Steady Flow through Gated Circular Culverts: Hydraulic Operation and Experiments

Inlet control flows often are characterized by the presence of supercritical flow within the culvert. In gated culverts, the main features affecting inlet control flow operations are: (1) inlet flow level, (2) sluice gate opening, (3) geometrical characteristics of the culvert determining the flow area, and (4) inlet elements for flow contraction used to accelerate the flow from the approach channel to the tube flow within the circular culvert, such as walls and/or wings (Normann et al. 1985; Schall et al. 2012). The flow at the inlet control section can be free or submerged. For inlet flow levels implying no culvert-entrance submergence, i.e., $h_u\le 1.2\min (D,W)$ (Chaudhry 2008), the flow entrance works similarly to a weir crest flow, with critical flow occurring just downstream from the tube entrance and gate. Flow from the inlet gate to the critical depth section involves high streamline curvature effects, a feature not accounted for in the present flow profile classification but occurring only in a short portion of the tube, close to the gate. When the gate is fully open ($W\ge D$) and the inlet flow levels involve culvert-entrance submergence, i.e., $1.2D<h_u\le 1.5D$ (Chaudhry 2008), hydraulically the entrance works as an orifice. These theoretical limits apply for dominating inlet head-losses and, thus, with negligible friction head losses within the culvert, which normally is the scenario in hydraulically short pipes. For hydraulically long culverts, the wall-dominated friction head losses within the culvert are significant, and, thus, the flow cases become outlet control. Generally, conditions corresponding to $h_u>1.5D$ imply pressurized flow from the inlet section (Chaudhry 2008), so these conditions fall into outlet control.

However, to be precise, under flow submergence conditions and partial gate opening ($W<D$), the flow is governed by the hydraulics of a more complex orifice shape formed by the combined shape of the pipe inlet and the gate (Fan 1985), hereon denoted flow under a sluice gate for practical purposes. The aforementioned theoretical limits for flow classification criteria appear as being adjusted below, according to the experiments in this work (see details in Appendix I). The subsequent steady flow states, which depend on the inlet and outlet submergences, are analyzed in the following subsections. Approach velocity effects were not considered in the ensuing discussion. In the plots, contraction effects past the sluice gate are not represented graphically.

For outlet control, the steady flow may involve a pressurized flow in combination with subcritical free-surface flow profiles, thus without a hydraulic jump formation. In gated culverts, the main features influencing the outlet control flow are: (1) the cross-sectional culvert area, (2) the shape of the culvert cross section, (3) the length of the culvert, (4) the flow resistance in the culvert, and (5) the flow depths at both culvert invert ends (Normann et al. 1985; Chaudhry 2008; Schall et al. 2012). For cases not implying submergence of the culvert outlet, the flow depth at the control section is $h_d$. If the tailwater level $h_d$ is below $h_c$, there is a critical depth control located upstream within the culvert. For $D>h_d>h_c$, the flow conditions are subcritical (tranquil) along the culvert, and there is no occurrence of the critical depth in any section (Schall et al. 2012; Chin 2013). The sluice gate action, as depicted for the inlet control flow states, affects the submergence conditions at the inlet, and, therefore, the hydraulic flow solution along the culvert. In the following subsections, outlet control steady flow cases are arranged in five major groups, in which the theoretical limits for the boundary variables have been adjusted based on experimental results when pertinent (Appendix I).

