Flow characteristics of a tangential vortex intake with steep-slope tapering section Open Access

B

approach channel width

D

dropshaft diameter

e

junction width

Fr_a

approach channel Froude number

g, g

gravitational acceleration

H_a, H_d

energy head at the approach channel and dropshaft, respectively

h_a, h_j

depth of approach channel and junction, respectively

h_ca, h_cj

critical depth of approach channel and junction, respectively

k

turbulent kinetic energy

n

Manning's coefficient

P

pressure

Q

discharge

Q’

non-dimensional discharge

r

radial distance from centre of dropshaft

R

dropshaft radius

Re_a

Reynolds number of the approach flow

t

time

U

(u, v, w) = Cartesian velocity field in CFD

U_a

average approach flow channel velocity

U_jx

average horizontal velocity at the junction

U_x, U_y

horizontal and transverse velocity at the junction, respectively

U_z

vertical velocity

U_θ

tangential velocity at the dropshaft

U*

uniform shaft velocity in the theory of Kellenberger (1988) 

x, y, z

Cartesian coordinates

z_a

approach channel vertical level

z*

determining height in the theory of Kellenberger (1988) 

α_a, α_w

air and water volume fractions, respectively

β

bottom slope of tapering section

θ

tapering angle of the width of tapering section

ε

dissipation rate of turbulent kinetic energy

Γ

vortex circulation

Γ₀

vortex circulation generated at the junction

λ

air core size ratio

λ_m

minimum air core size ratio along dropshaft

μ_t

turbulent viscosity

υ

kinematic viscosity of water

ρ

air-water mixture density

ρ_a, ρ_w

air and water densities, respectively

Vortex intakes are commonly used as the flow interception facilities for water supply, drainage and sewerage systems (e.g. Plant & Crawford 2016; Kumar et al. 2017). Different types of vortex intakes have been proposed. For intercepting supercritical inflow, tangential vortex intakes (Zhao et al. 2006; Yu & Lee 2009; Chan et al. 2018) and spiral vortex intakes (Pfister et al. 2018; Crispino et al. 2019; Fernandes & Jónatas 2019) are suitable. Retrofitting the approach channel of subcritical scroll-type intakes to cater for supercritical inflow was also reported (Del Giudice & Gisonni 2011). Tangential intakes are usually more preferable due to its simpler geometry for the ease of construction. A tangential vortex intake consists of an approach flow channel and a sloped tapering section connected to a dropshaft eccentrically (Figure 1(a)). The inflow enters tangentially into the dropshaft as an annular slot jet from the tapering section, and the flow swirls down the dropshaft, forming a stable air core that allows ventilation.

The hydraulic characteristics of a tangential intake depends heavily on its geometry. Extensive experimental, numerical and theoretical investigations were carried out to study the tangential vortex intake flow. Earlier studies focused on the use of hydraulic modelling to determine the intake performance (e.g. Brooks & Blackmer 1962; Jain & Kennedy 1983; Lee et al. 2006). Theoretical models, assuming one-dimensional flow, were developed to provide design guidelines of intake geometry for estimating the critical design parameters of flow depth and air core size (Jain 1984; Zhao et al. 2006; Yu & Lee 2009). Yu & Lee (2009), based on a series of hydraulic model tests, revealed possible undesirable flow instability of a tangential vortex intake, and proposed a stable design criterion based on a 1D theory with the assumption of symmetrical free vortex flow. Crispino et al. (2021) studied the energy head dissipation for tangential and spiral intakes and found that most of the energy dissipation occurs in the dissipation chamber below the dropshaft.

The prediction of a 1D model usually displays significant discrepancies compared to measurements. To better understand the vortex intake flow, detailed velocity measurement and three-dimensional (3D) numerical modelling are required. Detailed 3D velocity field of vortex intakes was measured using particle imaging velocimetry (PIV, Mulligan et al. 2018) and laser Doppler anemometry (LDA, Chan et al. 2018, 2021), greatly enhancing the understanding of such complex swirling flows. Nevertheless, swirling flow measurement is usually complicated due to the curved geometry, thus scant detailed data are available. On the other hand, 3D computational fluid dynamics (CFD) models with the volume-of-fluid (VOF) method have been increasingly used to tackle the air-water flow in complex vortex intake problems, including the modelling on tangential vortex intakes (Plant & Crawford 2016; Chan et al. 2018) and scroll vortex intakes (Mulligan et al. 2018; Chan et al. 2021), demonstrating the capability of 3D modelling approach in predicting vortex intake flows.

