A review of transient flow structure and unsteady mechanism of cavitating flow *

Biao Huang, Si-cong Qiu, Xiang-bin Li, Qin Wu, Guo-yu Wang

1. School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

2. Department of Nuclear Science and Engineering, North China Electric Power University, Beijing 102206,China

Abstract: The flow structure and the unsteady mechanism of the unsteady cavitating flow are reviewed in this paper. The flow patterns and structures in different cavitation regime, for the attached cavitation and the vortical cavitation, are shown with both the visualization and the quantitative information. The attached cavitating flow around the Clark-Y hydrofoil and the vortical cavitating flow around the Tulin hydrofoil are considered. In particular, the phenomena such as the large-scale vortex structure and the re-entrant flow associated with the cloud cavitation, and the cavitating vortex street's forming and crumbling are described. The evolution of the cavitation structure in the transient sheet/cloud cavity forming, along with the cavity collapse induced by the re-entrant flow and the shock wave propagation are discussed. The perspective future research of higher fidelity simulations, and the accurate identifications of the cavitating vortex structure is commented.

Key words: Cavitating flow, flow structure, unsteady mechanism

Biography

Biao Huang (1985-), Associate Professor of School of Mechanical Engineering, Beijing Institute of Technology (BIT). He obtained his bachelor and doctor degree of Power Machinery and Engineering in BIT (2007 and 2013 respectively). During his Ph. D.,he has spent one year in University of Michigan as a visiting Ph. D. Candidate. His research interests are in the fields of cavitating and supercavitating flow mechanism, multiphase hydrodynamics, advanced marine equipment, hydraulic machinery and system.He has published more than 60 pieces of high-level journal papers in Physics of Fluids, Journal of Multiphase Flow, Computers and Fluids and so on.He is also one of the editorial board member of Journal of Hydrodynamics. The scientific research achievement includes the second prize in the Progress of Science and Technology in the Ministry of Education (ranked the second) and the second prize in the Scientific and Technology Progress of China Ship Industry Co. (ranked the second).

Guo-yu Wang (1962-), Professor of School of Mechanical Engineering, Beijing Institute of Technology (BIT). He obtained his bachelor and master degree in China Agricultural University in 1983 and 1990 respectively. He got the Doctor degree in Tohoku University in 1999 and has been the Professor and doctoral tutor in BIT since 2003. His research is mainly on the hydrodynamics of vehicle power systems, the hydrodynamics of high-speed underwater vehicles, the hydraulic mechanical principle and optimization design of high-altitude performance. During the past 10 years, he is responsible for more than 30 scientific research projects, such as the National Natural Science Foundation's key projects.He is a committee member of the Chinese Society of Engineering Thermophysics (Fluid Machinery Branch)and a committee member of Water Professional Weapon Branch of Chinese Society of Naval Architecture and Marine Engineering. His research is mainly on the hydrodynamics of vehicle power systems, the hydrodynamics of high-speed underwater vehicles, the hydraulic mechanical principle and optimization design of high-altitude performance. He is also one of the editorial board member of Journal of Experiments in Fluid Mechanics, Journal of Drainage and Irrigation Machinery Engineering, etc..

Table 1 Summary of previous observations and measurements for cavitation structures

Introduction

The cavitation generally occurs when the local fluid pressure is reduced to the saturated vapor pressure and consequently bubbles filled with vapor are formed. It is a challenging issue in hydrodynamics and is a critical problem in many engineering applications. The cavitation involves many complex and interdisciplinary physical phenomena such as turbulence, multiphase flow, unsteady flow and compressibility[1-5]. In engineering applications, the cavitation is used to achieve drag reduction for underwater vehicles[6-11]and in various industries such as the chemical reaction, the extraction and wastewater treatment[12-14]. Meanwhile, the cavitation will have some undesirable effects including the pressure pulsation, the structural vibration and noise, the power loss and the decreased efficiency[15-22].

