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
  2.69310 e
S. Gleston, Udžbenik fizičke hemije, NKB 1967
Viscosity variation cont.
thermometer
heater
Velocity measurement
• A Pitot tube was used
v  2gH  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:
hrk   
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
hr  
r 
3 Q
2rv0
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
 vv   v  gzˆ
t

Inertial term
Convection
term
Viscosity
term
Gravitational
term (pressure)
Jump structure cont.
• laminar jump conditon:
 2
v
 v 
 vv

t
 small velocities
 Viscous liquids
 Steady rotation in jump region
(slika1)
Jump Structure cont.
• Stable turbulent jump:
 2
vv   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