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Numerical study on the transient behavior of water-entry supercavitating flow around a cylindrical projectile influenced by turbulent drag-reducing additives
Applied Thermal Engineering 104 (2016) 450–460
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Research Paper
Numerical study on the transient behavior of water-entry supercavitating flow around a cylindrical projectile influenced by turbulent drag-reducing additives Chen-Xing Jiang a, Zhi-Jun Shuai a, Xiang-Yuan Zhang a, Wan-You Li a, Feng-Chen Li b,⇑ a b
College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
h i g h l i g h t s Water-entry supercavitation with drag-reducing (DR) additives was simulated. DR additives could positively affect the formation of water-entry supercavitation. DR additives could further reduce the drag resistance on the projectile.
a r t i c l e
i n f o
Article history: Received 11 December 2015 Revised 25 April 2016 Accepted 17 May 2016 Available online 18 May 2016 Keywords: Water-entry supercavitation Projectile Drag-reducing additives Phase transition
a b s t r a c t Simulation of the unsteady behavior of water-entry supercavitating flows influenced by turbulent dragreducing additives is very complicated. This paper attempts to introduce a numerical simulation procedure to simulate such problems in viscous incompressible two-phase and three-phase media, respectively. Firstly we performed a numerical investigation on water-entry supercavities in water and turbulent drag-reducing solution, respectively, at an impact velocity of 28.4 m/s to confirm the accuracy of the numerical method. Based on the verification, projectile entering water and turbulent drag-reducing solution at relatively high velocity of 142.7 m/s (phase transition is considered) is simulated. The Cross viscosity equation was adopted to represent the shear-thinning characteristic of aqueous solution of drag-reducing additives. The configuration and dynamic characteristics of water-entry supercavity, flow resistance and the radial velocity of cavity boundary were discussed respectively. It was obtained that the numerical simulation results are in consistence with experimental data, verifying the established numerical procedures. Numerical results show that the supercavity length in drag-reducing solution is larger than that in water; the velocity attenuates faster at high impact velocity than at low impact velocity. Turbulent drag-reducing additives have the potential in enhancement of supercavitation and further drag reduction. Ó 2016 Published by Elsevier Ltd.
1. Introduction The water-entry phenomenon, intrinsically a transient process, is an important topic in naval hydrodynamic area. And the waterentry problem has been considered as a classic problem in the field of fluid mechanics. In the process of projecting into water, the behavior of a projectile comprises a series of complicated events both above and below the free surface. Past research activities mainly focused on three topics classified by different entry stages.
⇑ Corresponding author. E-mail address: [email protected] (F.-C. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.05.102 1359-4311/Ó 2016 Published by Elsevier Ltd.
The first topic is about the impact force on projectile at the initial stage of water entry. The second topic is about the dynamics of the supercavity formed behind the projectile after it penetrates into water at some distance. The third topic is about the trajectory of the projectile when it moves more deeply into water. The present study belongs to the second research topic of water-entry problem. Scientific research of water-entry phenomenon was initiated from the beginning of the 19th century. At early stage, the main method for investigation of water-entry phenomena is experiment. Worthington and Cole [1,2] first used single-spark photography to study the air cavity formed by the vertical entry of spheres into water. The study on this subject was intensified during World War II for the design of military projectiles entering water at high
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speed. The advent of high-speed cine-photography allowed for subsequent and quantitative measurements of cavitating flow. The most extensive one was conducted by May with a view to naval ordinance applications [3], in which particularly attention was paid to the air cavity formation during high-speed projectile impact on water and cavity running behavior. In addition to experiments, there have been also theoretical and numerical studies of the air cavity dynamics associated with water entry so far. Theoretical investigations of water-entry cavity dynamics have been conducted after the pioneering work by Birkhoff and Zarantonello [4], who proposed an approximate theory of cavity evolution. They considered the cavity to be extremely long and the motion of flow field around the cavity wall to be purely radial. The model only qualitatively predicts cavity shape and the dynamics of cavity formation induced by a high-speed projectile. Lee et al. [5] extended the model of Birkhoff and Zarantonello by developing a two-dimensional (2D) model for cavity formation and collapse induced by the high-speed vertical impact of a projectile on water. This extended model assumes that the kinetic energy loss of the projectile equals to the kinetic energy converted into the horizontal fluid section. Although great progress has been made in solving the waterentry problem with methods based on potential flow assumptions, it is difficult for these methods to treat a highly distorted or a breaking free surface. In practice, the effect of gravity is considerable, when the impact velocity is high enough, so that the phase change must be considered and the water surface includes complicated jets. In such cases, analytical solutions become very difficult or even impossible. These difficulties can be overcome by computational fluid dynamics (CFD) methods, which solve the governing equations of fluid flow behaviors numerically. With the development of CFD techniques, numerical simulation has become a reliable and powerful tool for the investigation of complex, unsteady and nonlinear flow phenomena. Panahi [6] introduced a finite volume-based moving mesh algorithm and simulated water-entry of wedges and water-exit of a cylinder. The algorithm employs a fractional step method to deal with the coupling between pressure and velocity fields. Interface is also captured by solving a volume fraction transport equation. A boundary-fitted body-attached mesh of quadrilateral control volumes is implemented to record hydrodynamic time histories of loads, motions and interfacial flow changes around the structure. Kim et al. [7] used the smoothed particle hydrodynamics method to simulate the water entry of 2D asymmetric bodies. In the smoothed particle hydrodynamics method, the fluid domain is divided into a finite number of mass carrying particles. The motion of the particles and pressure distribution in the fluid are obtained by solving the momentum equations and the continuity equation based on the Lagrangian method. Walhorn et al. [8] solved the 2D wave-body interaction problems with free surfaces using the level-set method. In the level-set method, the deformation and movement of the free surface can be captured by a smooth levelset function. Yang and Qiu [9] presented numerical solutions of slamming problems for 3D bodies entering calm water with vertical and oblique velocities, respectively. They solved the highly nonlinear water entry problems by a constrained interpolation profilebased finite difference method on a fixed Cartesian grid. In most of the previous researches, spheres or horizontal cylinders were usually the research models. Few vertical cylindrical projectile models, which resemble bullets, were used. On the other hand, most of the studies investigated projectile water-entry at low speed with significant gravitational effect or at high speed but neglecting phase transition effect. With the development of naval weapons, problems encountered in the water-entry process have been becoming more and more attractive. Investigations on water-entry supercavity and the underwater motion of the
projectile can provide significant data to support fundamental research in military field, such as the navigation of torpedoes and air drop underwater weapons. On the other hand, supercavitating projectile has been used for mine disarmament [10] and proposed for use as defense against torpedoes [11] as well. In our previous studies, we proposed a novel control strategy for supercavitating flows using turbulent drag-reducing additives. Fig. 1 shows a schematic of the experimental apparatus used in our previous work [13]. The projectile was projected vertically into water and aqueous solution of surfactant CTAC (cetyltrimethyl ammonium chloride) with weight concentrations of 100, 500 and 1000 ppm, respectively, using a pneumatic nail gun. The trajectories of the projectile and the supercavity configuration were recorded by a high-speed CCD camera. By using this simple experimental apparatus, we obtained important findings that at the same initial water-entry velocity: the size of supercavity in turbulent drag-reducing surfactant solution was larger than that in water; the navigating velocity of the launched projectile at the same depth in liquid was faster in drag-reducing surfactant solution; and the maintaining downstream distance of supercavity is longer in surfactant solution as well. The typical results are plotted in Fig. 2 for examples [12,13]. For more detailed information one refers to our paper [13]. In order to have a deep insight into the
Nail of pneumatic spear Barrel Projectile
Water surface
Upper part High-speed video camera Middle part
Lower part Air compressor
Water tank
Computer
Fig. 1. Schematic diagram of experiment set up.
20
15
10
5 Water 0.60MPa
100ppm 0.60MPa
500ppm 1000ppm 0.60MPa 0.60MPa
0 cm
Fig. 2. The configurations of supercavity formed around the water-entry projectile in aqueous solution of cetyltrimethyl ammonium chloride (CTAC) at different concentrations, taken at the upper part of water tank [12].
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characteristics of water-entry supercavitating flow influenced by turbulent drag-reducing activities, the water-entry supercavitations in drag-reducing solutions were numerically simulated. We at first adopt the case of bullet-shaped projectile downwardly entering drag-reducing solution in moderate velocity vertically and compared with the previous experimental data to confirm the accuracy of the employed numerical simulation procedure. And then we make further efforts in numerically investigating the water-entry supercavitating flow around a circular cylinder influenced by turbulent drag-reducing additives at high impact velocity. Since the phase transition is considered herein, it is a three-phase cavitating flow. 2. Numerical simulation procedures The forming process of water-entry supercavitation in the turbulent drag-reducing solution is realized through a commercially available CFD code ANSYS FLUENT. One has to deal with four distinct problems in CFD simulation of water-entry cavitating flow influenced by turbulent drag-reducing additives. The first problem to solve is how to represent the rheological properties of the dragreducing solution. The Cross viscosity model [14] is utilized herein to characterize the aqueous solution of drag-reducing additives with shear-thinning effect in the numerical procedure. This simple viscosity model has been used in numerical simulations of turbulent drag-reducing flows of viscoelastic fluid, such as by Li et al. [15] and Ro and Ryou [16], and also of supercavitating flows in drag-reducing solutions by ourselves [13,17]. The second problem is to solve the Navier–Stokes and continuity equations for the flow domain after mesh generation and parameters’ initialization. At this step, velocity and pressure fields are output to calculate loads acting on the projectile. In the third problem, the momentum equation for the projectile is solved in order to calculate the body motion and consequently to find out its new location. After this step, the procedure is continued to move the body of projectile as well as the computational mesh of the flow domain. This inner loop provides a strongly coupled solution in the whole computational domain between rigid body motions and flow field. At this stage, a strategy is required to introduce such movements into the computational domain and to update all the information for the next time step. Here, spring-based smoothing and local remeshing dynamic mesh models are used. For the fourth problem, the volume fraction transport equation is solved to represent the interface between liquid and gas phases in the position-updated mesh. Such an algorithm as mentioned above is continued to capture a desired time history of the water-entry cavitating flow. 2.1. Governing equations The volume of fluid (VOF) method is utilized in this study to capture the interface between fluid phases. This method treats the flow as a mixture of two phases (e.g. water and air) [18]. A volume fraction function is introduced with values between 0 and 1 to capture the free surface. The fraction function is reconstructed in each cell. The governing equations are the following continuity and momentum equations for the mixture, respectively,
@a @ ð1Þ þ ðaui Þ ¼ 0 @t @xi @ @ ðq Þ þ ðq ui Þ ¼ 0 ð2Þ @t m @xi m @ @ @P @ @ui @uj þ qm g þ F i ð3Þ l þ ðq uj Þ þ ðqm ui uj Þ ¼ þ @t m @xi @xj @xi m @xj @xi where a is the volume fraction of the air, u is velocity vector, P is pressure, qm ¼ ð1 aÞql þ aqg and lm ¼ ð1 aÞll þ alg are the
density and dynamic viscosity of the simulated mixture, respectively, Fi is body force.The standard k—e model with standard wall functions is adopted to provide turbulence closure: @ @ @ l @k ðqm kui Þ ¼ lm þ t ð4Þ þ qm G qm e ðqm kÞ þ rk @xi @t @xi @xi @ @ @ l @e q Ge q e2 ðq eui Þ ¼ lm þ t þ C e1 m C e2 m ð5Þ ðq eÞ þ re @xi k k @t m @xi m @xi
lt ¼ qm C l k2 =e
ð6Þ
where k and e are the turbulent kinetic energy and dissipation rate, respectively. The model parameters are chosen as follows: C l ¼ 0:09, C e1 ¼ 1:44, C e2 ¼ 1:92, rk ¼ 1:0, re ¼ 1:3. The values of these constants are set by referring to the literature [19]. These values have been determined based on experiments for fundamental turbulent flows including frequently encountered shear flows like boundary layers, mixing layers and jets as well as for decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows. Besides, by choosing these values, the numerical simulation results are shown to be in consistence with experimental data. 2.2. Cross viscosity model The turbulent drag-reducing solution is characterized through modeling the dynamic viscosity by the Cross viscosity equation, which shows the shear-thinning behavior,
l ¼ l0 =½1 þ ðkc_ Þm 2 ls M½l2 c k¼
5
R0 T
ð7Þ ð8Þ
where l0 is zero-shear viscosity; k is relaxation time; c_ is shear rate; m is Cross power index and m ¼ 0:3 is utilized here; M is the mean mole mass of the solute in g/mol; ls is viscosity of solvent; c is the solution concentration in g=cm3 ; R0 ¼ 8:31 J=mol K; T is the absolute temperature in K; ½l is intrinsic viscosity and
½l ¼ 1:03 102 M0:78 in ml=g. Together with the Cross equation, a decreased surface tension coefficient is used to characterize the drag-reducing solution. rs ¼ 0:036 N/m (half of that of water) is chosen for the solution. Note that, in our previous measurement for surface tension of turbulent drag-reducing CTAC surfactant solutions, we obtained that the surface tension coefficient of CTAC solution becomes less than half of that of water (0.033 N/m for solution compared with 0.071 N/m for water at room temperature) from a very dilute concentration (40 ppm) and changes only slightly with the increase of concentration: from 0.0326 N/m at 40 ppm concentration to 0.0322 N/m at 200 ppm concentration [20]. 2.3. Cavitation model When the initial velocity of the projectile is high enough, the local pressure near the projectile will drop below the saturation pressure, and consequently natural cavitation occurs. The formation of vapor leads to a significant interaction between different phases. In these cases, it is unreasonable to neglect the vapor phase. Thus, such flow case contains three phases. We have to track the interfaces between different phases and model the transition of liquid to vapor. The cavitation model is therefore employed to capture the phase transition from water to vapor. The Schnerr–Sauer cavitation model [21] is adopted here. The governing equation for vapor is as follows: ! @ ðav qv Þ þ r ðav qv V v Þ ¼ Re Rc @t
ð9Þ
where Re and Rc are the vapor evaporation and vapor condensation rates, respectively.
