Hydrodynamic Design and Analysis of a Swirling Flow Generator

Romeo SUSAN-RESIGA1, Sebastian MUNTEAN2 , Constantin TĂNASĂ1, Alin BOSIOC1,

(1) National Center for Engineering of Systems with Complex Fluids, “Politehnica” University of Timişoara, Bd. Mihai Viteazu, 1, 300222, Timişoara, Romania

(2) Center for Advanced Research in Engineering Sciences, Romanian Academy – Timişoara Branch, Bd. Mihai Viteazu, 24, 300223, Timişoara, Romania

Abstract The paper presents a hydrodynamic design and analysis of a swirling flow generator aimed at producing a flow similar to the one downstream a Francis turbine runner when operating at part load. This swirl generator is part of a swirling flow apparatus which is used for basic hydrodynamic studies of precessing spiral vortex and various flow control methods for mitigating the swirling flow instability and associated vortex breakdown phenomena.

We design swirling flows by considering either absolute or relative flow angles distributions from hub to tip in the annular bladed region. A special configuration is investigated, with fixed guide vanes followed by a free runner. The runner blades act like a turbine near the hub and as a pump near the tip, respectively, and rotate freely with an angular speed such that the total power vanishes. This free runner re-distributes the flux of angular momentum, thus decelerating the flow near the hub while accelerating it at the tip.

Analytical solutions are found for the axial and circumferential velocity profiles downstream fixed and rotating blades. A parametric study is presented in order to assess the influence of guide vanes angle and runner blade angle, respectively, on the swirling flow. Finally, the flow in the swirling flow apparatus is computed using an axisymmetric model, and the results are checked in two survey sections against actual data for a Francis turbine operated at partial discharge.

Nomenclature r , R Radius and wall radius, respectively

,α β Absolute and relative flow angles

,zV Vθ Axial and circumferential velocity components

0,p p Static pressure and stagnation (total) pressure, respectively ρ Liquid density Q Volumetric flow rate (discharge)

zV Average discharge velocity

ω Runner angular speed F Flux of circumferential momentum

(1) Quantities downstream guide vanes, upstream the runner (2) Quantities downstream the runner

hub , tip Quantities at the hub and tip (shroud), respectively

Introduction The swirling flow hydrodynamics, and in particular its stability properties and various approaches to control it is a very important topic in the context of hydraulic turbines. In particular, the decelerated swirling flow in the turbine draft tube cone becomes unstable when the Francis turbine is operated at part load, with the development of a precessing spiral vortex (so-called vortex rope) and associated severe pressure fluctuations, [1]. In order to investigate the swirling flow downstream the Francis runner blades, at operating points corresponding to partial discharge, one can obviously use a model turbine or choose a less expensive approach and build a suitable swirl generator, [3]. In this paper we investigate the swirling flow configuration that can be generated by an axial arrangement of guide vanes and runner blades, as shown in Fig. 1. The swirl apparatus designed and manufactured at the “Politehnica” University of Timisoara – National Center for Engineering of Systems with Complex Fluids has an upstream annular section with stationary and rotating blades for generating swirling flow, followed by a convergent section where the hub is ending with a nozzle. Downstream the throat section we have a conical diffuser similar to the hydraulic turbine draft tube cone. We consider four survey sections in the swirl apparatus: S1 – downstream the guide vanes, S2 – downstream the runner, S3 – similar to the Francis runner blade trailing edge, and S4 – just downstream the throat. Our goal is to find the suitable swirl (i.e. axial and circumferential velocity profiles) in section S2, which evolves further downstream at S3 and S4 similar to the Francis turbine investigated in [1]. More precisely, we investigate the swirling flow produced by guide vanes with a given distribution of flow

angle (1)α , further modified by a free runner (i.e. a runner with vanishing overall

torque) with given distribution of relative flow angle (2)β , where the absolute and relative flow angles are defined as in Fig. 2. We derive the differential equations for

the axial velocity profiles, and we find the suitable ( )(1) rα and ( )(2) rβ distributions

that produce the desired velocity profiles in survey sections S3 and S4 similar to the ones in Francis turbines operated at partial discharge. The results obtained in this paper are further used for the actual design of the guide vanes and runner blades.

