Numerical analysis of developed tip leakage cavitating flows using a new transport-based model

International Communications in Heat and Mass Transfer 78 (2016) 39–47

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Numerical analysis of developed tip leakage cavitating flows using a new transport-based model☆ Yu Zhao a,b, Guoyu Wang b,⁎, Yutong Jiang c, Biao Huang b a b c

Beijing Institute of Mechanical & Electrical Engineering, Beijing 100074, People's Republic of China School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, People's Republic of China School of Mechatronical Engineering, Beijing Institute of Technology, Beijing 100081, People's Republic of China

a r t i c l e

i n f o

Available online 16 August 2016 Keywords: Cavitation Tip leakage vortex Transport-based model Interface mass transfer

a b s t r a c t A new cavitation model, which takes into account the effects of vortex on mass transfer process in cavitation, is utilized for the computations of the developed tip leakage vortex cavitating flows. Compared with the conventional Zwart's model, better agreement is observed between the present model predicting cavity shape and the results of the experiments. Based on the computations, it is indicated that the rotating function in the cavitation source terms contributes to the improved modelling process of the liquid–vapor interface mass transfer. This is the main reason for the present model's better capability of predicting the cavity shape compared with the conventional Zwart's model. Furthermore, based on the predictions of the present model, it is found that as the decrease of the cavitation number, the leading edge attached cavity develops gradually and covers the suction side. This will result in the decrease of the hydrofoil lift and hence the TLV circulation. © 2016 Published by Elsevier Ltd.

1. Introduction The tip vortex is a phenomenon common to all 3D lifting surfaces, including open and ducted rotors, control surfaces, and hydrofoils. High velocities found in the cores of tip vortex can lead to cavitation in the wake and cause noise, erosion, vibration, and performance decay [1–6]. When a tip is located at a finitely-small distance from an endwall, the vortex is called a tip leakage vortex (TLV). Extremely complex vortex flows, consisting of a primary TLV and numerous secondary corotating and counter-rotating vortices, have been observed experimentally in the gap and the wake [7–10]. Moreover, in the multiphase cavitating flows, vortex–cavitation interactions would greatly enhance the complexity of these vortices [11–15]. The need to understand and control the dynamics of developed TLV cavitating flows has driven numerous researchers [16–24]. Significant challenges still exist with respect to the accuracy, stability, efficiency and robustness of the modelling strategies because of the complex interactions associated with the TLV cavitating flows. Rains [25] first proposed a “jet model” for single phase tip leakage flows based on the slender body approximation. Higashi et al. [26], Watanabe et al. [27] and Murayama et al. [28] introduced the thickness

☆ Communicated by P. Cheng and W.-Q. Tao. ⁎ Corresponding author at: 5 South Zhongguancun Street, Haidian District, Beijing, People's Republic of China. E-mail addresses: [email protected] (Y. Zhao), [email protected] (G. Wang), [email protected] (Y. Jiang), [email protected] (B. Huang).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2016.08.007 0735-1933/© 2016 Published by Elsevier Ltd.

of the hydrofoil and the bubble dynamics to the “jet model” to numerically predict the behaviour of the TLV cavitation of a rectangular hydrofoil. This series of modelling methods originates from the single phase flows, and hence the interactions between vortex structures and interface mass transfer are not included. Kinnas et al. [29,30] and Hanseong Lee [31] applied a low order potential based boundary element method for the numerical modelling of the developed tip vortex cavitation. It should be pointed out that a termination model must be applied at the end of the cavity, which is difficult to be physically defined in the developed tip vortex cavitating flows. The Navier–Stokesbased computations of turbulent cavitating flows have become quite popular due to advances in computational capabilities and in the physical modelling of cavitating problems. Among these approaches, the transport-based models are becoming increasingly popular because they include the physics of cavitating flows [32–38]. For example, the Zwart's model, which is proposed by Kubota et al. [32] and supplemented by Zwart et al. [38], as one of these typical transport-based models, has been well used to calculate the TLV cavitation [17,39,40]. However, its predictive capability has not been assessed. Cavitation–vortex interactions have received increased attention as they play an important role in cavitating flows. Gopalan et al. [14] observed that the collapse of the vapor structure is a primary mechanism of vorticity production. Iyer et al. [12] and Laberteaux et al. [13] further pointed out that the baroclinic torques were responsible for the production of vorticity during vapor cloud collapse since density gradients within the cloud cavitation are not necessarily aligned with the pressure gradients around the cloud during collapse. Additionally,

