Numerical study on the effect of tangential intake on vortex dropshaft assessment using pressure distributions

ABSTRACT Dropshafts with tangential intakes are common structures in urban water drainage systems. The plunging and vortex flowpatterns are the typical flow regimes, which are mainly affected by the tangential intake design and flood discharge. However, thehydraulic flow transition between vortex and plunging flow patterns has received little attention, which may affect the dropshaftoperation during extreme flood discharge events. In this study, the effects of the flow discharge and contraction ratio on the pressuredistributions of vortex dropshafts are analysed based on a series of numerical simulations. The pressure distributions of the plungingand vortex flow patterns are compared; two hydraulic factors are used to distinguish rotational and non-rotational dropshaftoperations. For appropriate vortex dropshaft designs, the minimum to maximum wall pressure ratio should be greater than 0.1 at theintake cross-section and greater than 0.2 during the falling process. Based on this principle, flows in the annular dropshaft areclassified into non-rotation, transition, and rotation regimes. The results agree with the theoretical rotation intensity of the vortex flow,and an empirical relation is proposed. The present study is easy to implement and practically useful to assess the vortex dropshaftdesign for local water drainage systems.


Introduction
Dropshafts are commonly used in urban water drainage systems to deliver water flows across different elevations. The tangential intake is one of the most common shapes and determines the dropshaft types, which are the vortex dropshaft and the plunging dropshaft (Jain & Kennedy, 1983). For plunging dropshafts, the water jet released from the tangential intake plunges directly into the opposite wall of the dropshaft, breaking and spreading along the vertical dropshaft wall (Granata et al., 2011). For vortex dropshafts, the approach water attaches along the dropshaft sidewall and moves as an annular swirling flow along the vertical shaft due to centrifugal force (Yu & Lee, 2009;Zhao et al., 2006). The hydraulic performances of different dropshafts are affected by the flow discharge capacity, intake chocking, and energy dissipation. With identical approach flow conditions, the transition between plunging and vortex flow occurs due to the design of tangential intake. The approach flow is deflected by the coupling effect between the intake and vertical dropshaft structures.
CONTACT Wangru Wei weiwangru@scu.edu.cn Because of the size limitation of the dropshaft, a stable annular flow pattern exists in the vortex dropshaft, and energy is effectively dissipated due to flow attachment and wall friction along the dropshaft (Guo, 1995;Vischer & Hager, 1995). To enhance the swirling effect of the annular flow, a contraction tangential intake is used. The flow clings along the wall and spirals down, forming an air core in the center of the dropshaft. Yu and Lee (2009) indicated that the contraction width of the tangential intake at the dropshaft junction was a key factor and provided a general design guideline for a tangential vortex intake that ensured smooth water flow. Conversely, increasing the contraction width can reduce the approach flow velocity under other identical geometrical conditions. Partial flows cannot be driven to cling along the dropshaft wall, impacting the opposite dropshaft wall. This causes the flow pattern to change to a plunging dropshaft flow (Ma et al., 2016). Due to the curved wall of the vertical dropshaft, a similar swirling flow is generated and attaches to the vertical shaft walls. Thus, the flow patterns in the transition process are generally similar, regardless of the tangential intake configuration. The plunging and vortex flows in the dropshaft resulted in different flow discharge capacity, energy dissipation and structural stress performances (Camino et al., 2009;Chanson, 2004;Rajaratnam et al., 1997). Despite the widespread use of vortex and plunging dropshafts, the hydraulic conditions between the two types of dropshafts have been scarcely studied in terms of flow transition assessment.
Recently, researchers showed that the numerical simulation was an effective way to attain reasonable pressure properties of fluid-structure systems (Ghalandari et al., 2019;Ma et al., 2020). Salih et al. (2019) emphasized the effect of fluid solid interaction on the characterization of the relationship between flowing fluids and solid walls. Based on a numerical study, Xiao et al. (2020) confirmed the boundary pressure distribution on the structure wall was related to the flow pattern variation during an intake tube operation. The attachment performance of the swirling flow in a vortex dropshaft is controlled by the centrifugal force imposed by the dropshaft wall. The wall-jet pattern for a typical plunging dropshaft in which the vertical velocity was dominant has previously been shown (Camino et al. 2015). Qiao et al. (2013) exhibited the vortex velocity and air core properties for a stable vortex dropshaft. Using three-dimensional computational fluid dynamics methods, Chan et al. (2018) showed typical annular pressure distributions for a stable vortex dropshaft. The pressure distribution pattern inside the vortex dropshaft was found to vary with the contraction ratio of the tangential intake and the approach flow conditions. This indicates that the flow transition between plunging and vortex flow may result in different pressure features, and the pressure distribution differences on the dropshaft wall should be investigated as a criteria for hydraulic assessment of the transition flow pattern. Consequently, as the flow pattern is the important indicator in the tangential intake design of the dropshaft, further study is needed to avoid misjudgement in the urban water drainage system given the complexity of flow transition operation.
Even though the pressure characteristics of dropshafts were widely studied, the pressure distributions representing the transition process between plunging and vortex flows have not been reported. In the present study, based on a series of three-dimensional (3D) computational fluid dynamics (CFD) models, the effects of the tangential intake contraction on the pressure distributions of vortex dropshafts are analysed. Specific wall pressure parameters are discussed for different approach flow discharges and tangential intake contractions. An optimum hydraulic factor is explored for determining the appropriate tangential intake performance in vortex dropshafts.

