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Problem 4. Hydraulic Jump 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? Experiment • Obtaining the effect • Parameters: • Liquid density • Liquid viscosity • Flow rate • Jet height 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 Apparatus container plate pump Apparatus cont. Viscosity variation • Water was heated from 20˚C to 60˚C • The achieved viscosity change was over 50% • Dependence of viscosity on temperature: 1730 6 T 2.69310 e S. Gleston, Udžbenik fizičke hemije, NKB 1967 Viscosity variation cont. thermometer heater Velocity measurement • A Pitot tube was used v 2gH H v – flow velocity H – water height in tube H ΔH – cappilary correction 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 Explanation cont. • Due to turbulence energy 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 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 Critical radius • Critical radius – jump formation radius • Condition for obtaining critical radius: hrk h – flow height rk – critical radius Δ – boundary layer thickness Critical radius cont. • Continuity equation: Q – flow rate h Q 2r vdz 0 v – flow velocity r – distance from jump centre z – vertical axis • Energy conservation: dJ dJ ot 0 dr dr J – kinetc energy pro unit time Jot – friction power Critical radius cont. • Flow velocity is approximately linear in height because of hte small flow height: v z ξ – constant z – vertical coordinate • The constant is obtained from continuity: Q 2 rh Q – flow rate r – radius h – flow height Critical radius cont. • Friction force is Newtonian due to flow thinness flow height equation: dh h 4 r dr r Q η – viscosity ρ – density v0 – initial velocity 4 2 Q hr r 3 Q 2rv0 Critical radius cont. • Free fall of the liquid causes the existence of initial velocity: v0 2gd g – free fall acceleration d – jet height Critical radius cont. • Boundary layer thickness is r v e.g. D. J. Acheson, ˝Boundary Layers˝, in Elementary Fluid Dynamics (Oxford U. P., New York, 1990) • Inserting: 3 rc 4 2 2g 1 3 2 1 1 Q 3 3 d 6 Result comparation • Theoretical scaling confirmed • Comparation of constant in flow rate dependence: η ρ 1.1·10-3 Pas 103 kg/m3 d 5 cm constant 41.16 s/m3 Experimental value: 41.0 ± 1.0 s/m3 Result comparation cont. 0,08 0,07 radius [m] 0,06 0,05 0,04 0,03 0,02 2e-5 3e-5 r(Q) experiment 4e-5 5e-5 6e-5 flow rate [m3/s] 7e-5 8e-5 Result comparation cont. 4,2 4,0 radius [cm] 3,8 3,6 3,4 3,2 3,0 2,8 0,0005 0,0006 r() experiment 0,0007 0,0008 viscosity [Pas] 0,0009 0,0010 Result comparation cont. 4,45 radius [cm] 4,40 4,35 4,30 4,25 4,20 0 2 r(d) experiment 4 6 jet height [cm] 8 10 Result comparation cont. water level in Pitot tube [m] 0,4 0,3 0,2 0,1 0,0 0,01 0,02 v(r) experiment 0,03 0,04 radius [m] 0,05 0,06 Jump structure • Main jump modes: • Laminar jump • Standing waves – wave jump • Oscillating/weakly turbulent jump • Turbulent jump Jump structure cont. • Decription of liquid motion – Navier - Stokes equation: v 2 vv v gzˆ t Inertial term Convection term Viscosity term Gravitational term (pressure) Jump structure cont. • laminar jump conditon: 2 v v vv t small velocities Viscous liquids Steady rotation in jump region (slika1) Jump Structure cont. • Stable turbulent jump: 2 vv v Large velocities Weakly viscous liquids Time – stable mode (slika3) 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 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