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Hindawi Publishing CorporationInternational Journal of Rotating MachineryVolume 2008, Article ID 125678, 8 pagesdoi:10.1155/2008/125678

Research ArticleTheoretical Analysis of Thermodynamic Effect of Cavitation inCryogenic Inducer Using Singularity Method

S. Watanabe,1 A. Furukawa,1 and Y. Yoshida2

1 Department of Mechanical Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan2 Space Transportation Propulsion Research and Development Center, Space Transportation Mission Directorate,Japan Aerospace Exploration Agency, 1 Koganezawa, Kimigaya, Kakuda 981-1525, Japan

Correspondence should be addressed to S. Watanabe, [email protected]

Received 1 April 2008; Accepted 11 June 2008

Recommended by Sung Ro

Vapor production in cavitation extracts the latent heat of evaporation from the surrounding liquid, which decreases the localtemperature, and hence the local vapor pressure in the vicinity of cavity. This is called thermodynamic/thermal effect of cavitationand leads to the good suction performance of cryogenic turbopumps. We have already established the simple analysis of partiallycavitating flow with the thermodynamic effect, where the latent heat extraction and the heat transfer between the cavity andthe ambient fluid are taken into account. In the present study, we carry out the analysis for cavitating inducer and compare itwith the experimental data available from literatures using Freon R-114 and liquid nitrogen. It is found that the present analysiscan simulate fairly well the thermodynamic effect of cavitation and some modification of the analysis considering the real fluidproperties, that is, saturation characteristic, is favorable for more qualitative agreement.

Copyright © 2008 S. Watanabe et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Vapor production in cavitation extracts the latent heat ofevaporation from the surrounding liquid, which decreasesthe local temperature, and hence the local vapor pressure inthe vicinity of cavity. This is called a thermodynamic/thermaleffect of cavitation. The thermodynamic effect of cavitationcould be ignored for usual applications especially with waterat ambient temperature, but is much more important forcryogenic fluids such as liquid oxygen and liquid hydrogen.For example, the suction performance of turbopump inducerfor liquid propellant rocket engine is much better if operatedwith cryogenic fluids than cold water (Yoshida et al. [1]);the development of cavity is suppressed because of localvapor pressure depression due to the thermodynamic effectof cavitation. Recently, Franc et al. [2] have visually observedthe thermodynamic effect in a turbopump inducer by usingFreon R-114 as a working fluid, and succeeded in evaluatingthe effective temperature depression and Stepanoff ’s B-factor(Stepanoff [3]) through the comparison of cold water andR-114. Yoshida et al. [1] have measured the casing wallpressure distribution in an inducer with liquid nitrogen, andestimated the cavity volume from the low pressure region

in the casing wall pressure distribution. Then, they haveevaluated the effective temperature depression and B-factorby comparing the cavitation number which gives the samecavity length between cold water and liquid nitrogen.

Many theoretical/numerical studies have been done toclarify the thermodynamic effect of cavitation. Focusing onthe recent studies, Kato [4] proposed a simple model express-ing the heat flow around the sheet cavity by one-dimensionalpartial differential equation of unsteady heat conduction,and Tokumasu et al. [5] investigated the thermodynamiceffect on closed sheet cavities by combining RANS simula-tion with Kato’s model. Tani and Nagashima [6] simulatedthe cavitating flow around a hydrofoil with cryogenic fluidsby the bubbly flow model based on the Rayleigh Plessetequation. Hosangadi et al. [7] have developed the compress-ible two-phase flow analysis considering the evaporation andthe condensation processes, and compared their results withexperiments using various test models and several cryogens(Hord [8]).

