Flow regime transition simulation incorporating ...

Flow regime transition simulation incorporating entrapped air pocket effects Jose G. Vasconcelos∗, Peter R. Klaver†and Daniel J. Lautenbach‡ November 21, 2013

Abstract Flow regime transition is observed in closed conduit when there is a transition between free surface and pressurized flows. This condition is often observed in stormwater storage tunnels undergoing rapid filling. Such tunnels provide stormwater control in urbanized areas, relieving the stormwater collection system during intense storms. Operational issues have been reported in such tunnels during intense rain events, some of which linked to air pocket entrapment. These issues motivated development of flow regime transition models and more recent models attempt to improve predictions by accounting for air effects in formulations. However no comparative study was performed between recent models that account for air pockets and earlier models that don’t include this feature. This work aims to address this gap, focusing on an actual tunnel geometry that was proposed for the city of Washington, DC. Results indicate that including air in model formulation yielded more realistic predictions of low pressures. Keywords: Flow regime transition; Stormwater Storage Tunnels; Air pockets; Numerical modeling

1

Introduction

Flow regime transition, which refers to a flow condition where there is a transition between free surface and pressurized flow regimes (Cunge and Wegner, 1964), is observed in various hydraulic systems, such as water mains and penstocks undergoing rapid filling, as well the filling of stormwater storage tunnels. The latter application is increasingly relevant, as use of such storage tunnels in urbanized areas becomes more widespread. Some of these tunnel systems have diameters on the order of 10 m and ∗ Assistant Professor, Dept. of Civil Engineering, Auburn University, 238 Harbert Engrg. Center, Auburn, AL, 36849. E-mail: [email protected] † Senior Project Engineer LimnoTech 501 Avis Drive Ann Arbor MI, 48108 . E-mail: [email protected] ‡ Project Engineer LimnoTech 501 Avis Drive Ann Arbor MI, 48108 . E-mail: [email protected]

1

lengths on the order of several kilometers (Song et al., 1983), which in turn render their implementation very costly to municipalities. Yet, operational issues have been observed in such stormwater systems undergoing rapid filling. Such issues include significant pressure surges (Guo and Song, 1990), manhole lids blowouts and structural damage (Zhou et al., 2002b), and even episodes of urban geysering (Guo and Song, 1991). The survey conducted by Lautenbach and Klaver (2010) with nine US cities that operate large stormwater storage tunnels has indicated that seven of these systems have presented operational issues of varying degrees of severity. Works by Zhou et al. (2002a); Vasconcelos and Wright (2011) among others have linked some of these occurrences to the presence of entrapped air pockets in closed conduits. These issues have motivated the development and improvement of numerical modeling techniques applied to flow regime transition. Pioneer contributions in flow regime transition simulation include the works by Cunge and Wegner (1964) and Wiggert (1972). The former has applied the Preissmann slot concept into a shock-capturing model that solved the Saint-Venant equations using the finite difference method and the semi-implicit Preissmann scheme in the context of penstock flow simulation. The work of Wiggert (1972) was based on a shock-fitting approach, explicitly tracking the pressurization interface so that a rigidcolumn (lumped inertia) approach would be used to simulate the pressurized flow regime while the free surface portion of the flow was simulated using the method of characteristics (MOC). The shock-fitting approach was advanced by Song et al. (1983), developing what is referred to as ”Full Dynamic Models” that applied MOC to both pressurized and free surface flows (Song et al., 1983; Cardle and Song, 1988; Politano et al., 2007). Shock-capturing models were also advanced using improved numerical techniques (Garcia-Navarro et al., 1994; Capart et al., 1997; Trajkovic et al., 1999); however, the Preissmann slot approach prevented the simulation of subatmospheric flows. The TPA approach proposed by Vasconcelos et al. (2006) enabled shock-capturing models, employing a single set of equations for both flow regimes, to simulate flow regime transition and sub-atmospheric flows. While the shock-fitting models were already able to simulate sub-atmospheric flows, they are more difficult to implement because the interface must be tracked explicitly. However, none of the aforementioned models is able to include the effects of entrapped air in their flow regime transition formulation. As pointed out earlier, some of the most severe operational issues in stormwater tunnels have been linked to the presence of entrapped air in the conduits. Incorporation of this feature in modeling efforts is thus important to improve modeling utility. One of the first instances of numerical modeling of air-water interactions in closed conduit flows is presented by Martin (1976), which proposed a model that simulated the advance of a water column into a region filled with air. Water flow was modeled using a rigid-column approach, whereas air pressurization was computed assuming ideal gas behavior. It was further assumed that the air pressure is uniform. Many flow regime transition models have adopted this strategy to represent air-water interactions in closed conduits. Among these models one includes the works by Li and McCorquodale (1999); Zhou et al. (2002a); Vasconcelos and Wright (2009) and Leon et al. (2010). Alternatives that represent air pressurization discretely include Arai and Yamamoto (2003) and Trindade and Vasconcelos (2013). Notably, the possibility of air pocket entrapment during rapid inflow conditions has the potential to adversely impact prediction results if a model does not explicitly account for the existence of the air phase. If flow conditions in a hydraulic system (e.g. tunnel system) are such that the transition from free surface into pressurized

