Equations and methodologies of inlet drainage system discharge coefficients: A review

1 Introduction

Poor planning, rapid urban development, and climate change are among the major drives behind urban flooding [1,2,3,4]. Urban drainage systems play a crucial role in managing storm water runoff in urban areas. They are designed to transport the excess surface water during the events of rain from urban areas and reduce floods to acceptable levels of service [5]. In recent years, the occurrence of urban flooding has been more frequent due to climate changes and rapid urbanization which resulted in intense rainfall events with longer durations and significant increases in the impervious layers especially in large cities [6,7,8,9]. The urban drainage system is composed of two subsystems called minor and major subsystems [10,11]. The major subsystem is composed of the roads and gutters that transport the surface runoff to the minor subsystem formed of the subsurface drainage network designed to carry the intercepted flow into designated areas (such as water treatment plants, rivers, lakes, or oceans) [12]. The minor and major subsystems are connected by inlets (manhole, gully) that work as linking elements. The major and minor subsystems with the linking element constitute the dual-drainage system. The discharge capacity of the linking elements can be determined experimentally or through weir/orifice discharge equations.

The hydraulic efficiency (E) of the drain inlet is defined as the ratio of the inlet intercepted flow (Q _i) to the upstream approaching flow (Q _a). In the last century, several studies have been conducted on inlets to investigate the storm water runoff and discharge coefficients as well as the major factors that influence the inlet’s hydraulic capacities [13,14,15,16,17,18,19,20,21,22,23,24,25] and to develop reliable urban flooding models [26]. The developed hydraulic efficiency and discharge coefficient equations have been evolving significantly in the last two decades to cope with the frequent urban flooding affected by urbanization and climate changes [27,28,29,30,31,32]. There are two main approaches for determining the inlet discharge capacity through the weir/orifice discharge coefficients. The first approach adopts constant weir/orifice discharge coefficients regardless of the inlet geometry or flow condition pattern while the second approach determines the inlet discharge coefficients experimentally. The first approach is mainly used in standards and manuals as well as numerical models. Many studies claimed that the weir/orifice discharge coefficients are not fixed and their values vary depending on many influential parameters such as the Froude number, water depth, and flow velocity [33,34]. Bazin et al. [35] tested the rectangular grate inlets under subcritical flow and reported weir discharge coefficients to be in the range of 0.53–0.67, whereas Martins et al.’s [36] experimental study on rectangular gully with no grating under supercritical flow showed a larger range of weir discharge coefficient of 0.22–1. However, the weir discharge coefficient range was distinct in the study of Rubinato et al. [37] on circular manholes with no lid under subcritical flow and it ranged between 0.493 and 0.587. This study distinguished three types of discharge coefficients, namely, weir, submerged, and orifice discharge coefficients. For circular grate inlets under subcritical flow, the reported weir discharge coefficient was quite lower than the latter study of non-grated circular inlets and it ranged between 0.115 and 0.372 [38]. This is attributed to the absence of the grate which can hinder the inlet interception and increase the water splash-over phenomenon over the grate resulting in lower efficiency and interception capacity for grated inlets. Studies on the orifice discharge coefficients of drain inlets revealed inconsistent margins for each study case. The orifice discharge coefficients reported by Rubinato et al. [37] for one circular manhole with no lid ranged between 0.159 and 0.174 which is quite lower than its counter values which ranged between 0.349 and 2.038 in the study by Rubinato et al. [38] for ten circular manholes with lids. The difference in the grate geometry and orientation in the latter study resulted in a larger range of discharge coefficients. This highlights the importance of considering grating geometry and its orientation for determining inlet discharge capacity. A similar trend of non-consistent orifice discharge coefficients was observed for rectangular inlets too. Even though orifice discharge coefficient margin of rectangular grated inlets was relatively similar to 0.13–0.41, 0.03–0.47, and 0.07–0.39 in the studies by Gómez et al. [39], Cosco et al. [34], and Zambrano et al. [40], respectively, some studies showed a smaller range such as that in the study by Sabtu [41] on two rectangular gullies under intermediate and terminal flow conditions where the orifice discharge coefficients ranged between 0.4 and 0.5. On the other hand, the rectangular inlet with no grate tested by Martins et al. [36] under supercritical flow possessed a quite large orifice discharge coefficient in the range of 1.67–2.68. This is attributed to the absence of the grate and this can increase the orifice effective area, reduce the hydraulic jump of the supercritical flow and increase the inlet discharge capacity and orifice discharge coefficients.

The large differences in the range of weir and orifice discharge coefficients reported above reveal that using a fixed weir/orifice discharge coefficient value would underestimate or overestimate the hydraulic performance of the urban drainage system and consequently, lead to sensible errors that can result in drastic urban drainage failure and urban flooding [34,42]. Due to the severe economic, health, urban, human, and social consequences of urban drainage failure, determining the accurate weir/orifice discharge coefficients is vital for assessing accurate flood hazard maps and ensuring sustainable urban drainage systems. From previous research, there is no uniform pattern or trend to determine the discharge coefficient of each inlet as it may vary significantly with the change in Froude number, inlet geometry, grate geometry, grate orientation, flow velocity, and water depth. Therefore, weir and orifice discharge coefficients have to be calibrated and adjusted experimentally taking into consideration the crucial factors that affect the inlet discharge capacity [43]. Previous research calibrated the experimental values of discharge coefficients and established new equations as a function of the main flow parameters such as water depth [36], the inlet effective perimeter and area [38], Froude number [34], and the approaching flow [40]. Even though the small uncertainty in the weir/orifice discharge coefficient might have a small effect on the interception capacity of a single inlet, this small uncertainty can lead to sensible errors when it is applied on a large scale such as large basin catchments in cities and urban areas which may contain thousands of hydraulic inlets. Underestimating the inlet hydraulic capacities can result in wasting national resources and economic losses, whereas overestimating its hydraulic capacities may lead to failure of the urban drainage system specifically under rainfall events of high intensities or longer durations and consequently, it may cause urban flooding with massive destruction and losses (heritage, social, economic, human, agricultural, and environmental). The severe consequences of urban drainage failure raise concerns among hydraulic engineers on the methods and techniques to design resilient sustainable drainage systems with minimal uncertainties. As a response to this serious problem, the aim of this study is to investigate, assess, and analyze the previous studies and establish correlations between the weir/orifice discharge coefficients and the main hydraulic parameters of drain inlets to provide designers and planners with reliable discharge coefficients to mitigate urban flooding and its severe consequences. Moreover, the outcomes of this study enable stakeholders and authorities to model flood events accurately, reduce the hydraulic efficiency uncertainty, enhance their planning and preparedness, and provide better guidelines/design specifications to aid in the design and fabrication of future street inlets as well toward sustainable drainage systems with minimal disruption to society, economy, and environment.

