Performance of a right-triangle stilling basin: a laboratory investigation

One must be sure that the hydraulic jump, as an energy dissipater, forms not only during all the possible flow conditions but its position remains constant (Hager & Sinniger 1985). To stabilize the situation of hydraulic jump downstream of hydraulic structures and to increase energy dissipation along hydraulic jump, different energy dissipation structures such as steps, stilling basins have been investigated over the years. A positive step is one of the basic types of stilling basins defined as a local rise in the horizontal channel bottom and the front of step is perpendicular to the flow direction (Hager & Sinniger 1985). Due to stability in jump position, especially on the small Froude numbers, attentions are pointed toward positive step by researchers (Hager & Bretz 1986). For hydraulic jump over a positive step in a horizontal bed, Rajaratnam (1965) stated that the flow rise is noticeable just over the step site and Karki et al. (1972) showed that the pressure on the front of the step is greater than the hydrostatic pressure. Applying the momentum equation, Hager & Sinniger (1985) derived theoretical equations for relating the sequent depth ratio of the hydraulic jump over the positive step into the inflow Froude number and relative step height. Four flow regimes over a positive step were experimentally identified by Hager & Bretz (1986). Apart from these substantial hydraulic studies, the elementary geometry of a positive step as a jump basin has a significant impact on the construction process and therefore the costs of the project. Due to its static stability, an abrupt rise as a stilling basin does not require further structures such as roughened bed, chute blocks (Demetriou & Dimitriou 2010). In recent years, a new theoretical model with great accuracy was developed based on a one-dimensional momentum equation to estimate the conjugate flow depth, jump length and, energy dissipation of hydraulic jump in a horizontal rectangular channel with an abrupt rise (Mohammadi et al. 2021; Pandey et al. 2023; Zolghadr et al. 2023). The important matter of the jump location and its displacement in a sill-controlled jump basin was investigated by Hafnaoui & Debabeche (2023). Meanwhile, Zhou et al. (2023) highlighted the obligation on avoiding wave-type flow occurring under no tailwater conditions in channels equipped with abrupt rise.

Despite many studies of hydraulic jump over the positive step in a horizontal bed, the issue of hydraulic jump on sloping beds with positive steps is less addressed. A hydraulic jump over a positive step in an inclined channel can result in high energy losses (Demetriou & Dimitriou 2010). The soil conservation service of the U.S. Department of Agriculture has advised the engineers to construct a channel with a floor slope of 1V:3H for the chute spillway. The hydraulic jump forming on a slope greater than 0.33 (18°) may act differently (Rajaratnam 1963). The jump on the sloping channel was investigated by Quraishi & Al-Brahim (1992), followed by Husain et al. (1994) and Negm (1996). Quraishi & Al-Brahim (1992) classified the hydraulic jump over the positive step in a sloping bed into A-jump and B-jump. For the B-jump, an uprising in flow depth is formed over the situation of a positive step, and the jump is formed by decreasing the tailwater depth of an A-jump. Formation of the hydraulic jump under proper topographic situation and tailwater depth on sloping channels results in considerable energy dissipation (Quraishi & Al-Brahim 1992). Several empirical equations are suggested by Husain et al. (1994) to estimate the conjugate flow depth and jump length for any jump type at a positive step. A modified Froude number depending on channel slope and step height is then derived by Negm (1996). Nevertheless, the formation of the hydraulic jump on steep slopes is a fundamental concern for engineers (Beirami & Chamani 2006). Working conditions of a positive step as a stilling basin in a steep channel with supercritical downstream flow regimes were examined experimentally by Carravetta & Vacca (2009).

To shed light on the matter of hydraulic jumps on sloping beds, researchers have mostly examined stilling basins with adverse slopes. Bateni & Yazdandoost (2009) used momentum equations and experimental results to drive relations predicting sequent depth ratio and roller length of jump on an adverse sloping floor. Nonetheless, the formation of the hydraulic jump on an adverse slope is still a challenging issue due to the unstable configuration of hydraulic jumps (Parsamehr et al. 2017). An experiment conducted by Pourabdollah et al. (2020) proved that the adverse slope equipped with a positive step increases energy loss. Results of Mazumder (2022) experiments on several stilling basins with varying adverse slopes combined with a positive step with different heights implied 13% higher energy loss than classical jump. Rising energy loss of the high kinetic energy of flow downstream of hydraulic structures with different methods is a crucial aspect to be considered for designers (Daneshfaraz et al. 2023).

