- Email: [email protected]
Experimental and numerical investigation of an unsteady supercavitating moving body
Ocean Engineering 59 (2013) 9–14
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental and numerical investigation of an unsteady supercavitating moving body B. Saranjam n Malek-Ashtar University of Technology, Shiraz, Iran
a r t i c l e i n f o
abstract
Article history: Received 25 June 2011 Accepted 1 December 2012 Available online 29 December 2012
In the last decade much research works has been carried out in order to improve the operation and speed of the underwater vehicles. In this study important supercavitation features including formation, evolution and loss of supercavitation and partial cavitation for an underwater moving body are both experimentally and numerically investigated. To achieve this goal the natural supercavitation experiment was conducted through a moving body and the trajectory of the body and the shape of supercavity profile were recorded by a high speed camera. For the numerical simulation of the underwater moving body the unsteady Reynolds averaged Navier–Stokes equations are coupled to a six-degree-of-freedom (6DOF) rigid body motion model. The numerical results are validated by comparing the predicted trajectory and velocity of the body with presented experimental data. A good agreement between the predicted and experimental values ensures the accuracy of the prepared numerical algorithm that used in the present study. The results reported in this paper illustrate that the above experimental method and numerical algorithm can be one of the most powerful and beneficial tools for the supercavitating underwater vehicles optimization and performance analysis. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Supercavitation Partial cavitation Dynamic mesh CFD
1. Introduction The maximum speed of underwater vehicles could not exceed a limit due to considerable drag force produced by the friction of body surface. Water is nearly an incompressible fluid and its properties change marginally even for a great change in pressure. However when its pressure drops below the saturated vapor pressure the cavitation phenomena occurs. When a underwater body travels at high speed, the water pressure drops below the saturated vapor pressure and a cavity is formed on the surface and behind of body. This phenomenon is called supercavitation. When supercavitation occurs on an underwater body, the skin friction drag is drastically reduced, therefore the supercavitating body can achieve higher speeds under water with the same amount of energy spent compared to that when the supercavitation does not appear. In the last decade, several researchers have numerically and experimentally investigated the supercavitating bodies. Most of these studies focused on the shape of cavitation, the velocity and pressure distributions of flow field, control and stability of the supercavitating vehicles. Savchenko et al. (1999) presented experimental results on high-speed motion of supercavitating objects in
n
Tel.: þ98 711 6231209; fax: þ98 711 6287508. E-mail address: [email protected]
0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.12.021
water and presented an empirical formula for calculating the shape of axisymmetric supercavities. Lindau et al. (2003) investigated the motion of a three-dimensional supercavitating vehicle numerically, by coupling a multiphase unsteady Reynolds averaged Navier– Stokes scheme with a six-degree-of-freedom rigid body motion model (URANS–6DOF dynamics model). In this study it has been shown that the numerical results independently support the concept that the forward motion of the vehicle, even in the absence of control surfaces, in the developed supercavity is fairly stable and immune to some magnitude of finite disturbances. Wang et al. (2005) investigated relations between cavity form and typical cavitator parameters and they presented empirical formula to calculate the length and thickness of natural supercavitation through projectiles tests. Ping et al. (2006) have numerically simulated the shape of ventilated cavity and the drag of an underwater body using commercial CFD code Fluent and they obtained empirical formula between ventilated cavity shape and ventilation rate. Zhang et al. (2007) performed series of projectile and closed loop water-tunnel experiments to study the shape characters of natural and ventilated supercavitation. The results of this research show that the shape of the natural and ventilated supercavitation are similar when the cavitation number is small and the asymmetry of supercavitation shape induced by gravity becomes clearer, when the Froude number decreases. Also it has been shown that the shedding frequency of the natural cavitation decreases with the increase of the natural cavitation number, and
10
B. Saranjam / Ocean Engineering 59 (2013) 9–14
an empirical formula has been presented forward to calculate it. Nouri and Eslamdoost (2009) studied supercavitating potential flow numerically by the boundary element method and they showed that this method is capable of predicting the asymptotic behavior of open wake and finite length cavity problems. Wang et al. (2010) investigated the gas-leakage rate of an unsteady ventilated supercavitating body. They calculated the rate of volume change of supercavity based on Logvinovich’s principle and the empirical formula and compared the simulation results with the results from experiment. The result demonstrated that the presented formula is effective. Bin et al. (2010) have numerically investigated both the natural and ventilated cavitation using a three-component model based on mass transfer equation. They observed a good agreement between the predicted cavitation characteristics including the cavity length, cavity diameter, cavity shape and experimental results. Recently Rabiee et al. (2011a,b,c) have numerically and experimentally investigated the unsteady supercavity phenomena recently but they did not use six-degreeof-freedom (6DOF) rigid body motion model. Most of the experimental investigations on supercavitation phenomenon are performed in closed loop water-tunnel and the model has been stationary and a few researchers have studied the moving supercavitating bodies experimentally. Also the numerical methods used in simulation of supercavitation are based on the potential flow or CFD multiphase models mostly and the shape of cavitation and the velocity and pressure distributions of flows at steady condition are investigated in these research works and the unsteady effects and dynamic behavior of the vehicle is neglected. In this study important supercavitation features including formation, evolution and loss of cavitation for an underwater moving object are both experimentally and numerically investigated with considering the unsteady effects and dynamic behavior of the body.
