Simulation of Cavitation Water Flows

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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 872573, 16 pages http://dx.doi.org/10.1155/2015/872573

Research Article Simulation of Cavitation Water Flows

Piroz Zamankhan1,2 1 Institute of Geothermal Energy, Dedan Kimathi University of Technology, 101000 Nyeri, Kenya 2Chemical Engineering Department, School of Engineering, Collage of Agriculture, Engineering and Science, University of KwaZulu-Natal, 4041 Durban, South Africa

Correspondence should be addressed to Piroz Zamankhan; [email protected]

Received 20 July 2014; Revised 4 September 2014; Accepted 13 September 2014

Academic Editor: Junuthula N. Reddy

Copyright © 2015 Piroz Zamankhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The air-water mixture from an artificially aerated spillway flowing down to a canyon may cause serious erosion and damage to both the spillway surface and the environment. The location of an aerator, its geometry, and the aeration flow rate are important factors in the design of an environmentally friendly high-energy spillway. In this work, an analysis of the problem based on physical and computational fluid dynamics (CFD) modeling is presented. The numerical modeling used was a large eddy simulation technique (LES) combined with a discrete element method. Three-dimensional simulations of a spillway were performed on a graphics processing unit (GPU). The result of this analysis in the form of design suggestions may help diminishing the hazards associated with cavitation.

1. Introduction changing the temperature at constant pressure. On the other hand,cavitationistheprocessofphasechangefromtheliquid In spillway engineering there are numerous challenging state to the vapor state by changing the local pressure at obstacles. One of the most determining factors is the geo- constant temperature. logicalgeometryinwhichthedamanditsspillwayhaveto Cavitation is a consequence of the reduction of the be built in. The geometry, such as length and slope of the pressure to a critical value, due to a flowing liquid or in an spillway, can range from short and steep to long and flat or acoustical field. Spillways are subject to cavitation. Inertial vice versa. Generally, the longer and steeper is the spillway cavitation is the process in which voids or bubbles rapidly the higher is the gradient of the energy head of the flowing collapse, forming shock waves. This process is marked by water. In some cases the velocity of the water might exceed intense noise. The shock waves formed by cavitation can 30 m/s and cavitation might occur. Used as an illustrative significantly damage the spillway face1 [ ]. Cavitation damage example, the high-energy spillway of the Karahnj´ ukar´ Dam on the spillway face is a complex process [2]. Richer concrete which supports the Halsson´ Reservoir in eastern Iceland mixes are used to increase the resistance to cavitation damage displays numerous problems in spillway engineering. An and erosion [3]. aerial perspective of this spillway can be seen in Figure 1. Falvey [1] suggested that impurities and microscopic air When a fluid changes state from liquid to vapor, its bubbles in the water are necessary to initiate cavitation. As volume increases by orders of magnitude and cavities (or canbeseenfromFigure 1, the color of water accumulated bubbles) can form. Indeed, cavities, which are filled with behind the dam indicates that it may contain various types water vapor, are formed by boiling at the local pressure which of impurities including suspended fine solid particles. The equals the water vapor pressure. If the cavity is filled with density of the water accumulated behind the dam was gases other than water vapor, then the process is named determined from a sample using a hydrometer. The relative gaseous cavitation [1]. density of the sample with respect to water has been found to There is a technical difference between boiling and be approximately 1.08. cavitation. On the one hand, boiling is defined as the process The melting water which transports suspended clay ofphasechangefromtheliquidstatetothevaporstateby particlesflowsintothesidechannelandthenfollowsan 2 Mathematical Problems in Engineering

1 Location of aerator 0.9 0.8

𝜎 0.7 0.6 0.5 0.4 0.3 Cavitation number, Cavitation 0.2 0.1 Grey: theoretical cavitation zone 0 350 300 250 200 150 100 50 0 Figure 1: Aerial view of Karahnj´ ukar´ spillway with bottom aerator, Distance from the outlet of Karahnj́ ukaŕ spillway (m) eastern Iceland. Figure 2: Cavitation numbers at different discharges on the Karahnj´ ukar´ spillway [4]. The location of the outlet is shown in 3 3 Figure 1. Here, blue diamonds: 200 m /s; red squares: 400 m /s; 3 3 green triangle: 800 m /s;bluesquares:950m /s; yellow square with 3 3 approximately 414 m long chute with a transition bend and green star: 1350 m /s; and orange dot: 2250 m /s. abottomaerator. The most important task is to reduce the energy head of the flowing water until it enters the river. In this example the As a precaution the cavitation zone has been set to 𝜎= jet should not hit the opposite canyon wall where it increases 0.25 because of the lower atmospheric pressure, low water erosion. An artificial aeration device “cushions” the water; temperature, and overall safety reasons. Thus, as a precaution, the widening at the chute’s end reduces the flow velocity the aerator was placed 125 m before the end of the chute. and increases the impact area in the plunge pool which in Nevertheless,incipientwhitewaterfarupstreamfromthe turn decreases rock scouring. Additionally, seven baffles and aerator allows the assumption that in such a long spillway lateral wedges at the main throw lip abet the jet disintegration. fully self-aerated flow may have already been developed. In general, every high-energy spillway has to be analyzed for Therefore, the purpose of this aerator is mainly to reduce the cavitation risk. impact energy in the plunge pool. The water in spillways contains air bubbles and impu- The main idea is to develop a fully 3D simulation of rities, such as suspended particles, in a range of sizes. As an aerated spillway in the future. But to achieve that the discussed later these air bubbles and impurities induce cav- physics of bubble growth due to cavitation and aeration and itation. Vaporization is the most important factor in bubble the formation and distribution of the bubbles have to be growth. At a critical combination of flow velocity, water pres- understoodandsimulationshavetobevalidated.Inthiswork sure, and the vapor pressure of the flowing water, cavitation it is shown that LES combined with a discrete element method starts. The cavitation number is used to define this starting on GPUs yields promising results. Cavitation is defined as the point. The equation for this parameter is derived from the formation of the vapor phase and the subsequent immediate Bernoulli equation for a steady flow between two points. implosion of small liquid-free zones in a liquid, called voids In dimensionless terms, the comparable equation results in or bubbles [1]. a pressure coefficient, 𝐶𝑝.Thevalueofthisparameterisa The first section deals with forced cavitation and the constant at every point until the minimum pressure at a dynamics of cavitation bubbles. An experiment was carried certain location is greater than the vapor pressure of water. out in which cavitation was forced to develop with clear The pressure at a certain location will not decrease any further water in glass and galvanized steel pipes. These findings were once the vapor pressure is reached. Taking everything into compared to computational simulations. The mathematical account, such as temperature, clarity of the water, and safety model and simulation results of the cavitation bubble dynam- reasons, the pressure coefficient is set to a minimum value ics will be described. andthencalledcavitationnumber𝜎.Thecavitationnumber Inthesecondpartofthepapertheflowbehaviorofan is a dimensionless number which expresses the relationship aerated spillway is computed. In this section, the reliability between the difference of a local absolute pressure from the of the predicted cavitation zone as shown in Figure 2 is vapor pressure and the kinetic energy per volume. It may be examined. used to characterize the potential of the flow to cavitate. It has been suggested that no cavitation protection for a 2. Experimental Methods and Observations spillway would be needed if the cavitation number is larger than 1.8. If the cavitation number is in the range of 0.12–0.17, Cavitation occurs when the fluid pressure is lower than the then the spillway should be protected by additional aeration local vapor pressure [5]. In the cavities, the vapor phase grooves. In Figure 2 the cavitation indices for different dis- replaces the liquid phase. The surrounding liquid then expe- chargeshavebeencalculated. riences evaporative cooling. For water these thermal effects Mathematical Problems in Engineering 3

