104557839 Vortex Dipoles Ordered Structures From Chaotic Flows

7/29/2019 104557839 Vortex Dipoles Ordered Structures From Chaotic Flows

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VORTEX DIPOLESOrdered structures from chaotic flows

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Abstract

This experiment involved the observation and analysis of unique structures known as vortexdipoles in a 2D fluid flow. The Reynolds number of the vortex dipoles was found fromestablished theory as a function of the volume flux into a stratified fluid. This was comparedwith the Reynolds number found experimentally as a function of the vortex dipole aspectratio, . The results agreed with those presented by Y. Afanasyev et al [1]. A well knownimaging technique, Particle Image Velocimetry was then employed to find the velocityfield of the dipole from which the vorticity at each point was found. The vorticity fieldsproduced were consistent with the polar solution of vortex dipoles first produced by SirH. Lamb in 1895 [2]. Other tertiary experiments were performed including observation ofvortex dipole collisions and the curious attraction of vortices to walls and boundaries.

Introduction

Vortex dipoles are fascinating jet flows that have two propagating, counter rotating vortices at theirfront. Although considerably rarer than the common single vortex, vortex dipoles still have applicationin an incredibly diverse range of scientific areas and have been observed in everything from cloudformations to Bose-Einstein condensates [3], an exotic state of matter where quantum effects areapparent on the macroscopic scale. In this report, the history and application of vortex dipoles (V-Ds) are discussed, with emphasis on evaluation of the various mathematical models used to explaintheir structure.

Figure 1: A vortex dipole of dyed water at t =50s after formation.

Typically, turbulent fluid flows occupy the full three dimensions of space. However, in certain cases

motion in one direction is suppressed and a two dimensional fluid system remains. This assumptionis common in explaining fluid flows and in particular for vortices, usually means that dependenceof polar co-ordinates is neglected [4]. Two dimensional turbulent experiments and simulations arecommon as simple models of atmospheric and oceanic turbulence. However, despite the simplicity ofthe models used, they accurately predict the properties of large scale dipoles, which regularly formin inlets and can even form on oceanic scales [5].

The earliest successful attempts to model straight vortex pairs was produced by Sir H. Lamb in1895. A modern revision of his original solution is provided in the theory section below. Althoughthe model reproduced the basic shape and properties of the V-Ds, it had no time dependence andindicated no V-D evolution. In 1905, Chaplykin reworked the original equations but produced resultswith only limited experimental significance. With the invention of jet propulsion aircraft in the 1940s,

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vortex dipoles became a topic of significant interest. This is because an accelerated flow past a wingproduces a vortex pair across the wingspan. The mathematics is exacting, but my interpretation ofthe equations produced by G. Saffman [7] is given here. Consider a rectangular elliptically loadedwing of span 2a, with a stream velocity W inclined at an angle (fig.2). The flow that develops isa vortex sheet after time time t = Z/W with U = W when the wing moves.

Figure 2: Vortex field behind a wing

This results because the vorticity, v, is a maximum at the wing tips since v = u . As the wingmoves, the flow going past it continues to be curled on each side of the aircraft, and so a spiral of flowcontinually grows at the tips with the motion of the wing. As the flow curls away from the wing tipit loses velocity which causes the flow to roll up into an approximately circularly symmetric vortex.

The distance between each consecutive turn limits to zero, as in fig.1 as the particle speed reducesalong its path from the wingtips. At each side of the aircraft a counter-rotating vortex is produced.As the craft accelerates the vortices interact in the plane perpendicular to the craft motion forminga V-D.

Before the invention of computer processors, only having graphical solutions meant that the equa-tions governing V-Ds had limited applicability. With the additional potential of numerical modeling,the equations could be heavily refined. The most comprehensive re-evaluation of the vortex dipolemodel was produced by Deem and Zabusky in 1977 [8]. The model included reworking to includetranslation of the dipoles and explained why they are not dispersive and the conditions under whichthey form. A good example of the benefit of the numerical model equations was presented in 2001 byL. Cortelezzi and A. Karagozian [9] who were able to model the V-Ds in 3D and gained significantinsight into the internal vortex pair topology. An example of the modelled V-Ds is shown in fig.3

Some of the places where vortex dipoles have been observed are genuinely surprising. Considerthat in this experiment, the V-Ds are produced by a delta function of impulse into a planar fluid.One simple example of this is rain water hitting the surface of a body of water at a high speed. If theconditions are right, the water droplet impulse should sink the droplet fast down into the water, sothat horizontal spread is suppressed. This essentially provides a stratified layering of the water. Thishas been done using dyed water droplets and dipoles can be readily observed. Attempts were madein this lab to replicate this phenomena but insufficient velocity was achieved.