2. Experimental setup and data processing As mentioned in the Section 1, the experimental investigations on supercavitation phenomenon are mostly performed in closed loop water-tunnel and the model has been stationary and the unsteady effects and dynamic behavior of the vehicle is neglected. Compared with water-tunnel experimental test, high-speed
moving bodies can form natural supercavitation easily and the trajectory of the object can be observed clearly, so this method is proper to investigate natural supercavitation. A water tank is designed and installed in order to perform unsteady supercavitation tests. Tank dimensions are such that the effect of walls on the flow field is small, i.e. a cross section of 1.2 1.2 m and 9 m in length. The experimental test set-up including a high speed camera, a computer for image processing, a test tank and a lighting system is shown in Fig. 1. The supercavitation experiment was conducted through a sphere head body that is shot by a pneumatic gun. The object is gradually accelerated by high-pressure air and achieved their maximum velocity at the exit of the gun, then decelerated gradually in a water tank. The trajectory of the object and the cavitation profile formed around the object is recorded by a highspeed camera with a 30,000 frames per second frame rate. In order to data processing, different parameters are derived from the pictures taken by the high speed camera. These quantities consist of the body position in horizontal and vertical directions and its orientation. After processing the data, using local differences, velocity components at various time and subsequently cavitation number can also be calculated. The model motion trajectory was located in the depth of 0.4 m from the water–air free surface. Photography of the flow pictures was taken through the glass windows in walls of the water tank.
3. Computational model 3.1. Model geometry and grid generation A model, including a spherical nose and cylindrical body, is considered. The model geometry and its cruciform fins, which are solely employed for stabilization are presented in Fig. 2 and some characteristics of the model have been summarized in Table 1. The three dimensional computation domain and grid distribution near the model are shown in Figs. 3 and 4, respectively. For ensuring computation accuracy, the mesh generation has been conducted carefully. As shown in Fig. 3, the computational domain separated to two stationary and deforming regions. The model moves in the deforming region using dynamics mesh and 6DOF scheme and considering computational cost, the stationary
Fig. 1. The experimental setup configuration.
B. Saranjam / Ocean Engineering 59 (2013) 9–14
11
rate of dissipation of energy from the turbulent flow: @ @ @ m @e þ r C 1 Se re þ reuj ¼ mþ t @t @xj @xj se @xj rC 2
e2
e
pffiffiffiffiffi þ C 1e C 2e Gb þ Se k k þ ne
ð4Þ
where the turbulent viscosity is 2
mt ¼ rC m
k
ð5Þ
e
The coefficient of dynamic viscosity is Cm ¼
1 A0 þ As kU=e
ð6Þ
In comparison with the standard k e model, the realizable k e model contains a new formulation of the turbulent viscosity where the dynamic viscosity coefficient is no longer constant. Singhal’s full cavitation model has been used in numerical simulations. The vapor mass fraction, f, is governed by a transport equation
Fig. 2. The three dimensional model of the moving underwater body.
Table 1 Some characteristics of the moving body. Mass
Length
Diameter
Number of fins
Center of mass
232 g
480 mm
26 mm
4
220 mm from nose
@ rf þ r ruf ¼ Re Rc @t where
region did not change during dynamic mesh process. The total number of the grid is about 1,800,000. This fine mesh size will be able to provide good spatial resolution for the distribution of most variables around the moving object. The parallel processing has been used in order to numerical simulations.