The research of cavitation is related with studies of the unsteady cavitating flows. Among the notable studies, Arndt[23]gave a broad overview of the physical phenomenon of cavitation in 1981 and pointed out the main factors triggering the cavitation,including such as the cavitation nuclei, the roughness

onthesurfaceandthefluidviscosity.AfterwardWang et al.[24]reviewed the stationary and nonstationary characteristics of the turbulent cavitating flows around solid objects via experiment and numerical simulations. Franc and Michel[25]developed a systematical physical theory of the cavitation,involving different types of cavitations, as the foundation for understanding the general nature of the cavitation. In the engineering aspects of the cavitation,Luo et al.[26]summarized the progress of the cavitation studies in the hydraulic machinery with focus on the methods for the cavitation simulations, and the details of the transient cavitating flow, using various experimental techniques. An overview of previous observations and measurements for the cavitation structures is given in Table 1.

Most experimental studies of the cavitation were generally focused on the external liquid flow,including the cavity shape, the external flow velocity and the pressure and the temperature. Arakeri and Acosta[27]developed the Schlieren flow visualization method to study the cavitation inception and development for two axisymmetric bodies. It can be observed that the cavitation inception occurs in the region of the separated flow and the turbulent reattachment of the separated flow has a great influence on the cavitation inception. Kueny[28]employed an image processing device to determine the shape of an attached cavity. Maeda[29]applied the holographic technique to measure the bubble population in the cloud cavitation. Guennoun et al.[30]applied an intensified light video camera for the observation of a specific type of bubble cavitation over a 2D NACA0009 hydrofoil. With the data measured with the embedded pressure transducers, it is found that the periodic bubble cavitation takes place in the minimum pressure area and originates from a local vaporization process. Washio et al.[31]provided vital images of the incipient cavities with a super high speed video camera in various constrictions and it is evident that the primary cavitation starts at the point of the flow separation.

For the determination of the velocity distribution,Kubota et al.[32]used the laser Doppler velocimetry(LDV) to measure the flow velocity around the cloud cavitation. It was found that the mean velocity in the cloud cavity was lower than the mean stream velocity with an increase of the vorticity. Liu et al.[33]observed the cavity flow behavior by means of a xenon flash lamp and a high-speed video, and measured the velocity fluctuations inside and outside the cavity by the LDV. The results demonstrated the strong fluctuation and intermittent reverse flow within the cavity. Ceccio and Brennen[34]used the silver epoxy electrodes flush-mounted on the surface of the two-dimensional hydrofoil to trace the individual bubble velocity. Gopalan and Katz[35]applied the high-speed photography and the particle image velocimetry (PIV) to measure the cavitating flow structure and demonstrated that the sheet cavity collapse is the main source of the vorticity production.Dreyer et al.[36]used the Stereo particle image velocimetry (SPIV) to measure the 3-D velocity fields in the clearance between the rotor and the stator of an axis hydro turbine. The tip leakage vortex under different working conditions was well described.

To study the variation of the local variables within the cavities, a double hot wire velocimetry was used at a fixed point on a foil section to measure the local vapor phase velocity[37]. Besides, optical probes were adopted to measure the flow velocity, the void fraction and the cavity size. Miller and Mitchie[38]first reported the working principle of the optical fibers and used a mono-fiber probe with a few hundred micron tips to measure the local void patterns in a large drum. Stutz and Rebound[39-40]used a double optical probe to measure the local void fraction and the velocities of the sheet cavitation in a Venturi-type section. They found that an extended reverse flow occurs along the solid surface with a significant influence on the vapor cloud shedding process.

With the development of the experimental techniques, different methods were combined to investigate the transient cavitation, focusing on the unsteady mechanics. Kawanami et al.[48]investigated the generation mechanism of the cloud cavitation on a hydrofoil via observations with high-speed videos and high-speed photos and pressure measurements with the pressure pickups and a hydrophone. They found that it is the re-entrant jet that triggers the collapse of a sheet cavity and gives rise to the cloud cavitation.Leroux et al.[49-50]demonstrated the unsteadiness of the partial cavitation based on the observation and the wall pressure measurement. It was indicated that the cavity instability originates from the interaction between the re-entrant jet and the cavity surface and a shock wave phenomenon due to the collapse of the large cloud cavitation. Chen et al.[51]used a simultaneous sampling technique to synchronize the observations of the cavitation images and the measurement of the wall pressure signals. Thus the transient cavitating flow structures can be closely related to the unsteady cavitation dynamics for a better understanding of the unsteady mechanism of the cavitation.

