Laboratory-Scale Investigation of the Pressurization of T-Junctions in Hydraulic Systems

Abstract

The increasing frequency of intense rain events will worsen the operational conditions of stormwater systems, including the frequency in which these systems experience pressurization. Unfortunately, there has been limited research on the issue, particularly the pressurization of junctions in stormwater systems that are subject to rapid filling. Past research provided valuable insights on flows in junctions operating either on pressurized or open-channel flow conditions, but did not focus on the transition between these two flow regimes. This work summarizes the results of an experimental investigation that focused on describing the pressurization processes in a junction undergoing rapid filling. The experimental program considered a total of 67 unique combinations, with variables including different slopes of the upstream and lateral pipes, as well as different inflow rates in each one of these conduits. Fast blockage of the flow led to the pressurization process, that was characterized through video-recording and pressure monitoring at selected points along in the apparatus. This innovative research identified for the first time five unique pressurization modes based on the video recordings of the pressurization. The pressurization modes were dependent on the experimental setup, including lateral and main branch flows as well as their slopes. An independent clustering-based analysis of the experimental data was used and confirmed this classification. These results are the first ones of its kind and clearly indicates potential limitations of numerical models in representing flows in the junctions when air pocket entrapment occurs due to rapid filling conditions. Future research should address the limitations of the present experimental work in terms of scale effects, including a wider range of tested flow conditions and slopes, besides different junction geometries with storage.

1. Introduction and Objectives

Stormwater systems are a fundamental part of urban infrastructure, and these have been undergoing increasing challenges created by the growing frequency of intense rainfall events and widespread urbanization [1,2,3,4,5]. In such conditions, sewers and tunnels are anticipated to convey more flows than they were designed for and, as a result, pressurization of these conduits may be anticipated [6]. Another issue, commonly observed in stormwater systems of developing countries is the no prediction of air-water mitigation devices like storage reservoirs, ventilation shafts, or manholes. It could lead to a no contact of the system with the atmospheric pressure, inducing dangerous hydraulic conditions, as pointed by [7]. Besides, sometimes failure of stormwater pumps, blockage created by solid waste, debris/sediment accumulation, lack of maintenance or even design/construction errors could lead to pressurization conditions [8,9]. The rapid filling of closed conduits in these conditions can result in the entrapment of air [10,11], which, in turn have adverse effects due to air-water interactions in stormwater systems which are well documented [12,13,14,15]. Different studies have shown the potential issues created by the compression and expansion of entrapped air in closed conduits [16,17], and the uncontrolled discharge of the entrapped air phase in stormwater systems leading to geysers [18,19], manhole cover displacements [20,21], among others.

Studies on the rapid filling of stormwater systems have approached the problem using various techniques. Among experimental investigations exploring the nature of rapid filling in stormwater systems, Wiggert [22] studied the advance of a pressurization wave in a closed conduit with a rectangular cross section, also proposing a shock-fitting numerical model to predict flow characteristics. Trajkovic et al. [23] studied the pressurization evolution due to a sudden blockage of a downstream valve and subsequent partial reopening, also applying a numerical model to explain the pressure changes. Vasconcelos and Wright [24] studied the development of upstream-moving and downstream-moving pressurization interfaces upon the introduction of an intermediate inflow in a system nearing pressurized flow conditions. Other authors have considered system-wide modeling of stormwater inflows and air-water interactions [25,26,27].

Most of the existing body of knowledge has focused on the description of the pressurization processes, air entrapment/motion/release, and dynamics in general focusing on the linear conduits. Remarkably, very little attention has been dedicated to the processes leading to the pressurization of stormwater junctions, which are ubiquitous in stormwater systems. Numerical modeling studies have mostly assumed that the behavior of these junctions can be described by the application of mass, momentum, and energy conservation principles. Although it seems an adequate approach, it neglects the fact that, during pressurization stages, air pockets entrapment can appear, invalidating several hypotheses related to single-phase flows frequently used in such numerical models [28].

