water Article
Numerical Simulation of Hydraulic Characteristics in A Vortex Drop Shaft Wenchuan Zhang , Junxing Wang *, Chuangbing Zhou, Zongshi Dong and Zhao Zhou State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China;
[email protected] (W.Z.);
[email protected] (C.Z.);
[email protected] (Z.D);
[email protected] (Z.Z.) * Correspondence:
[email protected]; Tel.: +86-137-0718-2138 Received: 18 July 2018; Accepted: 25 September 2018; Published: 28 September 2018
Abstract: A new type of vortex drop shaft without ventilation holes is proposed to resolve the problems associated with insufficient aeration, negative pressure (Unless otherwise specified, the pressure in this text is gauge pressure and time-averaged pressure) on the shaft wall and cavitation erosion. The height of the intake tunnel is adjusted to facilitate aeration and convert the water in the intake tunnel to a non-pressurized flow. The hydraulic characteristics, including the velocity (Unless otherwise specified, the velocity in this text is time-averaged velocity), pressure and aeration concentration, are investigated through model experiment and numerical simulation. The results revealed that the RNG k-ε turbulence model can effectively simulate the flow characteristics of the vortex drop shaft. By changing the inflow conditions, water flowed into the vertical shaft through the intake tunnel with a large amount of air to form a stable mixing cavity. Frictional shearing along the vertical shaft wall and the collisions of rotating water molecules caused the turbulence of the flow to increase; the aeration concentration was sufficient, and the energy dissipation effect was excellent. The cavitation number indicated that the possibility of cavitation erosion was small. The results of this study provide a reference for the analysis of similar spillways. Keywords: vortex drop shaft; turbulence model; spillway aerators; energy dissipation; cavitation
1. Introduction Compared with traditional energy dissipators, vortex drop shaft spillways can rapidly change the flow regime and form areas of turbulence or whirlpools to dissipate energy. These spillways provide various functions, such as flood discharge control, energy dissipation and drainage structure protection. Additionally, they transfer the energy dissipation task from outside to inside to avoid outlet atomization (in the traditional discharge energy dissipation process, water interacts with the air boundary to form an atomized flow [1]). However, within these spillways, the pressure near the vertical shaft wall gradually decreases because of gravity and wall friction, and generated negative pressure can easily cause cavitation erosion [2,3]. Therefore, considerable attention has been paid to negative pressure and cavitation erosion in the applications of vortex drop shaft spillways. To solve these problems, erosion reduction devices are often installed in vortex drop shaft spillways to ensure the stable operation of the discharge structures. Many studies have investigated the characteristics of vortex drop shaft spillways, mainly focusing on the diameter of the vertical shaft and the depth of the dissipation well; they found that the energy dissipation effect could be improved by optimizing the vertical shaft shape [4]. Design schemes of vortex chambers and vertical shafts were proposed based on previous research on vortex drop shaft spillways, and the design criteria was summarized [5,6]. Based on an analysis of the influence of the body shape on the hydraulic parameters of an energy dissipation well, one study found that a Water 2018, 10, 0; doi:10.3390/w10100000
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reasonable depth for an energy dissipation well should be 1.69 times the vertical shaft diameter [7]. Other studies focused on the tangential slot vortex intakes of vertical shafts for urban drainage, and they found that the flow in a tapering and downward-sloping vortex inlet channel was strongly dependent on the geometry of the inlet and vertical shaft and that the hydraulic instability was related to the discharge [8,9]. Del Giudice and Gisonni [10] found that an appropriate length of the lowered bottom could prevent free surface fluctuations at the intake structure, resulting in an appropriate vortex air core without choking problems for supercritical approach flows. In addition, the hydraulic characteristics of swirling flows and the energy dissipation rate were both related to the swirl number. The larger the swirl number was, the greater the energy dissipation. Furthermore, the tangential velocity ensured no negative pressure on the vertical shaft wall and enhanced the flow stability [11]. The radial distribution of the pressure in the swirl zone complied with the theoretical distribution of the pressure in combined eddies, and the pressure increased as the radius increased [12]. In previous studies, the velocity of a cavity swirl was assumed based on the theory of combined vortex and free vortex, and a corresponding pressure formula was obtained [13,14]. In another investigation, the need for aeration facilities was emphasized, and a method for calculating the aeration cavity length of annular aeration in a swirling vertical shaft was derived based on projectile theory [15]. Dong et al. [16] used the standard k-ε turbulence model to simulate a vortex drop shaft spillway and obtained the hydraulic parameters of spiral flow. Gao et al. [17] simulated the characteristics of turbulent flow through a vertical pipe inlet/outlet with a horizontal anti-vortex plate. However, compared with the standard k-ε turbulence model, the RNG k-ε turbulence model can more effectively process flows with high strain rates and streamline bending by correcting the turbulent viscosity and considering both rotation and swirling in the average flow regime. In addition, a previous study found that the water surface was more stable using this approach, and the simulation results agreed well with experimental data [18,19]. For traditional vortex drop shaft spillways, the water in the intake tunnel moves with pressure flow, and thus ventilation holes are set in the volute chamber or in front of the outlet tunnel [2,5,20]. Moreover, an annular aerator was set in the middle of the shaft to increase the aeration concentration and avoid erosion [15]. In practical engineering, aeration facilities often cause great difficulties during the process of construction. Therefore, to resolve the insufficient energy dissipation, cavitation of traditional spillways and construction difficulty, a vortex drop shaft with an increased-height intake tunnel and without ventilation holes is investigated in the present study. The pressure field, flow field and energy dissipation are simulated and analyzed by the renormalization group (RNG) k-ε turbulence model [21], and the results are compared with experimental data to better understand the distribution trends of the hydraulic parameters, the characteristics of the energy dissipation and the cavitation associated with a vortex drop shaft. 2. Physical Model A Froude similitude with a geometric scale ratio of 1:25 was applied in the physical model. The model scale has a slight effect on the time-averaged hydraulic characteristics such as the time-averaged water depth, pressure and velocity, and the scale effects could be negligible, so they can provide useful references for practical engineering [22]. Small-scale models based upon the Froude similitude may underestimate the air transport in the fluid because the relative effects of the surface tension and viscosity are over-represented, especially when the scale is smaller than 1:30 [23–26]. In summary, a model experiment can simulate the time-averaged water depth, pressure and velocity well, and the aeration concentration is underestimated so that it is larger in the prototype. A higher aeration concentration is more favourable for preventing cavitation corrosion, so the experimental results can be used as a reliable reference for engineering design. The vortex drop shaft investigated in this study consists of a tangential intake tunnel, a volute chamber, a gradient section, a vertical shaft, a dissipation well, and an outlet tunnel. The intake tunnel has an arched shape, with a slope and length of 1:7.5 and 0.68 m, respectively, and the section size