Fig. 1: Overall flow passage of the swirling flow apparatus from UPT-NCESCF.

Fig. 2: Velocity triangle, with absolute, α , and relative, β , flow angles.

Swirling Flow with Given Angle Distribution and Constant Total Pressure Let us consider a swirling flow produced by axial guide vanes with wrapping angle constant from hub to tip. Such vanes are easy to manufacture by milling. As a first approximation we assume a relatively large number of blades, and consequently the

swirling flow will have a constant angle (1)α defined as:

(1)(1)

(1)

( ) tan constant( )

zV rV rθ

α= = . (1)

The flow in the annular section upstream the guide vanes is axial uniform, thus the total pressure is constant from hub to tip. If hydraulic losses are neglected for the preliminary design and analysis, the Bernoulli equation holds and consequently the

total pressure downstream the stay vanes and upstream the runner, (1)0p is also

constant from hub to tip,

( ) ( )2 2(1) (1)(1) (1)0 constant

2 2zV Vp p θ

ρ ρ= + + = ,

or, using (1), ( )2(1)(1)

2 (1) constant2sin

zVpρ α

+ = .

(2)

After differentiating (2) with respect to the radius, and using the radial equilibrium equation,

2Vdpdr r

θρ= , (3)

we obtain the differential equation for the axial velocity profile from hub to tip,

(1) (1)2 (1)cosz zdV V

dr rα= − . (4)

The solution to Eq.(4) is, 2 (1)cos

(1) (1)hub

hub

( )z zrV r V

R

α−⎛ ⎞

= ⎜ ⎟⎝ ⎠

, (5)

where the integration multiplicative constant (1)hubzV is found from the overall

discharge condition

tip tip2 (1) 2 (1)

hub hub

2 (1)

(1) (1) cos sinhub hub

1 sin2 (1)hub hub tip

2 (1)hub

2 ( ) 2

21

1 sin

R R

z zR R

z

Q V r r dr V R r dr

R V RR

α α

α

π π

πα

+

= =

⎡ ⎤⎛ ⎞⎢ ⎥= −⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦

∫ ∫. (6)

From condition (6) we have

( )( )

( )

2 (1)

22 (1)

tip hub(1)hub 1 sin

tip hub

2 2tip hub

/ 1 1 sin2/ 1

where

z z

z

R RV V

R R

QVR R

α

α

π

+

− +=

−

=−

. (7)

It is useful to examine the extremum cases in Eq.(7). If (1) / 2α π= , i.e. there is no

swirl component, then (1)hubz zV V= . In fact, according to (5) we have in this case

(1) ( )z zV r V= , meaning that there is an uniform axial flow. However, for very large

swirl, (1) 0α → , we have

tip hub(1)hub

hub tip(1)tip

/ 12

/ 12

z z z

z z z

R RV V V

R RV V V

+= >

+= <

. (8)

Let us examine a numerical example corresponding to tip 75R mm= and

hub 45R mm= . The maximum axial velocity, at the hub, is 1.33 zV , while the

minimum axial velocity, at the shroud, is 0.8 zV . As a matter of fact, the swirling flow downstream the stay vanes with constant flow angle will allways have an axial velocity excess near the hub and an axial velocity deficit near the shroud, within the limit values given by (8).

A runner, rotating with the angular speed ω in the above incoming swirling flow will ingest a flux of angular momentum given by

( ) ( )

( )( )

tip tip

hub hub

2 (1)

2(1) (1) (1) (1) 2(1)

2 1 2sin(1) 3hub hub tip

2 (1) (1)hub

22 d dtan

21

1 2sin tan

R R

z zR R

z

F rV V r r V r r

V R RR

θ

α

πωω πα

πω

α α

+

= =

⎡ ⎤⎛ ⎞⎢ ⎥= −⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦

∫ ∫. (9)

If a variable flow angle is considered, (1) ( )rα , then the differential equation for the axial velocity profile downstream the guide vanes (4) has an additional term in the righ-hand side,

(1) (1) 2 (1)(1)

(1)

1 costan

zz

dV d Vdr dr r

α αα

⎛ ⎞= −⎜ ⎟⎝ ⎠

. (10)