2

Figure 1: Air pocket entrapment in stormwater tunnels at an certain time step in an actual simulation of rapid filling of a stormwater tunnel flows occurs at two different locations within the same conduit reach, an air pocket entrapment has taken place. This is shown in Figure 1, obtained from an actual inflow simulation by a TPA-based model using the geometry of a proposed stormwater tunnel. In such conditions, if the model formulation does not recognize the existence of air phase, this pocket region is erroneously simulated as if flow regime was in free surface mode, which is not the case since air pressure there is not atmospheric. As the simulation continues, the size of this ”void region” decreases and may eventually collapse entirely, generating a large pressure pulse as the water columns rejoin. Such a prediction is clearly incorrect, since (in the absence of perfect ventilation) pockets would be compressed by the surrounding flow, with no sudden collapse. Another topic in flow regime transition simulation is the value of the acoustic wave speed, which is important because pressure pulses scale directly with its magnitude. However, the magnitude and relevance of this parameter in cases when flow regime transition occur are uncertain. Magnitude of the acoustic wave speed is dependent on the air phase fraction in the pipe cross section Wylie and Streter (1993). Rapid filling condition favor air entrainment in pressurized flow regions within stormwater tunnels, and while one could anticipate a general decrease in the magnitude of the acoustic wave speed due to air entrainment, its temporal and spatial variation remain difficult to predict. With respect to relevance, it is clear that the acoustic wave speed has an important role in providing the velocity by which changes in the flow at key locations influence flow conditions elsewhere. However, with respect to the pressure predictions in tunnel undergoing flow regime transition, this relevance of this parameter is not as well established as a number of complicating factors exist. First, phenomena leading to waterhammer (e.g. sudden pump failure) are usually not present during rapid filling tunnel simulation. Second, propagation of pressure pulses would be attenuated by the existence of dropshafts and other structures that may provide pressure relief as noted by Vasconcelos and Leite (2012). And third, the presence of sizeable air pockets may provide further pressure attenuation if the pocket act as a hydropneumatic tank

3

Martin (1976). In summary, despite significant developments in the modeling strategies used in flow regime transition simulations, some important unanswered questions still remain. The objective of the present work is to study the effects of accounting for the presence of entrapped air pockets in the predictions of flow regime transition models. As presented, the inability to include entrapped air in model formulation has the potential to adversely impact simulation accuracy. This possibility is assessed by performing a comparison between two models that are exactly the same except in the approach used to handle the presence of entrapped air pockets. Simulations are conducted using an actual stormwater tunnel geometry proposed for the city of Washington, DC, for a variety of flow conditions that result in pocket formation. Also, another objective of this work is to study the relevance of the acoustic wave speed value in instances when air pockets may become entrapped. This is performed by systematically varying the value of this parameter while maintaining all other parameters unchanged in both of the models used in this work.

2 2.1

Methodology Numerical model

As noted, the present study compares approaches for handling the existence of air pockets in simulations in two models that are otherwise identical. These two finitevolume models, referred here as SHAFT (herein SH) and SHAFT-Air (herein SA) solve flow regime transition problems applying the Saint-Venant equations expressed in conservative, divergent format. These equations are further modified by the twocomponent pressure approach (TPA) proposed by Vasconcelos et al. (2006). The set of equations that are solved at each finite volume cell is presented below: ⃗) ⃗ ∂F (U ∂U ⃗) + = S(U ∂t ∂x [ ] [ ] ] [ Q 0 ⃗) = ⃗ = A , F (U ⃗) = 2 U , S( U Q Q gA(So − Sf ) + gA(hc + hs ) A

(1) (2)

in which: A is cross-sectional area of water flow, Q is the water flow rate, hc the distance between the free surface and the centroid of the flow cross-sectional area, hs is the pressurized flow pressure head, So is the bed slope and Sf is the energy slope (given by Manning equation in this implementation). As shown in Vasconcelos et al. (2006), by considering the pipe wall elasticity while neglecting the fluid compressibility, one demonstrates the complete identity between the Saint-Venant equations and the correspondent mass and momentum equations for closed-pipe flows. Compared to the traditional Saint-Venant equations, the TPA model introduces hs as a new term in the momentum flux that allows for the separation between the hydrostatic-like pressure term and the pressure term that is expected only in pressurized flows (including siphonlike flows). The expression for hs depends on A, on the wave celerity a (assumed constant for a given pipe reach) and on the conduit cross-sectional area Apipe : a2 A − Apipe hs = (3) g Apipe These equations are solved discretely using a finite volume method framework, by which the vector of conserved variables [A, Q]T are updated at each one of the cells in

4

the solution domain over time according to expression (Toro, 2001): ( ) ⃗ in+1 = U ⃗ in − ∆t (F ⃗ (U )n+1/2 − F ⃗ (U )n+1/2 + ∆tS ⃗in U i+1/2 i−1/2 ∆x

(4)

in which: n, i are the time and space indexes respectively. The fractional indexes in ⃗ (U )n+1/2 indicate cell-interface fluxes rather than the flux of conserved terms such as F i+1/2 variables presented in equation 2. These fluxes need to be computed according to the selected numerical scheme, and in this present work the non-linear, 1st order accurate Roe scheme was selected: n+1/2

F (U )i+1/2 =

2 ) 1∑ 1( ⃗ n ˜ (j) |∆˜ ⃗ )n − F (U )i+1 + F (U |λ v (j) r(j) i 2 2 j=1

(5)

