Research ArticleThe Correction and Evaluation of Cavitation Model consideringthe Thermodynamic Effect

Wei Li ,1 Yongfei Yang ,1 Wei-dong Shi ,2 Xiaofan Zhao,1 andWeiqiang Li1

1Research Center of Fluid Machinery Engineering and Technology, Jiangsu University, Zhenjiang 212013, China2College of Mechanical Engineering, Nantong University, Nantong 226019, China

Correspondence should be addressed to Wei Li; lwjiangda@ujs.edu.cn and Wei-dong Shi; wdshi2012@126.com

Received 5 October 2017; Revised 22 January 2018; Accepted 12 February 2018; Published 2 April 2018

Academic Editor: Matteo Aureli

Copyright © 2018 Wei Li et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Two cavitation models with thermodynamic effects were established based on the Rayleigh-Plesset equation to predict accuratelythe cavitation characteristics in the high-temperature fluid.The evaporation and the condensation coefficient of the cavitationmodelwere corrected. The cavitation flow of NACA0015 airfoil was calculated using the modified cavitation model, where the influenceof the thermodynamic effects of airfoil cavitation was analyzed.The result showed that the pressure coefficient distribution and thebubble volume fraction simulated have the same tendency of Zwart-Gerber-Belamri model’s result. According to the experimentaldata, the two models provide more accurate results. At the room temperature, the values of d𝑝V/d𝑇 obtained by the two improvedmodels are approximately equal.Thedifference between the twomodels’ results increases graduallywith the temperature increasing,but it is still small. The simulation results are consistent with the experimental data when the evaporation coefficient is 10 and 1.When the evaporation coefficient is 1, the bubble growth is inhibited, the volume fraction becomes lower, and the cavitation areabecomes flat. As the temperature increases, the cavitation area and the bubble volume fraction at airfoil front edge become larger,showing that the temperature plays a “catalytic” role in the cavitation process.

1. Introduction

Cavitation belongs to multiphase flow category. It is a phasetransition phenomenon when the internal fluid pressure ofthe pump is lower than the saturation pressure [1–3]. Atpresent, two main types of numerical models (homogeneousand the nonhomogeneous model) can solve the multiphaseflow. The thermal dynamic mechanism plays an importantrole in the cavitation process; it has already attracted theattention of researchers. In the theoretical analysis aspect, therule of bubble growth in water was deduced [4]. A methodto model the thermal dynamical process of cavitation wasproposed using thermal conduction in bubble surface. Low-temperature medium was applied to compare the tempera-ture drops in cavitation flows where the simulation resultsand experimental data are basically the same [5]. Consid-ering the thermal effect, the cavitation phenomenon in aninducer was studied. As the temperature was increasing, thelength of the cavity is reduced while cavitation performancewas improved [6, 7]. The influence of the dimensionless

thermodynamic coefficient on the critical cavitation numberin inducer and its cavitation states under different temper-ature was studied. When the thermodynamic coefficient islower than 0.54, the critical cavitation number decreasesalong with the increase of thermodynamic coefficient; andvice versa: when it is higher than 0.54, the critical cavitationnumber is not related to the thermodynamic coefficient [8].

The cavitation experiments at different temperatures in aninducer andNACA0015, respectively, were carried out.Whenthe temperature was increased under the same cavitationnumber, the pressure pulsation was reduced and the cavitythickness and length on the hydrofoil were increased [9,10]. Rotating cavitation at three different temperatures andthe distribution of cavity lengths and temperatures wereinvestigated [11]. The cavitation flow characteristics in fluori-nated ketone of NACA0015 hydrofoil at different speeds anddifferent attack angles were illustrated [12]. For the cavitationin low temperature, the dynamic characteristics of liquidcarbon dioxide cavitation near the critical point were studied.The vapor kinetic equation considering the thermodynamic

HindawiMathematical Problems in EngineeringVolume 2018, Article ID 7217513, 11 pageshttps://doi.org/10.1155/2018/7217513

2 Mathematical Problems in Engineering

effect and quantitatively predicting the cavitation of liquidcarbon dioxide near the critical point was also established.It was found that the thermodynamic boundary has animportant effect on the bubble dynamics [13]. A cavitationmodel was set up to capture the heat transfer process duringthe collapse of the cavity bubbles, which provides a theoreticalbasis for calculating the local temperature, pressure, and heattransfer parameters during the collapse of the cavitation [14].Also, a cavitation model considering the thermodynamiceffects based on the Zwart cavitation model was set up andapplied for simulating the cavitation state of the hydrothermalmanifold at different temperatures where the experimentswere carried out for calibrating parameters [15]. A heattransfer model of a bubble based on the Fourier heat transferlawwas used to calculate the effect of thermodynamics on theNACA0015 airfoil; then the experiments were carried out toverify the accuracy [16]. Based on the Plesset-Zwick formulaand the effect of temperature on bubble growth, a cavitationmodel was created to introduce the concept of the thermalboundary of the bubbles, numerical simulation about cavi-tation in NACA0015 hydrofoil, and the subminiature pump[17]. Also, a cavitation model based on Singhal full cavitationmodel including thermals effect was investigated. The modelwas verified by tested pressure coefficient −𝐶𝑝 of NACA66hydrofoil in hot water when 𝜎 = 0.91 [18].The vapor pressurebased on Singhal cavitation model and the thermal effecton cavitation under high-temperature conditions was studied[19]. A secondary development of technology was applied tointroduce modified Kunz and Merkle cavitation model andphysical parameters of nitrogen vary with the temperatureof the flow field. The cavitation flow in symmetrical rotorsand hydrofoils nitrogen was simulated [20, 21]. The mixedfluid model based on Rayleigh-Plesset fluid equations andthermals theory was applied to analyze the rotating cavitationphenomena in the two-dimensional cascades of inducer.Briefly, the research on cavitation under thermals effect hasmade some achievements, but the accuracy and applicabilityof the model have not been improved yet, and a largedeviation still exists between the numerical and experimentalresults [22].

