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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
- 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

Outlet pipe

Primary cooling system of the research reactor HIFAR

Reactor aluminium tank (RAT)

Outlet pipes

Top view

of the HIFAR cooling system

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.

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

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

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

Boundary conditions

h(ξ)

ξc

wφ(ξ)

ξ

Boundary-value problem with the vector eigenvalue:

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

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)

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 (εK2QR << 1):

is the exponential integral

(Miles, 1998; Stepanyants & Yeoh, 2007)

The surface tension effect

Correction to Miles’ solution due to surface tension

(εK2QR << 1, μQR/We << 1)(Stepanyants & Yeoh, 2007):

Depth of the whirlpool dent:

Corresponding approximate solution for the azimuthal velocity:

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.

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.

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.

Experimental data versus numerical modelling

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

J. Fluid Mech., 2006, 556, 121–146.

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.

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.

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.

- 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.