Lp D x l

Q Discharge Pa PMP

(high pressure pump) 4550

(a) y

x D

(b)

Figure 3: The experimental setup for investigating cavitation. (a) Schematic of the experimental setup. Inset (on the right-hand side): an image of the aquarium stone used in the cavitation test. Inset (lower): the minimum pressure measured was 4.55 kPa (absolute). (b) An image of sheet cavitation at the discharge Q =12.5L/s. can have a significant influence on the dynamics of bubbles in Table 1: Physical properties. the cavitation process. However, for simplicity, in this paper Dynamic Density Vaporization pressure the isothermal problem will be investigated which retains a Material viscosity (Pa s) (kg/m3) (mmHg) [18] number of interesting qualitative features of the cavitation 𝑃 = 8.07131 process. Water (liquid) 0.001 996.4 log10 −5 −1730.63/𝑇 − 39.724 Forwaterflowoverauniformroughnessinaspillway, Water (vapor) 1.34 × 10 0.5542 −5 incipient cavitation will occur randomly over the spillway Air 1.79 × 10 1.175 surface. If the cavitation number is lowered below incipient conditions, then cavitation will occur in sheets whose locationsareinfluencedbythemagnitudeandspectrumof 𝐷 = 0.0254 m and the length of 𝐿𝑝 = 0.3 m. Two very similar turbulent fluctuation within the boundary layer [1]. aquarium stones with a length of 𝑙=0.0215 m and a height To enhance the predictive reliability of computational of ℎ = 0.01 m were attached to the bottom of the pipe in the models for cavitation, which are still considered to be entrance section. The stones produced a sudden offset whose immature, further validation based on experimental data is leading edge is approximately an ellipse. The dimensions of required. Figure 3 shows a simple system in which sheet one of the stones are depicted in the inset of Figure 3(a). cavitation may be observed. Clear water was used in the experiments; its physical Surface roughness, cracks, and offsets may initiate cavita- properties are given in Table 1.Thedischarge𝑄 = 12.5 L/s tion. The local pressure on the step’s surfaces or directly at the was provided with a heavy-duty high-pressure wash-down step in a high velocity spillway can reach the vapor pressure. pump with a pressure range of 2.5 bar to 140 bars (maximum This may result in sheet cavitation similar to that shown in pressure). A high-pressure hose was used to connect the test Figure 3.Asstatedearlier,cavitationcancauseseveredamage tube to the high-pressure pump. to the spillway concrete. The extent of cavitation damage is a The average pressure at the pipe inlet was approximately function of the local cavitation number in the spillway chute 2.5 bars (absolute) and the minimum measured pressure and the duration of the flow. was 4.55 kPa (absolute). Figure 3(b) shows the monometer In brief, experiments were conducted in a test setup connected to the pressure measuring port (PMP) as shown as illustrated in Figure 3(a).Thetestswereperformedina in the second inset of Figure 3(a).Theportwaslocated horizontal galvanized steel pipe with an outer diameter of 𝐿𝑚 =0.048mdownstreamfromtheinletofthetesttube. 4 Mathematical Problems in Engineering

𝐿 𝑅 Here, 𝑐 represents the length of the sheet. The low pressure critical. Note that the bubble may contain a “contaminant” measured at the sampling port indicates that cavitation may gas which dissolves very slowly compared to the time scales have occurred in the test tube. associated with changes in its size. The above-mentioned test with the same discharge of Apparentlysmallbubbleswhoseradiiaresmallerthanthe Q = 12.5 L/s and the same average inlet pressure of 2.5 bar critical radius behave like rigid spheres in accelerated flows (absolute) was repeated in a cylindrical borosilicate glass pipe [7]. with a diameter of 𝐷 = 0.026 m. The pipe wall thickness was 0.001m.Acamerawasusedtotakeimagesofsheetcavitation 3.2. Bubble Dynamics in Accelerated Flows. For the present from the underside of the test tube. study the interaction of a small bubble whose size is char- Figure 3(b) shows an image of sheet cavitation with an acterized by the critical bubble radius with its neighboring average inlet channel pressure of approximately 2.5 bars bubbles is assumed to be subjected to the Lennard-Jones (LJ) (absolute). As can be seen from Figure 3(b),thesheetappears condition [8]. Figure 5(a) shows the Lennard-Jones 12-6 pair 𝐿 >5𝐷 to be attached to the stone with a length of 𝑐 .The potential. The force between two bubbles with diameter 𝑑𝐵 sheet consists of a number of fuzzy white clouds. A flash located at 𝑟𝑖 and 𝑟𝑗 is given as photograph reveals that a cloud consists of individual small 48𝜀 𝑑 12 1 𝑑 6 bubbles [1]. = [( 𝐵 ) − ( 𝐵 ) ] . In the following section, simulation results will be pre- F𝑖𝑗 2 r𝑖𝑗 (3) 𝑟𝑖𝑗 𝑟𝑖𝑗 2 𝑟𝑖𝑗 sented, giving a more detailed insight and showing an ongoing process of cavity formation and deformation. A simplified momentum equation for the 𝑖th spherical It is difficult to obtain accurate measurements using the bubble may be given as simple setup as shown in Figure 3.Thestonesattachedtothe 𝜋 𝑑k𝑖 bottom of the pipes may not be stable over long periods of 𝜌 𝑑3 𝐵 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟V 𝐵 time. 6 𝑑𝑡 Additional experimental studies in cavitation tunnels are momentum transfer required to supply a more reliable and convincing data basis 𝜋 2 󵄨 𝑖 󵄨 𝑖 = 𝜌𝑤𝑑 𝐶𝐷 (Re𝐵,𝐴𝑐)󵄨k − k𝑤󵄨 (k − k𝑤) for the validation of computational approaches to model ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 𝐵 󵄨 𝐵 󵄨 𝐵 cavitation. viscous 𝑖 𝜋 3 𝐷k𝑤 𝑑k𝐵 3 2 1/2 3. Mathematical Model and Simulation Results + 𝜌𝑤𝑑𝐵 ( − ) + 𝜌𝑤𝑑𝐵 (𝜋V𝑤) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟12 𝐷𝑡 𝑑𝑡 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟2 Basset If sheet cavitation, as described in the experiments in the added mass preceding section, occurs in the spillway chute, it can produce 𝑡 𝐷k 𝑑k𝑖 𝑑𝜏 𝜋 𝐷k extensive damage to its concrete linings [1]. Note that cavita- × ∫ ( 𝑤 − 𝐵 ) + 𝜌 𝑑3 𝑤 1/2 𝑤 𝐵 tion damage in tunnels and conduits is less likely to result in ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟−∞ 𝐷𝑡 𝑑𝑡 (𝑡−𝜏) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 𝐷𝑡 pressure term dam failure. histroy term Experimental observations such as those shown in Figure 3(b) are indispensable for investigating cavitation 𝜋 𝑁 − 𝑑3 (𝜌 −𝜌 ) + ∑ . problems. However, numerical simulation can provide possi- ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟6 𝐵 V 𝑤 g F𝑖𝑗 ⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑗=1 bilities to study cavity formation and deformation in greater buoyancy bubble-bubble detail than what is affordable by experiments. interaction (4) 3.1. Bubble Dynamics in a Quiescent Fluid. The importance of microscopic bubbles as cavitation nuclei has been known InthecurrentefforttheaimistocombinetheLagrangian for a long time [1]. Consider a single bubble, which contains (bubble-based) model (4) and large eddy simulation (LES) the vapor of the liquid, in equilibrium in anincompressible, in order to achieve more accurate simulations of cavita- Newtonianfluidthatisatrestatinfinity.Inthiscase,the tion water flows. The filtered continuity momentum for an bubble radius 𝑅𝐸 must satisfy the following condition [6]: isothermal 3D flow of water may be given as