To convey flow from high to low elevation efficiently, a chute spillway which requires lower excavation in hilly regions is necessary. For chute spillways, the steep slope of the chute bed is usually terminated by the horizontal bed of downstream channel and energy dissipators are essential elements at the spillway end to mitigate the damaging erosive impacts (Daneshfaraz et al. 2021b). Utilizing elements such as fishway on sloping bed can also act as an energy dissipator structure (Daneshfaraz et al. 2021a). But due to the development of the Roller Compacted Concrete (RCC) construction technique, stepped chutes have attracted the attention of engineers in recent years. Although United States Bureau of Reclamation (USBR) stilling basins have performed well under the experimental study of a stepped chute spillway (Valero et al. 2018; Hunt & Kadavy 2021), stepped chutes require 17% longer stilling basin lengths (Stojnic et al. 2022). While numerous investigations have been conducted on energy dissipation of stepped spillways, few have been conducted on energy dissipaters for smooth chute spillways.

Given the current global drive for more sustainable structures with smaller carbon footprints and alignment to the United Nations' sustainable development goals (SDGs), optimizing existing standardized designs are ongoing. The present study aims to introduce and investigate the application of a positive step at the intersection of the chute bed with the downstream horizontal channel, named right-triangle basin, as a new energy dissipation structure for dissipating the excess energy of supercritical flow downstream of the chute spillway located in hilly regions. Thus, in terms of scaling the experimental modeling to praxis, vulnerable island nations such as small island developing states (Chadee et al. 2021) can immediately benefit from hydraulic energy dissipation structures based primarily on size, cost, and location. Finally, previously, experimental study provided only at a point or on a plane chance for observation and measurement of the hydraulic jump's parameters. Now the sophisticated nature of jumps can be understood by a Computational Fluid Dynamics (CFD) simulation (De Padova & Mossa 2021). To this end, hydraulic jump properties over the right-triangle basin were investigated by using dimensional analysis, and experimental and numerical simulations.

Returning loose materials from downstream to upstream by backward flows would result in abrasion and destruction of right-triangle basins. This process would be avoided by establishing forward developing flow over the loose materials on downstream of basin. To this end, the length of L_d should be protected with appropriate materials such reinforced concrete. Constructing the front of basin with reinforced concrete is also advised.

In this study, ANSYS FLUENT software was applied to generate the computational mesh on the 3D simulated model of the right-triangle basin by AUTOCAD. Solution of the Navier Stokes equations is the basis of ANSYS FLUENT software package. These equations are based on mass and momentum flux balances. Meanwhile, only 4 m length of the channel is simulated to decrease computing time. A triangular unstructured mesh is selected in this study for computation. To determine the free surface of flow, a VOF model is utilized and for the numerical simulation the k–ε turbulence model is used. Boundary conditions include the air and water inlet for entrance section, pressure for outlet section, walls for bed and sides of the channel and pressure boundary for free surface of the flow. The Average Relative Error (ARE) criterion is used to evaluate the consistency between the numerical and experimental models. The mesh study analysis corresponding to the simulation of a test with Q = 0.0410 m³/s, z/d₁ = 3.17, and F₁ = 4.42 resulted in 1189524 number of nodes. The error analysis, which is based on the comparison of experimental water surface with numerical results, showed a value about 5.55% for the Average Relative Error (ARE).

The water flow system in the flume was recirculated and flow discharge was regulated using a butterfly valve situated on the transferring water pipe between the pump and flume entrance. Discharge values were measured using the volumetric method. Measurement of flow depth was conducted at the centerline of the flume using a point gauge with ±0.05 mm reading accuracy. The minimum required tailwater depth for establishing the hydraulic jump over the right-triangle basin was adjusted with a sliding rectangular sharp-crested weir installed at the flume outlet.