The necessity for this type of study comes from the fact that to the best knowledge of the author, there are no studies that compared, analyzed, and assessed the weir/orifice discharge coefficients of the recent hydraulic studies on drain inlets to propose the reliable weir/orifice discharge coefficient values that can be adopted based on the inlet parameters and hydraulic conditions to minimize the hydraulic uncertainties and mitigate urban flooding. This study provides a critical review of the most advanced techniques and equations for determining the hydraulic capacities of drain inlets and discusses the strengths, deficiencies, governing boundaries and applications of each method. Moreover, it provides the recommended ranges of weir and orifice discharge coefficients that can be used based on the hydraulic inlet and flow characteristics as well as it provides perspectives for future studies.

Figure 1 shows the flow chart of the steps for conducting this study. Initially, this review provides comprehensive literature on the experimental tests on grate inlets and their hydraulic capacities under different hydraulic conditions. Next it investigates the developed relationships between weir/orifice discharge coefficients and the approaching flow in previous studies and compares their established equations. After that, it compares and analyzes the relationships reported in previous studies on the correlations between the weir/orifice discharge coefficients and the Froude number for rectangular/circular inlets under different hydraulic conditions. Based on that, it provides an advanced range of weir/orifice discharge coefficients based on the inlet and hydraulic conditions of the drainage system (rectangular, circular, grated, non-grated, subcritical, supercritical, terminal flow, intermediate flow). Finally, it sums up the findings of this study and provides researchers with perspectives for future studies on aspects that have not been studied well.

Figure 1

Flow chart of the steps for conducting the study.

2 Discharge coefficients of hydraulic street inlets

The interaction of the linking element such as a manhole, inlet, or gully installed between the surface and subsurface drainage systems can be represented numerically through weir/orifice discharge equations. During rain flow events, the inlet initially intercepts the water as free discharge at low flow rates, and consequently, the flow is considered a weir flow. This weir flow transforms into an orifice flow at higher flow rates when the inlet gets submerged in water. The weir and orifice intercepted flows expressed in equations (1) and (2), respectively, are used in numerical computational fluid dynamics (CFD) models to represent the 1D and 2D flow behavior of the drainage systems [44]. Both discharge equations depend on the upstream water depth revealing the high influence of the inflow depth on the inlet interception efficiency as well as the flow type classification (weir/orifice). While the weir equation considers the inlet effective perimeter (P _e), the orifice equation considers the inlet effective area (A _e). The weir’s effective perimeter depends primarily on the flow pattern around the inlet. Several studies considered the effective weir perimeter as the sum of the inlet width and length since they are responsible for the frontal and side flow interceptions, respectively, specifically under supercritical flow [34,45,46]. In this case, the intercepted discharge contribution is determined from the upstream inlet width and the inlet side length, while the downstream inlet width does not contribute due to the relatively high flow velocity under supercritical flow conditions. However, Guo et al. [47] considered the effective weir length as the sum of the inlet length and twice its width under subcritical flow. Similarly, some studies considered the effective weir perimeter to be the sum of the inlet length and twice its width, especially under low flow velocity such as the case of subcritical flow where the inlet can intercept water at its upstream width, side length, and downstream width [44].

where Q _w is the weir intercepted flow (m³/s), P _e is the effective weir perimeter (m), g is the gravitational acceleration (m/s²), h _s is the surface water head (m), Q _o is the orifice intercepted flow (m³/s), A _e is the effective orifice area (m²), and h _p is the inlet water head (m).