It is easy to think that vortex dipoles are somehow confined to classical physics and are unlikelyto be found outside of predefined experiments. However, they have recently been observed in Bose-

Einstein condensates and in exotic superconducting materials near the critical temperature phase

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Figure 3: 3D vorticity contour produced by Cortelezzi et al.

transition [10]. Fleming et al. showed that the dissipation of energy near the superconducting region

of some materials is carried away by vortex dipoles which form in the material at a coupling rate Kwhich was highly temperature dependent and in particular - large around Tc. In fact, the equationsthey used to model the dipoles were based on the original work by Deem and Zabusky [8], with a fewvariations to include magnetic effects.

Although not as common as a singly rotating vortex, which is just the manifestation of shearstresses in a fluid, vortex dipoles are still found throughout all fluid systems. The original potentialsolution of V-Ds has been updated and numerical models have been employed and no doubt furtherwork will be done, as a complete revision has recently been offered by Y. Afasayev [11]. The keything is that, given how common they are in nature and the diversity of scientific fields in which theyare found, the work on V-Ds is in no way complete, and is a fascinating example of how somethingsimple can actually have a whole range of complex applications.

Theory

Modeling vortex dipoles

Unlike vortex arrays or systems containing large amounts of vortices, V-Ds do not require numeri-cal methods to model and can be analytically solved. Straight vortex pairs with equal magnitude butopposite sign were first mathematically modeled by Sir Horace Lamb in 1895 [2]. He showed that avortex pair with no net angular momentum can be modeled assuming its overall shape is circular, itsoverall motion is uniform and that it has evenly distributed velocity. Although more accurate modelshave been proposed, they usually limit themselves to specific cases, the only one of which we will

discuss is a delta function of impulse into a stratified fluid. A modernisation of Lambs original workwas produced in 2006 by Jie-Zhi Wu et al [4], which provides a simpler analysis, solving the potentialover a cylinder in polar co-ordinates, (r, ) with

u =1

r

, v =

r

= U

r

a2

r

sin, r > a

Here, a is the distance from the centre of the vortex dipole pair to its boundary at r = a and is the angle around the dipole origin. For simplicity, we impose an external flow ofU so that it

remains stationary with its centre on the origin. Also, since this is a 2d structure we neglect any

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dependence which would have to consider bottom and gravitational effects. We note that in generalin order to linearize the potential:

2 = f(, t), = k2

and so the general equation for vorticity yields

2

r2+

1

r

r+

1

r22

2= k2

The solution of which is involved but we can see some aspects of it straight away. From theprevious equation for it is clear that sin. Also, the equation takes the form of Besselsdifferential equation. Therefore the solution is a Bessel function

= CJ1(kr)sin

The first zero of J1(ka) is at ka = 3.83, which gives a closed circular streamline = 0 at r = a,the boundary of the dipole. More specifically, we can find C from velocity continuity noting that the

spatial derivative of at the boundary is

r= U

r

r

r2

r

sin,

From which the constant C is found to be

C =2U

k(J)1(ka)=

2U

kJ0(ka)

Subbing back in for the potential solution, inside the vortex dipole yields

= 2U J1(kr)

kJ0(ka)sin

= k2 = 2U kJ1(kr)

J0(ka)sin , r < a

This is the point at which the streamlines can be seen graphically and is probably a good finishingline on the general solution, showing that the vorticity magnitude is directly proportional to thepoint velocity everywhere inside the dipole. Also, the vorticity is largest at = /2 along theaxis between the dipole centers, a suggestion that my results disagree with by several degrees, mostlikely due to bottom forces. Fig.4 shows the streamlines in (r, ). Sergey Chaplykin elaborated on

this model [4] and determined that the vorticity has a maximum and minimum (due to symmetricalcounter rotation) at r0 = 0.48a. This result will be confirmed by producing a vorticity contour andobserving it across the plane, finding the maximum and minimum vorticity along a.

