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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d

Large Eddy Simulation of turbulent vortex-cavitation interactions in transient sheet/cloud cavitating ﬂows Biao Huang ⇑, Yu Zhao, Guoyu Wang School of Mechanical Engineering, Beijing Institute of Technology, 5 South Zhongguancun Street, Beijing 100081, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 4 June 2013 Received in revised form 10 December 2013 Accepted 28 December 2013 Available online 3 January 2014 Keywords: Sheet/cloud cavitation LES model Vortex structures Vorticity transport equation

a b s t r a c t The objectives of this study are to: (1) quantify the inﬂuence of sheet/cloud cavitation on the hydrodynamic coefﬁcients and surrounding ﬂow turbulent structures, (2) provide a better insight in the physical mechanisms that govern the dynamics and structure of a sheet/cloud cavity, (3) improve the understanding of the interaction between unsteady cavitating ﬂow, vortex dynamics and hydrodynamic performance. Results are presented for a 3D Clark-Y hydrofoil ﬁxed at an angle of attack of a = 8 degrees at a moderate Reynolds number, Re = 7 105, for both subcavitating (r = 2.00) and sheet/cloud cavitating conditions (r = 0.80). The experimental studies were conducted in a cavitation tunnel at Beijing Institute of Technology, China. The numerical simulations are performed via the commercial code CFX using a transport equation-based cavitation model, the turbulence model utilizes the Large Eddy Simulation (LES) approach with the Wall-Adapting Local Eddy-viscosity model. The results show that numerical predictions are capable of capturing the initiation of the cavity, growth toward the trailing edge, and subsequent shedding, in accordance with the quantitative features observed in the experiment. The detailed analysis of the vorticity transport equation shows strong correlation between the cavity and vorticity structure, the transient development of sheet/cloud cavitation has signiﬁcantly changed the interaction between the leading edge and trailing edge vortices, and hence the magnitude as well as the frequency of the hydrodynamic load ﬂuctuations. Compared to the subcavitating case, the sheet/cloud cavitation leads to much higher turbulent boundary layer thickness and substantial increase in velocity ﬂuctuation. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Cavitation is a dynamic phase-change phenomenon that occurs in liquids when the local pressure drops to reach the saturated vapor pressure of liquid [5,6]. It is well know that the unsteady cavitation in turbomachinery and marine control surfaces will lead to problems such material damage, vibration, noise and reduced efﬁciency [2,22]. In most industrial applications, cavitating ﬂows are turbulent and the dynamics of the interface formed involves complex interactions between phase-change and vortex structures [3], but these interactions are not well understood. In cavitating ﬂows, instabilities and turbulence often result in the formation of large-scale vortical structures. There have been a number of recent experimental and numerical studies examining turbulence/cavitation interactions, which serve to motivate the present study. Various experimental techniques have been developed to study the complex multiphase structures in developed cavitating ﬂows. Gopalan and Katz [14] used particle image velocimetry (PIV) and high-speed photography to measure the ﬂow structure at the closure region and downstream of vapor cav⇑ Corresponding author. Tel./fax: +86 10 68912395. E-mail address: [email protected] (B. Huang). 0045-7930/$ - see front matter Ó 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compﬂuid.2013.12.024

ities in the nozzle ﬂow. They observed that the collapse of the vapor structure is a primary mechanism of vorticity production leading to the generation of hair-pin vortices in the downstream region. Lyer and Ceccio [25] investigated the effects of the growth and collapse of cavitation on the dynamics of shear ﬂow. The streamwise velocity ﬂuctuations increased with cavitation, which are conﬁrmed that the collapse of the vapor cavities is a source of vorticity generation. Gopalan and Katz [14] used ﬂuorescent to measure the velocity ﬁeld in the wake of a cloud cavity. Reynolds shear stresses were found to have increase by 25–40% due to cavity. Cavitating ﬂows are generally relatively high Reynolds number ﬂows and hence the turbulence modeling plays an important role in the capture of unsteady behaviors. Usually, when unsteady ﬂows are required, Large Eddy Simulation (LES) has been proposed to replace RANS method. The LES approach, originally proposed by Smagorinsky [33] reﬁned by many researchers [28,27,26,31] is an actively pursued route to simulate turbulent ﬂow, as it is often more capable of reproducing large unsteadiness motion of the ﬂow ﬁeld, resulting in only the small unresolvable scales being modeled. In order to better capture the transient turbulence structures, Large Eddy Simulation (LES) model was used to simulate sheet/ cloud cavitation on a NACA0015 hydrofoil [36]. Ji et al. [20,21] and Luo and Ji [24] analyzed the three-dimensional cavity