In this study, based on the Rayleigh-Plesset equation,the influence of thermodynamic effects is introduced inthe evaporation and condensation coefficient of the cavi-tation model. Two cavitation models considering thermo-dynamic effects are proposed and were added to the cal-culation by commercial software CFX secondary develop-ment. The cavitation flow of the hydrofoil airfoil and theNACA0015 airfoil was calculated by adapting the Zwart-Gerber-Belamri cavitationmodel and themodified cavitationmodel. By contrast and analysis between the simulation resultand experimental data, the applicability of the modifiedcavitation model in high-temperature cavitation flow wasvalidated.

2. Geometric Model and Meshing

The geometry simulated was consistent with the model inthe experimentmentioned before [9].The specific parametersof the NACA0015 airfoil are chord length 115mm, wingspan

120

mm

140mm100 mm 115 mm

Figure 1: Parameters of experimental test.

80mm, and adjustable attack angle of 4∘, 5∘, 6∘, and 8∘. Duringthe test, three pressure holes were evenly arranged at 100mmin front and 140mmat the rear of the airfoil. At the same time,10 pressure holes were arranged on the airfoil suction surfacewhile two pressure holes were arranged on the airfoil pressuresurface, as shown in Figure 1.

In the numerical calculation, the average 𝑦+ of theairfoil suction plane and pressure surface are 21.2 and 18.5,respectively.Thewhole, the head, and the tail grids are shownin Figure 2. To exclude the effect of the grid number on thesimulation accuracy, grid independence study is carried outat the beginning of the numerical investigation. The flowfield around the NACA0015 hydrofoil was simulated usingmeshes with five different grid numbers, and the lift forcecalculated from different mesh is shown in Figure 3. Whenthe grid number is larger than 1 million the lift force is almostunchanged, and the error is still below 0.3%. Consequently,the grid number for the later investigation is selected as 1million.

The boundary conditions are consistent with those inthe experiment. The inlet boundary is set as velocity inlet,and the outlet boundary adapted static pressure under forcedoutflow. The airfoil suction and pressure surface are set asnonslipping solid wall and the roughness of the airfoil surfacewas ignored. The convergence accuracy was set to 1𝑒 − 5,where the cavitation number and the pressure coefficient aredefined as follows:

𝜎 = 𝑝∞ − 𝑝V(1/2) 𝜌V∞2 ,𝑐𝑝 = 𝑝 − 𝑝∞(1/2) 𝜌V∞2 ,

(1)

where 𝑝∞ is the reference pressure, V∞ is the referencevelocity, and 𝑝V is the vaporization pressure of water.

3. Governing Equationsand Simulation Method

3.1. Governing Equations. The homogeneous equilibriumflow model assumes that the medium is a mixture with thesame velocity and pressure field. The governing equationsare mainly set according to the continuity equation, themomentum equation, and the energy equation.

𝜕𝜌𝑚𝜕𝑡 + 𝜕 (𝜌𝑚𝑢𝑗)𝜕𝑥𝑗 = 0, (2)

Mathematical Problems in Engineering 3

Figure 2: Mesh of NACA0015 hydrofoil.

480000 720000 960000 1200000 1440000216

217

218

219

220

Lift

forc

e (N

)

Grid number

Figure 3: Lift force with different grid number.

𝜕 (𝜌𝑚𝑢𝑖)𝜕𝑡 + 𝜕 (𝜌𝑚𝑢𝑖𝑢𝑗)𝜕𝑥𝑗= − 𝜕𝑝𝜕𝑥𝑖+ 𝜕𝜕𝑥𝑗 [(𝜇 + 𝜇𝑡) ( 𝜕𝑢𝑖𝜕𝑥𝑗 +

𝜕𝑢𝑗𝜕𝑥𝑖 − 23 𝜕𝑢𝑖𝜕𝑥𝑗 𝛿𝑖𝑗)] ,(3)

𝜕𝜕𝑡 [𝜌𝑚𝑐𝑝𝑇] + 𝜕𝜕𝑥𝑗 [𝜌𝑚𝑢𝑗𝑐𝑝𝑇]= 𝜕𝜕𝑥𝑗 [( 𝜇

Pr𝐿+ 𝜇𝑡Pr𝑡

) 𝜕ℎ𝜕𝑥𝑗]− { 𝜕𝜕𝑡 [𝜌𝑚 (𝑓V𝐿)] + 𝜕𝜕𝑥𝑗 [𝜌𝑚𝑢𝑗 (𝑓V𝐿)]} ,

(4)

where 𝜌𝑚 = 𝜌𝑙𝛼𝑙 + 𝜌V(1 − 𝛼𝑙), 𝑡 is the time, 𝑢 is the mixingmedium velocity, 𝜌𝑚 is the density of the mixed medium, 𝑝 isthe pressure of the mixed medium, 𝜇 is the laminar viscositycoefficient, 𝜇𝑡 is the turbulent viscosity coefficient, 𝑐𝑝 is thespecific heat capacity, 𝑇 is the temperature, 𝑓V is the massfraction of water vapor, 𝐿 is the latent heat of vaporization,𝛼𝑙 is the liquid volume fraction, and ℎ is the enthalpy of thesubstance.