Vortex dipole global properties

We now leave the general case solution and look at some of the bulk properties of vortex dipoles.If we inject a short impulse of the water into a large body of fluid both the mass and linear momentumof the system will be conserved. Y. D. Afanasyev showed [5] that the time dependence of a dipoleslength and breadth, L, D are given by

L, D t1/2

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Figure 4: The graphical solution of the vorticity in (r, ) showing the closed streamlines and the mainfeatures of Lambs vortex dipole model [4].

implying L = D

where is a constant for a given dipole that depends on the incident velocity and therefore also theratio of the inertial to viscous forces in the fluid, Re. By plotting against time we can find its meanvalue after formation. We can then find the Reynolds number it coincides with using [1]. This givesan indication of the viscosity of the fluid and the level of turbulence of the system since

Re =

U2

d

/

U

d2

Where U is the fluid exit velocity, v = / is the kinematic viscosity of water and d is the diameter

of the injection nozzle. More specifically for the setup used we know that the velocity at the nozzleis the volume flux q, divided by the nozzle area (d2/4) so we have that

Re2 =J

v2

where J is defined as

J =4q2

d2

thus we see that Re = kq where k is just a constant of the setup that does not change for differentdipoles although it does have a small temperature dependence since v = v(T) as shown in Fig.5.

Volume flux calibration

We now must consider how the volume flux can be calculated from experimentally known quan-tities. In particular, given an initial fluid height, h0 and time interval of injection t we would liketo have q = q(h0, t). In fact, as will be shown, once a calibration has been performed the timeinterval has no bearing on the volume flux since q is a constant for a given initial height and a smallnet volume released. We start with the assumption that given a height of liquid with an opening atits bottom, the rate of change of its height is proportional to the height.

h

t h

which immediately implies

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Figure 5: Temperature dependence of v, the kinematic viscosity. In general v 0.012cm2/s in thedark lab used.

h(t) = h0ekt

The volume flux, q is the change in volume divided by the change in time. We can now write itas

q =h0 hf

t=

h0(1 ekt)

t

Let us now assume q is an experimentally found quantity, where we can set h0 hf = 1cm3 so

that we can solve for the constant k, which we will define as a magnitude only, with the sign outsideof the exponent.

1

h0 1 = ekt

k = ln{hfh0}/t

k was found to be 0.00183, details are given in the results. However using the k value found byexperiment we return to the equation for q and expand the exponent to first order

q = h0(1 e

kt)

t= h0(1 1 + kt)

t

q = kh0

The higher the initial height, the larger the volume released by the nozzle in one second, q.Further to this, we suggest that as we saw that Re = kq, the Reynolds number of the flow is directlyproportional to the height of the fluid (fig.6) provided we assume at h0 = 0 there is no volume flux.This imposes an effective range of Re which can be altered by changing the height of the burette butis typically 10 < Re < 80.

The first part of the experiment concluded with the estimation of Re via the aspect ratio , and

by the volume flux, q(h0).

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Figure 6: The Reynolds number as a function of water height

Particle Image Velocimetry (PIV)

In PIV, patterns of particles spanning a small sub-area are matched with a slightly shifted setof particles separated by a well-defined time interval. The resulting displacement gives an averagevelocity for that region [6]. The experimental considerations of PIV are detailed in the methodwhere as here, we provide an overview of the image processing entailed. The main principle of PIValgorithms is to take two images and perform cross-correlation on them to determine the motion ateach point in the image over the time interval between their capture. The most straight forwardapproach to cross-correlation of the images is to define the images as divided into sub-windows ofpixel side length N = 2n where the n used was typically 4, 5 or 6. Two dimensional cross correlationof each sub-window pair is performed.

Let us define sub-windows I and J within the first image F with indices i, j representing pixellocations within the sub-windows. Similarly, we define the same sub-windows in the second image

F

, we then have

R(s, t) =1

N2

N1i=0

N1j=0

FI,J(i, j)F

I,J(i + s, j + t)

where R is the cyclic cross-correlation between the windows I and J in each image pair. Theequation also has a concise fourier form which was not used in this experiment. This basic methodcan not interpret changes above N/2 and so more advanced algorithms are used. Also, it is necessaryto normalise the background which is not a complex procedure or equation, but is very large andbasically involves summing over the mean non determinate sub-window cross correlations and dividingeach cross correlation by this value of R(s, t). The actual program used was MatPIV16.1 written by

J. Kristian Sveen which is available as freeware online. The initialisation commands required an

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approximation of the mean vector per sub-window and an input of the time duration between theimages.