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When P v P P,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q q 3 2 ðPv PÞ Re ¼ v l av ð1 av Þ RB 3 qm ql When Pv 6 P,
q q 3 Rc ¼ v l av ð1 av Þ RB qm
ð10Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðP Pv Þ 3 ql
ð11Þ
Here, av is the vapor volume fraction, RB is the diameter of the 13 bubble, RB ¼ 1avav 43p 1n , n ¼ 1 1013 , P v is the saturation vapor pressure. 2.4. Numerical method and conditions Since the simulated water-entry projectile (a cylindrical bullet) is axially symmetric, the simulation model was then simplified to be 2D, as shown in Fig. 3. The simulated cylindrical projectile has exactly the same geometry and dimensions as used in the Refs. [12,22]. Fig. 3(a) and (b) are the mesh systems for two simulation cases, at lower initial velocity and higher initial velocity, respectively. Water-entry supercavitation in water and turbulent dragreducing solution, respectively, were numerically simulated based on unsteady Reynolds averaged Navier–Stokes scheme. The user
(a)
mm
Pressure-outlet Bullet
30
800
Penetration axis
Wall
150 7
(b) Wall
Projectile
Projectile motion
Two typical cases, namely one at a lower impact velocity without consideration of phase transition from water to vapor and the other at a higher impact velocity with consideration of phase transition, have been chosen for numerical simulations. Furthermore, for these two chosen cases, there were available experimental data to compare. Numerical simulations were at first performed for the water and drag-reducing surfactant solution (l0 ¼ 0:01 Pa s, k ¼ 1 s, m ¼ 0:3 and rs ¼ 0:036 N/m) flow cases at the initial velocity of V 0 ¼ 28:4 m/s, respectively. For these cases, only liquid phase and air phase are considered. On the base of results from these simulations, the higher initial velocity cases (impact velocity V 0 ¼ 142:7 m/s into water and drag-reducing solution, respectively) were then performed, which are three-phase flows with considerations of phase transition. Detailed discussions about the results for these two cases are presented as follows, respectively.
mm
The analytical calculations of supercavity profile for comparison in this part are the same as that used in Ref. [13]. Fig. 4 shows numerical simulation results for the radii of supercavity as compared with analytical calculations. In Fig. 4, R is the radius at the penetration depth z, and z0 represents the initial depth of the projectile. The positive value and negative value mean that the radii
Penetration axis
10
Pressure-outlet
100
Wall
3. Results and discussion
3.1. Cases at lower impact velocity of V 0 ¼ 28:4 m/s
Wall Projectile motion z
5
R (mm)
500
12.65
z
defined functions (UDF) embedded in the FLUENT platform were used to control the movement of the projectile. A segregated solver with SIMPLE scheme as the velocity–pressure coupling algorithm was selected. The pressure field was discretized using PRESTO scheme. The second order upwind scheme was used for the discretization of the velocity field. Geo-reconstruct method was used for the volume fraction. Inherently, the values of all the parameters were stored and calculated with double precision. The numerical simulations were for transient flows. The time step was set to be 0.00001 s, which gives 20 time steps per cycle. In all the simulations presented below, the working fluid was at 298 K, with liquid, air and vapor densities of 998.2, 1.225 and 0.5542 kg=m3 , respectively. An exit pressure was specified at the top (outlet) boundary (for Fig. 3(a)) and at the right boundary (for Fig. 3(b)). The both side-boundaries were set to be solid wall (no-slip for velocity and no penetration). A structured grid system was employed to discretize the region around the projectile with boundary layer clustering around the cylindrical body. The unstructured mesh was employed for the other region considering the dynamic mesh technology.
analytical calculation numerical simulation water numerical simulation solution
0
-5
-10 0
25.4
Fig. 3. Physical model, computational region and grid system.
5
10
15
20
25
(z-z0) (mm) Fig. 4. Comparison of numerical simulation results for the radii of supercavity at different axial locations of the projectile with the analytical model.
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E
N
1.56ms
E
N
3.12ms
E
N
4.68ms
E
N
6.24ms
E
N
7.80ms
Fig. 5. Comparisons of cavity configurations at water-entry between numerical simulations and experiments for the 1000 ppm CTAC solution case [12] (‘‘E” and ‘‘N” represent the experiments and numerical simulations, respectively).
are symmetrical about the axial plane. Excellent agreements between the present numerical simulations and analytical calculation have been obtained, verifying the currently conducted numerical simulation procedures. Furthermore, the shapes of supercavity in water and solution are almost the same, which means that, at very early stage of water-entry cavity, the drag-reducing additives have negligible influence on the cavity shape. Fig. 5 shows a comparison of supercavity forming process of water entry between numerical simulation and experiments for the 1000 ppm CTAC solution case [12]. A solid cylindrical projectile with density qs and length L vertically impacting a horizontal liquid surface with velocity V 0 is considered. The impact of a cylindrical projectile that creates a subsurface air cavity has several distinctive features. The first significant phenomenon is that, shortly after the impact of projectile on the liquid surface, an upward moving jetting and a laterally expanding splash are formed above the surface, and the projectile is suddenly surrounded by the splash (t = 1.56 ms). The second significant phenomenon is that, as the projectile penetrates into liquid, a cavity under the surface is formed. During this period, the air from atmosphere comes into the cavity continually. Initially, the air entrainment is due to gas flow induced by the projectile. Then the air is entrained into the cavity due to negative relative pressure of the cavity. The reason for the negative pressure is that in water entry, the volume of the cavity increases rapidly and the cavity cannot keep an equilibrium thermodynamic state. With time evolution, the airflow into the cavity is ended by the surface closure and the enclosed cavity is pulled away from the surface by the downward moving projectile. An axisymmetric cavity is evident below the surface with a splash curtain above the cavity. The evolution of splash curtain has been described by Aristoff and Bush [23]. At t = 3.12 ms for the ‘‘open-cavity phase”: the flow is separated from the projectile nose with generation of the cavity; air rushes in from above the liquid surface to fill the cavity; during this phase the projectile is cavity-running, i.e., a cavity is being generated around the projectile as it travels, but the cavity is still open to the atmosphere at its upper end (at the liquid surface). At t = 4.68 ms, it shows that the splash is forming a dome and closing over the entry point of the projectile to seal the cavity from the air above. As shown at t = 6.24 ms, ‘‘surface closure” occurs and the cavity is then pushed down from the liquid surface. From this stage, the cavity has closed from the liquid surface and envelops the projectile, forming the supercavity (t = 7.80 ms). In the process of water-entry, the cavity shape in solution is consistent well with the experimental observations. It again demonstrates that the presently conducted numerical simulation procedure is suitable for simulating the supercavitating projectile in drag-reducing solutions.
Fig. 6 compares cavity configurations between numerical simulation and experiments for water and 1000 ppm CTAC solution at t = 7.8 ms [12], respectively. It can be seen that the cavity shapes in water and drag-reducing solution are different. Inside the cavity, the volume of air component in solution is larger than that in water. In solution, the most part of the cavity is filled with air, and just in the cavity tail, the cavity is cloudy. But in water, the air distributes in an annular region at the rear part of the cavity, and the central region of the rear part of the cavity is water, which is in good consistence with the phenomena observed in the experiment. The reason may be due to the fact that when projectile penetrates into water, the re-entrained jet is more powerful in water than in solution. The smaller jet toward the air at the liquid surface in solution is another evidence. The non-dimensional length L ¼ Lc =Dn and diameter D ¼ Dc =Dn of the cavities (where Lc is the cavity length; Dc is the biggest cavity section diameter; Dn is the diameter of the cylindrical projectile) in water and drag-reducing CTAC solutions vs. time are compared in Figs. 7 and 8, respectively. As shown in Fig. 7, the non-dimensional length of cavity in solution keeps larger than that in water all the time; the length difference between water and solution cases becomes larger with time. The non-dimensional diameter varies with time complicatedly as shown in Fig. 8. In the first period (about 3 ms) the diameters of cavity increase sharply in both water and solution; then the diameters fluctuate for about 2 ms; after that, the cavity diameter in water decreases with time, and the
Solution E
N
Water E
N
Fig. 6. Comparisons of cavity configurations between numerical simulations and experiments for water and 1000 ppm CTAC solution case at t = 7.8 ms [12] (‘‘E” and ‘‘N” represent the experiments and numerical simulations, respectively).
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16
the projectile. The effect of g is negligibly small for high-speed water-entry case. Eq. (14) is then simplified,
Water Solution
m 12
dV p 1 ¼ qw A0 C d V 2p 2 dt
L
With definition of the velocity attenuation coefficient k ¼ qw A0 C d0 =2m, the following equations can be obtained by integrating Eq. (15):
8
4 0
2
4
6
8
10
t (ms)
Water Solution
4.0
3.6
3.2
2
4
6
8
10
30
t (ms) Fig. 8. Non-dimensional diameter vs. time for the cases at V 0 ¼ 28:4 m/s.
Dp q gh þ 0:5C a qa V 20 C a qa V 20 V2 ¼ w ¼ r0 02 2 2 2 qw V P 0:5qw V p 0:5qw V p VP
10 0
dV p X 1 m 2 ¼m ¼ F Z ¼ mg qw A0 C d V 2p 2 dt dt 2
ð14Þ
where m is the projectile mass, z is the penetration direction, t is the time, g is the gravitational acceleration, A0 is the projected area of
2
3
4
5
6
7
Fig. 9. Comparison between analytical calculation and numerical simulation for velocity attenuation of the projectile with supercavitation.
