Elsevier

Ocean Engineering

Volume 87, 1 September 2014, Pages 64-77
Ocean Engineering

Numerical simulation of three dimensional cavitation shedding dynamics with special emphasis on cavitation–vortex interaction

https://doi.org/10.1016/j.oceaneng.2014.05.005Get rights and content

Highlights

  • 3D cavitating turbulent structure around a twisted hydrofoil is simulated.

  • Three types of flow behavior along the hydrofoil suction side are illustrated.

  • The mechanism of cavitation–vortex interaction is discussed.

Abstract

Recent experiments showed that there is an interaction between the fluid vortex formation and cavitation, but the mechanism is still an open problem. In the present paper, the structure of the cavitating flow around a twisted hydrofoil was investigated numerically using the mass transfer cavitation model and the modified RNG k-ε model with a local density correction for turbulent eddy viscosity. The predicted three dimensional cavity structures and the shedding frequency agree fairly well with experimental observations. Three types of flow behavior along the suction side of the twisted hydrofoil are discussed. Further analysis of the flow field reveals that cavitation promotes vortex production and increases the boundary layer thickness with local separation and the flow unsteadiness. Finally, the influence of cavitation on the vorticity distribution is illustrated using the vorticity transport equation in a variable density flow and is demonstrated by the contribution of vortex stretching, vortex dilatation and baroclinic torque terms.

Introduction

The study of unsteady features of partial cavitation and shedding has been received great attention due to its strong background in the engineering application in pumps, turbine blades and ship propellers. Sheet cavitation shedding often leads to cloud cavitation, which strongly affects hydrodynamic performance and produces vibration, noise and cavitation erosion (Brennen, 1995, Franc and Michel, 2005). In most industrial applications, cavitation and turbulent flow often result in the formation of large-scale vortical structures, which will involve complex interactions between phase-change and vortex structures (Arndt, 2002). In order to control the unsteady shedding and the formation of cloud cavitation, it is very important to understand these interactions.

In the past, numerous experiments have been conducted to study partial cavitation structures especially on hydrofoils (Astolfi et al., 2000, Reisman et al., 1998, Tassin et al., 1995) or in Venturi-type sections (Barre et al., 2009, Stutz and Reboud, 1997a, Stutz and Reboud, 1997b). These experiments showed that partial sheet cavities are periodically broken-up and rolled up into bubble clouds. Although many interesting studies have been reported on these physical mechanisms, they are not yet fully understood due to the complex features of partial cavitating flows such as bubbly flow, laminar transition or turbulent flows, detached and reattached flows, shear layers and vortical structures. Kubota et al. (1989) successfully measured the unsteady structure of cloud cavitation using laser doppler anemometry (LDA) and matched the measurements with unsteady cavities photographed by a high-speed camera. Their results showed that the cloud cavitation observed in the experiment had a vorticity maximum at its center and a cluster containing many small cavitation bubbles. The structure of the two-phase flow inside the cavity was investigated by Stutz and Reboud, 1997a, Stutz and Reboud, 1997b. They succeeded in measuring the local void fraction and the velocity inside the cavities and confirmed the existence of reversed two-phase flow along the wall. Le et al. (1993) studied the global behavior of partial cavities, including cavitation patterns, cavity length, periodic shedding, and mean pressure in the cavity closure region. They found that the cavity unsteadiness is intimately related to cavity thickness and a re-entrant jet that results in vorticity production. Reliable estimates for the shedding rate of the circulation by the re-entrant jet mechanism for a periodic cavity can then be obtained. Kawanami et al. (1997) thoroughly investigated cloud cavitation in a series of detailed experiments on a two-dimensional Elliptic Nose Foil with high-speed photography as well as pressure measurements through pressure pick-ups and a hydrophone. They established a clear relationship between the re-entrant jet and the cloud cavity generation process. They then pointed out that a small obstacle attached at the mid-span near the termination of the sheet cavity can block the re-entrant jet, thereby preventing the generation of the cloud cavity. Pham et al. (1999) also conducted an experimental investigation of unsteady sheet cavitation using non-intrusive techniques to study the re-entrant jet dynamics and the interfacial instabilities. They found that the frequencies of the re-entrant jet surges are equal to the cloud shedding frequencies determined by unsteady pressure measurements, which demonstrated that the cloud shedding is actually driven by the re-entrant jet. A gravity effect analysis showed that the re-entrant jet role predominates over the interfacial instabilities in the generation of periodic cloud shedding. Arndt et al. (2000) used a two-dimensional NACA0015 hydrofoil to investigate the complex physics involved in the transition of sheet cavitation to cloud cavitation with an integrated experimental/numerical approach. They indicated that two competing mechanisms are found for the induced shedding of cloud cavitation. At high values of σ/2α, re-entrant jet physics dominate, while at low values of σ/2α, bubbly flow shock wave phenomena dominate. Watanabe et al. (2001) used a linearized free streamline theory using a singularity method to show that when the re-entrant jet is not taken into account, cavitation instability originates from the transitional cavity oscillation and the transition between partial and super cavities. Callenaere et al. (2001) experimentally investigated the instability of a partial cavity induced by the development of a re-entrant jet on a diverging step. They argued that the two parameters having the greatest effect on the re-entrant instability are: the adverse pressure gradient and the cavity thickness compared to the re-entrant jet thickness. Laberteaux and Ceccio (2001a) observed two types of partial cavities with closed partial cavities formed on a two dimensional NACA0009 hydrofoil and open partial cavities without re-entrant flow formed on a plano-convex hydrofoil.

