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Physical and numerical study on unsteady shedding behaviors of ventilated partial cavitating flow around an axisymmetric body
Ocean Engineering 197 (2020) 106884
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Physical and numerical study on unsteady shedding behaviors of ventilated partial cavitating flow around an axisymmetric body Guoyu Wang, Decai Kong, Qin Wu *, Taotao Liu **, Yanan Zheng, Biao Huang School of Mechanical Engineering, Beijing Institute of Technology, 100081, Beijing, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cavitation Ventilated partial cavitating flow Shedding mechanism Axisymmetric body
The objective is to investigate the unsteady ventilated partial cavitating flow characteristics with focus on the unsteady shedding behaviors via combined experimental and numerical methods. Experimental results are presented for an axisymmetric body in a water tunnel with the flow pattern recorded by a high speed camera. The numerical simulation is performed with a filter-based turbulence model. Good agreement can be obtained between the experimental and numerical results. The ventilated cavity around the axisymmetric body experi ences the rapid growth stage, the growth with small pulsation stage and the periodic shedding stage. The typical ventilated cavitating flow patterns at different angles of attack have also been studied to further investigate the effect of the asymmetry on the unsteady shedding behaviors of ventilated partial cavitating flow. With the in crease of the angle of attack, the ventilated cavity develops asymmetrically and the non-transparent gas-liquid mixture region is gradually concentrated on the opposing stream surface along with the transparent cavity region increasing on the confronted stream surface.
1. Introduction Cavitation generally occurs around high-speed underwater vehicles, which may lead to many problems such as pressure fluctuation, sudden change in load, vibration, noise, etc (Ausoni et al., 2007; Liu et al., 2018a; Wu et al., 2015, 2018a; Ma et al., 2018). Ventilated cavitation is to further agitate the cavitation via injecting non-condensable gas to form a continuous cavity covering the object surface, which is consid ered as an effective method to control the cavitating flow. As the pri mary benefit of ventilated cavitation versus natural cavitation is the friction drag reduction to enable higher speed for the underwater ve hicles, much effort has been conducted to achieve this, especially when huge amount of gas is injected into the cavity, a supercavity will be formed and the friction drag can be reduced up to 90% (Wang et al., 2017, 2018a; Yang et al., 2017; Hjartarson et al., 2009). There have been many experimental and numerical studies exam ining the ventilated cavitating flow characteristics. Wosonic et al. (Wosnik and Schauer, 2003) experimentally investigated the amount of ventilation gas required to sustain an artificial cavity and they found that the structural shape significantly affect the air demand through the interaction between the cavity wake and the structure. Javadpour et al.
(2017) presented a numerical and experimental study on the ventilated supercavitaing flow around a cone cavitator. They found that the maximum cavity length and diameter declines with the increase of the cavitation number. To date, four main ventilated supercavitation closure modes have been reported by many researchers, which are the re-entrant jet, twin vortex, quad vortex and pulsating vortex (Wosnik and Arndt, 2005; Kawakami and Arndt, 2012; Skidmore et al., 2016; Wang et al., 2018b). In addition, Karn et al. (Karn et al., 2016; Karn and Rosiejka, 2017) experimentally investigated the ventilated super cavitating flow physics and a number of new closure modes were found during the transition between the above modes. More recently, Jiang et al. (2018) conduct an experimental study of a new ventilation mechanism for supercavitation, gas jet ventilated supercavitation, over a broad range of ventilation and flow conditions. Different supercavity patterns were observed with different ventilation rate and distinct cavity regimes were investigated. Wu et al. (2019) investigate the internal flow of a ventilated supercavity by experimental method with fog flow visualization and particle image velocimetry (PIV) measurements. They found that the internal flow exhibits three distinct regions, including the ventilated influence region near the cavitator, the extended internal boundary layer along the liquid-gas interface and the reverse flow
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Q. Wu), [email protected] (T. Liu). https://doi.org/10.1016/j.oceaneng.2019.106884 Received 15 January 2019; Received in revised form 2 December 2019; Accepted 23 December 2019 Available online 7 January 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.
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region occupying a large centre portion of the supercavity. Previous studies are mainly related to ventilated supercavitation, while during the process of achieving supercavitation, the ventilated partial cavity is frequently observed, with the cavity covering part of the vehicle body and the exposing part under transient hydrodynamic load. Korpriva et al. (Kopriva et al., 2005) experimentally investigated the ventilated partial cavitation on the OK-2003 body and they found that the drag was effectively reduced and the lift to drag ratio was sharply increased. Xiang et al., 2011a, 2011b proposed a numerical scheme to model the development of the ventilated partial cavitation, with the liquid region upstream and the bubble dispersion region downstream were well simulated. The results showed that a remarkable drag reduction was achieved for the cavitation cases. Liu et al., 2017, 2018b performed an experimental and numerical study on the ventilated par tial cavitating flow structures. Four relatively stable flow patterns and three unstable ones were observed with different Froude numbers and gas entrainment coefficients. Wang et al. (2012) investigated the un steady shedding phenomenon of the ventilated partial cavitation around an axisymmetric projectile. They found that the cavity breaks off by the interaction between the re-entry jet and the gas injection in the middle of the projectile. Based on the understanding of the ventilated partial cavitating flow pattern, various factors which may affect the ventilated cavitating flow characteristics have been considered. Vlasenko and Savchenko (2011) experimentally investigated the ventilated super cavitation around axisymmetric bodies and explored the effect of body geometry on the unsteady cavity size. They found that the gas injection rate varied with the cavity length within the cylindrical part of the model. Salari (Salari et al., 2017) experimentally studied the effects of important parameters such as the cavitation number, upstream flow velocity and cone angle of cavitators on the drag coefficients as well as cavity shapes and relevant dimensions. Wang et al. (2018c) investigate the influence of ventilated cavitation on vortex shedding in the wake behind a bluff body combing high-speed camera and TR-PIV measure ment with POD analysis. The results show that three different vortex shedding behaviors exist in the wake. Among the above researches, much efforts have been done for the
global flow structures and corresponding hydrodynamic characteristics of the ventilated partial cavitation under different flow conditions. To better design and maneuver the high-speed underwater vehicles, further investigation of the unsteady ventilated partial cavitating flow charac teristics, with special focus on the unsteady shedding behaviors, is still necessary to improve the understanding of the unsteady flow mecha nism and the stability of ventilated partial cavitation. 2. Experimental setup The experimental setup, high-seed water tunnel in Beijing Institute of Technology (Wu et al., 2018b; Wang et al., 2018d), is schematically illustrated in Fig. 1, which consists of a reservoir, a suction tank, a test section, an axial flow pump, a ball valve, an electromagnetic flowmeter and pipes. The pressure tin is connected to the suction tank in order to adjust the system pressure and achieve different cavitating flow condi tions. The test section in the water tunnel is 0.7m (length) � 0.07m (width) � 0.19m (height). The high-speed water tunnel is able to operate at flow speed up to 20 m/s with a turbulence level of 2%. A conical axisymmetric body with the ventilation seam at the nose and the mounting at the rear part of the body is used in the experiments, as shown in Fig. 2. It is sting-mounted (hollow cylindrical strut) and the length is L ¼ 0.126m, and the diameter is D ¼ 0.02m (Fig. 3). Based on the diameter of the axisymmetric body, 0.02m, and the angle of attack 0� , is about 9%, so the blockage effect has been ignored in the present work. Further investigation of the blockage effect on the cavitating flow and corresponding hydrodynamic characteristics will be conducted in future work. The ventilation pipe is two fifths of the axisymmetric body in diameter and the ventilated flow speed and pressure are controlled by a control valve. To avoid the non-uniform of the ventilated gas flow due to the gravity effect, two rows of air injection holes are distributed circularly, with four holes in each row. For the visualization system, a ReDlake HG-LE high-speed camera is used to record the high-speed videos of cavity patterns, with a sampling frequency of 3000 fps. In general, the uncertainty of both the flow velocity and the upstream pressure is 2% (Huang et al., 2013). The uncertainties of the
Fig. 1. Schematic of the cavitation water tunnel. 2
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Fig. 2. Schematic of the axisymmetric body and the ventilation system.