This paper focuses on elucidating the flow features of a tangential vortex intake with a steep-slope tapering section (bottom slope angle β = 45°). Comparing to a tangential intake with a mild-slope tapering section (Chan et al. 2018, β = 13°), a tangential vortex intake with a steep-slope tapering section usually generates a hydraulic jump (the transition of supercritical flow initiated at the approach channel to subcritical flow at the junction) in the tapering section, which may lead to air entrainment. On the other hand, the interaction between the annular jet flow issued from tapering section and the dropshaft swirling flow becomes smaller due to the larger vertical velocity induced by the steep channel. The use of a tangential intake with a tapering section of β = 30° has been reported, showing satisfactory performance from experimental and numerical study (Plant & Crawford 2016). This paper presents an integrated experimental and numerical modelling study, aiming at providing insight to the flow structure for such intake design. The experimental and numerical model set-up will first be presented, followed by the discussion on the results and insights to the flow features.

The flow characteristics of a tangential vortex intake is determined by its geometric parameters: e = junction width, B = approach channel width, D = dropshaft diameter, β = bottom slope of tapering section, and θ = horizontal tapering angle of width of tapering section. For the model in this study, D = 0.124 m, e = 0.031 m, B = 0.124 m, β = 45°, and θ = 24° (Figure 1(a)).

The model was fabricated using transparent Perspex for easy flow visualization. A steady discharge was fed from an underground reservoir and delivered to an elevated water tank by a pump. The flow entered the head tank via a supply pipe. After circulating through the model, the flow was returned to the reservoir. The discharge was controlled by a regulating valve, and monitored by a Controlotron 1010P ultrasonic flowmeter.

The experiments were carried out in two series. In the first series of tests, the surface profiles of the approach-channel flow, and air core sizes of the vortex flow were measured. The average velocity U_a at the end of the approach channel was estimated from the measured discharge Q and measured water depth h_a at the channel centerline, as U_a = Q/(Bh_a). The shape and size of the air core at different levels in the dropshaft vortex flow were measured with a specially-designed eight-legged ruler with scales of 5 mm on each leg (Fig. S1a, Supplementary Material, Yu & Lee 2009; Chan et al. 2018, 2021). The ruler was lowered into the dropshaft to measure the flow thickness of each azimuth by reading the scale on each leg. The size of the highly asymmetrical air core was determined by summing the flow area subtended by each of the 8 arcs (Figure S1b). In this series, the tested discharges were Q = 1–20 L/s with increasing interval of 1 L/s (Table 1). The Reynolds number of the approach flow was large enough (Re_a = U_ah_a/υ = 8,000 − 160,600, υ = 10⁻⁶ m²/s is the kinematic viscosity of water) to maintain turbulent flow condition.

In the second series of the test, velocity field was measured using the non-intrusive laser Doppler anemometry (LDA) technique. The LDA system (Dantec Dynamics 2006) included a continuous argon ion laser, a three-dimensional traverse, a signal processor and a computer for traverse control and data acquisition. It measured the velocity field point-by-point at a single component each time at a frequency of around 5 Hz with a sampling duration of about 30 seconds. The Cartesian velocity components (U_x, U_y and U_z) at the junction (31 mm wide) were measured at three vertical transects from the side wall of the intake at y = 7, 18 (near centreline) and 29 mm. At each transect, velocities were measured at a vertical interval of 10 mm. Velocity of the swirling flow of the dropshaft was measured at different levels and azimuths (Table S1). For overcoming the distortion by the curved dropshaft wall on the laser beam, a special measurement window was installed. A Perspex box with an outer vertical planar surface was fitted outside the chamber wall and filled with water. Since the refraction indices of water and Perspex are similar, the refraction effect of the curved wall was minimized. The centreline of the LDA probe was directed perpendicular to the box surface for vertical and tangential velocity measurement in the dropshaft (Fig. S2). Similar technique was validated and used for measuring the swirling flow field of other types of vortex intakes (Guo 2012; Chan et al. 2018, 2021) with an accuracy of about 3%. The flow field was measured for Q = 4, 8 and 16 L/s, representative of different flow characteristics.

The governing Equations (1)–(3), and the k and ε equations of the turbulence model were solved numerically using the finite volume method in the commercial CFD code of ANSYS FLUENT 15 (ANSYS Inc. 2013). A second-order upwind advection scheme was used for momentum and density, while a first-order upwind advection scheme was used for k and ε. The volume fraction equation (Equation (1)) was spatially discretized using the Modified High Resolution Interface Capturing (HRIC) Scheme. The under-relaxation factors for the iterative solver were 0.5 for pressure and momentum, 0.2 for volume fraction, 0.8 for k and ε, and 1.0 for density and turbulent viscosity. Convergence for each time step was declared when the normalized residual is less than 10⁻⁴ for all variables. Some air entrainment is observed for discharges with a hydraulic jump at the tapering section, but it did not make much impact on the general flow features, thus air entrainment modelling is not included. The air-water interface is defined as 50% air volume fraction. CFD modelling using the VOF technique and k-ε turbulence model has been validated and applied in previous numerical studies, including the simulation of a stable tangential vortex intake (Chan et al. 2018) and a subcritical scroll vortex intake (Chan et al. 2021).