We have developed a simple analysis of unsteady cavi-tating flow combining a free streamline theory and a sin-gularity method, and succeeded in simulating the cavitationinstabilities of hydrofoil (Watanabe et al. [9]) as well as those

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2 International Journal of Rotating Machinery

Cavity 1st blade

0th blade

nth blade

Singularities

U

UUαα

iy

β

hC

x

l1

qn

γ1n γ2n

Figure 1: Model for present analysis.

of cascade such as rotating cavitation and cavitation surge(Watanabe et al. [10]). More recently, we have constructedan analytical method of the thermodynamic effect on steady,partially cavitating flow (Watanabe et al. [11]). In this work,a singularity analysis is combined with a heat transfer modelproposed by Kato [4] and described above.

In the present study, we apply our analysis for a cavitatingflow of two different working fluids, Freon R-114 and liquidnitrogen, in a cascade and compare the results with theexperimental ones in a turbopump inducer reported byFranc et al. [2] and Yoshida et al. [1]. Some modificationof the analysis is described, which seems to be necessaryto obtain more qualitative agreement with the experimentsusing cryogenic fluids.

2. FUNDAMENTAL FLOW FIELD

We consider a flat plate cascade with the chordlength C,spacing h, the stagger angle β, and the number of blade Nas shown in Figure 1. We assume that the flow far upstreamis uniform with the magnitude U and the angle of attackα. Each blade is represented by bound vortex distributionsγn(ξ) [n = 0, 1, . . . ,N−1], and the cavity with length ln whichdevelops on the suction surface of each blade is representedby source distributions qn(ξ) [n = 0, 1, . . . ,N − 1]. Thenthe complex conjugate velocity is expressed by the followingequation:

w(z) = u− iv

=Ue−iα+1

2π

N−1∑

n=0

[∫ ln

0qn(ξ)·

{fn(z, ξ) +

π

Nheiβ}dξ

+ i∫ ln

0γ1n(ξ)·

{fn(z, ξ)+

π

Nheiβ}dξ

+ i∫ C

lnγ2n(ξ)·

{fn(z, ξ)+

π

Nheiβ}dξ]

,

(1)

fn(z, ξ) = π

Nhe−i(π/2−β)· cot

[π

Nh(z − ξ)e−i(π/2−β) − nπ

N

].

(2)

We divide the velocity components into the uniformvelocity U and the deviation (us, vs), that is,

u = U + us, v = Uα + vs. (3)

In the present study, we linearize the equations under theassumptions of the small angle of attack α� 1 and the smallvelocity deviations |us|, |vs| � U .

3. BOUNDARY AND COMPLEMENTARY CONDITIONS

We assume that the cavity is sufficiently thin so that allboundary conditions are applied on the blade suctionsurface. In the following sections, boundary and comple-mentary conditions applied are described for nth blade.

3.1. Boundary condition on cavity surface

We assume that the pressure on the cavity surface is equalto the vapor pressure, which is locally different due to thetemperature depression around the cavity surface underthe presence of the thermodynamic effect of cavitation.Integrating the linearized momentum equation in the x-direction, we obtain the following equation:

ucsn(x)Uα

= σn(x)2α

, (4)

where ucsn(x) denotes the velocity on the cavity surface at x.The local cavitation number σn(x) has been defined using thelocal vapor pressure pvn(x),

σn(x) = p∞ − pvn(x)ρLU2/2

, (5)

where p∞ denotes the pressure far upstream and ρL denotesthe density of liquid phase.

3.2. Boundary condition on wetted surface

We employ the following flow tangency condition on thewetted blade surfaces:

Imag[w{nhe−i(π/2−β) + x − 0i

}]=0 (0 < x < ln),

Imag[w{nhe−i(π/2−β) + x ± 0i

}]=0 (ln < x < C).(6)

3.3. Kutta’s condition

We assume that the pressure difference across the blade van-ishes at the trailing edge. This condition is simply expressedas follows:

γ2n(ln) = 0. (7)

3.4. Cavity closure condition

We employ the closed cavity model for its simplicity.The cavity thickness ηn can be obtained by integratingthe following kinematic boundary condition on the cavitysurface:

Udηndx

= qn(x). (8)

Imposing the zero cavity thickness at the trailing edge ofcavity, we obtain the following cavity closure condition:

ηn(ln) = 1U

∫ ln

0qn(ξ)dξ = 0. (9)

S. Watanabe et al. 3

4. MODELING OF THERMODYNAMIC EFFECT

In the previous section, we have described about thekinematic and dynamic boundary conditions, assumingthat the local vapor pressure distribution along the cavitysurface is known. To close the problem, we have to obtainthe local vapor pressure distribution. Here, we model thethermodynamic effect of cavitation, using the following heatconduction model for the liquid flow around the cavity andthe evaporation model expressing the heat flux across thecavity surface due to evaporation.