Fig. 3. Computational domain and boundary conditions.
electromagnetic flowmeter, the pressure transducer and the gas entrainment coefficient are decided by the instrument accuracy (Wang et al., 2015). The uncertainty of the electromagnetic flowmeter is 0.5%, and the uncertainty of the pressure transducer is 0.25%. The maximum uncertainty in the gas entrainment coefficient is about 2%.
solves the transport of turbulent kinetic energy k and dissipation rate ε in the two partial differential equations: � �� � � ∂ρm ku ∂ ρm kuj ∂ μ ∂k ¼ μm þ t (5) þ Pt ρ m ε þ ∂xj ∂t ∂xj σ k ∂xj
3. Numerical methods
∂ρm ε ∂ ρm εuj ∂ ¼ þ ∂t ∂xj ∂xj
�
3.1. Navier-Stokes equations
μt ¼ C μ
The unsteady Reynolds Average Navier-Stokes (URANS) equations are used to maintain the balance between the accuracy and the computational cost. � ∂ρm ∂ ρm uj ¼0 (1) þ ∂t ∂xj �
∂ðρm ui Þ ∂ ρm ui uj ¼ þ ∂t ∂xj
�
�
∂p ∂ ∂ui ∂uj ðμ þ μt Þ þ þ ∂xi ∂xj ∂xj ∂xi
(2) (3)
μm ¼ μl αl þ μa αa
(4)
μm þ
�
�
μt ∂ε ε þ Cε1 Pt σε ∂xj k
ρm k 2 ε
Cε2
ρm ε2 k
(6) (7)
where σk and σ ε are the turbulence model constants, C1 and C2 are the turbulence model parameters that regulate respectively the production and destruction of the dissipation of turbulence kinetic energy, Pt is the turbulent production term and μt is the turbulent viscosity. In the RANS model, the eddy viscosity may be excessive and smear out the flow structures within a given grid resolution range. To offer a more satisfactory resolution, the effective viscosity be reduced by imposing a filter on the turbulence model. As set values of the filter size are smaller than the length scales returned by the conventional RANS model, the turbulent scales smaller than the filter size could not be resolved. In the paper, the filter-based turbulent model (FBM) is applied and the turbulent eddy viscosity is modified by comparing the local turbulence length scale, k3/2/ε, and the filter size λ, which is selected based on the local mesh size.
��
ρm ¼ ρl αl þ ρa αa
��
where u is the velocity, ρm is the mixture fluid density, ρl is the liquid density, ρa is the air density, p is the pressure, μm is the mixture laminar viscosity, μt is the turbulent viscosity, μl and μa are the liquid and vapor dynamic viscosity respectively. The subscripts i, j denote the directions of the Cartesian coordinates.
μt ¼ C μ
ρm k 2 f ; ε FBM
� � 1 fFBM ¼ min 1; ; lRANS
3.2. Turbulence model
(8)
Cμ ¼ 0:09
lRANS ¼
k3=2 λ⋅ε
(9)
when the λ≫k3/2/ε, such as very near the wall where the local mesh size
The k-ε turbulence model proposed by Launder and Spalding (1974) 3
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Ocean Engineering 197 (2020) 106884
is much greater than the turbulence length scale, μt ¼ Cμρmk2/ε, i.e. the standard k-ε model is recovered. When λ≪k3/2/ε, such as away from the wall where the local mesh size is smaller than the turbulence length scale, μt ¼ Cμρmλk1/2.
Table 1 Grid convergence study based on Richardson extrapolation (Ji et al., 2017; Long et al., 2019).
3.3. Numerical setup The free stream velocity U∞ ¼ Q/A, where Q is the flow rate measured by the electromagnetic flow meter and A is the section area of the test section. The ventilated flow pressure is 4atm and the ventilated flow rate Qin ranges from 0 to 800L/h. Thus the Froude number and the air entrainment coefficient are defined as: 1=2
Fr ¼ U∞ =ðgDÞ CQ ¼ Qin
�
U∞ D2
�
Mesh
I
II
III
Error (%)
Grid nodes Cd-mean
650000 0.114
920000 0.125
1300000 0.127
0.44
4. Results and discussion 4.1. Global multiphase structures in ventilated partial cavitating flows Figs. 5 and 6 shows the evolution of cavity length and corresponding typical ventilated partial cavitating flow patterns. Also, the pressure contours on the test body are shown. In the simulations, the isosurface of 10% gas volume fraction is used to illustrate the cavity boundary. Good agreement can be obtained between the experimental and numerical results. To better illustrate the comparison between the test and computational results, the mean square errors between the experimental and numerical results has been calculated, which is 0.08. According to the flow patterns, the ventilated cavity around the axisymmetric body
(13) (14)
where, U∞ is the average velocity, Qin is the volume flow of gas and D is the diameter of the body. The ventilated cavitating flow is simulated on the computing plat form of the CFX Solver. The computational domain and boundary con ditions are given according to the experimental setup. A no-slip boundary condition is imposed on the test body surface and boundaries of the tunnel. A constant turbulent intensity of 2% set at the inlet boundary, which is consistent to the measured value in the experiment. The inlet boundary of the domain is set as the opening boundary, both for the liquid and gas phases, with the inlet velocity of the gas injection set as the bulk mass flow rate. The ventilated cavitation simulations have been initialized with the steady-state, fully-wetted solutions, with den sity of the liquid (water at 25 � C) is ρl ¼ 998 kg/m3, viscosity μl ¼ ρlνl ¼ 1.139 � 10 3 Pa⋅s, and density of the air (air at 25 � C) is ρg ¼ 1.185 kg/ m3, viscosity μg ¼ 18.31 � 10 6 Pa⋅s. The 3D mesh, as shown in Fig. 4, is composed of 1300000 elements, and the value of y þ at the wall is smaller than 45. Three meshes, which give a constant refinement ratio r ¼ Nfine/Ncourse ¼ 1.414, are used in the grid convergence study, where N is the grid nodes number, as shown in Table 1. The relative error is estimated based on Richardson extrapolation (Ji et al., 2017; Long et al., 2019). As the relative errors are less than 1%, the results for mesh III are considered as mesh independent so the subsequent simulations are performed with the meshing scheme of mesh III. The time integration scheme is a second-order backward Euler algorithm, and the spatial derivatives are computed using a second-order upwind scheme. The time step chosen for the simulation is Δt ¼ 5 � 10 4s, based on convergence studies.
