- Email: [email protected]
Numerical study on the control mechanism of cloud cavitation by obstacles
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
750
2010, 22(5), supplement :792-797 DOI: 10.1016/S1001-6058(10)60032-7
Numerical study on the control mechanism of cloud cavitation by obstacles
Wei-guo Zhao, Ling-xin Zhang * , Xue-ming Shao, Jian Deng Institute of Fluid Engineering, Department of Mechanics, Zhejiang University, Hangzhou, China *Email: [email protected] ABSTRACT: Numerical simulations of cavitating flows on 2D NACA0015 hydrofoils with and without obstacle are performed. Cavitation model is based on a transfer equation for a void mass fraction and an improved RNG k-ε model is adopted to study the turbulent cavitating flows around the foils. Different arrangements and geometric parameters of the obstacles are investigated. Computational results show that cloud cavitation can be reduced effectively by an obstacle placed on the foil surface; also the performance of the hydrofoil is changed by the obstacle simultaneously. KEY WORDS: Cavitation; Re-entrant jet; NACA0015; Obstacle; Multiphase
1
INTRODUCTION
Cavitation is a special type of phenomena in liquid flow[1]. When the static pressure decreases below the saturated pressure in the local temperature, liquid becomes to vaporize and cavitation can be found in the flow[2]. In all kinds of cavitation, cloud cavitation is on the focus of researches because of its instability which can cause vibration and noise, even lead to structural damages[3]. Instability of the sheet cavitation, and the generation mechanism of cloud cavitation, has been performed a lot by many researchers[3,4]. Some mechanism put forward one after another, such as the shock wave theory[5], and the re-entrant jet theory[3].
Among them the re-entrant jet theory, is widely accepted. In 1997, Kawanami et al.[3] confirmed the major role of re-entrant jet in the instability of sheet cavitation, through setting up barriers in the hydrofoil surface. Since then, Pham et al.[4] further confirmed the effectiveness by setting up obstacles on the surface of hydrofoil to re-entrant jet suppression and cloud cavitation control. Experimental and numerical methods are the dominating research methods of cavitating flow. Since cavitation is a complex two-phase flow, so the experimental methods are special, in general, have a higher cost. Some experimental methods have been developed for cavitating research: PIV (Particle Imaging Velocimetry) - LIF (Laser Induced Fluorescence) method[6], LDV (Laser Doppler Velocimetry) method[7], and double optical probe method[8]. The numerical simulation is relatively less costly, but was limited by the cavitation model and the corresponding numerical method. Since cavitation is a two-phase flow, variable density and the density ratio between the two-phase flow changes abruptly, so the numerical method is different from the usual single-phase flow; and usually pressure-based approach is used to solve the equations[9]. Because of its relatively large density, stability and convergence of the algorithm is also not so perfect. Based on phase
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China transition of the cavitating flow, different researchers proposed many cavitation models. There are mainly two kinds of cavitation model: VOF method based on the interface tracking; the homogeneous mixture flow model which deals the two phase flow as a uniform mixture, and give the physical description of the phase transition through a volume fraction transport equation[10-12]. In this paper, numerical simulation is performed on a NACA0015 hydrofoil with and without obstacles on the suction surface, focusing on the instability of sheet cavitation and control of cloud cavitation. Several arrangements of obstacles were performed in order to reveal the influence to re-entrant jet and production mechanism of cloud cavitation. 2
751
cavitating flow is variable, the major difficulty for numerical simulation is to deal with the large density changes associated with phase change between water and vapor. Especially in cavitation area, the density of the mixture changes abruptly. When the conventional RNG k-ε turbulence model was used, the unstable cavitating behavior cannot be simulated correctly [2]. So modified RNG k-ε turbulence model by artificially reducing the turbulent viscosity of the mixture is essential to model the unsteady cavitating flow:
μ t = f ( ρ )C μ
k2
ε
f ( ρ ) = ρ v + (1 − α ) n ( ρ l − ρ v ) n > 1
（4） （5）
PHYSICAL MODEL
2.1 Multiphase model A homogeneous model approach is adopted to simulate the two phase cavitating flow: relative motion between liquid and vapor phase is neglected and the mixture comprised by liquid and water vapor is treated as a homogenous medium with variable density. Only one set of equations is solved for the mixture flow, while the void fraction is solved by a void mass transport equation. The continuity equation for the mixture flow is: JJG ∂ （1） ( ρ m ) + ∇ • ( ρVm ) = 0 ∂t Where ρ m = ρ vα + ρ l (1 − α ) and Vm denotes the density and velocity of the mixture respectively. The momentum equation for the mixture flow is:
JJGJJG JG ∂ JJG ( ρVm ) + ∇ • ( ρVm Vm ) = F − ∇p ∂t JJG JJG T +∇ • ⎡⎢ μ m (∇Vm + ∇Vm ) ⎤⎥ ⎣ ⎦
（2）
In cavitating flow, the liquid-vapor mass transfer is governed by the vapor transport equation: JJG ∂ （3） ( ρ m f ) + ∇ • ( ρ m f Vm ) = Re − Rc ∂t 2.2 Turbulence model Modified RNG k-ε turbulence model was introduced to close the control equations. Because density of the
Fig. 1 Function of density used for turbulence viscosity modification
The modification limits the turbulent viscosity in the region filled mainly by vapor phase and consequently allows the formation of a re-entrant jet and the cavitation cloud separation. Different values of n can affect the results, so several n values were investigated, and n=10 was adopted in the present computation. Fig.1 shows the modified value of density function. 2.3 Cavitation model Singhal’ cavitation model [11] was adopted in the present simulation. Experimental and numerical investigations have shown that turbulent pressure fluctuations have a significant effect on cavitating flows, so the saturation vapor pressure used in the cavitation model is modified accounting for:
′ /2 Pv = Psat + Pturb Where
（6）
′ = 0.39 ρ k is the turbulent pressure Pturb
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
752
fluctuation. Source terms Re and Rc are given by： 1/2
⎛2 P −P⎞ Re = Ce ρl ρv ⎜ v ⎟ σs ⎝ 3 ρl ⎠ (1 − f v − f g ) k
（7）
1/2
⎛2 P −P⎞ Rc = Cc ρl ρv ⎜ v ⎟ σs ⎝ 3 ρl ⎠ k
foil surface, non-slip wall boundary is given. On the upper and lower boundaries, slip wall boundary is adopted.
fv
（8）
Where Ce and Cc are empirical constants and the recommended value is Ce = 0.02 ， Cc = 0.01 ; the surface tension,
is
is the vapor mass fraction and
(a)
is the mass fraction of noncondensable gases. 3
COMPUTATIONAL MODEL AND GRID
A 2-D NACA0015 foil was used for the present simulation. The computational domain and grid system is shown in Fig 2. The foil chord length c is 0.1m. An arc shape inlet is located at 5 times chord upstream of the leading edge, while the outlet is located at 10 times chord length downstream of the trailing edge. The top and bottom boundary is located at 5 times chord length from the domain centre. The angle of attack is α =8 ° . Six arrangements of obstacle were set up for the simulation. The relative distance based on chord length between the left side of the obstacle to leading edge of the foil is indicated by cr, and three values of cr are 0.32, 0.37, 0.45. Width of the obstacle is 0.02c, and two heights are selected, 0.01c and 0.02c respectively. The total number of grid points is 231×200 and the first grid spacing near the wall based on y+ is about 50. For space discretization, the finite volume method is adopted, while for time discretization, a first order implicit scheme is applied. Convective terms of the governing equation are discretized with second-order upwind scheme, and diffusive terms are discretized using second-order central differencing scheme. The Simplec algorithm is used for the pressure and velocity coupling method. Uniform flow condition and pressure outlet are used at the inlet and outlet boundary respectively, while in the
(b) Fig. 2 Computational domain and grid. (a) without obstacle; (b) with an obstacle on the suction surface
4
RESULTS AND DISCUTION
Non-dimensional parameters simulation are defined as： Reynolds number： Re = Cavitating number： σ =
used
in
present
ρV∞ l μ
P∞ − Pv 0.5ρV∞2
Pressure coefficient： CP =
P − P∞ 0.5ρV∞2
For all the simulations in this paper, Reynolds number is kept const and Re=1×106. 4.1
Cavitation on the fil without obstacle
Cavitating flow over NACA0015 foil when the cavitation number σ = 1.5 is simulated to validate the computational code.
