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 effectObtaining the effect Parameters:Parameters: Liquid densityLiquid density Liquid viscosityLiquid viscosity Flow rateFlow rate Jet heightJet height

4. Experiment cont. Measurements:Measurements: Dependence of flow velocity on radiusDependence of flow velocity on radius Dependence of jump radius on flow rateDependence of jump radius on flow rate Dependence of jump radius on viscosityDependence of jump radius on viscosity Dependence of jump radius on jet heightDependence of jump radius on jet height Jump structure in dependence on velocityJump structure in dependence on velocity

5. Apparatus pump p l a t e container

6. Apparatus cont.

7. Viscosity variation Water was heated from 20˚C to 60˚CWater was heated from 20˚C to 60˚C The achieved viscosity change was over 50%The achieved viscosity change was over 50% Dependence of viscosity on temperature: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 usedA Pitot tube was used H v – flow velocity H – water height in tube ΔH – cappilary correction

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

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

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

13. Critical radius Critical radius – jump formation radiusCritical radius – jump formation radius Condition for obtaining critical radius:Condition for obtaining critical radius: h – flow height r k – critical radius Δ – boundary layer thickness

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

15. Critical radius cont. Flow velocity is approximately linear in height because of hte small flow height:Flow velocity is approximately linear in height because of hte small flow height: ξ – constant z – vertical coordinate The constant is obtained from continuity: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 thinnessFriction force is Newtonian due to flow thinness  flow height equation:  η – viscosity ρ – density v 0 – initial velocity

17. Critical radius cont. Free fall of the liquid causes the existence of initial velocity: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 isBoundary layer thickness is Inserting:Inserting: e.g. D. J. Acheson, ˝Boundary Layers˝, in Elementary Fluid Dynamics (Oxford U. P., New York, 1990)

19. Result comparation Theoretical scaling confirmedTheoretical scaling confirmed Comparation of constant in flow rate dependence:Comparation of constant in flow rate dependence: η1.1·10 -3 Pas ρ10 3 kg/m 3 d5 cm constant41.16 s/m 3 Experimental value: Experimental value: 41.0 ± 1.0 s/m 3

20. Result comparation cont.

24. Jump structure Main jump modes:Main jump modes: Laminar jumpLaminar jump Standing waves – wave jumpStanding waves – wave jump Oscillating/weakly turbulent jumpOscillating/weakly turbulent jump Turbulent jumpTurbulent jump

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

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

27. (slika1)

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

29. (slika3)

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

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