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Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

Nomenclature p Pressure Saturation vapor pressure pv Reference pressure p∞ ρ Density Radius of the central core r0 α Volume fraction t Time U Velocity Reference velocity U∞ Characteristic length scale L0 ϕ Mixture property of fluids μ Dynamic viscosity υ Kinematic viscosity S Shear strain rate Ω Vorticity k Turbulence kinetic energy ω Specific dissipation rate a1, a2, β⁎, βω, F1, F2,σk, σω, σω2 Coefficients of k–ω turbulence model Γ(Γ⁎) Circulation(dimensionless circulation) Mass of a single bubble mB Number of bubbles per unit volume NB _ − Source and sink terms in the cavitation model _ þ; m m CCond, CVap Coefficients of the present cavitation model ,rConst Coefficients of Zwart's cavitation model CZwart's Cond 0 Function in the present cavitation model Fr Rotating function fr Coefficient of rotating function Cr1 r⁎ Function in the present cavitation model Re Reynolds number σ Cavitation number C Chord length of hydrofoil s Span of hydrofoil AOA Angle of attack G(τ) Gap size(dimensionless gap size) L Tunnel section length H Tunnel section height B Tunnel section breadth X, Y, Z Space variables F(F⁎) Force(dimensionless force) δ Kronecker Delta function Dimensionless wall distance y+ Subscript i, j, k l v m L T

Component Liquid Vapor Mixture Laminar Turbulent

Hence, the goals of the present study are to (1) assess the predictive capability of the present model and conventional model for the developed TLV cavitation, (2) study the modelling of liquid–vapor interface mass transfer in the TLV cavitation, (3) and discuss the influence of cavitation on vortex in the developed TLV cavitating flows.

2. Mathematical model 2.1. Vortex cavitation model The authors proposed a new “vortex cavitation” model [44], which is utilized to simulate the developed TLV cavitating flows in the present study. (a) Firstly, a cavitation vortex is defined to help build the relations between the cavitation bubble radius and vortex effects. Fig. 1 shows the comparisons of the pressure distributions of free and cavitation vortices. In the figure, pv is the vapor pressure at the liquid temperature, p∞ is the pressure of the surroundings, ρ is the density, Γ is the circulation of vortex, and r0 is the radius of the bubble. For the free vortex, the pressure is continuous and decreases as the circulation (Γ) increases. For the cavitation vortex, however, the pressure in the central region (r ≤ r0) remains constant. Moreover, the pressure at the bubble boundary (r = r0) can be calculated as: pbubble

boundary

¼ pv ¼ p∞ −

ρΓ 2 1 : 8π2 r 20

ð1Þ

Then, the radius of the cavitation bubble can be determined by: r0 ¼

1 2π

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ  jΓj: 2ðp∞ −pv Þ

ð2Þ

This equation shows that the bubble radius is a function of the circulation in a static environment and the saturation vapor pressure. (b) Secondly, based on the relationships between a single bubble and bubble cluster proposed by Kubota et al. [32], the modelling of the condensation and vaporisation terms is given as follows. Neglecting the second-order terms and the surface tension, the Rayleigh–Plesset equations that describe the growth of a vapor bubble in a liquid the equation can reduce to: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr0 2 p∞ −pv : ð3Þ ¼ dt 3 ρl

r / r0

0.0 0.5

Dittakavi et al. [41] and Huang et al. [11,42] discussed the influence of cavitation on different terms of vorticity transport equation, and found that the baroclinic and the viscoclinic torques are important mechanisms for vorticity production and modification. Ji et al. [15,43] also found that even though the magnitude of the baroclinic torque term is smaller if compared with the vortex stretching term and dilatation term, the baroclinic torque is important for the production of vorticity and modifies the vorticity field along the liquid–vapor interface. It should be noted that these researches focused on the attached sheet/ cloud cavitating flows, with limited attention paid to the developed TLV cavitation.