The research studies on weir and orifice discharge coefficients of hydraulic inlets are summarized in Table 1, which includes the findings and developed formulae of each case. For instance, Mustaffa et al. [48] tested a reduced-scale manhole cover with four orifices (13 mm diameter) under subcritical and supercritical flow conditions (0.3 < Fr < 2.2) with a water depth of 6–18 cm high. The findings indicated that there is negligible disruption caused by nearby orifices and the orifice discharge coefficient (C _o) decreases with the increase in Froude number. The value of C _o = 0.616 is reported as an orifice discharge coefficient which is comparable to the 0.6 value adopted in several standards and numerical models. In a separate investigation, Guo et al. [47] examined different types of scaled grate inlets (rectangular bar, vane grate, curb opening, and combination). Initially, the flow started as a weir flow. As the water depth increased, it transformed into a transitional flow and then into an orifice flow. The weir discharge coefficient was C _w = 0.3 for bar and vane grates and C _w = 0.44–0.45 for curb inlets. On the other hand, the rectangular bar, vane, curb, and combination inlets showed a constant orifice discharge coefficient (C _o) of 0.67. However, the transitional flow discharge coefficient was C _t = 0.93 for different inlets, and the interception capacity (Q _p) at this stage was determined as the minimal between the inlet weir discharge (Q _w), transitional discharge (Q _t), and orifice discharge (Q _o) flow. Lee et al. [49] studied two-scaled rectangular grate inlets and classified the flow into weir/orifice flow based on the ratio of the total frontal head (H _t) to the short orifice dimension (b _o). The experimental results showed that for H _t/b _o less than or equal to 0.5, the intercepted flow can be determined from the weir equation. Contrarily, the flow is considered as orifice flow when H _t/b _o is greater than 0.5. According to this study, the weir and orifice discharge coefficients are 0.378 and 0.51, respectively. Therefore, using a constant value of C _o = 0.6 overestimates the inlet discharge capacity of this study. The interaction between the surface and subsurface drainage systems was investigated experimentally and numerically in the study of Djordjević et al. [50] on a full-scale circular gully (375 mm diameter) widely used in the UK. The gully was tested under terminal and intermediate flow conditions. Similar to the previous findings reported by Guo et al. [47], this study states that as the water head and flow rate increase, the flow changes from weir to orifice through an intermediate transitional stage of partial weir and orifice condition. The gully interception and discharge coefficient increased with the flow rate and water depth while having higher discharge coefficients under terminal flow conditions. At terminal inlets, flow is intercepted by the inlet with no by-passing flow, and therefore, terminal inlets are associated with larger water depth, inlet discharge coefficients, and flow discharge compared to intermediate inlets (with by-passing flow) [47,50,51,52].

Bazin et al. [35] tested 20 rectangular inlets (50 × 50 mm²) with 10 inlets on each side of the platform under subcritical flow and pressurized subsurface pipe. The weir and orifice discharge coefficients varied between 0.53–0.67 and 0.39–0.46, respectively. These reported orifice values are at least 23% lower than the 0.6 orifice coefficient adopted in manuals and standards [56,57,58,59], and therefore, using a 0.6 orifice discharge coefficient for these inlets overestimates their discharge capacity. In a more specific study, Martins et al. [36] studied real scale rectangular grate inlet (600 × 300 mm²) under supercritical flow (1.4 < Fr < 1.6) and correlated the weir/orifice discharge coefficients with the upstream water depth in the form of a power function. The range of experimental weir and orifice discharge coefficients were 0.22–1 and 1.67–2.68, respectively. The numerical values determined using OpenFOAM simulation were quite similar to 0.16–0.93 and 1.36–2.5 for weir and orifice discharge coefficients, respectively.

The terminal and intermediate inlets under subcritical flow conditions were investigated again in the study by Sabtu [41] on two full-scale rectangular grated gullies widely used in the UK. The experimental results showed that the gullies exhibited higher discharge coefficients under terminal conditions (sag location); therefore, the interception capacity of terminal gullies was higher than intermediate gullies. In addition, regression equations were developed to determine the correlation between the discharge coefficients and hydraulic parameters. Guo et al. [52] studied the depressed and leveled scaled inclined inlets (highway median inlets) with varied inclination angles between θ = 0° (horizontal grate) and θ = 90° (vertical/side grate). When the height of water was lower than the height of the inclined grate above the grade, the flow was classified as low weir/orifice flow, while the flow was considered as high weir/orifice flow when the water height was higher than the grate height. The corresponding weir/orifice formulae were derived for each flow condition and their discharge coefficients (C _d) were determined accordingly. The reported graphs showed that for θ = 0°, the discharge coefficient reach its maximum in the range of 0.6–0.84 and 0.74–0.96 for depressed and leveled grates, respectively. The discharge coefficient decreased to its minimal for 10° ≤ θ ≤ 15° before it started to increase again for inclination angle θ > 15°. Leveled grates showed higher discharge coefficients and therefore, higher interception capacity than depressed grates. This agrees with the experimental results reported by Wakif and Sabtu [60] where the efficiency and discharge coefficient were reduced by 6–10% when the rectangular grate inlet (1,100 × 500 mm²) had 20 mm vertical depression. The depressed grate experienced lower discharge coefficients and consequently lower efficiency due to the hydraulic jump and water splash over the depressed grate inlet.

