAB toolbox (PlgCirMap)presented in [20]. The trajectories of a point vortex in the multiply connected circular domain D can be written as the contours of the Hamiltonian, aka the Kirchhoff-Routh pathfunction [5]. The existence and uniqueness of such a function is established byLin [17].Let H D ( ζ, ζ ) denote the Hamiltonian for the motion of a single point vortex inthe circular domain D . An explicit formula for H D ( ζ, ζ ) has been derived by Crowdy& Marshall [5], H D ( ζ, ζ ) = − χ π log (cid:12)(cid:12)(cid:12)(cid:12) ζ ˆ ω ( ζ, ζ )ˆ ω (1 /ζ, /ζ ) ω ( ζ, /ζ )ˆ ω (1 /ζ, ζ ) (cid:12)(cid:12)(cid:12)(cid:12) , (1)where ζ is the complex conjugate of ζ , ω ( ζ, ξ ) is the Schottky-Klein prime function(prime function, for short) associated with the circular domain D , and the functionˆ ω ( ζ, ξ ) is defined through ω ( ζ, ξ ) = ( ζ − ξ )ˆ ω ( ζ, ξ ) , ζ, ξ ∈ D, and ˆ ω ( ζ, ξ ) = ˆ ω ( ζ, ξ ) . The Schottky-Klein prime function is an important special transcendental func-tion. Due to the works of Crowdy with various collaborators [4] in the last twodecades, the prime function has become a key tool to construct analytic formulas insolving several problems involving multiply connected circular domains. For moredetails on the properties of the prime function, its applications and numerical com-putation, the reader is referred to the review article [8] and the recent monograph [4].A simplified form of (1) has been proposed by Sakajo [25], H D ( ζ, ζ ) = − χ π log (cid:12)(cid:12)(cid:12)(cid:12) ˆ ω ( ζ, ζ ) ζω ( ζ, /ζ ) (cid:12)(cid:12)(cid:12)(cid:12) . In the two numerical methods presented in [8], the prime function is normalizedwith w (cid:48) ( ζ, ζ ) = 1, which yields H D ( ζ, ζ ) = χ π log (cid:12)(cid:12) ζω ( ζ, /ζ ) (cid:12)(cid:12) . (2)By making use of an important property of the prime function [4] that states ω ( ζ, /ζ ) = 0 for ζ ∈ m +1 (cid:91) k =1 C k , (3)it follows from (2) that H D ( ζ, ζ ) → −∞ as ζ approaches ∂D . By the help of conformal mappings, the above exact formula (2) can be extended tothe general multiply connected domains [5]. In fact, using the conformal mapping ζ = f ( z ) to map the polygonal domain G onto the circular domain D , the Hamil-tonian H G ( z, z ) of a single vortex motion of circulation χ in the bounded polygonaldomain G is given by [5] H G ( z, z ) = χ π log (cid:12)(cid:12) ζω ( ζ, /ζ ) (cid:12)(cid:12) + χ π log (cid:12)(cid:12) ( f − ) (cid:48) ( ζ ) (cid:12)(cid:12) , which can be simplified as H G ( z, z ) = χ π log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | f ( z ) | ω (cid:16) f ( z ) , /f ( z ) (cid:17) f (cid:48) ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)The formula (3) implies that H G ( z, z ) → −∞ as z approaches ∂G as well. In thesequel, we assume that the point vortex has a circulation χ = 1. Recently, a MATLAB toolbox (PlgCirMap) for computing the conformal mapping ζ = f ( z ), from the polygonal domain G onto the circular domain D , and its inverse z = f − ( ζ ) has been presented in [20]. The method used in [20] is based on afast implementation of Koebe’s iterative method [19] using the boundary integralequation with the generalized Neumann kernel [18] and fast multipole method [10].The toolbox works also if G is simply connected ( m = 1), and hence D is the unitdisk. On the other hand, to compute the prime function, two numerical methodsare presented in [8] with MATLAB codes available online. Their first procedureis a spectral method based on global relations, while the second depends on theboundary integral equation with the generalized Neumann kernel.In this paper, the Hamiltonian in (4) is computed by a combination of theMATLAB toolbox PlgCirMap and one or the other of the two numerical methodspresented in [8]. The outline of our procedure is summarized in Algorithm 1.Note that the point α in Step 2 of Algorithm 1 has no effect on the numericalcomputation of the Hamiltonian as long as it is sufficiently far from the boundaryof G . Algorithm 1
Computing the Hamiltonian H G for the polygonal domain G . Define the vertices of the polygons Γ k , k = 1 , . . . , m + 1, as a cell array ver{k} . Choose an auxiliary point alpha in G . Use the MATLAB function plgcirmap to compute f = plgcirmap(ver,alpha) where f is a MATLAB struct with several fields. Extract the centers and radii of the inner circles in D from f by f.cent(1:m) and f.rad(1:m) . Discretize the domain G by a matrix of points Z . Call the functions evalu and evalud from the toolbox PlgCirMap to computethe values of the conformal mapping fZ and its derivative dfZ on the meshgrid Z : fZ = evalu(f,Z,’d’); dfZ = evalud(f,Z,’d’) Compute the prime function by applying one of the two methods presentedin [8]. Use Equation (4) to compute the Hamiltonian.
