4.2. Scaling factor for fishway flows
It was noticed in the studies on the flow in a pool-and-weir fishway with a V-shaped sharp crested weir (Section 3.2), that the flow could not clearly be classified into distinct flow modes, that is into plunging or streaming mode or transitional mode. The flow phenomenon varies continuously (inside the defined boundaries) and the flow mode changes smoothly from one to the other. Actually this statement is valid for a number of open channel flows.
Dimensionless parameters are useful in many design processes in pre-evaluating the water depths and discharges in fishways. In this paper, a new scaling factor for creating dimensionless discharge equations for fishways have been established. Empirical laws for flows over weirs were taken as the basis, in addition to the assumptions that
fishways are hydraulically rough open channels,
flow is uniform and flow friction can be expressed by the channel width, the distance between the weirs, and the slope of the channel, and
flow always runs over sharp-crested weirs.
The constant boundary conditions in a fishway are channel width, distance between weirs, bottom slope, weir height and possible roughness elements on the channel walls. Flow is determined by the head over the weir.
As stated in Section 2.4, procedures developed for uniform flow can be used in theoretical considerations on the factors affecting flows in fishways. Velocity is neither accelerating nor decelerating in uniform flow, but is equal in every cross-section; the surface of the water is parallel to the channel bottom and the energy line (Fig. 16). The force of gravity producing the flow is equal to the friction resisting it:
where G is the weight of the fluid in a channel section of length of ΔL, β is the angle of inclination of the bottom and of the surface, τ is the mean shear stress in wetted perimeter p, A is the cross-sectional area of the channel, ρ is the water density, and g is the acceleration due to gravity.
From Eq. 11 follows
It has been shown empirically that τ /ρ is proportional to the second power of the mean velocity v. Thus, from Eq. 12 follows
where C is the coefficient of the proportionality.
If the channel slope I = tanβ is small (like in fishways), sinβ can be substituted by I. A/p is defined as the hydraulic radius R. Eq. 13 can be now written in the form
This equation is known as the Chezy’s equation for uniform open channel flow. According to Chezy, the coefficient C depends only on channel properties. In flows over weirs, discharge can be expressed by equation
where Q is the flow over the weir, C is a coefficient (discharge factor), b is the width of the weir crest, g is acceleration due to gravity and h is the head at the weir, which determines the discharge (Lakshmana Rao 1975).
The use of scaling factors is based on the empirical equation for flows over weirs. In fishways, the main factors affecting the discharge coefficient C are
the contraction ratio in the direction of breadth, which depends on the ratio b/B, where b is the contracted channel width and B is the uncontracted approach channel width,
the longitudinal contraction ratio, which depends on the ratio b/L, where b is the contracted channel width and L is the longitudinal distance between the contractions,
the Froude number at the contraction (approach section),
the submergence ratio.
Considering the effect of velocities in the approach channel, the commonly known Weisbach equation (e.g. Lakshmana Rao 1975)may be derived and expressed as
in which Cc is the contraction coefficient, Va is the approach velocity, Q is the rate of flow over the weir, g is the acceleration due to gravity, B is the length of the weir crest, and h is the head, or the upstream depth of flow, over the weir crest.
According to Lakshmana Rao (1975), Engel & Stainsby proposed in 1958 the generalized weir equation (17) applicable to all weirs:
in which C is the discharge coefficient including the approach velocity effect. This equation (17) has been taken as a basis for further considerations.
Fishways are formed of adjacent weirs with a sloping bottom. The effect of the slope should be taken into account in the formulas for dimensionless discharges. Lakshmana Rao (1975) stated that the effect of variable channel slope may be taken care of roughly by using a formula (18) suggested by Hiranandani and Chitale (1960):
where Q is the discharge rate, Se is the energy slope, and subscripts h and s refer to horizontal and sloping channels, respectively. In uniform flow, the energy slope and the bottom slope are the same, and thus energy slope Se can be replaced by bottom slope So.. From eq. 18 we can conclude that the effect of slope is proportional to the second power of discharge.
Each weir affects to the flow over the other weir. It has been stated in many studies that pool length is one essential factor that affects the rate of flow. Indications of this can be found in e.g. Veijalainen (1981) and Rajaratnam et al. (1987). Boiten (1990) stated that pool length is an important factor that affects the energy dissipation pattern and through this for the hydraulic operation of the fishway. The basis for the effect of the pool length on the fishway discharge lies on the energy dissipation scheme in pool-and-weir and Denil fishways: in both of the fishway types, energy dissipation is mainly created by the standing wave effect. The extent of the effect depends on discharge and pool length.
It was noticed in these studies on pool-and-weir fishways that, for a given discharge, the depth of flow at the weir decreased with the increase of the pool length. It can be stated that the extent of the effect of pool length on the discharge depends on the ratio y/L, where y is the upstream depth of flow over the weir crest and L is the distance between adjacent weirs.
Preferably, scaling factors for discharges should be dependent on only one changing hydraulic parameter, in this case, discharge. Other variables in the equation for the scaling factor for dimensionless discharge Q* should depend only on channel properties. It is known that the most important factors affecting the rate of flow are the contracted width of the weir b, and, for sloping channels, the bottom slope So. In the studies on pool-and-weir fishways with V-shaped sharp crested weirs, it was observed that the pool length L affects the flow, too. Thus, in theoretical analysis, it should be possible to replace the depth of flow over the weir y with the pool length L. For the analysis, it was presumed that the relation between Q and Q* is a following function:
Wu and Rajaratnam (1995) stated that wall jets and hydraulic jumps are actually similar flows, end states with the submerged jumps forming the transition in between. This statement is based on the similar velocity distribution of the flows. Accordingly, by choosing the right scaling factors, it can be shown that actually all so called overflow fishways (a variety of pool-and-weir fishways and Denil fishways) can be classified into the same category and a general equation for dimensionless discharge can be written.
With these assumptions, from equations (17), (18), and (19) we get the scaling factor (20) for dimensionless discharges in fishways:
where Q is measured discharge. The longitudinal distance between the weirs or baffles L is used for scaling distances and water depths. These choices were based on trials on several scaling factors for discharges and lengths, in addition to pure reasoning.