- Email: [email protected]
Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow
international journal of hydrogen energy xxx (xxxx) xxx
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow Xiaojun Li a, Tangjun Shen a, Pengcheng Li b, Xiaomei Guo c, Zuchao Zhu a,* a
National-Provincial Joint Engineering Laboratory for Fluid Transmission System Technology, Zhejiang Sci-Tech University, Hangzhou 310018, China b School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China c School of Mechanical and Automotive Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China
highlights An effective compressible cryogenic cavitation model is proposed. The inhibition of thermal effects on cavitation intensity is investigated. The indispensability of cavitation compressibility is analysed. The impacts of thermal and pressure effects on compressibility are discussed.
article info
abstract
Article history:
The compressibility of the vapoureliquid phase is indispensable in simulating liquid
Received 14 October 2019
hydrogen or liquid nitrogen cavitating flow. In this paper, a numerical simulation method
Received in revised form
considering compressibility and combining the thermal effects of cryogenic fluids was
7 January 2020
developed. The method consisted of the compressible thermal cavitation model and RNG k
Accepted 25 January 2020
eε turbulence model with modified turbulent eddy viscosity. The cavitation model was
Available online xxx
based on the ZwarteGerbereBelamri (ZGB) model and coupled the heat transfer and vapoureliquid two-phase state equations. The model was validated on cavitating hydrofoil
Keywords:
and ogive, and the results agreed well with the experimental data of Hord in NASA. The
Cryogenic flow
compressibility and thermal effects were correlated during the phase change process and
Cavitation model
compressibility improved the accuracy of the numerical simulation of cryogenic cavitating
Compressibility
flow based on thermal effects. Moreover, the thermal effects delayed or suppressed the
Thermal effects
occurrence and development of cavitation behaviour. The proposed modified compressible ZGB (MCZGB) model can predict compressible cryogenic cavitating flow at various conditions. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
* Corresponding author. E-mail address: [email protected] (Z. Zhu). https://doi.org/10.1016/j.ijhydene.2020.01.192 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
2
international journal of hydrogen energy xxx (xxxx) xxx
Nomenclature r u p m s a k ε l pv RB T K R Cp L hb C V x DT m_ þ m_
density velocity pressure dynamic viscosity cavitation number volume fraction turbulent kinetic energy turbulent eddy dissipation thermal conductivity saturated vapour pressure single spherical bubble radius temperature thermal conductivity constant pressure coefficient latent heat convection heat transfer coefficient heat capacity volume subscripts temperature drop condensation rate evaporation rate
Subscripts l liquid phase v vapour phase ∞ reference m mixture i, j,k directions of the Cartesian coordinates out outlet c empirical coefficient tur turbulence
Introduction Hydrogen energy is a clean energy source because of its low or zero carbon emission [1,2]. Moreover, hydrogen is easy to prepare and can be recycled [3,4]. Liquid hydrogen, which is often used as fuel for space launch vehicles, has the advantages of high energy and clean combustion [5,6]. Liquid hydrogen is indispensable in the development of the aerospace industry [7e9]. However, liquid hydrogen easily promotes cavitation in the turbo pump of space launch vehicles, causing flow passage components to oscillate and even resulting in spacecraft launch failure [10e13]. Compared with room-temperature cavitating flow, the cryogenic cavitating flows of liquid hydrogen and liquid nitrogen are more affected by thermodynamic properties [14e16]. Therefore, studying the thermal effects of cryogenic cavitating flow is essential to the cryogenic industry. Cavitating flow experiments using liquid hydrogen are extremely difficult to perform; thus, many researchers prefer to use other thermo-sensitive fluids, which are easy to control [17e19]. Franc et al. [20] experimentally investigated the
cavitating flow of Freon R114 in an inducer at different reference temperatures and discovered that cavity length near the inlet vane of the inducer decreases as reference temperature increased. Kikuta et al. [21] experimentally studied the influence of thermal effects on the performance and instability of inducers. They determined the cavity length at the impeller edge between liquid nitrogen and cold water and showed that thermal effects increase with cavity length. Kelly et al. [22] studied thermal cavitating flow by using fluoroketone instead of liquid hydrogen and used high-speed photography to study the unsteady state characteristics of cavitation at various freestream temperatures and cavitation numbers. Ito et al. [23] compared the cavitating flow characteristics of liquid nitrogen at 77.9 K and water at 292.5 K and 333.5 K and showed that the cavity in liquid nitrogen appears as misty bubbles compared with water. Zhu et al. [24] tested the liquid nitrogen cavitating flow in the venturi to obtain a clear picture of liquid nitrogen cavity shedding and further analysed the shedding process. Recently, Chen et al. [25] experimentally analysed the dynamic evolution of liquid nitrogen cavitating flow in a convergingediverging nozzle under a wide range of freestream conditions. The results showed that cavitation dynamics changed from the quasi-isothermal mode to the thermo-sensitive mode as throat temperature increased, and the transition temperature was 77e78 K. Numerical simulations are fundamental in the research of liquid hydrogen cavitating flow [26,27], and cavitation models play a vital role [28,29]. Hosangadi et al. [30] proposed a multiphase flow calculation formula, which considered compressibility, energy balance and thermodynamic property, to verify the influence of thermal effects on cavity interfaces. Cao et al. [31] developed a full cavitation model with the energy equation to simulate thermal cavitating flow and discussed cavitation mechanisms with thermal effects. Shi et al. [32] revised the Merkle cavitation model and performed numerical simulation on liquid nitrogen and liquid hydrogen cavitating flow. The result showed the thermal effects of liquid hydrogen is more obvious than liquid nitrogen. Zhang et al. [33] developed an effective cavitation model considering the thermodynamic transformation theory, the YoungeLaplace equation and the thermodynamic equilibrium hypothesis of the cavitation process. We also modified the ZwarteGerbereBelamri (ZGB) model by using the convective heat transfer approach and corrected the saturated vapour pressure based on the ClausiuseClapron equation [34,35]. The effectiveness were verified by the simulation of liquid nitrogen and fluoroketone cavitating flow. Numerous researchers have neglected vapoureliquid compressibility when simulating liquid hydrogen cavitating flow. In cavitating flow, the effect of compressibility is assignable, especially in cavitation dynamics [36e38]. In the comprehensive analysis of liquid hydrogen cavitating flow, considering gas and liquid compressibility is necessary [39]. Koukouvinis et al. [40] investigated cavitating flow in a diesel injector nozzle using the Tait equation of state. Budich et al. [41,42] simulated the cavitation flow of a sharp convergentedivergent wedge and propeller VP1304 by fully compressible numerical method, and analysed associated shock wave dynamics during cavity collapse. Wang et al. [43] considered vapoureliquid compressibility on the basis of the
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
Saito cavitation model. The numerical simulation method precisely predicted bubble dynamics closely related to bubble flow compressibility. Kyriazis et al.[44] used the Helmholtz energy equation of state to study flashing flows of liquid oxygen in a convergent-diverging nozzle. Le et al.[45] developed an effective numerical simulation method based on the modified Merkle cavitation model and vapoureliquid compressibility. The predicted pressure and temperature fields are more consistent with experimental data than those of other models. Clarify this purpose, a numerical simulation method considering compressibility and thermal effects effectively verified on cavitating hydrofoil and ogive is developed in this paper. The paper is divided into three sections. The first section introduces the derivation process of the numerical simulation method with compressibility and thermal effects. The second section verifies the numerical simulation method on cavitating hydrofoil and ogive, and the relationship between thermal effects and compressibility and their influence on cryogenic cavitating flow are analysed. The last section summarises the paper.
Numerical simulation method for compressible thermo-sensitive cavitating flow Governing equations The governing equation was the two-dimensional (2D) NaviereStokes equation (continuity, momentum and energy equation) with thermal effects and compressibility. The specific equations were as follows: vrm v þ r uj ¼ 0; vt vxj m
(1)
vðrm ui Þ v rm ui uj vp v vui vuj 2 vuk þ ¼ þ þ dij ; ðmm þ mt Þ vt vxi vxj vxi vxj vxi 3 vxk (2) v v v vT _ rm Cpm T þ rCpm Tuj ¼ ðlm þ lt Þ mL; vt vxj vxj vxj
(3)
(4)
rm ¼ arv þ ð1 aÞrl ;
(5)
mm ¼ amv þ ð1 aÞml ;
(6)
thermal effects of the cavitation region were calculated by solving the convective heat transfer equation. And the vapoureliquid twoephase state equation was used to explain compressibility.
