1 Introduction

Cavitation is usually defined as the breakdown of a liquid medium under the saturated vapor pressure[1]. This dynamic process can be found in marine propellers, pumps, hydrofoils and other hydraulic machine-ries by observing the formation, growth and collapse of vapor bubbles. A comprehensive understanding of the cavitation shedding is of crucial importance to explaining the vibration, erosion and power loss of hydraulic machinery operating under cavitating conditions.

The sharp changes in the fluid density, and the requirement of modelling phase change were the factors in numerical simulation of cavitation flows[2]. Accor-ding to the assumption of a homogenous equilibrium medium proposed by JI et al[3], the vapor and liquid phases were considered as homogeneous mixture in order to solve only one set of equations for the mass and momentum. SAUER[4] proposed the equations by ma-king some improvements of the Rayleigh equation, while MERKLE et al[5]and KUNZ et al[6]presented semi-analytical equations.

The interface between liquid and vapor was tracked by the volume of fluid (VOF) method, in which the volume fraction function γ was defined[7]. Many researchers have used this method to capture the cavity shape for its accuracy and robustness. For example, PENG et al[8] and BENSOW et al[9] used the VOF technology to simulate cavitation over hydrofoils and propellers respectively.

The simulation was associated with the experiments conducted by WANG et al[10], in which hairpin type of counter-rotating vapor vortices were observed at incipient cavitation, and large-scale vortex structure and rear re-entrant jet were found at cloud cavitation. HUANG et al[11], simulated the cavitating flow over the Clark-Y hydrofoil with Partially Averaged Navier-Stocks (PANS) model and proved the consistency between the simulation and experiments. Meanwhile, JOHAN et al[12] discussed the cavitation dynamics by different cavitating numbers and cavitation models with the same Large Eddy Simulation (LES) turbulence model. LES was based on resolving the large-scale eddies while modelling subgrid-scales. Many reviews indicated that both accurate discretization methods and the subgrid-scale models were important factors in LES[12].

硬件方面：一是外观设计标志化。对所有设计进行定标，使检测室具有食药监识别性。二是必备配置统一化。实现检测室“六个一”，包括一个门头、一块屏幕、一个检测设备、一个标准操作台、一套检测制度、一条文化标语。三是内部装饰可复制化。内部家具均来自于宜家，保证了后期检测室的持续复制。

In the present study, a finite volume, multiphase solver in the framework of OpenFOAM is also used to calculate the flow field of the cavitating flow over a Clark-Y hydrofoil by adopting two different subgrid-scale (SGS) models, namely, one equation eddy viscosity (oneEqEddy) and Smagorinsky in order to find a suitable model.

2 Numerical models

2.1 VOF Model

In the paper, γ is the fraction of water volume, γ=0 denotes vapor and γ=1 is liquid. Cells which lie between 0 and 1 contain interface regions. This can be written as[13]

车里只有一个男人，大概40多岁，他背着他的登山包，从楼梯上走下来，从包里拿出一个保温杯，朝着热水壶走去。随后，他坐在了父亲旁边的位子上。父亲随即将桌上放着的小包拿开，他朝父亲微微一笑，“你一个人过来爬山啊？”父亲开口问道；“是啊，呵呵”带着一股浓浓的川渝口音，他回答道。

 

(1)

An improved version of "The Compressive Interface Capturing Scheme for Arbitrary Meshes (CICSAM)" VOF technique[2] is used in OpenFOAM, which is defined as

+·(γv)+·[vcγ(1-γ)]=0,

(2)

 

(3)

where vc is interface-compressive velocity, and Cγ is a constant defining the intensity of the compression.

2.2 Transport based equation model

The mixture density ρ and the turbulent viscosity μ are defined, respectively, as follows

ρ=γρl+(1-γ)ρv,

(4)

μ=γμl+(1-γ)μv.

(5)

The transport equation for the phase change is

+

(6)

由国际商务专业毕业班组建的武汉自贸城班，其培养计划和教学内容、师资队伍大多是学校在原有的基础上制定的，企业并没有根据自身需求进行研究和调整。合作的关系主要依靠人脉和信誉，合作的内容局限在双方共建基地、顶岗实习等，很难做到将生产环节引入学校，或者在企业中实施教学。从合作教育整体来看，校企合作不够深入，关系不够稳定，一旦领导层有变动，就会造成合作关系的中断，学校和企业之间没有形成真正的依赖关系。合作的形式停留在表面，在实质上缺乏突破。

本研究中，对照组用常规护理，试验组用心理护理方法。结果显示，试验组满意程度、心理健康指标汉密尔顿相关指数、维持性血液透析配合程度、心理自我调节能力、维持性血液透析过程不良事件方面相比对照组更有优势，差异有统计学意义（P＜0.05）。