This additional term accounts for the variation of the flow angle (1) /d drα . Using the separation of variables, Eq.(10) can be written as

(1) (1) 2 (1)

(1) (1)

costan

z

z

dV d drV r

α αα

= − . (11)

After integrating (11) we obtain

2 (1)(1) (1) cos ( )ln lnsinz

rV drrαα= − ∫ . (12)

The integral in the righ-hand side of (12) requires special functions (Appendix C) or it could be in evaluated numerically. With the initial condition at the hub, the solution of Eq.(10) is

( )

hub

1 2 (1)(1) (1)

hub (1)hub

sin ( ) cos ( )( ) expsin

r

z zR

r xV r V dxx

α αα

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦∫ . (13)

Obviously, for constant (1)α Eq.(13) reduces to Eq.(5).

Swirling Flow Generated by a Free-Wheel Runner The swirling flow generated by the axial guide vanes has a constant total pressure. As a result, for constant flow angle there is an excess in the axial velocity near the hub, and a deficit near the tip, respectively, with respect to the average axial velocity. However, in actual hydraulic turbines operated at part load the swirling flow downstream the turbine runner usually displays an opposite distribution for the axial velocity profile, i.e. a velocity deficit near the hub and an excess at the tip. In order to obtain such a flow configuration, we have to decelerate the flow near the hub and accelerate it near the tip by re-distributing the total pressure. In doing so, the total pressure will have a deficit at the hub and an excess at the tip, similar to the Francis runner loading distribution at partial discharge.

The technical solution that meets the above requirement is to use a free-runner downstream the guide vanes, which ingests the swirl generated by the guide vanes, and modifies it according to the distribution chosen for the relative flow angle downstream the runner. Such a free-runner rotates with an angular speed such that the net torque vanishes. We will examine the swirl downstream a free runner, with a

given relative flow angle distribution , (2)β , defined as

(2)(2)

(2)tanzVV rθ ωβ

= − . (14)

The runner angular speed ω is also found from the condition of vanishing total power. This free-wheel runner is ingesting the above swirling generated by guide

vanes, with constant flow angle (1)α and constant total pressure (1)0p .

If the hydraulic losses are neglected, i.e. we consider and inviscid liquid, the relative total pressure is constant along a relative streamline, [5], p. 125,

( )22 2 2

02 2 2 2z VW U Vp p UV p UV fθ

θ θρρ ρ ρ ρ ψ+ − = + + − = − = , (15)

where the Stoke’s streamfunction ψ is defined by

1 ( )zV rr r

ψ∂=

∂. (16)

We assume here for simplicity that the flow is axisymmetric, thus the streamlines are helical curves on axisymmetric surfaces generated by constantψ = lines in the meridian half-plane. Using (15) between two sections upstream and downstream the runner gives

( ) ( )(2) (1)

0 0p pUV UVθ θψ ψρ ρ

⎡ ⎤ ⎡ ⎤− = −⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦. (17)

After differentiating Eq.(17) with respect to the streamfunction we obtain

( ) ( )(2) (1)(2)01 UV UVp θ θ

ρ ψ ψ ψ∂ ∂∂

− = −∂ ∂ ∂

, (18)

since (1)0p is constant. Since ψ is function only of r , we can rewrite Eq.(18) as

( ) ( )(2) (1)(2)0

(2) (1)

1 1 1

z z

UV UVprV r r rV r

θ θ

ρ

⎡ ⎤∂ ∂∂− = −⎢ ⎥

∂ ∂ ∂⎢ ⎥⎣ ⎦,

or, ( ) ( )(2) (1)(2) (2)

0(1)

1 z

z

d UV d UVdp Vdr dr V dr

θ θ

ρ− = − .

