Field measurements were performed to assess the performance of the proposed portable TCB flume. The tests were performed in the research farm of the Water Engineering Department at Imam Khomeini International University, IKIU, Qazvin, Iran.

A portable flume with b=24.4 cm (L_in=L_out=17 cm) was installed at the ditch entrance supplied by an old Qanat, a system for transporting water from an aquifer or water well to the surface used in Iran, TCB flume was placed in the ditch horizontally. It was pushed into the soil to level the ditch bed and flume floor (Figure 3A). In such a condition, the flume was stable, and the flume leakage was minimised. In the downstream section, the soil bed was slightly deeper to ensure a free overfall and a free flow condition, as shown in Figure 3B and C. The flow rate was determined by measuring the required time to fill a 32 L reservoir. The upstream flow depth was measured using a ruler at 5 cm upstream of the central baffle apex (Figure 3C). Finally, the associated laboratory- derived rating curve of the portable device was compared with the field measurements.

The free flow hydraulic of the central baffle flume with a floor height of p, equipped with an entrance ramp having a slope of tan b, could be expressed by the following functional relationship:

where j is a functional symbol, h is the upstream flow depth measured at 5 cm far from the upstream face of the central baffle, Q is discharge, B_c=B–b is the throat width, B is the approaching channel width, g is the acceleration due to gravity, ρ is water density and μ is the water viscosity.

Taking B_c, μ, and g, as reference variables and applying Buckingham’s theorem of dimensional analysis, the following dimensionless groups are obtained:

Rearranging Eq. (4f), the following equation is obtained:

In which Re is the Reynolds number.

Taking P₁ as the dependent dimensionless group, the dimensionless form of Eq. (3) is the following:

where f is a functional symbol.

For specific values of p/B_c, r, and tan b, when h/B_c → 0 then Q/[B_c ^5/2 g^1/2] → 0 and when h/B_c → ∞ then Q/[B_c ^5/2 g^1/2] → ∞. Therefore, according to the incomplete self-similarity (ISS) theory (Barenblatt, 1979, 1987), the group h/B_c can be extracted as a power type expression:

where n is a numerical constant to be obtained by experimental data and f₁ is a functional symbol. According to the experimental data of this study, the Reynolds number is in the range of 2.47×10⁶≤Re≤9.51×10⁷; therefore, viscous effects are small, and the Reynolds number may be neglected. Furthermore, the surface tension was neglected as previous studies showed that these effects are negligible except for very low values of the upstream flow depth (Rao and Shukla, 1971; Sargison 1972; Ranga Raju and Asawa, 1977; De Martino and Ragone, 1984).

Therefore, according to Eq. (6), the stage-discharge relationship is affected by the floor height ratio, p/B_c, the contraction ratio, r, and the entrance ramp slope, tan b.

Flume bed rise may occur during the flume installation. To study the effect of the floor height, for a given value of the contraction ratio, r, the stage-discharge curves associated with different values of p/B_c were compared in Figure 4. Note that p/B_c=0 is an accurate flume installation in which no bed rise occurs. As shown, the floor height ratio, p/B_c, did not affect the stage-discharge curve for r=0.17 and 0.39. Consequently, the floor height ratio affects the flow capacity when the contraction ratio increases.

For a specific value of h/B_c, Figure 4 indicated that for r=0.59 and 0.79, a flume with no floor height, i.e., p/B_c=0, had a higher discharge capacity than the cases having p/B_c>0. This might be attributed to the fact that for r=0.59 and 0.79, the head loss due to the effect of p/B_c increases significantly.

Therefore, according to the experimental results of this study, it is suggested to consider a floor height ratio of 0≤p/B_c≤1.76 with the contraction ratios of less than or equal to 0.39 to ensure that the floor height ratio does not influence the flow capacity. Consequently, the parameter p/B_c can be eliminated from the functional relationship Eq. (6) when r≤0.39.

Taking p/B_c in the range of 0.27 to 1.76 and r=0.17 and 0.39, the stage-discharge curves associated with different entrance ramps slopes (tan b=0, 0.2, and 1) were plotted in Figure 5. This figure demonstrated that for r<0.39, there was no significant effect of installing an entrance ramp on the stage-discharge curve. Consequently, the experimental investigation suggested taking b=0 to minimise the construction costs.

Taking r=0.59 and p/B_c=0.35, the effect of the entrance ramp slope on the stage-discharge curve was illustrated in Figure 6. As shown, for a specific value of h/B_c, the flume discharges associated with tan b=0.2 and tan b=1 were slightly higher than that obtained for tan b=0. In other words, an entrance ramp increases the discharge capacity when the contraction ratio is greater than 0.39.

In summary, the following recommendations are proposed: i) taking a contraction ratio in the range of r≤0.39, and a floor height ratio within the range of 0≤p/B_c≤1.76, neither the floor height nor the entrance ramp slope affects the flume flow capacity; ii) for r>0.39 (the cases r=0.59 and 0.76 are tested in this study), both floor height and entrance ramp slope affect the stage-discharge formula. An increased floor height has a negative impact, while an entrance ramp slope increases the discharge capacity. As a practical conclusion, from the construction point of view and to minimise the flume weight, it is suggested to take r≤0.39, and a floor height ratio of 0≤p/B_c≤1.76, for constructing the portable TCB flume. No entrance ramp installation is also suggested.

