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Apr 11, 2020

Experimental and Numerical Analysis of Two-Phase Flows in Plunge Pools

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Authors: J. M. Carrillo, Dr.Eng. https://orcid.org/0000-0003-4264-3269 [email protected], L. G. Castillo, Dr.Eng. [email protected], F. Marco https://orcid.org/0000-0002-2277-5515 [email protected], and J. T. García, Dr.Eng. [email protected]Author Affiliations
Publication: Journal of Hydraulic Engineering
Volume 146, Issue 6
https://doi.org/10.1061/(ASCE)HY.1943-7900.0001763
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Abstract

The current capacity of many spillways may be insufficient due to the increase in flood flows as a result of climate change. In those cases, the dam may be overtopped, generating new loading scenarios downstream of the dam. To date, information regarding the behavior of air-water flows in the plunge pool of free-falling jets is scarce. This study combines experimental and numerical approaches to analyze the submerged hydraulic jump generated downstream of overflow nappe impinging jets. A back-flushing Pitot tube and an optical probe were used to measure the velocity field and air entrainment rate. The velocity field allows to validate computational fluid dynamics (CFD) simulations. The choice of the turbulence model played a key role in the wall jet resolution. To the best of our knowledge, for the first time, it was possible to estimate the Sauter mean air bubble diameter in the entire submerged hydraulic jump. The results are in agreement with recordings obtained with a high-speed camera.

Introduction

In the last years, the increase in the magnitude of design floods due to climate change, the review of hydrologic data records, and/or the assumption of more demanding design methodologies [e.g., by adopting the probable maximum flood (PMF)] has promoted the re-evaluation of spillways capacity and the operational scenarios for large dams around the world. Several studies have indicated that the current capacity of many spillways is inadequate, and dams may overflow during extreme events (FEMA 2004, 2013, 2014). Increasing the capacity of spillways is costly and sometimes technically impracticable. In some situations, the overflow could be considered as an additional strategy of operation during extreme events. Overflow creates new loading scenarios, raising questions about the hydrodynamic actions and scour downstream of the dams (Wahl et al. 2008; FEMA 2014).
The choice of plunge pool type is usually a technical-economic decision between a deep and uncoated stilling basin and a shallow stilling basin with a lining. Their designers need to know the characteristics of the dynamic pressure at the bottom of the plunge pool. The required water cushion depth depends on the characteristics of the impingement jet, so most of the energy is dissipated, avoiding scour downstream of the dam (Annandale 2006). The energy dissipation mechanisms in free-falling jets can be divided into the following [Fig. 1(a)]: (1) spreading of the jet (aeration and atomization during the flight), (2) air entrainment by the entering jet, (3) diffusion in the pool, (4) impact on the pool bottom, and finally, (5) recirculation of flow in the plunge pool (Castillo and Carrillo 2017).
Fig. 1. (a) Scheme of overflow nappe impinging jets and receiving basin; and (b) infrastructure of free discharge in laboratory.
In recent years, several researchers have analyzed the near-flow field below the impingement point of falling jets. Table 1 summarizes the main characterizations of experimental and numerical settings. However, the current understanding on this highly-aerated-turbulent flow remains limited, and there is a lack of experimental studies providing detailed benchmark data (Wang et al. 2018; Carrillo et al. 2018).
Table 1. Flow conditions and instrumentation in recent studies of the near-flow field below the impingement point of free-falling jets
AuthorsNozzle (m)Free-falling distance (m)Impingement velocity (m/s)Instrumentation
Chanson et al. (2004)0.0250.13.5–4.4Prandtl-Pitot tube; hot-film probe; single-tip phase-detection probe
0.01250.052.42–3.46
0.02730.02731.79–2.49
Ma et al. (2010)0.0250.103.5–4.4Computational multiphase fluid dynamics (CMFD) code
Qu et al. (2013)0.060.102.0–2.5Particle image velocimetry (PIV)
Boualouache et al. (2017)0.0050.012.54ANSYS CFX and Fluent codes
Bertola et al. (2018)0.0120.102.49–7.43Dual-tip phase-detection probe; total pressure sensor; Prandtl-Pitot tube; acoustic displacement meter
Wang et al. (2018)0.0120.05–0.152.4–5.5Dual-tip phase-detection probe; total pressure sensor
Xu et al. (2018)0.020–0.05003.5–5.8Resistance air concentration probe
Although there has been progress in two-phase flows in hydraulic jumps, detailed theoretical and numerical models of the internal flow features have yet to be developed for a wide range of conditions (Bombardelli 2012).
The submerged hydraulic jump downstream of the impingement point of nappe flow is a particular case offering scant studies (Carrillo et al. 2018). This work seeks to improve this research field.
A nappe flow has been considered with a falling distance H=2.00  m and theoretical impingement jet velocity Vj=5.90  m/s [Fig. 1(b)]. The submerged hydraulic jump generated in the plunge pool has been analyzed with several procedures. The aeration has been measured with the optical fiber probe, yielding air concentrations, mean velocities, bubble frequencies, and Sauter mean bubble diameters. Moreover, the mean velocity has been measured with a back-flushing Pitot tube. In the quasi-unidirectional flow region of the submerged hydraulic jump, laboratory velocities have been compared with the numerical simulations carried out using ANSYS CFX (version 18.0).
The combination of experimental and numerical methodologies enabled the Sauter mean bubble diameter in the entire submerged hydraulic jump to be determined. The bubble diameter results were validated against images recorded with a high-speed camera.