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changed from 0.2 m × 0.24 m (width × height) to 0.176 m × 0.33 m (width × height). The diameter of the volute chamber isPEER 0.352 m, and the height from the gradient section to the top of volute chamber is Water 2018, 10, x FOR REVIEW 3 of 19 0.72 m. The diameter and depth of the vertical shaft are 0.216 m and 1.82 m, respectively. The depth of changed from 0.2ism0.28 × 0.24 (width × height)oftogradient 0.176 m ×section 0.33 m (width height). The diameter the dissipation well mmand the height is 0.24 ×m. The outlet tunnel of has an theshape, volute chamber is 0.352height, m, and the height from the of gradient to the of volute arched with a width, length and slope 0.2 m,section 0.24 m, 6.1 top m and 1:50,chamber respectively. 0.72no m.ventilation The diameter and depth the vertical shaft gradient are 0.216 m and 1.82orm,vertical respectively. Thereisare holes in theofvolute chamber, section, shaft,The anddepth air flows of the dissipation well is 0.28 m and the height of gradient section is 0.24 m. The outlet tunnel has an Water 2018, 10, xthe FOR intake PEER REVIEW of 19 with the water from tunnel into the vertical shaft. The sloped section and the3 straight plate arched shape, with a width, height, length and slope of 0.2 m, 0.24 m, 6.1 m and 1:50, respectively. are set at the inlet of the outlet tunnel to decrease fluctuations in the water and ensure that the flow Therechanged are no ventilation in(width the volute chamber, verticalThe shaft, and air from 0.2 m ×holes 0.24 m × height) to 0.176gradient m × 0.33 section, m (widthor × height). diameter offlows regime is the open-channel flow. The structure is shown in Figure 1. Totoreveal characteristics the water volute from chamber isintake 0.352 m, and the height from theshaft. gradient the top the of volute chamber plateof the with the tunnel into the vertical Thesection sloped section and the straight energy and within the shaft vortex three tests at flood frequencies 0.72 m.inlet The diameter and depth of the vertical are drop 0.216 mshaft, and 1.82 m, respectively. The depth aredissipation setisat the of the the cavitation outlet tunnel to decrease fluctuations in the water and ensure that the flow of of the dissipation well is 0.28 m and the height of gradient section is 0.24 m. The outlet tunnel has anof 5%, 2% and is 0.1% (expressed as PThe = 5%, 2% and 0.1% in following) were out in this regime open-channel flow. structure is shown in the Figure 1. To reveal thecarried characteristics thepaper. arched shape, with a width, height, length and slope of 0.2 m, 0.24 m, 6.1 m and 1:50, respectively. energy dissipation and the vortex0.18 drop shaft, at flood frequencies A fluviograph (the margins ofcavitation error are within plus orthe minus mm) is three used tests to control the upstreamofwater There are no ventilation holes in the volute chamber, gradient section, or vertical shaft, and air flows and 0.1% (expressed P = outlet 5%, 2% and 0.1% in the following) were carried out this paper. level,5%, and2% the tailwater depth inasthe is controlled by a section valve located inin the downstream with the water from the intake tunnel intotunnel the vertical shaft. The sloped and the straight plate A fluviograph (the margins of error are plus or minus 0.18 mm) is used to control the upstream channel. Eighteen are established in the wall ofand theensure vertical shaft, as shown are set at themeasurement inlet of the outletpoints tunnel to decrease fluctuations in the water that the flowwater level, regime and theis tailwater depth in the outlet tunnel is controlled by a valve located in the downstream open-channel The structure is shown in Figure 1. downstream To reveal the characteristics of thedrop shaft, in Figure 1, the odd and evenflow. numbers located upstream and of the vortex channel. Eighteen measurement pointswithin are established in the wall of the as shown in energy dissipation and the cavitation the vortex drop shaft, three testsvertical at flood shaft, frequencies of respectively; the measurement locations are close to wall. 0.1% (expressed as P = 5%, 2% andupstream 0.1% in theand following) were carried outvortex in this paper. Figure5%, 1, 2% theand odd and even numbers located downstream of the drop shaft, In theA present study, the experimental velocity was measured by a the small specially fluviograph (the margins oflocations error are plus or minus 0.18 mm) is used to control upstream water designed respectively; the measurement are close to wall. L-shapedIn tube [27] similar to a Pitot tube, which is 160 mm long and 2 mm in diameter with La short level, and the tailwater depth in the outlet tunnel is controlled by a valve located in the downstream the present study, the experimental velocity was measured by a small specially designed channel. Eighteen measurement points are established in the wall of the vertical shaft, as shown in inlet shaped as shown Figure 2. toThe margins of error are mm pluslong or minus 5 percent. Thewith L-shaped tubein[27] similar a Pitot tube, which is 160 and 2 mm in diameter a short tube Figure 1, the odd and even numbers located upstream and downstream of the vortex drop shaft, was aligned with the direction ofmargins incoming floware by plus visual measurement, the resulttube waswas carried inlet as shown in Figure 2. The of error or minus 5 percent. and The L-shaped respectively; the measurement locations are close to wall. aligned with the direction of incoming flow by visual measurement, and the result was carried out out after the liquid column height reached the maximum and was Time-averaged In the present study, the experimental velocity was measured by a stable. small specially designed L- pressure after the liquid height thewhich maximum and wasand stable. Time-averaged was was measured by acolumn liquid column manometer based on long hydrostatic principle, andpressure accuracy is shaped tube [27] similar to areached Pitot tube, is 160 mm 2 mm in diameter with athe short measured by a liquid column manometer based on hydrostatic principle, and the accuracy is 0.1mm. inletaeration as shown in Figure 2. The margins of error are plus or minus 5 percent.aeration The L-shaped tube was 0.1mm. The concentration was measured with a CQ6-2004 concentration meter aligned with the directionwas of incoming flowwith by visual measurement, and the result was carried The aeration concentration measured a CQ6-2004 aeration concentration meterout (China (China Institute of Water Resources and Hydropower Research, Beijing, China), and a single-chip after of the Water liquid column heightand reached the maximum and was stable. Time-averaged pressure was Institute Resources Hydropower Research, Beijing, China), and a single-chip microcomputer wasbyused to acquiremanometer and process data. Theprinciple, aeration concentration is determined by measured a liquid basedthe on hydrostatic the accuracy is microcomputer was usedcolumn to acquire and process the data. The aerationand concentration is 0.1mm. determined detecting the of water and electrodes. The resolution is 0.1% and Theresistance aeration concentration was aerated measuredwater with abetween CQ6-2004two aeration concentration meter (China by detecting the resistance of water and aerated water between two electrodes. The resolution is 0.1% Institute of Water Resources and Hydropower Research, Beijing, China), and a single-chip the sampling frequency is 1020isHz. and the sampling frequency 1020 Hz. microcomputer was used to acquire and process the data. The aeration concentration is determined by detecting the resistance of water and aerated water between two electrodes. The resolution is 0.1% 2.54 diversion sill x 0.200 and the sampling frequency is 1020 Hz. 2.42
1.50
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1.301.50 1.101.30 0.901.10 0.90
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Figure Vortex drop shaft profile. Figure Vortex drop drop shaft profile. Figure 1.1.1.Vortex shaft profile. 0.008 0.008 m m
0.002 m 0.16 m
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Figure 2. The structure of L-shaped tube. Figure ofL-shaped L-shapedtube. tube. Figure2.2.The Thestructure structure of
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3. Numerical Simulation 3.1. Turbulence Model Considering the advantage of the free interface capture and the specialized air entrainment model inside [28–30], Flow-3D (Flow Science, Santa Fe, NM, USA) was chosen for the numerical simulation. The RNG k-ε turbulence model and the volume of fluid (VOF) method [31] are used for the turbulence model and free surface tracking, respectively. Numerical discretization is performed with the finite difference method. The algebraic equations are solved using the generalized minimum residual (GMRES) method. The fluid is assumed to be incompressible, the governing equations are presented as follows: Continuity equation ∂ui =0 (1) ∂xi Momentum equation 1 ∂p ∂ fi − + ρ ∂xi ∂x j
∂u ν i ∂x j
!