An unstructured boundary-fitted mesh with 76,320 hexahedral cells was developed according to the experimental design (Figure 1(b)). The computational mesh was composed of grid cells with hexahedral cells for the approach channel and dropshaft, and triangular-prismatic cells for the tapering section. Cell sizes varied from about 1 mm close to the dropshaft wall, to a maximum of about 25 mm along the approach flow channel (Figure 1(b)). The computational model had three open boundaries – the approach channel inflow (prescribed with water flow rate), the dropshaft outflow and the top atmospheric boundary (both prescribed with zero gauge pressure). Grid convergence tests showed that the grid size is sufficient for resolving the flow details with discrepancy of less than 5%.

Numerical predictions were carried out for Q = 4, 8 and 16 L/s, corresponding to the cases with detailed velocity measurement. The model was initialized under dry-bed condition with air-phase only. Water-phase entered the intake from the approach channel inlet and the flow gradually develops into a steady-state (without further change in the predicted velocity) with a duration of about 30 s. A time step of 0.001 s was used. The run time for transient flow process for 30 s was about 40 h on a Dell workstation with an Intel i7-6700 3.4 GHz CPU with quad-core parallel computation.

The flow process in the model was observed with experimental and numerical modelling for different discharges. For Q = 4 L/s (Fig. S3), the flow was supercritical in the entire tapering section; a shock wave can be observed as the flow was constrained by the narrowing channel width. For Q = 8 L/s (Figure 2), the flow transformed from supercritical flow at the upper end of the tapering section to critical flow near the junction through an inclined hydraulic jump at the tapering section. The flow accelerated towards the junction, due to the narrowing and sloping of the tapering section. For Q = 16 L/s, the free surface became more stable and flatter. The predicted free surface compared well with the measurement (Figure 2).

The hydraulic conditions at the critical locations of the intake were analysed. The predicted free surface profile at the tapering section shows different characteristics for the tested discharges:

• The end of the approach flow channel forms a hydraulic control for the transition from (sub-)critical flow in the approach channel to supercritical flow for Q < 11 L/s (Q’ = = 0.65, non-dimensional discharge), as seen by the close match between the predicted critical depth (the approach flow Froude number ⁠) and measured depth h_a at the end of the approach channel (Figure 3(a)).

• At the junction, the flow was supercritical for Q < 9 L/s (Q’ = 0.53), as the measured depth h_j was greater than the critical depth at the junction ⁠. Due to the narrowing of the tapering section, an inclined shock wave formed in the tapering section. Unstable surface rollers were observed in some cases, causing breakup of the free surface and air bubbles entrained into the flow. The flow at the junction became subcritical for Q9 L/s (Figure 3(b)); the free surface of the approach channel was undulated as the inclined hydraulic jump generated was deflected upstream as discharge increased.

• For Q > 12 L/s (Q’ = 0.71), both the flow at the tapering section and the approach flow channel became subcritical. The water levels at the intake were relatively flat and stable, and rose more rapidly with increasing discharge.

Near the junction, the horizontal velocity U_x at the channel centreline was nearly constant in the vertical direction and about the same for the lower discharges of Q = 4 L/s (not shown) and 8 L/s (Figure 4(a)). The vertical velocity U_z increased from the surface to the bottom. For the higher discharge Q = 16 L/s, both the horizontal and vertical velocities increased with depth (Figure 4(b)), which was caused by narrowing in channel width and the increase in water depth. The observation was similar to that of a stable tangential intake (Chan et al. 2018). The difference in horizontal velocity profiles between low and high discharges can be attributed to the higher turbulence level generated by the hydraulic jump/shock wave at the tapering section at the lower flows, which increased the exchange of momentum at the junction. The predicted flow velocity compared satisfactorily with the point LDA measurement, despite the fluctuations due to the hydraulic jump.

The air core size predicted by CFD model is notably better than that using a 1D theory (e.g. Jain 1984; Yu & Lee 2009), which assumed that the air core is axisymmetric about the centre of dropshaft.