Experimental Setup

Physical Model

The experimental study was conducted on a relatively large-scale physical device designed for the study of turbulent jets and energy dissipation in overtopping weirs assembled at the Hydraulic Laboratory of the Universidad Politécnica de Cartagena (Spain).
The mobile device of the weir has been modified from that used by Carrillo (2014) in order to increase the range of flows and fall heights which can be tested. Some porous panels were placed to reduce the turbulence intensity before the weir to 0.02 (Tu=V′¯/V, where V′¯ is the root mean square of the velocity and V the mean velocity). In this study, the weir crest length was 0.85 m, and the weir crest elevation was located Pl=2.20  m from the bottom of the plunge pool [Fig. 1(b)].
The fixed stilling basin was 1.05 m wide, 3.00 m long, and 1.60 m high. It was made of methacrylate to enable observation of the flow inside. Different water cushions’ depths can be analyzed, from direct impact to depths of 1.00 m.
A three-dimensional flow pattern developed in the vicinity of the walls. However, the effects were not relevant in the center plane of the plunge pool.
The water discharge was measured with an electromagnetic flowmeter located in the supply line, with a measurement accuracy of ±0.5%. Discharges of up to 0.20  m3/s can be investigated.
The water level upstream of the weir was measured with a point gauge (accuracy of ±0.5  mm). The undulation of the free surface was estimated to be within the accuracy range of the point gauge (no remarkable undulation). The water level downstream of the physical device was obtained from the optical fiber probe measurements, considering the value y95 (distance to the bottom at which air concentration Cair=95%). This value was double-checked with wall observations.
The equipment was mounted on a trolley that enabled longitudinal and transverse translations. The accuracy of the position was estimated to be less than 1.0 mm using graduated rulers.
The experimental setup and test procedures are discussed in detail in Carrillo (2014).
In a physical model, scale effects may be presented. However, their effects may be reduced through careful choice of the parameters and interpretation of the results. Chanson (2009) and Heller (2011) considered that the Reynolds (R) and Weber (W) numbers should be larger than 105 and 103, respectively, to minimize the scale effects on Froude (F) similarity scale models of hydraulic jumps and vertical plunging jets.
A relatively large specific flow (0.085  m2/s) was considered in this study. Following Castillo et al. (2015), the dimensionless numbers may be obtained at issuance conditions of the nappe flow case, located at a vertical distance h downstream of the weir crest [Fig. 1(a)]: F=3.72; R=79,952; W=2,631; liquid parameter (also called Morton number) Z=2.57×10−11. These values accomplish the previous recommendations for the W number. However, as the R number in the issuance condition is smaller than 105, scale effects may affect the phenomenon at smaller scales.
With those considerations, a similitude lab based on the Froude number may be applied to the results. Hydrodynamic forces scaled from the physical device may be larger than those expected in the prototype, due to possible scale effects. To err on the side of caution, it may not be recommendable to extend the results for Froude scales larger than 1:10 (Castillo et al. 2015). However, further tests are required to corroborate this statement.