Although the cavitating flows around different objects were much studied, the focus was limited to the attached cavitation and the cavity development.The objectives of this paper are to review the transient flow structures and the unsteady mechanism of the unsteady cavitating flows. The present review includes:

(1) A summary of the detailed transient flow structures in both attached cavitation and vortical cavitation based on the comprehensive flow field data and the computational tools.

(2) A discussion of the cavitation structure evolution in the transient cloud cavitating flow, along with the re-entrant flow and the shock wave propagation induced cavity shedding behaviors.

(3) A proposal of the perspective future research based on a higher fidelity simulation and an accurate identification of the cavitating vortex structure.

1. Flow patterns and structures in different cavitation regimes

Since the first cavitation tunnel built by Parsons in England, numerous aspects of the cavitation were extensively studied for hydrofoils[52-61], as the fundamental element of the lifting surfaces. Among them, two types of cavitating flow patterns and structures can be obtained, that is, the attached cavitating flow and the vortical cavitating flow.

1.1 Attached cavitating flow around the Clark-Y hydrofoil

For an understanding of the transient cavitating flow mechanism, the cavitating flow structures were described and classified based on the experimental observations. According to the dynamic flow characteristics, Knapp et al.[62]defined the cavity patterns as the bubble cavitation, the attached cavitation, the vortex cavitation and the oscillated cavitation. The attached cavitation can be further divided into different stages according to the cavity appearance,which are, the incipient cavitation, the sheet cavitation,the cloud cavitation and the supercavitation. Based on the experimental visualization and the quantitative measurements, the cavitation regimes for the Clark-Y hydrofoil under different flow conditions and thecorrespondingnormalizedmaximumcavitylengthare outlined in Fig. 1, where L is the maximum cavity length measured in the mid-plane of the hydrofoil, and c is the chord length of the hydrofoil. For the incipient cavitation, the traveling bubbles can be observed around the hydrofoil within 0.1c from the leading edge of the hydrofoil.According to Wang et al.[24], the turbulent bursting and the hairpin-shaped structure are always accompanied by the incipient cavitation. With regard to the sheet cavitation, the initial traveling cavities are developed to a sheet cavity, which remains approximately at the same position as a larger cavity in the length scale, and more cavitating vortices appear at the rear part of the cavity. Similar findings were also reported by Gopalan and Katz[35], Stutz and Reboud[39]. With regard to the cloud cavitation, the sheet cavity further grows and the trailing edge of the cavity becomes increasingly unsteady accompanied by massive cloud cavity shedding. During the past decades, much attention was paid to the quasi-periodic cavity pattern, including the initiation of the cavity,the growth toward the trailing edge and the subsequent shedding, as shown in Fig. 2, and the unsteady characteristics of the cloud cavitation[63-70].With further decreasing the cavitation number, the final state of the cavitation comes into being, the supercavitation. As seen from Fig. 1, the relatively large cavity covers the whole hydrofoil and extends to the downstream of the hydrofoil.

Fig. 1 (Color online) Cavitation regimes for Clark-Y hydrofoil under different flow conditions

To have a further insight into the flow structures in more details, Fig. 3 shows the measured mean

Fig. 2 (Color online) Periodic development of the cloud cavitation (σ = 0.80)[71]

streamwise velocity and the vorticity distributionaround the Clark-Y hydrofoil in different cavitation regimes. It can be found that the cavitation has a significant effect on the flow structures. From the observed cavity patterns in Fig. 1, it is shown that in the core cavitation region, the velocity reduces and the vorticity increases significantly. The cavitation aggravates the mass transfer with remarkable vortex characteristics. In view of the severe turbulent characteristics of the cloud cavitation,, the cloud cavitating flow structures are very important. Figure 4 shows the measured mean, modal and standard deviations of the normalized averaged axial velocity profiles at different chord-wise locations in the non-cavitation and cloud cavitation regimes. For the non-cavitating flow, the modal averaged axial velocity agrees well with the mean averaged axial velocity, and the standard deviation is low due to the steady flow in the non-cavitation regimes. While for the cloud cavitating flow, the flow is dominated by the recirculating flow, as evident in the negative axial velocity. The standard deviation increases with the axial location, because of the increasing flow unsteadiness along the flow direction resulted by the cavity collapse. Besides, as shown in Fig. 4(b), the re-entrant jet can be clearly observed, which is the main factor of the cavity shedding mechanism and will be further discussed later on.

