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Validation of mathematical models for predicting the swirling flow and the vortex rope in a

Francis turbine operated at partial discharge

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2010 IOP Conf. Ser.: Earth Environ. Sci. 12 012051

(http://iopscience.iop.org/1755-1315/12/1/012051)

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eventually provide additional information with respect to the most amplified perturbations and their dominantfrequency, but it cannot estimate the level of pressure fluctuations associated with the flow unsteadiness.

A mathematical model to be used for analysis of swirling flow with spiral vortex core in a pipe has beendeveloped by Wang and Nishi [7]. Their quasi-three dimensional model considers a superposition of anaxisymmetric base flow with the velocity field induced by a helical vortex filament. The base flow is divided in amain flow in an annular section, with a central dead water region. The main flow has constant axial velocity andfree-vortex circumferential velocity distributions. On the other hand, Wang and Nishi consider a rigid bodyrotation (linear) for circumferential velocity and a parabolic variation for the axial velocity in the central deadwater region. After superimposing the helical vortex on this base flow, the theoretical results for the velocity

profiles, both with vortex rope and circumferentially averaged, as well as the precessing frequency, are shown in[7] to be in a reasonable agreement with experimental data.

The theory used in our investigations of precessing helical vortices in swirling flows was developed byAlekseenko et al. [8, 9] up to analytical solutions for velocity and pressure fields in cylindrical and conicalgeometries. A further step toward practical applications in hydraulic machines is presented by Kuibin et al. [10],where it is shown that the vortex rope geometry, precession frequency, as well as the wall pressure fluctuations

can be computed given a set of swirling flow integral quantities.In this paper we couple a theory for computing the axisymmetric swirling flow at the outlet of Francis turbinerunners operated far from the best efficiency regime with the theory of precessing helical vortices in order to

provide an integrated methodology that can be used in practice for turbine design and optimization, as well as forrapid evaluation of existing runners. First, we introduce the constrained variational problem for computing theaxial and circumferential velocity profiles at the Francis runner outlet, and provide an example for the FLINDTFrancis turbine operated at 70% from the best efficiency discharge. These velocity profiles are validated againstLaser Doppler Velocimetry measurements. Second, we employ the theory of helical vortices [10] to compute thevortex rope precessing frequency as well as the pressure fluctuations, and the results are validated againstcorresponding experimental data [3].

2. Swirling flow at the Francis runner outletThe swirling flow at the outlet of the hydraulic turbine runner essentially influences the overall pressure

recovery and hydraulic losses in the turbine draft tube. Susan-Resiga et al. [11] introduced an empiricalanalytical representation for both axial and circumferential components of the swirling flow ingested by the drafttube of a Francis turbine, leading to a simple parametric description of velocity experimental data. However,these formulae are valid and can accurately represent the experimental data only in the neighborhood of the bestefficiency point, i.e. 10% the best efficiency discharge. This approach is further refined by Tridon et al. [12]who examine the radial velocity component as well, besides the axial and circumferential components. When theturbine discharge is further decreased, say at 70% the best efficiency discharge, the swirling flow at runner outletchanges significantly, with a quasi-stagnant region near the symmetry axis and the flow is confined in an annularsection close to the wall [3, 11]. As a result, the analytical representation of the velocity profiles which uses asuperposition of Batchelor vortices [11, 12] cannot be used for operating regimes with partial discharge when the

precessing vortex rope is developed. Instead, Susan-Resiga et al. [13] introduce a mathematical model that allowthe computation of swirling flow at the Francis runner outlet for a large range of operating points, and we showthat this model is able to capture the main flow features even at low partial discharge.