Fig. 5. Evolution of the measured and predicted cavity lengths.
Fig. 4. Mesh distributions. 4
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(d). After that, the cavity development transits from the rapid growth to growth with small pulsation. During this stage, as shown in Fig. 5, the growth rate of the maximum cavity length declines with small pulsation. This is because the attached cavity develops, accompanied with the continuous shedding of the small-scale detached cavities at the rear part of the attached cavity, as shown in Fig. 6(e) and (f). Until t ¼ t0þ108.5 ms, the attached cavity develops to the maximum, following by a sig nificant fluctuation of the cavity length. During this stage, large-scale detached cavity sheds quasi-periodically, as shown in Fig. 6(g). It should be noted that the cavity shapes are extremely dissymmetrical along the circumference of the axisymmetric body. In other words, three-dimensional characteristics are extremely impressive in the un steady cavitating flows around an axisymmetric body. This is also mentioned by Lindau (Lindau et al., 2002) and Hu (Hu et al., 2015). Further investigation of the three-dimensional effect on the ventilated cavitating flows and corresponding hydrodynamic characteristics will be conducted in future work. Fig. 7 shows the evolution of the predicted pressure coefficient around the axisymmetric body at different sections. For the section at x/ D ¼ 0.5 and x/D ¼ 1, the evolution of pressure coefficients is markedly different from that at the other sections, which is affected by the highspeed gas flow from the ventilated hole. As for the section from x/D ¼ 1.5 to x/D ¼ 4, a pressure peak appears successively when the attached cavity gradually reached each section position and then drop dramatically to the saturated vapor pressure as the attached cavity covers the position. As for the pressure fluctuation occurred after the attached cavity covering the position, such as the section at x/D ¼ 1.5, x/D ¼ 2 and x/D ¼ 2.5, this is due to the shedding of the small-scale detached cavities at the rear part of the attached cavity. While for the pressure fluctuation before the pressure peaks, such as the section at x/ D ¼ 3.5 and x/D ¼ 4, this is because of the disturbance caused by the upstream detached shedding cavities. When it develops to the third stage, high-amplitude and low-frequency variation of the pressure co efficients can be obviously observed, which is corresponding to the pe riodical shedding of the large-scale detached cavities. To further investigate the relationship between re-entrant jet and cavity shedding, Fig. 8 shows the evolution of the velocity distribution inside the ventilated cavity, where the solid red line indicates the cavity boundary. It can be seen from Fig. 8(a) that the re-entrant jet mainly moves upstream along the wall of the axisymmetric body. When meets with the ventilation gas flow, it is pushed far away from the axisym metric body, resulting to the rising of the cavity diameter and the discontinuity of the cavity boundary curvature. The rolled-up re-entrant jet flow downstream with the main flow, meanwhile the concave cavity occurs at the position where the cavity boundary curvature changes significantly, as shown in Fig. 8(b). With the attached cavity develops, the concave position moves downstream and even deeper, as shown in Fig. 8(c). Until the concave cavity boundary reaches to the re-entrant jet Fig. 6. Evolution of the ventilated cavity shapes.
experiences three stages: rapid growth, growth with small pulsation and periodic shedding. During the first stage, the ventilated gas flow up stream and downstream from the ventilated hole and a ring-shape cavity is formed around the axisymmetric body, with the cavity length increasing dramatically, as shown in Fig. 5 and Fig. 6(a)–(b). Mean while, the re-entrant jet generates due to the pressure gradient at the rear part of the attached cavity and moves towards the shoulder of the axisymmetric body, resulting that the forepart of the attached cavity appears transparent and the rear part in a fuzzy shape, as shown in Fig. 6 (b). As the re-entrant jet encounters the main flow, part of the attached cavity breaks off due to the interaction of the two opposite flows, as shown in Fig. 6(c). Affected by the continuous ventilated gas flow, the concave deformation occurs on the gas-liquid interface of the cracked cavity and three-dimensional effect of the ventilated cavity can be obviously observed around the axisymmetric body, as shown in Fig. 6
Fig. 7. Predicted pressure distribution at different axial position. 5
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the measured and predicted cavity pattern is 0.18, 0.12 and 0.07 for the axisymmetric body with α ¼ 0� , 5� and 8� respectively. As shown in Fig. 9(a), at α ¼ 0� , symmetrical attached cavity is formed and develops around the axisymmetric body, which is filled with the gas-liquid mixture and appears non-transparent. With the increase of the angle of attack, the attached cavity develops asymmetrically, with the attached cavity length on the opposing stream surface is longer than that on the confronted stream surface. Moreover, the non-transparent gasliquid mixture region is gradually concentrated on the opposing stream surface and the attached cavity on the confronted stream surface ap pears transparent, as shown in Fig. 9(b). When the angle of attack is further increased to 8� , the cavity asymmetry is more obvious and the transparent cavity region increases, with the non-transparent gas-liquid mixture only existed at the end of the attached cavity, as shown in Fig. 9 (c). In order to further investigate the influence of the angle of attack on the ventilated cavity patterns and flow structures, Fig. 10 shows the water volume fraction at different cross-sections of the axisymmetric body with different angles of attack when the main attached cavity keeps steady. The cavity boundary is illustrated with the isosurface of 10% gas volume fraction. It can be seen that when α ¼ 0� , the gas volume fraction at different cross-sections distribute uniformly in the circum ferential direction, with the attached cavity diameter increasing along the flow direction until x/D ¼ 2.5, then fluctuating slightly between x/D ¼ 2.5 and x/D ¼ 3.5 and finally decreasing until the end of the attached cavity. When the angle of attack increases to α ¼ 5� , the gas volume fraction at the cross-section x/D ¼ 0.5 which is near the shoulder of the axisymmetric body remains uniform in the circumferential direction, as shown in Fig. 10(b). While the gas volume fraction at the latter crosssections is accumulated upwards and finally exhibits as the crescent shape. As the angle of attack further increases to α ¼ 8� , the axial po sition of the cross-section with a crescent-shape distribution of the gas volume fraction is much closer to the head of the axisymmetric body, indicating that the greater the angle of attack, the more asymmetrical cavity is obtained. What’s more, assuming that the pressure gradient has no component along the attached cavity closure line, the re-entrant jet velocity is directed in the opposite direction of the incident flow as the angle of attack α ¼ 0� . While as the angle of attack increases, the cavity closure line is inclined and with the increase of the angle of attack, the angle of inclination for the attached cavity closure line decreases. Ac cording to the conservation of the tangential momentum, the re-entrant jet velocity has two components, that are perpendicular and along the closure line. This is also mentioned by de Lange (De Lange et al., 1994)
Fig. 8. Velocity distribution during the periodic shedding stage.
near the wall, the latter part of the attached cavity with a rising cavity diameter breaks off with the attached cavity and large-scale detached cavity is formed and shed downstream, as shown in Fig. 8(d). 4.2. The influence of angle of attack on the ventilated partial cavitating flows To further investigate the effect of the asymmetry on the unsteady shedding behaviors of ventilated partial cavitating flow, Fig. 9 shows the typical ventilated cavitating flow patterns at different angles of attack α. In which, the Froude number Fr ¼ 15.66 and the gas entrainment co efficient CQ ¼ 0.06. Reasonable agreement can be obtained between the experimental and numerical results, with the mean square error between
Fig. 9. Typical cavity patterns at different angles of attack (Fr ¼ 15.66，CQ ¼ 0.06). 6
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Fig. 11. Pressure distribution on the surface of the axisymmetric body at different angles of attack.