实验室管理在创新人才培养过程中有着极其重要的作用。以“学生为主体，教师做引导，全方位培养，多层次交流”的管理理念，“导师负责制”为主体的管理模式和以创新能力培养为中心的管理制度，在实验室建设中取得了很好的实践效果，对学生综合素质提升具有重要的推动作用[1]。实验室管理的主要任务是对实验室所进行的一系列活动的组织、计划、协调和控制，以及对相关场所的管理，以保证实验任务的顺利完成[2]。

 

(7)

where R is the radius of the bubbles,pv is the saturated vapor pressure,Ce and Cc are coefficients related to processes of condensation and vaporization[14].

2.3 Large Eddy Simulation

整个上午，我们都关在屋里。中间去个茅厕，也是有个东洋兵紧跟着，站在茅厕外面。中饭时候儿，早上那个兵又来送饭。我刚要开口，他低头小声说：“你莫为难我。我帮不了你。”把东西搁桌上就着急走了。

 

(8)

Applying the filter (8) to the Navier-Stokes equations, the filtered equations are expressed as

+

(9)

(10)

where v+vT) is the rate-of-strain is the viscous stress tensor, is the subgrid stress tensor and can be computed in a wide variety of SGS models.

In Smagorinsky model and oneEqEddy model, the filters are carried out implicitly with the filter width Δ defined by the grid based on the assumption that the deviatoric part of the subgrid stress and the deviatoric part of the rate-of-strain tensor are aligned, such as

In Smagorinsky model vsgs is defined as

 

(11)

In oneEqEddy model vsgs is modeled as with k being the solution of

·(vsgs

(12)

where CD,Ck and Cε are model coefficients.

3 Mesh generation and calculation conditions

When decreasing the cavitation number, four different types of cavitation can be classified by the cavity length and shape, in which cloud cavitation accompanied by the highly unsteady trailing edge and vortex shedding has been a hot topic in hydrodynamic field. According to the experiments conducted by WANG et al[10], cloud cavitation was formed when the Reynolds number equals 7.0×105 and the cavitation number σ was 0.8, therefore, we can have the calculation conditions as described in Tab.1.

  

Fig.1 Geometry of computation domain

Structured grid was used in the whole domain and the C-type meshes were generated near the wall of the hydrofoil. The grid was created in ICEM software with 246 813 cells and then converted into the format for that OpenFOAM. The meshes around the hydrofoil were refined to ensure that the maximum value of y+ was less than one as shown in Fig.2.

  

Fig.2 Refined meshes and y+ around the hydrofoil

Two non-dimensional numbers, Reynolds number (Re) and cavitation number σ, define the calculation condition.

 

(13)

 

(14)

where u∞ is the free stream velocity, v is the kinematic viscosity of water at constant temperature of 293 K,ρ is the density of water, and the saturated vapor pressure of water pv equals 2.34 kPa.

As illustrated in Fig.1, the computation domain represents the test section part of the cavitation tunnel. The two-dimensional Clark-Y hydrofoil was clamped on both sides of the tunnel walls and certain cavitation number can be reached by a regulation system. The chord length of the hydrofoil is c=0.07 m and the attack angle equals 8°.

where u∞ is the free stream velocity, ρ is the density of water, L is the lift force exerted upon the hydrofoil and D is the drag force.

新华社讯 葡萄牙国防部长洛佩斯因去年发生的坦科斯军火库失窃案于近日辞职。据媒体报道，洛佩斯向总理科斯塔递交了辞职信，科斯塔接受了他的辞呈。洛佩斯在信中说，他之所以决定辞职是为了避免葡武装部队受到政治攻击的侵蚀以及政府针对他的指控。他还否认知晓任何企图掩盖和保护武器偷盗者的行动。

 

Tab.1 Boundary conditions of the simulation

  

PatchBoundaryconditionsInletVelocityinletu∞=10m/sOutletPressureoutletpr=42.3kPaTopandbottomNoslipFrontandbackEmptyHydrofoilNoslip

4) Cavity begins to disappear.

4 Results and discussion

The lift coefficient CL and the drag coefficient CD are defined as

 

(15)

 

(16)

对此，笔者在多年的小学数学教学实践中，依据学生的年龄特点，摸索出一套趣味审题“四部曲”，在实际教学中运用取得了一定的成效，尝试整理如下。

Since lift force is generated by the pressure diffe-rence between the cavity and the hydrodynamic pressure, the lift coefficient will increase in pace with the cavity length. However, after cavity length reaches its maximum, the pressure distribution at the trailing edge of the hydrofoil is influenced by the cavity shedding, which results in the oscillation of the lift coefficients. Fig.3 shows the variation of lift coefficients on one ca-vitation cycle using two models in addition to experimental data in reference [10].

