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Mathematical Basis and Validation of the Full Cavitation Model

Ashok K.Singhal

e-mail:aks@https://www.sodocs.net/doc/ea6268811.html, Mahesh M.Athavale

Huiying Li

Yu Jiang CFD Research Corporation,

Huntsville,AL35805Mathematical Basis and Validation of the Full Cavitation Model

Cavitating?ows entail phase change and hence very large and steep density variations in the low pressure regions.These are also very sensitive to:(a)the formation and transport of vapor bubbles,(b)the turbulent?uctuations of pressure and velocity,and(c)the magnitude of noncondensible gases,which are dissolved or ingested in the operating liquid.The presented cavitation model accounts for all these?rst-order effects,and thus is named as the‘‘full cavitation model.’’The phase-change rate expressions are derived from a reduced form of Rayleigh-Plesset equation for bubble dynamics.These rates de-pend upon local?ow conditions(pressure,velocities,turbulence)as well as?uid proper-ties(saturation pressure,densities,and surface tension).The rate expressions employ two empirical constants,which have been calibrated with experimental data covering a very wide range of?ow conditions,and do not require adjustments for different problems.The model has been implemented in an advanced,commercial,general-purpose CFD code, CFD-ACE?.Final validation results are presented for?ows over hydrofoils,submerged cylindrical bodies,and sharp-edged ori?ces.Suggestions for possible extensions of the model implementation,e.g.,to nonisothermal?ows,for ingestion and mixing of noncon-densible gases,and for predictions of noise and surface damage are outlined.

?DOI:10.1115/1.1486223

?

Introduction

The capability for multidimensional simulation of cavitating ?ows is of critical importance for ef?cient design and perfor-mance of many engineering devices.Some examples are:indus-trial turbomachinery,turbopumps in rocket propulsion systems, hydrofoils,marine propellers,fuel injectors,hydrostatic bearings, and mechanical heart valves.In most cases,cavitation is an unde-sirable phenomenon,causing signi?cant degradation in the perfor-mance, e.g.,reduced?ow rates,lower pressure increases in pumps,load asymmetry and vibrations and noise.Multidimen-sional simulations can enable a designer to eliminate,reduce or shift the cavitation regions.The objective of the present study is to develop a practical cavitation model capable of predicting major performance parameters.Its extensions to prediction of cavitation related surface damage,which affects the life of the equipment, may be considered in future.

Numerical simulation of cavitating?ows poses unique chal-lenges,both in modeling of the physics and in developing robust numerical methodology.The major dif?culty arises due to the large density changes associated with phase change.For example, the ratio of liquid to vapor density for water at room temperature is over40,000.Furthermore,the location,extent and type of cavi-tation are strongly dependent on the pressure?eld,which in turn is in?uenced by the?ow geometry and conditions.Therefore,in a practical modeling approach,a priori prescription?or assumption?of the location and/or size of cavitation region should not be re-quired.Likewise,the phase change correlations should have mini-mum essential empiricism so that diverse applications can be simulated without adjusting any constants or functions.

Over the last several decades,considerable effort from both experimental and analytical fronts has been devoted to under-standing cavitation.For example,References?1–12?include some recent reviews as well as attempts on modeling and application of cavitation.Unfortunately,all past models,including the two de-veloped by the principal author and his colleagues?10,11?,have had limited success,primarily due to:?a?the lack of robustness of numerical algorithms,and?b?lack of generality of the correlations or approach used.As a result,no cavitation model was routinely used for practical CFD-based design optimization studies.

The Full Cavitation Model described here meets all the above-mentioned requirements and is already beginning to get routinely used in industry for water and oil pumps,inducers,impellers,and fuel injection systems.

Description of the Full Cavitation Model

The basic approach consists of using the standard viscous?ow ?Navier-Stokes?equations for variable?uid density and a conven-tional turbulence model?e.g.,k-?model?.The?uid density is a function of vapor mass fraction f,which is computed by solving a transport equation coupled with the mass and momentum conser-vation equations.The?-f relationship is:

1

??

f

?v?

1?f

?l(1) and the vapor volume fraction?is deduced from f as:

??f

?