Vapour-liquid two-phase state equation Liquid hydrogen and liquid nitrogen share certain similarities at cryogenic temperatures, as shown in Fig. 1. Compared with liquid hydrogen, nitrogen has abundant data and was thus used in the analysis of the characteristics of cryogenic cavitating flow in the present study. The liquid phase was assumed to be a compressible fluid, and state equation was rooted in the Tammann equation [46]. Fig. 2 showed the liquid nitrogen density obtained by the liquid state equation was in good agreement with the liquid nitrogen density at different temperatures and pressure levels provided by the National Institute of Standards and Technology. The liquid state equation can be expressed as [47]: pl þ pc ¼ rl Rl ðTl þ Tc Þ;
(7)
where pc and Tc are the pressure and temperature empirical coefficients, respectively, of the liquid, and Rl is the liquid constant. Thermal effects significantly influenced the liquid densities of liquid nitrogen and liquid hydrogen. Therefore, considering the compressibility of cryogenic cavitating flow on the basis of thermal effects is necessary. The liquid state equation required a boundary condition with a small range. Such condition is precise because the influence of thermal effects is drastic. The liquid state equation was fitted separately for each case for the accuracy of the liquid nitrogen cavitating flow simulation. The pressure empirical coefficient pc, temperature empirical coefficient Tc and the liquid constant Rl can be written as pc ¼ 2:956 108 ; Tc ¼ 1:025 102 ; Rl ¼ 2:040 103 :
(8)
The gas phase can be regarded as an ideal gas
vðarv Þ v arv uj þ ; m_ ¼ m_ þ m_ ¼ vt vxj þ
3
where r, u and p are the density, velocity and pressure, respectively; v, l and m indicate the vapour, liquid and mixture phases, respectively; x and the subscripts i, j and k represent the coordinate axes; mm is the dynamic viscosity, and mt is the turbulent viscosity of the mixture; lm represents mixture thermal conductivity; lt represents turbulent thermal conductivity and T, Cpm, L and a denote temperature, specific heat, latent heat and volume fraction, respectively. The compressibility and thermal effects of the vapour and liquid phases in Eqs. (1)e(6) are explained in the subsequent sections. The
pv ¼ rv Rv Tv :
(9)
Rv ¼ 296.8 J/(kg$K) is the gas constant for nitrogen. In the present study, non-condensable gas was ignored in the gas phase. When local thermodynamic equilibrium was assumed, p ¼ pl ¼ pv and T ¼ Tl ¼ Tv. The mixture density rm can be written as apRl ðT þ Tc Þ þ Rv Tð1 aÞ p þ pc : rm ¼ Rv Rl TðT þ Tc Þ
(10)
The correctness of the vapoureliquid two-phase state equation was verified by fitting the sound speed of the mixture. Fig. 3 shows the sound speed of 293 K water/vapour mixture and 83.06 K liquid nitrogen/nitrogen mixture at different pressures, which are obtained by the mixture speed equation [45], respectively. The speeds were compared with the experimental sound speeds obtained by Karplus [48]. The results are in good agreement with the experimental values.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
4
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 1 e Liquid nitrogen and liquid hydrogen density for pressure and temperature.
Modification of the compressible Zwart-Gerber-Belamri (MCZGB) model The mass transfer equation integrated the heat transfer effect by considering the inertial effect in Eq. (4). In our previous study, the evaporation source term m_ þ and condensation source term m_ can be written as m_ þ ¼ Fvap
3anuc ð1 aÞrv RB
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 2 max pv ðTÞ p; 0 C0 hb pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 rl Kl rl Cl
3 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3arv 4 2 max p pv ðTÞ; 0 C0 hb 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi m_ ¼ Fcond 3 rl RB Kl rl Cl
(11)
(12)
where RB ¼ 1 106 m is the spherical bubble radius; anuc¼ 5 104 is the volume fraction of the gas core point, and Kl and Cl are liquid thermal conductivity and heat capacity, respectively. The thermal conductivity and heat capacity of liquid nitrogen can be regarded as constants in an extremely small temperature range. The variables hb and C0 represent convection heat transfer and experienced coefficients, respectively. The proposed value of C0hb is 1 W/(m2$K). The empirical coefficients (Fvap ¼ 5.0 and Fcond ¼ 0.001) were recalibrated by Sun et al. [50]. The vapoureliquid phase density was replaced by the vapoureliquid twoephase state equation for the cavitation model so that compressibility could be considered. Saturated
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
5
Fig. 2 e Liquid nitrogen density for NIST data and liquid state equation.
Fig. 3 e Relation between sound speed and vapour void fraction in water [47e49] and liquid nitrogen.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
6
international journal of hydrogen energy xxx (xxxx) xxx
vapour pressure pv(T) was expressed by polynomial fitting given that it was based on temperature [51]: pv ðTÞ ¼
4 X
ai T ; i
(13)
i¼1
Where a1 ¼ 7:962 109 ; a0 ¼ 4:796 108 ; a1 ¼ 1:154 107 ; a2 ¼ 1:386 105 ; a3 ¼ 8:275 102 ; a4 ¼ 1:951 100 :
(14)
Compressible correction of turbulence model In the present study, the filterebased density corrected model (FBDCM) was adopted, which can modify the turbulent viscosity and improve the correctness of numerical solution. The FBDCM combines the advantages of densityecorrected and filterebased models and is widely used in the numerical models of cavitation flow. The filterebased density corrected model was described as follows [52,53]: mTFBDCM ¼ Cm
rm k2 fhybrid ; ε
(15)
fhybrid ¼ ffFBM þ ð1 fÞfDCM ;
(16)
C1 ð0:6rm =rl C2 Þ ½2 tanhðC1 Þ; f ¼ 0:5 þ tanh 0:2ð1 2C2 Þ þ C2
(17)
fDCM ¼
rv þ ð1 aÞn ðrl rv Þ r*m ¼ ; rv þ ð1 aÞðrl rv Þ rm
D$ε fFBM ¼ min 1; 2=3 ; k
(18)
(19)
where Cm ¼ 0.0845, C1 ¼ 4 and C2 ¼ 0.2, and k and ε denote turbulent kinetic energy and turbulent eddy dissipation, respectively.
Verification of the numerical simulation method Hord et al. [54e56] performed subscale tests of cavitation in cryogenic fluids (liquid nitrogen and liquid hydrogen) in a transparent plastic blow-down tunnel. The hydrofoil and ogive were installed at the center of water tunnel. Several pressure and temperature sensors were arranged on the upper and lower surfaces of the hydrofoil and ogive. The accuracy of the proposed numerical method in the present study was verified using Hord’s results.
Results of compressible cavitating hydrofoil The 2D model with structured grid was adopted for the creation of a hydrofoil. Given that the hydrofoil was a symmetric structure, half of it was selected as the computational domain, which is shown in Fig. 4. The inlet and the hydrofoil were 12.7 and 3.96 mm in width, respectively. The boundary conditions are speed inlet and pressure outlet. The wall is a free slip wall, and the surface of the hydrofoil is no slip and smooth wall. Simulations are 2D and transient, and the isothermal simulation results without cavitation were used as the initial conditions for numerical calculations. The details of the computational mesh are shown in Fig. 5, and grid independence analysis was performed [34]. Fig. 6 compares the temperature coefficient between the numerical and the experimental results, and mesh 2 is better agreement with the experimental data. The applicability of the MCZGB model was verified by predicting pressure and temperature fields under four different temperatures. The main parameters of each case are listed in Table 1. The pressure and temperature data calculated by the ZGB, MZGB and MCZGB models (Fig. 7) were compared. At four different temperatures, the predicted values of the MCZGB model (blue line) showed better agreement with the experimental results, especially the temperature field. Large temperature and pressure depressions were observed at the leading edge of the hydrofoil. This finding was attributed to the occurrence of mass evaporation near the leading edge because of thermal effects, and the depression was caused by the dissipation of heat from the bulk liquid. Compared with the MZGB model used in our previous work, the MCZGB model showed a better ability to capture the temperature depressions in the cavity, and the temperature depressions are smaller and more consistent with the experiment. In terms of pressure depression, several models are not huge distinction. Therefore, compressibility is a further modification of the thermal effects in the cavitating flow. As liquid nitrogen flowed, the bubbles collapsed, and the temperature and pressure depressions disappeared gradually. The predicted temperature at the cavity tail was somewhat higher than the experimental data due to the blockage effect and the release of latent heat of vaporisation during condensation. The pressure in the cavity closure regions was relatively high because of the collapse of the bubbles. The existing cavitation models were shown in Fig. 8 and compared with the MCZGB model. Table 2 and Table 3 list relative errors of the temperature and pressure which calculated by the existing model and compared with the
Fig. 4 e Hydrofoil computational domain and boundary condition. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
7
Fig. 5 e Three different cases of mesh around the hydrofoil surface: (a) Mesh 1 (total 10,665 elements), (b) Mesh 2 (total 25,325 elements) and (c) Mesh 3 (total 50,423 elements) [42].