Yet, there has been no investigation on the behavior of stormwater junctions undergoing pressurization, despite various and important studies on flow characteristics at open-channel flow junctions. Pardee [29] presented a study to derive the free surface in hydraulic junctions considering various junction angles and conduit cross sections for both sub and supercritical flow regimes. Lin and Soong [30] performed a series of experimental investigations of flow characteristics in a 90-degree junction of a rectangular open channel and lateral channel. Lin and Soong [30] determined that the ratio between the lateral flow and the total flow was an important parameter to understand the flow characteristics in the junction. In addition, larger lateral flow rates provoked a backwater effect in the main channel upstream from the junction. Weber et al. [31] performed a careful characterization of flow characteristics, including the velocity field, in a 90-degree open-channel junction linking two reaches that had rectangular cross sections. Despite these important scientific contributions in the description of the main characteristics and flow in the junctions of open channels, these investigations did not characterize the pressurization process in a stormwater junction, leaving important open questions.

From the discussion above, it becomes clear that there is a knowledge gap in the description of how stormwater junctions may undergo pressurization. Among the open research questions are the following:

• Is it possible to derive a flow classification scheme that qualitatively describes the pressurization, and eventual air pocket entrapment?
• To what extent does an eventual air pocket entrapment pose challenges for the numerical modeling of junctions undergoing pressurization?
• Can the data mining from the raw experimental dataset yield useful information for flow classification that matches experimental observations of flow pressurization at the junction?

This work aims to address these questions through an experimental investigation of T-junction flow pressurization using video recordings and pressure monitoring. Clustering analysis of the experimental data was also performed, and results of this analysis are compared with a proposed flow classification based on experimental observations. Such results can support an assessment of whether existing numerical modeling tools for the simulation of rapid filling in stormwater systems are applicable. The manuscript is structured as follows: after this introductory section, the experimental data analyses approaches are presented and detailed in the Methodology section. The Results and Discussion section will focus on presenting the characteristics of the flow, proposing a flow classification scheme, and differences in pressure results for representative flow conditions. Subsequently, the results from the clustering analysis are compared with the proposed flow classification. Finally, the Final Remarks and Conclusions section reinforces the main conclusions, points out the limitations of the present study, and point to potential ways that the research in junction pressurization can continue.

2. Materials and Methods

The present work included an experimental program that created a range of flow conditions associated with different pressurization events in a T-junction, which is presented in Figure 1. The apparatus, while having elements that are common in stormwater systems, does not correspond to a planned or existing system. The experimental apparatus and procedure were designed to facilitate the development, observation, and characterization of the pressurization process in a stormwater junction. One aspect of the apparatus that is not typically observed in stormwater systems is the lack of some storage in junctions. Preliminary experiments indicated that the propagation of pressurization conditions upstream was delayed by storage in the junction. As a result, junction storage was disregarded in the present study, and a T-junction geometry was adopted.

These different pressurization events were created through a systematic variation of inflows and slopes within the apparatus. The section near the junction was built in acrylic, which enabled a visual classification of the junction pressurization. The classification scheme was subsequently associated with pressure recordings performed at selected locations within the apparatus. In parallel, an independent classification approach using an unsupervised clustering analysis was carried out using experimental parameters and measurements, which was compared with the visual classification scheme. The following sections describe details of the experimental program as well as the analyses performed in this study.

2.1. Experimental Program

The experimental apparatus used three pipe reaches with diameter $D_p=10$ cm laid out with varying slopes and used a 90-degree T junction, as is presented in Figure 1. Storage devices like manholes was not considered in the present appparatus because the goals here was to understand how a pressurization front arriving in a junction would spread to reaches at the upstream end. Therefore, storage devices was not used here in order to facilitate the propagation of the pressurization fronts. The conduits reaches were: (1) main upstream (referred to as U); (2) lateral upstream (referred to as L); and (3) downstream (referred to as D). The length of the reaches were 5.53 m, 5.42 m and 1.94 m for reaches U, L and D, respectively. All reaches were made in PVC, except for a length of 1.0 m in each reach where an acrylic pipe, also with 10 cm diameter, was connected with the junction. According to [32], 10-cm diameter pipes are large enough to avoid surface tension effects between air and water. Moreover, the range of flows that were aimed for the presented experiments would be achievable with such diameter size. The slopes for the U and L reaches were independently varied for the experiments to values of 0.01, 0.02 and 0.03; the slope of the reach D was set to a fixed value of 0.072 to ensure that during the early stages of the experiment flows were supercritical and presented no backwater effect.