4.1. Heat conduction model

We assume that the heat conduction in the main flowdirection (x) is negligibly small, compared to that in thedirection (y) normal to the cavity surface. The temperatureincrease (decrease) due to the formation of thin cavity ΔTn

should satisfy the following linearized energy equation forinviscid flow:

∂ΔTn

∂t+ U

∂ΔTn

∂x= εa

∂2ΔTn

∂y2, (10)

where a denotes a thermal diffusivity. Kato’s turbulentdiffusivity factor ε (Kato [4]) has been employed to takeaccount of the enhancement of thermal diffusion due to theturbulent flow around the cavity.

Temperature difference ΔTn between far upstream andthe cavity surface is expressed as follows:

ΔTn = Tn(x, 0)− T∞ = αT∞CTn(x, 0), (11)

where we have assumed that the temperature difference ΔTn

is sufficiently small and changes linearly with the change ofthe incidence α. Substituting the above equation into (10),we obtain the following equation:

U∂CTn

∂x= εa

∂2CTn

∂y2. (12)

We can find that this equation is equivalent to one-dimensional partial differential equation of unsteady heatconduction. Assuming no temperature depression at theleading edge of cavity (x = 0), we can analytically inte-grate the above equation using the normalized temperatureincrease along the cavity surface Cθn(x):

CTn(x, y) = y

2√πKP

∫ x

0Cθn(ξ)

exp(−y2/4KP(x − ξ))

(x − ξ)3/2 dξ,

(13)

where KP is defined here as KP = εa/U . Then, thetemperature gradient normal to the cavity surface is derivedas

limy→0

∂CTn(x, y)∂y

= − 1√πKP

∫ x

0

dCθn(ξ)dξ

1

(x − ξ)1/2 dξ. (14)

InterfaceLiquidflow

Vaporflow

U + u′csn

U + u′′csn

dxx

Vn

η′

η′′

ηn

(x)

Figure 2: Control volume for continuity equation.

4.2. Evaporation model

We assume that the velocity inside the cavity is uniform inthe y-direction which equals to the liquid velocity U + ucsnon the cavity surface as shown in Figure 2. Consideringthe continuity relation in the small control volume insidethe cavity, we obtain the following equation using the localevaporation velocity Vn of vapor phase:

Vndx =(U + u′′csn

)η′′n −

(U + u′csn

)η′n = Udηn. (15)

From this equation, we can calculate the local evapo-ration velocity Vn. However, it is experimentally observedthat, in general cases, the cavitation bubbles do not collapsecompletely at the trailing edge and some part of the cavityis flown away as a form of the cloud cavity shedding. Themagnitude of the cloud cavity shedding becomes larger asthe cavity becomes longer. Then, in order to take accountof the effect of the cloud cavity shedding, we modify theevaporation velocity by multiplying the cavity length (ln/C),which has been known to give a good approximation in ourprevious study (Watanabe et al. [11]). Then, the heat flux dueto the local latent heat of evaporation on the cavity surfacecan be expressed as follows:

qTn = −ρVL lnCVn = −ρVL ln

CUdηndx

= −ρVL lnCqn, (16)

where L denotes the latent heat of evaporation and ρV is thedensity of vapor phase. By using the expression Cqn = qn/Uα,which will be introduced in the next section, we obtain

qTn = −LρV lnCUαCqn. (17)

