Scaling of Tip Vortex Cavitation Inception Noise With a ... of Tip Vortex Cavitation Inception Noise...

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Chao-Tsung Hsiaoe-mail: [email protected]

Georges L. Chahinee-mail: [email protected]

Dynaflow, Inc.10621-J Iron Bridge Road, Jessup, MD 20794

Scaling of Tip Vortex CavitationInception Noise With a BubbleDynamics Model Accounting forNuclei Size DistributionThe acoustic pressure generated by cavitation inception in a tip vortex flow was simuin water containing a realistic bubble nuclei size distribution using a surface-averapressure (SAP) spherical bubble dynamics model. The flow field was obtained bReynolds-averaged Navier–Stokes computations for three geometrically similar scalesa finite-span elliptic hydrofoil. An ‘‘acoustic’’ criterion, which defines cavitation inceptias the flow condition at which the number of acoustical ‘‘peaks’’ above a pre-selepressure level exceeds a reference number per unit time, was applied to the threeIt was found that the scaling of cavitation inception depended on the reference v(pressure amplitude and number of peaks) selected. Scaling effects (i.e., deviationthe classicals i}Re

0.4! increase as the reference inception criteria become more string(lower threshold pressures and less number of peaks). Larger scales tend to deteccavitation inception events per unit time than obtained by classical scaling becaurelatively larger number of nuclei are excited by the tip vortex at the larger scale dusimultaneous increase of the nuclei capture area and of the size of the vortex coreaverage nuclei size in the nuclei distribution was also found to have an important imon cavitation inception number. Scaling effects (i.e., deviation from classical expressbecome more important as the average nuclei size decreases.@DOI: 10.1115/1.1852476#

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1 IntroductionScaling of the results of a propeller tip vortex cavitation ince

tion studies from laboratory to large scales has not always bvery successful. Aside from the problems associated with propscaling the flow field, existing scaling laws as derived or usedprevious studies, e.g.,@1–6#, lack the ingredients necessaryexplain sometimes major discrepancies between model andscale. One of the major aspects which has not been appropriincorporated in the scaling law is nuclei presence and nucleidistribution effects. Another issue which may cause scaling prlems is the means of detection of cavitation inception. Practicathe flow condition is considered to be at cavitation inception wheither an ‘‘acoustic’’ criterion or an ‘‘optical’’ criterion is met@7,8#. These two detection methods are known to provide differanswers in the most practical applications. Furthermore, for ptical reasons inception may be detected by one method at mscale and by another at full scale. To address this issue in a mconsistent manner for different scales, the present study consan ‘‘acoustic’’ criterion which determines the cavitation inceptioevent by counting the number of acoustical signal peaks thatceed a certain level in unit time.

To theoretically address the above issues in a practicalspherical bubble dynamics models were adopted in many stuin order to simulate the bubble dynamics and to predict tip vorcavitation inception@8–10#. In our previous studies@8,11#, animproved surface-averaged pressure~SAP! spherical bubble dy-namics model was developed and applied to predict single butrajectory, size variation and resulting acoustic signals. This mowas later shown to be much superior than the classical sphemodel through its comparison to a two-way fully thredimensional~3D! numerical model which includes bubble sha

Contributed by the Fluids Engineering Division for publication on the JOURNALOF FLUIDS ENGINEERING. Manuscript received by the Fluids Engineering DivisioSeptember 13, 2003; revised manuscript received August 13, 2004. Reviewducted by: S. Ceccio.

Copyright © 2Journal of Fluids Engineering

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deformation and the full interaction between the bubble andviscous flow field@11#. In the present study we incorporate thSAP spherical bubble dynamics model with a statistical nucdistribution in order to enable prediction of cavitation inceptiona practical liquid flow field with known nuclei size distributionThis is realized by randomly distributing the nuclei in space atime according to the given nuclei size distribution. Accordingprevious studies@12,13# the number of nuclei to use in the computation can be reduced by considering only the nuclei that pthrough a so-called ‘‘window of opportunity’’ and are capturedthe tip vortex.