Fig. 3 (Color online) Measured mean streamwise velocity distribution around the Clark-Y hydrofoil[72]

Fig. 4 Measured mean, modal and standard deviations of the normalized averaged axial velocity profiles

Fig. 5 (Color online) Vortical cavitating flow patterns under various conditions

Table 2 Vortex shedding parameters

Fig. 6 (Color online) Velocity and vorticity distributions around the Tulin hydrofoil (α = 15°) [80]

1.2 Vortical cavitating flow around the Tulin hydrofoil

The Tulin hydrofoil is of great interest in many different applications, such as for the support surfaces of super-high speed hydrofoil crafts, the support or active control surfaces of super-ventilated underwater or surface crafts, and supercavitating and surface piercing propellers[72-74]. Due to a sharp leading edge and a relatively thick trailing edge of the Tulin hydrofoil[75], the blade is allowed to cut through the fluid efficiently and to trigger the stable supercavity easily. Meanwhile, the unsteady cavitation behaviors and the vortex-cavitation interactions around the Tulin hydrofoil become more obvious and acuter. Based on the work of Li et al.[76], Shyy et al.[77], Zhang et al.[78]and Zhang et al.[79], we outline the vortical cavitating flow patterns under various conditions, with the Tulin hydrofoil at 15° angle-of-attack, as shown in Fig. 5.Different from the attached cavity, the flow structures experience the incipient cavitation, the vortex cavitation, the cloud cavitation, the mixture supercavitation and the developed supercavitation. During the first three stages, the cavity vortex is formed at the leading and trailing edges of the Tulin hydrofoil, which is enlarged with the decrease of the cavitation number.When the leading and trailing edge vortexes develop sufficiently, they shed alternatively and periodically.As the cavitation number reduces to 0.77 or lower, the supercavitation comes into being and a relative stable cavity covers the whole suction surface and extends to places behind the trailing edge. Different supercavity can be observed with decreasing the cavitation number, including the fluctuating supercavity with periodical vortex shedding, the vapor and water-vapor mixture coexisted supercavity, and the fully developed supercavity, filled with vapor along with a two-phase tail.

Fig. 7 (Color online) Unsteady development of the supercavity for the case with σ= 0.54[80]

For the vortex cavitation around the Tulinhydrofoil,themainfeatureisthevortexsheddingmechanism. Table 2 summarized the main flow parameters under different flow conditions. With the decrease of the cavitation number, the vortex shedding frequency increases. Meanwhile, the size of the vortex structures, as well as the development and shedding period, increase with the decrease of the cavitation number.

While for the supercavitation around the Tulin hydrofoil, various techniques were applied to have an insight of the flow structures in more details. Li[77,80]adopted a continuous laser beam sheet to capture the distinct interface between the liquid and the vapor.Figure 6 shows the measured mean streamwise velocity distribution around the Tulin hydrofoil. The velocity near the leading edge of the hydrofoil is similar to the main stream velocity, which is filled with the vapor, while the low velocity region is in the other part of the supercavity, which is filled with the water-vapor mixture. Moreover, with the decrease of the cavitation number, the upper and lower vortex strips stretch downstream with the decrease of the maximum vorticity value.

Here, it should be noted that, during the supercavitation stage, one sees some unsteady development of the supercavity patterns before the supercavity is fully developed. Figure 7 shows the unsteady development of the supercavity in the case ofIt can be seen that the interface between the front and rear zones is highly unstable. The water-vapor interface is formed at t=0 ms and continuously moves adversely. Although the cavity boundary is quite steady in the case of the supercavitation, the pressure fluctuation and mass transfer process can be substantial inside the cavity.