Fig. 6 e Comparison of the Tþ values between the simulation and the experiment.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
8
international journal of hydrogen energy xxx (xxxx) xxx
Table 1 e Condition for simulation of hydrofoil. Case
293F 290C 295D 296B
Inlet velocity Cavitation Inlet u∞/(m/s) number s∞ temperature T∞/K 77.90 83.06 83.20 88.54
23.9 23.9 24.3 23.7
1.70 1.70 1.67 1.61
Latent heat L/(J/ kg) 1.98 1.91 1.91 1.83
105 105 105 105
experimental data. In terms of pressure capture, the ability of each cavitation models was roughly the same in capturing the pressure depression processes. However, pressure recovery was unpunctual with Hosangadi’s model, which leaded to inaccurate cavity length prediction. Le’s and Utturkar’s models predicted the pressure at the cavity tail insufficiently, where the MCZGB model was over-predicted. For temperature field, each cavitation model had different degrees of temperature overflow at the cavity tail. Inside the cavity, Le’s model showed a certain temperature profile discontinuity (as indicated by the red points), but the MCZGB model performed excellent in this respect, and also showed a good ability to predict the temperature depression in the cavity. In terms of relative error, the MCZGB model performs relatively well under both temperature and pressure. Moreover, the first point of the pressure data and the fourth and fifth points of the temperature data listed in the Tables, the predictions of all modified models are extremely poor and far from the experimental data. This will be the direction of our future work. As shown in Fig. 9(a), thermal effects became increasingly inhibited as the reference temperature increased. Cavitation intensity gradually decreased, and the closed region of the cavity tail was retracted upstream along the hydrofoil surface. The predicted cavity interface was fuzzy because the MCZGB model considered thermal effects and saturated vapour pressure as functions of temperature. Therefore, this result was an improvement over that of our previous work. The influence of thermal effects on cavitation can be further shown by comparing the mass transfer rates. As shown in Fig. 9(b), the maximum mass transfer rate was obtained at the leading edge of the hydrofoil. This result indicated that vaporisation was relatively intense at this point and led to a significantly sharp temperature depression. The cavity tail captured the minimum mass transfer rate and interdicted the growth of the cavity. As the reference temperature increased, the maximum mass transfer rate at the leading edge of the hydrofoil increased, and the inhibition of thermal effects became increasingly evident. Eventually, the cavity length decreased. Fig. 10 shows the intact temperature field in the computational domain of 296B. The result did not show the inconsistency of the temperature profile described by Le et al. [45]. When liquid nitrogen flowed through the hydrofoil, temperature increased by about 0.1 K due to the blockage effect. By contrast, cavity temperature was reduced by about 2.0 K due to thermal effects. The lowest temperature point was observed at the leading edge of the hydrofoil, and the temperature changed regularly. Owing to the release of latent heat
of vaporisation during condensation and the blockage effect, the temperature abruptly increased at the cavity tail during the calculation. This result is in good agreement with the experimental data of the hydrofoil. Hence, the proposed method in the present study has good accuracy. Fig. 9(c) depicts the mixture density predicted at three temperatures by the MCZGB model. The mixture density outside the cavity was similar to liquid nitrogen density, but the mixture density inside was much greater than nitrogen density at the same temperature. This result indicated that the interior of the cavity was not completely nitrogen but a mixture of small nitrogen bubbles and liquid nitrogen. Utturkar et al. [7] verified this conclusion through numerical simulation on cryogenic fluids. As mentioned before, the relatively fuzzy phenomenon at the cavity interface is well reflected in the mixture density field. Furthermore, as the reference temperature rose from 77.9 K to 88.54 K, the mixture density field tended to be placid. This condition is related to the physical properties of nitrogen. Fig. 11(a) shows liquid nitrogen density field in the computational domain. As the passageway narrowed, the pressure became small, and the liquid nitrogen density decreased slightly. Meanwhile, the density minimum, which is the density value at saturated vapour pressure, was obtained outside the cavity. Within the cavity, liquid nitrogen density increased rapidly because the thermal effect was larger than the pressure effect. The maximum density of liquid nitrogen was attained at the point of maximum temperature depression. Fig. 11(b) shows the nitrogen density field within the cavity. The density at the leading edge of the hydrofoil was lower than nitrogen density at the reference temperature due to temperature and pressure depressions. Therefore, the relationship between thermal effects and compressibility inside the cavity was inseparable. The density field at the boundary of the cavity, whether the nitrogen or liquid nitrogen density field, resulting in foggy cryogenic fluid bubbles.
Simulation of compressible cavitating ogive The compressible thermal cavitation model was further reassessed, and confidence was instilled into the MCZGB model’s validity by simulating liquid nitrogen cavitating flow over an axisymmetric ogive geometry. The computational domain is shown in Fig. 12. The inlet and ogive were 14.7 and 4.53 mm wide, respectively. As shown in Fig. 13, the mesh details covered more than 25,000 grids, and grid density near the ogive wall was refined to guarantee yþ < 10. The main parameters of ogive cases are listed in Table 4. In Fig. 14, the pressure and temperature fields predicted by the ZGB and MCZGB models on the ogive were compared. The MCZGB model was superior. Similarly, the leading edge of the ogive had pressure and temperature depressions, which were more dramatic than those of the hydrofoil possibly because the leading edge of the ogive structure changed abruptly to cause substantial vaporisation. Identically, temperature and pressure were overpredicted near the cavity closure. Fig. 15 shows the contour of the cavity. The predicted pressure field
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
9
Fig. 7 e Numerical results along the hydrofoil in comparison with experimental data.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
10
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 8 e Comparison of MCZGB model with previous cavitation models (290C).
Table 2 e Numerical simulation error based on experimental data (209C, pressure). X(m) 0.005613 0.009042 0.014122 0.023012 0.033172 Average Error
MCZGB model
Le ‘s model
Hosangadi’s model
Utturkar’s model
7.94% 7.10% 4.52% 0.98% 1.62% 4.43%
8.24% 5.33% 4.01% 5.80% 4.66% 5.61%
11.31% 5.27% 7.57% 11.56% 1.33% 7.41%
21.78% 5.33% 3.90% 16.33% 8.81% 11.23%
Table 3 e Numerical simulation error based on experimental data (209C, Temperature). X 0.005994 0.010947 0.017297 0.026822 0.036474 Average Error
MCZGB model
Le ‘s model
Hosangadi’s model
Utturkar’s model
0.00% 0.06% 0.38% 0.52% 0.42% 0.28%
0.11% 0.23% 0.34% 0.45% 0.37% 0.30%
0.28% 0.09% 1.32% 0.36% 0.29% 0.47%
0.58% 0.18% 0.02% 0.36% 0.53% 0.34%
Fig. 9 e Comparisons of vapour volume fraction, mass transfer rate and mixture for different cases. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
11
Fig. 10 e Temperature field in the whole flow passage and cavity of 296B.
Fig. 11 e Numerical results along the hydrofoil (a) Liquid nitrogen density; (b) Nitrogen density.
Fig. 12 e Ogive computational domain and boundary condition. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
12
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 13 e Mesh generation details of ogive computational domain.
Table 4 e Condition for simulation of ogive. Case
312D 322E
Inlet Inlet velocity Cavitation temperature u∞/(m/s) number s∞ T∞/K 83.06 88.56
23.9 25.8
1.70 1.61
Latent heat L/(J/ kg) 1.91 105 1.83 105
was stairestepping due to the contour of the cavity tail. The cavity contour of 322E diminished due to increasing thermal effects. Nevertheless, the present numerical simulation
method produced a reasonable range of results for the ogive structure. The density field of nitrogen and liquid nitrogen in the ogive can be acquired simply from Fig. 16. Compared with the hydrofoil, the density of nitrogen showed more intense in the ogive. The density field of liquid nitrogen in 312D was similar to that in 296B. However, the liquid nitrogen density behind the cavity closure did not change as regularly as that of hydrofoil because the upper-wall surface on the flow field produced a small cavity. Liquid nitrogen density still decreased because of the release of the latent heat of vaporisation and
Fig. 14 e Numerical results along the ogive in comparison with the experimental data. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
13
Fig. 15 e The predicted vapour volume fraction contour of (a) 312D and (b) 322E.
Fig. 16 e Numerical results in the ogive. (a) Liquid nitrogen density field; (b) Nitrogen density field.
pressure depression caused by the narrowing of the passageway.
Summary This study develops an effective compressible thermal cavitation model (MCZGB model), which considers the compressibility of the vapoureliquid phase and the thermal effects of cryogenic fluids in the simulation of compressible cryogenic cavitating flow. Comparing with the existing cavitation models, the error between the simulated data of the MCZGB model and the experimental data is minimal. The MCZGB model can be accurate prediction of cavity temperature depression. This is an advantageous for studying the effects of thermal effects and compressibility in cryogenic cavitating flow. The main conclusions are as follows: (1) The prediction results of the MCZGB model are in good agreement with the experimental data of the liquid nitrogen cavitating hydrofoil and ogive. The predicted temperature field has not shown the inconsistency of the temperature profile, which presents a smaller temperature depression and is closer to the experimental data. The numerical simulation method can predict a compressible cryogenic cavitating flow well.
(2) Thermal effects can delay or suppress the occurrence and development of cavitation behaviour. In a cryogenic cavitating flow, the influence of thermal effects on the compressibility of the vapour-liquid phase is greater than that of the pressure effect. With the increase in reference temperature, temperature depression within the cavity becomes conspicuous, and the strength of the thermal effects increases. Hence, mass transfer rate increases, and the cavity shortens. (3) The effect of compressibility in the numerical simulation method is analysed. Compressibility further improves the accuracy of the numerical simulation of cryogenic cavitating flows on the basis of thermal effects. Compared with the existing models, MCZGB model which considers the compressibility on the basis of MZGB model performs better. The interior of the cavity comprises small nitrogen bubbles and liquid nitrogen. Under the influence of thermal effects and compressibility, the interfaces of the cryogenic fluid cavity are foggy. The model exhibiting high prediction accuracy can fully analyze the thermal effects and compressible effects in cryogenic cavitation flows. However, it only applies to certain type of structure (hydrofoil, ogive, and inducer) at the present stage. Commercial/industrial applications of this model will be our next work.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
14
international journal of hydrogen energy xxx (xxxx) xxx
Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant No. 51776189, 51976202), the Natural Science Foundation of Zhejiang Province (Grant No. LR20E090001). The supports are gratefully acknowledged.
