Research Article Simulation of Cavitation Water Flows

Research ArticleSimulation of Cavitation Water Flows

Piroz Zamankhan12

1 Institute of Geothermal Energy Dedan Kimathi University of Technology 101000 Nyeri Kenya2Chemical Engineering Department School of Engineering Collage of Agriculture Engineering and ScienceUniversity of KwaZulu-Natal 4041 Durban South Africa

Correspondence should be addressed to Piroz Zamankhan qpz002000yahoocom

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

Academic Editor Junuthula N Reddy

Copyright copy 2015 Piroz Zamankhan This is an open access article distributed under the Creative Commons Attribution Licensewhich 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 canyonmay cause serious erosion and damage to boththe spillway surface and the environmentThe location of an aerator its geometry and the aeration flow rate are important factors inthe design of an environmentally friendly high-energy spillway In this work an analysis of the problem based on physical and com-putational fluid dynamics (CFD)modeling is presentedThe 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 processingunit (GPU)The result of this analysis in the formof design suggestionsmay help diminishing the hazards associatedwith cavitation

1 Introduction

In spillway engineering there are numerous challengingobstacles One of the most determining factors is the geo-logical geometry in which the dam and its spillway have tobe built in The geometry such as length and slope of thespillway can range from short and steep to long and flat orvice versa Generally the longer and steeper is the spillwaythe higher is the gradient of the energy head of the flowingwater In some cases the velocity of the water might exceed30ms and cavitation might occur Used as an illustrativeexample the high-energy spillway of the Karahnjukar Damwhich supports the Halsson Reservoir in eastern Icelanddisplays numerous problems in spillway engineering Anaerial perspective of this spillway can be seen in Figure 1

When a fluid changes state from liquid to vapor itsvolume increases by orders of magnitude and cavities (orbubbles) can form Indeed cavities which are filled withwater vapor are formed by boiling at the local pressure whichequals the water vapor pressure If the cavity is filled withgases other than water vapor then the process is namedgaseous cavitation [1]

There is a technical difference between boiling andcavitation On the one hand boiling is defined as the processof phase change from the liquid state to the vapor state by

changing the temperature at constant pressure On the otherhand cavitation is the process of phase change from the liquidstate to the vapor state by changing the local pressure atconstant temperature

Cavitation is a consequence of the reduction of thepressure to a critical value due to a flowing liquid or in anacoustical field Spillways are subject to cavitation Inertialcavitation is the process in which voids or bubbles rapidlycollapse forming shock waves This process is marked byintense noise The shock waves formed by cavitation cansignificantly damage the spillway face [1] Cavitation damageon the spillway face is a complex process [2] Richer concretemixes are used to increase the resistance to cavitation damageand erosion [3]

Falvey [1] suggested that impurities and microscopic airbubbles in the water are necessary to initiate cavitation Ascan be seen from Figure 1 the color of water accumulatedbehind the dam indicates that it may contain various typesof impurities including suspended fine solid particles Thedensity of the water accumulated behind the dam wasdetermined from a sample using a hydrometer The relativedensity of the sample with respect to water has been found tobe approximately 108

The melting water which transports suspended clayparticles flows into the side channel and then follows an

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 872573 16 pageshttpdxdoiorg1011552015872573

2 Mathematical Problems in Engineering

Figure 1 Aerial view of Karahnjukar spillway with bottom aeratoreastern Iceland

approximately 414m long chute with a transition bend anda bottom aerator

The most important task is to reduce the energy head ofthe flowing water until it enters the river In this example thejet should not hit the opposite canyon wall where it increaseserosion An artificial aeration device ldquocushionsrdquo the waterthe widening at the chutersquos end reduces the flow velocityand increases the impact area in the plunge pool which inturn decreases rock scouring Additionally seven baffles andlateral wedges at themain throw lip abet the jet disintegrationIn general every high-energy spillway has to be analyzed forcavitation risk

The water in spillways contains air bubbles and impu-rities such as suspended particles in a range of sizes Asdiscussed later these air bubbles and impurities induce cav-itation Vaporization is the most important factor in bubblegrowth At a critical combination of flow velocity water pres-sure and the vapor pressure of the flowing water cavitationstarts The cavitation number is used to define this startingpoint The equation for this parameter is derived from theBernoulli equation for a steady flow between two pointsIn dimensionless terms the comparable equation results ina pressure coefficient 119862119901 The value of this parameter is aconstant at every point until the minimum pressure at acertain location is greater than the vapor pressure of waterThepressure at a certain locationwill not decrease any furtheronce the vapor pressure is reached Taking everything intoaccount such as temperature clarity of the water and safetyreasons the pressure coefficient is set to a minimum valueand then called cavitation number 120590 The cavitation numberis a dimensionless number which expresses the relationshipbetween the difference of a local absolute pressure from thevapor pressure and the kinetic energy per volume It may beused to characterize the potential of the flow to cavitate

It has been suggested that no cavitation protection for aspillway would be needed if the cavitation number is largerthan 18 If the cavitation number is in the range of 012ndash017then the spillway should be protected by additional aerationgrooves In Figure 2 the cavitation indices for different dis-charges have been calculated

1

09

08

07

06

05

04

03

02

01

0350 300 250 200 150 100 50 0

Distance from the outlet of K arahnjukar spillway (m)

Cavi

tatio

n nu

mbe

r120590

Location of aerator

Grey theoretical cavitation zone

Figure 2 Cavitation numbers at different discharges on theKarahnjukar spillway [4] The location of the outlet is shown inFigure 1 Here blue diamonds 200m3s red squares 400m3sgreen triangle 800m3s blue squares 950m3s yellow square withgreen star 1350m3s and orange dot 2250m3s

As a precaution the cavitation zone has been set to 120590 =

025 because of the lower atmospheric pressure low watertemperature and overall safety reasonsThus as a precautionthe aerator was placed 125m before the end of the chuteNevertheless incipient white water far upstream from theaerator allows the assumption that in such a long spillwayfully self-aerated flow may have already been developedTherefore the purpose of this aerator is mainly to reduce theimpact energy in the plunge pool

The main idea is to develop a fully 3D simulation ofan aerated spillway in the future But to achieve that thephysics of bubble growth due to cavitation and aeration andthe formation and distribution of the bubbles have to beunderstood and simulations have to be validated In this workit is shown that LES combinedwith a discrete elementmethodonGPUs yields promising results Cavitation is defined as theformation of the vapor phase and the subsequent immediateimplosion of small liquid-free zones in a liquid called voidsor bubbles [1]

The first section deals with forced cavitation and thedynamics of cavitation bubbles An experiment was carriedout in which cavitation was forced to develop with clearwater in glass and galvanized steel pipes These findings werecompared to computational simulations The mathematicalmodel and simulation results of the cavitation bubble dynam-ics will be described

In the second part of the paper the flow behavior of anaerated spillway is computed In this section the reliabilityof the predicted cavitation zone as shown in Figure 2 isexamined

2 Experimental Methods and Observations

Cavitation occurs when the fluid pressure is lower than thelocal vapor pressure [5] In the cavities the vapor phasereplaces the liquid phase The surrounding liquid then expe-riences evaporative cooling For water these thermal effects

Mathematical Problems in Engineering 3

h

x

Q

l

PMPDischarge(high pressure pump) 4

550

Pa

Lp

Lm

D

(a)

y

x

Lc

D

(b)

Figure 3 The experimental setup for investigating cavitation (a) Schematic of the experimental setup Inset (on the right-hand side) animage of the aquarium stone used in the cavitation test Inset (lower) the minimum pressure measured was 455 kPa (absolute) (b) An imageof sheet cavitation at the discharge Q = 125 Ls

can have a significant influence on the dynamics of bubbles inthe cavitation process However for simplicity in this paperthe isothermal problem will be investigated which retains anumber of interesting qualitative features of the cavitationprocess

For water flow over a uniform roughness in a spillwayincipient cavitation will occur randomly over the spillwaysurface If the cavitation number is lowered below incipi-ent conditions then cavitation will occur in sheets whoselocations are influenced by the magnitude and spectrum ofturbulent fluctuation within the boundary layer [1]