Optical Fiber Probe

The air concentration was analyzed using double-tip optical fiber equipment from RBI-Instruments. The diffraction in a Descartes prism is different in air and water. The diffraction index of the probe is n=1.42 (Stutz and Reboud 1997b); clear water has an index of n1=1.33, whereas dry air has an index of n2=1.00. Considering those values, the Fresnel formula may be employed to detect air-water differences.
Optical fiber equipment uses light diffraction to estimate the phase change between air and water. The rise and fall of the signal detected are, respectively, the arrival and departure of the change phase at the sensor tip. The threshold values were set at 1.0 and 2.5 V (Boes and Hager 1998).
The local air concentration Cair may be defined as the ratio between the total time the probe is in air (ΣtGi) and the sample duration t (e.g., Boes and Hager 1998; RBI-Instrumentation 2012).
According to Stutz and Reboud (1997a, b), this equipment allows to measure air-water flows with velocities up to 20  m/s. The relative uncertainty of the air concentration is estimated at approximately 15% of the measured value, and the sensitivity to a threshold variation of 1% is less than 1% (Stutz and Reboud 1997a, b).
A source of error in estimating the presence of air in the flow is due to the statistical count of the number of air bubbles in contact with the tips of the probe (Stutz 1996). Therefore, a short duration of the measurement would contribute to less accurate results.
Boes and Hager (2003) carried out experiments with 4,000 air bubbles and samplings of 30 s. The authors considered that the accuracy of the air concentration and velocity measurements is related to the variation of the phase, air-water variation, or the inverse, rather than the sample duration t.
To evaluate the minimum duration of the measurements, André et al. (2005) analyzed the time required to stabilize the mean value during the measurement and that the quasi-stationary values were statistically representative of the air concentration. Based on the sensitivity study of the probe behavior, the authors recommend a 60 s sampling sequence as a good compromise between precision and duration of the experiments.
Following those ideas, a sample sequence of 90 s was considered in this study. Fig. 2 shows the evolution of the air concentration until a relative uncertainty of about 1% is reached, as well as the cumulative number of bubbles detected during the test.
Fig. 2. Evolution of the air concentration and number of bubbles detected by the optical fiber equipment during the test.
The two-phase flow characteristics were measured in the cross section of the free-falling jet used as a boundary condition of the numerical model and in different cross sections spaced 0.10 m downstream of the stagnation point of the plunge pool.

Back-Flushing Pitot Tube

A Pitot tube was used for measuring velocity profiles of the plunge pool in the same locations measured with the optical fiber equipment. The external diameter of the Pitot tube is 12 mm. The Pitot tube measures the pressure difference between the stagnation pressure at the tip of the tube (through a 2.3 mm hole) and the static pressure, measured by a ring of ports located in the circumference of the Pitot tube. GE Druck UNIK 5000 (General Electric, Billerica, Massachusetts) pressure transducers (range between −200 and +800  mbar and a precision of ±0.04% of the full scale) were used to record pressures variations. The sensors were located below the bottom of the plunge pool. After carrying out a static calibration, the pressure accuracy of the transducers was ±0.01  m. The output signal of both ports was scanned at 20 Hz for 60 s.
In order to avoid air entering the Pitot tube and connecting tubing during the tests, continuous back-flushing was provided by means of a constant head source which fed both the static pressure and the total head ports of the Pitot tube (Matos and Frizell 2000; Matos et al. 2002; Bombardelli et al. 2011). The back-flushing flow rate to each port was limited to near zero by needle valves.
The time-averaged velocity from the measured pressure of the back-flushing Pitot tube was determined by Wood (1983)
V=2ΔPρw(1−λCair)
(1)
where V is the velocity, ΔP is the difference between the total pressure head and the static pressure head, ρw is the density of water, Cair is the local air concentration (volume of air per total volume), and λ is the tapping coefficient which accounts for the nonhomogeneous behavior of the air-water flow approaching the stagnation point of the Pitot tube. As in some previous studies (Chamani and Rajaratnam 1999; Matos et al. 2002), Eq. (1) was applied, considering homogeneous air-water flow near the tip of the Pitot tube (tapping coefficient λ=1). This assumption was found to notably underestimate the velocity for Cair>0.6 to 0.7, based on data gathered by Lai (1971) and by Frizell et al. (1994) (Matos 2000).
The recirculating region may affect the measurements of the Pitot tube. For this reason, local velocities were limited to the wall jet near the bottom region.

High Speed Camera

A high-speed camera FASTCAM SA3 Model 120K (Photron, Tokyo) was used to identify the average bubble size. The characteristics of the camera during the experiments were the following: (1) Nikkor zoom lens with 50 mm focal length, (2) lens aperture f/4, (3) 512×512 pixels resolution, (4) 8 bits→255 shades, and (5) a horizontal distance from the camera to the submerged hydraulic jump of 0.50 m. With those data, the pixel dimensions were 0.00016  m/pixel.
Illumination of the experiment was achieved with six regular 1,200 W light bulbs with reflecting mounts in front of the flow. Video footage was recorded at 2,000 Hz. Frames were taken at several distances from the stagnation point of the jet and at several elevations from the bottom.
The images were analyzed using the Imaging Processing toolbox functions of MATLAB R2014a. A technique of edge detection from the intensity gradient between pixels was used as a first approximation to the contour of each bubble. As a result, a binary image was obtained with multiple polylines that delimit the contour of the bubbles. The existence of inflection points allowed to separate contours that belong to different bubbles. From this, an ellipse was adjusted to each contour using the technique of least squares from a given set of points. The bubble size was finally calculated by averaging the two axes of the adjusted ellipse. A similar process had been previously considered by other authors (e.g., Cuevas et al. 2013; Zafari et al. 2015).