When the mean particle displacement is obtained at each window over the whole image, weeffectively have obtained the velocity field. Taking the curl of the velocity field, U(u, v) gives thespatially resolved vorticity of the fluid. Using basic Matlab commands it can be shown that the totalvorticity of the dipole is near zero and that the y-directional momentum has a maximum along thecentre line of the dipole. Given R C subwindows of a vorticity matrix M we can say.

M+ = sum(M.*(M> 0)

M = sum(M.*(M< 0)

Net vorticity = M+ + M

and also to find net vorticity at each X and Y level:

for n = 1:R , for m = 1:C

X(n) = sum( M(R,:) )

Y(m) = sum( M(:,C) )

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Method

The equipment was set up as shown. A web camera with a maximum resolution of 640 480 pixelswas connected to a laptop using the Logitech drivers available online. The .avi videos recorded wereviewed frame by frame using virtualdub.exe (www.virtualdub.org), a freeware .avi viewer which givesthe frame number and time in ms for each image. The frame rate of the camera was set to 30

frames per second. This was confirmed by counting the number of frames over a set time interval.The diammeter of the nozzle was found by inserting thin wires into the injection point and thenmeasuring the wires with a m gauge.

Figure 7: The experimental setup for non PIV imaging.

The water tank and burette were prepared separately.

The tank had a top surface area of 1250cm2. We required a salt water bottom layer of 2-3cm.

Thus, in order to produce 2.5cm of bottom layer, 3L of salt water was included, requiring 350g ofsalt per refresh of the layer to hit a reasonable ( 50%) level of salt saturation. The salt is addedand stirred, left for ten minutes and then stirred again to maximise distribution. When this was notdone, the top layer was seen to be uneven on the surface of the salt water layer. A sheet of A3 paperwas placed on top of the salt layer and pure water was sprayed onto the surface. The paper givensuggest a thin as possible surface layer however, this was found to produce significant bottom effectsdue to interference of the bottom and top layers which affected the dipoles produced. A larger layerof 0.5cm layer was used on top, requiring 600ml of de-ionised water. The A3 sheet was then carefullyremoved and the layers were given time to settle.

During this time, the burette was prepared which contained up to 25ml of red dyed pure water.The burette was carefully inserted into the top surface layer. A retort stand was used to keep the

height of the needle to the pure water layer constant. An additional control retort stand was usedto re-balance the needle angle in the fluid, since the needle tended to point up slightly when placedinto the water due to upthrust.

The volume flux was calibrated using q = kh0. k was found experimentally using known volumefluxes at increments of 1ml along the burette liquid height. Typical dipoles were 0.1ml or less involume and so h0 hf was highly inaccurate and so q was found indirectly using k. Generally, therewas a pay off between pixel resolution and the time during which the dipole was in the camera view.Also, the scaling of cm to pixels was recorded with each run since any movement of the camera causedit to vary. Surprisingly, the x and y planes had the same calibration factor. t and q were varied.During a run, the following procedure was used. More than 40 video runs are available.

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Run procedure

The initial height h0 was recorded and its corresponding Reynolds number was found (fig.6).

The light was turned on and camera was set to view the nozzle and 12cm in front of it.

The video was started and a short injection of fluid was allowed through the burette.

Several frames of the dipoles development were isolated using Virtualdub.exe

The pixel-cm scale was calibrated in paint and L(t),D(t) and t were recorded for each frame.

was found for the dipole and the corresponding Reynolds number was found graphically

Figure 8: Time evolution of a vortex dipole at a) 0.5s , b)6s c) 8s and d 14s)

Generally, the dipoles were initially long and narrow, only spreading out once their velocity haddecreased below a threshold that was usually about 50% of the injection velocity. In terms of theanalysis, this meant that varied with time, specifically it was seen to have an asymptotic behaviour,quickly reducing to a near constant value where it stayed for t > 1min. This meant that a bestfit wasrequired which would omit the first few high values. An example is included in the results. Runswere also performed where a continuous dye stream was injected behind the dipole. This gave larger,faster moving dipoles. Also, when this was performed in a non density stratified fluid, a dipole wasseen to form into a more spherical dipole shape.

