CAV2009 Paper No. 9. Unsteady Dynamics of Cloud Cavitating Flows around a Hydrofoil

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1 Proceedings of the 7 th Internationa Symposium on aitation AV009 August 17-, 009, Ann Arbor, Michigan, USA AV009 Paper No. 9 Unsteady Dynamics of oud aitating Fows around a Hydrofoi Guoyu Wang Emai: wangguoyu@bit.edu.cn Beijing Institute of Technoogy, hina Bo Zhang Emai: wangguoyu@bit.edu.cn Beijing Institute of Technoogy, hina Biao Huang Emai: wangguoyu@bit.edu.cn Beijing Institute of Technoogy, hina Mindi Zhang Emai: wangguoyu@bit.edu.cn Beijing Institute of Technoogy, hina ABSTRAT The unsteady dynamics of coud caitating fow around a hydrofoi are inestigated by joint experimenta and numerica methods. Experiments are carried out in a rectanguar test section of a caitation tunne. A high-speed ideo camera is used to isuaize the unsteady fow structures. The isuaized data are anayzed by using a home made soft ware. The drag and ift under the caitation condition are measured. The spectra anaysis for the measured date is conducted. The computations are conducted on the two-dimensiona hydrofoi section, based on a singe-fuid mode of the caitation: the iquid/apor mixture is considered as a homogeneous fuid whose composition is reguated by mass transfer equation. The RNG k turbuence mode with modified eddy iscosity coefficient is used for the computations, and the modified coefficient is reated to the apor and iquid densities in caitated regions for simuating the caitating fow. A good agreement is obtained between experimenta data and numerica simuations. The coud caitating area is diided in two parts: attached apor sheet in the foreside of the caity, and unsteady two-phase mixture in the rear region in the process of caity breaking off. The oca pressure increasing induced by the re-entrant jet is the main reason to ead the coud caity. The aderse pressure gradient in the rear area of the caity is mainy responsibe for the generation of the re-entrant jet. INTRODUTION aitation occurs in a wide ariety of engineering systems: pumps, hydrofois and underwater bodies. In certain caitation number conditions, a the aporized area becomes unstabe with periodica shedding of arge bubbe custer. This caitation configuration is usuay caed coud caitation. Phenomenoogica, coud caitation often inoes compex interactions of turbuence and phase-change dynamics, arge density ariation between phases, fast and mutipe time scaes, and pressure fuctuations, which is responsibe for seera instabiities that affect the goba behaior of the hydrauic systems. For exampe, in rocket engine turbopump, coud caitation is associated with seere ibrations that may ead to bade destruction. The physica mechanisms of sef-osciatory behaiour of the coud caitation are not we understood due to the compex, unsteady fow structures associated with caitation dynamics and turbuence. In the ast decade or two, arious adanced experimenta techniques hae been deeoped to study the physica mechanisms and fow structures of the coud around a hydrofoi. Up to now, most of the work point out a re-entrant jet that fows under the caity from its rear part to its upstearm end. When this jet reaches the caiity sheet interface, the caity breaks off and its downstream part is conerted by the main fow[1]. Furness and Hutton[] first suggest that re-entrant jet is the principa mechanism if coud caitation, which is confirmed by Kubota et a.[3], Le et a.[4] and Kawanami et a.[5]. More oer, the occurrence of the re-entrant jet has ed to seera conjectures: aenaere point out that the aderse pressure gradient in the caity wake pays an impotant roe in the progression [6]. Gopaan and Katz find that the coapse of apor caities in the cosure region of attached caitation is the primary mechanism of orticity production [7]. More generay, Leroux et a. measured the pressure distribution in partia caitation, and discussed the unsteady characteristics [8]. Wang et a. proided the dynamics of attached caitation, imposing 1

