Ocean Engineering 78 (2014) 22–34
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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Numerical simulation of flow around two- and three-dimensional partially cavitating hydrofoils Fahri Celik a,n, Yasemin Arikan Ozden a, Sakir Bal b a b
Yildiz Technical University, Department of Naval Architecture and Marine Engineering, 34349 Besiktas, Istanbul, Turkey Istanbul Technical University, Department of Naval Architecture and Marine Engineering, 34469 Maslak, Istanbul, Turkey
art ic l e i nf o
a b s t r a c t
Article history: Received 30 October 2012 Accepted 29 December 2013
A new method is developed for the prediction of cavity on two-dimensional (2D) and three-dimensional (3D) hydrofoils by a potential-based Boundary Element Method (BEM). In the case of specified cavitation number and cavity length, the iterative solution method proceeds by addition or subtraction of a displacement thickness on the cavity surface of the hydrofoil. The appropriate cavity shape is obtained by the dynamic boundary condition on the cavity surface and the kinematic boundary condition on the whole foil surface including the cavity. For a given cavitation number the cavity length of 2D hydrofoil is determined according to the minimum error criterion among different cavity lengths. In the 3D case, the prediction of cavity is exactly the same as in the case of 2D method in span wise locations by the transformation of the pressure distribution from analysis of 3D to 2D. The 3D effects at each span-wise location are considered by the multiplication of the cavitation number by a coefficient. The pressure recovery and termination wall models are used as cavity termination. For the 2D case the NACA 16006 and NACA 16012 hydrofoil sections are investigated for two angles of attack using different cavity termination models. For 3D analysis an application for a rectangular hydrofoil with NACA16006 section is carried out. The results are compared with those of other potential based boundary element codes and a commercial CFD code (FLUENT). The effects of different Reynolds numbers (Rn) on the cavitation behavior are also investigated. The results developed from present method are in a good agreement with those obtained from the others. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Boundary element method Sheet cavitation CFD Hyrofoil Cavity termination model
1. Introduction Cavitation appears as 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 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. In the past the two dimensional 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 increasing the foil thickness in the same flow conditions. On the other hand the
n
Corresponding author. Tel.: þ 90 212 383 2856; fax: þ 90 212 236 4165. E-mail addresses:
[email protected] (F. Celik),
[email protected] (Y. Arikan Ozden),
[email protected] (S. Bal). 0029-8018/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2013.12.016
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 increasing the foil thickness. Some studies dealing with leading-edge corrections to linear theory for cavitating hydrofoils and blades have been presented by Tulin (1964); Amromin et al. (1990) and Kinnas (1991). With the beginning of the use of the boundary element methods, a fast approach for the cavitation analysis has been provided. Important studies by using the boundary element methods for the flow analysis of two (2D) and three dimensional (3D) cavitating hydrofoils can be found (Fine and Kinnas, 1993; Kinnas and Fine, 1992, 1993b; Kinnas et al., 1994; Kinnas, 1998; Kim et al., 1994; 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 for 2D and 3D cavitating hydrofoils have been also investigated. In addition the studies (Bal and Kinnas, 2003; Bal, 2007, 2008a, 2008b, 2011) where the iterative BEM methods are applied to cavitating surface
F. Celik et al. / Ocean Engineering 78 (2014) 22–34
Nomenclature 2D 3D
α λ Φ ϕ s ρ
AR B BEM
Two dimensional Three dimensional Angle of attack Length of the transition zone Total potential Perturbation potential Cavitation number Density Aspect ratio Half wing width Boundary Element Method
piercing bodies, and to 2D and 3D cavitating hydrofoils with finite depth and wave tank effects are other important studies in this area. Another approach for the cavitation simulation methods including the viscous effects is based on the numerical solution of RANS and Euler equations. A method based on the solution of the Navier–Stokes equations (Deshpande et al., 1993) and the Euler equations (Deshpande et al., 1994) has been applied by Deshpande et al. They have predicted that the presence of viscous effects on a cavitating flow over 2D hydrofoils have little impact on the cavitating region in regard to Euler analysis. The cavitation erosion risk has been predicted on a 2D hydrofoil (Li et al., 2010) for steady and unsteady conditions and it is concluded that a RANS code, FLUENT, shows promising results in 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 Huang et al. (2010). They have concluded that the RNG turbulence model with enhanced near-wall function offer more satisfactory results than other models. To predict the early stages of the cavity development on the hydrofoils a CFD model has been used by 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. Regardless of all successes of the RANS/LES methods, the potential theory remains to be an important tool at least in the design and optimization stages of the hydrofoils because the consumption of computer time is extremely less than that of other methods. In this study an iterative nonlinear method based on potential theory is developed for the cavity