The predicted and measured tangential velocity U_θ at different vertical levels of the dropshaft generally decreased with increasing radial distance from the shaft centre (Figure 7). The flow thickness was the largest at θ= 45° and decreased around the circumference. U_θ increased linearly from near the air core in the centre to a maximum at r/R = 0.6 and then decreased with increasing radius. The reduced velocity near the air core was due to the viscous dissipation and the increase in downward vertical velocity. The maximum U_θ was relatively constant at different levels in the dropshaft. As observed from the vortex circulation Γ = U_θr (Fig. S4), the flow resembled a Rankine combined vortex (e.g. Hite & Mih 1994) with the nature of a forced vortex (increasing Γ) close to the air core and free vortex (constant Γ) at the majority of the flow thickness. The average Γ was similar to that induced by the eccentricity at the junction ⁠, indicating that there was little loss of angular momentum of the swirling flow. The measured turbulent intensity (as root-mean-square fluctuation from the mean) of the swirling flow was approximately 10–20% of the mean velocity.

The pressure of the swirling flow can be found by integrating the above equation with respect to r, from the surface of the air core to the wall. It is assumed that the air core is under atmospheric pressure (P = 0). With the measured tangential velocity distribution U_θ(r) at certain level, the pressure distribution of the swirling flow was evaluated in the radial direction and in the tangential direction. It can be seen that the CFD predicted pressure compared well with that estimated from Equation (5) (Figure 8), showing an increasing trend towards the dropshaft wall.

The vertical velocity U_z remained more or less constant for the thickness of the flow, although a slight decreasing trend can be seen near the shaft wall. The vertical velocity increased as the flow swirls down the dropshaft (Figure 9(a)). The predicted U_z agreed reasonably well with LDA measurement in the dropshaft. Similar swirling flow feature at the dropshaft was also found in a stable tangential vortex intake design (Chan et al. 2018), despite there being no hydraulic jump at the tapering section.

The Manning's coefficient n is assumed to be 0.01 for Perspex used in the experiment set-up; U* is the uniform shaft velocity that establishes under a sufficiently long shaft. It is surprising that, except for the region close to the junction, the vertical velocity profiles were generally well predicted by the theory. The theory was originally developed for intakes with a horizontal bottom (e.g. scroll-type intake), with an initial zero vertical velocity. The asymmetric air core and non-uniform velocity of the tangential vortex flow is very different from that of a scroll-type intake. The steep-slope tapering channel induced an initial vertical momentum, which gradually dissipated, and the friction and gravity forces dominated the flow in the dropshaft. It can be seen that the vertical velocity increased with downward distance.

Reduction in total energy head can be clearly seen in Figure 10. For Q = 8 L/s, there was a significant energy dissipation due to the hydraulic jump formation at the tapering section. The energy head was reduced by about 30% as the flow entered the dropshaft. Further energy dissipation is induced by the friction loss of the flow at the dropshaft. The energy dissipation of the intake at higher flow (Q = 16 L/s) is seen to be relatively smaller due the subcritical nature of the flow without a hydraulic jump. The energy dissipation was mainly resulted from the frictional loss of the flow at the dropshaft. The predicted energy head generally agreed well with the measurement (Figure 10). The percentage of energy dissipation in the intake and dropshaft is similar to that reported by Crispino et al. (2021) for tangential intakes.

An experimental and numerical study of a tangential vortex intake with a steep-slope tapering section has been performed to elucidate the flow characteristics of the intake. The numerical CFD model has been validated and compared well with experimental measurement on flow profile, air core size and velocity field.

The intake flow is determined by the hydraulic controls at the end of approach flow channel and/or the junction between the tapering section and the dropshaft. For lower flows, the transition of supercritical to subcritical flow at the tapering section results in an inclined hydraulic jump with energy dissipation. For higher flows, subcritical flow prevails at both the tapering section and the approach flow channel.

The air core of the tangential intake is highly asymmetrical about the dropshaft axis. The swirling flow at the dropshaft resembling a Rankine vortex. The vertical velocity increases as the flow swirls down the dropshaft. The pressure gradient of the swirling flow is caused by the centrifugal acceleration, and can be predicted reasonably using the tangential velocity. Flow energy dissipation of the intake is mainly caused by the hydraulic jump at the tapering section and the friction loss at the dropshaft, which could result in about 30% of energy loss to that induced at the inflow.

The study has offered comprehensive insights of tangential vortex intake flow, which will be a useful basis for formulating better design theories and guidelines. Also, this study demonstrated that 3D CFD modelling is capable of predicting the air core size and the velocity field, which is superior to previous 1D theories.

This research was supported by the Hong Kong University of Science and Technology Faculty Initiation Grant (IGN17EG05). The experiments were carried out in the Croucher Laboratory of Environmental Hydraulics of the University of Hong Kong with the support from the Hong Kong Research Grants Council. The assistance of the laboratory staff is gratefully acknowledged.

All relevant data are included in the paper or its Supplementary Information.