Numerical modeling and simulation of the shedding mechanism and vortex structures at the development stage of ventilated partial cavitating flows

European Journal of Mechanics / B Fluids 76 (2019) 223–232

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European Journal of Mechanics / B Fluids journal homepage: www.elsevier.com/locate/ejmflu

Numerical modeling and simulation of the shedding mechanism and vortex structures at the development stage of ventilated partial cavitating flows ∗

Tiezhi Sun a,b , Xiaoshi Zhang c , Chang Xu d , Guiyong Zhang a,b,e , , Shengchao Jiang a , Zhi Zong a,b,e a

School of Naval Architecture, Dalian University of Technology, Dalian 116024, China State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China Beijing Institute of Machinery Equipment, Beijing 100854, China d Department of Ocean Engineering, Texas A&M University, College Station 77843, USA e Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China b c

highlights • • • •

An effective computation strategy for ventilated partial cavitating flows is established. Unsteady flow patterns are revealed at the development stage of ventilated partial cavitation. The shedding mechanism of ventilated partial cavitating flows is analyzed. The mechanism of the vortex-ventilated cavitation is revealed.

article

info

Article history: Received 21 August 2018 Received in revised form 14 January 2019 Accepted 25 February 2019 Available online 9 March 2019 Keywords: Shedding mechanism Vortex structures Development stage Ventilated partial cavitating flows Numerical modeling

a b s t r a c t Ventilated partial cavitation is a complex multi-phase turbulent flow due to the strong interactions between gas and liquid. In the present work, we specially focus on the numerical modeling and simulation of the shedding mechanism and vortex structures. The Reynolds Averaged Navier–Stokes (RANS) method combined with a filter-based turbulence model (FBM) is proposed to explore the physical mechanism of the ventilated partial cavitating flows. Experimental results of cavity evolution and pressure are utilized to assess the prediction ability of the proposed method. Good agreements are observed between experimental measurements and numerical predictions, including the ventilated cavity growth, break off, shedding and the transient dynamic pressure inside the cavity. Based on the model strategy, the cavity dynamic evolution and shedding mechanism are analyzed. The results indicate that the re-entrant flow gives birth to the gas leakage at the cavity interface and is responsible for the ventilated cavity shedding. In addition, streamline vortex is presented to reveal the ventilated cavity shedding characteristics. Moreover, based on the vorticity transport equation, the influence of velocity gradient, fluid volumetric expansion/contraction, pressure gradient and the viscous dissipation factors on the vortex production in ventilated cavitating flows is examined. The present study can provide important basis to better understand the shedding mechanism and vortex structures on the development stage of ventilated partial cavitating flows. © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction During the high-speed movement of underwater vehicles, cavitation will occur when the ambient pressure around the vehicle is lower than the vapor pressure of the local thermodynamic ∗ Corresponding author at: School of Naval Architecture, Dalian University of Technology, Dalian 116024, China. E-mail address: [email protected] (G. Zhang). https://doi.org/10.1016/j.euromechflu.2019.02.011 0997-7546/© 2019 Elsevier Masson SAS. All rights reserved.

state [1,2]. The natural properties of cavitation, such as growth, shedding, break off and collapse will lead to large pressure fluctuations on the surface of the vehicle and can affect the motion stability of the vehicle [3–9]. Ventilated cavitation is a special case of cavitation by injecting non-condensable gas into the cavitation area, it is considered to be an effective way to improve flow field and motion stability [10–12]. However, ventilated cavitation is unsteady in nature and inevitably involves complex multiphase turbulence flows [13]. Better understanding of the ventilated