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pared with the experimental data. Dittakavi [9] and Dittakavi et al. [10] used the LES method to investigate the turbulent-cavitation interaction in the venture nozzle geometry. They also analyzed the inﬂuence of cavitation on different terms of vorticity transport equation, such as vortex stretching, baroclinic term and dilatation term. The objective of this paper is to investigate turbulent vortexcavitation interactions in transient sheet/cloud cavitating ﬂows. The aims are to (1) quantify the inﬂuence of sheet/cloud cavitation on the hydrodynamic coefﬁcients and surrounding ﬂow turbulent structures, (2) provide a better insight in the physical mechanisms that govern the dynamics and structure of a sheet/cloud cavity, (3) improve the understanding of the interaction between unsteady cavitating ﬂow, vortex dynamics and hydrodynamic performance. In present paper, the numerical models are presented in Section 2. Summary of the experimental and numerical setups are shown in Sections 3 and 4, respectively. In Section 5, comparisons of numerical predictions with experimental visualizations and measurements are ﬁrst presented, followed by detailed analysis of time evolution of unsteady cavitation and vortex structures, the effects of sheet/cloud cavitation on the dynamic load coefﬁcients and ﬂow structures. Finally, the major ﬁndings and future work are summarized in Section 6. Fig. 1. 3D boundary conditions and meshes used in the CFD computations.

structures around a twisted hydrofoil with the LES method, and the unsteady behaviors of cavity shedding and the induced pressure ﬂuctuation are discussed, and the detail analysis using the vorticity transport equation shows the cavitation accelerates the vortex stretching and dialation and increase the baroclinic torque as the major source of vorticity generation [20,21]. Roohi et al. [30] applied the implicit LES model and VOF technique to simulate the cavitating ﬂow over the Clark-Y hydrofoil, and the predicted cavitation dynamic, cavity’s diameter and force coefﬁcients are com-

2. Numerical models 2.1. Governing equations and Large Eddy Simulation In the mixture model of the vapor/liquid two-phase ﬂow, the multiphase components are assumed to have the same velocity and pressure. The governing equations consist of the mass and momentum conservation equation as follows:

@ qm @ðqm uj Þ þ ¼ 0; @xj @t

ð1Þ

Fig. 2. Comparison of the predicted vortex structures (the counter-clockwise is positive) obtained using (a) RANS model and (b) LES method for r = 2.00, Re = 7 105, a = 8 degrees.

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Fig. 3. Comparisons of the time-averaged experimentally observed (left) and numerically (right) cavitation patterns at the mid-span of the foil for r = 0.80, Re = 7 105, a = 8 degrees.

Experimental

Simulation (Q-criterion)

Simulation (Vapor)

(a) t1=0.20 Cycle

(b) t2=0.50 Cycle

(c) t3=0.70 Cycle

(d) t4=0.90 Cycle

Fig. 4. Comparisons of the experimentally observed cavitation pattern (left), numerically predicted vapor fraction contours and ﬂow streamlines (middle) and Q-criterion contours (right) at the mid-span of the foil for r = 0.80, Re = 7 105, a = 8 degrees.

@ðqm ui Þ @ðqm ui uj Þ @p @ þ ¼ þ @t @xj @xi @xj @ ql al @ðql al uj Þ _ þþm _ ; þ ¼m @xj @t

lm

@ui ; @xj

ð2Þ

ð3Þ

where u is the velocity, p is the pressure, ql is the liquid density, qv is the vapor density, av is the vapor fraction, al is the liquid fraction. The subscripts (i, j, k) denote the directions of the Cartesian coordi_ þ , and the sink term m _ , in Eq. (3) reprenates. The source term m sent the condensation and evaporation rates, respectively. The mixture density qm and dynamic viscosity lm are deﬁned as:

Large Eddy Simulation (LES) is about ﬁltering of the equations of movement and decomposition of the ﬂow variables into a large scale (resolved) and a small scale (unresolved) parts. By performing the volume averaging and neglecting density ﬂuctuations, the ﬁlter Navier–Stokes equations become:

i Þ @ðqm u i u j Þ @ðqm u @p @ þ ¼ þ @t @xj @xi @xj

lm

i @u @xj

@ sij ; @xj

ð6Þ

sij is the sub-grid scale (SGS) stress, which is deﬁned as:

sij ¼ qm ui uj ui uj :

ð7Þ

qm ¼ ql al þ qv av ;

ð4Þ

The SGS stress is assumed to be proportional to the modulus of the strain rate tensor Sij of the ﬁltered large-scale ﬂow [33]:

lm ¼ ll al þ lv av :

ð5Þ

sij skk dij ¼ 2lt Sij :

1 3

ð8Þ

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Fig. 5. Comparisons of time evolution of experimentally measured and predicted the lift coefﬁcient Cl for r = 0.80, Re = 7 105, a = 8 degrees.

Fig. 6. Comparisons of the experimentally measured and predicted power spectral densities of lift coefﬁcient ﬂuctuations for r = 0.80, Re = 7 105, a = 8 degrees.

Table 1 Comparisons of the time averaged measured and predicted lift coefﬁcient Cl and primary oscillating frequencies St for r = 0.80, Re = 7 105, a = 8 degrees.

RANS [39] RANS with modiﬁcation [39] LES by present study Experimental data [37,38]

Primary St

Cl

Cd

0.190 0.245 0.175 0.168

0.682 0.543 0.690 0.760

0.118 0.121 0.117 0.119

Fig. 7. Time evolution of predicted lift coefﬁcient Cl for r = 0.80, Re = 7 105, a = 8 degrees.

