Investigation on the Water Depth of Choked Flow due to Bottom Blockages in Circular Open Channels

Both the water supply and the wastewater collection networks are critical infrastructure for communities. Due to the sheer scale of the pipe networks and the fact that most of them are buried underground, water utilities in most countries are facing challenges in managing aging water and wastewater assets. In the past two decades, substantial research has been conducted for non-destructive and noninvasive defect detection in water networks, such as blockage detection (Duan et al. 2013; Meniconi et al. 2013), leak detection (Covas et al. 2005; Almeida et al. 2018; Wang et al. 2018; Zeng et al. 2020), and pipe condition assessment (Zeng et al. 2018; Zhang et al. 2019; Zeng et al. 2023). The research and development have resulted in the industry application of smart water networks (Rousso et al. 2023). In contrast, research on defect detection in wastewater collection networks (i.e., gravity sewer networks), is very limited. Fundamentals, such as hydraulic modeling (Liu and Chen 2022) and air-water interactions (Qian et al. 2022), are still active research topics for sewer networks.

Sewer blockages pose a serious problem for wastewater network operators since the resultant spills contaminate the environment, endanger public health, and impact the economy (Fenner et al. 2007; DeSilva et al. 2011; Tizmaghz et al. 2022). Consequences of spillages and the costs of excessive preventative maintenance have encouraged innovation directed at developing economical sewer blockage detection techniques (Rodríguez et al. 2012; Mustafa Abro et al. 2019). Advances in sensing and telemetry technology have resulted in an influx of cost-effective devices designed to monitor and report water depths in free-surface flow conditions (Häck and Wiese 2006; Hill et al. 2014). Despite these increasing technological capabilities enabling data-driven and/or threshold analysis of water depths in open channel applications (Edmondson et al. 2018; Webber et al. 2022; Do et al. 2023), the applications are yet to be supported by fundamental knowledge of choking transitions (Henderson 1966) caused by partial blockages.

Transitions and critical flow have traditionally been used for the measurement of flow rate in open channels (Clemmens et al. 2001). For circular open channels, e.g., sewer pipes, Palmer and Bowlus (1936) proposed a Venturi flume with a rectangular or trapezoidal throat for flow measurement. In the short discussion letter on the original paper by Palmer and Bowlus (1936), Stevens (1936) presented a curved slab without any side restrictions (streamlined along the axial direction with no extended flat surface) for flow measurement in sewer pipes. A chart describing the relationship between the depth of approach and the flow rate for a few selected slab heights was provided; however, simplifications were involved in the analysis (a key simplification is that the total energy head at the cross-section corresponding to the highest point of the slab was assumed as the depth of water upstream of the slab for the purposes of calculating the upstream flow area and velocity head). Diskin (1963) developed a pier-shaped structure for flow measurement in sewer pipes. The structure was installed vertically in the center of the channel, and flow was forced to pass through the two sides of the pier-shaped structure where critical flow was established due to the restriction. Wenzel Harry (1975) developed another type of critical flow meter or Venturi flume by introducing one or two smooth and extended restrictions on the sides of the sewer pipe. It could also act as a Venturi meter for full pipe flow. Clemmens et al. (1984) detailed a design of a broad-crested weir for flow measurement in circular pipes flowing partially full. The structure was a horizontal weir sill with a designed length and a ramp at the upstream to minimize energy loss. A table describing the relationship between the specific energy at the upstream of the structure and the flow rate was given. However, the upstream depth was not calculated. Clemmens et al. (1984) stated that an iterative trial-and-error process using two look-up tables could be taken to estimate the upstream depth, which was similar to the approach used in Stevens (1936). Nevertheless, Clemmens et al. (1984) did not follow this approach. Instead, they proposed to use an empirical formula to describe the complex relationship between the flow rate and the upstream depth under choked conditions. The empirical formula was to be calibrated by experimental data.

For circular channels with the channel bed raised due to a change in position of the centerline (the geometry of the cross section does not change), or due to a decrease in channel diameter (the geometry of the cross section is proportionally reduced), Vittal (1978) conducted dimensionless analysis and presented graphical solutions to both choke-free flow and choked flow conditions; Dey (1998) studied the maximum permissible limits of bed elevation rises for choke-free flows and presented graphical solutions; Vatankhah and Bijankhan (2010) further extended the work and provided explicit equations for the maximum allowable bed rise. For circular channels with a bottom blockage (raised channel bed with a flat base but continuous centerline, such that the geometry of the cross section changes with the height of the blockage), most research has focused on sediment transportation (Skipworth Peter et al. 1999; Banasiak and Verhoeven 2008; Seco et al. 2018). Although Dey (1998) analyzed the effect of raised channel bed, which effectively was a partial bottom blockage, the work focused on determining the critical height of the bottom blockage that would just induce critical flow on top of the blockage but no choking to the upstream flow yet, and only the subcritical flow condition was studied. The previous literature review has demonstrated that there is a lack of modern solutions on determining how the height of an extended bottom blockage affects the immediate upstream water depth. This has become a limiting factor for the successful application of smart sewer technologies for blockage detection and spill early warning.