5. Conclusions Experimental and numerical studies have been conducted for an axisymmetric body to investigate the unsteady shedding behaviors of ventilated partial cavitating flow. High speed visualization technique has been adopted to present the global multiphase structures and nu merical simulation has been combined to further investigate the cavity shedding mechanism. The primary findings include: (1) The ventilated cavity around the axisymmetric body experiences three stages: rapid growth, growth with small pulsation and pe riodic shedding. During the rapid growth stage, the ventilated gas flow upstream and downstream from the ventilated hole and a ring-shape cavity is formed around the axisymmetric body, along with the interaction between the main attached cavity and the reentrant jet. During the growth with small pulsation stage, the attached cavity develops, accompanied with the continuous shedding of the small-scale detached cavities at the rear part of the attached cavity, resulting that the growth rate of the maximum cavity length declines with small pulsation. During the periodic shedding stage, large-scale detached cavity sheds quasiperiodically accompanied with a significant fluctuation of the attached cavity length. (2) The re-entrant jet has a significant role on the unsteady cavity shedding behaviors. The re-entrant jet generated due to the pressure gradient and moves upstream along the wall of the axisymmetric body. When meets with the ventilation gas flow, the re-entrant jet is pushed far away from the axisymmetric body, resulting to the rising of the attached cavity diameter and the discontinuity of the cavity boundary curvature, which leads to the attached cavity depressed inward. Until the concave cavity boundary reaches to the re-entrant jet near the wall, the latter part of the attached cavity with a rising cavity diameter breaks off with the attached cavity and large-scale detached cavity is formed and shed downstream. (3) The angle of attack influence the ventilated partial cavitating flow patterns. Symmetrical cavity is formed and develops around the axisymmetric body with the angle of attack α ¼ 0� . The attached cavity develops asymmetrically with the angle of attack α ¼ 5� and 8� . The non-transparent gas-liquid mixture region is gradually concentrated on the opposing stream surface and the
Fig. 10. Gas distribution at different cross sections.
and Franc (2001). To further investigate the effect of the angles of attack on the ventilated partial cavitation, Fig. 11 shows the predicted pressure dis tribution at both confronted and opposing stream surface of the axisymmetric body with different angles of attack. As the angle of attack increases, the high pressure region corresponding to the attached cavity closure at the confronted stream surface moves upstream, and that at the opposing stream surface remains almost steady. This is because the angle of attack mainly affect the attached cavity length at the confronted stream surface of axisymmetric body and has little effect on the attached cavity length at the opposing stream surface. As the relative thickness of the attached cavity and the re-entrant jet is an important factor for the detached cavity shedding, in order to reveal the cavity shedding mechanism of the axisymmetric body at different angles of attack, Fig. 12 shows the attached cavity and reentrant jet boundary on the cross section at x/D ¼ 2 with different an gles of attack. When the angle of attack α ¼ 0� , the attached cavity is much thicker than the re-entrant jet layer and the interaction between the re-entrant jet and the ventilated gas flow is weak so that the reentrant jet can reach to the head of the attached cavity resulting to the partial shedding of the ventilated cavity. With the increases of the angle of attack of the axisymmetric body, the maximum thickness of the re-entrant jet is close to the attached cavity boundary and there is a strong interaction between the re-entrant jet and ventilation gas flow, resulting that the re-entrant jet mainly exists in the rear region of the cavity, which is corresponding to the small-scale detached cavity shedding at the end of the attached cavity.
7
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Fig. 12. Relationship between re-entrant jet and cavity boundary layer at different angles of attack.
transparent cavity region increases on the confronted stream surface.
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Author contributions section Guoyu Wang: Resources, Supervision, Funding acquisition. Decai Kong: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing-Original Draft. Qin Wu: Software, Validation, Writing-Review & Editing, Supervi sion, Funding acquisition. Taotao Liu: Software, Validation, Writing-Review & Editing, Super vision, Funding acquisition. Yanan Zheng: Project administration. Biao Huang: Conceptualization, Software, Validation, WritingReview & Editing, Supervision, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant Nos: 51839001, 51909002, 51679005 and 91752105), the National Natural Science Foundation of Beijing (Grant No: 3172029) and the Open Foundation of Key Laboratory of Fluid and Power Ma chinery, Ministry of Education of China in Xihua University (Grant Nos: szjj2018-124, szjj2019-024). References Ausoni, P., Farhat, M., Escaler, X., et al., 2007. Cavitation influence on von K� arm� an vortex shedding and induced hydrofoil vibrations. J. Fluids Eng. 129 (8), 966–973. De Lange, D.F., de Bruin, G.J., van Wijngaarden, L., 1994. On the mechanism of cloud cavitation - experiment and modelling. Proceeding of the Second International Symposium on Cavitation 45–49. Franc, J.P., 2001. Partial Cavity Instabilities and Re-entrant Jet. CAV2001. Hjartarson, A., Mokhtarzadeh, H., Kawakami, E., et al., 2009. A Dynamic Test Platform for Evaluating Control Algorithms for a Supercavitating Vehicle 110, 1–4. Hu, C.L., Wang, G.Y., Chen, G.H., et al., 2015. Three-dimensional unsteady cavitating flows around an axisymmetric body with a blunt headform[J]. J. Mech. Sci. Technol. 29 (3), 1093–1101. Huang, B., Yong, Y.L., Wang, G.Y., Shyy, W., 2013. Combined experimental and computational investigation of unsteady structure of sheet/cloud cavitation. J. Fluids Eng. 135, 071301-1-071301-16. Javadpour, S.M., Farahat, S., Ajam, H., et al., 2017. Experimental and numerical study of ventilated supercavitation around a cone cavitator. Heat Mass Transf. 53 (5), 1491–1502. Ji, B., Long, Y., Long, X., Qian, Z., Zhou, J., 2017. Large eddy simulation of turbulent attached cavitating flow with special emphasis on large scale structures of the hydrofoil wake and turbulence-cavitation interactions. J. Hydrodyn. 29 (1), 27–39.
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Wu, Q., Huang, B., Wang, G., Gao, Y., 2015. Experimental and numerical investigation of hydroelastic response of a flexible hydrofoil in cavitating flow. Int. J. Multiph. Flow 74, 19–33. Wu, Q., Huang, B., Wang, G., Cao, S., 2018. The transient characteristics of cloud cavitating flow over a flexible hydrofoil. Int. J. Multiph. Flow 99, 162–173. Wu, Q., Wang, C.C., Huang, B., Wang, G.Y., Cao, S.L., 2018. Measurement and prediction of cavitating flow-induced vibrations. J. Hydrodyn. 30 (6), 1064–1071. Wu, Y., Liu, Y., Shao, S.Y., et al., 2019. On the internal flow of a ventilated supercavity. J. Fluid Mech. 862, 1135–1165.
