Decomposition of unsteady sheet/cloud cavitation dynamics in fluid-structure interaction via POD and DMD methods

doi:10.1016/j.ijmultiphaseflow.2021.103690

International Journal of Multiphase Flow

Volume 142, September 2021, 103690

https://doi.org/10.1016/j.ijmultiphaseflow.2021.103690 Get rights and content

Highlights

l
Analyze the cavitation-structure interaction with focus on the dominant coherent structures.
l
Improve the understanding of frequency characteristics of the cavitation-vortex interactions.
l
Investigate the energy and modal features via POD and DMD.

Abstract

The objective of this paper is to study sheet/cloud cavitation dynamics in fluid-structure interaction by experimental and numerical methods. The high-speed camera is applied to observe the cavitating flow structures and the Laser Doppler Vibrometer is used to characterize the vibration. A hydrodynamic load cell is applied to measure the lift and drag static force. The results present different cavitating patterns of the flexible hydrofoil and the vibration amplitude is enhanced when the cloud cavitation occurs. The hybrid coupled fluid structure algorithm is adopted to simulate the cloud cavity shedding downstream due to the re-entrant jet from the cavity closure to the hydrofoil's leading edge. The vibration analysis shows that the frequency spectrum of the flexible hydrofoil is much more complicated than the rigid one, the main cavitating flow-induced vibration frequency of the flexible hydrofoil is due to the cavity shedding, others are corresponding to vortex shedding frequency and natural frequency in water. The Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) methods are used to investigate the dominant coherent structures of the cavitating flow. With the POD method, it reveals that large-scale cloud cavities occupy a large amount of energy in the flow field. The DMD method accurately extracts the dominant frequency and modal characteristic, with the first mode corresponding to the mean flow field, the second mode being cavity shedding and the third and fourth mode being vortex shedding.

Introduction

Cavitation generally occurs when the local pressure drops below the saturated vapor pressure with the gas filled or gas and vapor filled cavities formed (Wang et al., 2017; Batchelor, 1967). Some problems such as vibration, noise, erosion, even structural failure is unavoidable with the occurrence of cavitation, so it has become a great issue for many applications, such as hydraulic machinery, marine engineering and naval industry (Peters et al., 2018; Arndt, 1981; Yang et al., 2019) With using lightweight structures in these fields, the elastic deformation of structure cannot be ignored, so the cavitation-structure interaction has become more complex.

Much work has been conducted to study the cavitating flow over rigid hydrofoils. Recent works have studied the unsteady cavitation structures of hydrofoils via experimental (Wu et al., 2018b; Peng et al., 2016; Harwood et al., 2019) and numerical studies (Wang et al., 2018a,2018b; Wu et al., 2016; Saito et al., 2007; Sun et al., 2019). Wang et al. (2001) experimentally investigated different cavitation regimes around hydrofoils with visualization and measurement. They showed a distinctly quasi-periodic pattern that in cloud cavitation. To further investigate the cloud cavitation, Kubota et al. (1992) employed experimental and numerical methods to study the cloud cavitation around a hydrofoil. They pointed out that the cloud cavities shed from the trailing edge of the cavities along with large-scale vortices. Huang et al. (2019) reviewed the cavitating flow structures and flow mechanisms. They discussed the evolution of sheet/cloud cavity, along with cavitating vortex street's forming. Leroux et al. (2005) found that re-entrant jet was mainly responsible for the cavity break off. Ji et al. (2013) simulated cloud cavitation around a twisted hydrofoil. The results showed that the cavitation-vortex interaction is the principal mechanism for the evolution of horse-shoe vortexes. The re-entrant jet mechanism is one of the most important factors of the cloud cavitation shedding. Che et al. (2019) experimentally investigated the dynamic behaviors of re-entrant jet in the partial cavity and transitional cavity oscillation. The results showed that in the partial cavity oscillation, the sheet cavitation grew with good spanwise uniformity, but in the transitional cavity oscillation, the cavity grew with a concave line. Pelz et al. (2017) investigated the transition from sheet to cloud cavitation analytically, and a physical model was introduced. They presented the evidence of nucleation and bubble collapse for the growth of the sheet cavity and underlined the role of wall friction for the evolution of the re-entrant jet. Smith et al. (2020a) studied the influence of fluid-structure interaction on cloud cavitation about a stiff hydrofoil in experimental, numerical and SPOD methods. They found the FSI was only reflected in lock-in phenomena and a strong correction between the tip deformations and spanwise cavity oscillations. So, they claimed that the influence of FSI on the cavitation behavior with the stainless model is almost inconsequential.

