Cavitation occurs whenever the pressure in the flow of water drops to the value of the pressure of the saturated water vapour, pv (at the prevailing temperature); cavities filled by vapour, and partly by gases excluded from the water as a result of the low pressure, are formed. When these bubbles are carried by the flow into regions of higher pressure, the vapour quickly condenses and the bubbles implode, the cavities being filled suddenly by the surrounding water. Not only is this process noisy, with disruption in the flow pattern, but more importantly if the cavity implodes against a surface, the violent impact of the water particles acting in quick succession at very high pressures (of the order of 1000 atm), if sustained over a period of time, causes substantial damage to the (concrete or steel) surface, which can lead to a complete failure of the structure. Thus cavitation corrosion (pitting) and the often accompanying vibration is a phenomenon that has to be taken into account in the design of hydraulic structures, and pre- vented whenever possible 30
Low pressures well below atmospheric pressure will occur at points of separation of water flowing alongside fixed boundaries, particularly if the flow velocity is high. Thus there are two factors, pressure p and velocity u, which influence the onset of cavitation. They are combined with density! in the cavitation number, ":! = p! p v "u 2 Cavitation occurs if the cavitation number falls below a critical value " c which is a function of the geometry and can vary widely. As an example, the incipient cavitation number for sloping offsets and triangular protrusions, as determined from data by Wang and Chou Incipient cavitation number for sloped protrusions 31
According to Ball and Johnson (Cassidy and Elder, 1984), a 3mm perpendicular offset into the flow can cause cavitation at velocities as low as 11m/s; for an equally high recess from the flow the critical velocity is about 32 m/s. A value of " c of around 0.2 is sometimes considered (Falvey, 1990) when assessing the critical velocity on smooth concrete surfaces 32
Overfall spillway The basic shape of the overfall (ogee) spillway is derived from the lower envelope of the overall nappe flowing over a high vertical rectangular notch with an approach velocity V 0 = 0 and a fully aerated space beneath the nappe (p = p0), 33
For a notch of width b, head h, and coefficient of discharge C d, the discharge equation is which, for V 0 =0, reduces to C d is about 0.62 34
Scimeni (1937) expressed the shape of the nappe in coordinates x and y, measured from an origin at the highest point, for a unit value of H as with K=0.5 and n=1.85. As the nappes for other values of H are similar in shape, equation can be rewritten as: Or: 35
for K=0.5 (curve a, in slide 27) the pressures acting on the surface defined by equation are atmospheric (p0) for K>0.5 (curve b) the pressures acting on the spillway will be negative (p<p0) for K<0.5 (curve c) the pressures acting on the spillway will be positive (p > p0). For an overflow spillway we can thus rewrite the discharge equation as 36
There are three possibilities for the choice of the relationship between the design head Hd used for the derivation of the spillway shape and the maximum actual head Hmax H d > H max H d = H max H d < H max For Hd=Hmax the pressure is atmospheric and Cd=0.745. For Hd>Hmax the pressure on the spillway is greater than atmospheric and the coefficient of discharge will be 0.578 < Cd < 0.745. The lower limit applies for broad- crested weirs with Cd=1/3 0.5, and is attained at very small values of Hmax/Hd (say, 0.05). For Hd < Hmax negative pressures result, reaching cavitation level for H=2Hd with Cd=0.825. For safety it is recommended not to exceed the value Hmax=1.65Hd with Cd=0.81, in which case the intrusion of air on the spillway surface must be avoided, as otherwise the overfall jet may start to vibrate. 37
For gated spillways, the placing of the gate sills by 0.2H downstream from the crest substantially reduces the tendency towards negative pressures for outflow under partially raised gates. The discharge through partially raised gates may be computed from: with C d1 =0.6 or, better, from: where a is the distance of the gate lip from the spillway surface, and He the effective head on the gated spillway ( H) (0.55 <Cd2 < 0.7) 38
in the equations b refers to the spillway length. In the case of piers on the crest (e.g. gated spillways) this length has to be reduced to be =b!2(n "p +"a) # where n is the number of contractions and Kp and Ka are the pier and abutment contraction coefficients 39
40
Shaft spillway For ratios H/D < 0.225 the spillway is free-flowing (i.e. with crest control) and for H/D >0.5 the overfall is completely drowned For the free overfall the discharge is: for the drowned (submerged) régime 41
