4.3. The dimensionless discharge equation for fishways

In this analysis, the relation between the discharge and the depth of flow is considered. The discharge is supposed to be a relation of the depth of flow over the weir instead of the head over the weir. For practical reasons (easiness of calculations), a power regression equation was selected for further development for the dimensionless discharge equation. This gives equations that are structurally similar to that of the conveyance factor K, which is a measure of the carrying capacity of the channel section (Chow 1959). Like the developed scaling factor Q_*, the conveyance factor K is also directly proportional to discharge Q and inversely proportional to bottom slope S_o. Conveyance K is a function of the depth of flow y and is defined by equation

Equation 21.

where C is a coefficient, Y is the depth of flow, and N is a parameter called the hydraulic exponent for uniform-flow computation. This is actually also structurally similar to the general form of discharge rating curve. The general form of the discharge rating curve is Q=CY^N where C is a constant dependent on the channel properties, Y is the depth of flow, and N is an exponent dependent on the cross sectional shape.

The head at the weir, h, can be replaced by the vertical water depth at the weir y. This simplification can be justified with some reasons:

• the approach velocity in fishways is usually so small that the effect can be neglected.

• the head at the weir can not be defined exactly, and the error due to this simplification is in this analysis negligible.

• the weir is in real structures essentially the most unequivocal place to measure the depth of flow.

The use of the water depth instead of the head at the weir has its own disadvantages. In pool-and-weir fishways, in the so called transitional stages of flow, i.e. when the stage of flow changes from about plunging to the so called streaming flow mode, the depth of flow may have at least two different values depending on the flow mode. In the transitional stage, the water surface fluctuates and the fluctuations in depth of flow along the length of the fishway do not cope with the location of weirs. In pool-and-weir fishways, when the flow is in the streaming mode, the approach velocity begins to take effect increasing the water velocity at the weir and decreasing the depth of flow. All this causes scatter and distortion in the results.

The scaling factor (20) was applied to the author’s study results on different pool-and-weir, vertical slot, and Denil fishway types. After scaling, the calculated dimensionless discharges were plotted against y_o/L, where y_o is the depth of flow over the weir and L is the pool length (Fig. 17). Based on this data (given in App. 7), a regression power equation was obtained:

Equation 22.

It can be seen that dimensionless discharge Q_* - dimensionless depth y_o/L –pairs calculated from the measured values fit quite well into the developed fitting curve except the data on pool-and-weir fishways. The regression coefficient for the fishway discharge is R²=0.96. The equation was formed based on a total of almost 450 separate discharge - water depth pairs.

Regression lines for data sets of each fishway type are included in Fig. 17. One can see that regression lines for different pool-and-weir and Denil fishways have almost the same slopes. The slope of the regression line for the data set on vertical slot fishways differs from those of the other regression lines.

The generated curve was evaluated with Student’s paired t-test. The test is commonly used to test the absence of bias in discharge rating curves. Further considerations and other equations for dimensionless discharges are presented in Section 4.4.