To enhance the predictive reliability of computationalmodels for cavitation which are still considered to beimmature further validation based on experimental data isrequired Figure 3 shows a simple system in which sheetcavitation may be observed

Surface roughness cracks and offsets may initiate cavita-tionThe local pressure on the steprsquos surfaces or directly at thestep in a high velocity spillway can reach the vapor pressureThis may result in sheet cavitation similar to that shown inFigure 3 As stated earlier cavitation can cause severe damageto the spillway concrete The extent of cavitation damage is afunction of the local cavitation number in the spillway chuteand the duration of the flow

In brief experiments were conducted in a test setupas illustrated in Figure 3(a) The tests were performed in ahorizontal galvanized steel pipe with an outer diameter of

Table 1 Physical properties

Material Dynamicviscosity (Pa s)

Density(kgm3)

Vaporization pressure(mmHg) [18]

Water (liquid) 0001 9964 log10

119875 = 807131

minus173063119879 minus 39724Water (vapor) 134 times 10minus5 05542

Air 179 times 10minus5 1175

119863 = 00254m and the length of 119871119901 = 03m Two very similaraquarium stones with a length of 119897 = 00215m and a heightof ℎ = 001m were attached to the bottom of the pipe in theentrance sectionThe stones produced a sudden offset whoseleading edge is approximately an ellipse The dimensions ofone of the stones are depicted in the inset of Figure 3(a)

Clear water was used in the experiments its physicalproperties are given in Table 1 The discharge 119876 = 125 Lswas provided with a heavy-duty high-pressure wash-downpump with a pressure range of 25 bar to 140 bars (maximumpressure) A high-pressure hose was used to connect the testtube to the high-pressure pump

The average pressure at the pipe inlet was approximately25 bars (absolute) and the minimum measured pressurewas 455 kPa (absolute) Figure 3(b) shows the monometerconnected to the pressure measuring port (PMP) as shownin the second inset of Figure 3(a) The port was located119871119898 = 0048m downstream from the inlet of the test tube


Here 119871119888 represents the length of the sheet The low pressuremeasured at the sampling port indicates that cavitation mayhave occurred in the test tube

The above-mentioned test with the same discharge ofQ = 125 Ls and the same average inlet pressure of 25 bar(absolute) was repeated in a cylindrical borosilicate glass pipewith a diameter of 119863 = 0026mThe pipe wall thickness was0001m A camera was used to take images of sheet cavitationfrom the underside of the test tube

Figure 3(b) shows an image of sheet cavitation with anaverage inlet channel pressure of approximately 25 bars(absolute) As can be seen from Figure 3(b) the sheet appearsto be attached to the stone with a length of 119871119888 gt 5119863 Thesheet consists of a number of fuzzy white clouds A flashphotograph reveals that a cloud consists of individual smallbubbles [1]

In the following section simulation results will be pre-sented giving a more detailed insight and showing anongoing process of cavity formation and deformation

It is difficult to obtain accurate measurements using thesimple setup as shown in Figure 3 The stones attached to thebottom of the pipes may not be stable over long periods oftime

Additional experimental studies in cavitation tunnels arerequired to supply a more reliable and convincing data basisfor the validation of computational approaches to modelcavitation

3 Mathematical Model and Simulation Results

If sheet cavitation as described in the experiments in thepreceding section occurs in the spillway chute it can produceextensive damage to its concrete linings [1] Note that cavita-tion damage in tunnels and conduits is less likely to result indam failure

Experimental observations such as those shown inFigure 3(b) are indispensable for investigating cavitationproblems However numerical simulation can provide possi-bilities to study cavity formation and deformation in greaterdetail than what is affordable by experiments

31 Bubble Dynamics in a Quiescent Fluid The importanceof microscopic bubbles as cavitation nuclei has been knownfor a long time [1] Consider a single bubble which containsthe vapor of the liquid in equilibrium in anincompressibleNewtonian fluid that is at rest at infinity In this case thebubble radius 119877119864 must satisfy the following condition [6]

(119901V minus 119901(0)

infin) 1198773

119864minus 2120574119877

2

119864+ 119866 = 0 (1)

The bubble radius at the limit point where (119901V minus 119901(0)

infin) =

(32120574327119866)

12 can be expressed in terms of material param-eters as [5]

119877critical = (3119866

2120574)

12

(2)

If 119877119864 le 119877critical then the bubble is stable against infinitesimalchanges in its radius The bubble becomes unstable if 119877119864 gt

119877critical Note that the bubble may contain a ldquocontaminantrdquogas which dissolves very slowly compared to the time scalesassociated with changes in its size

Apparently small bubbles whose radii are smaller than thecritical radius behave like rigid spheres in accelerated flows[7]

32 Bubble Dynamics in Accelerated Flows For the presentstudy the interaction of a small bubble whose size is char-acterized by the critical bubble radius with its neighboringbubbles is assumed to be subjected to the Lennard-Jones (LJ)condition [8] Figure 5(a) shows the Lennard-Jones 12-6 pairpotential The force between two bubbles with diameter 119889119861

located at 119903119894 and 119903119895 is given as

F119894119895 =48120576

1199032119894119895

[(119889119861

119903119894119895

)

12

minus1

2(

119889119861

119903119894119895

)

6

] r119894119895 (3)

A simplified momentum equation for the 119894th sphericalbubble may be given as

120587

6120588V1198893

119861

119889k119894119861

119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

momentum transfer

=120587

61205881199081198892

119861119862119863 (Re119861 119860119888)

10038161003816100381610038161003816k119894119861

minus k11990810038161003816100381610038161003816

(k119894119861

minus k119908)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

viscous

+120587

121205881199081198893

119861(

119863k119908119863119905

minus119889k119894119861

119889119905)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

added mass

+3

21205881199081198892

119861(120587V119908)

12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Basset

times int

119905

minusinfin

(119863k119908119863119905

minus119889k119894119861

119889119905)

119889120591

(119905 minus 120591)12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

histroy term

+120587

61205881199081198893

119861

119863k119908119863119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

pressure term

minus120587

61198893

119861(120588V minus 120588119908) g

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

buoyancy

+

119873

sum

119895=1

F119894119895⏟⏟⏟⏟⏟⏟⏟⏟⏟

bubble-bubbleinteraction

(4)

In the current effort the aim is to combine the Lagrangian(bubble-based) model (4) and large eddy simulation (LES)in order to achieve more accurate simulations of cavita-tion water flows The filtered continuity momentum for anisothermal 3D flow of water may be given as

120597120588

120597119905+ nabla sdot (120588k119908) = minus119878V (5)

120597120588k119908120597119905

+ nabla sdot (120588k119908k119908) = minus1

120588119908

nabla (120588119901) +1

120588119908

nabla sdot (120588) + nabla sdot 120591

+ 119891119887119908 + 120588g

(6)

The viscous stress tensor in (6) is defined as

120590 = minus2

3120583119908nabla sdot k119908I + 120583119908 (nablak119908 + (nablak119908)

119879) (7)


In (6) 120591may be given as

120591 = minus120588 (k119908k119908 minus k119908 k119908) minus 120588 (k10158401015840119908k119908 +

k119908k10158401015840

119908) minus 120588 (

k10158401015840119908k10158401015840119908

)

(8)

The SGS stress tensor 120591 is required to close the equationsfor the large-scale fields on a grid small enough (but muchlarger than the Kolmogorov scale) to provide reasonableresolutions In contrast to the filtered single-phase equationsa conceptual restriction arising in the present approach is thatthe filter width Δ should strictly be larger than the lengthscale characteristic of the small bubblesThus an appropriatechoice ofΔ should provide a sufficiently large-scale resolutionwhile not violating the aforementioned restriction

Following Sauer and Schnerr [9] the mass flow ratethrough the surface of a bubble with the radius of 119877 119878V maybe given as

119878V = 41205871198991198611198772

(1 +4

31205871198991198611198773)

minus2120588119908120588V

120588 (9)