When D is the unit disk, the associated prime function is simply ω ( ζ, α ) = ζ − α. Thus, the Hamiltonian (2) of a single vortex motion of circulation χ in the unit disk D reduces to [5] H D ( ζ, ζ ) = 14 π log | − ζ ζ | = 14 π log(1 − | ζ | ) . Hence, the contour lines of the Hamiltonian are circles with a center point at theorigin. It is clear that 0 < − | ζ | ≤ ζ is in the unit disk D and 1 − | ζ | → + as ζ approaches ∂D . Hence H D ( ζ, ζ ) → −∞ as ζ approaches ∂D and H D (0 ,
0) = 0is the unique global maximum.Similarly, the Hamiltonian (4) in the bounded simply connected polygonal do-main G is given by [5] H G ( z, z ) = 14 π log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | f ( z ) | (cid:16) f ( z ) − /f ( z ) (cid:17) f (cid:48) ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 14 π log 1 − | f ( z ) | | f (cid:48) ( z ) | , where f is the above described conformal mapping from G onto D . Note also that H G ( z, z ) → −∞ as z approaches ∂G .For convex domains, Gustafsson [11] proved that the Hamiltonian is strictlyconcave, and therefore the point vortex has a unique equilibrium point, which isthe global maximum of H G . In this case, contour plots of the Hamiltonian H G ,which correspond to vortex trajectories span the domain G by encircling the uniquecritical point in a similar topological way as in the disk D . When passing to nonconvex domains, the uniqueness of critical points does not hold in general as noticedby Gustafsson [12] who provided a counter-example for star-shaped domains. In thenext subsection we present a numerical example that illustrates this fact. The vertices of the star-shaped domain in Figure 1 are: 1+0 . , , . , a i , − . , − , − . , − b, − − . , − − i , − . − i , − a i , . − i , − i , − .
5i and b for threedifferent values of ( a, b ). The X-shaped polygon in Figure 1a has ( a, b ) = (0 . , . .
5. Similar to the unit disk case, the Hamiltonian in this polygonal domainhas a unique critical point, which is a center at the origin surrounded by closedorbits.If the two vertices on the imaginary axis are displaced by 0 .
25 unit toward eachother, so that they are not anymore located on the same circle with the other twovertices on the real axis, then we have more than one critical point; see Figure 1b.Indeed, the neutrally stable center point becomes a saddle, which is connected toitself by a pair of symmetric homoclinic orbits forming together a separatrix. Twonew vortex centers appear on the real axis at the same distance from the saddlepoint zero. The Hamiltonian has four other centers where each point is surroundedby a homoclinic orbit connecting a saddle point. These four hyperbolic points arethemselves connected by a heteroclinic loop. Observe also how all equilibria of thissystem are distributed in a way that reflects line symmetries in the domain.Another worth commenting dynamical feature in this example is the reappear-ance of the center point at the origin instead of the homoclinic structure whenthe four concave corners are placed again on a circle of center zero and radius 0 . (a) ( a, b ) = (0 . , .
5) (b) ( a, b ) = (0 . , .
