Corresponding author: Ji Pei. jpei@ujs.edu.cn

Comparison of various turbulence models applied to a twisted hydrofoil 1,2Tingyun Yin; 2Giorgio Pavesi, 1Ji Pei*, 1Shouqi Yuan, 1Nana Adu Daniel

1National Research Center of Pumps, Jiangsu University, Zhenjiang 212013, China; 2Department of Industrial Engineering, University of Padova, Padova 35131, Italy

Abstract In the paper, various turbulence models are compared and different modifications employed to investigate more in deep to what extent the RANS are applicable to simulate the unsteady cavitating flow around a twisted foil. The predicted vapor shedding frequency and lift force coefficient are compared with the experimental data. Moreover, the dynamics behavior induced by the cavitation is discussed in detail. The results show that frequency predicted by density correction based model (DCM) SST is much closer to the experimental data, but the model fails to predict the second shedding vapor. In general, DCM RNG shows a better prediction ability. Firstly, a vapor cloud is sheared toward downstream due to the effect of main flow and re-entrant flow. After that, the second shedding vapor catches up with the first one and becomes the main vapor. The oscillation of the lift coefficient changes consistently with the vapor fluctuations and when a second vapor shedding is highlighted, a second peak value occurs in the time history of lift coefficient. Keywords: turbulence model; cavitation; twisted hydrofoil; unsteady shedding

1. Introduction Cavitation, considered as a complex three-dimensional, two phases flows phenomena, always influences design, operation range and reliability of hydraulic machines[1-2]. Recent increases in available computational resources and improvements in turbulence modelling capabilities means with the correct tools and knowledge, cavitation can now be predicted reliably and accurately. Current research is concerned with cavitation simulations of Delft twisted hydrofoil using ANSYS CFX. Various turbulence models are tested and the results are compared with the experimental data obtained by Foeth et al. [3] in a cavitation tunnel. Moreover, the dynamics behavior induced by the cavitation will be discussed in detail.

2. Mathematical model In present multiphase flow simulation approach, any physical variable is assumed to be consistent on each phase component. The mixture made up of liquid and vapor phases are regarded as a kind of one-fluid compressible medium, where no slip between the phases is considered. The density and viscosity of mixture are expressed by the vapor volume fraction αv,

µ = µvα v + µl 1−α v( ) , (1)

ρ = ρvα v + ρl 1−α l( ) . (2)

In the current study, k-ε, RNG k-ε, SST k-ω model and PANS model [4] are compared. To account the large density changes caused by cavitation, Delgosha et al. [5] have considered the compressibility of vapor-liquid mixture phases and proposed a density correction based model (DCM), which effectively reduces the mixture turbulent viscosity. However, in this method, the expression and the constants of k and ε equations are unchanged. The modified turbulent viscosity µt is defined as follows:

µt = f ρm( )Cµ

k 2

ε, (3)

f ρm( ) = ρv + ρm − ρv( )n

ρl − ρv( )n−1

, (4)

For cavitating flow, the use of f(ρm) decreases the turbulent viscosity in flow fields with high vapor volume fraction. However, for non-cavitating liquid flows, the formula of µt follows the original form. Here, n is equal to 10. Since Johansen et al. [6] formulated a filter-based model (FBM), which combines the standard k-ε model with large eddy simulation, many researchers have applied it to cavitation simulation and extended k-ε to RNG k-ε model. In Johansen’s approach, the expression and the constants of k and ε equations are unchangeable; but the formula for turbulent viscosity µt is computed by a filtering procedure:

µt = FilρmCµ

k 2

ε, (5)

Fil = Min 1,C3Δεk 3 2⎡⎣ ⎤⎦ , (6)

where Fil is the filter function with C3 = 1. To ensure the filter process in the present study, the filter size is chosen to be not less than the largest grid scale adopted in the calculation namely, ∆≥fgrid*max((∆x∆y∆z)1/3), ∆x, ∆y and ∆z which are the length of the grid in the three coordinate directions. fgrid is the constant, which is equal to 1.05. Considering compressibility of cavitating flow, a mixed filter-based model (DCM FBM) is presented, which has been discussed by Zhang et al. [7]. The modified turbulent viscosity µt is defined by equation (5) and in equation (6) the constant 1 is replaced by f(n), which is defined as follow:

(7)

As listed in table 1, three modifications are applied to k-ε, RNG k-ε, SST k-ω model. The cavitation process is governed by the mass transfer equation for the conservation of the vapor volume fraction, which is defined as

∂ ρvα v( )∂t

+ ∂∂x j

ρvα vu j( ) = !m = !m+ − !m− (8)

The source terms for the specific mass transfer rate corresponding to the vaporization ( !m+ ) and condensation ( !m- ) are given by

!m+ = Fevap

3α nuc (1−α v )ρv

Rb

23

Max( pv − p,0)pl

(9)

!m- = Fcond

3α vρv

Rb

23

Max( p − pv ,0)ρl

(10)

where Fevap and Fcond are empirical coefficients for the vaporization and the condensation processes with recommended values of 50 and 0.01[8]. αnuc has the value of 5×10-4 and Rb is the radius of typical bubble size with the value of 1×10-6.

