Fahri Celik, Yasemin Arikan Ozden and Sakir Bal

1. Department of Naval Architecture and Marine Engineering, Y?ld?z Technical University, Istanbul 34349, Turkey

2. Department of Naval Architecture and Marine Engineering, Istanbul Technical University, Istanbul 34469, Turkey

1 Introduction1

Cavitation appears to be an unavoidable phenomenon especially in water devices like pumps, marine propellers and hydrofoils where fast flow regimes exist. It can cause undesirable results like performance losses, structural failure,corrosion, noise and vibration. The prediction of performance losses caused by the sheet cavitation which is common particularly on hydrofoils is very important in their design stages. So the development of computational analyses methods is very important since the cavitation occurrence is extensively unavoidable for modern high speed marine vehicles with hydrofoils.

In the past the two-dimensional (2D) cavitating hydrofoil flows were basically formulated under linear theories (Tulin,1953; Acosta, 1955; Geurst and Timman, 1956). The linear theory predicts an increase in the cavity size and volume with the increasing of the foil thickness in the same flow conditions. On the other hand the numerical nonlinear surface vorticity method developed by Uhlman (1987)where the cavity surface is obtained by an iterative manner until both the kinematic and dynamic boundary conditions are satisfied on the cavity surface, predicts that the cavity size should decrease with the increasing of the foil thickness.This defect of the linear theory has been corrected by the introduction of the nonlinear leading-edge correction(Kinnas, 1991).

With the beginning of the use of boundary element methods (BEM), a fast approach for the cavitation analysis has been provided. After the first applications of the boundary element methods (Hess and Smith, 1966; Hess,1973; Hess, 1975), various studies using BEM have been conducted for the flow analysis of 2D non-cavitating hydrofoils (Katz and Plotkin, 2001; Chow, 1979). For the cavity prediction of partial and supercavitating 2D and 3D hydrofoils based on the potential theory through the use of boundary element methods, a nonlinear analysis method has been developed (Kinnas and Fine, 1993).

Some important studies on the use of the boundary element methods for the flow analysis of 2D and 3D cavitating hydrofoils can be found (Fine and Kinnas, 1993;Kinnas and Fine, 1992; Kinnaset al., 1994; Kinnas, 1998;Kimet al., 1994; Kim and Lee, 1996; Dang and Kuiper,1998; Dang, 2001; Krishnaswamy, 2000). In these studies,non-linear analysis methods for viscous and inviscid flows around 2D and 3D cavitating hydrofoils and the prediction methods for unsteady sheet cavitation around 3D cavitating hydrofoils are presented. The effects of various cavity termination models on 2D and 3D cavitating hydrofoils have been also investigated. In addition the studies (Bal and Kinnas, 2003; Bal, 2007, 2008, 2011) where the iterative BEM methods are applied to the finite depth and wave tank effects on cavitating 2D and 3D hydrofoil analyses(including the surface piercing type of hydrofoils) are other important studies on this subject.

Another approach for the cavitation simulation methods including the viscous effects is based on the numerical solution of the RANS and Euler equations. A method based on the solution of the Navier-Stokes equations (Deshpandeet al., 1993) and the Euler equations have been applied by Deshpandeet al. (1994). They have predicted that the presence of the viscous effects on a cavitating flow over 2D hydrofoils have little impact on the cavitating region in regard to the Euler analysis. A different study on the analysis of the partial cavity flow on 2D hydrofoils with the use of thek-εturbulence model has been carried out (Dupont and Avellan, 1991) and has been extended (Hirschiet al., 1997)for the cavity prediction on a twisted hydrofoil and a pump impeller.

CFD methods also offer effective approaches to model the cavitating flows. The cavitating flow can be modeled as steady or unsteady phenomena. An unsteady flow approach has been studied (Berntsenet al., 2001) where the cavity length is calculated with a high level of accuracy. The cavitation erosion risk has been predicted on a 2D hydrofoil(Liet al., 2010) for steady and unsteady conditions and it is concluded that a RANS code, FLUENT, shows promising results with the prediction of unsteady cavitation phenomena.The effects of different turbulence models and wall treatments on the cavitating flow around hydrofoils have been compared by Huanget al. (2010). They have concluded that the RNG turbulence model with enhanced near-wall functioning offers more satisfactory results than those of the other models. To predict the impact of the early stages of the cavity development on the hydrofoils, a CFD model has been used in Hoekstra and Vaz (2009). They have concluded that the re-entrant jet model must be considered as an invalid approach for the modeling of steady partial cavities.