5) (c) ( a, b ) = (0 . , . Figure 1: Vortex trajectories in a star-shaped domain. The vertices of the polygonare: 1 + 0 . , , . , a i , − . , − , − . , − b, − − . , − − i , − . − i , − a i , . − i , − i , − .
5i and b . We consider in Figures 2-4 three other examples of general bounded simply con-nected polygonal domains that are non convex. The C-shaped polygon in Figure 2has eight vertices at 1 + i , − , − − i , − i , − . , a − . , a + 0 . , . , . , . a i , − . a i , − . , − , − − i , − . − i , − . − a i , . − a i , . − i , − i, and the twelvevertices of the E-shaped polygon in Figure 4 are 1 − i , − . , − . , − . , − . , . , . , . , . , , − a + i , − a − i.The common dynamical feature between the three shapes is that the uniquestable equilibrium point is always located at the center of the maximal rectanglethat has antiknobs in the boundary. When this maximal rectangle gets narrowerin either the C-shaped polygon or the H-shaped domain, the center point changesto a saddle connected by two homoclinic trajectories that surround two new vortexcenters. Observe that this not the case in the E-shaped domain, as this center pointstill occurs but it is now surrounded by a heteroclinic orbit. In this example, twonew saddle points and two new vortex centers are born making each group of ellipticand hyperbolic points vertically aligned. (a) a = 0 (b) a = − .
25 (c) a = − . Figure 2: Vortex trajectories in a C-shaped polygon of vertices: 1 + i , − , − − i , − i , − . , a − . , a + 0 . , . (a) a = 0 . a = 0 . a = 0 . Figure 3: Vortex trajectories in a H-shaped polygon of vertices: 1 + i , . , . a i , − . a i , − . , − , − − i , − . − i , − . − a i , . − a i , . − i , − i. (a) a = 1 (b) a = 0 . a = 0 . Figure 4: Vortex trajectories in a E-shaped polygon of vertices: 1 − i , − . , − . , − . , − . , . , . , . , . , , − a + i , − a − i. Sakajo [25] studied the motion of a single point vortex in doubly connected circulardomains. If the center of the inner circle is not the origin, then the Hamiltonianalways has one elliptic center point and one saddle point connected to itself bytwo homoclinic vortex trajectories; see Figure 5b. In this section, we discuss whathappens to the dynamics of point vortex when we replace circular domains withpolygonal domains. (a) p = 0 (b) p = 0 . Figure 5: Vortex trajectories in doubly connected circular domains. The outer circleis the unit circle | ζ | = 1 and the inner circle is | ζ − p | = 0 . G is composed ofthe square with vertices ± ± i. For the inner obstacle shape, we consider four cases:a square, a rotated square, an equilateral triangle and a hexagon. In Figures 6-11,we show vortex motion in these doubly connected polygonal domains by plottingsome contours of the Hamiltonian for different positions of the polygonal obstaclecenter. We first consider the case of concentric domains in Subfigures (a) of Figures 6, 7, 8and 11. Unlike the annulus domain (Figure 5a) where there is no center or saddle,the polygonal structure gives rise to several equilibria of this type. The number ofvortex centers of the Hamiltonian is the same as that of saddle points, which is againin accordance with Morse theory. The elliptic equilibria are surrounded by closedvortex trajectories and the hyperbolic ones are connected by either homoclinic orheteroclinic orbits.In Table 1, we show the number of saddle points and how they are linked byvortex trajectories in concentric polygonal doubly connected domains. By looking atthese numbers there are aspects to be highlighted regarding the topology of vortextrajectories near the obstacle boundaries. For a square obstacle (standard androtated), we observe a pure heteroclinic structure, which is not the case for triangleand hexagon obstacles. Near the square obstacle in Figure 6a, there occur eightsaddle points that are connected to each other by eight heteroclinic loops. At firstthis number seems to result from the total number of reflectional and rotationalsymmetries or the number of corner points in the domain. However, when werotate the square obstacle by an angle of π/ After breaking the symmetry with respect to the outer square in Figure 6b by a per-turbation of the center p of the square obstacle with a translation of 0 .