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
753
Fig.3 shows the relatively stable sheet cavity appearance when σ = 1.5 . The angle of attack is 8° and Reynolds number based on the foil chord length is 3×105. Experimental observations[12] show that the bubble clusters mainly located at the leading edge, and there are fewer bubbles located in the middle of the hydrofoil. Kubota’s model [12] and Wang’s simulation [7, 13] captured the character well. The present simulation agrees well with the experimental results.
Fig. 5 Period of lift and drag coefficients on NACA0015 foil without obstacle, σ = 1.2 Fig. 3 Sheet cavity appearance on NACA0015 hydrofoil in normal state, α=8°， Re=3×105, σ = 1.5 .
Cloud shape cavitation without obstacle on the foil surface is simulated when cavitation number σ = 1.2 . Fig. 4 shows the periodic shedding process of the unsteady cloud cavity. Through the lift and drag coefficient variation shown in Fig5, the fluctuation frequency of the cavity shedding can be evaluated. In present state, the frequency is about 60Hz. The mean length of the cavity is about 0.49c. Strouhal number based on the mean cavity length and reference velocity equals to 0.27, which is close to the classical Strouhal number 0.3. A typical shedding shape is observed in Fig. 6, re-entrant jet can be found in the bottom of the cavity.
Fig. 4 Cloud cavitation shape indicated by vapor void fraction on NACA0015 foil without obstacle,
σ = 1.2
Fig. 6 Cavity shedding and Re-entrant jet on NACA0015 foil without obstacle, σ = 1.2
4.2
Cavitation on the foil with obstacles h=1%c
Fig.7 shows the variation of lift and drag coefficients on NACA0015 foil without and with obstacle which extrudes 1%c from the foil surface. It is observed that lift of the foil with an obstacle drops comparing with the foil without obstacle. The reason is the existence of an obstacle on the foil surface change the pressure distribution on the suction side. Also it can be found that the drag coefficient drops. Experiment by Kawanami et al[3]. found this phenomenon too. Fig.8 is the frequency spectrum analysis of the lift coefficient. Except when cr=0.32, frequency pulsations of the other cases decreased obviously. When cr=0.37 and cr=0.45, shedding area of cloud structure also decreases. Fig. 9 shows the cavity appearance when cr=0.45, where we can see that the length of the sheet caivity and the shedding area are less than that of the foil without the obstacle.
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
754
the ratio has a reversed tendency and the lift drop is larger, which means performance of the foil has a greater change. Fig.11 is the frequency spectrum analysis of the lift coefficient. Except when cr=0.37, frequency pulsations of the other cases decreased and have a relative lower amplitude.
Fig. 7 Variation of lift and drag coefficients on NACA0015 foil with obstacles，h=1%c
Fig. 10 Variation of lift and drag coefficients on NACA0015 foil with obstacles，h=2%c Table 1 Time averaged force coefficients with and without obstacle cr
Fig. 8 Frequency spectrum analysis of lift on NACA0015 foil with obstacles h=1%c
h=1%c Cl
Cd
h=2%c Cl/Cd
Cl
Cd
Cl/Cd
0.32
0.659 0.089 7.370
0.681
0.091
7.474
0.37
0.641 0.084 7.626
0.665
0.091
7.307
0.45
0.656 0.086 7.669
0.578
0.085
6.810
no obs 0.731 0.115 6.372
0.731
0.115
6.372
Fig. 9 Cavitation shape indicated by vapor void fraction on NACA0015 foil witht obstacle, cr=0.45,h=1%c, σ = 1.2
4.3
Cavitation on the foil with obstacles h=2%c
Fig.10 shows the variation of lift and drag coefficients on NACA0015 foil without and with obstacle which extrudes 2%c from the foil surface. Same tendency of the variation of lift and drag is shown. The time averaged force coefficients and lift to drag ratio is given in table 1. We can see that the lift to drag ratio increased with an obstacle arranged on the foil. When the height of the obstacle is 1%c, the ratio increases with the distance from the leading edge becomes longer. While when the height of the obstacle is 2%c,
Fig. 11 Frequency spectrum analysis of lift on NACA0015 foil with obstacles h=2%c
5
CONCLUSIONS
Cavitating flow of a 2D NACA0015 foil with and without obstacles is performed. Six arrangements of obstacles are used for the present simulation in order