(0) 3 2 𝜕𝜌 (𝑝 −𝑝 ) 𝑅 −2𝛾𝑅 +𝐺=0. +∇⋅(𝜌k̃)=−𝑆, (5) V ∞ 𝐸 𝐸 (1) 𝜕𝑡 𝑤 V (𝑝 −𝑝(0))= 𝜕𝜌k̃ 1 1 The bubble radius at the limit point, where V ∞ 𝑤 +∇⋅(𝜌k̃k̃)= − ∇(𝜌𝑝) + ∇⋅(𝜌𝜎̃)+∇⋅𝜏 3 1/2 𝑤 𝑤 (32𝛾 /27𝐺) , can be expressed in terms of material param- 𝜕𝑡 𝜌𝑤 𝜌𝑤 eters as [5] + 𝑓𝑏𝑤 + 𝜌g. 3𝐺 1/2 (6) 𝑅 =( ) . (2) critical 2𝛾 The viscous stress tensor in (6) is defined as 2 𝑇 If 𝑅𝐸 ≤𝑅 , then the bubble is stable against infinitesimal critical 𝜎 =− 𝜇𝑤∇⋅k𝑤I +𝜇𝑤 (∇k𝑤 +(∇k𝑤) ). (7) changes in its radius. The bubble becomes unstable if 𝑅𝐸 > 3 Mathematical Problems in Engineering 5

In (6), 𝜏 may be given as 3.3. Simulation Results

𝜏 =−𝜌(k̃k̃ − k̃k̃) − 𝜌(k̃󸀠󸀠k̃ + k̃k󸀠󸀠) − 𝜌(k̃󸀠󸀠k󸀠󸀠) . 3.3.1. Air Bubbles. Figures 4(b) and 4(c) show a computed 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 𝑤 largebubbleinanaeratedtank[7]. The bubble is a cluster (8) of small bubbles with a size of the critical air bubble radius. Figure 4(d) depicts the denoised version of the bubble using The SGS stress tensor, 𝜏,isrequiredtoclosetheequations multiscale image denoising based on a multiscale representa- for the large-scale fields on a grid small enough (but much tion of the images [14]. The denoised version of the bubble- larger than the Kolmogorov scale) to provide reasonable resolutions. In contrast to the filtered single-phase equations, like structure in Figure 4(b) is very similar to the small a conceptual restriction arising in the present approach is that bubblesobservedinanaeratedtank[7]. the filter width, Δ, should strictly be larger than the length Figures 4(e) and 4(f) are snapshots of bubbles separated 𝑡=0.3 scale characteristic of the small bubbles. Thus, an appropriate by sec.AscanbeseenfromFigure 4(f), three bubbles choice of Δ should provide a sufficiently large-scale resolution in Figure 4(e) coalesce to form one large bubble. The denoised while not violating the aforementioned restriction. version depicted in Figure 4(g) is very similar to the bubbles Following Sauer and Schnerr [9], the mass flow rate with complex geometries shown in [7]. through the surface of a bubble with the radius of 𝑅, 𝑆V,may To further assess the quality of the model described in be given as the preceding section, the cavitation water flow in Figure 3 issimulatedinthefollowingsection. −2 𝜌 𝜌 2 4 3 𝑤 V ̇ 𝑆V =4𝜋𝑛𝐵𝑅 (1 + 𝜋𝑛𝐵𝑅 ) 𝑅. (9) 3 𝜌 3.3.2. 3D Model Using 2D Images. Apparently, it is not possible to reconstruct a 3D model from a single image Here, the changes in the bubble radius may be calculated ofthestoneasshownintheinsetofFigure 5(a).Toget using the famous Rayleigh-Plesset equation [6] around this limitation, an optical processing algorithm can be 𝑝 + 𝐺𝑇/𝑅3 −𝑝 1 2𝛾 4𝜇 3 developed that employs multiple photographs taken at every V ∞ − ( + 𝑤 𝑅)̇ = 𝑅𝑅+̈ 𝑅̇2. 𝜌 𝜌 𝑅 𝑅 2 (10) 15 degrees, from the above and from the side to create a 3D 𝑤 𝑤 model of the stone. Figure 5(a) illustrates the positions of the The vapor transport equation may be given as camera for taking about 40 photographs of the stone. The images acquired are the input of the Autodesk 123D Catch 𝜕 3D scanning software15 [ ]. (𝜙 𝜌 )+∇⋅(𝜙𝜌 𝑉 )=𝑆. (11) 𝜕𝑡 V V V V 𝐵 V Figure 5(b) shows the basics of the stitching process for the accurate creation of a 3D model of the stone. Figure 5(c) It is known that the presence of bubbles contributes to the represents a medium quality 3D mesh of the stone which is processes of energy removal from the resolved scales of the used to simulate sheet cavitation in the following section. liquid phase. This two-way coupling effect may be modeled by superposing bubble-induced SGS energy dissipation to 3.3.3. Sheet Cavitation that induced by shear. A tentative first attempt at closure, the momentum equation (6),maybeclosedwithanSGSmodel Aspects of the Simulation. As mentioned earlier, turbulence for, 𝜏,givenas[10] modeling has a critical role in cavitation prediction. Cavita- tion flow in the liquid-vapor region is locally compressible [9, ̃ ̃ ̃ 𝜏 mod =−𝜌(k̃𝑤k̃𝑤 − k̃𝑤k̃𝑤) 16]. To capture the shedding dynamics and the unsteadiness of cavitation the modified form of LES as detailed in the 1/2 󵄨 󵄨 + 𝜌[2(𝐶 Δ)2 (̃ : ̃) +(𝐶Δ) 󵄨k̃󵄨] preceding section would be required. Note that (5) and (6) 𝑠 S S 𝑠 󵄨 slip󵄨 (12) look like the equations of motion of a fluid with variable 1 density 𝜌=𝜌𝑤(1 − 𝜙V). ×(̃ − ̃ ). S 3 tr SI In this section, the mathematical model described in the preceding section will be used to analyze sheet cavitation in a Here, the value of the backscatter parameter, 𝐶𝑏0,issetto pipe flow as shown in Figure 3.Thedischargewassetto𝑄= 𝐶𝑏0 = 0.2.Themodelfor𝜏 has been validated for dense gas- 12.5 L/s. The tube inner diameter was 𝐷 = 0.024 mandits particle flows [11–13]. length was 𝐿 = 0.3 m. During each time step of the simulation, the forces The length and the height of the stones in the simulations acting on each particle should be calculated. This requires are exactly the same as those used in the experiments. knowledge of the local values of the fluid velocity components The model of the stone is shown in Figure 5(c).Itislikely at the position of the bubbles. These variables are only known that cavitation as shown in Figure 3(b) starts at minute at spatial grid points in the computational domain. A tricubic cracks on the otherwise smooth surface of the stones. The interpolation has been used for calculating the fluid velocity aforementioned system may be characterized as belonging to components at the centers of the bubbles. The bubbles are the singular roughness category [1]. propagated using the generalized Verlet algorithm, whose In brief, the Lagrangian (bubble-based) and the Eulerian parallelization for the GPU is described in detail in [10]. (grid-based) methods are used to simulate the cavitation 6 Mathematical Problems in Engineering