01 unit alongthe imaginary axis, the topology pattern of vortex motion changes dramatically.Only two saddle points which are situated on the side of the obstacle movementstill occur. The saddle connection is made by two heteroclinic loops encircling twocenter points, which are symmetrically located on the imaginary axis with one abovethe obstacle and the other below it. The same configuration appears if we continuemoving the square obstacle northward as shown in Figures 6c and 6d.On the other hand, if we choose to break the symmetry by shifting the obstaclecenter to the upper right corner of the outer square as in Figure 6e, then a smallperturbation changes again the topology pattern, but in a slightly different way.Besides the heteroclinic cycle containing two saddles in the direction of the obstaclemovement as in the previous case, the other elliptic point which is situated on the0 (a) p = 0 (b) p = 0 .
01i (c) p = 0 .
5i (d) p = 0 . p = 0 .
01 + 0 .
01i (f) p = 0 .
11 + 0 .
11i (g) p = 0 .
65 + 0 .
65i (h) p = 0 . . Figure 6: Vortex trajectories in doubly connected square domains where the squareobstacle has a center p and a horizontal radius of 0 . y = x is now the center of a new heteroclinic loopitself joining two new saddles. Each of the latter hyperbolic points belongs to onehomoclinic orbit surrounding a vortex center, which gives a total of eight criticalpoints.When displacing the center of the obstacle to p = 0 .
11 + 0 .
11i as shown inFigure 6f, the vortex motion loses all its rest points except one saddle and onecenter, which are respectively located on the upper right and lower left sides ofthe diagonal line y = x . Observe also that a topological configuration similar tothe vortex motion in Figures 6c and 6d appears but on the diagonal instead ofthe imaginary axis when p = 0 .
65 + 0 . We now examine the effect of symmetry breaking when the obstacle is the rotatedsquare in Figure 7. The small displacement of the center along the imaginary axisin Figure 7b does not reduce the number of equilibria as in the square obstacle.However, this breakdown destroys the heteroclinic structure seen in Figure 7a, whichconfirms the structurally instability of heteroclinic cycles. Indeed, none of the foursaddles is anymore the intersection of two heteroclinic loops. However, each saddlegives rise to the bifurcation of two homoclinic orbits that form a separatrix.On the other hand, a perturbation of √ × .
01 toward the top right corner of theouter square, as done in Figure 7c, seems to damage less the heteroclinic structure1 (a) p = 0 (b) p = 0 .
01i (c) p = 0 .
01 + 0 . p = 0 .
11i (e) p = 0 .
8i (f) p = 0 . . Figure 7: Vortex trajectories in doubly connected polygonal domains where thevertices of the rotated square obstacle are on the circle of center p and radius 0 . y = x .Figure 7d reveals that when we move the obstacle center to p = 0 . In Figure 8, we show the effect of obstacle displacement on vortex motion for the caseof the equilateral triangle. We note first that the imaginary axis is the unique lineof reflectional symmetry in the starting configuration of this example; that is when2the obstacle center is at p = 0; see Figure 8a. This configuration seems structurallymore stable to obstacle movement to the upper side external boundary. Indeed, asmall perturbation by a 0 .
01 unit does not change at all the topology pattern ofvortex trajectories.In Figure 9, we illustrate how a small perturbation of the triangle obstacle loca-tion in different directions impacts the topology pattern of vortex trajectories. Wecan see that the configuration does not change for a vertical translation of 0 .
05 whileit does with the same translation on the opposite direction. On the other hand, weobserve a change to the same configuration when a small perturbation is appliedalong the horizontal and diagonal directions; see Figure 9d-9f.Next, we explore closely how the topological pattern of vortex motion changesafter displacement of the obstacle triangle. First, for the vertical translation to theupper side boundary, the bifurcation from the two lower saddles and three centers toa unique center point does not happen until an approximate displacement of 0 . p = − . .
11 unit in Figure 8e. The upper saddle changesto a center point for an approximate bifurcation displacement of 0 .
145 yielding thepattern given in Figure 10c. This configuration stays the same all the way as theobstacle approaches the lower boundary; see Figure 8f.Now, when we break the line symmetry with respect to the imaginary axis bymoving the obstacle triangle with 0 .