  

Fig.3 Variation of lift coefficient with time

As shown in Fig.3, two SGS models over predict the maximum lift coefficients although the minimum lift coefficients are nearly equal. Besides, oneEqEddy model predicts more complex oscillatory peaks and the result of Smagorinsky model has a better consistence with that of experiments. The averages of lift and drag coefficients on one cavitation cycle between experiments and the simulation are presented in Tab.2. Results of the two SGS models indicate that different SGS models predict different averages of force coefficients and the drag coefficient of the simulation is inaccurate because the maximum error is about 20%. Remar-kably, the maximum error of oneEqEddy model in reference [2] is 16.66%, this disparity may be caused by the different parameters used in the simulation and measured in the experiment such as average diameter of nuclei and number of nuclei per volume.

 

Tab.2 Averages of lift and drag coefficients

  

CLError/%SmagorinskyoneEqEddyExperiment[10]0.6900.7200.7609.25.3-CDError/%SmagorinskyoneEqEddyExperiment[10]0.1430.1410.12019.217.5-

Time-averaged velocity distribution at different locations is one of the important factors to study the flow details. The LDV measurements start from the surface point of the hydrofoil which are located at x/c=20%, 40%, 60%, 80%, 100%, 120% and end near y=0.025 m. When we consider the difficulties in measu-ring the velocity profiles near the closure region, it is more representative to compare the CFD results with experiment data in the middle part of the hydrofoil. Detailed information of velocity profiles at x/c=40% and x/c=60% is illustrated in Fig.4 and Fig.5 respectively. From the numerical results, the ordinate of the maximum velocity is about y=0.01 m, Smagorinsky model predicts higher velocity than oneEqEddy model when the value is higher than 0.01. According to the velocity profiles in Fig.5, there is reverse flow and the velocity of OneEqEddy model is slightly higher than that of Smagorinsky model in the near-wall region.

  

Fig.4 Time-averaged velocity at x/c=40%

  

Fig.5 Time-averaged velocity at x/c=60%

Comparing the two SGS models with the experiment data, it is clear that there is a great difference near the inflection point of the curve. This region is the interface between liquid and vapor phases based on the estimated thickness in reference[10].This disparity between the simulation and experiment is greater along the direction of flow in the middle part of the hydrofoil. In the near-wall region, the velocity profiles calculated by these two SGS models are consistent with experiment data and the difference between out-of-cavity flow velocities with these two models is minimal.

According to the experimental results of reference[10], time evolution of cloud cavitation from the side view is shown in Fig.6. The frames in Fig.6 show the cavitation cycle as follows[2]

1) Cavity appears at the leading edge of the hydrofoil.

前面所描述的行为逻辑表示任意的动态事件，不仅改变了变元指派或者计算机状态。逻辑更适合处理自然语言，而且适用于会话、博弈策略等。所有涉及语言动态效果的意识，都可以认为是基于初始状态的可计算情况，自然语言是基于认知行为的程序语言。

2) Cavity grows until it occupies most of the hydrofoil.

The specific interphase mass transfer rate in Schnerr-Sauer Model is given as below

3) The massive cavity shedding appears and a re-entrant jet can be observed.

加强县乡人大工作和建设是坚持和完善人民代表大会制度、加强基层国家政权建设的可靠保证。在中国特色社会主义进入新时代这一新的历史条件下，县乡人大只有与时俱进、创新发展，才能为社会主义民主法治建设作出新的更大贡献。

The cavitation mass transfer model in numerical simulations was Schnerr-Sauer. The only difference between simulations was the application of oneEqEddy subgrid-scale model and Smagorinsky subgrid-scale model. The pressure implicit with splitting of operators (PISO) algorithm was used to solve the governing equations and the maximum courant number was set to be 0.6 by adopting adaptive time step.

The cutoff filter can be written as

  

Fig.6 Time evolution of the cloud cavitation from the side view[10]

Fig.7 presents cavitation dynamics of oneEqEddy (left) model and Smagorinsky (right) model.

  

Fig.7 Contour of the fraction of liquid γ, oneEqEddy (left) and Smagorinsky (right)

Based on the results, the following features can be concluded:

1) Different SGS models predict different cavity structures and the difference between oneEqEddy mo-del and Smagorinsky model is greater over time.

2) The re-entrant jet can be observed in both SGS models.

3) The cavity structure at the trailing edge of OneEqEddy model is more complicated.