Xiang, M., Cheung, S.C.P., Yeoh, G.H., et al., 2011. On the numerical study of bubbly flow created by ventilated cavity in vertical pipe. Int. J. Multiph. Flow 37, 756–768. Xiang, M., Cheung, S.C.P., Tu, J.Y., Zhang, W.H., 2011. Numerical research on drag reduction by ventilated partial cavity based on two-fluid model. Ocean. Eng. 38, 2023–2032. Yang, D., Xiong, Y.L., Guo, X.F., 2017. Drag reduction of a rapid vehicle in supercavitating flow. International Journal of Naval Architecture and Ocean Engineering 9 (1), 35–44.
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Physical and numerical study on unsteady shedding behaviors of ventilated partial cavitating flow around an axisymmetric body Guoyu Wang, Decai Kong, Qin Wu *, Taotao Liu **, Yanan Zheng, Biao Huang School of Mechanical Engineering, Beijing Institute of Technology, 100081, Beijing, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cavitation Ventilated partial cavitating flow Shedding mechanism Axisymmetric body
The objective is to investigate the unsteady ventilated partial cavitating flow characteristics with focus on the unsteady shedding behaviors via combined experimental and numerical methods. Experimental results are presented for an axisymmetric body in a water tunnel with the flow pattern recorded by a high speed camera. The numerical simulation is performed with a filter-based turbulence model. Good agreement can be obtained between the experimental and numerical results. The ventilated cavity around the axisymmetric body experi ences the rapid growth stage, the growth with small pulsation stage and the periodic shedding stage. The typical ventilated cavitating flow patterns at different angles of attack have also been studied to further investigate the effect of the asymmetry on the unsteady shedding behaviors of ventilated partial cavitating flow. With the in crease of the angle of attack, the ventilated cavity develops asymmetrically and the non-transparent gas-liquid mixture region is gradually concentrated on the opposing stream surface along with the transparent cavity region increasing on the confronted stream surface.
1. Introduction Cavitation generally occurs around high-speed underwater vehicles, which may lead to many problems such as pressure fluctuation, sudden change in load, vibration, noise, etc (Ausoni et al., 2007; Liu et al., 2018a; Wu et al., 2015, 2018a; Ma et al., 2018). Ventilated cavitation is to further agitate the cavitation via injecting non-condensable gas to form a continuous cavity covering the object surface, which is consid ered as an effective method to control the cavitating flow. As the pri mary benefit of ventilated cavitation versus natural cavitation is the friction drag reduction to enable higher speed for the underwater ve hicles, much effort has been conducted to achieve this, especially when huge amount of gas is injected into the cavity, a supercavity will be formed and the friction drag can be reduced up to 90% (Wang et al., 2017, 2018a; Yang et al., 2017; Hjartarson et al., 2009). There have been many experimental and numerical studies exam ining the ventilated cavitating flow characteristics. Wosonic et al. (Wosnik and Schauer, 2003) experimentally investigated the amount of ventilation gas required to sustain an artificial cavity and they found that the structural shape significantly affect the air demand through the interaction between the cavity wake and the structure. Javadpour et al.
(2017) presented a numerical and experimental study on the ventilated supercavitaing flow around a cone cavitator. They found that the maximum cavity length and diameter declines with the increase of the cavitation number. To date, four main ventilated supercavitation closure modes have been reported by many researchers, which are the re-entrant jet, twin vortex, quad vortex and pulsating vortex (Wosnik and Arndt, 2005; Kawakami and Arndt, 2012; Skidmore et al., 2016; Wang et al., 2018b). In addition, Karn et al. (Karn et al., 2016; Karn and Rosiejka, 2017) experimentally investigated the ventilated super cavitating flow physics and a number of new closure modes were found during the transition between the above modes. More recently, Jiang et al. (2018) conduct an experimental study of a new ventilation mechanism for supercavitation, gas jet ventilated supercavitation, over a broad range of ventilation and flow conditions. Different supercavity patterns were observed with different ventilation rate and distinct cavity regimes were investigated. Wu et al. (2019) investigate the internal flow of a ventilated supercavity by experimental method with fog flow visualization and particle image velocimetry (PIV) measurements. They found that the internal flow exhibits three distinct regions, including the ventilated influence region near the cavitator, the extended internal boundary layer along the liquid-gas interface and the reverse flow
* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (Q. Wu), [email protected] (T. Liu). https://doi.org/10.1016/j.oceaneng.2019.106884 Received 15 January 2019; Received in revised form 2 December 2019; Accepted 23 December 2019 Available online 7 January 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved.
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region occupying a large centre portion of the supercavity. Previous studies are mainly related to ventilated supercavitation, while during the process of achieving supercavitation, the ventilated partial cavity is frequently observed, with the cavity covering part of the vehicle body and the exposing part under transient hydrodynamic load. Korpriva et al. (Kopriva et al., 2005) experimentally investigated the ventilated partial cavitation on the OK-2003 body and they found that the drag was effectively reduced and the lift to drag ratio was sharply increased. Xiang et al., 2011a, 2011b proposed a numerical scheme to model the development of the ventilated partial cavitation, with the liquid region upstream and the bubble dispersion region downstream were well simulated. The results showed that a remarkable drag reduction was achieved for the cavitation cases. Liu et al., 2017, 2018b performed an experimental and numerical study on the ventilated par tial cavitating flow structures. Four relatively stable flow patterns and three unstable ones were observed with different Froude numbers and gas entrainment coefficients. Wang et al. (2012) investigated the un steady shedding phenomenon of the ventilated partial cavitation around an axisymmetric projectile. They found that the cavity breaks off by the interaction between the re-entry jet and the gas injection in the middle of the projectile. Based on the understanding of the ventilated partial cavitating flow pattern, various factors which may affect the ventilated cavitating flow characteristics have been considered. Vlasenko and Savchenko (2011) experimentally investigated the ventilated super cavitation around axisymmetric bodies and explored the effect of body geometry on the unsteady cavity size. They found that the gas injection rate varied with the cavity length within the cylindrical part of the model. Salari (Salari et al., 2017) experimentally studied the effects of important parameters such as the cavitation number, upstream flow velocity and cone angle of cavitators on the drag coefficients as well as cavity shapes and relevant dimensions. Wang et al. (2018c) investigate the influence of ventilated cavitation on vortex shedding in the wake behind a bluff body combing high-speed camera and TR-PIV measure ment with POD analysis. The results show that three different vortex shedding behaviors exist in the wake. Among the above researches, much efforts have been done for the
global flow structures and corresponding hydrodynamic characteristics of the ventilated partial cavitation under different flow conditions. To better design and maneuver the high-speed underwater vehicles, further investigation of the unsteady ventilated partial cavitating flow charac teristics, with special focus on the unsteady shedding behaviors, is still necessary to improve the understanding of the unsteady flow mecha nism and the stability of ventilated partial cavitation. 2. Experimental setup The experimental setup, high-seed water tunnel in Beijing Institute of Technology (Wu et al., 2018b; Wang et al., 2018d), is schematically illustrated in Fig. 1, which consists of a reservoir, a suction tank, a test section, an axial flow pump, a ball valve, an electromagnetic flowmeter and pipes. The pressure tin is connected to the suction tank in order to adjust the system pressure and achieve different cavitating flow condi tions. The test section in the water tunnel is 0.7m (length) � 0.07m (width) � 0.19m (height). The high-speed water tunnel is able to operate at flow speed up to 20 m/s with a turbulence level of 2%. A conical axisymmetric body with the ventilation seam at the nose and the mounting at the rear part of the body is used in the experiments, as shown in Fig. 2. It is sting-mounted (hollow cylindrical strut) and the length is L ¼ 0.126m, and the diameter is D ¼ 0.02m (Fig. 3). Based on the diameter of the axisymmetric body, 0.02m, and the angle of attack 0� , is about 9%, so the blockage effect has been ignored in the present work. Further investigation of the blockage effect on the cavitating flow and corresponding hydrodynamic characteristics will be conducted in future work. The ventilation pipe is two fifths of the axisymmetric body in diameter and the ventilated flow speed and pressure are controlled by a control valve. To avoid the non-uniform of the ventilated gas flow due to the gravity effect, two rows of air injection holes are distributed circularly, with four holes in each row. For the visualization system, a ReDlake HG-LE high-speed camera is used to record the high-speed videos of cavity patterns, with a sampling frequency of 3000 fps. In general, the uncertainty of both the flow velocity and the upstream pressure is 2% (Huang et al., 2013). The uncertainties of the
Fig. 1. Schematic of the cavitation water tunnel. 2
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Fig. 2. Schematic of the axisymmetric body and the ventilation system.