2.1 Flow kinematics downstream a fixed-pitch runnerTraditionally, the geometry of runner blades in the neighborhood of the trailing edge is chosen such that the

relative flow at runner outlet has a certain distribution from hub to shroud of the relative flow angle, , definedas the angle between the relative velocity vector and the tangential direction,

tan z U

R U =

(1)

where U z and U are the axial and circumferential absolute velocity components at the radius R, and is therunner angular velocity. For fixed pitch runners, the relative flow angle remains practically constant over arather large operating range, particularly for Francis turbines where the large number of blades prevents severeflow detachment at trailing edge. From the velocity triangle at runner outlet, the tangential velocity U is positiveat partial discharge, i.e. co-rotating swirl with respect to runner rotation, and is becomes negative at full load, i.e.the swirl rotates in opposite direction with respect to the runner. In between, there is a regime where the

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tangential velocity vanishes, thus we call the corresponding axial velocity the swirl-free velocity U 0, which isdefined according to eq. (1) as tan = U 0/( R ). As a result, instead of using the relative flow angle to describe

the swirling flow kinematics at runner outlet, we are going to use the swirl-free velocity profile U 0 ( R ), definedas

( )0 1 z U U

U R=

(2)

The above definition can be re-written in dimensionless form using a reference radius Rref and a referencevelocity Rref ,

( ) ( )0 z u ru r u = where u = U /( Rref ) and r = R / Rref are the dimensionless velocity and radius, respectively. From eq. (3), thecircumferential velocity can be written as function of the axial velocity profile, for given swirl-free velocity

profile,

( )01 z u r u u = (4)Equation (4) has the advantage of avoiding the singularity at the axis, where u0 remains finite.

As an example, the swirl-free velocity profile can be determined using the measured axial andcircumferential velocity at runner outlet for a Francis turbine for several operating points within the dischargerange of 70%...110% from the best efficiency discharge [3, 5, 101]. Figure 1 shows the experimental data for 7operating points, translated in u0 values using eq. (3). It can be seen that a simple parabolic fit can correctlyrepresent the u0(r ) profile, which depends directly on the runner blades design at the trailing edge.

0 0.2 0.4 0.6 0.8 1dimensionless radius

0

0.1

0.2

0.3

0.4

0.5

0.6

d i m e n s i o n l e s s s w i r l f r e e v

e l o c i t y

experimental dataparabolic fit

Fig. 1 Swirl-free velocity distribution at runner outlet.

0 0.2 0.4 0.6 0.8 1dimensionless radius

0

0.1

0.2

0.3

0.4

d i m e n s i o n l e s s v e

l o c i t y c o m p o n e n t s

axial velocity measuredswirl velocity measuredaxial velocity computedswirl velocity computed

Fig. 2 Axial and circumferential velocity profilesat runner outlet. LDV measurements [3] correspo

nd to the survey section located at z = 1.4262.2 Integral quantities for swirling flow

The swirling flow if characterized by two main integral quantities. The first one is the volumetric flow rate Q defined as

w

0

2 R

z Q U R dR (5)where Rw is the wall radius. In dimensionless form we introduce the discharge coefficient,

( ) ( )w

22

ref ref 0

r

z

Qu d r

R R =

(6)The second integral quantity for swirling flows is the flux of moment of momentum,

( )w

0

2 R

z RU U R dR (7)which can be written in dimensionless form using eq. (4) as,

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( )( ) ( ) ( )

w w

2 2 23 2

00 0ref ref

1r r

z z z

u M m ru u d r u r d r

u R R

= = (8)For a given operating point of the turbine, both and m can be determined, and these two values characterizeglobally the swirling flow, in addition to the local kinematical condition from eq. (4).

2.3 Constrained variational problem for axial velocity profileBenjamin [14] introduced the concept of the flow force, which we define for swirling flow in a pipe as,

( )w

2w

0

2 R

z F U P P R dR + (9)where P is the static pressure, with the corresponding value at the wall P w. Benjamin showed that the flow forcedefined in eq. (9) is minimized for admissible swirling flow configurations. The radial distribution of the static

pressure is related to the circumferential velocity through the radial equilibrium equation,21 U dP

dR R

= (10)which, using eq. (4), can be rewritten in dimensionless form as

( ) ( )

2

22

0ref

d 11

2 z u P dp d r

u R

= (11)

By integrating this differential equation we obtain the dimensionless pressure departure with respect to the wall pressure,

( ) ( )

w2

2w2

0ref

11

2

r z

r

P P ud r

u R

= (12)Using eq. (12), the flow force can be set in dimensionless form involving only the axial velocity profile.