Regarding the classification for hydraulic jump on sloping channels with positive or negative step Quraishi & Al-Brahim (1992) introduced two types of jumps known as A-jump and B-jump. Based on described flow condition by Quraishi & Al-Brahim (1992) and measured water surface in this study, the B-jump was formed in experiments. Hydraulic jump characteristics over three models of right-triangle basin (models 1 through 3) were measured. Due to the high turbulence in the jump, flow depth and bed pressure were highly fluctuated, as well as on basin front. Therefore, flow depth and bed pressure were quantified by time-averaged method. To indicate the efficiency of hydraulic jump over the right-triangle basin, some A-jumps on sloping bed without a downstream step was also tested for models 4 and 5 by establishing the initial supercritical depth of jump at the toe of chute models. Summary of flow conditions and bed geometry of tested hydraulic jumps are presented in Table 1.

Table 1

Summary of flow conditions and bed geometry of tested hydraulic jumps

Model .	z (cm) .	P (cm) .	Q (m3/s) .	d1 (cm) .	F1 (–) .	Re1 (–) .	d2 (cm) .	Ld (cm) .	.	ω (–) .	ωL (–) .
1	12.0	39.6	0.04386	3.98	4.4	109,650	17.58	76	54.03	0.10	0.61
			0.04099	3.79	4.42	102,475	15.58	56	56.55	0.09	0.59
			0.03316	3.13	4.78	82,900	13.95	51	59.46	0.08	0.49
			0.02542	2.42	5.39	63,550	12.03	51	64.20	0.06	0.36
			0.01774	1.73	6.24	44,350	8.98	41	71.07	0.04	0.22
			0.01342	1.33	7	33,550	8.01	26	73.69	0.03	0.12
2	12.0	54.0	0.04572	3.89	4.75	114,300	16.3	60	60.72	0.10	0.57
			0.03885	3.32	5.12	97,125	14.9	56	63.62	0.08	0.48
			0.03514	3.13	5.06	87,850	14.1	50	62.82	0.08	0.47
			0.02736	2.47	5.63	68,400	12.5	45	66.31	0.06	0.35
			0.02338	2.18	5.78	58,450	11.8	41	66.31	0.05	0.31
			0.01894	1.76	6.5	47,350	10.9	28	69.45	0.04	0.21
			0.01299	1.31	6.92	32,475	8.8	21	70.91	0.03	0.12
3	6.0	54.0	0.04664	3.98	4.68	116,600	22.4	78	50.05	0.10	0.58
			0.04361	3.77	4.71	109,025	21.5	73	50.23	0.09	0.56
			0.03669	3.23	5.05	91,725	18.8	65	54.94	0.08	0.48
			0.03312	2.94	5.24	82,800	17.2	56	57.57	0.07	0.43
			0.02572	2.37	5.62	64,300	14.7	47	60.63	0.06	0.34
			0.0221	1.99	6.27	55,250	12	40	68.22	0.05	0.25
			0.01825	1.8	6.02	45,625	10.3	33	67.18	0.05	0.24
			0.01194	1.33	6.22	29,850	8.5	23	66.26	0.03	0.16
4	0.0	39.6	0.05183	5	3.7	129,575	23.7	–	35.70	0.13	0.76
			0.04191	4	4.18	104,775	22.2	–	40.11	0.10	0.63
			0.03706	3.8	3.99	92,650	19.9	–	38.41	0.10	0.62
			0.03344	3.4	4.26	83,600	18.6	–	42.63	0.09	0.56
			0.02922	3	4.49	73,050	16.9	–	46.28	0.08	0.50
			0.02351	2.6	4.48	58,775	15.5	–	43.32	0.07	0.45
			0.02017	2.2	4.93	50,425	14.1	–	49.07	0.06	0.37
			0.01881	2.1	4.93	47,025	13.4	–	49.28	0.05	0.35
			0.01089	1.4	5.24	27,225	9.8	–	50.68	0.04	0.22
5	0.0	54.0	0.04777	3.8	5.15	119,418	26.1	–	49.82	0.10	0.53
			0.04421	3.6	5.17	110,523	24.7	–	50.19	0.09	0.51
			0.03791	3.1	5.54	94,779	21.7	–	55.32	0.08	0.43
			0.02654	2.5	5.36	66,357	18.2	–	50.85	0.06	0.38
			0.01894	1.9	5.77	47,352	15.1	–	53.51	0.05	0.27
			0.01299	1.4	6.26	32,467	11.3	–	59.32	0.04	0.17