In addition to the cavitating flow around rigid hydrofoils, numerous researchers have investigated the effects of cavitation on the flexible hydrofoils. Ducoin and Young (2013) numerically investigated the viscous effects, like transition and stall, on the hydroelastic stability of hydrofoils. They found that viscous effects tend to delay or suppress divergence because the effective angle of attack decreased due to the center of pressure moved toward the midchord at large-scale separating flow. Lelong et al. (2018) conducted an experimental procedure to analyze the cavitating flow around a lightweight hydrofoil with fluid-structure interaction considered. They found that with the development of cavitation, the cavity shedding frequency and the bending mode frequency controlled the structure response. Ducoin et al. (2012a) experimentally investigated the boundary-layer transition induced vibrations on a flexible hydrofoil, the results showed that foil vibrations characteristics in terms of frequency and amplitude depend on the vortex shedding frequency. They (Ducoin et al, 2012b) further found that cavitation induced a large increase of the vibration level due to hydrodynamic load unsteadiness and change of modal response for specific frequencies. Not only the effects of cavitation on the foil, but also the influence of foil on cavitation has been investigated. Chae et al. (2016) presented numerical studies of flow-induce vibration of flexible hydrofoils. The foil vibrations are found to be dominated by the natural frequencies in absence of large-scale vortex shedding due to flow separation. Wu et al. (2018a) compared the cavitation patterns and cavitating flow-induced vibration of steel and flexible hydrofoils. They stated that cavities around the flexible hydrofoil appeared to be fragmentized due to its vibration and the main vibration frequency was dominated by the cavity shedding frequency. Young (2018) conducted experimental and numerical work to study the load-depend bend-twist coupling effects on the hydroelastic response of composite hydrofoils. They pointed at material bend-twist coupling that led to nose-up twist accelerated stall and static divergence. While the opposite is true for material bend-twist coupling led to nose-down twist. Smith et al. (2018) compared the rigid and compliant hydrofoil experimentally to explain many observed phenomena and the extensive research into fluid-structure interaction. They found that the compliance dampened the higher frequency force fluctuations while showed strong correction between normal force and tip deflection. Subsequently, Smith et al. (2020b) further analyze the high-speed photography and force measurements on stiff and flexible hydrofoils, they found that the flexibility led to high frequency attenuation of the forces, frequency modulation, accelerated cavitation regime transition as well as multiple lock-in modes.