3.2. The Selection of Turbulence Model. For the turbulenceclosure to solve 𝜇𝑡 in (3) and (4), a turbulent model is nec-essarily solved in combination with the governing equations.According to the experimental investigations, turbulence isfound to have a significant effect on the cavitation intensity[23, 24]. Thus, to make a better prediction for cavitatingflows, it is important to simulate the flow field accurately anddeal with the turbulent terms appropriately. The commonturbulent models usually approximate all the length scales,which may create higher eddy viscosity field. Due to thenumerical dissipation, some large periodic turbulence flowshave the possibility of being ignored [25]. Consequently,filter functions are proposed to limit the eddy viscosity to areasonable degree [26, 27].

In the current research, the Rayleigh-Plesset cavitationmodel is solved in combination with four common turbu-lence models (standard k-𝜀 turbulence model, standard k-𝜔 turbulence model, RNG k-𝜀 turbulence model, and SSTturbulence model) and also two different filter turbulentmodels (k-𝜀 FBM turbulence model and RNG k-𝜀 FBMturbulence model). Simulation results are compared to selectthe suitable turbulentmodel for predicting the cavitation flowfield. NACA0015 airfoil at 5∘ attack angle and 1.5 cavitationnumber at 25∘C is simulated.

As shown in Figure 4, the pressure coefficient distribu-tions of the airfoil obtained by numerical calculation usingtypical turbulence model and the filter turbulence modelshave the same tendency. There are two mutations between𝑥/𝑐 = 0.25 and 0.4. When 𝑥/𝑐 = 0.4∼1.0, the pressurecoefficients simulated by different turbulence models are notso different and well agree with the experimental data. In the

4 Mathematical Problems in Engineering

0.2 0.4 0.6 0.8 1.0

−0.5

0.0

0.5

1.0

1.5

x/c

−C

p

SSTk-２．＇ k- ＆＂－

２．＇ k-k- ＆＂－

k-Experiment

0.0

Figure 4: −𝐶𝑝 of NACA0015 acquired with different turbulencemodels.

airfoil head cavitation area, where 𝑥/𝑐 = 0∼0.25, there is a bigdifference between the predicted value and tested values.Thecavity length simulated using the filtered turbulence modelis shorter than the typical turbulence model and the cavitylength of the k-𝜔 turbulence model is the longest one amongthe typical turbulence models, and the cavity length of theSST turbulence model is the shortest one. By comparing thesimulation results using different turbulence models withthe experiment data, it can be judged that the calculatedresult using RNG k-𝜀 turbulence model is the closest to theexperimental value. In the position where chord length ratio𝑥/𝑐 = 0.25∼0.4, the value of −𝐶𝑝 calculated using the filtermodels is lower than the result of the k-𝜀model and RNG k-𝜀model. It is indicated that the simulation result is sensitive tothe filter function of turbulent viscosity 𝜇𝑡 and more robustfilter function or more suitable filter scale is necessary forthe simulation of turbulent cavitating flows. The result of thecommon turbulencemodel without filter functions is similar,while the k-𝜀 based models are closer to the experimentaldata than the k-𝜔 based models, and RNG k-𝜀model is moreaccurate than standard k-𝜀models as it replaced the constant𝐶𝜀1 with more detailed function 𝐶𝜀RNG. Therefore, the RNGk-𝜀 turbulence model is chosen in this study.

3.3.TheModified CavitationModels. The common cavitationmodels are mostly based on the bubble growth equation pro-posed byRayleigh-Plesset (R-P) [28].TheR-Pmodel providesthe basic equations for controlling the vapor generation andcondensation and describes the growth of bubbles in theliquid; the details are as follows:

𝑅𝐵 d2𝑅𝐵d𝑡2 + 32 (d𝑅𝐵d𝑡 )2 + 2𝑆𝑅𝐵 + 4𝜐𝑅 d𝑅d𝑡

= 𝑝V (𝑇𝐵) − 𝑝∞𝜌𝑙 ,(5)

where 𝑅𝐵 is the bubble radius, 𝑝V is the bubble pressure, 𝑝 isthe liquid pressure, 𝑆 is the surface tension coefficient, and 𝜐is liquid kinematic viscosity.

Assuming that the bubble growth process is a sphericalmodel, the bubble volume change rate is as follows:

d𝑉𝐵d𝑡 = d

d𝑡 (43𝜋𝑅3𝐵) = 4𝜋𝑅2𝐵√23 𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 . (6)

The mass change rate of the bubble is

d𝑚𝐵d𝑡 = 𝜌V d𝑉𝐵d𝑡 = 4𝜋𝑅2𝐵𝜌V√23 𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 . (7)

If there are 𝑁𝐵 bubbles per unit volume, the volumefraction 𝛼V can be expressed as

𝛼V = 𝑉𝐵𝑁𝐵 = 43𝜋𝑅3𝐵𝑁𝐵. (8)

Then the total interphase mass transfer rate (evaporationprocess) per unit is

𝑚− = 𝑁𝐵 d𝑚𝐵d𝑡 = 3𝛼V𝜌V𝑅𝐵 d𝑅d𝑡 . (9)

Similarly, the condensation coefficient is deduced asfollows:

𝑚+ = 𝐶vap3𝛼nuc (1 − 𝛼V) 𝜌V𝑅𝐵 d𝑅

d𝑡 . (10)

The thermodynamic parameter is a function of tempera-ture, theTaylor series of vaporization pressure𝑝V is expanded,and the higher order terms (including the quadratic term),the surface tension term, and the viscosity term in the R-P(5) are ignored. The reduced model is as follows:

d𝑅d𝑡 = √23 [𝐼 + 𝐼𝐼]

= √ 23 [𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 + d𝑝Vd𝑇 (𝑇 − 𝑇∞)𝜌𝑙 ],

(11)

where 𝐼 is the pressure difference term and II is the thermo-dynamic term.