Moreover, one can see from eqs. (6), (8), and (12) that we can introduce a new radial variable corresponding tothe dimensionless radius squared, y r 2. In addition, the flow force can be made dimensionless as f F /[( Rref ) 2 2ref R ].

We arrived at the variational formulation for the swirling flow downstream the Francis turbine runner, asfollows: given the and m, as well as the swirl-free velocity profile u0, find the axial velocity profile u z whichminimizes the flow force

w w w

s s

2

2

0

11

2

y y y z

z y y y

u f u dy dx dy

u

= + (13a)

subject to the variational constraints,w

s

y

z

y

u dy = ,w

s 0

1 y

z z

y

um u y dy

u

=

(13b) Note that in eqs. (13) the lower limit of the integrals has been set to ys instead of 0. This accounts for the possibility that the swirling flow develops a central stagnant region, and ys 0 is an additional unknown of the problem.

2.4 Numerical example and validationThe nonlinear constrained variational problem (13) is solved numerically using the NCONF subroutine from

the IMSL library, together with the QDAG and TWODQ subroutines for simple and double integrals evaluation.The unknown function u z ( y), defined on the interval ys y yw, is approximated using a B-spline interpolationwith N equally spaced nodal values.

Figure 2 shows an example of swirling flow configuration for a Francis turbine operated at 70% thedischarge at best efficiency point, and nominal head. The swirl-free velocity profile is the one shown in Fig. 1,u0 = 0.312+0.096 y, and the integral quantities for the swirl have the values, computed from experimental data, = 0.264 and m = 0.0483. The dimensionless wall radius is r w = 1.063 since the survey section where velocity

profiles are measured is a bit downstream the turbine throat / runner outlet. The resulting central stalled region

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has the radial extent of r s = 0.363.From Fig. 2 we can conclude that the above simplified mathematical model, which does not account for

viscous and three-dimensional effects in the inter-blade channels of the runner, can capture the main features ofthe swirling flow at runner outlet. First, in correctly predicts the extent of the stalled region. Second, it agreeswell with the axial and circumferential velocity profiles. The increase in both axial and tangential velocity shownin measurements is associated with the inter-blade vortex developed at the junction of the blade with the band.Of course, the present model cannot account for these secondary flows. On the other hand, we can provide agood approximation of the swirl at runner outlet, without actually computing the runner flow, thus evaluating theflow behavior in the discharge cone for a wide range of operating range. This is quite useful in the early designand optimizations stage, since it provides a quantitative tool to assess various design choices with respect to the

blade trailing edge.

3. Precessing vortex ropeThe axi-symmetric approach can correctly predict the circumferentially averaged velocity profiles, but it

cannot describe three-dimensional vortex structure arising behind the turbine runner operating at partial load. A proper model can be developed using the theory of helical vortices [8, 9]. The possibility for realization of suchidea is evident from paper by Kuibin et al. [10] where both frequency and amplitude of pressure pulsationsgenerated by the precessing vortex rope were found through given integral characteristics: vortex intensity, liquidflow rate, momentum and moment of momentum fluxes. Here we present a method for evaluating the vortexrope parameters close to the mentioned one. Instead of using the integral fluxes we will estimate the geometricaland dynamical vortex characteristics from direct comparison of the measured circumferentially averagedvelocity profiles with dependencies inherent to the model of helical vortex.

3.1 Helical symmetryThe first step in finding the vortex rope configuration for a given circumferentially averaged swirling flow is

to examine the helical symmetry property of the flow. It is shown in [8-1.5.2] that the helical symmetrycondition implies

axis constant z z r u u ul + = = (14)where u z axis is the axial velocity value at the axis, and the characteristic length l is related to the axial pitch h bythe relationship h = 2l . In our case, both the experimental data and the simplified model display a stagnantregion near the axis, thus the right-hand side in eq. (14) vanishes, u z axis = 0.

