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.