In order to study scale effects in a simple vortex flow filed wconsider the tip vortex flows generated by a set of three geomcally similar elliptic hydrofoils. The flow fields are obtained bsteady-state Navier–Stokes computations which provide thelocity and pressure fields for the bubble dynamics computatioThe SAP spherical model is then used to track all nuclei relearandomly in time and space from the nuclei release area anrecord the acoustic signals generated by their dynamics andume oscillations.

2 Numerical Models

2.1 Navier–Stokes Computations. To best describe the tipvortex flow field around a finite-span hydrofoil, the Reynoldaveraged Navier–Stokes~RANS! equations with a turbulencemodel are solved. These have been shown to be successfaddressing tip vortex flows@14# and general propulsor flows@15,16#. The three-dimensional unsteady Reynolds-averagedcompressible continuity and Navier–Stokes equations in nomensional form and Cartesian tensor notations are written as

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005 by ASME JANUARY 2005, Vol. 127 Õ 55

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whereui5(u,v,w) are the Cartesian components of the velocxi5(x,y,z) are the Cartesian coordinates,p is the pressure,Re

5ru* L* /m is the Reynolds number,u* andL* are the charac-teristic velocity and length selected to be, respectively, thestream velocity,V` and root chord length,C0 . r is the liquiddensity, andm is its dynamic viscosity. The effective stress tenst i j is given by

t i j 51

ReF S ]ui

]xj1

]uj

]xiD2

2

3d i j

]uk

]xkG2uiuj (3)

whered i j is the Kronecker delta andui8uj8 is the Reynolds strestensor resulting from the Reynolds averaging scheme.

To numerically simulate the tip vortex flow around a finite-sphydrofoil, a body-fitted curvilinear grid is generated and Eqs.~1!and~2! are transformed into a general curvilinear coordinate stem. The transformation provides a computational domain thabetter suited for applying the spatial differencing scheme andboundary conditions. To solve the transformed equations, wethe three-dimensional incompressible Navier–Stokes flow solDFIUNCLE, derived from the code UNCLE developed at Misssippi State University. The DFIUNCLE code is based on theartificial-compressibility method@17# which a time derivative ofthe pressure multiplied by an artificial-compressibility factoradded to the continuity equation. As a consequence, a hypersystem of equations is formed and is solved using a time marcscheme in pseudo-time to reach a steady-state solution.

The numerical scheme in DFIUNCLE uses a finite volume for-mulation. First-order Euler implicit differencing is applied to thtime derivatives. The spatial differencing of the convective teruses the flux-difference splitting scheme based on Roe’s me@18# and van Leer’s MUSCL method@19# for obtaining the first-order and the third-order fluxes, respectively. A second-order ctral differencing is used for the viscous terms which are simplifiusing the thin-layer approximation. The flux Jacobians requirethe implicit scheme are obtained numerically. The resulting stem of algebraic equations is solved using the Discretized NewRelaxation method@20# in which symmetric block Gauss–Seidsub-iterations are performed before the solution is updated atNewton interaction. Ak2« turbulence model is used to model thReynolds stresses in Eq.~3!.

All boundary conditions in DFIUNCLE are imposed implicitly.Here, a free stream constant velocity and pressure conditiospecified at all far-field side boundaries. The method of characistic is applied at the inflow boundary with all three componeof velocities specified while a first-order extrapolation for all vaables is used at the outflow boundary. On the solid hydrofoil sface, a no-slip condition and a zero normal pressure gradientdition are used. At the hydrofoil root boundary, a plane symmecondition is specified.

2.2 Statistical Nuclei Distribution Model. In order to ad-dress a realistic liquid condition in which a liquid flow field contains a distribution of nuclei with different sizes, a statistical nclei distribution is used. We consider a liquid with a known nucsize density distribution function,n(R). n(R) is defined as thenumber of nuclei per cubic meter having radii in the range@R,R1dR#. This function has a unitm24 and is given by

n~R!5dN~R!

dR(4)

whereN(R) is the number of nuclei of radiusR in a unit volume.This function can be obtained from experimental measuremsuch as light scattering, cavitation susceptibility meter and AAcoustic Bubble Spectrometer® measurements@21# and can beexpressed as a discrete distribution ofM selected nuclei sizesThus, the total void fraction,a, in the liquid can be obtained by

56 Õ Vol. 127, JANUARY 2005

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whereNi is the discrete number of nuclei of radiusRi used in thecomputations. The position and timing of nuclei released inflow field are obtained using random distribution functions, aways ensuring that the local and overall void fraction satisfy tnuclei size distribution function.

