Scientific Bulletin of the “Politehnica” University of Timisoara

Transactions on Mechanics Tom 52(66), Fascicola 6, 2007

2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems

Timisoara, Romania October 24 - 26, 2007

STABILITY OF INFINITELY LONG ASYMMETRICAL VORTEX ROPE

Vladimír HABÁN

V. Kaplan Department of Fluids Engineering, Brno University of Technology

Pavel RUDOLF * V. Kaplan Department of Fluids Engineering,

Brno University of Technology František POCHYLÝ

V. Kaplan Department of Fluids Engineering, Brno University of Technology

Jiří KOUTNÍK Voith-Siemens Hydro Power Generation,

Heidenheim Klaus KRÜGER

Voith-Siemens Hydro Power Generation, Heidenheim

*Corresponding author: Brno University of Technology, Technická 2, Brno, Czech Republic 61669 Tel.: +420541142336, Fax: +420541142347, E-mail: rudolf@fme.vutbr.cz

ABSTRACT

Eigen value analysis of cavitating vortex rope motion was carried out. Analysis is based on Euler equation, continuity equation and Laplace equation of stress in shell. Instability, which depends on angular velocity of the fluid rotation and radii of vortex rope and draft tube, is judged according to sign of the real part of eigen value. Resulting eigen mode shapes indicate existence of vortex rope shape with ellipti-cal cross-section as was previously documented experimentally.

KEYWORDS Stability, vortex rope, perturbation, eigen mode shape

NOMENCLATURE

0c [m.s-1] steady part of axial velocity H [Pa] eigen mode shape of pressure k magnitude of wave vector L [m] draft tube length n unit normal vector p [Pa] pressure

0p [Pa] steady part of pressure r [m] radius R [m] vortex rope radius

0R [m] steady part of vortex rope radius 1R [m] vortex rope radius (in figures and visu

alization) 2R [m] draft tube diameter

s eigen value t [s] time

iu [m.s-1] eigen mode shape of velocity iv [m.s-1] perturbation of velocity iw [m.s-1] relative velocity vector

z [m] axial coordinate α real part of eigen value δ [m] eigen mode shape of deformation Δ [m] deformation of radius φ [rad] angular coordinate Λ [m] wave length ρ [kg.m-3] density σ [Pa] pressure perturbation

pσ [N.m-1] surface tension ω imaginary part of eigen value Ω [rad.s-1] angular velocity

INTRODUCTION Flow with circumferential velocity component

enters draft tube, when turbine is operated out of best efficiency point. Region of low pressure and axial flow reversal appears along draft tube axis due to the flow rotation. Stability of such flow can be studied using eigen value analysis, see [1]. Instabil-ity in form of vortex breakdown occurs for higher swirl rates. Symmetric form of the vortex break-down [2] transforms into helical mode with further increase of circumferential velocity [3], [4].

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 102

Cavitating vortex rope originates, if the pressure within the vortex rope drops below vapor pressure, see Fig.1. Vortex rope boundary is impermeable and Laplace equation for stress in shell, which includes the effect of surface tension, forms the dynamic boundary condition. Aim of the presented study is to find stability conditions of the cavitating vortex rope in cylindrical domain. It is assumed that the vortex rope is infinitely long and the steady velocity field is composed of two components: axial velocity com-ponent 0c , which is constant in whole investigated domain, and circumferential component rc Ω=ϕ0 , which characterizes solid body-like rotation of the fluid. Small perturbations are superimposed to these steady velocity and pressure fields. Their stability is judged according to magnitude of real part of the eigen value. Eigen value analysis also enables to determine stability conditions and eigen mode shapes for different parameters of examined system.

Figure 1. Cavitating vortex rope in cylindrical domain

MATHEMATICAL MODEL

Incompressible inviscid fluid and small pulsations of the cavitating vortex rope are assumed. Motion of the fluid is investigated in rotating cylindrical frame of reference. Cavitating vortex rope rotates with angu-lar speed Ω . Euler equations under these assumptions have following form:

zp

dtdwz

∂∂

−=ρ1

(1)

rprw

rw

dtdwr

∂∂

−=Ω−Ω−−ρφ

φ 12 22

(2)

φρ

φφ

∂∂

−=Ω++p

rw

rww

dtdw

rr 112 (3)

Continuity equation:

( ) 011=

∂

∂+

∂∂

+∂∂

φφw

rrw

rrzw

rz (4)

Solution is performed using Euler approach to continuum mechanics, therefore material derivative

dtd is introduced:

φ

φ

∂∂

+∂∂

+∂∂

+∂∂

=r

wr

wz

wtdt

drz (5)

Velocity zw in direction of the cylinder axis is assumed to be composed of two parts: steady part

0c and unsteady part ( )trzvz ,,, φ .

zz vcw += 0 (6)

Considering (6) and neglecting the non-linear terms we may rewrite eqs. (1) - (4):

zvc

tv

dtdv zzz

∂∂

+∂∂

= 0 (7)

z

wct

wdt

dw rrr

∂∂

+∂∂

= 0 (8)

z

wc

tw

dtdw

∂

∂+

∂

∂= φφφ

0 (9)

After inserting to eqs. (1) - (4):

zp

zvc

tv zz

∂∂

−=∂∂

+∂∂

ρ1

0 (10)

rprw

zwc

tw rr

∂∂

−=Ω−Ω−∂∂

+∂∂

ρφ12 2

0 (11)

φρ

φφ

∂∂

−=Ω+∂∂

+∂∂ p

rw

zw

ct

wr

1120 (12)

( ) 011=

∂∂

+∂∂

+∂∂

φφw

rrw

rrzw

rx (13)

Small pressure changes and small deformations may be assumed due to small changes of velocity. Perturbations of pressure and radius have form: ( ) Δ>>Δ+= 001 ,,, R trzRR φ (14)

( )trzpp ,,,0 φσ+= (15)

Considering previous relations in eqs. (10) - (13):

rrp 20 Ω=∂∂ ρ

after integration:

220 2

rKp Ω+=ρ

(16)

zz

vct

v zz

∂∂

−=∂∂

+∂∂ σ

ρ1

0 (17)

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 103

r

wz

wct

w rr

∂∂

−=Ω−∂∂

+∂∂ σ

ρφ120 (18)

φσ

ρφφ

∂∂

−=Ω+∂∂

+∂∂

rw

zw

ct

wr

1120 (19)

( ) 011=

∂

∂−+

∂∂

+∂∂

φφw

rrw

rrzw

rx (20)

Eigen value problem Infinitely long vortex rope is assumed and solu-

tion of following equations in form presented in [1]. φα nkzst −−= ; ωα is += Re, ∈ωα

( )( )( )

( )rHH He

ruu euwruu euw

ruu euv

i

irr

irr

xxi

zz

==

==

==

==

;

;

;

;

α

φφα

φφ

α

α

σ

(21)

Following system of equations is obtained after inserting to eqs. (17) - 20) :

( )rHuukcsi r ∂∂

−=Ω−−ρφ120 (22)

( ) nHr

iuukcsi r120 ρφ −=Ω+− (23)

( ) 01=−

∂∂

+− φurniru

rriku rz (24)

( ) HkukHiukcsi zz Ω=⇒=−

λρρ0 (25)

Eigen mode shapes Huuu zr ,,, φ can be deter-mined now.

( )

rukcsirHu

Ω−

+∂∂

Ω=

221 0

ρφ

Substitution is introduced to simplify the relation:

kcs 0

2−Ω

=λ (26)

λρφ

ruirHu +∂∂

Ω=

21

(27)

Inserting to eq.(23):

( ) nHr

iuuirH kcsi rr

122

10 ρλρ

=Ω+⎥⎦

⎤⎢⎣

⎡+

∂∂

Ω−

And after rearrangement:

nHr

ivrHi r

12

112

12 Ω

=⎟⎠⎞

⎜⎝⎛ −+

∂∂

Ω ρλρλ

( ) ruHrn

rHi 111

22

2 −−=⎥⎦⎤

⎢⎣⎡ −

∂∂

Ωλ

λλρ

⎥⎦⎤

⎢⎣⎡ −

∂∂

−Ω−= H

rn

rHiur λλρ

λ 11

12 2

2

(28)

Equations (24), (25) are rearranged:

( ) ( ) ( ) 0100 =−⎥⎦

⎤⎢⎣⎡ −

∂∂

+−− kcsurniru

rrukcsik rz φ

( ) ( ) 010

2

=−⎥⎦⎤

⎢⎣⎡ −

∂∂

+− kcsurniru

rrHik

r φρ (29)

Equation (28) is inserted to (27):

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

∂∂

−Ω−

∂∂

Ω= H

rn

rHii

rHu

λλρλ

λρφ1

11

221

2

2

⎟⎠⎞

⎜⎝⎛ −

∂∂

−Ω+

∂∂

Ω= H

rn

rH

rHu

λλλ

ρρφ1

121

21

2

Hrn

rHu

121

)1(1

21

22 −Ω−

∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+Ω

=λλ

ρλλ

ρφ (30)

