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Hydrodynamic characteristics of blowing and suction on sheet-cavitating flows around hydrofoils
Ocean Engineering 114 (2016) 25–36
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Hydrodynamic characteristics of blowing and suction on sheet-savitating flows around hydrofoils Pooria Akbarzadeh a,n, Ebrahim Akbarzadeh b a b
Department of Mechanical Engineering, Shahrood University of Technology, P.O.X:3619995161 Daneshgah Boulevard, Shahrood, Iran Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2015 Accepted 16 January 2016
In this study the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows over hydrofoils is investigated. A computational program is developed for simulation of inviscid cavitating flows. In this simulation, the Jameson's finite volume technique and the progressive power-law preconditioning method for analyzing the cavitating flows are utilized. For cavitation modeling the barotropic cavitation pattern is used. The numerical stabilization is achieved via the second and fourth-order dissipation terms. Explicit four-step Runge–Kutta time integration is applied to achieve the steady-state condition. Cavitating flows over NACA-hydrofoils for different cavitation numbers are investigated. To apply the blowing and suction, a jet is placed on the hydrofoil's upper surface. Four blowing (and suction) parameters i.e., velocity ratio, jet location, width of jet, and jet angle are scrutinized. The effects of these parameters on surface pressure distribution, length of cavitation, lift and pressure drag coefficients are examined. The results indicate that blowing jets often decrease the length of cavity and reduce the lift and pressure drag coefficients, while suction jets behave reversely. Also, by increasing the blowing amplitude or the width of jet, the lift and pressure drag coefficients and the length of cavity decrease while in suction cases, they increase. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Blowing and suction Cavitating flow Power-law preconditioning Lift Drag
1. Introduction Cavitation is a complex physical phenomenon which occurs when the local liquid pressure is lower than its saturated vapor pressure. Cavitating flows are important issues in the design and performance of many engineering devices such as underwater projectiles, marine propellers, hydro-machineries, fuel injectors, hydrofoils, to name but a few. It is well known that cavitation often causes undesirable problems such as load asymmetry, erosion, noise, vibration, efficiency reduction of hydro-turbines, head and efficiency drop of pumps, to name but a few (Esfahanian et al., 2012; Shang, 2013). However cavitation has a wide range of industrial application, such as surface cleaning, drag reduction of underwater moving body by using supercavitation, medicine production, chemical mixing, transfer large molecules into biological cells, and etc. (Shang, 2013). Hence, a large number of studies have been conducted into cavitation and cavitating flows (Geurest, 1960; Fine and Kinnas, 1993; Coutier-Delgosha et al., 2003; Wang and n
Corresponding author. Tel.: þ 98 9122022776. E-mail addresses: [email protected] (P. Akbarzadeh), [email protected] (E. Akbarzadeh). http://dx.doi.org/10.1016/j.oceaneng.2016.01.015 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
Starzewski, 2007; Li et al., 2010; Morgut et al., 2011; Esfahanian et al., 2012; Ji et al., 2012; Roohi et al., 2013; Shang, 2013; Celik et al., 2014; Ji et al., 2014, and etc.). Cavitation control has tremendous importance in various fields in industry and it is performed in two major ways. One way of controlling the cavitation (especially sheet-cavitating flows around hydrofoils) emphasizes passive methods such as modifying geometric shapes (For example Shen and Eppler (1981), Ukon et al. (1994), Amromin et al. (2006), and etc.) or using obstacles placed on the foil surface (For example Zhao et al. (2010)) to maneuver the pressure gradient and change the cavitation regime. Another approach is active flow control, which changes the lift and the drag by injecting a small quantity of fluid (e.g. a fluid with lower viscosity such as air or micro-bubbles, a fluid with higher viscosity such as polymers, and using the same fluid around the object) or energy (e.g. acoustic or ultrasonic waves) into the system to control the boundary layer and the cavitation incident. Indeed, suction and blowing fluid or injecting energy can actively modify the pressure distribution over a body surface. Kato et al. (1987) investigated experimentally, the suppression effect of water discharge (using the same fluid around the object) on incipient and developed sheet cavitation on a hemispherical headform. They found that, by increasing the discharge rate, the
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length of the sheet cavity becomes shorter and then the cavity is suppressed completely. Chahine et al. (1993) injected a dragreducing polymer solution (a fluid with higher viscosity) into the tip vortex region of propeller blades and significantly delayed tip vortex cavitation. Arndt et al. (1995) investigated the use of air injection (a fluid with lower viscosity) through small holes in the leading edge of a NACA0015 hydrofoil to mitigate undesirable cavitation effects. Chatterjee and Arakeri (1997) and Chatterjee (2003) examined the feasibility of using ultrasonic waves for hydrodynamic cavitation control. Pham et al. (1999) examined the control of cloud cavitation using a tiny obstacle (barrier) on the foil surface and air injection through a slit situated in the vicinity of the leading edge. Chen et al. (2008) simulated numerically the cavity shape and its characteristics of steady ventilated cavitating flows in water tunnels. In their simulations, a symmetric underwater vehicle model, which consists of a disk cavitator, two ventilation bowls and a slender body was considered. When very low Mach number flows (or almost incompressible fluid flow) are simulated, the large condition number (or degree of stiffness), i.e. the ratio of the largest to the smallest characteristic speed (eigenvalues of system of equations), leads to destroy the convergence rate in the case of marching methods. Indeed, the allowable time step is limited by the fastest moving wave. During one time step, the slowest wave propagates only over a fraction of cell width. Thus, a large condition number reduces the wavepropagation efficiency (slows down the convergence). Besides the incompressible flow, for numerical simulation of multiphase flows (particularly cavitating flows) due to change of sound speed from one phase to another, it is required to modify the sound speed (or characteristic speed) artificially in order to come close the stiffness of governing equations to unity. In fact, convergence fails or accuracy weakens when the speed of sound is much greater than the fluid flow velocity (or when the stiffness of governing equations is much greater than unity). The preconditioning methods (PMs) can do this modification by adding some terms into the continuity and momentum governing equations. Therefore many researchers have used PMs to simulate the cavitating flow such as Refs. Kunz et al. (2000), Ahuja et al. (2001), Venkateswaran et al. (2002), Huang (2007), Esfahanian et al. (2012) and Akbarzadeh (2013). It is worth to note that, their results showed that PMs have a tremendous effect on reducing the iteration steps of the numerical convergence. In this study, the hydrodynamic behavior of sheet-cavitating flows over hydrofoils by considering the blowing and suction is investigated for the first time. For this purpose, a computational code is developed in order to simulate the inviscid cavitating flows. Indeed, cavitation is a 3-dimensional, unsteady and discontinuous or periodic phenomenon of formation, growth and collapse of bubbles. Due to the natural complexity of cavitation, it is neither reliably assessable nor fully understood yet. Hence, a large number of computational grids and small time steps have to be considered to simulate the cavitation event accurately from microscopic to macroscopic level (for example using direct numerical simulation or DNS). The mathematical models describing the cavitation mechanisms based on certain assumptions and simplifications can avoid such difficulties (Shang, 2013). Therefore, in this code, the implemented cavitation model is based on the barotropic state law initiated by Delannoy and Kueny (1990). The validity of this simple model has been approved by many researching works such as Merkle et al. (1998), Song and He (1998), Coutier-Delgosha et al. (2003), Coutier-Delgosha et al. (2005), Esfahanian et al. (2012), Akbarzadeh (2013), and etc. The governing equations are modified by the progressive power-law preconditioning method introduced by Esfahanian et al. (2012). This physical model is integrated with the cell-centered Jameson's finite volume algorithm proposed by Jameson et al. (1981). The
numerical stabilization is achieved by the second and fourth-order dissipation terms. To obtain the steady state condition, explicit four step Runge–Kutta method is utilized. Cavitating flows over NACA-hydrofoils for various cavitation numbers and angle of attacks are examined. To simulate the blowing and suction (as an active control method of cavitation), a continuous water jet is placed on the hydrofoil's upper surface. Four important parameters with relation to the surface blowing and suction i.e., velocity ratio (jet amplitude), jet position, width of jet and jet angle are studied in cavitating flows. In this study the effects of uniform surface blowing and suction on the length of cavitation, the lift and pressure drag coefficients, and the surface pressure distribution are demonstrated. The results prove that continuous blowing jets often decrease the length of sheet-cavity and reduce the lift and pressure drag coefficients. While continuous suction jets behave reversely. Also, the numerical data indicates that by increasing the blowing velocity ratio or the width of jet, the lift and pressure drag coefficients and the length of cavity decrease. While when the jet sucks the fluid with higher amplitude or through a widened width, the mentioned hydrodynamic parameters increase.
2. Preconditioning governing equations The non-dimensional preconditioning governing equations (inviscid fluid flow) are expressed as follows (Esfahanian et al., 2012):
Γ1
! ! ! ∂Q ∂ F ∂ E þ þ ¼0 ∂t ∂x ∂y
ð1Þ
T ! ! ! E ¼ where, Q ¼ p u v ; F ¼ ð ρu p þ ρu2 ρuv ÞT , ð ρv ρuv p þ ρv2 ÞT , and Γ is the preconditioning matrix defined as follows: 0 1 β2 0 0 B C ð2Þ Γ¼B 0 C @ σ u=ρ 1=ρ A σ v=ρ
0
1=ρ
In Eq. (1), u ¼ u=U 1 and v ¼ v=U 1 are velocity components in the x and y-directions, respectively. x ¼ x=L and y ¼ y=L are the Cartesian coordinates, p ¼ p=ρ1 U 21 is the static pressure, ρ ¼ ρ=ρ1 is the density and t ¼ tU 1 =L is the time. U 1 is the reference velocity, L is the reference length, and ϕ represents the dimensional form of quantity ϕ. β is the artificial compressibility coefficient which is calculated similar to the references (Esfahanian et al., 2012; Malan et al., 2002). σ is called preconditioning factor which is defined based on the power-law preconditioning method as follows (Esfahanian et al., 2012; Akbarzadeh, 2013):
σ ¼ ð1 Au Þm
ð3Þ where, Au ¼ limx-x þ ∇uðx0 Þ ∇uðxÞ= ∇uðx0 Þ ∇uðxÞ is a local 0 velocity sensor, ∇uðx0 Þ ¼ limx-x0 ½uðxÞ uðx0 Þ=½x x0 is the velocity gradient at x ¼ x0 , and m is an integer number. In Eq. (3) when m is set to zero, σ becomes similar to the standard preconditioning method introduced by Turkel (1987) and when m is set to one, σ becomes equal to the relation introduced by Malan et al. (2002). In addition, the case σ ¼ 0 is similar to the standard artificial compressibility method initiated by Chorin (1967). In this study, the value of m ¼ 2 or m ¼ 3:, which has been defined in reference (Esfahanian et al., 2012) as a best selection, is used to facilitate the convergence of the numerical simulation.