=
∂ui ∂u + uj i ∂t ∂x j
(2)
k equation
ε equation
∂k ∂ ∂(ρk) ∂(ρui k) αk µe f f + Gk + ρε + = ∂t ∂xi ∂xi ∂xi
(3)
C∗ ∂ ∂(ρε) ∂(ρui ε) ∂ε ε2 = + αε µe f f + 1ε Gk − C2ε ρ ∂t ∂xi ∂xi ∂xi k k
(4)
The fluid configurations are defined in terms of a VOF function F(x,y,z,t), which represents the VOF per unit volume and satisfies the following equation: ∂F ∂ + ( Fui ) = 0 ∂t ∂xi
(5)
where ui and xi are the time-averaged velocity and coordinate components, respectively. f i is gravity component, t is the time, µ, ν and ρ are coefficient of dynamic viscosity, kinematic viscosity and density, respectively. Gk is the generation of turbulent energy caused by the average velocity gradient, p is the gauge pressure, µe f f is the revisionary coefficient of dynamic viscosity, F is the VOF function, η 1 η 1− η ∂u j ∂ui ∂ui k2 ∗ 0 µe f f = µ + µt , Gk = µt ∂x + ∂x ∂x , µt = ρCµ ε , C1ε = C1ε − 1+ βη 3 , η = 2Eij × Eij 2 kε , Eij = j i j ∂u j 1 ∂ui 2 ∂x + ∂x . There are some constants provided by Launder [32] and verified by a large number of j
i
experiments: Cµ = 0.0845, αk = αε = 1.39, C1ε = 1.42, C2ε = 1.68, η0 = 4.377 and β = 0.012. 3.2. Air Entrainment Model The air entrainment model in FLOW-3D was first proposed by Hirt in 2003 [33,34]. This model assumes that air entrainment at the free surface is caused by an instability force Pt produced by the turbulence of the free surface. When the level of turbulence exceeds the stability force Pd , which is associated with gravity and surface tension, air with a volume δV will be entrained into the water. The governing equations are as follows: LT =
CNU 0.75 k1.5 ε
Pt = ρk; Pd = ρgn L T +
(6) σ LT
(7)
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( δV =
Cair AS
h
2( Pt − Pd ) ρ
i
0
if if
Pt > Pd
(8)
Pt < Pd
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where L T denotes the turbulence length scale, CNU is a constant equal to 0.09, k and ε are the turbulent kinetic energy and turbulent dissipation rate, gn is the component of gravity normal to 2 𝑃 respectively. −𝑃 𝑖𝑓 𝑃 𝑃 𝐶 𝐴 the free surface, σ is the coefficient δV is the volume of air entrained 𝛿𝑉 =of surface tension, (8) per unit time, 𝜌 0 As represents 𝑖𝑓 𝑃the𝑃surface area. Cair is a coefficient of proportionality and where 𝐿 denotes the turbulence length scale, 𝐶𝑁𝑈 is a constant equal to 0.09, 𝑘 and 𝜀 are the
3.3. Computational Grid andand Boundary turbulent kinetic energy turbulentConditions dissipation rate, respectively. gn is the component of gravity
normal to the free surface, 𝜎 is the coefficient of surface tension, 𝛿𝑉 is the volume of air entrained per
For the experimental results, the upstream (Y-Min) unitcomparison time, 𝐶 is a with coefficient of proportionality and 𝐴 represents the surface area. and top (Z-Max) boundaries are set as pressure boundaries, and the water elevation at the upstream boundary is added, 3.3. Computational Grid and Boundary Conditions the downstream (Y-Max) boundary is set as the outflow. The left (X-Min), right (X-Max) and bottom For comparison with experimental results, the upstream (Y-Min) and top (Z-Max) of the nested (Z-Min) boundaries are set as the solid, non-slip wall boundaries. Moreover, the boundaries boundaries are set as pressure boundaries, and the elevationincludes at the upstream grids are set as symmetry, as shown in Figure 3a. water The model 2 meshboundary blocks, isand the nested added, the downstream (Y-Max) boundary is set as the outflow. The left (X-Min), right (X-Max) and grids are used to ensure the accuracy of grid segmentation in the calculation, as shown in Figure 3b. bottom (Z-Min) boundaries are set as solid, non-slip wall boundaries. Moreover, the boundaries of To evaluate thegrids gridare independence ofasthe results, the3a. discharges were undertaken the nested set as symmetry, shown in Figure The model tests includes 2 mesh blocks, andwith different the nested grids are used to ensure the accuracy of grid segmentation in the calculation, as shown inbetween grids grids. Four schemes were carried out to verify grid accuracy in Table 1, the difference Figure 3b. To evaluate the grid independence of the results, the discharges tests were undertaken 3 and 4 was 1.12%, which suggested that grid 4 was already sufficient for the simulation. The time with different grids. Four schemes were carried out to verify grid accuracy in Table 1, the difference step isbetween variable, and the initial time step and residual error are 1 × 10−7 and 1 × 10−6 , respectively. grids 3 and 4 was 1.12%, which suggested that grid 4 was already sufficient for the −7 and Additionally, theThe termination time is and set as time which the rate of change the1total volume of simulation. time step is variable, the the initial timeatstep and residual error are 1 × 10in × 10−6is , respectively. Additionally, the termination time set sasin thethis timepaper. at which the rate of change in the fluid less than 0.1%, and simulation time is is200 the total volume of the fluid is less than 0.1%, and simulation time is 200 s in this paper.
(a) Boundary conditions
(b) Grid mesh
Figure 3. Boundary conditions and grid mesh in the numerical model.
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Figure 3. Boundary conditions and grid mesh in the numerical model. Table Table 1. 1. The The results results of of grid grid independence independence (P (P == 0.1%). 0.1%).
Grid Grid 1
1
2
2
3
3
4
4
Size (X (X ××YY××Z)Z) Size 1515mm mm××1515mm mm××15 15mm mm mm××1010mm mm××10 10mm mm 1010mm mm××1010mm mm××10 10mm mm 1010mm mm××5 5mm mm××55mm mm 5 5mm mm××5 5mm mm××55mm mm 5 5mm 3 mm × 3 mm × 3 mm 3 mm × 3 mm × 3 mm mm××4 4mm mm××44mm mm 4 4mm 2 mm × 2 mm × 2 mm 2 mm ×2 mm × 2 mm
Description Description containing block block containing nested nested block block containing containing block block nested nested block block containing containing block block nested nested block block containing containing block block nested block nested block
Discharge (L/s) Discharge (L/s)
54.47 54.47 58.25 58.25 60.28 60.28 60.96 60.96
4. Results and and Discussion Discussion 4. Results 4.1. Model Verification Verification 4.1. Model Liu et et al. al.[19] [19]found foundthat thatboth boththe the standard and RNG turbulence models could simulate Liu standard and RNG k-εk-ε turbulence models could simulate the the flood movement of the vortex drop shaft spillway well and reproduced the water flow flood movement of the vortex drop shaft spillway well and reproduced the water flow state ofstate the of the swirling flow in cavity in the vertical shaft; however, k-ε turbulence model predicted swirling flow cavity the vertical shaft; however, the RNGthe k-εRNG turbulence model predicted pressure pressure flowbetter velocity than thek-ε standard model. Figure shows that the and flow and velocity thanbetter the standard model. k-ε Figure 4 shows that4 the pressure andpressure velocity and velocity distributions the shaft vortex drop shaft calculated thehave two similar models patterns. have similar distributions of the vortex of drop calculated using the twousing models The patterns. The pressure and velocity values calculated in the vertical shaft were close and in rather good pressure and velocity values calculated in the vertical shaft were close and in rather good agreement agreement with the experiment; k-ε turbulence model generated to with the experiment; however, however, the RNG the k-ε RNG turbulence model generated results results closer closer to those those obtained in the experiment, especially in the dissipation well. The relative deviations for pressure obtained in the experiment, especially in the dissipation well. The relative deviations for pressure of of standard and RNG turbulencemodels modelswith withrespect respecttotothe themodel modeltest test results results were were 18% 18% and thethe standard and RNG k-εk-εturbulence and 11%, respectively, and the relative deviations for velocity were 8% and 5%, respectively. In general, 11%, respectively, and the relative deviations for velocity were 8% and 5%, respectively. In general, the drop shaft shaft than than the the standard standard the RNG RNG k-ε k-ε turbulence turbulence model model can can better better simulate simulate the the flow flow of of the the vortex vortex drop k-ε turbulence model, indicating that the RNG k-ε turbulence model is more suitable for dealing a k-ε turbulence model, indicating that the RNG k-ε turbulence model is more suitable for dealingwith with distorted flow or rotational flow with a high strain rate. Therefore, the following analysis is based on a distorted flow or rotational flow with a high strain rate. Therefore, the following analysis is based the RNG k-ε k-ε model. on the RNG model. 2.0