˜ (j) and r(j) are respectively the approximate eigenvalues and eigenvectors in which λ obtained from a linearized Jacobian matrix A˜ whereas ∆˜ v (j) is a measurement of the strength of the j th wave that is generated at the cell interface (Toro, 2001; Guinot, 2003). The implementation of the Roe scheme into the finite volume update formulation yields: [ ] ∑ (j) (j) n ⃗ in+1 = U ⃗ in − ∆t (F ⃗i+1 ⃗in ) − U +F |λ |(∆v (j) )i+1/2⃗ri+1/2 2∆x j (6) [ ] ∑ (j) ∆t (j) n n ⃗i−1 + F ⃗i ) − ⃗in (F |λ |(∆v (j) )i−1/2⃗ri−1/2 + ∆tS + 2∆x j The solution is updated at regular time steps that are limited by the CFL condition. The Courant number used in simulations are usually in the range of 0.90 to 0.95. The size of the finite volume cells ∆x varied for every tunnel reach and was selected to match the conduit diameter D. Some final remarks on the numerical modeling strategy used here pertain to the initial conditions and boundary conditions used in the simulation. For initial conditions two possible alternatives are used: the tunnel is either initially empty, or a constant, quiescent water surface is present (that is, the tunnel is partially full). To avoid computational difficulties in handling dry fronts, which are not the focus of the simulations, ”empty” conduits are initially assigned a flow depth of 0.1% of the tunnel diameter. At tunnel system boundaries, such as tunnel junctions and dropshafts, the numerical solution outlined above cannot be applied. Instead the approach used here is to combine a relevant characteristic equation (associated with a characteristic line) that would be generated in the solution domain, and combine this with expressions enforcing mass conservation and energy conservation as required by the specific boundary condition. Due to the large diversity of geometries encountered in such stormwater tunnel junctions, a detailed description of the solution strategy is not provided here. However, the interested reader is referred to Sturm (2010) for examples on how these boundary conditions may be constructed.

2.2

Air phase modeling

The description presented earlier demonstrates the method by which SH model updates flows at internal model cells. One notices that there is no special handling of the air phase. However, because the model solves the free surface without introducing any

5

assumption with regards to its shape, it has the ability of detecting the onset of a pocket formation, as shown in Figure 1. Because there is no specific consideration of air phase, SH model continues the simulation with the assumption that this air would ventilate ideally, and not pressurize while the pocket volume is filled by water. At the moment when this pocket volumes becomes zero, the water pressure spikes much in the same way as a vapor pocket collapse observed in transient flows involving cavitation. Upon pocket collapse, the associated change in the pressure head ∆hs can be computed in terms of shrinkage rate of air pocket volume Vair as: ∆hs =

dVair a 2gApipe dt

(7)

However, rather than a sudden pocket collapse, the entrapped air pocket should undergo a series of expansion and compression cycles as it interacts with the surrounding water flow. The SA model attempts to reproduce this general behavior and draws some of its idea from the work by Li and McCorquodale (1999). The modeling assumptions used for the air phase are: 1. Air pocket length and correspondent coordinates are unchanged upon pocket entrapment. 2. Air pocket thickness is uniform. 3. There is only a single entrapped air pocket per reach. The first assumption, albeit limiting, is necessary since at this stage there are no definitive guidelines on how to provide 1-D modeling estimates of pocket displacement considering the combined effects of varying background flow velocities and conduit slopes for circular pipes. The second assumption implies that the proposed modeling framework does not consider any details of the shape of the entrapped pocket. The third assumption is made largely for simplification of the computations, noting that concurrent air pocket entrapments in a given reach have not been observed in previous SH model applications. It has been observed in experiments by the authors that pockets may undergo small to moderate fragmentation following entrapment; however, it is assumed that this phenomenon would not have a major impact in the modeling description considering the focus is to represent the pressure surges caused by pocket compression/expansion. Despite these simplifying assumptions, the proposed conceptual model to simulate air phase compared well to experimental data on sudden air pocket entrapment presented by Vasconcelos and Leite (2012). The SA model thus has all the computational steps of SH model with the addition of two more: 1) identification of whether pocket entrapment has taken place anywhere in the conduits; and 2) solution of the region of the flow domain that contains the air pocket using a separate set of equations based on lumped inertia, air phase continuity and the ideal gas law: dQcol g.Acol ( Lcol Qcol |Qcol | ) = . Hups − Hdws − f (8) dt Lcol D 2gA2col dHair Hair dVair = −k. . (9) dt Vair dt dVair = Qdws − Qups (10) dt Equation 8 represents the momentum conservation in the column underneath the air pocket, with Qcol the flow rate under the air pocket, Lcol the length of the column,

6

Acol the water flow area in the region of the pocket, f the friction factor, Hups and Hdws corresponding to the pressure heads upstream and downstream from the pocket region. Equation 9 is the implementation of the ideal gas law for the pocket, with Hair the absolute pressure head for the air phase within the pocket, Qups and Qdws corresponding to the flow rates upstream and downstream from the pocket region, and k as the polytropic exponent (adopted here as 1.2). Equation 10 is the expression of the air pocket volume change over time. A schematic of the conceptual model is presented in Figure 2. The procedure for the initialization of variables Qcol , Lcol and Acol is as follows. The flow rates in the computational cells under the air pocket are averaged, and this average is assigned as Qcol to the rigid column region under the air pocket. The pocket length Lcol is determined by tracking the location of the pressurization interfaces, and this length is assumed constant during simulation. The summation of the area of the air in each computational cell at the moment of the pocket entrapment yields a total pocket volume Vair , and the variable Acol is determined with the expression Acol = (Lcol Apipe −Vair )/Lcol . Acol is updated as new air phase volumes are computed every time step. Fundamentally, each tracked air pocket behaves in the simulation as an internal boundary condition. Head and flow rates upstream and downstream from the pocket (Hups , Hdws , Qups , Qdws ) are obtained from cells computed using the St. Venant equations, beyond the buffer zone in Figure 2. This buffer zone has a length comprised by 1 or 2 computational cells. Upon the solution of equations 8 to 10, one can obtain the pressure head in the pocket region as well the water flow rate underneath the air pocket. It is assumed that these values calculated for the air pocket region are also valid for the buffer zone. At the buffer zones flow fills the pipe cross section, and it is possible to calculate at these zones the conserved variables (A, Q) so that the air pocket pressurization effects to flow and pressure are transmitted to the remainder of the model. One reiterates that equation 10 does not account for any ventilation. However, in the event that a ventilation point exists, the change in pocket volume could be simulated by the addition of a head-discharge relationship to the system of equations, following Zhou et al. (2002a). One notices that the lumped inertia adopted for the air phase prevents the representation of water compressibility/pipe elasticity effects, which is a shortcoming of the modeling approach. However, we recall that these lumped inertia calculations take place only in regions within reaches that have entrapped air pockets. Simulations performed to date indicate that these regions do not correspond to a large portion of the solution domain, so it is estimated that these impacts are limited. A final limitation with regards to air modeling, specifically the neglect of early air pressurization during rapid filling, needs to be discussed. Earlier work from the authors (Vasconcelos and Wright, 2009) has indicated that air pressure head in experiments of rapid filling with limited ventilation can exceed twice the pipeline diameter. However, the translation of these findings to models that involve more complex system geometry is very hard. To consider air pressurization during filling one would need to be able to develop headdischarge relations at air release points (e.g. shafts) during the filling process. To the authors’ knowledge, studies to collect field measurements of air pressurization have not been performed, which makes the development of such relationships difficult. The work by Li and McCorquodale (1999) present such a head-discharge relation, but a question is whether this would hold for larger, prototype scales.