Further upstream the apparatus was connected to a water supply system that provided steady flow rates to both reaches U and L during a given experimental run. Along with the slopes, such inflows $Q_U$ and $Q_L$ were systematically varied for each test to generate 67 unique conditions. Following [14], the values for the inflows were normalized by $\sqrt{g{D}_p^5}$, with g as the gravity acceleration. The normalized inflows admitted in reach U during the runs were 0.0, 0.040 or 0.187, and are referred to as $Q_U^{*}$. Similarly, for the lateral reach L, the normalized inflows $Q_L^{*}$ were $0.0,0.042$ or $0.166$. The summation of these normalized inflows reached a level comparable to the peak flows reported in [33] that were used to create surging in stormwater systems. Given that the junction had no storage, the summation of $Q_U$ and $Q_L$ represented the flow that was freely discharged through the reach D.

Table 1. Experimental variables and dimensional values considered.

Experimental Variable	Values
	Min.    Med.    Max.
$Q_U$ (L/s)	$0.00$    $0.37$    $1.63$
$Q_L$ (L/s)	$0.00$    $0.37$    $1.45$
$S_U$ (%)	$1.00$    $2.00$    $3.00$
$S_L$ (%)	$1.00$    $2.00$    $3.00$

presents all dimensional variables used in the present work.

At the very end of the reach D, a 10-cm knife gate (KG) valve was suddenly closed (under 0.5 s) during the experimental runs to initiate the filling of the reach D which led to the pressurization of the junction. This approach is similar to what was used in previous experimental studies to trigger pipe pressurization, such as [23,34]. This pressurization was monitored through a video camera Canon PowerShot SX50 HS with a 1080p resolution carefully located to record the acrylic portion of the apparatus at a frequency of 23 frames per second. The acrylic section of the apparatus enabled to measure the flow perimeter and, therefore, calculate the flow depth and the flow cross-sectional area prior the sudden closure of the knife gate valve.

In addition to the monitoring of the flow features, pressure recordings were obtained with four MPX5010dp pressure sensors, represented as $P{S}_{KG}$, $P{S}_L$, $P{S}_U$, and $P{S}_L$ in Figure 1. Sensor $P{S}_{KG}$ was placed at the invert of the discharge of reach D near the knife gate valve. $P{S}_J$ was placed at the invert of the junction; $P{S}_U$ and $P{S}_L$ were placed at the inverts in reaches U and L respectively, each about 1.0 m upstream from the junction at the location where the acrylic and PVC pipes joined. All pressure sensors were connected to a data acquisition board, sampling pressure at a frequency of 10 Hz.

The procedure used in the experimental runs can be summarized as follows:

• The apparatus was set to a desired combination of slopes and flow rates $S_U$, $S_L$, $Q_U^{*}$ and $Q_L^{*}$, and it were maintained until a steady-state condition was attained. This was ensured by observing the flows in the transparent portion of the apparatus.
• Video recording was initiated.
• Pressure sensors were activated and started recording data.
• The knife gate valve was suddenly closed and the apparatus pressurization initiated.
• Observations regarding the nature of the pressurization of the junction were made during the data collection.
• Conditions were maintained until the system was full. At that time instant, there was no more flows in the system and the piezometric head was uniform among all sensors, signaling the end of the experimental run.
• Every experimental condition was repeated at least three times to ensure consistency in the data collection.

2.2. Experimental Data Treatment and Exploratory Clustering-Based Analysis

An exploratory statistical analysis was accomplished for extracting useful patterns from the experimental raw dataset [35]. Before the visual pressurization analysis the whole dataset was evaluated for detecting possible experimental errors and inconsistency. The Grubbs’ test [36] for outliers detection was applied at a 5% significance level. Eventual outliers were carefully evaluated and eventually excluded from the data set (See

Table 2. Summary of the observed pressurization modes grouped by inflows versus slope branches.