We can also calculate the local heat flux from the temperaturegradient (14) as

qTn = −ελ∂Tn

∂y

∣∣∣∣y=0

= ελαT∞√πKP

∫ x

0

dCθn(ξ)dξ

1

(x − ξ)1/2 dξ,

(18)


where λ denotes the thermal conductivity for the liquidphase. By equating this equation with (17), we obtain thefollowing one:

∫ x

0

dCθn(ξ)/dξ

(x − ξ)1/2 dξ = −(lnC

)L

CPT∞

(ρVρL

)√CU

a

√π

CεCqn,

(19)

where CP denotes the specific heat of liquid phase. As we cansee from (17), the heat flux across the cavity surface is directlyrelated to the source term Cqn = qn/Uα; Cqn > 0 correspondsto the evaporation with the heat absorption, and Cqn < 0corresponds to the condensation with the heat generation.Then, we have heat absorption from the surrounding liquidnear the leading edge and heat generation near the trailingedge of the cavity. In the present closed cavity model,the cavity terminates with a rapid condensation near thetrailing edge, but it is again reasonable to imagine that someportion of vapor flows away forming a cloud cavitation andcondensates far downstream of the cascade. Then, as wehave done in our previous study (Watanabe et al. [11]),we hereafter take account only of the heat flux due toevaporation and neglect that due to condensation near thetrailing edge of cavity.

4.3. Vapor pressure

We have assumed that the pressure on the cavity surfaceis equal to the vapor pressure, which is locally differentdue to the temperature depression around the cavity sur-face under the presence of the thermodynamic effect ofcavitation. In order to relate the vapor pressure PVn(x)withthe temperatureTn(x), we apply the Clapeyron-Clausiusequation as follows:

PVn(x)− PV∞ = dPVdT

(Tn(x, 0)− T∞

)

= L

T∞ρVρL

ρL − ρV

(Tn(x, 0)− T∞

)

= ρVLαCTn(x, 0),

(20)

where we have assumed that ρV � ρL. Then the cavitationnumber based on the local vapor pressure σn(x) defined in(5) can be expressed as follows:

σn(x)2α

= σ

2α−(ρVρL

)L

U2CTn(x, 0), (21)

where σ = 2(P∞ − Pv∞)/ρLU2 is a usual cavitation numberbased on the far upstream values.

5. ANALYTICAL METHOD

Discretization of singularities distributed along the bladesand cavities are made in the same manner as Horiguchiet al. [12], where nodes are distributed more densely nearthe leading and trailing edges of the blades and cavities. Thetemperatures along the cavity surface are evaluated at themidpoints between each node as well as at the leading edge.

The control points, where boundary conditions are applied,are also placed at the midpoints between each node.

We define the normalized strength of singularities asfollows:

Cqn(ξ) = qn(ξ)Uα

,

Cγ1n(ξ) = γ1n(ξ)Uα

,

Cγ2n(ξ) = γ2n(ξ)Uα

.

(22)

Discretizing the boundary and complementary condi-tions (4)–(9), (19), (21), we obtain the following set of linearequations:[A(ln)

]{Q} = {B}{Q} =

{Cqn(ξ), . . . ,Cγ1n(ξ), . . . ,Cγ2n(ξ), . . . ,Cθn(ξ), . . . ,

σ

2α

}T,

(23)

where A(ln) and B are a coefficient matrix and a constantvector, respectively. From this equation, we can obtain thecavity length ln as well as the other unknowns for the givenvalue of σ/2α.

6. RESULTS AND DISCUSSIONS

The present analysis treats the cavity on each blade individu-ally, so that it can be applied to the analysis of cavitating flowwith different cavity shapes for each blade such as alternateblade cavitation, which is known to occur for inducers witheven blade count (Horiguchi et al. [12]). However, in thepresent paper, we concentrate on the cavitating flow withidentical cavities for each blade.