r
3.2. Numerical model and fluid flow modeling The URANS–6DOF dynamics model (unsteady Reynolds averaged Navier–Stokes equations coupled with a six-degree-offreedom rigid body motion model) has been used as numerical method in order to simulate and predictunsteady cavitation shape, dynamics behavior and trajectory of the moving object and acted forces on the body. The Reynolds-averaged Navier–Stokes equations were solved for the supercavitating flow field in a computational domain based on the finite volume method and the pressure-based segregate algorithm, using mixture multiphase model, associated with dynamic mesh method. The continuity equation for the mixture is @rm @ rm uj þ ¼0 ð1Þ @t @xj the momentum equations for the mixture is @ui @uj @ @ @p @ rm ui þ rm ui uj ¼ þ mm þ mt þ þ rm g i @t @xj @xi @xj @xj @xi ð2Þ where uj and xj represent the velocity component and the coordinate axis, respectively, t is the time, p denotes the local pressure, the density rm and the viscosity mm of the mixture are computed in a volume–fraction–average manner, and mt indicates the turbulent viscosity. The viscous equation set is closed by using the two equation realizable k e turbulence model. Turbulent kinetic transport equation @ @ @ rk þ rkuj ¼ @t @xi @xi
mþ
mt @k þ Gk þ Gb reY M þ Sk sk @xj ð3Þ
pffiffiffi k
2 ppv 1=2 ð1f Þ 3 pl s pffiffiffi 1=2 k 2 ppv f rr Rc ¼ C c s l v 3 pl fg 1f v f g 1 f ¼ v þ þ Re ¼ C e
rv
rl rv
rg
rl
The source terms Re and Rc denote vapor generation evaporation and condensation rates, and can be functions of flow parameters (pressure, flow character or velocity) and fluid properties (liquid and vapor phase densities, saturation pressure, and liquid– vapor surface tension), where rv, rl, pv and fg are vapor density, liquid density, saturation pressure and mass fraction of noncondensable gas, respectively. As mentioned above, a segregated solution algorithm with a finite volume-based technique has been used as the numerical method. This technique consists of an integration of the governing equations of mass, momentum, and energy on the individual cells within the computational domain to construct algebraic equations for each unknown dependent variable. The pressure and velocity are coupled using the SIMPLE (semi-implicit method for pressure linked equations) algorithm which uses a guess-andcorrect procedure for the calculation of pressure on the staggered grid arrangement. The CFD simulation convergence is judged upon the residuals of all governing equations. This ‘‘scaled’’ residual is defined as P P cells p9 nb anb fnb þ bap fp 9 P Rf ¼ ð7Þ cells p9ap fp 9 where fp is a general variable at a cell p, ap is the center coefficient, anb are the influence coefficients for the neighboring cells, and b is the contribution of the constant part of the source term. The results reported in this paper are achieved when the residuals are smaller than 1.0 10 5.
4. Results and discussion In this work series of numerical and experimental investigations have been performed to study important supercavitation features. The supercavitation experiment was conducted through a sphere head body that is shot by a pneumatic gun.
12
B. Saranjam / Ocean Engineering 59 (2013) 9–14
Fig. 3. The three dimensional computational domain and boundary conditions.
Fig. 4. The grid generation near the moving underwater object at initial time.
Fig. 5 presents the experimental results for the development of the natural supercavitation. In this figure the formation, evolution and loss of supercavitation have been shown clearly. As shown in this figure, the high speed photography was performed in two test section. In the first test sections, the object enters the water and its velocity rapidly reaches to maximum value and supercavitation is formed then the body velocity decreases due to the drag force acted in the opposite direction and the dimension of supercavitation profile decreases. The velocities of the body that could be used to calculate natural cavitation number were deduced from the information recorded in photographs. When the velocity of body reaches to its maximum value (about 80 m/s) the cavitation number is about 0.03 and when the velocity is reduced to 60 m/s the interface of supercavitation profile begins to disappear and transits to partial cavitation. The reduction of supercavitation to partial cavitation has been observed in second test section. With decreasing velocity of the body, partial cavitation vanishes gradually and the body velocity reduces to its minimum value in this test section. In Figs. 6 and 7 the comparison of the experimental measurements and numerical predictions of the moving underwater body trajectory and velocity at various time has been presented.
Fig. 5. Experimental photography of formation, evolution and loss of supercavitation around the moving body.
In these figures the dimensionless displacement of the object from the initial position, the dimensionless velocity and the dimensionless time were defined as x/L , umax =u and t umax =L, where x, L, umax, u and t are referred to the displacement of the object from the initial position, the object length, the maximum velocity of the object, the local velocity of the object and the duration from the initial time (shooting the object) ,respectively. As it has been shown in these figures, a good agreement between the numerical predictions and experimental measurements is observed at supercavitation regime (t umax =L o 3) and a deviation is detected after this time at partial cavitation regime. Fig. 8 shows the variations of the natural cavitation number during the body movement both numerically and experimentally. According to this figure, the cavitation number increases gradually during the shooting process due to reduction of the object velocity. When the cavitation number is smaller than 0.1 the
B. Saranjam / Ocean Engineering 59 (2013) 9–14
13
Fig. 6. Comparison of the experimental and numerical predictions of the moving underwater body trajectory at various times.