1.0

1.5

-0.5

-1.0

Free Vortex Cavitation Vortex -1.5

Fig. 1. Comparisons of the pressure distributions of free and cavitation vortices.

Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

2.2. Governing equations and turbulence model

Then, the rate of change of a single bubble mass is dmB dr0 ¼ ρg 4πr20 ¼ ρg 4πr 20 dt dt

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 p∞ −pv : 3 ρl

ð4Þ

Meanwhile, the number of bubbles per unit volume, NB, may be expressed as: 3 αg ; NB ¼ 4 πr 30

ð5Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dmB 3α g ρg 2 p∞ −pv _ ¼ NB  : ¼ m dt r0 3 ρl

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffi 2 p∞ −pv 2 2π 3α v ρv jp∞ −pv j ¼ : 3 ρl 3 jΓ j ρl

ð6Þ

ð7Þ

This expression has been derived assuming bubble growth. Meanwhile, as the vapor volume fraction increases, the vaporisation must decrease accordingly because there is less liquid. Thus, the condensation and vaporisation terms can be generalised as: 8 α ρ maxðp−pv ; 0Þ > > _ − ¼ C Cond F r  v v : >m _ þ ¼ C Vap F r  : ρl

ð8Þ

Here, Fr = 2π/|Γ| represents the local vortex effect. CCond = 45 and CVap = 90,000 are empirical constants. (c) Finally, the function Fr is modelled. In the present study, a simple format based on dimensional analysis is used. 1 maxð f r ; 0:1Þ: U ∞ L0

ð9Þ

Here, U∞ is the reference velocity scale and L0 is the characteristic length scale. The lower limit is introduced for numerical stability reasons. fr is called the rotating function. Based on Spalart and Shur's research [45], the rotating function can be modelled as: f r ¼ ð1 þ C r1 Þ

  ∂ρm ∂ ρm U j ¼ 0; þ ∂x j ∂t

ð11Þ

" ! #   ∂ðρm U i Þ ∂ ρm U i U j ∂p ∂ ∂U i ∂U j 2 ∂U k þ ¼− þ ðμ L þ μ T Þ þ δij : − 3 ∂xk ∂x j ∂xi ∂t ∂xi ∂x j ∂x j

2r  −C r1 : 1 þ r

The mixture property ϕm can be expressed as: ϕm ¼ ϕl α l þ ϕv ð1−α l Þ;

Considering the relation between the bubble radius and the vortex structure shown in Eq. (2), the total interphase mass transfer rate due to cavitation per unit volume may reduce to:

Fr ¼

The set of governing equations, written in Cartesian coordinates for ease of presentation, are as follows:

ð12Þ

where α g represents the vapor volume fraction. Then, the total interphase mass transfer rate due to cavitation per unit volume can be expressed as:

3α v ρv _ ¼ m r0

41

ð10Þ

Here, the Galilean-invariant factor r⁎ is calculated as r⁎ = | S/Ω | (S and Ω represent the shear strain rate and vorticity magnitude, respectively). The reason for this formation is that when r⁎ = 1.0, the constraint fr = 1.0 is satisfied for flows without strong vortex effect. Cr1 is the empirical constant and a simple format Cr1 = 0.0 is used in the present study. The difference between the present model and the conventional Zwart's model should be pointed out. In the conventional Zwart's model, the bubble radius r0 is constant. However, in the developing process of the present model, r0 comes from the cavitation vortex model and contains local vortex flow information.

ð13Þ

where ρm is the mixture density; U is the velocity; p is the pressure; μ L and μT are the laminar and turbulent viscosity, respectively; the subscripts i, j, k are the directions of the axes; the subscripts l and v denote liquid and vapor, respectively; and ϕ can be the density, viscosity, etc. A nominal density ratio of 1000 is used. For the cavitation modelling, a transport equation with source terms is solved as follows:   ∂ρv α v ∂ ρv α v U j _ þ þm _ −; ¼m þ ∂t ∂x j