From previous studies@12,13#, we know that only nuclei that‘‘enter’’ a given region or ‘‘window of opportunity’’ are actuallycaptured by the vortex and generate strong acoustic signTherefore, it is economical to consider only nuclei emitted frothis ‘‘window of opportunity.’’ This is similar to considering afictitious volume of cross area equal to the window area andlength equal toV`Dt, whereV` is the free stream velocity andDtis the total time of signal acquisition~see Fig. 1!.

2.3 Bubble Dynamics. The nuclei convected in the flowfield are treated using a spherical bubble dynamics model. Toso, we use the Rayleigh–Plesset equation modified to accouna slip velocity between the bubble and the host liquid, and fornonuniform pressure field along the bubble surface@10#. The re-sulting modified surface-averaged pressure~SAP! Rayleigh–Plesset equation can be written as:

Fig. 1 The location and size of a fictitious volume for ran-domly distributing the nuclei

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where Ris the time varying bubble radius,R0 is the initial orreference bubble radius,g is the surface tension parameter,pv isthe vapor pressure.pg0 is the initial or reference gas pressuinside the bubble, andk is the polytropic compression law constant. u is the liquid convection velocity andub is the bubbletravel velocity. Pencounter is the ambient pressure ‘‘seen’’ by thbubble during its travel. In the SAP methodPencounteris defined asthe average of the liquid pressures over the bubble surface@11#.

The bubble trajectory is obtained using the following motiequation@22#

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where the drag coefficientCD is given by an empirical equationsuch as that of Haberman and Morton@23#

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The pressure at a distancel from the bubble center generated bthe bubble dynamics is given by the expression

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Whenl @R, Eq. ~9! becomes the expression for the acoustic prsurepa of Fitzpatrick and Strasberg@24# after introduction of thedelayed timet8 due to a finite sound speed,c

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To determine the bubble motion and its volume variationRunge–Kutta fourth-order scheme is used to integrate Eqs.~6!and ~7! through time. The liquid velocity and pressures are otained directly from the RANS computations. The numericallution of the RANS equations, however, offers the solutionrectly only at the grid points. To obtain the values for aspecified location (x,y,z) on the bubble we need to interpolafrom the background grid. To do so, an interpolation stencil ainterpolation coefficients at any specified location are determiat each time step. We use a three-dimensional point-locascheme based on the fact that the coordinates (x,y,z) of thebubble location are uniquely represented relative to the eightner points of the background grid stencil by

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where

N15~12f!~12c!~12w!, N25f~12c!~12w!,

N35~12f!c~12w!, N45fc~12w!,(12)

N55~12f!~12c!w, N65f~12c!w,

N75~12f!cw, N85fcw.

f, c, w are the interpolation coefficients, and (xi ,yi ,zi) are thecoordinates of the eight corner points of a grid stencil in the baground grid. Equation~11! is solved using a Newton–Raphsomethod. For a bubble point to be inside the grid stencil requthat the correspondingf, c, w satisfy 0<f<1, 0<c<1, 0<w<1.

Journal of Fluids Engineering

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Once the interpolation stencil and interpolation coefficientsdetermined, the pressure and velocities can be obtained by ussimilar equation to Eq.~11!.