Fig. 7. Comparison of the experimental and numerical predictions of the moving underwater body velocity at various times.
Fig. 8. Comparison of the numerical prediction and experimental measurement of the cavitation number at various times.
supercavitation appears (t umax =L o3) and the supercavitation reduces to partial cavitation when the cavitation number is greater than this value. In Figs. 9 and 10 the numerical predictions and experimental measurements of the both supercavitation profiles and partial cavitation profiles are presented respectively. As shown in these figures, the numerical predictions of length and diameter of unsteady cavitation profiles and formation, evolution and loss of supercavitation are in very good conformity with the experimental observations. This suitable agreement ensures the accuracy of the prepared numerical algorithm that used in the present study.
5. Conclusion In this paper the investigation of important unsteady supercavitation features is presented both numerically and experimentally. To achieve this goal the unsteady supercavitation experiments were conducted through a moving body that is shot into a water tank and the trajectory of the body and the cavity shape were recorded by a high speed camera and the URANS– 6DOF dynamics model is used as numerical method in order to simulate unsteady supercavitation shape and trajectory of the moving object. The experimental observations present a suitable
14
B. Saranjam / Ocean Engineering 59 (2013) 9–14
Fig. 9. Comparison of the numerical prediction and experimental measurements of the supercavitation profile at various dimensionless times.
Fig. 10. Comparison of the numerical prediction and experimental measurements of the partial cavitation profile at various dimensionless times.
and informative feature of unsteady supercavitation phenomenon and formation, evolution and loss of supercavitation were shown clearly. The trajectory and dynamic behaviors of the object are calculated successfully. For the numerical simulation of the unsteady supercavitation around a moving underwater body, the predicted supercavitation characteristics including the formation, evolution and loss of supercavitation, trajectory and velocity of the object and cavity shape agrees satisfactorily with the experimental results. Finally the results reported in this paper illustrate that this experimental method and numerical algorithm can be one of the most powerful tools for the supercavitating underwater vehicles design and optimization. References Bin, J., Lue, X.W., Peng, X.X., Zhang, Y., Wu, Y.L., Xu, H.Y., 2010. Numerical investigation of the ventilated cavitating flow around an under-water vehicle based on a three-component cavitation model. J. Hydrodyn. 22 (6), 753–759.
Lindau, J.W., Kunz, R.F., Mulherin, J.M., Dreyer, J.J., Stinebring, D.R., 2003. Fully coupled 6-DOF to URANS modeling of cavitating flows around a supercavitating vehicle. In: Proceedings of the Fifth International Symposium on Cavitation (CAV2003) November 1–4, Osaka, Japan. Nouri, N.M., Eslamdoost, A., 2009. An iterative scheme for two-dimensional supercavitating flow. Ocean. Eng. 36, 708–715. Ping, J.L., Wang, C., Wei, Y.J., 2006. Numerical simulation of artificial ventilated cavity. J. Hydrodyn. Ser. B 18 (3), 273–279. Rabiee, A., Alishahi, M.M., Emdad, H., Saranjam, B., 2011a. Experimental investigation of unsteady supercavitation flows. IJST, Trans. Mech. Eng. 35, 15–29, N0. M1. Rabiee, A., Alishahi, M.M., Emdad, H., Saranjam, B., 2011b. Numerical investigation of unsteady supercavitation flows. IJST, Trans. Mech. Eng. 35, 31–46, N0. M1. Rabiee, A., Alishahi, M.M., Emdad, H., Saranjam, B., 2011c. Experimental investigation of bounce phenomenon. Sci. Iranica, B 18 (3), 416–422. Savchenko, Y.N., Vlasenko, Y.D., Semenenko, V.N., 1999. Experimental study of high-speed cavitated flows. Int. J. Fluid. Mech. Res. 26 (3), 365–374. Wang, H.B., Zhang, J.Z., Wei, Y.J., 2005. Study on relations between cavity form and typical cavitator parameters. J. Hydrodyn. Ser. A 20 (2), 251–257. Wang, Z., Yu, K.P., Wan, X.H., 2010. Research on the gas-leakage rate of unsteady ventilated supercavity. J. Hydrodyn. 22 (5), 778–783. Zhang, W.W., Wei, Y.J., Zhang, J.Z., Wang, C., Yu, K.P., 2007. Experimental research on the shape characters of natural and ventilated supercavitation. J. Hydrodyn. Ser. B 19 (5), 564–571.