z (m)
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standard standard
velocity velocity (m/s) (m/s) 0.0 0.0
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(c)Upstream Upstreamvelocity velocity (c)
0.5 0.5 0.0 0.0
velocity velocity (m/s) (m/s) 0.0 0.0
2.0 2.0
4.0 4.0
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(d)Downstream Downstreamvelocity velocity (d)
Figure4.4.Pressure Pressureand differentturbulence turbulencemodels models(P (P===0.1%). andvelocity velocitycalculated calculatedusing usingdifferent different turbulence models (P 0.1%). Figure
4.2. Flow Regime Regime 4.2.Flow Flow 4.2. Regime The discharges, which depend on the flood elevation upstream, are are obtained through via via flow flow Thedischarges, discharges,which whichdepend dependon on theflood floodelevation elevationupstream, upstream, obtainedthrough through The the are obtained via flow meter and baffle (the baffle is a tool in Flow-3D). Because of measurement error in flow meter and meterand andbaffle baffle(the (thebaffle baffleisisaatool toolin inFlow-3D). Flow-3D).Because Becauseof ofmeasurement measurementerror errorin inflow flowmeter meterand and meter computational grid, small differences were unavoidable. Overall, the errors between experiment and computational grid, small differences were unavoidable. Overall, the errors between experiment and computational grid, small differences were unavoidable. Overall, the errors between experiment and simulation were less than 3% (the maximum error is 2.14% at the flood frequency of 0.1%) as shown simulationwere wereless lessthan than3% 3%(the (themaximum maximumerror errorisis2.14% 2.14%at atthe theflood floodfrequency frequencyof of0.1%) 0.1%)as asshown shown simulation in Table 2. The numerical results were similar to the experimental results, which suggested that the inTable Table2. 2.The Thenumerical numericalresults resultswere weresimilar similarto tothe theexperimental experimentalresults, results,which whichsuggested suggestedthat thatthe the in numerical simulation results were reliable. Figure 5 shows the Froude number (the intake tunnel has an numerical simulation results were reliable. Figure 5 shows the Froude number (the intake tunnel has numerical simulation results were reliable. Figure 5 shows the Froude number (the intake tunnel has arched shape andand the flow was non-pressure flow; flow; because the flow be can viewed as the open-channel anarched arched shape theflow flow wasnon-pressure non-pressure because thecan flow beviewed viewed asthe theopenopenp an shape and the was flow; because the flow can be as flow in a rectangular channel, the Froude number is calculated by F = V/ g h,V/where h is water r channel flow in a rectangular channel, the Froude number is calculated by 𝐹 = g h, where channel flow in a rectangular channel, the Froude number is calculated by 𝐹 = V/ g h, where hhisis depth) in the intake tunnel. The Froude number was larger than 1 and increased with the discharge, water depth) depth) in the the intake intake tunnel. tunnel. The Froude Froude number number was larger than and increased increased with the the water in The was larger than 11 and with which indicated the flow was draining freely into the drop shaft and was supercritical along the entire discharge, which indicated the flow was draining freely into the drop shaft and was supercritical discharge, which indicated the flow was draining freely into the drop shaft and was supercritical intake tunnel. Even though thereEven were though some errors experiment and simulation due toand the along the the entire intake tunnel. therebetween were some some errors between between experiment along entire intake tunnel. Even though there were errors experiment and surface fluctuations aftersurface aerationfluctuations and unavoidable instrument error, in general, instrument the numerical model simulation due to to the the after aeration aeration and and unavoidable error, in simulation due surface fluctuations after unavoidable instrument error, in was able to model the main characteristics of the vortex drop shaft. general, the numerical model was able to model the main characteristics of the vortex drop shaft. general, the numerical model was able to model the main characteristics of the vortex drop shaft. Table 2. Comparison of discharge between experiment and simulation. Table2.2.Comparison Comparisonof ofdischarge dischargebetween betweenexperiment experimentand andsimulation. simulation. Table Flood Frequency
FloodFrequency Frequency Flood 5% 5% 5% 2% 2% 2% 0.1% 0.1% 0.1%
Discharge (L/s) (L/s) Discharge
Discharge (L/s) Deviation Deviation Deviation Simulation Simulation Simulation 30.45 29.92 29.92 1.74% 30.45 29.92 1.74% 30.45 1.74% 38.21 38.01 0.52% 38.21 38.01 0.52% 38.21 38.01 0.52% 62.29 60.96 2.14% 62.29 60.96 2.14% 62.29 60.96 2.14%
Experiment Experiment Experiment
Figure 5. Froude number number profile profile in intake tunnel. Figure Froude inintake intaketunnel. tunnel. Figure 5.5.Froude number profile in
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In theexperiment, experiment, thewater water enteredthe the intaketunnel tunnel inaastate state ofopen-channel open-channel flow,and and it In In the the experiment, the the water entered entered the intake intake tunnel in in a state of of open-channel flow, flow, and itit provided a height of 5.0-cm for the air inlet under the flood frequency of 0.1% (as shown in Figure 6). provided providedaaheight heightof of5.0-cm 5.0-cmfor forthe theair airinlet inletunder underthe theflood floodfrequency frequencyof of0.1% 0.1%(as (asshown shownin inFigure Figure6). 6). The dischargedwater water flowedinto into thevertical vertical shaftwith with a largeamount amount ofair air toform form a stablemixing mixing The Thedischarged discharged waterflowed flowed intothe the verticalshaft shaft withaalarge large amountof of airto to formaastable stable mixing cavity through the intake tunnel. The water surface at the downstream wall of the vortex chamber cavity cavity through through the the intake intake tunnel. tunnel. The The water water surface surface at at the the downstream downstream wall wall of of the the vortex vortex chamber chamber washigher higherthan thanthat thatat atthe theupstream upstreamwall, wall,and andthe thesurface surfaceexhibited exhibitedsome somefluctuations, fluctuations,as as shownin in was was higher than that at the upstream wall, and the surface exhibited some fluctuations, as shown shown in Figures 6 and7.7.The The cavitythroat throat (i.e.,the the narrowestcavity cavity section)occurred occurred inthe the gradientsection section Figures Figures 66 and and 7. The cavity cavity throat (i.e., (i.e., the narrowest narrowest cavity section) section) occurred in in the gradient gradient section because of the decreases in the section size and discharge area. As the axial velocity increased in the because because of of the the decreases decreases in in the the section section size size and and discharge discharge area. area. As As the the axial axial velocity velocity increased increased in in the the vertical shaft, the the water layer became thinner. Then, water dropped into the dissipation well at the end vertical vertical shaft, shaft, the water water layer layer became became thinner. thinner. Then, Then, water water dropped dropped into into the the dissipation dissipation well well at at the the of the vertical shaft andand formed a water cushion, which exhibited a large-scale swirling motion and end end of of the the vertical vertical shaft shaft and formed formed aa water water cushion, cushion, which which exhibited exhibited aa large-scale large-scale swirling swirling motion motion intense mixing. An aeration phenomenon was obvious, and the flow was full of milky white bubbles. and and intense intense mixing. mixing. An An aeration aeration phenomenon phenomenon was was obvious, obvious, and and the the flow flow was was full full of of milky milky white white Highly turbulent flow was observed in the sloped section, air bubbles gradually drifted upward and bubbles. bubbles. Highly Highly turbulent turbulent flow flow was was observed observed in in the the sloped sloped section, section, air air bubbles bubbles gradually gradually drifted drifted out from theout surface inthe the outlet in tunnel,outlet and the flowand was stable. upward upward and and out from from the surface surface in the the outlet tunnel, tunnel, and the the flow flow was was stable. stable.