6.1. For Freon R-114

Figure 3 shows Franc’s experimental results by using FreonR-114 and cold water as working fluids (Franc et al.[2]). As we can see from the plot of the cavity lengthagainst cavitation number (parameter) in Figure 3(a), thedevelopment of the cavity is suppressed for R114 with thehigher temperature compared with those for cold water andR114 with the lower temperature due to the thermodynamiceffect of cavitation. Franc et al. have estimated the effectivetemperature depression due to the thermodynamic effect ofcavitation by using the following equation:

12ρLU

2(σc − σ) = dP

dTΔT , (24)

where σc denotes the cavitation number without the ther-modynamic effect. In cold water, the thermodynamic effectcan usually be negligible, then the cavitation number incold water is herein used for σc. Under the assumption thatthe flow around the cavity for the given cavity length issimilar even in the different working fluids, we can estimate


0

0.2

0.4

0.6

0.8C

avit

yle

ngt

h/b

lade

spac

ing

0 0.02 0.04 0.06

Cavitation parameter, σ

R114-20◦C-5000 min−1

R114-40◦C-5000 min−1

Cold water-5500 min−1

Rotating

cavitation

Alternate bladecavitation

End of rotating cavitation

Onset of rotating cavitation

Onset of alternate blade cavitation

Water

R114-20◦C

R114-40◦C

(a) Cavity length

0

1

2

3

4

5

6

7

8

9

Tem

per

atu

rede

pres

sion

ΔT

(◦C

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Cavity length/blade spacing

20◦C-5000 min−1

40◦C-5000 min−1

20◦C-4000 min−1

30◦C-4000 min−1

40◦C-4000 min−1

20◦C-3000 min−1

20◦C30◦C

40◦C

(b) Temperature depression

Figure 3: Experimental results in the case of R-114 and cold water(Franc et al. [2]).

the effective temperature depression via (24) by comparingthe cavitation number with the same cavity length betweencold water and the other fluid. We can see from Figure 3(b)that the temperature depression is more significant in thehigher temperature cases with the higher rotational speed.The temperature depression becomes larger as the cavitybecomes longer, which is simply because the amount of theevaporation is larger for longer cavitation.

Figure 4 shows the present analytical result for the R-114 and cold water cases. The analysis has been done forthe cascade with the stagger angle of 78.8◦, the solidity of2.0, and the dimensional chord length is 0.203 mm as anumerical configuration. The incidence angle and the main

0

0.2

0.4

0.6

0.8

Stea

dyca

vity

len

gth

,l/h

0 0.2 0.4 0.6 0.8

Cavitation number, σ

W/o thermodynamic effectR-114 290 KR-114 310 K

(a) Cavity length

0

10

20

30

40

Tem

per

atu

rede

pres

sion

,ΔT

(K)

0 0.2 0.4 0.6 0.8

Steady cavity length, l/h

R-114 310 K 5000 rpmR-114 310 K 4000 rpmR-114 290 K 5000 rpmR-114 290 K 4000 rpm


Figure 4: Numerical results in the case of R-114 and cold water.

flow velocity are chosen as α = 4.32◦ and U = 34.0 m/swith the rotational speed of 5000 min−1. The turbulencediffusion factor ε is unknown but is set to be 1000, whichgives comparable figures with the experimental ones. Wecan see from the comparisons between Figures 3 and 4 thatthe present analysis can simulate the phenomena fairly well;the thermodynamic effect is more apparent with the largertemperature depression for the higher temperature cases.

6.2. For liquid nitrogen

Figure 5 shows Yoshida’s experimental results by using liquidnitrogen (LN2) with 80 K and cold water as a working fluid(Yoshida et al. [1]). In their experiments, they succeededin measuring the cavity length indirectly from the pressuremeasurement, which is plotted in Figure 5(a). They also


0.4

0.6

0.8

1

1.2

1.4

1.6

1.8C

avit

yle

ngt

h/b

lade

spac

ing

Ccl

0 0.01 0.02 0.03 0.04 0.05 0.06

Cavity number, σ

Nitrogen (80 K)Water (296 K) by visualizationWater (296 K) by pressure sensorFit curve ± standard deviation