Here the changes in the bubble radius may be calculatedusing the famous Rayleigh-Plesset equation [6]

119901V + 1198661198791198773

minus 119901infin

120588119908

minus1

120588119908

(2120574

119877+

4120583119908

119877) = 119877 +

3

22 (10)

The vapor transport equation may be given as

120597

120597119905(120601V120588V) + nabla sdot (120601V120588V119881119861) = 119878V (11)

It is known that the presence of bubbles contributes to theprocesses of energy removal from the resolved scales of theliquid phase This two-way coupling effect may be modeledby superposing bubble-induced SGS energy dissipation tothat induced by shear A tentative first attempt at closure themomentum equation (6) may be closed with an SGS modelfor 120591 given as [10]

120591 mod = minus120588 (k119908k119908 minus k119908 k119908)

+ 120588 [2 (119862119904Δ)2

(S S)12

+ (119862119904Δ)10038161003816100381610038161003816kslip

10038161003816100381610038161003816]

times (S minus1

3tr SI)

(12)

Here the value of the backscatter parameter 1198621198870 is set to1198621198870 = 02 The model for 120591 has been validated for dense gas-particle flows [11ndash13]

During each time step of the simulation the forcesacting on each particle should be calculated This requiresknowledge of the local values of the fluid velocity componentsat the position of the bubblesThese variables are only knownat spatial grid points in the computational domain A tricubicinterpolation has been used for calculating the fluid velocitycomponents at the centers of the bubbles The bubbles arepropagated using the generalized Verlet algorithm whoseparallelization for the GPU is described in detail in [10]

33 Simulation Results

331 Air Bubbles Figures 4(b) and 4(c) show a computedlarge bubble in an aerated tank [7] The bubble is a clusterof small bubbles with a size of the critical air bubble radiusFigure 4(d) depicts the denoised version of the bubble usingmultiscale image denoising based on a multiscale representa-tion of the images [14] The denoised version of the bubble-like structure in Figure 4(b) is very similar to the smallbubbles observed in an aerated tank [7]

Figures 4(e) and 4(f) are snapshots of bubbles separatedby 119905 = 03 sec As can be seen from Figure 4(f) three bubblesin Figure 4(e) coalesce to formone large bubbleThedenoisedversion depicted in Figure 4(g) is very similar to the bubbleswith complex geometries shown in [7]

To further assess the quality of the model described inthe preceding section the cavitation water flow in Figure 3is simulated in the following section

332 3D Model Using 2D Images Apparently it is notpossible to reconstruct a 3D model from a single imageof the stone as shown in the inset of Figure 5(a) To getaround this limitation an optical processing algorithm can bedeveloped that employs multiple photographs taken at every15 degrees from the above and from the side to create a 3Dmodel of the stone Figure 5(a) illustrates the positions of thecamera for taking about 40 photographs of the stone Theimages acquired are the input of the Autodesk 123D Catch3D scanning software [15]

Figure 5(b) shows the basics of the stitching process forthe accurate creation of a 3D model of the stone Figure 5(c)represents a medium quality 3D mesh of the stone which isused to simulate sheet cavitation in the following section

333 Sheet Cavitation

Aspects of the Simulation As mentioned earlier turbulencemodeling has a critical role in cavitation prediction Cavita-tion flow in the liquid-vapor region is locally compressible [916] To capture the shedding dynamics and the unsteadinessof cavitation the modified form of LES as detailed in thepreceding section would be required Note that (5) and (6)look like the equations of motion of a fluid with variabledensity 120588 = 120588119908(1 minus 120601V)

In this section the mathematical model described in thepreceding section will be used to analyze sheet cavitation in apipe flow as shown in Figure 3 The discharge was set to 119876 =

125 Ls The tube inner diameter was 119863 = 0024m and itslength was 119871 = 03m

The length and the height of the stones in the simulationsare exactly the same as those used in the experimentsThe model of the stone is shown in Figure 5(c) It is likelythat cavitation as shown in Figure 3(b) starts at minutecracks on the otherwise smooth surface of the stones Theaforementioned systemmay be characterized as belonging tothe singular roughness category [1]

In brief the Lagrangian (bubble-based) and the Eulerian(grid-based) methods are used to simulate the cavitation


10 15 2520

r

(a) (b)

(c) (d)

(e) (f) (g)

5

4

3

2

1

0

x

y

z

x

y

z

x

y

z

y

zx

V(r)

Figure 4The interaction of small bubbles in an aerated tank (a)The Lennard-Jones 12-6 pair potential (b) Computed bubble-like structurein 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 bubbleis approximately 1 cm and its height is 065 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 03 sec (g) The denoised version of (f)

singular roughness in a tube flow Figure 6(a) illustrates themodel used in the simulations Here the dimensions are 119871119901 =03m 119863 = 0024 cm and ℎ119904 = 0004m Figure 6(b) depictsa top view of the entrance region of the pipe in which thepositions of the stones are clearly shown The total length

of the pipe covered by stones is 1198972 = 0037m and the pipersquoslength covered by the first stone is 1198971 = 00185m The lengthand height of the stones are exactly the same as those shownin Figure 3(a) Figure 6(c) shows the grid which consists ofmore than 5 times 10

6 tetrahedral meshes


z

x

y

(a) (b)

h

y

l

z

(c)

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

y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)

Figure 6 A CFDmodel 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


The filtered Navier-Stokes equations (5) and (6) arediscretized with the finite-volume method on a staggeredCartesian grid Convective and diffusive fluxes are approxi-mated with central differences of second-order accuracy andtime advancement is achieved by a second-order explicitAdams-Bashforth scheme The equation for coupling thepressure to the velocity field is solved iteratively using theSIP method [17] The subgrid-scale stresses appearing in thefilteredNavier-Stokes equations are computed using (12) No-slip boundary conditions are employed on the surface of thestone and the tube walls The physical properties used inthe simulations are listed in Table 1 The temperature wasassumed to be constant and it was set to 119879 = 302K Note thatthe vaporization pressure [18] at that temperature is 4500 Pa(absolute) The acceleration due to gravity is 119892 = 981ms2

The Lagrangian (bubble-based) simulation was per-formed to calculate the motion of bubbles in the liquid using(4) In brief equivalent spherical bubbles were determinedfrom irregular vapor structure bymarking the cells with 120601119908 lt

095 Equation (4) was integrated using the generalized Verletalgorithm The time step used in the simulation was Δ119905 =5 times 10

minus7 s The linear interpolation routines were used tocommunicate the information from grid nodes to particlepositions and vice versa The simulations were performed byusing GPU computing [10]

Results Obviously a single parameter such as the cavitationnumber cannot describe many of the complexities of cavita-tion Falvey [1] suggested that for flow past stones as shown inFigure 7(a) cavitation will not occur if the cavitation numberis greater than about 18 Figure 7(a) shows that cavitationbubbles form within the flow at the discharge 119876 = 125 LsThis indicates that the cavitation number was below 18Figure 7(b) represents a view from below of an instantaneousconfiguration of bubbles in the cavitation water flow In thiscase a sheet consists of a large number of small bubbleswhich are extensively developed downstream of the stepformed by the stones in the pipe Here cavitation is formedby turbulence in the shear zone which is produced by thesudden change of flow direction at the stone face Figure 7(b)also shows large nonspherical cavities with patterns similar tothose shown in Figure 3(b)

Following Falvey [1] it may be estimated that the systemin Figure 7(a) operates at a cavitation number of 120590ℎ = 09For this case the reference location is immediately upstreamof the first stone and at its maximum height The computedlength of the sheet is quite comparable with that shownin Figure 3(b) It would be expected that for even lowervalues of the cavitation number the clouds form one longsupercavitating pocket

