When a falling jet of fluid strikes a horizontal fluid layer, a hydraulic jump arises downstream of the point of impact, provided a critical flow rate is exceeded. We here examine a phenomenon that arises below this jump threshold, a circular deflection of relatively small amplitude on the free surface that we call the hydraulic bump. The form of the circular bump can be simply understood in terms of the underlying vortex structure and its height simply deduced with Bernoulli arguments. As the incoming flux increases, a breaking of axial symmetry leads to polygonal hydraulic bumps. The relation between this polygonal instability and that arising in the hydraulic jump is discussed. The coexistence of hydraulic jumps and bumps can give rise to striking nested structures with polygonal jumps bound within polygonal bumps. The absence of a pronounced surface signature on the hydraulic bump indicates the dominant influence of the subsurface vorticity on its instability.
REFERENCES
1.
2.
Lord
Rayleigh
, “On the theory of long waves and bores
,” Proc. R. Soc. London, Ser. A
90
, 324
–328
(1914
).3.
I.
Tani
, “Water jump in the boundary layer
,” J. Phys. Soc. Jpn.
4
, 212
–215
(1949
).4.
E.
Watson
, “The radial spread of a liquid jet over a horizontal plane
,” J. Fluid Mech.
20
, 481
–499
(1964
).5.
T.
Bohr
, P.
Dimon
, and V.
Putkaradze
, “Shallow-water approach to the circular hydraulic jump
,” J. Fluid Mech.
254
, 635
–648
(1993
).6.
T.
Bohr
, C.
Ellegaard
, A. E.
Hansen
, and A.
Haaning
, “Hydraulic jumps, flow separation and wave breaking: An experimental study
,” Physica B
228
, 1
–10
(1996
).7.
T.
Bohr
, V.
Putkaradze
, and S.
Watanabe
, “Averaging theory for the structure of hydraulic jumps and separation in laminar free-surface flows
,” Phys. Rev. Lett.
79
, 1038
–1041
(1997
).8.
J. W. M.
Bush
and J. M.
Aristoff
, “The influence of surface tension on the circular hydraulic jump
,” J. Fluid Mech.
489
, 229
–238
(2003
).9.
A. R.
Kasimov
, “A stationary circular hydraulic jump, the limits of its existence and its gasdynamic analogue
,” J. Fluid Mech.
601
, 189
–198
(2008
).10.
S.
Watanabe
, V.
Putkaradze
, and T.
Bohr
, “Integral methods for shallow free-surface flows with separation
,” J. Fluid Mech.
480
, 233
–265
(2003
).11.
A.
Andersen
, T.
Bohr
, and T.
Schnipper
, “Separation vortices and pattern formation
,” Theor. Comput. Fluid Dyn.
24
, 329
–334
(2009
).12.
K.
Yokoi
and F.
Xiao
, “Mechanism of structure formation in circular hydraulic jumps: Numerical studies of strongly deformed free-surface shallow flows
,” Physica D
161
, 202
–219
(2002
).13.
X.
Liu
and J. H.
Lienhard
V, “The hydraulic jump in circular jet impingement and in other thin liquid films
,” Exp. Fluids
15
, 108
–116
(1993
).14.
J. W. M.
Bush
, J. M.
Aristoff
, and A. E.
Hosoi
, “An experimental investigation of the stability of the circular hydraulic jump
,” J. Fluid Mech.
558
, 33
–52
(2006
).15.
C.
Ellegaard
, A.
Hansen
, A.
Haaning
, K.
Hansen
, A.
Marcussen
, T.
Bohr
, J.
Hansen
, and S.
Watanabe
, “Creating corners in kitchen sinks
,” Nature (London)
392
, 767
–768
(1998
).16.
C.
Ellegaard
, A. E.
Hansen
, A.
Haaning
, K.
Hansen
, A.
Marcussen
, T.
Bohr
, J. L.
Hansen
, and S.
Watanabe
, “Cover illustration: Polygonal hydraulic jumps
,” Nonlinearity
12
, 1
–7
(1999
).17.
E.
Martens
, S.
Watanabe
, and T.
Bohr
, “Model for polygonal hydraulic jumps
,” Phys. Rev. E
85
, 036316
(2012
).18.
J.
Plateau
, Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only
(Gauthier-Villars
, Paris
, 1873
), Vol. 1
.19.
Lord
Rayleigh
, “On the instability of jets
,” Proc. London Math. Soc.
10
, 4
–13
(1878
).20.
L. M.
Hocking
and D. H.
Michael
, “The stability of a column of rotating liquid
,” Mathematika
6
, 25
–32
(1959
).21.
T. J.
Pedley
, “The stability of rotating flows with a cylindrical free surface
,” J. Fluid Mech.
30
, 127
–147
(1967
).22.
J.
Kubitschek
and P.
Weidman
, “The effect of viscosity on the stability of a uniformly rotating liquid column in zero gravity
,” J. Fluid Mech.
572
, 261
–286
(2007
).23.
Sir W.
Thomson
, “On vortex atoms
,” Philos. Mag.
34
, 15
–24
(1867
).24.
25.
S. E.
Widnall
and C. Y.
Tsai
, “The instability of the thin vortex ring of constant vorticity
,” Philos. Trans. R. Soc. London, Ser. A
287
, 273
–305
(1977
).26.
T.
Maxworthy
, “Some experimental studies of vortex rings
,” J. Fluid Mech.
81
, 465
–495
(1977
).27.
P.
Saffman
, “The number of waves on unstable vortex rings
,” J. Fluid Mech.
84
, 625
–639
(1978
).28.
O. M.
Knio
and A. F.
Ghoniem
, “Numerical study of a three-dimensional vortex method
,” J. Comput. Phys.
86
, 75
–106
(1990
).29.
S.
Perrard
, Y.
Couder
, E.
Fort
, and L.
Limat
, “Leidenfrost levitated liquid tori
,” Europhys. Lett.
100
, 54006
(2012
).30.
A. K.
Biń
, “Gas entrainment by plunging liquid jets
,” Chem. Eng. Sci.
48
, 3585
–3630
(1993
).31.
E.
Lorenceau
, D.
Quéré
, and J.
Eggers
, “Air entrainment by a viscous jet plunging into a bath
,” Phys. Rev. Lett.
93
, 254501
(2004
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