01 unit eastward in Figure 8g, the heteroclinicloop disappears as each of the three saddles becomes an intersection point of homo-clinic orbits. The inner saddle point disappears first at an approximate bifurcationdisplacement of 0 .
063 (Figure 10d), and then the other inner saddle changes to acenter point at almost p = 0 .
102 (Figure 10e). Afterwards, the vortex motion main-tains a unique unstable rest point until the obstacle comes close the right boundary;see Figure 8h-8i.Note also that as we said before, a small perturbation along the line y = x yieldsa topologically equivalent pattern to the horizontal translation case. However, whenthe obstacle triangle approaches the upper right corner of the outer square, we get adifferent configuration to the horizontal movement case as the vortex motion keepstwo unstable equilibria; see Figures 8j-8k. It is worth mentioning that this does nothappen when the obstacle comes close to the lower right corner. The configurationin this case is topologically equivalent to the horizontal translation as shown inFigure 8l. In this example we consider how the displacement of the hexagon obstacle in Fig-ure 11 impacts the point vortex motion.3First, the upper heteroclinic loop connecting the two saddles on the real axisgets splitted into two homoclinic orbits after a small perturbation of the center p of the hexagon by 0 .
01 unit along the imaginary axis; see Figure 11b. The uppersaddle on the imaginary axis becomes unique after the three other points disappearat a vertical displacement of 0 .
11 unit. When approaching the upper boundary side,this saddle changes to a center point of a heteroclinic loop linking two new saddles.When we displace the center of the triangle by 0 .
01 unit eastward in Figure 11e,a topologically equivalent pattern to the (6 ,
1) homoclinic-heteroclinic structure wegot after a small vertical perturbation of p appears again. The only difference in thehorizontal displacement is that the pattern appearing at p = 0 .
11 stays the sameeven after approaching the right side boundary.On the other hand, a diagonal displacement of the hexagon obstacle changesthe dynamics of the point vortex in a different manner. A small perturbation asin Figure 11h destroys the heteroclinic structure completely, and all four saddlesbecome intersection points of homoclinic orbits. At p = 0 .
11 + 0 . (a) p = 0 (b) p = 0 .
01i (c) p = 0 . p = − .
01i (e) p = − .
11i (f) p = − . p = 0 .
01 (h) p = 0 .
11 (i) p = 0 . p = 0 .
01 + 0 .
01i (k) p = 0 . .
8i (l) p = 0 . − . Figure 8: Vortex trajectories in doubly connected square domains where the verticesof the triangle obstacle are p + 0 . e π i / , p + 0 . e − π i / , p + 0 . e − π i / .5 (a) p = 0 (b) p = β i (c) p = − β i (d) p = β (e) p = βe π i4 (f) p = βe − π i4 Figure 9: Critical vortex trajectories after a small displacement ( β = 0 .
05) ofthe triangle obstacle in different directions. The vertices of the triangle are p + 0 . e π i / , p + 0 . e − π i / , p + 0 . e − π i / . (a) p = 0 . p = − . (c) p = − . p = 0 .
063 (e) p = 0 . Figure 10: Critical vortex trajectories for bifurcation locations of the triangle ob-stacle. The vertices of the triangle are p + 0 . e π i / , p + 0 . e − π i / , p + 0 . e − π i / .6 (a) p = 0(b) p = 0 .
01i (c) p = 0 .
11i (d) p = 0 . p = 0 .
01 (f) p = 0 .
11 (g) p = 0 . p = 0 .
01 + 0 .
01i (i) p = 0 .
11 + 0 .