In this study an iterative nonlinear method based on the potential theory was developed for the cavity prediction around partially cavitating 2D hydrofoils. The present analyses method is based on the method developed by the same authors in Ar?kanet al.(2012). For the 2D cavitation analysis, the NACA 16006, NACA 16012 and NACA 16015 sections were selected to predict the cavity shapes and pressure distributions, including lift and drag coefficients.The analyses were made for two different incoming flow angles of attack (4 and 6 degrees). The results obtained from the present method were compared with those of another potentially based BEM method (PCPAN) (Kinnas and Fine,1993a) and a commercial CFD code (FLUENT).

2 Mathematical formulation for partially cavitating 2D hydrofoil analysis

The flow over of the hydrofoil is assumed to be incompressible, inviscid and irrotational according to the potential theory. The submerged 2D hydrofoils are subject to a uniform inflow (U) (see Fig. 1).

Fig. 1 2D hydrofoil

The total potential for the flow around the hydrofoil can be described for the 2D case as (Bal and Kinnas, 2003):

The total and the perturbation velocity potential must satisfy the Laplace equation:

The following boundary conditions should be satisfied on the foil surface.

1) Kinematic boundary condition:

The flow must be tangential to the foil and the cavity surfaces,

wherenis the unit vector normal to the foil surface and cavity surface.

2) Dynamic boundary condition:

The dynamic boundary condition requires that the pressure on the cavity surface is constant and equal to the vaporization pressure. This condition is satisfied by the deformation of the foil geometry so that on the cavity surface the pressure coefficient (－Cp) values are equal to the cavitation number (s),

whereP∞is the total,Pvis the vaporization andPmis the static pressure.

3) Kutta condition:

The velocity in the trailing edge of the hydrofoil must be finite,

4) Cavitation condition:

This requires the cavity to close at the cavity trailing edge(Kinnas and Fine, 1993).

3 Numerical implementation

An iterative solution method is developed for the prediction of cavitation over 2D hydrofoils. The cavitating flow analyses are carried out in an iterative manner using an analysis method for non-cavitating hydrofoils. For the flow analysis around a 2D hydrofoil, a potential based boundary element method with a constant-strength source and doublet distributions (Dirichlet type of boundary condition) is used.The details of the method can be found in Katz and Plotkin,(2001). For a fixed cavity length and a constant cavitation number, the cavity shape is searched iteratively by deforming the foil surface in the cavity region with small intervals (Dh). The flow around the hydrofoil is analyzed for the new foil geometries (deformed by cavitation) in each of the iteration stages by the BEM method. When the dynamic boundary condition is provided over the cavity surface with an acceptable tolerance value, in the last iteration, the final cavity shape and the pressure distributions on the cavitating hydrofoil are obtained (Fig. 2). For a constant cavitation number, cavity shapes are searched for various fixed cavity lengths on the foil surface, and the appropriate cavity length and shape is selected according to the minimum error criteria where the sum of |-Cp+σ| along the cavity length is the minimum.

Fig. 2 Cavity shapes at different iteration stages

3.1 Cavity detachment

An important parameter regarding the cavitation analysis is the location of the detachment point. While the cavitation begins in the leading edge in cases with a sharp leading edge,the cavity detaches from a point downstream of the laminar boundary layer separation point in cases with a smooth curved leading edge. The location of the cavity detachment point can be determined by using the smooth detachment condition (Brillouin-Villat condition). In this condition it is assumed that the cavity does not intersect with the foil surface at its leading edge and that the pressure on the wetted foil surface in front of the cavity is not lower than the vapor pressure (Krishnaswamy, 2000; Kinnas and Fine,1991). In the present study the detachment point is assumed to be located on the leading edge of the hydrofoil. The search algorithm similar to that of Kinnas and Fine (1991)for the cavity detachment point is under progress.