ð14Þ

_ þ and m _ − are the liquid–vapor evaporation and condensation where m rates, respectively; the modelling of these terms is based on Eq. (8). Considering it is generally observed to perform well in an adverse pressure gradient and to predict boundary-layer separation well, the two-equation k–ω Shear-Stress Transport (SST) model with wall function treatment is used for turbulence closure. The transport equations for the turbulence kinetic energy (k) and specific dissipation rate (ω) are given as: " #    ∂ðρm kÞ ∂ ρm kU j ∂ μ T ∂k  ; ¼ P k −β ρm kω þ μL þ þ σ k ∂x j ∂t ∂x j ∂x j

ð15Þ

     ∂ðρm ωÞ ∂ ρm ωU j ω ∂ μ ∂ω ¼ a2 P k −βω ρm ω2 þ μL þ T þ σ ω ∂xi k ∂t ∂xi ∂x j 1 ∂k ∂ω : þ 2ρm ð1−F 1 Þ σ ω2 ω ∂xi ∂xi

ð16Þ

The turbulence viscosity is defined as: μT ¼

ρm a1 k : maxða1 ω; S F 2 Þ

ð17Þ

Details of the constants in k–ω model are available in Menter's research [46].

Table 1 Summary of test conditions. Foil Foil section C (chord length) s (span) AOA G (gap size) τ (G/thickness)

Section Rectangular Clark-Y 70 mm 68 mm 4o 2 mm 0.244

L (tunnel section length) H (tunnel section height) B (tunnel section breadth) U∞ (velocity of main flow) Re (chord-based) σ (cavitation number)

440 mm 120 mm 70 mm 7.8 m/s 546,000 3.22–0.75

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Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

Inlet

hydrofoil

Gap

Outlet No-slip Wall Y Z

X

Fig. 2. Domain and boundary conditions.

0.2080

0.078 Circulation (all vortices, X/C=0.5) Force (in span direction)

0.2078 0.076 0.2076

0.075

Dimensionless force

Dimensionless circulation

0.077

0.074 0.2074 0.073 20

40

60

80

100

Node number across the gap Fig. 3. Mesh refinement (single phase flow).

3. Experimental and computational setup Two important dimensionless parameters are the Reynolds number Re and the cavitation number σ, which are defined based on the outlet pressure p∞, the saturated vapor pressure pv, and the inlet velocity U∞: Re ¼

ρl U ∞ C p −p ;σ ¼ ∞ 2 v : μl ρl U ∞ =2

ð18Þ

Fig. 2 schematically depicts the computational domain and boundary conditions. A constant-velocity inlet is specified, and a static pressure condition (p∞) is imposed at the outlet, which is specified by the cavitation number. No-slip wall conditions are specified at the hydrofoil surface and the sides of the domain. A structural mesh is used in the bulk of the domain. O—H-type grids are generated near the hydrofoil. A y+ less than five is ensured in the boundary layers of the hydrofoil and end-wall to preserve the zeroslip condition and velocity profile, both of which are critical in the high-shear region of the gap. To highlight the flow details, the mesh inside the gap is carefully checked with various node numbers. The investigation is performed by monitoring the circulation of the vortices (Γ⁎ = Γ/(2 U∞ C)) located at X = 0.5C from the leading edge in the single phase flows. It should be noted that this circulation is calculated by integrating the X-vorticity on a given X-plane (Δ Y × Δ Z = 35 mm × 10 mm) in the gap region, and contains not only the primary TLV but also numerous secondary vortices. Meanwhile, the dimensionless the Z-direction force (F⁎ = F/(0.5ρlU2∞Cs)) on the hydrofoil is also monitored. Fig. 3 demonstrates the validity of performing the mesh refinement sequentially. As shown in the figure, the final mesh with 70 nodes in the gap is used for the subsequent simulations. The total node count of the whole domain is about 3.64 million. A commercial CFD code (ANSYS CFX) is used in the present study. Steady calculations are conducted with the convergence criterion that RMSs of the variable values equal to 1e − 5.

4. Model validations The computational domain and the boundary conditions are given according to the experiments, which are conducted in a closed-loop cavitation tunnel at Beijing Institute of Technology [11]. The critical parameters of the experiments and calculations are given in Table 1.

The Q-criterion introduced by Hunt et al. [47] is used to capture the 3D vortex structures. Typical numerically predicting vortex structures of the non-cavitating case (σ = 3.22) near the hydrofoil tip represented by

Fig. 4. Iso-surface of Q-criterion (value of 5 × 105 s−2) and streamlines of typical flows near hydrofoil tip, the present model predictions, σ = 3.22.

Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

(a) Experimental results

(b) Predictions with the Zwart’s model

43

(c) Predictions with the present model

(i) = 1.24

(ii)

= 1.08

(iii) = 0.91

(iv) = 0.75 Fig. 5. Comparisons of experimentally observed and numerically predicted cavity shape.

the iso-surface of Q-criterion (value of 5 × 105 s−2) are shown in Fig. 4. Streamlines with different colours are also shown to help identify different kinds of vortex structures. In the figure, a primary TLV with red streamlines can be observed. It starts at the suction side of the tip located near the front part of the hydrofoil and extends downstream. Meanwhile, some separation vortices marked with green colour streamlines can also be captured in the predictions. One separation vortex can be observed at the pressure side of the tip near the leading edge. Another can be found in the gap that starts at the suction side of the hydrofoil tip. Both of them are generated by the boundary layer separation [48], and finally involved in the primary TLV, which can be indicated by the mixing of streamlines. Moreover, a typical secondary vortex with blue streamlines can also be found near the primary TLV. It is induced by the strong primary TLV and contains a counter-rotative motion [48]. To conclude, the present calculations are capable of capturing the complex vortex structures near the tip region, including the primary TLV, the separation vortices and the main secondary vortex. The present study focuses on the primary TLV, as the vortex cavitation mainly happens inside. To highlight the cavitation patterns, the image processing is utilized to deal with the experimental 2D pictures taken by the high speed camera (the frequency is set to 2000 fps). Three main steps are given as follows: 1) one particular picture of non-cavitation case is chosen as the background, which is to be removed from the original pictures in various cavitating cases; 2) the filtered pictures are then converted to binary pictures, based on a specified threshold, to highlight the cavitation; 3) regarding the unsteadiness of the TLV cavitating flows, 200 continuous binary pictures in 0.1 s of one case are finally put together to form a 3D time-dependent cavitation image. In Fig. 5(a), both the front view and upward view (only the flows after the middle of the hydrofoil are shown to better capture the vortex cavitation) of the 3D time-dependent cavitation images are shown. It can be found that when cavitation number is large (σ = 1.24), isolated cavitation patterns in upward view indicate the

dissociated inception TLV cavitation near the trailing edge. As cavitation number decreases (σ = 1.08 ~ 0.75), the TLV cavitation becomes continuous and extends further downstream gradually.

Fig. 6. The location of the “tip region”.

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Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

Numerically predicting iso-surfaces of vapor volume fraction (value of 0.1) with different models are also shown in Fig. 5(b) and (c). When cavitation number decreases to σ = 1.08, the Zwart's model predicting TLV cavitation does not extend beyond the trailing edge, while the present model predicting TLV cavitation extends until about 0.5C downstream from the trailing edge, which is in better accordance with the experimental observations. As the cavitation number declines to σ = 0.91 and σ = 0.75, the Zwart's model predicting vortex cavitation ends up near the trailing edge. However, those predicted by the present model extend much further downstream (approximately 0.9C and 1.2C from the trailing edge), which better matches the experimental observations. The following discussions concentrate on the relations between vortex and the liquid–vapor interface mass transfer to find out the reasons for the different predictions with these two models.

model predictions, vaporization terms mainly concentrate in the upstream region from the trailing edge. On the contrary, as for the condensation terms, distributions of the Zwart's model and the present model predictions differ significantly. Those predicted by the Zwart's model mainly concentrate on the region with − 0.5C b X b 0.5C from the trailing edge, while those predicted by the present model concentrate on the region that starts near the trailing edge and extends much further downstream. This particular distributions of the present model predicting condensation terms imply that the interface mass transfer happens even further downstream, which contributes to the better predictions of the developed TLV cavitation extending downstream away from the trailing edge. To make better comparisons between the modelling of interface mass transfer, difference between the condensation terms of the Zwart's and the present models is highlighted as follows:

5. Modelling of interface mass transfer

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 u > u2 > > u maxðp−pv ; 0Þ > > 0 > s 3α v ρv t3 0 > _ − ¼ C Zwart  < Zwart s model : m Cond Const ρl r0 > > > > > α ρ maxðp−pv ; 0Þ > > _ − ¼ C Cond F r  v v : The present model : m : ρl