Model .	z (cm) .	P (cm) .	Q (m3/s) .	d1 (cm) .	F1 (–) .	Re1 (–) .	d2 (cm) .	Ld (cm) .	.	ω (–) .	ωL (–) .
1	12.0	39.6	0.04386	3.98	4.4	109,650	17.58	76	54.03	0.10	0.61
			0.04099	3.79	4.42	102,475	15.58	56	56.55	0.09	0.59
			0.03316	3.13	4.78	82,900	13.95	51	59.46	0.08	0.49
			0.02542	2.42	5.39	63,550	12.03	51	64.20	0.06	0.36
			0.01774	1.73	6.24	44,350	8.98	41	71.07	0.04	0.22
			0.01342	1.33	7	33,550	8.01	26	73.69	0.03	0.12
2	12.0	54.0	0.04572	3.89	4.75	114,300	16.3	60	60.72	0.10	0.57
			0.03885	3.32	5.12	97,125	14.9	56	63.62	0.08	0.48
			0.03514	3.13	5.06	87,850	14.1	50	62.82	0.08	0.47
			0.02736	2.47	5.63	68,400	12.5	45	66.31	0.06	0.35
			0.02338	2.18	5.78	58,450	11.8	41	66.31	0.05	0.31
			0.01894	1.76	6.5	47,350	10.9	28	69.45	0.04	0.21
			0.01299	1.31	6.92	32,475	8.8	21	70.91	0.03	0.12
3	6.0	54.0	0.04664	3.98	4.68	116,600	22.4	78	50.05	0.10	0.58
			0.04361	3.77	4.71	109,025	21.5	73	50.23	0.09	0.56
			0.03669	3.23	5.05	91,725	18.8	65	54.94	0.08	0.48
			0.03312	2.94	5.24	82,800	17.2	56	57.57	0.07	0.43
			0.02572	2.37	5.62	64,300	14.7	47	60.63	0.06	0.34
			0.0221	1.99	6.27	55,250	12	40	68.22	0.05	0.25
			0.01825	1.8	6.02	45,625	10.3	33	67.18	0.05	0.24
			0.01194	1.33	6.22	29,850	8.5	23	66.26	0.03	0.16
4	0.0	39.6	0.05183	5	3.7	129,575	23.7	–	35.70	0.13	0.76
			0.04191	4	4.18	104,775	22.2	–	40.11	0.10	0.63
			0.03706	3.8	3.99	92,650	19.9	–	38.41	0.10	0.62
			0.03344	3.4	4.26	83,600	18.6	–	42.63	0.09	0.56
			0.02922	3	4.49	73,050	16.9	–	46.28	0.08	0.50
			0.02351	2.6	4.48	58,775	15.5	–	43.32	0.07	0.45
			0.02017	2.2	4.93	50,425	14.1	–	49.07	0.06	0.37
			0.01881	2.1	4.93	47,025	13.4	–	49.28	0.05	0.35
			0.01089	1.4	5.24	27,225	9.8	–	50.68	0.04	0.22
5	0.0	54.0	0.04777	3.8	5.15	119,418	26.1	–	49.82	0.10	0.53
			0.04421	3.6	5.17	110,523	24.7	–	50.19	0.09	0.51
			0.03791	3.1	5.54	94,779	21.7	–	55.32	0.08	0.43
			0.02654	2.5	5.36	66,357	18.2	–	50.85	0.06	0.38
			0.01894	1.9	5.77	47,352	15.1	–	53.51	0.05	0.27
			0.01299	1.4	6.26	32,467	11.3	–	59.32	0.04	0.17

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Equation (8) indicates that F₁ and z/d₁ are dimensionless influencing parameters on the sequent depth ratio of hydraulic jump over right-triangle basin.