2. The unsteady mechanism of cavitating flow

The unsteady mechanism of the cavitating flow is of great practical interest since the high unsteadiness is accompanied with large fluctuations of the hydrodynamic forces and strong vibrations of the structures. Arndt[81]firstly observed these two types of behaviors for the NACA0015 hydrofoil based on the parameter σ/2 α and concluded that the re-entrant jet was responsible for the cavity destabilization whenwhile a bubbly shock dominated the flow when. Leroux et al.[82]observed the quasi stable partial sheet cavity and the sheet to cloud cavitation for the NACA66 hydrofoil and obtained a similar conclusion based on the parameter σ/2(α-where0α is the angle of zero lift and it is zero for symmetric hydrofoils such as the NACA0015.Based on the visualization of the cavity patterns for various flow velocities and incidence angles, the transition between the stable and unstable cavities around a NACA66 hydrofoil is summarized in Fig. 8.The transition occurs whenand

Fig. 8 (Color online) Transition between stable and unstable cavities[83]

Based on the overall data obtained in various cavitating flow regimes, Fig. 9 further shows the maximum cavity length and the Strouhal number under different cavitation conditions. With the decrease ofthe cavity length increases up to a normalized cavity length of 0.8 and the corresponding Strouhal number is about 0.3, as shown in the Type I in Fig. 9. When,the transition between the regions of stability and instability can be observed. In this region, the maximum cavity length ranges between 0.8c and 1.2c, and the periodical formation and the shedding of the large-scale cloud cavity can be observed,however, in two different cloud cavitation patterns.The demarcation between the stable cloud cavitation and the unstable cloud cavitation can be seen for the cavity length. When the cavity length is less than 1.0, the cloud cavitation occurs with the Strouhal number of 0.3-0.4, while as the cavity extends to the downstream of the hydrofoil, the Strouhal number decreases to 0.10-0.15 accompanied with the intense vibration and noise. For a lower value of, as the Type III in Fig. 9, the cavity length increases with the decrease ofin an unstable pattern and the Strouhal number increases ranging from 0.15 to 0.20. Further decreasing, the flow turns to the stable supercavitation with a smooth interface formed between the liquid and the vapor, with a reduction of the flow resistance and the flow-induced vibrations. The unsteady mechanics for these two cloud cavitation patterns will be analyzed in detail later on.

Fig. 9 (Color online) Typical cavitation around NACA66 hydrofoil[83]

2.1 Re-entrant flow induced shedding behaviors

The pictures of the re-entrant flow induced shedding are presented in Fig. 10, as the main cause for the cavitation cloud separation and mainly responsible for the quasi-periodic characteristics of the cloud cavitation[84]. The partial sheet cavity develops from the leading edge of the foil to the position ofas shown in Fig. 10(a). During the cavity developing process, the re-entrant jet forms at the rear end of the cavity and moves from both sidesofthetrailingedgeofthecavitytowardsthecenter of the leading edge of the cavity. The collision of these two bundles of the upward flow cuts off the downstream part of the cavity with the shedding of the cloud cavity, and a wedge-shape hole at the rear part of the remaining sheet cavity, as shown in Fig. 10(b).The cloud cavity rolls up, moves downstream and finally collapses, as shown in Fig. 10(c). According to Yang et al.[85], the re-entrant flow and the spanwise velocity along the hydrofoil suction side are critical for the three dimensional cavitation. It is also suggested by De Lange et al.[86], Franc et al.[87]that the re-entrant jet has two components, perpendicular and along the cavity closure line, when the three dimensional cavity is inclined according to the conservation of the tangential momentum. Foeth[88]even pointed out that the direction and the momentum of the re-entrant jet need to be accurately simulated for its significant influence on the cavity shedding.Since the re-entrant jet always reflects from the rear end of the cavity, the length scale of the disturbance along the cavity closure is more important than the length of the re-entrant jet in the flow direction[89].

Fig. 10 Re-entrant flow induced shedding behaviors[83]

From the velocity and pressure distributions around the hydrofoil, as shown in Fig. 11, it can be seen that in a periodic development process of the cavities, the relatively great adverse pressure gradient occurs at the rear end of the cavity when the cavity size goes to the maximum, triggering the re-entrant flow. Meanwhile, a local high pressure is found at the head of the re-entrant flow, which becomes stronger gradually due to the adverse pressure gradient. The main factor for the re-entrant jet to induce the shedding of the cloud cavity is the local high pressure at the head of the re-entrant jet. As the cavity shedding process is periodic, the development of the re-entrant jet is responsible for triggering the cavity detachment and hence unsteadiness.

