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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
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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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Primary cooling system of the research reactor HIFAR
Outlet pipe
Reactor aluminium tank (RAT)
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Top viewof the HIFAR cooling system
Outlet pipes
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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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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
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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
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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
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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
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(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)
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(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
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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:
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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)
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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
)
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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
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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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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
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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.
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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
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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.