	${\mathit{Q}}_{\mathit{L}}^{*}$		0.000			0.042			0.166
${\mathit{Q}}_{\mathit{U}}^{*}$		${\mathit{S}}_{\mathit{L}}$	0.01	0.02	0.03	0.01	0.02	0.03	0.01	0.02	0.03
	${\mathit{S}}_{\mathit{U}}$
0.000	0.01		-	-	-	NHFS	NHFS	NHFS	Excluded	LPFB	TNLB
	0.02		-	-	-	NHFS	NHFS	NHFS	TNLB	TNLB	LPFB
	0.03		-	-	-	NHFS	NHFS	NHFS	TNLB	LPFB	TNLB
0.040	0.01		NHFS	NHFS	NHFS	NHFS	NHFS	NHFS	Excluded	DPFB	DPFB
	0.02		NHFS	NHFS	NHFS	NHFS	NHFS	NHFS	DPFB	DPFB	DPFB
	0.03		NHFS	NHFS	NHFS	NHFS	NHFS	NHFS	DPFB	DPFB	DPFB
0.187	0.01		Excluded	TNUB	TNUB	Excluded	TNUB	TNUB	Excluded	EJP	EJP
	0.02		UPFB	UPFB	UPFB	UPFB	UPFB	UPFB	EJP	EJP	EJP
	0.03		UPFB	UPFB	UPFB	UPFB	UPFB	UPFB	EJP	EJP	EJP

).

Following, a data mining technique based on hierarchical cluster analysis [37,38] was accomplished by considering the flow rates in each reach. For this task, the flow rates were normalized by the flow rate that would be anticipated if the reaches were flowing in full capacity (i.e., depth equal to $D_p$), and that the energy slope matched the reach slope. Thus, this alternative normalization yielded the following expressions:

$$Q_{U_n}=Q_U·\frac{2^{10/3}n}{\pi D_p^{8/3}\sqrt{S_U}}$$

(1)

$$Q_{L_n}=Q_L·\frac{2^{10/3}n}{\pi D_p^{8/3}\sqrt{S_L}}$$

(2)

$$Q_{D_n}=Q_D·\frac{2^{10/3}n}{\pi D_p^{8/3}\sqrt{S_D}}$$

(3)

The clustering analysis was performed in the software R 4.1.0, using the stats package. The complete-linkage method was used for clustering, and Euclidean distance was used as the measure of distance. This technique is used in several areas [39] including flow studies [35,40,41] allowing the grouping of all experimental cases and respective repetitions runs into classes of high level of similarity [42], being a stand-alone tool to get insight into data distribution. The number of classes for clustering was defined by using the R package Nbclust [43] which provides 30 clustering validity indices to determine the most suitable number of clusters in a data set.

Pressurization patterns identified and classified through a visual analysis were then assessed within the clusters, aiming to identify whether these patterns were also clustered due to the experimental configuration. The clustering analyses also enabled to assess the consistency of the run’s triplicates. Whenever the triplicate runs did not cluster, the set of triplicates was inspected.

3. Results and Discussion

3.1. Description of Flow Conditions Prior to Pressurization

Prior to the closure of the knife gate valve, flows in each reach were in gradually-varied steady-state mode. For the experiments involving low ($Q_D^{*}$: $0.040$ to $0.082$) or intermediate ($Q_D^{*}$: $0.166$ to $0.229$) flow rates, it was possible to observe that free surface flow conditions existed in the apparatus, even with the higher water depth in the junction that was created by the energy loss in that point. For experimental runs involving maximum flow rate ($Q_D^{*}$=$0.353$), it was possible to notice that the upstream reaches U and L approached an incipient pressurization, with the water level barely touching the pipe crown prior to reaching the junction. However, even in these cases, the downstream reach operated in free surface flow mode due to the steeper slope in reach D. Figure 2 illustrates the two typical initial conditions observed in the experimental runs.

Immediately prior to the closure of the knife gate valve, the flow depth $H_D$ was measured within reach D, and it was compared with various inflows $Q_D$. As indicated in Figure 3, for all tested flows the measured water depth (normalized by the pipe diameter $D_p$) did not exceeded 40% of the diameter for the largest tested flows. Closer to the junction, and within reach D, the water depth neared 55% of the pipe diameter, but still not near pressurized conditions. As expected, the flow depth could reach values as low as 10% of the pipe diameter with decreasing value of $Q_D$.