A laboratory investigation was conducted at the hydraulic facilities of the Institut Agro, Montpellier (France). A gated circular culvert made of polymethyl methacrylate (PMMA), i.e., transparent acrylic, for both the pipe and gate was installed in a prismatic, tilting flume 7 m long, 0.29 m wide, and 0.5 m high (Figs. 13 and 14). The circular culvert was 0.15 m in diameter and 2 m long. This is about half the size of typical outlet structures in rice irrigation schemes in southern Spain. The sluice gate consisted of a $0.2\text{-}\mathrm{m}/\mathrm{side}$ square PMMA plate fixed with clamps to the wall of the flume and operated manually. The pipe was installed 3 m downstream from the flume inlet, leaving a distance of 2 m from the pipe end to the flume tailgate. The experimental setup included eight Baumer U500.DA0-AA1B.720 ultrasonic sensors (Frauenfeld, Switzerland), operated at a frequency of 30 Hz, to measure flow levels at four points upstream from the entrance of the pipe and four points downstream from the exit. The sensors were installed using a rack that maintained an interval distance of 0.16 m between each; the sensors closest to the upstream and downstream pipe ends were 0.04 m from the inlet and outlet culvert sections (the minimum distance required according to the technical specifications). The discharge was measured using an electromagnetic flowmeter (Krohne WATERFLUX 3100 C; discharge range $0-70\mathrm{l}/\mathrm{s}$, accuracy 99.7%, Duisburg, Germany) installed in the flume recirculation pipe. Pressure and water level inside the culvert pipe were measured using 15 piezometers installed equidistant on the bottom edge of the culvert pipe along the first 1.2 m of the inlet. The piezometers were read manually with an approximate uncertainty of $\pm 1\mathrm{mm}$ in the presence of hydraulic jumps when the fluctuating pressure was averaged; otherwise the estimated uncertainty diminished to approximately $\pm 0.5\mathrm{mm}$, i.e., for steady flow cases without a hydraulic jump.

The experimental setup permitted five groups of conditions to be established, namely I3 for inlet control, and O2, O3, O4, and O5 for outlet control. Other groups, most of them associated with steep slope conditions, could not be generated in the flume because the inlet and outlet walls of the culvert were fixed perpendicularly to the flume direction (Fig. 14), so that the high slopes would not resemble real conditions in which the inlet and outlet walls are vertical. Photographs of the laboratory observation of the control groups explored are provided in Fig. 15. A total of 230 tests were conducted: 190 with $S=0.000167$, 22 with $S=0.001$, and 18 with $S=0.00436$. The case of $S=0.000167$ corresponded to the residual slope of the flume at its horizontal position. In terms of the reproducibility of the tests for scientific purposes, we have maintained this value in the manuscript rather than horizontal ($S=0$) given that the $h_n$ level is sensitive to $S$ and is relevant to the classification of steady flow solutions. Each test lasted at least 10 min, the time to reach steady flow conditions. All Reynolds numbers were in the interval ${10}^6>\mathsf{R}>{10}^4$. The data set is given in Appendix II.

A general relationship between the discharge $Q$ and the flow and geometry variables for flow through gated circular culverts can be written as follows:where $n$ = Manning roughness coefficient; $\mu$ = dynamic viscosity; $\rho$ = flow density; and $g$ = gravitational acceleration. For the analysis of our data, constant variables during experimentation were dropped in Eq. (1), resulting in

The relevant variables in Eq. (2) are $h_u$, $W$, and $S$ for inlet control, and $h_u-h_d$, $W$, and $S$ for outlet control. $W$ is an operational variable, whereas $S$ is a design variable, and therefore both are fixed. Then discharge can be represented versus $h_u$ for inlet control data analysis and versus $h_u-h_d+SL$ for outlet control data analysis, while keeping $W$ and $S$ constant. The datum reference level for outlet control is set at the culvert exit, so the upstream water level is $h_u+SL$. Rating equations were derived from Eq. (2) for inlet and outlet control conditions.

