4.3 SURGES AND BORES
4.3 Surges and bores 65
Fig. 4.7 Photograph of tidal bore. Tidal bore at Truro, Nova Scotia, Canada (Courtesy of Larry Smith) - Cobequod Bay (Indian name), being called Salmon River upstream - looking downstream at the incom- ing bore.
Positive surge (as seen by an observer
standing on the bank) Gate
Control volume
Initial water level
QUASI-STEADY FLOW ANALOGY Positive surge
(as seen by an observer travelling at the surge velocity)
d^
V, + V, srg
Gf2
V^ + Vsrg
Gate opening
New water level
d2
(b) Initial water level
"srg
J^
- ^
VsrQ-V2 <
Fig. 4.8 Positive surges and wave front propagation: (a) advancing front moving upstream and (b) advancing front moving downstream.
For a rectangular horizontal channel and considering a control volume across the front of the surge travelling at a velocity F^rg (Fig. 4.8a), the continuity equation is:
{V^ + V,,^)d, = {V, + V,,^)d2 (4.9a) where Fg^g is the surge velocity, as seen by an observer immobile (standing) on the channel
bank, the subscript 1 refers to the initial flow conditions and the subscript 2 refers to the new flow conditions.
By analogy with the hydraulic jump (Fig. 4.4), the momentum equation for the control volume, neglecting friction loss, yields (see Section 4.2.2):
2^'^Fr,
^^2 = ' ,3/2 (4.6)
i
1 + SFr^ - 1 where the Froude numbers Frj and Fr2 are defined as (Fig. 4.8a):Fn
Fr^
•\lgdi
This is a system of two equations (4.9) and (4.6) with five variables (i.e. d^, di, Vi, F2, V^^^.
Usually the upstream conditions Vi, d^ are known and the new flow rate QilQx is determined by the rate of closure of the gate (e.g. complete closure Q2 = 0).
Note that the continuity equation provides an estimate of the velocity of the surge:
"•^ (^2 -d^)B
^sr, = . f TL (4.10)
Equations (4.9) and (4.6) can be solved graphically or numerically to provide the new flow depth d2 and velocity F2, and the surge velocity V^^^ as functions of the initial flow conditions (i.e. di, Vl) and the new flow rate Q2 (e.g. Henderson, 1966: pp. 75-77).
Notes
1. A stationary surge (i.e. V^^g = 0) is a hydraulic jump.
2. A surge can be classified as for a hydraulic jump as a fiinction of its 'upstream' Froude number Fri. As an example, a surge with Frj = (Fi + V^^^ljgdx = 1.4 is called an undular surge.
3. Equations (4.9) and (4.10) are valid for a positive surging moving upstream (Fig. 4.8a). For a positive surging moving downstream (Fig. 4.8b), equation (4.9) becomes:
(^l-^'srg)rfl = (^'2-^srg)'^2 (4.9b)
Note that Fg^g must be larger than Vx since the surge is moving downstream in the direction of the initial flow.
4.3 Surges and bores 67 Application
Considering the flow upstream of a gate (Fig. 4.8, top), the gate suddenly closes. The initial flow conditions were Q = 5000 m^/s, ^ = 5 m, ^ = 100 m. The new discharge isQ = 3000 m^/s. Compute the new flow depth and flow velocity.
Solution
The surge is an advancing wave front (i.e. positive surge). Using the quasi-steady flow assumption, the flow conditions upstream of the surge front are (notation defined in Fig. 4.8) di = 5m,Vi = 10 m/s, Qi = 5000 w?/s (i.e. Fr= 1.43). The flow conditions downstream of the front surge are Q2 = 3000 w?/s mdB= 100 m.
To start the calculations, it may be assumed V^^^ = 0 (i.e. stationary surge or hydraulic jump). In this particular case (i.e. V^rg = 0), the continuity equation (4.9) becomes:
Using the definition of the Froude number, it yields:
For an initialization step where Fg^g = 0, the above equation is more practical than equation (4.10).
Notation Equation 1st iteration 2nd iteration 3rd iteration Solution
V '^srg
0.0 2 3 3.26
Fr,
1.43 1.71 1.86 1.89
d2
Equation (4.10) 6.34^
15 11.7 11.1
Fr2 Equation 0.72 0.62 0.58 0.57
(4.6) Definition of Fr2 V2 10.9
5.5 3.2 2.70
V2d2B Continuity equation 2540 8229 3730 3000 Note: ^During the initialization step, it is assumed: dj = d^. That is, the positive surge would be a steady hydraulic jump. Note that with an upstream Froude number close to unity, the downstream flow depth is slightly greater than the critical depth (equation (4.5b)).