7

Figure 2: Conceptual model for the entrapment of air pockets in proposed for SA model

2.3

Description of tunnel system and modeling parameters

The proposed tunnel configuration studied here is based on a portion of the system currently being designed as part of the District of Columbia Water and Sewer Authority’s (DC Water) combined sewer overflow Long Term Control Plan, referred to as the Clean Rivers Project. The project includes several phases of deep tunnel construction; the first phase consists of a 7-meter diameter tunnel running approximately 9.8 km along the Anacostia River from near RFK Stadium to the Blue Plains Wastewater Treatment Plant, at depths from 24 to 30 meters below grade. This tunnel, which includes a 1.2-km spur tunnel running from Poplar Point to the Main & O Pump Station, will intercept 14 CSOs that currently discharge to the Anacostia River, and provide about 430,000 m3 of storage. A second phase of construction will extend the tunnel along the Northeast Boundary Sewer, along with additional spur tunnels that together provide another 291,000 m3 of storage. For this study, a slightly simplified representation of the first phase was modeled, consisting of the Anacostia River and Main & O spur. A profile view of this configuration is shown in Figure 3. Boundary conditions employed in this study were generally of the type described above, combining a characteristic line from each reach connected to a particular junction with a continuity expression for volume within the junction. For situations where tunnel reaches are surcharged, the free surface in the junction shaft is represented by writing energy and volume balance equations that consider the tunnel reaches to act as rigid columns. Some pressurized boundary conditions that may be implemented with the MOC (e.g. pumps) were not simulated in the present study, although MOC approaches have been used successfully in the SH model framework in other applications. Inflows to the proposed tunnel system were derived from a 1-dimensional linknode type of collection system model representing the entire combined sewer service area. The tunnels are conservatively sized and provide a high degree of control with

8

Figure 3: Selected tunnel system geometry applied in Washington, DC Long Term Control Plan and used in this work respect to the number of overflow events that are completely contained. Low-frequency, extreme events, on the other hand, are expected to produce rapid filling effects along with concomitant risks, which are important to protect against and thus must be considered in the design process. Inflows resulting from a storm event with a 100-year return period were computed with the collection system model and used, in conjunction with the SH model, to inform the design process of the various phases of the tunnel system (Lautenbach et al., 2008, 2012). This 100-year event was also used in the present study, as it is known to induce rapid filling phenomena of interest with respect to the comparison of the two modeling frameworks. Inflows occur at eight locations in the modeled system; Figure 4 shows a composite hydrograph for all locations for the first four hours of the event. Although the event continues past this point, the entire system is full and overflowing at several locations by this point, and transient effects are largely absent. The investigation goals stated for this work included the assessment of various flow rates introduced to the tunnel system through ten of the tunnel junctions. The underlying idea of varying flow rates was an attempt to generate conditions whereby air pockets with different volumes would develop in the simulation. With these different pocket conditions, the next step in the analysis was to assess the difference in the simulation results yielded by SH and SA models. To achieve this, 100-year return period hydrographs derived from the collection system modeling were multiplied by 70%, 100% or 130%. These varying inflow conditions are herein referred to as Q∗ = 0.7, 1.0 and 1.3 respectively. Another goal was to assess the effect of the selected value for the acoustic wave speed in the simulation. This parameter is considered key in closed-pipe unsteady flow applications, and thus would be anticipated that it would impact the solution. It was uncertain whether the effect of varying the acoustic wave speed would be more pronounced for SH or SA model, and whether there was a correlation between this

9

Figure 4: (A) Inflows hydrographs used in the simulations at all junctions; and (B) example of the effect of the inflow multiplier Q∗ applied to CSO19 parameter and the existence of entrapped air pockets in the modeling results. As pointed out by Vasconcelos et al. (2009), the use of large values of acoustic wave speed yield post-shock oscillations at the pressurized side of pipe-filling bore fronts. In order to limit these oscillations, the range of values for the acoustic wave speed a used in this work was a = 100 m/s, a = 200 m/s and a = 400 m/s. Yet, as discussed earlier, with the possibility of air bubble entrainment in these flows, the proposed range of acoustic wave speed, particularly 200 m/s and 400 m/s, may be feasible to occur in such tunnels. Considering the three range of inflows used in the simulation, the three values of the acoustic wave speed used in the simulation and the use of both SA and SH models, a total of 18 numerical simulations were performed and the results are presented and discussed in the following section.