Figs. 16(a, c, and e) show the inlet-control data in $Q$ versus $h_u$ charts for $S=0.000167$, 0.001, and 0.00436, respectively. Similarly, Figs. 16(b, d, and f) show the outlet-control data in $Q$ versus $h_u-h_d+SL$ charts for the same slopes. Data not affected by the gate ($W\ge D$) also are plotted in Fig. 16(b). The data in the charts are arranged by groups with similar $W$. Discharge data with similar $W$ depicts trends followingwhere $\alpha$ and $\beta$ = scaling and exponent parameters, respectively, depending on $W$ and $S$; and $H$ = head given by $h_u$ for inlet control and $h_u-h_d+SL$ for outlet control. In each case, rating parameters include all effects, whether or not friction head losses are significant, considering that both local and friction head losses are related to kinetic energy. However, hydraulically long culverts may have very different flow profiles, e.g., due to the presence (or absence) and location of a hydraulic jump within the pipe, and the variability of the wetted area within the pipe. Because the evaluation of all head losses in the tests was beyond the scope of this study, specific rating curves for hydraulically long tests were not adjusted here, but are included in the data set for outlet control. The discharge rating analysis conducted is shown in Figs. 16 and 17 for the inlet and outlet control groups, respectively, using Eq. (3). The $\alpha$ and $\beta$ values in Figs. 16 and 17 were determined by the best power fit of the data as grouped by narrow ranges of $W$. The average $W$ of each range is labeled in Figs. 16 and 17 over each corresponding rating curve. As expected for the operation of an orifice-controlled flow, $\beta$ was in the range 0–1, with typical values around 0.5 for inlet control [Figs. 16(a, c, and e)]. Thus the latter applies to hydraulically short culverts. In addition, $Q$ data were found to be directly proportional to the gate opening, i.e., large $W$ produced flows that exceeded those of small $W$, with an evident direct dependency of $\alpha$ and $\beta$ on $W$ for each $S$ values. This arrangement was clear for inlet control, for which all data points corresponded to a single flow state, I3, with $\alpha =0.054(W/D{)}^2+0.057W/D-0.0028$ with $R^2=0.99$, and $\beta =-3.126(W/D{)}^2+3.459W/D-0.034$ with $R^2=0.99$, for $S=0.000167$. However, the outlet-control data points corresponded to four different flow states (O2, O3, O4, and O5), so that the different $W$ groups overlapped, especially for $W/D>0.3$. These flow cases describe situations for which the gate may exert little control on the flow, with the outlet and culvert friction head-losses becoming significant, as in hydraulically long culverts. Only if the frictional head-losses within the culvert become significant, should an addend appear on the right-hand side of Eq. (3) (Zeng et al. 2020). Fig. 17 shows the data points in Fig. 16(b) as separated by outlet control states, as well as their rating analysis using Eq. (3). The arrangement of data points by $W$ then becomes evident in the domain $(h_u-h_d+SL)\ge 0.09\mathrm{m}$, whereas overlapping remains in the region near the origin, namely when $(h_u-h_d+SL)<0.09\mathrm{m}$. In Fig. 17, data with no gate action for O4 and O5 groups are not analyzed by means of Eq. (3) because they show overlapping with gate action data, and so do the resulting rating curves. This rating analysis was not intended to provide a modeling tool, but just to discuss the results and arrangement of data.

Using Eq. (3), flow transition regions in the discharge rating, i.e., between data pertaining to different groups of conditions, can be discussed. Exploring all 41 steady flow cases in Figs. 2–11, the main flow transition regions identified between groups were: (1) I1 to I3, (2) I2 to I4, (3) O1 to O2, and (4) O5 to I3. Other likely transitions between flow cases or groups are not accounted for here. Due to the lack of data for groups I1, I2, I4, and O1 in this work, flow transition regions in Transition regions 1–3 could not be analyzed. However, these transitions can be analyzed theoretically. For example, as $h_u$ increases in Transition region 3, the flow states evolve describing a reduction of the length of the likely developed $M_1$-type curve in the O1 group leading to fully pressurize culvert in the O2 group. However, the occurrence of Transition region 4 was confirmed by conducting an experiment in which the discharge was increased progressively by maintaining all other conditions in the flume and gated circular culvert model. Fig. 18 shows the analysis of the flow transitions zones between the O5 outlet and I3 inlet groups for main gate openings of 8.75 and 6.25 cm. To compare both type of rating curves, the $H$-axis of the O5 curves was scaled using the observed averaged $h_d/h_u$ ratios: 0.53 for $W=8.75\mathrm{cm}$, and 0.47 for $W=6.25\mathrm{cm}$. Thereby, the head in the outlet discharge rating curves [Eq. (3)], which is $H=h_u-h_d+SL$, was approximated as $H=0.47h_u$ for $W=8.75\mathrm{cm}$ and as $H=0.53h_u$ for $W=6.25\mathrm{cm}$, neglecting the $SL$ term. Thus, the $H$-axis in the O5 curves originally plotted in Fig. 17 was converted into $h_u=H/0.47$ and $h_u=H/0.53$ for 8.75- and 6.25-cm gate openings, respectively. Snapshots of the video recorded for the discharge incrementing experiment that motivated this analysis also are shown in Fig. 18, in which the transition from O5 outlet control conditions to I3 inlet control conditions is evident in time. According to Bodhaine (1968), straight lines can be used to connect two rating curves. We propose parabolic transitions to maintain continuity between curves. However, more experimental data are required to help define these flow transition regions accurately. This gains importance at low stages if critical flow may occur upstream from the culvert inlet section, i.e., in the approach flow, and at stages in which high head flow is likely to occur (Bodhaine 1968).