It's well known that the cavitating flows are quasi periodic instead of real periodic, one cycle is insufficient to describe the hydrodynamic characteristics of the hydrofoil in the computational domain. To extract the dominant coherent features and statistical data of the cavitation flow, POD and DMD have been proposed and applied in a wide of applications, such as orifice plate jet (Alenius, 2014), open cavity flow (Vinha et al., 2016), wave packet (Pan et al., 2015). The POD method is to acquire the most energetic modes of the flow field and use them to reconstruct the flow field. Miyanawala and Jaiman (2019) utilized POD method to analyze the wake flow of a square cylinder under different flow conditions. They utilized POD mode to capture the organized motions of wake flow, the vortex street, the shear layer and the near-wake bubble. Wang et al. (2018) used POD method to study the ventilated cavitating flow around a bluff body, with focus on the vortex shedding behavior in the wake. The results showed that the first and second mode frequencies, associated with the vortex shedding frequency reduced as the gas entrainment coefficient Q_v increase, the energy of large-scale structures decreased with the increase of Q_v in terms of spatial scales. POD method is effective to determine the most energetic flow structures, however, for the complex cavitating flow, the evolution of cavitation structures is characterized by the obvious dominant frequency, which also includes the low-energy features but with a great influence on the dynamic characteristics of the flow. Dynamic Mode Decomposition method is proposed to present the contribution of the flow structure with different frequencies to the flow field by extracting the dominant frequency of the system. Liu et al. (2019) employed DMD method to analyze the cavitating flow coherent structures around a hydrofoil, they found that the cavitating flow remained stable at the long side due to the dynamic mode mainly occurred on the short side. They also accurately extracted the frequency characteristics in DMD methods. Prothin et al. (2016) used POD and DMD methods to analyze the hydrodynamic instability of sheet cavities around a hydrofoil. They showed that the 3D effects were caused by the re-entrant jet instability or propagating shock wave mechanism at the beginning of the cloud cavitation shedding process. Liang et al (2020) employed the POD and DMD method to investigate the liquid nitrogen cavitating flows with emphasis on the vortex structures. They compared the thermal modes with the isothermal modes, the results showed the significant difference between the two typical cavitation modes is the positions.

Although the phenomenon and mechanism of cavitation have been investigated, a concise review of previous works shows that the explanation of complex cavitation-induced vibration frequency needs to be further investigated. In addition, the effect of the flexibility on the cavitation regime transition, the forces, deformation, and vibration has not been well explained yet. For this reason, the paper presents the experimental and numerical study of the cloud cavitating flow structures and cavitation-induced vibration on a NACA66 hydrofoil. By comparing the rigid and flexible hydrofoil, the effect of the flexibility on the fluid-structure interaction has been further investigated. Then POD and DMD methods are employed to investigate the association among the unsteady cloud cavity shedding, vortex shedding and corresponding vibrations.

Section snippets

Experimental setup

Measurements are conducted in the cavitation tunnel in Beijing Institute of Technology (Wu et al., 2017), as shown in Fig. 1. It is mainly composed of the inlet pipe, the constricted section, the test section, the diffusion section and the return pipe, with the test section 0.7 m long, 0.07 m wide and 0.19 m high, as shown in Fig. 2. The adjustable flow parameters are the upstream pressure (with range of 0-0.095 MPa and control accuracy of 5.3%) and the flow velocity (with range of 0-20 m/s and

Governing equations

The incompressible, unsteady Reynolds Average Navier-Stokes (URANS) equations without heat transfers and body force can be described as follow: $\frac{\partial (ρ_{m})}{\partial t} + \frac{\partial (ρ_{m} μ_{j})}{\partial x_{j}} = 0$ $\frac{\partial (ρ_{m} u_{j})}{\partial t} + \frac{\partial (ρ_{m} u_{i} u_{j})}{\partial x_{j}} = - \frac{\partial p}{\partial x_{i}} + \frac{\partial}{\partial x_{j}} [(μ_{m} + μ_{t}) (\frac{\partial u_{i}}{\partial x_{j}} + \frac{\partial u_{j}}{\partial x_{i}})]$ $ρ_{m} = ρ_{l} α_{l} + ρ_{v} α_{v}$ $μ_{m} = μ_{l} α_{l} + ρ_{v} α_{v}$ where $ρ_{l}$ , $ρ_{v}$ and $ρ_{m}$ are the liquid, vapor and mixture densities, u is the velocity, p is the pressure, $μ_{l}$ , $μ_{v}$ and $μ_{m}$ are the liquid, vapor and mixture laminar dynamic viscosity, $μ_{t}$ is the turbulent viscosity, $α_{v}$ is the vapor fraction, $α_{l}$ is the liquid

Mode decomposition methodology

The POD and DMD methods are applied to extract the cavitating flow coherent structures around the flexible hydrofoil.