Figure 7(c) depicts the computed instantaneous contoursof the averaged vapor volume fraction in the xz-plane Inthis figure the locations at which cavitation occurred on thesurfaces of the stones are presented by the use of color codingAs can be seen from Figure 7(c) the process starts with theoccurrence of swarm of bubbles in the small regions of thefirst stone at sampling port 3 located at119909 = 0015mHoweverno evidence of cavitation can be found at sampling port 4located at 119909 = 0022m in a narrow cleft between the stones

where the local pressure ismuch higher than that of the vaporpressure This figure also shows that a large number of smallbubbles are extensively produced on the trailing edge of thesecond stone close to sampling port 8 located at 119909 = 0045mSampling ports 1 through 16 are all located in the 119909119911-planeThe distances of these ports from the inlet of the pipe are 119909 =

5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055

0065 009 012 02 025 029 and 0298m respectivelyFigure 7(d) illustrates the averaged vapor volume frac-

tion 120601V as a function of the vertical distance from the pipewall 119897119908 at sampling ports 1 through 4 Here the squarescircles diamonds and left triangles represent the computedvapor volume fraction at ports 1 2 3 and 4 respectivelyThis figure indicates the occurrence of a swarm of bubbles atsampling port 3

The squares circles diamonds and left triangles inFigure 7(e) represent variations of 120601V as a function of 119897119908 atsampling ports 5 6 7 and 8 respectivelyThis figure indicatesthat cavitation reoccurred at sampling port 7 and becameextensive at port 8 These complexities cannot be describedwith a single parameter such as cavitation number

Figure 7(f) shows variations of 120601V as a function of 119897119908 atsampling ports 9 10 11 and 12 using the squares circlesdiamonds and left triangles respectively As can be seen fromFigure 7(f) the vapor bubbles roll up into a larger volume andbecome cloudy as they are transported further downstreamand leave the entrance region of the pipe

Figure 7(g) represents the final stage of the process whichis an ongoing process of bubble formation and deformationThis figure depicts variations of 120601V as a function of 119897119908 atsampling ports 13 14 15 and 16 using the squares circlesdiamonds and left triangles respectively

In brief the appearance of visible cavitation in the flowingwater in the pipe as shown in Figure 7 was preceded by theoccurrence of small bubbles in the small area on the surfaceof the stones This observation highlights the importance ofbubbles as cavitation nuclei which has been known for a longtime [6]

Figure 7(d) indicated that a swarm of small bubblesoccurred at sampling port 3 However no evidence of cavita-tion was found at sampling port 4 which was located furtherdownstream from port 3 To address this issue the pressurefield is illustrated in the xz-plane in Figure 8(a) Here thedimensionless pressure Π

lowast is defined as Ln(119901)Ln(119901V) where119901 represents the local pressure Figure 8(b) depicts variationsof the dimensionless pressure 119901

lowast (defined as 119901119901V) as afunction of 119897119908 at sampling ports 1 2 3 and 4 using the squarescircles diamonds and left triangles respectively This figureindicates that the local pressure in the narrow cleft betweenthe stones is approximately 50 times higher than that of thevapor pressure

Figure 8(c) illustrates variations of the dimensionlesspressure 119901

lowast as a function of 119897119908 at sampling ports 5 6 7and 8 using squares circles diamonds and left trianglesrespectively As can be seen from this figure the local pressureat port 7 on the surface of the second stone reached the vaporpressure Consequently the small bubbleswere formed on thetrailing edge of the second stone as shown in Figure 7(e)The


yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)

Figure 7 Computer experiments at a cavitation number of 120590ℎ = 09 in a duct (a) A top view of the instantaneous configuration of bubblesin a cavitating water flow of water through a pipe with inner diameter of 119863 = 0024m at the discharge of 119876 = 125 Ls and at the cavitationnumber of 120590ℎ = 09 (b) A view from below of the bubbles in the cavitating flow of water through the pipe Here the large cavities consistingof small individual bubbles are illustrated (c) Computed contours of the averaged vapor volume fraction on the xz-plane for the cavitatingwater 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 thesurfaces of the stones (d)ndash(g) Variations of 120601V as a function of 119897119908 at the sampling ports 1 through 16 respectively

appearance of visible cavitation as shown in Figure 7(a) waspreceded by the occurrence of these bubbles in the small areaaround sampling port 5 on the surface of the second stoneThe computed low pressure at port 8 (which is the same as

thatmeasured at the PMP in Figure 3(a)) provides a favorablesituation for a sheet which consists of a large number of smallbubbles to be developed and transported downstream asdepicted in Figures 7(a) and 7(b)The low pressure condition


156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)

Figure 8 Cavitation in a tube (a) The instantaneous computed contours of the dimensionless pressure Πlowast on the xz-plane bubbles in a

cavitation flow of water at the cavitation number of 120590ℎ = 09 The locations of the sampling ports are the same as those in Figure 7(c) (b)ndash(e) Variations of the dimensionless pressure 119901

lowast as a function of 119897119908 (f) The instantaneous computed velocity field of a cavitating water flowthrough the pipe with inner diameter of 119863 = 0024m at the discharge of 119876 = 125 Ls

extends from sampling port 9 to sampling port 14 as shownin Figures 8(d) and 8(e) The squares circles diamonds andleft triangles in Figure 8(d) represent variations of 119901

lowast as afunction of 119897119908 at sampling ports 9 10 11 and 12 respectively

In the aforementioned region the vapor bubbles roll upinto a larger volume and become less extensive as they aretransported further downstream The volume fraction of thevapor bubble is lower at the upper part of the pipe betweenports 9 and 14 due to the fact that the pressure slightlyincreases with 119897119908

The squares circles diamonds and left triangles inFigure 8(e) represent variations of 119901

lowast as a function of 119897119908 atsampling ports 13 14 15 and 16 respectively This figureindicates that the pressure increased from ports 14 to 16 and

reached the atmospheric pressure at the pipe outlet at whichthe vapor bubbles vanished

The area-weighted average pressure at the inlet of the pipewas found to be 257 kPaThe results show agreement betweensimulation and experimentThemodel is therefore deemed tobe sufficiently flexible to capture a number of the interestingqualitative features of the cavitation process

Figure 8(f) shows the computed velocity vector filed inthe xz-plane This figure indicates a complex shearing flowaround sampling port 8 at which cavitation was formed Asstated earlier the system in Figure 7(a) operates at a cavitationnumber of 120590ℎ = 09 Note that the computed pressure atsampling port 1 immediately upstream of the first stone andat its maximum height is approximately 300 kPa In this case


(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)

Figure 9 Cavitation in a spillway chute (a) Schematic of a hypothetical hole with a diameter 119889ℎ = 002 on the surface of a model ofKarahnjukar spillway at a distance from the inlet of 119871ℎ = 300m The total length of the spillway is 119871119879 = 414m and its slope is 120572 = 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 300and 414m respectively (b) The averaged air volume fraction as a function of 119897119908 at the sampling ports 1 2 3 and 4 Here the squares circlesdiamonds and left triangles represent the air volume fraction at the sampling ports 1 through 4 respectively (c)The computed velocity vectorfield 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 values ofthe local pressure

Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)

Figure 10 A CFD model for the Karahnjukar bottom aerator (a) Perspective view of the simulation model and selected nomenclature (b)The front view of the spillway with bottom aerator Here 120575 = 002m 119871 = 242m 119867119886 = 044m 119867119908 = 011m 1198711 = 011m 1198712 = 215m ℎ0 =0022 119867out = 055m 1205791 = 20∘ and 1205792 = 30∘

the computed average water velocity is 25ms The area-weighted average pressure at the inlet of the pipe was foundto be 257 kPa

In the following section the model is employed to checkthe reliability of the predicted cavitation zone as shown inFigure 2

334 Cavitation in a Spillway Chute The previous sectiondescribed the complexity of the occurrence of cavitation Asstated earlier Figure 2 indicates that the cavitation numbershould be quite low at approximately 135m from the end ofthe chute of Karahnjukar spillway Apparently the designersof the Karahnjukar spillway decided to use an aerator at that


location (as shown in Figure 1) in order to protect the spillwaysurface from cavitation