3.2 Cavity termination

The cavity closure region consists of a two phase turbulent zone where a very complex flow occurs. The difficulties in obtaining the flow characteristics in this region require the use of the cavity termination models.There are various cavity termination models to simulate the cavity closure region such as the end plate Riabouchinsky or termination wall model, the re-entrant jet model, Tulin’s spiral vortex model, the open wake model and the pressure recovery model. The most physically realistic cavity termination model is the re-entrant jet model due to its best representation of the streamlines around a cavitating body.Another physically realistic termination model is the termination wall (the Riabouchinsky) model which is similar to the re-entrant jet model. The wall termination model in which the cavity region is closed with a vertical wall is shown in Fig. 3. With both the termination wall model and the re-entrant jet models, a stagnation point which has been observed experimentally, occurs in the trailing edge of the cavity (Krishnaswamy, 2000). Another cavity termination model is the pressure recovery model, and its details are given in the following paragraph; the use of this model has been described by Kinnas and Fine (1993).

In this study two different termination models are considered for the cavitation analysis of the 2D hydrofoil:First, the pressure recovery model and second the termination wall model.

Fig. 3 Termination wall model

The pressure recovery model requires that in the transition zone (T-L), on the cavity surface, the pressure is not equal to the vaporization pressure in order to close the cavity region (Fig. 4).

Fig. 4 Pressure recovery model for partially cavitating the two-dimensional hydrofoil

For the application of the pressure recovery termination model the equations below are employed (Kinnas and Fine,1993).

here,sfis the arc-length of the foil measured beneath the cavity, measured from the cavity’s leading edge. The(0>lt;A>lt;1) andv(v>gt;0) are arbitrary constants. The cavity length is shown withland the abbreviationsDstands for the detachment point,Tfor the beginning of the transition zone andLfor the trailing edge of the cavity. In the present method the cavitation number (σT) is taken as:

Through the comparison of the pressure coefficient (－Cp)with the new cavitation number (σT), the dynamic boundary condition on the foil surface (D-L) is provided.

In the present method, for the 2D analysis, the termination wall model is applied automatically if thef(Sf)value in Eq. (7) in the termination model is selected as zero for the entire cavity region (D-L).

4 CFD Analysis Method

In the computational method, the governing equations are discretized using a first-order upwind interpolation scheme,and the discretized equations are solved using the SIMPLEC algorithm. The related equations are solved for the flow,vapor and turbulence. The convergence criterion for the solution is selected as 10-6. The mixture model is chosen as the multiphase model and the cavitation model is included in the calculations (Fluent 6.3 User’s Guide, 2006).

For the mathematical model the incompressible,time-averaged, unsteady mean flow equations of continuity and momentum which are used in the mixture multiphase model are given below (Fluent 6.3 User’s Guide, 2006):

whereθis the mass averaged velocity,ρmis the mixture density andαkis the volume fraction of phasek.

herenis the number of the phases,Fis a body force,μmis the viscosity of the mixture andvdr.kis the drift velocity for secondary phasek.

Thek-ε(RNG) turbulence model with enhanced wall treatment is selected. Turbulence kinetic energy,k(Eq. (11))and its dissipation,ε(Eq. (12)) for the mixture model are:

In these equations,Gkrepresents the generation of the turbulence kinetic energy due to the mean velocity gradients,Gbis the generation of the turbulence kinetic energy due to buoyancy,YMis the contribution of the fluctuating dilatation in the compressible turbulence to the overall dissipation rate,αkandαεare the inverse effective Prandtl numbers forkandε,SkandSεare the source terms.

The cavitation effects for the two-phase mixture flows are included by using the standard cavitation model in Fluent.The cavitation model implemented in Fluent is based on the“full cavitation model” (Singhalet al., 2001). The model can be used with all of the turbulence models and with the mixture multiphase model. For the calculations, the first phase is selected as liquid water and the second phase is selected as water vapor. The vapor transport equation with“f” as the vapor mass fraction is given as (Hoekstra and Vaz,2009):

whereρis the mixture density,vvis the velocity vector of the vapor phase,γis the effective exchange coefficient,ReandRcare the vapor generation and condensation rate terms.The final phase rate expressions obtained after accounting for the effects of the turbulence-induced pressure fluctuations and non condensable gases are forp>lt;pv:

The suffixes (”l” and “v”) denote the liquid and the vapor phases,σis the surface tension coefficient of the liquid,pvis the liquid vaporization pressure andCeandCcare empirical constants.