Rubinato et al. [37] studied the sewer-to-surface and surface-to-sewer flow of scaled circular manholes with 240 mm diameter and no lid under steady and unsteady flows. Four different flow scenarios were considered which are weir inlet, submerged inlet, orifice with surface flow, and orifice with dry surface (no surface flow). The reported discharge coefficients ranged between 0.493 and 0.587, 0.054 and 0.057, and 0.159 and 0.174 for weir, submerged, and orifice flows, respectively. The numerical model showed a good agreement with the experimental results for steady flow, whereas it overestimated the surcharge flow rate due to the accompanied turbulence of unsteady flow which resulted in larger head losses during sewer-to-surface outflow. Similarly, Beg et al. [53] tested a surcharged real-scale rectangular gully (600 × 240 mm²) connected to a manhole and classified the surcharge conditions into three different categories low, intermediate, and high surcharges based on the surcharge level. A CFD model developed using OpenFOAM was validated with the previous experimental results and it showed that at low surcharge conditions with a plunging jet flow, the orifice discharge coefficient was 0.677, whereas it increased to 0.755 at intermediate surcharge when the gully discharge was submerged jet. Finally, the orifice discharge coefficient reached 0.82 for high surcharge when the gully started to reverse the flow. Overall, the orifice discharge coefficient varied in the range of 0.677–0.82. This reveals that the orifice discharge coefficient can be higher than the classical values of 0.6/0.67 used in manuals and standards that can underestimate the overflow. On the other hand, the orifice discharge coefficients in the experimental investigation of Rubinato et al. [38] on 10 circular scaled grate inlets (240 mm diameter) were quite higher than 0.82 (Figure 2). The weir discharge coefficient was in the range of 0.115–0.372 and the orifice coefficient margin was quite large as it ranged between 0.349 and 2.038. The observed ranges of the discharge coefficients are primarily influenced by the void area ratio, inlet effective perimeter, and inlet orientation. For instance, inlets F and G of the same grate inlet type (same geometry, effective perimeter, and effective area) and different orientations experienced different inlet exchange flows. The exchange flow at inlet G was slightly higher than inlet F for both weir and orifice flow. This is due to the inlet orientation where the openings of inlet G are more parallel to the flow direction than the openings of inlet F which reduces the water splash over the grate bars of inlet G and enhances its interception capacity. Similar results are reported in previous research on the efficiency superiority of inlets with grates parallel to the flow direction compared to inlets with grates perpendicular to the flow direction [8]. Even though grates perpendicular to the flow direction can increase the lateral flow interception, they hinder the frontal flow interception which can account for 60 to 80% of the inlet total interception as reported by Gómez et al. [61]. This highlights the need for additional studies to develop equations that can account for all the effective geometrical and hydraulic parameters to reduce the uncertainties of the current equations and to avoid using constant discharge coefficients that can result in underestimated or overestimated inlet hydraulic behavior. Rubinato et al. [38] formulated power equations for determining the weir and orifice discharge coefficients as a function of the effective weir perimeter (P _w) and the effective orifice void area (A _o), respectively. Moreover, this study showed the validity of the weir/orifice linking equations to mimic the surface-to-sewer drainage behavior.

Figure 2

Ten grate inlets with the black arrow indicating the flow direction [38].

Gómez et al. [39] tested experimentally one rectangular full-scale inlet under a longitudinal slope of 0–10% and surcharge flow condition. Results showed that the influence of the longitudinal slope on the orifice discharge capacity was minimal and the orifice discharge coefficient was between 0.13 and 0.41. Therefore, using a 0.6 orifice coefficient determined from tank emptying will overestimate the overflow of this dual drainage system by 46–360%. These results are limited to the case of no surface flow and additional studies are recommended to investigate the influence of the upstream flow on the inlet overflow under surcharging conditions. In a recent study, Cosco et al. [34] conducted a full-scale experimental and numerical study to consider the supercritical flow hydrodynamic conditions specifically the flow depth and flow velocity while determining the inlet discharge coefficient. Three rectangular grate inlets Barcelona1, E-25, and Meridiana were tested under supercritical flow with Froude number between 1 and 14 and a wide range of flow of 25–200 l/s. The reported weir and orifice discharge coefficients were in the range of 0.004–0.66 and 0.03–0.47, respectively which are quite lower than the commonly used values in literature. Recently, Cárdenas-Quintero and Carvajal-Serna [46] reported a similar power relationship between the weir discharge coefficient and the Froude number for three rectangular grate inlets under supercritical flow. Similarly, Zhuk et al. [55] developed a power relationship between the weir discharge coefficients and the Froude number of two square grate inlets under subcritical flow. The developed correlations are discussed in detail in Sections 2.1 and 2.2.

In a recent study on inlet clogging, Hao et al. [54] tested clean and clogged scaled rectangular inlets (250 × 150 mm²) under different clogging conditions and positions (half and quarter clogging conditions of parallel and perpendicular positions). Intuitively, the clogged inlets experienced lower discharge coefficients than clean inlets. The weir discharge coefficient of clogged inlets was 3.9–11.76% lower than the clean inlets of 0.051. This difference was more distinct in the case of orifice flow where the orifice clogged coefficient was 27–45% lower than the clean orifice coefficient of 1.262. Even though the influence of the clogging condition dominated the clogging pattern influence, both showed significant influence on the clogged discharge coefficients specifically under the overflow condition.

Tellez-Alvarez et al. [42] used experimental and numerical approaches (Flow 3D) to determine the orifice discharge coefficients for three full-scale rectangular grate inlets under surcharging conditions. The experimental findings showed high agreement with the numerical results. The range of the experimental and numerical orifice discharge coefficients was 0.14–0.41 and 0.15–0.46, respectively. The reported orifice discharge coefficients were quite far from the 0.6 recommended by practitioners. The orifice discharge coefficient changed depending on the inlet type and surcharge rate. For instance, the orifice discharge coefficient increased when the surcharge rate increased whereas, the effect of the longitudinal slope on the orifice coefficient was negligible. Similarly, Zambrano et al. [40] used the Flow 3D CFD model to study the discharge coefficients of seven rectangular full-scale inlets widely used in Tunja, Colombia. Results showed that the orifice discharge coefficients were in the range of 0.07–0.39 which is quite similar to the reported values of Tellez-Alvarez et al. [42]. Xia et al. [43] studied a scaled rectangular inlet (16S518) under subcritical flow and proposed a novel approach to determine the inlet discharge capacity as a function of the Froude number and the upstream flow velocity. However, the flow was categorized into weir/orifice flow based on the water depth. Hence, the flow was considered as a weir flow for flow height (h) below 45 mm with a weir discharge coefficient of 0.44. On the other side, the flow was classified as an orifice flow with an orifice discharge coefficient of 0.54 when the water depth exceeded 64 mm. The flow regime between 45and 64 mm water depth exhibited a complex transitional phase involving weir and orifice flow, which requires further investigation to accurately determine the inlet discharge capacity during this stage.