Sdij ¼

1 1 2 gij þ g2ji dij g2kk ; 2 3

gij ¼

i @u ; @xj

Ls ¼ minðkd; C s DÞ;

ð11Þ

where Ls is the mixing length for sub-grid scale, k is von Karman constant, d is the distance to the closest wall, with the WALE constant Cs = 0.5, which is the default values in CFX [27], D is the local grid size, namely,

Dgrid;3D ¼ ðDx Dy DzÞ1=3 ;

ð12Þ

where Dx, Dy and Dz are the mesh sizes in each direction. The wall-adapted local eddy-viscosity (WALE) model [27] is formulated locally and uses the following equation to computer the eddy viscosity. Compared with the classic Smagorinsky formulation, the main advantage of the WALE model are all the turbulence structures relevant for the kinetic energy dissipation could be detected with this model, and also, it is able to reproduce the laminar to turbulent transition process trough the growth of linear models.

lt

3=2 Sdij Sdij ¼ qL2s 5=2 5=4 ; Sij Sij þ Sdij Sdij

i @ u j 1 @u ; Sij ¼ þ 2 @xj @xi

ð9Þ

ð10Þ

2.2. Physical cavitation model In Kubota model [23], the growth and collapse of the bubble clusters are assumed to be governed by the simpliﬁed Rayleigh– Plesset equation [6]. The cavitation process is governed by the mass transfer equation given in Eq. (3), and the source and sink terms are deﬁned as follows:

_ ¼ C dest m

_ þ ¼ C prod m

1=2 3anuc ð1 av Þqv 2 pv p ; 3 ql RB

1=2 3av qv 2 p pv ; 3 ql RB

p > pv ;

p < pv ;

ð13Þ

ð14Þ

B. Huang et al. / Computers & Fluids 92 (2014) 113–124

117

Main flow LE

TE

Fig. 8. Time sequence of photographs of cavity patterns in the ﬁrst phase of ﬂow cycle obtained via high-speed video for r = 0.80, Re = 7 105, a = 8 degrees, viewed from the bottom (foil suction-side) of the test section.

Fig. 9. The predicted 3D cavity structures, vapor fraction, vorticity contours, dilation terms and baroclinic torque terms at the mid-span of the foil at (a) t1 = 11% cycle and (b) t2 = 27% cycle for r = 0.80, Re = 7 105, a = 8 degrees.

Main flow LE

TE

Fig. 10. Time sequence of photographs of cavity patterns in the second phase of ﬂow cycle obtained via high-speed video for r = 0.80, Re = 7 105, a = 8 degrees, viewed from the bottom (foil suction-side) of the test section.

where anuc is the nucleation volume fraction, RB is the bubble diameter, pv is the saturated liquid vapor pressure, and p is the local ﬂuid pressure. Cdest is the rate constant for vapor generated from the liquid in a region where the local pressure is less than the vapor pressure. Conversely, Cprod is the rate constant for re-conversion of vapor back into liquid in regions where the local pressure exceeds the vapor pressure. In this work, the assumed model constants are anuc = 5 104, RB = 1 106 m, Cdest = 50, and Cprod = 0.01, which are the default values in CFX [41], and are used because of their supposed general applicability. Validation of the Kubota cavitation model with the assumed constants for the case of unsteady cavitating ﬂow around the 2D Clark-Y hydrofoil have been presented by Huang et al. [18].

3. Experimental setup and description To investigate the dynamics of transient cavitating ﬂows, numerical predictions are compared with experimental measurements of a Clark-Y hydrofoil conducted at the cavitation tunnel at Beijing Institute of Technology [38,18]. The test section is 0.133 m2 squared and 0.7 m long. The cavitation tunnel is capable of generating free stream velocity ranging from 2 to 15 m/s with a minimum cavitation number of r ¼ 2ðp1 pv Þ=ðqU 21 Þ ¼ 0:30, where p1 is the reference static pressure, and U1 is the free stream velocity. The tunnel inﬂow turbulence intensity, deﬁned as U1rms/ U1 at the inlet of the test-section, is about 2%. The foil has a uniform cross-section with a Clark-Y thickness distribution with

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Fig. 11. The predicted 3D cavity structures, vapor fraction, vorticity contours, dilation terms and baroclinic torque terms at the mid-span of the foil at (a) t3 = 48% cycle, (b) t4 = 55% cycle, (c) t5 = 60% cycle and (d) t6 = 63% cycle for r = 0.80, Re = 7 105, a = 8 degrees.

B. Huang et al. / Computers & Fluids 92 (2014) 113–124

Fig. 12. The generation and development of the re-entrant jet at the mid-span of the foil for r = 0.80, Re = 7 105, a = 8 degrees.