?v(2) The vapor mass fraction,f,is governed by a transport equation:?

?t??f??????V

?f???????f??R

e

?R c(3)

The source terms R e and R c denote vapor generation?evapora-tion?and condensation rates,and can be functions of:?ow param-eters?pressure,?ow characteristic velocity?and?uid properties ?liquid and vapor phase densities,saturation pressure,and liquid-vapor surface tension?.

The above formulation employs a homogenous?ow approach, also known as Equal-Velocity-Equal-Temperature?EVET?ap-

Contributed by the Fluids Engineering Division for publication in the J OURNAL OF F LUIDS E NGINEERING.Manuscript received by the Fluids Engineering Division April20,2001;revised manuscript received February28,2002.Associate Editor:J. Katz.

proach.For the objective of a practical and general model of cavitating?ows,this is a fairly good simpli?cation because of the following reasons:

1In most engineering devices,the low-pressure regions,where cavitation occurs,are also the regions of relatively high velocities. In such high-velocity regions,the velocity slips between the liquid and vapor phases are rather small.

2Most often,the generated vapor takes the form of small bubbles.While such?ows can be characterized by a more rigor-ous two-?uid approach,which allows for velocity slip between the liquid and vapor phases,the computed?ow?elds strongly depend upon the physical models used for the computation of local bubble sizes and interface drag forces.Unfortunately,there are no general or reliable physical models for these parameters,and therefore the extra computational effort in the two-?uid approach is of little practical value.

The present model focuses on the use of simple rational formula-tions for phase change rates?R e and R c?.

Bubble Dynamics Consideration.We assume that,in most engineering situations,there are plenty of nuclei for the inception of cavitation.Thus,our primary focus is on proper account of bubble growth and collapse.In a?owing liquid with zero velocity slip between the?uid and bubbles,the bubble dynamics equation can be derived from the generalized Rayleigh-Plesset equation as ?1,12?:

R B D2R B

Dt2

?

3

2?D R B Dt

?2??P B?P?l??4?l R B R˙B?2S?l R B(4)

This equation provides a physical approach to introduce the ef-fects of bubble dynamics into the cavitation model.In fact,it can be considered to be an equation for void propagation and,hence, mixture density.

To obtain an expression of the net phase change rate,the two-phase continuity equations are written as follows:Liquid phase:?

?t??1????l??????1????l V

????R(5)

Vapor phase:

?

?t???v???????v V

???R(6) Mixture:

?

?t????????V

???0(7)

where R is the net phase change rate?(R e?R c),and?is the mixture https://www.sodocs.net/doc/ea6268811.html,bining Eqs.?5?–?7?yields a relation between the mixture density and void fraction?:

D?Dt ????l??v?

D?

Dt

(8)

The vapor volume fraction?can be related to the bubble number density,‘‘n’’and radius of bubble R B as

??n 4

3

?R B3(9)

substituting Eq.?9?into Eq.?8?we obtain

D?Dt ????l??v??n4??1/3?3??2/3

D R B

Dt

(10)

Using the Rayleigh-Plesset Equation,Eq.?4?,without the viscous damping and surface tension terms?the2nd and3rd term on r.h.s.?,and combining Eqs.?5?,?6?,?8?,and?10?,the expression for the net phase change rate R is?nally obtained as:

R??n4??1/3?3??2/3

?v?l

??23

?P B?P?l??23R B D2R B Dt2?1/2

(11) Using Eqs.?3?and?11?,and ignoring the second-order derivative of R B?important mainly during initial bubble acceleration?,we get the following simpli?ed equation for vapor transport:?

?t??f??????f V???n4??

1/3?3??2/3

?v?l

??23

?P B?P?l??1/2

(12) where the right side of the equation represents the vapor genera-tion or‘‘evaporation’’rate.Though we expect the bubble collapse process to be different from that of the bubble growth,as a?rst approximation,Eq.?12?is also used to model the collapse?con-densation?,when P?P B,by using the absolute value of the pres-sure difference and treating the right side as a sink term.The local far-?eld pressure P is taken to be the same as the cell center pressure.The bubble pressure P B is equal to the saturation vapor pressure in the absence of dissolved gas,mass transport and vis-cous damping,i.e.,P B?P v.