Fig. 3. Computational domain and boundary conditions.
electromagnetic flowmeter, the pressure transducer and the gas entrainment coefficient are decided by the instrument accuracy (Wang et al., 2015). The uncertainty of the electromagnetic flowmeter is 0.5%, and the uncertainty of the pressure transducer is 0.25%. The maximum uncertainty in the gas entrainment coefficient is about 2%.
solves the transport of turbulent kinetic energy k and dissipation rate ε in the two partial differential equations: � �� � � ∂ρm ku ∂ ρm kuj ∂ μ ∂k ¼ μm þ t (5) þ Pt ρ m ε þ ∂xj ∂t ∂xj σ k ∂xj
3. Numerical methods
∂ρm ε ∂ ρm εuj ∂ ¼ þ ∂t ∂xj ∂xj
�
3.1. Navier-Stokes equations
μt ¼ C μ
The unsteady Reynolds Average Navier-Stokes (URANS) equations are used to maintain the balance between the accuracy and the computational cost. � ∂ρm ∂ ρm uj ¼0 (1) þ ∂t ∂xj �
∂ðρm ui Þ ∂ ρm ui uj ¼ þ ∂t ∂xj
�
�
∂p ∂ ∂ui ∂uj ðμ þ μt Þ þ þ ∂xi ∂xj ∂xj ∂xi
(2) (3)
μm ¼ μl αl þ μa αa
(4)
μm þ
�
�
μt ∂ε ε þ Cε1 Pt σε ∂xj k
ρm k 2 ε
Cε2
ρm ε2 k
(6) (7)
where σk and σ ε are the turbulence model constants, C1 and C2 are the turbulence model parameters that regulate respectively the production and destruction of the dissipation of turbulence kinetic energy, Pt is the turbulent production term and μt is the turbulent viscosity. In the RANS model, the eddy viscosity may be excessive and smear out the flow structures within a given grid resolution range. To offer a more satisfactory resolution, the effective viscosity be reduced by imposing a filter on the turbulence model. As set values of the filter size are smaller than the length scales returned by the conventional RANS model, the turbulent scales smaller than the filter size could not be resolved. In the paper, the filter-based turbulent model (FBM) is applied and the turbulent eddy viscosity is modified by comparing the local turbulence length scale, k3/2/ε, and the filter size λ, which is selected based on the local mesh size.
��
ρm ¼ ρl αl þ ρa αa
��
where u is the velocity, ρm is the mixture fluid density, ρl is the liquid density, ρa is the air density, p is the pressure, μm is the mixture laminar viscosity, μt is the turbulent viscosity, μl and μa are the liquid and vapor dynamic viscosity respectively. The subscripts i, j denote the directions of the Cartesian coordinates.
μt ¼ C μ
ρm k 2 f ; ε FBM
� � 1 fFBM ¼ min 1; ; lRANS
3.2. Turbulence model
(8)
Cμ ¼ 0:09
lRANS ¼
k3=2 λ⋅ε
(9)
when the λ≫k3/2/ε, such as very near the wall where the local mesh size
The k-ε turbulence model proposed by Launder and Spalding (1974) 3
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is much greater than the turbulence length scale, μt ¼ Cμρmk2/ε, i.e. the standard k-ε model is recovered. When λ≪k3/2/ε, such as away from the wall where the local mesh size is smaller than the turbulence length scale, μt ¼ Cμρmλk1/2.
Table 1 Grid convergence study based on Richardson extrapolation (Ji et al., 2017; Long et al., 2019).
3.3. Numerical setup The free stream velocity U∞ ¼ Q/A, where Q is the flow rate measured by the electromagnetic flow meter and A is the section area of the test section. The ventilated flow pressure is 4atm and the ventilated flow rate Qin ranges from 0 to 800L/h. Thus the Froude number and the air entrainment coefficient are defined as: 1=2
Fr ¼ U∞ =ðgDÞ CQ ¼ Qin
�
U∞ D2
�
Mesh
I
II
III
Error (%)
Grid nodes Cd-mean
650000 0.114
920000 0.125
1300000 0.127
0.44
4. Results and discussion 4.1. Global multiphase structures in ventilated partial cavitating flows Figs. 5 and 6 shows the evolution of cavity length and corresponding typical ventilated partial cavitating flow patterns. Also, the pressure contours on the test body are shown. In the simulations, the isosurface of 10% gas volume fraction is used to illustrate the cavity boundary. Good agreement can be obtained between the experimental and numerical results. To better illustrate the comparison between the test and computational results, the mean square errors between the experimental and numerical results has been calculated, which is 0.08. According to the flow patterns, the ventilated cavity around the axisymmetric body
(13) (14)
where, U∞ is the average velocity, Qin is the volume flow of gas and D is the diameter of the body. The ventilated cavitating flow is simulated on the computing plat form of the CFX Solver. The computational domain and boundary con ditions are given according to the experimental setup. A no-slip boundary condition is imposed on the test body surface and boundaries of the tunnel. A constant turbulent intensity of 2% set at the inlet boundary, which is consistent to the measured value in the experiment. The inlet boundary of the domain is set as the opening boundary, both for the liquid and gas phases, with the inlet velocity of the gas injection set as the bulk mass flow rate. The ventilated cavitation simulations have been initialized with the steady-state, fully-wetted solutions, with den sity of the liquid (water at 25 � C) is ρl ¼ 998 kg/m3, viscosity μl ¼ ρlνl ¼ 1.139 � 10 3 Pa⋅s, and density of the air (air at 25 � C) is ρg ¼ 1.185 kg/ m3, viscosity μg ¼ 18.31 � 10 6 Pa⋅s. The 3D mesh, as shown in Fig. 4, is composed of 1300000 elements, and the value of y þ at the wall is smaller than 45. Three meshes, which give a constant refinement ratio r ¼ Nfine/Ncourse ¼ 1.414, are used in the grid convergence study, where N is the grid nodes number, as shown in Table 1. The relative error is estimated based on Richardson extrapolation (Ji et al., 2017; Long et al., 2019). As the relative errors are less than 1%, the results for mesh III are considered as mesh independent so the subsequent simulations are performed with the meshing scheme of mesh III. The time integration scheme is a second-order backward Euler algorithm, and the spatial derivatives are computed using a second-order upwind scheme. The time step chosen for the simulation is Δt ¼ 5 � 10 4s, based on convergence studies.