Hence, in the analysis of the re-entrant flow instability, two parameters are shown to be of the most importance: the adverse pressure gradient andthe cavity thickness as compared with the re-entrant flow thickness[92]. The cavity thickness is directly related with the angle of attack, and the ratio of the thickness of the cavity to that of the re-entrant flow is very important in the estimation of the unsteady behavior of the partial cavity. Whenat, the cavity is thick enough to limit the interaction between the re-entrant flow and the cavity interface, which allows the re-entrant flow to reach the cavity leading edge for the formation of a large scale vapor structures. However, when =0.40σ at, because of the smaller cavity thickness, there is a strong interaction between the cavity interface and the re-entrant flow. The re-entrant flow can only reach the rear part of the hydrofoil, as shown in Fig. 12,unlike the case of the large scale cloud shedding whenat α=8° , small vapor structures are formed.

Fig. 11 (Color online) The re-entrant flow mechanism. (a) The re-entrant jet induced cavity shedding, (b) Velocity distribution around the hydrofoil[90], (c) Evolution of reverse velocity, (d) Pressure distribution around the hydrofoil, (e) Evolution of the pressure gradient[91]

For better understanding the unsteadiness of the re-entrant flow, the development of the cavitating vortical structures is important. The detailed processes of the large-scale cavity formation, shedding, breakage and collapse are summarized in Fig. 13. During the shedding process of the large-scale cloud cavity,the U shape cavitating vortex structure is formed.Then the cavitating vortex moves downstream and breaks from the head, with two small cavitating vortexes observed. Finally, the small cavitating vortexes collapse from both sides. Further numerical simulations show that the instantaneous high-pressure is the main factor for the vortex breakage at the head of the cavity. Ceccio and Ceccio[93-94]investigated the cloud cavitation of the hydrofoil NACA0009 through experiments, and observed that the re-entrant jet is the cause of the periodic characteristics of the cloud cavitation. More recently, Huang et al.[95]combined the experimental and numerical results to show the strong correlation between the cavity and vorticity structures through an analysis of the vorticity transport equation. The interaction between the re-entrant jet and the cavity can motivate the liquid-vapor phase change and increase the vorticity near the wall.

Fig. 12 (Color online) Schematic diagram of the re-entrant flow in the region of an attached cavity

2.2 Shock wave induced shedding

Unlike the re-entrant jet flow mentioned above,when the maximum cavity length is larger than 1.0c,the collapse of the cavity occurs in the downstream of the hydrofoil, as shown in Fig. 14(a), and the pressure difference between the suction side and the pressure sideofthehydrofoil increases thepressuregradientbetween the inner and outer regions of the cavity. As the cavity collapses, as shown in Figs. 14(b), 14(c) the residual cavity stops to grow almost at the same time,then it shrinks dramatically and even disappears entirely on the suction side. From the numerical results with consideration of the compressibility of both the water and the vapor[97], it can be found that the shrink of the attached residual cavity is caused by the shock wave propagation induced by the cloud cavity collapse. During the propagation process, the local pressure is increased to make the attached cavity collapse. According to Brennen[98], the vapor fraction has a significant effect on the shock wave propagation speed and the sound speed within the cavity can have a sharp decrease with a certain vapor fraction,resulting in a sudden increase of the local Mach number. Meanwhile, the collapse of the large-scale bubble clusters is supposed to release a largeamplitude pressure pulsation, which indicates the possible existence of the shock wave induced shedding, as shown in Fig. 15. To further explain the shock wave dynamics, the vorticity transport equation is adopted to analyze the cavitation-vortex interaction in the compressible turbulent flow region[99]. During the re-entrant flow process, the baroclinic torque term plays a major role as it is mainly induced by the adverse pressure gradient, while during the shock wave propagation process, the vortex dilatation term dominates due to the fact that it is highly related to the compressibility effect.