Fig. 19 depicts the influence of $S$ on the $Q$ versus head relationship. Inlet and outlet control data for $W=2.4\mathrm{cm}$ (the most numerous subgroups of data including the maximum of values of $S$ in both the inlet and outlet control tests) are displayed in Figs. 19(a and b), respectively; the three different values of $S$ tested are indicated by different shades. With the selected gate opening, no differences were observed in the $Q$ versus head relationship due to the variation in $S$ in the interval of $S$ tested at both groups. Thus, the influence of $S$ on the $Q$ versus head relationship is unclear in the range of $S$ tested in our experiments.

The preceding results reveal the complexity of the hydraulics within gated culverts. Such complexity could be tackled empirically using dimensionless models like the one proposed by Zeng et al. (2020). That model relies on four dimensionless flow rating equations for gated culverts to model different flow conditions groups, namely (1) mild slope culvert orifice or full pipe flow, (2) mild slope culvert open channel flow, (3) steep-slope culvert full pipe flow, and (4) steep slope culvert unsubmerged or submerged with inlet control, which are, respectivelywhere $G=\min (W,D)$; $A_G$ and $A_{hd}$ = cross-sectional flow area beneath gate and $h_d$, respectively; $A_D$ = cross-sectional area of culvert; and $C_1$, $C_2$, $C_3$ = empirical parameters. Eqs. (4)–(7) relate the dimensionless head to the dimensionless discharge by means of three empirical, case-dependent parameters. Whereas Eqs. (4) and (5) are derived for mild slopes (i.e., $h_n>h_c$) with $G<h_d$ and $G\ge h_d$, respectively, Eqs. (6) and (7) apply for steep slopes (i.e., $h_n<h_c$) with $h_d\le D$ and $h_d\le D$, respectively (Zeng et al. 2020). According to Zeng et al. (2020), $C_3$ accounts for the roughness effect; thus, for the experiments in this study, it was assumed that $C_3=0$, which is consistent for short barrels with negligible friction losses. Calibrating $C_1$ and $C_2$ for the data collected in the present experiments, Figs. 20(a and b) show the performance of Eq. (4) for our experimental data in the Zeng et al.’s (2020) subgroup “mild slope culvert orifice or full pipe flow.” The combination of $C_1=0.8$, $C_2=2$, and $C_3=0$ produced the best fit, with errors in the prediction of the dimensionless discharge bounded by $+60\%$ and $-40\%$ limits [Fig. 20(b)]; $C_2$ was set equal to 2 in Fig. 20(a) to maintain the energy balance analogy with Eq. (4), which is an appropriate value for outlet control flow (Zeng et al. 2020). Figs. 20(c and d) depict the application of Eq. (5) to our data in the subgroup “mild slope culvert open channel flow.” The best fit was found using $C_1=0.7$, $C_2=1.5$, and $C_3=0$, which had modeling errors bounded by $+20\%$ and $-30\%$ limits [Fig. 20(d)]. Lastly, the results of applying Eq. (7) to our data in the subgroup “steep slope culvert unsubmerged or submerged with inlet control” are shown in Figs. 20(e and f). Here, the combination of $C_1=10$, $C_2=2$, and $C_3=0$ led to the best fit, with estimation errors within $\pm 20\%$ limits [Fig. 20(f)]. Eq. (6) could not be tested here because none of our data satisfied the criteria of Zeng et al.’s (2020) subgroup “steep-slope culvert full pipe flow.” Numerical tests with Eqs. (4), (5), and (7) showed that the uncertainty of level measurements (which was very little in our experiments) could explain only a very small part of the error in predictions. According to Zeng et al. (2020), the discharge coefficient $C_D$ for a flow under a sluice gate in inlet control conditions is an independent variable for $C_1$ in Eq. (7). Considering the results in Figs. 20(e and f), this suggests that $C_D$ should not be approximated by a single value for a group of data, but its modeling is rather complex and deserves further investigation. This issue also occurred in the analysis of errors by Eqs. (4) and (5), given that the scaling parameter is defined as $C_1=f(C_b)$, where $C_b$ is the discharge coefficient expressing contraction associated with culvert entrance edge and gate conditions (Zeng et al. 2020). Eqs. (4) and (5) are case-dependent, and thus require site-specific calibration.