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China to describe the influence of the geometry of the obstacles to cavitating flow. Comparing with cavitating flow on the foil without obstacles, cavitating flow with different arrangements of obstacles shows some variable characteristics. (1) For all arrangements of obstacles on the foil surface, the foil performance is changed. Lift and drag force decrease, but comparing with the foil performance without obstruction, the lift and drag ratio increase. (2) The frequency characteristic based on lift coefficient is changed, and with a lower obstacle, obstacle located in cr=0.45 and cr=0.37 have better influence to the cavitation, while with a higher obstacle, obstacle located in cr=0.32 and cr=0.45 works well for the control of cloud cavitation. (3) Re-entrant jet can be restrained by obstacles on the foil surface. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 10602030 and 10802075) and the National Key Basic Research Program of China (No. 2009CB724303). REFERENCES [1] Knapp R T, Daily J W. Hammit F.G. Cavitation [M]. McGraw-Hill, 1970. [2] Coutier-Derlgosha O, Fortes-Patella R, Reboud J L. Evaluation of the Turbulence Model Influence on the Numerical Simulations of Unsteady Cavitation [J]. Journal
755
of Fluids Engineering, 2003, 125: 38-45. [3] Kawanami Y, Kato H, Yamaguchi H, et al. Mechanism and control of cloud cavitation [J]. J. Fluids Eng, 1997, 119(4):788-794. [4] Pham T M, Larrate F, Fruman D H. Investigation of Unsteady Sheet Cavitation and Cloud Cavitation Mechanisms [J]. J. Fluids Eng, 1999, 121(2): 289-296. [5] Reisman G E, Wang Y C, Brennen C E. Observations of shock waves in cloud cavitation. Journal of Fluid Mechanics, 1998, 355:255-283 [6] Dular M, Bachert R, Stoffel B, et al. Experimental evaluation of numerical simulation of cavitating flow around hydrofoil [J]. European Journal of Mechanics/B fluids, 2005, 24:522-538. [7] Wang G Y, Senocak I, Shyy W, et al. Dynamics of attached turbulent cavitating flow [J]. Progress in Aerospace Sciences, 2001, 37:551-581. [8] Barre S, Rolland J, Boitel G, et al. Experiments and modeling of cavitating flows in venture: attached sheet cavitation [J]. European Journal of Mechanics - B/Fluids, 2009, 28(3):444-464. [9] Senocak I, Shyy W. A Pressure-based method for turbulent cavitating flow computations. Journal of Computational Physics [J]. 2002, 176:363-383. [10] Lindau J W, Kunz R F, Venkateswaran S, et al. Application of preconditioned, multiple-species, Navier-stokes models to cavitating flows [C]. Fourth International Symposium on Cavitation, Pasadena, CA USA, 2001. [11] Singhal A K, Athavale M, Li H Y, et al. Mathematical basis and validation of the full cavitation model [J]. Journal of Fluids Engineering, 2002, 124(3):617-624. [12] Kubota A, Kato H, Yamaguchi H. A new modeling of cavitating flows: a numerical study of unsteady cavitation on a hydrofoil section [J]. J. Fluid Mech, 1992, 240:59–96. [13] Wang G, Ostoja-Starzewski M. Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil [J]. Applied Mathematical Modeling. 2007, 31:417-447.
750
2010, 22(5), supplement :792-797 DOI: 10.1016/S1001-6058(10)60032-7
Numerical study on the control mechanism of cloud cavitation by obstacles
Wei-guo Zhao, Ling-xin Zhang * , Xue-ming Shao, Jian Deng Institute of Fluid Engineering, Department of Mechanics, Zhejiang University, Hangzhou, China *Email: [email protected] ABSTRACT: Numerical simulations of cavitating flows on 2D NACA0015 hydrofoils with and without obstacle are performed. Cavitation model is based on a transfer equation for a void mass fraction and an improved RNG k-ε model is adopted to study the turbulent cavitating flows around the foils. Different arrangements and geometric parameters of the obstacles are investigated. Computational results show that cloud cavitation can be reduced effectively by an obstacle placed on the foil surface; also the performance of the hydrofoil is changed by the obstacle simultaneously. KEY WORDS: Cavitation; Re-entrant jet; NACA0015; Obstacle; Multiphase
1
INTRODUCTION
Cavitation is a special type of phenomena in liquid flow[1]. When the static pressure decreases below the saturated pressure in the local temperature, liquid becomes to vaporize and cavitation can be found in the flow[2]. In all kinds of cavitation, cloud cavitation is on the focus of researches because of its instability which can cause vibration and noise, even lead to structural damages[3]. Instability of the sheet cavitation, and the generation mechanism of cloud cavitation, has been performed a lot by many researchers[3,4]. Some mechanism put forward one after another, such as the shock wave theory[5], and the re-entrant jet theory[3].
