A simple cavitation model for unsteady simulation and its application to cavitating flow in twodimensional convergent-divergent nozzle

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1 IOP Conference Series: Materials Science and Engineering OPEN ACCESS A simple caitation model for unsteady simulation and its application to caitating flow in twodimensional conergent-diergent nozzle To cite this article: Y Yamamoto et al 2015 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. Related content - A Thermodynamic Caitation Model for Caitating Flow Simulation in a Wide Range of Water Temperatures Zhang Yao, Luo Xian-Wu, Jibin et al. - Suitability research on the caitation model and numerical simulation of the unsteady pulsed caitation jet flow S Y Chen, X F Yu, D Y Luan et al. - A Computational Study of Caitation Model Validity Using a New Quantitatie Criterion Hagar Alm El-Din, Zhang Yu-Sheng and Medhat Elkelawy This content was downloaded from IP address on 07/06/2018 at 00:21

2 A simple caitation model for unsteady simulation and its application to caitating flow in two-dimensional conergentdiergent nozzle Y Yamamoto 1, S Watanabe 2 and S I Tsuda 2 1 Graduate School of Engineering, Kyushu Uniersity, 744 Motooka, Nishi-ku, Fukuoka, Fukuoka , Japan 2 Department of Mechanical Engineering, Kyushu Uniersity, 744 Motooka, Nishi-ku, Fukuoka, Fukuoka , Japan fmnabe@mech.kyushu-u.ac.jp Abstract. In this paper, a simple caitation model is deeloped under the framework of homogeneousone-fluid model, in which the perfect mixture of liquid and apor phases is assumed. In most of conentional models, the apor phase is considered as a dispersed phase against the liquid phase as a continuous phase, while in the present model, two extreme conditions are considered: for low oid fraction, dispersed apor bubbles in continuous liquid phase, while for high oid fraction, dispersed droplets in continuous apor phase. The growth of bubbles and droplets are taken into account in the mass transfer between apor and liquid phases, and are switched according to the local oid fraction. The model is applied for the simulation of caitating flow in a two-dimensional conergent-diergent nozzle, and the result is compared with that using a conentional model. To enhance the unsteadiness of caitation due to the instability at the caity interphase, the turbulent shear stress is modified depending upon the continuous phases in combination with the proposed caitation model, which drastically reduces the turbulent iscosity for high oid fraction region. As a result, the unsteadiness of caitation obsered in experiments is well reproduced. 1. Introduction The recent rapid progress of computer science has enabled us to simulate the caitating flow and achiee qualitatie, and in some extent, quantitatie agreements with actual obsered flows. Howeer, such caitation CFD (Computational Fluid Dynamics) still often fails to predict the caitation performance een in simple cases of caitating hydrofoil (Kato [1]). Moreoer, still due to the limited computer resources, Reynolds-Aeraged Naier-Stokes (RANS) simulation is often used to simulate the unsteady caitating flow. Howeer, the reproduction of the unsteadiness of caitation is ery limited, and key unsteady caitation phenomena such as a re-entrant jet which deelops beneath the sheet caity and the formation of cloud caity can neer be simulated with RANS model deeloped based on incompressible flow. On the other hands, Reboud et al. [2]considered that the incompressible turbulence model oerestimates the turbulent iscosity in the two-phase region, and has proposed the well-known Reboud correction to cut the turbulence iscosity in most of two phase flow region. With the tuned parameter, this correction method has successfully reproduced the major unsteadiness of caitation in cases of a two-dimensional conergent-diergent nozzle [2], a two-dimensional hydrofoil [], and a three-dimensional twisted hydrofoil [4]. Content from this work may be used under the terms of the Creatie Commons Attribution.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd 1