Fig. 5. Evolution of the measured and predicted cavity lengths.
Fig. 4. Mesh distributions. 4
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(d). After that, the cavity development transits from the rapid growth to growth with small pulsation. During this stage, as shown in Fig. 5, the growth rate of the maximum cavity length declines with small pulsation. This is because the attached cavity develops, accompanied with the continuous shedding of the small-scale detached cavities at the rear part of the attached cavity, as shown in Fig. 6(e) and (f). Until t ¼ t0þ108.5 ms, the attached cavity develops to the maximum, following by a sig nificant fluctuation of the cavity length. During this stage, large-scale detached cavity sheds quasi-periodically, as shown in Fig. 6(g). It should be noted that the cavity shapes are extremely dissymmetrical along the circumference of the axisymmetric body. In other words, three-dimensional characteristics are extremely impressive in the un steady cavitating flows around an axisymmetric body. This is also mentioned by Lindau (Lindau et al., 2002) and Hu (Hu et al., 2015). Further investigation of the three-dimensional effect on the ventilated cavitating flows and corresponding hydrodynamic characteristics will be conducted in future work. Fig. 7 shows the evolution of the predicted pressure coefficient around the axisymmetric body at different sections. For the section at x/ D ¼ 0.5 and x/D ¼ 1, the evolution of pressure coefficients is markedly different from that at the other sections, which is affected by the highspeed gas flow from the ventilated hole. As for the section from x/D ¼ 1.5 to x/D ¼ 4, a pressure peak appears successively when the attached cavity gradually reached each section position and then drop dramatically to the saturated vapor pressure as the attached cavity covers the position. As for the pressure fluctuation occurred after the attached cavity covering the position, such as the section at x/D ¼ 1.5, x/D ¼ 2 and x/D ¼ 2.5, this is due to the shedding of the small-scale detached cavities at the rear part of the attached cavity. While for the pressure fluctuation before the pressure peaks, such as the section at x/ D ¼ 3.5 and x/D ¼ 4, this is because of the disturbance caused by the upstream detached shedding cavities. When it develops to the third stage, high-amplitude and low-frequency variation of the pressure co efficients can be obviously observed, which is corresponding to the pe riodical shedding of the large-scale detached cavities. To further investigate the relationship between re-entrant jet and cavity shedding, Fig. 8 shows the evolution of the velocity distribution inside the ventilated cavity, where the solid red line indicates the cavity boundary. It can be seen from Fig. 8(a) that the re-entrant jet mainly moves upstream along the wall of the axisymmetric body. When meets with the ventilation gas flow, it is pushed far away from the axisym metric body, resulting to the rising of the cavity diameter and the discontinuity of the cavity boundary curvature. The rolled-up re-entrant jet flow downstream with the main flow, meanwhile the concave cavity occurs at the position where the cavity boundary curvature changes significantly, as shown in Fig. 8(b). With the attached cavity develops, the concave position moves downstream and even deeper, as shown in Fig. 8(c). Until the concave cavity boundary reaches to the re-entrant jet Fig. 6. Evolution of the ventilated cavity shapes.
experiences three stages: rapid growth, growth with small pulsation and periodic shedding. During the first stage, the ventilated gas flow up stream and downstream from the ventilated hole and a ring-shape cavity is formed around the axisymmetric body, with the cavity length increasing dramatically, as shown in Fig. 5 and Fig. 6(a)–(b). Mean while, the re-entrant jet generates due to the pressure gradient at the rear part of the attached cavity and moves towards the shoulder of the axisymmetric body, resulting that the forepart of the attached cavity appears transparent and the rear part in a fuzzy shape, as shown in Fig. 6 (b). As the re-entrant jet encounters the main flow, part of the attached cavity breaks off due to the interaction of the two opposite flows, as shown in Fig. 6(c). Affected by the continuous ventilated gas flow, the concave deformation occurs on the gas-liquid interface of the cracked cavity and three-dimensional effect of the ventilated cavity can be obviously observed around the axisymmetric body, as shown in Fig. 6
Fig. 7. Predicted pressure distribution at different axial position. 5
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the measured and predicted cavity pattern is 0.18, 0.12 and 0.07 for the axisymmetric body with α ¼ 0� , 5� and 8� respectively. As shown in Fig. 9(a), at α ¼ 0� , symmetrical attached cavity is formed and develops around the axisymmetric body, which is filled with the gas-liquid mixture and appears non-transparent. With the increase of the angle of attack, the attached cavity develops asymmetrically, with the attached cavity length on the opposing stream surface is longer than that on the confronted stream surface. Moreover, the non-transparent gasliquid mixture region is gradually concentrated on the opposing stream surface and the attached cavity on the confronted stream surface ap pears transparent, as shown in Fig. 9(b). When the angle of attack is further increased to 8� , the cavity asymmetry is more obvious and the transparent cavity region increases, with the non-transparent gas-liquid mixture only existed at the end of the attached cavity, as shown in Fig. 9 (c). In order to further investigate the influence of the angle of attack on the ventilated cavity patterns and flow structures, Fig. 10 shows the water volume fraction at different cross-sections of the axisymmetric body with different angles of attack when the main attached cavity keeps steady. The cavity boundary is illustrated with the isosurface of 10% gas volume fraction. It can be seen that when α ¼ 0� , the gas volume fraction at different cross-sections distribute uniformly in the circum ferential direction, with the attached cavity diameter increasing along the flow direction until x/D ¼ 2.5, then fluctuating slightly between x/D ¼ 2.5 and x/D ¼ 3.5 and finally decreasing until the end of the attached cavity. When the angle of attack increases to α ¼ 5� , the gas volume fraction at the cross-section x/D ¼ 0.5 which is near the shoulder of the axisymmetric body remains uniform in the circumferential direction, as shown in Fig. 10(b). While the gas volume fraction at the latter crosssections is accumulated upwards and finally exhibits as the crescent shape. As the angle of attack further increases to α ¼ 8� , the axial po sition of the cross-section with a crescent-shape distribution of the gas volume fraction is much closer to the head of the axisymmetric body, indicating that the greater the angle of attack, the more asymmetrical cavity is obtained. What’s more, assuming that the pressure gradient has no component along the attached cavity closure line, the re-entrant jet velocity is directed in the opposite direction of the incident flow as the angle of attack α ¼ 0� . While as the angle of attack increases, the cavity closure line is inclined and with the increase of the angle of attack, the angle of inclination for the attached cavity closure line decreases. Ac cording to the conservation of the tangential momentum, the re-entrant jet velocity has two components, that are perpendicular and along the closure line. This is also mentioned by de Lange (De Lange et al., 1994)
Fig. 8. Velocity distribution during the periodic shedding stage.