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cavitating flow mechanism is the goal that researchers have long been pursuing due to its great practical application in ocean engineering and scientific relevance in multiphase fluid dynamics [9,14,15]. In the past few decades, many efforts have been made to study the supercavitating flow as the friction drag can be significantly reduced by over 90% when the vehicle completely covered by cavity [9,15–25]. These studies mainly focused on cavity evolution mechanism, velocity and pressure distributions of flow field as well as the stability of supercavitating vehicles and provided valuable insights into the mechanism of ventilation supercavitation. Actually, ventilated partial cavitation is an indispensable process before supercavitation. To improve the understanding of the complex flow structures of ventilated partial cavitating flow, considerable work has been exerted using experimental studies [26–28]. It is known that the ventilated partial cavitation always involves complex interactions between turbulent flow structures and multiphase dynamics with large variations in local pressure and density [7,29,30]. These results help to provide better insight into the mechanism of ventilated cavitating flows. Despite decades of vigorous research into ventilated cavitating flows, these physical mechanisms are not well understood based on the limited experimental measurements and studies. Hence, numerical simulation is an effective and complementary strategy to investigate the inherent complexity of the cavitating flows. Amromin [31] developed and validated computational fluid dynamics models for initial stages of cavitation. The advantages of multimodel CFD analysis of cavitation are manifested in his paper. Gnanaskandan and Mahesh [32] investigated a large eddy simulation of the transition from sheet to cloud cavitation over a wedge. They found that the attached sheet cavity grew up to a critical length, after which it broke into a cloud cavity which with highly three-dimensional and vortical in nature. More recently, Wang et al. [33] and Wu et al. [34] performed fruitful numerical calculations on transient characteristics of cloud cavitating flows and shock wave dynamics. They provided valuable results about the influence of water/vapor compressibility in the cavitation instabilities and the relation between the cavity development and the pressure fluctuations. More importantly, the numerical results also provide detailed flow field information, which could help us to condense valuable references for the ventilated cavitating flows. Early studies of numerical modeling for partially ventilated cavitating flow were primarily focused on capturing the ventilated cavity [35–38]. In the numerical modeling of cavitating flows, the turbulence model plays an important role in the prediction of the unsteady complex behavior. The Reynolds Averaged Navier–Stokes (RANS) method has been widely used to simulate turbulent cavitating flows. However, the unsteady multiscale cavities cannot be captured well due to the over-prediction of turbulent eddy viscosity in multiphase mixture region [39–41]. Recently, the large eddy simulation (LES) is used to study the cavitating flows, which can predict considerable details of large-scale turbulent eddies in the cavitation flow field. [42–44]. However, the high demand for the grid density especially in the boundary layer and the high computational cost make it unpractical in most applications. Hence, some hybrid RANS–LES turbulence models have been developed to investigate cavitating flows due to its advantages in calculation accuracy and efficiency [45,46]. Johansen et al. [47] proposed a filter-based model (FBM) combining the standard k − ε model [48] and LES model [49]. Considering the multi-scale characteristics in both space and time in unsteady cavitating flows, more attention should be paid on applying the FBM model to partially ventilation cavitating flows. Although ventilated cavitating flows has received much attention for decades, the physical mechanisms of partial ventilated

cavitation is still not well understood, especially in the development stage. Meanwhile, research effort of numerical modeling is endeavored to gain an effective computational method based on an acceptable computational cost. The aims of the present study are to (1) establish an effective numerical method to predict the ventilation cavitation mechanism, (2) improve the understanding of the shedding mechanism of the partial ventilation cavitating flows, (3) gain a broad and improved understanding of the physical mechanism of vortex-ventilated cavitation. The present paper is structured as follows: Section 2 provides the details of numerical methodology including the governing equations, turbulence model and numerical setup. Subsequently, in Section 3, we present results and discussion on the cavity dynamics evolution, mechanisms of shedding and vortex-ventilated cavitation in detail. Finally, the main findings are summarized in Section 4. 2. Numerical methodology The main goal of the present study is to establish a numerical method and address the vortex structures inside the ventilated cavity based on the homogeneous equilibrium flow model and filter-based turbulence model. The basic mathematical formulations and numerical method are described in Sections 2.1 and 2.2. The governing equations were discretized by the finite volume method (FVM). The second-order implicit method is used for time integration scheme. 2.1. Governing equations In the present study, the homogeneous equilibrium flow model is used to capture the liquid–gas interface of the ventilated cavitating flows. The homogeneous equilibrium flow model assumes strong coupling between phases, and various fluid components are assumed to share the same velocity [50]. In addition, the phases are assumed to be in mechanical and thermal equilibrium: they share the same pressure and temperature. Hence, the evolution of the two-phase flow can be described by the conservation laws that employ the representative flow characteristics as unknowns just as in a single-phase problem [51]. In the computation process, the density in the mixed area is calculated based on the volume rate of each phase in each control element. The mixture density ρ is expressed as:

ρ = ρl αl + ρg αg

(1)

where ρg is the volume fraction of the vapor; αl , αg are the volume fractions of the liquid and vapor, respectively. The equations for conservation of mass and momentum are given by the following:

∂ρ ∂ ( ) + ρ uj = 0 (2) ∂t ∂ xj [ ( )] ) ∂ ∂ ( ∂p ∂ ∂ ui ∂ uj ρ ui uj = − + + (ρ ui ) + (µ + µt ) ∂t ∂ xj ∂ xi ∂ xj ∂ xj ∂ xi (3) where t is time, xi is the Cartesian coordinates, ui is the velocity components (i = 1, 2, 3); p is the pressure; u is the velocity; uj is the dynamic viscosity; µt is the turbulent eddy viscosity. 2.2. Filter-based turbulence model (FBM) The original two-equation models (such as k − ε and k − ω models) trend to over predict the turbulent eddy viscosity in the rear part of the cavity, which significantly modifies the cavitating flows behavior [39,52]. Johansen et al. [47] proposed a filterbased turbulence model (FBM) to improve a better prediction

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on the turbulent eddy viscosity. The derivation of the governing equations in FBM is based on the k − ε model [48]:

∂ µt ∂ k ∂ (ρ k) ∂ (ρ kuj ) + = [(µ + ) ] + Pk − ρε + Pkb ∂t ∂ xj ∂ xj σk ∂ xj ∂ (ρε ) ∂ (ρεuj ) ∂ µt ∂ε + = [(µ + ) ] ∂t ∂ xj ∂ xj σε ∂ xj ε + (Cε1 Pk − Cε2 ρε + Cε1 Pεb ) k

(4)

(5)

Fig. 1. Computational domain and boundary conditions.

The turbulence viscosity µt is a function related to the total turbulence kinetic energy k and dissipation ε :

µt = ρ Cµ

k2

(6)

ε

In Eqs.(8), Pk represents the turbulence production due to viscous forces, and it can be expressed by: Pk = µt

(

∂ uj ∂ ui + ∂ xj ∂ xi

)

∂ ui 2 ∂ uk − ∂ xj 3 ∂ xk

(

) ∂ uk 3µt + ρk ∂ xk

(7)

Fig. 2. Computational geometry model.

where Cε1 , Cε2 , Cµ , σk and σε are constants, Cε1 = 1.44, Cε2 = 1.92, Cµ = 0.09, σk = 1.0 and σε = 1.3; Pkb and Pεb represent the influence of the buoyancy forces. In the filter-based turbulence model, the expression of turbulence kinetic energy k and dissipation ε are the same form as Eqs.(8) and (9). However, the turbulent eddy viscosity is expressed as:

µt =

Cµ ρ k2

ε

fFBM

(8)

The filter function fFBM is determined by the filter size λ and turbulence length scale. It is given as:

[

fFBM = min 1, C3

λε k3/2

]

, C3 = 1.0

(9)