references
lu TN. “Green” path from fossil[1] Muradov NZ, Vezirog based to hydrogen economy: an overview of carbonneutral technologies. Int J Hydrogen Energy 2008;33:6804e39. [2] Coutier-Delgosha O, Patella RF, Reboud JL. Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation. ASME J Fluid Eng 2008;125:38e45. n G, Valde s-Solı´s T. Towards the hydrogen economy? [3] Marba Int J Hydrogen Energy 2007;32:1625e37. [4] Ball M, Wietschel M. The future of hydrogen e opportunities and challenges. Int J Hydrogen Energy 2009;34:615e27. [5] Liu M, Tan L, Cao S. Cavitationevortexeturbulence interaction and one-dimensional model prediction of pressure for hydrofoil ALE15 by large eddy simulation. ASME J Fluid Eng 2019;141:21103e17. [6] Merouani S, Hamdaoui O, Rezgui Y, Guemini M. Computational engineering study of hydrogen production via ultrasonic cavitation in water. Int J Hydrogen Energy 2016;41:832e44. [7] Zheng X, Lin Z, Xu BY. Thermal conductivity and sorption performance of nano-silver powder/FAPO-34 composite fin. Appl Therm Eng 2019;160:114055. [8] Abderrezzak B, Yong H. A contribution to the understanding of cavitation effects on droplet formation through a quantitative observation on breakup of liquid jet. Int J Hydrogen Energy 2016;41:15821e8. [9] Takakuwa O, Soyama H. Suppression of hydrogen-assisted fatigue crack growth in austenitic stainless steel by cavitation peening. Int J Hydrogen Energy 2012;37:5268e76. [10] Smirnov N, Betelin V, Shagaliev R. Hydrogen fuel rocket engines simulation using LOGOS code. Int J Hydrogen Energy 2014;39:10748e56. [11] Jothi S, Croft T, Wright L. Multi-phase modelling of intergranular hydrogen segregation/trapping for hydrogen embrittlement. Int J Hydrogen Energy 2015;40:15105e23. [12] Tao J, Lin Z, Ma C, Ye J, Zhu Z, Li Y, Mao W. An experimental and numerical study of regulating performance and flow loss in a V-port ball valve. ASME J Fluids Eng 2020;142(2):021207. [13] Sanad A. Modelling of flow of cryogenic fluids (oxygen) utilised in rocket engines. City University; 2018. [14] Pei J, Osman MK, Wang W. Unsteady flow characteristics and cavitation prediction in the double-suction centrifugal pump using a novel approach. Proc IME J Power Energy 2019. https://doi.org/10.1177/0957650919863636. [15] Huang B, Qiu SC, Li XB, Wu Q, Wang GY. A review of transient flow structure and unsteady mechanism of cavitating flow. J Hydrodyn 2019;31:429e44. [16] Dular M, Coutier-Delgosha O. Thermodynamic effects during growth and collapse of a single cavitation bubble. J Fluid Mech 2013;736:44e66. [17] Li X, Chen B, Luo X, Zhu Z. Effects of flow pattern on hydraulic performance and energy conversion characterisation in a centrifugal pump. Renew Energy 2019. https://doi.org/10.1016/j.renene.2019.11.049.
[18] Dular M, Khlifa I, Fuzier S, Maiga MA, Coutier-Delgosha O. Scale effect on unsteady cloud cavitation. Exp Fluid 2012;53:1233e50. [19] Yang H, Zhang W, Zhu Z. Unsteady mixed convection in a square enclosure with an inner cylinder rotating in a bidirectional and time-periodic mode. Int J Heat Mass Tran 2019;136:563e80. [20] Franc JP, Janson E, Morel P, Rebattet C, Riondet M. Visualizations of leading edge cavitation in an inducer at different temperatures. Fourth International Symposium on Cavitation 2001:124e30. [21] Kikuta K, Yoshida Y, Watanabe M, Hashimoto T, Nagaura K, Ohira K. Thermodynamic effect on cavitation performances and cavitation instabilities in an inducer. Nihon Kikai Gakkai Ronbunshu B Hen/transactions of the Japan Society of Mechanical Engineers Part B 2008;130:1349e57. [22] Kelly S, Segal C. Experiments in thermosensitive cavitation of a cryogenic rocket propellant. Gainesville: A Dissertation of University of Florida; 2012. p. 83e216. [23] Ito Y, Tsunoda A, Nagasaki T. Experimental comparison of backflow-vortex cavitation on pump inducer between cryogen and water. J Phys Conf 2015:656. [24] Zhu J, Xie H, Feng K, Zhang X, Si M. Unsteady cavitation characteristics of liquid nitrogen flows through venturi tube. Int J Heat Mass Tran 2017;112:544e52. [25] Chen T, Chen H, Liang W, Huang B, Le Xiang. Experimental investigation of liquid nitrogen cavitating flows in converging-diverging nozzle with special emphasis on thermal transition. Int J Heat Mass Tran 2019;132:618e30. [26] Cheng H, Bai X, Long X, Ji B, Peng X, Farhat M. Large eddy simulation of the tip-leakage cavitating flow with an insight on how cavitation influences vorticity and turbulence. Appl Math Model 2020;77:788e809. [27] Li X, Jiang Z, Zhu Z, Si Q, Yi L. Entropy generation analysis for the cavitating head-drop characteristic of a centrifugal pump. Proc IME C J Mech Eng Sci 2018;232:4637e46. [28] Dular M, Stoffel B, Sirok B. Development of a cavitation erosion model. Wear 2006;261:642e55. [29] Li X, Li B, Yu B, Ren Y, Chen B. Calculation of cavitation evolution and associated turbulent kinetic energy transport around a NACA66 hydrofoil. J Mech Sci Technol 2019;33:1231e41. [30] Hosangadi A, Ahuja V. Numerical study of cavitation in cryogenic fluids. ASME J Fluid Eng 2005;127:267e81. [31] Cao X, Zhang X, Li M. Validation of full cavitation model in cryogenic fluids. Chin Sci Bull 2009;54:1633e40. [32] Shi S. Numerical calculation of thermal effect on cavitation in cryogenic fluids. Chin J Mech Eng 2012;25:1176e83. [33] Zhang X, Zhao W, Xiang S, Qiu L. Modeling cavitation flow of cryogenic fluids with thermodynamic phase-change theory. Sci Bull 2013;58:567e74. [34] Zhang S, Li X, Zhu Z. Numerical simulation of cryogenic cavitating flow by an extended transport-based cavitation model with thermal effects. Cryogenics 2018;92:98e104. [35] Zhang S, Li X, Hu B, Liu Y, Zhu Z. Numerical investigation of attached cavitating flow in thermo-sensitive fluid with special emphasis on thermal effect and shedding dynamics. Int J Hydrogen Energy 2019;44:3170e84. [36] Schnerr G, Sezal I, Schmidt S. Numerical investigation of three-dimensional cloud cavitation with special emphasis on collapse induced shock dynamics. Phys Fluids 2008;20:040703. [37] Wei Z, Bai XD, Zheng M, Gang C, Yong W. Compressible effect on the cavitating flow: a numeric study. J Hydrodyn 2017;29:1089e92. [38] Schmidt S, Mihatsch M, Thalhamer M. Assessment of erosion sensitive areas via compressible simulation of unsteady cavitating flows. Advanced experimental and
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
[39]
[40]
[41]
[42]
[43]
[44] [45]
[46]
numerical techniques for cavitation erosion prediction 2014:329e44. Vidal A, Rodriguez C, Koukouvinis P. Modelling of diesel fuel properties through its surrogates using perturbed-chain, statistical associating fluid theory. Int J Engine Res 2018:1468087418801712. Koukouvinis P, Naseri H, Gavaises M. Performance of turbulence and cavitation models in prediction of incipient and developed cavitation. Int J Engine Res 2017;18:333e50. Budich B, Schmidt S, Adams N. Numerical simulation and analysis of condensation shocks in cavitating flow. J Fluid Mech 2018;838:759e813. Budich B, Schmidt S, Adams N. Numerical investigation of a cavitating model propeller including compressible shock wave dynamics. Proceedings of the Fourth International Symposium on Marine Propulsors 2015;31. Wang C, Huang B, Wang G, Duan Z, Ji B. Numerical simulation of transient turbulent cavitating flows with special emphasis on shock wave dynamics considering the water/vapor compressibility. J Hydrodyn 2018;30:37e55. Kyriazis N, Koukouvinis F, Karathanassis I. A tabulated data technique for cryogenic two-phase flows. 2018. Le AD, Junosuke O, Iga Y. Modification of energy equation for homogeneous cavitation simulation with thermodynamic effect. ASME J Fluid Eng 2018;141:1e12. Chen HT, Collins R. Shock wave propagation past an ocean surface. J Comput Phys 1971;7:89e101.
15
[47] Iga Y, Nohml M, Goto A, Ikohagi T. Numerical analysis of cavitation instabilities arising in the three-blade cascade. ASME J Fluid Eng 2004;126:110e6. [48] Karplus HB. Velocity of sound in a liquid containing gas bubbles. J Acoust Soc Am 1957;29:1261e2. [49] Henry RE, Grolmes MA, Fauske HK. Pressure-pulse propagation in two-phase one- and two-component mixtures. Technical Report Archive & Image Library; 1971. [50] Sun TZ, Xiangfu MA, Wei YJ, Cong W. Computational modeling of cavitating flows in liquid nitrogen by an extended transport-based cavitation model. Sci China Technol SC 2016;59:337e46. [51] Li PS, Ren H. The fitting of the saturation pressure on hydrogen and oxygen. Structure & Environment Engineering 2010;38:60e3. [52] Johansen ST, Wu J, Wei S. Filter-based unsteady RANS computations. Int J Heat Fluid Flow 2004;25:10e21. [53] Zhu Z, Qian Z, Guo Z. Numerical simulation of cavitation around a Clark-Y hydrofoil with different LES models in OpenFOAM. J Drain Irrig Mach Eng 2018;36(4):277e82. [54] Hord J. Cavitation in liquid cryogens. 1: venturi. NASA CR2054 1972. [55] Hord J. Cavitation in liquid cryogens. 2: hydrofoil. NASA CR2056 1973. [56] Hord J. Cavitation in liquid cryogens. 3: ogives. NASA CR-2242 1973.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow Xiaojun Li a, Tangjun Shen a, Pengcheng Li b, Xiaomei Guo c, Zuchao Zhu a,* a
National-Provincial Joint Engineering Laboratory for Fluid Transmission System Technology, Zhejiang Sci-Tech University, Hangzhou 310018, China b School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China c School of Mechanical and Automotive Engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China
highlights An effective compressible cryogenic cavitation model is proposed. The inhibition of thermal effects on cavitation intensity is investigated. The indispensability of cavitation compressibility is analysed. The impacts of thermal and pressure effects on compressibility are discussed.