Recall that Semenkov and Saranchev [19] suggested thatthe beginning of the spillway is the cavitation-hazardous sec-tion Particularly for large discharges natural air entrainmentis absent in more than 60m from the inlet [19] The bottomturbulent layer is shielded above by the undisturbed potentialflow core which reduces diffusion air entrainment [19] Con-sequently on the entire length of the cavitation-hazardoussection of the spillway passage the volume air concentrationin the boundary layer with thickness 02ndash025mwould be lessthan 7-8 In this case artificial air entrainment is needed toprotect the surface of the spillway [20]

In this section the usefulness of using artificial airat 125m from the end of the chute of the Karahnjukarspillway is investigated using the mathematical model asdetailed in the preceding section To this end a simplifiedmodel of Karahnjukar spillway with a hypothetical hole withdiameter of 002m is used in the simulation Figure 9(a)shows the hypothetical hole which is essential to initiatecavitation at the cavitation number of 025 The simplifiedmodel of Karahnjukar spillway is 414m long with a slope ofapproximately 120572 = 11∘

The inlet boundary condition at 119909 = 0 was set to119881 = 20ms The height of the water at the inlet was 3mFurthermore it was assumed that the top surface of themodelis open to the atmosphereTheoutlet boundary conditionwasset to 119901 = 119901atm Periodic boundary conditions were used inthe 119911-direction The no-slip boundary condition was appliedon the walls of the model To capture the effects of viscousboundary layers a wall-function was used to specify velocityat the forcing points as [21]

119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)

where 119910+ represents dimensionless sublayer-scaled distance

and Κ is the von Karman constant (typically the value 041 isused)

The inset of Figure 9(a) represents the computed velocitycontours in the model of the Karahnjukar spillway Thecomputed bubble volume fraction as a function of verticaldistance from the wall 119897119908 at four sampling ports is illustratedin Figure 9(b) In this case the bubble is an air bubble whichmay contain vapor As can be seen from Figure 9(a) thehypothetical hole is located at 135m from the outlet of themodel Figure 9(b) indicates that the thickness of the air-water mixture increases with 119909 Here the squares circlesdiamonds and left triangles represent the bubble volumefraction at sampling ports 1 2 3 and 4 respectively Ascan be seen from Figure 9(b) the bubble volume fraction at

381

286

190

095

000

lowast

Figure 11 Computed contours of the velocity field for the bottomaerator of Karahnjukar spillway Inset a pair of vortices in theventilation zone

sampling port 2 is lower than 7 This finding highlights theneed of artificial aerators between sampling ports 1 and 2 inorder to protect the surface of the spillway

Figure 9(b) indicates that the bubble volume fraction atthe hypothetical hole (ie at sampling port 3) is approx-imately 38 which is in accordance with Semenkov andSaranchev [19] and very much above the ldquocavitation saferdquovalue of 7 of entrained air In this case the vapor volumefraction is quite low

Figure 9(c) illustrates the computed velocity vector fieldA color code as illustrated on the left hand side of the hole inFigure 9(c) is used to indicate the values of the local pressureThis figure indicates that the water flows mainly over thehole A small part of the water stream hits the holersquos face andreticulates inside the hole As can be seen from Figure 9(c)the pressure reaches a maximum of 2500 kPa and divertsthe flow to an upward direction over a little edge and in adownward direction which initiates the water swirl in thehole The local pressure in the center of the hole is slightlylower than 5 kPa This finding indicates that cavitation mightoccur at 125m from the end of the chute of the Karahnjukarspillway Indeed cavitation will occur for the flow past thesudden into-the-flow offset introduced by the hypotheticalhole as noted in Figure 9(c) This figure indicates a low-pressure region and thus potential cavitationmight also occurshortly after passing the holersquos edge However the air volumefraction at the hole as shown in Figure 9(b) is approximately38 This finding suggests that the use of an aerator at port3 in order to protect the surface of the spillway might not beneeded Adding extra air at sampling port 3 would producea very thick air-water mist flow at the outlet of the spillwayThis will be further analyzed in the following section

Figure 9(b) also indicates that in the absence of anartificial aerator the computed thickness of the air-watermixture at the outlet of the spillway is approximately 8m Inaddition the average bubble volume fraction is higher than50

335 Artificial Aeration The designers of the Karahnjukarspillway were challenged in their task due to the lack ofliterature on air entrainment for spillways As mentionedin the preceding section an aerator must be introduced toprevent cavitation in the Karahnjukar spillway at high dis-charges The air volume fraction in the spillway downstreamfrom sampling port 2 in Figure 9(b) is well above the 7


186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)

Figure 12 The computed flow field for the aerator of Karahnjukar spillway (a) Computed contours of velocity field of a more optimizeddesign for the aerator of Karahnjukar spillway Inset vortical structures in the ventilation zone (b) Instantaneous configuration of bubbles ina spillway with ski jump aerator Here 119871119899119887 = 062m (c) Air concentration distribution versus 119911 in the aeration region Here right trianglessquares left triangles and diamonds represent the air concentration at the sampling ports S1 S2 S3 and S4 respectively

minimum requirement of its protection against cavitationHence artificial aeration would only be needed betweensampling ports 1 and 2 illustrated in Figure 9(a)

In this section the mathematical model described in thepreceding section will be used to analyze the Karahnjukarspillway with a bottom aerator Figure 10 illustrates a CFDmodel which can be imagined as a 1 15 Froude scale of theKarahnjukar bottom aerator Figure 1 shows an aerial view ofthe Karahnjukar spillway and its bottom aerator located ineastern Iceland

By implementing periodic boundary conditions in the119910-direction a small part of the system that is far from thevertical side walls was simulated The model flow rate wasset to V1199080 = 20ms at the inlet with a size of 119867119908 = 011mcorresponding to 119876 = 2250m3s

As can be seen from Figure 11 the water stream isdeflected by a deflector in order to ventilate the nappe to theatmosphere The connection to the atmosphere is made viaa channel under the ski jump of the aerator Note that theair entraining capacity of the aerator depends on the take-off

distance from the free flowThe air pressure is atmospheric atthe inlets and the outlet

Figure 12(a) and its insets illustrate the computed con-tours and vector velocity field of a more optimized design forthe aerator of the Karahnjukar spillway Here vortices canbe seen to be created at the ventilation gate and to detachperiodically from its other side A pair of vortices ismagnifiedand replotted in the inset The air enters the ventilation zonewith a normalized velocity of Vlowast = 381 Here the velocity isnormalized with the water velocity at inlet V1199080 Figure 12(b)represents an instantaneous configuration of bubbles in thespillway with an optimized bottom aerator The high streamof air tends to move the water stream up In this case the airconcentration in the ventilated zone appears to be more thanthat needed to prevent cavitation Figure 1 shows a highlyaerated water stream which is likely to serve as a source foran intense canyon erosion

When water flows over a spillway vortex sheet insta-bility at the air-water interface known as Kelvin-Helmholtzinstability can occur The instability will be in the form of


z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)

Figure 13 The computed flow field for a hypothetical case with a discharge of 119876 = 3150m3s (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 Vlowast as a function of 119911Squares circles diamonds left triangles gradients and deltas represent the normalized velocity at the sampling ports S1 S6 S5 S2 S3 andS4 respectively


waves being generated on the water surface which can initiatenatural air entrainment

The air entrainment process is strengthened when agrowing boundary layer in the water flow reaches the freesurface In this case the entrained air from the free surfacecan reach the spillway surface to minimize the danger ofcavitation As can be seen from Figure 12 the details of airentrainment through the upper air-water interface of the jetcan be predicted by the model In addition the details ofair entrainment through the lower interface are very similarto those presented in [10] Here 119871119899119887 represents the lengthat which the air entrainment through the lower interfacereaches that through the upper interface Figure 12(c) depictsthe variations of air concentration as a function of 119911 at fourdifferent sampling ports as shown in (b)

The region atwhich the natural air entrainment effectivelyeliminates the risk of damage due to cavitation is called thefully aerated region The artificial aeration process discussedabove would not be needed in the fully aerated region