Figure 3 shows the plot of ru versus u z in order to determine the geometrical parameter l in eq. (14). In particular we are interested in the region close to the axis with largest radial gradients of both axial andcircumferential velocity components, where the vortex rope is developed. By fitting the experimental data withinthe vortex rope region with a line passing through origin we obtain l exp = 0.329, as shown with the black dashedline. On the other hand, for the simplified model presented in Section 2 eq. (14) is trivial in the stagnant regionwith r < r s. However, we can employ an analytical continuation of the numerical results for the simplifiedswirling flow model, as shown with the red dashed line, resulting in l num = 0.311. The corresponding

dimensionless pitch values of the helical symmetry are hexp = 2.070 and hnum = 1.952, respectively.On the other hand, the swirling flow with vortex rope modeled in this paper has been investigated experimentally by Ciocan and Iliescu [15] using a 3D-PIV method. They identified the vortex rope core from velocitymeasurements and proposed a geometrical description as a conical logarithmic spiral of the vortex rope shape, invery good agreement with experimental data. As one can see from Fig. 4 the actual vortex rope develops in thedraft tube cone, and it is wrapped on a cone with half-cone angle of rope = 17. This cone originates on therunner crown, where it has a radius of r 0 = 0.09 for the axial coordinate value z 0 = 0.615. As a result, thedistance between the vortex rope core and cone axis increases downstream as r rope ( z ) = r 0 ( z z 0) tan rope . Thedischarge cone shown in Fig. 4 starts at z = 1 and ends at z = 1 3 = 2.732, and has a half-cone angle ofcone = 8.5, [5, 15].

For the survey section located at z = 1.426 where velocity measurements shown in Fig. 2 were performed,the dimensionless radial distance of the vortex rope core from the axis is r rope = 0.338, quite close to the stagnantregion radius in the simplified model r s = 0.363. This confirms the conclusion of Susan-Resiga et al. [5] that thevortex rope is located at the boundary between the main swirling flow and the central stagnant region.

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0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.0 axial velocity

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

r a d i u s

* c

i r c u m

f e r e n

t i a

l v e

l o c

i t y

experimental datal = 0.329485 in Eq.(14)simplified modell = 0.310629 in Eq.(14)

Fig. 3 Helical symmetry in the vortex rope region. Points correspond to the data in Fig. 2.

Fig. 4 Vortex rope core measured by Ciocan andIliescu [15] and fitted as a conical

logarithmic spiral.

The axial pitch of the conical vortex rope shown in Fig. 4 increases as the flow evolves downstream into theturbine discharge cone. From the mathematical model of the conical vortex rope [15] we find that the pitchincreases with the rope radius as hrope ( z ) = r rope ( z ) ln( b) / tan rope = r 0, where b = 3.2 is the rate of radial growthfor a complete rotation. With the data above, the rope pitch increases linearly as hrope ( z ) = | 0.373 + 1.163 z |.However, one should bear in mind that the vortex rope model employed in this paper considers a helical vortexin a cylindrical pipe, and does not account for the conical shape of the vortex rope actually developed in theturbine discharge cone. As a result, for the purpose of comparison with the present simplified model it seemsreasonable to consider a mean value for the actual vortex rope pitch, computed within the discharge cone,hrope = 1.8, which is slightly smaller than the values obtained from the helical symmetry considerations above.3.2 Model of helical vortex

At the next step we evaluate the vortex rope parameters by the comparison of the measured velocity profiles withmodel ones [8, 9]

axis( ), ( )2 2 z z u G r u u G r

r l = = +

,0,1

( )1,

z z S

r r G r dS

r r S

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0 0.2 0.4 0.6 0.8 1dimensionless radius

0

0.1

0.2

0.3

0.4

d i m e n s i o n l e s s v e l o c i t y c o m p o n e n t s

axial velocity measuredswirl velocity measuredaxial velocity vortex rope modelswirl velocity vortex rope model

Fig. 5 Axial and circumferential velocity profiles at the runner outlet and computed

with the model of helical vortex. Experimental data are the same as in Fig. 2.

The corresponding velocity profiles and their comparison with the measured ones shown in Fig. 5 give goodenough fitting of the experimental data only in the inner area of the flow. It looks as the flow induced by thehelical vortex immersed into ambient axi-symmetric flow. Strictly speaking, such superposition would be validonly in the case when the pitch of vortex lines remains the same in whole area of the flow. Unfortunately, onecan see from Fig. 3 that there exist zones with different pitch. Thus, one should consider this model as someengineering approach rather than a rigorous one.