Fig.13 (Color online) Detailed processes for the large-scale cavity formation, shedding, breakage and collapse[72, 96]

More recently, the shock wave propagating phenomena were captured in the convergent-divergent channels by Saito and Sato[100]with a high speed video, by Wang et al.[101]with a simultaneous sampling technique to synchronize the transient cavitationbehaviors and the wall-pressure signals and by Ganesh et al.[102]with the X-ray densitometry. The presence of the propagation discontinuity in the void fraction was clearly observed and the pressure peak appears at the front of the shock wave with a much larger amplitude than that in the re-entrant flow.

3. Perspective future work

The future researches will further advance the understanding of the transient flow structures and the unsteady mechanism of the cavitating flows:

(1) Varieties of methods were applied to simulate the unsteady turbulent cavitating flows. Most of them were based on the RANS turbulence model to capture the turbulence fluctuations. It should be noted the above methods are model-dependent and timeaveraged, so the multiscale unsteadiness in the cavitating flow field cannot be revealed. They are not accurate with respect to the physical interaction between the cavity shedding, the turbulent fluctuations and the vortex dynamics for the unsteady cavitating flows. Higher fidelity simulations via the LES and the DNS may be better for the flow physics. Furthermore,the verification and validation (V&V) procedures of the LES and the DNS for simulating the unsteady turbulent cavitating flow are very important. The V&V procedures of the LES and the DNS will come up with many difficulties due to the close coupling of grids and numerical models, and the unsteady characteristics of the cavitation flow field also make the V&V procedures more complicated. Long et al.[103-104]investigated the V&V procedures of the URANS simulations for the turbulent cavitating flow around a hydrofoil. Anyway, more investigations of the V&V procedures are needed for the LES and DNS simulations.

Fig. 14 Shock wave propagation induced shedding[83]

Fig. 15 (Color online) Illustration of shock wave propagation induced shedding

(2) The vortex structure was widely identified in the cavitating flow field and was found to play a critical role in the turbulence generation and evolution.The time evolution of the cavitating flow is closely associated with the transient vortex structure. The accurate identification of the vortex structure would probably help to improve the understanding of the interplay between the formation, the development, the shedding of cavities and the vortex dynamics.Although several vortex identification methods, such as the Δ- method, the Q and2λ criteria were widely used to capture the vortex structure in the highly-unsteady and massively-separated flows, a threshold is generally required and a proper choice of the threshold is still quite illusive with many limitations. Although the Ω method is more robust in the identification of the vortex structures, it is just a post-processing method for calculating results,without much effect in improving the precision.Recently, Liu et al.[105]proposed a new vortex identification method, the “Omega and Liutex/Rortex based Systems”. The vortex is not only considered as a flow phenomenon but also with a clear mathematical definition, which becomes a driving force in the flow field. It is a promising way for the further research.

(3) Machine learning is a field of computer science, using statistical techniques to “l(fā)earn” from the data. The machine-learning-assisted turbulence modeling starts with a dialogue with the traditional turbulence modeling community. Zhang et al.[106]optimized the Reynolds stress model of the Channel Flow based on the reduced-order modeling of the DNS data with the deep learning method. Although from the existing researches, it is not possible to predict the unknown flow regimes through the deep machine learning, the preliminary success of the RANS based turbulence modeling, implies that the machine learning has a potential capability for the turbulent cavitating modeling. Optimizing vortex parameters of the numerical model through the machine learning helps the numerical simulations to be closer to the real flow fields. Besides, the various vortex identification methods, mentioned above, can also be combined and be trained with the machine learning method, so that more intelligent and comprehensive identifications of the vortex structures can be realized. These would also be the future development directions of the machine leaning combined with the fluid mechanics.

(4) With the continuous development of the hydraulic machinery in the directions of high power,high speed, and high loadings, the problems of the noise induced by the cavitation flow are increasingly outstanding. At present, some achievement has been made for the prediction of the hydrofoil and propeller noises[107-109]. The difficulty is the cavitation noise reduction and vibration reduction, which is the task of future work.

Acknowledgments

Some of the experimental results in this paper are obtained in EPFL-LMH with the help of Prof. Farhat.This work was supported by the Open Foundation of Key Laboratory of Fluid and Power Machinery,Ministry of Education of China, Xihua University.