Experimental data of water Experimental data of solution (1000 ppm) Analytical calculation Numerical simulation of water Numerical simulation of solution
240
160
z (mm)
ð13Þ
1
t (ms)
ð12Þ
where C d0 is the drag coefficient at r ¼ 0 (C d0 0:82 for the disk cavitator). For a navigating projectile normally penetrated into water, the deceleration reads,
d z
20
15
where qw is water density, h is water depth (which is negligible here), qa is air density, C a is airflow pressure drop coefficient ðC a ¼ 5—15Þ, V 0 is the initial velocity of the projectile. At sufficiently small r, the drag coefficient of a blunted cavitator can be approximated as [13]
C d ¼ C d0 ð1 þ rÞ
ð17Þ
25
V (m/s)
cavity diameter in solution keeps stable for about 3 ms and then decreases. Before about 6.5 ms, the cavity diameter in water is larger than that in solution, and after 6.5 ms, the cavity diameter in solution becomes larger. From the above observation, we can conclude that the cavity in solution is more stable than that in water. It has well been documented that, at its initial stage, the waterentry supercavity opens to the atmosphere [13]. During the waterentry process, the pressure difference Dp between inside and outside the supercavity can be assumed to be constant; thus, the cavitation number only depends on the projectile velocity Vp, as shown in the following equation [13]:
r¼
ð16Þ
analytical calculation numerical simulation of water numerical simulation of solution
2.8 0
pffiffiffiffiffiffi
1
The numerical simulation results for the variation of projectile velocity as compared with the analytical predictions are shown in Fig. 9. It can be seen that, for water-entry supercavitation at initial velocity of V 0 ¼ 28:4 m/s, the velocity derivative (and thus the drag) obtained from the numerical simulation is comparable to that predicted using the analytical model, with 25% maximum discrepancy. Note that, in the impact process, the projectile will lose some energy as compared with the analytical formulation. This energy loss contributes to the under-estimation of velocity in numerical simulation as compared with that in theoretical calculation. From Fig. 9, we can also see that the velocity in solution decays slower than in water, which means that, at the same instant, the projectile in solution flies faster than in water. Fig. 10 shows numerical simulation results for the penetration distance (z) as compared with the analytically predicted value
Fig. 7. Non-dimensional length vs. time for the cases at V 0 ¼ 28:4 m/s.
4.4
pffiffiffiffiffiffi
r0 V 0 tan arctan pffiffiffiffiffiffi k r0 V 0 t r0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 1 þ r0 1 1 pffiffiffiffiffiffi cos arctan pffiffiffiffiffiffi k r0 V 0 t z ¼ ln k r0 r0
Vp ¼
D
ð15Þ
80
0 0
2
4
6
8
10
t (ms) Fig. 10. Comparison of experiments, analytical calculation and numerical simulation for penetration distance.
C.-X. Jiang et al. / Applied Thermal Engineering 104 (2016) 450–460
and the experimental data. It can be seen that the projectile in solution travels to a longer distance than in water at the same instant. Actually, the projectile cannot move as far as the analytical calculation due to the loss of projectile kinetic energy during the water-entry process. From the results mentioned above, it is clear that the numerical simulation approach developed in the present study allows one to investigate the characteristics of water-entry induced supercavitation influenced by turbulent drag-reducing additives. The drag force of projectile during water entry is predicted based on the method suggested by von Karman [24]. This method assumes that momentum is conserved during the water-entry process. The momentum loss of the body when it slows down is converted into the momentum of the ‘‘added mass” of water. When a body moves through still liquid, a liquid flow is induced in the vicinity of the body. The ‘‘added mass” is the mass that should have the same kinetic energy as that of the flow. The problem can then be treated approximately in the following manner (as schematically shown in Fig. 11), and the force acted between the body and the water can be calculated. Let m = the mass of the body, z = the vertical distance through which the body travels within time t, r = the radius of the cylindrical body, dm = the added mass. The fictitious added water mass dm, which changes as the body penetrates the water, is thought of as having the speed of body at each instant. For a given body at a given entry angle it is assumed that the added mass is a function of only the body penetration distance, and further that dm is proportional to q and to the cube of the linear scale of the body; but is not a function of the mass of the body or its speed [3]. We proceed to calculate the force by application of the momentum theorem.
mV 0 ¼ ðm þ dmÞV
ð18Þ
dm ¼ K qpr 2 f ðzÞ
ð19Þ
mV 0 ¼ ðm þ K qpr f ðzÞÞV 2
2 1 1 m DE ¼ dmV 2 ¼ ðK qpr 2 f ðzÞÞ V 0 2 2 m þ K qpr 2 f ðzÞ
ð20Þ ð21Þ
So, if we can know the definition of K and f ðzÞ, the loss of the projectile energy during the water entry can be calculated. Fig. 12 plots drag coefficients versus time for both water and drag-reducing solution cases. The drag coefficient C d is defined as below:
Cd ¼
F
ð22Þ
0:5qV 2p A0
where F is the overall drag force.
Projectile
z
Surface
r
Fig. 11. Schematic diagram of the water-entry problem.
1.8 Water Solution 1.2
Cd
456
0.6
0.0 0
1
2
3
4
5
6
7
t (ms) Fig. 12. Time history of drag coefficients for projectile in water and in dragreducing solution cases at V 0 ¼ 28:4 m/s, respectively, obtained by numerical simulation.
It can be seen that the drag coefficients rise sharply when the projectile impacts the liquid surface, and then decrease, which can quantitatively explain why the projectile loses much kinetic energy and its velocity decreases much when impacting the liquid surface. The drag coefficient is obviously smaller in the dragreducing solution than that in water. It can be noticed that, the largest drag coefficient point is larger in solution than in water at the beginning of water entry, which means that when the projectile impacts the liquid surface, the projectile loses more kinetic energy in solution. This is because the viscosity of the CTAC aqueous solutions is larger than that of water (the viscosity of the CTAC solution increases with the increase of the concentration). The larger viscosity has retardation on the water-entry projectile making the energy loss more in solution. But after that point, the projectile resists smaller drag in solution, which indicates the turbulent dragreducing effect. After the surface impact, the Reynolds number of liquid flow relative to the projectile has reached the valid range of the turbulent drag-reducing effect, which makes the turbulent frictional resistance smaller in CTAC solution. From these results, it can be conjectured that adding small amount of drag-reducing additives, the water-entry projectile can move faster in the dragreducing solution than in water provided the initial driving power is the same. It is expected that the actual drag coefficient would be greater than that predicted for a disk cavitator because there is drag force on the projectile body. The occurrence of tail-slap is probably the main contributor. The working fluid used our previous experimental study [13] was a viscoelastic aqueous solution of surfactant with turbulent drag-reducing effect. The viscoelastic effect is the most remarkable feature for turbulent drag-reducing solutions. Whether or not the surfactant solution displays drag-reducing effect depends on whether or not the shear-induced rod-like and then network microstructures can be formed in the solution. Such microstructures are essentially micelles composed of small surfactant molecules. The ability to form micelle structures is the typical feature that the surfactant solution displays after the solution concentration exceeds the critical micelle concentration (CMC). Above the CMC, the surfactant monomer molecules and the formed micelle structures are always in a thermodynamic equilibrium state [25]. Mehrabadi et al. simulated drag reduction phenomenon using polymeric additives in turbulent pipe flows [26]. Fig. 13 schematically shows a simplified phase diagram for surfactant solution [27]. Fig. 14 shows the comparison of the viscosity of CTAC aqueous solutions between experimental data and Cross viscosity model estimations. In addition to the variation of the rheological properties, from constant viscosity to shear-shinning characteristic, we
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Concentratoin
Rod-like micelles
CMC II
Micelles CMC I
Monomers Temperature Fig. 13. The change of micelle structures of surfactant solution when the concentration, and temperature of fluid increased.
Experimental data for 200 ppm Experimental data for 500 ppm Experimental data for 1000 ppm Cross model result for 200 ppm Cross model result for 500 ppm Cross model result for 1000 ppm
Shear Viscosity (Pa•s)
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20
40
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Shear Rate (s-1) Fig. 14. Comparison of the viscosity of CTAC aqueous solutions between experimental data and Cross viscosity model.
have experimentally obtained that the surface tension coefficient of aqueous solution of CTAC/sodium salicylate (CTAC/NaSal) is about half of that of water [20]. Thereby, in the numerical simulations, together with the Cross equation, a decreased surface tension coefficient (rs ¼ 0:036 N/m, half of that of water) was employed to characterize the drag-reducing solution. Fig. 15 illustrates the instantaneous radial velocity ðV r Þ of water-entry cavity boundary at various depths (z) in water and solution cases, respectively. At t = 0.5 ms, the radial velocity of the cavity in solution is larger than that in water from the depth deeper than 1 mm, and only near the liquid surface the radial velocity of the cavity is larger in water. With increase of the projectile penetration, the radial velocity of the cavity decreases both in water and in solution, but still it is larger in water near the liquid surface (seeing t = 1.0 ms and t = 1.5 ms). At t = 2 ms, the radial
Water: t = 0.5 ms Water: t = 1.0 ms Water: t = 1.5 ms Water: t = 2.0 ms
8
3.2. Higher impact velocity of V 0 ¼ 142:7 m/s In this section we shall discuss the unsteady behavior of waterentry supercavitating flows in water and aqueous solution of turbulent drag-reducing additives at a higher impact velocity of V 0 ¼ 142:7 m/s. The numerical simulation results are compared with experimental data [22] and analytical calculations. Fig. 16 shows the attenuation of projectile velocity in water and in solution as compared to the analytical predictions and experimental data [22]. It can be seen that, for water-entry supercavitation at initial velocity V 0 ¼ 142:7 m/s, the velocity attenuation obtained from the numerical simulation is comparable to that predicted using the analytical model and experimental data, demonstrating the accuracy of the established numerical simulation method for high velocity water-entry with phase transition. Furthermore, we can see that the projectile velocity in solution decays slower than in water, which is the same as for the lower impact velocity case. Again, it is proved that the projectile moves faster in drag-reducing solution than in water, no matter the impact velocity is high or low. Compared with Fig. 9, it can be seen that the velocity attenuates faster at higher water-entry velocity. At t = 4 ms, the velocity difference ðDVÞ between in water and in solution normalized with the impact velocity DV/ V0_low = 2.7/28.4 = 9.51% and DV/V0_high = 3.3/142.7 = 2.31%. So it can be concluded that the influence of drag-reducing solution is more obvious at lower impact velocity. Fig. 17 shows the penetration distance (z) versus time. It is seen that the projectile in solution moves to a longer distance than in water at the same instant, which is the same as that happened to the case of lower impact velocity of V0 = 28.4 m/s.
Solution: t = 0.5 ms Solution: t = 1.0 ms Solution: t = 1.5 ms Solution: t = 2.0 ms
4
Analytical calculation Numerical simulation of water Numerical simulation of Solution Experimental data of water [22]
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Vr (m/s)
6
velocity of the cavity in solution becomes larger than that in water. Fig. 15 shows that the radial velocity of the cavity boundary in solution keeps larger in the cavity forepart. When the projectile impacts the liquid surface, the higher viscosity of the solution prevents the cavity from opening to the air, resulting in a lower radial velocity of the cavity near the surface. When the projectile penetrates into the liquid, the smaller interface tension between solution and the air makes the cavity boundary expands faster under the liquid surface, which leads to the larger size of the cavity in solution. As illustrated above, we have numerically realized that the turbulent drag-reducing additives do have evident and positive effects on the formation of water-entry supercavitation, resulting in further decrease of flow resistance of the projectile, as obtained in the experiment.
100
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z (mm) Fig. 15. Instantaneous radial velocity of water-entry cavity boundary at various depths in water and solution.
0
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t (ms) Fig. 16. Numerical simulation results for velocity attenuation of the projectile with supercavitation as compared with analytical calculation and experimental data [22].