Equation?12?is referred to here as the Reduced Bubble Dynam-ics Formulation.

Phase Change Rates.In Eq.?12?,all terms except‘‘n’’are either known constants or dependent variables.In the absence of a general model for estimation of the number density,the Phase Change Rate expression is rewritten in terms of bubble radius, R B,as follows:

R e?

3?

R B

??v?l??23P v?P?l?1/2(13)

For simplicity,the typical bubble size R B is taken to be the same as the limiting?maximum possible?bubble size.Then,R B is de-termined by the balance between aerodynamic drag and surface tension forces.A commonly used correlation in the nuclear indus-try is?13?:

R B?

0.061We?

2?l v rel2

(14)

For bubbly?ow regime,V rel is generally fairly small,e.g.,5–10% of liquid velocity.By using various limiting arguments,e.g.,R B →0as?→0,and the fact that the per unit volume phase change rates should be proportional to the volume fractions of the donor phase,the following expressions for vapor generation/ condensation rates are obtained in terms of the vapor mass frac-tion f:

R e?C e

V ch

??l?v

?2

3

P v?P

?l?1/2?1?f?(15)

R c?C c

V ch

??l?l

?2

3

P?P v

?l?1/2f(16) Here C e and C c are two empirical coef?cients and V ch is a char-acteristic velocity,which re?ects the effect of the local relative velocity between liquid and vapor.These relations are based on the following assumptions:

1.In the bubble?ow regime,the phase change rate is propor-

tional to V rel2;however,in most practical two-phase?ow conditions,the dependence on velocity is found/assumed to be linear.

2.The relative velocity between the liquid and vapor phase is

of the order of1to10%of the mean velocity.In most turbulent?ows,the local turbulent velocity?uctuations are also of this order.Therefore,as a?rst pragmatic approxima-tion,V ch in Eqs.?15?and?16?can be expressed as the square root of local turbulent kinetic energy?k.

The Effect of Turbulence.Several experimental investiga-tions have shown signi?cant effect of turbulence on cavitating ?ows?e.g.,references?3,14??.Also,Singhal et al.?11?reported a numerical model,using a probability density function?PDF?ap-proach for accounting the effects of turbulent pressure?uctua-tions.This approach required:?a?estimation of the local values of the turbulent pressure?uctuations as?15?:

P turb

??0.39?k(17) and?b?computations of time-averaged phase-change rates by in-tegration of instantaneous rates in conjunction with assumed PDF for pressure variation with time.In the present model,this treat-ment has been simpli?ed by simply raising the phase-change threshold pressure value as:

P v??P sat?P turb?/2?(18) This practice has been found to be much simpler,robust and al-most as good as the more rigorous practice of ref.?11?.

Effect of Noncondensable Gases…NCG….In most engineer-ing equipment,the operating liquid contains a?nite amount of non-condensable gas?NCG?in dissolved state,or due to leakage or by aeration.Even a small amount?e.g.,10ppm?of NCG can have signi?cant effects on the performance of the machinery ?16,17?.The primary effect is due to the expansion of gas at low pressures which can lead to signi?cant values of local gas volume fraction,and thus have considerable impact on density,velocity and pressure distributions.The secondary effect can be via in-creases in the phase-change threshold pressure.This has been ne-glected due to lack of a general correlation.

Final Form of Full Cavitation Model.The working?uid is assumed to be a mixture of liquid,liquid vapor and NCG.The calculation of the mixture density?Eq.?1??is modi?ed as:

1??f v

?v?

f g

?g?

1?f v?f g

?l(19)

Non-condensable gas density?g is calculated as:

?g?WP

RT

(20)

V olume fractions of NCG and liquid are modi?ed as:

?g?f g ?

?g;(21)

?l?1??v??g(22) Finally,with the consideration of the NCG effect,and also using ?k to replace V ch,Eqs.?15?and?16?are rewritten as:

R e?C e ?k

??l?v

?2

3

P v?P

?l?1/2?1?f v?f g?(23)

R c?C e ?k

??l?l

?2

3

P?P v

??1/2f v(24)

where the phase-change threshold pressure P v is estimated from Eqs.?17?and?18?.The recommended values of the empirical constants C e and C c are0.02and0.01,respectively.The basis for these values is described below.