2.1 Relationship between discharge coefficients and approaching flow

In the experimental and numerical investigation of Martins et al. [36], a full-scale rectangular gully (0.6 × 0.3 m²) with no grating was tested under a supercritical flow rate of 5–60 l/s (1.4 < Fr < 1.6). A power function was developed for determining the weir and orifice discharge coefficients from the upstream flow depth (h _in) as shown in Table 1. The findings showed high agreement between the experimental and numerical (OpenFOAM) results, and therefore, proved the validity of the developed numerical model to simulate the hydraulic behavior of urban drainage systems under similar hydraulic conditions. Similarly, Tellez-Alvarez et al. [42] investigated the orifice discharge coefficients of full-scale rectangular inlets under surcharging conditions (two inlets were tested experimentally and three inlets were modeled numerically). The experimentally tested inlets Barcelona-1 (diagonal grate) and Barcelona-2 (fishbone grate) had the same dimensions of 0.745 × 0.26 m² and similar effective void areas of 0.0852 and 0.0857 m², respectively (Figure 3).

Figure 3

Inlets Barcelona-1 to the left and Barcelona-2 to the right [62].

The weir and orifice discharge coefficients in the study conducted by Martins et al. [36] on non-grated gully, as well as the orifice discharge coefficients obtained under surcharge conditions by Tellez-Alvarez et al. [42], are plotted as a function of the approaching flow as shown in Figure 4. The weir and orifice discharge coefficients of the first study are denoted by the letters “W” and “O,” respectively. Notably, the numerical labels “1” and “2” of the latter study correspond to the distinct inlet configurations referred to as Barcelona-1 and Barcelona-2, respectively. Table 2 enlists the regression equations that describe the functional relationship between the weir (C _w) and orifice (C _o) discharge coefficients as a function of the upstream approaching flow (Q _a). These regression equations can be used for predicting discharge coefficients and optimizing design parameters in practical applications, thereby facilitating efficient and effective engineering solutions. Graphically, the non-grated gully of Martins et al. [36] experienced a smooth descending weir discharge coefficient curve from 0.99 at 10 l/s to 0.22 at 60 l/s (Figure 4). This is reflected in Table 2 by the high adequacy of the second degree polynomial to represent C _w as a function of Q _a with R ² = 0.97. On the other hand, the orifice discharge coefficient increased sharply from 1.67 at Q _a = 5 l/s to 2.68 at Q _a = 10 l/s. Thereafter, it declined smoothly to its minimum of 1.81 at Q _a = 60 l/s. The second degree polynomial showed a weak correlation between C _o and Q _a with R ² = 0.25 between 5 and 60 l/s, whereas it showed a high correlation of R ² = 0.99 between 10 and 60 l/s. The weir and orifice discharge coefficients can have a positive correlation with the approaching flow until the inlet attains its maximum discharge capacity where discharge coefficients drop afterwards. Therefore, the discontinuity of the orifice discharge coefficient curve between 5 and 10 l/s is due to the inlet reaching its maximum surcharge efficiency at a flow rate of 10 l/s and consequently, attaining its maximum orifice discharge coefficient of 2.68. Several studies reported similar findings where the inlet efficiency increases with the inflow depth until a certain threshold and drops thereafter [36,63]. Additional research is required on this transitional stage between the ascending and descending patterns of the discharge coefficient curves. Overall, these studies show that to establish a good correlation for the weir/orifice discharge coefficients with their approaching flow, their curve equation have to be divided into two equations (pre-maximum and post-maximum discharge coefficient equations).

Figure 4

Weir/orifice discharge coefficients (C _w/C _o) vs approaching flow (Q _a).

Table 2

Relationship between discharge coefficients and the approaching flow (Q _a)

Reference	Equation	Weir/orifice equation	R 2
Martins et al. [36]	Gully (0.6 × 0.3 m2)	C w = 0.0004 Q a 2 − 0.0378 Q a + 1.2307	0.97
		C o = − 0.0003 Q a 2 + 0.0121 Q a + 2.0591	0.25
Tellez-Alvarez et al. [42]	Barcelona-1 (0.745 × 0.26 m2)	C o = 0.0068 Q a + 0.08	0.99
	Barcelona-2 (0.745 × 0.26 m2)	C o = 0.0059 Q a + 0.121	0.99

The effective void area of the non-grated gully studied by Martins et al. [36] is 0.18 m² which is 2.1 times higher than inlets Barcelona-1 and Barcelona-2 and its unit flow of 10–120 l/s is 3–7.2 times higher than its counterpart of inlets Barcelona-1 and Bracelona-2 of 3.3–16.6 l/s. Therefore, the orifice discharge coefficient graph of the non-grated gully increased with the approaching flow to its maximum and started its descending pattern afterwards, whereas the orifice discharge coefficients of inlets Baercelona-1 and 2 increased gradually as the approaching flow increased and did not reach their peak. They may reach their peak and start to drop if the upstream flow is increased over 16.6 l/s unit flow rate. The orifice discharge coefficients of inlet Barcelona-2 are 6.9–28.6% higher than inlet Barceona-1, and therefore, resulted in higher interception capacity and efficiency. This difference is attributed to the fact that inlets with fishbone grates (case of Barcelona-1) acquire higher efficiency than inlets with diagonal grates (case of Barcelona-2) [64]. The orifice discharge coefficients of inlets Barcelona-1 and 2 are expected to reach the peak at higher flow rates and start dropping beyond that threshold.