This concluded the main analysis of the dipole momentum. The other most significant vortexdipole property, vorticity, was explored using particle image velocimetry (PIV).

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PIV

The setup was altered in three ways. Firstly, pure non-dyed water was used as the injectionfluid. Secondly, a projector was position parallel to the water surface and a thin sheet of light wasspread across the top layer. Thirdly, 50m balls were seeded into the water using a spray, wherethey floated. These had a tendency to clump together even at relatively low particle densities. Tocompensate for this, the balls were then sprinkled by a spatula over the surface, this gave a muchbetter distribution. The seeded particles typically took 10 minutes to settle. A video was started froma pre-chosen height with known cm-pixel scale. The water was injected as before and frames wereisolated with a known time between them using virtuadub.exe. The time interval was usually between0.1s < t < 1s. The two images (im1, im2) were then saved in a directory containing MatPIV.m, thecore file of MatPIV16.1. The command structure for cross correlation of the images was

function [x,y,u,v]=matpiv(im1,im2,winsize,Dt,overlap,method,wocofile,mask,ustart,vstart)

where winsize was the sub-window size in pixels, which is a multiple of 2n, overlap was the requiredoverlap size of the windows, which could be set to zero. Method was cross correlation and nomasking of areas was usually performed. Vorticity was then found using the velocity map [x,y,u,v]by taking its curl, the code is easy to follow and is based on the idea of least squares:

{

DeltaX=x(1,2)-x(1,1);

DeltaY=y(1,1)-y(2,1);

for i=3:1:size(x,2)-2

for j=3:1:size(x,1)-2

vor(j-2,i-2)= -(2*v(j,i+2) +v(j,i+1) -v(j,i-1) -2*v(j,i-2))/(10*DeltaX)...

+ (2*u(j+2,i) +u(j+1,i) -u(j-1,i) -2*u(j-2,i))/(10*DeltaY);

end

end

outp=vor;

xa=x(3:end-2,3:end-2);

ya=y(3:end-2,3:end-2);

}

We end in a matrix output outp which can be plotted as a surf plot or as required. The mainpractical issue with this method was the web camera resolution. Inevitably, there was a pay offbetween the resolution of individual particles and viewing of the entire dipole. Small sections of adipole could be viewed giving excellent velocity maps, but larger viewed areas lost resolution, requiringmasking of null velocity areas where the particles were not picked up. A compromise was reachedby generally producing small dipoles (Dmax = 2cm) although the larger ones were more impressive.

Additionally, it was possible to perform PIV by turning the bottom projector on, taking theimages as usual. The images were then gray-scaled and the colours were inverted and the brightnesswas reduced. The effect of this was to simplify the setup while still maintaining essentially similarpictures of white seeded particles against a dark background.

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Results

The flux calibration against height is shown below in fig.9 (a). Although the time intervals t(h)were non linear with height, the flux, q(h) was seen to be proportional to height of the liquid. Fromthis, the constant k was determined for each height, its variation is shown in fig.9 (b). In c.g.s units,if we consider that the height of the burette correlates to cm3 k was 0.00182cm3/s per cm of height.

Figure 9: a) The averaged flux as a function of height. b) the constant k and its bestfit.

The graph provided for fitting against Reynolds number was logarithmic, squared and generallydid not give a clear indicator of the experimental error. A genfit was performed in MathCAD usingdata from the bestfit (Re2) line so that a direct correlation between and Re could be foundgraphically.

Figure 10: The alpha and Reynolds number values found for 16 dipoles

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This avoided the tedious conversion from logarithmic to linear scaled data. The line took the formof the exponential of a polynomial, exp{f(x1, x2, x3, x4...)}. Poly-Gaussian fit was not considerednecessary as the line was well formed. The list of dipoles analyzed is shown graphically with thetheoretical fit in fig.10. Another way of interpreting this data is to find the Reynolds number byboth methods and to find their ratio against height (fig.11). This proves to be useful as it showsthe values were typically within 90% accuracy and that since most of the points are below unity, theRe(h) values were larger than the Re() values. It is unlikely that the flux was overestimated, as thevalues found were surprisingly small, which suggests that the actual graph provided with the papermay have been off the true value by some constant. In fact, it is possible that the fluxes were notoverestimated but underestimated since q(0) = 0 was assumed.