3

Modeling results and discussion

A general description of the predicted tunnel filling process as a consequence of the inflows admitted in the system junctions is presented below. All times are related to simulations cases performed when the inflow multiplier was Q∗ = 1.0. These would be either larger or smaller depending on the selected Q∗ value. 1. The simulation was initiated assuming an entirely empty tunnel as the initial condition. 2. The initial inflows are observed in the upstream reaches of the tunnel, from

10

junctions CSO19 to CSO05, with the former as the junction that contributes most inflows into the system. 3. For the scenarios when Q*=1.0, after 1.7 hours inflows have reached the lowest point of the tunnel, junction BP-TDPS (Blue Plains Tunnel Dewatering Pumping Station). 4. After 1.7 hours, an open-channel bore forms at BP-TDPS point and advances toward the upstream reaches of the tunnel. This bore loses strength, however, and at about 2.2 hours into the simulation it is no longer clearly detected. 5. After 2.46 hours, a gradual transition into pressurized flow is observed at the BP-TDPS junction. This is a gradual flow regime transition, with no bore present initially. 6. As the filling proceeds, the pressurized region expands more rapidly, catching up with the backward front that was generated at 1.7 hours at BP-TDPS junction. During this process, the pressurization front becomes steeper, and at 2.53 hours into the simulation an open-channel bores re-forms in the simulation. At 2.67 hours, as indicated in Figure 5, a pipe-filling bore forms in the system. 7. This pipe-filling bore, with a height of approximately 3.0 m, advances rapidly at about 6.2 m/s toward the three-way junction at Poplar Point. Both upstream branches at that junction still flow in free surface mode. 8. At 2.75 hours the pipe-filling bore reaches Poplar Point junction, where it loses strength as it splits into two moving fronts, one advancing toward the main branch of the tunnel toward CSO19, the other toward MPS-DS junction at the the upstream end of the lateral tunnel. 9. As filling continues, the water level will eventually reach pressurization levels at Poplar Point. Both SH and SA models identified that when pressurization occurred at Poplar Point, air pocket entrapment soon followed in tunnel reaches adjacent to Poplar Point. In the majority of the simulated conditions this entrapment occurred in the lateral tunnel branch. Prior to this air pocket entrapment, results from both SH and SA models are indistinguishable. After the air pocket entrapment the differences between the model predictions began to be noticed. 10. Filling will continue and a pipe-filling bore continues to advance until it reaches CSO19 at time 2.9 hours. As the inflow continues the whole tunnel becomes pressurized. Results presented here focus on comparing and contrasting the predictions by SH and SA models for all tested cases. A comparison between the models in terms of the simulated pressure head hydrographs in CSO19 into the conduits is presented in Figure 6. One notices that differences between SH and SA model predictions at that point are negligible, possibly due to the distance between this point and the location where pockets formed in simulations. Note also that the differences in tested values for the acoustic wave speed had minimum impact in the resulting flows. As anticipated, larger Q∗ led to more rapid tunnel pressurization, which in turn caused earlier pressurization at CSO19. Results between the models show some discrepancy when the selected station is closer to the point where air pockets appear in simulation. Results of inflows into BAFB junction incoming from Poplar Point are presented in Figure 7 from both SH and SA models. Simulation results indicate that flows at that station are positive until the arrival of the inflow bore, which causes the flow rate to drop. Later in

11

Figure 5: HGL profile of the tunnel at the time when a pipe-filling bore forms during a simulation

Figure 6: Pressure heads at CSO19 predicted by SH and SA models for all tested Q∗ and a values

12

Figure 7: Flows from Poplar Point into BAFB junction predicted by SH and SA models for all tested Q∗ and a values simulation, when an air pocket is eventually captured by the increasing water depth at that tunnel region, model results begin to diverge, particularly for A∗ values of 0.7 and 1.0. SH model predictions, based on the incorrect assumption of perfect ventilation, indicate the pocket will disappear and the water column collapse over itself, generating waterhammer-like pressures. SA model on the other hand preserves the air pocket phase, which may undergo compression/expansion cycles depending on the modeling condition. These results also indicate that for both SH and SA model the effect of the acoustic wave speed in the flow hydrograph simulation is minor. An issue related to the misrepresentation of air pocket behavior is noticeable in the numerical oscillations presented in Figure 7. Such oscillations may be described as pseudo-physical: they arise from pocket collapse, which is a non-physical artifact of the assumption of perfect ventilation, but their behavior is physically the same as a pressure pulse in a closed pipe MOC model. In this sense they should be distinguished from the numerical post-shock oscillations that are characteristic of shock-capturing models when pipe-filling bores are represented. Such post-shock oscillations in the simulated HGL average out and have no net effect in shaft water levels/diameters, as shown in Vasconcelos et al. (2009), for both the SH and SA models. The pseudophysical oscillations in the SH model, on the other hand, are absent from the SA model because pocket collapse does not occur.. A key distinction between SH and SA models predictions is observed looking at the predicted minimum HGL profiles at the reaches near to entrapped air pockets. As indicated in the results presented in Table 1, SH model predictions tend to present values for the minimum pressure head at these reaches that are smaller than the predictions yielded by the SA model. As explained, SH model assumes perfect ventilation. As result, following the entrapment of an air pocket, what is predicted by the SH