article info
abstract
Article history:
The compressibility of the vapoureliquid phase is indispensable in simulating liquid
Received 14 October 2019
hydrogen or liquid nitrogen cavitating flow. In this paper, a numerical simulation method
Received in revised form
considering compressibility and combining the thermal effects of cryogenic fluids was
7 January 2020
developed. The method consisted of the compressible thermal cavitation model and RNG k
Accepted 25 January 2020
eε turbulence model with modified turbulent eddy viscosity. The cavitation model was
Available online xxx
based on the ZwarteGerbereBelamri (ZGB) model and coupled the heat transfer and vapoureliquid two-phase state equations. The model was validated on cavitating hydrofoil
Keywords:
and ogive, and the results agreed well with the experimental data of Hord in NASA. The
Cryogenic flow
compressibility and thermal effects were correlated during the phase change process and
Cavitation model
compressibility improved the accuracy of the numerical simulation of cryogenic cavitating
Compressibility
flow based on thermal effects. Moreover, the thermal effects delayed or suppressed the
Thermal effects
occurrence and development of cavitation behaviour. The proposed modified compressible ZGB (MCZGB) model can predict compressible cryogenic cavitating flow at various conditions. © 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
* Corresponding author. E-mail address: [email protected] (Z. Zhu). https://doi.org/10.1016/j.ijhydene.2020.01.192 0360-3199/© 2020 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
2
international journal of hydrogen energy xxx (xxxx) xxx
Nomenclature r u p m s a k ε l pv RB T K R Cp L hb C V x DT m_ þ m_
density velocity pressure dynamic viscosity cavitation number volume fraction turbulent kinetic energy turbulent eddy dissipation thermal conductivity saturated vapour pressure single spherical bubble radius temperature thermal conductivity constant pressure coefficient latent heat convection heat transfer coefficient heat capacity volume subscripts temperature drop condensation rate evaporation rate
Subscripts l liquid phase v vapour phase ∞ reference m mixture i, j,k directions of the Cartesian coordinates out outlet c empirical coefficient tur turbulence
Introduction Hydrogen energy is a clean energy source because of its low or zero carbon emission [1,2]. Moreover, hydrogen is easy to prepare and can be recycled [3,4]. Liquid hydrogen, which is often used as fuel for space launch vehicles, has the advantages of high energy and clean combustion [5,6]. Liquid hydrogen is indispensable in the development of the aerospace industry [7e9]. However, liquid hydrogen easily promotes cavitation in the turbo pump of space launch vehicles, causing flow passage components to oscillate and even resulting in spacecraft launch failure [10e13]. Compared with room-temperature cavitating flow, the cryogenic cavitating flows of liquid hydrogen and liquid nitrogen are more affected by thermodynamic properties [14e16]. Therefore, studying the thermal effects of cryogenic cavitating flow is essential to the cryogenic industry. Cavitating flow experiments using liquid hydrogen are extremely difficult to perform; thus, many researchers prefer to use other thermo-sensitive fluids, which are easy to control [17e19]. Franc et al. [20] experimentally investigated the
cavitating flow of Freon R114 in an inducer at different reference temperatures and discovered that cavity length near the inlet vane of the inducer decreases as reference temperature increased. Kikuta et al. [21] experimentally studied the influence of thermal effects on the performance and instability of inducers. They determined the cavity length at the impeller edge between liquid nitrogen and cold water and showed that thermal effects increase with cavity length. Kelly et al. [22] studied thermal cavitating flow by using fluoroketone instead of liquid hydrogen and used high-speed photography to study the unsteady state characteristics of cavitation at various freestream temperatures and cavitation numbers. Ito et al. [23] compared the cavitating flow characteristics of liquid nitrogen at 77.9 K and water at 292.5 K and 333.5 K and showed that the cavity in liquid nitrogen appears as misty bubbles compared with water. Zhu et al. [24] tested the liquid nitrogen cavitating flow in the venturi to obtain a clear picture of liquid nitrogen cavity shedding and further analysed the shedding process. Recently, Chen et al. [25] experimentally analysed the dynamic evolution of liquid nitrogen cavitating flow in a convergingediverging nozzle under a wide range of freestream conditions. The results showed that cavitation dynamics changed from the quasi-isothermal mode to the thermo-sensitive mode as throat temperature increased, and the transition temperature was 77e78 K. Numerical simulations are fundamental in the research of liquid hydrogen cavitating flow [26,27], and cavitation models play a vital role [28,29]. Hosangadi et al. [30] proposed a multiphase flow calculation formula, which considered compressibility, energy balance and thermodynamic property, to verify the influence of thermal effects on cavity interfaces. Cao et al. [31] developed a full cavitation model with the energy equation to simulate thermal cavitating flow and discussed cavitation mechanisms with thermal effects. Shi et al. [32] revised the Merkle cavitation model and performed numerical simulation on liquid nitrogen and liquid hydrogen cavitating flow. The result showed the thermal effects of liquid hydrogen is more obvious than liquid nitrogen. Zhang et al. [33] developed an effective cavitation model considering the thermodynamic transformation theory, the YoungeLaplace equation and the thermodynamic equilibrium hypothesis of the cavitation process. We also modified the ZwarteGerbereBelamri (ZGB) model by using the convective heat transfer approach and corrected the saturated vapour pressure based on the ClausiuseClapron equation [34,35]. The effectiveness were verified by the simulation of liquid nitrogen and fluoroketone cavitating flow. Numerous researchers have neglected vapoureliquid compressibility when simulating liquid hydrogen cavitating flow. In cavitating flow, the effect of compressibility is assignable, especially in cavitation dynamics [36e38]. In the comprehensive analysis of liquid hydrogen cavitating flow, considering gas and liquid compressibility is necessary [39]. Koukouvinis et al. [40] investigated cavitating flow in a diesel injector nozzle using the Tait equation of state. Budich et al. [41,42] simulated the cavitation flow of a sharp convergentedivergent wedge and propeller VP1304 by fully compressible numerical method, and analysed associated shock wave dynamics during cavity collapse. Wang et al. [43] considered vapoureliquid compressibility on the basis of the
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
Saito cavitation model. The numerical simulation method precisely predicted bubble dynamics closely related to bubble flow compressibility. Kyriazis et al.[44] used the Helmholtz energy equation of state to study flashing flows of liquid oxygen in a convergent-diverging nozzle. Le et al.[45] developed an effective numerical simulation method based on the modified Merkle cavitation model and vapoureliquid compressibility. The predicted pressure and temperature fields are more consistent with experimental data than those of other models. Clarify this purpose, a numerical simulation method considering compressibility and thermal effects effectively verified on cavitating hydrofoil and ogive is developed in this paper. The paper is divided into three sections. The first section introduces the derivation process of the numerical simulation method with compressibility and thermal effects. The second section verifies the numerical simulation method on cavitating hydrofoil and ogive, and the relationship between thermal effects and compressibility and their influence on cryogenic cavitating flow are analysed. The last section summarises the paper.
Numerical simulation method for compressible thermo-sensitive cavitating flow Governing equations The governing equation was the two-dimensional (2D) NaviereStokes equation (continuity, momentum and energy equation) with thermal effects and compressibility. The specific equations were as follows: vrm v þ r uj ¼ 0; vt vxj m
(1)
vðrm ui Þ v rm ui uj vp v vui vuj 2 vuk þ ¼ þ þ dij ; ðmm þ mt Þ vt vxi vxj vxi vxj vxi 3 vxk (2) v v v vT _ rm Cpm T þ rCpm Tuj ¼ ðlm þ lt Þ mL; vt vxj vxj vxj
(3)
(4)
rm ¼ arv þ ð1 aÞrl ;
(5)
mm ¼ amv þ ð1 aÞml ;
(6)
thermal effects of the cavitation region were calculated by solving the convective heat transfer equation. And the vapoureliquid twoephase state equation was used to explain compressibility.