Numerical Model

Computational fluid dynamics (CFD) models may be used to simulate two-phase flows such as air-water flows. However, the numerical models still present accuracy issues when modeling some hydraulic phenomena (Bombardelli 2012). In those situations, it is necessary to validate numerical results with experimental data.
There are different approaches to solve hydraulic jumps. Viti et al. (2018) made a review of CFD simulations for free hydraulic jumps. Numerical studies regarding submerged hydraulic jumps are scarce. Ma et al. (2001) considered that the inclusion of air entrainment, the streamline curvature effects, and more accurate free surface models would improve the numerical results of submerged hydraulic jumps.
In this study, the ANSYS CFX program (version 18.0) was used. It solves a finite-volume method. Solution variables and fluid properties are stored at the nodes (mesh vertices), whereas control volumes are constructed around the mesh nodes.
The simulations performed solve the differential Reynolds-averaged Navier-Stokes (RANS) equations of the flow in the mesh of the fluid domain, retaining the reference quantity in the three directions. The equations for the conservation of mass and momentum for the air-water mixture may be written in compact form as
∂(ρØ)∂t+∂∂xj[ρUjØ−Γ∂Ø∂xj]=S
(2)
where t is the time, Ø is the transported quantity, i and j are indices which range from 1 to 3, and xi represents the coordinate directions. For equations ρ=∑k=1Nprkρk, Uj=1ρ∑k=1NprkρkUkj, and Γ=∑k=1NprkΓk, rk indicates the volume fraction of kth fluid, Γk is the diffusion coefficient associated with the transported quantity for phase k, Np is the number of phases, and S is the sources/sinks for the transported quantity (ANSYS version 18.0).

Numerical Model Implementation

Simulations with two continuous fluids were considered in this study.
To solve the air-water two-phase flow, the Eulerian-Eulerian multiphase flow homogeneous model was selected. With this model, phases share the same velocity fields, as well as other relevant fields such as pressure and turbulence.
Transient simulations of 60 s were simulated using a time step of 0.05 s. The average of the different parameters was obtained after 40 s of simulation, once the quasi-steady-state conditions were reached.
The convergence of the solution was judged by monitoring the residuals for each equation at the end of each time step. The root mean square residual values were set at 10−4 for the mass and momentum variables.
The free surface model was selected to track the interphase. By default, it was assumed that the free surface was on the 0.5 air volume fraction (ANSYS version 18.0). A compressive discretization scheme was used in the volume fraction advection scheme to keep the interface sharp. This control reduces smearing at the free surface (ANSYS version 18.0). A surface tension model was considered with a surface tension coefficient of 0.072  N/m.
The numerical model was previously validated to obtain relatively good agreement solving rectangular free falling jets and their plunge pool (Carrillo 2014; Castillo et al. 2014, 2017). Those studies considered the evaluation of the mesh size and the choice of the turbulence model.