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Analytical calculation Experimental data of water [22] Water Solution
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z (mm)
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solution
t = 1 ms
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t = 2 ms
100
t = 3 ms 0 0
1
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Fig. 18 compares cavity configurations between numerical simulation and experimental results for water case at t = 1.25 ms. It can be seen that the cavity shape are essentially the same. More detailed information can be seen from the numerical simulation result. At the aft part, the cavity boundary is saw-toothed, which is the result of the phase transition from water to vapor. More explanations about this phenomenon will be given later. Besides, at the center of the cavity, there exist small droplets coming from the reentrant jet. At this moment, the free surface is closed and a liquid film forms to cut off the air from entering into the cavity. At the free surface, there are two jets formed: one comes into the cavity due to the pressure difference inside and outside the cavity, and the other splashes upward into the air. Fig. 19 shows a comparison of the supercavity forming process of the projectile entering water and drag-reducing solution respectively. In consideration of the phase transition from water to vapor, we use the contour of liquid phase to represent the cavity shape. In other words, the fluid region where there is no water represents the cavity. At t = 1 ms, the free surface is already closed in water due to surface closure, and a jet is formed toward the air. For the drag-reducing solution case, the cavity is still open to the air and the generated jet is toward the cavity. At t = 2 ms, the reentrant jet is formed in water, and still the intensity of the jet is weaker than that in solution and the aft part of the cavity is cupped in solution. On the other hand, from t = 1 ms to t = 4 ms the size of the cavity is continually increasing. This is an interesting phenomenon. As abovementioned, at t = 1 ms, the surface closure is already formed in water. So there must be some other component coming
N
E
Fig. 18. Comparisons of cavity configurations between numerical simulation and experiments for water at t = 2 ms [21] (‘‘E” and ‘‘N” represent the experiments and numerical simulations, respectively).
t = 4 ms Fig. 19. Comparisons of cavity configurations of supercavity between that in water and in drag-reducing solution.
into the cavity to enlarge the size of cavity, which is the vapor generated from phase change as is discussed below. Paying more attention to the cavity evolution process from t = 2 ms to t = 3 ms in solution, we can find that the surface closure is different. The actual process is that the closure is disturbed after t = 2 ms, and more air rushes into the cavity and then the cavity is closed again, which results in a larger sized cavity. This cavity closure process shows that the cavity formed in turbulent drag-reducing solution is not closed at one time, but first is closed and then is disturbed and then is closed again. When the impact velocity is high enough, the phenomenon of natural cavitation must be considered. Note that, natural cavitation can be defined as the breakdown of a liquid medium under very low pressure. When the speed of an underwater body reaches a certain value, water pressure near the cavitator nose will become lower than the saturated vapor pressure corresponding to the local temperature, and cavitation happens [28]. Cavitation number,
r ¼ 2ðp1 pv Þ=qV 21 , is the key parameter characterizing cavitating flow [29]. In this paper, p1 and pv are the ambient liquid pressure and the saturation pressure, respectively; V 1 is the velocity of submerged projectile; q is the liquid density. Theoretically, when the local pressure is lower than the saturation pressure (2350 Pa) corresponding to 20 °C water temperature, natural cavitation happens. Fig. 20 shows the gas-phase components inside the cavity in water and in drag-reducing solution, respectively. It can be seen that the cavity is composed of vapor and air. When the projectile impacts on the free surface, the projectile will transfer energy to the water, and the flow begins to be separated from the projectile nose. The air first rushes into the cavity, so most part of the cavity is immediately filled with air. When the projectile enters into the water, at every instant, the kinetic energy loss of the projectile is equal to the kinetic and potential energy transferred to the liquid, resulting in liquid-to-vapor phase change, and so the cavity will expand. So far, in most of the published literatures about natural supercavitation, the projectiles are launched underwater or fixed in a water tunnel. In these cases only liquid phase and vapor phase are involved. But there are few published literatures about waterentry induced three-phase cavitation, which is more complicated. If the compressibility is ignored, the air and the projectile can be seen as a whole ‘‘imaginary solid”. And then the problem becomes a natural cavitation phenomenon. With the movement of the ‘‘imaginary solid”, the pressure near the nose decreases. So the vapor will be first generated near the nose of the ‘‘imaginary solid”. From Fig. 20 (a), it can be seen that the vapor is mainly distributed near the cavity boundary. Stinebring et al. [30] has reported that the rear of these cavities is a complex and often highly turbulent
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(a)
water
solution
t = 1 ms
The non-dimensional length L and diameter D of the cavities in water and drag-reducing solutions vs. time are compared in Figs. 21 and 22, respectively. Fig. 21 shows that the non-dimensional length of cavity in solution keeps larger than that in water all the time; the length difference between water and solution becomes larger with the increase of time, which is the same as obtained for the case of lower impact velocity V0 = 28.4 m/s. Fig. 22 shows that, in the first period (before about 1 ms), the diameters of cavity increase sharply
t = 2 ms 18
Water Solution
t = 3 ms
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L
t = 4 ms
9
(b)
water
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t = 1 ms
3 0
1
2
3
4
5
t (ms)
t = 2 ms
Fig. 21. Non-dimensional length vs. time for the cases at V 0 ¼ 142:7 m/s.
5.4
Water Solution
t = 3 ms
t = 4 ms
D
4.8
Fig. 20. The gas-phase components inside the cavity in water and in drag-reducing solution. (a) Vapor component, (b) air component.
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3.0 0
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t (ms) Fig. 22. Non-dimensional diameter vs. time for the cases at V 0 ¼ 142:7 m/s.
Water Solution
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region of the flow where the contents of the cavity are carried or entrained away. However, in the case of vaporous cavitation where the cavity contains mostly vapor, the loss of vapor is immediately made up by the evaporation from the walls of the cavity. After the formation of the reentrant jet into the cavity, the low pressure transforms the liquid jet into vapor; this is where the vapor behind the projectile comes from. Comparing the vapor region in water and in solution, it can be seen that the vapor region in solution is larger than that in water, which indicates that the drag-reducing solution enhances the formation of vapor (enhances phase change). Besides, there have been some evidences showing that, during the process of growth or collapse of a gas bubble in liquid, or the initiation stage of supercavitation, the surface tension may play a significant role. He [31] reported that the growth of a laser-induced cavitation bubble was suppressed by the surface tension and the increase of surface tension enhanced the collapse of the bubble. Shi et al. [32] stated that the surface tension keeps the process of cavity growth stable and keeps the cavity orbicular, and accelerates the process of bubble collapse. The abovementioned examples imply that the cavitation process can be enhanced with a decrease of surface tension: the bubble being larger in size and longer in life-time. Consequently, it can be conjectured naturally that the cavitating process might be effectively controlled by changing the local surface tension. In order to demonstrate such idea, natural supercavitation influenced by rheological characteristics of turbulent drag-reducing additives was discussed in detail in our previous paper very recently [17].
4.2
1.0
0.5
0.0 0
1
2
3
4
5
t (ms) Fig. 23. Time history of drag coefficients for projectile in water and in dragreducing solution cases at V 0 ¼ 142:7 m/s, respectively, obtained by numerical simulation.
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in both water and solution and the diameter in water is smaller than that in solution; after this period the diameter in water becomes larger than that in solution; then the diameters increase slowly from about 2.5 ms. When the time is larger than 4 ms, the diameters in both water and solution begin to decrease. Fig. 23 plots drag coefficients versus time for both water and drag-reducing solution cases. It can be seen that the drag coefficients rise sharply when the projectile impacts the liquid surface, and then decrease. The drag coefficient is slightly smaller in drag-reducing solutions than that in water. Compared with Fig. 10 for the case of lower impact velocity V0 = 28.4 m/s, it indicates that the dragreducing additives may have more significantly positive effects on the formation of water-entry supercavitation and further drag reduction of the projectile at lower impact velocity. 4. Conclusions This paper introduced a numerical simulation procedure to simulate water-entry supercavitating flows influenced by a dragreducing solution at typically low velocity and high velocity. The configuration and dynamic characteristics of water-entry supercavity, flow resistance and radial velocity of water-entry cavity boundary were discussed, respectively. It was obtained that the numerical simulation results are in consistence with experimental data, verifying the established numerical simulation procedures. The following conclusions are drawn from this study. (1) Comparisons of the cavity lengths, velocity attenuations, penetration distances and drag coefficients for water and drag-reducing solution cases at the higher and lower impact velocity cases indicated that the influence of drag-reducing additives on water-entry supercavitating flow, i.e., positive effects on the formation of water-entry supercavitation and further drag reduction of the projectile, is more significant at lower impact velocity. (2) The viscosity of the solution prevents the cavity from opening to air, resulting in a lower radial velocity of the cavity near the surface. The smaller surface tension of dragreducing solution makes the cavity boundary expands faster under the liquid surface, which leads to the larger size of the cavity in solution. (3) At higher impact velocity, the natural cavitation must be considered; due to the existence of the vapor rushing into the cavity from the atmosphere, the vapor is mainly distributed near the cavity boundary. 5. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this article. Acknowledgements This work is supported by National Natural Science Foundation of China under Grant Nos. 51276046 and 51421063. References [1] A.M. Worthington, R.S. Cole, Impact with a liquid surface, studied by the aid of instantaneous photography, Phil. Trans. Roy. Soc. Lond. A 189 (1897) 137–148.