Vapour-liquid two-phase state equation Liquid hydrogen and liquid nitrogen share certain similarities at cryogenic temperatures, as shown in Fig. 1. Compared with liquid hydrogen, nitrogen has abundant data and was thus used in the analysis of the characteristics of cryogenic cavitating flow in the present study. The liquid phase was assumed to be a compressible fluid, and state equation was rooted in the Tammann equation [46]. Fig. 2 showed the liquid nitrogen density obtained by the liquid state equation was in good agreement with the liquid nitrogen density at different temperatures and pressure levels provided by the National Institute of Standards and Technology. The liquid state equation can be expressed as [47]: pl þ pc ¼ rl Rl ðTl þ Tc Þ;
(7)
where pc and Tc are the pressure and temperature empirical coefficients, respectively, of the liquid, and Rl is the liquid constant. Thermal effects significantly influenced the liquid densities of liquid nitrogen and liquid hydrogen. Therefore, considering the compressibility of cryogenic cavitating flow on the basis of thermal effects is necessary. The liquid state equation required a boundary condition with a small range. Such condition is precise because the influence of thermal effects is drastic. The liquid state equation was fitted separately for each case for the accuracy of the liquid nitrogen cavitating flow simulation. The pressure empirical coefficient pc, temperature empirical coefficient Tc and the liquid constant Rl can be written as pc ¼ 2:956 108 ; Tc ¼ 1:025 102 ; Rl ¼ 2:040 103 :
(8)
The gas phase can be regarded as an ideal gas
vðarv Þ v arv uj þ ; m_ ¼ m_ þ m_ ¼ vt vxj þ
3
where r, u and p are the density, velocity and pressure, respectively; v, l and m indicate the vapour, liquid and mixture phases, respectively; x and the subscripts i, j and k represent the coordinate axes; mm is the dynamic viscosity, and mt is the turbulent viscosity of the mixture; lm represents mixture thermal conductivity; lt represents turbulent thermal conductivity and T, Cpm, L and a denote temperature, specific heat, latent heat and volume fraction, respectively. The compressibility and thermal effects of the vapour and liquid phases in Eqs. (1)e(6) are explained in the subsequent sections. The
pv ¼ rv Rv Tv :
(9)
Rv ¼ 296.8 J/(kg$K) is the gas constant for nitrogen. In the present study, non-condensable gas was ignored in the gas phase. When local thermodynamic equilibrium was assumed, p ¼ pl ¼ pv and T ¼ Tl ¼ Tv. The mixture density rm can be written as apRl ðT þ Tc Þ þ Rv Tð1 aÞ p þ pc : rm ¼ Rv Rl TðT þ Tc Þ
(10)
The correctness of the vapoureliquid two-phase state equation was verified by fitting the sound speed of the mixture. Fig. 3 shows the sound speed of 293 K water/vapour mixture and 83.06 K liquid nitrogen/nitrogen mixture at different pressures, which are obtained by the mixture speed equation [45], respectively. The speeds were compared with the experimental sound speeds obtained by Karplus [48]. The results are in good agreement with the experimental values.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
4
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 1 e Liquid nitrogen and liquid hydrogen density for pressure and temperature.
Modification of the compressible Zwart-Gerber-Belamri (MCZGB) model The mass transfer equation integrated the heat transfer effect by considering the inertial effect in Eq. (4). In our previous study, the evaporation source term m_ þ and condensation source term m_ can be written as m_ þ ¼ Fvap
3anuc ð1 aÞrv RB
"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 2 max pv ðTÞ p; 0 C0 hb pffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 rl Kl rl Cl
3 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 3arv 4 2 max p pv ðTÞ; 0 C0 hb 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi m_ ¼ Fcond 3 rl RB Kl rl Cl
(11)
(12)
where RB ¼ 1 106 m is the spherical bubble radius; anuc¼ 5 104 is the volume fraction of the gas core point, and Kl and Cl are liquid thermal conductivity and heat capacity, respectively. The thermal conductivity and heat capacity of liquid nitrogen can be regarded as constants in an extremely small temperature range. The variables hb and C0 represent convection heat transfer and experienced coefficients, respectively. The proposed value of C0hb is 1 W/(m2$K). The empirical coefficients (Fvap ¼ 5.0 and Fcond ¼ 0.001) were recalibrated by Sun et al. [50]. The vapoureliquid phase density was replaced by the vapoureliquid twoephase state equation for the cavitation model so that compressibility could be considered. Saturated
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
5
Fig. 2 e Liquid nitrogen density for NIST data and liquid state equation.
Fig. 3 e Relation between sound speed and vapour void fraction in water [47e49] and liquid nitrogen.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
6
international journal of hydrogen energy xxx (xxxx) xxx
vapour pressure pv(T) was expressed by polynomial fitting given that it was based on temperature [51]: pv ðTÞ ¼
4 X
ai T ; i
(13)
i¼1
Where a1 ¼ 7:962 109 ; a0 ¼ 4:796 108 ; a1 ¼ 1:154 107 ; a2 ¼ 1:386 105 ; a3 ¼ 8:275 102 ; a4 ¼ 1:951 100 :
(14)
Compressible correction of turbulence model In the present study, the filterebased density corrected model (FBDCM) was adopted, which can modify the turbulent viscosity and improve the correctness of numerical solution. The FBDCM combines the advantages of densityecorrected and filterebased models and is widely used in the numerical models of cavitation flow. The filterebased density corrected model was described as follows [52,53]: mTFBDCM ¼ Cm
rm k2 fhybrid ; ε
(15)
fhybrid ¼ ffFBM þ ð1 fÞfDCM ;
(16)
C1 ð0:6rm =rl C2 Þ ½2 tanhðC1 Þ; f ¼ 0:5 þ tanh 0:2ð1 2C2 Þ þ C2
(17)
fDCM ¼
rv þ ð1 aÞn ðrl rv Þ r*m ¼ ; rv þ ð1 aÞðrl rv Þ rm
D$ε fFBM ¼ min 1; 2=3 ; k
(18)
(19)
where Cm ¼ 0.0845, C1 ¼ 4 and C2 ¼ 0.2, and k and ε denote turbulent kinetic energy and turbulent eddy dissipation, respectively.
Verification of the numerical simulation method Hord et al. [54e56] performed subscale tests of cavitation in cryogenic fluids (liquid nitrogen and liquid hydrogen) in a transparent plastic blow-down tunnel. The hydrofoil and ogive were installed at the center of water tunnel. Several pressure and temperature sensors were arranged on the upper and lower surfaces of the hydrofoil and ogive. The accuracy of the proposed numerical method in the present study was verified using Hord’s results.
Results of compressible cavitating hydrofoil The 2D model with structured grid was adopted for the creation of a hydrofoil. Given that the hydrofoil was a symmetric structure, half of it was selected as the computational domain, which is shown in Fig. 4. The inlet and the hydrofoil were 12.7 and 3.96 mm in width, respectively. The boundary conditions are speed inlet and pressure outlet. The wall is a free slip wall, and the surface of the hydrofoil is no slip and smooth wall. Simulations are 2D and transient, and the isothermal simulation results without cavitation were used as the initial conditions for numerical calculations. The details of the computational mesh are shown in Fig. 5, and grid independence analysis was performed [34]. Fig. 6 compares the temperature coefficient between the numerical and the experimental results, and mesh 2 is better agreement with the experimental data. The applicability of the MCZGB model was verified by predicting pressure and temperature fields under four different temperatures. The main parameters of each case are listed in Table 1. The pressure and temperature data calculated by the ZGB, MZGB and MCZGB models (Fig. 7) were compared. At four different temperatures, the predicted values of the MCZGB model (blue line) showed better agreement with the experimental results, especially the temperature field. Large temperature and pressure depressions were observed at the leading edge of the hydrofoil. This finding was attributed to the occurrence of mass evaporation near the leading edge because of thermal effects, and the depression was caused by the dissipation of heat from the bulk liquid. Compared with the MZGB model used in our previous work, the MCZGB model showed a better ability to capture the temperature depressions in the cavity, and the temperature depressions are smaller and more consistent with the experiment. In terms of pressure depression, several models are not huge distinction. Therefore, compressibility is a further modification of the thermal effects in the cavitating flow. As liquid nitrogen flowed, the bubbles collapsed, and the temperature and pressure depressions disappeared gradually. The predicted temperature at the cavity tail was somewhat higher than the experimental data due to the blockage effect and the release of latent heat of vaporisation during condensation. The pressure in the cavity closure regions was relatively high because of the collapse of the bubbles. The existing cavitation models were shown in Fig. 8 and compared with the MCZGB model. Table 2 and Table 3 list relative errors of the temperature and pressure which calculated by the existing model and compared with the
Fig. 4 e Hydrofoil computational domain and boundary condition. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
7
Fig. 5 e Three different cases of mesh around the hydrofoil surface: (a) Mesh 1 (total 10,665 elements), (b) Mesh 2 (total 25,325 elements) and (c) Mesh 3 (total 50,423 elements) [42].