Methods
When water flows from the tangential intake into the dropshaft, the flow rotates and drops along the partial and vertical tunnels. This swirling and dropping behavior is typical of a 'stratified flow regime' (Brennen & Brennen, 2005) in a partial tunnel with a specific air-water interface, as shown in Figure 1. The flow pattern in the dropshaft is mainly determined by the tangential intake design parameters, including the dropshaft diameter D, the contraction width e, and the tapering section slope β.  The dropshaft diameter D = 0.24 m is calculated by the equation D = σ (Q 2 /g) 0.2 (Hager, 1990;Jain, 1984) with the factor σ = 1.25 for a discharge capacity of 50 L/s, where Q is the approach flow rate and g is the gravitational acceleration (g = 9.81 m/s 2 ). The horizontal chute width is equal to D. The tapering section is located between the chute and the dropshaft and has a constant bottom slope (β) of 13°. Different contraction widths are designed with contraction ratios (e/D) ranging from 0.1 to 0.8 to investigate the influence of the tangential intake design on the flow pattern in the dropshaft. Water flows in the tunnel system are simulated with flow discharges (Q) ranging from 5 L/s to 40 L/s. The summary of the inflow condition of the simulation scenarios are summarized in Table 1. The Volume of Fluid (VOF) method was used to track the air-water interface. Recent studies (Chau & Jiang, 2004;Yang et al., 2018) have confirmed that the implicit VOF method provides high simulation accuracy. The RNG (Renormalization Group) k-ε two-equation turbulence model is used as it is more accurate and reliable for a wider class of flows than the standard k-ε model (Ma et al., 2020;Xiao et al., 2020;Zhang & Chen, 2004). An implicit numerical solver was used due to its quicker convergence and larger time step in the calculation process. 3D meshes were used throughout the computational domain ( Figure 2). Sensitivity tests were performed on the grid size and time step. Fine grids were used near the taper section and inside the vertical tunnel, whereas coarser grids were used in other regions. The computational domain had a total of 50,958, 106,990, or 152,958 cells, and the predicted velocities in the dropshaft were within 1% when the cell number increased from 106,990 to 152,958. Therefore, the numerical model adopted a total of 106,990 unstructured hexahedral cells (Figure 2), with a minimum grid size of 1 mm close to the tapering section and walls. A typical time step of 0.002 s was used in all simulations. The steady state was reached after approximately 10 s, which took approximately 20 h to compute on an Intel i7-4790 3.6 GHz CPU computer with quad-core parallel computation. The predicted maximum and minimum dropshaft wall pressure at z = −0.2 m (e/D = 0.25, Q = 20 L/s) are plot as a function of simulation time in Figure 3. Both the minimum and maximum pressure become stable after 10 s simulation time, and the differences between 10 and 12 s are within 1%.
As the convergence condition by Courant-Friedrichs-Lewy (CFL) is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. The maximum CFL presents at the tangential part of the dropshaft with the minimum cell size ( x) of 0.001 m and velocity u of  about 2 m/s, therefore, the maximum CFL is calculated to be C max = u t/ x = 2.0. Implicit solvers are usually less sensitive to numerical instability and so larger values of C max may be tolerated. In our simulation, the implicit solver is used, so it is okay to have a maximum CFL number of 2.0. As the flow is highly turbulent, the non-equilibrium wall function is used. The key elements of the wall function are the Launder and Spalding's loglaw for mean velocity is sensitized to pressure-gradient effects, and the two-layer-based concept is adopted to compute the budget of turbulence kinetic energy in the wall-neighboring cells. The y + ( = ρu τ y/μ) value should be between 30 and 300. In our cases, the y + ranges from about 50-255 in all the cases.
At the tangential intake upstream, an approach flow discharge was introduced, and the entrance of the air flow for the vertical dropshaft was in contact with the atmosphere, where the total pressure was equal to one standard atmospheric pressure. As mass flow inlet boundary condition is used, the phase velocity ratio of the two phases is set as 1.0, meaning the non-slip velocity of the two phases. For the solid wall boundary condition, no slip was applied, and the velocity on the solid wall was set to zero. In terms of model validation, the predicted free water surfaces in the approach channel and the tapering section with an e/D ratio of 0.25 were compared with the measurement data. A lower flow rate of 5 L/s and a medium flow rate of 20 L/s were used for validation, and the prediction results of the CFD model were consistent with the measurements.