119

carefully when LES method is applied to simulate the cavitating ﬂow. The ﬁne 3D ﬂuid mesh (shown in Fig. 1b is composed of more than 3,000,000 is applied to capture the detail structure of the cavitating structures. There is more than 120 structured elements across the foil boundary layer, which is selected to ensure y+ = yus/ml = 1 where y is the thickness of the ﬁrst cell from the foil surface, and us is the wall frictional velocity. In the computations, Dt = 2 105 s is chosen based on convergence studies, which ensures an average courant number base on physical time is less than 1.0. Fig. 2 shows the comparison of the typical predicted vorticity contours with LES model and RANS model (SST k–x model is used as RANS model) for subcavitating ﬂow (r = 2.00), respectively. As shown in Fig. 2a, the periodic shedding vortices predicted with the LES method are somewhat similar to the bottom half of a Karman vortex street in pattern and their shedding Strouhal number St is about 0.23, which is similar to that from a turbulent circular cylinder, when the projected chord length is used St = fc sin a/U1 (f is the shedding frequency, a is the angle of attack) [29]. As demonstrated in Fig. 1, the predicted vorticity patterns are different between the LES method and RANS model. For the LES method, the vortex structure becomes more dispersed and less conﬁned in the wake region. In contrast, for the RANS model, the impact of turbulence viscosity is dominant, and the results exhibits less ﬂuctuation in space. 5. Results and discussion

a maximum thickness-to-chord ratio of 11.7%, the chord length is c = 0.07 m, the span length is s = 0.07 m. The hydrofoil is mounted horizontally in the tunnel test section at a ﬁxed angle of attack of a = 8 degrees. In the experiments, high-speed video and Particle Image Velocimetry (PIV) technique are used to observe the cavitation patterns and measure the ﬂow velocity and vorticity ﬁelds, in the measurements of PIV, the ﬂow ﬁeld at the mid-span of the foil is illuminated by a continuous laser beam sheet (LSB), 500 PIV image pairs captured at 2000 Hz is applied to record more than six full cavity development cycle, the 500 instantaneous velocity ﬁelds are averaged to obtain the ensemble averaged in-plane velocity and vorticity ﬁelds at the middle span of the foil, meanwhile, a dynamic measure system is applied to measure the hydrodynamic coefﬁcients [18].

4. Numerical setup and description The computational domain and boundary conditions are given according to the experimental setup in [40,38,18], which are shown in Fig. 1a. The Clark-Y hydrofoil is placed in the center of water tunnel with a = 8 degrees. A no-slip boundary condition is imposed on the hydrofoil surface, and wall conditions are imposed on the top and bottom boundaries of the tunnel. The inlet velocity is set to be U1 = 10 m/s and the outlet pressure is set to vary according to the cavitation number r. Dittakavi [8] and Ji et al. [20,21] suggest that the mesh resolution at the place where cavitation occurs should be checked

5.1. Experimental validation studies Fig. 3 shows the comparison of the predicted time-averaged vapor fraction contour at the mid-span of the foil along with the experimental visualization for r = 0.80, Re = 7 105, a = 8 degrees. The white vapor fraction represents the pure vapor and the black represents the water. As shown in Fig. 3, a fair comparison is observed between the observed and predicted time-averaged cavity shapes. Sheet/cloud cavitation is highly turbulent and unsteady, in order to better validate the numerical models to capture the unsteady dynamics, Fig. 4 shows the comparison between the predicted time evolutions of vapor fraction contours with the experimental photographs presented in [17]. It is observed that cloud cavitation has a distinct quasi-periodic pattern, the cavity visualizations are placed side by side according to 20%, 50%, 70%, and 90% of each corresponding cycle, the period of the cloud cavitation is estimated to be 40 ms. The ﬂow patterns visualized based on the Q-criterion (Q > 0) are also shown in Fig. 4 to identify the vortex structures. Regions of high vorticity are not necessarily a vortex [1], as shear ﬂow can experience high shear. In order to differentiate between vorticity due to a vortex or due to shear, the Q-criterion is introduced by Hunt et al. [19], which is deﬁned as:

Q ¼ 1=2ðjXj2 jSj2 Þ;

ð15Þ

where X is the vorticity tensor, and S is the rate-of strain tensor. As for the numerical results predicted with the LES method in Fig. 4,

Main flow

LE

TE

Fig. 13. Time sequence of photographs of cavity patterns in the third phase of ﬂow cycle obtained via high-speed video at r = 0.80, Re = 7 105, a = 8 degrees, viewed from the bottom (foil suction-side) of the test section.

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Fig. 14. The predicted 3D cavity structures, vapor fraction, vorticity contours, dilation terms and baroclinic torque terms at the mid-span of the foil (a) t7 = 81% cycle, (b) t8 = 86% cycle, (c) t9 = 100% cycle and (d) t6 = 63% cycle for r = 0.80, Re = 7 105, a = 8 degrees.

the features of every stage of unsteady cavitation in experimental visualizations could be well-captured, including the initiation of the attached cavity, growth toward the trailing edge, and subsequent cloud shedding: the attached cavity expanded up to the trailing edge of the hydrofoil at t1 and at t2, followed by the breakup near the foil leading edge due to the re-entrant jet at t3, then completely convection of the cloud cavity into wake is observed at t4. It is shown that the shape of cavity shapes for the