Cosco et al. [34] investigated experimentally the weir/orifice discharge coefficients of grate inlet Barcelona-1 (0.745 × 0.26 m²) under different longitudinal (S _L: 0–10%) and transversal (S _T: 0–4%) slopes (Figure 5). The reported results of the weir/orifice discharge coefficient vs the approaching flow are plotted in Figure 6. The weir and orifice discharge coefficients ranged between 0.022–0.481 and 0.071–0.428, respectively. At the same inflow of 100 l/s, C _o varied between 0.087 (S _L = 8% and S _T = 2%) and 0.428 (S _L = 1% and S _T = 4%). This difference in the orifice discharge coefficient (C _o) is attributed to the larger interception efficiency associated with higher transversal slopes (larger water flow conveyed to the gutter and inlet) and lower longitudinal slopes (lower flow velocity and consequently, lower splash-over grate bars phenomenon) [65–69]. The longitudinal slope effect on the inlet discharge coefficients is minimal [39] and consequently, the large range of the orifice discharge coefficients (C _o) at Q _a = 100 l/s is mainly influenced by the transversal slope. It is noticeable that the range of the orifice discharge coefficient values in Figure 6 is quite far from the 0.6 value recommended in manuals such as Australia [59], WVDOH [57], QUDM [58], and Neenah [56] or the 0.67 value suggested by other manuals such as by Brown et al. [70], Dakota [71], DID [72], USDCM [73], and Texas [74]. These fixed orifice discharge coefficients (0.6 or 0.67) are determined from the case of tank emptying where the flow has no velocity [34]. The case of tank emptying differs from the case of surface runoff drainage where the flow velocity parameter has to be considered while determining the efficiency or discharge coefficients of hydraulic inlets specifically under supercritical flow. The values of the weir/orifice discharge coefficients are influenced by the inlet geometry (shape and dimensions of the inlet and its grate) and the approaching flow characteristics (flow depth and Froude number). Several studies in the last two decades correlated the equations of the efficiency and discharge coefficients to significant flow parameters such as the flow depth, Froude number, and the flow velocity to increase the accuracy of determining the hydraulic characteristics of urban drainage systems. For instance, under supercritical flow conditions dominated by the Froude number (water depth and flow velocity), Martins et al. [36] and Cosco et al. [34] formulated power functions to express the weir/orifice discharge coefficients as a function of water depth and Froude number, respectively.

Figure 5

Grate inlet Barcelona-1 [34].

Figure 6

Weir and orifice discharge coefficients of inlet Barcelona-1 studied by Cosco et al. [34].

The range of weir (C _w) and orifice (C _o) discharge coefficients with the approaching flow (Q _a) reported in previous studies are plotted in Figures 7 and 8, respectively. The weir discharge coefficient under supercritical flow was investigated in the study of Martins et al. [36] and Cosco et al. [34] for rectangular inlets without and with grates, respectively. Figure 7 shows that the weir discharge coefficient was in the range of 0.22–1 and 0.004–0.66 for inlets without and with grates, respectively. Therefore, the range of 0.22 < C _w < 0.66 can be considered for grated and non-grated rectangular inlets under supercritical flow. On the other hand, the range of weir discharge coefficients under subcritical flow was in the range of 0.53 < C _w < 0.67 for non-grated rectangular inlets and in the range of 0.4–0.6 for grated rectangular inlets as reported by Bazin et al. [35] and Sabtu [41], respectively. Based on that, a weir discharge coefficient in the range of 0.53 < C _w < 0.6 can be adopted for the cases of grated and non-grated rectangular inlets under subcritical flow. The larger margin of the weir coefficient (0.4 < C _w < 0.6) under supercritical flow compared to subcritical flow (0.53 < C _w < 0.67) with low flow velocity is attributed to the larger influence of flow velocity on the hydraulic behavior of drainage systems under supercritical flow which can alter the inlet efficiency and discharge rates significantly. On the other hand, the flow interception shows a more stable behavior under subcritical flow characterized by higher water depths and lower flow velocities.

Figure 7

Relationship of the suggested range of weir discharge coefficients vs approaching flow.

Figure 8

Relationship of the suggested range of orifice discharge coefficients vs approaching flow.

Based on the study of Martins et al. [36] and Cosco et al. [34], it is recommended to use an orifice discharge coefficient of 1.67–2.68 for rectangular non-grated inlets and 0.03–0.47 for rectangular grated inlets under supercritical flow condition. Therefore, the orifice discharge coefficient of the non-grated rectangular inlet is found to be over 3.5 folds higher than that of the grated rectangular inlet. The orifice discharge coefficient superiority for inlets with no grate is due to the hydraulic jump in the case of surcharge grated inlets when the water splashes on the grated bars and consequently, larger energy loss and lower inlet surcharge rate. This hydraulic jump head loss is more pronounced in the case of supercritical flow characterized by larger flow velocity. For rectangular grated inlets under subcritical flow, Bazin et al. [35] reported that the orifice discharge coefficient (C _o) is in the range of 0.39–0.46, whereas Sabtu [41] reported that the C _o varies between 0.4 and 0.5 (Figure 8). Consequently, an orifice discharge coefficient in the range of 0.4–0.46 is recommended for rectangular grated inlets under subcritical flow. In the full-scale studies conducted by Gómez et al. [39], Tellez-Alvarez et al. [42], and Zambrano et al. [40], rectangular grate inlets were tested under surcharge conditions without mentioning the flow type (whether subcritical or supercritical). Beg et al. [53] carried out a similar full-scale study on rectangular inlets but with no lid cover. The orifice discharge coefficients of the studies of Beg et al. [53], Gómez et al. [39], Tellez-Alvarez et al. [42], and Zambrano et al. [40] were in the range of 0.677–0.82, 0.13–0.41, 0.14–0.41, and 0.07–0.39, respectively. Therefore, when the Froude number is unknown, the suggested orifice discharge coefficient is in the range of 0.14–0.39 and 0.677–0.82 for rectangular inlets with and without grates, respectively.