Governing equations
The present study used the VOF method by combining the RNG k-ε turbulence model to simulate stratified air-water flow and estimate the air discharge demand. The governing equations are as follows: Transport equation: Momentum equation: k equation: where ρ and μ are the average density of the volume fraction and the molecular viscosity, respectively; p is the pressure; and μ t is the turbulent viscosity, which can be deduced a turbulent kinetic energy k and energy dissipation rate ε as: The detailed values are shown in Table 2. The governing equations for the water fraction α w are: where u i is the velocity component and x i is the coordinate (i = 1, 2, 3). The air-water interface was tracked by solving the continuity equations. The sum of the water and air volume fractions was 1 in the controlling body. ρ w and ρ a are the density of water and air, respectively, and μ w and μ a are the viscosity of water and air, respectively.

Model validation
The predicted free water surfaces in the approach channel and the tapering section with an e/D ratio of 0.25 were compared with the measurement data, as shown in Figure 4. The prediction results of the CFD model were in good agreement with the measurements, with an average relative error of 6.5% in the four validation cases. The two flow rates of 5 and 20 L/s are illustrated in the figure. The prediction results of the CFD model agree well with the measurements. Constant water depths are maintained in the approach channel before the water approaches the tapering section. A significant drop in the free surface is observed at the junction point, which is more obvious at a lower flow rate (5 L/s). The free water surface changes more smoothly from the junction point to the vertical shaft at a flow rate of 20 L/s.

Radial pressure distributions
Hydrodynamic pressure profiles on the vertical dropshaft wall represent the water distribution and rotation. The tangency point of the horizontal channel and the dropshaft is α = 0°, and the rotation angle increases clockwise from 0 to 360. The pressure distributions along the dropshaft wall in a design with a contraction ratio e/D of 0.3    In Figure 6, for a design with a contraction ratio e/D of 0.3, typical annular pressure distributions are shown at different depths for two flow rates, 5 and 40 L/s. At the initial dropshaft cross-section (z/D = 0), the Q = 5 L/s flow condition results in a partial pressure distribution along the entire annular wall (α = 0°-90°). As the water drops to z/D = 1 along the vertical direction, the wall pressure decreases, and the pressure distribution area changes to α = 90°-180°. When z/D = 2-3, the distribution range expands slightly, reaching approximately half of the entire dropshaft, and the wall pressure decreases continuously. As the water drops to z/D = 5, the pressure profile becomes loose and discontinuous, and the distribution region does not considerably differ from the cross-section at z/D = 3. This indicates that the water rotation along the annular wall has become weak, and the flow pattern has changed to a falling flow along the vertical dropshaft. For a large water discharge condition (Q = 40 L/s), there is a nearly complete distribution of the hydrodynamic pressure at the initial cross-section z/D = 0, with an annular air core in the middle of the dropshaft. The high-pressure area is located at approximately α = 0°-180°, indicating that the vortex flow pattern is generated when the water clings to the dropshaft through the tangential intake. When the falling distance is z/D = 5, the pressure gradually decreases, and the entire annular distribution remains stable. In addition, a relatively high-pressure area can be observed during the falling process, such as from α = 90°-180°at z/D = 1 and α = 150°-210°at z/D = 2-5. This indicates that the centrifugal force is effective at driving the vortex flow.
In addition to the pressure distribution differences caused by flow discharges in the vortex flow pattern, the effect of the tangential intake contraction ratio e/D on the pressure distribution is analyzed and shown in Figure 7. Considering an initial vertical dropshaft cross-section z/D = 0 with Q = 40 L/s, the pressure is evenly distributed when e/D = 0.1, and the impact area caused by the clinging flow is not noticeable. As the ratio e/D increases to 0.5, the impact areas with relatively higher pressures are generally stable, ranging from α = 0°-90°, whereas the pressure gradually decreases from α = 90°-270°. This indicates that the vortex capacity weakens along the latter half of the dropshaft. For larger contraction ratios of e/D = 0.7 and 0.8, the entire pressure profile represents typical plunging and falling flow. Thus, the radial pressure distribution on the dropshaft wall can reflect the effects of Q and e/D on the flow pattern transition from the plunging dropshaft to the vortex dropshaft.