(19)

The right-hand side term is

( )(1)(2) (1)(2)

(1)

sin 22

zz

z

d rVV VV dr

θω αω− = − . (20)

For variable (1) ( )rα angle, the right-hand side term in Eq.(19) becomes

( )(1)(2) (1) (1)(2)

(1)

sin 22

zz

z

d rVV dV rV dr dr

θω α αω⎛ ⎞

− = − −⎜ ⎟⎝ ⎠

. (21)

The left-hand side term can be expanded as

( )

( )

(2)(2)0

(2) (2)(2) (2)(2) (2) (2)

(2) (2)(2)(2) (2)

(2) (2) (2) (2)(2)

(2) (2)

(2)2 (2

1

1

12tan tan

1sin

zz

zz

z z z zz

z

d UVdpdr dr

dV dVdp dVV V V rdr dr dr dr

dV VdVV V rdr dr r

dV V dV VVdr dr r

V

θ

θ θθ θ

θ θθ

ρ

ω ωρ

ω

ωβ β

β

− =

= + + − −

⎛ ⎞= + − +⎜ ⎟

⎝ ⎠⎡ ⎤⎛ ⎞

= − − +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

=(2) (2)

) 2 (2) (2)

1 2tan tan

z zdV Vdr r

ωβ β

⎡ ⎤+ −⎢ ⎥

⎣ ⎦

(22)

Combining (20) and (22) results in the differential equation for the axial velocity

profile downstream the runner with constant relative flow angle (2)β ,

(2) (2)2 (2) (2) (1) 2 (2)1cos sin 2 sin 2 sin

2z zdV V

dr rβ ω β α β⎡ ⎤+ = −⎢ ⎥⎣ ⎦

. (23)

If variable relative flow angle, (2) ( )rβ is considered downstream the free runner, then Eq.(22) becomes

( ) ( )(2)(2) (2) (2)(2)

(2) (2)0

(2) (2) (2) (2)2 (2) (2)

2 (2) (2)

1

cos sin 2sin tan

zz

z z z

d UVdp dV VdVV V rdr dr dr dr r

V dV V r ddr r dr

θ θ θθ ω

ρ

ββ ω ββ β

⎛ ⎞− = + − + =⎜ ⎟

⎝ ⎠⎡ ⎤⎛ ⎞

= + − −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

(24)

As a result, Eq.(23) includes and additional term in the left hand side,

(2) (2) (2)2 (2)

(2)

(2) (1) 2 (2)

( )cos ( )tan ( )

1sin 2 ( ) sin 2 sin ( )2

z zdV r d r Vrdr r dr r

r r

βββ

ω β α β

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

⎡ ⎤= −⎢ ⎥⎣ ⎦

. (25)

Finally, combining (24) and (21) we obtain the differential equation for both (1)α and (2)β variable from hub to tip:

(2) (2) (2)2 (2)

(2)

(1) (1)(2) 2 (2)

( )cos ( )tan ( )

sin 2 ( ) ( )sin 2 ( ) sin ( )2

z zdV r d r Vrdr r dr r

r d rr r rdr

βββ

α αω β β

⎛ ⎞+ − =⎜ ⎟⎝ ⎠

⎡ ⎤⎛ ⎞= − −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

. (26)

Note that for 0ω = Eq.(23) reduces to Eq.(4). The analytical solution of Eq.(23) with

initial condition at the hub (2) (2)hub hub( )z zV R V= is (see Appendix A)

2 ( 2 )

2 ( 2 )

cos(2) (2)

hubhub

(2) (1) 2 (2)1 cos

hub2 (2)

( )

1sin 2 sin 2 sin2 1

1 cos

z zrV r V

R

Rrr

β

ββ α βω

β

−

+

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎡ ⎤− ⎡ ⎤⎢ ⎥ ⎛ ⎞⎣ ⎦+ −⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦

. (27)

The two parameters in (27), (2)hubzV and ω , are found by simultaneously satisfying the

conservation of mass and of the angular momentum flux. The first condition corresponds to the given discharge value

tip

hub

(2) ( )2

R

zR

QV r r drπ

=∫ . (28)

The integrals in (28) can be evaluated analytically as shown in Appendix B. The second condition is,

( )tip

hub

(2) (2) (2) (1)2 dR

zR

F rV V r r Fθω π≡ =∫ , (29)

with (1)F given by (9). As a result, Eq.(29) can be rewritten as

( )( )

2 (1)tip

hub

2 1 2sin(1) 3hub hub tip(2) (2) 22 (1) (1)

hub

d 11 2sin tan

Rz

zR

V R RV V r r

R

α

θ α α

+⎡ ⎤⎛ ⎞⎢ ⎥= −⎜ ⎟+ ⎢ ⎥⎝ ⎠⎣ ⎦∫ . (30)

The system of equations (28) and (30) is solved numerically using the NEQNF subroutine (solve a system of nonlinear equations using a modified Powell algorithm and a finite-difference approximation of the Jacobian) from the IMSL library to obtain

the (2)hubzV and ω values.