11i (j) p = 0 . . Figure 11: Vortex trajectories in doubly connected polygonal domains where thevertices of the hexagonal obstacle are p + 0 . , p + 0 . e − π i / , p + 0 . e − π i / , p − . , p +0 . e − π i / , p + 0 . e − π i / .7 In his paper [25], Sakajo studied also the motion of a single point vortex in multiplyconnected circular domains. The centers of the inner circles are assumed to be onthe real axis. The contour plot of the Hamiltonian for these domains is given inFigure 12. Crowdy and Marshal [5] have also considered several circular domains. (a) p = 0 . p = 0 . p = 0 . p = 0 . Figure 12: Vortex trajectories in triply connected circular domains. The outer circleis the unit circle | ζ | = 1 and the inner circles are | ζ ± p | = 0 . ± ± i. The contour plot of the Hamiltonian in this case is then shown inthe top row of Figure 13. The topology pattern of vortex motion looks similar to thecircular domain case. Looking at how the point vortex trajectories interact with thepolygonal obstacles, we observe that this interaction depends on the distance be-tween these obstacles. When they are close to each other, as in Figure 13a, then wehave two saddle points between them connected by a heteroclinic loop, compared toone saddle for the circular case. As the separation between the two square obstaclesincreases, we get a vortex merging from two saddles and one center to one saddle atwhich two homoclinic orbits intersect and surround the previous upper and lowercenters separately; see Figure 13b. Increasing further the separation distance, likefor example to 0 . y = x , as shown in the bottom row of Figure 13. When these centersare located at ± p for p = 0 . .
2i or p = 0 . .
3i in Figures 13e and 13f, the samedynamical configuration occurs which is topologically equivalent to the pattern inFigure 13b. Observe that all three saddle points are located on the main diagonalline. After displacing the square obstacles toward the top right and bottom leftouter corners in Figure 13g, the homoclinic structure at the center disappears, anda unique vortex center is born in a similar way to the horizontal separation. Asexpected, when the obstacles gets closer to the corners in Figure 13h, the phaseportrait between each obstacle and the external boundary becomes similar to thecase of one square obstacle. The Hamiltonian has four saddles near each side andall eight points are joined by heteroclinic connections.8 (a) p = 0 . p = 0 . p = 0 . p = 0 . p = 0 . .
2i (f) p = 0 . .
3i (g) p = 0 . .
5i (h) p = 0 . . Figure 13: Vortex trajectories in triply connected polygonal domains where theinner squares have side length equal to 0 . p and − p .We examine now the effect of reducing reflectional symmetry to a unique lineby placing the obstacle at different locations, as shown in the top row of Figure 14.We first place one obstacle centered at zero and the other once in the north andthen in the right east; see Figures 14a and 14b. In these two examples, the dy-namics restricted near the lower side of the line of reflectional symmetry consistsof a (2 ,
1) homoclinic-heteroclinic structure similar to the one square obstacle case.The Hamiltonian has also one saddle between the two obstacles and two saddlesconnected by a heteroclinic loop between the obstacle and the external boundary.In Figures 14c and 14d, we consider two cases where the two obstacles are placed atsymmetric locations with respect to the line y = x and the imaginary axis, respec-tively. The behavior of vortex trajectories in these two examples is the same. Thereis a vortex center located almost at zero and two saddles between each obstacleand the external boundary. Each two points of the four saddles are connected by aheteroclinic loop.In the bottom row of Figure 14 we demonstrate the effect of symmetry breakingon vortex motion. As the patterns in Figures 14a and 14b are topologically equiv-alent, we break the vertical line symmetry only in the first example. First, aftera small horizontal perturbation of 0 .
01 unit on the center of the lower obstacle, asshown in Figure 14e, the (2 ,
1) homoclinic-heteroclinic pattern in the lower side ofthis obstacle is no longer present, and a new vortex center is born near the lower leftcorner of the same obstacle. The other center and three saddles are still present inthe phase space with the same patterns. However, a similar horizontal perturbationof the upper obstacle does not cause the two lower saddles to disappear, but it doessplit their heteroclinic connections. Observe also that the upper critical points stilloccur; see Figure 14f. Looking at Figures 14g and 14h, we see the effect of breakingthe line symmetry in the two examples shown in Figures 14c and 14d. Indeed, ifwe move one obstacle and keep the other at the same location, then the two saddle9points near one obstacle get disconnected from the two saddles near the other. (a) p = 0 , q = 0 .
6i (b) p = 0 , q = 0 . . (c) p = 0 . , q = 0 .
6i (d) p, q = ± . . p = 0 . , q = 0 .
6i (f) p = 0 , q = 0 .
05 + 0 . (g) p = 0 . , q = 0 .
6i (h) p = 0 . , q = 0 . . Figure 14: Vortex trajectories in triply connected polygonal domains where theinner squares have side length equal to 0 .