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental and numerical investigation of an unsteady supercavitating moving body B. Saranjam n Malek-Ashtar University of Technology, Shiraz, Iran
a r t i c l e i n f o
abstract
Article history: Received 25 June 2011 Accepted 1 December 2012 Available online 29 December 2012
In the last decade much research works has been carried out in order to improve the operation and speed of the underwater vehicles. In this study important supercavitation features including formation, evolution and loss of supercavitation and partial cavitation for an underwater moving body are both experimentally and numerically investigated. To achieve this goal the natural supercavitation experiment was conducted through a moving body and the trajectory of the body and the shape of supercavity profile were recorded by a high speed camera. For the numerical simulation of the underwater moving body the unsteady Reynolds averaged Navier–Stokes equations are coupled to a six-degree-of-freedom (6DOF) rigid body motion model. The numerical results are validated by comparing the predicted trajectory and velocity of the body with presented experimental data. A good agreement between the predicted and experimental values ensures the accuracy of the prepared numerical algorithm that used in the present study. The results reported in this paper illustrate that the above experimental method and numerical algorithm can be one of the most powerful and beneficial tools for the supercavitating underwater vehicles optimization and performance analysis. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Supercavitation Partial cavitation Dynamic mesh CFD
1. Introduction The maximum speed of underwater vehicles could not exceed a limit due to considerable drag force produced by the friction of body surface. Water is nearly an incompressible fluid and its properties change marginally even for a great change in pressure. However when its pressure drops below the saturated vapor pressure the cavitation phenomena occurs. When a underwater body travels at high speed, the water pressure drops below the saturated vapor pressure and a cavity is formed on the surface and behind of body. This phenomenon is called supercavitation. When supercavitation occurs on an underwater body, the skin friction drag is drastically reduced, therefore the supercavitating body can achieve higher speeds under water with the same amount of energy spent compared to that when the supercavitation does not appear. In the last decade, several researchers have numerically and experimentally investigated the supercavitating bodies. Most of these studies focused on the shape of cavitation, the velocity and pressure distributions of flow field, control and stability of the supercavitating vehicles. Savchenko et al. (1999) presented experimental results on high-speed motion of supercavitating objects in
n
Tel.: þ98 711 6231209; fax: þ98 711 6287508. E-mail address: [email protected]
0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.12.021
water and presented an empirical formula for calculating the shape of axisymmetric supercavities. Lindau et al. (2003) investigated the motion of a three-dimensional supercavitating vehicle numerically, by coupling a multiphase unsteady Reynolds averaged Navier– Stokes scheme with a six-degree-of-freedom rigid body motion model (URANS–6DOF dynamics model). In this study it has been shown that the numerical results independently support the concept that the forward motion of the vehicle, even in the absence of control surfaces, in the developed supercavity is fairly stable and immune to some magnitude of finite disturbances. Wang et al. (2005) investigated relations between cavity form and typical cavitator parameters and they presented empirical formula to calculate the length and thickness of natural supercavitation through projectiles tests. Ping et al. (2006) have numerically simulated the shape of ventilated cavity and the drag of an underwater body using commercial CFD code Fluent and they obtained empirical formula between ventilated cavity shape and ventilation rate. Zhang et al. (2007) performed series of projectile and closed loop water-tunnel experiments to study the shape characters of natural and ventilated supercavitation. The results of this research show that the shape of the natural and ventilated supercavitation are similar when the cavitation number is small and the asymmetry of supercavitation shape induced by gravity becomes clearer, when the Froude number decreases. Also it has been shown that the shedding frequency of the natural cavitation decreases with the increase of the natural cavitation number, and
10
B. Saranjam / Ocean Engineering 59 (2013) 9–14
an empirical formula has been presented forward to calculate it. Nouri and Eslamdoost (2009) studied supercavitating potential flow numerically by the boundary element method and they showed that this method is capable of predicting the asymptotic behavior of open wake and finite length cavity problems. Wang et al. (2010) investigated the gas-leakage rate of an unsteady ventilated supercavitating body. They calculated the rate of volume change of supercavity based on Logvinovich’s principle and the empirical formula and compared the simulation results with the results from experiment. The result demonstrated that the presented formula is effective. Bin et al. (2010) have numerically investigated both the natural and ventilated cavitation using a three-component model based on mass transfer equation. They observed a good agreement between the predicted cavitation characteristics including the cavity length, cavity diameter, cavity shape and experimental results. Recently Rabiee et al. (2011a,b,c) have numerically and experimentally investigated the unsteady supercavity phenomena recently but they did not use six-degreeof-freedom (6DOF) rigid body motion model. Most of the experimental investigations on supercavitation phenomenon are performed in closed loop water-tunnel and the model has been stationary and a few researchers have studied the moving supercavitating bodies experimentally. Also the numerical methods used in simulation of supercavitation are based on the potential flow or CFD multiphase models mostly and the shape of cavitation and the velocity and pressure distributions of flows at steady condition are investigated in these research works and the unsteady effects and dynamic behavior of the vehicle is neglected. In this study important supercavitation features including formation, evolution and loss of cavitation for an underwater moving object are both experimentally and numerically investigated with considering the unsteady effects and dynamic behavior of the body.