5 x 4 z 3

2 y V(r)

1.0 1.52.0 2.5 r

(a) (b)

x z

(e) (f) (g)

Figure 4: The interaction of small bubbles in an aerated tank. (a) The Lennard-Jones 12-6 pair potential. (b) Computed bubble-like structure in an aerated tank at the Reynolds number of 29500. (c) A different view of (b). (d) The denoised version of (c). The length of the bubble is approximately 1 cm and its height is 0.65 cm. (e) Configuration of a three bubble-like structure in an aerated tank. (f) The bubbles in (e) coalesce to form a larger bubble after 0.3 sec. (g) The denoised version of (f).

singular roughness in a tube flow. Figure 6(a) illustrates the of the pipe covered by stones is 𝑙2 =0.037m,andthepipe’s model used in the simulations. Here, the dimensions are 𝐿𝑝 = length covered by the first stone is 𝑙1 = 0.0185 m. The length 0.3 m, 𝐷 = 0.024 cm, and ℎ𝑠 =0.004m.Figure 6(b) depicts and height of the stones are exactly the same as those shown a top view of the entrance region of the pipe in which the in Figure 3(a). Figure 6(c) shows the grid which consists of 6 positions of the stones are clearly shown. The total length more than 5×10 tetrahedral meshes. Mathematical Problems in Engineering 7

y x (a) (b)

z h

y l

(c)

Figure 5: A CFD model for a stone in Figure 3. (a) Taking 2D images of the stone by moving camera’s position at regular intervals. (b) Automatically aligning the images with the others. (c) Reconstruction of a 3D model from 2D image.

z D l1 y x x Lp l2 l

hs y

(a) (b) z

(c)

Figure 6: A CFD model for the tube in Figure 3. (a) Schematic of the pipe used in sheet cavitation simulations with some nomenclatures. (b) A top view of the entrance region of the pipe. (c) The grid used in the cavitation simulations. Insets: the grid is magnified and replotted. 8 Mathematical Problems in Engineering

The filtered Navier-Stokes equations (5) and (6) are where the local pressure is much higher than that of the vapor discretized with the finite-volume method on a staggered pressure. This figure also shows that a large number of small Cartesian grid. Convective and diffusive fluxes are approxi- bubbles are extensively produced on the trailing edge of the mated with central differences of second-order accuracy and second stone close to sampling port 8 located at 𝑥 = 0.045 m. time advancement is achieved by a second-order, explicit Sampling ports 1 through 16 are all located in the 𝑥𝑧-plane. Adams-Bashforth scheme. The equation for coupling the Thedistancesoftheseportsfromtheinletofthepipeare𝑥= −5 pressure to the velocity field is solved iteratively using the 5×10 , 0.0077, 0.015, 0.022, 0.025, 0.029, 0.035, 0.045, 0.055, SIP method [17]. The subgrid-scale stresses appearing in the 0.065, 0.09, 0.12, 0.2, 0.25, 0.29, and 0.298 m, respectively. filtered Navier-Stokes equations are computed using (12).No- Figure 7(d) illustrates the averaged vapor volume frac- slipboundaryconditionsareemployedonthesurfaceofthe tion, 𝜙V, as a function of the vertical distance from the pipe stoneandthetubewalls.Thephysicalpropertiesusedin wall, 𝑙𝑤, at sampling ports 1 through 4. Here, the squares, the simulations are listed in Table 1.Thetemperaturewas circles, diamonds, and left triangles represent the computed 𝑇 = 302 assumed to be constant and it was set to K. Note that vapor volume fraction at ports 1, 2, 3, and 4, respectively. the vaporization pressure [18]atthattemperatureis4500Pa This figure indicates the occurrence of a swarm of bubbles at 𝑔 = 9.81 2 (absolute).Theaccelerationduetogravityis m/s . sampling port 3. The Lagrangian (bubble-based) simulation was per- The squares, circles, diamonds, and left triangles in formed to calculate the motion of bubbles in the liquid using Figure 7(e) represent variations of 𝜙V as a function of 𝑙𝑤 at (4). In brief, equivalent spherical bubbles were determined samplingports5,6,7,and8,respectively.Thisfigureindicates from irregular vapor structure by marking the cells with 𝜙𝑤 < 0.95 that cavitation reoccurred at sampling port 7 and became .Equation(4) was integrated using the generalized Verlet extensive at port 8. These complexities cannot be described algorithm. The time step used in the simulation was Δ𝑡 = 5×10−7 with a single parameter such as cavitation number. s. The linear interpolation routines were used to 𝜙 𝑙 communicate the information from grid nodes to particle Figure 7(f) shows variations of V as a function of 𝑤 at positions and vice versa. The simulations were performed by sampling ports 9, 10, 11, and 12 using the squares, circles, using GPU computing [10]. diamonds, and left triangles, respectively. As can be seen from Figure 7(f),thevaporbubblesrollupintoalargervolumeand Results.Obviously,asingleparametersuchasthecavitation become cloudy as they are transported further downstream number cannot describe many of the complexities of cavita- andleavetheentranceregionofthepipe. tion. Falvey [1] suggested that for flow past stones as shown in Figure 7(g) represents the final stage of the process which Figure 7(a),cavitationwillnotoccurifthecavitationnumber is an ongoing process of bubble formation and deformation. is greater than about 1.8. Figure 7(a) shows that cavitation This figure depicts variations of 𝜙V as a function of 𝑙𝑤 at bubbles form within the flow at the discharge 𝑄 = 12.5 L/s. sampling ports 13, 14, 15, and 16 using the squares, circles, This indicates that the cavitation number was below 1.8. diamonds, and left triangles, respectively. Figure 7(b) represents a view from below of an instantaneous In brief, the appearance of visible cavitation in the flowing configuration of bubbles in the cavitation water flow. In this water in the pipe as shown in Figure 7 was preceded by the case,asheetconsistsofalargenumberofsmallbubbles occurrence of small bubbles in the small area on the surface which are extensively developed downstream of the step of the stones. This observation highlights the importance of formed by the stones in the pipe. Here, cavitation is formed bubbles as cavitation nuclei which has been known for a long by turbulence in the shear zone which is produced by the time [6]. sudden change of flow direction at the stone face. Figure 7(b) Figure 7(d) indicated that a swarm of small bubbles also shows large nonspherical cavities with patterns similar to occurred at sampling port 3. However, no evidence of cavita- those shown in Figure 3(b). tion was found at sampling port 4 which was located further Following Falvey [1], it may be estimated that the system downstream from port 3. To address this issue, the pressure 𝜎ℎ = 0.9 field is illustrated in the xz-plane in Figure 8(a).Here,the in Figure 7(a) operates at a cavitation number of . ∗ For this case, the reference location is immediately upstream dimensionless pressure Π is defined as Ln(𝑝)/Ln(𝑝V),where 𝑝 represents the local pressure. Figure 8(b) depicts variations of the first stone and at its maximum height. The computed ∗ lengthofthesheetisquitecomparablewiththatshown of the dimensionless pressure 𝑝 (defined as 𝑝/𝑝V)asa in Figure 3(b). It would be expected that for even lower function of 𝑙𝑤 atsamplingports1,2,3,and4usingthesquares, values of the cavitation number, the clouds form one long circles, diamonds, and left triangles, respectively. This figure supercavitating pocket. indicates that the local pressure in the narrow cleft between Figure 7(c) depicts the computed instantaneous contours the stones is approximately 50 times higher than that of the of the averaged vapor volume fraction in the xz-plane. In vapor pressure. this figure, the locations at which cavitation occurred on the Figure 8(c) illustrates variations of the dimensionless ∗ surfaces of the stones are presented by the use of color coding. pressure, 𝑝 , as a function of 𝑙𝑤 atsamplingports5,6,7, AscanbeseenfromFigure 7(c), the process starts with the and 8 using squares, circles, diamonds, and left triangles, occurrence of swarm of bubbles in the small regions of the respectively. As can be seen from this figure, the local pressure first stone at sampling port 3 located at 𝑥 = 0.015 m. However, at port 7 on the surface of the second stone reached the vapor no evidence of cavitation can be found at sampling port 4 pressure.Consequently,thesmallbubbleswereformedonthe located at 𝑥 = 0.022 m in a narrow cleft between the stones trailing edge of the second stone as shown in Figure 7(e).The Mathematical Problems in Engineering 9