Effect of tangential intake on rotation generation
Based on the calculated pressure distributions on the dropshaft wall, the maximum pressure P max and minimum pressure P min at the initial cross-section of the dropshaft (z/D = 0) can be extracted. The effect of e/D on the variation trends of P min /P max is shown in Figure 8. For the small flow discharge condition (Q = 5 L/s), the values of P min /P max are much smaller than 0.1 due to the sharp pressure profile. This illustrates that the plunging flow pattern dominates, and that vortex rotation cannot be effectively generated in the dropshaft. When Q is increased to 10 L/s-40 L/s, the values of P min /P max are typically greater than 0.1 when e/D < 0.4 and less than 0.1 when e/D > 0.4. Thus, P min /P max ≥ 0.1 for z/D = 0 is used as the assessment condition for rotation generation when water enters the dropshaft through the tangential intake. Figure 9 shows the variation trends of P min /P max at different locations (z/D) along the dropshaft with various e/D values during the falling process. Generally, for moderate e/D ratios of 0.4 and 0.5, P min /P max increases to greater values than 0.2 when z/D increases to 2 due to the restraint of the annular wall. For a small flow rate of Q = 5 L/s, the P min /P max ratios are less than 0.2 in both cases, indicating that the plunging flow pattern does not change. As the flow spirals down further, the value of P min /P max increases gradually. However, it remains smaller than 0.2, and subsequently decreases as z/D increases to 5. This illustrates that the rotation capacity is reduced for large e/D and that the vortex pattern cannot remain in the entire dropshaft. The phenomenon of increase and subsequent decrease can be considered as a transition flow pattern that occurs when the vortex and plunging flow patterns coexist in the dropshaft. Using Q = 40 L/s, Figure 9 (3) plunging flow when e/D = 0.8. It should be noted that although an e/D ratio of 0.1 can result in a stable P min /P max greater than 0.2, the increasing trend is slower than with an e/D ratio 0.4. This shows that a moderate e/D ratio can effectively promote vortex flow development in the dropshaft. Consequently, to assess the rotation movement of water flow spiraling down in the dropshaft, P min /P max ≥ 0.2 for z/D > 1 is obtained to ensure a stable vortex flow pattern.