Energy dissipation Energy dissipation at dams and weirs is closely associated with spillway design, particularly with the chosen specific discharge q, the difference between the upstream and downstream water levels (H*) and the downstream conditions In the design of energy dissipation, environmental factors have to be considered; some of the most important ones are the effect of dissolved gases supersaturation on fish in deep plunge pools, and of spray from flip bucket jets which can result in landslides and freezing fog 42
Consider the energy dissipation process in five separate stages, some of which may be combined or absent: 1. on the spillway surface; 2. in a free-falling jet; 3. at impact into the downstream pool; 4. the stilling basin; 5. at the outflow into the river. 43
Energy dissipation on the spillway surface The energy head loss on the spillway surface can be expressed as where V is the (supercritical) velocity at the end of the spillway, # is the Coriolis coefficient and $ is the head loss coefficient, related to the velocity coefficient % (the ratio of actual to theoretical velocity) by The ratio of the energy loss, e, to the total energy E (i.e. the relative energy loss) is 44
For S/H<30 The value of $ could be increased (and % decreased) by using a rough spillway or by placing baffles on the spillway surface. However, unless aeration is provided at these protrusions, the increased energy dissipation may be achieved only by providing an opportunity for cavitation damage 45
In many modern spillway designs, increased energy dissipation is achieved by using free-falling jets, either at the end of a ski-jump or downstream of a flip bucket The energy loss on a ski-jump spillway can be substantially enlarged by splitting the overfall jet into several streams or by using two spillways with colliding jet 46
The combined energy loss in the first three phases of energy dissipation can be expressed from a velocity coefficient % 1 3, which can be determined in model tests from the theoretical supercritical flow depth conjunctive to the subcritical depth needed to form a stable jump downstream of a (ski-jump) spillway. Even if this value of % may be subject to scale effects (the prototype % is likely to be smaller because of increased aeration), the model studies give a very good indication of the relative merits of various designs. Generally % 1 3 = f(s /S, q, geometry) where S is the height of the take-off point above the reference datum. 47
The increase of % (and decrease of relative energy loss) with q is again demonstrated. By using a suitable design, the values of % have been reduced throughout by a factor about 0.7; this results in substantial energy losses, e.g. for % 1 3 = 0.5, e/e=75% The disintegration of a falling circular jet of diameter D was studied, who showed that a complete decay of the solid inner core occurs after a length of fall L, with L/D in the range 50 100. For flat jets, which are more relevant to spillway design, established experimentally that the length of fall from the crest required for total jet disintegration (for q in m 3 s -1 m -1 ) is 48
Stilling basins The stilling basin is the most common form of energy dissipator converting the supercritical flow from the spillway into subcritical flow compatible with the downstream river régime. The straightforward and often best method of achieving this transition is through a simple submerged jump formed in a rectangular cross-section stilling basin 49
Sky-jump bucket type energy dissipator 50
A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface. The rapidly flowing liquid is abruptly slowed and increases in height, converting some of the flow's initial kinetic energy into an increase in potential energy, with some energy irreversibly lost through turbulence to heat. In an open channel flow, this manifests as the fast flow rapidly slowing and piling up on top of itself similar to how a shockwave forms. 51
Basic equation (rectangular section): Energy Depth; Depth of the stilling basin Length of the stilling basin where " and K are coefficients (derived from laboratory and field experiments). 52
For a suitably chosen % and a value of E corresponding to the total energy available above the stilling basin floor, y1, y2and y can be computed from equations (from a chosen value of safety coefficient). The values of the coefficients " and K in equations can be taken (Novak and Cábelka, 1981) as 1.1<& <1.25 and 4.5<K<5.5, where the lower value of K applies for Fr 1 >10 and the higher for Fr 1 < 3 53
54
The energy loss in the fourth and fifth phases of energy dissipation can be expressed as Downstream of the jump at the outflow from the basin there is still a substantial proportion of excess energy left, mainly due to the high turbulence of flow which can be expressed where # is the increased value of the Coriolis coefficient reflecting the high degree of turbulence and uneven velocity distribution; 2<# <5 for 3<Fr 1 <10, while # 1. 55
Equation shows that the efficiency of energy dissipation in the jump itself within the stilling basin decreases with the Froude number, leaving up to 50% of the energy to be dissipated downstream of the basin at low Froude numbers 56