prediction around partially cavitating 2D and 3D hydrofoils. The present analyses method is based on the method developed by the same authors in Arıkan et al. (2012). The potential based panel methods developed for the flow analysis of cavitating 2D or 3D hydrofoils usually gets the result by using a cavity solution approach. In fully wetted flow analysis of the hydrofoil geometry deformed by cavity, in some cases it could be shown that the pressure on the cavity surface is not strictly equal to the vapor pressure of water especially for 3D cases (Kinnas and Fine, 1993b). In the present method, the cavity shape is searched iteratively by deforming the foil surface on the cavity region, and the flow analysis is carried out for the new foil geometry deformed by cavity surface. This new geometry can now be computed as in the fully wetted case. When the dynamic boundary condition is provided over the cavity surface with an acceptable tolerance value, the final cavity shape can be obtained. So it is assured that the determined cavity shape is accurate so that the pressure on the cavity surface being equal to the vapor pressure of water. For 2D cavitation analysis, the NACA 16006 and 16012 sections and for 3D cavitation analysis a 3D rectangular hydrofoil (wing) (the selected aspect ratio AR is 2) with NACA 16006 sections are
c CD CL Cp JB l p Pv Pm Po Rn U
23
Chord length Drag coefficient Lift coefficient Pressure coefficient Strip number on the half wing Cavity length Pressure Vaporization pressure Static pressure Reference Pressure Reynolds number Inflow velocity
used to predict the cavity shapes and pressure distributions, including lift and drag coefficients. Whereas the analyses for the 2D case is made with an incoming flow angles of attack (41 and 61), the analysis for the 3D case is made with an angle of attack (41). The 2D analyses results are compared with those of a potential based BEM (PCPAN) (Kinnas and Fine, 1993a) and a commercial CFD code (FLUENT), while for the 3D cavitating hydrofoil another potential based boundary element method (MXPAN3D) (Fine and Kinnas, 1993) is used for comparison. In addition 2D NACA 16006 hydrofoil section is investigated by CFD to determine the effects of different Reynolds numbers (Rn) on the cavitation behavior.
2. Mathematical formulation for partially cavitating 2D and 3D hydrofoil analysis The flow over the hydrofoil is assumed to be incompressible, inviscid and irrotational according to the potential theory. The submerged 2D and 3D hydrofoils are subject to a uniform inflow (U). The total potential for the flow around the hydrofoil can be described for the 2D case as (Bal and Kinnas, 2003):
Φðx; zÞ ¼ Ux þ ϕðx; zÞ
ð1Þ
The total and the perturbation velocity potential must satisfy the Laplace equation ∇2 Φðx; zÞ ¼ ∇2 ϕðx; zÞ ¼ 0
ð2Þ
The following boundary conditions should be satisfied on the foil surface: (1) Kinematic boundary condition The flow must be tangent to the foil and the cavity surfaces !! ∂ϕ ¼ U:n ð3Þ ∂n ! where n is the unit vector normal to the foil 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 ( C p ) values are equal to the cavitation number (s).
s¼
P1 Pv 1=2 ρ U 2
CP ¼
Pm P1 1=2ρ U 2
ð4Þ
ð5Þ
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F. Celik et al. / Ocean Engineering 78 (2014) 22–34
where P 1 is the total, P v the vaporization and P m is the static pressure. (3) Kutta condition The velocity in the trailing edge of the hydrofoil must be finite. ∇ϕT:E: ¼ finite
ð6Þ
(4) Cavity termination condition This requires the cavity to close at the cavity trailing edge (Kinnas and Fine, 1993b). The cavity termination condition is explained in detail in Section 3.
here, C p1 is the pressure coefficient of the two dimensional section and C p2 is the pressure coefficient of the same section in the three dimensional wing. The cavity shape in 3D is searched as in the same in 2D. For a fixed cavity length in each section, the cavity shape is searched iteratively by deforming the foil surface on the cavity region with small intervals.
The cavity region is determined for the entire wing by repeat
The boundary conditions mentioned above are also valid for the flow analysis of 3D hydrofoils. The y component of the coordinate system should now be included into the formulations (Bal, 2011).
3. Numerical implementation An iterative solution method is developed for the prediction of cavitation over 2D and 3D hydrofoils. The cavitating flow analyses are carried out in an iterative manner using an analysis method for noncavitating hydrofoils. For the flow analysis around a 2D hydrofoil a potential based boundary element method with constant-strength source and doublet distributions (Dirichlet 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 on the cavity region with small intervals. The flow around the hydrofoil is analyzed for the new foil geometries (deformed by cavitation) in each 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. 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 j C p þ sj along the cavity length is minimum. In the three-dimensional hydrofoil problem the prediction of the cavity shape is exactly the same as in the case of the twodimensional method. For the flow analysis of 3D hydrofoil a 3D BEM method with constant-strength source and doublet distributions given in (Katz and Plotkin, 2001) is used. In spanwise locations of the wing the pressure coefficient values ( Cp) obtained from the three-dimensional analyses are used by the transformation to two dimensional case. The cavitation analysis for a constant cavitation number (s) and incoming flow conditions are carried out as follows:
The pressure distributions on the non-cavitating 3D wing are obtained.