Condition assessment of vortex dropshaft
Because the tangential intake directs the chute flow into a vertical dropshaft, flows with a small discharge and low tangential restraint can decrease, plunging into the dropshaft wall and bottom areas. Vortex dropshafts with tangential intakes are intended to deflect chute flows to annularly swirl down to low elevations. The hydraulic condition for the vortex flow pattern must therefore be checked for the flow discharge and contraction design of the tangential intake. Plunging and transition flow patterns are observed for small Q and large e/D conditions. When using P min /P max ≥ 0.1 with z/D = 0 for the intake cross-section and P min /P max ≥ 0.2 with z/D > 1 for the flow development in the dropshaft, the flow differs in the non-rotation, transition and rotation regimes. The data from the current study are plotted in Figure 10, showing the straight line P min /P max = 0.04 z/D + 0.1 as the limit between the non-rotation and rotation flow patterns.
For the tangential vortex intake, Yu and Lee (2009) classified the approach flow discharge as the critical flow discharge Q C and the free drainage discharge Q f , where both assessment discharges are a function of the tangential intake and dropshaft geometry parameters. As Yu and Lee (2009) suggested, the critical flow discharge Q C represents the smooth flow transition from the upstream to the downstream in the tangential intake chute, and the free drainage discharge Q f , represents the critical flow condition where the hydraulic jump can take place in the tangential intake once the flow discharge is larger than Q f . Based on the tangential intake design parameters, all present tests satisfy Q C < Q f , as shown in Figure 11. This illustrates that the flow is stable at the junction between the tapering chute and the annular dropshaft. Thus, the effects of blocking and fluctuating water conditions on the plunging and vortex flow patterns are eliminated. Moreover, for various tangential intake e/D conditions, the present flow pattern transition from plunging to vortex dropshafts shows that plunging flow (and the transition flow) is expected to occur in the dropshaft if Q < Q f , and Q > Q f can be adopted as the heuristic criterion for vortex dropshaft operation. According to the theoretical analysis of the free drainage discharge (Jain & Ettema, 2017), the rotation intensity of the vortex flow in the dropshaft increases with Q. Because Q f increases with e/D under other constant conditions, the rotation intensity of the vortex flow in the dropshaft decreases; consequently, the minimum flow discharge for the rotation operation increases with e/D.
According to the present investigation, the vortex flow pattern in the dropshaft is mainly determined by the e/D of the tangential intake, and the index of P min /P max with z/D = 0 for the intake cross-section is the essential factor for the flow rotation performance. Based on the simulation data, an empirical relation is suggested as P min /P max = (6Q+0.05)·(1-e/D), which can be referred for the quantitative relationship between the approach flow discharge and e/D, shown in Figure 12(a). By setting different criteria values of P min /P max for the tangential intake design, the critical flow discharge Q V for the vortex flow in dropshafts can be obtained in Figure 12(b). With the increase of criteria values of P min /P max from 0.10 to 0.30, the critical Q V for the vortex flow generation gets greater. This indicates that for a wide range flow discharge operation, a small e/D of the tangential intake should be considered rigorously for a vortex dropshaft in hydraulic design applications.
Using the pressure distribution differences around the dropshaft wall, the present simulation works show the transition flow pattern between plunging and vortex flows, which is related to the flow discharge operation and contraction ratio design of the tangential intake. This highlights the importance of the tangential intake parameter for the vortex dropshaft design. The P min /P max can be used as a criterion for hydraulic assessment of the transition flow pattern. On the other hand, extreme rainfall  Effects of P min /P max variation on the hydraulic design of the tangential intake: (a) an empirical relation for P min /P max ; (b) Q C variation for different P min /P max criterion. events happened frequently in the urban area. Considering that the dropshaft tunnel are set underground, the present hydraulic factor P min /P max can be used as a monitoring index for the prototype applications, because the over level flood discharge may occur in the urban water drainage systems.

Conclusions
The three-dimensional pressure field of an annular dropshaft with a tangential intake is numerically studied for various approach flow discharges and intake contraction designs. The numerical simulation results reveal that a large flow discharge combined with a moderate contraction design of the tangential intake can cause water to rotate and spiral down stably in the annular dropshaft. A decrease in the flow discharge and contraction intensity results in a pattern transition to plunging flow in the dropshaft. Based on radial pressure distribution differences on the annular wall, the ratio of minimum to maximum pressures P min /P max is used to distinguish between the rotation and non-rotation flow patterns in the dropshaft. The heuristic criteria for determining the dropshaft flow pattern are P min /P max ≥ 0.1 at the tangential intake section and P min /P max ≥ 0.2 downstream, which shows a stable vortex flow spiraling down along the annular wall. The present study confirms that the P min /P max can be used as a criterion for hydraulic assessment of the transition flow pattern, offering additional insights into tangential intake flow transitions. In the further research, the detailed effects of dropshaft design parameters on the flow pattern in the flood discharge operation should be investigated. It will be a certain application value to clarify the influence of local flow pattern transitions on the entire water drainage system operation, and to provide a basis for hydraulic design in water supply, drainage and sewerage systems.

Data availability statement
All data that support the findings of this study are available from the corresponding author upon reasonable request. All models used during the study appear in the published article.

Disclosure statement
No potential conflict of interest was reported by the author(s).