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3. Barotropic cavitation model In this study, the homogenous barotropic cavitation model is applied for simulation the cavitation flows. This model which considers the vapor–liquid mixture as a single fluid, was initiated by Delannoy and Kueny (1990). According to Eq. (4), in this model, when the fluid equals the liquid phase, ρl , all the computational cell is full of liquid (incompressible), and if it equals the vapor phase, ρv , the computational cell is occupied by vapor (incompressible). Between these two extreme values, the cell is occupied by a two-phase mixture that is still considered as one single fluid. In this zone, the fluid density is controlled by a barotropic state law that links the fluid density variations to the local static pressure. This type of cavitation model has been applied recently by many researchers such as Merkle et al. (1998), Song and He (1998), Coutier-Delgosha et al. (2003, 2005), Esfahanian et al. (2012), Akbarzadeh (2013), and etc. 8 ρl ; p pvap 4 Δpvap > > > < ρ þρ ρ ρ 2 p pvap l v l v ; p pvap o Δpvap sin ρ ρ C min ð4Þ ρðpÞ ¼ 2 þ 2 l v > > > :ρ ; p p o Δp v
vap
27
local velocity sensor i.e. Au, is numerically computed as follows: 4ui;j ui þ 1;j ui 1;j ui;j þ 1 ui;j 1 ðAu Þi;j ¼ ð7Þ ui;j ui þ 1;j þ ui;j ui 1;j þ ui;j ui;j þ 1 þ ui;j ui;j 1
5. Blowing and suction control parameters In this study, as shown in Fig. 1, four basic parameters of blowing and suction are used for the numerical investigation. These parameters are jet location Lj , velocity ratio Aj ¼ V j =U 1 , blowing (or suction) angle θj , and width of jet dj , where, V j is the jet entrance velocity. It should be noted that positive value of θj represents the blowing condition and negative value of θj indicates the suction condition. The effects of various values of the jet location (Lj ¼ 0:3–0:9 of the chord length after leading edge), the velocity ratio (Aj ¼ 0.01–0.1), the blowing (suction) angle (θj ¼ 30–90°), and the width of jet (dj ¼ 2:5% to 1:0% of chord length) on hydrodynamic behavior of sheet-cavitating flows over hydrofoils are discussed.
vap
In Eq. (4), ρl is the liquid-phase density, ρv is the vapor-phase density, pvap is the vapor liquid pressure, Δpvap ¼ π C 2min = ρl ρv and C min is defined as the minimum speed of sound wave in the liquid–vapor mixture. Based on the Coutier-Delgosha et al. (2005) recommendation, C min is chosen between 1.5 m/s and 4.0 m/s and ρv =ρl is set to 0.001. In order to consider the vapor–liquid pressure in the code, the cavitation number is introduced via the relation: s ¼ ðp pvap Þ=ð1=2ρ1 U 21 Þ.
4. Finite volume numerical method The integral form of the preconditioned governing equations specified by relation (1) over a control volume Ω with surface of ∂Ω is written as (Esfahanian et al., 2012): Z Z ! ! ∂ ! Q dA þ Γ F dx E dy ¼ 0 ð5Þ ∂t Ω ∂Ω According to the references (Jameson et al., 1981; Esfahanian et al., 2012), the discretized form of Eq. (5) over the (i,j)th computational cell-volume accompanied by adding Jameson's dissipative terms becomes Ai;j ∂Q i;j;k =∂t ¼ Gi;j;k þ Di;j;k , where Ai;j is the area of (i,j)th cell, k ¼ 1; 2; 3 is the index of vector's components, Di;j;k is the artificial dissipative expression composed of second and fourth order terms initially suggested by Coutier-Delgosha et al. (2003) for cavitating flows, and Gi;j;k is the numerical flux related to (i,j)th cell which 2P 2 is defined as Gi;j;1 ¼ β edge ðF 1 Δx E 1 ΔyÞi;j ; Gi;j;2 ¼ ð σ ui;j Gi;j;1 =β Þ P 2 P Gi;j;3 ¼ ð σ vGi;j;1 =β Þ edge ðF 3 Δx E3 ΔyÞi;j . edge ðF 2 Δx E 2 ΔyÞi;j ; The optimal fourth stage Runge–Kutta method utilized by Malan et al. (2002) and Esfahanian et al. (2012) is applied to solve governing Eq. (5) as follows: 8 ð0Þ ðnÞ > > > Q i;j;k ¼ Q i;j;k > < Δt 1Þ ð0Þ Q ðKÞ ¼ Q ð0Þ αK Ai;ji;j Q ðK Q i;j;k ð6Þ i;j;k i;j;k i;j;k > > > ðn þ 1Þ ð4Þ > : Q i;j;k ¼ Q i;j;k where K ¼ 1; 2; 3; 4, the coefficients αK are set to 4, 3, 2 and max 1 respectively, Δt i;j ¼ CFL ΔLmin is the local time step, ΔLmin i;j =λi;j i;j is the minimum length of (i,j)th computational cell's sides, CFL is max the Courant–Friedrichs–Lewy number, and the parameter λi;j indicates the maximum eigenvalue of the governing equations. The
6. Boundary conditions The velocity boundary condition on the body surface is set as ! ! V n ¼ 0, where V n is the perpendicular component of fluid velocity to the solid surface. The pressure boundary condition is set as ∂p=∂n ¼ 0 where n is the unit vector perpendicular to the body surface. Classical incompressible type of the outlet and inlet boundary is applied: imposed velocities at the inlet and imposed static pressure at the outlet. For the jet boundary condition, constant velocity is applied at the entrance and pressure is achieved via inside of flow field.
7. Result In order to investigate the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows over hydrofoils, many cases are considered. NACA16009 and NACA66(MOD) hydrofoils are selected for reporting in this paper. For this subject, many grids with different mesh-size have been examined and finally two O-gird geometries with the size of 150 64 for NACA16009 hydrofoil and 180 64 for NACA66(MOD) are chosen as shown in Fig. 2 (The chord length is considered as unity, c ¼ 1:0). The position of the outlet computational domain is set around 10 chord length far away from the hydrofoils’ surface. For these test cases, the uniform flow (i.e., p1 ¼ 1:0, U 1 ¼ 1:0, and ρ1 ¼ 1:0) is used as the initial condition. To validate the present numerical computations, the results for NACA16009 are compared with the numerical data reported by Krishnaswamy (2000) and the results for NACA66(MOD) are checked with the numerical works of Deshpande et al. (1994) and the experimental works of Shen and Dimotakis (1989). Fig. 3a shows the surface pressure distribution, C p ¼ p1 p = 0:5ρ1 U 21 ,
Fig. 1. Blowing and suction control parameters.
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P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
NACA66(MOD)Hydrofoil
0.6
0.6
0.4
0.4
0.2
0.2
y
y
NACA16009 Hydrofoil
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
-0.2
1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
x Fig. 2. Grid for NACA16009 and NACA66(MOD) hydrofoils.
1.5
1.5 Deshpande et al.(1994) Present work
Krishnaswamy (2000) Present work
1.0
1.0
Pressure Side Pressure Side Cp
0.5
Cp
0.5
0.0
0.0
-0.5
Suction Side
-1.0
-1.5
-1.0
Cavity Zone 0.0
0.2
0.4
Suction Side
-0.5
Cavity Zone 0.6
0.8
1.0
x
-1.5
0.0
0.2
0.4
0.6
0.8
1.0
x
Fig. 3. Surface pressure distributions and cavity zone: (a) sheet-cavitating flow over NACA66(MOD) hydrofoil at AOA ¼ 4° and s ¼ 0:84. (b) Sheet-cavitating flow over NACA16009 hydrofoil at AOA ¼ 3° and s ¼ 0:66.
of sheet-cavitating flow over NACA66(MOD) hydrofoil at the angle of attack (AOA) of 4° and the cavitation number of s ¼ 0:84. Also, Fig. 3b illustrates the wall pressure distribution of sheet-cavitating flow over NACA16009 hydrofoil at AOA¼3° and s ¼ 0:66. These results are closely comparable to those of Deshpande et al. (1994) and Krishnaswamy (2000), respectively (for both pressure field and length of cavity). However, the observed differences in cavity closure zone (reattachment/detachment regions) between the present simulation and their works are because of the flow (and cavitation) modeling differences. For example, the methodology of Krishnaswamy (2000) is based on cavitation interface tracking approach and potential flow model. In such methods, a large sharp change in pressure distribution (in cavity closure zone) is seen. The other sample that confirms the accuracy of the numerical method is
shown in Fig. 4. This figure presents the calculated lift (CL) and pressure drag (CD) coefficients for sheet-cavitating flows around the NACA66(MOD) hydrofoil at AOA¼4° and 6° as a function of cavitation number. The results are compared with the experimental works performed by Shen and Dimotakis (1989). The coefficients agree quite well with the experimental measurements. Both the experiments and calculations show that lift increases with decreasing the cavitation number, as expected. However, It should be noted that the experiments of Shen and Dimotakis (1989) were conducted with the hydrofoil centered in a High-Speed Water Tunnel whose walls were five chord lengths apart at Re ¼ 2 106 . Therefore, the observed deviation in Fig. 4 may be due to the viscous, turbulence, and wall effects in the experimental measurements.