3.2. Characterization of the Junction Pressurization Resulting from the Rapid Filling

Each experimental condition was evaluated and a qualitative characterization of the junction pressurization was performed taking into consideration the pressurization advance and air-water interactions. Through careful evaluation of the recorded experimental conditions, five representative pressurization modes were identified. Such modes, also identified in markers within Figure 3, are linked with distinct processes by which the pressure changes took place in the junction upon the sudden closure of the knife gate valve at the downstream end of the apparatus. Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 illustrate the pressurization of the reaches and the junction observed in each mode. The images correspond to a time a few seconds after the knife gate valve closure, and the free surface was highlighted in blue for better visualization. Table 2 presents a summary of all observed pressurization modes grouped by inflows versus slope branches.

A general description of the filling process is presented with each of the identified junction pressurization modes:

• Near-Horizontal Free Surface (NHFS) mode: This was observed as the most gradual pressurization mode, and was observed for all cases when $Q_U^{*}\le 0.040$ and $Q_L^{*}\le 0.042$. As the knife gate valve was closed, a slow filling process with a near-horizontal air-water interface was observed in the D reach (Figure 4). Upon reaching the junction, the advance of the pressurization interface stalled for a brief time as the hydraulic grade line increased in reaches U and L. After some time, the pressurization front continued to advance and eventually pressurized the junction.
• Upstream Pipe-Filling Bore (UPFB) mode: This was one of the three pressurization modes that were associated with the development of a moving hydraulic jump (i.e., bore) following the closure of the knife gate valve. These bores were similar to the ones reported by [23], and varied in their strength according to the $Q_D$ values. These bores advanced within reach D toward the junction, and upon reaching the junction it continued to advance toward reach U, as is shown in Figure 5. There was no identifiable bore advance within reach L. Such conditions were typically observed when $Q_U^{*}=0.187$ and $Q_L^{*}\le 0.042$. A limiting pressurization mode between NHFS and UPFB, referred to as Transitional Near-horizontal Upstream Bore (TNUB), occurred in a few cases when $S_U=0.01$ and when slopes were such that $S_L>S_U$. These pressurization interfaces had a mode undulatory-like format, and as it touched the pipe crown these advanced leaving a trail of air pockets in the pipe.
• Lateral Pipe-Filling Bore (LPFB) mode: This is similar to the UPFB condition, however the observed hydraulic jump that was created by the valve closure, upon advancing toward the junction, continues into the lateral reach rather than moving upstream. This condition was observed in certain cases when the $Q_L^{*}=0.166$ and $Q_U^{*}=0.0$, and it is presented in Figure 6). Another limiting pressurization mode between NHFS and LPFB was observed, depending on the apparatus slope setup. This intermediate mode was referred to as Transitional Near-horizontal Lateral Bore (TNLB), which occurred in the remainder of the cases when $Q_L^{*}=0.166$ and $Q_U^{*}<0.040$, also with an undulatory-like pressurization front.
• Dual Pipe-Filling Bore (DPFB) mode: This pressurization mode is characterized by the propagation of two pipe-filling bores in the two reaches U and L after the initial pressurization bore arrived at the junction. All of these conditions occurred when $Q_L^{*}=0.166$ and $Q_U^{*}=0.040$. The pressurization mode is presented in Figure 7.
• Early Junction Pressurization (EJP) mode: For this case, there was initially free surface downstream from the junction, and incipient pressurization at the upstream and lateral reaches. After the knife gate valve closure, there was a rapid air displacement from the D branch towards L and U branches. This air pocket could not be released due to the blockage created by the pressurization of U and L branches. There is an appearance of a near-instantaneous air pocket that is displaced upstream. As is discussed later, pressure spikes are observed both at the knife gate valve and at the junction. The relative motion of the air over the water appears to create waves in the free surface that is similar to a Kelvin-Helmholtz instability (Figure 8). This condition was presented in all cases when $Q_U^{*}=0.187$ and $Q_L^{*}=0.166$.

Analysing the plot in Figure 3, for values of $H_D^{*}$ lower than ∼0.2, only the NHFS mode is observed. Between values of flow depth ∼0.2 and ∼0.35 the modes UPFB, LPFB, DPFB and the transitional modes were observed. Finally, for $H_D^{*}$ up to ∼0.35 only the EJP mode is presented. This means that both NHFS and EJP modes have some independence of the apparatus slopes U and L. Further analysis on the experimental conditions is presented in Section 3.4, considering a flow normalization based on $Q_{U_n}$, $Q_{L_n}$, and $Q_{D_n}$ in order to understand the slope dependence of the intermediate modes.