(a) (a) Cavity Cavity in in the the drop drop shaft shaft
(b) (b) Dissipation Dissipation well well
Figure Figure 6. Flow conditions in the experiment (P 0.1%). Figure6. 6.Flow Flowconditions conditionsin inthe theexperiment experiment(P (P===0.1%). 0.1%).
(a) (a) Longitudinal Longitudinal profile profile
(b) (b) Transverse Transverse profile profile
Figure Figure 7. Volume fractionof offluid fluidin inthe thenumerical numericalsimulation simulation(P (P==0.1%). 0.1%). Figure7. 7.Volume Volumefraction
4.3.Velocity VelocityDistribution Distribution 4.3. 4.3. Velocity Distribution Theflow flowvelocity velocityis isan an important importantparameter parameterwhen whenanalyzing analyzingthe theenergy energydissipation dissipationeffect effectand and The The flow velocity is an important parameter when analyzing the energy dissipation effect and calculating the energy dissipation ratio. The velocity in the vortex drop shaft was mainly affected by calculating calculating the the energy energydissipation dissipation ratio. ratio.The The velocity velocity in in the the vortex vortex drop drop shaft shaft was was mainly mainlyaffected affected by by gravity, friction and shear force, which first increased and then decreased, and the velocity peaked gravity, gravity, friction friction and and shear shear force, force, which which first first increased increased and and then then decreased, decreased, and and the the velocity velocity peaked peaked
at at the the connection connection with with the the sloped sloped section section (z (z == 0.70 0.70 m). m). The The maximum maximum velocities velocities were were 2.50 2.50 m/s, m/s, 2.68 2.68 m/s, m/s, 3.45 3.45 m/s m/s (at (at flood flood frequencies frequencies of of 5%, 5%, 2% 2% and and 0.1%, 0.1%, respectively) respectively) in in the the experiment experiment and and 2.56 2.56 m/s, m/s,
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at the connection with the sloped section (z = 0.70 m). The maximum velocities were 2.50 m/s, 9 of 19 2.68 m/s, 3.45 m/s (at flood frequencies of 5%, 2% and 0.1%, respectively) in the experiment and 2.56 m/s, 2.89 m/s, 3.49 m/s in the simulation, as shown in Figure 8. At the bottom of the vertical 2.89 m/s, 3.49 m/s in the simulation, as shown in Figure 8. At the bottom of the vertical shaft, the shaft, the upper and lower flows collided, resulting in a rapid decrease in the velocity, indicating that upper and lower flows collided, resulting in a rapid decrease in the velocity, indicating that the the energy dissipation effect was sufficient. energy dissipation effect was sufficient. Water 2018, 10, x FOR PEER REVIEW
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(b) Downstream velocity Figure Figure8.8.Velocity Velocityin inthe thevortex vortexdrop dropshaft. shaft.
The velocity field of of 0.1%) is shown in Figure 9. At 9. theAt same the velocity The field(flood (floodfrequency frequency 0.1%) is shown in Figure theelevation, same elevation, the closer tocloser the wall waswall smaller to friction thealong vertical wall. When the waterthe fellwater from velocity to the was due smaller due to along friction the shaft vertical shaft wall. When the from wall into well, the maximum was velocity approximately 5.0 m/s. Subsequently, fell the the walldissipation into the dissipation well, thevelocity maximum was approximately 5.0 m/s. the discharged flow collided with the water cushion in the well, and the velocity rapidly decreased to Subsequently, the discharged flow collided with the water cushion in the well, and the velocity 1.0 m/sdecreased at the bottom of m/s the well approximately m/s at the tunnel 2.0 outlet. energy was rapidly to 1.0 at theand bottom of the well2.0 and approximately m/sThus, at thethe tunnel outlet. effectively dissipated. Thus, the energy was effectively dissipated. The direction of the velocity varied within the vertical shaft, and the average velocity was calculated based on a theoretical analysis to quantify the cross-sectional kinetic energy distribution. The velocity of a swirling flow can be decomposed into a tangential velocity, an axial velocity and a radial velocity. The radial velocity is often ignored because it is smaller than the tangential and axial velocities (generally by more than two orders of magnitude). Additionally, it is often assumed that swirling flow is associated with a free eddy and meets the conditions for the conservation of angular momentum.
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(a) Longitudinal profile
(b) Transverse profile
Figure Figure9.9.Resultant Resultantvelocity velocityfield fieldin inthe thenumerical numericalsimulation simulation(P (P==0.1%). 0.1%).
Twodirection sections of in the velocity vertical shaft used the to establish the Bernoulli as shown in The variedare within vertical shaft, and the equation, average velocity was Figure 10: based on a theoretical analysis to quantify the cross-sectional kinetic energy distribution. calculated 2 dS V 2 velocity, an axial velocity and a (V + dV )2a tangential The velocity of a swirling flow canVbe decomposed into + dZ = +λ 0 (9) 2g radial velocity. The radial velocity2g is often ignored2gbecause it4R is smaller than the tangential and axial velocities (generally by more than two orders of magnitude). Additionally, it is often assumed that dZ where the hydraulic radius is R0 = A = 2πRVQcos θ and the water trajectory is dS = cos and Q θ; R swirling flow is associated with a free2πR eddy and meets the conditions for the conservation of angular are the radius of vertical shaft and the flow discharge, respectively; g is the constant of gravitational momentum. acceleration; θ is the angle between the direction of the velocity and the vertical direction; and λ Two sections in the vertical shaft are used to establish the Bernoulli equation, as shown in Figure denotes the factor of friction loss. Additionally, V is the resultant velocity, and dV is considered 10: negligible in this study. The hydraulic radius and the water trajectory are substituted into Equation (9) 𝑉 𝑉 + 𝑑𝑉 𝑑𝑆 𝑉 to obtain Equation (10): = + 𝑑Z (9) +𝜆 3 2 2g V 2g πλR V4𝑅 2g = 1− dZ (10) d 2g 2Q 2g where the hydraulic radius is 𝑅 = = and the water trajectory is 𝑑𝑆 = ; 𝑅 and 𝑄 are 2 3/2 2/3 2 πλR V πλR V 3/2 g the constant gravitational the of vertical shaftand and discharge, respectively; withradius G = K the = flow , we have G = K 2g . Thisisformula can beofsubstituted into 2g 2Q 2Q acceleration; 𝜃 is the angle between the direction of the velocity and the vertical direction; and 𝜆 Equation (10), and the new equation can then be integrated as follows: denotes the factor of friction loss. Additionally, 𝑉 is the resultant velocity, and 𝑑𝑉 is considered Z Z Z G Z G Z G0 dG and the dG trajectory negligible in this study. hydraulic radius water aredG substituted into Equation K ThedZ = = − (11) 1 − G3/2 0 G0 1 − G3/2 0 1 − G3/2 0 (9) to obtain Equation (10):
𝑉 H (G) = H (G𝜋𝜆𝑅 KZ 0) + 𝑉 = 1− 𝑑Z √ 2𝑄2 2g 1 2g √ 2 G+1 𝑑
!