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
7.1. Blowing jet
represent the lift and the pressure drag coefficients, respectively in the case of Aj ¼ 0:0 (no blowing or suction). The figures show that blowing reduces the lift and the pressure drag coefficients. Indeed, the coefficients decrease as blowing amplitude increases above the amplitude of 0:01 and this reduction is seen at every jet location. In order to interpret this issue, the wall pressure distribution, C p , and the contour of density for NACA16009 hydrofoil at the mentioned blowing amplitudes are plotted in Figs. 6 and 7 at jet location of Lj ¼ 0:7 and Lj ¼ 0:3, respectively. It can be concluded
Predicted lift and pressure drag coefficients at the blowing angle of 90°, blowing amplitude of Aj ¼ 0:01; 0:05; 0:1, and various jet locations for sheet-cavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, and dj ¼ 0:025) and NACA66(MOD) (AOA ¼ 4°, s ¼ 0:84, and dj ¼ 0:025) are shown in Fig. 5a and b, respectively. The coefficients are normalized by their corresponding values in the baseline case (no blowing or suction). In this paper, C Lb and C Db
1.40
29
Present work, Lift Present work, Pressure drag Shen and Dimotakis (1989)
0.20
1.20
0.80
CL
0.12 o AOA=6
0.60
0.40
CD
0.16
1.00
AOA=4
0.08
o
0.04 0.20
0.00
0.00 0.8
1.0
1.2
1.4
1.6
1.8
2.0
s Fig. 4. Lift and pressure drag prediction for sheet-cavitating flows around the NACA66(MOD) hydrofoil at AOA ¼4° and 6°.
1.4
1.2
A = 0.01 A = 0.05 A = 0.10
1.2
1.1
CL / CLb
CL / CLb
1.0 0.8 0.6
A = 0.01 A = 0.05 A = 0.10
0.4
1.0
0.9
0.2 0.0 0.2
0.4
0.6
0.8
0.8 0.2
1.0
1.4
0.4
0.6
0.4
0.6
0.8
1.0
0.8
1.0
1.1
1.2
1.0
CD / CDb
CD / CDb
1.0 0.8 0.6
0.9 0.8
0.4
0.7
0.2 0.0 0.2
0.4
0.6
0.8
Lj
1.0
0.6 0.2
Lj
Fig. 5. Lift and pressure drag coefficients of sheet-cavitating flows .during blowing when, dj ¼ 0:025, 0:1 o Aj o 0:1, and 0:1o Lj o 0:9: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66. (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84.
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Aj=0.0 (No Blowing) Aj=0.01 Aj=0.05 Aj=0.10
1.2
Aj=0.0
0.8
Aj=0.01
Cp
0.4
0.0 Aj=0.05
-0.4 Aj=0.10
-0.8 0.0
0.2
0.4
0.6
0.8
1.0
x Fig. 6. Effect of blowing amplitude on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼3°, s ¼ 0:66, dj ¼ 0:025, and Lj ¼ 0:7: (a) surface pressure distribution. (b) Contour of density.
Aj=0.0 (No Blowing) Aj=0.01 Aj=0.05 Aj=0.10 Aj=0.15
1.2
0.8
Aj=0.0
Aj=0.01
Cp
0.4 Aj=0.05
0.0 Aj=0.10
-0.4
-0.8 0.0
0.2
0.4
x
0.6
0.8
1.0
Aj=0.15
Fig. 7. Effect of blowing amplitude on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼3°, s ¼ 0:66, dj ¼ 0:025, and Lj ¼ 0:3: (a) surface pressure distribution. (b) Contour of density.
from these computation results that the length of cavity decrease as blowing amplitude increases. According to Figs. 6b and 7b, by reducing the cavity length, the difference between hydrofoil's upper and lower surfaces pressure reduces. So the lift and pressure drag coefficients decrease significantly in comparison to the nobowing case as seen in Fig. 5 (the main reason). The inferior reason
is that, by increasing the blowing amplitude while the width jet _ ¼ ρdj V j ) is has been fixed, the mass flow rate exhaust (i.e. m _ j ) helps increased. Hence, the reaction of momentum force (i.e. mV to push the hydrofoil in opposite direction of blowing and reduce the lift coefficient, accordingly. One of the noticeable conclusions of fluid blowing is seen in the last image of Fig. 7b. From the figure,
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.6
A j=0.0 (No B lowing) A j=0.01 A j=0.05 A j=0.10
1.2
31
Aj=0.0
0.8 Aj=0.01
Cp
0.4 0.0 Aj=0.05
-0.4 -0.8
Aj=0.10
0.0
0.2
0.4
x
0.6
0.8
1.0
Fig. 8. Effect of blowing amplitude on sheet-cavitating flows over NACA88(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, dj ¼ 0:025, and Lj ¼ 0:7: (a) surface pressure distribution. (b) Contour of density.
1.6
Aj=0 (No Blowing) Aj=0.01 (b) Aj=0.05 Aj=0.10 Aj=0.18
1.2 0.8
Aj=0.0
Aj=0.01
0.4 Cp
Aj=0.05
0.0 Aj=0.10
-0.4 -0.8 0.0
0.2
0.4
0.6
0.8
1.0
Aj=0.18
x Fig. 9. Effect of blowing amplitude on sheet-cavitating flow over NACA88(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, dj ¼ 0:025, and Lj ¼ 0:3: (a) surface pressure distribution. (b) Contour of density.
it is evident that at Lj ¼ 0:3 and Aj ¼ 0:15, the cavitating region is broken into two smaller regions. This phenomenon always occurs at high blowing amplitude and the jet location near the leading edge. The same results of wall pressure distribution and the contour of density for NACA66(MOD) hydrofoil at jet location of Lj ¼ 0:7 and Lj ¼ 0:3 are plotted in Figs. 8 and 9, respectively. The cavity length reduction, decrease of the difference of hydrofoil's upper and lower surfaces pressure, and the break of the cavity zone are also observed when the blowing amplitude increases step by step.
To explore the effect of blowing angle, a jet at location of Lj ¼ 0:7 and various blowing angles are simulated for sheet-cavitating flows over NACA16009 (AOA¼3°, s ¼ 0:66, and dj ¼ 0:025) and NACA66 (MOD) (AOA¼4°, s ¼ 0:84, and dj ¼ 0:025) and the calculated lift and pressure drag are presented in Fig. 10a and b, respectively. Also, the surface pressure distributions for these cases are shown in Fig. 11a and b. Note from Fig. 11 that, by increasing the blowing angle, the length of cavity becomes shorter and therefore the lift and pressure drag coefficients decrease as shown in Fig. 10. Nonetheless, these reductions are not significant for NACA66(MOD) hydrofoil.
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.16
1
1.04
1.12
0.9
0.9
1.02
1.08
0.8
0.8
1.00
1.04
0.7
0.7
0.98
1.00
0.96
0.96
CL / CLb
1
0.6
0.6
0.5 15
0.5 30
45
60
75
90
j
CL / CLb
1.06
CD / CDb
1.1
1.1
0.94 15
CD / CDb
32
0.92 30
45
60
75
90
j
Fig. 10. Effect of blowing angle on the lift and pressure drag coefficients: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
Fig. 11. Effect of blowing angle on surface pressure distribution: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
In this section, the last investigation in the case of blowing jet belongs to the effect of jet width on the cavity length and the lift and pressure drag coefficients. This examination is presented only for sheet-cavitating flows over NACA16009 (AOA¼3°, s ¼ 0:66, Lj ¼ 0:7, and Aj ¼ 0:05) for various jet widths (0:025 o dj o 0:1) whose results are plotted in Fig. 12. The wall pressure distributions (Fig. 12b) illustrate that the length of cavitation zone decreases significantly when the width jet increases from dj ¼ 0:025 to dj ¼ 0:1, and this is the main reason of the lift and pressure drag coefficients reduction as one can see in Fig. 12a. Also, by increasing the width jet while the blowing _ ¼ ρdj V j ) amplitude has been fixed, the mass flow rate exhaust (i.e. m _ j ) helps to is increased. Hence, the reaction of momentum force (i.e. mV push the hydrofoil in opposite direction of blowing and reduce the lift coefficient, accordingly. 7.2. Suction jet In the following, sheet-cavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, and dj ¼ 0:025) at the suction angle of 90°,
suction amplitude of Aj ¼ 0:01; 0:05; 0:1, and various jet locations are calculated. The computed normalized lift and pressure drag coefficients are displayed in Fig. 13a and b, respectively. From the results it is evident that suction intensifies the lift and the pressure drag coefficients and this intensification is observed at each jet location. In order to explain this event, the wall pressure distribution and the contour of density at the mentioned suction amplitudes are depicted with Fig. 14 at the jet location of Lj ¼ 0:3, as an example. From Fig. 14, the immediate observation regarding suction effects is that as suction amplitude increases, the flow becomes more attached to the surface and a larger cavity can be formed. Greater cavity length causes increase of the surfaces pressure difference between hydrofoil's upper and lower side. So the lift and pressure drag coefficients increase significantly in comparison to the no-suction case as seen in Fig. 13. Fig. 15a and b mainly focus on the effects of suction angle on sheet-cavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, and
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.4
0.9
1.6
0.8
1.2
1.2
33
dj=0.025 dj=0.050 dj=0.075 dj=0.10
0.7
0.5
0.8
0.4
Cp
CL / CLb
0.6
CD / CDb
0.8 1.0
0.0
0.4 0.6
0.4 0.00
0.03
0.05
0.08
0.10
0.3
-0.4
0.2 0.13
-0.8
0.0
0.2
0.4
x
dj
0.6
0.8
1.0
Fig. 12. Effect of the width of blowing jet on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Lj ¼ 0:7, and Aj ¼ 0:05: (a) Lift and pressure coefficients. (b) Surface pressure distributions.