2.4 Computational Domain and Grid Generation. Tocompute the flow around the finite-span elliptic hydrofoil we geerated an H–H type grid with a total of 2.7 million grid pointswhich 19131013101 grid points were created in the streamwisspanwise and normal direction, respectively, and 81361 gridpoints were used to discretize the hydrofoil surface. The gridsubdivided into 12 blocks for a computational domain which hall far-field boundaries located six~6! chord lengths away fromthe hydrofoil surface~see Fig. 2!. Grid resolution was determinedaccording to previous numerical studies@14,25# in which exten-sive investigations of the grid resolution for the tip vortex floshowed that the minimum number of grid points needed for goresolution is at least 15 grid points across the vortex core. Hthe grid resolution for the tip vortex was optimized through rpeated computations and regridding to align grid clusteraround the tip vortex centerline. The final refined grid selectedthe results shown below had at least 16 grid points in the spandirection and 19 grid points in the crosswise direction within tvortex core. The first grid above the hydrofoil surface was locasuch thaty1'1 in order to properly apply the turbulence mode

3 Results

3.1 3D Steady-State Tip Vortex Flow. The selected finite-span elliptic foil has a NACA16020 cross section with an aspratio of 3~based on semispan!. The flow field at an angle of attack

Fig. 2 Computational domain and grid for the current study

JANUARY 2005, Vol. 127 Õ 57

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Fig. 3 Pressure coefficient variations along the NACA16020elliptic foil for three values of the Reynolds number
Table 1 Characteristics of the three NACA16020 foil used

Small scale Medium scale Large scale

C0 0.144 m 0.288 m 0.576 mV` 10 m/s 10 m/s 10 m/sRe 1.443106 2.883106 5.763106

2Cpmin 3.34 4.34 5.48


of 12 deg was computed for three foil sizes or three differReynolds numbers in order to study cavitation scaling effeThese correspond to the three scales shown in Table 1. In all tcases a steady-state solution was considered achieved wh¹

•V<131024. The resulting pressure coefficients along thevortex centerline are shown in Fig. 3. It is seen that the locatiof the minimum pressure for all three cases are very close tohydrofoil tip and are located atx/C050.085, 0.075, and 0.075The corresponding minimum pressure coefficients are showTable 1. If the cavitation inception number is assumed to2Cpmin , then these values correlate with the power formulatios i}Re

0.36.To validate the steady state computations an additional case

computed at an angle of attack equal to 10° andRe54.753106.The results were compared to the available experimental measments of@2# by considering the tangential and axial velocity components across the tip vortex core at two streamwise locationsseen in Fig. 4, the comparison indicates that the tip vortex flowwell predicted in the near-field region in which the pressure coficient along the vortex center reaches its minimum. Howevover-diffusion in vortex core size and over-dissipation in veloties are seen for the numerical solution further downstream ecially for the axial velocity component whose velocity profichanges from excess to deficit. Notice, however, that this occbeyond the region of interest here for bubble dynamics studIndeed, the bubble dynamics simulations show that the bubgrowth and collapse durations are relatively very short~see Fig. 5!and occur beforex/C050.1. In this region, our numerical solutio

Fig. 4 Comparison of tangential and axial velocity components across the tip vortex core at x ÕC0Ä0.1 and 0.3between present numerical result and experimental measurements „Fruman et al. 1992 …


Journal of Fluids Engin

Fig. 5 Example computation of bubble dynamics for bubble radius, encoun-tered pressure, and emitted acoustic pressure versus time during bubble cap-ture in the tip vortex

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agrees quite well with the experimental measurements. Therethe present Navier–Stokes computations are reliable for studthe bubble behavior in theCpmin region of interest.

3.2 Window of Opportunity. The ‘‘window of opportu-nity’’ can be determined by releasing nuclei upstream of the foand tracking their trajectories to see if they enter into the lpressure areas in the tip vortex flow. A release plane locatex/C0520.1 ahead of the hydrofoil tip (x/C050) was used. Nu-clei were released from this plane at various locations, trackand the minimum pressure they encountered is recorded acorresponding release point.

Initially, 300 nuclei of a given size were released from trelease plane. All properties are defined at 20°C. The cavitanumber was specified high enough such that the maximum grosize of nucleus was less than 10%. Figure 6 shows a contourof the minimum pressure coefficient encountered for each relelocation for different nuclei sizes in the small scale. The contoare blanked out for the release points where the nuclei collide wthe hydrofoil surface. It is seen that the size of the ‘‘windowopportunity’’ becomes smaller and its location shifts closer tohydrofoil surface of pressure side when the nuclei sizes decreThe contours of minimum encounter pressure coefficient for

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ferent scales with the same initial nuclei size are shown in FigIt is seen that the size of the ‘‘window of opportunity’’ increasas the scale increases. This implies that larger scales capturenuclei into the vortex for the same nuclei sizes and durationobservation time when compared to smaller scales.