[2] A.M. Worthington, R.S. Cole, Impact with a liquid surface, studied by the aid of instantaneous photography: paper 2, Phil. Trans. Roy. Soc. Lond. A 194 (1900) 175–199. [3] May A. Water entry and the cavity-running behavior of missiles. NAVSEA Hydroballistics Advisory Committee, NTIS, Silver Spring, MD, Technical Report No. SEAHAC/TR 75-2, 1975. [4] G. Birkhoff, E.H. Zarantonello, Jets, Wakes, and Cavities, Academic Press, 1957. [5] M. Lee, R.G. Longoria, D.E. Wilson, Cavity dynamics in high-speed water entry, Phys. Fluids 9 (1997) 540–550. [6] R. Panahi, Simulation of water-entry and water-exit problems using a moving mesh algorithm, J. Theor. Appl. Mech. 42 (2012) 79–92. [7] Y.W. Kim, Y. Kim, Y.M. Liu, D. Yue, On the water-entry impact problem of asymmetric bodies, in: Proceedings of 9th International Conference on Numerical Ship Hydrodynamics, Michigan, USA, 2007. [8] E. Walhorn, A. Kölke, B. Hübner, D. Dinkler, Fluid–structure coupling within a monolithic model involving free surface flows, Comput. Struct. 83 (2005) 2100–2111. [9] Q. Yang, W. Qiu, Numerical solution of 3-D water entry problems with a constrained interpolation profile method, J. Offshore Mech. Arctic Eng. 134 (2012) 041101-1. [10] A. Jenkins, T. Evans, Sea mine neutralization using the AN/AWS-2 rapid airborne mine clearance system, in: IEEE Aerospace Conference Proceedings, 2004. [11] M. Schaffar, C. Rey, G. Boeglen, Behavior of supercavitating projectiles fired horizontally in a water tank: theory and experiments, in: CFD Computations with the OTi-HULL Hydrocode, 35th AIAA Fluid Dynamics Conference and Exhibit, Toronto, Canada, 2005. [12] F.C. Li, Z.L. Zou, W.H. Cai, J.C. Yang, H.N. Zhang, Experimental study on the characteristics of bullet water-entry supercavity in drag-reducer solution, J. Eng. Thermophys. 31 (2010) 857–862 (in Chinese). [13] C.X. Jiang, F.C. Li, Experimental and numerical study of water entry supercavity influenced by turbulent drag-reducing additives, Adv. Mech. Eng. 2014 (2014) 280643. [14] R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Fluid Mechanics, vol. 1, John Wiley & Sons, 1987. [15] C.F. Li, S.D. Xu, X.D. Feng, Y.H. Wang, Numerical simulation on heat transfer characteristics for turbulent drag-reducing flow with additives, J. Drain. Irrig. Mach. Eng. 27 (2009) 196–199 (in Chinese). [16] K.C. Ro, H.S. Ryou, Development of the modified k–e turbulent model of powerlaw fluid for engineering applications, Sci. China – Technol. Sci. 55 (2012) 276– 284. [17] C.X. Jiang, F.C. Li, Numerical study of natural supercavitation influenced by rheological characteristics of turbulent drag-reducing additives, Adv. Mech. Eng. 2014 (2014) 275316. [18] C.R. Hirt, B.D. Nichols, Volume of fluid method for the dynamics of free boundaries, J. Comput. Phys. 39 (1981) 201–225. [19] B.E. Launder, D.B. Spalding, Lectures in Mathematical Models, Academic Press, London, England, 1972. [20] F.C. Li, Y. Dong, Y. Kawaguchi, M. Oshima, Experimental study on swirling flow of dilute surfactant solution with deformed free-surface, Exp. Therm. Fluid Sci. 33 (2008) 161–168. [21] G.H. Schnerr, J. Sauer, Physical and numerical modeling of unsteady cavitation dynamics, in: Fourth International Conference on Multiphase Flow, New Orleans, USA, 2001. [22] W. Zhang, Z.T. Guo, X.K. Xiao, C. Wang, Experimental investigations on behaviors of projectile high-speed water entry, Explos. Shock Waves 31 (2011) 579–584. [23] J.M. Aristoff, J.W.M. Bush, Water entry of small hydrophobic spheres, J. Fluid Mech. 619 (2009) 45–78. [24] T. von Karman, The Impact of Seaplane Floats during Landing, NACA, Transl. TN 321, 1929. [25] F.C. Li, B. Yu, J.J. Wei, et al., Turbulent Drag Reduction by Surfactant Additives, John Wiley & Sons, 2012. [26] M.A. Mehrabadi, K. Sadeghy, Simulating drag reduction phenomenon in turbulent pipe flows, Mech. Res. Commun. 35 (8) (2008) 609–613. [27] J.L. Zakin, H.W. Bewersdorff, Surfactant drag reduction, Rev. Chem. Eng. 14 (1998) 253–318. [28] J.P. Franc, J.M. Michel, Fundamentals of Cavitation, Springer Science & Business Media, 2006. [29] R. Knapp, J. Daily, F. Hammitt, Cavitation, McGraw-Hill, New York, 1970. [30] D.R. Stinebring, M.L. Billet, J.W. Lindau, et al., Developed Cavitation-cavity Dynamics, Pennsylvania State Univ University Park Applied Research Lab, 2001. [31] J. He, Experimental study on the influence of surface tension on dynamic characteristics of laser-induced bubble Master Thesis, Nanjing University of Science and Technology, Nanjing, China, 2008 (in Chinese). [32] H.H. Shi, M. Itoh, T. Takami, Optical observation of the supercavitation induced by high-speed water entry, ASME J. Fluids Eng. 122 (2000) 806–810.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Numerical study on the transient behavior of water-entry supercavitating flow around a cylindrical projectile influenced by turbulent drag-reducing additives Chen-Xing Jiang a, Zhi-Jun Shuai a, Xiang-Yuan Zhang a, Wan-You Li a, Feng-Chen Li b,⇑ a b
College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China
h i g h l i g h t s Water-entry supercavitation with drag-reducing (DR) additives was simulated. DR additives could positively affect the formation of water-entry supercavitation. DR additives could further reduce the drag resistance on the projectile.
a r t i c l e
i n f o
Article history: Received 11 December 2015 Revised 25 April 2016 Accepted 17 May 2016 Available online 18 May 2016 Keywords: Water-entry supercavitation Projectile Drag-reducing additives Phase transition
a b s t r a c t Simulation of the unsteady behavior of water-entry supercavitating flows influenced by turbulent dragreducing additives is very complicated. This paper attempts to introduce a numerical simulation procedure to simulate such problems in viscous incompressible two-phase and three-phase media, respectively. Firstly we performed a numerical investigation on water-entry supercavities in water and turbulent drag-reducing solution, respectively, at an impact velocity of 28.4 m/s to confirm the accuracy of the numerical method. Based on the verification, projectile entering water and turbulent drag-reducing solution at relatively high velocity of 142.7 m/s (phase transition is considered) is simulated. The Cross viscosity equation was adopted to represent the shear-thinning characteristic of aqueous solution of drag-reducing additives. The configuration and dynamic characteristics of water-entry supercavity, flow resistance and the radial velocity of cavity boundary were discussed respectively. It was obtained that the numerical simulation results are in consistence with experimental data, verifying the established numerical procedures. Numerical results show that the supercavity length in drag-reducing solution is larger than that in water; the velocity attenuates faster at high impact velocity than at low impact velocity. Turbulent drag-reducing additives have the potential in enhancement of supercavitation and further drag reduction. Ó 2016 Published by Elsevier Ltd.
1. Introduction The water-entry phenomenon, intrinsically a transient process, is an important topic in naval hydrodynamic area. And the waterentry problem has been considered as a classic problem in the field of fluid mechanics. In the process of projecting into water, the behavior of a projectile comprises a series of complicated events both above and below the free surface. Past research activities mainly focused on three topics classified by different entry stages.
⇑ Corresponding author. E-mail address: [email protected] (F.-C. Li). http://dx.doi.org/10.1016/j.applthermaleng.2016.05.102 1359-4311/Ó 2016 Published by Elsevier Ltd.
The first topic is about the impact force on projectile at the initial stage of water entry. The second topic is about the dynamics of the supercavity formed behind the projectile after it penetrates into water at some distance. The third topic is about the trajectory of the projectile when it moves more deeply into water. The present study belongs to the second research topic of water-entry problem. Scientific research of water-entry phenomenon was initiated from the beginning of the 19th century. At early stage, the main method for investigation of water-entry phenomena is experiment. Worthington and Cole [1,2] first used single-spark photography to study the air cavity formed by the vertical entry of spheres into water. The study on this subject was intensified during World War II for the design of military projectiles entering water at high
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speed. The advent of high-speed cine-photography allowed for subsequent and quantitative measurements of cavitating flow. The most extensive one was conducted by May with a view to naval ordinance applications [3], in which particularly attention was paid to the air cavity formation during high-speed projectile impact on water and cavity running behavior. In addition to experiments, there have been also theoretical and numerical studies of the air cavity dynamics associated with water entry so far. Theoretical investigations of water-entry cavity dynamics have been conducted after the pioneering work by Birkhoff and Zarantonello [4], who proposed an approximate theory of cavity evolution. They considered the cavity to be extremely long and the motion of flow field around the cavity wall to be purely radial. The model only qualitatively predicts cavity shape and the dynamics of cavity formation induced by a high-speed projectile. Lee et al. [5] extended the model of Birkhoff and Zarantonello by developing a two-dimensional (2D) model for cavity formation and collapse induced by the high-speed vertical impact of a projectile on water. This extended model assumes that the kinetic energy loss of the projectile equals to the kinetic energy converted into the horizontal fluid section. Although great progress has been made in solving the waterentry problem with methods based on potential flow assumptions, it is difficult for these methods to treat a highly distorted or a breaking free surface. In practice, the effect of gravity is considerable, when the impact velocity is high enough, so that the phase change must be considered and the water surface includes complicated jets. In such cases, analytical solutions become very difficult or even impossible. These difficulties can be overcome by computational fluid dynamics (CFD) methods, which solve the governing equations of fluid flow behaviors numerically. With the development of CFD techniques, numerical simulation has become a reliable and powerful tool for the investigation of complex, unsteady and nonlinear flow phenomena. Panahi [6] introduced a finite volume-based moving mesh algorithm and simulated water-entry of wedges and water-exit of a cylinder. The algorithm employs a fractional step method to deal with the coupling between pressure and velocity fields. Interface is also captured by solving a volume fraction transport equation. A boundary-fitted body-attached mesh of quadrilateral control volumes is implemented to record hydrodynamic time histories of loads, motions and interfacial flow changes around the structure. Kim et al. [7] used the smoothed particle hydrodynamics method to simulate the water entry of 2D asymmetric bodies. In the smoothed particle hydrodynamics method, the fluid domain is divided into a finite number of mass carrying particles. The motion of the particles and pressure distribution in the fluid are obtained by solving the momentum equations and the continuity equation based on the Lagrangian method. Walhorn et al. [8] solved the 2D wave-body interaction problems with free surfaces using the level-set method. In the level-set method, the deformation and movement of the free surface can be captured by a smooth levelset function. Yang and Qiu [9] presented numerical solutions of slamming problems for 3D bodies entering calm water with vertical and oblique velocities, respectively. They solved the highly nonlinear water entry problems by a constrained interpolation profilebased finite difference method on a fixed Cartesian grid. In most of the previous researches, spheres or horizontal cylinders were usually the research models. Few vertical cylindrical projectile models, which resemble bullets, were used. On the other hand, most of the studies investigated projectile water-entry at low speed with significant gravitational effect or at high speed but neglecting phase transition effect. With the development of naval weapons, problems encountered in the water-entry process have been becoming more and more attractive. Investigations on water-entry supercavity and the underwater motion of the
projectile can provide significant data to support fundamental research in military field, such as the navigation of torpedoes and air drop underwater weapons. On the other hand, supercavitating projectile has been used for mine disarmament [10] and proposed for use as defense against torpedoes [11] as well. In our previous studies, we proposed a novel control strategy for supercavitating flows using turbulent drag-reducing additives. Fig. 1 shows a schematic of the experimental apparatus used in our previous work [13]. The projectile was projected vertically into water and aqueous solution of surfactant CTAC (cetyltrimethyl ammonium chloride) with weight concentrations of 100, 500 and 1000 ppm, respectively, using a pneumatic nail gun. The trajectories of the projectile and the supercavity configuration were recorded by a high-speed CCD camera. By using this simple experimental apparatus, we obtained important findings that at the same initial water-entry velocity: the size of supercavity in turbulent drag-reducing surfactant solution was larger than that in water; the navigating velocity of the launched projectile at the same depth in liquid was faster in drag-reducing surfactant solution; and the maintaining downstream distance of supercavity is longer in surfactant solution as well. The typical results are plotted in Fig. 2 for examples [12,13]. For more detailed information one refers to our paper [13]. In order to have a deep insight into the
Nail of pneumatic spear Barrel Projectile
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Computer
Fig. 1. Schematic diagram of experiment set up.
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Fig. 2. The configurations of supercavity formed around the water-entry projectile in aqueous solution of cetyltrimethyl ammonium chloride (CTAC) at different concentrations, taken at the upper part of water tank [12].