Numerical Results The theory for swirling flow presented in the above two sections is used to determine

a suitable ( )(1) rα and ( )(2) rβ distributions for the swirl generator developed at the

UPT-NCESCF. The annular section with guide vanes and runner blades, Fig. 1, has the hub and tip radii tip 75R mm= and hub 45R mm= . The overall discharge is

30.03 /Q m s= . We have shown that a constant (1)α angle produces an axial velocity profile downstream the guide vanes with velocity excess at the hub and

velocity deficit at the tip. As a result, we have chosen a linear distribution ( )(1) rα ,

starting with (1)hub 45α = , then increasing the flow angle as we approach the tip. It is

well known that a free vortex has a corresponding constant axial velocity for a constant pressure flow. In order to obtain approximately a free vortex in section S1

we must have (1) (1)hub tip hub tiptan / tan /R Rα α = . The flow angle at the tip should be 59

according to this condition, and we will choose for our design (1)tip 60α = . The

corresponding axial and circumferential velocity profiles on section S1 are shown in the left-hand picture of Fig. 8. One can see that the free vortex approximation (dashed lines) is in very good agreement with the swirling flow obtained with a linear

variation of the flow angle from (1)hub 45α = to (1)

tip 60α = .

In order to design the flow downstream the free runner blades, we perform a

parametric study for ( )(2) (2)hub tip, 20 ,90 20 ,90β β ⎡ ⎤ ⎡ ⎤∈ ×⎣ ⎦ ⎣ ⎦ . The main goal is to

achieve a maximum axial velocity deficit at the hub, with the related axial velocity excess at the tip. As a result, in Fig. 3 we plot the axial velocity variation at the hub,

(2) (1)hub hub hubz z zV V VΔ = − and (2) (1)

tip tip tipz z zV V VΔ = − , respectively. A similar parametric

investigation is shown in Fig. 4 for the circumferential velocity variation, (2) (1)

hub hub hubV V Vθ θ θΔ = − and (2) (1)tip tip tipV V Vθ θ θΔ = − , respectively. In addition, we

investigate within the same parameter space the relative flow angle variation in the

runner, Fig. 5, (2) (1)hub hub hubβ β βΔ = − and (2) (1)

tip tip tipβ β βΔ = − , respectively, in order to

check that an acceptable relative flow deflection is required through the runner blades. Finally, the free runner angular speed is shown in Fig. 6. After examining the above results, we choose the relative flow angle at the hub, downstream the runner

blades, to be (2)hub 25β = . The (2)

tipβ value can be chosen by maximizing the weighted

axial velocity variation ( ) ( ) ( )2 2 2

hub tip tip hub/z zV V R RΔ + Δ , as shown in Fig. 7. We

obtain (2)tip 55β = , and the corresponding swirl configuration is shown in the right-

hand side of Fig. 8.

Fig. 3: Axial velocity variation in the runner, for upstream flow angle from (1)hub 45α = to

(1)tip 60α = , and downstream relative flow angle from (2)

hubβ to (2)tipβ .

Fig. 4: Swirl velocity variation in the runner, for upstream flow angle from (1)hub 45α = to

(1)tip 60α = , and downstream relative flow angle from (2)

hubβ to (2)tipβ .

Fig. 5: Relative flow angle variation in the runner, for upstream flow angle from (1)hub 45α = to (1)

tip 60α = , and downstream relative flow angle from (2)hubβ to (2)

tipβ .

Fig. 6: Free runner angular speed,

upstream flow angle from (1)hub 45α = to

(1)tip 60α = , and downstream relative flow

angle from (2)hubβ to (2)

tipβ .