Cavitation patterns associated with the structural vibrations

The measured mean values of lift coefficients and moment coefficients for the rigid and flexible hydrofoils as a function of σ at α = 8°, Re = 10 × 10⁵ are shown in Fig. 5(a) and (b). The results show that the trend of lift and moment coefficients for the rigid hydrofoil and the flexible one is approximately the same, the measured mean lift and moment coefficients slightly increase from incipient cavitation (σ =1.9) to sheet cavitation (σ =1.5), which is attributed to the increase in effective

Conclusions

In this work, experimental and numerical methods, as well as the reduced order model are used to analyze the effect of flexibility on the cavitating flow structures and vibration characteristic, the complex discrepancy is presented by the comparison of a rigid and a flexible NACA66 hydrofoil. The primary finds are including:

(1)
Compared with rigid hydrofoil, the cloud cavitation of the flexible hydrofoil experienced a greater range of cavitation numbers in the measured lift and moment coefficients

Discussion and future work

It should be noted that the present work is inadequate, additional experimental and numerical investigations are needed. The experimental study is based on the FFT analysis on a one point laser vibration measurement, the position of the LDV would adjust to sweep more points along the surface of the hydrofoil to investigate the vibration distinction along the chordwise and spanwise. Also, the inadequacy of conventional turbulent model and cavitation model employed in the present simulation to

CRediT authorship contribution statement

Yunqing Liu: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration, Funding acquisition. Qin Wu: Software, Writing - review & editing, Supervision, Funding acquisition. Biao Huang: Writing - review & editing, Supervision, Funding acquisition. Hanzhe Zhang: Software, Investigation. Wendong Liang: Software, Investigation. Guoyu Wang:

Declaration of Competing Interest

None.

Acknowledgments

The authors gratefully acknowledge the support by the National Natural Science Foundation of China (Nos. 51839001, 51909002, 52079004), Beijing Natural Science Foundation (Nos. 3204056, 3212023), Defense Industrial Technology Development Program (JCKY2018203B025).

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Decomposition of unsteady sheet/cloud cavitation dynamics in fluid-structure interaction via POD and DMD methods

Highlights

Abstract

Introduction

Section snippets

Experimental setup

Governing equations

Mode decomposition methodology

Cavitation patterns associated with the structural vibrations

Conclusions

Discussion and future work

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Comput. Fluids

J. Fluids Struct.

Eur. J. Mech. B. Fluids

J. Fluids Struct.

Comput. Fluids

Int. J. Multiph. Flow

Int. J. Heat Mass Transfer

Int. J. Multiph. Flow

Wear

Int. J. Multiph. Flow

Aerosp. Sci. Technol.

Ocean Eng.

Prog. Aerosp. Sci.

Exp. Therm. Fluid Sci.

Int. J. Multiph. Flow

Ocean Eng.

Int. J. Multiph. Flow

Eur. J. Mech. B. Fluids

Cavitation in fluid machinery and hydraulic structures

Annu. Rev. Fluid Mech.

An Introduction to Fluid Dynamics

The cavitation instability induced by the development of a re-entrant jet

J. Fluid Mech.

Numerical and experimental investigation of natural flow-induced vibrations of flexible hydrofoils

Phys. Fluids

Dynamic behaviors of re-entrant jet and cavity shedding during transitional cavity oscillation on NACA0015 hydrofoil

J. Fluids Eng.- Trans. ASME

Numerical simulation of the unsteady behavior of cavitating flows

Int. J. Numer. Methods Fluids

Re-entrant jet modeling of partial cavity flow on three dimensional hydrofoils

J. Fluids Eng.

Sheet cavitation and cloud cavitation, re-entrant jet and three-dimensionality

Appl. Sci. Res.