Boundary Conditions

Symmetry in the y-direction was observed in the laboratory device. For simplicity, only one half of the model was simulated. The symmetry condition in the longitudinal plane of the plunge pool was used.
In order to avoid smearing in the nappe flow resolution, the inlet condition of the numerical model was considered in the cross section of the free-falling located 0.55 m from the bottom; 0.22 m from the water cushion free surface (Fig. 3).
Fig. 3. Schematic of fluid domain and boundary conditions.
As the weir crest length was smaller than the stilling basin width, the impingement jet thickness was estimated using the parametric methodology for rectangular free-falling jets proposed by Castillo et al. (2015)
Bj=q2gH+4φh[2H−2h]
(3)
where q is the specific flow, H is the height between the upstream and the downstream water levels, h is the energy head over the weir crest, g is the gravitational acceleration, and φ is the turbulence parameter in the nappe flow case.
With this methodology, the impingement jet thickness was estimated to be Bj=0.029  m for a specific flow of 0.085  m2/s.
The transversal jet width was estimated from photography. During the fall of this specific flow, the jet does not suffer contraction/expansion effects in the transversal direction. The transversal jet width before the impingement point was measured as bj=0.85  m.
The jet did not enter the plunge pool vertically. The jet angle was estimated from the
Scimemi (1930) formulation
x*=2.155(z*+1)12.33−1
(4)
where x*=x/h, and z*=−z/h, with x e z being the coordinates axis considering the origin in the weir crest.
This formula was previously corroborated as a good estimator of rectangular jet trajectory (Castillo et al. 2014; Carrillo et al. 2018). The impingement jet angle was 11.8° from the vertical.
The impingement jet velocity and air concentration were measured with a dual-tip optical fiber probe. The jet velocity was slightly lower than the gravitational velocity from the upper tank (Vj=5.90  m/s). The mean air concentration of the jet obtained was Cair=0.50. Those jet velocity and air concentration values are in agreement with the jet transversal area if the continuity equation is considered.
The outlet condition was considered with flow normal to the boundary and hydrostatic pressure. The water level height at the outlet was modified according to the water cushion depth, Y, measured in the laboratory device. The open area in the outlet was consistent with the baffle used in the laboratory to generate the water cushion.
For all walls of the upper tank, the weir, and the dissipation basin, no slip smooth wall conditions were considered. The roughness of methacrylate was indicated in the walls.
To validate the inlet boundary condition, the mean pressure at the stagnation point was compared with laboratory measurements. GE Druck UNIK 5000 pressure transducers were used.
Considering a specific flow of 0.085  m2/s, a crest height of 2.20 m, and a water cushion depth of 0.32 m, the mean pressure obtained in the laboratory was 0.83 m. Using the boundary conditions observed in Fig. 4, the numerical model provided a mean pressure of 0.81 m in the stagnation point (a relative difference of 2.2% with laboratory data). This level of agreement may be considered accurate despite air concentrations being calculated with a homogeneous-type model (Jha and Bombardelli 2010; Bombardelli 2012).
Fig. 4. Mesh based on 0.010 m hexahedral elements, with elements of 0.0005 m near the bottom.

Mesh Convergence

ANSYS CFX has different near-the-wall treatments. ω-based turbulence models use automatic wall treatments which switch between regular wall functions (Pope 2000) and low-Reynolds wall treatment (Menter 1994). Wall functions are used when the wall adjacent vertices are in the log-law layer (y+=yu*/ν≈20–200). The low-Reynol ds wall treatment is used when the wall adjacent vertices are in the viscous sublayer (ANSYS version 18.0). Considering the wall treatment used by ANSYS CFX, the mesh sizes close to the solid boundary were smaller than in the rest of the domain. For all simulations, values of y+ were in the r an ge of the log-law layer.
The grid convergence index (GCI) (Roache 1997) may be used to ensure with a level of confidence that the solution is approaching the mesh convergence solution. With three mesh sizes, ASCE (2009) recommends a factor of safety, Fs=1.25; if two mesh sizes are compared, a very conservative safety factor, Fs=3.0, may be used. When testing the mesh convergence, the GCI values were computed using different mesh sizes. The analysis was based on the maximum velocity in the cross section located 0.10 m downstream of the stagnation point, giving GCI values of less than 2.0% for the test carried out (Table 2).
Table 2. Mesh convergence for maximum velocity in the cross section located 0.10 m downstream of the stagnation point
Mesh size near the wall (m)Maximum velocity Vx,max (m/s)Relative error (%)Less conservative GCI (%)Very conservative GCI (%)
0.00103.69———
0.00053.680.270.111.08
Considering those results and previous studies, a mesh based on 0.010 m hexahedral elements was considered. Special refinement was required near the bottom to obtain the recommended y+ values (Fig. 4).

Turbulence Models

To solve the Navier-Stokes equations in a reasonable time frame, four of the most usual two-equation turbulence models were tested for the free-falling jet and basin resolution: the standard k-ε model (Launder and Sharma 1974), the re-normalization group (RNG) k-ε model (Yakhot and Orszag 1986; Yakhot and Smith 1992), the k-ω turbulence model (Wilcox 2006), and the shear stress transport (SST) turbulence model (Menter 1994). The k-ε model was previously employed to solve the characteristics of submerged hydraulic jumps using two-dimensional predictions (Long et al. 1991).