Q-criterion contours is more complicated than that of the vapor fraction contours, the vortexs in Fig. 4c and d have already shed downstream into the wake, which is more consistent with the observation experimentally. The oscillating partial sheet cavity and the shedding of the cloud cavity discussed earlier signiﬁcantly affect the hydrodynamic load. Furthermore, Fig. 5 shows the experimentally measured and predicted time evolution of the lift coefﬁcient (C l ¼ L=ð1=2qL U 21 scÞ,

B. Huang et al. / Computers & Fluids 92 (2014) 113–124

Fig. 15. Comparisons of the measured and predicted normalized ensemble averaged amplitudes of the axial velocity at the selected monitoring locations along the foil at the mid-span of the foil for r = 2.00 and r = 0.80 at Re = 7 105 and a = 8 degrees.

where s is the span length, L is lift). First of all, both the lift signals are seen to exhibit periodic behaviors within the time-span during which the force signals were processed. The time average predicted lift coefﬁcient in present calculation is Cl = 0.690, which is 9% lower than the experimentally measured value of 0.760, and are more accurate than the RANS results of Wei et al. [39]. Compared with the lift coefﬁcient, the agreements between CFD and experimental data in terms of drag coefﬁcient are all good. Spectral analysis on the lift is also shown in Fig. 6, the primary oscillating frequencies are seen to be at St = 0.168 and 0.175 with the measured and predicted results, respectively, when the frequency is normalized by the chord length c and the upstream velocity U1, St = fc/U1. As listed in Table 1, there is a better agreement between the experimental and numerical data in present study, than the RANS results of Wei et al. [39]. The primary frequency of the hydrodynamic ﬂuctuations is induced by the unsteadiness of the cavity, and is in agreement with the main cavity shedding frequency, which is conﬁrmed in present study and presented by other researchers [38,4,29]. It should be noted that following the primary frequencies, another medium amplitude peaks are observed to be St = 0.340 and 0.385 with the measured and predicted results, respectively, as shown in Fig. 6. It may be due the instability of the vortical structures which is interacted with cavitation, which will be discussed following.

121

Fig. 18. Comparisons of the measured and predicted normalized ensemble averaged amplitudes of the turbulent ﬂuctuations at the selected monitoring locations along the foil at the mid-span of the foil for r = 2.00 and r = 0.80 at Re = 7 105 and a = 8 degrees.

Fig. 19. Comparisons of the measured and predicted normalized averaged zvorticity, xz/xo proﬁles at the selected monitoring locations along the foil at the mid-span of the foil for r = 2.00 and r = 0.80 at Re = 7 105 and a = 8 degrees.

Overall, from the experimental validations regarding the force analysis, frequency, and the cavity visualization, reasonable agreements are observed between measurements and computations in sheet/cloud cavitation. Detail investigation on the physical mechanisms that govern the dynamics and structure of the sheet/cloud

Fig. 16. amplitude of the horizontal components of Reynold stress (a) u0 u0 and (b)

v0v0

at the mid-span of the foil for r = 2.00, Re = 7 105, a = 8 degrees.

Fig. 17. amplitude of the horizontal components of Reynold stress (a) u0 u0 and (a)

v0v0

at the mid-span of the foil for r = 0.80, Re = 7 105, a = 8 degrees.

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cavity, and the interaction between unsteady cavity, vortex dynamics and hydrodynamic performance will be discussed in the following subsections. 5.2. Time evolution of unsteady cavitation and vortical structures Fig. 7 shows the time evolution of the lift coefﬁcient for r = 0.80 with LES model. Figs. 9, 11 and 14 shows the evolution of predicted 3D cavity structures and vapor fraction contours, vorticity contours at the mid-span of the foil for nine representative times (from t1 to t9) in one ﬂow cycle, as shown in Fig. 7b. To get a better understanding of the effect of cavitation on vorticity dynamics, it is useful to examine the vorticity transport equation for a three-dimensional ﬂow [20,21,11,14]:

Dx ¼ ðx rÞV xðr VÞ þ rqm rp=q2m þ tr2 x: Dt

ð16Þ

The RHS consists for four terms namely the vortex stretching, the dilatation (volumetric expansion/contraction), baroclinic torque, and viscous diffusion. The baroclinic term acts whenever pressure and density gradients are not aligned. Large gradients in density and pressure do not lead to any vorticity production if the gradients are aligned, such as in a barotropic ﬂow. The ﬁnal term is called the viscoclinic torque [12] and [13], and it represents the laminar and turbulent diffusion of the vorticity. In order to examine the effects of cavitation on the production/alteration of vorticity around the cavitating ﬂow, the contour plots of the vortex stretching term V(x r), dilatation term x(r V) and baroclinic torque ðrqm rp=q2m Þ at the mid-span of the foil are also shown in Figs. 9, 11 and 14 at selected time instances. For the unsteady cloud cavitating ﬂows (r = 0.80), concerning the detail developments of the cavity and the corresponding hydrofoil response, as shown in Fig. 7b, are characterized by the following three phases: 5.2.1. Initiation of the attached cavity, growth toward the trailing edge (TE) Fig. 8 shows the evolution of cavity patterns in the ﬁrst phase of ﬂow cycle obtained via high-speed camera [18]. As shown in Fig. 8, for r = 0.80, the ﬁrst phase of the ﬂow cycle is the growth of the attached cavity immediately after a vortex shedding of cloud cavitation event takes place, the attached cavity gradually grows to its maximum and it has a smooth interface. From t = 0% to t = 27% cycle of the numerical results, the ﬂow is characterized by the initiation of the attached cavity on the leading edge (LE), growth toward the trailing edge (TE). As shown in Fig. 7b, the overall trend of the lift coefﬁcient increases because of the expansion of the attached LE cavity and LEV between t1 = 11% cycle (formation of the LE cavity on the suction side) and t2 = 27% cycle (maximum extent of the LE cavity while fully attached on the foil suction side). The lift coefﬁcient reaches a local maxima at t2, when corresponds to the time when the LE cavity is the largest and before it merges with the ‘‘’’ TEV. The results show strong correlation between the attached cavity and vorticity structures, which suggest cavitation as an important mechanism for vorticity generation, although the magnitude of baroclinic torque ðrqm rp=q2m Þ are smaller than the vortex stretching term V(xr) and dilatation term x(r V) as shown in Fig. 9. As expected, for both t1 and t2, the baroclinic term of the vorticity is important along the liquid–vapor interface, but negligible inside the attached cavity region, especially when the cavity reaches its maximum at t2. Compared with the ﬂow structures, it shows that the vortexstretching term and the dilation term are highly dependent on the cavity evolution. The dilation term x(r V) represents the vortex stretching due to the ﬂow compressibility and is zero for the

non-cavitation region. It should be noted that the results yield very different levels of dilation term in the attached cavity region, it has relative higher level of dilation term near the foil, and the signiﬁcant difference of dilatation term implies that the compressibility of cavity region closed to the foil surface is much stronger. 5.2.2. The forms and development of the re-entrant ﬂow Fig. 10 shows the evolution of cavity patterns and development of re-entrant jet in the second phase of ﬂow cycle. When the stable attached sheet cavity grows to its maximum length, at which time the cavity interface becomes wavy/bubbly, the adverse pressure gradient is strong enough to overcome the weaker momentum of the ﬂow conﬁned by the near-wall region, a re-entrant jet forms and moves upstream, the re-entrant ﬂow impinges on the cavity interface and lead to periodic production of cloud cavitation. Compared with the experimental visualizations in the second phase within a typical ﬂow cycle, the numerical prediction can basically capture these dynamic behaviors. Complex vortex structures can be observed due to the unsteady partial breakup and the recirculation created by the re-entrant jet, as illustrated in the ﬁrst rows in Fig. 11a–d. The formation of re-entrant jet and its relationship to the cavity visualizations imply that the re-entrant jet plays a key role to trigger the unsteady dynamics of cavitation. In this phase, the medium amplitude peaks are observed are observed due to the vortices that are associated with the interaction of the LE attached cavity, LEV and TEVs. The interaction and convection of LEV and TEV lead to about three times load ﬂuctuations. The local maxima at t3, as shown in Fig. 7b corresponds to the time when the LEV is the largest and before it merges with the ‘‘+’’ TEV, as shown on the right side in the ﬁrst row in Fig. 11a. t4 should be half way between local maxima at t3 and the next local minima at t5: The lift coefﬁcient drops between t3 and t4, as the adverse pressure gradient caused by the growth of the counterclockwise ‘‘+’’ TEV force the partial LEV to shed downstream, which allowed the re-entrant jet to reach the midchord point of the foil. The trough at t5 corresponds to times just as the ‘‘+’’ TEV has shed in the wake, immediately after t5, the ‘‘+’’ TEV shed downstream, as the lift coefﬁcients rises again at this phase, as shown in Fig. 7b. The evolutions of the predicted effective terms of the vorticity transport equation are shown in the second rows in Fig. 11a–d. Fig. 12 highlights the signiﬁcance of the re-entrant jet to show the representative unsteady behaviors, the re-entrant jet interacts with the cavity as it moves upstream, which motivates the phasechange process and lead to an increase of vorticity near the wall, the strength of the vortex stretching term and dilatation term changed relatively signiﬁcantly along with the moving of the reentrant jet. The baroclinic term modiﬁes the vorticity ﬁeld in regions with high density and pressure gradients, i.e. along the liquid–vapor interface and near the cavity closure, the affected region of the baroclinic torque increase with the development of the re-entrant jet, because the dynamics of re-entrant jet tends to promote the unsteady phase-change in the cavities. Consequently, the vorticity ﬁeld is modiﬁed by the dynamics of re-entrant jet, as evidence via the vorticity, vortex stretching and dilatation term contours. 5.2.3. Large scale cloud cavity sheds downstream As the re-entrant ﬂow reaches the vicinity of cavity leading edge, the sheet cavity is lifted away from the hydrofoil surface and sheds downstream in the form of a cloud cavity, as shown in Fig. 13. As shown in Fig. 7b, the loads drops dramatically after t7 due to the shedding of the large cloud cavity structure, as shown on the left and middle sides in the ﬁrst row in Fig. 14a–c. An interaction between the bubbly cavity mixture and the trailing edge vortex is observed at t7 and t8, and followed by complete

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convection of the cloud cavity into the wake at t9. Notice that immediately after the ﬁrst cavity completely sheds downstream as a cloud cavity, another leading edge cavity forms. Comparison of the vorticity patterns in Figs. 11 and 14 are to suggest that the complete cloud sheds has more signiﬁcant impacts on wake dynamics, a relative increase in baroclinic torque is observed in the regions where the vapor structure collapse. The large scale of vortex structures are observed due to the interaction between the vortex structures associated with the cavity and the TE vortex, and the vortices are convected in the ﬂow direction and diffused by viscosity.