a
4.0
1.8 Aj=0.01 Aj=0.05 Aj=0.10
1.6
Aj=0.01 Aj=0.05 Aj=0.10
3.5 3.0
CD / CDb
CL / CLb
1.4
1.2
2.5 2.0 1.5
1.0 1.0 0.8 0.3
0.4
0.5
0.6
0.7
0.8
0.9
Lj
0.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Lj
Fig. 13. Lift and pressure drag coefficients of sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼ 3° and s ¼ 0:66 during suction when, dj ¼ 0:025, 0:1 o Aj o 0:1, and 0:1 o Lj o 0:9.
dj ¼ 0:025) and NACA66(MOD) (AOA ¼4°, s ¼ 0:84, and dj ¼ 0:025), respectively. In these simulations a suction jet at location of Lj ¼ 0:7 and amplitude of Aj ¼ 0:05 is placed on the upper surface and the calculated lift and pressure drag are plotted. In addition, the surface pressure distributions for these cases are demonstrated in Fig. 16a and b. It can be correspondingly seen in Fig. 16 that the length of cavity does not change meaningfully when the suction angle increase from θj ¼ 30 to θj ¼ 90. Therefore, there is no significant change in the lift and pressure drag coefficients as can be easily seen in Fig. 15. It means that suction angle has no serious effect on attaching flow to the surface in comparison to the suction amplitude as discussed in Fig. 14. The last investigation in the case of suction jet belongs to the effect of jet width on the cavity length and the lift and pressure
drag coefficients. This exploration is exhibited only for sheetcavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, Lj ¼ 0:7, and Aj ¼ 0:05) for various jet widths (0:025 o dj o0:1) whose results are clarified in Fig. 17. The wall pressure distributions (Fig. 17b) elucidate that the length of cavitation region decreases slowly when the width jet increases from dj ¼ 0:025 to dj ¼ 0:1, and this is one of the evidence of the lift and pressure drag coefficients reduction as noted in Fig. 17a. In addition, by increasing the width jet as long as the suction amplitude has been fixed, the mass flow _ ¼ ρdj V j ) is increased. rate transferred to the hydrofoil (i.e. m _ j ) helps to push Hence, the reaction of momentum force (i.e. mV the hydrofoil in opposite direction of the suction and strengthen the lift coefficient, correspondingly.
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P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.6
Aj=0.0 (No Suction)) Aj=0.01 Aj=0.05 Aj=0.10
1.2
Aj=0.0
0.8 Aj=0.01
Cp
0.4 0.0
Aj=0.05
-0.4 -0.8 Aj=0.10
0.0
0.2
0.4
x
0.6
0.8
1.0
Fig. 14. Effect of suction amplitude on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, dj ¼ 0:025, and Lj ¼ 0:3: (a) surface pressure distribution. (b) Contour of density.
1.20
1.06
1.4
2.0
1.04
1.3
1.8
1.2
1.6
1.1
1.4
1.0
1.2
0.9
1.0
1.15
1 0.95 0.98
0.90 0.85 0.80 15
CL / CLb
CL / CLb
1.00
CD / CDb
1.02
1.05
CD / CDb
1.10
0.96 30
45
60
75
90
j
0.8 15
0.8 30
45
60
75
90
j
Fig. 15. Effect of suction angle on the lift and pressure drag coefficients: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
8. Conclusion In this paper the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows past NACA16009 and NACA66(MOD) hydrofoils is scrutinized. A numerical program is developed for computer simulation of inviscid sheet-cavitating flows. In this simulation, the Jameson's finite volume technique and the power-law preconditioning method are utilized. The single-phase barotropic cavitation pattern is employed for cavitation modeling. In order to stabilize the numerical solver, the second and fourth-order dissipation terms are employed. Explicit four-step Runge–Kutta time integration is utilized to obtain the steady-state condition. To implement the blowing and suction effects, a jet is situated on the hydrofoil's upper surface. Four types
of blowing (and suction) parameters i.e., jet amplitude, jet position, width of jet and jet angle are investigated. In this study, surface pressure distribution, length of cavitation, lift and pressure drag coefficients are examined when the mentioned parameters are changed. The numerical simulations indicate that blowing jets often reduce the length of cavity because of detaching the flow from the surface and consequently decrease the lift and pressure drag coefficients, while suction jets behave reversely. In addition, by increasing the blowing amplitude and the width of jet, the lift and pressure drag coefficients and the length of cavity decrease. In contrast, suction shows an increase of the mentioned hydrodynamic coefficients as well as the length of sheet-cavity in the case of jet magnitude or width of jet enhancement.
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
35
Fig. 16. Effect of suction angle on surface pressure distribution: (a) NACA16009 hydrofoil at AOA ¼3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
1.5
2.0
1.3
1.8
dj=0.025 dj=0.05 dj=0.075 dj=0.10
1.0
1.2
0.5
Cp
1.4
CD / CDb
CL / CLb
1.6 1.1
0.0
1.0 1.2
-0.5
0.9 0.000
1.0 0.025
0.050
0.075
0.100
0.125
dj
0.0
0.2
0.4
0.6
0.8
1.0
x
Fig. 17. Effect of the suction jet width on cavitating flows over NACA16009 hydrofoil NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7: (a) lift and pressure drag coefficients. (b) surface pressure distribution.
Acknowledgments The authors would like to thank the Shahrood University of Technology for supporting this study.
References Ahuja, V., Hosangadi, A., Arunajatesan, S., 2001. Simulations of cavitating flows using hybrid unstructured meshes. J. Fluid Eng. – T. ASME 123, 331–340. Akbarzadeh, P., 2013. Cavitating/non-cavitating flows simulation by third-order finite volume scheme and power-law preconditioning method. Appl. Math. Mech. – Engl. Ed. 34 (2), 209–228. Amromin, E.L., Kopriva, J., Arndt, R.E.A., Wosnik, M., 2006. Hydrofoil drag reduction by partial cavitation. J. Fluid Eng. – T. ASME 128, 931–936. Arndt, R.E.A., Ellis, C.R., Paul, S., 1995. Preliminary investigation of the use of air injection to mitigate cavitation erosion. J. Fluid Eng. – T. ASME 117, 498–504. Celik, F., Ozden, A.Y., Bal, S., 2014. Numerical simulation of flow around two- and three-dimensional partially cavitating hydrofoils. Ocean Eng. 78, 22–34. Chahine, G.L., Frederick, G.F., Bateman, R.D., 1993. Propeller tip vortex cavitation suppression using selective polymer injection. J. Fluid Eng. – T. ASME 115 (3), 497–503.
Chatterjee, D., Arakeri, V.H., 1997. Towards the concept of hydrodynamic cavitation control. J. Fluid Mech. 332, 377–394. Chatterjee, D., 2003. Use of ultrasonics in shear layer cavitation control. Ultrasonics 41, 465–475. Chen, X., Lu, C., Li, J., Pan, Z., 2008. The wall effect on ventilated cavitating flows in closed cavitation tunnels. J. Hydrodyn. 20 (5), 561–566. Chorin, A.J., 1967. A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26. Coutier-Delgosha, O., Reboud, J.L., Delannoy, Y., 2003. Numerical simulation of the unsteady behaviour of cavitating flows. Int. J. Numer. Methods Fluids 42, 527–548. Coutier-Delgosha, O., Fortes-Patella, R., Reboud, J.L., Hakimi, N., Hirsch, C., 2005. Numerical simulation of cavitating flow in 2D and 3D inducer geometries. Int. J. Numer. Methods Fluids 48, 135–167. Delannoy, Y., Kueny, J.L., 1990. Two phase flow approach in unsteady cavitation modeling, Cavitation and Multiphase Flow Forum. ASME-FED 98, 153–158. Deshpande, M., Feng, J., Merkle, C.L., 1994. Cavity flow predictions based on the Euler equations. J. Fluid Eng. – T. ASME 116, 36–44. Esfahanian, V., Akbarzadeh, P., Hejranfar, K., 2012. An improved progressive preconditioning method for steady non-cavitating and sheet-cavitating flows. Int. J. Numer. Methods Fluids 68, 210–232. Fine, N.E., Kinnas, S.A., 1993. A boundary element method for the analysis of the flow around 3-D cavitating hydrofoils. J. Ship Res. 37, 213–224. Geurest, J.A., 1960. Linearized theory for fully cavitated hydrofoils. Int. Shipbuild. Prog. 7, 48–54.