2. Experimental setup and data processing As mentioned in the Section 1, the experimental investigations on supercavitation phenomenon are mostly performed in closed loop water-tunnel and the model has been stationary and the unsteady effects and dynamic behavior of the vehicle is neglected. Compared with water-tunnel experimental test, high-speed
moving bodies can form natural supercavitation easily and the trajectory of the object can be observed clearly, so this method is proper to investigate natural supercavitation. A water tank is designed and installed in order to perform unsteady supercavitation tests. Tank dimensions are such that the effect of walls on the flow field is small, i.e. a cross section of 1.2 1.2 m and 9 m in length. The experimental test set-up including a high speed camera, a computer for image processing, a test tank and a lighting system is shown in Fig. 1. The supercavitation experiment was conducted through a sphere head body that is shot by a pneumatic gun. The object is gradually accelerated by high-pressure air and achieved their maximum velocity at the exit of the gun, then decelerated gradually in a water tank. The trajectory of the object and the cavitation profile formed around the object is recorded by a highspeed camera with a 30,000 frames per second frame rate. In order to data processing, different parameters are derived from the pictures taken by the high speed camera. These quantities consist of the body position in horizontal and vertical directions and its orientation. After processing the data, using local differences, velocity components at various time and subsequently cavitation number can also be calculated. The model motion trajectory was located in the depth of 0.4 m from the water–air free surface. Photography of the flow pictures was taken through the glass windows in walls of the water tank.
3. Computational model 3.1. Model geometry and grid generation A model, including a spherical nose and cylindrical body, is considered. The model geometry and its cruciform fins, which are solely employed for stabilization are presented in Fig. 2 and some characteristics of the model have been summarized in Table 1. The three dimensional computation domain and grid distribution near the model are shown in Figs. 3 and 4, respectively. For ensuring computation accuracy, the mesh generation has been conducted carefully. As shown in Fig. 3, the computational domain separated to two stationary and deforming regions. The model moves in the deforming region using dynamics mesh and 6DOF scheme and considering computational cost, the stationary
Fig. 1. The experimental setup configuration.
B. Saranjam / Ocean Engineering 59 (2013) 9–14
11
rate of dissipation of energy from the turbulent flow: @ @ @ m @e þ r C 1 Se re þ reuj ¼ mþ t @t @xj @xj se @xj rC 2
e2
e
pffiffiffiffiffi þ C 1e C 2e Gb þ Se k k þ ne
ð4Þ
where the turbulent viscosity is 2
mt ¼ rC m
k
ð5Þ
e
The coefficient of dynamic viscosity is Cm ¼
1 A0 þ As kU=e
ð6Þ
In comparison with the standard k e model, the realizable k e model contains a new formulation of the turbulent viscosity where the dynamic viscosity coefficient is no longer constant. Singhal’s full cavitation model has been used in numerical simulations. The vapor mass fraction, f, is governed by a transport equation
Fig. 2. The three dimensional model of the moving underwater body.
Table 1 Some characteristics of the moving body. Mass
Length
Diameter
Number of fins
Center of mass
232 g
480 mm
26 mm
4
220 mm from nose
@ rf þ r ruf ¼ Re Rc @t where
region did not change during dynamic mesh process. The total number of the grid is about 1,800,000. This fine mesh size will be able to provide good spatial resolution for the distribution of most variables around the moving object. The parallel processing has been used in order to numerical simulations.