Among them the re-entrant jet theory, is widely accepted. In 1997, Kawanami et al.[3] confirmed the major role of re-entrant jet in the instability of sheet cavitation, through setting up barriers in the hydrofoil surface. Since then, Pham et al.[4] further confirmed the effectiveness by setting up obstacles on the surface of hydrofoil to re-entrant jet suppression and cloud cavitation control. Experimental and numerical methods are the dominating research methods of cavitating flow. Since cavitation is a complex two-phase flow, so the experimental methods are special, in general, have a higher cost. Some experimental methods have been developed for cavitating research: PIV (Particle Imaging Velocimetry) - LIF (Laser Induced Fluorescence) method[6], LDV (Laser Doppler Velocimetry) method[7], and double optical probe method[8]. The numerical simulation is relatively less costly, but was limited by the cavitation model and the corresponding numerical method. Since cavitation is a two-phase flow, variable density and the density ratio between the two-phase flow changes abruptly, so the numerical method is different from the usual single-phase flow; and usually pressure-based approach is used to solve the equations[9]. Because of its relatively large density, stability and convergence of the algorithm is also not so perfect. Based on phase
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China transition of the cavitating flow, different researchers proposed many cavitation models. There are mainly two kinds of cavitation model: VOF method based on the interface tracking; the homogeneous mixture flow model which deals the two phase flow as a uniform mixture, and give the physical description of the phase transition through a volume fraction transport equation[10-12]. In this paper, numerical simulation is performed on a NACA0015 hydrofoil with and without obstacles on the suction surface, focusing on the instability of sheet cavitation and control of cloud cavitation. Several arrangements of obstacles were performed in order to reveal the influence to re-entrant jet and production mechanism of cloud cavitation. 2
751
cavitating flow is variable, the major difficulty for numerical simulation is to deal with the large density changes associated with phase change between water and vapor. Especially in cavitation area, the density of the mixture changes abruptly. When the conventional RNG k-ε turbulence model was used, the unstable cavitating behavior cannot be simulated correctly [2]. So modified RNG k-ε turbulence model by artificially reducing the turbulent viscosity of the mixture is essential to model the unsteady cavitating flow:
μ t = f ( ρ )C μ
k2
ε
f ( ρ ) = ρ v + (1 − α ) n ( ρ l − ρ v ) n > 1
（4） （5）
PHYSICAL MODEL
2.1 Multiphase model A homogeneous model approach is adopted to simulate the two phase cavitating flow: relative motion between liquid and vapor phase is neglected and the mixture comprised by liquid and water vapor is treated as a homogenous medium with variable density. Only one set of equations is solved for the mixture flow, while the void fraction is solved by a void mass transport equation. The continuity equation for the mixture flow is: JJG ∂ （1） ( ρ m ) + ∇ • ( ρVm ) = 0 ∂t Where ρ m = ρ vα + ρ l (1 − α ) and Vm denotes the density and velocity of the mixture respectively. The momentum equation for the mixture flow is:
JJGJJG JG ∂ JJG ( ρVm ) + ∇ • ( ρVm Vm ) = F − ∇p ∂t JJG JJG T +∇ • ⎡⎢ μ m (∇Vm + ∇Vm ) ⎤⎥ ⎣ ⎦
（2）
In cavitating flow, the liquid-vapor mass transfer is governed by the vapor transport equation: JJG ∂ （3） ( ρ m f ) + ∇ • ( ρ m f Vm ) = Re − Rc ∂t 2.2 Turbulence model Modified RNG k-ε turbulence model was introduced to close the control equations. Because density of the
Fig. 1 Function of density used for turbulence viscosity modification
The modification limits the turbulent viscosity in the region filled mainly by vapor phase and consequently allows the formation of a re-entrant jet and the cavitation cloud separation. Different values of n can affect the results, so several n values were investigated, and n=10 was adopted in the present computation. Fig.1 shows the modified value of density function. 2.3 Cavitation model Singhal’ cavitation model [11] was adopted in the present simulation. Experimental and numerical investigations have shown that turbulent pressure fluctuations have a significant effect on cavitating flows, so the saturation vapor pressure used in the cavitation model is modified accounting for:
′ /2 Pv = Psat + Pturb Where
（6）
′ = 0.39 ρ k is the turbulent pressure Pturb
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
752
fluctuation. Source terms Re and Rc are given by： 1/2
⎛2 P −P⎞ Re = Ce ρl ρv ⎜ v ⎟ σs ⎝ 3 ρl ⎠ (1 − f v − f g ) k
（7）
1/2
⎛2 P −P⎞ Rc = Cc ρl ρv ⎜ v ⎟ σs ⎝ 3 ρl ⎠ k
foil surface, non-slip wall boundary is given. On the upper and lower boundaries, slip wall boundary is adopted.