In particular, the FBM model will help the turbulent eddy viscosity in the cavitating wake region and can improve the prediction of unsteady cavitating flows dynamics. Note that the FBM model was successfully applied in numerical studies of cavitating flows by Wu et al. [53], Tseng and Shyy [54] and Huang et al. [55]. 2.3. Numerical setup and description In the present study, the computational model was established according to our experiment [56]. The computational domain size and boundary conditions are given according to the experimental setup. The length of the test section is 1 m, and the cross section is 260×260 mm square. The inlet and outlet boundaries were set, respectively, at 7.25D and 17.25D based on the reference position of the leading edge. Meanwhile, velocity inlet and pressure outlet boundaries were used to impose the computational flows. The tunnel walls and test model were treated as non-slip walls with wall-functions applied. To match the experimental conditions, the water and air densities were 997 and 0.0185 kg m−3 , respectively, and dynamic viscosities of 8.899×10−4 and 1.831×10−5 kg m−1 s−1 . Fig. 1 presents a 2D sketch of the computational domain and boundary conditions. The schematic of the computational geometry model is shown in Fig. 2. It is a conical axisymmetric body made with a ventilation slot at the nose. The diameter is D = 0.04 m, and the length is L = 315 mm. Two pressure monitor points are set according to the experiment system to validate the numerical method. The important dimensionless parameters cavitation number σ , gas ventilation rate CQ and pressure coefficient Cp used in the present 2 study are defined as σ = (p∞ − pc )/(0.5ρl V∞ ), CQ = Qin /(V∞ D2 ) 2 and Cp = (pc − p∞ )/(0.5ρl V∞ ), where p∞ is the free stream

Fig. 3. Mesh generation.

pressure, pc is the pressure inside cavity, ρl is the liquid density, V∞ is the free stream velocity, Qin is the volume flow rate, D is the diameter of the test body. Fig. 3 shows the mesh generation of the computational domain. A structured hexahedral mesh was created and consisted of 160 million cells. Additional refinement was applied around the head and tail of the vehicle and the ventilation area. Meanwhile, an O–H type grid was applied around the foil surface to improve the mesh quality. Care was taken to ensure that the y+falls within the logarithmic boundary layer region in order for the wall function models to be valid (y+ ∈ ⟨30, 100⟩). During the solution process, the time-dependent governing equations were discretized in both time and space with the direct pressure–velocity coupling method [57]. In each time step, excess iterations cost too much computational time with little improvement in accuracy, while too few iterations may lead to bad convergence. In the present simulation, finally, 25 iterations per time step were used to balance the accuracy and efficiency. In order to capture the transient characteristic of cavitating flows, in the computations, ∆t = 1.6 × 10−5 is chosen based on convergence studies, which ensures the average Courant number base on physical time is less than 1.0. 3. Results and discussions 3.1. Cavity dynamic evolution and numerical method evaluation In this section, the unsteady and three-dimensional cavitating flow over the axisymmetric body is investigated to validate the accuracy of the employed numerical method. The numerical results are compared with the experimental results by Zhang et al. [56]. In the experiment, the velocity at the inlet of the test section was V∞ = 8.07 m/s, the pressure of the test section was

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Fig. 4. Comparison of instantaneous cavity shape around the conical axisymmetric body (a) experimental observations, (b) predicted 3D cavity evolution, (c) predicted volume of fraction on the XOY plane.