The pressure distribution of any section (from the middle to tip) of the non-cavitating 3D wing in the spanwise direction is taken. For the same section the pressure distribution from the 2D flow analysis is determined. To calculate the cavity shape on the section geometry, the 2D flow analysis is used. However on the cavity region with a fixed length, the cavitation number values (s) which will be compared with the pressure coefficient ( C p ) values are recalculated to reflect the 3D wing effects as
s1 ¼ c s
ð7Þ
C p1 c ¼ C p2
ð8Þ
ing the calculations in each section of the span-wise locations beginning from the wing midsection to the tips. Before searching the cavity shape in any section in the spanwise direction, the pressure distributions on this section are found by repeating the 3D flow analysis for the last wing geometry deformed from the cavitation analysis of the previous sections. So the pressure distributions found from the noncavitating 3D wing are used only for the section in the midspan. In all the other sections the pressure distributions are recalculated. The cavitation search process is repeated iteratively for all sections in the span direction so that the minimum error criteria mentioned for 2D cavity analysis is satisfied for the cavity shapes of all sections of the 3D wing.
The algorithm of the cavitation prediction method stated above for 3D hydrofoils can be seen in Fig. 1. 3.1. Cavity detachment An important parameter in the cavitation analysis is the location of the detachment point. While the cavitation begins in the leading edge in cases with sharp leading edge, the cavity detaches from a point downstream of the laminar boundary layer separation point in cases with smooth curved leading edge. In the studies of Arakeri (1975) and Arakeri and Acosta (1976), the relation between the separation point and cavity detachment point has been investigated. Experimental studies by Franc and Michel (1985) have shown that the position of the cavity detachment point has an important effect on the cavity extent and cavity volume. The numerical results obtained by Pellone and Rowe (1988) show that the position of the cavity detachment point has a non-negligible effect and the major effect of the position of the detachment point is on the cavity length. 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. However in experiments the pressure on the foil surface in front of the cavity detachment point has been measured lower than the vapor pressure. From experimental studies, Arakeri (1975) has proposed a criterion to predict the position of detachment of a cavity on an arbitrarily smooth body. In the case of cavitating hydrofoils, the results obtained by Franc and Michel (1985) also determined that the location of the cavity detachment point is just downstream of the boundary-layer separation, but is not where the pressure is minimum in subcavitating conditions. They have proposed a criterion to determine a cavity detachment point. In the present study the detachment point is assumed to be located on the leading edge of the hydrofoil. In the next study, a search algorithm can be applied for mid-chord cavity and face cavity on the hydrofoil surface and the effect of detachment point on the results (cavity length, shape etc.) can be discussed. This study is now under progress.
F. Celik et al. / Ocean Engineering 78 (2014) 22–34
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Fig. 1. Algorithm for the prediction of the cavity shape for a cavitating 3D hydrofoil.
3.2. Cavity termination The cavity termination region consists of a two phase turbulent zone where a very complex flow occurs. The difficulties in obtaining the flow characteristics in this region bring along the use of cavity termination models. The termination model and the cavity detachment point have significant effects on the cavity shape and length of hydrofoils (Franc and Michel, 1985) and (Kinnas and Fine, 1989). There are various cavity termination models to model the cavity termination region such as the termination wall model (the Riabouchinsky model), the reentrant jet model, Tulin0 s spiral vortex models, 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. In this model, a jet flow enters into the cavity from the termination region. Another physically realistic termination model is the termination wall (the Riabouchinsky) model which is similar
Fig. 2. Termination wall model.
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. 2. In the open wake model, it is assumed that a semi infinite wake exists downstream of the real body. In the Tulin spiral vortex models, the free streamlines terminate in a vortex. In both the termination wall model and the re-entrant jet models, a stagnation point which has been observed experimentally by Meijer (1959), occurs in the trailing edge of the cavity. Tulin0 s spiral vortex model and the open wake model do not have a stagnation point in the aft end of the cavity (Krishnaswamy, 2000). Another cavity termination model is the pressure recovery model; the use
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Fig. 3. Pressure recovery model for partially cavitating two-dimensional hydrofoil.
of this is described by Kinnas and Fine (1993b). In this model whose details given below, the pressure on the cavity varies in a region near the end of the cavity so the cavity closes at the cavity trailing edge. In this study two different termination models are considered for cavitation analysis of 2D hydrofoil: the pressure recovery model and the termination wall (Riabouchinsky) model. Due to the effectiveness of the adaptation of both models into the developed method, these models have been used as cavity termination model. Another reason that the pressure recovery model is selected is that the other BEMs in literature applied the same model. So the comparison of the results from the present method is possible with those of other BEMs. Meanwhile in the three dimensional hydrofoil cases only the pressure recovery model is considered. The pressure recovery model requires that in the transition zone (T–L), on the cavity surface, pressure is not equal to the vaporization pressure in order to close the cavity region (Fig. 3). For the application of the pressure recovery termination model the equations below are employed (Kinnas and Fine, 1993b). 8 if sf o sT