Figure 13(a) illustrates the computed contours of thevelocity field for a hypothetical case in which the dischargeof the Karahnjukar spillway is 119876 = 3150m3s Figure 13(b)indicates that the air enters the ventilation zone with anormalized velocity of Vlowast = 56 The vortex shedding of theaeration gate is quite intenseTherefore the air-watermixtureentering the canyon could cause extensive erosion Recall thatthe canyon shown in Figure 1 suffered a disastrous erosion in2009

4 Conclusions

A previous study revealed that the cavitation number ofthe Karahnjukar spillway is below 025 at a discharge of2250m3s Hence the spillway was protected by additionalaeration grooves located some 300m from its inlet In thiswork an analysis of the problem based on physical andcomputational fluid dynamics (CFD) modeling is presentedThenumericalmodeling is based on the large eddy simulationtechnique (LES) combinedwith the discrete elementmethod

A three-dimensional simulation of a cavitation tube flowat cavitation number of ca 09 was performed to highlightthe importance of artificial aeration for protecting a spillwayMore simulations were performed using a CFDmodel whichis a 1 15 Froude scale of the Karahnjukar bottom aeratorin order to gain insights into how to diminish the hazardsassociated with cavitation and minimize canyon erosion Itis recommended to use the artificial aerators only at thedistances less than 100m from the inlet in order to minimizecanyon erosion

The main finding was that the design of the aerator ofthe Karahnjukar spillway should be improved It is highlyunlikely that the artificial aerators would be needed at thedistances of 300m from the inlet of the spillway

Nomenclature

119860119888 Archimedes number1198621198870 Backscatter parameter

119862119863 Drag coefficient of the bubble which is afunction of the Reynolds number of thebubble

119862119901 Pressure coefficient119862119904 Smagorinsky constant given as

119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12

119863 Pipe diameter119889119861 Bubble diameter119889ℎ Hole diameter119865 Force between two bubbles119891119887119908 Forces exerted by the liquid on the bubbles

per unit volumeℎ Height of aquarium stoneℎ119904 Step height119866 Gas constant119892 Gravitational accelerationI Second-order identity tensorK Kolmogorov constant119871119888 Length of the sheet119871ℎ Distance of the hole from the inlet119871119898 Axial distance of the pressure measuring

port for the pipe inlet119871119901 Pipe length119871 119905 Total length of spillway chute119897 Length of aquarium stone1198971 Length of the stone in the axial direction1198972 Total length of the region covered by

stones119897119908 Vertical distance from the pipe wall119899119861 Bubble number density119875lowast Dimensionless pressure defined as

119875lowast

= 119901119901V119901V Vapor pressure at ambient temperature119901(0)

infin Constant in (1)

119901infin Ambient pressure in the fluid at a largedistance from the bubble

119901 Filtered pressure119876 Discharge119877critical Bubble critical radius119903119894 Location of the 119894th bubble119903119894119895 Distance between the centers of two

bubbles119877119864 Bubble equilibrium radiusRe119861 Bubble Reynolds number defined as

Re119861 = |V119894119861

minus V119908|119889]119908119877119896 Ratio of the mesh spacing and the

Kolmogorov length scale so that119877119896 = minus1 + (120578Δ)

43

119877 Bubble radius Bubble interface velocityS Resolved rate of strain tensor119878V Source term in (5) to account for the mass

transfer between liquid and vapor phase119879 Temperature119905 Time119881 Mixture velocityV119894119861 Velocity of the 119894th bubble with a diameter

119889119861


V119908 Water velocity in absence of the bubblek10158401015840119908 Subgrid-scale part of v119908 based on Favre

filteringk119908 Density-weighted Favre filter velocity

defined as v119908 = 120588v119908120588

kslip Slip velocityVlowast Normalized velocity

Greek Symbols

120572 Chute angle for chute slopeΔ Filter width120576 Constant in (3)120574 Surface tension120583119908 Dynamic viscosity of waterΠlowast Dimensionless pressure defined as

Πlowast

= Ln119875Ln119875V120588 Filtered density defined as 120588 = 120588119908120601119908

120601 Bubble volume fraction120601V Vapor volume fraction defined as

120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)

120601119886 Air volume fraction120601119908 Water volume fraction120588119886 Air density120588V vapor density120588119908 Water density120590 Cavitation number120590ℎ Cavitation number120591 Subgrid scale (SGS) stress tensor

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] H T Falvey Cavitation in Chutes and Spillways US Depart-ment of the Interior Bureau of Reclamation 1990

[2] W Lee and J A Hoopes ldquoPrediction of cavitation damage forspillwaysrdquo Journal of Hydraulic Engineering vol 122 no 9 pp481ndash488 1996

[3] P Novak A I B Moffat C Nalluri and R NarayananHydraulic Structures Spon Press London UK 2004

[4] Landsvirkjun Karahnjukar HEP Iceland Physical Model Inves-tigation on the Karahnjukar Dam Spillway VAW Laboratoryof Hydraulics and Glaciology of the Swiss Federal Instituteof Technology Zurich 2006 httpwwwvawethzchpeoplewbarchivewb 251 karahnjukar dam spillway

[5] D D Joseph ldquoCavitation in a flowing liquidrdquo Physical Review Evol 51 Article ID R1649(R) 1995

[6] L G LealAdvanced Transport Phenomena Cambridge Univer-sity Press Cambridge UK 2007

[7] P Zamankhan ldquoLarge eddy simulation and PIV experiments ofair-water mixing tanksrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 15 no 6 pp 1511ndash1525 2010

[8] M P Allen and D J Tildesley Computer Simulations of LiquidsOxford University Press Oxford UK 1987

[9] G H Sauer and J Schnerr ldquoPhysical and numerical modelingof unsteady cavitation dynamicsrdquo in Proceedings of the 4th

International Conference on Multiphase Flow New Orleans LaUSA 2001

[10] P Zamankhan ldquoSolid structures in a highly agitated bed ofgranular materialsrdquo Applied Mathematical Modelling vol 36no 1 pp 414ndash429 2012

[11] P Zamankhan ldquoFormation patterns at the air-grain interfaces inspinning granular films at high rotation ratesrdquoCommunicationsinNonlinear Science andNumerical Simulation vol 17 no 11 pp4224ndash4240 2012

[12] P Zamankhan ldquoSimulation of self-segregation of a low densityspherical particle in a bubbling bedrdquo Chemical EngineeringJournal vol 181-182 pp 842ndash845 2012

[13] P Zamankhan ldquoSinking and recirculation of large intruders invertically vibrated granular bedsrdquoAdvanced Powder Technologyvol 24 no 6 pp 1070ndash1085 2013

[14] C Chaux L Duval A Benazza-Benyahia and J-C PesquetldquoA nonlinear Stein-based estimator for multichannel imagedenoisingrdquo IEEE Transactions on Signal Processing vol 56 no8 pp 3855ndash3870 2008

[15] httplabsautodeskcomtechnologies123d[16] O Coutier-Delgosha R Fortes-Patella and J L Reboud

ldquoEvaluation of the turbulencemodel influence on the numericalsimulations of unsteady cavitationrdquo Journal of Fluids Engineer-ing vol 125 no 1 pp 38ndash45 2003

[17] H L Stone ldquoIterative solution of implicit approximations ofmultidimensional partial differential equationsrdquo SIAM Journalon Numerical Analysis vol 5 pp 530ndash558 1968

[18] C Antoine ldquoTensions des vapeurs nouvelle relation entre lestensions et les temperaturesrdquo Comptes Rendus des Seances delrsquoAcademie des Sciences vol 107 p 681 1888

[19] V M Semenkov and V O Saranchev ldquoComputer program forhydraulic calculation of structures for controlling aeration ofthe flow on spillwaysrdquo Hydrotechnical Construction vol 30 no5 pp 234ndash242 1996

[20] R S Galrsquoperin A G Oskolkov V M Semenkov and GN Tsedrov Cavitation on Hydraulich Structures EnergiyaMoscow Russia 1977 (Russian)