A further analysis of the velocity profiles shown in Fig. 5 clearly reveals that for radius values larger than theregion with large velocity gradients which corresponds to the vortex rope location the swirling flow quicklyapproaches the irrotational free vortex. The thin black dashed lines from Fig. 5 correspond to the constant axialvelocity u z = /2l and u = /2r , according to eq. (14), with = 0.531 and l = 0.329 for r a = 0.349. It isclear that the helical vortex model correctly captures the region of the steep gradient in both axial andcircumferential velocity components, while in the main swirl up to the wall it captures only the irrotational partof the swirling flow with constant specific energy. The rotational part of the swirling flow depends on the bladerunner design, described by the swirl-free velocity u0 in Section 2, and it is not accounted for in the helicalvortex model.

3.3 Frequency of helical vortex precession Now we can calculate the frequency of the vortex precession with formula derived by Kuibin and Okulov

[16] rewritten here in the following form

( ) ( ) ( ) ( )3 3

axis 2 2 21 22 2 3 3

1 1 3 9 7log 1 1 log

2 1 12 2 z v l a g g l l a

= + + + + + + %

% (15)

where ( )2 3 4

1 2 3 4

1 1.455 1.723 0.711 0.616 10.486 1.176 4

g + + + + =

+ + +

The parameter = l /a means the relative torsion of a helical line; = (1 + 2)1/2 , = l /r , = (1 + 2)1/2 , a% = exp[( )/ ( )/]( + )/( + ). The function g 2() describes the impact into thefrequency from the non-uniform vorticity distribution in the core,

( )2

22 2

0

1 44

w d =

when a fraction of the total vorticity is concentrated inside the core. Here w denotes the local circumferentialvelocity in the vortex core. At uniform vorticity distribution g 2() 0. For the Scully vortex [9] with localdistribution of type ( r 2 + 2) 2 we have

( ) ( )21 1

log 14 2 2

g = + + (16)

One can consider some other structures of the vortex core. Nonetheless, the Scully vortex is the most apropriatemodel for highly turbulent flow [17] which takes place in the hydroturbine draft tube.

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good agreement with the experimental data available for a Francis model turbine. It is remarkable that we do notneed to actually compute the three-dimensional flow in the turbine runner in order to assess the swirling flow atrunner outlet.

Second, we employ a precessing helical vortex model in order to assess the precession frequency and wall pressure fluctuation level associated with the vortex rope developed a Francis turbine discharge cone at partialdischarge. We show that this helical vortex model can correctly predict the swirling flow unsteadiness, withrespect to precession frequency and level of pressure fluctuation at the wall, in comparison with availableexperimental data. Further investigations will consider the evaluation of the above parameters for swirling flowconsiderations encountered downstream the runners of Francis turbines with other specific speed values.

Acknowledgments

Prof. P. Kuibin and Prof. V. Okulov were supported by RFBR (projects No. 10-08-01096, 10-08-01093) andMinistry of Education and Science of Russian Federation (Program Scientific potential development of highschool, project No. 2.1.2/1270). Prof. R. Susan-Resiga and Dr. S. Muntean were supported by CNCSIS UEFISCSU, PNII IDEI 799/2008.

Nomenclature

Symbolsa Dimensionless distance from axis to vortex

core center (radius of helix) y Dimensionless modified radial

coordinate (dimensionless radiussquared)

F , f Flow force [ N ], dimensionless Discharge coefficientG, g 1,

g 2 Functions, dimensionless Relative flow angle

h Dimensionless axial pitch of the vortexrope

Dimensionless vortex intensity

l Dimensionless characteristic length forhelical symmetry

Vortex core radius, dimensionless

M , m Flux of moment of momentum [ m5/ s3],dimensionless

, Angular speed [ rad / s], dimensionless

P , p Pressure [ Pa ], dimensionless Energy coefficientQ Volumetric discharge [ m3/ s] Density [ kg /m3]