3.3 Statistical Nuclei Size Distribution. Nuclei size distri-bution studies in water tunnels, lakes and oceans@26,27# show apower-law distribution for the number density distribution funtion, with n(R)'1/Rb, where the exponentb lies between 2.5and 4. In the present study we consider a nuclei size distriburanging from 10 to 100mm with a void fractiona'131026 asshown in Fig. 8. In order to consider a same bubble populationall scales, we have accounted for the fact that a bubblechange its radius in a static equilibrium fashion when the ambpressure is changed. Therefore, for the same scaled cavitnumber, initial nuclei sizes are reduced for the larger scales whthe ambient pressure would be larger. This is not a major chain the values since gas pressure inside the bubble varies likecube of the radius, while surface tension which is predominvaries like the inverse of the radius. This results in nuclei si

Fig. 6 Contours of the minimum pressure coefficient encountered at high cavitation number for differentnuclei size in the small foil scale


Fig. 7 Contours of the minimum pressure coefficient encountered at high cavitation number for R0Ä20 mm and for the mediumand large foil scale

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ranging from 10 to 100mm for the small scale, 9.2–92mm for themedium scale, and from 8.5 to 85mm for the large scale. Thescurves are used to generate the nuclei field.

With the void fraction and size distribution provided, the tonumber of nuclei released for each scale is then determined bon the length of signal acquisition time and the size of the relearea. To determine an appropriate statistically meaningful obvation time we tested two different signal acquisition timesDt50.2 and 1 s. Both cases were conducted for the small scacavitation numbers53.0. The number of nuclei released and tnumber of nuclei reaching critical~cavitating! condition versusnuclei size for these two cases are shown in Fig. 9. In this figa nucleus is considered to be a cavitation bubble whenPencounter,Pcr , where the critical pressure is defined as

Pcr5pv2~3k21!S 2g

3k D 3k/3k21

~pg0R03k!21/3k21 (13)

with k51.4. Comparison between these two cases shows thasmaller acquisition time only results in a slightly smaller probabity for cavitation. Therefore,Dt50.2 second is statistically sufficient and was used for the other tests. For the release windowconsider an area to be large enough to cover the ‘‘windowsopportunity’’ for all nuclei sizes released. Here, the size ofrelease area is specified as 7 mm35 mm, 14 mm310 mm, and 28

Fig. 8 Nuclei size number density distributions applied at thethree scales


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mm320 mm for the small, medium, and large scale, respectivAs a result, the number of nuclei in each population is 142, 5and 2272 for the three scales, respectively.

3.4 Scaling of Cavitation Inception Noise. As nuclei travelin the computational domain, the resulting acoustic pressurmonitored. The acoustic pressure was computed at a locationm away from the hydrofoil tip for all cases. A series of compu

Fig. 9 The number of nuclei released and the number of nucleireaching critical pressure „cavitating … versus nuclei size ob-tained at sÄ3.0 for two different acquisition times


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tions were conducted at different cavitation numbers for the thscales to obtain the acoustic signals for conditions above andlow cavitation inception. Figures 10–12 illustrate the acoustic snals for three different scales at three different cavitation numbHigh-level peaks of acoustic signals are clearly seen when

Fig. 10 The acoustic signals for the small scale at three differ-ent cavitation numbers


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cavitation number is near the cavitation inception number. Iseen that, as expected, for all scales the number of high-lpeaks increases as the cavitation number decreases. Howevelarger scale is more sensitive to cavitation number changes s

Fig. 11 The acoustic signals for the medium scale at threedifferent cavitation numbers


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the number of peaks increases much faster than for the smscale as the cavitation number decreases. Figure 13 showresulting frequency spectra for the acoustic signals shown in F

Fig. 12 The acoustic signals for the large scale at three differ-ent cavitation numbers


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10–12. A peak in the frequency range 30–40 kHz is seen ascales. The amplitude of this peak increases as the cavitation nber decreases.