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characteristics of water-entry supercavitating flow influenced by turbulent drag-reducing activities, the water-entry supercavitations in drag-reducing solutions were numerically simulated. We at first adopt the case of bullet-shaped projectile downwardly entering drag-reducing solution in moderate velocity vertically and compared with the previous experimental data to confirm the accuracy of the employed numerical simulation procedure. And then we make further efforts in numerically investigating the water-entry supercavitating flow around a circular cylinder influenced by turbulent drag-reducing additives at high impact velocity. Since the phase transition is considered herein, it is a three-phase cavitating flow. 2. Numerical simulation procedures The forming process of water-entry supercavitation in the turbulent drag-reducing solution is realized through a commercially available CFD code ANSYS FLUENT. One has to deal with four distinct problems in CFD simulation of water-entry cavitating flow influenced by turbulent drag-reducing additives. The first problem to solve is how to represent the rheological properties of the dragreducing solution. The Cross viscosity model [14] is utilized herein to characterize the aqueous solution of drag-reducing additives with shear-thinning effect in the numerical procedure. This simple viscosity model has been used in numerical simulations of turbulent drag-reducing flows of viscoelastic fluid, such as by Li et al. [15] and Ro and Ryou [16], and also of supercavitating flows in drag-reducing solutions by ourselves [13,17]. The second problem is to solve the Navier–Stokes and continuity equations for the flow domain after mesh generation and parameters’ initialization. At this step, velocity and pressure fields are output to calculate loads acting on the projectile. In the third problem, the momentum equation for the projectile is solved in order to calculate the body motion and consequently to find out its new location. After this step, the procedure is continued to move the body of projectile as well as the computational mesh of the flow domain. This inner loop provides a strongly coupled solution in the whole computational domain between rigid body motions and flow field. At this stage, a strategy is required to introduce such movements into the computational domain and to update all the information for the next time step. Here, spring-based smoothing and local remeshing dynamic mesh models are used. For the fourth problem, the volume fraction transport equation is solved to represent the interface between liquid and gas phases in the position-updated mesh. Such an algorithm as mentioned above is continued to capture a desired time history of the water-entry cavitating flow. 2.1. Governing equations The volume of fluid (VOF) method is utilized in this study to capture the interface between fluid phases. This method treats the flow as a mixture of two phases (e.g. water and air) [18]. A volume fraction function is introduced with values between 0 and 1 to capture the free surface. The fraction function is reconstructed in each cell. The governing equations are the following continuity and momentum equations for the mixture, respectively,
@a @ ð1Þ þ ðaui Þ ¼ 0 @t @xi @ @ ðq Þ þ ðq ui Þ ¼ 0 ð2Þ @t m @xi m @ @ @P @ @ui @uj þ qm g þ F i ð3Þ l þ ðq uj Þ þ ðqm ui uj Þ ¼ þ @t m @xi @xj @xi m @xj @xi where a is the volume fraction of the air, u is velocity vector, P is pressure, qm ¼ ð1 aÞql þ aqg and lm ¼ ð1 aÞll þ alg are the
density and dynamic viscosity of the simulated mixture, respectively, Fi is body force.The standard k—e model with standard wall functions is adopted to provide turbulence closure: @ @ @ l @k ðqm kui Þ ¼ lm þ t ð4Þ þ qm G qm e ðqm kÞ þ rk @xi @t @xi @xi @ @ @ l @e q Ge q e2 ðq eui Þ ¼ lm þ t þ C e1 m C e2 m ð5Þ ðq eÞ þ re @xi k k @t m @xi m @xi
lt ¼ qm C l k2 =e
ð6Þ
where k and e are the turbulent kinetic energy and dissipation rate, respectively. The model parameters are chosen as follows: C l ¼ 0:09, C e1 ¼ 1:44, C e2 ¼ 1:92, rk ¼ 1:0, re ¼ 1:3. The values of these constants are set by referring to the literature [19]. These values have been determined based on experiments for fundamental turbulent flows including frequently encountered shear flows like boundary layers, mixing layers and jets as well as for decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows. Besides, by choosing these values, the numerical simulation results are shown to be in consistence with experimental data. 2.2. Cross viscosity model The turbulent drag-reducing solution is characterized through modeling the dynamic viscosity by the Cross viscosity equation, which shows the shear-thinning behavior,
l ¼ l0 =½1 þ ðkc_ Þm 2 ls M½l2 c k¼
5
R0 T
ð7Þ ð8Þ
where l0 is zero-shear viscosity; k is relaxation time; c_ is shear rate; m is Cross power index and m ¼ 0:3 is utilized here; M is the mean mole mass of the solute in g/mol; ls is viscosity of solvent; c is the solution concentration in g=cm3 ; R0 ¼ 8:31 J=mol K; T is the absolute temperature in K; ½l is intrinsic viscosity and
½l ¼ 1:03 102 M0:78 in ml=g. Together with the Cross equation, a decreased surface tension coefficient is used to characterize the drag-reducing solution. rs ¼ 0:036 N/m (half of that of water) is chosen for the solution. Note that, in our previous measurement for surface tension of turbulent drag-reducing CTAC surfactant solutions, we obtained that the surface tension coefficient of CTAC solution becomes less than half of that of water (0.033 N/m for solution compared with 0.071 N/m for water at room temperature) from a very dilute concentration (40 ppm) and changes only slightly with the increase of concentration: from 0.0326 N/m at 40 ppm concentration to 0.0322 N/m at 200 ppm concentration [20]. 2.3. Cavitation model When the initial velocity of the projectile is high enough, the local pressure near the projectile will drop below the saturation pressure, and consequently natural cavitation occurs. The formation of vapor leads to a significant interaction between different phases. In these cases, it is unreasonable to neglect the vapor phase. Thus, such flow case contains three phases. We have to track the interfaces between different phases and model the transition of liquid to vapor. The cavitation model is therefore employed to capture the phase transition from water to vapor. The Schnerr–Sauer cavitation model [21] is adopted here. The governing equation for vapor is as follows: ! @ ðav qv Þ þ r ðav qv V v Þ ¼ Re Rc @t
ð9Þ
where Re and Rc are the vapor evaporation and vapor condensation rates, respectively.
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When P v P P,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q q 3 2 ðPv PÞ Re ¼ v l av ð1 av Þ RB 3 qm ql When Pv 6 P,
q q 3 Rc ¼ v l av ð1 av Þ RB qm
ð10Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðP Pv Þ 3 ql
ð11Þ
Here, av is the vapor volume fraction, RB is the diameter of the 13 bubble, RB ¼ 1avav 43p 1n , n ¼ 1 1013 , P v is the saturation vapor pressure. 2.4. Numerical method and conditions Since the simulated water-entry projectile (a cylindrical bullet) is axially symmetric, the simulation model was then simplified to be 2D, as shown in Fig. 3. The simulated cylindrical projectile has exactly the same geometry and dimensions as used in the Refs. [12,22]. Fig. 3(a) and (b) are the mesh systems for two simulation cases, at lower initial velocity and higher initial velocity, respectively. Water-entry supercavitation in water and turbulent dragreducing solution, respectively, were numerically simulated based on unsteady Reynolds averaged Navier–Stokes scheme. The user
(a)
mm
Pressure-outlet Bullet
30
800
Penetration axis
Wall
150 7
(b) Wall
Projectile
Projectile motion
Two typical cases, namely one at a lower impact velocity without consideration of phase transition from water to vapor and the other at a higher impact velocity with consideration of phase transition, have been chosen for numerical simulations. Furthermore, for these two chosen cases, there were available experimental data to compare. Numerical simulations were at first performed for the water and drag-reducing surfactant solution (l0 ¼ 0:01 Pa s, k ¼ 1 s, m ¼ 0:3 and rs ¼ 0:036 N/m) flow cases at the initial velocity of V 0 ¼ 28:4 m/s, respectively. For these cases, only liquid phase and air phase are considered. On the base of results from these simulations, the higher initial velocity cases (impact velocity V 0 ¼ 142:7 m/s into water and drag-reducing solution, respectively) were then performed, which are three-phase flows with considerations of phase transition. Detailed discussions about the results for these two cases are presented as follows, respectively.
mm
The analytical calculations of supercavity profile for comparison in this part are the same as that used in Ref. [13]. Fig. 4 shows numerical simulation results for the radii of supercavity as compared with analytical calculations. In Fig. 4, R is the radius at the penetration depth z, and z0 represents the initial depth of the projectile. The positive value and negative value mean that the radii
Penetration axis
10
Pressure-outlet
100
Wall
3. Results and discussion
3.1. Cases at lower impact velocity of V 0 ¼ 28:4 m/s
Wall Projectile motion z
5
R (mm)
500
12.65
z
defined functions (UDF) embedded in the FLUENT platform were used to control the movement of the projectile. A segregated solver with SIMPLE scheme as the velocity–pressure coupling algorithm was selected. The pressure field was discretized using PRESTO scheme. The second order upwind scheme was used for the discretization of the velocity field. Geo-reconstruct method was used for the volume fraction. Inherently, the values of all the parameters were stored and calculated with double precision. The numerical simulations were for transient flows. The time step was set to be 0.00001 s, which gives 20 time steps per cycle. In all the simulations presented below, the working fluid was at 298 K, with liquid, air and vapor densities of 998.2, 1.225 and 0.5542 kg=m3 , respectively. An exit pressure was specified at the top (outlet) boundary (for Fig. 3(a)) and at the right boundary (for Fig. 3(b)). The both side-boundaries were set to be solid wall (no-slip for velocity and no penetration). A structured grid system was employed to discretize the region around the projectile with boundary layer clustering around the cylindrical body. The unstructured mesh was employed for the other region considering the dynamic mesh technology.
analytical calculation numerical simulation water numerical simulation solution
0
-5
-10 0
25.4
Fig. 3. Physical model, computational region and grid system.
5
10
15
20
25
(z-z0) (mm) Fig. 4. Comparison of numerical simulation results for the radii of supercavity at different axial locations of the projectile with the analytical model.
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E
N
1.56ms
E
N
3.12ms
E
N
4.68ms
E
N
6.24ms
E
N
7.80ms
Fig. 5. Comparisons of cavity configurations at water-entry between numerical simulations and experiments for the 1000 ppm CTAC solution case [12] (‘‘E” and ‘‘N” represent the experiments and numerical simulations, respectively).
are symmetrical about the axial plane. Excellent agreements between the present numerical simulations and analytical calculation have been obtained, verifying the currently conducted numerical simulation procedures. Furthermore, the shapes of supercavity in water and solution are almost the same, which means that, at very early stage of water-entry cavity, the drag-reducing additives have negligible influence on the cavity shape. Fig. 5 shows a comparison of supercavity forming process of water entry between numerical simulation and experiments for the 1000 ppm CTAC solution case [12]. A solid cylindrical projectile with density qs and length L vertically impacting a horizontal liquid surface with velocity V 0 is considered. The impact of a cylindrical projectile that creates a subsurface air cavity has several distinctive features. The first significant phenomenon is that, shortly after the impact of projectile on the liquid surface, an upward moving jetting and a laterally expanding splash are formed above the surface, and the projectile is suddenly surrounded by the splash (t = 1.56 ms). The second significant phenomenon is that, as the projectile penetrates into liquid, a cavity under the surface is formed. During this period, the air from atmosphere comes into the cavity continually. Initially, the air entrainment is due to gas flow induced by the projectile. Then the air is entrained into the cavity due to negative relative pressure of the cavity. The reason for the negative pressure is that in water entry, the volume of the cavity increases rapidly and the cavity cannot keep an equilibrium thermodynamic state. With time evolution, the airflow into the cavity is ended by the surface closure and the enclosed cavity is pulled away from the surface by the downward moving projectile. An axisymmetric cavity is evident below the surface with a splash curtain above the cavity. The evolution of splash curtain has been described by Aristoff and Bush [23]. At t = 3.12 ms for the ‘‘open-cavity phase”: the flow is separated from the projectile nose with generation of the cavity; air rushes in from above the liquid surface to fill the cavity; during this phase the projectile is cavity-running, i.e., a cavity is being generated around the projectile as it travels, but the cavity is still open to the atmosphere at its upper end (at the liquid surface). At t = 4.68 ms, it shows that the splash is forming a dome and closing over the entry point of the projectile to seal the cavity from the air above. As shown at t = 6.24 ms, ‘‘surface closure” occurs and the cavity is then pushed down from the liquid surface. From this stage, the cavity has closed from the liquid surface and envelops the projectile, forming the supercavity (t = 7.80 ms). In the process of water-entry, the cavity shape in solution is consistent well with the experimental observations. It again demonstrates that the presently conducted numerical simulation procedure is suitable for simulating the supercavitating projectile in drag-reducing solutions.
