Experimental Investigation of the Threshold Submergence in Combined Throat Flumes

Introduction
There are generally two main methods for measuring fluid flow in open channels. The first method involves measuring the average velocity and cross-sectional area and multiplying them to calculate the discharge. The second method involves creating a controlled depth using a structure and establishing a direct relationship between the depth and flow rate (known as the stage-discharge relationship), where the flow rate can be directly determined by measuring the depth (Potter et al., 2012). Flow passing through a structure for measurement is categorized into two main types: free or modular flow and submerged or non-modular flow. Since the hydraulic behavior of these structures differs under free and submerged conditions, determining the boundary between these two flow states is essential. The threshold of each structure represents the boundary between these two flow states. So far, no studies on this topic have been specifically for flumes with combined cross-sections. Therefore, this research aims to cover this subject. This study aims to experimentally investigate the threshold of submergence in trapezoidal-rectangular and triangular-rectangular combined cross-section flumes. For this purpose, the submergence threshold of these flumes is examined for various flow rates and geometries. Additionally, empirical relationships for both proposed flume types have been developed after dimensional analysis and non-dimensionalities of influential parameters for engineers to use in the design phase. Finally, the submergence thresholds of these two types of flumes are compared.

Methodology
The experiments in this study were conducted on horizontal and rectangular flumes with dimensions of 20, 0.6, and 0.5 m for length, width, and height. The flumes have a closed water flow system, and an end gate to control of downstream depth. In this research, different models of flumes were tested at various flow rates. In total, 170 experiments were performed for the trapezoidal-rectangular flume, and 101 experiments were conducted for the triangular-rectangular flume in the threshold submergence state.
The effective parameters on the threshold submergence of a trapezoidal-rectangular flume include geometrical parameters, fluid characteristics, and gravity acceleration. The geometric parameters are the height of the flume P, the amount of floor opening a, and the width of flume B. The angles of upstream and downstream transitions are not considered because they are fixed. Fluid-related parameters include dynamic viscosity µ and specific mass ρ. Another parameter affecting the flow in open channels is the acceleration of gravity g. Also, the parameters related to the flow, including the upstream depth h, and the downstream depth ht, are effective variables on the threshold submergence. By using Buckingham's Π theory, choosing the parameters µ, g, and h as repetition variables, and using the theory of incomplete self-similarity (Barenblatt, 1987), finally, two Eqs. (1) and (2) are as follows.
h_t/P=(h/B)^m f_1 ( a/B,Z) (1)
h_t/P=(h/B)^m f_2 ( Z) (2)
where m is a numerical constant that is determined based on experimental data.

Results and Discussion
The results indicate that although the ht/B index shows a consistent increasing trend, it effectively fails to differentiate between flows with different heights. The ht/h index demonstrates that this index does not exhibit a consistent trend compared to the h/B parameter. Therefore, based on the analyses conducted in this study, ht/P was considered a suitable threshold index for both types of flows under investigation.
The index increases as the height of the flume increases for all three triangular prism side lengths (5, 10, and 15 cm). It is worth mentioning that the changes for low h/B ratios are negligible, and the difference between different flume depths increases with an increase in this ratio, especially for structures with side lengths of 10 and 15.
The results show that with an increase in this parameter, the threshold submergence index decreases, indicating an increase in the sensitivity of the flow to the upstream depth. For a specific flow height and discharge, the upstream depth of the structure decreases with an increase in the parameter a. Therefore, the upstream energy of flows with higher opening ratios is lower, resulting in their submergence at an earlier stage.
Similarly, similar to the trapezoidal-rectangular flow, in this type of flume, with an increase in height, the threshold submergence index increases, and the flume submerges later.
Equations (3) and (4) demonstrate the threshold submergence index for the combined trapezoidal-rectangular and triangular-rectangular flumes. The results show that more than 80% of the data have an error of less than 5%. The provided empirical equations have achieved satisfactory accuracy in estimating the experimental results.
h_t/P=4(h/B)^(1.1) 〖(a/B)^(0.088) (Z)〗^(0.096) (3)
h_t/P=1.443(h/B)^(1.254) (Z)^(1.29) (4)

Conclusion
The investigation of various dimensionless parameters has shown that the ratio of the downstream depth to the flume height (ht/P) is a suitable indicator for the threshold submergence for both types of flumes. In both types of flumes, as the flow depth increases, the threshold submergence index also increases, indicating a decrease in the sensitivity of the flume to the downstream depth. The study of the effect of the trapezoidal base on the threshold submergence index has shown that as the base width increases, the threshold submergence decreases. A comparison between the two types of flumes has shown that in low flow rates, the triangular-rectangular combined flume reaches the threshold submergence earlier than the trapezoidal-rectangular flume. In contrast, in high flow rates, the opposite is true. Statistical analyses have demonstrated that the proposed relationships accurately predict the experimental results, with over 80% of the predictions having an error of less than 5%.