2 the eoement of caitation ortices under arious caitation regimes [9]. On the computationa modeing side, unsteady Naier- Stokes equations-based techniques hae been deeoped to simuate caitation characteristics, incuding pressure, eocity, and phase change characteristics. Aternatie computationa modeing approaches hae been proposed. These studies can be cassified into two categories, namey, interface tracking methods [10] and homogeneous equiibrium fow modes. Here, we concentrate on the homogeneous modeing approach. Reiews of this approach can be found in the references [11]. Based on the homogeneous equiibrium fow theory, the mixture concept can be introduced, and mass and momentum equations aong with turbuence and caitation modes can be estabished for the entire fow fied. Specificay, two approaches hae been utiized to mode the caitation dynamics. The first one is the arbitrary barotropic equation mode, which suggests that the reationship between density and pressure is f( p), and the second one is the transport equation-based mode (TEM). Barotropic equations were proposed by Deannoy and Kueny [1]. They assumed that density is a continuous function of pressure where both pure phases were incompressibe, and the phase change coud be fitted by a sine cure. Arbitrary barotropic equation modes (density is ony a function of pressure) can t capture barocinic orticity production because the barocinic term of the orticity transport equation yieds zero by definition [13]. onsistent with the experimenta study [7], Senocak and Shyy hae demonstrated computationay that the barocinic orticity generation is important in the cosure region [14]. In the TEM, a transport equation for either mass or oume fraction, with appropriate source terms to reguate the mass transfer between apor and iquid phases, is adopted. An adantage of this mode comes from the conectie character of the equation, which aows modeing of the impact of inertia forces on caities ike eongation, detachment and drift of caity bubbes, especiay in compex 3-D interface situations [9]. Numericay, Singha et a. [15], and Wu et a. [10] utiized pressure-based agorithms, whie Kunz et a. [16] empoyed the artificia compressibiity method. For fows with arge property ariations and high Reynods number, the conection treatments and the boundary treatment are of importance as we. Some of the issues regarding the conection treatment can be found in Shyy [13]. As for the turbuence mode, different cosures hae been utiized to treat unsteady caitating fows[17]. Since the high eddy iscosity in the standard k-ε mode dampens the unsteady characteristics dramaticay [18], aternatie approaches hae been adopted by modifying the eddy iscosity [19]. It seems that especiay for unsteady fow computations,,satisfactory resuts more criticay depend on the turbuence mode. The unsteady dynamics of coud caitating fow around a hydrofoi are inestigated by joint experimenta and numerica methods. A high-speed ideo camera is used to isuaize the unsteady fow structures. The isuaized data are anayzed by using a home made soft ware. The drag and ift under coud caitation conditions are measured. The spectra anaysis for the measured date is conducted. The computations are conducted on the two-dimensiona hydrofoi section, based on a singe-fuid mode of the caitation: the iquid/apor mixture is considered as a homogeneous fuid whose composition is reguated by mass transfer equation. The RNG turbuence mode with modified eddy iscosity coefficient is used for computing, and the modified coefficient is reated to the apor and iquid densities in caitated regions for simuating the caitating fow. Based on the computationa and experimenta resuts, seera key probems reated to the unsteady dynamics of coud caitating fows are address. EXPERIMENTAL APPROAH The experiments are carried out in a cosed-oop caitation tunne,as show in Figure 1. A tank with a oume of 5m 3 is paced upstream of the test section to separate undesired free stream bubbes. The top of the tank is connected to a acuum pump for controing the pressure in the tunne. Between the test section and the tank, a corner ane and a straightening ane are used to reduce the turbuence ee of the fow in the test section. The detaied position of a hydrofoi ocated in the test section, with 10c in ength,.7c in height and 1c in width, is gien in Figure. The fowing patterns can be obsered from three windows, one on the top, one on the bottom and one on the side, which are made of perspex for optica access. In the whoe test section, the aerage eocities are we-distributed, and the turbuence intensity ees are smaer than %. The caitation number is controed to within 5% uncertainty. Figure 1: Schematic of the caitation tunne Figure : Geometries of the test section In this study, the reference eocity U is fixed at 10m/s, the caitation number, the Reynods number Re and the ift/drag coeffieient are defined:

3 p p (1) 1 U U Re t () p p p 0.5U (3) d F (4) y U 0.5 Fx 0.5U (5) A ark-y hydrofoi, as shown in Figure 3, is adopted in the present study. The hydrofoi, with 1 in spanwise direction, is made of stainess stee, and highy surface-poished. The suction side of the foi is mounted toward the bottom for the conenience of iewing the fowing fied. The definition of incidence ange α is presented in Figure 4. Figure 5: Schematic of the ayout of the experimenta setup Figure 6: The schematic diagram of the dynamic measure system Figure 3: The picture of a ark-y hydrofoi Figure 4: The definition of incidence ange The caitation phenomena are documented by a high-speed digita camera (HG-LE, by Redake), up to a rate of 10 5 frames per second (fps). In order to maintain desirabe spatia resoutions, much ower recording speed is adopted. Specificay, depending on the focus of the inestigation, three rates, namey, 50, 000, and 500 fps are used in this study, respectiey. The experimenta setup is iustrated in Figure 5. The ift and drag are measured using a dynamic measure deice in our aboratory,as show in Figure 6. Signas are ampified, fitered and coected through a 8-channe 16-bit A/D digitizer, at simutaneous sampe with a maximum aaiabe sampe frequency of 51. khz. The contro and the measurement data storage are performed by a P. Finay, the signa processing is deeoped using MATLAB software. OMPUTATIONAL APPROAH Goerning equations The Fare aeraged form of the mass conseration equation and Naier-Stokes equation for the momentum conseration are written in artesian co-ordinates: ( u) 0 (6) t ( u) ( uu) P [( t ) u] t (7) 1 [( t ) u ] 3 here, 1 ) (8) ( Turbuence mode The RNG k-ε mode proposed by Yakhot et a. [0] is as foows: ( k) t ( ku) k pk t krng (9) ( ) t ( u) t RNG ( 1RNG pk RNG ) k where, the turbuent iscosity is defined as: (10) 3

4 t f k (11) RNG It was obsered that the RNG k-ε mode resuted in shorter caity engths compared to the experiments[17]. Accordingy, a modified RNG k-ε turbuence cosure mode, proposed by outier-degosha et a.[17], is adopted in this study. Here, the major difference is the definition of the function f. ompared to f in the origina mode, it was defined as: n 1 f n 1 (1) where the effect of apor phase was added. aitation mode The caitation process is goerned by the thermodynamics and the kinetics of the phase change dynamics occurring in the system. Equation 13 gies the conseration equation of apor oume fraction. Here, the source terms m and m, represent eaporation and condensation of the phases [3], respectiey. ( u) ( ) m m (13) t nuc Fe (14) RB 3 m 3 (1 ) P P Fc (15) RB 3 m 3 P P Noting that Eqs. (14) and (15) are different from the origina mode proposed by Kubota et a. [3]. Since the eaporation rate is much higher than the condensation one- condensation usuay occurs sowy and aporization occurs quicky, different coefficients are imposed. Further, seera experimenta studies hae shown significant effect of turbuence on caitating fows. Specificay, the incipient caitation number increases with the turbuence intensity of the fow [1]. It seems that the oca turbuent pressure fuctuation shoud be incorporated into the definition of the apor pressure, so that the phase change responds to both thermodynamic and fuid dynamic conditions. In this study, we foow the mode proposed by Singha et a.[15]: P turb 0.39k (16) P P ( turb Psat ) (17) This approach has been found to be simpe and yied more faorabe outcome (Singha et a. [15]). Grid, boundary and initia conditions The two-dimensiona computationa domain is chosen to correcty foow the geometry of the experimenta test section. An orthogona mesh with 56,000 ces is generated as shown in Figure 7. Aternatie meshe systems, with 41,540 and 65,580 ces, respectiey, were aso chosen to eauate the grid sensitiity of the computation. Fig. 8 gies the grid distribution, finer mesh are ocated around the hydrofoi and the caitating region as the experiments indicated, to obtain more accurate caitating resuts. The nondimensiona distance to soid was y + is gien between 0 and 50, which can satisfy the requirement of wa functions. Figure 7: The schematic diagram of the dynamic measure system Figure 8: omputationa grids around the hydrofoi Tabe 1 shows the cacuated ift/drag coefficients for a three meshes under no caitation condition. In can be found that two resuts, computed with and ces, are simiar basicay. onsidering the computationa economy, ces are secected as shown in Fig. 8. Gird size Tabe. 1 and d ersus gird size.5 3 EXP d d d The numerica simuation conditions are adopted according to the experimenta processes, as aso shown in Fig. 7. In this study, the infow streamwise eocity, oume fractions and turbuence quantities are specified at the inet boundary, and the cross-sectiona-aeraged static pressure is imposed