0.0010

0.00050

0.0005

0.00025

0.0

0.5

0.00000 1.0

0.0010

0.00050

0.0005

0.00025

0.00000 -0.5

0.0

0.5

(a) = 1.24

(b) = 1.08

Vaporization term_Zwart's model Vaporization term_present model Condensation term_Zwart's model Condensation term_present model

0.004

0.006

0.003

0.004

0.002

0.002

0.001

0.000 0.0

0.5

1.0

1.0

Vaporization term_Zwart's model Vaporization term_present model Condensation term_Zwart's model Condensation term_present model

= 0.75 0.005

0.008

-0.5

0.00075

X/C

= 0.91

0.00100

0.0015

X/C

0.010

0.000 -1.0

0.0020

0.0000 -1.0

Integration of vaporization term(kg/(m s))

-0.5

Vaporization term_Zwart's model Vaporization term_present model Condensation term_Zwart's model Condensation term_present model

0.010

0.005

0.008

0.004

0.006

0.003

0.004

0.002

0.002

0.001

0.000 -1.0

0.000 -0.5

0.0

X/C

X/C

(c) = 0.91

(d) = 0.75

Fig. 7. Stream-wise distributions of integrations of the cavitation source terms in “tip region”.

0.5

Integration of condensation term(kg/(m s))

0.00075

Integration of vaporization term(kg/(m s))

0.0015

0.0000 -1.0

Integration of vaporization term(kg/(m s))

0.00100

Integration of condensation term(kg/(m s))

Integration of vaporization term(kg/(m s))

0.0020

It is found that the function (Fr) in that of the present model plays an important role in the modelling of the condensation process. Therefore, stream-wise contours of the rotating function (fr) in Fr, as well as those

= 1.08

Integration of condensation term(kg/(m s))

Vaporization term_Zwart's model Vaporization term_present model Condensation term_Zwart's model Condensation term_present model

= 1.24

ð19Þ

1.0

Integration of condensation term(kg/(m s))

To better analyse the TLV cavitation, a rectangular “tip region” in X-plane is defined as shown in Fig. 6. The size of this region is set as Δ Y × Δ Z = 35 mm × 10 mm to ensure that TLV cavitation in all cases would not develop beyond this region. Both numerically predicting vaporization and condensation terms on the “tip regions” located at different X-distances are integrated, as shown in Fig. 7, to help get an insight of the interface mass transfer of developed vortex cavitation with various cavitation numbers (σ = 1.24, 1.08, 0.91 and 0.75). For both the Zwart's model and the present

Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

45

6. Interactions between cavitation and vortex

(a) X-vorticity contours

(b) the rotating function (fr) contours Fig. 8. Stream-wise contours of X-vorticity and the rotating function (fr) in the “tip region”, the present model predictions, σ = 3.22.

of X-vorticity in the “tip region” predicted by the present model are shown in Fig. 8. To point out, the computing result of non-cavitating case (σ = 3.22) is specified, and hence the effect of cavitation on vortex is neglected, which will be discussed in the following section. It can be found that the predicted rotating function (fr) in Fig. 8(b) is able to capture the primary vortex structures represented by the X-vorticity in Fig. 8(a). This means that the interface mass transfer in the primary TLV away from the trailing edge can be effectively modelled by the condensation term in the present model. This is the main reason for the better capability of the present model to capture the developed TLV cavitation.

σ = 3.22 σ = 1.24 σ = 1.08 σ = 0.91 σ = 0.75 σ = 0.59

0.06

0.04

0.02

0.00 -0.5

0.0

0.5

X/C

Fig. 9. Stream-wise distributions of the dimensionless TLV circulation in the “tip region”, the present model predictions.