The current research focuses on describing the changes in the hydraulic behavior of circular open channels due to changes in the cross-sectional geometry, with a particular interest on subcritical cases with extended bottom steps to simulate extended bottom blockages (e.g., deposits) in sewer pipes. A numerical framework based primarily on the analysis of the continuity and specific energy conservation was developed to estimate the water depth upstream to a bottom blockage in a circular open channel. The numerical framework extended the analysis by Dey (1998) from choke-free flow to choked flow scenarios for various combinations of blockage, channel and flow conditions. Except for neglecting energy losses, simplifications and empirical analysis used in early studies (Stevens 1936; Clemmens et al. 1984) were avoided in the current work. The numerical solutions for the upstream depth estimation were verified through a controlled set of experiments using a full-scale rig where various conditions (e.g., flow, blockage height, slope) were tested. The numerical solutions can be used for modeling of simple sewer blockages, improving the capability of smart sewer systems for partial blockage detection and sewer spill early warning. Normalized charts describing the relationships are also provided for the ease of practical applications.

Transition problems are traditionally analyzed using the energy relationship (Chaudhry 2022), where the specific energy $E$ is defined aswhere subscript $i$ = position in the circular open channel with respect to the blockage (i.e., 1 for upstream and 2 for top of the bottom blockage, used throughout this paper); $y$ = depth of water (m); $V$ = mean velocity (m/s); and $g$ = gravitational acceleration ($\mathrm{m}/{\mathrm{s}}^2$).

Neglecting energy losses, $E$ upstream ($E_1$) and on top of a bottom blockage ($E_2$) should satisfy $E_1=E_2+z$, where $z$ is the height of the blockage. With $z$ increasing from zero, $E_2$ decreases. When $z$ is large enough for the flow over the blockage to reach its critical condition, $z$ is determined as the critical height of the blockage $z_c$, where subscript “c” represents the critical flow condition and used throughout this paper. When $z\ge z_c$, the flow over the blockage remains at critical condition such that $E_2=E_{2c}$, and $E_1$ is updated to a new value ${\widehat{E}}_1$ to satisfy $E_{2c}+z$. Depending on the original flow regime (i.e., supercritical or subcritical), $y_1$ may increase or decrease to a new value ${\widehat{y}}_1$ to match the new energy.

In circular open channels with a bottom blockage, the cross-sectional geometry of the channel changes with $z$. As a result, the $E\text{-}y$ curve on top of the blockage changes as a function of $z$ for the same $Q$. Similarly, the critical flow regime on top of the blockage (i.e., $E_{2c}$ and $y_{2c}$) varies with $z$ for the same $Q$.

A numerical framework is developed to analyze the impact of extended bottom blockage to the upstream flow depth. Energy losses at the transition are neglected, which is applicable to bottom blockages with a smooth ramp on the upstream end (e.g., sediment deposits).

This study has formulated, implemented, and validated a numerical framework to quantify the water depth in partially filled circular open channels subject to downstream bottom blockages (raised channel bed with a flat base but continuous channel centerline). The numerical estimations of the water depth upstream of the blockage have been validated through a series of experiments using a full-scale setup, showing an excellent agreement between numerical estimations and experimental measurements (average error $<2\%$).

The proposed numerical framework allows not only the identification of the critical bottom blockage height for any specific circular open channel system, but also a fast and reliable quantification of the new water depth upstream to the bottom blockage for both supercritical and subcritical regimes under choked flow conditions. Graphical solutions have been presented for practitioners to estimate the critical bottom blockage height that would induce choking (Fig. 2), and to determine the new upstream depth under choked but still free-surface flow conditions (Figs. 3 and 4). For systems with subcritical flow regimes, the scenario of particular interest to practitioners managing sewer systems, the upstream water depth proportionally increases with increasing blockage height values and discharge rates. For systems with supercritical flow regimes, the upstream water depth increases with increasing flow rates for a given blockage height but decreases with increasing blockage height for a given flow rate.

This study is limited to free surface flow conditions and bottom blockages. Future work can expand to pressurized conditions and other types and shapes of blockages (i.e., porosity, location in the cross-sectional area of the pipe). The impact of energy losses in supercritical flow conditions also needs further investigation since it might induce hysteresis behavior to the flow depth.

This section demonstrates how the normalized charts presented in Figs. 2–4 can be used to estimate the possible impact of bottom blockages in specific sewer pipe systems. For demonstration, an intact gravity sewer with an internal diameter $D=0.2\mathrm{m}$, a designed dry-weather flow rate of $Q=0.005{\mathrm{m}}^3/\mathrm{s}$ and a corresponding flow depth of $y_1=0.08\mathrm{m}$ is considered. It is also known that the sewer is designed to have subcritical flow, e.g., the corresponding Froude number $F_1<1$. Note that all these parameters can be measured from real sewer systems.