②Kingsley Davis，“The Forms of Illegitimacy”，Social Forces，77(18)，1939，pp．78 ～84．

Pressure contours of oneEqEddy (left) model and Smagorinsky (right) model are illustrated in Fig.8. Based on the comparison and analysis, some findings are listed as below:

增加零点对系统有什么影响，零点的大小对系统的影响如何，这是自动控制原理教学中的难点。将例题中的传递函数改为Ф相当于零点值是变化的。当α=1,0.5,1,2时，系统的单位阶跃响应如图6所示。

1) Low-pressure area mainly distributes along the hydrofoil, there is little difference before cavity shedding between oneEqEddy model and Smagorinsky model.

2) A local-high pressure region exists near the trailing edge of the hydrofoil.

3) The Smagorinsky model predicts higher pressure than oneEqEddy model.

  

Fig.8 Contours of pressure, oneEqEddy (left) and

 

Smagorinsky (right)

5 Conclusions

Two subgrid-scale models of LES including the Smagorinsky model and the oneEqEddy model were implemented for predicting the cavitation flow over the Clark-Y hydrofoil. According to the results of the simulation, lift force performs an oscillatory behavior with time and oneEqEddy model predicts a little more accurate lift coefficients than Smagorinsky model, and these two models can be used to predict the velocity profiles in the cloud cavitation regime.

Acknowledgements

The research was supported by the National Natural Science Foundation of China(No. 51422906).

References

[ 1 ] FRANC J P, MICHEL J M. Fundamentals of cavitation[M]. Netherlands:Springer, 2005.

[ 2 ] ROOHI E, ZAHIRI A P, PASSANDIDEH-FARD M. Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model[J]. Applied mathematical modelling, 2013, 37(9):6469-6488.

[ 3 ] JI B, LUO X W, ARNDT R E A, et al. Large eddy simulation and theoretical investigations of the transient cavitating vortical flow structure around a NACA66 hydrofoil[J]. International journal of multiphase flow, 2015, 68(68):121-134.

[ 4 ] SAUER J. Instationaren kaviterendeStromung — Ein neues Modell, baserend auf Front Capturing (VOF) and Blasendynamik[D]. Karlsruhe: Karlsruhe Universitat, 2000.

[ 5 ] MERKLE C L, FENG J, BUELOW P E O. Computational modeling of the dynamics of sheet cavitation[C]//Third International Symposium on Cavitation, Grenoble, France, 1998.

[ 6 ] KUNZ R F, BOGER D A, STINEBRING D R, et al. A preconditioned Navier-Stokes method for two-phase flows with application to cavitation prediction[J]. Computers & fluids, 2000, 29(8):849-875.

[ 7 ] HIRT C W, NICHOLS B D. Volume of fluid (VOF) method for the dynamics of free boundaries[J]. Journal of computational physics, 1981, 39(1):201-225.

[ 8 ] PENG X X, JI B, CAO Y, et al. Combined experimental observation and numerical simulation of the cloud cavitation with U-type flow structures on hydrofoils[J]. International journal of multiphase flow, 2016, 79:10-22.

[ 9 ] BENSOW R E, BARK G. Simulating cavitating flows with LES in openFOAM[C]//Proceedings of Computational Fluid Dynamics, Lisbon, Portugal, 2010:14-17.

[10] WANG G, SENOCAK I, WEI S, et al. Dynamics of attached turbulent cavitating flows[J]. Progress in aerospace sciences, 2001, 37(6):551-581.

[11] HUANG B, WANG G Y. Partially averaged Navier-Stokes method for time-dependent turbulent cavitating flows[J]. J. Hydrodyn, 2011, 23(1):26-33.

[12] JOHAN Meyers, MARTINE Baelmans. Determination of subfilter energy in large-eddy simulations[J]. Journal of turbulence, 2004, 527(5):1468-5248.

[13] ASNAGHI A. Interphase change foam tutorial and PANS turbulence model[D]. Chalmers: Chalmers University of Technology, 2013.

[14] BIN J I, LUO X W, PENG X X, et al. Numerical investigation of the ventilated cavitating flow around an under-water vehicle based on a three-component cavitation model[J]. J. Hydrodyn. Ser.B, 2010, 22(6):753-759.

[15] FUREBY C. On the justification and extension of mixed models in LES[J]. Journal of turbulence, 2007, 8(8):1-17.

然而，由“看袜”数钱数到手抽筋，绝非酒铺老板王嬷嬷能预料到的。那时，尚无“眼球经济”一说。但名人效应的道理，王嬷嬷并不陌生。加之，看客围观、起哄、架秧子，是“看热闹不嫌事大”。李肇《唐国史补》一书，载有王嬷嬷发迹经过，上“命高力士缢贵妃于佛堂前梨树下。马嵬店媪收得锦靿一只。相传过客每一借翫，必须百钱，前后获利极多，媪因至富”。