13

Figure 8: Pressure hydrographs at junction 6 (close to Poplar Point) following the collapse of an air pocket for SH model. Results of SA model (no pocket collapse) are presented for comparison for two a values model is that the region occupied by the pocket shrinks as the filling progresses. At some point in the simulation the pocket will vanish, as the model predicts the filling water columns collapsing on one another much in the same way as column rejoining following column separation caused by cavitation in water pipelines. This is an unphysical outcome considering that there is no air ventilation at that point, and this faulty assumption results in the wrong representation of air-water interactions during rapid filling. Strong, non-physical oscillations and negative pressures were observed in SH model, and the magnitude of these negative pressures increased with the acoustic wave speed a, as shown in Table 1. A pressure time series at a location near the air pocket collapse downstream of Poplar Point is shown for both models in Figure 8. One notes that the period of these oscillations is explained by the acoustic wave speed and distance between the nearest drop shafts, which is about 2650 meters. These oscillations have period values that are consistent with the flow rate hydrographs for the SH model results presented in Figure 7. SA model results in Figure 8 show no sign of such pressure oscillations. Modeling results yielded by the SA model generally show larger values for the minimum piezometric pressure heads, and there is no identifiable relation between the pressure heads and the acoustic wave speed values. This is because the SA model accounts for the mass of the air in the system, and will not allow for a collapse of the pocket. Instead, the pocket presence is somewhat similar to a hydropneumatic tank within the conduit. SA model indicates that near these pockets pressures may oscillate slightly in response of varying flow conditions, and may even become slightly sub-atmospheric. While the relative variation between SH and SA model results may not be significant in absolute

14

Table 1: Minimum piezometric head reached near pocket entrapment SH Model a=100 m/s Q*=0.7 0.68 Q*=1.0 2.38 Q*=1.3 2.64 SA Model Q*=0.7 Q*=1.0 Q*=1.3

a=100 m/s 1.64 2.48 4.66

(m) predicted by SH and SA models at a=200m/s -7.49 1.18 -1.22

a=400m/s Vapor pressure Vapor pressure Vapor pressure

a=200m/s -2.92 4.73 3.25

a=400m/s 3.31 3.72 -0.44

terms (particularly for smaller a values), the cause for these minimum pressures in SH model is non-physical, and can for larger a values even lead to vapor pressure levels. The practical outcome of these predictions of sub-atmospheric pressures are very significant. There are concerns that very low pressures may damage the lining of stormwater tunnels, which in turn may lead to further structural damage to the tunnel walls. For this reason, accurate predictions of sub-atmospheric pressure are of interest to design engineers. Careful observation of SH modeling results, however, indicate that these low pressures may result from the inability of the model to properly incorporate the effect of an entrapped air pocket in the conduits, and may be unrealistically low. An important point to reiterate is that both modeling frameworks, SA or SH models, are not able to simulate the onset of distributed cavitation. To the authors knowledge there is no approach that has been able to include air pressurization effects and distributed cavitation in the same mixed flow modeling framework. Recent developments (Vasconcelos and Marwell, 2011) may provide a means by which this integration may be achieved. An interesting comparison between the predicted minimum HGL yielded by SH and SA models for the case when Q∗ = 1.0 and a = 400 m/s is presented in Figure 9. One notices that strong negative pressures in model reach 6 (between Poplar Point and BAFB junctions) are predicted by SH model, whereas SA modeling results predict milder negative pressures resulting from the nearby entrapped air pocket. These results are expected to be more useful for tunnel designers in that they provide a realistic depiction of the potential effects on tunnel linings. Discussion now focuses on the results presented by SA model with regards to the resulting entrapped air pockets. Most of the simulations indicated that there would be air pocket entrapment at reaches connected to Poplar Point. This is an interesting and practical outcome in terms of design, since it provides an indication that this junction requires careful consideration with regards to air ventilation, so as to avoid operational issues such as geysering. The model provides detailed results on predicted characteristics of entrapped air pockets and related water flows, and these results indicate that sizes of entrapped air pockets are very dependent on the shape of the free surface flow at the moment when two pressurization interfaces develop in a reach, enclosing an air pocket. This shape will clearly depend on the selected Q∗ as is indicated in Figure 10. This figure also shows the evolution of the predicted air pocket volumes following their entrapment, as well the changes in water flows underneath air pockets. An would be anticipated, predicted air pocket volumes decreased with

15

Figure 9: Minimum pressure head profiles predicted for Q∗ = 1.0 and a = 400m/s at the reach next to Poplar Point three-way junction time as result of overall pressure increase as the filling process continued. There was no correlation between the inflows that were admitted in the system and the flow rate under the air pockets, which depends on the pressure gradient in the tunnel segment. This may explain why flow rate fluctuations do not match oscillations in pocket volume/pressure, and why the amplitude of flows rate variations is not linked with the amplitude of pressure variations. Theoretically, acoustic wave speed values should not be linked to the resulting air pocket volume. However, results in Figure 10 point to such a relationship, even though no clear trend was identifiable. The authors speculate that characteristics of shock capturing models applied to flow regime transition problems (either Preissmann slot or TPA conceptual models) may be the cause of such behavior. Such models establish a link between surcharge head and storage in pressurized flows. Considering the range of surcharge pressures heads that are observed in simulations (around 25 m), the cross sectional storage due to pipe elasticity can reach values ranging from 2.4% of the pipe cross sectional area (a=100 m/s) down to 0.60 % and 0.15 % of the pipe cross section (a=200 m/s and 400 m/s respectively). This added storage slightly delays the advance of pipe-filling bores, which in turn have an impact on the exact location of the air pocket boundaries in the moment when two pressurization interfaces are formed. Given the complexity of the inflows in the system, it becomes impossible to anticipate the relationship between acoustic wave speed and air pocket volume for a given inflow scenario. However, because the effects of elastic storage for larger acoustic wave speeds are minimized for larger a values, the authors anticipate that the results obtained for larger a values should represent more accurately entrapped air pocket volumes.