If the thermodynamic effect is not taken into account, thepressure difference between liquid phase and the vapor phasedrives the interphase mass transfer; then 𝐼𝐼 = 0; that is, 𝑇 =𝑇∞. The bubble radius growth rate without thermodynamiceffect is reduced as

d𝑅d𝑡 = √ 23 𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 . (12)

However, in the variable temperature medium, the latentheat of vaporization absorbed during the vaporization pro-cess forces the surrounding liquid to form a temperaturegradient outside the bubble, which results in a temperaturedifference between the bubble and the outer liquid. The

Mathematical Problems in Engineering 5

temperature difference will affect the growth of the bubble.Therefore, in order to predict the cavitation phenomenonmore accurately, the existing cavitation model must beproperly modified by adding the source term considering thethermodynamic effect.

Method 1. According to the function of pressure and temper-ature, the relationship of d𝑝V/d𝑇 is given directly [29]:

d𝑝Vd𝑇 = 6∑

𝑛=0

𝑛𝑝𝑛𝑇𝑛−1. (13)

Then dR/dt was obtained:

d𝑅d𝑡 = √23 [𝐼 + 𝐼𝐼]= √ 23 [𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 + ( 6∑

𝑛=0

𝑛𝑝𝑛𝑇𝑛−1) (𝑇 − 𝑇∞)𝜌𝑙 ].(14)

In the formula, 𝑝0 = 611.2, 𝑝1 = 44.692, 𝑝2 =1.3777, 𝑝3 = 0.0297, 𝑝4 = 2𝑒−3, 𝑝5 = 3𝑒−6, 𝑝6 = −2𝑒−9.Method 2. d𝑝V/𝑑𝑇 was obtained by using the Clapeyronformula:

Δ𝑃VΔ𝑇 = 𝜌𝑙𝜌V(𝜌𝑙 − 𝜌V) 𝑇𝐿. (15)

Then dR/dt was obtained:

d𝑅d𝑡= √ 23 𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 + (23 𝜌𝑙𝜌V(𝜌𝑙 − 𝜌V) 𝑇∞)𝐿(𝑇 − 𝑇∞)𝜌𝑙 . (16)

Equations (14) and (16) are put into (9) and (10), respec-tively, to obtain two kinds of the cavitation models con-sidering thermodynamic effects (Thermal Cavitation Model,TCM). These two cavitation models are named TCM1 andTCM2, respectively. The specific expression is as follows:

𝑚+ = 𝐶vap3𝛼nuc (1 − 𝛼V) 𝜌V𝑅𝐵

⋅ √ 23 𝑝V (𝑇∞) − 𝑝∞𝜌𝑙 + 𝑅thermal,𝑚− = 𝐶cond

3𝛼V𝜌V𝑅𝐵 √23 𝑝∞ − 𝑝V (𝑇∞)𝜌𝑙 + 𝑅thermal,(17)

where𝑅thermal is the factors of thermodynamic effects and thetwo forms of 𝑅thermal are (2/3)(∑6𝑛=0 𝑛𝑝𝑛𝑇𝑛−1)((𝑇 − 𝑇∞)/𝜌𝑙)and (2/3)(𝜌𝑙𝜌V/(𝜌𝑙 − 𝜌V)𝑇∞)𝐿((𝑇 − 𝑇∞)/𝜌𝑙), respectively.

The results show that turbulent kinetic energy may havean important effect on cavitation. The above models use the

Table 1: Comparison between TCM1 and TCM2.

Temperature (K) d𝑝V/d𝑇 (kg/(K m s2))TCM1 TCM2

298 186.1 189.1323 591.51 612.2343 1281.61 1398.8373 3376.5 3622.3

modified method to reduce the effect of turbulent kineticenergy on cavitation [19]. The formula is as follows:

𝑝V (𝑇𝑙) = (𝑝sat + 12 × 0.39𝜌𝑚𝑘) , (18)

where 𝑝V(𝑇𝑙) is the local saturated vapor pressure and 𝑘 is thelocal turbulent kinetic energy.

4. Validation and Modification of CavitationModel considering Thermodynamic Effects

4.1. Comparison and Analysis of TCM1 and TCM2. The sur-face pressure coefficient distributions of the NACA0015airfoil predicted by the three cavitation models (the Defaultcavitation model, TCM1, and TCM2) at 25∘C are shown inFigure 5. It can be seen that, with different cavitation models,the position of important turning point on the suction surfaceof the airfoil is almost the same as that in the experimentalresult, which indicates that the thermodynamic effect ofcavitation at this temperature is not prominent.Therefore, thenumerical simulation of cavitation under room temperaturecan ignore the effect of thermodynamic effects. However,on the airfoil leading edge, the numerical simulation resultspredicted by the three cavitation models are quite differentfrom the test experiment data, and the error of the pressuredistribution coefficient of the suction surface at the chordlength ratio 0.1 is about 15%. The pressure coefficient distri-bution and bubble volume fraction distribution map are verysimilar in two kinds of modified cavitation models. Also, theregion and the location where bubble occurred in these twomodels are almost the same.