z y

g x x

(a) (b) 0.35 1

0.3 0.8 0.25

0.2 0.6 𝜙 1.0 0.15 𝜙 0.4 0.1 x

0.7 𝜙 w 0.05 0.2 y z 0 g 0 𝜙 0.5 0.01 0.02 0.01 0.02 0 0.5 1 lw (cm) lw (cm) (c) (d) (e) 1

0.8 0.6

0.6

0.4 𝜙 𝜙 0.4 0.2

0.2

0 0 0.01 0.02 0.01 0.02

lw (cm) lw (cm) (f) (g)

Figure 7: Computer experiments at a cavitation number of 𝜎ℎ = 0.9 in a duct. (a) A top view of the instantaneous configuration of bubbles in a cavitating water flow of water through a pipe with inner diameter of 𝐷 = 0.024 m at the discharge of 𝑄 = 12.5 L/s and at the cavitation number of 𝜎ℎ = 0.9. (b) A view from below of the bubbles in the cavitating flow of water through the pipe. Here, the large cavities consisting of small individual bubbles are illustrated. (c) Computed contours of the averaged vapor volume fraction on the xz-plane for the cavitating water flow in (a). Here, a color code as illustrated below the stones is used to indicate the values of the averaged vapor volume fraction on the surfaces of the stones. (d)–(g) Variations of 𝜙V as a function of 𝑙𝑤 at the sampling ports 1 through 16, respectively.

appearance of visible cavitation as shown in Figure 7(a) was that measured at the PMP in Figure 3(a)) provides a favorable preceded by the occurrence of these bubbles in the small area situation for a sheet, which consists of a large number of small aroundsamplingport5onthesurfaceofthesecondstone. bubbles, to be developed and transported downstream as The computed low pressure at port 8 (which is the same as depicted in Figures 7(a) and 7(b). The low pressure condition 10 Mathematical Problems in Engineering

1 1

1.56 ∗

∗ 10 1.42 P

P 10 x lw 1.29 ∗ y z Π

1.14

Lsp SP 100 1 100 0.01 0.02 0.01 0.02

lw (cm) lw (cm) (a) (b) (c)

1 1

27.5 ∗ ∗ P 5 2 P

3 13.7 (m/s) V 4 5 40 0.01 0.02 0.01 0.02

lw (cm) lw (cm) 0.0 (d) (e) (f)

∗ Figure 8: Cavitation in a tube. (a) The instantaneous computed contours of the dimensionless pressure Π on the xz-plane bubbles in a cavitation flow of water at the cavitation number of 𝜎ℎ = 0.9.ThelocationsofthesamplingportsarethesameasthoseinFigure 7(c).(b)– ∗ (e) Variations of the dimensionless pressure 𝑝 as a function of 𝑙𝑤. (f) The instantaneous computed velocity field of a cavitating water flow through the pipe with inner diameter of 𝐷 = 0.024 m at the discharge of 𝑄 = 12.5 L/s.

extends from sampling port 9 to sampling port 14 as shown reached the atmospheric pressure at the pipe outlet at which in Figures 8(d) and 8(e).Thesquares,circles,diamonds,and the vapor bubbles vanished. ∗ left triangles in Figure 8(d) represent variations of 𝑝 as a The area-weighted average pressure at the inlet of the pipe function of 𝑙𝑤 at sampling ports 9, 10, 11, and 12, respectively. was found to be 257 kPa. The results show agreement between In the aforementioned region the vapor bubbles roll up simulation and experiment. The model is therefore deemed to into a larger volume and become less extensive as they are be sufficiently flexible to capture a number of the interesting transported further downstream. The volume fraction of the qualitative features of the cavitation process. vapor bubble is lower at the upper part of the pipe between Figure 8(f) shows the computed velocity vector filed in ports9and14duetothefactthatthepressureslightly the xz-plane. This figure indicates a complex shearing flow increases with 𝑙𝑤. around sampling port 8 at which cavitation was formed. As The squares, circles, diamonds, and left triangles in stated earlier, the system in Figure 7(a) operates at a cavitation ∗ Figure 8(e) represent variations of 𝑝 as a function of 𝑙𝑤 at number of 𝜎ℎ = 0.9. Note that the computed pressure at sampling ports 13, 14, 15, and 16, respectively. This figure sampling port 1 immediately upstream of the first stone and indicates that the pressure increased from ports 14 to 16 and at its maximum height is approximately 300 kPa. In this case, Mathematical Problems in Engineering 11

1.0

0.8

0.6 a 𝜙 0.4

0.2

0 0510

lw (m) (a) (b) 4.5

1400

y (kPa)

2500 (c)

Figure 9: Cavitation in a spillway chute. (a) Schematic of a hypothetical hole with a diameter 𝑑ℎ = 0.02 on the surface of a model of ∘ Karahnj´ ukar´ spillway at a distance from the inlet of 𝐿ℎ = 300 m. The total length of the spillway is 𝐿𝑇 = 414 manditsslopeis𝛼 =11. Inset: the computed contours of velocity in the spillway. The distances of the sampling ports 1, 2, 3, and 4 from the inlet are 0, 107, 300, and 414 m, respectively. (b) The averaged air volume fraction as a function of 𝑙𝑤 at the sampling ports 1, 2, 3, and 4. Here, the squares, circles, diamonds, and left triangles represent the air volume fraction at the sampling ports 1 through 4, respectively. (c) The computed velocity vector field around the hole at the sampling port 3. Here, a color code as illustrated on the left-hand side of the hole is used to indicate the valuesof the local pressure.