as the reference pressure at the outet. A no-sip boundary condition is used at both the upper and ower was. Aso, the no-sip wa condition with a wa function is proided at the foi surface. RESULTS AND DISUSSIONS The unsteady caitating fow around a hydrofoi are inestigated by joint experimenta and numerica methods. The focus is on the discernibe charactristics of coud caitaton, incouding the unsteady dynamics, re-entrant jet generation, and the roe of re-entrant jet in the sheding process of the coud structures. In the resuts discussed beow, the Reynods number is fixed at aue of , the attack ange 8, and the caitaton number 0.8, whie the upstream fow eocity U =10m/s. in Eoution of coud caittion patterns Figure.9 presents one cyce aspects of the coud caity obtained by the experimenta obseration and numerica simuation respectiey. The eft ones resut from isuaizations. Fig 9(b) and Fig 9(c) present the oid fraction and the fow 4

5 density eoution during one osciation cyce predicted by the cacuation respectiey. Both the resuts obtained by obserations and numerica simuations describe a quasiperiodic process of coud caitation deeopment distincty. When t=t0, immediatey after a caitating ortex shedding eent takes pace, a thin caity occurs near the eading edge of the foi. The caity grows with time, whie traeing downstream. The coud caitating area consists of two parts: attached apor sheet in the foreside of the caity, and unsteady two-phase mixture in the rear region. When t=t 0 +16ms, the caity coers the whoe suction surface of the foi, then it can be found a re-entrant jet is generated in the rear of caity. The re-entrant jet runs from downstream to upstream. When t=t 0 +8ms, the re-entrant jet reaches the attached sheet of the caity, with the caity break off two parts: attached apor sheet in the foreside of the caity, and unsteady two-phase mixture ortex structure in the rear region. Then the sheet caity is compressed and begins to coapse, the two-phase mixture ortex structure fow downstream with a cockwise rotation. The rotating direction can be deduced by the infuence of the main fow direction. The unsteady features of the coud caitation are changed periodicay. The periodica apor coud shedding is mainy induced by the re-entrant jet. high pressure area becomes arge, and which induces the coapse of the attached apor sheet in the foreside. It is cear that the raising of the oca pressure is the main reason to ead the breaking off and shedding of the coud structure. It shoud be pointed out that the high pressure is just ocated at the head of the re-entrant jet, as shown in Fig.9, that is, the raising of the oca pressure is generated by the re-entrant jet. Fig.11 shows the time eoution of the pressure coefficient around the foi suction surface, and Fig.1 shows the contour pots of water apor and eocity ectors around the foi at the corresponding time. The head of re-entrant jet at different time is marked in the Fig.1. It can be found that a oca high pressure is generated by the re-entrant jet. The oca pressure is just at the head of the re-entrant jet. When t=t 0 +8ms, the maxima oca pressure at point c can be found, which is corresponding to the caity breaking off as shown in Fig.9. It is cear that the main reason for re-entrant jet to induce the shedding of the coud caity is the oca high pressure at the head of the re-entrant jet. Figure 10: Pressure fieds around the foi during the caity breaking-off Figure 9: Time eoutions of the caity shape ( ) Unsteady fow structure in the rear of a coud caity Figure.10 indictes the pressure distribution around the foi in the progress of the two-phase mixture ortex structure fowing downstreamy. The time in the figure corresponds to Fig.9. It can be found that in t=t 0 +0ms, the distribution on the suction surface of the foi is amost uniform, but there are gradient of pressure in the rear part of the caity, and the caity is unsteady in the area. When t=t 0 +4ms, a oca high pressure is generated in the midde of the caity, which is foowed with the caity breaking off as mentioned aboe. The high pressure induces the condensation of apor ocay, and forms the breaking off of the coud caity. With the eoution of time, the Figure 11: Time eoution of the p on the suction side In order to know the behaior of the fow in the rear of the coud caity at the time when re-entrant jet is generated in detai, Fig.13 shows the time eoution of the pressure coefficient in ine AB, which is ocated in the rear of the caity. Fig.14 gies Time eoution of caitating fow structure in the rear area of the foi. Fig.14 (a) is time eoution of caitation patterns obtained by high-speed obseration. Fig.14 (b) is time eoution of eocity ector distributions in the corresponding area obtained by the numerica simuation. It can be obsered that the pressure gradient in the ine AB is sma at the stage of 5