r
3.2. Numerical model and fluid flow modeling The URANS–6DOF dynamics model (unsteady Reynolds averaged Navier–Stokes equations coupled with a six-degree-offreedom rigid body motion model) has been used as numerical method in order to simulate and predictunsteady cavitation shape, dynamics behavior and trajectory of the moving object and acted forces on the body. The Reynolds-averaged Navier–Stokes equations were solved for the supercavitating flow field in a computational domain based on the finite volume method and the pressure-based segregate algorithm, using mixture multiphase model, associated with dynamic mesh method. The continuity equation for the mixture is @rm @ rm uj þ ¼0 ð1Þ @t @xj the momentum equations for the mixture is @ui @uj @ @ @p @ rm ui þ rm ui uj ¼ þ mm þ mt þ þ rm g i @t @xj @xi @xj @xj @xi ð2Þ where uj and xj represent the velocity component and the coordinate axis, respectively, t is the time, p denotes the local pressure, the density rm and the viscosity mm of the mixture are computed in a volume–fraction–average manner, and mt indicates the turbulent viscosity. The viscous equation set is closed by using the two equation realizable k e turbulence model. Turbulent kinetic transport equation @ @ @ rk þ rkuj ¼ @t @xi @xi
mþ
mt @k þ Gk þ Gb reY M þ Sk sk @xj ð3Þ
pffiffiffi k
2 ppv 1=2 ð1f Þ 3 pl s pffiffiffi 1=2 k 2 ppv f rr Rc ¼ C c s l v 3 pl fg 1f v f g 1 f ¼ v þ þ Re ¼ C e
rv
rl rv
rg
rl
The source terms Re and Rc denote vapor generation evaporation and condensation rates, and can be functions of flow parameters (pressure, flow character or velocity) and fluid properties (liquid and vapor phase densities, saturation pressure, and liquid– vapor surface tension), where rv, rl, pv and fg are vapor density, liquid density, saturation pressure and mass fraction of noncondensable gas, respectively. As mentioned above, a segregated solution algorithm with a finite volume-based technique has been used as the numerical method. This technique consists of an integration of the governing equations of mass, momentum, and energy on the individual cells within the computational domain to construct algebraic equations for each unknown dependent variable. The pressure and velocity are coupled using the SIMPLE (semi-implicit method for pressure linked equations) algorithm which uses a guess-andcorrect procedure for the calculation of pressure on the staggered grid arrangement. The CFD simulation convergence is judged upon the residuals of all governing equations. This ‘‘scaled’’ residual is defined as P P cells p9 nb anb fnb þ bap fp 9 P Rf ¼ ð7Þ cells p9ap fp 9 where fp is a general variable at a cell p, ap is the center coefficient, anb are the influence coefficients for the neighboring cells, and b is the contribution of the constant part of the source term. The results reported in this paper are achieved when the residuals are smaller than 1.0 10 5.
4. Results and discussion In this work series of numerical and experimental investigations have been performed to study important supercavitation features. The supercavitation experiment was conducted through a sphere head body that is shot by a pneumatic gun.
12
B. Saranjam / Ocean Engineering 59 (2013) 9–14
Fig. 3. The three dimensional computational domain and boundary conditions.
Fig. 4. The grid generation near the moving underwater object at initial time.
Fig. 5 presents the experimental results for the development of the natural supercavitation. In this figure the formation, evolution and loss of supercavitation have been shown clearly. As shown in this figure, the high speed photography was performed in two test section. In the first test sections, the object enters the water and its velocity rapidly reaches to maximum value and supercavitation is formed then the body velocity decreases due to the drag force acted in the opposite direction and the dimension of supercavitation profile decreases. The velocities of the body that could be used to calculate natural cavitation number were deduced from the information recorded in photographs. When the velocity of body reaches to its maximum value (about 80 m/s) the cavitation number is about 0.03 and when the velocity is reduced to 60 m/s the interface of supercavitation profile begins to disappear and transits to partial cavitation. The reduction of supercavitation to partial cavitation has been observed in second test section. With decreasing velocity of the body, partial cavitation vanishes gradually and the body velocity reduces to its minimum value in this test section. In Figs. 6 and 7 the comparison of the experimental measurements and numerical predictions of the moving underwater body trajectory and velocity at various time has been presented.
Fig. 5. Experimental photography of formation, evolution and loss of supercavitation around the moving body.
In these figures the dimensionless displacement of the object from the initial position, the dimensionless velocity and the dimensionless time were defined as x/L , umax =u and t umax =L, where x, L, umax, u and t are referred to the displacement of the object from the initial position, the object length, the maximum velocity of the object, the local velocity of the object and the duration from the initial time (shooting the object) ,respectively. As it has been shown in these figures, a good agreement between the numerical predictions and experimental measurements is observed at supercavitation regime (t umax =L o 3) and a deviation is detected after this time at partial cavitation regime. Fig. 8 shows the variations of the natural cavitation number during the body movement both numerically and experimentally. According to this figure, the cavitation number increases gradually during the shooting process due to reduction of the object velocity. When the cavitation number is smaller than 0.1 the
B. Saranjam / Ocean Engineering 59 (2013) 9–14
13
Fig. 6. Comparison of the experimental and numerical predictions of the moving underwater body trajectory at various times.