Fig. 13 Amplitude spectra for all three scales at three differentcavitation numbers


Journal of Fluids Engin

Fig. 14 Number of pressure peaks versus cavitation number deduced at twocriteria of acoustic level for the three scales considered

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Based on the results shown in Figs. 10–12, we can defincavitation inception number based on the number of acoussignal peaks per unit time that exceed a certain level. To dedthe cavitation inception number based on this criterion, a curvethe number of pressure peaks higher than a give acoustic prelevel is created for each cavitation number and for the thscales. Figure 14 shows such curves with two acoustic preslevels, 10 and 40 Pa, are chosen for each scale. Given a selcriterion based on the number of peaks and acoustic preslevel, one can determine the cavitation inception number frFig. 14.

The deduced cavitation inception numbers of the three scfor the criteria: 10 peaks/s over 10 Pa and 50 peaks/s over 40are shown in Table 2. The deduced cavitation numbers2Cpmin are fitted with the classical power formulas i}Re

g , andthe fitted values ofg are also shown in Table 2. It is seen thdifferent criteria for defining the cavitation inception event clead to different cavitation inception numbers and different scing laws. The scaling effect due to the nuclei can be demonstrby comparing the deduced inception number with2Cpmin . Theresults in Table 2 show that cavitation inception scaling deviamore from 2Cpmin when the reference inception criterion bcomes less stringent~higher reference pressure amplitude alarger number of peaks!. Furthermore, the predicted value ofg iscloser to the classical value~g50.4!, as the reference inceptiocriterion becomes less stringent. This agrees with many exp

Table 2 Cavitation inception numbers obtained from the nu-merical study using various criteria, and power law fit deducedfrom these results

Numerical computedvalues fors i Re

g curve fit

Smallscale

Mediumscale

Largescale g

Square ofcorrelationcoefficient

2Cpmin 3.34 4.34 5.48 0.357 0.99910 peaks/sover 10 Pa

3.28 4.33 5.47 0.369 0.998

50 peaks/sover 40 Pa

3.12 4.28 5.44 0.401 0.994

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mental studies usually established in laboratory conditions whbackground noise and detection techniques lead to high valuethe pressure amplitude for inception detection.

3.5 Nuclei Size Distribution Effect. To illustrate how dif-ferent nuclei size distributions influence the prediction of cavition inception, a much finer nuclei size distribution ranging fromto 10mm is tested. In the computations the total number of nucreleased in each case was kept the same. This results in a msmaller void fraction (a'131029) than in the previous caseFigure 15 shows the acoustical signal obtained ats53.0 and thenumber of nuclei cavitating for each prescribed nuclei sizeshown in Fig. 16. It is seen that, as expected, the number of pis dramatically reduced for the smaller nuclei size range whcomparing the results to those of the larger nuclei size range. Tis because as shown in Fig. 16 near inception the nuclei conuting to the high-level peaks are only the larger bubble sizes.

A series of computations were also conducted at different ctation numbers for the small and the large scale with the sma

Fig. 15 The acoustic signals for the small scale at sÄ3.0 us-ing the smaller nuclei size range „1–10 mm…


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nuclei size range. Two acoustic pressure levels, 10 and 40 Paalso selected to determine the number of pressure peaks forcavitation number. The resulting curves of number of presspeaks versus cavitation numbers for the small and large scaleshown in Fig. 17. The deduced cavitation inception numbersshown in Table 3. By comparing with Table 2 we can see veimportant differences for the small scale when the cavitationception criteria are less stringent. Also, scaling effects and detion from classical formula due to nuclei size distribution are seto significantly increase when nuclei sizes~or void fraction!decreases.

Fig. 16 The number of nuclei released and the number of nu-clei cavitating versus nuclei size obtained for the 1–10 mmsmall nuclei size distribution at sÄ3.0

Fig. 17 Number of pressure peaks versus cavitation numberdeduced at two criteria of acoustic level for the small and largescales considered

Table 3 Cavitation inception numbers obtained from the nu-merical study using various criteria, and power law fit deducedfrom these results. For smaller void fraction „aÉ1Ã10À9

….