Taking r=0.39 and B_c=15.6 cm, the validity of Eq. (10) was evaluated for the proposed portable flume field application. For this aim, the field observations were compared with the experimental stage-discharge curve obtained by Eq. (10).

As shown in Figure 9, the proposed portable TCB flume could be used effectively to obtain the flow rate through an unlined ditch. The figure also revealed that Eq. (10) was applicable to accurately determine the flow rate with a mean absolute relative error of 3.8%. The discharge varied in the range of 0.64 to 3.05 L/s. The maximum relative error of –6.3% was observed for Q=0.63 L/s located significantly out of the calibrated range of the TCB flume. Detailed field data points and the relative errors associated with Eq. (10) were listed in Table 4.

The fundamental assumptions used to develop Eq. (10) were: i) it was valid when r≤0.39 and 0≤p/B_c≤1.76; ii) neither the floor height nor an entrance ramp slope would affect the rating curve; iii) the flow must be critical in the throat. To this end, tailwater depth must be limited according to the distinguishing condition curve; iv) the flume must be installed horizontally.

Note that, in all field measurements, the flume’s bed was levelled with the ditch bed (Figure 3A), i.e., p/B_c=0. The acceptable accuracy of Eq. (10) revealed that the assumptions made in laboratory tests could also be extended to the field application of the portable TCB flume.

Tailwater increase may affect the flow through a TCB flume. Such a flow condition is classified as a submerged regime. For a given flow rate, a unique tailwater depth h_t exists beyond which the submerged flow condition occurs. The floor height value would affect the submergence threshold of the portable TCB flume. A distinguishing condition curve, also known as the ‘modular limit’, is a relationship between upstream and downstream flow depths which is employed to obtain the submergence threshold (see Bos, 1989, page 29). The following functional relationship is used to describe the maximum tailwater depth for which the free flow regime occurs, h_t:

in which α is a coefficient and f₃ is a functional symbol.

For tan b=0, the pairs (Q/(Bc^2.5g^0.5), h_t/B_c) are plotted in Figure 10. As shown, for a given r, the required maximum tailwater depth for which free-flow condition reveals, i.e., h_t/B_c, significantly increases with higher values of p/B_c.

For tan b=0 and applying the incomplete self-similarity condition, Eq. (11) takes the following form:

in which a, l, c, and d, are coefficients to be estimated by the available measurements. Using the experimental data of this study, the obtained coefficients were listed in Table 5.

A comparison between the calculated and observed values of h_t/B_c is shown in Figure 11. Accordingly, using Eq. (12), 80% and 97% of the data points can be respectively estimated for p/B_c=0 and 0.27≤p/B_c≤1.76 with relative errors of ±10%.

Eq. (12) was used to estimate values of h_t/B_c associated with the triangular baffle flumes with an entrance ramp, i.e., tan b>0. As shown in Figure 12, Eq. (12) can be applied to predict h_t/B_c accurately, even if an entrance ramp is constructed. The associated mean absolute relative error is 5.5%, and 90% of the data points can be calculated with the relative errors limiting in the range of ±10%.

In other words, taking tan b>0 does not affect the maximum tailwater depth for which the free flow condition is anticipated.

Substituting Q/[B_c^2.5g^0.5] from Eq. (10) into Eq. (12), the following distinguishing condition curve was obtained for the portable TCB flume when p/B_c ranges from 0.64 to 0.83:

The relative error distribution associated with Eq. (13) was depicted versus h/B_c in Figure 13. As shown, the submergence threshold was estimated within a range of ±10% with a mean absolute relative error of 5.8%.

Although the triangular shape of the central baffle minimises the flume size and weight, according to Kolavani et al. (2019), it might be more sensitive to submergence than the typical cases with a longer baffle length. Such a disadvantage should be considered, especially when the tailwater depth cannot be adjusted. Hence, to ensure a free flow condition, during the installation of the portable TCB flume, it is suggested to make the soil bed slightly deeper at the tailwater section to see a free overfall and a free flow condition, as shown in Figure 3B and C.

a, c, d, n, l, and α = empirical coefficients;

B = the approaching channel width;

b = the baffle width;

B_c=B-b;

g = acceleration due to gravity;

h = Upstream depth;

h_th = the maximum permitted tailwater depth to allow the free flow condition;

H₁= total upstream head;

H_c= specific energy at the critical flow section;

L_s = Sill length;

p= floor height;

Q = discharge;

Q_V= flow rate of a venturi flume;

Q_BR= flow rate of a bed-rise structure;

Re= Reynolds number;

r (B_c/B)= contraction ratio;

b= entrance ramp slope angle;

f, f₁, and f₂ = functional symbols;

μ = water viscosity.