The complexity of cavitating flow is due to the strong coupling between cavitation and turbulence as well as the strong compressibility effects. With the development of computational fluid dynamics, numerical simulation of cavitating flows is becoming an important tool in cavitation research. The cavitation model and turbulence model in the simulation of cavitating flows are the key factors to obtain the reasonable results. A homogeneous model based on the assumption that the cavitation area can be considered as only one fluid with the mixture of water and vapor is commonly used to simulate the cavitation. Two kinds of cavitation models are often used in the simulations of cavitation flow, the State Equation Model (SEM) (Coutier-Delgosha et al., 2003, Goncalves and Patella, 2009) and the Transport Equation Model (TEM) (Kunz et al., 2000, Schnerr and Sauer, 2001, Singhal et al., 2002). Recent experimental results have confirmed that vorticity production in the cavity closure region is an important aspect of cavitating flows due to the baroclinic torque term (Gopalan and Katz, 2000, Laberteaux and Ceccio, 2001a). However, SEM is not able to properly reflect this term because the gradients of density and pressure in SEM are always parallel, which leads to zero baroclinic torque. TEM may be a better choice to simulate the complex cavitating flow, which introduces an additional equation for the vapor (or liquid) volume fraction including source terms for evaporation and condensation processes. Comparison studies of different TEM models have been shown in Refs. (Ducoin et al., 2012, Morgut et al., 2011). For simulation of cavitating flow accurately, the turbulence model is crucial because the cavitation process is basically unsteady in nature and there must be strong interactions between the cavity interface and the boundary layer during the cavitation development. Though the Reynolds Average Navier-Stokes (RANS) equation approach has been widely used to model turbulent flows (Karim and Ahmmed, 2012, Seif et al., 2010), the capability of the RANS model with eddy viscosity turbulence models to simulate unsteady cavitating flows is limited due to its over-prediction of eddy viscosity at the rear part of cavitation (Coutier-Delgosha et al., 2003). Conversely, the more accurate large-eddy simulations (LES) and direct numerical simulations (DNS) are restricted in their application because of their high requirements in computing power (Luo et al., 2012, Roohi et al., 2013, Zhang and Khoo, 2013). Recent research efforts are directed towards development of some hybrid RANS/LES model (Huang and Wang, 2011, Ji et al., 2012b, Wu et al., 2005, Zhang and Chen, 2013) or RANS model with some consideration of the local compressibility effect of two-phase mixtures on turbulent closure models (Decaix and Goncalves, 2013, Liu et al., 2013, Wang et al., 2012).