Model Implementation

The full cavitation model has been implemented into an ad-vanced,general purpose,commercial CFD code,CFD-ACE??18?.The relevant features of CFD-ACE?include:unstructured/

adaptive/hybrid grids,a?nite volume,pressure-based formulation for incompressible and compressible?ows,a variety of turbulence models,multi-media heat transfer,steady-state and time-accurate solution,arbitrary sliding interface treatment,and moving grids for deforming/sliding domains.

Some points to be noted about the current cavitation model are: 1.The cavitation model can be applied to any geometric sys-

tem?3D,2D planar,or2D axisymmetric?;all grid cell types ?quad,tri,hex,tet,prism,poly?and arbitrary interfaces are supported;

2.Concurrent use of the turbulence,grid deformation and/or

structures solution modules are fully supported;

3.Flow is assumed isothermal and?uid properties are taken as

constant at a given temperature for the entire?ow domain.

Due to this assumption,the cavitation module currently is decoupled from heat transfer and radiation modules.

4.Noncondensible gas mass fraction f g is assumed to be con-

stant in the?ow?eld.An appropriate value of f g,estimated based on the operating liquid and conditions,is prescribed as

a part of the model input.

The simpli?cations listed in items3and4above can be removed in future as outlined at the end of the paper.

Determination of Empirical Constants C e and C c.The two constants,C e and C c,have been determined by performing sev-eral series of computations for sharp-edged ori?ce and hydrofoil ?ows.Both of these?ows have excellent data,covering a wide range of operating conditions.Numerical computations were ini-tially performed assuming C e?C c,and nominal values were found to be in the range0.01–0.1.The assessment criteria in-cluded:

?a?Comparison of computed mass?ow rates,discharge coef-?cients,and?ow pattern?location and extent of cavitation

zone?;and

?b?Special attention to the calculated minimum pressures,and their sensitivity to the assumed values of coef?cients. The primary objectives of this exercise were to:completely elimi-nate negative pressure regions,obtain minimum pressures close to saturation pressures and obtain minimal sensitivity to pressure variations.It was found that to reproduce experimental trends, C c?C e.Several other postulations for slowing down the conden-sation?vapor destruction?process were also tried.None of these were found to be very general or robust.Therefore,C c values were varied in the range of C e to0.1C e.A large number of combinations of C e and C c values were tried for several ori?ce ?ow conditions?upstream total pressure?2,3,5,50,and500 bar?and for selected hydrofoil?ow cases?representative low and high?ow rates for two angles of attack,i.e.,for leading and mid-chord cavitation?.After many hundreds of permutations and com-binations,the most satisfactory values were found to be

C e?0.02and C c?0.01(25) These values were then used for many other problems,including ?ows past submerged cylindrical bodies,inducers,impellers and axial pumps.All these simulations produced satisfactory results, i.e.,good convergence rates,no negative pressures,and reason-able comparison with available data and/or?ow patterns.There-fore,the present set of values,C e?0.02and C c?0.01seems quite satisfactory for general use.

Validation of Full Cavitation Model

This section presents some of the validation results for?ow over a hydrofoil,over a submerged cylindrical body,and?ow in a sharp-edged ori?ce.In each case experimental data is available for wide range of conditions.Good agreement has been obtained in all cases without adjusting any coef?cient values.

In all the simulations presented below,the working?uid was water at300K,with liquid and vapor densities of1000and 0.02558kg/m3,saturation pressure of3540Pa and surface tension

??0.0717N/m.A second-order upwind scheme was used to dis-cretize the convective ?uxes,and turbulence was treated using the standard k-?model

1Cavitating Flow Over a Hydrofoil.Effects of leading edge and mid-chord cavitation on the hydrodynamic forces on a hydrofoil were experimentally investigated by Shen and Dimot-akis ?19?.A NACA66?MOD ?airfoil section with camber ratio of 0.02,mean line of 0.8and thickness ratio of 0.09was used.A 2-D working section of the hydrofoil was mounted in a water tunnel.Static pressures on hydrofoil surface were measured at different angles of attack and Reynolds numbers.The non-dimensional pa-rameters of interest were:

Re ?