(a) Cavity length

0

2

4

6

8

10

12

14

16

18

Tem

per

atu

rede

pres

sion

ΔT

(K)

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Cavity length/blade spacing Ccl

5

10

15

20

25

B-f

acto

r


Figure 5: Experimental results in the case of LN2 and cold water(Yoshida et al. [1]).

estimated the effective temperature depression using Franc’smethod with (24), which is plotted in Figure 5(b). As we cansee from Figure 5(a), the development of the cavity is sup-pressed in the LN2 cases due to the thermodynamic effect ofcavitation, which is similar to the cases with R-114 (Figure 3).In Figure 5(b), the effective temperature depression is plottedfor longer cavitation in Yoshida’s LN2 experiment than thatin Franc’s R-114 experiment, and we can see from this figurethat the larger the cavity length becomes, the more thetemperature depression also increases, which is similar tothe case with R-114. However, when the cavity trailing edgereaches the throat section of the inducer (the cavity lengthequals to the pitch h), the temperature depression decreasesslightly, which is probably due to the interaction between thecavity trailing edge and the flow around the leading edge

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cav

ity

len

gth

,l/h

0.01 0.1 1 10

σ/2α

WaterNitrogen (80 K)Nitrogen (76 K)

(a) Cavity length

0

2

4

6

8

10

12

14

16

18

20

Tem

per

atu

rede

pres

sion

,ΔT

(K)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Cavity length, l/h

Nitrogen (80 K)Nitrogen (76 K)


Figure 6: Numerical results in the case of LN2 and cold water.

of the adjacent blade. Further increase of the cavity lengthleads to the increase of the temperature depression again,whereas the temperature depression is being saturated whenthe cavity trailing edge approaches the trailing edge of theinducer blade. At that moment, the effective temperaturedepression is about 14 K. Because the temperature range inthe liquid nitrogen is between 63 K (triple point) and 126 K(critical point) and the test temperature is 80 K, the valueof 14 K is supposed to be almost the maximum possibletemperature depression (80−63 = 17 K), which probablyexplains why the temperature depression seems to be limitedin Figure 5(b).

Figure 6 shows the present analytical result for the LN2with 76 K and 80 K and cold water cases. The analysis hasbeen done for the cascade with the stagger angle of 75◦,the solidity of 2.0. The incidence angle is set to be 5◦ andthe turbulence diffusion factor ε is again set to be 1000.We can see from the comparisons between Figures 5 and6 that the present analysis can simulate the phenomenon


0

0.5

1V

apor

pres

sure

,PV

(MPa

)

60 70 80 90 100 110 120Temperature, T (K)

Liquid nitrogen

Clapeyron-Clausiusequation

•

Figure 7: Saturation curve for nitrogen.

fairly well except that the effective temperature depressioncontinues to increase even when the cavity trailing edgereaches the trailing edge of the blade in our analysis.This is probably because we have employed the Clapeyron-Clausius equation to estimate the vapor pressure for thelocal temperature, where the vapor pressure is assumedto change linearly with the change in the temperature;for the liquid nitrogen, the temperature range is narrow.The linear assumption is probably inappropriate in suchcases.

Figure 7 shows the saturation curve in the nitrogen, fromwhich we can find the large deviation of the Clapeyron-Clausius equation from the real saturation curve. Then, wemade the analysis using the saturation curve in Figure 7instead of (20). The results are shown in Figure 8. Wecan see the qualitatively good agreement between presentanalysis and the experiments shown in Figure 5. Then, in thecases with the fluids with narrower liquid temperature rangeand/or the cases where the large temperature depression isexpected, the assumption of Clapeyron-Clausius equation asused in the estimate of B-factor seems to be insufficient topredict the thermodynamic effect of cavitation; at least, thereal saturation curve should be taken into account for morequalitative/quantitative analysis of the thermodynamic effectof cavitation.