68.4 kPa and the gas ventilation rate CQ = 0.149. Fig. 4 compares the instantaneous cavity shape between the experimental observations and numerical predictions. To better analyze the cavity evolution, the corresponding volume fraction distribution of the XOY plane is also illustrated in Fig. 4(c). It can be seen that the numerical predictions agree reasonably well with the experimental observations. In Fig. 4, from t0 to t0 + 15 ms, the ventilated cavity develops towards downstream. Meanwhile, the re-entrant flow appears inside the cavity and begins to induce shedding. At t0 + 30 ms, as shown in Fig. 4(c), a collision occurs between the re-entrant flow and the cavity surface and thus a subsequent shedding of large cavities from the whole cavity over the body. This process is the primary shedding stage. It is interesting to note that a secondary shedding appears at t0 + 80 ms in both numerical calculations and experimental observations. Thus, the present simulation reasonably predicts the cavitation patterns and their evolution over the body. In order to further verify the numerical prediction ability, the predicted pressure along the conical axisymmetric body were compared with the experimental measurement, which is presented in Fig. 5. It can be seen that the computed pressure fluctuations is in good agreement with the experimental results. Thus the present numerical method can effectively predict the flow field evolution and dynamic characteristics of the ventilated cavitating flow. 3.2. Shedding mechanism of ventilated partial cavitating flows In order to present the transient behavior of the flow structures in re-entrant flow regime clearly. Fig. 6 shows the cavity shape, gas volume fraction and velocity vector distribution around the body. The upper left figure is the morphology of the cavity, the upper right is the gas volume fraction on the XOY plane and the lower part is the velocity vector. It can be seen that the re-entrant flow inside the attached cavity moves upstream and trends to pinch off the cavity. To be observed clearly, this region is highlighted by dashed blue and red line. It also indicates that the formation of the re-entrant flow appears at a rather early

Fig. 5. Comparison between the measured and predicted pressure of the body surface.

stage, while the ventilated cavity is still growing. When the reentrant flow touches the upper cavity interface, the ventilated cavitation shedding will be induced. Moreover, the velocity vector distribution shows that the approximate stagnation flow appears in shoulder of the body. In order to better understand the flow characteristics of the ventilated cavity near the closure position, Fig. 7 illustrates eight typical predicted cavity shape, the zoomed in velocity vector and pressure field over the body. As mentioned in Section 3.1, two types of shedding are observed. The first type is characterized by a relatively large, horseshoe-shaped cavity as shown in Fig. 7(d) and (e). The velocity vector in the shedding region reveals that the re-entrant flow is the cause of the cavity surface disturbance. Moreover, the pressure gradient is small at the closure region. For the secondary shedding cavity shown in Fig. 7(h), the cavity shape is relatively stable, with only the region at the tail of the cavity shedding due to the re-entrant flow. Compared with the primary shedding, the pressure gradient is larger at the closure region.

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Fig. 6. The cavity shape (Upper left), air volume fraction on the XOY plane (Upper right) and velocity vector around the body (Bottom) at t0 .

Fig. 7. The predicted instantaneous cavity shape (Top picture), enlarged view of velocity vector and pressure distribution near the cavity closure region (Bottom picture).

As the experimental observation and numerical prediction in Fig. 4, we can find that the ventilated cavitating flow involves a special shedding mechanism, which integrates both ventilated cavity and bubble around the body. Herein, we present

a schematic of shedding mechanism that determined entrant flow models of a ventilated cavity in Fig. 8. seen that a clear continuous cavity is formed at the of the test body. Meanwhile, in the cavity closure

by the reIt can be front part point, the

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Fig. 8. Schematic of shedding mechanism of the partial ventilated cavitating flow.

continuous cavity is broken up by the re-entrant flow. Then gas leakage appears at the cavity base near the re-entrainment point, which gives birth to numerous small bubbles due to the cavity of high turbulence. Subsequently, small-scale bubble groups are then re-circulated with the re-entrant flow in the vortex region, while the majority of bubbles entrain along the test body surface downstream by the main flow. For a better understanding of the shedding dynamics, four typical instants are picked up to present the flow details in Fig. 9. The upper figures of each time instant in Fig. 9 is threedimensional ventilated cavity morphology, and the lower figures are the instantaneous velocity vector (left) and streamline (right) through the body surface. In Fig. 9, the three-dimensional structures of the main shedding cavity and secondary shedding cavity are analyzed. It can be seen from the cavity morphology that, before the ventilated cavity shedding, at t0 +5 ms, there is a large-scale vortex structure on both sides of the front part of the cavity. The streamline indicates the re-entrant flow of the jet. On the cavity surface on both sides of the body, the velocity vector and streamline distribution is nearly symmetric. From the velocity vector diagram, it can be seen that the local flow direction in the cavity contains both clockwise and counterclockwise, forming a small-scale vortex