?l U ?C

?1

,??

P ??P v

12?l U ?

2,

C p ?

P ?P ?12

?l U ?2(26)

A two-block grid consisting of 30?130cells/block ?7800cells ?is shown in Fig.1.Two other grids consisting of 4250and 14,700cells were also used to check grid sensitivity of solutions.Calcu-lated C p values for the two higher cell count grids were found to differ less than 1%.Velocities,turbulence quantities and NCG mass fraction were speci?ed at the left ?inlet ?boundary and an exit pressure was speci?ed at the right ?exit ?boundary.The ?ow rate was varied to change the ?ow Reynolds number and the angle of attack was changed by airfoil section rotation.The NCG level was set to f g ?1ppm.

1.1Leading Edge Cavitation.Simulations were performed at Re ?2?106and an angle of attack of 4deg;under these con-ditions,the cavitation is con?ned to the front of the hydrofoil.The exit pressure was varied to yield ?values of 1.76,1.0,0.91and 0.84.Calculated Cp values on hydrofoil top surface for two of the four cases are shown in Figs.2and 3together with experimental data,and good correlation is seen.A typical vapor mass fraction distribution is shown in Fig.4,which shows the cavitation zone on the hydrofoil surface.

1.2Mid-Chord Cavitation.Simulations were performed at Re ?3?106and an angle of attack of 1deg.Cavitation inception was seen at ??0.415;simulations were performed at ??0.43,0.38and 0.34.Calculated and experimental plots of Cp on the hydrofoil top surface for two of the three cases are shown in Figs.5and 6.A cavitation zone exists in the mid-chord region

and Fig.3Pressure variation on the suction side of a hydrofoil;??

0.91

Fig.4Computed total volume fraction distributions at cavita-tion number ?

0.91

Fig.5Pressure variation on the suction side of a hydrofoil;??

0.43

Fig.1Computational domain and grid,and grid distribution near the hydrofoil for ??4

deg

Fig.2Pressure variation on the suction side of a hydrofoil;??1.76

extends towards the trailing edge with decreasing ?.A view of the

cavitation zone for ??0.34is shown in Fig.7.

2Front Cavitation Over Submerged Cylindrical Bodies The present cavitation model was applied and assessed for cavi-tating ?ows over cylindrical submerged bodies with different types of head shapes.Extensive experimental data are reported by Rouse and McNown ?20?.The experiments were conducted in a water tunnel with cylindrical test objects 0.025m in diameter and 0.3048m in length ?1.0in.and 12in.?.The ?ow was characterized using the parameters de?ned as:

Re ?

?1U ?d

?1

,??

P ??P v

0.5?1U ?

2

,

C p ?

P ?P ?

0.5?1U ?

2

(27)

2-D axisymmetric computational grids were built for these problems.All simulations were performed at a ?xed inlet U ??10m/s and exit pressure levels were varied to achieve the proper inlet pressure P ?.The NCG level was set to f g ?1ppm for the deaerated water used in the https://www.sodocs.net/doc/ea6268811.html,putations were performed on bodies with hemispherical,45deg conical,and blunt heads.Details of the body with a 45deg conical head are shown here.

The computational grid shown in Fig.8has two blocks with 61?39and 124?39cells ?total of 7200cells ?.Two other grids with 3375and 12,700cells were also used to check grid-independence of the solutions,and the C p results for the two larger grids again differed by less than 1%.Results were obtained at ??0.3,0.4,0.5,0.7,1.0and 1.3.Calculated and experimental distributions of Cp for four of the ?values are shown in https://www.sodocs.net/doc/ea6268811.html,puted results match well with the experimental data.Pressure coef?cients along the conical head,inside the cavitation zone and the recovery zone show very good agreement.Results for the other two cases ?hemispherical and blunt heads ?also showed similar agreements.

3Cavitating Flow in a Sharp-Edged Ori?ce.Pressure-driven ?ow in a sharp-edged ori?ce is typically encountered in fuel-injectors,and has received a lot of attention.This is a very challenging ?ow computation,because the pressure differentials involved can be very high ?up to 2500bar ?,which drive a ?ow through a very small ori?ce,and the problem tests the robustness of the numerical and physical models.