[21] P Zamankhan ldquoRollers in low-head damsmdashchallenges andsolutionsrdquoCommunications in Nonlinear Science andNumericalSimulation vol 17 no 12 pp 5273ndash5285 2012

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Figure 1 Aerial view of Karahnjukar spillway with bottom aeratoreastern Iceland

approximately 414m long chute with a transition bend anda bottom aerator

The most important task is to reduce the energy head ofthe flowing water until it enters the river In this example thejet should not hit the opposite canyon wall where it increaseserosion An artificial aeration device ldquocushionsrdquo the waterthe widening at the chutersquos end reduces the flow velocityand increases the impact area in the plunge pool which inturn decreases rock scouring Additionally seven baffles andlateral wedges at themain throw lip abet the jet disintegrationIn general every high-energy spillway has to be analyzed forcavitation risk

The water in spillways contains air bubbles and impu-rities such as suspended particles in a range of sizes Asdiscussed later these air bubbles and impurities induce cav-itation Vaporization is the most important factor in bubblegrowth At a critical combination of flow velocity water pres-sure and the vapor pressure of the flowing water cavitationstarts The cavitation number is used to define this startingpoint The equation for this parameter is derived from theBernoulli equation for a steady flow between two pointsIn dimensionless terms the comparable equation results ina pressure coefficient 119862119901 The value of this parameter is aconstant at every point until the minimum pressure at acertain location is greater than the vapor pressure of waterThepressure at a certain locationwill not decrease any furtheronce the vapor pressure is reached Taking everything intoaccount such as temperature clarity of the water and safetyreasons the pressure coefficient is set to a minimum valueand then called cavitation number 120590 The cavitation numberis a dimensionless number which expresses the relationshipbetween the difference of a local absolute pressure from thevapor pressure and the kinetic energy per volume It may beused to characterize the potential of the flow to cavitate

It has been suggested that no cavitation protection for aspillway would be needed if the cavitation number is largerthan 18 If the cavitation number is in the range of 012ndash017then the spillway should be protected by additional aerationgrooves In Figure 2 the cavitation indices for different dis-charges have been calculated

1

09

08

07

06

05

04

03

02

01

0350 300 250 200 150 100 50 0

Distance from the outlet of K arahnjukar spillway (m)

Cavi

tatio

n nu

mbe

r120590

Location of aerator

Grey theoretical cavitation zone

Figure 2 Cavitation numbers at different discharges on theKarahnjukar spillway [4] The location of the outlet is shown inFigure 1 Here blue diamonds 200m3s red squares 400m3sgreen triangle 800m3s blue squares 950m3s yellow square withgreen star 1350m3s and orange dot 2250m3s

As a precaution the cavitation zone has been set to 120590 =

025 because of the lower atmospheric pressure low watertemperature and overall safety reasonsThus as a precautionthe aerator was placed 125m before the end of the chuteNevertheless incipient white water far upstream from theaerator allows the assumption that in such a long spillwayfully self-aerated flow may have already been developedTherefore the purpose of this aerator is mainly to reduce theimpact energy in the plunge pool

The main idea is to develop a fully 3D simulation ofan aerated spillway in the future But to achieve that thephysics of bubble growth due to cavitation and aeration andthe formation and distribution of the bubbles have to beunderstood and simulations have to be validated In this workit is shown that LES combinedwith a discrete elementmethodonGPUs yields promising results Cavitation is defined as theformation of the vapor phase and the subsequent immediateimplosion of small liquid-free zones in a liquid called voidsor bubbles [1]

The first section deals with forced cavitation and thedynamics of cavitation bubbles An experiment was carriedout in which cavitation was forced to develop with clearwater in glass and galvanized steel pipes These findings werecompared to computational simulations The mathematicalmodel and simulation results of the cavitation bubble dynam-ics will be described

In the second part of the paper the flow behavior of anaerated spillway is computed In this section the reliabilityof the predicted cavitation zone as shown in Figure 2 isexamined

2 Experimental Methods and Observations

Cavitation occurs when the fluid pressure is lower than thelocal vapor pressure [5] In the cavities the vapor phasereplaces the liquid phase The surrounding liquid then expe-riences evaporative cooling For water these thermal effects


h

x

Q

l


550

Pa

Lp

Lm

D

(a)

y

x

Lc

D

(b)









Density(kgm3)



119875 = 807131

















(119901V minus 119901(0)

infin) 1198773

119864minus 2120574119877

2

119864+ 119866 = 0 (1)


infin) =

(32120574327119866)


119877critical = (3119866

2120574)

12

(2)






F119894119895 =48120576

1199032119894119895

[(119889119861

119903119894119895

)

12

minus1

2(

119889119861

119903119894119895

)

6

] r119894119895 (3)


120587

6120588V1198893

119861

119889k119894119861

119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

momentum transfer

=120587

61205881199081198892

119861119862119863 (Re119861 119860119888)

10038161003816100381610038161003816k119894119861

minus k11990810038161003816100381610038161003816

(k119894119861

minus k119908)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

viscous

+120587

121205881199081198893

119861(

119863k119908119863119905

minus119889k119894119861

119889119905)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

added mass

+3

21205881199081198892

119861(120587V119908)

12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Basset

times int

119905

minusinfin

(119863k119908119863119905

minus119889k119894119861

119889119905)

119889120591

(119905 minus 120591)12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

histroy term

+120587

61205881199081198893

119861

119863k119908119863119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

pressure term

minus120587

61198893

119861(120588V minus 120588119908) g

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

buoyancy

+

119873

sum

119895=1

F119894119895⏟⏟⏟⏟⏟⏟⏟⏟⏟


(4)


120597120588


120597120588k119908120597119905


120588119908

nabla (120588119901) +1

120588119908


+ 119891119887119908 + 120588g

(6)


120590 = minus2


119879) (7)




k119908k10158401015840

119908) minus 120588 (

k10158401015840119908k10158401015840119908

)

(8)



119878V = 41205871198991198611198772

(1 +4

31205871198991198611198773)

minus2120588119908120588V

120588 (9)


119901V + 1198661198791198773

minus 119901infin

120588119908

minus1

120588119908

(2120574

119877+

4120583119908

119877) = 119877 +

3

22 (10)


120597

120597119905(120601V120588V) + nabla sdot (120601V120588V119881119861) = 119878V (11)



+ 120588 [2 (119862119904Δ)2

(S S)12

+ (119862119904Δ)10038161003816100381610038161003816kslip

10038161003816100381610038161003816]

times (S minus1

3tr SI)

(12)
















10 15 2520

r

(a) (b)

(c) (d)

(e) (f) (g)

5

4

3

2

1

0

x

y

z

x

y

z

x

y

z

y

zx

V(r)






z

x

y

(a) (b)

h

y

l

z

(c)


y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)











5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h

x

Q

l


550

Pa

Lp

Lm

D

(a)

y

x

Lc

D

(b)









Density(kgm3)



119875 = 807131

















(119901V minus 119901(0)

infin) 1198773

119864minus 2120574119877

2

119864+ 119866 = 0 (1)


infin) =

(32120574327119866)


119877critical = (3119866

2120574)

12

(2)






F119894119895 =48120576

1199032119894119895

[(119889119861

119903119894119895

)

12

minus1

2(

119889119861

119903119894119895

)

6

] r119894119895 (3)


120587

6120588V1198893

119861

119889k119894119861

119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

momentum transfer

=120587

61205881199081198892

119861119862119863 (Re119861 119860119888)

10038161003816100381610038161003816k119894119861

minus k11990810038161003816100381610038161003816

(k119894119861

minus k119908)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

viscous

+120587

121205881199081198893

119861(

119863k119908119863119905

minus119889k119894119861

119889119905)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

added mass

+3

21205881199081198892

119861(120587V119908)

12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Basset

times int

119905

minusinfin

(119863k119908119863119905

minus119889k119894119861

119889119905)