5 Numerical results and discussions

In this section some numerical results of the present method for the cavity prediction on 2D hydrofoils are presented. The calculations where the cavity shapes and pressure distributions are determined for the 2D hydrofoils are carried out for three different sections (NACA 16006,NACA 16012 and NACA 16015) at two angles of attack(α=4oandα=6o). For all the cases a constant cavity length(x/c=0.5) is considered, and the cavitation numbers (s) are obtained from another potential based BEM (PCPAN) with the pressure recovery model. Furthermore, the influences of the different cavity termination models on the cavity shapes and pressure distributions of the cavitating foil are investigated. The fluctuations on pressure distribution is avo?ided in a similar method (filtering technique) given in(Longuet-Higgins and Cookelet, 1976). The 2D analyses results are compared with those of PCPAN (Kinnas and Fine,1993a) and a CFD technique (FLUENT).

Cavitation prediction for the NACA 16006, NACA 16012 and NACA 16015 hydrofoils

With the 2D BEM analysis a total number of 129 panels are implemented on the entire hydrofoil (Fig. 5). 129 panels are distributed in equal numbers on the pressure side [DM],on the cavity surface [DL] and on the non-cavitating part of the suction side [LM]. The panels are placed with full cosine spacing. The panel numbers are determined according to the best converged results obtained from the numerical studies.For all of the applications of the present method, the deformation amount non-dimensionalized by the chord length of the section on the cavity surface in each iteration step is Dh/c=0.00001 and the convergence rate is reached in a maximum 1 000-iterations in each case investigated here.The pressure recovery termination and the wall termination models implemented in the analyses are explained in the previous sections. With the present method, when a termination model is not selected, the wall termination model is applied by default. For comparison with similar conditions, the constants in the pressure recovery termination model are selected as the same as in PCPAN(n= 1,l= 0.1,A=0.5 as suggested for NACA 16006 section)(Kinnas and Fine, 1993a).

Fig. 5 Panel arrangement of the 2D hydrofoil (height magnified by 5)

In Figs. 6-17 the comparisons of the cavity shapes and pressure distributions by the present method and PCPAN for NACA 16006 (σ=0.896 and 1.349;α=4oand 6o), NACA 16012 (σ=0.878 and 1.330;α=4oand 6o) and NACA 16015(σ=0.863 and 1.311;α=4oand 6o) foil sections (height magnified by 5) are demonstrated. Furthermore, the results obtained from the present method with two different cavity termination models are compared.

Fig. 6 Pressure distributions and cavity shapes (pressure recovery termination model) (2D hydrofoil, height magnified by 5)

Fig. 7 Pressure distributions and cavity shapes from present method for the pressure recovery and wall termination models

Fig. 8 Pressure distributions and cavity shapes (pressure recovery termination model)

Fig. 9 Pressure distributions and cavity shapes from present method for the pressure recovery and wall termination models

Fig. 10 Pressure distributions and cavity shapes (pressure recovery termination model)

Fig. 11 Pressure distributions and cavity shapes from present method for the pressure recovery and wall termination models

Fig. 12 Pressure distributions and cavity shapes (pressure recovery termination model)

Fig. 13 Pressure distributions and cavity shapes from present method for the pressure recovery and wall termination models

Fig. 14 Pressure distributions and cavity shapes (pressure recovery termination model)

Fig. 15 Pressure distributions and cavity shapes from present method for the pressure recovery and wall termination models

Fig. 16 Pressure distributions and cavity shapes (pressure recovery termination model)

Fig. 17 Pressure distributions and cavity shapes from present method for the pressure recovery and wall termination models

In Figs. 6-17 the non-dimensional cavity lengths (x/c) are obtained approximately as 0.5 from the present method with the pressure recovery model, and as 0.475 with the wall termination model.