Nurick ?21?has published extensive experimental data for cavi-tation in a sharp-edged circular ori?ce.Geometrical parameters of the ori?ce are D /d ?2.88and L /d ?5,where D ,d ,and L are inlet diameter,ori?ce diameter,and ori?ce length,respectively.Experi-ments were done with a ?xed exit pressure,P b ?0.95bar,and the upstream total pressure,P 0,was varied to generate different ?ow rates.High ?ow velocities near the ori?ce entrance generate a zone of very low pressure right after the constriction,where the ?ow cavitates.This reduces the ?ow rate ?choking type phenom-enon ?and can lead to surface damage downstream of the ori?ce.The discharge coef?cient for the ori?ce,C d ,is of interest and the cavitation number ?characterizes the ?ow:

??

P o ?P v

P o ?P b

,

C d ?

m ˙actual m ˙ideal ?m ˙actual

A o ?2?1?P o ?P b ?

(28)C d ?C c ??

(29)

where C c ,the contraction coef?cient,was evaluated at 0.62.The ?ow is 2-D axisymmetric,and a 2-block structured grid with 2800cells ?20?20cells in the ?rst block and 20?120cells in the second ?was employed to discretize the geometry with grid clustering around the sharp-edged corner ?Fig.10?.The other grids used for grid sensitivity check had 1300and 5400cells.The predicted mass ?ow rates from the 2800and 5400cell grids var-ied by less than 1%.A large number of cases were computed,with the inlet total pressure ranging from 1.9to 2500bars;the inlet pressures and the corresponding cavitation numbers are listed in Table 1.NCG level f g was set to 15ppm.

Figure 11shows the comparison between the predicted dis-charge coef?cients C d with Nurick’s correlation,Eq.?29?.The calculated values are in very close agreement with the experimen-tal data.The model correctly predicts the inception of cavitation at ??1.7.The discharge coef?cient Cd is constant in the non-cavitating ?ow (??1.7),while it clearly shows a square-root dependence on ?in the cavitation regime.The cavitation easily handles the ?ows at very low ?values,where the upstream pres-sures are very high,over 2000bar.Simulation of ?ows with such high pressure-ratios is a dif?cult task even for single-phase ?ow;but there were no dif?culties in treating this ?ow with the full cavitation model,indicating the robustness of the numerical procedure.

Solution and Convergence Characteristics.In all the vali-dation cases presented above,the computed minimum pressures are fairly close to the saturation pressures,and all error residuals drop by at least 4orders of magnitude.Figures 12?a ?and 12?b ?show typical

convergence plots for the hydrofoil and ori?ce cases

Fig.6Pressure variation on the

suction side of a hydrofoil;??0.34

Fig.7Volume

fractions for ??0.34,showing mid-chord cavitation

Fig.8Computational domain and grid,and grid distribution near a 45-degree conical fore-body

respectively.Convergence for the ori?ce case shows a plateau initially while the initial condition errors in the ?ow are convected out after which the solution converges rapidly.

Applications of the Full Cavitation Model

While the results presented here focused on the validation as-pects of the cavitation model,this model has also been used suc-cessfully on a variety of different problems for research as well as commercial applications.These include:

1.Cavitation in diesel fuel injectors with complex multi-port geometries and time-varying geometries and pressure load-ing

2.Cavitation in rocket turbomachinery, e.g.,cavitation in rocket inducers and impellers has been analyzed,and results validated against experiments.This work is being published separately ?22,23?.

3.Automotive Vane and Gear pump oil pump design optimiza-tion.

4.Cavitation in automotive thermostatic valves.

In all of these applications,the basic set of equations and con-stants described in previous sections have been found to generate accurate solutions with robust convergence

characteristics.

Fig.10Computational grid used for the sharp-edged

ori?ce

Fig.9Comparison between computed and measured Cp over a fore-body with a 45-degree conical head

Table 1Total inlet pressure and cavitation number P o ?105(Pa)

1.9

2.0 2.5

3.0 3.75 5.0?