It follows from Figure 3 that the sequent depth ratio of hydraulic jump over right-triangle basin is increased with increasing inflow Froude number. For z/d₁ < 3, increasing in relative basin front length resulted in reduction in sequent depth ratio of hydraulic jump over right-triangle basin. But after that (3 < z/d₁), more reduction in d₂/d₁ is not nearly achieved by increasing in z/d₁. Therefore, to achieve the smallest required sequent depth for hydraulic jump over right-triangle basin, the relative step height range of 3 < z/d₁ is advised for design of right-triangle basin. For 3 < z/d₁, sequent depth ratio of hydraulic jump over right-triangle basin is averagely obtained 30% smaller than that of the classical jump. It is also obtained 15% smaller than that of hydraulic jump data over USBR type Ⅲ stilling basin, indicating usefulness of right-triangle basin for practical purposes.

The Mean Absolute Percentage Error (MAPE) of Equation (9) for estimating the sequent depth ratio of hydraulic jump over right-triangle basin models was calculated as 5.3%, indicating high accuracy of the developed equation.

Regarding the jump type classification, A-jump has been described as a jump that entirely located at upstream zone of an abrupt rise over a sloping bed. It means that the termination section of its roller length is exactly at the starting point of the positive rise (Quraishi & Al-Brahim 1992). However, B-jump is characterized by its roller length location which is partly upstream and partly downstream of an abrupt rise and decreasing the tailwater in B-jump results in two distinct portions for roller length (Quraishi & Al-Brahim 1992). Assessment of jump type based on the comparison of Quraishi & Al-Brahim (1992) classifications and longitudinal profile of water surface in this study (Figure 4) substantiates the formation of B-jump over right-triangle basin.

As it can be observed from Figure 5, energy dissipation of hydraulic jump over right-triangle basin is significantly larger than that of the classical hydraulic jumps with horizontal and sloping inflows. For 3 < z/d₁, the relative energy dissipation data of hydraulic jumps over right-triangle basin is very close to the maximum limit of energy dissipation curve, indicating intense energy dissipation that resulted in smaller sequent depth of hydraulic jump over right-triangle basin compared to classical hydraulic jump, as previously observed from Figure 3. Relative energy loss increased with rise in z/d₁ of the right-triangle basin by 37% compared to classical jump on horizontal floor downstream of chute.

The MAPE of Equation (10) for estimation of L_d/d₂ was calculated as 10.5%, indicating acceptable accuracy of the developed equation.

This study introduced a new conceptual model of a right-triangle basin to dissipate the excess energy of supercritical flow on downstream of chute spillway. In comparison to the classical hydraulic jump models, energy dissipation of hydraulic jump over right-triangle basin is significantly larger. The formation of two large recirculation regions over both the jump roller and jump bed at beginning of the downstream horizontal channel is the main reason for large energy dissipation. Due to this larger energy dissipation, a smaller tailwater depth is required for stabilizing hydraulic jump in the proposed stilling basin design.

To achieve the smallest required sequent depth for hydraulic jump over right-triangle basin, the range of the relative basin front length of 3 < z/d₁ is advised for design optimality of the right-triangle stilling basin. However, the results of this study and suggested relationships are applicable for tested hydraulic jumps over right-triangle basin under geometric and hydraulic conditions of θ = 18.43°, 4.4 < F₁ < 7.0, and z/d₁ < 9.2. This design achieves an increased energy loss of approximately 37% as compared to classical jump on horizontal floor downstream of chute. Thus, this design can provide an improved and efficient energy dissipator in stilling basin construction and allow for a less maintenance issues due to erosion and scour during the lifecycle of the structure.

To practically implement this design, contemporary construction practices will have to be modified to accommodate extra excavation and concrete works to position the right-triangle basin, which is the downstream of chute spillway. It is also envisaged that the structure of right-triangle basin could act as an end sill at the toe of chute and, consequently, reduce the discharge seepage from the chute foundation. This is the subject for future studies to justify the economics and sustainability of the right-triangle basin for practical applications.

The authors extend their appreciation to the deanship of scientific research at King Khalid University for funding this work through large group research project under grant number (RGP2/94/44).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.