Analysis of Results

Air Concentration in the Plunge Pool

Air concentration is one of the key parameters to be analyzed in the submerged hydraulic jump generated in the basin. The effects of entrained air may be essential for the safe operation of hydraulic structures (Wood 1991).
To know the air concentration, an optical fiber probe was employed. Different cross sections were analyzed downstream of the stagnation point of the rectangular free-falling jet (X = horizontal distance to the stagnation point). In each cross section, more than 50 points were collected. Fig. 5 shows the air concentration profiles (Cair=ΣtGi/t) obtained in the laboratory. The vertical axis has been normalized, considering the y95 value (distance at which Cair=95%).
Fig. 5. Air concentration profiles downstream of the stagnation point.
The process of self-aeration seems to be important only near the impingement point. Downstream of this region of entrainment, the transport capacity of the flow is limited, and it seems to be a detrainment region where the entrained air tends to escape.
The largest values of air concentration were obtained close to the stagnation point of the rectangular jet, reaching values relatively near the bottom of around 15%–20% (0.15<y/y95<0.35). As the flow moves downstream from the impact zone of the jet, the air concentration tends to decrease. The profiles located downstream of X=0.60  m from the stagnation point show air concentrations lower than 10%, reaching values around 7% in the profile located at X=1.00  m (for y/y95 between 0.2 and 0.8).
For y/y95>0.80−85, the air entrainment values increase rapidly. This may indicate the layer of the unsteady flow depth at a given location.

Bubble Frequency in the Plunge Pool

The frequency of detecting bubbles has been analyzed by different authors in free hydraulic jumps (e.g., Murzyn et al. 2005; Murzyn and Chanson 2007). However, there are few references to submerged hydraulic jumps downstream of nappe flows (Carrillo et al. 2018).
Fig. 6 shows the bubble frequency as detected by the optical probe.
Fig. 6. Bubble detection frequency profiles downstream of the stagnation point.
Following Chanson (1995), different flow regions may be identified in the hydraulic jump: (1) the entrapped air packets are broken up into smaller air bubbles in the turbulent shear region, with the highest values being obtained in the vicinity of the bottom (y/y95<0.40) in most of the profiles; (2) the bubble frequency tends to reduce in the boiling flow region; and (3) a foam layer appears near the free-surface that tends to increase the bubble frequency (y/y95≈0.90).
In the profile located closest to the stagnation point, maximum values of around 120 Hz were obtained. The maximum values tend to reduce as the flow moves away from the stagnation point. In the profile located at X=1.00  m, the frequency tends to be around 20–30 Hz for almost the entire depth (0.15<y/y95<0.90).
In this study, the impingement Froude number was around 11.08. The values recorded were similar to the results obtained by Murzyn and Chanson (2007) who measured maximum frequencies of around 120 Hz for free hydraulic jumps with a Froude number of 8.30, and relatively larger than those of Murzyn et al. (2005) who obtained maximum frequencies of around 85 Hz in free hydraulic jumps with a Froude number of 4.82.
As in Murzyn et al. (2005), the bubble frequency profiles near the stagnation point show two different bubble frequency peaks. The first one corresponds to the wall jet, whereas the second one is located at the beginning of the recirculation region.

Distribution of Velocities in the Dissipation Basin

Velocities at the plunge pool located downstream of the stagnation point were measured with two different types of intrusive equipment. Those data were essential to achieve a validated numerical model. Results for the same cross sections were obtained from the CFD simulations using four different turbulence models. Fig. 7 shows the results obtained in the cross section located 0.10 m downstream of the stagnation point.
Fig. 7. Comparison of horizontal mean velocity in X=0.10  m of the plunge pool.
The laboratory equipment and the numerical models allow to analyze the wall jet. Near the bottom, the streamlines are parallel, and the laboratory equipment seems to be accurate. The maximum velocity values of the numerical simulations obtained with k-ε and SST turbulence models are in agreement with the measurements obtained with the back-flushing Pitot tube and with the optical probe, with relative differences lower than 2% for the measurement point located 0.01 m from the bottom (y/y95=0.02). However, the k-ω turbulence model provides around 8% smaller maximum velocities, and the RNG k-ε turbulence model yields relative differences in the maximum velocity of around 26% in the vicinity of the bottom.
Outside the wall jet, the behavior of the turbulence models is similar. However, in that region, the streamlines are not parallel to the bottom, and laboratory measurements may be affected by the velocity vector angle. For that reason, laboratory velocities were not recorded in the roller.
The velocity profiles in the forward flow of hydraulic jumps can be compared if they are normalized with a velocity scale equal to the maximum velocity Vmax at any section, and with a length scale δl equals to the vertical distance y from the bottom where the local velocity V=Vmax/2, and the velocity gradient being negative (Rajaratnam 1965).
Using the results obtained by diverse researchers, Wu and Rajaratnam (1996) considered the length scale δl for the submerged jumps as a function of the distance to the beginning of the jump. Those authors deduced that most of the observations for submerged jumps are contained within one standard deviation of the mean value of the wall jet, and only the data points near the end of the jump show an accelerated growth rate.
Table 3 summarizes the equations proposed by several authors for hydraulic jumps.
Table 3. Formulae of the velocity distribution in wall jets and hydraulic jumps
AuthorFormula
Görtler (1942), cited by Liu et al. (1998)VxVmax=1−tanh2(0.881yδl)
Rajaratnam (1976)VxVmax=e−0.693(yδl)2
Lin et al. (2012)VxVmax=2.3(yδl)0.42(1−erf(0.886yδl))
De Dios et al. (2017)VxVmax=2.0(yδl)1/7(1−erf(0.55yδl))−0.39
Castillo et al. (2017)VxVmax=1.48(yδl)1/7(1−erf(0.66yδl))

Note: erf(u)=2π∫0ue−t2dt.