(12) (10)
2 1 1 √ + ln G + G + 1 − √ arctan √ ln − arctan √ (13) / 1− / 3 3 3 3 3 G and 𝐾 = , we have 𝐺 = 𝐾 . This formula can be substituted into with 𝐺 / = g g where K is a coefficient, G is a function related to R, Q, λ and V, H is a function related to G, Z is the Equation (10), and the new equation can then be integrated as follows: length of the shaft from the gradient section. 𝑑𝐺 in the vertical 𝑑𝐺shaft (at z = 0, as shown The initial velocity was obtained by𝑑𝐺 measuring the velocity (11) 𝐾 𝑑Z = = − / / in Figure 10) several times. Then, the average velocity in 𝐺 each section was by combining 1− 𝐺 1− 1 − calculated 𝐺 / Equations (12) and (13). In the numerical simulation, the average velocity was measured by establishing a flux plane (baffle) in the measurement section. The average velocities obtained based on the theoretical calculations and simulations𝐻 were the (12) 𝐺 = compared 𝐻 𝐺 + 𝐾Zin Figure 11. This figure shows that analytically and numerically obtained resultant velocities were in good agreement. Both results also had similar patterns of variation and the mean error between the theoretical and simulated values were 3.81%, 3.63% and 5.65% frequencies of 5%, 2% 2and 0.1%, respectively). In general, there was a 2 (at flood 1 1 2√𝐺 + 1 1 (13) 𝐻 𝐺 = 𝑙𝑛 + 𝑙𝑛 𝐺 + √𝐺 + 1 − 𝑎𝑟𝑐𝑡𝑎𝑛 − 𝑎𝑟𝑐𝑡𝑎𝑛 3 1 − √𝐺 3 √3 √3 √3 H (G) =
2r
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Vt ds
V θ Vz
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high consistency between𝐺the and numerical where 𝐾 is a coefficient, is atheoretical function related 𝜆 and 𝑉,The 𝐻 isdeveloped a functiontheoretical related tocalculation 𝐺, Z is the 2Rto 𝑅, 𝑄, results. method for flow velocity can function as a reference for vertical shaft design. length of the shaft from the gradient section.
2r
0
dz
Vt V
ds
Figure 10. Diagram of the average velocity in the shaft section. θ
Vz
The initial velocity was obtained by measuring the velocity in the vertical shaft (at z = 0, as shown V+dV z section was calculated by combining in Figure 10) several times. Then, the average velocity in each Equations (12) and (13). In the numerical simulation, the average velocity was measured by 2R establishing a flux plane (baffle) in the measurement section. The average velocities obtained based on the theoretical calculations and simulations were compared in Figure 11. This figure shows that the analytically and numerically obtained resultant velocities were in good agreement. Both results also had similar patterns of variation and the mean error between the theoretical and simulated values were 3.81%, 3.63% and 5.65% (at flood frequencies of 5%, 2% and 0.1%, respectively). In general, there was a high consistency between the theoretical and numerical results. The developed Figure 10. Diagram of the average average velocity in inasthe the shaft section. section. theoretical calculationFigure method for flow of velocity can function a reference for vertical shaft design. 10. Diagram the velocity shaft 2.0velocity The initial vertical shaft (at z = 0, as shown theoretical z (m)was obtained by measuring the velocity in the0.1% 0.1% in Figure 10) several times. Then, the average velocity in each section wassimulation calculated by combining 2% theoretical Equations (12) velocity was measured by 1.5 and (13). In the numerical simulation, the average 2% simulation establishing a flux plane (baffle) in the measurement section. The average velocities obtained based 5% theoretical 5% 11. simulation on the theoretical calculations and simulations were compared in Figure This figure shows that 1.0 the analytically and numerically obtained resultant velocities were in good agreement. Both results also had similar patterns of variation and the mean error between the theoretical and simulated 0.5 values were 3.81%, 3.63% and 5.65% (at flood frequencies of 5%, 2% and 0.1%, respectively). In velocity general, there was a high consistency between the theoretical and numerical results. (m/s)The developed 0.0 theoretical calculation method for flow velocity can function as a reference for vertical shaft design. 1.0
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2.0 0.1% theoretical zComparison (m) Figure of ininthe Figure11. 11.Comparison ofthe thesimulated simulatedand andtheoretical theoreticalaverage averagevelocity velocity theshaft shaftsection. section. 0.1% simulation 2% theoretical 4.4. Distribution 4.4.Pressure Pressure1.5 Distribution 2% simulation 5% theoretical Cavitation is a phenomenon that may occur when the local pressure extends Cavitation is a phenomenon that may occur when the local pressure extendsbelow belowthe thevapor vapor 5% simulation pressure of the liquid at the operating temperature. When cavitation occurs, collapsing cavitation 1.0 pressure of the liquid at the operating temperature. When cavitation occurs, collapsing cavitation
bubbles bubblescan cancause causematerial materialerosion, erosion,thereby therebydecreasing decreasingthe thelifespan lifespanand andperformance performanceof ofthe thevortex vortex drop shaft spillway [35]. Therefore, the pressure is one of the most important indicators of whethera 0.5 drop shaft spillway [35]. Therefore, the pressure is one of the most important indicators of whether velocity avortex vortexdrop dropshaft shaftcan canstably stablyoperate; operate;consequently, consequently,negative negativepressure pressurezones zonesshould should avoided bebe avoided oror as (m/s) as small as possible. The time-averaged pressure along the vortex drop shaft exhibited an obvious small as possible. The time-averaged pressure along the vortex drop shaft exhibited an obvious 0.0 distribution and but 1.0 1.5 being 2.0 large 2.5 at 3.0 top 4.0 4.5 small 5.0 at 5.5middle 6.0 as distribution characteristic, characteristic, being large at the the top3.5 and bottom bottom but small atthe the middle asshown shownin in Figure 12. The minimum pressures were 0.04 kPa, 0.13 kPa, 0.23 kPa (at flood frequencies of 5%, Figure 12. The minimum pressures were 0.04 kPa, 0.13 kPa, 0.23 kPa (at flood frequencies of 5%, 2% Figurerespectively) 11. Comparison theexperiment simulated and theoretical average velocity the shaft section. 2% and 0.1%, inofthe and 0.07 kPa, 0.12 kPa, 0.18inkPa in the simulation at z = 0.7 m. Additionally, the pressures in the gradient section and dissipation well were obviously 4.4. Pressure Distribution higher than that in the vertical shaft. The wall pressure in the vertical shaft was caused by a centrifugal force,Cavitation and duringisthe falling process, themay tangential velocity centrifugal gradually a phenomenon that occur when theand local pressure force extends below decreased, the vapor which subsequently ledattothe theoperating decreasedtemperature. pressure. pressure of the liquid When cavitation occurs, collapsing cavitation bubbles can cause material erosion, thereby decreasing the lifespan and performance of the vortex drop shaft spillway [35]. Therefore, the pressure is one of the most important indicators of whether a vortex drop shaft can stably operate; consequently, negative pressure zones should be avoided or as small as possible. The time-averaged pressure along the vortex drop shaft exhibited an obvious distribution characteristic, being large at the top and bottom but small at the middle as shown in Figure 12. The minimum pressures were 0.04 kPa, 0.13 kPa, 0.23 kPa (at flood frequencies of 5%, 2%
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and 0.1%, respectively) in the experiment and 0.07 kPa, 0.12 kPa, 0.18 kPa in the simulation at z = 0.7 m. Additionally, the pressures in the gradient section and dissipation well were obviously higher than that in the vertical shaft. The wall pressure in the vertical shaft was caused by a centrifugal force, Water 2018, 10, 0 and during the falling process, the tangential velocity and centrifugal force gradually decreased,12 of 18 which subsequently led to the decreased pressure.