Periodic Ha L L1 Hw 𝜃1 ho z 𝜃2 g z x y

L2 x H t out

(a) (b)

Figure 10: A CFD model for the Karahnj´ ukar´ bottom aerator. (a) Perspective view of the simulation model and selected nomenclature. (b) The front view of the spillway with bottom aerator. Here, 𝛿 = 0.02 m, 𝐿 =2.42m,𝐻𝑎 = 0.44 m, 𝐻𝑤 = 0.11 m, 𝐿1 = 0.11 m, 𝐿2 =2.15m,ℎ0 = 𝐻 𝜃 ∘ 𝜃 ∘ 0.022, out = 0.55 m, 1 =20,and 2 = 30 . the computed average water velocity is 25 m/s. The area- 3.3.4. Cavitation in a Spillway Chute. The previous section weighted average pressure at the inlet of the pipe was found described the complexity of the occurrence of cavitation. As to be 257 kPa. stated earlier, Figure 2 indicates that the cavitation number In the following section, the model is employed to check should be quite low at approximately 135 m from the end of the reliability of the predicted cavitation zone as shown in the chute of Karahnj´ ukar´ spillway. Apparently, the designers Figure 2. of the Karahnj´ ukar´ spillway decided to use an aerator at that 12 Mathematical Problems in Engineering location (as shown in Figure 1) in order to protect the spillway 3.81 surface from cavitation. Recall that Semenkov and Saranchev [19]suggestedthat 2.86

the beginning of the spillway is the cavitation-hazardous sec- ∗

1.90 tion. Particularly for large discharges, natural air entrainment is absent in more than 60 m from the inlet [19]. The bottom 0.95 turbulent layer is shielded above by the undisturbed potential flow core, which reduces diffusion air entrainment19 [ ]. Con- 0.00 sequently, on the entire length of the cavitation-hazardous Figure 11: Computed contours of the velocity field for the bottom section of the spillway passage the volume air concentration aerator of Karahnj´ ukar´ spillway. Inset: a pair of vortices in the in the boundary layer with thickness 0.2–0.25 m would be less ventilation zone. than 7-8%. In this case, artificial air entrainment is needed to protect the surface of the spillway [20]. In this section, the usefulness of using artificial air at 125 m from the end of the chute of the Karahnj´ ukar´ sampling port 2 is lower than 7%. This finding highlights the spillway is investigated using the mathematical model as need of artificial aerators between sampling ports 1 and 2in detailed in the preceding section. To this end, a simplified order to protect the surface of the spillway. model of Karahnj´ ukar´ spillway with a hypothetical hole with Figure 9(b) indicates that the bubble volume fraction at diameter of 0.02 m is used in the simulation. Figure 9(a) the hypothetical hole (i.e., at sampling port 3) is approx- shows the hypothetical hole which is essential to initiate imately 38% which is in accordance with Semenkov and cavitation at the cavitation number of 0.25. The simplified Saranchev [19]andverymuchabovethe“cavitationsafe” model of Karahnj´ ukar´ spillway is 414 m long with a slope of valueof7%ofentrainedair.Inthiscase,thevaporvolume ∘ approximately 𝛼 =11. fraction is quite low. The inlet boundary condition at 𝑥=0was set to Figure 9(c) illustrates the computed velocity vector field. 𝑉=20m/s. The height of the water at the inlet was 3 m. A color code as illustrated on the left hand side of the hole in Furthermore, it was assumed that the top surface of the model Figure 9(c) is used to indicate the values of the local pressure. is open to the atmosphere. The outlet boundary condition was This figure indicates that the water flows mainly over the 𝑝=𝑝 hole. A small part of the water stream hits the hole’s face and set to atm. Periodic boundary conditions were used in the 𝑧-direction. The no-slip boundary condition was applied reticulates inside the hole. As can be seen from Figure 9(c), on the walls of the model. To capture the effects of viscous the pressure reaches a maximum of 2500 kPa and diverts boundary layers, a wall-function was used to specify velocity the flow to an upward direction over a little edge andina at the forcing points as [21] downward direction which initiates the water swirl in the hole. The local pressure in the center of the hole is slightly 𝑢 lower than 5 kPa. This finding indicates that cavitation might 𝑢∗ occur at 125 m from the end of the chute of the Karahnj´ ukar´ spillway.Indeed,cavitationwilloccurfortheflowpastthe 𝑦+ 𝑦+ ≤5 { sudden into-the-flow offset introduced by the hypothetical { {−1.15576 + 1.47869𝑦+ hole as noted in Figure 9(c).Thisfigureindicatesalow- { = pressure region and thus potential cavitation might also occur { −0.0527848𝑦+2 + 0.000655444𝑦+3 5<𝑦+ ≤30 { shortly after passing the hole’s edge. However, the air volume { + fractionattheholeasshowninFigure 9(b) is approximately { (𝑦 ) ln + 5.1 𝑦+ > 30, { 38%.Thisfindingsuggeststhattheuseofanaeratoratport Κ 3inordertoprotectthesurfaceofthespillwaymightnotbe (13) needed. Adding extra air at sampling port 3 would produce + where 𝑦 represents dimensionless, sublayer-scaled distance, averythickair-watermistflowattheoutletofthespillway. and Κ is the von Karm´ an´ constant (typically the value 0.41 is This will be further analyzed in the following section. used). Figure 9(b) also indicates that in the absence of an The inset of Figure 9(a) represents the computed velocity artificial aerator the computed thickness of the air-water contours in the model of the Karahnj´ ukar´ spillway. The mixture at the outlet of the spillway is approximately 8 m. In computed bubble volume fraction as a function of vertical addition, the average bubble volume fraction is higher than 50%. distance from the wall 𝑙𝑤 at four sampling ports is illustrated in Figure 9(b).Inthiscase,thebubbleisanairbubblewhich maycontainvapor.AscanbeseenfromFigure 9(a),the 3.3.5. Artificial Aeration. The designers of the Karahnj´ ukar´ hypothetical hole is located at 135 m from the outlet of the spillway were challenged in their task due to the lack of model. Figure 9(b) indicates that the thickness of the air- literature on air entrainment for spillways. As mentioned water mixture increases with 𝑥. Here, the squares, circles, in the preceding section, an aerator must be introduced to diamonds, and left triangles represent the bubble volume prevent cavitation in the Karahnj´ ukar´ spillway at high dis- fraction at sampling ports 1, 2, 3, and 4, respectively. As charges. The air volume fraction in the spillway downstream canbeseenfromFigure 9(b),thebubblevolumefractionat from sampling port 2 in Figure 9(b) is well above the 7% Mathematical Problems in Engineering 13

Lnb

1.86

S 1.40 x y 1 S2 S3 ∗

0.93 S4 0.47

0.00 (a) (b) 1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6 a a a a 𝜙 𝜙 𝜙 𝜙