Comparing (13) and (15), it is found that the maindifference between TCM1 and TCM2 is in the definitionof d𝑝V/d𝑇. TCM1 is obtained based on the derivative ofthe continuous function 𝑝V(𝑇), and the function describingrelationship between vaporization pressure and the temper-ature can be deduced by polynomial fitting. In TCM2, theterm d𝑝V/d𝑇 is obtained based on the Clapeyron equationapproximation.

The specific values d𝑝V/d𝑇 of the two improved cavitationmodels TCM1 and TCM2 at different temperatures areshown in Table 1. The values of d𝑝V/d𝑇 obtained in roomtemperature of two improved models are basically the same.With the increase of temperature, the difference between thetwo shows a gradually increasing trend, but the overall gap isnot large.

It can be seen from the above analysis that the improvedthermodynamic cavitation models TCM1 and TCM2 are of

6 Mathematical Problems in Engineering

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

x/c

ExperimentDefault

TCM1TCM2

−0.5

−C

p

(a) Comparison of NACA0015 −𝐶𝑝

Default

TCM1

TCM2

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.2

0.3

0.1

0.0

(b) Comparison of vapor volume fractions

Figure 5: Comparison of −𝐶𝑝 and vapor volume fractions among default cavitation model and TCM.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

x/c

ExperimentTCM2

−C

p

−0.5

Figure 6: Comparison between exp. and TCM2.

the same essence and only different in the expression form.The effect of cavitation thermodynamics at room temperatureis negligible. When the temperature is under 100∘C, thedifference in d𝑝V/d𝑇 values between TCM1 and TCM2 issmall. Therefore, at 70∘C, the cavitation model TCM2 isselected for verification.

4.2. Modification and Validation of TCM2. Figure 6 showsthe comparison chart of pressure coefficient distribution onNACA0015 airfoil surface at 70∘C of modified cavitationmodel TCM2 and the test value. It can be seen from thefigure that the simulated values in the airfoil rear area arebetter identified with the test value, but in the airfoil headarea where chord ratio ranges from 0.1 to 0.4 the differenceis large. Therefore, the cavitation model TCM2 still needs tobe corrected.

0.0 0.2 0.4 0.6 0.8 1.0x/c

ExperimentCvap = 1 Ccond = 0.0002

Cvap =10 Ccond = 0.002Cvap = 50 Ccond = 0.01

0.0

0.5

1.0

1.5

−0.5

−C

p

Figure 7: Comparison with different coefficients.

Evaporation and condensation coefficients both have aneffect on cavitation, so the evaporation and condensationcoefficients need to be corrected after the cavitation modelis modified. The default evaporation and condensation coef-ficients are 50 and 0.01. According to experience, the othertwo sets of coefficients are defined as 10, 0.002 and 1, 0.0002,respectively. Figure 7 shows the comparison chart of theairfoil pressure distribution coefficient of cavitation modelTCM2 at 70∘C under three evaporation and condensationcoefficients. It can be seen from the figure that when theevaporation coefficient is 10 and 1, the simulation valueagrees well with the experimental value. With the decrease ofevaporation and condensation coefficient, the pressure coef-ficient distribution curve becomes smooth. However, whenthe evaporation coefficient is 1, several turning points of someimportant chord ratios are not shown, the bubble growth issuppressed, the bubble volume fraction becomes lower, and

Mathematical Problems in Engineering 7

16.91ms−1

Cavitation zoneYplus

203.49

180.88

158.27

135.66

113.05

90.44

67.83

45.22

22.61

0.00

XY

Z

Figure 8: Sketch of domain mesh.

25∘C

50∘C

Figure 9: Hydrofoil bulk interface mass transfer rate at differenttemperatures.

the shape of the bubble area becomes flat. Therefore, thecavitation model TCM2 is chosen in the following cavitationsimulation, and the evaporation and condensation coefficientare set as 10 and 0.002, respectively, which is the same as in[17].

5. Cavitation Thermodynamic Effectsin Hydrofoil

Based on the cavitation model TCM2 with evaporation andcondensation coefficients of 10 and 0.002, respectively, thecavitation conditions in hydrofoil airfoil and thermodynamiceffects in NACA0015 airfoil were simulated, respectively.

5.1.The Cavitation in Hydrofoil at 25∘C and 50∘C. The hydro-foil airfoil is first simulated, and its boundary conditions areset as follows: the inlet velocity is 16.91m/s and the outletstatic pressure is 51957 Pa. Airfoil working conditions andgrid are shown in Figure 8; the airfoil grid average y+ = 89.2.

The interphase mass exchange rate distribution diagramof airfoils at 25∘C and 50∘C is shown in Figure 9. Theinterphase mass exchange rate (vapor to liquid) reflects thespeed of interphase mass exchange, which indirectly reflectsthe generation and development of bubbles. It can be seenfrom Figure 9 that, at different temperatures, the regionswhere bubble is generated and disappears are similar toeach other. The bubble is mainly generated downstreamof the middle of airfoil chord on the suction surface anddisappears after it reaches the high-pressure zone. However,

compared to the interphase mass exchange rate distributionat 25∘C, the bubble generation region at 50∘C is wider, andthe area of the vaporization region and the bubble volumefraction are larger. At 50∘C, the process of the bubble tobecome smaller or disappear is more obvious and the areaof collapse region becomes larger, which both will causeimportant consequence. It can be seen that the effect ofthermodynamics on the cavitation process of hydrofoil airfoilis obvious, and the higher the temperature, the more obviousthe development trend of cavitation.

