J.
Fluid
Mech.
(1995),
1101.
287,
pp.
299-316
Copyright
0
1995 Cambridge
University
Press
299
The
flow
field downstream
of
a hydraulic jump
By
HANS G. HORNUNG, CHRISTIAN WILLERT
AND
STEWART TURNER?
Graduate Aeronautical Lab., California Institute
of
Technology, Pasadena, CA 91 125, USA
(Received
15
April 1994 and in revised form 13 August 1994)
A control-volume analysis of a hydraulic jump is used to obtain the mean vorticity
downstream of the jump as a function of the Froude number. To do this it is necessary
to include the conservation of angular momentum. The mean vorticity increases from
zero as the cube of Froude number minus one, and, in dimensionless form, approaches
a constant at large Froude number. Digital particle imaging velocimetry was applied
to travelling hydraulic jumps giving centre-plane velocity field images at a frequency
of
15
Hz over a Froude number range of
2-6.
The mean vorticity determined from
these images confirms the control-volume prediction to within the accuracy of the
experiment. The flow field measurements show that a strong shear layer is formed
at the toe of the wave, and extends almost horizontally downstream, separating from
the free surface at the toe. Various vorticity generation mechanisms are discussed.
1.
Introduction
At sufficiently high Froude number, the flow downstream of a hydraulic jump
is
so
obviously rotational that it can be seen with the naked eye. The mechanism of
vorticity generation is, however, still controversial. There is, of course, a source of
vorticity at the solid boundary below the bottom fluid. The sign of the vorticity
generated there depends on whether this boundary is stationary relative to the jump
or relative to the upstream lower fluid. This vorticity stays within the boundary layer,
however, and is not the concern here.
In a recent publication, Yeh
(1991)
discusses mechanisms of vorticity generation
in a steady-flow hydraulic jump.
Three contributions are identified: viscous shear
at the interface between the two fluids, the baroclinic torque brought about by the
static pressure gradient in the upper fluid, and the baroclinic torque brought about
by the dynamic pressure gradient associated with a suitable velocity field in the upper
fluid. The last of these is shown to dominate the other two and is proportional to the
density ratio. It requires the vertical component of the pressure gradient in the upper
fluid to be of opposite sign to that corresponding to the static gradient,
so
that a
significant velocity field with prescribed features has to be present in the upper fluid.
Since this velocity field will depend on whether the far-field upper fluid is at rest
relative to the wave or relative to the upstream lower fluid, the vorticity generation
would be different in these two cases.
When the density ratio is very large, such as for an air/water interface, where
it is approximately
1000,
such a dependence on the motion of the tenuous upper
fluid seems too sensitive. One might carry the argument to the extreme case of a
t
Permanent address:
Research School of Earth Sciences, Australian National University,
Canberra, Australia
300
H.
G.
Hornung,
C.
Willert and
S.
Turner
mercury vapour/liquid interface, and ask whether the vapour motion would be able
to influence the vorticity downstream of a hydraulic jump in the liquid. In that case
the density ratio is 0(107), and Yeh’s theory would imply that the dimensionless
vorticity would be 10000 times as large in the mercury case than in the water/air
case at the same Froude number, which is not physically reasonable.
The fact that the
momentum vector
of
the fluid entering the jump and the momentum vector
of
the
fluid leaving it are not collinear suggested the application of the conservation of
angular momentum to a control volume surrounding the jump. In this manner it was
hoped that, just as in the classical derivation of the jump conditions, the omission of
the conservation
of
mechanical energy would allow
unresolved
dissipative processes
in the control volume to occur, and yet permit the jump conditions and the mean
downstream vorticity to be determined. It also does not preclude unsteady processes
within the control volume. The aim
of
the first part of this work is thus to consider
a steady-flow hydraulic jump in a constant-density fluid on a horizontal, frictionless,
solid surface, when the fluid has a density very much larger than that of the overlying
fluid,
so
that the pressure at the free surface may be considered to be uniform, with
a view to determining the mean vorticity downstream of the jump by the application
of the conservation of angular momentum. This approach leaves the mechanism of
vorticity generation unspecified. With these assumptions, the velocities of the far-field
upper fluid and of the solid bottom relative to the jump are irrelevant. This part
of
the work was previously presented at a conference by Hornung (1992).
In a second part, the aim
is
to test the predictions of this theoretical analysis
by experiment. The digital particle-imaging velocimetry set-up developed by Gharib
and his co-workers was made available for this purpose. The hydraulic jump was
generated in the water channel operated by Dr
F.
Raichlen at the Keck Laboratory
at Caltech, which is ideal for the purpose.
The paper concludes with a discussion of possible mechanisms for the generation
of vorticity in the hydraulic jump.
The present investigation was motivated by this difficulty.
2.
Control-volume analysis
2.1.
