Australian Nuclear Science & Technology Organisation BATHTUB VORTICES IN THE LIQUID DISCHARGING FROM THE BOTTOM ORIFICE OF A CYLINDRICAL VESSEL Yury A.

Australian Nuclear Science & Technology Organisation

BATHTUB VORTICES IN THE LIQUID

DISCHARGING FROM THE BOTTOM

ORIFICE OF A CYLINDRICAL VESSEL

Yury A. Stepanyants and Guan H. Yeoh

Motivation

• Bathtub vortices is a very common phenomenon

- vortices are often observed at home conditions (kitchen sinks, bathes)

- appear in the undustry and nature (liquid drainage from big reservoirs,

water intakes from natural estuaries, vortices forming in the cooling

systems of nuclear reactors)

• Intence vortices cause some undesirable and negative effects due to

gaseos cores entrainment into the drainage pipes

- produce vibration and noise

- reduce a flow rate

- cause a negative power transients in nuclear reactors, etc.

• A theory of bathtub vortices was not well-developed so far – a challenge

for the theoretical study

Primary cooling system of the research reactor HIFAR

Outlet pipe

Reactor aluminium tank (RAT)

Top viewof the HIFAR cooling system

Outlet pipes

Laboratory experiment

(R. Bandera, G. Ohannessian, D. Wassink)

Vortex visualisation and characterization

Bathtub vortices in a rotating container

Andersen A., Bohr T., Stenum B., Rasmussen J.J., Lautrup B.

J. Fluid Mech., 2006, 556, 121–146.

Objectives

• Develope a theoretical/numerical model for bathtub vortices

• Construct stationary solutions decribing vortices in laminar

viscous flow with the free surface and surface tension effect

• Investigate different regimes of drainage including:

- subcritical regime, when small-dent whirlpoos may exist

- critical regime, when vortex heads reach the vessel bottom

- supercritical regime, when vortex cores penetrate into the drainage system

subcritical regime supercritical regime

Theory

Basic set of hydrodynamic equations for stationary motions:

10r z

d w w

d

21 1

Rerr

r

w d wdw P dw

d d d

1 1ln

Rer

d dw w

d d

2 2

2 2

1 11

Re

z z z z zr z

w w P w w ww w

– the continuity equation

– Navier–Stokesequations

where ξ = r/H0, = z/H0 , {wr, wφ, wz} = {ur, uφ, uz}/Ug,

P = p/(Ug2), Reg = H0Ug/ν, Ug = (gH0)1/2

LABSRL model(Lundgren, 1985; Andersen et al., 2003; 2006;

Lautrup, 2005; Stepanyants & Yeoh, 2007)

R = r0/H0, QR = UH0/(2ν), We = Ug2H0/σ – the Weber number

2

2

1, 0 ;1

ln, ; RQ R

Rd wd

hRd d

21

21

We dhd

dhd

wd dh

d d

Main assumptions:

1. Radial and azimuthal velocity components are independent of the vertical coordinate z;

2. Reg >> 1

Boundary conditions

Boundary-value problem with the vector eigenvalue:

2

21, 0, 0, 0, 0.

dwdh d hh w

d d d

2

0 020 00

0 , 0, , 0 0, .h h wdwdh d h

h wd d d

0 0, ,h h w

Possible simplifications: i) ξc << R; ii) We = ; …

ξ

h(ξ)

wφ(ξ)

ξc

(Burgers, 1948; Rott, 1958)

Zero-order approximation: h(ξ) 1,

Burgers–Rott vortex and generalisations

2K1 exp

2 2RQ

w

Burgers vortex (solid red line) and its approximation by the inviscid Rankine vortex (dashed blue line)

(Miles, 1998; Stepanyants & Yeoh, 2007)

When surface tension is neglected (We = ), the equation for the liquid surface can be integrated:

21

21

We dhd

dhd

wd dh

d d

Miles’ approximate solution

2

1w

h d

By substitution here the Burgers solution for the azimuthal velocity,Miles’ solution can be obtained (ε K2QR << 1):