Fig. 6 compares cavity configurations between numerical simulation and experiments for water and 1000 ppm CTAC solution at t = 7.8 ms [12], respectively. It can be seen that the cavity shapes in water and drag-reducing solution are different. Inside the cavity, the volume of air component in solution is larger than that in water. In solution, the most part of the cavity is filled with air, and just in the cavity tail, the cavity is cloudy. But in water, the air distributes in an annular region at the rear part of the cavity, and the central region of the rear part of the cavity is water, which is in good consistence with the phenomena observed in the experiment. The reason may be due to the fact that when projectile penetrates into water, the re-entrained jet is more powerful in water than in solution. The smaller jet toward the air at the liquid surface in solution is another evidence. The non-dimensional length L ¼ Lc =Dn and diameter D ¼ Dc =Dn of the cavities (where Lc is the cavity length; Dc is the biggest cavity section diameter; Dn is the diameter of the cylindrical projectile) in water and drag-reducing CTAC solutions vs. time are compared in Figs. 7 and 8, respectively. As shown in Fig. 7, the non-dimensional length of cavity in solution keeps larger than that in water all the time; the length difference between water and solution cases becomes larger with time. The non-dimensional diameter varies with time complicatedly as shown in Fig. 8. In the first period (about 3 ms) the diameters of cavity increase sharply in both water and solution; then the diameters fluctuate for about 2 ms; after that, the cavity diameter in water decreases with time, and the
Solution E
N
Water E
N
Fig. 6. Comparisons of cavity configurations between numerical simulations and experiments for water and 1000 ppm CTAC solution case at t = 7.8 ms [12] (‘‘E” and ‘‘N” represent the experiments and numerical simulations, respectively).
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16
the projectile. The effect of g is negligibly small for high-speed water-entry case. Eq. (14) is then simplified,
Water Solution
m 12
dV p 1 ¼ qw A0 C d V 2p 2 dt
L
With definition of the velocity attenuation coefficient k ¼ qw A0 C d0 =2m, the following equations can be obtained by integrating Eq. (15):
8
4 0
2
4
6
8
10
t (ms)
Water Solution
4.0
3.6
3.2
2
4
6
8
10
30
t (ms) Fig. 8. Non-dimensional diameter vs. time for the cases at V 0 ¼ 28:4 m/s.
Dp q gh þ 0:5C a qa V 20 C a qa V 20 V2 ¼ w ¼ r0 02 2 2 2 qw V P 0:5qw V p 0:5qw V p VP
10 0
dV p X 1 m 2 ¼m ¼ F Z ¼ mg qw A0 C d V 2p 2 dt dt 2
ð14Þ
where m is the projectile mass, z is the penetration direction, t is the time, g is the gravitational acceleration, A0 is the projected area of
2
3
4
5
6
7
Fig. 9. Comparison between analytical calculation and numerical simulation for velocity attenuation of the projectile with supercavitation.
Experimental data of water Experimental data of solution (1000 ppm) Analytical calculation Numerical simulation of water Numerical simulation of solution
240
160
z (mm)
ð13Þ
1
t (ms)
ð12Þ
where C d0 is the drag coefficient at r ¼ 0 (C d0 0:82 for the disk cavitator). For a navigating projectile normally penetrated into water, the deceleration reads,
d z
20
15
where qw is water density, h is water depth (which is negligible here), qa is air density, C a is airflow pressure drop coefficient ðC a ¼ 5—15Þ, V 0 is the initial velocity of the projectile. At sufficiently small r, the drag coefficient of a blunted cavitator can be approximated as [13]
C d ¼ C d0 ð1 þ rÞ
ð17Þ
25
V (m/s)
cavity diameter in solution keeps stable for about 3 ms and then decreases. Before about 6.5 ms, the cavity diameter in water is larger than that in solution, and after 6.5 ms, the cavity diameter in solution becomes larger. From the above observation, we can conclude that the cavity in solution is more stable than that in water. It has well been documented that, at its initial stage, the waterentry supercavity opens to the atmosphere [13]. During the waterentry process, the pressure difference Dp between inside and outside the supercavity can be assumed to be constant; thus, the cavitation number only depends on the projectile velocity Vp, as shown in the following equation [13]:
r¼
ð16Þ
analytical calculation numerical simulation of water numerical simulation of solution
2.8 0
pffiffiffiffiffiffi
1
The numerical simulation results for the variation of projectile velocity as compared with the analytical predictions are shown in Fig. 9. It can be seen that, for water-entry supercavitation at initial velocity of V 0 ¼ 28:4 m/s, the velocity derivative (and thus the drag) obtained from the numerical simulation is comparable to that predicted using the analytical model, with 25% maximum discrepancy. Note that, in the impact process, the projectile will lose some energy as compared with the analytical formulation. This energy loss contributes to the under-estimation of velocity in numerical simulation as compared with that in theoretical calculation. From Fig. 9, we can also see that the velocity in solution decays slower than in water, which means that, at the same instant, the projectile in solution flies faster than in water. Fig. 10 shows numerical simulation results for the penetration distance (z) as compared with the analytically predicted value
Fig. 7. Non-dimensional length vs. time for the cases at V 0 ¼ 28:4 m/s.
4.4
pffiffiffiffiffiffi
r0 V 0 tan arctan pffiffiffiffiffiffi k r0 V 0 t r0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi 1 þ r0 1 1 pffiffiffiffiffiffi cos arctan pffiffiffiffiffiffi k r0 V 0 t z ¼ ln k r0 r0
Vp ¼
D
ð15Þ
80
0 0
2
4
6
8
10
t (ms) Fig. 10. Comparison of experiments, analytical calculation and numerical simulation for penetration distance.
C.-X. Jiang et al. / Applied Thermal Engineering 104 (2016) 450–460
and the experimental data. It can be seen that the projectile in solution travels to a longer distance than in water at the same instant. Actually, the projectile cannot move as far as the analytical calculation due to the loss of projectile kinetic energy during the water-entry process. From the results mentioned above, it is clear that the numerical simulation approach developed in the present study allows one to investigate the characteristics of water-entry induced supercavitation influenced by turbulent drag-reducing additives. The drag force of projectile during water entry is predicted based on the method suggested by von Karman [24]. This method assumes that momentum is conserved during the water-entry process. The momentum loss of the body when it slows down is converted into the momentum of the ‘‘added mass” of water. When a body moves through still liquid, a liquid flow is induced in the vicinity of the body. The ‘‘added mass” is the mass that should have the same kinetic energy as that of the flow. The problem can then be treated approximately in the following manner (as schematically shown in Fig. 11), and the force acted between the body and the water can be calculated. Let m = the mass of the body, z = the vertical distance through which the body travels within time t, r = the radius of the cylindrical body, dm = the added mass. The fictitious added water mass dm, which changes as the body penetrates the water, is thought of as having the speed of body at each instant. For a given body at a given entry angle it is assumed that the added mass is a function of only the body penetration distance, and further that dm is proportional to q and to the cube of the linear scale of the body; but is not a function of the mass of the body or its speed [3]. We proceed to calculate the force by application of the momentum theorem.
mV 0 ¼ ðm þ dmÞV
ð18Þ
dm ¼ K qpr 2 f ðzÞ
ð19Þ
mV 0 ¼ ðm þ K qpr f ðzÞÞV 2
2 1 1 m DE ¼ dmV 2 ¼ ðK qpr 2 f ðzÞÞ V 0 2 2 m þ K qpr 2 f ðzÞ
ð20Þ ð21Þ
So, if we can know the definition of K and f ðzÞ, the loss of the projectile energy during the water entry can be calculated. Fig. 12 plots drag coefficients versus time for both water and drag-reducing solution cases. The drag coefficient C d is defined as below:
Cd ¼
F
ð22Þ
0:5qV 2p A0
where F is the overall drag force.
Projectile
z
Surface
r
Fig. 11. Schematic diagram of the water-entry problem.
1.8 Water Solution 1.2
Cd
456
0.6
0.0 0
1
2
3
4
5
6
7
t (ms) Fig. 12. Time history of drag coefficients for projectile in water and in dragreducing solution cases at V 0 ¼ 28:4 m/s, respectively, obtained by numerical simulation.
It can be seen that the drag coefficients rise sharply when the projectile impacts the liquid surface, and then decrease, which can quantitatively explain why the projectile loses much kinetic energy and its velocity decreases much when impacting the liquid surface. The drag coefficient is obviously smaller in the dragreducing solution than that in water. It can be noticed that, the largest drag coefficient point is larger in solution than in water at the beginning of water entry, which means that when the projectile impacts the liquid surface, the projectile loses more kinetic energy in solution. This is because the viscosity of the CTAC aqueous solutions is larger than that of water (the viscosity of the CTAC solution increases with the increase of the concentration). The larger viscosity has retardation on the water-entry projectile making the energy loss more in solution. But after that point, the projectile resists smaller drag in solution, which indicates the turbulent dragreducing effect. After the surface impact, the Reynolds number of liquid flow relative to the projectile has reached the valid range of the turbulent drag-reducing effect, which makes the turbulent frictional resistance smaller in CTAC solution. From these results, it can be conjectured that adding small amount of drag-reducing additives, the water-entry projectile can move faster in the dragreducing solution than in water provided the initial driving power is the same. It is expected that the actual drag coefficient would be greater than that predicted for a disk cavitator because there is drag force on the projectile body. The occurrence of tail-slap is probably the main contributor. The working fluid used our previous experimental study [13] was a viscoelastic aqueous solution of surfactant with turbulent drag-reducing effect. The viscoelastic effect is the most remarkable feature for turbulent drag-reducing solutions. Whether or not the surfactant solution displays drag-reducing effect depends on whether or not the shear-induced rod-like and then network microstructures can be formed in the solution. Such microstructures are essentially micelles composed of small surfactant molecules. The ability to form micelle structures is the typical feature that the surfactant solution displays after the solution concentration exceeds the critical micelle concentration (CMC). Above the CMC, the surfactant monomer molecules and the formed micelle structures are always in a thermodynamic equilibrium state [25]. Mehrabadi et al. simulated drag reduction phenomenon using polymeric additives in turbulent pipe flows [26]. Fig. 13 schematically shows a simplified phase diagram for surfactant solution [27]. Fig. 14 shows the comparison of the viscosity of CTAC aqueous solutions between experimental data and Cross viscosity model estimations. In addition to the variation of the rheological properties, from constant viscosity to shear-shinning characteristic, we
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Concentratoin
Rod-like micelles
CMC II
Micelles CMC I
Monomers Temperature Fig. 13. The change of micelle structures of surfactant solution when the concentration, and temperature of fluid increased.
Experimental data for 200 ppm Experimental data for 500 ppm Experimental data for 1000 ppm Cross model result for 200 ppm Cross model result for 500 ppm Cross model result for 1000 ppm
Shear Viscosity (Pa•s)
0.020
0.015
0.010
0.005
0.000 0
20
40
60
80
100
Shear Rate (s-1) Fig. 14. Comparison of the viscosity of CTAC aqueous solutions between experimental data and Cross viscosity model.
have experimentally obtained that the surface tension coefficient of aqueous solution of CTAC/sodium salicylate (CTAC/NaSal) is about half of that of water [20]. Thereby, in the numerical simulations, together with the Cross equation, a decreased surface tension coefficient (rs ¼ 0:036 N/m, half of that of water) was employed to characterize the drag-reducing solution. Fig. 15 illustrates the instantaneous radial velocity ðV r Þ of water-entry cavity boundary at various depths (z) in water and solution cases, respectively. At t = 0.5 ms, the radial velocity of the cavity in solution is larger than that in water from the depth deeper than 1 mm, and only near the liquid surface the radial velocity of the cavity is larger in water. With increase of the projectile penetration, the radial velocity of the cavity decreases both in water and in solution, but still it is larger in water near the liquid surface (seeing t = 1.0 ms and t = 1.5 ms). At t = 2 ms, the radial
Water: t = 0.5 ms Water: t = 1.0 ms Water: t = 1.5 ms Water: t = 2.0 ms
8
3.2. Higher impact velocity of V 0 ¼ 142:7 m/s In this section we shall discuss the unsteady behavior of waterentry supercavitating flows in water and aqueous solution of turbulent drag-reducing additives at a higher impact velocity of V 0 ¼ 142:7 m/s. The numerical simulation results are compared with experimental data [22] and analytical calculations. Fig. 16 shows the attenuation of projectile velocity in water and in solution as compared to the analytical predictions and experimental data [22]. It can be seen that, for water-entry supercavitation at initial velocity V 0 ¼ 142:7 m/s, the velocity attenuation obtained from the numerical simulation is comparable to that predicted using the analytical model and experimental data, demonstrating the accuracy of the established numerical simulation method for high velocity water-entry with phase transition. Furthermore, we can see that the projectile velocity in solution decays slower than in water, which is the same as for the lower impact velocity case. Again, it is proved that the projectile moves faster in drag-reducing solution than in water, no matter the impact velocity is high or low. Compared with Fig. 9, it can be seen that the velocity attenuates faster at higher water-entry velocity. At t = 4 ms, the velocity difference ðDVÞ between in water and in solution normalized with the impact velocity DV/ V0_low = 2.7/28.4 = 9.51% and DV/V0_high = 3.3/142.7 = 2.31%. So it can be concluded that the influence of drag-reducing solution is more obvious at lower impact velocity. Fig. 17 shows the penetration distance (z) versus time. It is seen that the projectile in solution moves to a longer distance than in water at the same instant, which is the same as that happened to the case of lower impact velocity of V0 = 28.4 m/s.