The first step is to convert the dimensional flow rate $Q$ and flow depth $y_1$ to their corresponding normalized values $\overline{Q}$ and ${\overline{y}}_1$, and it is achieved by Eqs. (21) and (22)

For the specific gravity sewer under consideration, the normalized flow $\overline{Q}=0.005/({0.2}^{2.5}\sqrt{9.81})=0.09$, and the normalized depth ${\overline{y}}_1=0.08/0.2=0.4$. To determine the critical bottom blockage height that would just induce choking, the chart in Fig. 2 is used. The normalized flow is $\overline{Q}=0.09$, such that the curve with circular symbols (“o” shaped symbol) should be used (refer to the legend of Fig. 2). Since the normalized depth ${\overline{y}}_1=0.4$, based on this value on the x-axis, the corresponding value of the normalized critical blockage height ${\overline{z}}_c$ (on the y-axis in the chart) is found to be very close to 0.1 from the curve with circular symbols. Converting this dimensionless height to the corresponding dimensional value by multiplying the diameter, the critical blockage height $z_c={\overline{z}}_{c}D=0.1\times 0.2=0.02\mathrm{m}$. This means, for this specific sewer system with the designed flow $Q=0.005{\mathrm{m}}^3/\mathrm{s}$, a bottom blockage with a heigh of 0.02 m or higher will induce choking.

For a bottom blockage with a heigh of 0.02 m or higher, its impact on the upstream flow depth can be predicted using the charts in Fig. 3. Since the normalized flow is $\overline{Q}=0.09$ and the flow regime is subcritical ($F_1<1$), the curve with circular symbols in the top chart should be used. Note that only the part of the curve with $\overline{z}>{\overline{z}}_c$ is meaningful. For instance, for a blockage with a height $z=0.08\mathrm{m}$, the corresponding normalized height is $\overline{z}=z/D=0.08/0.2=0.4$. Now referring to the curve with circular symbols in the top chart, the normalized new upstream depth is expected to be ${\overline{\widehat{y}}}_1=0.66$. Converting to the corresponding dimensional value, this means the new upstream depth under the choked flow condition is expected to be ${\widehat{y}}_1={\overline{\widehat{y}}}_{1}D=0.66\times 0.2=0.13\mathrm{m}$.

For the same blockage ($z=0.08\mathrm{m}$, $\overline{z}=0.4$) in the sewer system, its impact under other flow conditions can be assessed using the charts in Fig. 4. Since the flow regime is expected to stay as subcritical, the curve with cross symbols (“+”) in the top chart should be used. For the original designed flow $Q=0.005{\mathrm{m}}^3/\mathrm{s}$ (normalized flow $\overline{Q}=0.09$), the corresponding normalized new upstream depth is read as ${\overline{\widehat{y}}}_1=0.66$ from the curve. This is consistent with the finding from using Fig. 3. For a higher flow rate, $Q=0.015{\mathrm{m}}^3/\mathrm{s}$ for instance, the corresponding normalized flow is $\overline{Q}=0.27$. Using the aforementioned curve, the corresponding normalized new upstream depth can be found by reading the value on the y-axis, and it is ${\overline{\widehat{y}}}_1=0.89$. Converting to the corresponding dimensional value, the new upstream depth for the sewer system with a blockage $z=0.08\mathrm{m}$ and a flow rate $Q=0.015{\mathrm{m}}^3/\mathrm{s}$ is expected to be ${\widehat{y}}_1={\overline{\widehat{y}}}_{1}D=0.89\times 0.2=0.18\mathrm{m}$.

Fig. 7 demonstrates how the water depth can be determined by the arclength. Water depths were calculated by reading the wrapped measuring bands on both sides of the pipe (${\alpha }_1$ and ${\alpha }_2$) and calculating the arclength ($l$) of the unfilled section above the flow [Eq. (23)], which was then used to calculate the apothem ($x$) while considering pipe wall thickness [Eq. (24)]. Based on $l$, the water depth was estimated by adding or subtracting the apothem to the inside radius, $r_i$ [Eq. (25)]where ${\alpha }_1$ and ${\alpha }_2$ are opposite measuring points on the perimeter of the pipe, $l$ is the arclength of the section not filled with water, $x$ is the apothem, $r_i$ and $r_o$ are respectively the inside and outside radii of the pipe and $y$ is the water depth.

The experimental data that support the findings of this study are available from the corresponding author upon reasonable request.

The research has been supported by the Barwon Region Water Corporation (Geelong, Australia) and Intelligent Water Networks (https://www.iwn.org.au/) through a collaborative research project (Project Code: PJ06416).