3.3. Piezometric Pressure Results

Pressure measurements were performed at sensors $P{S}_{KG}$, located at the knife gate valve, $P{S}_J$ located at the junction, and $P{S}_U$ and $P{S}_L$ located upstream from the junction reaches U and L respectively. Representative pressure results from the modes NHFS, UPFB, DPFB and EJP are presented in Figure 9, and results may be summarized as follows:

• NHFS mode: The piezometric pressure results by $P{S}_{KG}$ indicate a gradual rise immediately after the valve closure. At about 9 s, the rate of pressure rise slows down as the inflow front reached the junction and there is a gradual rise of the hydraulic grade line in reaches U and L. Pressures at the other three sensors show a very gradual rise as well until the complete pressurization is attained after many minutes, according to $Q_U$ and $Q_L$ values.
• UPFB mode: The piezometric pressure results by $P{S}_{KG}$ indicate a more rapid after the valve closure, with the inflow front reaching the junction in under 4 s. At about 6 s, results from $P{S}_L$ indicate a rise that marked the arrival of the gradual flow regime transition (GFRT) pressurization interface, as described by [44]. The hydraulic jump propagates more slowly toward $P{S}_U$ due to the larger values of $Q_U$, and arrive at the sensor at about 11 s. While the pressure result variation are more abrupt than the NHFS case, these pressure changes should be possible to represent in 1D hydraulic models that are capable of tracking the motion of bores.
• DPFB mode: In general these results are similar to the UPFB mode. With the larger lateral inflow and energy losses at the junction, the initial pressure at $P{S}_J$ is higher than the previous two modes. Compared to the UPFB mode, the advance of the hydraulic bore in reach U is faster due to the smaller inflow rate in the junction, enabling to create a pressure rise at the location prior to 6 s after the valve closure. The second hydraulic jump advances slightly slower within reach L, arriving at $P{S}_L$ prior to 9 s and triggering a pressure rise. Similarly to UPFB mode, such conditions can be modeled in 1D hydraulic that are able to track the motion of bores.
• EJP mode: This mode is markedly different from the previous results in that the valve closure created an immediate pressure surge that was detected in $P{S}_{KG}$ and in $P{S}_J$. The air pocket that forms immediately initiates the advance toward reaches U and L, with the wedge-shaped pressurization front followed by the leading edge of the discrete air pocket. The pressurization at $P{S}_U$ and $P{S}_L$ take place at about the same time, 2 s after the valve closure. The pressure rise at the $P{S}_J$ after 3 s corresponds to the time when the tail of the air pocket leaves the junction. The strong air-water interaction created by the entrapment of the air pocket, and the immediate pressure spike created by the air valve is similar to what was reported in [34]. Such conditions would pose important difficulties for single-phase 1D hydraulic models.

It may be noticed that certain pressurization modes (LPFB, TNLB, and TNUB) were not represented in Figure 9 for the sake of clarity. The pressure results for the LPFB mode was similar to the DPFB mode in that the pressure rise in the upstream reach occurs sooner and is more gradual than the results in the lateral branch. Regarding the transitional modes (TNLB and TNUB), the pressure changes are slightly faster than the NHFS mode, but still retaining a similar behavior. Most importantly, these three cases are also likely to be accurately represented by single-phase 1D hydraulic models.

3.4. Exploratory Clustering-Based Analysis

A cluster analysis considering the normalized flow rates has been done to better understand the modes observed from the video recordings. Figure 10a shows the resulting dendrogram from the 201 runs that correspond to all three repetitions from the 67 experimental conditions. The different colors indicate the four main clusters (C1, C2, C3 and C4) according to Nbclust analysis [43].

Table 3. Experimental configuration and pressurization pattern according to video record analysis.