The results obtained for these cavity lengths from the present method are in good agreement with the results of PCPAN by means of the cavity shapes and pressure distributions for all cases (Figs. 6, 10, 12, 14, 16) except for the case of NACA 16006 atα=6o(Fig. 8). It is considered that the differences seen in the transition zone in Fig. 8 are due to the section with a small thickness and a high flow angle of attack. It is observed from Figs. 7, 9, 11, 13, 15, and 17 where the cavity closure models of the present method are compared,that the pressure distributions are in accordance except for the cavity closure region. In the comparison of the present method with the wall termination model, the occurrence of a small slip on the peak values of the pressure coefficient (-Cp)is seen due to the characteristics of the termination model in the transition zone.

In the CFD analysis a NACA 16006 foil is used for the sheet cavity prediction in the 2D calculations. The CFD solver FLUENT 6.3 is used and the grid is generated with GAMBIT 2.4.6. For the 2D case a structured mesh with 46,800 elements is created. The near foil surface region is meshed denser for a correct cavitation simulation. The boundary conditions and the grid over the foil are shown in Figs. 18 and 19, respectively. The following boundary conditions are considered for the calculations, the inlet and the side surfaces are set as the velocity inlet and the outlet is set as the pressure-outlet. A first order upwind scheme is selected for the discretization. The mixture fluids for the cavitation modelling are selected as liquid water and water vapour. The density of the liquid water is selected as 1 000 kg/m3and the density of the water vapour is selected as 0.025 58 kg/m3. The non-condensable mass fraction is taken as 1.6×10-5. As for the turbulence model, thek-ε(RNG)turbulence model is set. The cavitation analyses are made by adjusting the vaporisation pressure, the flow velocity and the gauge pressure in the outlet to provide the intended cavitation number. Calculations for the two dimensional hydrofoil are made forα=4oandα=6owithσ=0.896,σ=1.35,and for the Reynolds numberRe= 7.9×106.

Fig. 18 Mesh view and boundary conditions

Fig. 19 Mesh around the hydrofoil

In Figs 20 and 22, the pressure distributions for the 2D hydrofoil (NACA 16006) are shown forα=4o,σ=0.896 andα=6o,σ=1.349, respectively. The results from the present method with the pressure recovery termination model are in comparison with the results of CFD.

Fig. 20 Pressure distributions from the present method and CFD

Fig. 21 Pressure distributions from the present method and CFD

Fig. 22 Pressure distributions from the present method and CFD

Fig. 23 Pressure distributions from the present method and CFD

Fig. 25 Pressure distributions from the present method and CFD

In Figs. 20-25, the accordance of the pressure distributions (-Cp) with the present method and CFD is observed apart from the region of the transition zone where the closure model is used in the present method. So the comparison of the results of CFD including the viscous effects with the BEM results is reasonable except for near the region of the cavity termination. Further results are supported by the results presented in Krishnaswamy (2000)and Huanget al. (2010).

As it can be seen in Fig. 21, especially near the region of the cavity termination, the pressure distributions obtained from the present method differ from the results of CFD due to the fact that the thinner section and the higher angles of attack of the incoming flow give thicker cavity shapes. This can also be seen in the case of the cavity shapes given in

Figs. 26-30. If the angle of attack increases, the cavity becomes thicker and the difference between the cavity lengths of the present method and those of CFD becomes larger (Figs. 26 and 27). If the thickness of the foil increases,the difference now between the cavity lengths of the present method and those of CFD becomes smaller (Figs. 26 and 30).

The cavity shape from the present method with the pressure recovery termination model and CFD are demonstrated in Figs. 26 and 31.

From Figs. 22 and 23 it is observed that the cavity length obtained by CFD is shorter than the length from the present method, but the cavity shape is in accordance except for near the zone of the cavity termination. Furthermore, from Fig. 22 (α= 4o) it can be seen that the cavity length obtained by CFD is approximately equal to the length obtained by the present method in regard to the pressure distribution (-Cp)on the hydrofoil. It is assumed that the shorter cavity lengths obtained by CFD (which is more realistic because of the including of the viscous effects) occur due to the effects of the break off cycle. These effects are presented in de Lange and de Bruin (1998) based on the results of the experiments.

For the NACA 16006, NACA 16012 and NACA 16015 sections and for the angles of attackα=4oandα=6o, the lift and drag coefficients (CL,CD) from the present method,PCPAN with the pressure recovery cavity closure model and CFD are given in Table 1.