Fig. 6 e Comparison of the Tþ values between the simulation and the experiment.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
8
international journal of hydrogen energy xxx (xxxx) xxx
Table 1 e Condition for simulation of hydrofoil. Case
293F 290C 295D 296B
Inlet velocity Cavitation Inlet u∞/(m/s) number s∞ temperature T∞/K 77.90 83.06 83.20 88.54
23.9 23.9 24.3 23.7
1.70 1.70 1.67 1.61
Latent heat L/(J/ kg) 1.98 1.91 1.91 1.83
105 105 105 105
experimental data. In terms of pressure capture, the ability of each cavitation models was roughly the same in capturing the pressure depression processes. However, pressure recovery was unpunctual with Hosangadi’s model, which leaded to inaccurate cavity length prediction. Le’s and Utturkar’s models predicted the pressure at the cavity tail insufficiently, where the MCZGB model was over-predicted. For temperature field, each cavitation model had different degrees of temperature overflow at the cavity tail. Inside the cavity, Le’s model showed a certain temperature profile discontinuity (as indicated by the red points), but the MCZGB model performed excellent in this respect, and also showed a good ability to predict the temperature depression in the cavity. In terms of relative error, the MCZGB model performs relatively well under both temperature and pressure. Moreover, the first point of the pressure data and the fourth and fifth points of the temperature data listed in the Tables, the predictions of all modified models are extremely poor and far from the experimental data. This will be the direction of our future work. As shown in Fig. 9(a), thermal effects became increasingly inhibited as the reference temperature increased. Cavitation intensity gradually decreased, and the closed region of the cavity tail was retracted upstream along the hydrofoil surface. The predicted cavity interface was fuzzy because the MCZGB model considered thermal effects and saturated vapour pressure as functions of temperature. Therefore, this result was an improvement over that of our previous work. The influence of thermal effects on cavitation can be further shown by comparing the mass transfer rates. As shown in Fig. 9(b), the maximum mass transfer rate was obtained at the leading edge of the hydrofoil. This result indicated that vaporisation was relatively intense at this point and led to a significantly sharp temperature depression. The cavity tail captured the minimum mass transfer rate and interdicted the growth of the cavity. As the reference temperature increased, the maximum mass transfer rate at the leading edge of the hydrofoil increased, and the inhibition of thermal effects became increasingly evident. Eventually, the cavity length decreased. Fig. 10 shows the intact temperature field in the computational domain of 296B. The result did not show the inconsistency of the temperature profile described by Le et al. [45]. When liquid nitrogen flowed through the hydrofoil, temperature increased by about 0.1 K due to the blockage effect. By contrast, cavity temperature was reduced by about 2.0 K due to thermal effects. The lowest temperature point was observed at the leading edge of the hydrofoil, and the temperature changed regularly. Owing to the release of latent heat
of vaporisation during condensation and the blockage effect, the temperature abruptly increased at the cavity tail during the calculation. This result is in good agreement with the experimental data of the hydrofoil. Hence, the proposed method in the present study has good accuracy. Fig. 9(c) depicts the mixture density predicted at three temperatures by the MCZGB model. The mixture density outside the cavity was similar to liquid nitrogen density, but the mixture density inside was much greater than nitrogen density at the same temperature. This result indicated that the interior of the cavity was not completely nitrogen but a mixture of small nitrogen bubbles and liquid nitrogen. Utturkar et al. [7] verified this conclusion through numerical simulation on cryogenic fluids. As mentioned before, the relatively fuzzy phenomenon at the cavity interface is well reflected in the mixture density field. Furthermore, as the reference temperature rose from 77.9 K to 88.54 K, the mixture density field tended to be placid. This condition is related to the physical properties of nitrogen. Fig. 11(a) shows liquid nitrogen density field in the computational domain. As the passageway narrowed, the pressure became small, and the liquid nitrogen density decreased slightly. Meanwhile, the density minimum, which is the density value at saturated vapour pressure, was obtained outside the cavity. Within the cavity, liquid nitrogen density increased rapidly because the thermal effect was larger than the pressure effect. The maximum density of liquid nitrogen was attained at the point of maximum temperature depression. Fig. 11(b) shows the nitrogen density field within the cavity. The density at the leading edge of the hydrofoil was lower than nitrogen density at the reference temperature due to temperature and pressure depressions. Therefore, the relationship between thermal effects and compressibility inside the cavity was inseparable. The density field at the boundary of the cavity, whether the nitrogen or liquid nitrogen density field, resulting in foggy cryogenic fluid bubbles.
Simulation of compressible cavitating ogive The compressible thermal cavitation model was further reassessed, and confidence was instilled into the MCZGB model’s validity by simulating liquid nitrogen cavitating flow over an axisymmetric ogive geometry. The computational domain is shown in Fig. 12. The inlet and ogive were 14.7 and 4.53 mm wide, respectively. As shown in Fig. 13, the mesh details covered more than 25,000 grids, and grid density near the ogive wall was refined to guarantee yþ < 10. The main parameters of ogive cases are listed in Table 4. In Fig. 14, the pressure and temperature fields predicted by the ZGB and MCZGB models on the ogive were compared. The MCZGB model was superior. Similarly, the leading edge of the ogive had pressure and temperature depressions, which were more dramatic than those of the hydrofoil possibly because the leading edge of the ogive structure changed abruptly to cause substantial vaporisation. Identically, temperature and pressure were overpredicted near the cavity closure. Fig. 15 shows the contour of the cavity. The predicted pressure field
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
9
Fig. 7 e Numerical results along the hydrofoil in comparison with experimental data.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
10
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 8 e Comparison of MCZGB model with previous cavitation models (290C).
Table 2 e Numerical simulation error based on experimental data (209C, pressure). X(m) 0.005613 0.009042 0.014122 0.023012 0.033172 Average Error
MCZGB model
Le ‘s model
Hosangadi’s model
Utturkar’s model
7.94% 7.10% 4.52% 0.98% 1.62% 4.43%
8.24% 5.33% 4.01% 5.80% 4.66% 5.61%
11.31% 5.27% 7.57% 11.56% 1.33% 7.41%
21.78% 5.33% 3.90% 16.33% 8.81% 11.23%
Table 3 e Numerical simulation error based on experimental data (209C, Temperature). X 0.005994 0.010947 0.017297 0.026822 0.036474 Average Error
MCZGB model
Le ‘s model
Hosangadi’s model
Utturkar’s model
0.00% 0.06% 0.38% 0.52% 0.42% 0.28%
0.11% 0.23% 0.34% 0.45% 0.37% 0.30%
0.28% 0.09% 1.32% 0.36% 0.29% 0.47%
0.58% 0.18% 0.02% 0.36% 0.53% 0.34%
Fig. 9 e Comparisons of vapour volume fraction, mass transfer rate and mixture for different cases. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
11
Fig. 10 e Temperature field in the whole flow passage and cavity of 296B.
Fig. 11 e Numerical results along the hydrofoil (a) Liquid nitrogen density; (b) Nitrogen density.
Fig. 12 e Ogive computational domain and boundary condition. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
12
international journal of hydrogen energy xxx (xxxx) xxx
Fig. 13 e Mesh generation details of ogive computational domain.
Table 4 e Condition for simulation of ogive. Case
312D 322E
Inlet Inlet velocity Cavitation temperature u∞/(m/s) number s∞ T∞/K 83.06 88.56
23.9 25.8
1.70 1.61
Latent heat L/(J/ kg) 1.91 105 1.83 105
was stairestepping due to the contour of the cavity tail. The cavity contour of 322E diminished due to increasing thermal effects. Nevertheless, the present numerical simulation
method produced a reasonable range of results for the ogive structure. The density field of nitrogen and liquid nitrogen in the ogive can be acquired simply from Fig. 16. Compared with the hydrofoil, the density of nitrogen showed more intense in the ogive. The density field of liquid nitrogen in 312D was similar to that in 296B. However, the liquid nitrogen density behind the cavity closure did not change as regularly as that of hydrofoil because the upper-wall surface on the flow field produced a small cavity. Liquid nitrogen density still decreased because of the release of the latent heat of vaporisation and
Fig. 14 e Numerical results along the ogive in comparison with the experimental data. Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
13
Fig. 15 e The predicted vapour volume fraction contour of (a) 312D and (b) 322E.
Fig. 16 e Numerical results in the ogive. (a) Liquid nitrogen density field; (b) Nitrogen density field.
pressure depression caused by the narrowing of the passageway.
Summary This study develops an effective compressible thermal cavitation model (MCZGB model), which considers the compressibility of the vapoureliquid phase and the thermal effects of cryogenic fluids in the simulation of compressible cryogenic cavitating flow. Comparing with the existing cavitation models, the error between the simulated data of the MCZGB model and the experimental data is minimal. The MCZGB model can be accurate prediction of cavity temperature depression. This is an advantageous for studying the effects of thermal effects and compressibility in cryogenic cavitating flow. The main conclusions are as follows: (1) The prediction results of the MCZGB model are in good agreement with the experimental data of the liquid nitrogen cavitating hydrofoil and ogive. The predicted temperature field has not shown the inconsistency of the temperature profile, which presents a smaller temperature depression and is closer to the experimental data. The numerical simulation method can predict a compressible cryogenic cavitating flow well.
(2) Thermal effects can delay or suppress the occurrence and development of cavitation behaviour. In a cryogenic cavitating flow, the influence of thermal effects on the compressibility of the vapour-liquid phase is greater than that of the pressure effect. With the increase in reference temperature, temperature depression within the cavity becomes conspicuous, and the strength of the thermal effects increases. Hence, mass transfer rate increases, and the cavity shortens. (3) The effect of compressibility in the numerical simulation method is analysed. Compressibility further improves the accuracy of the numerical simulation of cryogenic cavitating flows on the basis of thermal effects. Compared with the existing models, MCZGB model which considers the compressibility on the basis of MZGB model performs better. The interior of the cavity comprises small nitrogen bubbles and liquid nitrogen. Under the influence of thermal effects and compressibility, the interfaces of the cryogenic fluid cavity are foggy. The model exhibiting high prediction accuracy can fully analyze the thermal effects and compressible effects in cryogenic cavitation flows. However, it only applies to certain type of structure (hydrofoil, ogive, and inducer) at the present stage. Commercial/industrial applications of this model will be our next work.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
14
international journal of hydrogen energy xxx (xxxx) xxx
Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant No. 51776189, 51976202), the Natural Science Foundation of Zhejiang Province (Grant No. LR20E090001). The supports are gratefully acknowledged.