7. CONCLUSIONS

In the present study, we have carried out the singularityanalysis considering the thermodynamic effect of cavitationfor the cavitating cascade in Freon R-114 and liquid nitro-gen. Through the detailed comparisons with the existingexperiments done by Franc et al. [2] and Yoshida et al.[1], it is found that the present analysis can qualitativelywell reproduce the thermodynamic effect of cavitation.The major conclusions obtained here are summarized asfollows.

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Cav

ity

len

gth

,l/h

0.01 0.1 1 10

σ/2α

WaterNitrogen (80 K)Nitrogen (76 K)

(a) Cavity length

0

1

2

3

4

5

6

7

8

9

10

Tem

per

atu

rede

pres

sion

,ΔT

(K)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Cavity length, l/h

Nitrogen (80 K)Nitrogen (76 K)


Figure 8: Modified results in the case of LN2 and cold water.

(1) The development of the cavity is suppressed due tothe thermodynamic effect of cavitation. This effect ismore significant for higher temperature.

(2) The thermodynamic effect becomes more apparent,and then the temperature depression becomes largeras the cavity becomes longer, because the largeramount of latent heat of evaporation is needed.

(3) When the cavity trailing edge reaches the throatsection of the inducer, the temperature depressionslightly decreases. This is probably due to the inter-action between the cavity trailing edge and the flowaround the leading edge of the adjacent blade.

(4) In the cases with the fluids with narrower liquidtemperature range and/or the cases where the largetemperature depression is expected, the temperaturedepression might be limited by the triple point. Tosimulate this effect, the real saturation curve shouldbe at least taken into account.


The present study analytically models the thermody-namic effect of cavitation, whereas some parts such as thedetermination of the turbulence diffusion factor ε are stilldifficult. The quantitative estimates of them are crucial todevelop this analysis for more practical applications. Furtherexperimental data with cryogenic fluids and sophisticationof the model are needed for this purpose. Moreover, thereal flow in the cavitating inducer is so complicated withthe tip leakage and the back flow cavitations in additionto the blade surface cavitation treated here. The analysisfor the other forms of cavitation still remains for futurestudy.

NOMENCLATURE

A: Coefficient matrix in (23)a: Thermal diffusivityB: Constant vector in (23)C: ChordlengthCq,Cγ1,Cγ2: Normalized strength of singularitiesCP : Specific heatCT ,Cθ : Normalized temperature increasef : Kernel function, defined by (2)h: Blade spacingi: Imaginary unitKP : KP = εa/UL: Latent heat of evaporationl: Cavity lengthN : Number of bladesn: Blade indexPV : Vapor pressureQ: Unknown vector in (23)q: Source representing cavityqT : Heat flux on cavity surfaceT : TemperatureU : Main flow velocityV : Local evaporation velocityu, v: Flow velocity components in x and y

directionsucs: Velocity deviation on cavity surfaceus, vs: Velocity deviations in x and y directionsw: Complex conjugate velocityx, y: Coordinatesz: Complex coordinate, = x + iyα: Angle of attackβ: Stagger angleγ1, γ2: Bound vortices representing bladeΔT : Temperature depressionε: Turbulent diffusion factorη: Cavity thicknessλ: Thermal conductivityρL, ρV : Densities of liquid and vapor phasesσ : Cavitation number.

SUBSCRIPTS

n: Blade index∞: Upstream infinity.

ACKNOWLEDGMENT

This study is partly supported by the Grant-in-Aid forScientific Research for the Ministry of Education, Science,Sports and Culture (no. 19760119).

REFERENCES

[1] 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.

[2] 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.

[3] A. J. Stepanoff, “Cavitation properties of liquids,” Journal ofEngineeing and Power, vol. 86, pp. 195–200, 1964.

[4] H. Kato, “Thermodynamic effect on incipient and developedsheet cavitation,” in Proceedings of the International Sym-posium on Cavitation Inception, vol. 16, pp. 127–136, NewOrleans, La, USA, December 1984.

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