2, and centers at p and q , respectively.In Figure 15, we show the effect of increasing the size of the square obstacles.Since the squares become too close, a new saddle point appears between them inFigures 15a and 15b. The dynamics of the point vortex motion is more remarkablein the case of Figure 15c. Two new saddles connected by a heteroclinic loop appearnear the top right corner of the outer square. Moreover, on the other side, the (2 , ,
1) pattern. (a) p = 0 and q = 0 .
6i (b) p = 0 .
45 and q = 0 .
6i (c) p = 0 and q = 0 . . Figure 15: Vortex trajectories in triply connected polygonal domains where the largeand small inner squares have side length equal to 0 . .
4, and centers at p and q , respectively.Finally, when we consider a quintuply connected polygonal domain with foursquare obstacles in Figure 16, we notice the same dynamical feature as in the cor-0responding case of triply connected domains. It is worth mentioning that the pre-sented method can also be efficiently used to compute vortex trajectories in multiplyconnected polygonal domains of high connectivity as illustrated in Figure 17. (a) p = 0 . , q = 0 . p = 0 . , q = 0 .
75 (c) p = 0 . , q = 0 .
4i (d) p = 0 . , . Figure 16: Vortex trajectories in quintuply connected polygonal domains where theinner squares have side length equal to 0 . ± p and ± q .Figure 17: Vortex trajectories in a multiply connected polygonal domain with 15obstacles. We have numerically computed the Hamiltonian of a point vortex motion in simplyand multiply connected bounded polygonal domains. We have first conformallymapped the polygonal domain onto a circular domain, and then applied the analyticformula presented in [5] for the Hamiltonian in multiply connected circular domains.For simply connected domains, we have considered four polygonal shapes that arenon convex. In these four examples, we have studied first the case when the system istopologically equivalent to the unit disk where the Hamiltonian has a unique criticalpoint. Then keeping the same general shape and changing the distance between thecenter point location and boundaries has yielded a lost of this uniqueness property.In all cases, the number of critical points satisfies the Morse Theorem; that is thenumber of centers is equal to that of saddles plus one.1We have then examined the case of doubly connected polygonal domains withfour examples of an obstacle in a square. For concentric domains, we have noticeda pure heteroclinic structure of vortex trajectories around the standard and rotatedsquares obstacle, which was not the case for triangle and hexagon obstacles. Af-terwards, we have concerned ourselves with the manner the vortex motion changesdue to small perturbations of the obstacle location and its displacement toward theexternal boundary. Through various examples, the effect of symmetry breaking, andobstacle shape and location on vortex trajectories has been well demonstrated.In triply connected polygonal domains, we have looked at how vortex trajecto-ries interact with the square obstacles and the outer square boundaries. We haveobserved that this interaction depends on the distance between these obstacles, andbetween each obstacle and the external boundary. The common dynamical featureat close boundaries interaction involves two saddles for side to side and one saddlefor corner to side or corner to corner. As the separation between the two obstaclesor between one obstacle and the outer boundaries increases, a vortex merging hasbeen noticed. Another worth commenting dynamical feature is that heteroclinicconnections between saddle points are quickly broken by lost of symmetry, whichconfirms their structural instability.We have also demonstrated that our method can also be efficiently used to studysingle point vortex motion in multiply connected polygonal domains of high con-nectivity. Indeed, we have shown vortex trajectories in a square domain with foursquare obstacles, and a rectangle with fourteen rectangular obstacles.Finally, as an outlook on future perspectives, we mention three directions. Wehave seen in this paper that heteroclinic loops is generally associated with the pres-ence of symmetries in the model. Breaking such symmetries often break heteroclin-icity and produce new homoclinic orbits. The formation of homoclinic orbits mightbe important for the mixing of passive tracers [26]. Future work could also focuson the case in which the Hamiltonian is not a log function but follows a power law,which are an important new frontier for applied mathematics and physics [2]. Onthe other hand, in this paper, we have considered the multiply connected polygonaldomains. With the help of the method presented in [18, 19, 21], our method canbe extended to multiply connected domains other than polygonal domains such asdomains exterior to rectilinear slits.
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