Following those ideas, the maximum velocity Vmax and the length scale δ1 were calculated in each cross section of the submerged hydraulic jump. Fig. 8 shows the nondimensional velocity profiles in cross sections located from 0.10 to 1.00 m downstream of the stagnation point. Laboratory measurements were obtained with the back-flushing Pitot tube and optical fiber equipment. In the same cross sections, the results of the CFD simulations were obtained with two different turbulence models.
Fig. 8. Profiles of horizontal mean velocity in different sections of the plunge pool.
In general, the results of the numerical simulations show the same behavior as the values obtained in the laboratory. Few differences were detected between the results obtained with the k-ε and SST turbulence models. Among them, the k-ε model results seem to be slightly more similar to the laboratory measurements (Fig. 7).
The optical probe data tends to be more disperse than the Pitot measurements. This may be related with the uncertainty concerning the mean velocity measurement in the optical fiber equipment when data out of the wall jet were measured (the accuracy of the optical probe is based on the successive detection of air-water interfaces by the two tips aligned with the flow direction). These results are in line with those obtained in chute spillways, where the flow curvature seemed to have an effect on the accuracy of the measurements of the optical fiber probe (Matos et al. 2002). According to the manufacturer, the results should be considered with caution when the cross-correlation coefficient is lower than 0.7 (RBI-Instrumentation 2012).
The results are also in agreement with nondimensional velocity distribution formulae obtained by several authors.
In the upper part of the plot, there is a recirculation area. The laboratory results tend to differ from those obtained with the numerical models and with the nondimensional distribution obtained by Castillo et al. (2017) in submerged hydraulic jumps downstream of free-falling jets. Those data are from points out of the wall jet, where streamlines are not parallel to the bottom, and laboratory measurements may be inaccurate.

Mean Bubble Size in the Plunge Pool

Assuming the hypothesis that bubbles are spherical and equally distributed in time, the size of the bubbles detected by the optical fiber equipment can be characterized by the Sauter mean diameter, Dsm (Clift et al. 1978; RBI-Instrumentation 2012). This is the diameter of the bubbles whose volume/surface ratio is the same as that calculated for all the bubbles detected during the test. The Sauter mean diameter can be calculated as
Dsm=3CairV2F
(5)
where Cair is the air concentration, V is the mean velocity of the bubbles, and F is the bubble detection frequency.
As the mean velocity measured by the optical fiber equipment may be inaccurate when the flow is non one-directional, the CFD velocities obtained with the k-ε turbulence model were employed in this section. This enables the bubble size distribution in the entire submerged hydraulic jump to be obtained.
Fig. 9 shows the bubble size calculated in different sections of the plunge pool. Most values are around 2 mm, with maximum values below 4 mm. These diameters are slightly lower than the results obtained by Murzyn et al. (2005) and by Murzyn (2010) in hydraulic jumps.
Fig. 9. Mean diameter of bubbles detected downstream of the stagnation point.
Mean diameter results were compared with the values obtained from a high-speed camera. Images were recorded at two different distances from the stagnation point of the jet, 0.10 m and 0.60 m. In each of these locations, three different elevations from the bottom were measured (0.10, 0.20, and 0.25 m). The diameters were analyzed with a bubble fitting algorithm. The contours of bubbles were detected, considering a contour validation through curvature analysis. Each contour was adjusted to ellipse fits. A total of 140 images with a window size of 0.080×0.080  m were analyzed. Fig. 10 shows the mean diameter size obtained in the cross section located 0.10 m downstream of the stagnation point at an elevation of 0.20 m from the bottom (y/y95≈0.60). At this location, the mean bubble diameter obtained as the average of the two axes of the ellipses was around 3.0 mm. The results are in agreement with Fig. 9.
Fig. 10. Detail of bubbles in the cross section located 0.10 m downstream of the stagnation point at elevation of 0.20 m from the bottom.