Numerical computedvalues fors i Re

g curve fitgSmall Scale Large Scale

2Cpmin 3.34 5.48 0.35710 peaks/s over 10 Pa 3.20 5.40 0.37750 peaks/s over 40 Pa 2.0 5.18 0.687


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4 ConclusionsThe study of the behavior of a realistic distribution of nuclei

the tip vortex flow field of a NACA16020 foil at three scales henabled observation of several effects:

1. Comparison of the size of the bubble capture area or ‘‘wdow of opportunity’’ at the various scales shows that the larscale results in more cavitation events by allowing more nuper unit time to be captured by the tip vortex;

2. the numerical results show that different criteria for definithe cavitation inception can lead to a different cavitation inceptnumbers as well as different scaling laws. By comparing the pdicted cavitation inception number with2Cpmin , we found thatscaling effects~i.e., deviation from2Cpmin) due to nuclei in-crease as the reference inception criteria become less strin~higher reference pressure amplitude and larger number of pe!;

3. the predicted value ofg in the power formula (s i}Reg) is

closer to the classical value~g50.4!, as the reference inceptiocriterion becomes less stringent;

4. the range of nuclei sizes was shown to have an imporeffect on the prediction of cavitation inception. Differences btween predicted cavitation inception number and2Cpmin increaseas nuclei sizes~or void fractions! decrease. This implies that scaing effects due to nuclei size distribution are stronger whenwater contains only small nuclei~or for low void fraction!.

AcknowledgmentsThis work was conducted at Dynaflow, Inc.~www.dynaflow-

inc.com! and has been supported by the Naval Surface WarCarderock Division and the Office of Naval Research. This sport of Dr. Ki-Han Kim, ONR, and Dr. Young Shen, NSWCCD,greatly appreciated.

References@1# McCormick, B. W., 1962, ‘‘On Cavitation Produced by a Vortex Trailing Fro

a Lifting Surface,’’ ASME J. Basic Eng.,84, pp. 369–379.@2# Fruman, D. H., Dugue, C., Pauchel, A., and Cerrutti, P., 1992, ‘‘Tip Vort

Roll-Up and Cavitation,’’Eighteenth Symposium on Naval Hydrodynamic,Seoul, Korea.

@3# Arndt, R. E., and Dugue, C., 1992, ‘‘Recent Advances in Tip Vortex CavitatResearch,’’Proc. International Symposium on Propulsors Cavitation, Ham-burg, Germany, pp. 142–149.

@4# Farrell, K. J., and Billet, M. L., 1994, ‘‘A Correlation of Leakage VorteCavitation in Axial-Flow Pumps,’’ ASME J. Fluids Eng.,116, pp. 551–557.

@5# Maines, B., and Arndt, R., 1997, ‘‘Tip Vortex Formation and CavitationASME J. Fluids Eng.,119, pp. 413–419.

@6# Arndt, R. E. A., 2002, ‘‘Cavitation in Vertical Flows,’’Annu. Rev. Fluid Mech.34, pp. 143–175.

@7# Copalan, S., Liu, H. L., and Katz, J., 2000, ‘‘On the Flow Structure, TLeakage Cavitation Inception and Associated Noise,’’ 23rd SymposiumNaval Hydrodynamics.

@8# Hsiao, C.-T., Chahine, G. L., and Liu, H. L., 2003, ‘‘Scaling Effects on Prdiction of Cavitation Inception in a Line Vortex Flow,’’ ASME J. Fluids Eng125, pp. 53–60.

@9# Ligneul, P., and Latorre, R., 1989, ‘‘Study on the Capture and Noise of Sphcal Nuclei in the Presence of the Tip Vortex of Hydrofoils and PropellerAcustica,68, pp. 1–14.

@10# Hsiao, C.-T., and Pauley, L. L., 1999, ‘‘Study of Tip Vortex Cavitation Incetion Using Navier–Stokes Computation and Bubble Dynamics Model,’’ASMJ. Fluids Eng.,121, pp. 198–204.

