Article published by EDP Sciences and available at http ... · Ghoniem ortic it y of a v ortex tub...

ESAIM� Proceedings� Vol� �� pp� ��http�www�emath�frprocVol��

Filament Surgery and Temporal Grid Adaptivity

Extensions to a Parallel Tree Code for Simulation

and Diagnosis in �D Vortex Dynamics

Victor M� Fernandez� Norman J� Zabusky

Department of Mechanical and Aerospace Engineering and CAIP Center

Rutgers University

Piscataway� NJ �� USA

Pangfeng Liu�� Sandeep Bhatt�� Apostolos Gerasoulis�

�Bell Communications Research

Morristown� NJ �� USA�Department of Computer Science

Rutgers University� Piscataway� NJ �� USA

Abstract

The vortex collapse�reconnection process presents behaviors commonly observed in turbu�lent �ows� multiple spatial and temporal scales� rapid vorticity and strain�rate ampli�cationand dissipation through generation of small scales� In this work we reduce the computationalcomplexity of our problem by using hierarchical methods �tree codes�� introducing a time ex�trapolation framework for each particle� and applying a �lament surgery algorithm� basedon the energy density� as a regularization for vortex collapse� The high performance parallelimplementation of Barnes�Hut algorithm permit us to increase by one order of magnitude theresolution of the vortex collapse simulations� The use of the time extrapolation for slow movingparticles helps in concentrating the computational e�ort in the important dynamic domains�The vortex �lament surgery regularizes e�ectively the growth of the number of particles in thecollapse regions of the �ow� The reduction in complexity achieved will contribute to optimizethe use of the numerical simulations in the reduced model building process�

� Introduction

The study of small scales formation in high Reynolds number �ows is of fundamental importancebecause of the relationship this process has with important areas of engineering applications� includ�ing drag reduction and mixing �� The study of these types of �ows tends to require substantialamount of computational resources� In multi�lament models� the On�� nature of the computa�tional expense of the Biot�Savart direct method where n is the number of grid points� severelylimits the resolution of the simulations �� The objective of our work is to adapt a high per�formance On log n� parallel implementation of N�Body algorithms � � � � to the vortex dynamicsproblem and extend it by including a lament surgery algorithm based on the energy density ��We also exploit the multiple time scales in the �ow to concentrate the computational e�ort inregions with higher dynamic activity� We emphasize the use of the fast summation techniques forthe computation of diagnostics�

��

Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:1996010

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http://dx.doi.org/10.1051/proc:1996010

V�M� Fernandez et al� � Filament Surgery and Temporal Grid Adaptivity Extensions � � � ��

� A tree code for vortex dynamics

The version of the vortex method we use was developed by Knio and Ghoniem �� The vortic�ity of a vortex tube is represented by a bundle of vortex laments �i�� t

�� each of them withcirculation �j

�x� t��

NfXj��

�j

ZCd��j � t

�� f�jx� ��j � t��j� � ��

The total number of laments forming the vortex tube is Nf and �j is the initial arc length alongeach lament� The core function f� is given by the third�order Gaussian

f��

�� t��exp �� t��

where � is the core �radius��The ordering of the grid points in the laments is only important when computing the di�er�

entials

d� � ��j ��

��j�� j��

This is a characteristic of the lament approach in �D vortex methods� In contrast with the �vortexarrow� algorithm �� updating the �strength� of the vortex elements in the lament methoddoes not require the evaluation of the velocity gradient� which involves the computation of anadditional integral over all of the particles� Also� laments with form of closed curves� satisfy thedivergence free condition of the vorticity eld� This is not always the case at all times in the vortexarrow approach�

After computing the di�erentials the order of the summation is unimportant� Therefore thevorticity distribution may be discretized by

�x� t��

NpXp��

�p f�jx� �pt��j� � ��

where�p � �p ��p � ��

With this vorticity distribution the stream function becomes

�x� �

NpXp��

�p �jx � �pj� � �

and the velocity eld is

ux� � �

NpXp��

�p �r�jx� �pj� � ��

where

��

Z �

�

ds

s�

Z s

�

f�t� t� dt � ��

We use the dimensionless time t � t��ts � t� �C � ��a� � � where �C and a are dened in ��

ESAIM� Proceedings� Vol� �� pp� ��


A fast evaluation of the stream function � and the velocity eld u can be obtained by usingthe Taylor�s expansion of the function �� Using the notation by Band �� and Schwartz �� for thepowers �� r�n� we obtain the multi�pole expansion of the stream function

�x� �m�j�k�nXn��j�k

Mj� k� n � j � k� Dj� k� n � j � k� �x� �� m ��

and the multi�pole expansion of the velocity eld

ux� � �m�j�k�nXn��j�k

Mj� k� n � j � k� � Dj� k� n � j � k� r�x� �� r��m � ��

where

Mj� k� n � j � k� �

NpXp��

�p p � ��j� p � ��

k� p � ��

n�j�k� � ��

Dj� k� n � j � k� ��n

n� j � k�� j� k�j�

k�

n�j�k� ��

and

�m �

NpXp��

�p Km � ��

with

Km�x��p �

j�k�m��Xj�k

�m� � � ��p � �� j� ��p � ��

k� ��p � ��

m��j�k�

�m� �� j � k � j� k��j� �

k� �

m��j�k� ��x� �� t ��p � ��

��

and � � t � ��

As a diagnostics for vortex collapse� we compute the strain�rate and also the energy density�which can be evaluated fast by using the multi�pole expansion of ��

The error of the truncated multi�pole expansion has been estimated for the gravitational prob�lem� among others� by Salmon and Warren �� We have checked that their estimate applies tothe multi�pole expansion in the vortex dynamics problem� For the point vortex Green�s function� � �� the weak bounds �� of the error are� for the vector stream function

j�mx�j � C� b

d�m��

and for the velocity eld

jr ��mx�j � C� b

d�m��

The constants are

C� ��

d

�

�� b�d

NpXp��

j�pj � ��

and

C� ��

d��

� � b�d��m � ��

b

dm � ��

NpXp��

j�pj � ��



where b � max j�p � ��j is the size of the cluster and d � jx � ��j is the distance to thecluster� More precise estimates are also available by Salmon and Warren �� and more recently byWinckelmans et al� �� for the vortex dynamics problem�

To reduce the computational complexity in the sums in Eq� �� we use � � the hierarchicalmethod by Barnes and Hut �� for gravitational elds� To organize a hierarchy of clusters� we rstcompute an oct�tree partition of the three�dimensional box enclosing the vortex elements� Oncethe oct�tree has been built� the moments of the internal nodes are computed in one phase up thetree� starting at the leaves� The next step is to compute induced velocities� each particle traversesthe tree in depth�rst manner starting at the root� For any internal node� if the distance D fromthe corresponding box to the particle exceeds the quantity R�� where R is the side�length of thebox and � is an accuracy parameter� then the e�ect of the subtree on the particle is approximatedby a particle�to�cluster interaction� where the cluster is represented by point vortex multi�poleslocated at the geometric center of the tree node� The tree traversal continues� but the subtree isbypassed� Once the induced velocities of all the bodies are known� the new positions and vortexelement strengths are computed� The entire process� starting with the construction of the oct�tree�is repeated for the desired number of time steps� For convenience we refer to the set of nodes whichcontribute to the velocity on a particle as the essential nodes for the particle � �� Each particlehas a distinct set of essential nodes which changes with time� One remark concerning distancemeasurements is in order� There are several ways to measure the distance between a particle anda box� Salmon �� discusses various alternatives in some detail� For consistency� we measuredistances from bodies to the perimeter of a box in the L� metric� In our experiments we vary � totest the accuracy of the code compared to solutions computed using the direct method at specicpoints� The tree code has been implemented on a CM� with SPARC vector units� The simulationsustains an overall rate of over �� M�op � sec per node � ��

Our initial condition is a multi�lament vortex ring constructed around the center line

�� a cos �� b sin �� c sin ��

where � � � � �� and which we call the �Lissajous�elliptic� ring �� The thickness of the multi�lament ring is �C � The circulation distribution �p� and the initial lament core radius �p also needto be specied� Low aspect ratio elliptic rings a � b� c � � have periodic behavior that can be usedfor dynamic testing� The initial testing of the code has been carried out with two cases� The rstis a low aspect ratio elliptic ring with a � �� b � �� and c � �� The second case tested correspondsto a collapsing vortex ring with a � �� b � �� and c � �� The collapse state is shown in Fig� ��This ring has Nf � �� laments with a total number of particles Np � ��

� Multiple time scales and temporal grid adaptivity

It has been natural in astrophysics to recognize and exploit the hierarchical structures on di�erenttime scales that occur in gravitational systems� In �� Aarseth �� introduced the use of variableand individual time step for each particle star�� Some time later �� Ahmad and Cohen ��linked this idea to a hierarchy in space� They separated force contributions in a slow varyingforce due to distant stars regular component� and a highly �uctuating force induced by immediateneighbors irregular component�� More recently� temporal extrapolations have been introduced inthe multi�pole expansion by Jernigan and Porter �� McMillan and Aarseth �� and Sundaram��

The immediate options to exploit di�erent time scales are either to extrapolate force at slowmoving particles or extrapolate contribution by distant particles extrapolation of multi�pole mo�



ments� i�e� temporal multi�pole expansion�� An optimal scheme is obtained for well separated timescales like in turbulent �ows�� We started considering these ideas by exploring the rst approach�

We keep the original time framework of our previous algorithm� which we call now the globaltime� The extrapolation of the velocity eld for the slow moving particles is carried out by intro�ducing a local particle time reference system Fig� �� In the global time framework� the velocity isevaluated using the full Biot�Savart integral� The second order predictor�corrector scheme requirestwo velocity evaluations� Using these� plus one evaluation from the previous time step three evalu�ations in total�� we construct a cubic polynomial extrapolation for the position in the local particletime

k � ��h � � �� k � ��h �u��

�k � ��h� �u��

�

k � ��h� �u�� dk��

where �u� and �u�� are central di�erences obtained using un�� un and un�� in the global time frame� u represent position and velocity components�� This extrapolation for the particle positionis equivalent to a third order Runge�Kutta integration scheme with velocity evaluated using aquadratic extrapolation�

The local truncation error in the extrapolation

dk��

�� k � �� k � �� h� u�� Oh��

is the sum of the error of the third order Runge�Kutta scheme and the error of the central di�erences�u� and �u�� which corresponds to the error in the velocity extrapolation� It is important to remark�that even though the central di�erences preserved the order of the local truncation error of theRunge�Kutta scheme� the local truncation error in the extrapolation is proportional to the fourthpower of the number of extrapolated time steps k � �� This means� the extrapolation can onlybe used at slow moving particles� where a small value of u�� balances the error�

The two�times frame algorithm is as follows we compute the extrapolated new position �k��If the normed displacement of the particle from the position at the local initial particle time is lessthan the accuracy parameter e

j�k�� j

� e � ��

then �k�� is accepted as the new particle position� Otherwise we compute �n�� with full velocityevaluation� update �u� and �u�� and reset the local particle time to zero� The normalization ofthe particle displacement with the particle core size � recognizes this parameter as a measure ofresolution� A more appropriate scheme would evaluate the third derivative of the velocity� in orderto use the error estimate �� to select the particles to be moved using the interpolation� The costof this scheme could be higher since more velocity evaluations would be required for the evaluationof the third derivative of the velocity�

One example of how this scheme may be used to increase accuracy in selected regions withless expense is presented in Fig� �� In this case we present the inter�lament distance sd and themaximum curvature � ��max vs� time� as we decrease the time step from dt � ��e�� to �e�� fora collapse run with the accuracy parameter e � �� In the case of the inter�lament distance sd�convergence is observed for the two smaller time steps and a larger error for the coarse time step�The error is decreased in the half time step by multiplying the CPU time by a factor of �� insteadof � required by the case with no extrapolation�


V�M� Fernandez et al� � Filament Surgery and Temporal Grid Adaptivity Extensions � � � �

� Filament surgery

Surgery for vortex contours �D� and laments �D� has been used in the past� In �D� it was usedby Berk and Roberts �� in the �water�bag� model for Vlasov plasmas and also by Overman andZabusky �� in Contour Dynamics �� for the �D Euler equations with simple topologies� TheContour Dynamics�Contour Surgery algorithm by Dritschel �� based on proximity and curvature�has been applied successfully in large scale simulations with complex topologies� For �D �ows�Leonard �� performed lament surgery based on proximity and orientation antiparallel congu�ration�� More recently� Chorin �� introduced a hairpin removal and lament surgery algorithmbased on proximity and orientation in the theoretical framework of the �innite temperature� state�See also Schwarz� Phys� Rev B ��

In previous studies of vortex collapse �� it is found that the energy density of the collapsingregion in the lament tends to zero as the collapse time is approached� This signature of collapseoccurs despite the observed increase in the self�energy� The reason for this behavior is that theantiparallel region of the vortex lament produces also a peak�like contribution that cancels the self�energy growth� Collapsed regions in the lament don�t contribute much to the total energy of the�ow they are low or zero energy density regions�� but they produce a very large growth of vortexelements in algorithms with variable spatial resolution� due to the large vortex stretching present�The growth of the number of particles can be so large� that the simulation has to be stoppedbecause of the computational expense� In order to regularize the particle growth it is necessary tointroduce a lament surgery algorithm to eliminate the collapsed portions of the vortex laments�

In our algorithm the energy

E ��

�

nfXj��

�j

ZCj

Esj� d sj ��

is computed using the approximation of the energy density

�Es� �d�

ds� �s� � ��

where � is dened in Eq� � After computing the energy density in the laments� we identify theregions below an specied energy density threshold d� The next step is to nd pairs of reconnectingregions by checking orientation of the thresholded regions� This permits the identication of �local�and �nonlocal� collapse regions� which are dealt with di�erently for bookkeeping purposes� The laststep is to eliminate low energy density regions by reconnecting and deleting particles bookkeepingoperations�� The variable resolution subroutine implemented in the past provides some roundingof the edges left by the surgery as it interpolates new grid points with a cubic spline� At presentwe restrict reconnection to regions in the same lament� This is only to simplify the bookkeeping�In the future we will remove this restriction� leaving only the condition that the �ow of vorticitythrough the vortex laments remains continuous vorticity must be divergence free�� This will bedone by reconnecting laments only if they have the same circulation�

Surgery is a regularization that produces punctuated or intermittent dissipation� This means�it operates only when a collapse event has occurred� and at the level of the small scales generatedby this event� We are still investigating how the rate of energy dissipation obtained by this algo�rithm scales with respect to the enstrophy of the �ow� Another related aspect we are examining�corresponds to the energy density threshold to be used� which we believe should be the minimumenergy density at the validity limit determined by the local core size� of the simulation�

Our numerical results are as follows� In Fig� � we present the e�ect obtained by using di�erentenergy density thresholds d for a single�lament collapsing vortex ring at t � �� with the constant



core model �� Whereas for d � � there is practically no small scales removal� these are removedmore e!ciently as the energy density threshold is increased� It can be noticed that the only regiona�ected by the surgery corresponds to the collapse region� In Fig� �a�� we observe how the growthof the number of particles Np is reduced as the energy density threshold is increased� The largestparticle growth corresponds to the no surgery case� In Fig� �b� we see that the inter�lamentdistance Sd increases as vortex reconnection takes place� This occurs earlier in time for higherenergy density threshold d�

In Figs� a� and b� we have the non�zero component of the linear impulse and the energyfor the simulation with energy density threshold d � �� In parts c� and d� of the same gurewe observe the inter�lament distance and the maximum normalized curvature � ��max vs� timerespectively� The continuous line corresponds to the combination of the surgery� with four pointsmoothing of regions with high normalized curvature � � � �� Even though the appearance of thecurves is smoother for this case� the cost in terms if the energy loss is higher� In part d� of thegure we observe that as the reconnection takes place� the surgery algorithm leaves high curvatureregions in the laments� These regions are eliminated by the surgery only as they evolve intohairpins with the pairing characteristic of antiparallel regions�� In the other hand� the curvaturesmoothing eliminates the high curvature regions more e�ectively� We are examining currently themeaning of these results in terms of energy dissipation and expect to have quantitative results inthe near future�

We started applying the lament surgery to multi�lament simulations with the constant coremodel� The no surgery case is presented in Fig� � for a ring with �� laments� The same caseat the energy density threshold d � �� with no curvature smoothing� is presented in Fig� �� Amore detailed view of the reconnection process at t � �� and t � �� is presented in Fig� �� Thediagnostics box surrounds the region with the maximum normalized vortex stretching magnitude�The last time shown corresponds to the end of the reconnection� The isosurfaces correspond tothe vorticity magnitude� In Fig� �� we have the non�zero component of the linear impulse andthe energy vs� time for the multi�lament simulation� In Fig� ��a� we observe how the surgeryalgorithm e�ectively regularizes the very steep growth of the number of particles observed in the nosurgery case� In part b� of the same gure we observe the evolution of the maximum curvature ofthe laments for these simulations� which were performed without the curvature smoothing process�

� Conclusions

In this work we rst reduce the computational complexity of our simulations via a high performanceparallel implementation of the Barnes�Hut method� Then we introduce a temporal extrapolationscheme that permits higher resolution in selected regions at lower computational expense� Finally�we introduce a lament surgery algorithm based on the energy density� This is an e�ective reg�ularization that produces punctuated dissipation� as it eliminates small scales whenever they aregenerated during collapse events�

This work is part of the Hypercomputing and Design HPCD� project� supported by ARPA�contract DABT� ��C�� Parallel simulations were performed on the CM� at NCSA� Futureimplementations will be carried out on the SP� at CTC�



References

�� Aarseth� S�J�� Dynamical Evolution of Clusters of Galaxies� I�� Monthly Notices Roy� Astro�

nomical Soc�� pp� ��"��

�� Ahmad� A� and Cohen� L�� A Numerical Integration Scheme for the N �Body GravitationalProblem�� J� Comp� Phys�� pp� ��"��

�� Band� W�� Introduction to Mathematical Physics� University Physics Series D� Van NostrandCompany� Inc�� Princeton� New Jersey� pp� ��"��

�� Barnes� J� and Hut� P�� A Hierarchical ON logN� Force�Calculation Algorithm�� Nature��

�� Berk� H� and Roberts� K�� The Watter�Bag Model�� Meth� Comp� Phys�� pp� ��

� � Bhatt� S�� Liu� P�� Fernandez� V� and Zabusky� N�� Tree Codes for Vortex Dynamics Applicationof a Programming Framework� Workshop on Solving Irregular Problems on Parallel Machines�International Parallel Processing Symposium� Santa Barbara� ��

�� Chorin� A�J�� Hairpin Removal in Vortex Interactions�� J� Comp� Phys�� pp� �"��

�� Chorin� A�J�� Hairpin Removal in Vortex Interactions II�� J� Comp� Phys�� pp� �"��

�� Dritschel� D�G�� Contour Surgery A Topological Reconnection Scheme for Extended Integra�tions using Contour Dynamics�� J� Comp� Phys�� pp� ��"� � ��

�� Fernandez� V�M�� Zabusky� N�J� and Gryanik� V�M�� Vortex Intensication and Collapse of theLissajous�Elliptic Ring Single and Multi�Filament Biot�Savart Simulations and Visiometrics��J� Fluid Mech�� pp� ��"��

�� Fernandez� V�M�� Zabusky� N�J� and Gryanik� V�M�� Near�Singular Collapse and Local Inten�sication of a Lissajous�Elliptic Vortex Ring Non�Monotonic Behavior and Zero�ApproachingLocal Energy Densities�� Phys� Fluids A� � �� pp� ��"��

�� Jernigan� J�G� and Porter� D�H�� A Tree Code with Logarithmic Reduction of the Force Terms�Hierarchical Regularization of all Variables� and Explicit Accuracy Controls�� Astrophysical J�

Suppl� Series� �� pp� ��"��

�� Knio� O�M� and Ghoniem� A�F�� Numerical Study of a Three�Dimensional Vortex Method��J� Comp� Phys�� pp� ��"��

�� Leonard� A�� Numerical Simulation of Interacting� Three�Dimensional Vortex Filaments��Proc� �th International Conference in Numerical Methods in Fluid Dynamics� number �� inLecture Notes in Physics� Springer�Verlag� pp� ��"��

�� Leonard� A�� Computing Three�Dimensional Incompressible Flows with Vortex Elements��Ann� Rev� Fluid Mech�� pp� ��"��

�� Liu� P�� The Parallel Implementation of N �Body Algorithms�� DIMACS Technical Report��



�� Majda� A�J� �Vorticity� Turbulence and Acoustics in Fluid Flow�� SIAM Review� �� pp��"��

�� McMillan� S�L�W� and Aarseth� S�J�� An ON logN� Integration Scheme for Collisional StellarSystems�� Astrophysical J�� pp� ��"��

�� Overman� E�A� and Zabusky� N�J�� Evolution and Merger of Isolated Vortex Structures��Phys� Fluids� �� pp� ��"��

�� Salmon� J�K�� Parallel Hierarchical N�Body Methods� Ph�D� Thesis� California Institute ofTechnology� Pasadena� California ��

�� Salmon� J�K� and Warren� M�S�� Skeletons from the Treecode Closet�� J� Comp� Phys�� pp� �� "��

�� Salmon� J�K�� Warren� M�S� and Winckelmans� G�S�� Fast Parallel Tree Codes for Gravita�tional and Fluid Dynamical N �Body Problems�� Int�l� Supercomputer Appl��

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Annual Forum on Vortex Methods for Engineering Applications� Sandia National Laboratories�Albuquerque� NM� pp� ��"��

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Figure � Global and local particle time frames�

Figure � High resolution Np � �� particles� simulation of vortex collapse�



10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.0

1.0

2.0

3.0

4.0

5.0

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.0

1.0

2.0

3.0

4.0

5.0

dt = 10.0e-4

dt = 5.0e-4

dt = 7.5e-4

validity limit

κ δδ

e = 0.1

dt = 10.0e-4, cpu = 45.2 secdt = 7.5e-4, cpu = 54.8 secdt = 5.0e-4, cpu = 63.4 sec

( a ) ( b ) ( c )

S /d

Figure � a� Inter�lament distance Sd vs� time� b� maximum curvature � ��max vs� time and c�top view of vortex ring for di�erent time steps� Use of time extrapolation with e � �� producessavings in cpu time as the time step is decreased�

d = 0.000d = 0.005d = 0.010d = 0.030d = 0.050

d = 0.000 d = 0.005 d = 0.010 d = 0.030 d = 0.050

Figure � Surgery of single�lament ring at di�erent energy density thresholds d�



10 t1.2 1.4 1.6 1.8 2.0 2.2 2.40

100

200

300

400

d = 0.010

d = 0.030

no surgery

d = 0.050

( a )

N p

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

d = 0.010

no surgery

d = 0.050

d = 0.030

0.0

3.0

6.0

9.0

12.0

15.0

( b )

δdS /

Figure � a� Evolution of the growth of the number of particles Np vs� time for the single�lamentring with surgery for di�erent energy density thresholds d� b� Inter�lament distance Sd vs� time

0.95

1.00

1.05

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

curvature smoothing

no smoothing

E / E 0

0.95

1.00

1.05

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

P / P 0

0

4

8121620κ δ

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

( a ) ( b )

( c ) ( d )

δdS /

0.0

3.0

6.0

9.0

12.0

15.0

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

Figure a� Non�zero component of linear impulse P and b� energy E are normalized withthe initial values P� and E� respectively� c� Inter�lament distance Sd vs� time� d� maximumnormalized curvature � ��max vs� time� The continuous line combines surgery with smoothing ofhigh curvature regions�



t = 0.22 t = 0.21 t = 0.20 t = 0.19 t = 0.14

Top view

Side view

Figure � Evolution of multi�lament ring with no surgery�

Side view

Top view

t = 0.22 t = 0.21 t = 0.20 t = 0.19

Figure � Multi�lament ring with surgery at the energy density threshold d � �� no curvaturesmoothing��



Figure � Diagnostics box in region with maximum normalized vortex stretching magnitude at twodi�erent times in the reconnection process�



10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.95

1.00

1.05

d = 0.02

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

no surgery

d = 0.01

0.95

1.00

1.05

d = 0.02

E / E 0P / P 0

( b )( a )

Figure �� a� Linear impulse non�zero component� and b� energy for the vortex reconnectionmulti�lament simulation�

10 t1.2 1.4 1.6 1.8 2.0 2.2 2.4

0.0

2.0

4.0

6.0

8.0

10.0κ δ

no surgery

d = 0.02

( b )10 t

1.2 1.4 1.6 1.8 2.0 2.2 2.4

d = 0.01

0

2500

5000

7500

10000

no surgery

d = 0.02

( a )

N p

Figure �� a� Growth of the number of particles Np vs� time for the multi�lament ring with andwithout surgery� b� Maximum normalized curvature � ��max vs� time�


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Transcript of Article published by EDP Sciences and available at http ... · Ghoniem ortic it y of a v ortex tub...