fv
（8）
Where Ce and Cc are empirical constants and the recommended value is Ce = 0.02 ， Cc = 0.01 ; the surface tension,
is
is the vapor mass fraction and
(a)
is the mass fraction of noncondensable gases. 3
COMPUTATIONAL MODEL AND GRID
A 2-D NACA0015 foil was used for the present simulation. The computational domain and grid system is shown in Fig 2. The foil chord length c is 0.1m. An arc shape inlet is located at 5 times chord upstream of the leading edge, while the outlet is located at 10 times chord length downstream of the trailing edge. The top and bottom boundary is located at 5 times chord length from the domain centre. The angle of attack is α =8 ° . Six arrangements of obstacle were set up for the simulation. The relative distance based on chord length between the left side of the obstacle to leading edge of the foil is indicated by cr, and three values of cr are 0.32, 0.37, 0.45. Width of the obstacle is 0.02c, and two heights are selected, 0.01c and 0.02c respectively. The total number of grid points is 231×200 and the first grid spacing near the wall based on y+ is about 50. For space discretization, the finite volume method is adopted, while for time discretization, a first order implicit scheme is applied. Convective terms of the governing equation are discretized with second-order upwind scheme, and diffusive terms are discretized using second-order central differencing scheme. The Simplec algorithm is used for the pressure and velocity coupling method. Uniform flow condition and pressure outlet are used at the inlet and outlet boundary respectively, while in the
(b) Fig. 2 Computational domain and grid. (a) without obstacle; (b) with an obstacle on the suction surface
4
RESULTS AND DISCUTION
Non-dimensional parameters simulation are defined as： Reynolds number： Re = Cavitating number： σ =
used
in
present
ρV∞ l μ
P∞ − Pv 0.5ρV∞2
Pressure coefficient： CP =
P − P∞ 0.5ρV∞2
For all the simulations in this paper, Reynolds number is kept const and Re=1×106. 4.1
Cavitation on the fil without obstacle
Cavitating flow over NACA0015 foil when the cavitation number σ = 1.5 is simulated to validate the computational code.
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
753
Fig.3 shows the relatively stable sheet cavity appearance when σ = 1.5 . The angle of attack is 8° and Reynolds number based on the foil chord length is 3×105. Experimental observations[12] show that the bubble clusters mainly located at the leading edge, and there are fewer bubbles located in the middle of the hydrofoil. Kubota’s model [12] and Wang’s simulation [7, 13] captured the character well. The present simulation agrees well with the experimental results.
Fig. 5 Period of lift and drag coefficients on NACA0015 foil without obstacle, σ = 1.2 Fig. 3 Sheet cavity appearance on NACA0015 hydrofoil in normal state, α=8°， Re=3×105, σ = 1.5 .
Cloud shape cavitation without obstacle on the foil surface is simulated when cavitation number σ = 1.2 . Fig. 4 shows the periodic shedding process of the unsteady cloud cavity. Through the lift and drag coefficient variation shown in Fig5, the fluctuation frequency of the cavity shedding can be evaluated. In present state, the frequency is about 60Hz. The mean length of the cavity is about 0.49c. Strouhal number based on the mean cavity length and reference velocity equals to 0.27, which is close to the classical Strouhal number 0.3. A typical shedding shape is observed in Fig. 6, re-entrant jet can be found in the bottom of the cavity.
