1. Problem 4. Hydraulic Jump

2. Problem When a smooth column of water hits a horizontal plane, it flows out radially. At some radius, its height suddenly rises. Investigate the nature of the phenomenon. What happens if a liquid more viscous than water is used?

3. Experiment • Obtaining the effect • Parameters: • Liquid density • Liquid viscosity • Flow rate • Jet height

4. Experiment cont. • Measurements: • Dependence of flow velocity on radius • Dependence of jump radius on flow rate • Dependence of jump radius on viscosity • Dependence of jump radius on jet height • Jump structure in dependence on velocity

5. Apparatus container p l a t e pump

6. Apparatus cont.

7. Viscosity variation • Water was heated from 20˚C to 60˚C • The achieved viscosity change was over 50% • Dependence of viscosity on temperature: S. Gleston, Udžbenik fizičke hemije, NKB 1967

8. Viscosity variation cont. thermometer heater

9. Velocity measurement • A Pitot tube was used v – flow velocity H – water height in tube ΔH – cappilary correction H

10. Explanation • Hydraulic jump – sudden slow-down and rising of liquid because of turbulence • The turbulence appears when the viscous boundary layer reaches the flow surface • Boundary layer detachment appears and a vortex is formed • The vortex spends flow energy and slows it

11. Explanation cont. • Due to turbulenceenergy is lost in the jump •  Flow before the jump is slower than behind • Water level is higher due to continuity jump ˝Nonviscous˝ layer Boundary layer

12. Explanation cont. • Tasks for the theory: • Dependence of jump radius on parameters • Dependence of flow velocity on radius • Jump structure • Governing equations: • Continuity and energy conservation • Navier – Stokes equation

13. Critical radius • Critical radius – jump formation radius • Condition for obtaining critical radius: h – flow height rk – critical radius Δ– boundary layer thickness

14. Critical radius cont. • Continuity equation: • Energy conservation: Q – flow rate v – flow velocity r – distance from jump centre z – vertical axis J – kinetc energy pro unit time Jot – friction power

15. Critical radius cont. • Flow velocity is approximately linear in height because of hte small flow height: ξ – constant z – vertical coordinate • The constant is obtained from continuity: Q – flow rate r – radius h – flow height

16. Critical radius cont. • Friction force is Newtonian due to flow thinness •  flow height equation: •  η – viscosity ρ – density v0 – initial velocity

17. Critical radius cont. • Free fall of the liquid causes the existence of initial velocity: g – free fall acceleration d – jet height

18. Critical radius cont. • Boundary layer thickness is • Inserting: e.g. D. J. Acheson, ˝Boundary Layers˝, in Elementary Fluid Dynamics (Oxford U. P., New York, 1990)

19. Result comparation • Theoretical scaling confirmed • Comparation of constant in flow rate dependence: Experimental value: 41.0 ± 1.0s/m3

20. Result comparation cont.

21. Result comparation cont.

22. Result comparation cont.

23. Result comparation cont.

24. Jump structure • Main jump modes: • Laminar jump • Standing waves – wave jump • Oscillating/weakly turbulent jump • Turbulent jump

25. Jump structure cont. • Decription of liquid motion – Navier - Stokes equation: Gravitational term (pressure) Inertial term Convection term Viscosity term

26. Jump structurecont. • laminar jump conditon: •  small velocities • Viscous liquids • Steady rotation in jump region

27. (slika1)

28. Jump Structure cont. • Stable turbulent jump: • Large velocities • Weakly viscous liquids • Time – stable mode

29. (slika3)

30. Struktura skoka cont. • The remaining time – dependent modes are • Difficult to obtain • Unstable • Mathematical cause: the inertial term in the equation of motion • Observing is problematic

31. Conclusion • We can now answer the problem: • The jump is pfrmed because of boundary layer separation and vortex formation • Energy is lost in the jump, so the flow height is larger after the jump • The jump in viscous liquids is laminar or wavelike, without turbulence