20 30 40 50 60 70 80 90beta_tip [dgr]

1

1.5

2

2.5

3

3.5

axia

l vel

ocity

cha

nge

[m/s

]

Fig. 7: Optimization of the (2)tipβ angle for

(2)hub 25β = , and guide vanes with flow

angle from (1)hub 45α = , (1)

tip 60α = .

0.045 0.05 0.055 0.06 0.065 0.07 0.075radius [m]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

velo

city

com

pone

nt [m

/s]

axial velocityswirl velocity

0.045 0.05 0.055 0.06 0.065 0.07 0.075radius [m]

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

velo

city

com

pone

nt [m

/s]

axial velocityswirl velocity

Fig. 8: Axial and circumferential velocity profiles downstream the guide vanes (survey

section S1, left) with (1)hub 45α = , (1)

tip 60α = , and downstream the free runner (survey

section S2, right) with (2)hub 25β = , (2)

tip 55β = .

The axial and circumferential velocity profiles obtained downstream the free runner, right picture in Fig. 8, are used as inlet conditions for an axisymmetric turbulent swirling flow simulation in the domain shown in Fig. 1. The numerical results obtained for our swirling flow apparatus are compared against the numerical [4] and experimental [1] corresponding to a Francis turbine model investigated in the FLINDT project, and operated at partial discharge.

One can see from Fig. 9 that the dimensionless meridian and circumferential velocity profiles in the survey section S3 (immediately downstream the runner blades for the

Francis turbine model) are in good qualitative agreement. Our swirl velocity is larger, but one should keep in mind that this preliminary hydrodynamic design does not account for the viscous and 3D effects in the actual blades.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1dimensionless radius [−]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

dim

ensi

onle

ss m

erid

ian

velo

city

[−]

FLINDT numericswirl apparatus

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1dimensionless radius [−]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

dim

ensi

onle

ss s

wirl

vel

ocity

[−]

FLINDT (numeric)swirl apparatus

Fig. 9: Meridian and circumferential velocity profiles downstream on section S3, with (1)hub 45α = , (1)

tip 60α = , and downstream the free runner with (2)hub 25β = , (2)

tip 55β = .

Further downstream, in the survey section S4, Fig. 10, the comparison of the swirling flow computed in our apparatus against experimental data flow FLINDT project, [1][4], show a good agreement as well.

1.1 0.9 0.7 0.5 0.3 0.1 0.1 0.3 0.5 0.7 0.9 1.1dimensionless radius [−]

−0.10

0.10.20.30.40.50.60.70.80.9

11.11.21.31.41.5

axia

l vel

ocity

com

pone

nt [−

]

FLINDT LDV measurementsswirl apparatus

1.1 0.9 0.7 0.5 0.3 0.1 0.1 0.3 0.5 0.7 0.9 1.1dimensionless radius [−]

−1.5−1.3−1.1−0.9−0.7−0.5−0.3−0.1

0.10.30.50.70.91.11.31.5

dim

ensi

onle

ss s

wirl

vel

ocity

[−]

FLINDT LDV measurementsswirl apparatus

Fig. 10: Axial and circumferential velocity profiles downstream on section S4, with (1)hub 45α = , (1)

tip 60α = , and downstream the free runner with (2)hub 25β = , (2)

tip 55β = .

The comparison shown in Fig. 9 and Fig. 10 shows that the swirling flow from Fig. 8 produces upstream and downstream the throat section a flow configuration similar to the one encountered downstream the Francis runner when operating at part load.

Conclusion The paper investigates the basic hydrodynamic design of a swirling flow generator. The main goal is to generate a swirl which flows through a convergent-divergent test section and behaves similar to the flow downstream a Francis turbine runner operated at partial discharge. More precisely, we would like to generate a spiral vortex breakdown in the conical part of the test section, similar to the well known precessing vortex rope in hydraulic turbines.

It is found that we need a combination of guide vanes and free runner in order to generate the suitable axial and circumferential velocity profiles. Our parametric study identified that the linear absolute flow angle variation downstream the guide vanes,

from (1)hub 45α = to (1)

tip 60α = , and linear relative flow angle variation downstream the

free runner, from (2)hub 25β = to (2)

tip 55β = , are a suitable combination to generate

the spiral vortex breakdown downstream the throat. This preliminary conclusion is supported here by comparing the flow in the swirl apparatus with numerical and experimental data for a Francis turbine operated at part load, with well defined precessing vortex rope in the draft tube cone.