Fig. 4 Cloud cavitation shape indicated by vapor void fraction on NACA0015 foil without obstacle,
σ = 1.2
Fig. 6 Cavity shedding and Re-entrant jet on NACA0015 foil without obstacle, σ = 1.2
4.2
Cavitation on the foil with obstacles h=1%c
Fig.7 shows the variation of lift and drag coefficients on NACA0015 foil without and with obstacle which extrudes 1%c from the foil surface. It is observed that lift of the foil with an obstacle drops comparing with the foil without obstacle. The reason is the existence of an obstacle on the foil surface change the pressure distribution on the suction side. Also it can be found that the drag coefficient drops. Experiment by Kawanami et al[3]. found this phenomenon too. Fig.8 is the frequency spectrum analysis of the lift coefficient. Except when cr=0.32, frequency pulsations of the other cases decreased obviously. When cr=0.37 and cr=0.45, shedding area of cloud structure also decreases. Fig. 9 shows the cavity appearance when cr=0.45, where we can see that the length of the sheet caivity and the shedding area are less than that of the foil without the obstacle.
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China
754
the ratio has a reversed tendency and the lift drop is larger, which means performance of the foil has a greater change. Fig.11 is the frequency spectrum analysis of the lift coefficient. Except when cr=0.37, frequency pulsations of the other cases decreased and have a relative lower amplitude.
Fig. 7 Variation of lift and drag coefficients on NACA0015 foil with obstacles，h=1%c
Fig. 10 Variation of lift and drag coefficients on NACA0015 foil with obstacles，h=2%c Table 1 Time averaged force coefficients with and without obstacle cr
Fig. 8 Frequency spectrum analysis of lift on NACA0015 foil with obstacles h=1%c
h=1%c Cl
Cd
h=2%c Cl/Cd
Cl
Cd
Cl/Cd
0.32
0.659 0.089 7.370
0.681
0.091
7.474
0.37
0.641 0.084 7.626
0.665
0.091
7.307
0.45
0.656 0.086 7.669
0.578
0.085
6.810
no obs 0.731 0.115 6.372
0.731
0.115
6.372
Fig. 9 Cavitation shape indicated by vapor void fraction on NACA0015 foil witht obstacle, cr=0.45,h=1%c, σ = 1.2
4.3
Cavitation on the foil with obstacles h=2%c
Fig.10 shows the variation of lift and drag coefficients on NACA0015 foil without and with obstacle which extrudes 2%c from the foil surface. Same tendency of the variation of lift and drag is shown. The time averaged force coefficients and lift to drag ratio is given in table 1. We can see that the lift to drag ratio increased with an obstacle arranged on the foil. When the height of the obstacle is 1%c, the ratio increases with the distance from the leading edge becomes longer. While when the height of the obstacle is 2%c,
Fig. 11 Frequency spectrum analysis of lift on NACA0015 foil with obstacles h=2%c
5
CONCLUSIONS
Cavitating flow of a 2D NACA0015 foil with and without obstacles is performed. Six arrangements of obstacles are used for the present simulation in order
9th International Conference on Hydrodynamics October 11-15, 2010 Shanghai, China to describe the influence of the geometry of the obstacles to cavitating flow. Comparing with cavitating flow on the foil without obstacles, cavitating flow with different arrangements of obstacles shows some variable characteristics. (1) For all arrangements of obstacles on the foil surface, the foil performance is changed. Lift and drag force decrease, but comparing with the foil performance without obstruction, the lift and drag ratio increase. (2) The frequency characteristic based on lift coefficient is changed, and with a lower obstacle, obstacle located in cr=0.45 and cr=0.37 have better influence to the cavitation, while with a higher obstacle, obstacle located in cr=0.32 and cr=0.45 works well for the control of cloud cavitation. (3) Re-entrant jet can be restrained by obstacles on the foil surface. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 10602030 and 10802075) and the National Key Basic Research Program of China (No. 2009CB724303). REFERENCES [1] Knapp R T, Daily J W. Hammit F.G. Cavitation [M]. McGraw-Hill, 1970. [2] Coutier-Derlgosha O, Fortes-Patella R, Reboud J L. Evaluation of the Turbulence Model Influence on the Numerical Simulations of Unsteady Cavitation [J]. Journal
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