The dimensionless circulation (Γ⁎ = Γ/(2 U∞ C)) is used to quantitatively represent the vortices in the “tip region”, in which the circulation (Γ) is calculated by integrating the X-vorticity in the “tip region”. This indicates that the circulation contains not only the primary TLV, but also the secondary and separation vortices as shown in Fig. 4. Stream-wise distributions of the normalized circulation in cases with various cavitation numbers are shown in Fig. 9. Generally speaking, the TLV circulation in all cases has the similar trends: it increases gradually along the X-direction and reaches its maximum point at about 0.15C upstream from the trailing edge. Then it drops with a much smaller slope compared with that of the increasing process. Similar trends can be observed in Oweis et al.'s research [7]. Moreover, the circulation decreases as the decrease of the cavitation number. The circulation in the case with small cavitation number (σ = 0.75) is smaller than that in the non-cavitating case (σ = 3.22), especially near the trailing edge. Experimental observed instantaneous leading edge attached cavity shape and corresponding numerical predicted pressure coefficient (−Cp = (p∞ − p) / (0.5ρlU2∞)) at representative span direction locations (Z/s = 0.50, 0.90, 0.95) on the hydrofoil surface are shown in Fig. 10. Firstly, as shown in the figure, numerical predicted pressure coefficients well match the experimental observed leading edge attached cavity. In the case with σ = 1.08, the attached cavity, which is represented by the locations with small value of the pressure, covers the leading part of the hydrofoil. It develops as the decrease of the cavitation number, and covers almost the whole suction side in the case with σ = 0.59. Secondly, the difference between the pressures at different spans indicates the influence of the TLV on the leading edge attached cavity. This influence is significant in the case with σ = 1.08, while becomes ignorable in the case with σ = 0.59. This means that in the present study, as the decrease of the cavitation number, the attached cavity gradually becomes dominating near the suction side of the “tip region”. Finally, based on above discussions, the lifting-line theory can be used to explain the reasons for the decrease of the TLV circulation. It can be found in Fig. 10 that as the decrease of the cavitation number, the pressure drop from the hydrofoil pressure side to the suction side decreases, resulting in the decrease of the hydrofoil lift. This drop of the lift is the main reason for the decrease of the TLV circulation. 7. Conclusions In this study, calculations of the developed tip leakage vortex cavitating flows around the tip of a rectangular Clark-Y hydrofoil are conducted to assess the predictive ability of a new transport-based cavitation model. Based on the computations, modelling of liquid– vapor interface mass transfer and the interrelations between cavity and vortex are discussed. The following conclusions can be drawn: (1) The present calculations are capable of capturing the complex vortex structures near the tip region, which include the primary TLV, separation vortices and main secondary vortex. Compared with the conventional Zwart's model, better agreement is observed between the present model predicting TLV cavitation and the experimental results. (2) The results indicate that the rotating function in the cavitation source terms contributes to the improved modelling process of the liquid–vapor interface mass transfer. This is the main reason for the present model's better capability of predicting the TLV cavity shape compared with the conventional Zwart's model. (3) Based on the predictions of the present model, it is found that as the decrease of the cavitation number, the leading edge attached cavity develops gradually and covers the suction side. This will result in the decrease of the hydrofoil lift and hence the TLV circulation.

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Y. Zhao et al. / International Communications in Heat and Mass Transfer 78 (2016) 39–47

σ =1.08 (Z/s = 0.50) σ =1.08 (Z/s = 0.90) σ =1.08 (Z/s = 0.95)

0

1

-Cp

-Cp

1

-1 0.0

σ =0.91 (Z/s = 0.50) σ =0.91 (Z/s = 0.90) σ =0.91 (Z/s = 0.95)

0

-1 0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

X/C

(a)

= 1.08

0.8

1.0

0

= 0.91

σ =0.59 (Z/s = 0.50) σ =0.59 (Z/s = 0.90) σ =0.59 (Z/s = 0.95)

1

-Cp

-Cp

(b)

σ =0.75 (Z/s = 0.50) σ =0.75 (Z/s = 0.90) σ =0.75 (Z/s = 0.95)

1

0

-1

-1 0.0

0.6 X/C

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

(c) = 0.75

0.6

0.8

1.0

X/C

X/C

(d)

= 0.59

Fig. 10. Experimental observed instantaneous leading edge attached cavity shape and corresponding numerical predicted pressure coefficient on the hydrofoil surface, the present model predictions.

Acknowledgment This work was supported by the National Natural Science Foundation of China (Grant No. 51239005 and 51479002).

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