5.2. The Cavitation in NACA0015 Hydrofoil at 25∘C, 50∘C,and 70∘C. Numerical simulations were carried out at 25∘C,50∘C, and 70∘C, respectively, based on the NACA0015 airfoil,at 8-degree attack angle and 2.5 cavitation number. Thedistribution of bubble volume fraction and the isosurfaceof 0.5 is shown in Figure 10. It can be seen from Figure 10that the degree of cavitation in NACA0015 airfoil at differenttemperatures is different and cavitation region occurs mainlyin the wing head where chord ratio ranges from 0.01 to 0.25.With the increase of temperature, the bubble volume fractionin the cavitation region becomes higher and the cavitationthermodynamic effect becomes more obvious, showing thatthe temperature plays a “catalytic” role in the cavitationprocess.

Figure 11 shows the pressure coefficient distribution ofairfoil surface at 25∘C, 50∘C, and 70∘C with 8-degree attackangle. It can be seen from the figure that the positions ofthe important turning points on three curves are similar. Thepressure distribution curve can be divided into three regions.The first region is the cavitation generation area, where thepressure coefficient value of suction surface (−𝐶𝑝) decreasesslowly along the chord.The second region corresponds to thecavitation weakening process, and the −𝐶𝑝 value decreasesrapidly in this region. In the third region, the cavitationeffect almost ends and the pressure decreases slowly alongthe chord length ratio. Compared with the calculation resultsat three different temperatures, it can be found that thethree regions of the pressure coefficient curve have obviousdemarcation points at 70∘C, and these demarcation pointscorrespond to the position where the chord length ratio is

8 Mathematical Problems in Engineering

1.00.90.80.70.60.50.40.2 0.30.10.0

25∘C

50∘C

70∘C

(a) Bubble volume fraction

25∘C

50∘C

70∘C

Pressurecontour 1

[Pa]

1.300

e+00

5

1.17

4e+00

5

1.04

8e+00

5

9.220

e+00

4

7.960e

+00

4

6.700

e+00

4

5.440e

+00

4

4.180

e+00

4

2.920

e+00

4

1.660e

+00

4

4.000

e+00

3

(b) Isosurface with 0.5 bubble volume fraction

Figure 10: Comparison of bubble volume fractions at different temperatures.

0.0 0.2 0.4 0.6 0.8 1.0−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

x/c

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.401.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

x/c

−C

p

−C

p

25∘C50∘C70∘C

25∘C50∘C70∘C

Figure 11: Comparison of −𝐶𝑝 at different temperatures.

𝑥/𝑐 = 0.1 and 0.18, respectively, while at 25∘C and 50∘C,that curve drops comparatively gently and there is no obviousdemarcation point to distinguish three different regions. At70∘C, the first region corresponds to the area where thechord length ratio is about 𝑥/𝑐 = 0∼0.1, and the length ofthe region where cavitation occurred is greater than that in25∘C and 50∘C situation. In the area where chord length

ratio is in the range from 0 to 0.07, the pressure coefficientsat three temperatures are relatively close, while in the areawhere chord length ratio is in the range from 0.07 to 0.18,the pressure coefficients at three temperatures show a largedifference. When the chord length ratio ranges from 0.07 to0.14, the lowest value of the pressure in the airfoil surfaceoccurs at 70∘C while the highest value occurs at 25∘C. Then

Mathematical Problems in Engineering 9

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400.0

0.2

0.4

0.6

0.8

1.0

Volu

me f

ract

ion

of v

apou

r

x/c

25∘C50∘C70∘C

Figure 12: Comparison of volume fractions at different tempera-tures.

the pressure at 70∘C rises rapidly, while the pressure at 25∘Crises slowly. When the chord length ratio reaches 0.14–0.18,the highest value of the pressure in the airfoil surface occursat 70∘Cwhile the lowest value occurs at 25∘C.When the chordlength ratio 𝑥/𝑐 is larger than 0.18, the pressure values inthe airfoil surface at three different temperatures all increasegradually and the difference between the three temperaturesdecreases.

Figure 12 shows the distribution of vacuolar volumefraction along the chord length ratio in the suction surfaceat three temperatures. It can be seen that when the chordlength ratio 𝑥/𝑐 ranges from 0 to 0.14, the highest value ofvapor volume fraction occurs at 70∘Cwhile the smallest valueoccurs at 25∘C. But when the vapor volume fraction decreasesdownstream along the chord, the higher the temperature, thefaster the decrease speed, which indicates that, in the areawhere cavitation disappears, higher temperature promotescondensation process.

In summary, the bubbles in the growth process willtransfer downstream along with the flow field; in the areawhere chord length ratio is in the range from 0.07 to 0.18, thedifference of pressure coefficient under different temperatureismost obvious.Thehigher the temperature, themore seriousthe cavitation. In cavitation inception stage (where chordlength ratio 𝑥/𝑐 = 0∼0.07), the pressure coefficient at threetemperatures is basically the same, which indicates thatcavitation has occurred at all temperatures in this region. Inthe cavitation development stage (where chord length ratio𝑥/𝑐 = 0.07∼0.1), the effect of cavitation on the pressure wassignificantly different at each temperature and the highertemperature means larger cavity length. But in the cavitationdisappearance stage (where chord length ratio 𝑥/𝑐 = 0.1∼0.18), the pressure is restoredmuch faster; when the cavitationeffect disappears, the pressure in the airfoil surface risesslowly at all temperatures, and the difference between themis narrowed gradually.