Group	Number of Runs Clustered	Experimental Configuration **	Pressurization Pattern Clustered
C1 (blue)	81	two lowest $Q_U^{*}$ (0.0 and 0.04) and $Q_L^{*}$ (0.0, 0.042) flows	NHFS → $Q_{U_n}$ = 0% to 6% $Q_{L_n}$ = 0% to 6%, $S_L$ (0.01, 0.02, 0.03) and $S_U$ (0.01, 0.02, 0.03)
C2 (red)	24	the highest $Q_U^{*}$ flow (0.187), ranging from 14% to 25% full-pipe capacity, and the highest $Q_L^{*}$ flow (0.166)	EJP → $Q_{U_n}$ = 14% to 25% $Q_{L_n}$ = 13% to 22%, $S_L$ (0.01, 0.02, 0.03) and $S_U$ (0.01, 0.02, 0.03)
C3 (green)	48	the highest $Q_U^{*}$ flow (0.187) and the 2 lowest $Q_L^{*}$ flows (0.0 and 0.042)	TNUB (12) when $Q_{U_n}$ = 25%, $Q_{L_n}$ = 0% to 4% and $S_L$ (0.02, 0.03) > $S_U$ (0.01) UPFB(36) when $Q_{U_n}$ = 14% to 18%, $Q_{L_n}$ = 0% to 6% for $S_L$ (0.01, 0.02, 0.03) > $S_U$ (0.02, 0.03)
C4 (pink)	48	two lowest $Q_U^{*}$ (0.0 and 0.04), and the highest $Q_L^{*}$ flow (0.166)	DPFB(24) when $Q_{U_n}$ = 3% to 6%, $Q_{L_n}$ = 13% to 22% and any $S_U$ and $S_L$ combination LPFB(9) when $Q_{U_n}$ = 0%, $Q_{L_n}$ = 13% to 16% and $S_U$ (0.01, 0.02, 0.03) and $S_L$ (0.02, 0.03) TNLB (15) when $Q_{U_n}$ = 0%, $Q_{L_n}$ = 13% to 22% and any $S_U$ and $S_L$ combination

${}^{**}$ slopes assessed were 0.01, 0.02 and 0.03.

shows the main characteristics of the experimental configuration found in the clusters (inflow and slope conditions), in addition to the six pressurization patterns defined independently, based on the qualitative characterization of the video recordings. Figure 10b shows the scatter plot of the normalized flows ($Q_{U_n}$, $Q_{L_n}$ and $Q_{D_n}$) selected for assessing the clustering analysis where is possible to verify the spatial grouping with the identification of the pressurization pattern within the clusters.

Following the cluster analysis is possible to observe:

• Overall, group C1 contained the greatest number of experimental runs and was characterized by low U and L flow rates, and the three experimental slopes assessed in each reach. This configuration was produced for both U conveyance lower and higher than L conveyance over the experimental runs. The 81 runs comprised in this group were visually classified as NHFS pressurization pattern and have more than 90% of similarity.
• The group C2 had a smaller number of experimental runs and corresponds to the experimental condition where high U and L flows were evaluated together with the three experimental slopes. This configuration also was produced for both U conveyance lower and higher than L conveyance. The 24 runs comprised in this group were visually classified as EJP pressurization pattern and have more than 88% similarity.
• Group C3 comprised 48 experimental runs with more than 66% of similarity, which were visually classified as UPFB and TNUB. The runs within this group where related to the highest U flow combined with the two lowest L flows, and the three experimental slopes in each reach. In any case, U conveyance was higher than L conveyance. TNUB belong to a C3 subgroup with 12 runs with more than 85% similarity. It corresponds to experimental runs with U flow higher than L flow, and U slope lower than L slope. UPFB pressurization pattern was related to subgroups with 36 experimental runs with similarity higher than 89% and comprises configurations with U flow and U slope higher than L flow and L slope.
• Finally, the group C4 comprises 48 experimental runs with more than 67% of similarity, which were visually classified as DPFB, LPFB and TNLB. The runs within this group where related to the two lowest U flows combined with the highest L flow, and the 3 experimental slopes in each reach. These configurations produced U conveyance lower than L conveyance in all runs assessed. The modes TNLB, DPFB and LPFB are distributed within the C4 subgroups with more than 80% similarity. However, it was not possible to identify a specific pressurization pattern associated to the slopes evaluated during the runs.
• The transitional modes, as well the DPFB mode were clustered in C3 and C4 groups. In these cases, the main difference between the runs that generated the transitional modes and DPFB can be explained by using the conveyance. In group C3 configurations with U conveyance higher than L conveyance generated TNUB and UPFB, while in the group C4 the L conveyance higher than U conveyance produce the TNLB and DPFB.

