Page 519 - Fair, Geyer, and Okun's Water and wastewater engineering : water supply and wastewater removal

JWCL344_ch13_457-499.qxd 8/7/10 8:49 PM Page 477

13.8 Hydraulic Jumps and Discontinuous Surge Fronts 477
13.8 HYDRAULIC JUMPS AND DISCONTINUOUS SURGE FRONTS

When a conduit steep enough to discharge at supercritical velocities and depths is followed by
a relatively flat channel in which entering velocities and depths cannot be maintained, a more
or less abrupt change in velocity and depth takes the form of a hydraulic jump (Fig. 13.6).
Whereas alternate depths are characterized by equal specific energies (d h v ), sequent depths
are characterized by equal pressure plus momentum. In accordance with the momentum prin-
2
2
ciple, illustrated in Fig. 13.7, the force producing momentum changes, 1>2(rgd 1 rgd 2 ),
when equated to the momentum change per unit volume, qr(v 2 v 1 ) and q and v 2 are elimi-
nated by the continuity equation q v 1 d 1 v 2 d 2 , leads directly to the relationship:
1
2 2
(v >1gd ) = (d >d )31 + (d >d )4 = F (13.26)
1
1
2
1
1
2
2
where r is the mass density of the water, g is the gravity constant, q is the rate of flow, F is
the Froude number, and 1gd 1 is the celerity of an elementary gravity wave, or the ratio of
the sequent depths d 2 (upper) and d 1 (lower), as determined by
1 2 1/2
d 2 >d 1 [(l 8F ) 1] (13.27)
2
As shown by Rouse (1950), depths change (a) with substantially no loss of head in a
series of undulations when 2
F
1 and (b) with appreciable head loss and a breaking
wave when F
2. For cross-sections other than rectangles of unit width, all terms in Eq.
13.27 have numerical coefficients that must be determined experimentally.

2
v 1 v d 2
d 1
h
v 2
h e
h 1 h v 1
d 2 h 2
d 1
(a) F
2. Breaking-wave jump
h 0
e

hv 2
h 1 h v 1
d 2 h 2
d 1

(b) 2
F
1. Undulating jump
Figure 13.6 Naturally Occurring Hydraulic Jump. Figure 13.7 Profiles of Hydraulic Jumps.
(Source: Wikipedia, http://en.wikipedia.org/wiki/Image:
Hydraulic-Jump-on-Upper-Spokane-Falls.jpg.)