Conclusions

The analysis of two-phase flows in hydraulic structures is very complicated because of the undiluted nature of the flow. Under these conditions, the experimental data and the simulations results may not be easy to compare clearly.
The main challenges to perform numerical simulations of submerged hydraulic jumps downstream of a nappe flow jet are associated to the free-falling jet definition, the changing flow direction, and the air entrainment modeling.
In the present work, laboratory and numerical approaches have been used together to reduce the limitation of each technique.
Optical fiber and back-flushing Pitot tube equipment were mainly used in the quasi-unidirectional flow region of the submerged hydraulic jump. Those data allowed to verify the CFD simulations. The numerical model provided results close to the values measured in the laboratory, despite having used a simple two-phase flow model.
The choice of the turbulence model seems to play a key role to solve the wall jet. The k-ε model was relatively more accurate.
In general, the characteristics of the air-water flow (air concentration, bubble frequency, and Sauter mean diameter) are similar to those observed in free hydraulic jumps with different Froude numbers. This allows us to consider that our findings may describe the submerged hydraulic jump downstream of a nappe flow case.
The nondimensional numerical velocity flow profiles were in line with those gathered in previous relevant studies of free and/or submerged hydraulic jumps. Hence, the results may be considered representative of the phenomenon, limiting possible scale effects.
Considering the observations of Wood (1991), a better knowledge of the air concentrations downstream of a nappe flow case would be important to designers as it may affect their designs.

Notation

The following symbols are used in this paper:
Bj
impingement jet thickness;
bj
transversal jet width before the impingement point;
Cair
air concentration;
Dsm
Sauter mean bubble diameter;
F
bubble detection frequency;
F
Froude number;
Fs
safety factor;
g
gravitational acceleration;
H
height between the upstream water level and the downstream water;
h
energy head over the weir crest;
k=1/2ui′ui′¯
turbulent kinetic energy;
Np
number of phases;
n
diffraction index;
Pl
weir crest elevation;
q
specific flow;
R
Reynolds number;
rk
volume fraction of kth fluid;
S
sources/sinks for the transported quantity;
t
time;
tGi
time the probe is in air;
tu
turbulence intensity;
Ui
mean velocity;
u*
friction velocity at the nearest wall;
V
mean velocity;
Vj
impingement jet velocity;
Vmax
maximum velocity of the velocity profile;
V′¯
root mean square of the velocity;
W
Weber number;
X
horizontal distance to the stagnation point;
x, y, z
coordinates axis;
Y
water cushion depth;
y+
dimensionless wall distance;
y95
distance to the bottom at which Cair=95%;
Z
liquid parameter (Morton number);
ΔP
difference between the total pressure head and the static pressure head;
δl
length scale for the hydraulic jumps;
ε
rate of dissipation of turbulence energy;
ρw
density of water;
ν
kinematic viscosity of the fluid;
λ
tapping coefficient;
ρ
flow density;
Ø
transported quantity;
φ
turbulence parameter in nappe flow; and
ω
turbulent frequency.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

The researchers express their gratitude for the financial aid received from the Ministerio de Ciencia, Innovación y Universidades (MCIU), the Agencia Estatal de Investigación (AEI), and the Fondo Europeo de Desarrollo Regional (FEDER), through “The flow aeration in the free surface overtopping of dams in prototype situations and its effect in energy dissipation plunge pools” Project (RTI2018-095199-B-I00).

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Journal of Hydraulic Engineering
Volume 146 • Issue 6 • June 2020

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.

History

Received: Jan 15, 2019
Accepted: Dec 16, 2019
Published online: Apr 11, 2020
Published in print: Jun 1, 2020
Discussion open until: Sep 11, 2020

Authors

Affiliations

J. M. Carrillo, Dr.Eng. https://orcid.org/0000-0003-4264-3269 [email protected]
Associate Professor, Hydraulic Engineering Area, Universidad Politécnica de Cartagena, Cartagena 30.203, Spain (corresponding author). ORCID: https://orcid.org/0000-0003-4264-3269. Email: [email protected]
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L. G. Castillo, Dr.Eng. [email protected]
Titular Professor, Hydraulic Engineering Area, Universidad Politécnica de Cartagena, Cartagena 30.203, Spain. Email: [email protected]
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F. Marco https://orcid.org/0000-0002-2277-5515 [email protected]
Ph.D. Student, Hydraulic Engineering Area, Universidad Politécnica de Cartagena, Cartagena 30.203, Spain. ORCID: https://orcid.org/0000-0002-2277-5515. Email: [email protected]
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J. T. García, Dr.Eng. [email protected]
Lecturer, Hydraulic Engineering Area, Universidad Politécnica de Cartagena, Cartagena 30.203, Spain. Email: [email protected]
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