(a) Upstream pressure
(b) Downstream pressure Figure12. 12.Pressure Pressure trends trends in shaft. Figure inthe thevertical vertical shaft.
An area of negative pressureformed formed at at the the connection thethe vertical shaftshaft and and outletoutlet An area of negative pressure connectionbetween between vertical tunnel because the swirling flow moved away from the vertical shaft wall as shown in Figure 13 tunnel because the swirling flow moved away from the vertical shaft wall as shown in Figure 13 (flood (flood frequency of 0.1%). Moreover, the pressure in the dissipation well was larger, approximately frequency of 0.1%). Moreover, the pressure in the dissipation well was larger, approximately 10 kPa, 10 kPa, due to the influences of the hydrostatic pressure and discharge. The maximum pressure in due to influences of the hydrostatic pressure discharge. The maximum pressure in the entire thethe entire vortex drop shaft was observed at theand bottom of the dissipation well, with experimental vortex drop shaft was observed at the bottom of respectively the dissipation well, with experimental simulated and simulated values of 13.1 kPa and 12.6 kPa, (flood frequency of 0.1%). The and similarity values of 13.1 kPa and 12.6 kPa, respectively (flood frequency of 0.1%). The similarity between the between thex FOR experimental and simulated values further verified the accuracy of the numerical Water 2018, 10, PEER REVIEW 13 of 19 experimental and simulated values further verified the accuracy of the numerical simulation. simulation.
(a) Longitudinal profile
(b) Transverse profile
Figure Pressureprofiles profilesbased based on on the (P =(P0.1%). Figure 13.13. Pressure the numerical numericalsimulations simulations = 0.1%).
4.5. Aeration Concentration In the vortex drop shaft without ventilation holes, the water flowed into the vertical shaft with a large amount of air and created a stable mixing cavity through the intake tunnel. Due to the rotation of the flow and the existence of the cavity, air was continuously drawn into the cavity, and the velocity was high in the cavity area. In the experiment, the water layer was thin, and there was an
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(b) Transverse profile
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Figure 13. Pressure profiles based on the numerical simulations (P = 0.1%).
4.5. Aeration Concentration 4.5. Aeration Concentration In the vortex drop shaft without ventilation holes, the water flowed into the vertical shaft with a In the vortex drop shaft without ventilation holes, the water flowed into the vertical shaft with large amount of air and created a stable mixing cavity through the intake tunnel. Due to the rotation of a large amount of air and created a stable mixing cavity through the intake tunnel. Due to the rotation the flow and the existence of the cavity, air was continuously drawn into the cavity, and the velocity of the flow and the existence of the cavity, air was continuously drawn into the cavity, and the was high in the cavity area. In the experiment, the water layer was thin, and there was an obvious velocity was high in the cavity area. In the experiment, the water layer was thin, and there was an aeration phenomenon. The aerated water rushed into the dissipation well at a high speed and there obvious aeration phenomenon. The aerated water rushed into the dissipation well at a high speed was considerable velocity fluctuation in the water cushion. The aeration concentration was measured and there was considerable velocity fluctuation in the water cushion. The aeration concentration was with a CQ6-2004 aeration concentration meter, and the results are shown in Figure 14. measured with a CQ6-2004 aeration concentration meter, and the results are shown in Figure 14. 2.0
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(b) Downstream aeration concentration Figure 14. 14. Aeration Aeration concentration concentration in in the the vortex vortex drop dropshaft. shaft. Figure
Aerating a large amount of air at a low-pressure is an effective way to prevent cavitation erosion. When the air content in the water increases, the compressibility of the mixture of water and vapor correspondingly increases, which can mitigate the impact of cavitation collapse and reduce the risk of erosion. Experimental data [36,37] showed that when the aeration concentration in the flow reached 1% to 2%, cavitation erosion at the solid boundary was reduced. Moreover, when the aeration concentration reached 5% to 7%, cavitation erosion would not occur. In this paper, the data showed that the aeration concentration ranged approximately from 5% to 25% and decreased with the increasing discharge, which reflected a low probability of cavitation erosion. However, the values obtained in the experiment were almost larger than the results of the numerical simulation, suggesting that the numerical simulation of aeration concentration needs further study. Because of the effects of surface tension and viscosity, the aeration concentration is underestimated so that the aeration concentration is larger in the prototype [23–26]. Therefore, the results of aeration concentration could be used as a reference for engineering design
of erosion. Experimental data [36,37] showed that when the aeration concentration in the flow reached 1% to 2%, cavitation erosion at the solid boundary was reduced. Moreover, when the aeration concentration reached 5% to 7%, cavitation erosion would not occur. In this paper, the data showed that the aeration concentration ranged approximately from 5% to 25% and decreased with the increasing discharge, which reflected a low probability of cavitation erosion. However, the values Water obtained 2018, 10, 0 in the experiment were almost larger than the results of the numerical simulation,14 of 18 suggesting that the numerical simulation of aeration concentration needs further study. Because of the effects of surface tension and viscosity, the aeration concentration is underestimated so that the 4.6. Cavitation Number aeration concentration is larger in the prototype [23–26]. Therefore, the results of aeration concentration could be used as a reference engineering The probability of cavitation is lower for when there is design a greater difference between the absolute
pressure and the vapor pressure of a liquid at a certain point. In addition, the larger the velocity is, 4.6. Cavitation Number the more easily cavitation occurs. Therefore, the cavitation number is often used to describe the degree The probability of cavitation is lower there is a greater difference between the absolute of flow cavitation [38] and can be defined aswhen follows: pressure and the vapor pressure of a liquid at a certain point. In addition, the larger the velocity is, − pv the more easily cavitation occurs. Therefore, thep0cavitation number is often used to describe the (14) N= 2 degree of flow cavitation [38] and can be defined as follows: 0.5ρV 𝑝 −𝑝
𝑁= where p0 is the absolute pressure, pv is the elevation-dependent vapor pressure, V is the(14) resultant 0.5𝜌𝑉 velocity close to the wall, and ρ is the density. The smaller the cavitation number From Equation (14) where 𝑝 is the absolute pressure, 𝑝 is the elevation-dependent vapor pressure, 𝑉 is the resultant is, the more easily cavitation occurs. velocity close to the wall, and 𝜌 is the density. The smaller the cavitation number From Equation (14) In the hydraulic structures, negative is, more easily cavitation occurs. pressure exists in sections of abrupt changes, thereby increasing the probability of cavitation. In study, pressure the bodyexists changed abruptlyofatabrupt the gradient section and the In hydraulic structures,this negative in sections changes, thereby connection between the vertical shaft and outlet andbody therefore cavitation in these increasing the probability of cavitation. In this tunnel, study, the changed abruptlymight at theoccur gradient section the connection the vertical shaft tunnel, therefore sections. Theand cavitation numberbetween was calculated based onand theoutlet pressure andand velocity, and cavitation the results are might occur in shown in Figure 15.these sections. The cavitation number was calculated based on the pressure and velocity, and the results occurs are shown in Figure 15. Generally, cavitation easily in practical applications when the cavitation number is less Generally, cavitation occurs easily in practical applications the cavitation number is less well than 0.2 [39]. In this study, the pressures in the vortex chamber, when gradient section and dissipation than 0.2 [39]. In this study, the pressures in the vortex chamber, gradient section and dissipation well were large, and the velocities were small, thus, the associated cavitation numbers were larger than 1.5. were large, and the velocities were small, thus, the associated cavitation numbers were larger than Although negative pressure and large velocity appeared in the vertical shaft and resulted in smaller 1.5. Although negative pressure and large velocity appeared in the vertical shaft and resulted in cavitation from 0.40 to 0.80, they werethey all larger than the than critical 0.2, so smallernumbers cavitation numbers from 0.40 to 0.80, were all larger thevalue criticalofvalue of there 0.2, sowas a smaller probability of cavitation. there was a smaller probability of cavitation. 2.0
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(a) Upstream cavitation number
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(b) Downstream cavitation number Figure Cavitation number number in drop shaft. Figure 15.15.Cavitation inthe thevortex vortex drop shaft.