Fig. 7. Comparison of the experimental and numerical predictions of the moving underwater body velocity at various times.
Fig. 8. Comparison of the numerical prediction and experimental measurement of the cavitation number at various times.
supercavitation appears (t umax =L o3) and the supercavitation reduces to partial cavitation when the cavitation number is greater than this value. In Figs. 9 and 10 the numerical predictions and experimental measurements of the both supercavitation profiles and partial cavitation profiles are presented respectively. As shown in these figures, the numerical predictions of length and diameter of unsteady cavitation profiles and formation, evolution and loss of supercavitation are in very good conformity with the experimental observations. This suitable agreement ensures the accuracy of the prepared numerical algorithm that used in the present study.
5. Conclusion In this paper the investigation of important unsteady supercavitation features is presented both numerically and experimentally. To achieve this goal the unsteady supercavitation experiments were conducted through a moving body that is shot into a water tank and the trajectory of the body and the cavity shape were recorded by a high speed camera and the URANS– 6DOF dynamics model is used as numerical method in order to simulate unsteady supercavitation shape and trajectory of the moving object. The experimental observations present a suitable
14
B. Saranjam / Ocean Engineering 59 (2013) 9–14
Fig. 9. Comparison of the numerical prediction and experimental measurements of the supercavitation profile at various dimensionless times.
Fig. 10. Comparison of the numerical prediction and experimental measurements of the partial cavitation profile at various dimensionless times.
and informative feature of unsteady supercavitation phenomenon and formation, evolution and loss of supercavitation were shown clearly. The trajectory and dynamic behaviors of the object are calculated successfully. For the numerical simulation of the unsteady supercavitation around a moving underwater body, the predicted supercavitation characteristics including the formation, evolution and loss of supercavitation, trajectory and velocity of the object and cavity shape agrees satisfactorily with the experimental results. Finally the results reported in this paper illustrate that this experimental method and numerical algorithm can be one of the most powerful tools for the supercavitating underwater vehicles design and optimization. References Bin, J., Lue, X.W., Peng, X.X., Zhang, Y., Wu, Y.L., Xu, H.Y., 2010. Numerical investigation of the ventilated cavitating flow around an under-water vehicle based on a three-component cavitation model. J. Hydrodyn. 22 (6), 753–759.
Lindau, J.W., Kunz, R.F., Mulherin, J.M., Dreyer, J.J., Stinebring, D.R., 2003. Fully coupled 6-DOF to URANS modeling of cavitating flows around a supercavitating vehicle. In: Proceedings of the Fifth International Symposium on Cavitation (CAV2003) November 1–4, Osaka, Japan. Nouri, N.M., Eslamdoost, A., 2009. An iterative scheme for two-dimensional supercavitating flow. Ocean. Eng. 36, 708–715. Ping, J.L., Wang, C., Wei, Y.J., 2006. Numerical simulation of artificial ventilated cavity. J. Hydrodyn. Ser. B 18 (3), 273–279. Rabiee, A., Alishahi, M.M., Emdad, H., Saranjam, B., 2011a. Experimental investigation of unsteady supercavitation flows. IJST, Trans. Mech. Eng. 35, 15–29, N0. M1. Rabiee, A., Alishahi, M.M., Emdad, H., Saranjam, B., 2011b. Numerical investigation of unsteady supercavitation flows. IJST, Trans. Mech. Eng. 35, 31–46, N0. M1. Rabiee, A., Alishahi, M.M., Emdad, H., Saranjam, B., 2011c. Experimental investigation of bounce phenomenon. Sci. Iranica, B 18 (3), 416–422. Savchenko, Y.N., Vlasenko, Y.D., Semenenko, V.N., 1999. Experimental study of high-speed cavitated flows. Int. J. Fluid. Mech. Res. 26 (3), 365–374. Wang, H.B., Zhang, J.Z., Wei, Y.J., 2005. Study on relations between cavity form and typical cavitator parameters. J. Hydrodyn. Ser. A 20 (2), 251–257. Wang, Z., Yu, K.P., Wan, X.H., 2010. Research on the gas-leakage rate of unsteady ventilated supercavity. J. Hydrodyn. 22 (5), 778–783. Zhang, W.W., Wei, Y.J., Zhang, J.Z., Wang, C., Yu, K.P., 2007. Experimental research on the shape characters of natural and ventilated supercavitation. J. Hydrodyn. Ser. B 19 (5), 564–571.