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
Reactor aluminium tank (RAT)
Outlet pipe
Top view
of 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:
1 d  wr  wz

 0 – the continuity equation
 d

2
dwr w
P 1 d  1 d  wr  
wr





d

 Re d    d  
wr 

1 d 1 d
ln 

w
  
Re d   d

wz
wz
P 1   2 wz 1 wz  2 wz
wr
 wz



 2 


 Re  
   2
where ξ = r/H0,
– Navier–Stokes
equations

 1

 = 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:
1. Radial and azimuthal velocity components are
independent of the vertical coordinate z;
2. Reg >> 1
 1,


d

w
d
1  

h  
ln 
  QR  R 2
d   d 
 2 ,




d 
We 1 d
h   

d
 d


dh
d
1   ddh 
2
0    R;
  R;

2
 w
 

R = r0/H0, QR = UH0/(2ν), We = Ug2H0/σ – the Weber number
Boundary conditions
1.0
1
h(ξ)
1
w t ( x)
ξc
0.5
wφ(ξ)
h an ( z)
ξ
0.
0
0.015
x x z
0
h     1,
h  0   h0 ,
dh
d
 0,
 
dh
d
 0,
 0
0.03
0.03
d 2h
 0, w     0,
d 2  
d 2h
 h0, w  0   0,
d  2  0
Boundary-value problem with the vector eigenvalue:
Possible simplifications:
i) ξc << R;
dw
d
 0.
 
dw
d
 w .
 0
  h0 , h0, w 
ii) We = ; …
Burgers–Rott vortex and generalisations
Zero-order approximation: h(ξ)  1,
w 
K
2

 QR 2 
1

exp
   

 2


(Burgers, 1948; Rott, 1958)
1
0.8
0.6
0.4
0.2
0
2
4
6
8
10
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:

d 
We 1 d
h   

d
 d


dh
d
1   ddh 
2

2
 w
 


w2
h    1  
d 

 
By substitution here the Burgers solution for the azimuthal velocity,
Miles’ solution can be obtained (ε  K2QR << 1):
 
QR 2
2



Q

1
h    1  2  E1  R   E1 QR 2 
1 e 2
2 
8   2 
QR 



E1  x   
x
e 

d

is the exponential integral
(Miles, 1998; Stepanyants & Yeoh, 2007)




2




The surface tension effect
Correction to Miles’ solution due to surface tension
(ε  K2QR << 1, μ  QR/We << 1) (Stepanyants & Yeoh, 2007):
 
QR 2
2


Q 

1
h    1  2  E1  R   E1 QR 2 
1 e 2
2 
8   2 
QR 



2


  4 f  QR 






Depth of the whirlpool dent:
2
  ln 2
 K QR  ln 2 QR 
  1  h(0)  2 



2 
4  2
 4  2 We 
Corresponding approximate solution for the azimuthal velocity:
    0     1   +2   
4
Liquid surface, h(x)
1.00
0.99
0.98
3
a)
2
1
0.97
0
1
2
3
4
Radial distance, x
5
Azimuthal velocity component, (x)
The surface tension effect
1
1.0
2
0.8
0.6
0.4
0.2
0.0
0
b)
1
2
3
4
5
Radial distance, x
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.
Analytical versus numerical solutions
Liquid surface, h(x)
1.00
0.99
0.98
0.97
0.96
0
1
2
3
4
5
Radial distance, x
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: H0 
1 
Q
46.154 r0 2 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 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.