Comments
It must be noted that the initial flow conditions are supercritical (i.e. Fj /Jgd^ = 1.43). The surge is a large disturbance travelling upstream against a supercritical flow. After the passage of the surge, the flow becomes subcritical (i.e. V2 /Jgd^ = 0.3). The surge can be characterized as a weak surge (Fri = 1.89). For a surge flow, engineer should not be confused between the surge Froude numbers (^1 + ^srg) /\/g^i (and) (F2 + Fsrg) /Jgd^, and the initial and new channel Froude numbers (Viligd[ and
V2/ylgd2 respectively). Positive surge calculations are performed with the surge Froude numbers.
4.3.3 Discussion
Considering the simple case of a positive surge travelling upstream of a sluice gate (after the gate closure), the flow sketch is sketched in Fig. 4.8 (top). Several important results derive fi*om the basic equations and they are summarized as follov^s:
(a) For an observer travelling with the flow upstream of the surge front (i.e. at a velocity Fj), the celerity of the surge (relative to the upstream flow) is:
_ \\ d^ (, . do
' srg ^
'', + ''., = M , ( ^ ^ > + ^ l (4.'»
Note that ifd2 > dx then (Fi + V^^^ > Jgd^. For a small wave (i.e. ^2 ^ ^i + ^d\ the term Vx + Kgjg tends to the celerity of a small disturbance Jgd^ (see Section 3.3.3, critical condi- tions and wave celerity).
(b) As (Fi + Fsrg) > Jgdx a surge can move upstream even if the upstream (initial) flow is super- critical. Earlier in the monograph it was stated that a small disturbance celerity C cannot move upstream in a supercritical flow. However, a large disturbance can make its way against supercritical flow provided it is large enough and in so doing the flow becomes subcritical (see example in previous section). A positive surge is a large disturbance.
(c) Considering the flow upstream of the surge fi-ont: for an observer moving at the same speed as the upstream flow, the celerity of a small disturbance travelling upstream is Jgdx. For the same observer, the celerity of the surge (i.e. a large disturbance) is: (Ki + V^^^ > Jgdlx. As a result, the surge travels faster than a small disturbance. The surge overtakes and absorbs any small disturbances that may exist at the free surface of the upstream water (i.e. in front of the surge).
Relative to the downstream water the surge travels more slowly than small disturbances as (^2 + Fsrg) < Vg^2- Any small disturbance, downstream of the surge fi*ont and moving upstream toward the wave front, overtakes the surge and is absorbed into it. So the wave absorbs random disturbances on both sides of the surge and this makes the positive surge stable and self-perpetuating,
(d) A lower limit of the surge velocity Fg^g is set by the fact that (Vx + V^rg) > Jgdx. It may be used as the initialization step of the iterative process for solving the equations. An upper limit of the surge celerity exists but it is a function of ^^2 and F2, the unknown variables.
Note
A positive surge can travel over very long distance without losing much energy because it is self- perpetuating. In natural and artificial channels, observations have shown that the wave fi*ont may travel over dozens of kilometres.
In water supply channels, brusque operation of controls (e.g. gates) may induce large surge which might overtop the channel banks, damaging and eroding the channel. In practice, rapid operation of gates and controls must be avoided. The theoretical calculations of positive surge development are detailed in Chapter 17.
4.3.4 Positive and negative surges Definitions
Positive surges
A positive surge is an advancing wave front resulting from an increase of flow depth moving upstream (i.e. closure of a downstream gate) or downstream (i.e. opening of an upstream gate and dam break) (Fig. 4.8).
Negative surges
A negative surge is a retreating wave front resulting from a decrease in flow depth moving upstream (e.g. opening of a downstream gate) or downstream (e.g. closure of an upstream gate) (Fig. 4.9) (e.g. Henderson, 1966; Montes, 1988).
4.4 Flow resistance in open channels 69
Negative surge
(as seen by an observer standing on the bank)
Gate opening
J S -
////////////////////////////////////
Fig. 4.9 Negative surge.
Discussion
Positive surge flows are solved using the quasi-steady flow analogy (Section 4.3.2). For negative surges, the flow is unsteady and no quasi-steady flow analogy exists. The complete unsteady analysis is necessary (e.g. Henderson, 1966; Liggett, 1994). This will be developed in Chapters 16 and 17.
For practising engineers, it is important to recognize between positive and negative surge cases. The table below summarizes the four possible cases:
Surge wave:
Moving
Positive surge Negative surge
downstream upstream downstream upstream Front of deep water deep water shallow water shallow water Energy balance loss of energy loss of energy gain of energy gain of energy Wave front stability stable stable unstable unstable Analysis quasi-steady quasi-steady unsteady unsteady
Notes
1. A positive surge is characterized by a steep advancing front. It is easy to recognize. Negative surges are more difficult to notice as the free-surface curvature is very shallow.
2. For a negative surge, the quasi-steady analysis is not valid (Chapters 16 and 17).
4.4 FLOW RESISTANCE IN OPEN CHANNELS