6. Conclusions

To select the suitable turbulent model for cavitation predic-tion, four common turbulent models and two filter turbulentmodels are solved in combination with Rayleigh-Plessetcavitation model, and their accuracies are compared. Basedon the Rayleigh-Plesset equation, the source consideringthermodynamic effects is added to the Zwart cavitationmodel using the Taylor formula and the Clapeyron formula.Two cavitation models considering thermodynamics effects,TCM1 and TCM2, are proposed and tested by comparingthe result with the experimental data of NACA0015. Toimprove the accuracy of the currently proposed cavitationmodel, the evaporation and condensation coefficient arefurther corrected. The following are found from the currenttheoretical and numerical research:

(i) The accuracy of a simulation result for cavitationflows is affected by the turbulent model in largedegree. The cavity length simulated using the filteredturbulence model is shorter than that of the typicalturbulence model, and the result of RNG k-𝜀 turbu-lence model is most close to the experimental value.

(ii) According to the simulation results using TCM1 andTCM2 models, the fluid temperature has an effect onthe cavitation degree around the hydrofoil, and thecavitation becomes more intensive with the increaseof the temperature. The results calculated using thesetwo cavitation models are similar, the differencebetween them shows a gradually increasing trendwith the increase of temperature, but the overall gapis small.

(iii) The evaporation and condensation coefficient has agreat effect on the simulation result of the cavita-tion flow under different temperature, and the valueof 10 and 0.002 for evaporation and condensationcoefficients are most suitable for the currently pro-posed cavitation model. The simulation result usingthe modified cavitation model agrees well with theexperimental result, which provides a new method tosimulate the cavitation flow considering the thermo-dynamic effects.

Nomenclature

𝑝∞: Reference pressure [Pa = N/m2]𝑝V: Vaporization pressure [Pa = N/m2]𝜌: Density [kg/m3]𝜎: Cavitation number = (𝑝∞ − 𝑝V)/(𝜌V∞2/2)𝑐𝑝: Pressure coefficient = (𝑝 − 𝑝∞)/(𝜌V∞2/2)V∞: Reference velocity [m/s]𝑡: Time [s]𝑥: Position [m]𝑢: Velocity [m/s]𝑇: Temperature [K]𝑓V: Mass fraction of water vapor𝐿: Latent heat of vaporization𝛼: Volume fraction

10 Mathematical Problems in Engineering

ℎ: Enthalpy [kJ/kg]𝜇: Laminar viscosity𝜇𝑡: Turbulent viscosity coefficient𝛿: Unit tensorPr: Prandtl number𝑅𝐵: Bubble radius𝑆: Surface tension coefficient𝜐: Liquid kinematic viscosity.

Subscripts

𝑚: Mixture𝑖, 𝑗: Dimension of tensors𝑙: LiquidV: Vapor.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was sponsored by the National Natural Sci-ence Foundation of China (no. 51679111, no. 51579118,and no. 51409127), PAPD, Six Talents Peak Project ofJiangsu Province (no. HYZB-002), Key R&D Projects inJiangsu Province (BE2015119, BE2015001-4, BE2016319, andBE2017126), the Natural Science Foundation of JiangsuProvince (no. BK20161472), Science and Technology SupportProgram of Changzhou (no. CE20162004), and ScientificResearch Start Foundation Project of Jiangsu University (no.13JDG105).

References

[1] B. Ji, X. W. Luo, R. E. Arndt, X. Peng, and Y. Wu, “Largeeddy simulation and theoretical investigations of the transientcavitating vortical flow structure around a NACA66 hydrofoil,”International Journal of Multiphase Flow, vol. 68, pp. 121–134,2015.

[2] W. Li, C. Wang, and W. Shi, “Numerical calculation andoptimization designs in engine cooling water pump,” Journal ofMechanical Science & Technology, vol. 31, no. 5, pp. 2319–2329,2017.

[3] W. Li, W. Shi et al., “Influence of different impeller diameteron cavitation performance in an engine cooling water pump,”in Proceedings of the SME/JSME/KSME 2015 Joint Fluids Engi-neering Conference, American Society ofMechanical Engineers,2015.

[4] M. S. Plesset andA. Prosperetti, “Cavitation and bubble dynam-ics,” Annual Review of Fluid Mechanics, vol. 9, pp. 145–185, 1977.

[5] D. H. Fruman, J.-L. Reboud, and B. Stutz, “Estimation ofthermal effects in cavitation of thermosensible liquids,” Inter-national Journal of Heat and Mass Transfer, vol. 42, no. 17, pp.3195–3204, 1999.

[6] J.-P. Franc, C. Rebattet, and A. Coulon, “An experimentalinvestigation of thermal effects in a cavitating inducer,” Journalof Fluids Engineering, vol. 126, no. 5, pp. 716–723, 2004.

[7] J.-P. Franc and C. Pellone, “Analysis of thermal effects in acavitating inducer using Rayleigh equation,” Journal of FluidsEngineering, vol. 129, no. 8, pp. 974–983, 2007.

[8] J. Kim and S. J. Song, “Measurement of Temperature Effectson Cavitation in a Turbopump Inducer,” Journal of FluidsEngineering, vol. 138, no. 1, 2016.

[9] A. Cervone, C. Bramanti, E. Rapposelli, and L. D’Agostino,“Thermal cavitation experiments on a NACA 0015 hydrofoil,”Journal of Fluids Engineering, vol. 128, no. 2, pp. 326–331, 2006.

[10] A. Cervone, R. Testa, C. Bramanti, E. Rapposelli, and L.D’Agostino, “Thermal effects on cavitation instabilities in helicalinducers,” Journal of Propulsion and Power, vol. 21, no. 5, pp.893–899, 2005.