3. Geometry The models, tested on the Delft, is the Twist-11 hydrofoil shown in Fig. 1. The hydrofoil is a wing having a rectangular platform of a NACA0009 section with varying attack angles from 0° at the side section to 11° in the mid-section. The chord length of the foil is c = 0.15 m and the span length is 0.3 m. The attack angle of the entire hydrofoil was -2°. A standard right-hand coordinate system with the X-axis (red mark) in flow direction, the Y-axis (green mark) in span wise direction and the Z-axis (blue mark) directed upwards are used as reference coordinate systems. The computational domain is shown in Fig. 2. The hydrofoil was located in a channel with height 2c, a length of 2c upstream of the leading edge, a length of 5c downstream of the leading edge and a width of 2c. The flow simulations discretized only half of the hydrofoil due to its geometric symmetry.

f n( ) = ρv +α ln ρl − ρv( )

ρv +α l ρl − ρv( )

Fig. 1. Three-dimensional twisted hydrofoil.

Fig. 2. Diagram of the computational domain.

Table 1 Turbulence model used in the present research.

Modified method k-ε RNG k-ε SST k-ω PANS

DCM √ √ √ fk = 0.2 fε = 1

FBM √ √

DCMFBM √ √

4. Mesh An O-H type grid was generated for the domain with sufficient refinement towards the foil surface, as shown in Fig. 3. To better resolve the 3D cavity structure, the mesh along the span wise direction was carefully checked in non-cavitation condition with three node numbers. The investigation was performed by monitoring the minimum and maximum pressures around hydrofoil surface and the values of the lift Cl and drag Cd coefficients. From the results shown in Table 2, the differences can be neglected between the medium and fine resolution meshes. Thus, the medium resolution mesh with about 3.3 million nodes was selected as the final grid.

5. Boundary Conditions The inflow velocity was set to Vin = 6.97 m/s. The static pressure at the outlet plane of the domain, i.e. pout, was assigned according to the cavitation number 1.07. The mid-plane was a symmetry plane. The hydrofoil surface was a non-slip wall while the tunnel walls were used free slip walls. To accelerate convergence, k-ε model was used to simulate non-cavitating flow. 20 iterations in each time step was fixed and the iterations stopped when the RMS residual was less than 10-

4 within each time step. The time step was chosen equal to 1×10-4s through the simulation.

6. Results and discussion 6.1 Transient vapor behavior The total vapor volume, Vcav, is a convenient parameter to evaluate the transient behavior of cavitating flows. The total vapor volume calculated at each time step indicates that the vapor volume variation due to the cavity shedding from the twisted hydrofoil is periodic (Fig. 4). It is found that unmodified RANS models fail to predict the unsteady cavitation characteristic. Moreover, DCM has no effect on improving the prediction capability of k-ε model. No significant differences are highlighted on the variation of the vapor volume among all modified models. In Fig. 5, the vapor fraction iso-surfaces αv = 0.1 together with pressure colored foil surface are plotted and compared with the top view pictures taken by Foeth. All modified models are able to simulate the process of vapor shedding. The sheet cavity reaches maximum length and has a convex shape at the rear in instant I. It can be found that only FBM k-ε (Fig. 5.I.b) and DCM FBM RNG (Fig. 5.I.d) model fail to capture the convex shape. The pressure distribution, on the hydrofoil surface highlights a strong adverse pressure gradient at the ending of the cavity that forces the re-entrant flow into the vapor

(a) k-ε model (b) RNG k-ε model

Fig. 4.a Time history of vapor volume.

Fig. 3. Meshing in the computational domain (80nodes).

Table 2 Steady results of non-cavitation flow. Span wise

nodes Grid

number pmin pmax Cl Cd

Angle °

Determinant 2*2*2

40 1,628,400 -54,463.7 54,860.8 0.4244 0.01549 36 0.721

80 3,256,800 -54,679.1 54,847.3 0.4246 0.01546 36 0.754

120 4,885,200 -54,715.6 54,860.9 0.4246 0.01546 36 0.766

(c) SST model (d) PANS model

Fig. 4.b Time history of vapor volume. region. This re-entrant flow moves upstream along the suction surface of the hydrofoil and causes the primary shedding of the cavity (Fig. 5.II). After that, the shedding cavity develops into a second-vapor structure, as shown in Fig .5.III. Not all the modified models showed a good prediction of this structure, for example, PANS (Fig. 5.III.c) and FBM RNG (Fig. 5.III.f) model.

DCMFBM k-ε FBM k-ε PANS DCMFBMRNG DCMRNG FBMRNG DCMSST Experiment

Fig. 5 Cavitation patterns during one cavity shedding cycle.