@11# Hsiao, C.-T., and Chahine, G. L., 2004, ‘‘Prediction of Vortex Cavitation Iception Using Coupled Spherical and Nonspherical Models and UnRAComputations,’’ J. Marine Sci. Technol.,8, No. 3, pp. 99–108.

@12# Arndt, R. E. A., and Maines, B. H., 2000, ‘‘Nucleation and Bubble Dynamin Vortical Flows,’’ ASME J. Fluids Eng.,122, pp. 488–493.

@13# Hsiao, C.-T., and Pauley, L. L., 1999, ‘‘Study of Tip Vortex Cavitation Incetion Using Navier–Stokes Computation and Bubble Dynamics Model,’’ASMJ. Fluids Eng.,121, pp. 198–204.

@14# Hsiao, C.-T., and Pauley, L. L., 1998, ‘‘Numerical Study of the Steady-StTip Vortex Flow Over a Finite-Span Hydrofoil,’’ ASME J. Fluids Eng.,120,pp. 345–349.

@15# Hsiao, C.-T., and Pauley, L. L., 1999, ‘‘Numerical Calculation of Tip VorteFlow Generated by a Marine Propeller,’’ ASME J. Fluids Eng.,121, pp. 638–645.

@16# Taylor, L. K., Pankajakshan, R., Jiang, M., Sheng, C., Briley, W. R., WhitfieD. L., Davoudzadeh, F., Boger, D. A., Gibeling, H. J., Gorski, J., Haussli


e

u

a

e

G

s

of2,

d,’’

95,

i-

H., Coleman, R., and Buley, G., 1998, ‘‘Large-Scale Simulations for Manvering Submarines and Propulsors,’’ AIAA Paper 98-2930.

@17# Chorin, A. J., 1967, ‘‘A Numerical Method for Solving Incompressible ViscoFlow Problems,’’ J. Comput. Phys.,2, pp. 12–26.

@18# Roe, P. L., 1981, ‘‘Approximate Riemann Solvers, Parameter Vectors,Difference Schemes,’’ J. Comput. Phys.,43, pp. 357–372.

@19# van Leer, B., 1979, ‘‘Towards the Ultimate Conservative Difference SchemA Second Order Sequel to Godunov’s Method,’’ J. Comput. Phys.,32, pp.101–136.

@20# Vanden, K., and Whitfield, D. L., 1993, ‘‘Direct and Iterative Algorithms fothe Three-Dimensional Euler Equations,’’ AIAA-93-3378.

@21# Chahine, G. L., Kalumuck, K. M., Cheng, L.-Y., and Frederick, G., 200‘‘Validation of Bubble Distribution Measurements of the ABS Acoustic BubbSpectrometer With High Speed Video Photography,’’4th International Sympo-sium on Cavitation, California Institute of Technology, Pasadena, CA.

@22# Johnson, V. E., and Hsieh, T., 1966, ‘‘The Influence of the Trajectories of


u-

s

nd

V.

r

1,le

as

Nuclei on Cavitation Inception,’’Sixth Symposium on Naval Hydrodynamic,pp. 163–179.

@23# Haberman, W. L., and Morton, R. K., 1953, ‘‘An Experimental Investigationthe Drag and Shape of Air Bubbles Rising in Various Liquids,’’ Report 80DTMB.

@24# Fitzpatrick, N., and Strasberg, M., 1958, ‘‘Hydrodynamic Sources of Soun2nd Symposium on Naval Hydrodynamics, pp. 201–205.

@25# Dacles-Mariani, J., Zilliac, G. G., Chow, J. S., and Bradshaw, P., 19‘‘Numerical/Experimental Study of a Wingtip Vortex in the Near Field,’’AIAAJ., 33, No. 9, pp. 1561–1568.

@26# Billet, M. L., 1984, ‘‘Cavitation Nuclei Measurements,’’International Sympo-sium on Cavitation Inception, FED-Vol. 16, pp. 33–42.

@27# Franklin, R. E., 1992, ‘‘A Note on the Radius Distribution Function for Mcrobubbles of Gas in Water,’’ASME Cavitation and Mutliphase Flow Forum,FED-Vol. 135, pp. 77–85.


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