It is noted that many researchers have studied cavitating flows around 3D hydrofoils to discuss the instability of partial cavitation. Dang and Kuiper (1999) numerically studied a re-entrant jet using a hydrofoil with various angles of attack in the spanwise direction. They found that the direction of the re-entrant jet was strongly influenced by the cavity topology and the change in the cavity shape was determined not by the sweep angle but by the loading. Laberteaux and Ceccio (2001b) showed for a series of swept wedges that the cavity instability was strongly influenced by the span-wise pressure gradients and the re-entrant jet may be directed away from the cavity interface, allowing sheet cavitation to form a cloud cavity far downstream. Dular et al. (2007) numerically and experimentally investigated re-entrant jet reflection at an inclined cavity closure line around a hydrofoil with an asymmetric leading edge. Saito et al. (2007) investigated cavitating flows around a three dimensional hydrofoil with uniform profiles and uniform attack angles along the span wise direction and pointed out that the sidewall effect is the main reason for generation of the U-shaped cavitation. Foeth (2008) and Foeth et al., 2006, Foeth et al., 2008 used time-resolved PIV and a high speed camera to study fully developed sheet cavitation on a hydrofoil with a spanwise varying angle of attack and clarified that the shedding of a sheet cavity was governed by the direction and momentum of the re-entrant and side-entrant jets and their impingement on the free surface of the cavity. The cavitating flow around the Delft twisted hydrofoil was used as benchmark data in two workshops, VIRTURE WP4 and SMP11 (Hoekstra et al., 2011), because it resembles propeller cavitation with well defined experimental data that is easily studied. This case was selected to evaluate and validate current CFD codes to predict the complicated cavitating flows (Bensow, 2011, Ji et al., 2013a, Ji et al., 2013b, Li et al., 2010, Park and Rhee, 2013, Schnerr et al., 2008).

Till now, most research of partial cavities has focused on the structure of the cavitating flow and its shedding dynamics. However, there has been little attention given to the interaction between the vortices and the cavities in turbulent cavitating flow. Recently Foeth (2008) and Foeth et al., 2006, Foeth et al., 2008 found from their experiment that the shedding of 3D sheet cavitation is in fact a mixing layer. Its characteristic vortical structure is clearly visible on the images presented for a large scale cavity. However, it is difficult to obtain the quantitative data of these cavitating vortices using PIV due to various limitations in the measurement techniques. Inspired by their work, the present paper will analyze the shedding vapor clouds over a three dimensional twisted hydrofoil, with special emphasis placed on the three dimensional vortex cavitation caused by vorticity shedding into the flow field. The cavitating flow around the twisted hydrofoil is simulated using the renormalization-group (RNG) k-ε model with local density correction to capture the evolution of vortical flow induced by cavitation development.

Section snippets

Governing equations

In the uniformity assumption of the mixture of water and vapor in the cavity flow, the multiphase fluid components are assumed to share the same velocity and pressure. The continuity and momentum equations for the mixture flow areρt+xj(ρuj)=0(ρui)t+(ρuiuj)xj=pxi+xj[(μm+μt)(uixj+ujxi23ukxkδij)]where ui is the velocity component in the i direction, p is the mixture pressure, μm is the laminar viscosity and μt is the turbulent viscosity which is closed by the

Results and discussion

The flow around the Delft Twist-11 hydrofoil has been investigated by a numerical method described above with an inlet velocity of V=6.97 m/s. The static pressure at the outlet plane of the domain, pout, was assigned according to the cavitation number, σ=(poutpv)/(0.5ρlV2)=1.07. Note that the steady solution with non-cavitating flow was calculated and used as initial condition to simulate the unsteady situation. From the results of unsteady cavitating flow, the oscillation period of cavity

Conclusions

In this paper, unsteady cavitating turbulent flow around a twisted hydrofoil (σ=1.07, α=−2°) was simulated using a mass transfer cavitation model and the modified RNG k-ε model with a local density correction for turbulent eddy viscosity. The interaction between cavitation and vortical flow is discussed and analyzed. Based on the present study, the following conclusions can be drawn.

The unsteady cavitation shedding structure around the twisted hydrofoil was successfully simulated compared with

Acknowledgment

The authors would like to appreciate Dr. Foeth and Prof. & Dr. Tom van Terwisga, Delft University of Technology, for opening geometry of the Delft Twist-11 hydrofoil and the experimental data.

This work was financially supported by the National Natural Science Foundation of China (Project nos. 51206087 and 51179091), the Major National Scientific Instrument and Equipment Development Project (Project no. 2011YQ07004901), and the China Postdoctoral Science Foundation funded projects (2011M500314

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