structure. At t0 + 15 ms, the cavity continues to grow. From the velocity vector and streamline, it can be seen that there are vortex structures on both sides of the ventilated cavity, due to the effect of vortex, the re-entrant flow to the front of the body moves along the circumferential direction. From the velocity vector diagram, there are both clockwise and counterclockwise vectors, this again indicates that the movement direction of the jet contains both clockwise and counterclockwise. At t0 + 25 ms, The tailing edge of the ventilated cavity breaks off, at the cavity closure position, the large-scale vortex structure in the cavity splits into two symmetrical vortex structures with different distribution scales along the main flow direction. One pair of large-scale symmetric vortex structures is located in the detached cavities and the other pair is in the front part of the cavities. From the trajectory of the streamline, it can be seen that the shedding cavities move in the opposite direction and gradually move away from the front part of cavity on the vehicle. From t0 + 30 ms, the cavity starts to shedding and separates from the front part of the cavity. The scale of vortex corresponding to the front part of the cavity gradually becomes larger, and the vortex core gradually moves towards the front of the cavity under the influence of the re-entrant flow. The vortex corresponding to the downstream shedding cavities gradually collapses while moving downwards under the effect

Fig. 9. The predicted time sequences of cavity shape (Upper) around the body, velocity vector (Bottom left-top view) and streamline near the cavity closure region (Bottom right-top view).

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Fig. 10. Comparison of different term in vorticity transport equation at plane Z = 0, (a) 2D view of vortex stretching contour, (b) 2D view of vortex dilation contour, (c) 2D view of baroclinic torque contour, (d) 2D view of viscous diffusion contour.

of the main flow. The strength of the vortex gradually decreases with the collapse, and the vortex core gradually moves away from the front part of the cavity. Hence, there is close relationship between the streamline vortex and the cavity shedding during the development stage of ventilated cavitating flows. 3.3. The mechanism of the vortex-ventilated cavitation In order to understand the interactions mechanism of the vortex-ventilated cavitation, the vorticity transport equation is employed as follows

→ D− ω Dt

∇ρm × ∇ p − → → − → → = (− ω · ∇) U − − ω (∇ · U ) + ρm2 → +(ν + ν )∇ 2 − ω m

t

(10)

− → → The first term on the right hand side (− ω · ∇ ) U is the vortex stretching term, which indicates the bending and stretching of the vortex caused by the velocity gradient in the flow field, this term also causes variation in vorticity direction and size, and shows unsteady flow characteristics during the development of ventilated cavitation. The velocity gradient in the flow field near the ventilated cavity varies significantly. With the development of the ventilated cavity, the gradient distribution has a strong unsteady nature, as a result, the velocity gradient production term also ex− → → hibits strong unsteadiness. The second term − ω (∇· U ), represents the vortex dilatation due to volumetric expansion/contraction; 2 the third term (∇ρm × ∇ p)/ρm represents baroclinic torque due to non-parallel pressure gradient and density gradient; the fourth → term (νm + νt )∇ 2 − ω represents the vorticity variation due to viscous dissipation. To examine the influence of ventilated cavitation on vortex production as well as its alteration over the conical axisymmetric body, the results are mainly shown in the plane Z = 0 in the following discussion. Meanwhile, the corresponding vorticity transport equations and the associated quantities are given as Dωz Dt

=

] [ [( [ ( − )− ∇ρm × ∇ p →] →)] − → → ω ·∇ U − − ω ∇· U + z z ρm2 z [ ] → 2− + (νm + νt ) ∇ ω z (11)