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0.0 0.0 0.0 0.0 0.0 0.35 0.0 0.25 0.0 0.4 0.0 0.2 l (m) l (m) l (m) l (m) (c)

Figure 12: The computed flow field for the aerator ofKarahnj´ ukar´ spillway. (a) Computed contours of velocity field of a more optimized design for the aerator of Karahnj´ ukar´ spillway. Inset: vortical structures in the ventilation zone. (b) Instantaneous configuration of bubbles in a spillway with ski jump aerator. Here, 𝐿𝑛𝑏 = 0.62 m. (c) Air concentration distribution versus 𝑧 in the aeration region. Here, right triangles, squares, left triangles, and diamonds represent the air concentration at the sampling ports S1,S2,S3,andS4,respectively.

minimum requirement of its protection against cavitation. distance from the free flow. The air pressure is atmospheric at Hence, artificial aeration would only be needed between the inlets and the outlet. sampling ports 1 and 2 illustrated in Figure 9(a). Figure 12(a) and its insets illustrate the computed con- In this section, the mathematical model described in the tours and vector velocity field of a more optimized design for preceding section will be used to analyze the Karahnj´ ukar´ the aerator of the Karahnj´ ukar´ spillway. Here, vortices can spillway with a bottom aerator. Figure 10 illustrates a CFD be seen to be created at the ventilation gate and to detach modelwhichcanbeimaginedasa1:15Froudescaleofthe periodically from its other side. A pair of vortices is magnified Karahnj´ ukar´ bottom aerator. Figure 1 showsanaerialviewof andreplottedintheinset.Theairenterstheventilationzone ∗ the Karahnj´ ukar´ spillway and its bottom aerator located in with a normalized velocity of V =3.81.Here,thevelocityis eastern Iceland. normalized with the water velocity at inlet V𝑤0. Figure 12(b) By implementing periodic boundary conditions in the represents an instantaneous configuration of bubbles in the 𝑦-direction,asmallpartofthesystemthatisfarfromthe spillway with an optimized bottom aerator. The high stream vertical side walls was simulated. The model flow rate was of air tends to move the water stream up. In this case, the air set to V𝑤0 = 20 m/s at the inlet with a size of 𝐻𝑤 =0.11m, concentration in the ventilated zone appears to be more than 3 corresponding to 𝑄=2250 m /s. that needed to prevent cavitation. Figure 1 shows a highly AscanbeseenfromFigure 11,thewaterstreamis aerated water stream which is likely to serve as a source for deflected by a deflector in order to ventilate the nappe tothe an intense canyon erosion. atmosphere. The connection to the atmosphere is made via Whenwaterflowsoveraspillway,vortexsheetinsta- a channel under the ski jump of the aerator. Note that the bility at the air-water interface known as Kelvin-Helmholtz air entraining capacity of the aerator depends on the take-off instability can occur. The instability will be in the form of 14 Mathematical Problems in Engineering

5.6

z 4.2 l ∗

x 2.8 g S5 S 6 S2 1.4 S3 S4

0.0 (a) (b) 2 2 2 ∗ ∗ ∗ 1 1 1

0 0 0 0.0 0.2 0.4 0.0 0.25 0.5 0.0 0.25 0.5 l (m) l (m) l (m) (c)

5 5 5

4 4 4

3 3 3 ∗ ∗ ∗ 2 2 2

1 1 1

0 0 0 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.0 0.1 0.2 0.3 l (m) l (m) l (m) (d)

3 Figure 13: The computed flow field for a hypothetical case with a discharge of 𝑄 = 3150 m /s. (a) Computed contours of velocity field. (b) ∗ Computed vector plot of vertical structures in the ventilation zone. (c) and (d) Variations of the normalized velocity V as a function of 𝑧. Squares, circles, diamonds, left triangles, gradients, and deltas represent the normalized velocity at the sampling ports S1,S6,S5,S2,S3,and S4,respectively. Mathematical Problems in Engineering 15

waves being generated on the water surface which can initiate 𝐶𝐷: Drag coefficient of the bubble, which isa natural air entrainment. function of the Reynolds number of the The air entrainment process is strengthened when a bubble growing boundary layer in the water flow reaches the free 𝐶𝑝: Pressure coefficient surface. In this case, the entrained air from the free surface 𝐶𝑠: Smagorinsky constant, given as √ 2 3 1/2 can reach the spillway surface to minimize the danger of 𝐶𝑠 =2𝜋/3 3(−𝑅(1 +𝑏0 𝐶 𝑅 )/Κ ) cavitation. As can be seen from Figure 12, the details of air 𝐷:Pipediameter entrainment through the upper air-water interface of the jet 𝑑𝐵:Bubblediameter can be predicted by the model. In addition, the details of 𝑑ℎ:Holediameter air entrainment through the lower interface are very similar 𝐹:Forcebetweentwobubbles 𝐿 to those presented in [10]. Here, 𝑛𝑏 represents the length 𝑓𝑏𝑤: Forcesexertedbytheliquidonthebubbles at which the air entrainment through the lower interface per unit volume reaches that through the upper interface. Figure 12(c) depicts ℎ:Heightofaquariumstone 𝑧 the variations of air concentration as a function of at four ℎ𝑠:Stepheight different sampling ports as shown in (b). 𝐺:Gasconstant The region at which the natural air entrainment effectively 𝑔: Gravitational acceleration eliminates the risk of damage due to cavitation is called the I: Second-order identity tensor fully aerated region. The artificial aeration process discussed K: Kolmogorov constant abovewouldnotbeneededinthefullyaeratedregion. 𝐿𝑐: Length of the sheet Figure 13(a) illustrates the computed contours of the 𝐿ℎ: Distance of the hole from the inlet velocity field for a hypothetical case in which the discharge 𝐿 3 𝑚: Axial distance of the pressure measuring of the Karahnj´ ukar´ spillway is 𝑄=3150 m /s. Figure 13(b) port for the pipe inlet indicates that the air enters the ventilation zone with a 𝐿𝑝: Pipe length V∗ = 5.6 normalized velocity of .Thevortexsheddingofthe 𝐿𝑡: Total length of spillway chute aeration gate is quite intense. Therefore, the air-water mixture 𝑙:Lengthofaquariumstone entering the canyon could cause extensive erosion. Recall that 𝑙1: Lengthofthestoneintheaxialdirection thecanyonshowninFigure 1 suffered a disastrous erosion in 𝑙2: Total length of the region covered by 2009. stones 𝑙𝑤:Verticaldistancefromthepipewall 𝑛𝐵: Bubble number density ∗ 4. Conclusions 𝑃 : Dimensionless pressure defined as 𝑃∗ = 𝑝/𝑝 A previous study revealed that the cavitation number of V 𝑝 the Karahnj´ ukar´ spillway is below 0.25 at a discharge of V: Vapor pressure at ambient temperature 3 𝑝(0) 2250 m /s. Hence, the spillway was protected by additional ∞ :Constantin(1) 𝑝 aeration grooves located some 300 m from its inlet. In this ∞: Ambient pressure in the fluid at a large work,ananalysisoftheproblembasedonphysicaland distance from the bubble 𝑝 computational fluid dynamics (CFD) modeling is presented. : Filtered pressure 𝑄 The numerical modeling is based on the large eddy simulation :Discharge 𝑅 technique (LES) combined with the discrete element method. critical: Bubble critical radius 𝑟 𝑖 A three-dimensional simulation of a cavitation tube flow 𝑖:Locationoftheth bubble 𝑟 at cavitation number of ca. 0.9 was performed to highlight 𝑖𝑗 : Distance between the centers of two the importance of artificial aeration for protecting a spillway. bubbles 𝑅 More simulations were performed using a CFD model which 𝐸:Bubbleequilibriumradius isa1:15FroudescaleoftheKarahnj´ ukar´ bottom aerator Re𝐵: Bubble Reynolds number defined as =|V𝑖 − V |𝑑/] in order to gain insights into how to diminish the hazards Re𝐵 𝐵 𝑤 𝑤 𝑅 associated with cavitation and minimize canyon erosion. It 𝑘: Ratioofthemeshspacingandthe is recommended to use the artificial aerators only at the Kolmogorov length scale, so that 4/3 distances less than 100 m from the inlet in order to minimize 𝑅𝑘 = −1 + (𝜂/Δ) canyon erosion. 𝑅: Bubble radius The main finding was that the design of the aerator of 𝑅̇: Bubble interface velocity the Karahnj´ ukar´ spillway should be improved. It is highly S: Resolved rate of strain tensor unlikely that the artificial aerators would be needed at the 𝑆V:Sourcetermin(5) to account for the mass distancesof300mfromtheinletofthespillway. transfer between liquid and vapor phase 𝑇:Temperature 𝑡:Time Nomenclature 𝑉:Mixturevelocity V𝑖 𝑖 𝐴𝑐: Archimedes number 𝐵:Velocityoftheth bubble with a diameter 𝑑 𝐶𝑏0: Backscatter parameter 𝐵 16 Mathematical Problems in Engineering