near the wall, the latter part of the attached cavity with a rising cavity diameter breaks off with the attached cavity and large-scale detached cavity is formed and shed downstream, as shown in Fig. 8(d). 4.2. The influence of angle of attack on the ventilated partial cavitating flows To further investigate the effect of the asymmetry on the unsteady shedding behaviors of ventilated partial cavitating flow, Fig. 9 shows the typical ventilated cavitating flow patterns at different angles of attack α. In which, the Froude number Fr ¼ 15.66 and the gas entrainment co efficient CQ ¼ 0.06. Reasonable agreement can be obtained between the experimental and numerical results, with the mean square error between
Fig. 9. Typical cavity patterns at different angles of attack (Fr ¼ 15.66，CQ ¼ 0.06). 6
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Fig. 11. Pressure distribution on the surface of the axisymmetric body at different angles of attack.
5. Conclusions Experimental and numerical studies have been conducted for an axisymmetric body to investigate the unsteady shedding behaviors of ventilated partial cavitating flow. High speed visualization technique has been adopted to present the global multiphase structures and nu merical simulation has been combined to further investigate the cavity shedding mechanism. The primary findings include: (1) The ventilated cavity around the axisymmetric body experiences three stages: rapid growth, growth with small pulsation and pe riodic shedding. During the rapid growth stage, the ventilated gas flow upstream and downstream from the ventilated hole and a ring-shape cavity is formed around the axisymmetric body, along with the interaction between the main attached cavity and the reentrant jet. During the growth with small pulsation stage, the attached cavity develops, accompanied with the continuous shedding of the small-scale detached cavities at the rear part of the attached cavity, resulting that the growth rate of the maximum cavity length declines with small pulsation. During the periodic shedding stage, large-scale detached cavity sheds quasiperiodically accompanied with a significant fluctuation of the attached cavity length. (2) The re-entrant jet has a significant role on the unsteady cavity shedding behaviors. The re-entrant jet generated due to the pressure gradient and moves upstream along the wall of the axisymmetric body. When meets with the ventilation gas flow, the re-entrant jet is pushed far away from the axisymmetric body, resulting to the rising of the attached cavity diameter and the discontinuity of the cavity boundary curvature, which leads to the attached cavity depressed inward. Until the concave cavity boundary reaches to the re-entrant jet near the wall, the latter part of the attached cavity with a rising cavity diameter breaks off with the attached cavity and large-scale detached cavity is formed and shed downstream. (3) The angle of attack influence the ventilated partial cavitating flow patterns. Symmetrical cavity is formed and develops around the axisymmetric body with the angle of attack α ¼ 0� . The attached cavity develops asymmetrically with the angle of attack α ¼ 5� and 8� . The non-transparent gas-liquid mixture region is gradually concentrated on the opposing stream surface and the
Fig. 10. Gas distribution at different cross sections.
and Franc (2001). To further investigate the effect of the angles of attack on the ventilated partial cavitation, Fig. 11 shows the predicted pressure dis tribution at both confronted and opposing stream surface of the axisymmetric body with different angles of attack. As the angle of attack increases, the high pressure region corresponding to the attached cavity closure at the confronted stream surface moves upstream, and that at the opposing stream surface remains almost steady. This is because the angle of attack mainly affect the attached cavity length at the confronted stream surface of axisymmetric body and has little effect on the attached cavity length at the opposing stream surface. As the relative thickness of the attached cavity and the re-entrant jet is an important factor for the detached cavity shedding, in order to reveal the cavity shedding mechanism of the axisymmetric body at different angles of attack, Fig. 12 shows the attached cavity and reentrant jet boundary on the cross section at x/D ¼ 2 with different an gles of attack. When the angle of attack α ¼ 0� , the attached cavity is much thicker than the re-entrant jet layer and the interaction between the re-entrant jet and the ventilated gas flow is weak so that the reentrant jet can reach to the head of the attached cavity resulting to the partial shedding of the ventilated cavity. With the increases of the angle of attack of the axisymmetric body, the maximum thickness of the re-entrant jet is close to the attached cavity boundary and there is a strong interaction between the re-entrant jet and ventilation gas flow, resulting that the re-entrant jet mainly exists in the rear region of the cavity, which is corresponding to the small-scale detached cavity shedding at the end of the attached cavity.
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Fig. 12. Relationship between re-entrant jet and cavity boundary layer at different angles of attack.
transparent cavity region increases on the confronted stream surface.
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Author contributions section Guoyu Wang: Resources, Supervision, Funding acquisition. Decai Kong: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing-Original Draft. Qin Wu: Software, Validation, Writing-Review & Editing, Supervi sion, Funding acquisition. Taotao Liu: Software, Validation, Writing-Review & Editing, Super vision, Funding acquisition. Yanan Zheng: Project administration. Biao Huang: Conceptualization, Software, Validation, WritingReview & Editing, Supervision, Funding acquisition. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work is supported by the National Natural Science Foundation of China (Grant Nos: 51839001, 51909002, 51679005 and 91752105), the National Natural Science Foundation of Beijing (Grant No: 3172029) and the Open Foundation of Key Laboratory of Fluid and Power Ma chinery, Ministry of Education of China in Xihua University (Grant Nos: szjj2018-124, szjj2019-024). References Ausoni, P., Farhat, M., Escaler, X., et al., 2007. Cavitation influence on von K� arm� an vortex shedding and induced hydrofoil vibrations. J. Fluids Eng. 129 (8), 966–973. De Lange, D.F., de Bruin, G.J., van Wijngaarden, L., 1994. On the mechanism of cloud cavitation - experiment and modelling. Proceeding of the Second International Symposium on Cavitation 45–49. Franc, J.P., 2001. Partial Cavity Instabilities and Re-entrant Jet. CAV2001. Hjartarson, A., Mokhtarzadeh, H., Kawakami, E., et al., 2009. A Dynamic Test Platform for Evaluating Control Algorithms for a Supercavitating Vehicle 110, 1–4. Hu, C.L., Wang, G.Y., Chen, G.H., et al., 2015. Three-dimensional unsteady cavitating flows around an axisymmetric body with a blunt headform[J]. J. Mech. Sci. Technol. 29 (3), 1093–1101. Huang, B., Yong, Y.L., Wang, G.Y., Shyy, W., 2013. Combined experimental and computational investigation of unsteady structure of sheet/cloud cavitation. J. Fluids Eng. 135, 071301-1-071301-16. Javadpour, S.M., Farahat, S., Ajam, H., et al., 2017. Experimental and numerical study of ventilated supercavitation around a cone cavitator. Heat Mass Transf. 53 (5), 1491–1502. Ji, B., Long, Y., Long, X., Qian, Z., Zhou, J., 2017. Large eddy simulation of turbulent attached cavitating flow with special emphasis on large scale structures of the hydrofoil wake and turbulence-cavitation interactions. J. Hydrodyn. 29 (1), 27–39.
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