[11] Y. Yoshida, K. Kikuta, S. Hasegawa, M. Shimagaki, and T.Tokumasu, “Thermodynamic effect on a cavitating inducer inliquid nitrogen,” Journal of Fluids Engineering, vol. 129, no. 3,pp. 273–278, 2007.

[12] J. P. R. Gustavsson, K. C. Denning, and C. Segal, “Hydrofoilcavitation under strong thermodynamic effect,” Journal ofFluids Engineering, vol. 130, no. 9, pp. 0913031–0913035, 2008.

[13] H. Pham S, N. Alpy, S. Mensah et al., “A numerical study ofcavitation and bubble dynamics in liquid CO2, near the criticalpoint,” International Journal of Heat & Mass Transfer, vol. 102,pp. 174–185, 2016.

[14] Z. Qin and H. Alehossein, “Heat transfer during cavitationbubble collapse,” Applied Thermal Engineering, vol. 105, pp.1067–1075, 2016.

[15] T. Sun, X. Ma, Y. Wei, and C. Wang, “Computational modelingof cavitating flows in liquid nitrogen by an extended transport-based cavitation model,” Science China Technological Sciences,vol. 59, no. 2, pp. 337–346, 2016.

[16] T. Chen, B. Huang, G. Wang, and X. Zhao, “Numerical studyof cavitating flows in a wide range of water temperatures withspecial emphasis on two typical cavitation dynamics,” Interna-tional Journal of Heat and Mass Transfer, vol. 101, pp. 886–900,2016.

[17] Y. Zhang, X.-W. Luo,H.-Y. Xu, andY.-L.Wu, “The improvementand numerical applications of a thermodynamic cavitationmodel,” Kung Cheng Je Wu Li Hsueh Pao/Journal of EngineeringThermophysics, vol. 31, no. 10, pp. 1671–1674, 2010.

[18] B. Ji, X. Luo, Y. Wu et al., “Cavitation flow simulation for high-temperature water based on thermodynamic effects,” TsinghuaScience and Technology, vol. 50, no. 2, pp. 262–265, 2010.

[19] W. Wang, Y. Lin, X. Wang, and Y. Wang, “Thermodynamicseffect on cavitation for high temperature water,” Paiguan JixieGongcheng Xuebao/Journal of Drainage and Irrigation Machin-ery Engineering, vol. 32, no. 10, pp. 835–839, 2014.

[20] S. Shi, G. Wang, C. Hu, and D. Gao, “Experimental studyon hydrodynamic characteristics of cavitating flows aroundhydrofoil under different water temperatures,” Jixie GongchengXuebao/Journal of Mechanical Engineering, vol. 50, no. 8, pp.174–181, 2014.

[21] S. Shi, G. Wang, and R. Ma, “Numerical calculation of thermaleffect on cavitation in cryogenic fluids,” Engineering Mechanics,vol. 5, pp. 61–67, 2012.

[22] F. Tang, J. Li, Y. Li et al., “Influence of thermodynamics effect oninducer rotating cavitation under low-temperature condition,”Journal of Rocket Propulsion, vol. 39, no. 2, pp. 29–34, 2013.

[23] A. Asnaghi, A. Feymark, and R. Bensow, “Improvement of cav-itation mass transfer modeling based on local flow properties,”International Journal of Multiphase Flow, vol. 93, pp. 142–157,2017.

Mathematical Problems in Engineering 11

[24] B. Stoffel and W. Schuller, “Investigations concerning the influ-ence of pressure distribution and cavity length on the hydro-dynamic cavitation intensity,” Journal of Fluids Engineering, vol.226, pp. 51–58, 1995.

[25] C.-C. Tseng and L.-J. Wang, “Investigations of empirical coef-ficients of cavitation and turbulence model through steady andunsteady turbulent cavitating flows,” Computers & Fluids, vol.103, pp. 262–274, 2014.

[26] C. Tseng and S. Wei, “Turbulence modeling for isothermaland cryogenic cavitation,” International Journal of Heat & MassTransfer, vol. 53, no. 1, pp. 513–525, 2013.

[27] A. K. Singhal, M. M. Athavale, H. Li, and Y. Jiang, “Mathemati-cal basis and validation of the full cavitation model,” Journal ofFluids Engineering, vol. 124, no. 3, pp. 617–624, 2002.

[28] M. S. Plesset, “The Dynamics of Cavitation Bubbles,” Journal ofApplied Mechanics, vol. 16, no. 3, pp. 277–282, 1949.

[29] Y. Zhang, Modification The Cavitation Model of HydraulicMachinery considering The Thermodynamic Effects, JiangsuUniversity, 2015.

Hindawiwww.hindawi.com Volume 2018

MathematicsJournal of

Hindawiwww.hindawi.com Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwww.hindawi.com Volume 2018

Probability and StatisticsHindawiwww.hindawi.com Volume 2018

Journal of

Hindawiwww.hindawi.com Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwww.hindawi.com Volume 2018

OptimizationJournal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Engineering Mathematics

International Journal of

Hindawiwww.hindawi.com Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwww.hindawi.com Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwww.hindawi.com Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwww.hindawi.com Volume 2018

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Hindawiwww.hindawi.com Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com

Di�erential EquationsInternational Journal of

Volume 2018

Hindawiwww.hindawi.com Volume 2018

Decision SciencesAdvances in

Hindawiwww.hindawi.com Volume 2018

AnalysisInternational Journal of

Hindawiwww.hindawi.com Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwww.hindawi.com