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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

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Australian nuclear science technology organisation

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


Australian nuclear science technology organisation

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


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Outlet pipe

Primary cooling system of the research reactor HIFAR

Reactor aluminium tank (RAT)


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Outlet pipes

Top view

of the HIFAR cooling system


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Laboratory experiment

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

Vortex visualisation and characterization


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Bathtub vortices in a rotating container

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

J. Fluid Mech., 2006, 556, 121–146.


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subcritical regime

supercritical regime

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


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Theory

Basic set of hydrodynamic equations for stationary motions:

– the continuity equation

– Navier–Stokes

equations

whereξ= r/H0,  = z/H0 , {wr, wφ, wz} = {ur, uφ, uz}/Ug,

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


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LABSRL model

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

Lautrup, 2005; Stepanyants & Yeoh, 2007)

  • Main assumptions:

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

  • Reg >> 1

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


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Boundary conditions

h(ξ)

ξc

wφ(ξ)

ξ

Boundary-value problem with the vector eigenvalue:

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


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Burgers–Rott vortex and generalisations

Zero-order approximation: h(ξ) 1,

(Burgers, 1948; Rott, 1958)

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


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Miles’ approximate solution

When surface tension is neglected (We = ),

the equation for the liquid surface can be integrated:

By substitution here the Burgers solution for the azimuthal velocity,

Miles’ solution can be obtained (εK2QR << 1):

is the exponential integral

(Miles, 1998; Stepanyants & Yeoh, 2007)


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The surface tension effect

Correction to Miles’ solution due to surface tension

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

Depth of the whirlpool dent:

Corresponding approximate solution for the azimuthal velocity:


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The surface tension effect

a)

b)

  • 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∙10-2andμ = 1.647∙10-1.

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


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Analytical versus numerical solutions

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.


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Numerical solutions for subcritical vortices

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.


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Experimental data versus numerical modelling

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

J. Fluid Mech., 2006, 556, 121–146.


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Numerical solution for the critical vortex

without surface tension

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.


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Critical regime of discharge

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

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 corrections

to the corresponding dependencies.


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Numerical solution for the supercritical vortex

Vortex profile (a) and azimuthal velocity component (b) as calculated

within the LABSRL model with QR = 5∙104 and K = 9.91∙10-4.


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  • 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.


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