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Huang, D., 2007. Preconditioned dual-time procedures and its application to simulating the flow with cavitations. J. Comput. Phys. 223, 685–689. Jameson, A., Schmidt, W., Turkel, E., 1981. Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge–Kutta Time-stepping Schemes AIAA Paper. (AIAA-1981-1259). Ji, B., Luo, X., Peng, X., Wu, Y., Xu, H., 2012. Numerical analysis of cavitation evolution and excited pressure fluctuation around a propeller in non-uniform wake. Int. J. Multiphase Flow 43, 13–21. Ji, B., Luo, X., Wu, Y., Peng, X., Duan, Y., 2014. Numerical analysis of unsteady cavitating turbulent flow and shedding horse-shoe vortex structure around a twisted hydrofoil. Int. J. Multiphase Flow 51, 33–43. Kato, H., Yamaguchi, H., Okada, S., Kikuchi, K., Miyanaga, M., 1987. Suppression of sheet cavitation inception by water discharge through slit. J. Fluid Eng. – T. ASME 109 (1), 70–74. Krishnaswamy, P., 2000. Flow Modelling for Partially Cavitating Hydrofoils (Ph.D. thesis), Technical University of Denmark, Lyngby, Denmark. Kunz, R., Boger, D., Stinebring, D., Chyczewski, T., Lindau, J., Gibeling, H., Venkateswaran, S., Govindan, T., 2000. A preconditioned Navier–Stokes method for two-phase flows with application to cavitation prediction. Comput. Fluids 29, 849–875. Li, Dq, Grekula, M., Lindell, P., 2010. Towards numerical prediction of unsteady sheet cavitation on hydrofoils. J. Hydrodyn. Ser. B 22, 741–746. Malan, A.G., Lewis, R.W., Nithiarasu, P., 2002. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I. Theory and implementation. Int. J. Numer. Methods Eng. 54, 659–714. Merkle, C.L., Feng, J., Buelow, P.E.O., 1998. Computational modelling of the dynamics of sheet cavitation. In: Proceedings of The Third International Symposium on Cavitation. Grenoble, France.
Morgut, M., Nobilea, E., Bilušb, I., 2011. Comparison of mass transfer models for the numerical prediction of sheet cavitation around a hydrofoil. Int. J. Multiphase Flow 37 (6), 620–626. Pham, T.M., Larrarte, F., Fruman, D.H., 1999. Investigation of unsteady sheet cavitation and cloud cavitation mechanisms. J. Fluid Eng. – T. ASME 121 (2), 289–296. Roohi, E., Zahiri, A.P., Passandideh-Fard, M., 2013. Numerical simulation of cavitation around a two-dimensional hydrofoil using VOF method and LES turbulence model. Appl. Math. Model. 37, 6469–6488. Shang, Z., 2013. Numerical investigations of supercavitation around blunt bodies of submarine shape. Appl. Math. Model. 37, 8836–8845. Shen, Y.T., Eppler, R., 1981. Wing section for hydrofoils. J. Ship Res. 35, 191–200. Shen, Y.T., Dimotakis, P.E., 1989. The influence of surface cavitation on hydrodynamic forces. In: Proceedings of The 22nd American Towing Tank Conference. St. Johns. pp. 44–53. Song, C., He, J., 1998. Numerical simulation of cavitating flows by single-phase flow approach. In: Proceedings of The Third International Symposium on Cavitation, Grenoble, France. Turkel, E., 1987. Preconditioning methods for solving the incompressible and low speed Compressible equations. J. Comput. Phys. 72, 227–298. Ukon, Y., Kudo, T., Kurobe, Y., Hoshino, T., 1994. Design and evaluation of new supercavitating propellers. In: Kato, H., (Ed.) Second International Symposium on Cavitation. Tokyo, Japan, April 5–7: 395–402. Venkateswaran, S., Lindau, J., Kunz, R., Merkle, C., 2002. Computation of multiphase mixture flows with compressibility effects. J. Comput. Phys. 180, 54–57. Wang, G., Starzewski, O.M., 2007. Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil. Appl. Math. Model. 31, 417–447. Zhao, W., Zhang, L., Shao, X., Deng, J., 2010. Numerical study on the control mechanism of cloud cavitation by obstacles. J. Hydrodyn. 22 (5), 792–797.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Hydrodynamic characteristics of blowing and suction on sheet-savitating flows around hydrofoils Pooria Akbarzadeh a,n, Ebrahim Akbarzadeh b a b
Department of Mechanical Engineering, Shahrood University of Technology, P.O.X:3619995161 Daneshgah Boulevard, Shahrood, Iran Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 31 May 2015 Accepted 16 January 2016
In this study the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows over hydrofoils is investigated. A computational program is developed for simulation of inviscid cavitating flows. In this simulation, the Jameson's finite volume technique and the progressive power-law preconditioning method for analyzing the cavitating flows are utilized. For cavitation modeling the barotropic cavitation pattern is used. The numerical stabilization is achieved via the second and fourth-order dissipation terms. Explicit four-step Runge–Kutta time integration is applied to achieve the steady-state condition. Cavitating flows over NACA-hydrofoils for different cavitation numbers are investigated. To apply the blowing and suction, a jet is placed on the hydrofoil's upper surface. Four blowing (and suction) parameters i.e., velocity ratio, jet location, width of jet, and jet angle are scrutinized. The effects of these parameters on surface pressure distribution, length of cavitation, lift and pressure drag coefficients are examined. The results indicate that blowing jets often decrease the length of cavity and reduce the lift and pressure drag coefficients, while suction jets behave reversely. Also, by increasing the blowing amplitude or the width of jet, the lift and pressure drag coefficients and the length of cavity decrease while in suction cases, they increase. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Blowing and suction Cavitating flow Power-law preconditioning Lift Drag
1. Introduction Cavitation is a complex physical phenomenon which occurs when the local liquid pressure is lower than its saturated vapor pressure. Cavitating flows are important issues in the design and performance of many engineering devices such as underwater projectiles, marine propellers, hydro-machineries, fuel injectors, hydrofoils, to name but a few. It is well known that cavitation often causes undesirable problems such as load asymmetry, erosion, noise, vibration, efficiency reduction of hydro-turbines, head and efficiency drop of pumps, to name but a few (Esfahanian et al., 2012; Shang, 2013). However cavitation has a wide range of industrial application, such as surface cleaning, drag reduction of underwater moving body by using supercavitation, medicine production, chemical mixing, transfer large molecules into biological cells, and etc. (Shang, 2013). Hence, a large number of studies have been conducted into cavitation and cavitating flows (Geurest, 1960; Fine and Kinnas, 1993; Coutier-Delgosha et al., 2003; Wang and n
Corresponding author. Tel.: þ 98 9122022776. E-mail addresses: [email protected] (P. Akbarzadeh), [email protected] (E. Akbarzadeh). http://dx.doi.org/10.1016/j.oceaneng.2016.01.015 0029-8018/& 2016 Elsevier Ltd. All rights reserved.
Starzewski, 2007; Li et al., 2010; Morgut et al., 2011; Esfahanian et al., 2012; Ji et al., 2012; Roohi et al., 2013; Shang, 2013; Celik et al., 2014; Ji et al., 2014, and etc.). Cavitation control has tremendous importance in various fields in industry and it is performed in two major ways. One way of controlling the cavitation (especially sheet-cavitating flows around hydrofoils) emphasizes passive methods such as modifying geometric shapes (For example Shen and Eppler (1981), Ukon et al. (1994), Amromin et al. (2006), and etc.) or using obstacles placed on the foil surface (For example Zhao et al. (2010)) to maneuver the pressure gradient and change the cavitation regime. Another approach is active flow control, which changes the lift and the drag by injecting a small quantity of fluid (e.g. a fluid with lower viscosity such as air or micro-bubbles, a fluid with higher viscosity such as polymers, and using the same fluid around the object) or energy (e.g. acoustic or ultrasonic waves) into the system to control the boundary layer and the cavitation incident. Indeed, suction and blowing fluid or injecting energy can actively modify the pressure distribution over a body surface. Kato et al. (1987) investigated experimentally, the suppression effect of water discharge (using the same fluid around the object) on incipient and developed sheet cavitation on a hemispherical headform. They found that, by increasing the discharge rate, the
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P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
length of the sheet cavity becomes shorter and then the cavity is suppressed completely. Chahine et al. (1993) injected a dragreducing polymer solution (a fluid with higher viscosity) into the tip vortex region of propeller blades and significantly delayed tip vortex cavitation. Arndt et al. (1995) investigated the use of air injection (a fluid with lower viscosity) through small holes in the leading edge of a NACA0015 hydrofoil to mitigate undesirable cavitation effects. Chatterjee and Arakeri (1997) and Chatterjee (2003) examined the feasibility of using ultrasonic waves for hydrodynamic cavitation control. Pham et al. (1999) examined the control of cloud cavitation using a tiny obstacle (barrier) on the foil surface and air injection through a slit situated in the vicinity of the leading edge. Chen et al. (2008) simulated numerically the cavity shape and its characteristics of steady ventilated cavitating flows in water tunnels. In their simulations, a symmetric underwater vehicle model, which consists of a disk cavitator, two ventilation bowls and a slender body was considered. When very low Mach number flows (or almost incompressible fluid flow) are simulated, the large condition number (or degree of stiffness), i.e. the ratio of the largest to the smallest characteristic speed (eigenvalues of system of equations), leads to destroy the convergence rate in the case of marching methods. Indeed, the allowable time step is limited by the fastest moving wave. During one time step, the slowest wave propagates only over a fraction of cell width. Thus, a large condition number reduces the wavepropagation efficiency (slows down the convergence). Besides the incompressible flow, for numerical simulation of multiphase flows (particularly cavitating flows) due to change of sound speed from one phase to another, it is required to modify the sound speed (or characteristic speed) artificially in order to come close the stiffness of governing equations to unity. In fact, convergence fails or accuracy weakens when the speed of sound is much greater than the fluid flow velocity (or when the stiffness of governing equations is much greater than unity). The preconditioning methods (PMs) can do this modification by adding some terms into the continuity and momentum governing equations. Therefore many researchers have used PMs to simulate the cavitating flow such as Refs. Kunz et al. (2000), Ahuja et al. (2001), Venkateswaran et al. (2002), Huang (2007), Esfahanian et al. (2012) and Akbarzadeh (2013). It is worth to note that, their results showed that PMs have a tremendous effect on reducing the iteration steps of the numerical convergence. In this study, the hydrodynamic behavior of sheet-cavitating flows over hydrofoils by considering the blowing and suction is investigated for the first time. For this purpose, a computational code is developed in order to simulate the inviscid cavitating flows. Indeed, cavitation is a 3-dimensional, unsteady and discontinuous or periodic phenomenon of formation, growth and collapse of bubbles. Due to the natural complexity of cavitation, it is neither reliably assessable nor fully understood yet. Hence, a large number of computational grids and small time steps have to be considered to simulate the cavitation event accurately from microscopic to macroscopic level (for example using direct numerical simulation or DNS). The mathematical models describing the cavitation mechanisms based on certain assumptions and simplifications can avoid such difficulties (Shang, 2013). Therefore, in this code, the implemented cavitation model is based on the barotropic state law initiated by Delannoy and Kueny (1990). The validity of this simple model has been approved by many researching works such as Merkle et al. (1998), Song and He (1998), Coutier-Delgosha et al. (2003), Coutier-Delgosha et al. (2005), Esfahanian et al. (2012), Akbarzadeh (2013), and etc. The governing equations are modified by the progressive power-law preconditioning method introduced by Esfahanian et al. (2012). This physical model is integrated with the cell-centered Jameson's finite volume algorithm proposed by Jameson et al. (1981). The
numerical stabilization is achieved by the second and fourth-order dissipation terms. To obtain the steady state condition, explicit four step Runge–Kutta method is utilized. Cavitating flows over NACA-hydrofoils for various cavitation numbers and angle of attacks are examined. To simulate the blowing and suction (as an active control method of cavitation), a continuous water jet is placed on the hydrofoil's upper surface. Four important parameters with relation to the surface blowing and suction i.e., velocity ratio (jet amplitude), jet position, width of jet and jet angle are studied in cavitating flows. In this study the effects of uniform surface blowing and suction on the length of cavitation, the lift and pressure drag coefficients, and the surface pressure distribution are demonstrated. The results prove that continuous blowing jets often decrease the length of sheet-cavity and reduce the lift and pressure drag coefficients. While continuous suction jets behave reversely. Also, the numerical data indicates that by increasing the blowing velocity ratio or the width of jet, the lift and pressure drag coefficients and the length of cavity decrease. While when the jet sucks the fluid with higher amplitude or through a widened width, the mentioned hydrodynamic parameters increase.