Finally, Rubinato et al. [37] tested a non-grated circular inlet under subcritical flow with a Froude number of 0.151–0.691. Similarly, Rubinato et al. [38] tested ten circular grated inlets under subcritical flow with a Froude number of 0.495–0.612. Figure 7 shows that under similar hydraulic conditions, it is recommended to use a weir discharge coefficient (C _w) in the range of 0.493–0.587 and 0.115–0.372 for circular inlets without and with grate, respectively. Similarly, an orifice discharge coefficient (C _o) in the range of 0.159–0.174 and 0.349–2.038 can be adopted for circular inlets without and with grate, respectively as shown in Figure 8.

2.2 Relationship between discharge coefficients and Froude number

Cosco et al. [34] tested three rectangular grated inlets (Barcelona-1, E-25, and Meridiana) under supercritical flow and formulated power functions to correlate the weir/orifice discharge coefficients (C _w/C _o) with the Froude number (Fr) as shown in Table 3. The graphs of the experimental weir and orifice discharge coefficients vs the Froude number (Fr) are plotted using the formulae reported by Martins et al. [36], Cosco et al. [34], Cárdenas-Quintero and Carvajal-Serna [46], and Zhuk et al. [55] as shown in Figure 9. The experimental and numerical orifice discharge coefficients of Martins et al. [36] exhibited high agreement showing the validity of numerical software (OpenFOAM) in simulating the hydraulic behavior of dual drainage systems accurately. Therefore, only the experimental results of the latter study are plotted in Figure 9. Graphically, the weir and orifice discharge coefficients of the three inlets investigated by Cosco et al. [34] showed a negative correlation with the Froude number (Fr). This agrees with the reported results of studies which show that weir and orifice discharge coefficients decrease at higher Froude number [48,75]. For instance, Figure 9 shows that at high flow velocity with Froude number Fr = 14, the weir and orifice discharge coefficients of inlet Barcelona-1 were as low as 0.004 and 0.03, respectively. As the Froude number increases, the flow is characterized by higher flow velocity, and therefore, the flow overpasses the inlet and splashes over its grate bars, and consequently, larger head loss, lower inlet discharge, lower efficiency, and lower discharge coefficients were reported. Moreover, the water films developed on the inlet surface are more intense under larger flow velocities, and therefore, lower discharge capacity. The water film over the inlet is reported to reduce the inlet efficiency significantly especially for inlets with short lengths where the water film may cover over 50% of the inlet length reducing its frontal and lateral interceptions as reported by Kim et al. [65]. In the study by Cosco et al. [34], the weir and orifice discharge coefficients showed a significant decline between Fr = 1 and Fr = 2, and thereafter, they decreased gradually with the Froude number increase. Similarly, the weir discharge coefficient of the non-grated inlet (0.6 × 0.3 m²) tested by Martins et al. [36] dropped significantly from C _w = 0.99 at Fr = 1.42 to C _w = 0.22 at Fr = 1.57. On the other hand, its orifice discharge coefficient increased sharply from 1.67 to 2.68 when the Froude number increased from 1.42 to 1.46 and dropped significantly beyond Fr = 1.46. This behavior can be due to the efficiency increasing to reach its optimum at Fr = 1.46 before it starts to drop with the Froude number increase.

Table 3

Power correlation between the discharge coefficients and Froude number

Reference	Grate	L × W (m2)	C w	C o
Cosco et al. [34]	Barcelona-1	0.745 × 0.26	C w = 0.6593 F r ‒ 1.899	C o = 0.4239 F r ‒ 1.006
	E-25	0.64 × 0.3	C w = 0.505 F r ‒ 1.692	C o = 0.4737 F r ‒ 0.826
	Meridiana	0.8 × 0.3	C w = 0.4237 F r ‒ 1.794	C o = 0.418 F r ‒ 0.906
Cárdenas-Quintero and Carvajal-Serna [46]	Dalian 4	0.549 × 0.338	C w = 0.4857 F r ‒ 0.892	NA
	Dalian 5	0.461 × 0.21	C w = 0.5612 F r ‒ 0.9258	NA
	Dalian 6	0.435 × 0.238	C w = 0.5102 F r ‒ 0.8948	NA
Zhuk et al. [55]	Inlet 1	0.245 × 0.245	C w = 1.34 F r 0.67
	Inlet 2	0.325 × 0.325	C w = 1.54 F r 0.821

Figure 9

Discharge coefficient vs Froude number for (a) weir flow and (b) orifice flow.

The grate inlet E-25 studied by Cosco et al. [34] and the non-grated gully of Martins et al. [36] have similar inlet dimensions of 0.64 × 0.3 m² and 0.6 × 0.3 m², respectively. However, the void area of the non-grated gully is 0.18 m² which is 2.2 times higher than the effective void area of E-25 (0.0808 m²). Initially, the weir discharge coefficient of the non-grated gully was quite higher than E-25 until Fr = 1.55 where it dropped below the E-25 curve as shown in Figure 9(a). This behavior change may be attributed to the larger perimeter involved in the flow interception at higher Froude numbers and consequently, higher weir discharge coefficient. For instance, under weir flow, the inlet perimeter dominates and the effective perimeter of inlet E-25 is slightly higher than the non-grated gully of Martins et al. [36]. The larger effective perimeter can enhance the weir interception capacity of E-25 at higher Froude number flow with larger velocity where a larger portion of the grate surface/perimeter increases the frontal/lateral flow interception. Additionally, larger flow speed and Froude number are attributed to larger water film formed on the inlet and consequently, inlets with larger lengths surpass the performance of inlets with shorter lengths. On the other hand, the orifice effective area dominates under orifice discharge. Therefore, the orifice discharge coefficient of the non-grated gully of Martins et al. [36] was higher than its counterpart in inlet E-25 as shown in Figure 9(b). The inlet effective perimeter dominates for weir flow of free discharge inlets, whereas the inlet effective area governs the orifice flow of submerged inlets.