references
lu TN. “Green” path from fossil[1] Muradov NZ, Vezirog based to hydrogen economy: an overview of carbonneutral technologies. Int J Hydrogen Energy 2008;33:6804e39. [2] Coutier-Delgosha O, Patella RF, Reboud JL. Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation. ASME J Fluid Eng 2008;125:38e45. n G, Valde s-Solı´s T. Towards the hydrogen economy? [3] Marba Int J Hydrogen Energy 2007;32:1625e37. [4] Ball M, Wietschel M. The future of hydrogen e opportunities and challenges. Int J Hydrogen Energy 2009;34:615e27. [5] Liu M, Tan L, Cao S. Cavitationevortexeturbulence interaction and one-dimensional model prediction of pressure for hydrofoil ALE15 by large eddy simulation. ASME J Fluid Eng 2019;141:21103e17. [6] Merouani S, Hamdaoui O, Rezgui Y, Guemini M. Computational engineering study of hydrogen production via ultrasonic cavitation in water. Int J Hydrogen Energy 2016;41:832e44. [7] Zheng X, Lin Z, Xu BY. Thermal conductivity and sorption performance of nano-silver powder/FAPO-34 composite fin. Appl Therm Eng 2019;160:114055. [8] Abderrezzak B, Yong H. A contribution to the understanding of cavitation effects on droplet formation through a quantitative observation on breakup of liquid jet. Int J Hydrogen Energy 2016;41:15821e8. [9] Takakuwa O, Soyama H. Suppression of hydrogen-assisted fatigue crack growth in austenitic stainless steel by cavitation peening. Int J Hydrogen Energy 2012;37:5268e76. [10] Smirnov N, Betelin V, Shagaliev R. Hydrogen fuel rocket engines simulation using LOGOS code. Int J Hydrogen Energy 2014;39:10748e56. [11] Jothi S, Croft T, Wright L. Multi-phase modelling of intergranular hydrogen segregation/trapping for hydrogen embrittlement. Int J Hydrogen Energy 2015;40:15105e23. [12] Tao J, Lin Z, Ma C, Ye J, Zhu Z, Li Y, Mao W. An experimental and numerical study of regulating performance and flow loss in a V-port ball valve. ASME J Fluids Eng 2020;142(2):021207. [13] Sanad A. Modelling of flow of cryogenic fluids (oxygen) utilised in rocket engines. City University; 2018. [14] Pei J, Osman MK, Wang W. Unsteady flow characteristics and cavitation prediction in the double-suction centrifugal pump using a novel approach. Proc IME J Power Energy 2019. https://doi.org/10.1177/0957650919863636. [15] Huang B, Qiu SC, Li XB, Wu Q, Wang GY. A review of transient flow structure and unsteady mechanism of cavitating flow. J Hydrodyn 2019;31:429e44. [16] Dular M, Coutier-Delgosha O. Thermodynamic effects during growth and collapse of a single cavitation bubble. J Fluid Mech 2013;736:44e66. [17] Li X, Chen B, Luo X, Zhu Z. Effects of flow pattern on hydraulic performance and energy conversion characterisation in a centrifugal pump. Renew Energy 2019. https://doi.org/10.1016/j.renene.2019.11.049.
[18] Dular M, Khlifa I, Fuzier S, Maiga MA, Coutier-Delgosha O. Scale effect on unsteady cloud cavitation. Exp Fluid 2012;53:1233e50. [19] Yang H, Zhang W, Zhu Z. Unsteady mixed convection in a square enclosure with an inner cylinder rotating in a bidirectional and time-periodic mode. Int J Heat Mass Tran 2019;136:563e80. [20] Franc JP, Janson E, Morel P, Rebattet C, Riondet M. Visualizations of leading edge cavitation in an inducer at different temperatures. Fourth International Symposium on Cavitation 2001:124e30. [21] Kikuta K, Yoshida Y, Watanabe M, Hashimoto T, Nagaura K, Ohira K. Thermodynamic effect on cavitation performances and cavitation instabilities in an inducer. Nihon Kikai Gakkai Ronbunshu B Hen/transactions of the Japan Society of Mechanical Engineers Part B 2008;130:1349e57. [22] Kelly S, Segal C. Experiments in thermosensitive cavitation of a cryogenic rocket propellant. Gainesville: A Dissertation of University of Florida; 2012. p. 83e216. [23] Ito Y, Tsunoda A, Nagasaki T. Experimental comparison of backflow-vortex cavitation on pump inducer between cryogen and water. J Phys Conf 2015:656. [24] Zhu J, Xie H, Feng K, Zhang X, Si M. Unsteady cavitation characteristics of liquid nitrogen flows through venturi tube. Int J Heat Mass Tran 2017;112:544e52. [25] Chen T, Chen H, Liang W, Huang B, Le Xiang. Experimental investigation of liquid nitrogen cavitating flows in converging-diverging nozzle with special emphasis on thermal transition. Int J Heat Mass Tran 2019;132:618e30. [26] Cheng H, Bai X, Long X, Ji B, Peng X, Farhat M. Large eddy simulation of the tip-leakage cavitating flow with an insight on how cavitation influences vorticity and turbulence. Appl Math Model 2020;77:788e809. [27] Li X, Jiang Z, Zhu Z, Si Q, Yi L. Entropy generation analysis for the cavitating head-drop characteristic of a centrifugal pump. Proc IME C J Mech Eng Sci 2018;232:4637e46. [28] Dular M, Stoffel B, Sirok B. Development of a cavitation erosion model. Wear 2006;261:642e55. [29] Li X, Li B, Yu B, Ren Y, Chen B. Calculation of cavitation evolution and associated turbulent kinetic energy transport around a NACA66 hydrofoil. J Mech Sci Technol 2019;33:1231e41. [30] Hosangadi A, Ahuja V. Numerical study of cavitation in cryogenic fluids. ASME J Fluid Eng 2005;127:267e81. [31] Cao X, Zhang X, Li M. Validation of full cavitation model in cryogenic fluids. Chin Sci Bull 2009;54:1633e40. [32] Shi S. Numerical calculation of thermal effect on cavitation in cryogenic fluids. Chin J Mech Eng 2012;25:1176e83. [33] Zhang X, Zhao W, Xiang S, Qiu L. Modeling cavitation flow of cryogenic fluids with thermodynamic phase-change theory. Sci Bull 2013;58:567e74. [34] Zhang S, Li X, Zhu Z. Numerical simulation of cryogenic cavitating flow by an extended transport-based cavitation model with thermal effects. Cryogenics 2018;92:98e104. [35] Zhang S, Li X, Hu B, Liu Y, Zhu Z. Numerical investigation of attached cavitating flow in thermo-sensitive fluid with special emphasis on thermal effect and shedding dynamics. Int J Hydrogen Energy 2019;44:3170e84. [36] Schnerr G, Sezal I, Schmidt S. Numerical investigation of three-dimensional cloud cavitation with special emphasis on collapse induced shock dynamics. Phys Fluids 2008;20:040703. [37] Wei Z, Bai XD, Zheng M, Gang C, Yong W. Compressible effect on the cavitating flow: a numeric study. J Hydrodyn 2017;29:1089e92. [38] Schmidt S, Mihatsch M, Thalhamer M. Assessment of erosion sensitive areas via compressible simulation of unsteady cavitating flows. Advanced experimental and
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192
international journal of hydrogen energy xxx (xxxx) xxx
[39]
[40]
[41]
[42]
[43]
[44] [45]
[46]
numerical techniques for cavitation erosion prediction 2014:329e44. Vidal A, Rodriguez C, Koukouvinis P. Modelling of diesel fuel properties through its surrogates using perturbed-chain, statistical associating fluid theory. Int J Engine Res 2018:1468087418801712. Koukouvinis P, Naseri H, Gavaises M. Performance of turbulence and cavitation models in prediction of incipient and developed cavitation. Int J Engine Res 2017;18:333e50. Budich B, Schmidt S, Adams N. Numerical simulation and analysis of condensation shocks in cavitating flow. J Fluid Mech 2018;838:759e813. Budich B, Schmidt S, Adams N. Numerical investigation of a cavitating model propeller including compressible shock wave dynamics. Proceedings of the Fourth International Symposium on Marine Propulsors 2015;31. Wang C, Huang B, Wang G, Duan Z, Ji B. Numerical simulation of transient turbulent cavitating flows with special emphasis on shock wave dynamics considering the water/vapor compressibility. J Hydrodyn 2018;30:37e55. Kyriazis N, Koukouvinis F, Karathanassis I. A tabulated data technique for cryogenic two-phase flows. 2018. Le AD, Junosuke O, Iga Y. Modification of energy equation for homogeneous cavitation simulation with thermodynamic effect. ASME J Fluid Eng 2018;141:1e12. Chen HT, Collins R. Shock wave propagation past an ocean surface. J Comput Phys 1971;7:89e101.
15
[47] Iga Y, Nohml M, Goto A, Ikohagi T. Numerical analysis of cavitation instabilities arising in the three-blade cascade. ASME J Fluid Eng 2004;126:110e6. [48] Karplus HB. Velocity of sound in a liquid containing gas bubbles. J Acoust Soc Am 1957;29:1261e2. [49] Henry RE, Grolmes MA, Fauske HK. Pressure-pulse propagation in two-phase one- and two-component mixtures. Technical Report Archive & Image Library; 1971. [50] Sun TZ, Xiangfu MA, Wei YJ, Cong W. Computational modeling of cavitating flows in liquid nitrogen by an extended transport-based cavitation model. Sci China Technol SC 2016;59:337e46. [51] Li PS, Ren H. The fitting of the saturation pressure on hydrogen and oxygen. Structure & Environment Engineering 2010;38:60e3. [52] Johansen ST, Wu J, Wei S. Filter-based unsteady RANS computations. Int J Heat Fluid Flow 2004;25:10e21. [53] Zhu Z, Qian Z, Guo Z. Numerical simulation of cavitation around a Clark-Y hydrofoil with different LES models in OpenFOAM. J Drain Irrig Mach Eng 2018;36(4):277e82. [54] Hord J. Cavitation in liquid cryogens. 1: venturi. NASA CR2054 1972. [55] Hord J. Cavitation in liquid cryogens. 2: hydrofoil. NASA CR2056 1973. [56] Hord J. Cavitation in liquid cryogens. 3: ogives. NASA CR-2242 1973.
Please cite this article as: Li X et al., Extended compressible thermal cavitation model for the numerical simulation of cryogenic cavitating flow, International Journal of Hydrogen Energy, https://doi.org/10.1016/j.ijhydene.2020.01.192