2. Preconditioning governing equations The non-dimensional preconditioning governing equations (inviscid fluid flow) are expressed as follows (Esfahanian et al., 2012):
Γ1
! ! ! ∂Q ∂ F ∂ E þ þ ¼0 ∂t ∂x ∂y
ð1Þ
T ! ! ! E ¼ where, Q ¼ p u v ; F ¼ ð ρu p þ ρu2 ρuv ÞT , ð ρv ρuv p þ ρv2 ÞT , and Γ is the preconditioning matrix defined as follows: 0 1 β2 0 0 B C ð2Þ Γ¼B 0 C @ σ u=ρ 1=ρ A σ v=ρ
0
1=ρ
In Eq. (1), u ¼ u=U 1 and v ¼ v=U 1 are velocity components in the x and y-directions, respectively. x ¼ x=L and y ¼ y=L are the Cartesian coordinates, p ¼ p=ρ1 U 21 is the static pressure, ρ ¼ ρ=ρ1 is the density and t ¼ tU 1 =L is the time. U 1 is the reference velocity, L is the reference length, and ϕ represents the dimensional form of quantity ϕ. β is the artificial compressibility coefficient which is calculated similar to the references (Esfahanian et al., 2012; Malan et al., 2002). σ is called preconditioning factor which is defined based on the power-law preconditioning method as follows (Esfahanian et al., 2012; Akbarzadeh, 2013):
σ ¼ ð1 Au Þm
ð3Þ where, Au ¼ limx-x þ ∇uðx0 Þ ∇uðxÞ= ∇uðx0 Þ ∇uðxÞ is a local 0 velocity sensor, ∇uðx0 Þ ¼ limx-x0 ½uðxÞ uðx0 Þ=½x x0 is the velocity gradient at x ¼ x0 , and m is an integer number. In Eq. (3) when m is set to zero, σ becomes similar to the standard preconditioning method introduced by Turkel (1987) and when m is set to one, σ becomes equal to the relation introduced by Malan et al. (2002). In addition, the case σ ¼ 0 is similar to the standard artificial compressibility method initiated by Chorin (1967). In this study, the value of m ¼ 2 or m ¼ 3:, which has been defined in reference (Esfahanian et al., 2012) as a best selection, is used to facilitate the convergence of the numerical simulation.
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3. Barotropic cavitation model In this study, the homogenous barotropic cavitation model is applied for simulation the cavitation flows. This model which considers the vapor–liquid mixture as a single fluid, was initiated by Delannoy and Kueny (1990). According to Eq. (4), in this model, when the fluid equals the liquid phase, ρl , all the computational cell is full of liquid (incompressible), and if it equals the vapor phase, ρv , the computational cell is occupied by vapor (incompressible). Between these two extreme values, the cell is occupied by a two-phase mixture that is still considered as one single fluid. In this zone, the fluid density is controlled by a barotropic state law that links the fluid density variations to the local static pressure. This type of cavitation model has been applied recently by many researchers such as Merkle et al. (1998), Song and He (1998), Coutier-Delgosha et al. (2003, 2005), Esfahanian et al. (2012), Akbarzadeh (2013), and etc. 8 ρl ; p pvap 4 Δpvap > > > < ρ þρ ρ ρ 2 p pvap l v l v ; p pvap o Δpvap sin ρ ρ C min ð4Þ ρðpÞ ¼ 2 þ 2 l v > > > :ρ ; p p o Δp v
vap
27
local velocity sensor i.e. Au, is numerically computed as follows: 4ui;j ui þ 1;j ui 1;j ui;j þ 1 ui;j 1 ðAu Þi;j ¼ ð7Þ ui;j ui þ 1;j þ ui;j ui 1;j þ ui;j ui;j þ 1 þ ui;j ui;j 1
5. Blowing and suction control parameters In this study, as shown in Fig. 1, four basic parameters of blowing and suction are used for the numerical investigation. These parameters are jet location Lj , velocity ratio Aj ¼ V j =U 1 , blowing (or suction) angle θj , and width of jet dj , where, V j is the jet entrance velocity. It should be noted that positive value of θj represents the blowing condition and negative value of θj indicates the suction condition. The effects of various values of the jet location (Lj ¼ 0:3–0:9 of the chord length after leading edge), the velocity ratio (Aj ¼ 0.01–0.1), the blowing (suction) angle (θj ¼ 30–90°), and the width of jet (dj ¼ 2:5% to 1:0% of chord length) on hydrodynamic behavior of sheet-cavitating flows over hydrofoils are discussed.
vap
In Eq. (4), ρl is the liquid-phase density, ρv is the vapor-phase density, pvap is the vapor liquid pressure, Δpvap ¼ π C 2min = ρl ρv and C min is defined as the minimum speed of sound wave in the liquid–vapor mixture. Based on the Coutier-Delgosha et al. (2005) recommendation, C min is chosen between 1.5 m/s and 4.0 m/s and ρv =ρl is set to 0.001. In order to consider the vapor–liquid pressure in the code, the cavitation number is introduced via the relation: s ¼ ðp pvap Þ=ð1=2ρ1 U 21 Þ.
4. Finite volume numerical method The integral form of the preconditioned governing equations specified by relation (1) over a control volume Ω with surface of ∂Ω is written as (Esfahanian et al., 2012): Z Z ! ! ∂ ! Q dA þ Γ F dx E dy ¼ 0 ð5Þ ∂t Ω ∂Ω According to the references (Jameson et al., 1981; Esfahanian et al., 2012), the discretized form of Eq. (5) over the (i,j)th computational cell-volume accompanied by adding Jameson's dissipative terms becomes Ai;j ∂Q i;j;k =∂t ¼ Gi;j;k þ Di;j;k , where Ai;j is the area of (i,j)th cell, k ¼ 1; 2; 3 is the index of vector's components, Di;j;k is the artificial dissipative expression composed of second and fourth order terms initially suggested by Coutier-Delgosha et al. (2003) for cavitating flows, and Gi;j;k is the numerical flux related to (i,j)th cell which 2P 2 is defined as Gi;j;1 ¼ β edge ðF 1 Δx E 1 ΔyÞi;j ; Gi;j;2 ¼ ð σ ui;j Gi;j;1 =β Þ P 2 P Gi;j;3 ¼ ð σ vGi;j;1 =β Þ edge ðF 3 Δx E3 ΔyÞi;j . edge ðF 2 Δx E 2 ΔyÞi;j ; The optimal fourth stage Runge–Kutta method utilized by Malan et al. (2002) and Esfahanian et al. (2012) is applied to solve governing Eq. (5) as follows: 8 ð0Þ ðnÞ > > > Q i;j;k ¼ Q i;j;k > < Δt 1Þ ð0Þ Q ðKÞ ¼ Q ð0Þ αK Ai;ji;j Q ðK Q i;j;k ð6Þ i;j;k i;j;k i;j;k > > > ðn þ 1Þ ð4Þ > : Q i;j;k ¼ Q i;j;k where K ¼ 1; 2; 3; 4, the coefficients αK are set to 4, 3, 2 and max 1 respectively, Δt i;j ¼ CFL ΔLmin is the local time step, ΔLmin i;j =λi;j i;j is the minimum length of (i,j)th computational cell's sides, CFL is max the Courant–Friedrichs–Lewy number, and the parameter λi;j indicates the maximum eigenvalue of the governing equations. The
6. Boundary conditions The velocity boundary condition on the body surface is set as ! ! V n ¼ 0, where V n is the perpendicular component of fluid velocity to the solid surface. The pressure boundary condition is set as ∂p=∂n ¼ 0 where n is the unit vector perpendicular to the body surface. Classical incompressible type of the outlet and inlet boundary is applied: imposed velocities at the inlet and imposed static pressure at the outlet. For the jet boundary condition, constant velocity is applied at the entrance and pressure is achieved via inside of flow field.