Finally eqs. (28) and (30) are inserted to (29). After rearrangement:

( ) 0112

222

2

2

=⎥⎦

⎤⎢⎣

⎡−−+

∂∂

+∂∂ H

rnk

rH

rrH λ (31)

Let us denote:

( )12 −= λκ k (31a)

k – wave vector magnitude, Λ

=π2k

Λ is the wave length (chosen according to pitch of the helix)

Solution of eq. (31) has form:

( ) ( ){ }rYBrJAH nnnnnn κκ += (32)

where nJ and nY are Bessel functions. (2.32) Taking into account eqs. (15), (16) we may write:

( ) ( ){ } ni

nnnnnnn erYBrJArKp ακκρ ∑

∞

=

++Ω−=0

22

2 (33)

Integration constants nn BA , and nκ are deter-mined from boundary conditions.

Boundary conditions on vortex rope boundary Force equillibrium on vortex rope surface is

maintained by surface tension pσ and described by Laplace equation.

ppdiv vp −=nσ (34) where vp is the vapor pressure inside the vortex rope and p is outside pressure.

Unit normal vector ( )3,, nnnR φ=n is given by components:

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 104

z

RDRn

DRn

DRnR ∂

∂−=

∂∂

−== 113

11 ;1;φφ (35)

2/1

12

12

121

21 1

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎠⎞

⎜⎝⎛∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛∂∂

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

=z

RRz

RRRDφφ

(36) Changes of radius 1R are small, if the vortex rope

deformation is small:

1RD = (37)

Shape of the vortex rope surface is described by divergence of the unit normal vector:

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+=z

RDR

zDR

RDdiv 111

1

111φφ

n (38)

Equation (34) can be re-written, if no volume changes occur for small changes of radius 1R :

σφ

σ =⎟⎟⎠

⎞⎜⎜⎝

⎛∂Δ∂

−∂Δ∂

2

2

20

2

2 1Rzp (39)

Impermeability boundary condition for case of small deformations must be prescribed on vortex rope boundary.

t

cz

wr ∂Δ∂

+∂Δ∂

= 0 (40)

Deformation is expressed in sense of eigen value analysis:

tie αδ=Δ (41)

Following boundary condition in position 1Rr = is obtained after inserting to eqs. (39), (40).

HRkRn

Rp −=⎟⎟

⎠

⎞⎜⎜⎝

⎛+ δ

σ0

2

0

2

0 (42)

( )01

2

2 11

12

kcsHRn

rH

−=⎟⎟⎠

⎞⎜⎜⎝

⎛−

∂∂

−Ω− δ

λλρλ

(43)

Boundary conditions on draft tube wall Zero radial velocity is on the draft tube wall:

2Rr = : 0=rw

This condition has impact on eq.(28) in this posi-tion

01

2

=∂∂

−rHH

Rn

λ (44)

Boundary conditions for nκ and λ (eq.(31a)) have following form, when eq.(42) is inserted to eq. (43) and eq.(32) is considered.

( ) ( )

( ) ( )00

0

00

=⎥⎦

⎤⎢⎣

⎡∂

∂Γ−Θ+

+⎥⎦

⎤⎢⎣

⎡∂

∂Γ−Θ

nnn

nnnn

nnn

nnnn

Br

RYRY

Ar

RJRJ

κκ

κκ

(45)

( ) ( )

( ) ( )02

2

22

=⎥⎦

⎤⎢⎣

⎡∂

∂−+

+⎥⎦

⎤⎢⎣

⎡∂

∂−

nnn

nnn

nnn

nnn

Br

RYRY

Ar

RJRJ

κκε

κκε

(46)

( )λ

λρσλ 142

20

220

222 −Ω++=Θ RnRknpn (47)

( )20

220 RknR pn +=Γ σλ (48)

2R

nn λε = (49)

Eigen values s are determined from eq.(26) for given λ .

Visualization Results in form of eigen mode shapes of velocity

are illustrated on series of pictures. The first solid red line on the left is draft tube axis in picture of meridional section, the next one is vortex rope bound-ary (i.e. constant pressure boundary condition) 1. Eigen mode shape of velocity for 0=n is symmet-

rical, characterized by radial motion of the vortex rope surface. This eigen mode shape is stable for any magnitude of damping.

