2. Literature Review
2.1 Introduction
Research into the components of the hydraulic characteristics found at low-head dam
structures has taken place as far back as about a half century ago. Early investigations of
the subject were chiefly concerned with the characterization of the submerged hydraulic
jump which forms at most of these low-head dam sites. The early investigations of this
subject do not seem to acknowledge, knowingly or not, the life-threatening nature of
these structures. They simply try to characterize the phenomena occurring in these
locations.
More recently, in the last decade or two, attention has been drawn to the dangers that
exist at low-head dams throughout the country. State governments have published
several brochures and papers warning recreational users of the dangers which can
hazardous to the recreational user. Newspaper articles show up all too often detailing the
tragic drownings that take place at low-head dams every summer. Year after year these
tragedies are detailed in local newspapers and yet to date little has been done to rectify
the problem. Some states, such as Minnesota, have started keeping track of low-head
dam accidents. During the 29-year period ending in 2002, The Boat and Water Safety
Section of the Minnesota Department of Natural Resources reported 53 deaths and 50
injuries at low-head dams throughout the state (Tschantz 2003). While this is an
alarming figure, many states do not keep such specific statistics, so the aggregate effect
of these dangerous hydraulic structures can not be adequately quantified. Though the
effects can not be positively identified it is clear that safety concerns at low-head dams
must be addressed.
This section is focused on detailing previous research on hydraulic jumps and the
recirrculating currents produced at low-head dams. Attempts will be made to outline
previous research and present pertinent characteristic equations used in the understanding
and control of these hydraulic characteristics. As well, seven retrofits or alteration plans
have been found and will be explained in terms of the information available from the
source. Information on numerical and physical modeling will also be presented. In
addition, several accounts of specific accidents at low-head dams have been collected and
will be discussed in order to highlight the problem occurring at these sites.
2.2 Hydraulic Jump
Hydraulic jumps occur most commonly in man-made channels as a way to dissipate
energy, usually gained from dropping over an overflow structure. A hydraulic jump
occurs when flow changes from supercritical to subcritical. According to Hwang and
Houghtalen (1996), critical flow is the flow at which a flow rate, [Q], can be passed with
minimum energy. This occurs at the critical depth. Therefore, it follows that if the water
level in the structure drops, the velocity must increase in order to convey the same flow.
This situation is called supercritical flow. When the water depth is greater than the
critical depth the flow is called subcritical, which results in a lower velocity necessary to
handle the same Q. The flow regime can be characterized by a comparison of the unit
inertial reaction to the unit gravitational force or Froude number, F, (Forester & Skrinde
1949). It is defined by Foster & Skrinde (1949), Leutheusser & Birk (1991), and
Leutheusser & Fan (2001) as follows:
F
V1
gd1
By definition when F=1 the flow is critical, when F>1 supercritical flow has developed,
and when F<1 the flow is subcritical. The water levels before and after the hydraulic
jump, the change from super critical to subcritical, is defined by the Belenger equation
(Foster & Skrinde (1949), Leutheusser & Birk (1991), and Leutheusser & Fan (2001) :
d2 1 
2
  1  8F1  1

d1 2 
The flow rate per unit width of overflow [q] can be determined using the head on the
overflow (Leutheusser & Birk (1991), and Leutheusser & Fan (2001) :
q
where
3
2
Cw 2g H 2
3
C w  .611  .075
H
P
According to Foster & Skrinde (1949) and Leutheusser & Birk (1991) a hydraulic jump
will form when the downstream depth [d2] satisfies equation 2 meaning that the
downstream velocity and subcritical Froude number are the important characteristics in
determining the structure height and optimal hydraulic jump location (Leutheusser &
Birk 1991). From this equation it can seen that there is an ideal manner for the jump to
form. In reality these conditions do not occur readily in the field.
2.3 Submerged Hydraulic Jump
The ideal situation does not usually occur at low-head dams. The phenomenon which
takes place at these structures is referred to as a submerged hydraulic jump. When the
tail water [d2] raises to become higher than the ideal condition would require in eq.2 the
jump becomes submerged. A submerged hydraulic jump sweeps back on itself and
creates a vortex (Leutheusser & Fan 2001). According to Leutheusser and Fan this
vortex swirls on a horizontal axis creating a strong upstream surface velocity pushing
whatever it comes in contact with back into the dam (2001). Rajaratnam (1965) and
Leutheusser and Fan (2001) have the described the behavior in terms of submergence of
the jump using the relationship:
S
dt  d2
d2
The optimal jump occurs when S = 0, the jump is swept downstream if S < 0, and
dangerous submerged jump happens when S > 0 (Leutheusser & Fan 2001). This relation
illustrates the fact that the submerged jump occurs if the tailwater depth downstream of
an overflow structure exceeds the subcritical depth of the hydraulic jump (Leutheusser &
Fan 2001).
The horizontal surface velocity of the upstream directed wave was modeled first by
Leutheusser and Birk (1991). With their initial investigation they developed a rough
estimate for the surface velocity. In accordance with predicted results the velocity
directed upstream decreased as the tailwater increased (Leutheusser and Birk 1991).
They also state that the maximum human swimming velocity is about 2m/s, which will be
used for comparison wherever necessary. It is also important to remember that a 2m/s
swimming velocity is only achievable by Olympic class athletes and would probably not
be possible over the extended period of time necessary to escape the reciculating current.
In 2001 Leutheusser and Fan developed a more comprehensive way to predict the
maximum upstream directed velocity. The velocity can be predicted using the equation:

Vs 
16E dm d1


V1  S  1 1  8F12  1 F12 



1
3

The experimentation of Leutheusser and Fan support their statement that the maximum
upstream velocity, [Vs], is about one-third the unsubmerged jump supercritical inflow
velocity V1 (2001). Using general hydraulic methods as well as the relationships
determined by previous research it should be possible to quantify the dangerous hydraulic
features occurring at low-head dam structures.
Notation:
The following are used in the paper:
g = acceleration due to gravity
F = Froude number
F1 = Froude number at jump inflow
d1= supercritical depth
d2 = subcritical depth
H = head on weir
q = flow rate per unit width
S = submergence
dt = local tailwater depth of channel
Vs = surface velocity
Edm = change in energy defined in Leutheusser & Fan 2001
 = Experimental constant found in Table 1 of Leutheusser & Fan 2001