7. Result In order to investigate the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows over hydrofoils, many cases are considered. NACA16009 and NACA66(MOD) hydrofoils are selected for reporting in this paper. For this subject, many grids with different mesh-size have been examined and finally two O-gird geometries with the size of 150 64 for NACA16009 hydrofoil and 180 64 for NACA66(MOD) are chosen as shown in Fig. 2 (The chord length is considered as unity, c ¼ 1:0). The position of the outlet computational domain is set around 10 chord length far away from the hydrofoils’ surface. For these test cases, the uniform flow (i.e., p1 ¼ 1:0, U 1 ¼ 1:0, and ρ1 ¼ 1:0) is used as the initial condition. To validate the present numerical computations, the results for NACA16009 are compared with the numerical data reported by Krishnaswamy (2000) and the results for NACA66(MOD) are checked with the numerical works of Deshpande et al. (1994) and the experimental works of Shen and Dimotakis (1989). Fig. 3a shows the surface pressure distribution, C p ¼ p1 p = 0:5ρ1 U 21 ,
Fig. 1. Blowing and suction control parameters.
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NACA66(MOD)Hydrofoil
0.6
0.6
0.4
0.4
0.2
0.2
y
y
NACA16009 Hydrofoil
0.0
0.0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
-0.2
1.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
x
x Fig. 2. Grid for NACA16009 and NACA66(MOD) hydrofoils.
1.5
1.5 Deshpande et al.(1994) Present work
Krishnaswamy (2000) Present work
1.0
1.0
Pressure Side Pressure Side Cp
0.5
Cp
0.5
0.0
0.0
-0.5
Suction Side
-1.0
-1.5
-1.0
Cavity Zone 0.0
0.2
0.4
Suction Side
-0.5
Cavity Zone 0.6
0.8
1.0
x
-1.5
0.0
0.2
0.4
0.6
0.8
1.0
x
Fig. 3. Surface pressure distributions and cavity zone: (a) sheet-cavitating flow over NACA66(MOD) hydrofoil at AOA ¼ 4° and s ¼ 0:84. (b) Sheet-cavitating flow over NACA16009 hydrofoil at AOA ¼ 3° and s ¼ 0:66.
of sheet-cavitating flow over NACA66(MOD) hydrofoil at the angle of attack (AOA) of 4° and the cavitation number of s ¼ 0:84. Also, Fig. 3b illustrates the wall pressure distribution of sheet-cavitating flow over NACA16009 hydrofoil at AOA¼3° and s ¼ 0:66. These results are closely comparable to those of Deshpande et al. (1994) and Krishnaswamy (2000), respectively (for both pressure field and length of cavity). However, the observed differences in cavity closure zone (reattachment/detachment regions) between the present simulation and their works are because of the flow (and cavitation) modeling differences. For example, the methodology of Krishnaswamy (2000) is based on cavitation interface tracking approach and potential flow model. In such methods, a large sharp change in pressure distribution (in cavity closure zone) is seen. The other sample that confirms the accuracy of the numerical method is
shown in Fig. 4. This figure presents the calculated lift (CL) and pressure drag (CD) coefficients for sheet-cavitating flows around the NACA66(MOD) hydrofoil at AOA¼4° and 6° as a function of cavitation number. The results are compared with the experimental works performed by Shen and Dimotakis (1989). The coefficients agree quite well with the experimental measurements. Both the experiments and calculations show that lift increases with decreasing the cavitation number, as expected. However, It should be noted that the experiments of Shen and Dimotakis (1989) were conducted with the hydrofoil centered in a High-Speed Water Tunnel whose walls were five chord lengths apart at Re ¼ 2 106 . Therefore, the observed deviation in Fig. 4 may be due to the viscous, turbulence, and wall effects in the experimental measurements.
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
7.1. Blowing jet
represent the lift and the pressure drag coefficients, respectively in the case of Aj ¼ 0:0 (no blowing or suction). The figures show that blowing reduces the lift and the pressure drag coefficients. Indeed, the coefficients decrease as blowing amplitude increases above the amplitude of 0:01 and this reduction is seen at every jet location. In order to interpret this issue, the wall pressure distribution, C p , and the contour of density for NACA16009 hydrofoil at the mentioned blowing amplitudes are plotted in Figs. 6 and 7 at jet location of Lj ¼ 0:7 and Lj ¼ 0:3, respectively. It can be concluded
Predicted lift and pressure drag coefficients at the blowing angle of 90°, blowing amplitude of Aj ¼ 0:01; 0:05; 0:1, and various jet locations for sheet-cavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, and dj ¼ 0:025) and NACA66(MOD) (AOA ¼ 4°, s ¼ 0:84, and dj ¼ 0:025) are shown in Fig. 5a and b, respectively. The coefficients are normalized by their corresponding values in the baseline case (no blowing or suction). In this paper, C Lb and C Db
1.40
29
Present work, Lift Present work, Pressure drag Shen and Dimotakis (1989)
0.20
1.20
0.80
CL
0.12 o AOA=6
0.60
0.40
CD
0.16
1.00
AOA=4
0.08
o
0.04 0.20
0.00
0.00 0.8
1.0
1.2
1.4
1.6
1.8
2.0
s Fig. 4. Lift and pressure drag prediction for sheet-cavitating flows around the NACA66(MOD) hydrofoil at AOA ¼4° and 6°.
1.4
1.2
A = 0.01 A = 0.05 A = 0.10
1.2
1.1
CL / CLb
CL / CLb
1.0 0.8 0.6
A = 0.01 A = 0.05 A = 0.10
0.4
1.0
0.9
0.2 0.0 0.2
0.4
0.6
0.8
0.8 0.2
1.0
1.4
0.4
0.6
0.4
0.6
0.8
1.0
0.8
1.0
1.1
1.2
1.0
CD / CDb
CD / CDb
1.0 0.8 0.6
0.9 0.8
0.4
0.7
0.2 0.0 0.2
0.4
0.6
0.8
Lj
1.0
0.6 0.2
Lj
Fig. 5. Lift and pressure drag coefficients of sheet-cavitating flows .during blowing when, dj ¼ 0:025, 0:1 o Aj o 0:1, and 0:1o Lj o 0:9: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66. (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84.
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Aj=0.0 (No Blowing) Aj=0.01 Aj=0.05 Aj=0.10
1.2
Aj=0.0
0.8
Aj=0.01
Cp
0.4
0.0 Aj=0.05
-0.4 Aj=0.10
-0.8 0.0
0.2
0.4
0.6
0.8
1.0
x Fig. 6. Effect of blowing amplitude on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼3°, s ¼ 0:66, dj ¼ 0:025, and Lj ¼ 0:7: (a) surface pressure distribution. (b) Contour of density.
Aj=0.0 (No Blowing) Aj=0.01 Aj=0.05 Aj=0.10 Aj=0.15
1.2
0.8
Aj=0.0
Aj=0.01
Cp
0.4 Aj=0.05
0.0 Aj=0.10
-0.4
-0.8 0.0
0.2
0.4
x
0.6
0.8
1.0
Aj=0.15
Fig. 7. Effect of blowing amplitude on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼3°, s ¼ 0:66, dj ¼ 0:025, and Lj ¼ 0:3: (a) surface pressure distribution. (b) Contour of density.
from these computation results that the length of cavity decrease as blowing amplitude increases. According to Figs. 6b and 7b, by reducing the cavity length, the difference between hydrofoil's upper and lower surfaces pressure reduces. So the lift and pressure drag coefficients decrease significantly in comparison to the nobowing case as seen in Fig. 5 (the main reason). The inferior reason
is that, by increasing the blowing amplitude while the width jet _ ¼ ρdj V j ) is has been fixed, the mass flow rate exhaust (i.e. m _ j ) helps increased. Hence, the reaction of momentum force (i.e. mV to push the hydrofoil in opposite direction of blowing and reduce the lift coefficient, accordingly. One of the noticeable conclusions of fluid blowing is seen in the last image of Fig. 7b. From the figure,
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.6
A j=0.0 (No B lowing) A j=0.01 A j=0.05 A j=0.10
1.2
31
Aj=0.0
0.8 Aj=0.01
Cp
0.4 0.0 Aj=0.05
-0.4 -0.8
Aj=0.10
0.0
0.2
0.4
x
0.6
0.8
1.0
Fig. 8. Effect of blowing amplitude on sheet-cavitating flows over NACA88(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, dj ¼ 0:025, and Lj ¼ 0:7: (a) surface pressure distribution. (b) Contour of density.
1.6
Aj=0 (No Blowing) Aj=0.01 (b) Aj=0.05 Aj=0.10 Aj=0.18
1.2 0.8
Aj=0.0
Aj=0.01
0.4 Cp
Aj=0.05
0.0 Aj=0.10
-0.4 -0.8 0.0
0.2
0.4
0.6
0.8
1.0
Aj=0.18
x Fig. 9. Effect of blowing amplitude on sheet-cavitating flow over NACA88(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, dj ¼ 0:025, and Lj ¼ 0:3: (a) surface pressure distribution. (b) Contour of density.
it is evident that at Lj ¼ 0:3 and Aj ¼ 0:15, the cavitating region is broken into two smaller regions. This phenomenon always occurs at high blowing amplitude and the jet location near the leading edge. The same results of wall pressure distribution and the contour of density for NACA66(MOD) hydrofoil at jet location of Lj ¼ 0:7 and Lj ¼ 0:3 are plotted in Figs. 8 and 9, respectively. The cavity length reduction, decrease of the difference of hydrofoil's upper and lower surfaces pressure, and the break of the cavity zone are also observed when the blowing amplitude increases step by step.