Appendix A. Linear First-Order Ordinary Differential Equations Equation (23) is a linear first-order ordinary differential equation, [2] p. 381, of the form

ddy a y bx x+ = , (31)

where a and b are constant coefficients. In order to find its solution we multiply

Eq.(31) by the integrand factor ax , [2], and we obtain

( )dd

a ax y b xx

= . (32)

Integrating Eq.(32) gives,

1

1

aa xx y b c

a

+

= ++

, (33)

where c is an integration constant. The solution to Eq.(31) finally is

1a bxy c x

a−= +

+. (34)

The constant c can be found from the initial condition 0 0( )y x y= , which introduced in (33) gives

10

0 0 1

aa xx y b c

a

+

= ++

. (35)

After substracting (35) from (33) we obtain

1 10

0 0 1

a aa a x xx y x y b

a

+ +−= +

+,

giving the solution of the form

10

00

( ) 11

a axx bxy x yx a x

− +⎡ ⎤⎛ ⎞ ⎛ ⎞= + −⎢ ⎥⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎣ ⎦. (36)

Appendix B. Flow Rate Integrals The following integrals are used for computing the volumetric flow rate downstream the guide vanes and runner, respectively. Essentially, we have to integrate each term in (36), multiplied by the independent variable.

1

0

220 0 1

00 0

d 12

a ax

x

y xx xy x xx a x

− −⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟− ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∫ , (37)

1

0

3 21 330 0 0 01

1 1

1 d 1 11 1 3 2

aax

x

x x x xbx b xx xa x a x a x

−+ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎛ ⎞ ⎛ ⎞⎪ ⎪⎛ ⎞− = − − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟+ + −⎝ ⎠ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭∫ . (38)

Appendix C. Integral in Eq.(12) The integral in the right-hand side of Eq.(12) can be evaluated analytically once we

assume a simple variation for the flow angle (1) ( )rα , say linear.

( )

( ) ( ) ( ) ( ) ( )

22 (1) coscos ( )

1 1 1Si 2 sin 2 Ci 2 cos 2 ln2 2 2

a brr dr drr r

br a br a br

α += =

= − + +

∫ ∫, (39)

where ( ) ( )(1) (1)hub tip tip hub tip hub/a R R R Rα α= − − and ( ) ( )(1) (1)

tip hub tip hub/b R Rα α= − − .

The sine and cosine integrals, respectively, are defined as:

( )0

sinSix tx dt

t= ∫ , ( )

0

cos 1Ci lnx tx x dt

tγ −

= + + ∫ , (40)

where 0.5772156649...γ = is the Euler’s constant, and are available in the IMSL Math Library Special Functions. As required in Eq.(13) we need the definite integral

( ) ( ) ( )

( ) ( ) ( )

2

1

2

2 11

22 1

cos ( )exp

sin 2Si 2 Si 2

2expcos 2

Ci 2 i 22

R

R

a br drr

abR bRR

R abR C bR

⎡ ⎤+− =⎢ ⎥⎢ ⎥⎣ ⎦

⎧ ⎫− −⎡ ⎤⎪ ⎪⎣ ⎦⎪ ⎪

⎨ ⎬⎪ ⎪− −⎡ ⎤⎣ ⎦⎪ ⎪⎩ ⎭

∫

. (41)

Note that in the limit 0b→ , the expression in (41) becomes

( ) 21 cos 2cos

21 2 2 2

2 1 1 1

cos(2 )exp ln2

R R R RR R R R

αα

α+

− −⎡ ⎤ ⎛ ⎞ ⎛ ⎞− = =⎜ ⎟ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠

, (42)

thus recovering the constant angle case.

Acknowledgements The present research has been supported by the Romanian National Authority for Scientific Research through the CEEX-C2-M1-1185 (C64/2006) “iSMART-flow” project, and by the Swiss National Science Foundation through the SCOPES Joint Research Project IB7320-110942/1.

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