Fig. 26 Cavity shapes, present method (solid line) and CFD(colored), NACA 16006, s= 0.89 and α= 4o

Fig. 27 Cavity shapes, present method (solid line) and CFD(colored), NACA 16006, s= 1.349 and α= 6o

Fig. 28 Cavity shapes, present method (solid line) and CFD(colored), NACA 16012, s= 0.878 and α= 4o

Fig. 29 Cavity shapes, present method (solid line) and CFD(colored), NACA 16012, s= 1.330 and α= 6o

Fig. 30 Cavity shapes, present method (solid line) and CFD(colored), NACA 16015, s= 0.864 and α= 4o

Fig. 31 Cavity shapes, present method (solid line) and CFD(colored), NACA 16015, s= 1.311 and α= 6o

The 2D hydrofoil’s lift coefficient (CL) and drag coefficient (CD) are as follows:

Table 1 The lift and drag coefficients (CL, CD) from different methods

It is shown in Table 1 that the lift coefficients (CL) from this method show good agreement with those of PCPAN and CFD while the differences in the drag coefficients are relatively larger. TheCLvalues obtained from the present method show an approximately 2% deviation for all sections and angles in comparison to those of PCPAN. While the lift coefficients obtained from CFD for the NACA 16006 section differ about 3% from the PCPAN, the error increases for the section thickness (NACA 16012 and NACA 16015).It is assumed that these results are caused by the viscous effects which are included in the CFD analysis and by the decrease of the cavity thickness while the section thickness increases.

6 Conclusions

An iterative method based on the potential theory has been presented for the prediction of cavities around partially cavitating 2D hydrofoils. For a specified cavitation number and cavity length, the surface part of the hydrofoil in the cavity length is deformed iteratively until the dynamic boundary condition has been satisfied with a fully wetted analysis within an acceptable accuracy. The appropriate cavity length and shape of the 2D hydrofoil have been determined according to the minimum error criterion among different cavity lengths.

For the 2D analysis, the NACA 16006, 16012 and NACA 16015 sections have been selected. The analyses have been performed with incoming flow angles of attack of 4 and 6 degrees. The results obtained from the 2D analysis have been compared by means of cavity shapes and pressure distributions with the results obtained by PCPAN and by the CFD technique. The results have shown good consistency with the results of PCPAN and CFD.

As for the cavity closure model, the pressure recovery and wall termination models are used. From the applications carried out in the study, it has been found that the cavity lengths with the wall termination models are less than those with the pressure recovery model for all cases as expected.The other termination models can also be included with the present method.

Since the present method developed in this study is based on a fully wetted analysis, the dynamic boundary condition over the cavity surface is satisfied automatically with the kinematic boundary condition. So the present method gives more realistic cavity shapes. The present method gives a faster approach for the cavity prediction than especially the CFD or experimental methods in terms of time.

In future studies the methods presented here for the analysis of 2D cavitating hydrofoils can be extended to cavitating 3D hydrofoils and marine propellers. Furthermore,these methods can also be applied to supercavitating cases and to cases where face cavitation occurs.

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Author biographies

Fahri ?elik, graduated from the Department of Naval Architecture and Marine Engineering, ?stanbul Technical University, Turkey in 1994. He completed his M.Sc., PhD in 1997, 2005, respectively. He has been working as a Professor at Yildiz Technical University,Turkey.

Yasemin Arikan ?zden, graduated from the Department of Naval Architecture and Marine Engineering, Yildiz Technical University, Turkey in 2008. Shecompleted her M.Sc in 2010 and has been studying for her PhD since 2010 at Yildiz Technical University, Turkey. She has also been working as a Research Assistant at the NAME of YTU.

Sakir Bal was born in 1967. He is a full professor at the Department of Naval Architecture and Marine Engineering and superintendent of the Ata Nutku Ship Model Testing Laboratory,Istanbul Technical University. He was a visiting research scholar at the University of Texas, Austin, USA and the University of Newcastle upon Tyne (UK). He gave lectures at the University of Liege, Belgium. His current research interests include ship hydrodynamics, and marine propellers.