2 2

22 2

1 12 2

11 E E 1

8 2

RQR

RR

Qh Q e

Q

is the exponential integral 1Ex

ex d

Corresponding approximate solution for the azimuthal velocity:

Correction to Miles’ solution due to surface tension

(ε K2QR << 1, μ QR/We << 1) (Stepanyants & Yeoh, 2007):

The surface tension effect

2 2

22 2

1 12 24

11 E E 1

8 2

RQR

RR

R

Qh Q e

Qf Q

2

2 2

Kln 2 ln 21 (0)

4 2 4 2 WeR RQ Q

h

0 1 2+

Depth of the whirlpool dent:

a) Vortex profile versus dimensionless radial coordinate x for ε = 1.71∙10-2. Line 1 – Miles’ solution without surface tension (μ = 0 ); line 2 – corrected solution with small surface tension (μ = 5.64∙10-2); line 3 – corrected solution with big surface tension (μ = 1.647∙10-1).

b) Azimuthal velocity component versus radial coordinate for ε = 1.71∙10-2 and μ = 1.647∙10-1.

Line 1 – the Burgers vortex), line 2 – corrected solution.

The surface tension effect

0 1 2 3 4 5Radial distance, x

0.97

0.98

0.99

1.00

Liqu

id s

urfa

ce, h

(x)

21

3

4

0 1 2 3 4 5Radial distance, x

0.0

0.2

0.4

0.6

0.8

1.0

Azi

mut

hal v

eloc

ity c

ompo

nent

, (x

)

2

1

a) b)

Vortex profile versus dimensionless radial coordinate x.Red lines – ε = 1.71∙10-2, μ = 5.64∙10-2;Blue lines – ε = 5.76∙10-2, μ = 0.24.

Solid lines – approximate theory, dotted lines – numerical calculations within the LABSRL model.

Analytical versus numerical solutions

0 1 2 3 4 5Radial distance, x

0.96

0.97

0.98

0.99

1.00Li

quid

sur

face

, h(x

)

Vortex profile (a) and azimuthal velocity component (b)as calculated within the LABSRL model.

Red lines – results obtained with surface tension;Blue lines – results obtained without surface tension.

QR = 106, K = 3.05∙10-3; We = 3.4∙104.

Numerical solutions for subcritical vortices

Experimental data versus numerical modelling

Andersen A., Bohr T., Stenum B., Rasmussen J.J., Lautrup B.

J. Fluid Mech., 2006, 556, 121–146.

Vortex profile (a) and azimuthal velocity component (b).Red lines – results of numerical calculations within the LABSRL model;Blue line – Burgers solution.

QR = 5∙104, K = 0.206.

Numerical solution for the critical vortexwithout surface tension

Critical regime of discharge

K = 46.154QR-1/2 or in the dimensional form: 0

0

1

46.154 2

QH

r g

The same functional dependency, K ~ QR-1/2, follows from different

approximate theories (Odgaard, 1986; Miles, 1998; Lautrup, 2005)

and from the empirical approach developed by Hite & Mih (1994)

Kolf number versus QR: circles – results of numerical calculations;line 1 – best fit approximation;line 2 – Odgaard’s and Miles’ results;line 3 – the dependency that follows from Lautrup (2004);line 4 and 5 – surface tension correctionsto the corresponding dependencies.

Vortex profile (a) and azimuthal velocity component (b) as calculatedwithin the LABSRL model with QR = 5∙104 and K = 9.91∙10-4.

Numerical solution for the supercritical vortex

Conclusion

• The relevant set of simplified equations adequately describing stationary vortices in the laminar flow of viscous fluid with a free surface is derived.

• Approximate analytical solution describing the free surface shape and velocity field in bathtub vortices is obtained taking into account the surface tension effect.

• The simplified set of equations is solved numerically, and three different regimes of fluid discharge are found: subcritical, critical and supercritical. This is in accordance with experimental observations.

• The relationship between flow parameters when the critical regime of discharge occurs is found.