Draft Tube Flow

Swirl at the outlet from Francis runners

ω

c1 w1

u1

c2w2

u2

c2

w2

u2β2

c2w2

u2β2

c2m

c2u

c2

w2

u2β2

c2m

c2u

Phenomenon in the draft tube flow

– Swirl flow– Flow in bend– Positive pressure gradient in the diffuser - separation

• Strong coupling between the flow field and the pressure gradients

rpF

zvv

rv

rv

rv

rvv r

rz

rrr ∂

∂−=

∂∂⋅⋅ρ+⋅ρ−

∂∂⋅⋅ρ+

∂∂⋅⋅ρ θθ

2

Swirl flow in draft tubes

Anisotropic turbulence• The turbulence is influenced by the geometry and

the velocity• The draft tube flow is sensitive to the inlet

conditions (velocity and pressure)• A vortex filament is present

Swirl flow

∫

∫

⋅⋅ρ⋅⋅

⋅⋅⋅ρ⋅== R

z

R

zr

drUrR

drUUr

MomentumAxialMomentumAngularnumberSwirl

0

2

0

2

0,0

0,3

0,6

0,9

1,2

1,5

-1,0 -0,5 0,0 0,5 1,0

Radius [ - ]

Vel

ocity

[ -

]

S=0,1

S=0,4

S=0,7

S=0,95

Mean Axial Velocity

Swirl flow

Vortex breakdown

∫

∫

⋅⋅ρ⋅⋅

⋅⋅⋅ρ⋅== R

z

R

zr

drUrR

drUUr

MomentumAxialMomentumAngularnumberSwirl

0

2

0

2

Vortex breakdown is present when a negative axial velocity occurs in the center of the flow.

Vortex breakdown occurs when S > 1

0,0

0,3

0,6

0,9

1,2

1,5

-1,0 -0,5 0,0 0,5 1,0

Radius [ - ]

Vel

ocity

[ -

]

S=0,1

S=0,4

S=0,7

S=0,95

Rankine Vortex

Swirl flow

Swirl flow

Swirl flow

Vortex filament at part load Vortex filament at full load

Flow in bends

A

A

A - A

StreamlineStreamline

Rcdbdsdndbdsdn

np 2

⋅⋅⋅⋅ρ=⋅⋅⋅∂∂−

Flow in bends

0ncc

np1

=∂∂⋅+

∂∂⋅

ρ

.konstcR =⋅⇓

Free Vortex

From Bernoulli’s equation

Newton’s 2 law

Positive pressure gradient in the diffuser

Location of recirculation zones

Results:

The hydraulic design of the draft tube gives secondary flow and therefore a reduced efficiency

The Navier Stokes equations in Cylindrical coordinates

[ ]

+−+

+−=+−++ 2

2

22

2

2

2 21111zUU

rU

rrU

rrrrpg

zUU

rUU

rU

rUU

tU rr

rrr

zrr

rr

∂∂

∂θ∂

∂θ∂

∂∂

∂∂

ρ∂∂

ρ∂∂

∂θ∂

∂∂

∂∂ θθθ

[ ]

+++

+−=+−++ 2

2

22

2

2

2111zUU

rU

rrU

rrrpg

zUU

rUUU

rU

rUU

tU r

zr

r ∂∂

∂θ∂

∂θ∂

∂∂

∂∂

ρµ

∂θ∂

ρ∂∂

∂θ∂

∂∂

∂∂ θθ

θθθθθθθθ

++

+−=+++ 2

2

2

2

2

111zUU

rrUr

rrzpg

zUUU

rU

rUU

tUz zzz

zz

zzz

r ∂∂

∂θ∂

∂∂

∂∂

ρµ

∂∂

ρ∂∂

∂θ∂

∂∂

∂∂ θ

r-direction:

z-direction:

θ-direction:

Euler equations

rpg

zUU

rUU

rU

rUU

tU

rr

zrr

rr

∂∂

ρ∂∂

∂θ∂

∂∂

∂∂ θθ 12

−=+−++

∂θ∂

ρ∂∂

∂θ∂

∂∂

∂∂

θθθθθθθ pgzUU

rUUU

rU

rUU

tU

zr

r1

−=+−++

zpg

zUUU

rU

rUU

tUz

zz

zzz

r ∂∂

ρ∂∂

∂θ∂

∂∂

∂∂ θ 1

−=+++

r-direction:

z-direction:

θ-direction:

r-direction

• Assume steady state solution 0=⇒tU r

∂∂

• Assume axis symmetry 0=⇒∂θ∂θ rU

rU

zUU

rU

rUU

rp r

zr

r ∂∂ρρ

∂∂ρ

∂∂ θ ⋅−+⋅−=

2

rpg

zUU

rUU

rU

rUU

tU

rr

zrr

rr

∂∂

ρ∂∂

∂θ∂

∂∂

∂∂ θθ 12

−=+−++

• Assume g-force to be neglectible 0=⇒ rg

Pressure distribution at the inlet

Low pressure zones

P re s

s ur e

[Pa ]

drdUU r

r ⋅⋅− ρrU 2

θρ ⋅+dzdUU r

z ⋅⋅− ρ0,

1 m

zUU

rU

rUU

rp r

zr

r ∂∂ρρ

∂∂ρ

∂∂ θ ⋅−+⋅−=

2

P re s

s ur e

[Pa ]

drdUU r

r ⋅⋅− ρrU 2

θρ ⋅+dzdUU r

z ⋅⋅− ρ0,

1 m

Radius [m]zUU

rU

rUU

rp r

zr

r ∂∂ρρ

∂∂ρ

∂∂ θ ⋅−+⋅−=

2

400

mm

Pressure distribution at theinlet

P re s

s ur e

[Pa ]

drdUU r

r ⋅⋅− ρrU 2

θρ ⋅+dzdUU r

z ⋅⋅− ρ0,2

m

P re s

s ur e

[Pa ]

drdUU r

r ⋅⋅− ρrU 2

θρ ⋅+dzdUU r

z ⋅⋅− ρ0,2

m

Radius [m]

drdUU r

r ⋅⋅− ρrU 2

θρ ⋅+dzdUU r

z ⋅⋅− ρ

P re s

s ur e

[Pa ]

0,4

m

drdUU r

r ⋅⋅− ρrU 2

θρ ⋅+dzdUU r

z ⋅⋅− ρ

P re s

s ur e

[Pa ]

0,4

m

Radius [m]

Static Pressure at the inlet

Velocity at the inlet to the draft tube

Velocity