5.3. Inﬂuence of cavitation on the turbulent ﬂow structure Ensemble averaging may be used to better quantify the nature of cavitating ﬂows. Fig. 15 compares the predicted mean values of the normalized ensemble averaged axial (u) velocity proﬁles at the mid-span of the foil for r = 2.00 (subcavitating) and r = 0.80 (sheet/cloud cavitating), respectively. For r = 2.00, the numerical predictions compare well with experimental measurements except for small differences at the foil trailing edge (x/c = 1.0) due to turbulent ﬂuctuations induced by the trailing edge vortex. The results suggest that the unsteady sheet/cloud cavitation case with r = 0.80 has a much thicker turbulent boundary layer than the subcavitating case with r = 2.00. In fact, for the sheet/cloud cavitating case, recirculation breaks up and lifts the cavity upward, which signiﬁcantly increases the thickness of the boundary layer, compared to the subcavitating case at r = 2.00. In general, reasonable agreement is observed between the numerical predictions and experimental measurements for r = 0.80, although greater difference could be observed compared to the results for the subcavitating case (r = 2.00). In order to highlight the effect of turbulent-cavitation interactions on velocity ﬂuctuation and turbulence generation due to the sheet/cloud cavitation, Figs. 16 and 17 show the normalized amplitude of the predicted statistical Reynolds stresses u0 u0 and v 0 v 0 at the middle span of the foil for r = 2.00 and r = 0.80, respectively, the Reynolds Stress components are generated using running statistics of the instantaneous velocity ﬁeld. Here, u0 and v 0 are the horizontal and vertical components of the turbulent velocity ﬂuctuations. For r = 2.00, the magnitude of the vertical components of the turbulent velocity ﬂuctuations is small compared with the horizontal components, which is due to unsteady shedding of the trailing edge vortex, as evidence via the contours of spanwise vorticity in Fig. 1. For the cloud cavitating case at r = 0.80 shown in Fig. 1a, large vertical components of the turbulent velocity ﬂuctuations occurs in the cavity region and continues into the wake, indicating unstable large-scale ﬂuctuations due to the formation and breakup of the sheet cavity, and the subsequent unsteady shedding of the cloud cavity. Fig. 18 compares the measured and predicted normalized amplitudes of the averaged of the turbulence statistics pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u0 2 þ v 0 2=U 1 along the selected monitor locations. As expected, the ﬂuctuating velocities are much higher for the cloud cavitating case (r = 0.80), which also lead to much thicker turbulent boundary layer. As shown in Fig. 18, the turbulence statistics are overall better predicted with LES model, especially near the cavity closure for r = 0.80, compared with the numerical results predicted by RANS model [18]. The application of LES instead of RANS would help to improve the prediction of the temporal variation of the turbulent statistics, which may be attributed that LES model could resolve more of the turbulence scale and better resolution of the turbulent cavitating ﬂow structures surrounding the hydrofoil is obtained with LES model.

Fig. 19 shows the comparisons of the predicted and measured normalized out-of-plane (z-component) vorticity ﬁeld, xz/xo, for r = 2.00 and r = 0.80 at the selected monitoring locations along the foil. Here, xz is deﬁned as:

xz ¼ @ v [email protected] @[email protected];

ð17Þ 3

1

and xo = U0/d 1.43 10 [s ] (d 0.1, c = 0.007m is the approximated turbulent boundary layer thickness at the foil trailing edge based on the experimental measurements of the ﬂow velocity for r = 2.00). In general, reasonable agreement is observed between the measured and predicted values. However, some differences are observed between the predicted and measured values, especially near the cavity closure region. Compared with the subcavitating and cavitating cases, the vorticity proﬁle is drastically different, and the magnitude of the vorticity is much greater than the subcavitating cases, which clearly demonstrate that unsteady sheet/cloud cavitation is an important source of vorticity production and modiﬁcation.