Figure 11: The alpha and Reynolds number values found for 16 dipoles

The behaviour of and the dipole velocity with time was explored, an example is given below(fig.12) for run7. The velocity had a strong, asymptotic behaviour, contradicting the suggestion by

Y. Afanasyev [11] that the steady state velocity is constant at Uconst = Ujet/2. The behaviour wasfound repeatedly. Typically the initial velocity was above 5cm/s for half of a second, within 5 secondsthe jet typically travelled at (0 1.5)cm/s slowing to (1 2)mm/s in the steady state. Typicallythere was a 10-100 fold decrease in velocity.

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Figure 12: Velocity and time dependence for one dipole. Other runs gave very similar resultsprovided as an appendix.

Also, dipoles were collided to observe the effects (fig.13). Normal process was that the slowerdipole would remain unchanged while the faster dipole would smear out over the front of the other,forming small, isolated vortices. This behaviour has been well documented in numerous papers by Y.Afanasyev.

Figure 13: Vortex dipole collision

PIV provided interesting results. The velocity fields were well formed for small image areas buthad aberrations over large areas. These were then converted into vorticity fields using the codeprovided in the theory section. Fig.14 shows the vorticity field for a large dipole. Limited maskingwas employed at the corners of the image. The circulation is counter-rotational in each vortex, andso we see one side of the dipole as negative and the other as positive. Summing over the whole area ofthe dipole gave a near zero value for vorticity, 41/(478 + 437) 4.5%. The y axis momentum had apeak at the central sub-window row. Also, recalling the result of the theoretical model, rmax = 0.48a,

in (b) we see maxima and minima at 0.487a and 0.52a, both within 10% of rmax.

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Figure 14: Vorticity field for a steady state V-D.

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Figure 15: Velocity field for a large V-D flowline.

Figure 16: Velocity field detail of the off vortex velocity. The circulation is non zero outside thedipole.

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Conclusion

In this experiment vortex dipoles were produced using a dye injection into a layered fluid. TheReynolds number of each dipole was found by two methods; as a function of incident volume flux, q,and as a function of the vortex dipole aspect ratio, . The results were as expected in [1]. ParticleImage Velocimetry was used to find the velocity field of the dipole from which the vorticity at each

point was found. The vorticity fields produced were consistent with the analytical models presented.The velocities found were consistent with the asymptotic time dependence observed using frame byframe analysis. Other tertiary experiments were performed, an example being dipole production offof a main jet using a wind force above the water surface, shown in fig.17.

Figure 17: A body of dyed water was blown with a 10cm/s wind gust, producing a dipole.

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References

[1] Y. Afanasyev Investigating vortical dipolar flows using particle image velicometry Am.J.Phys.70 (1) January 2002

[2] Sir Horace Lamb Mathematical Theory of the Motion of Fluids 2nd Edition (1932)

[3]T. W. Neely et al. Dec 2009 Observation of vortex dipoles in an oblate Bose-Einstein condensatehttp://www.citebase.org/abstract?id=oai:arXiv.org:0912.3773

[4] Jie-Zhi Wu, Hui-Yang Ma, Ming-De Zhou - 2006 Vorticity and vortex dynamics p288 - 291

[5] Y. Afanasyev Investigating vortical dipolar flows using particle image velicometry Am.J.Phys.70 (1) January 2002 section III, data analysis.

[6] J. Kristian Sveen et al. Jan. 22nd 2004. Quantitative Imaging Techniques and their applicationto wavy flows Dep. of Mathematics, University of Oslo, Norway [email protected]

[7] P.G. Saffman, Vortex dynamics, Cambridge University press, Ch5, Ch 6.

[8] G. S. Deem , N. J. Zabusky Vortex Waves: Stationary V states, interaction recurrence andbreaking Physical review letters, Vol. 40 N.13 Dec. 1977.

[9] Cortelezzi et al. J. Fluid Mech. (2001), vol. 446, pp. 347, 2001 Cambridge University Press, Onthe formation of the counter-rotating vortex pair in transverse jets

[10] F. Fleming et al. Physical Review Letters, Vol.62, No. 6, 18Dec 2009, Vortex Pair Excitationnear the Superconducting transition of Bi2Sr2CaCu2O8 Crystals

[11] Physics of fluids, 18 037103 (2006) Y. Afanasyev Formation of vortex dipoles

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104557839 Vortex Dipoles Ordered Structures From Chaotic Flows

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