1.0004 1.001 1.009 1.019 1.101 1.226P o ?105(Pa)

105010050010002500?

1.327

1.446

1.590

1.704

1.871

1.963

Fig.11Ori?ce cavitation:comparison of cavitation model pre-dictions with Nurick’s correlation

Potential Extensions and Collaborations

The current limitations in the implementation of Full Cavitation Model in CFD-ACE ?include isothermal ?ow assumption,and a ?xed,uniform mass concentration of NCG.Both of these assump-tions can be easily relaxed by solving appropriate additional trans-port equation,and modifying corresponding parameters like P sat and f g .The present model provides many useful ?ow character-istics such as local gradients of pressure,density and volume frac-tions,general location and approximate extents of vapor regions,and approximate values of turbulence intensity.Approximate bubble size variations can also be deduced if desired.Since the model seems to be reasonably accurate for predictions of perfor-mance parameters over a wide range of conditions,it is very likely that the detailed ?ow characteristics are also in the realistic ranges.Such details can provide a sound foundation for the de-velopment of correlations for cavitation induced noise levels.A preliminary module based on integration of Lighthill equation us-ing a Kirchoff-Ffowacs-Williams-Hawking ?KWFH ?solver has already been developed and used on vane pump noise predictions.Likewise,appropriate additional equations and modules can be incorporated for the predictions of approximate location and mag-nitude of cavitation induced surface damage.

Because of the intricate inter-coupling of various physical mod-els and numerical solution procedures and computer software data structures,the model extensions mentioned above will be best performed by universities and/or interested R&D groups working in close collaboration with the authors of the present paper.Summary and Conclusions

A comprehensive model for cavitating ?ows has been devel-oped and incorporated into an advanced CFD code for perfor-mance predictions of engineering equipment.This CFD code and cavitation model was applied to a number of validation and dem-onstration problems to verify the accuracy of the model and to assess the convergence performance on dif?cult engineering prob-lems.Presented here were validation results for high-speed ?ow cavitation on hydrofoil and submerged cylindrical bodies,and in both cases the predictions from the cavitation model were in very good agreement with the experimental data.The model was also applied to cavitating ?ow through an ori?ce and computed results compared well with experimental data,even for very severe ?ow conditions involving very high pressure differentials across the ori?ce.The full cavitation model,coupled with CFD-ACE ?code,can be applied to a wide range of problems,and be a valuable prediction tool for design veri?cation and optimization.Collabo-rative efforts are encouraged to extend this model,e.g.,to include thermal effects and the prediction of cavitation damage.

Acknowledgments

This work was funded in part under NSF SBIR Grant No.DMI-9801239;this support is gratefully acknowledged.The authors would like to thank:

1.Dr.N.Vaidya and Ram Avva for their contributions in the development of two earlier models,which laid the basis for the current model;and

2.Mr.Dennis Gibson of Caterpillar,Inc.,for his introduction and initial support for the development of cavitation model ?even though it has taken over eight years to meet the chal-lenge of developing a practical capability ?.

Nomenclature

C ?hydrofoil chord length

C e ,C c ?constants in vapor generation condensation rate

expression

D ,d ?diameter

f v ,f

g ?vapor,gas mass fraction

k ?turbulence kinetic energy m actual ?actual ori?ce mass ?ow m ideal ?ideal ori?ce mass ?ow

n ?bubble number density P ?pressure

P v ?vapor pressure

P turb

??turbulence pressure ?uctuation Q ??ow rate

R ?universal gas constant

R e ,R c ?vapor generation,condensation rates

Re ?Reynolds number R B ?bubble radius S ?surface tension T ?temperature

U ??freestream velocity V ??uid velocity vector

V ch ??ow characteristic velocity

W ?molecular weight of non-condensible gas We ?Weber number Greek

?

?angle of attack

?e ,?v ,?g ?liquid,vapor,gas volume fraction ?,?e ,?v

?density of mixture,liquid,vapor ??surface tension

?,??dynamic,kinematic viscosities ??cavitation

number

Fig.12Convergence characteristics for two of the validation cases presented above;…a …hydrofoil,…b …ori?ce

References

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