3 The caitating flows in two-dimensional conergent-diergent nozzles hae been studied by many researchers, because of their simple structures but reealing important unsteady natures of caitating flow such as re-entrant jet eolutions and ertical cloud caity shedding. Stutz and Reboud [5], [6] hae measured, by using a double optical probe, two-phase structures inside the caity on the throat section of conergent diergent nozzle. Their nozzles hae a common conergent angle of 4ºbut two different dierging angles of 4ºand 8º. Keil et al. [7] hae studied an aggressieness of collapse of cloud caitation in this kind of nozzle for the wide ariety of Reynolds number. In our preious study [8], we inestigated the unsteady character of caitation in two-dimensional conergent-diergent nozzles with diergent angles of 4.0 and 8.4º. In both nozzles, we could not obsere the re-entrant jet nor large scale cloud caities, probably due to low Reynolds number flow compared to the other studies, while we obsered small cloud caity shedding from the trailing edge of sheet caity as an essential character of caitating flow in this kind of nozzle. In the present study, we propose a simple caitation model which considers irtually the interface between liquid and apour phases in the framework of homogeneous one-fluid model, in which a subgrid-scale model of bubbles in the liquid phase and that of droplets in the apour phase are employed. Mass source terms of liquid/apour phases are considered through the growths and collapses of bubbles and droplets which are locally switched depending upon the local oid fraction. In conjunction with this treatment of continuum phase, we also switch the eddy iscosity as well as the molecular iscosity depending upon whether the continuum phase is apour or liquid. By doing so, it is expected to reproduce the strong unsteadiness of caitating flow such as the generations of cloud caities from the trailing edge of sheet caity een ifincompressibletwo-equation RANS turbulence model is employed.in this paper, the proposed model is applied for the simulation of caitating flow in a simple two-dimensional conergent-diergent nozzle. The effects of the choices of the mass source terms for bubbles and droplets as well as the turbulence model modification on the global motion of unsteady caitating flow are inestigated. 2. Numerical method A numerical simulation was carried out byusing an opensource software, OpenFOAM. Seeral caitation models are implemented in OpenFOAM, among which we employ interphasechangefoam (IPCF) as a base soler. This soler is an incompressible Naier-Stokes one with homogeneous caitation model considering the phase change between liquid and apour phases.besides the momentum equation of the mixture and the transport equations of turbulence properties, IPCF soles the following continuity equation and the mass conseration of liquid phase as follows u j 1 1 m m x j l (1) l 1 lu j m m t x (2) j l where isa Cartesian coordinate, is elocity componentin the direction, and are densities of liquid and apour phases, is a olume fraction of liquid phase. The source terms in the aboe equations, and, are mass transfer rates between two phases due to eaporation and condensation, which are modelled in the following section Caitation model Schnerr-Sauer model (SS). Schnerr-Sauer (SS) model [9] implemented in OpenFOAM is employed as a base model for this study, which has been used in many studies(for recent examples, [10] and [11]).In SS model, the apour bubbles are always considered as dispersed phase in continuum 2

4 liquid phase een for ery large oid fraction region. The SS model is based on the Rayleigh equation as follows 2 2 d R dr p p 2 2 l R dt dt where is a bubble radius, is pressure, is saturation pressure. If we neglect the second deriatieterm, the bubble radius growth can be represented by the following equation. dr 2 p p sign p p (4) dt In this model, the oid fraction represents the apour olume per unit mixture olume and the bubble radius is deried from as follows. n 1 n R 4 nuc n 4 R 4 R nuc nuc l () (5) 1 (6) where is the number of bubbles per unit liquid olume.from those equations, the mass source terms of SS model can be obtained as where C c and C e 2 p p l m Cc 1 ( )? p p (7) R 2 p p l l m Ce 1 ( )? p p R (8) are condensation and eaporation coefficients and 1 l is the l mixture density.in the present study, the basic parameters in this model are set as and Bubble-Droplet1 model (BD1). Since applying the dispersed bubble models for large oid fraction regions seems to be inappropriate, we consider another extreme case in which apour phase contains more or less liquid droplets. In this study, this extreme condition near the oid fraction is unity is taken into account in our model, at which we treat apour phase as a continuum media. Figure 1 shows the conceptual drawing of our model. This model irtually considers the interface between liquid and apour phases as the iso-surface of oid fraction, and in this study is set to be 0.5 for simplicity. When the local oid fraction satisfies, the apour phase is treated as dispersed phase, while, the liquid phase is treated as dispersed phase. The mass transfer between apour and liquid are dominated by that occurs at the surfaces of the bubbles/droplets, then the mass transfer rates and are switched depending upon the local oid fraction. In the model Bubble-Droplet 1(BD1), the SS model, i.e. equation (7) and (8), is adopted for, while for the phase change at the surface of droplet is considered using Schrage s mass flux [12], expressed as follows