In the study by Sabtu [41], two full-scale gully inlets (gully A: 0.325 × 0.437 m² and gully B: 0.4 × 0.432 m²) were tested experimentally under intermediate and terminal flow conditions. For the intermediate inlet, a part of the approaching flow (Q _a) is captured/intercepted by the inlet (Q _in) and the other part continues to the next inlet (Q _b). Therefore, the approaching flow is equal to the sum of the intercepted flow, bypassed flow, and system loss (Q _a = Q _i + Q _b + Q _{system loss}). On the other hand, all the approaching flow is captured by the inlet and will not continue to the next inlet (sag inlets as an example) under terminal inlet conditions. The approaching flow is equal to the sum of the intercepted flow (Q _i) and the system loss flow (Q _ap = Q _i + Q _{system loss}). The findings of this study show that the discharge coefficients of the intermediate gully were lower than the terminal gully. This difference is due to the larger flow velocity and splash-over phenomenon on the grate bars that are more pronounced in the case of intermediate gully. This study formulated equations to correlate the discharge coefficient with the gully length, gully width, flow velocity, water depth, and Froude number. A second degree polynomial was used to represent the correlation between the discharge coefficient (C _d) and the Froude number (Fr) for each gully under different flow conditions as shown in Table 4.

Table 4

Equations of discharge coefficient (C _d) vs Fr for gullies A and B [41]

Flow condition	Gully	Equation
Terminal	A (0.325 × 0.437 m2)	Gully A: C d = −1.7282 Fr2 + 2.1827 Fr + 0.0149
	B (0.4 × 0.432 m2)	Gully B: C d = −0.5252 Fr2 + 1.4319 Fr + 0.1997
Intermediate	A (0.325 × 0.437 m2)	Gully A: C d = −11.087 Fr2 + 6.1246 Fr – 0.4554
	B (0.4 × 0.432 m2)	Gully B: C d = 3.616 Fr2 + 0.1917 Fr + 0.1413

3 Conclusion and perspectives

This review sums up the main methodologies and equations of determining the inlet weir and orifice discharge coefficients and analyzes the values and correlations reported in recent studies to reduce the weir/orifice discharge uncertainty and find the accurate weir/orifice discharge coefficients that can be selected based on the inlet parameters and flow conditions. The result analysis and comparison led to the following conclusions:

• The curve of discharge coefficient vs approaching flow vs Froude number shows a bell-shaped trend where the discharge coefficients increase with the approaching flow/Froude number increase to reach a maximum and decline afterwards. As the approaching flow increases, the flow passes from weir flow free discharge to partial-weir-partial-orifice (transitional stage) when the inlet is partially submerged and into orifice flow when it is fully submerged in water. Similarly, as the Froude number increases, the flow changes from a subcritical flow (slow flow) into a supercritical flow (rapid shallow flow) when the flow velocity and depth exceed the critical values of the transitional stage where the flow reaches its splash-over velocity threshold and the inlet becomes more prone to the influence of the formed water film on its grate, and therefore, its efficiency and discharge coefficients drop afterwards.

• The effect of the approaching flow on the inlet hydraulic characteristics is more influential under subcritical flow, whereas the Froude number effect is more pronounced under supercritical flow. Therefore, the discharge coefficient correlation with the approaching flow is suggested under subcritical flow. On the other hand, the discharge coefficient correlation with the Froude number is recommended under supercritical flow characterized by larger flow velocity with a more pronounced flow splash-over phenomenon and reduced inlet interception capacity.

• For rectangular inlets under subcritical flow, the weirs and orifices discharge coefficients are between 0.53 and 0.6 (for grated and non-grated inlets), and between 0.4 and 0.46 (for grated inlets), respectively. For rectangular inlets under supercritical flow, the weir discharge coefficient of grated and non-grated inlets is in the range of 0.22–0.66. For rectangular inlets under supercritical flow, the orifice discharge coefficient of grated and non-grated inlets is in the range of 0.03–0.47 and 1.67–2.68, respectively.

• For flow with an unknown Froude number, the orifice discharge coefficient of grated and non-grated inlets is in the range of 0.14–0.39 and 0.677–0.82, respectively.

• For circular grated inlets under subcritical flow, the weir and orifice discharge coefficients are in the range of 0.115–0.372 and 0.349–2.038, respectively. Whereas, the weir and orifice discharge coefficients of non-grated circular inlets under subcritical flow are in the range of 0.493–0.587 and 0.159–0.174, respectively.

• Additional studies on the hydraulic limits and parameters governing the transitional stage from weir to orifice flow and from subcritical to supercritical flow are essential for understanding the hydraulic interaction between the surface and subsurface drainage systems to improve inlet design guidelines and mitigate urban flooding hazards.

• Additional research is recommended on the partially-weir-partially-orifice transitional stage and its characteristics (critical depth/critical flow velocity) to determine the equations of the ascending and descending curves of the discharge coefficient vs approaching flow and Froude number.

• There is a need for studies on orifice discharge coefficient equations and values for non-grated inlets under subcritical flow. Similarly, further research is needed on the weir and orifice discharge coefficients of circular grated and non-grated inlets under supercritical flow conditions.

• Additional research is required on the hydraulic interaction of rectangular and circular inlets and the effect of the inlet geometrical shape on its hydraulic behavior as well as the influence of the topographic location of the drain inlet (centered, curb side) on its hydraulic characteristics.