The classical
jump
conditions
The classical equations connecting the conditions upstream and downstream of a
hydraulic jump are derived from a consideration of the conservation
of
mass and
momentum in a control volume reaching to regions upstream and downstream where
the flow is considered to be uniform. This derivation is repeated here as a form of
introduction of the variables of the problem. Figure
1
shows the control volume of
the classical situation, with uniform velocity profiles upstream and downstream of the
jump.
In terms of the quantities defined by Figure
1,
the conservation of mass across the
jump is ensured if
hi
Ui
=
h2
U2,
or
Similarly, the conservation of momentum requires (in the absence of friction on the
The flow field downstream
of
a
hydraulic jump
t
-
301
-
c
c
c
t
.h2
-
c
-
c
I
bottom) that
t
or, manipulating this by using (l),
Now introduce the definitions of the Froude number
F:
F=-
u:
g
hl’
and the height ratio
H:
in order to rewrite equation (2.2) in the form
F=C
2
(l+&),
(2.3)
with the limits
H
-+
1
as
F
-+
1, and
H
-+
(2F)1/2
as
F
-+
co.
2.2.
Jump conditions with downstream vorticity
We now anticipate that the velocity profile downstream of the jump will be rotational
and give it not only
a
mean velocity
U2,
but in addition
a
mean vorticity
o
(figure 2).
Thus the velocity distribution on the downstream side
of
the jump is now written as
where
y
is the distance from the horizontal solid bottom, measured vertically upward.
Note that this does not mean that the control-volume analysis is only valid if the
downstream vorticity is uniform. It merely means that the control-volume analysis
can only determine the
average
value
of
the vorticity, whatever its distribution may be.
This change does not affect the mass balance, but the momentum balance has to
be modified. It now requires that
302
H.
G. Hornung,
C.
Willert and
S.
Turner
tY
FIGURE
2.
Schematic sketch
of
hydraulic jump with control volume
for
the case with finite mean vorticity downstream
where
20
h
1
52=
I=25
h2
1'
[c
(f-y)+$
(";.)'I
d(g)
=6H,
(2.7)
where
Substituting this back into the momentum balance and writing the new result in the
dimensionless variables, we obtain the new relation
522
2F
12(H
-
1)'
The difference between the previous result, equation
(2.5)
and this is that a new term
in
Q2
appears on the right.
Q
is of course not known until
a
new condition is applied.
The appropriate new condition is the conservation of angular momentum, which
demands that the torque applied to the control volume by external forces be equal
to the rate of change of the angular momentum of the fluid contained in it. The
procedure is considerably more complex than in the conservation of linear momentum,
because the vertical forces on the control volume, which are balanced in the linear
momentum equation, produce a torque.
2.2.1.
The torque arising
from
vertical forces
If the vertical components of the inertial forces were zero, the bottom pressure
would exactly balance the weight of the fluid,
so
that no net torque would be exerted
by the vertical force components. In the left half of the control volume there is
a mean concave-up streamline curvature.
To
provide this curvature a transverse
pressure gradient is required. Since the pressure at the free surface is independent
of streamwise distance (assuming the density of the overlying fluid to be negligible)
the pressure on the bottom must exceed the static pressure corresponding to the
height of liquid above it. The opposite situation occurs in the downstream half
of
the control volume, where the mean streamline curvature is convex up,
so
that the
bottom pressure is lower than it would be without this inertial force.
The additional bottom pressure distribution brought about by the vertical acceler-
ation of the fluid is thus antisymmetric in
x,
and will exert a clockwise torque
on
the
fluid. This also has the required feature that it disappears at
F
-+
1,
since the mean
The jow jield downstream
of
a
hydraulic jump
303
streamline curvature disappears, and increases as
F
increases. Unfortunately, it is not
possible to obtain it without some further assumptions. Let this torque be
t
per unit
lateral distance and introduce the dimensionless torque
To determine
T,
consider the differential form of the continuity and vertical
momentum equations:
au
av
ax ay
-+-=o,
u-++-
av
av
=----
1
aP
g
ax
ay
P
aY
where the symbols have their usual meaning. By using the continuity equation to
replace au/ay in the momentum equation, and replacing p with the excess pressure p'
over the static pressure according to
P'
=
P
-
pg(h
-
Y),
the momentum equation becomes
(2.10)
At the free surface,
dh
dx
V(X,
h)
=
U(X,
h)
-.
Assume that
u
h(x)
=
U1
hi,
i.e. independent of y, and
Y
dh
Y
v
=
V(X,h)-
=
U1
hl
--,
h
dx
h2
i.e.
a
linear profile satisfying the bottom condition v(x,O)
=
0.
Substituting these in
equation (2.10), we obtain
lap'
U:h:y d ldh
P
aY
h2
dx
hdx
-
This may be integrated over y from
0
to
h
to give the excess bottom pressure:
2p6
=
-p
U:
h:
-
Consequently the clockwise torque per unit transverse distance exerted by the excess
bottom pressure on the fluid is
(2.11)
Figure
3
plots the excess pressure as a function of dimensionless distance for the
However, it is in fact not necessary to assume
a
particular wave shape, because
case of a hyperbolic-tangent wave shape.