2. Circumferentially asymmetrical eigen mode shape is typical of 1=n , represents vortex rope preces-sion. Instability occurs for certain angular velocity Ω in combination with magnitudes of k, 0R , 2R .

3. Circumferentially asymmetrical eigen mode shape is typical of 2=n , represents vortex rope rotation around its axis with biscuit-like shape. Instability occurs for certain angular velocity Ω in combina-tion with magnitudes of k, 1R , 2R . Superposition of these eigen ( 2,1,0=n ) mode

shapes forms precessing vortex rope with elliptical cross-section, see also [1], [5].

CONCLUSIONS

It is obvious that axial velocity 0c only influences natural frequency ω of infinitely long vortex rope, whereas stability ( 0>α means instability) depends on angular velocity Ω and wave vector magnitude (i.e. pitch of the helix).

Eigen mode shape of velocity and pressure is de-termined from eqs. (28), (30), (25), (32) with con-sidering of eq. (21).

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 105

a) Phase angle = 0

b) Phase angle = π /3

c) Phase angle = 4 /3 π

Figure. 2 Eigen mode shape of velocity for 0=n , mR 015.01 = , mR 15.02 = , mL 1= ,

cross-section at draft tube inlet, downstream view

a) Phase angle = 0

b) Phase angle = π /3

c) Phase angle = 4 /3 π

Figure 3. Eigen mode shape of velocity for 0=n , mR 015.01 = , mR 15.02 = , mL 1= ,

meridional cross-section

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 106

a) Phase angle = 0

b) Phase angle = π /3

c) Phase angle = 2 /3 π

Figure 4. Eigen mode shape of velocity for 1=n , mR 015.01 = , mR 15.02 = , mL 1= ,

cross-section at draft tube inlet, downstream view

a) Phase angle = 0

b) Phase angle = π /3

c) Phase angle = 2 /3 π

Figure 5. Eigen mode shape of velocity for 1=n , mR 015.01 = , mR 15.02 = , mL 1= ,

meridional cross-section

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 107

a) Phase angle = 0

b) Phase angle = π /3

c) Phase angle = 2 /3 π

Figure 6. Eigen mode shape of velocity for 2=n , mR 015.01 = , mR 15.02 = , mL 1=

cross-section at draft tube inlet, downstream view

a) Phase angle = 0

b) Phase angle = π /3

c) Phase angle = 2 /3 π

Figure 7. Eigen mode shape of velocity for 2=n , mR 015.01 = , mR 15.02 = , mL 1=

meridional cross-section

Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems 108

Stability conditions change, when boundary con-ditions are considered. Steady axial velocity 0c influ-ences not only natural frequency, but also vortex rope stability, if boundary conditions are introduced. Stability conditions will also change for different circumferential distribution )(rww φφ = given for example by Rankine or Lamb vortex or by interpo-lation of measured data [6]. Therefore it is of inter-est to know boundary condition on draft tube inlet:

)(00 rcc = , )(rww φφ = , respectively ( )zwpp = .

ACKNOWLEDGEMENTS Grant Agency of the Czech Republic is gratefully

acknowledged for support of this research under project No 101/06/0152 "Unsteady motion of solid body in liquid with large motion of boundaries".

REFERENCES [1] Pochylý, F., Rudolf, P., Habán, V., Koutník, J.,

Krüger, K., 2007, "Stability of Rotationally Symmetrical Flow", Proc. IAHR Int. Meeting of WG on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Timisoara

[2] Leibovich, S., 1978, "The Structure of Vortex Break-down", Ann. Rev. Fluid Mech., 10, pp. 221--246

[3] Alekseenko, S.V., Kuibin, P.A., Okulov, V.L., Shtork, S.I., 1999, "Helical vortices in swirl flow", Journal of Fluid Mechanics, 382, pp. 195-243

[4] Okulov, V., Sorensen, J.N., Voigt, L.K., 2001, "L-transition from right- to left-handed helical vortices", Proceedings of the NATO RAW, Zakopane

[5] Rudolf, P., Habán, V., Pochylý, F., Koutník, J., Krüger, K., 2007, "Collapse of Cylindrical Cavitating Region and Conditions for Existence of Elliptical Form of Cavitating Vortex Rope", Proc. IAHR Int. Meeting of WG on Cavitation and Dynamic Problems in Hydrau-lic Machinery and Systems, Timisoara

[6] Resiga, R., Ciocan, G.D., Anton, I., Avellan, F., 2006, "Analysis of the Swirling Flow Downstream a Francis Turbine Runner, Journal of Fluids Engineering, 128, pp. 177-189