∂ Uy ∂ Ux ωz = − ∂x ∂y [( )− ∂ Uz ∂ Uz ∂ Uz →] − → ω · ∇ U = ωx + ωy + ωz z ∂x ∂y ∂z ( ) [ ( − ∂ Ux ∂ Uy ∂ Uz →)] − → ω ∇· U = ωz + + z ∂x ∂y ∂z [ ] ( ) ∇ρm × ∇ p 1 ∂ρm ∂ p ∂ρm ∂ p = · − · ρm2 ρm2 ∂x ∂y ∂y ∂x z

(12) (13) (14) (15)

( 2 ) ] [ ∂ ωz ∂ 2 ωz ∂ 2 ωz → ω z = (νm + νt ) + + (νm + νt ) ∇ 2 − ∂ x2 ∂ y2 ∂ z2

(16)

Fig. 10 shows the vorticity distribution of each term in the vorticity transport equation on plane Z = 0 at t = t0 + 25 ms. It is noted that the influence of the viscous dissipation is smaller compared to other three terms on the right hand side of Eq. (14). In Fig. 10, we can also find that the baroclinic torque is very important for the production of vorticity during the development and shedding of the ventilated cavitation. This is consistent with the experimental results by Laberteaux and Ceccio [58] and the numerical work by Ji et al. [59]. Moreover, it should be noted that, although the baroclinic torque has significant effect during the development and shedding of the ventilated cavitation, the main contributor in the vorticity transport equation is the vortex stretching term. However, the vortex dilatation term plays a major role in the natural cavitating flows [59]. Specially, in the cloud region, vorticity dilatation leads to an increase in vorticity there Gnanaskandan and Mahesh [32]. Hence, the vorticity analysis shows that the vorticity distribution and evolution characteristics of ventilated cavitation and natural cavitation flows are different. Fig. 11 shows the predicted contours of vortex stretching term and vortex dilatation term in vorticity Z transport equation at typical cavitating flow instances, including the attached ventilated cavity growth, the re-entrant flow movement, ventilated cavity collapse and cavity shedding, to study the ventilated cavitation–vortex interaction in more detail. It can be seen that the development of the vortex stretching term has obvious unsteady characteristics. For the ventilated cavity around the conical axisymmetric body, the negative vorticity is mainly distributed in the upper half of the Z = 0 plane and the positive vorticity is mainly distributed in the lower part of Z = 0. From t0 to t0 + 5 ms, the velocity gradient at the air–water interface of the attached cavity is relatively large. Thus, the value of the first term at the gas–water interface is also relatively large. With the continuous development of ventilated cavity, a re-entrant flow is formed near the closure region of the cavity. At this time, the direction of flow in the cavity closure region is complex and varied, resulting in a wide value range of the vortex stretching term. As the re-entrant flow develops upstream along the surface, it reaches the front of the cavity and interacts with the ventilation. As a result, the internal velocity gradient of the cavity changes greatly, and the positive vorticity and negative vorticity appear alternately in the cavity. At t0 + 15 ms, when the attached cavity is fully developed, the cavity internal ventilation momentum is greater than the reentrant flow momentum. At this time, the internal flow direction of the cavity is relatively more uniform, and the internal flow velocity of the cavity forms a small pressure gradient. The value of the vortex stretching term in the interior region approaches zero, and there is still a large flow velocity gradient at the gas–liquid interface, resulting in a large value at gas–liquid interface area.

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Fig. 11. The contours of vorticity transport equation at the plane Z = 0 for vortex stretching term (left) and vortex dilatation term (right).