The shedding cavity quickly changes from a smooth pocket vapor into a highly turbulent vapor cloud, as shown in Fig. 5.IV. Moreover, a new sheet cavity grows from the leading edge of the hydrofoil with a concave shape of the closure line, while the shedding vapor become more turbulent and is entrained downstream by the main flow. After that, the sheet cavity grows slowly and the shedding vapor cloud quickly shrinks and finally collapses in Fig. 5.V. Only DCM SST model fails to predict the secondary shedding structure (Fig. 5.V.g). The predicted shedding frequencies are listed in Table 3. It can be found that frequency predicted by DCM SST is closer to the experimental data, but other modified models show very good results as well. 6.2 Lift force coefficient The predicted lift force coefficients over several cycles are show in Fig 6 and the time averaged values are listed in Table 4. Among all modified models, DCM RNG result is closer to the measurement data. The lift force coefficient evolution using DCM RNG model is compared with the cavity volume in several cycles in Fig. 7. The cavity volume reaches its maximum value contextually with the maximum lift coefficient, which is marked by pink circles. Likewise, when cavity volume drops to the minimum, force coefficient reaches the minimum values, which is marked by a red triangle. When a second vapor shedding occurs, a secondary lift coefficient peak, marked by a blue rectangle, is highlighted.

6.3 Second-vapor structure development The comparison carried out in the sections 6.1 and 6.2 highlighted that DCM RNG model is able to predict the cavitation flow field more accurately. Consistently, the development of second-vapor structure is discussed using

DCM RNG model. Many researchers (Ji et al[9], Bensow[10], Park et al[11]) have successfully captured this phenomenon, but few details were reported.

(a) k-ε model (b) RNG k-ε mode

(c) SST model (d) PANS model

Fig. 6. Time history of lift force coefficient.

Table 4 Time-averaged lift coefficients predicted by various model.

Turbulence model Cal. Exp. Error (%)

RNG k-ε - 0.5167 -

FBMRNG k-ε 0.4053 0.5167 21.56

DCMRNG k-ε 0.4384 0.5167 15.15

DCMFBMRNG k-ε 0.4090 0.5167 20.84

k-ε - 0.5167 -

FBM k-ε 0.4158 0.5167 19.53

DCM k-ε - 0.5167 -

DCMFBM k-ε 0.4180 0.5167 19.10

SST - 0.5167 -

DCMSST 0.4375 0.5167 15.32

PANS 0.3904 0.5167 24.45

Table 3 Shedding frequency predicted by various model.

Turbulence model Cal. Exp.(Hz) Error (%)

RNG k-ε - 32.2 -

FBMRNG k-ε 33.90 32.2 5.28

DCMRNG k-ε 31.29 32.2 2.83

DCMFBMRNG k-ε 34.60 32.2 7.45

k-ε - 32.2 -

FBM k-ε 33.22 32.2 3.17

DCM k-ε - 32.2 -

DCMFBM k-ε 33.90 32.2 5.28

SST - 32.2 -

DCMSST 32.36 32.2 0.50

PANS 34.97 32.2 8.59

Fig 8 shows the vapor volume fraction and velocity vector in the mid-plane. A vapor cloud is sheared towards the downstream due to the effect of main flow and re-entrant flow. This vapor is called first vapor and moves along the foil surface slowly. During this process, the attached vapor in the leading edge is ready to be shed, as shown in instant 2. After the primary shedding occurs nearby the leading edge, a strong attached flow develops quickly. Together with re-entrant flow, this strong flow shears off another vapor, which is called the second vapor. The second vapor moves away from the foil surface and rotates quickly towards the downstream, as shown in instant 3. After that, the second shedding vapor catches up with the first one and becomes a main vapor. Moreover, the leading vapor begins to grow up and reproduces this process, as shown in instant 4-5.

(a) Instant 1 (b) Instant 2

(c) Instant 3 (d) Instant 4

Fig. 8. Evolutions of the αv distribution and velocity vector on the mid-plane.

(e) Instant 5

7. Conclusions Several turbulence models and different modifications were used to simulate the unsteady cavitating flow around a twisted foil in this work. The results can be summarized in the following conclusions: 1. Frequency predicted by DCM SST is much closer to the experimental data, but it fails to predict the second shedding vapor. In general, DCM RNG show a better prediction ability. 2. The oscillation of the force coefficient has some distinct features. The force changes consistently with the vapor fluctuates but a secondary peak value occurs in the time history of lift coefficient when there is a second vapor shedding. 3. A second-vapor structure was observed in the leading edge. Firstly, A vapor cloud is sheared towards the downstream due to the effect of main flow and re-entrant flow. After that, the second shedding vapor catches up with the first one and becomes a main vapor.

Fig. 7. The lift and drag coefficients in contrast to the cavity volume

fluctuation in several cycles.

Acknowledgement This study is supported by National Natural Science Foundation of China (Grant No. 51409123), Natural Science Foundation of Jiangsu Province (Grant No. BK20140554), Qing Lan Project, Training Project for Young Core Teacher of Jiangsu University. The author also gratefully acknowledges the financial support provided by the China Scholarship Council (No. 201708320237). References 1. Kumar P, Saini R P. Study of cavitation in hydro turbines—A review, (2010), Renewable and

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