5 k M p p (9) 2 RT g where is a eaporate/condensate coefficient, is a gas constant, and is temperature. The alue of k was set to 0.4 throughout this computation. By considering the phase change through the total surface area of droplets, the mass transfer rate can be obtained as k 1 m m m p p (10) 2 RT R g d where is a radius of droplet, which can be calculated by R d l nnuc 1 l nnuc Since we do not consider any collisions and fissions of droplets/bubbles and any nucleation and destructions, is constant and common for bubbles and droplets. Since the caity interface can be irtually treated, the mass transfer at the caity interface is supposed to be possibly treated, which remains for our future study. (11) Figure 1. Conceptual drawing of bubble drop model Bubble-Droplet2 model (BD2) Schrage s mass flux can also be applied for the bubbly flow region. In the model Bubble-Droplet 2 (BD2), the following mass transfer rate deried from the phase change through the bubble surfaces is applied for the region with instead of equation (7) and (8) in SS model. k m m m p p 2 RT R (12) g Bubble-Droplet1 iscosity filtering model(bd1vf). Throughout the present computations, the standard k- model was used. Howeer, it is known that key unsteady caitation phenomena such as a re-entrant jet which deelops beneath the sheet caity and the formation of cloud caity can neer be simulated with incompressible RANS turbulence model.to enhance the unsteadiness due to the instability on the sheet caity interface, we switch the eddy iscosity as well as the molecular iscosityby referring only the continuum phase. This treatment may look similar to well-known Reboud correction [2], while in this study the turbulent and molecular iscositiesare modified based 4

6 on the fluid properties of continuum phase only, which is suitably applied in combination with BD1 model Computational method Numerical simulations using aboe models were carried out for the unsteady caitating flow in a twodimensional conergent-diergent nozzle shownin Figure 2. The height of throat h is 5mm. The nozzle consists of a top straight wall and a bottom inclined wall with conergent and diergent angles of 4 and 8.4 degrees, respectiely. This nozzle shape is similar to that used in our preious experiment [8], in which we obsere a continuous cloud caity shedding from the trailing edge of the sheet caity. As the boundary conditions, the elocity is fixed at the inlet ( ), and the static pressure is fixed at the outlet. The non-slip flow condition is applied on the upper and lower wall of the nozzle. The number of nodes is1,200 (along the wall) x 60 (perpendicular to the wall). For the conection scheme, second order upwind scheme is basically used except the oid fraction; for the conection scheme in equation (2), TVD scheme [1] is used to well capture the caity interface. As for the time integration, implicit Euler scheme is employed with the ariable small time step satisfying local CFL number less than 0.8. Figure 2. Computational domain (main flow direction is from left to right). Result and discussion Figure shows typical caity shapes at four instants with the time interal of 4ms, simulated by using four models described before. The caitation number defined using the area-aeraged nozzle throat elocity as takes similar alue of 1.6in all cases except SS model (0.49), resulting in the time aeraged caity length of. The similar caitation obsered in the preious experiment [8] is also shown in figure (a). By comparing the results between SS, BD1 and BD2 models, it is found that the sheet caity is slightly longer for BD1 and BD2, but the unsteady behaior of caitation is ery similar: there are no cloud caities obsered but the sheet caity slightly fluctuates. Howeer, by applying the iscosity filtering model along with BD1 (BD1VF), we can clearly see the cloud caity shedding similar to that obsered in the experiment. These results suggest that the treatment of the turbulent shear stress is important rather than the caitation model, whereas the concept of the iscosity filtering treatment in this study is conceptually suitable to the proposed caitation model BD1 and BD2. Figure 4(a) shows the results of the FFT analysis of pressure fluctuation measured2h downstream on the upper wall in the case with. In this figure, the strong peak around 40Hz (60Hz in SS model) and its harmonics can be clearly seen for SS, BD1 and BD2. This is associated with the caity olume change obsered in figure. (b)-(d), which numericallycauses surge-like oscillation of liquid column downstream of the sheet caity. For the BD1VF model, the eery frequency component is significantly larger than those for SS, BD1 and BD2, but we can clearly see the broad-banded frequency peaks around 45Hz, which is caused by the continuous cloud caity shedding from the trailing edge of the sheet caity. Similar tendency could be found for the other conditions with arious sheet caity lengths. Figure 4 (b) shows the comparisons of Strouhal number of the cloud caity shedding predicted by BD1VF model with that obtained by the experiment. Although the numerical 5