6. Conclusions Physical numerical analysis with WALE LES model is presented for a Clark-Y hydrofoil to investigate turbulent vortex-cavitation interactions in transient sheet/cloud cavitating ﬂows. In general, the predicted cavity dynamics, velocity and vorticity distributions by the numerical models compared well with experimental measurements and observations for both the subcavitating (r = 2.00) and sheet/cloud cavitating (r = 0.80) cases. Detail investigations on the physical mechanisms that govern the dynamics and structure of the sheet/cloud cavity, and the interaction between unsteady cavitating ﬂow, vortex dynamics and hydrodynamic performance have been conducted. The primary ﬁndings include: (1) Both the numerical and experimental results show selfoscillatory behaviors as due to periodic cavity shedding caused by sheet/cloud cavitation at r = 0.80. The results show strong interactions between the cavity evolution and the vortical structure. Concerning the detail developments of the cavity and the corresponding hydrofoil response at r = 0.80, are characterized by the three phases in one typical cycle. The initiation of the attached cavity, growth toward the trailing edge is observed in the ﬁrst phase, lift coefﬁcient increases because of the expansion of the attached LE cavity. Complex vortex structures can be observed due to the unsteady partial breakup and the recirculation created by the re-entrant jet in the second phase, the vortex structure generated by the partial shedding of the LE cavity intensiﬁed the vortical interaction at the foil TE, which lead to higher frequency load ﬂuctuations. For the last phase, the amplitude of the hydrodynamic loads drop more suddenly once the cavity completely sheds and transforms to a cloud cavity. (2) The recirculating ﬂow and the re-entrant jet are responsible for the breakup and lift off of the partial sheet cavity, which is then shed downstream in the form of a cloud cavity. The results suggest that the formation and evoultion of the re-entrant jet which tends to promote the unsteady phase-change in the sheet cavitity, as well as the associated baroclinic, vortex stretching term and dilatation term are important mechanisms for vorticity production and transportation, and the vortex stretching term and dilatation term are highly dependent on the cavitation evolution. (3) Cavitation also leads to signiﬁcant inﬂuence the ﬂows structures. The averaged velocity distributions show that the unsteady collapse of the cavities in the closure region

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involves a substantial increase in turbulence, momentum, and displacement thickness. A substantial increase in turbulent velocity ﬂuctuations occurs in the cavity region and continues into the wake, indicating unstable large-scale ﬂuctuations due to the formation and breakup of the sheet cavity, and the subsequent unsteady shedding of the cloud cavity, which will in turn lead signiﬁcantly different velocity and vorticity distribution compared with the subcavitating ﬂows. It should be noted that for the greater discrepancy between the predicted and measured velocity and vorticity ﬁelds for the sheet/ cloud cavitation case compared to the subcavitating case maybe contributed to the overprediction of the vapor volume fraction in the cavitation region. As shown in the recent X-ray measurements within the unsteady cavitation [34,7], the measured vapor fraction is typically limited to 50%, which is less than the predicted value shown in this work. Furthermore, as investigated by and Tseng and Shyy [35] and Shyy et al. [32], uncertainties associated with the inlet conditions, model parameters and the overall turbulence characteristics can be better addressed using a global sensitivity evaluation based on surrogate modeling techniques [15,16]. These are great opportunities to improve the fundamental understanding, and predictive capabilities of unsteady cavitating ﬂows. Regarding future direction, additional research work are also needed to investigate the interplay between unsteady cavitation and vorticity dynamics for nonstationary hydrofoils, particularly for the case with spatially/temporally varying inﬂow, and for cases with rigid/elastic body motion, which is very important when analyzing the hydroelastic stability and vibration performs of hydraulic machineries. Acknowledgements The authors gratefully acknowledge support by the National Natural Science Foundation of China (NSFC, Grant No.: 51306020), Researh Fund for the Doctoral Program of Higher Education of China (RFDP, Grant No.: 20131101120014) and Excellent young scholars Research Fund of Beijing Institue of Technology). References [1] Adrian RJ, Meinhart CD, Tomkins CD. Vortex organization in the outer region of the turbulent boundary layer. J Fluids Mech 2000;422:1–54. [2] Arndt REA. Cavitation in ﬂuid machinery and hydraulic structures. Ann Rev Fluid Mech 1981;13:273–328. [3] Arndt REA. Cavitation in vortical ﬂows. Ann Rev Fluid Mech 2002;34:143–75. [4] Arndt REA, Song CCS, Kjeldsen M, He J, Keller A. Instability of partial cavitation: a numerical/experimental approach. In: Proceedings of 23th symposium on naval hydrodynamics, Val de Renil, France; 2000. [5] Batchelor GK. An introduction to ﬂuid mechanics. New York: Cambridge University Press; 1967. [6] Brennen CE. Cavitation and bubble dynamics. Oxford: Oxford University Press; 1995. [7] Coutier-Delgosha O, Devillers J-F, Pichou T, Vabre A, Woo R, Legoupil S. Internal structure and dynamics of sheet cavitations. Phys Fluids 2006;18:017103. [8] Dittakavi N. Computational acoustics of wall-bounded turbulent ﬂows: swirl combustor and venturi cavitation. PhD thesis, Purdue University; 2008. [9] Dittakavi N. Numerical modeling and simulation of turbulence-cavitation interactions in a venture geometry. Thesis of master degree, Purdue University; 2009. [10] Dittakavi N, Chunekar A, Frankel S. Large eddy simulation of turbulentcavitation interactions in a venture nozzle. J Fluids Eng 2010;132:121301.

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