119889120591

(119905 minus 120591)12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

histroy term

+120587

61205881199081198893

119861

119863k119908119863119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

pressure term

minus120587

61198893

119861(120588V minus 120588119908) g

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

buoyancy

+

119873

sum

119895=1

F119894119895⏟⏟⏟⏟⏟⏟⏟⏟⏟


(4)


120597120588


120597120588k119908120597119905


120588119908

nabla (120588119901) +1

120588119908


+ 119891119887119908 + 120588g

(6)


120590 = minus2


119879) (7)




k119908k10158401015840

119908) minus 120588 (

k10158401015840119908k10158401015840119908

)

(8)



119878V = 41205871198991198611198772

(1 +4

31205871198991198611198773)

minus2120588119908120588V

120588 (9)


119901V + 1198661198791198773

minus 119901infin

120588119908

minus1

120588119908

(2120574

119877+

4120583119908

119877) = 119877 +

3

22 (10)


120597

120597119905(120601V120588V) + nabla sdot (120601V120588V119881119861) = 119878V (11)



+ 120588 [2 (119862119904Δ)2

(S S)12

+ (119862119904Δ)10038161003816100381610038161003816kslip

10038161003816100381610038161003816]

times (S minus1

3tr SI)

(12)
















10 15 2520

r

(a) (b)

(c) (d)

(e) (f) (g)

5

4

3

2

1

0

x

y

z

x

y

z

x

y

z

y

zx

V(r)






z

x

y

(a) (b)

h

y

l

z

(c)


y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)











5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















(119901V minus 119901(0)

infin) 1198773

119864minus 2120574119877

2

119864+ 119866 = 0 (1)


infin) =

(32120574327119866)


119877critical = (3119866

2120574)

12

(2)






F119894119895 =48120576

1199032119894119895

[(119889119861

119903119894119895

)

12

minus1

2(

119889119861

119903119894119895

)

6

] r119894119895 (3)


120587

6120588V1198893

119861

119889k119894119861

119889119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

momentum transfer

=120587

61205881199081198892

119861119862119863 (Re119861 119860119888)

10038161003816100381610038161003816k119894119861

minus k11990810038161003816100381610038161003816

(k119894119861

minus k119908)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

viscous

+120587

121205881199081198893

119861(

119863k119908119863119905

minus119889k119894119861

119889119905)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

added mass

+3

21205881199081198892

119861(120587V119908)

12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

Basset

times int

119905

minusinfin

(119863k119908119863119905

minus119889k119894119861

119889119905)

119889120591

(119905 minus 120591)12

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

histroy term

+120587

61205881199081198893

119861

119863k119908119863119905⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

pressure term

minus120587

61198893

119861(120588V minus 120588119908) g

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

buoyancy

+

119873

sum

119895=1

F119894119895⏟⏟⏟⏟⏟⏟⏟⏟⏟


(4)


120597120588


120597120588k119908120597119905


120588119908

nabla (120588119901) +1

120588119908


+ 119891119887119908 + 120588g

(6)


120590 = minus2


119879) (7)




k119908k10158401015840

119908) minus 120588 (

k10158401015840119908k10158401015840119908

)

(8)



119878V = 41205871198991198611198772

(1 +4

31205871198991198611198773)

minus2120588119908120588V

120588 (9)


119901V + 1198661198791198773

minus 119901infin

120588119908

minus1

120588119908

(2120574

119877+

4120583119908

119877) = 119877 +

3

22 (10)


120597

120597119905(120601V120588V) + nabla sdot (120601V120588V119881119861) = 119878V (11)



+ 120588 [2 (119862119904Δ)2

(S S)12

+ (119862119904Δ)10038161003816100381610038161003816kslip

10038161003816100381610038161003816]

times (S minus1

3tr SI)

(12)
















10 15 2520

r

(a) (b)

(c) (d)

(e) (f) (g)

5

4

3

2

1

0

x

y

z

x

y

z

x

y

z

y

zx

V(r)






z

x

y

(a) (b)

h

y

l

z

(c)


y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)











5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












k119908k10158401015840

119908) minus 120588 (

k10158401015840119908k10158401015840119908

)

(8)



119878V = 41205871198991198611198772

(1 +4

31205871198991198611198773)

minus2120588119908120588V

120588 (9)


119901V + 1198661198791198773

minus 119901infin

120588119908

minus1

120588119908

(2120574

119877+

4120583119908

119877) = 119877 +

3

22 (10)


120597

120597119905(120601V120588V) + nabla sdot (120601V120588V119881119861) = 119878V (11)



+ 120588 [2 (119862119904Δ)2

(S S)12

+ (119862119904Δ)10038161003816100381610038161003816kslip

10038161003816100381610038161003816]

times (S minus1

3tr SI)

(12)
















10 15 2520

r

(a) (b)

(c) (d)

(e) (f) (g)

5

4

3

2

1

0

x

y

z

x

y

z

x

y

z

y

zx

V(r)






z

x

y

(a) (b)

h

y

l

z

(c)


y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)











5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10 15 2520

r

(a) (b)

(c) (d)

(e) (f) (g)

5

4

3

2

1

0

x

y

z

x

y

z

x

y

z

y

zx

V(r)






z

x

y

(a) (b)

h

y

l

z

(c)


y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)











5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

x

y

(a) (b)

h

y

l

z

(c)


y

hs

Lp

z

x

D

(a)

y

l

l1

l2

x

(b)

y

z

x

(c)











5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















5 times 10minus5 00077 0015 0022 0025 0029 0035 0045 0055













yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










yz

xg

(a)

y

x

(b)

10

07

05

05

120601

10

120601

y z

x

g

lw

(c)

035

03

025

02

015

01

005

001 002

120601

lw (cm)

0

(d)

001 002

120601

08

06

04

02

0

lw (cm)

1

(e)

001 002

120601

lw (cm)

08

06

04

02

0

1

(f)

001 002

120601

lw (cm)

06

04

02

0

(g)





156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










156

142

129

114

1

Πlowast

y z

xlw

Lsp SP

(a)

1

10

100

001 002

Plowast

lw (cm)

(b)

1

10

100

001 002

Plowast

lw (cm)

(c)

1

2

3

4

5001 002

Plowast

lw (cm)

(d)

1

5

40001 002

Plowast

lw (cm)

(e)

275

137

00

V(m

s)

(f)













(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

10

08

06

04

02

0

0 5 10

120601a

lw (m)

(b)

y

x

45

1400

2500

(kPa

)

(c)


Periodic

z

yxg

t

(a)

1205792

1205791

z

x

ho

Ha

Hw

Hout

L1

L2

L

(b)










119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119906

119906lowast

=

119910+

119910+

le 5

minus115576 + 147869119910+

minus00527848119910+2

+ 0000655444119910+3

5 lt 119910+

le 30

ln (119910+

)

Κ+ 51 119910

+gt 30

(13)




381

286

190

095

000

lowast








186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










186

140

093

047

000

lowast

(a)

nb

x

z

yS1

S2S3

S4

L

(b)

0000 02

10

08

06

04

02

120601a

l (m)0400

10

08

06

04

02

00120601a

l (m)00 035

00

10

08

06

04

02

120601a

l (m)00 025

00

10

08

06

04

02

120601a

l (m)

(c)










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z

x

l

g

S1

S2S3

S5

S4

S6

(a)

56

42

28

14

00

lowast

(b)

00 02 04 00 025 05 00 025 05

lowast

lowast lowast

2

1

0

2

1

0

2

1

0

l (m)l (m)l (m)

(c)

00 01 02 03

5

4

3

2

1

0

lowast

00 01 02

5

4

3

2

1

0

lowast

00 01 02 03

5

4

3

2

1

0

lowast

l (m)l (m)l (m)

(d)







4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














4 Conclusions




Nomenclature




119862119904 = 21205873radic3(minus119877(1 + 11986211988701198772)Κ3)12





119875lowast






Re119861 = |V119894119861



43



119889119861




defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












defined as v119908 = 120588v119908120588


Greek Symbols


Πlowast



120601V = (43)1205871198991198611198773(1 + (43)120587119899119861119877

3)




References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Simulation of Cavitation Water Flows