6 caity eongating as shown in Fig.13 when t=t ms. When the caity coers the whoe suction surface of the foi, there is distinct increasing for the pressure gradient, as shown when t=t ms. At the time, there is reerse fow which can obsered at the rear of foi as shown in Fig.14. With the increasing of the pressure gradient, the reerse becomes stronger, and a re-entrant jet is generated as shown in Fig.14 when t=t ms. It can be concuded that the pressure gradient in the rear area of the foi is one of the reasons to induce the re-entrant jet. Figure 14: Time eoution of coud caitation in the rear area of the foi Figure 1: The contour pot of water apor with corresponding eocity ector Frequency characteristics of coud caitating The frequency anaysis of coud caitation pattern is aso conducted. Fig.15 (a) presents the time eoution of caity area obtained by experiment. The caity area is defined the obsered resuts on a aser beam sheet as shown in Fig.9 (a). Fig.15 (b) is the power spectra density of the caity area obtained by FFT anaysis. The caity area changes with time periodicay. The frequency is about Hz as which is corresponding to the shedding frequency of the caity S/Sc T/Tref (a) Figure 13: Time eoution of the p at the trai of the foi (b) Figure 15: Power Spectra density of the caity area 6

7 The eoutions of the ift coefficient obtained both by experiment and numerica simuation are presented in Figure16 (a). Figure 16(b) presents the power spectra density of the ift signa. A good good agreement is obtained between experimenta data and numerica resuts. A strong spectra peak data occurs at a frequency f=hz. It is cear that the frequency of the coud caity shedding is equiaent to the frequency of ift coefficient. (b) Figure 16: Power Spectra density of the ift coefficient (a) ONLUSION The coud caitation around a hydrofoi is inestigated in this paper both numericay and experimentay. Foowing is a summary of the main findings: (1) A good agreement is obtained between experimenta data and numerica simuations, concerning both the quasiperiodic deeopment process of coud caitation, the deeopment of the re-entrant jet and main frequency of the ift acted on the foi. () The coud caitating area is diided in two parts: attached apor sheet in the foreside of the caity, and unsteady twophase mixture in the rear region, in the process of caity breaking off, whie the attached apor sheet is compressed unti to coapse, and the two-phase mixture ortex structure fow downstreamy with a cockwise rotation. (3) The oca high pressure at the head the re-entrant jet is the main reason to ead to sheding of the coud caity. There is a high pressure in the head of the re-entrant jet. With the running upstream of the jet, the high pressure increase unti to break off the caity. (4) The aderse pressure gradient in the rear area of the caity is mainy responsibe for the generation of the re-entrant jet. AKNOWLEDGMENTS The authors gratefuy acknowedge support by the Nationa Natura Science Foundation of hina (NSF, Grant No.: ) NOMENLATURE chord ength of hydrofoi Re n S U U V d m m d Reynods number empirica constant bubbe surface tension reference eocity eocity component in x-direction eocity component in y-direction ift coefficient drag coefficient condensation rate eaporation rate caitation number density of iquid-apor mixture density of iquid density of apor ift coefficient drag coefficient krng, empirica constants RNG 1RNG, RNG, RNG empirica constants F, e F, c empirica constants nuc P oca turbuent pressure fuctuating turb P phase-change threshod pressure of apor P B P pressure in the apor bubbe reference static pressure. P sat saturated apor pressure K turbuent kinetic energy turbuent dissipation rate kinematic iscosity iminar iscosity t R B turbuent iscosity apor bubbe radius apor oume fraction REFERENES [1] Leroux, J B, outier-degosha, O. Astofi, J A. 005, A joint experimenta and numerica anaysis of mechanisms associated to unsteady partia caitation, Phys. Fuids. 17(5): [] Furness, R. A., and Hutton, S.P. 1975, Experimenta and Theoretica Study of Two-Dimensiona Fixed-Type aities, Journa of Fuids Engineering. 97(4): [3] Kubota, A., Kato,H. 1989, Unsteady Structure Measurement of oud caitation on a foi section using 7

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