V𝑤: Water velocity in absence of the bubble International Conference on Multiphase Flow,NewOrleans,La, 󸀠󸀠 k𝑤: Subgrid-scale part of v𝑤 basedonFavre USA, 2001. filtering [10] P. Zamankhan, “Solid structures in a highly agitated bed of k̃𝑤: Density-weighted Favre filter velocity granular materials,” Applied Mathematical Modelling,vol.36, defined as ṽ𝑤 = 𝜌v𝑤/𝜌 no. 1, pp. 414–429, 2012. k :Slipvelocity [11] P.Zamankhan, “Formation patterns at the air-grain interfaces in ∗slip V :Normalizedvelocity. spinning granular films at high rotation rates,” Communications in Nonlinear Science and Numerical Simulation,vol.17,no.11,pp. Greek Symbols 4224–4240, 2012. [12] P. Zamankhan, “Simulation of self-segregation of a low density 𝛼: Chuteangleforchuteslope spherical particle in a bubbling bed,” Chemical Engineering Δ:Filterwidth Journal,vol.181-182,pp.842–845,2012. 𝜀:Constantin(3) [13] P. Zamankhan, “Sinking and recirculation of large intruders in 𝛾: Surface tension vertically vibrated granular beds,” Advanced Powder Technology, 𝜇𝑤:Dynamicviscosityofwater vol. 24, no. 6, pp. 1070–1085, 2013. ∗ Π : Dimensionless pressure defined as [14] C. Chaux, L. Duval, A. Benazza-Benyahia, and J.-C. Pesquet, ∗ Π = Ln𝑃/Ln𝑃V “A nonlinear Stein-based estimator for multichannel image denoising,” IEEE Transactions on Signal Processing,vol.56,no. 𝜌: Filtered density defined as 𝜌=𝜌𝑤𝜙𝑤 𝜙: Bubble volume fraction 8, pp. 3855–3870, 2008. [15] http://labs.autodesk.com/technologies/123d/. 𝜙V: Vapor volume fraction defined as 3 3 𝜙V = (4/3)𝜋𝑛𝐵𝑅 /(1 + (4/3)𝜋𝑛𝐵𝑅 ) [16]O.Coutier-Delgosha,R.Fortes-Patella,andJ.L.Reboud, “Evaluation of the turbulence model influence on the numerical 𝜙𝑎:Airvolumefraction simulations of unsteady cavitation,” Journal of Fluids Engineer- 𝜙𝑤:Watervolumefraction ing,vol.125,no.1,pp.38–45,2003. 𝜌𝑎: Air density [17] H. L. Stone, “Iterative solution of implicit approximations of 𝜌V: vapor density multidimensional partial differential equations,” SIAM Journal 𝜌𝑤: Water density on Numerical Analysis,vol.5,pp.530–558,1968. 𝜎:Cavitationnumber [18] C. Antoine, “Tensions des vapeurs, nouvelle relation entre les 𝜎ℎ:Cavitationnumber 𝜏 tensions et les temperatures,”´ Comptes Rendus des Seances´ de : Subgrid scale (SGS) stress tensor. l’Academie´ des Sciences, vol. 107, p. 681, 1888. [19] V. M. Semenkov and V. O. Saranchev, “Computer program for Conflict of Interests hydraulic calculation of structures for controlling aeration of the flow on spillways,” Hydrotechnical Construction,vol.30,no. The author declares that there is no conflict of interests 5,pp.234–242,1996. regarding the publication of this paper. [20] R. S. Gal’perin, A. G. Oskolkov, V. M. Semenkov, and G. N. Tsedrov, Cavitation on Hydraulich Structures, Energiya,´ References Moscow,Russia,1977,(Russian). [21] P. Zamankhan, “Rollers in low-head dams—challenges and [1] H. T. Falvey, Cavitation in Chutes and Spillways, U.S. Depart- solutions,” Communications in Nonlinear Science and Numerical ment of the Interior, Bureau of Reclamation, 1990. Simulation, vol. 17, no. 12, pp. 5273–5285, 2012. [2] W. Lee and J. A. Hoopes, “Prediction of cavitation damage for spillways,” JournalofHydraulicEngineering,vol.122,no.9,pp. 481–488, 1996. [3]P.Novak,A.I.B.Moffat,C.Nalluri,andR.Narayanan, Hydraulic Structures, Spon Press, London, UK, 2004. [4] Landsvirkjun, Karahnj´ ukar´ HEP Iceland, Physical Model Inves- tigation on the Karahnj´ ukar´ Dam Spillway, VAW, Laboratory of Hydraulics and Glaciology of the Swiss Federal Institute of Technology Zurich, 2006, http://www.vaw.ethz.ch/people/ wb/archive/wb 251 karahnjukar dam spillway. [5] D. D. Joseph, “Cavitation in a flowing liquid,” Physical Review E, vol. 51, Article ID R1649(R), 1995. [6] L.G.Leal,Advanced Transport Phenomena, Cambridge Univer- sity Press, Cambridge, UK, 2007. [7] P.Zamankhan, “Large eddy simulation and PIV experiments of air-water mixing tanks,” Communications in Nonlinear Science and Numerical Simulation,vol.15,no.6,pp.1511–1525,2010. [8]M.P.AllenandD.J.Tildesley,Computer Simulations of Liquids, Oxford University Press, Oxford, UK, 1987. [9] G. H. Sauer and J. Schnerr, “Physical and numerical modeling of unsteady cavitation dynamics,” in Proceedings of the 4th Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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