7 data are limited to longer sheet caities, the tendency of the predicted Strouhal number against the caity length is ery similar to the experiment. Therefore, we beliee that the present model can well simulate the unsteadiness of the cloud caity shedding in this kind of nozzle flow. (a) Experiment [8] (=1.14) (b) SS (L c/h=11., =0.49) (c) BD1(L c/h=14.4, =1.62) (d) BD2(L c/h=15.7, =1.64) (e) BD1VF (L c/h=11.5, =1.6) Figure. Comparisons of caity behaiours among four numerical models and experiment Amplitude of pressure coefficient [-] SS BD1 BD2 BD1VF Frequency [Hz] (a) Typical FFT results withl c/h =11-16 Strouhal number, St=fh/U throat Experiment BD1VF Normalized caity length L c /h (b) Strouhal number of cloud caity shedding Figure 4. Results of FFT analysis of pressure fluctuation caused by unsteady caitation 4. Concluding remarks In this paper, a simple caitation model was deeloped under the framework of homogeneous onefluid model. The model is constructed consideringtwo extreme conditions; for low oid fraction, dispersed apour bubbles in continuous liquid phase, while for high oid fraction, dispersed droplets in continuous apour phase, and according to the local oid fraction, the growth of bubbles and droplets are taken into account in the mass transfer between apour and liquid phases. To enhance the unsteadiness of caitation due to the instability at the caity interphase, the turbulent shear stress is modified depending upon the continuous phases in combination with the proposed caitation model. It is found that the proposed model can well predict the major unsteadiness of caitation, i.e. continuous cloud caity shedding from the sheet caity, in a two-dimensional conergent-diergent nozzle flow. 6

8 References [1] Kato C 2011 Proc. ASME-JSME-KSME Joint Fluids Eng. Conf.( (Hamamatsu, Shizuoka, Japan, 24-29, July,2011) AJK [2] Reboud J. L., Stutz B, and Coutier O 1998 Proc. rd Int. Sym. on Caitation (Grenoble, France, 7-10, April, 1998) [] Coutier-Delgosha O, Fortes-Patella R and Reboud J L 200 ASME J. Fluids Eng [4] Ji B, Luo X W, Arndt R E A and Wu Y L 2014 Ocean Eng [5] Stutz B and Reboud J L 1997 Physics of Fluids [6] Stutz B and Reboud J L 2000 Experiments in Fluids [7] Keil T, Pelz P F, Cordes U and Ludwig G 2011 Proc. WIMRC rd International Caitation Forum 2011(Warwick, UK, 4 6 July 2011) [8] Watanabe S, Enomoto K, Hara Y and Furukawa A 2012 Proc. 5th Int. Symp. on Fluid Machinery and Fluid Eng. (Jeju, Korea, Oct 2012,) [9] Schnerr G and Sauer J 2001 Proc. 4th International Conference on Multiphase Flow (New Orleans, USA, 27 May - 1 June 2001) [10] Young Y L and Saander B R 2011 Ocean Engineering [11] Roohi E, Zahiri A P, and Passandideh-Fard M 201 Applied Mathematical Modelling [12] Schrage R W 195 Theoretical Study of Interphase Mass Transfer (Columbia: Columbia Uniersity Press) [1] Bram V L 1974 J. of Computational Physics