Australian Nuclear Science & Technology Organisation

1. 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

2. 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

3. Outlet pipe Primary cooling system of the research reactor HIFAR Reactor aluminium tank (RAT)

4. Outlet pipes Top view of the HIFAR cooling system

5. Laboratory experiment (R. Bandera, G. Ohannessian, D. Wassink) Vortex visualisation and characterization

6. Bathtub vortices in a rotating container Andersen A., Bohr T., Stenum B., Rasmussen J.J., Lautrup B. J. Fluid Mech., 2006, 556, 121–146.

7. 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

8. 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

9. 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

10. Boundary conditions h(ξ) ξc wφ(ξ) ξ Boundary-value problem with the vector eigenvalue: Possible simplifications: i)ξc << R; ii)We = ; …

11. 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)

12. 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)

13. 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:

14. 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.

15. 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.

16. 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.

17. Experimental data versus numerical modelling Andersen A., Bohr T., Stenum B., Rasmussen J.J., Lautrup B. J. Fluid Mech., 2006, 556, 121–146.

18. 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.

19. 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.

20. 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.

21. 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.