To explore the effect of blowing angle, a jet at location of Lj ¼ 0:7 and various blowing angles are simulated for sheet-cavitating flows over NACA16009 (AOA¼3°, s ¼ 0:66, and dj ¼ 0:025) and NACA66 (MOD) (AOA¼4°, s ¼ 0:84, and dj ¼ 0:025) and the calculated lift and pressure drag are presented in Fig. 10a and b, respectively. Also, the surface pressure distributions for these cases are shown in Fig. 11a and b. Note from Fig. 11 that, by increasing the blowing angle, the length of cavity becomes shorter and therefore the lift and pressure drag coefficients decrease as shown in Fig. 10. Nonetheless, these reductions are not significant for NACA66(MOD) hydrofoil.
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1.16
1
1.04
1.12
0.9
0.9
1.02
1.08
0.8
0.8
1.00
1.04
0.7
0.7
0.98
1.00
0.96
0.96
CL / CLb
1
0.6
0.6
0.5 15
0.5 30
45
60
75
90
j
CL / CLb
1.06
CD / CDb
1.1
1.1
0.94 15
CD / CDb
32
0.92 30
45
60
75
90
j
Fig. 10. Effect of blowing angle on the lift and pressure drag coefficients: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
Fig. 11. Effect of blowing angle on surface pressure distribution: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
In this section, the last investigation in the case of blowing jet belongs to the effect of jet width on the cavity length and the lift and pressure drag coefficients. This examination is presented only for sheet-cavitating flows over NACA16009 (AOA¼3°, s ¼ 0:66, Lj ¼ 0:7, and Aj ¼ 0:05) for various jet widths (0:025 o dj o 0:1) whose results are plotted in Fig. 12. The wall pressure distributions (Fig. 12b) illustrate that the length of cavitation zone decreases significantly when the width jet increases from dj ¼ 0:025 to dj ¼ 0:1, and this is the main reason of the lift and pressure drag coefficients reduction as one can see in Fig. 12a. Also, by increasing the width jet while the blowing _ ¼ ρdj V j ) amplitude has been fixed, the mass flow rate exhaust (i.e. m _ j ) helps to is increased. Hence, the reaction of momentum force (i.e. mV push the hydrofoil in opposite direction of blowing and reduce the lift coefficient, accordingly. 7.2. Suction jet In the following, sheet-cavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, and dj ¼ 0:025) at the suction angle of 90°,
suction amplitude of Aj ¼ 0:01; 0:05; 0:1, and various jet locations are calculated. The computed normalized lift and pressure drag coefficients are displayed in Fig. 13a and b, respectively. From the results it is evident that suction intensifies the lift and the pressure drag coefficients and this intensification is observed at each jet location. In order to explain this event, the wall pressure distribution and the contour of density at the mentioned suction amplitudes are depicted with Fig. 14 at the jet location of Lj ¼ 0:3, as an example. From Fig. 14, the immediate observation regarding suction effects is that as suction amplitude increases, the flow becomes more attached to the surface and a larger cavity can be formed. Greater cavity length causes increase of the surfaces pressure difference between hydrofoil's upper and lower side. So the lift and pressure drag coefficients increase significantly in comparison to the no-suction case as seen in Fig. 13. Fig. 15a and b mainly focus on the effects of suction angle on sheet-cavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, and
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.4
0.9
1.6
0.8
1.2
1.2
33
dj=0.025 dj=0.050 dj=0.075 dj=0.10
0.7
0.5
0.8
0.4
Cp
CL / CLb
0.6
CD / CDb
0.8 1.0
0.0
0.4 0.6
0.4 0.00
0.03
0.05
0.08
0.10
0.3
-0.4
0.2 0.13
-0.8
0.0
0.2
0.4
x
dj
0.6
0.8
1.0
Fig. 12. Effect of the width of blowing jet on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Lj ¼ 0:7, and Aj ¼ 0:05: (a) Lift and pressure coefficients. (b) Surface pressure distributions.
a
4.0
1.8 Aj=0.01 Aj=0.05 Aj=0.10
1.6
Aj=0.01 Aj=0.05 Aj=0.10
3.5 3.0
CD / CDb
CL / CLb
1.4
1.2
2.5 2.0 1.5
1.0 1.0 0.8 0.3
0.4
0.5
0.6
0.7
0.8
0.9
Lj
0.5
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Lj
Fig. 13. Lift and pressure drag coefficients of sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼ 3° and s ¼ 0:66 during suction when, dj ¼ 0:025, 0:1 o Aj o 0:1, and 0:1 o Lj o 0:9.
dj ¼ 0:025) and NACA66(MOD) (AOA ¼4°, s ¼ 0:84, and dj ¼ 0:025), respectively. In these simulations a suction jet at location of Lj ¼ 0:7 and amplitude of Aj ¼ 0:05 is placed on the upper surface and the calculated lift and pressure drag are plotted. In addition, the surface pressure distributions for these cases are demonstrated in Fig. 16a and b. It can be correspondingly seen in Fig. 16 that the length of cavity does not change meaningfully when the suction angle increase from θj ¼ 30 to θj ¼ 90. Therefore, there is no significant change in the lift and pressure drag coefficients as can be easily seen in Fig. 15. It means that suction angle has no serious effect on attaching flow to the surface in comparison to the suction amplitude as discussed in Fig. 14. The last investigation in the case of suction jet belongs to the effect of jet width on the cavity length and the lift and pressure
drag coefficients. This exploration is exhibited only for sheetcavitating flows over NACA16009 (AOA ¼3°, s ¼ 0:66, Lj ¼ 0:7, and Aj ¼ 0:05) for various jet widths (0:025 o dj o0:1) whose results are clarified in Fig. 17. The wall pressure distributions (Fig. 17b) elucidate that the length of cavitation region decreases slowly when the width jet increases from dj ¼ 0:025 to dj ¼ 0:1, and this is one of the evidence of the lift and pressure drag coefficients reduction as noted in Fig. 17a. In addition, by increasing the width jet as long as the suction amplitude has been fixed, the mass flow _ ¼ ρdj V j ) is increased. rate transferred to the hydrofoil (i.e. m _ j ) helps to push Hence, the reaction of momentum force (i.e. mV the hydrofoil in opposite direction of the suction and strengthen the lift coefficient, correspondingly.
34
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
1.6
Aj=0.0 (No Suction)) Aj=0.01 Aj=0.05 Aj=0.10
1.2
Aj=0.0
0.8 Aj=0.01
Cp
0.4 0.0
Aj=0.05
-0.4 -0.8 Aj=0.10
0.0
0.2
0.4
x
0.6
0.8
1.0
Fig. 14. Effect of suction amplitude on sheet-cavitating flows over NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, dj ¼ 0:025, and Lj ¼ 0:3: (a) surface pressure distribution. (b) Contour of density.
1.20
1.06
1.4
2.0
1.04
1.3
1.8
1.2
1.6
1.1
1.4
1.0
1.2
0.9
1.0
1.15
1 0.95 0.98
0.90 0.85 0.80 15
CL / CLb
CL / CLb
1.00
CD / CDb
1.02
1.05
CD / CDb
1.10
0.96 30
45
60
75
90
j
0.8 15
0.8 30
45
60
75
90
j
Fig. 15. Effect of suction angle on the lift and pressure drag coefficients: (a) NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
8. Conclusion In this paper the effect of blowing and suction on hydrodynamic behavior of sheet-cavitating flows past NACA16009 and NACA66(MOD) hydrofoils is scrutinized. A numerical program is developed for computer simulation of inviscid sheet-cavitating flows. In this simulation, the Jameson's finite volume technique and the power-law preconditioning method are utilized. The single-phase barotropic cavitation pattern is employed for cavitation modeling. In order to stabilize the numerical solver, the second and fourth-order dissipation terms are employed. Explicit four-step Runge–Kutta time integration is utilized to obtain the steady-state condition. To implement the blowing and suction effects, a jet is situated on the hydrofoil's upper surface. Four types
of blowing (and suction) parameters i.e., jet amplitude, jet position, width of jet and jet angle are investigated. In this study, surface pressure distribution, length of cavitation, lift and pressure drag coefficients are examined when the mentioned parameters are changed. The numerical simulations indicate that blowing jets often reduce the length of cavity because of detaching the flow from the surface and consequently decrease the lift and pressure drag coefficients, while suction jets behave reversely. In addition, by increasing the blowing amplitude and the width of jet, the lift and pressure drag coefficients and the length of cavity decrease. In contrast, suction shows an increase of the mentioned hydrodynamic coefficients as well as the length of sheet-cavity in the case of jet magnitude or width of jet enhancement.
P. Akbarzadeh, E. Akbarzadeh / Ocean Engineering 114 (2016) 25–36
35
Fig. 16. Effect of suction angle on surface pressure distribution: (a) NACA16009 hydrofoil at AOA ¼3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7 (b) NACA66(MOD) hydrofoil at AOA ¼ 4°, s ¼ 0:84, Aj ¼ 0:05, and Lj ¼ 0:7.
1.5
2.0
1.3
1.8
dj=0.025 dj=0.05 dj=0.075 dj=0.10
1.0
1.2
0.5
Cp
1.4
CD / CDb
CL / CLb
1.6 1.1
0.0
1.0 1.2
-0.5
0.9 0.000
1.0 0.025
0.050
0.075
0.100
0.125
dj
0.0
0.2
0.4
0.6
0.8
1.0
x
Fig. 17. Effect of the suction jet width on cavitating flows over NACA16009 hydrofoil NACA16009 hydrofoil at AOA ¼ 3°, s ¼ 0:66, Aj ¼ 0:05, and Lj ¼ 0:7: (a) lift and pressure drag coefficients. (b) surface pressure distribution.
Acknowledgments The authors would like to thank the Shahrood University of Technology for supporting this study.
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