The occurrence of the observed modes can be explained by the combination of the flow rates and slope values imposed to each branch during the experimental runs, as summarized in Table 3. NHFS and EJP modes are associated, respectively, with the lower (in U and L) and higher (in U and L) flow rates, regardless of the slopes assessed in these branches. On the other hand, the other modes of pressurization were directly influenced by the slopes of the branches. Overall, the bore propagation upward of the junction (in U or L branch) was observed in the branch with the higher values of inflow and slope.

4. Final Remarks and Conclusions

This work presents a first systematic description and characterization of the processes leading up to the pressurization of stormwater T-junctions. Besides the important insights related to the flow features linked to the pressurization, this work indicates that certain flow conditions will lead to early pressurization of junctions, which is caused by the entrapment of an air pocket. Air pocket entrapments are typically neglected in single-phase hydraulic models that are used in the description of the rapid filling of stormwater systems. By pointing out to conditions in which the predictions of single-phase models may fail, we hope to guide the improvements of future numerical tools used in stormwater hydraulics analysis.

Overall, five different types of flow pressurization modes were observed, with an additional two transitional types of pressurization that have characteristics that are common to two of those. Although the most experimental tested conditions could be successfully represented by single-phase models—i.e., neglecting air phase interactions—the cases that were characterized by air pocket entrapment were the ones with the largest pressure rises in the junction. These pressure spikes were similar to the ones reported in [34], albeit with a more complex experimental apparatus. Different types of pressurization interfaces were observed including wedge-like gradual flow regime transitions, pipe-filling bores, and undulatory front that reached the pipe crown and pressurized flows leaving a trail of air behind.

The pressurization modes visually observed were also consistent with the pressure variations gathered by the pressure sensors, proving the occurrence of the phenomenon. In addition, the use of a greater number of pressure sensors near the junction, as well as the measurement of velocity variations at the bottom of the junction, can allow a better understanding of the pressurization phenomena and should be considered in future studies.

The unsupervised clustering analysis allowed to confirm, without a prior criteria, whether the patterns of a previous pressurization grouping done based on the visual interpretation of the pressurization of a T-junction were clustered by the experimental configurations. The procedure proved to be effective in clustering the runs according to the normalized flows which embrace the different flow rates and slopes, the main experimental features assessed, and which led to the different pressurization behavior at T-junction during the infilling process. Therefore, the multivariate analysis based on hierarchical clustering can be an interesting tool to identify the prevalence of different pressurization modes at junctions by considering the main characteristics of flow conveyance during the rapid filling process, mainly when a dynamic and large set of configurations is been assessed.

Besides achieving a better understanding of the flow characteristics associated with the pressurization of a stormwater junction, this research has contributions on the numerical modeling of such flows. Most flow combinations results in very gradual filling (i.e., NHFS pressurization mode), or in conditions that led to the development of hydraulic bores. These flow features can be adequately represented by existing hydraulic models that are build with the Saint-Venant equations. In particular, these models would need to be constructed with adequate spatial discretization and the simulation carried out with Courant number near unity (to minimize numerical diffusion/dispersion). Yet, some cases shown the development of early junction pressurization and the entrapment of a large air pocket at the junction. In such cases, the air pocket presence will violate assumptions used in most existing 1-D hydraulic models, and would only be modeled using 3-D computational fluid dynamic tools.

This research, being the first one of its kind, has its limitations and still left many important questions to be addressed in future works. First, a systematic evaluation of the scale effects associated with air-water were not attained given that all tests were performed using a single pipe diameter. Past studies, such as [45,46], have clearly indicated the importance of air entrainment in bores, and that large diameters will influence the nature of air bubbles entrained in these bores. Since some pressurization modes also are characterized by moving bores, scale effects associated with junction pressurization should be a focus of future investigation.

In order to address the limitations of the present work, future research should consider a wider combination of $Q_U$ and $Q_L$, particularly between the positive values in this research, should be tested to access the transition between the pressurization modes identified in this work. For lower inflow fronts, shallower slopes $S_U$ and $S_L$ should also be applied. Improved observation of flow features will be attained if the junction was constructed in transparent material, showing more clearly the early stages of the air pockets formation in the junction. Finally, another point to be considered is the addition of different geometry configurations like additional reaches, junctions with angles lower than 90 degrees or even local storage at the junctions, which could emulate a manhole or a shaft, and assess to what extent these vertical structures impact the development of pressurization in stormwater junctions.