Solution: t = 0.5 ms Solution: t = 1.0 ms Solution: t = 1.5 ms Solution: t = 2.0 ms
4
Analytical calculation Numerical simulation of water Numerical simulation of Solution Experimental data of water [22]
150
125
V (m/s)
Vr (m/s)
6
velocity of the cavity in solution becomes larger than that in water. Fig. 15 shows that the radial velocity of the cavity boundary in solution keeps larger in the cavity forepart. When the projectile impacts the liquid surface, the higher viscosity of the solution prevents the cavity from opening to the air, resulting in a lower radial velocity of the cavity near the surface. When the projectile penetrates into the liquid, the smaller interface tension between solution and the air makes the cavity boundary expands faster under the liquid surface, which leads to the larger size of the cavity in solution. As illustrated above, we have numerically realized that the turbulent drag-reducing additives do have evident and positive effects on the formation of water-entry supercavitation, resulting in further decrease of flow resistance of the projectile, as obtained in the experiment.
100
75
2 50
0
0
10
20
30
40
50
z (mm) Fig. 15. Instantaneous radial velocity of water-entry cavity boundary at various depths in water and solution.
0
1
2
3
4
t (ms) Fig. 16. Numerical simulation results for velocity attenuation of the projectile with supercavitation as compared with analytical calculation and experimental data [22].
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Analytical calculation Experimental data of water [22] Water Solution
400
z (mm)
300
water
solution
t = 1 ms
200
t = 2 ms
100
t = 3 ms 0 0
1
2
3
4
5
t (ms) Fig. 17. Comparisons of experiments [22], analytical calculation and numerical simulation for penetration distance.
Fig. 18 compares cavity configurations between numerical simulation and experimental results for water case at t = 1.25 ms. It can be seen that the cavity shape are essentially the same. More detailed information can be seen from the numerical simulation result. At the aft part, the cavity boundary is saw-toothed, which is the result of the phase transition from water to vapor. More explanations about this phenomenon will be given later. Besides, at the center of the cavity, there exist small droplets coming from the reentrant jet. At this moment, the free surface is closed and a liquid film forms to cut off the air from entering into the cavity. At the free surface, there are two jets formed: one comes into the cavity due to the pressure difference inside and outside the cavity, and the other splashes upward into the air. Fig. 19 shows a comparison of the supercavity forming process of the projectile entering water and drag-reducing solution respectively. In consideration of the phase transition from water to vapor, we use the contour of liquid phase to represent the cavity shape. In other words, the fluid region where there is no water represents the cavity. At t = 1 ms, the free surface is already closed in water due to surface closure, and a jet is formed toward the air. For the drag-reducing solution case, the cavity is still open to the air and the generated jet is toward the cavity. At t = 2 ms, the reentrant jet is formed in water, and still the intensity of the jet is weaker than that in solution and the aft part of the cavity is cupped in solution. On the other hand, from t = 1 ms to t = 4 ms the size of the cavity is continually increasing. This is an interesting phenomenon. As abovementioned, at t = 1 ms, the surface closure is already formed in water. So there must be some other component coming
N
E
Fig. 18. Comparisons of cavity configurations between numerical simulation and experiments for water at t = 2 ms [21] (‘‘E” and ‘‘N” represent the experiments and numerical simulations, respectively).
t = 4 ms Fig. 19. Comparisons of cavity configurations of supercavity between that in water and in drag-reducing solution.
into the cavity to enlarge the size of cavity, which is the vapor generated from phase change as is discussed below. Paying more attention to the cavity evolution process from t = 2 ms to t = 3 ms in solution, we can find that the surface closure is different. The actual process is that the closure is disturbed after t = 2 ms, and more air rushes into the cavity and then the cavity is closed again, which results in a larger sized cavity. This cavity closure process shows that the cavity formed in turbulent drag-reducing solution is not closed at one time, but first is closed and then is disturbed and then is closed again. When the impact velocity is high enough, the phenomenon of natural cavitation must be considered. Note that, natural cavitation can be defined as the breakdown of a liquid medium under very low pressure. When the speed of an underwater body reaches a certain value, water pressure near the cavitator nose will become lower than the saturated vapor pressure corresponding to the local temperature, and cavitation happens [28]. Cavitation number,
r ¼ 2ðp1 pv Þ=qV 21 , is the key parameter characterizing cavitating flow [29]. In this paper, p1 and pv are the ambient liquid pressure and the saturation pressure, respectively; V 1 is the velocity of submerged projectile; q is the liquid density. Theoretically, when the local pressure is lower than the saturation pressure (2350 Pa) corresponding to 20 °C water temperature, natural cavitation happens. Fig. 20 shows the gas-phase components inside the cavity in water and in drag-reducing solution, respectively. It can be seen that the cavity is composed of vapor and air. When the projectile impacts on the free surface, the projectile will transfer energy to the water, and the flow begins to be separated from the projectile nose. The air first rushes into the cavity, so most part of the cavity is immediately filled with air. When the projectile enters into the water, at every instant, the kinetic energy loss of the projectile is equal to the kinetic and potential energy transferred to the liquid, resulting in liquid-to-vapor phase change, and so the cavity will expand. So far, in most of the published literatures about natural supercavitation, the projectiles are launched underwater or fixed in a water tunnel. In these cases only liquid phase and vapor phase are involved. But there are few published literatures about waterentry induced three-phase cavitation, which is more complicated. If the compressibility is ignored, the air and the projectile can be seen as a whole ‘‘imaginary solid”. And then the problem becomes a natural cavitation phenomenon. With the movement of the ‘‘imaginary solid”, the pressure near the nose decreases. So the vapor will be first generated near the nose of the ‘‘imaginary solid”. From Fig. 20 (a), it can be seen that the vapor is mainly distributed near the cavity boundary. Stinebring et al. [30] has reported that the rear of these cavities is a complex and often highly turbulent
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(a)
water
solution
t = 1 ms
The non-dimensional length L and diameter D of the cavities in water and drag-reducing solutions vs. time are compared in Figs. 21 and 22, respectively. Fig. 21 shows that the non-dimensional length of cavity in solution keeps larger than that in water all the time; the length difference between water and solution becomes larger with the increase of time, which is the same as obtained for the case of lower impact velocity V0 = 28.4 m/s. Fig. 22 shows that, in the first period (before about 1 ms), the diameters of cavity increase sharply
t = 2 ms 18
Water Solution
t = 3 ms
15 12
L
t = 4 ms
9
(b)
water
solution
6
t = 1 ms
3 0
1
2
3
4
5
t (ms)
t = 2 ms
Fig. 21. Non-dimensional length vs. time for the cases at V 0 ¼ 142:7 m/s.
5.4
Water Solution
t = 3 ms
t = 4 ms
D
4.8
Fig. 20. The gas-phase components inside the cavity in water and in drag-reducing solution. (a) Vapor component, (b) air component.
3.6
3.0 0
1
2
3
4
5
t (ms) Fig. 22. Non-dimensional diameter vs. time for the cases at V 0 ¼ 142:7 m/s.
Water Solution
2.0
1.5
Cd
region of the flow where the contents of the cavity are carried or entrained away. However, in the case of vaporous cavitation where the cavity contains mostly vapor, the loss of vapor is immediately made up by the evaporation from the walls of the cavity. After the formation of the reentrant jet into the cavity, the low pressure transforms the liquid jet into vapor; this is where the vapor behind the projectile comes from. Comparing the vapor region in water and in solution, it can be seen that the vapor region in solution is larger than that in water, which indicates that the drag-reducing solution enhances the formation of vapor (enhances phase change). Besides, there have been some evidences showing that, during the process of growth or collapse of a gas bubble in liquid, or the initiation stage of supercavitation, the surface tension may play a significant role. He [31] reported that the growth of a laser-induced cavitation bubble was suppressed by the surface tension and the increase of surface tension enhanced the collapse of the bubble. Shi et al. [32] stated that the surface tension keeps the process of cavity growth stable and keeps the cavity orbicular, and accelerates the process of bubble collapse. The abovementioned examples imply that the cavitation process can be enhanced with a decrease of surface tension: the bubble being larger in size and longer in life-time. Consequently, it can be conjectured naturally that the cavitating process might be effectively controlled by changing the local surface tension. In order to demonstrate such idea, natural supercavitation influenced by rheological characteristics of turbulent drag-reducing additives was discussed in detail in our previous paper very recently [17].
4.2
1.0
0.5
0.0 0
1
2
3
4
5
t (ms) Fig. 23. Time history of drag coefficients for projectile in water and in dragreducing solution cases at V 0 ¼ 142:7 m/s, respectively, obtained by numerical simulation.
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in both water and solution and the diameter in water is smaller than that in solution; after this period the diameter in water becomes larger than that in solution; then the diameters increase slowly from about 2.5 ms. When the time is larger than 4 ms, the diameters in both water and solution begin to decrease. Fig. 23 plots drag coefficients versus time for both water and drag-reducing solution cases. It can be seen that the drag coefficients rise sharply when the projectile impacts the liquid surface, and then decrease. The drag coefficient is slightly smaller in drag-reducing solutions than that in water. Compared with Fig. 10 for the case of lower impact velocity V0 = 28.4 m/s, it indicates that the dragreducing additives may have more significantly positive effects on the formation of water-entry supercavitation and further drag reduction of the projectile at lower impact velocity. 4. Conclusions This paper introduced a numerical simulation procedure to simulate water-entry supercavitating flows influenced by a dragreducing solution at typically low velocity and high velocity. The configuration and dynamic characteristics of water-entry supercavity, flow resistance and radial velocity of water-entry cavity boundary were discussed, respectively. It was obtained that the numerical simulation results are in consistence with experimental data, verifying the established numerical simulation procedures. The following conclusions are drawn from this study. (1) Comparisons of the cavity lengths, velocity attenuations, penetration distances and drag coefficients for water and drag-reducing solution cases at the higher and lower impact velocity cases indicated that the influence of drag-reducing additives on water-entry supercavitating flow, i.e., positive effects on the formation of water-entry supercavitation and further drag reduction of the projectile, is more significant at lower impact velocity. (2) The viscosity of the solution prevents the cavity from opening to air, resulting in a lower radial velocity of the cavity near the surface. The smaller surface tension of dragreducing solution makes the cavity boundary expands faster under the liquid surface, which leads to the larger size of the cavity in solution. (3) At higher impact velocity, the natural cavitation must be considered; due to the existence of the vapor rushing into the cavity from the atmosphere, the vapor is mainly distributed near the cavity boundary. 5. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this article. Acknowledgements This work is supported by National Natural Science Foundation of China under Grant Nos. 51276046 and 51421063. References [1] A.M. Worthington, R.S. Cole, Impact with a liquid surface, studied by the aid of instantaneous photography, Phil. Trans. Roy. Soc. Lond. A 189 (1897) 137–148.
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