From t0 + 25 ms to t0 + 80 ms, with the collapse and shedding of ventilated cavity, the distribution of velocity gradients in the cavity area is complicated, and the vorticity accumulation zone is distributed in a strip shape near the model surface. Due to the collapse and shedding of the downstream cavity, the positive and negative vorticity areas of the model surface appear alternately and move to the downstream. Fig. 12 presents the baroclinic torque term and viscous diffusion term in the vorticity transport equation. For better comparison, the same scale range is used for the four terms in vorticity transport equation. The value of the baroclinic torque term appears to change in positive or negative values alternately in the ventilated cavity air–water interface. This is due to the existence of an adverse pressure gradient in the gas–water interface, and the gas–water two phases are strongly mixed and form a large density gradient, causing the change of vorticity. This indicates that, after the mixing of gas–vapor, the density gradient and the adverse pressure gradient in the gas–liquid interface together caused the vorticity variation. The range of the viscous dissipation effects term is smaller than that of the other three terms. Specially, for the ventilated cavitating turbulent flows under high Reynolds number conditions. This numerical prediction characteristic is consistent with the numerical results of Huang et al. [60] and Ji et al. [59].

4. Conclusions In the present study, numerical modeling and simulation is conducted to address the shedding mechanism and vortex evolution patterns at the development stage of ventilated partial cavitating flows based on the Reynolds Averaged Navier–Stokes (RANS) method. The main findings are summarized as follows (1) The RANS method based on filter-based turbulence model (FBM) is an effective computational strategy for understanding the physical mechanism of ventilated cavitating flows. The predicted ventilated cavity evolution and pressure inside cavity agree reasonably well with the experimental results. Especially, the important physical mechanism of re-entrant flow and shedding can be captured well during the development stage of ventilated partial cavitating flows. (2) Ventilated partial cavitating flows during the development stage involve a special shedding mechanism. The re-entrant flow plays an important role in the stability of ventilated regimes and is responsible for the ventilated cavity shedding generation. The primary shedding cavity appears at the early development stage is characterized by a relatively large, horseshoe-shaped. However, compared with the secondary shedding instant, the pressure gradient is smaller at the cavity closure region. (3) A close relationship between the streamline vortex and the cavity shedding exists during the development stage of ventilated

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Fig. 12. The contours of vorticity transport equation at the plane Z = 0 for baroclinic torque term (left) and viscous diffusion term (right).

cavitating flows. Before the primary shedding occurs, there is a pair of small-scale vortex structure on both sides of the front part of the ventilated cavity. Interestingly, when the primary shedding occurs, the vortex structure inside the cavity is splitted into two symmetrical vortex structures with different scales along the main flow direction. (4) The evolution characteristics of the ventilated cavitating flows over the conical axisymmetric body are closely related to the vortex structure. Due to the growth, break off, shedding and collapse of the ventilated cavitating flows, the positive and negative vorticity bends appear alternately and move to the downstream. An important finding is that the influence of the viscous dissipation is smaller than the vortex stretching term, the vortex dilatation term and the baroclinic torque term. Especially, the main contributor in the vorticity transport equation is the vortex stretching term for the production of vorticity during the development and shedding of the ventilated cavitation.

Acknowledgments The authors gratefully acknowledge support by the National Natural Science Foundation of China (Nos: 51709042, 51579042, 51639003), the Natural Science Foundation of Liaoning Province of China (No: 20180550619), the China Postdoctoral Science Foundation (No: 2018M631791), the Recruitment Program of Global Young Experts, China (No. D1007001) and the Fundamental Research Funds for the Central Universities (No: DUT18RC(4) 018). References [1] C.E. Brennen, Cavitation and Bubble Dynamics, in: Oxford Engineering & Sciences Series 44, Oxford University Press, Oxford, UK, 1995, p. 90, Chapter 3. [2] T. Sun, Y. Wei, L. Zou, Y. Jiang, C. Xu, Z. Zong, Numerical investigation on the unsteady cavitation shedding dynamics over a hydrofoil in thermo-sensitive fluid, Int. J. Multiph. Flow. 111 (2019) 82–100. [3] D.D. Joseph, Cavitation in a flowing liquid, Phys. Rev. E 51 (3) (1995) R1649.

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