16

Figure 10: Predicted values for air pocket volume and water flows underneath air pockets predicted by SA model

17

4

Conclusions

Flow regime transition models can be valuable tools in the hydraulic analysis of complex stormwater tunnel systems as a means of anticipating operational issues linked to surges. Important theoretical developments have gradually improved the ability of these models to represent more diverse flow conditions in these systems, enabling a broader range of simulation conditions to be considered during tunnel design. While the majority of current flow regime transition models are unable to explicitly incorporate some key air phase interactions in their formulations, recent research has linked the occurrence of relevant operational issues to the presence of large entrapped air pockets that apparently form as these tunnels undergo rapid filling. New models formulations have been proposed to address this limitation, but no study has performed a direct comparison of the advantages of incorporating air pockets effects into the dynamics of the water filling process. This is one of the motivations for this work, along with an attempt to assess the importance of the adopted value for the acoustic wave speed in predictions. The introduction of more sophisticated flow modeling approaches combining finitevolume and non-linear numerical schemes for the simulation of tunnel systems is relatively new. To date, such models have been successful representing laboratory observations (Trajkovic et al., 1999; Vasconcelos et al., 2006). More limited number of studies were performed applying field scale measurements (Trindade and Vasconcelos, 2013), and thus some important up-scaling questions still persist, particularly because of air-water interactions. It is believed these questions will persist as long as studies with adequate sampling frequency and/or spatial detail are not performed. Predictions from a TPA-based model that neglects air pockets in the formulation (SH model) were compared to a similar model that conserved air pocket mass and incorporated effects of air pocket compression/expansion (SA model) for a variety of flow conditions using a realistic stormwater tunnel geometry. The formation of air pockets led to discrepancies between the two modeling approaches predictions, particularly for points near the points where pocket formed. Two main types of discrepancies were noted: a) pressure and flow oscillations were much more intense in SH predictions following the instant when air pocket vanished from modeling simulation; and b) minimum pressures predicted by SH model were much stronger than SA model, leading to cavitation in the conduits at certain tested cases. These discrepancies arise from a non-physical assumption of ideal ventilation that SH model, and many other existing flow regime transition models, implicitly adopt. These low pressures seem thus to be strongly overestimated, which in turn may have important practical implications for designers as this may affect the design of the lining of these tunnels. Albeit not a mature model, SA offers a glimpse on the behavior of these systems when there is pocket formation. In practice, a combined analysis involving both SH and SA models will present a deeper insight on potential issues during the rapid filling of these stormwater tunnels. Thus, an important conclusion is that the incorporation of air phase effects when simulating rapidly filling tunnels is a critical step in obtaining useful predictions of anticipated flow conditions. While the assumptions inherent in one-dimensional modeling frameworks limit their ability to precisely resolve certain features of rapidly varied flow, such models still provide insights into the risk of potentially significant phenomena such as bores and entrapped air pockets, and their refinement is of use to designers. There are many other improvements that can be implemented in future models that incorporate air pressurization effects. One would be the tracking of en-

18

trapped air pockets as they move and spread in tunnel reaches. Another envisioned improvement would be to incorporate the effect of air entrainment through pipe-filling bores in decreasing the acoustic wave speed values.

19

A

Notation

The following symbols a = A = Acol = Apipe = D = f = ⃗) = F (U g = hc = hs = Hair = Hups , Hdws = I = k = L = Lcol = M OC = p = p∗ = Q = Q∗ = Qcol = Qups , Qdws = ⃗r = ⃗) = S(U SA = SH = ⃗) = S(U So = Sf = t = TPA = ⃗ = U ∀ = ∀g = ∀l = V = Vair = y = z = α = ∆x = ∆t = ∆v = λ =

are used in this paper. Acoustic wave speed Cross-sectional area of the flow Water flow area underneath air pocket Cross-sectional area of pipe Conduit diameter Friction factor Vector of conserved variables fluxes Gravitational acceleration Distance between free surface and centroid of flow cross-sectional area Surcharge head Absolute air pressure pressure head Pressure heads upstream and downstream from the pocket region 1st momentum of inertia of the flow cross section Polytropic exponent Pipeline length Length of the air pocket/water column underneath air pocket Method of characteristics Pressure Absolute pressure Discharge Inflow multiplier factor Flow rate under the air pocket Flow rates upstream and downstream from the pocket region approximate eigenvector Vector of source terms SHAFT-Air model SHAFT model Vector of source terms Pipeline slope Energy slope Time from flow startup Two-component Pressure Approach Vector of conserved variables = [A, Q]T Volume of discretization cell Volume of free gas within discretization cell Volume of water within discretization cell Flow velocity Air pocket volume Piezometric head Elevation from datum Fraction of free gas content in the flow Dimension of space discretization Dimension of time discretization Strength of the wave crossing cell interface Approximate eigenvalue