4.7. Energy Dissipation Rate The energy dissipation within a vortex drop shaft is mainly reflected by two factors. First, the centrifugal force generated by the swirling flow increases the pressure on the wall and consequently increases the frictional resistance. Simultaneously, as the flow swirls along the wall, the trajectory becomes longer and further increases the head loss. Second, the flow fluctuates with aeration and
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4.7. Energy Dissipation Rate The energy dissipation within a vortex drop shaft is mainly reflected by two factors. First, the centrifugal force generated by the swirling flow increases the pressure on the wall and consequently increases the frictional resistance. Simultaneously, as the flow swirls along the wall, the trajectory becomes longer and further increases the head loss. Second, the flow fluctuates with aeration and combined with the large velocity gradient, results in swirling, which causes shearing between flow layers. In addition, the shearing, rotation and collision caused by a flow jet that enters the dissipation well can also dissipate energy. In the present study, the calculated section is shown in Figure 1, where the section of volute chamber A-A at z = 1.9 m and the section of outlet tunnel B-B at Y + 1.8 m. The reference position of the potential energy is z = 0.0 m. The energy dissipation rate of the traditional energy dissipators is generally between 40% and 50%; any rate over 50% is considered excellent. In this paper, the energy dissipation rates exceeded 70% as shown in Table 3, indicating that the vortex drop shaft without ventilation holes sufficiently dissipated energy and met the needs of the actual project. As shown in Figure 16, energy dissipation mainly occurred in the dissipation well and sloped section, and the dissipation rate of turbulent kinetic energy was generally between 20 and 30 m2 ·s−3 with a maximum of 32 m2 ·s−3 . In addition, the wall friction dissipated some energy, and the dissipation rate of turbulent kinetic energy near the wall in the same cross-section was larger than that in other parts of the structure. Table 3. The calculation results of the energy dissipation rate. Section A-A Condition
Section B-B
V2 /2g (m)
H + p/fl (m)
V2 /2g (m)
H + p/fl (m)
Energy Dissipation Rate (%)
5%
Experiment Simulation
0.16 0.17
1.65 1.69
0.19 0.20
0.29 0.28
73.48 74.12
2%
Experiment Simulation
0.18 0.19
1.82 1.81
0.23 0.24
0.31 0.33
73.00 71.66
2.12 2.13
0.24 0.30
0.39 0.41
72.95 70.13
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Notes: H is the elevation head, γ is the specific weight of water, p is the gauge pressure.
(a) Dissipation well
(b) Section C-C (z = 0.14 m)
Figure 16.16. Dissipation rate ofof turbulent kinetic energy inin thethe numerical simulation (P(P = 0.1%). Figure Dissipation rate turbulent kinetic energy numerical simulation = 0.1%).
5. Conclusions 5. Conclusion To address the negative pressure and cavitation erosion problems in vortex drop shaft spillways, To address the negative pressure and cavitation erosion problems in vortex drop shaft spillways, a new vortex drop shaft without ventilation holes is proposed in this paper. An increased height intake a new vortex drop shaft without ventilation holes is proposed in this paper. An increased height tunnel is used to facilitate aeration. The vortex drop shaft is simulated using the RNG k-ε turbulence intake tunnel is used to facilitate aeration. The vortex drop shaft is simulated using the RNG k-ε model. By comparing the experimental and simulation results, the hydraulic characteristics of the turbulence model. By comparing the experimental and simulation results, the hydraulic vortex drop shaft are studied in detail. The results of the research are as follows: characteristics of the vortex drop shaft are studied in detail. The results of the research are as follows:
1.
The RNG k-ε turbulence model effectively simulated the flow characteristics of the vortex drop shaft. The hydraulic parameters, including the velocity and pressure, agreed well with the experimental data and showed the same trends. Thus, similar swirling problems can be simulated by the RNG k-ε turbulence model.
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The RNG k-ε turbulence model effectively simulated the flow characteristics of the vortex drop shaft. The hydraulic parameters, including the velocity and pressure, agreed well with the experimental data and showed the same trends. Thus, similar swirling problems can be simulated by the RNG k-ε turbulence model. The flow regime in the vertical shaft was stable, but the flow in the dissipation well was swirling and turbulent. The energy dissipation rate exceeded 70% from the gradient section to the outlet tunnel, indicating the occurrence of sufficient energy dissipation. The velocity was small at the top and bottom but large in the middle, and the pressure was large at the top and bottom but small in the middle. Small negative pressure areas were observed. By increasing the clearance height of the intake tunnel and changing the flow conditions, the water flowed into the vertical shaft with a large amount of air. There was a clear phenomenon of aeration and the aeration concentration could provide some reference for engineering design. Additionally, the cavitation number was larger than the critical value, which indicated a low probability of cavitation.
However, our study has some limitations. For example, the diameters of the vertical shaft, vortex chamber and slope section need to be optimized based on existing body sizes to determine the best body shape for safe and stable operation. In addition, considering the scale effect, different scales of experiments and numerical simulations need to be performed. Author Contributions: Conceptualization, W.Z. and J.W.; Methodology, W.Z.; Software, Z.D.; Validation, W.Z., J.W. and C.Z.; Formal Analysis, W.Z.; Investigation, W.Z.; Resources, J.W.; Data Curation, Z.Z.; Writing-Original Draft Preparation, W.Z.; Writing-Review & Editing, W.Z., Z.D. and Z.Z.; Supervision, J.W. Funding: This research received no external funding. Acknowledgments: The numerical calculations in this paper were performed using the supercomputing system at the Supercomputing Center of Wuhan University. Conflicts of Interest: The authors declare no conflict of interest.
Notations As C Cair ε F Fr fi g gn Gk h H i k LT N P Pd Pt p p0 pv ρ
surface area (m2 ) aeration concentration coefficient of proportionality turbulent dissipation rate m2 ·s−3 volume of fluid (VOF) function Froude number gravity component (m·s−2 ) gravity acceleration (m·s−2 ) component of gravity normal to the free surface (m·s−2 ) generation of turbulent energy caused by the average velocity gradient water depth (m) elevation head (m) slope turbulent kinetic energy (kg·m2 ·s−2 ) turbulence length scale (m) cavitation number flood frequency disturbance energy per unit volume (N·m−2 ) stabilising forces per unit volume (N·m−2 ) gauge pressure (Pa) absolute pressure (Pa) vapor pressure (Pa) density of water (kg·m−3 )
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discharge (L·s−1 ) radius of the cavity (m) specific weight of water (N·m−3 ) radius of vertical shaft (m) hydraulic radius (m) water trajectory (m) time (s) velocity component (m/s) coefficient of dynamic viscosity (kg·m−1 ·s−1 ) revisionary coefficient of dynamic viscosity (kg·m−1 ·s−1 ) coefficient of kinematic viscosity (m2 ·s−1 ) resultant velocity (m·s−1 ) tangential velocity (m·s−1 ) vertical velocity (m·s−1 ) volume of air entrained per unit time m3 coefficient of surface tension (N·m−1 ) angle between the velocity direction and the vertical direction factor of friction loss coordinate component
Q r γ R R0 S t ui µ µe f f ν V Vt Vz δV σ θ λ xi
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