R, r Radius [ m], dimensionless Dimensionless local radius in the coreU , u Velocity [ m/ s], dimensionless , Relative torsion of helical lines,

dimensionlessw Local circumferential velocity in the vortex

core, dimensionless

Subscript and superscript

0 Swirl-free conditions w Walls Stagnant region z Axial directionref Reference values Circumferential direction

References

[1] Ruprecht A, Helmrich T, Aschenbrenner T and Scherer 2002 Simulation of Vortex Rope in a TurbineDraft Tube Proc. of the 21st IAHR Symp. on Hydr. Mach. and Syst. vol 1 (Lausanne, Switzerland) pp259-76

[2] Sick M, Stein P, Doerfler P, White P and Braune A 2005 CFD Prediction of the Part-Load Vortex inFrancis Turbines and Pump-Turbines Int. J. of Hydropower and Dams 12(85)

[3] Ciocan G D, Iliescu M S, Vu T C, Nennemann B and Avellan F 2007 Experimental Study and NumericalSimulation of the FLINDT Draft Tube Rotating Vortex J. Fluids Eng. 129 pp 146-58

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[4] Zhang R K, Mao F, Wu J Z, Chen S Y, Wu Y L and Liu S H 2009 Characteristics and Control of the Draft-Tube Flow in Part-Load Francis Turbine J. of Fluids Engineering 131 021101-1-13

[5] Susan-Resiga R, Muntean S, Stein P and Avellan F 2009 Axisymmetric Swirling Flow Simulation of theDraft Tube Vortex in Francis Turbines at Partial Discharge Int. J. Fluid Mach. and Syst. 2(4) 295-302

[6] Nishi M, Matsunaga S, Okamoto M, Uno M and Nishitani K 1988 Measurement of three-dimensional periodic flow in a conical draft tube at surging condition Flows in Non-Rotating TurbomachineryComponents eds Rohatgi U S et al (FED vol 69) pp 81-88

[7] Wang X and Nishi M 1998 Analysis of Swirling Flow with Spiral Vortex Core in a Pipe JSME Int. J.B 41(2) 254-61

[8] Alekssenko SV, Kuibin P A, Okulov V L and Shtork S I 1999 Helical vortices in swirl flow J. Fluid Mech. 382 195-243

[9] Alekseenko SV, Kuibin P A and Okulov V L 2007 Theory of Concentrated Vortices: An Introduction (Berlin Springer)

[10] Kuibin P A, Okulov V L and Pylev I M 2006 Simulation of the Flow Structure in the Suction Pipe of aHydroturbine by Integral Characteristics Heat Transfer Research 37(8) 675-84

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Susan-Resiga R, Ciocan G D, Anton I and Avellan F 2006 Analysis of the Swirling Flow Downstream aFrancis Turbine Runner J. Fluids Eng. 128 177-89[12] Tridon S, Barre S, Ciocan G. D and Tomas L 2010 Experimental analysis of the swirling flow in a Francis

turbine draft tube: Focus on radial velocity component determination European J. of Mech. B/Fluids doi:10.1016/j.euromechflu.2010.02.004

[13] Susan-Resiga R, Muntean S, Avellan F and Anton I 2010 Mathematical Modeling of Swirling Flow atFrancis Runners Outlet for the Full Turbine Operating Range J. of Fluids Eng. (submitted)

[14] Benjamin T J 1962 Theory of the Vortex Breakdown Phenomenon J. Fluid Mech. 14 593-629[15] Ciocan G D and Iliescu M S 2007 Vortex Rope Investigation by 3D-PIV Method Proc. of the 2 nd IAHR

International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery andSystems (Timi oara, Romania) (Scientific Bulletin of the Politehnica University of Timisoara,Transactions on Mechanics, Vol. 52(66), Issue 6) pp 159-72

[16] Kuibin P A and Okulov V L 1998 Self-Induced Motion and Asymptotic Expansion of the Velocity Field in

the Vicinity of Helical Vortex Filament Phys. of Fluids 10(3) 607-14[17] Murakhtina T O and Okulov V L 2000 The influence of the core vorticity distribution on a possibility ofthe spontaneous change of swirling flow regimes Thermophysics and Aeromechanics 7(1) 63-8

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