20

References Arai, K. and Yamamoto, K. (2003). Transient Analysis of Mixed Free-surfacePressurized Flows with Modified Slot Model (Part 1: Computational Model and Experiment). Proc. 4th ASME-JSME Joint Fluids Engineering Conference. Honolulu, Hawaii Cardle, J.A. and Song, C.S.S. (1988) Mathematical modeling of unsteady flow in Storm Sewers. Int. J. Eng. Fluid Mech. 1(4) 495-518. Capart H, Sillen X, and Zech Y. (1997). Numerical and experimental water transients in sewer pipes. J. Hydr. Res., 35(5): 65972, 1997. Cunge, J.A. and Wegner, M.(1964). Integration numerique des equations d’ecoulement de Barre de St. Venant par un schema implicite de differences finies. Application au cas d’une galerie tantot en charge tantot a surface libre.” La Houille Blanche, 1: 33–39, 1964. Garcia-Navarro, P. , Priestley, A. and Alcrudo F. (1994). Implicit method for water flow modelling in channels and pipes. J. Hydr. Res., 32(5): 721-742, 1994. Guinot,V.(2003) Godunov-type schemes: An introduction for engineers, 1st Edition. Elsevier, Amsterdam, Netherlands Guo, Q. and Song, C. S. S. (1991). Surging in Urban Storm Drainage Systems. J. Hydr. Eng., 116(12): 1523–1537, 1990. Guo, Q. and Song, C. S. S. (1991). Dropshaft Hydrodynamics under Transient Conditions. J. Hydr. Eng., 117(8): 1042–1055, 1991. Lautenbach, D.J.; Vasconcelos, J.G.; Wright, S.J.; Wolfe, J.R.; Cassidy, J.F.; Klaver, P.R. and Benson, L.R.(2008). Analysis of Transient Surge in the Proposed District of Colombia Water and Sewer Authority Deep Tunnel System. Paper presented at 72nd Annual Conference of the Indiana Water Environment Association, Indianapolis, IN Lautenbach, D.J.; Klaver, P.R.; Vasconcelos, J.G. and Wright, S.J.(2012). Numerical and Experimental Analysis of Surge, Transient, and Trapped Air Phenomena in Two Large Wastewater Storage Tunnels. Proc. 2012 North American Tunneling edited by M. Fowler, R. Palermo, R. Pintabona and M. Smithson Jr. Englewood, CO; Society for Mining, Metallurgy and Exploration.. Lautenbach, D. J. and Klaver, P. (2010). Experience of built tunnel systems with surges, transients and geysering. Memorandum from LimnoTech Inc. to London Tideway Tunnels Delivery Team, 2010. Leon,A.S., Ghidaoui,M.S., Schmidt A.R. and Garcia M.H.(2010) A robust twoequation model for transient-mixed flows. J. Hydr. Res., 48(1): 44–56, 2010. Li, J. and McCorquodale, A. (1999). Modeling Mixed Flow in Storm Sewers. J. Hydr. Eng., 125(11): 1170–1180, 1999. Martin, C. S. (1976). Entrapped Air in Pipelines. Proc. Second Int. Conf. on Pressure Surges London, UK

21

Politano,M., Oodgard,A.J. and Klecan, W.(2007) Case Study: Numerical Evaluation of Hydraulic Transients in a Combined Sewer Overflow Tunnel System. J. Hydr. Engrg., 133(10): 1103-1110, 2007. Song,C.S.S., Cardle,J.A., and Leung,K.S.(2004) Transient Mixed-flow Models for Storm Sewers. J. Hydr. Engrg., 109(11): 1487–1504, 1983. Sturm,T.(2010) Open Channel Hydraulics 2nd edition. McGraw-Hill, New York, NY Toro,E.F.(2001) Shock-Capturing Methods for Free-Surface Shallow Flows John Wiley and Sons, Chichester, UK Trajkovic, B.; Ivetic, M.; Calomino, F.; D’Ippolito, A.(1999). Investigation of transition from free surface to pressurized flow in a circular pipe Water Sci. and Tech. , 39(9): 105-112, 1999 Trindade, B.C. and Vasconcelos, J.G.(2013). Modeling of water pipeline filling events accounting for air phase interactions J. Hydr. Engrg., 139(9), 2013. Vasconcelos,J.G., Wright,S.J. and Roe,P.L.(2006). Improved simulation of flow regime transition in sewers: The two-component pressure approach J. Hydr. Engrg. , 132(6): 553-562, 2006 Vasconcelos,J.G., Wright,S.J. and Roe,P.L.(2009). Numerical oscillations in pipe-filling bore predictions by shock capturing models J. Hydr. Engrg. , 135(4): 296-305, 2009 Vasconcelos,J.G., Leite,G.M.(2012). Pressure surges following sudden air pocket entrapment in storm-water tunnels J. Hydr. Engrg. , 138(12): 1081-1089, 2012 Vasconcelos,J.G. and Wright,S.J.(2009). Investigation of rapid filling of poorly ventilated stormwater storage tunnels J. Hydr. Res. , 47(5): 547-558, 2009 Vasconcelos,J.G. and Marwell,D.T.B.(2011). Innovative simulation of unsteady lowpressure flows in water mains J. Hydr. Engrg. , 137(11): 1490-1499, 2011 Vasconcelos,J.G. and Wright,S.J.(2011). Geysering generated by large air pockets released through water-filled ventilation shafts J. Hydr. Engrg. , 137(5): 543-555, 2011 Wiggert, D.C. (1972) Transient flow in free-surface, pressurized systems. J. Hydr. Div., 98(1): 11–27, 1972. Wylie,E.B. and Streeter,V.L.(1993) Fluid Transient in Systems Prentice Hall Zhou, F.; Hicks, F. E. ;Steffler P. M. (2002a) Transient Flow in a Rapidly Filling Horizontal Pipe Containing Trapped Air. J. Hydr. Engrg., 128(6): 625–634, 2002. Zhou, F.; Hicks, F. E. ;Steffler P. M. (2002b) Observations of AirWater Interaction in a Rapidly Filling Horizontal Pipe. J. Hydr. Engrg., 128(6): 635–639, 2002.

22