Investigation of cavitation noise using Eulerian-Lagrangian multiscale modeling

Graphical abstract

Highlights

• •A new Eulerian-Lagrangian model is developed for hydrodynamic cavitation.
• •Characteristics of multiscale cavitation features and noise are well revealed.
• •High-frequency noise induced by dispersed cavitation bubbles is identified.

Abstract

We have employed the large eddy simulation (LES) approach to investigate the cavitation noise characteristics of an unsteady cavitating flow around a NACA66 (National Advisory Committee for Aeronautics) hydrofoil by employing an Eulerian-Lagrangian based multiscale cavitation model. A volume of fluid (VOF) method simulates the large cavity, whereas a Lagrangian discrete bubble model (DBM) tracks the small bubbles. Meanwhile, noise is determined using the Ffowcs Williams-Hawkings equation (FW-H). Eulerian-Lagrangian analysis has shown that, in comparison to VOF, it is more effective in revealing microscopic characteristics of unsteady cavitating flows, including microscale bubbles, that are unresolvable around the cloud cavity, and their impact on the flow field. It is also evident that its evolution of cavitation features on the hydrofoil is more consistent with the experimental observations. The frequency of the maximum sound pressure level corresponds to the frequency of the main cavity shedding for the noise characteristics. Using the Eulerian-Lagrangian method to predict the noise signal, results show that the cavitation noise, generated by discrete bubbles due to their collapse, is mainly composed of high-frequency signals. In addition, the frequency of cavitation noise induced by discrete microbubbles is around 10 kHz. A typical characteristic of cavitation noise, including two intense pulses during the collapsing of the cloud cavity, is described, as well as the mechanisms that underlie these phenomena. The findings of this work provide for a fundamental understanding of cavitation and serve as a valuable reference for the design and intensification of hydrodynamic cavitation reactors.

1. Introduction

Hydrodynamic cavitation is a common phenomenon in fluid machinery [1], [2], [3], [4] which primarily occurs at locations where local pressures are lower than saturated vapour pressures, resulting in several adverse effects, like pressure loss, erosion, vibration, and noise [5], [6], [55]. For example, cavitation noise caused by propellers has emerged as a major source of underwater radiated noise emanating from ships, posing a serious threat to the marine environment [7], [8], [56], [63]. On the contrary, hydrodynamic cavitation can be utilized to enhance the reaction process of various chemical and environmental applications [58], [59], [60], [62]. Full-scale measurements are the most accurate method of measuring noise; however, these measurements are limited by experimental facilities and scaling assumptions. It is, therefore, imperative to develop practical computational methods for predicting noise signatures that address these concerns.

There has been a continuous development of numerical models over the last few years, and the noise characteristics of cavitation have attracted increasing attention [9], [10], [11], [12], [13], [61]. A method of inverse tracing was developed by Kao [14] to address radiated noise resulting from unsteady cavitation of propeller blades. A significant difference was observed between the predicted and measured radiated noise due to an overestimation of the size of the blade cavity. A computational fluid dynamics (CFD) method has been demonstrated by McIntyre et al. [15] to accurately predict the qualitative trend of sound spectra produced by propeller cavitation. Nevertheless, the Schnerr-Sauer cavitation model overestimated the amount of radiated noise, indicating that the cavitation models need to be improved. Dong et al. [16] investigated composite hydrofoils' dynamic response and acoustic properties using cavitation-induced vibrations. In their studies, the composite materials were found to suppress cavitation, decrease pressure pulsations in the flow field, and increase lift-to-drag ratios. Kalikatzarakis et al. [17] developed a hybrid model that integrated physical and data-driven simulations to model propeller cavitation and noise. Using the semi-empirical formula, the near-field propeller cavitation noise prediction results were also relatively accurate [18]. Additionally, several studies have demonstrated that combining CFD and acoustic analogies based on the Williams and Hawkings (FW-H) equation is effective [19]. To investigate unsteady cavitation around a NACA0015 hydrofoil and its associated noise, Yu et al. [20] combined a filter-based turbulence model with the Zwart-Gerbera-Belamri (ZGB) cavitation model. To predict hydrodynamic noise, caused by tip-vortex cavitation (TVC) of submarine propellers, Ku et al. [21] employed delayed detached eddy simulation (DDES) with adaptive mesh refinement and the FW-H equation, and concluded that a higher pitch angle resulted in more noise. It has been reported by Testa [22] that an improved hydroacoustic formulation based on the standard FW-H equation can accurately predict noise magnitude and direction based on the emitted noise and sheet cavitation dynamics. An investigation by Sezen et al. [23] examined the influence of biofouling roughness on propeller cavitation and underwater radiation noise using the Reynolds-averaged Navier-Stokes (RANS) solver coupled with the porous FW-H equation.

The combination of CFD and FW-H equations can effectively predict hydrodynamic noise caused by cavitation, but accurate prediction of the noise spectrum still has some limitations. In an attempt to predict the acoustic behaviour of TVC, Sezan et al. [24] combined the RANS-DES approach with the porous FW-H equation. RANS solver prediction was found to be insufficient compared to DES solver prediction. Using the multiphase DDES model, Li et al. [25] investigated the underwater radiated noise of a ship operating at its design speed, and they found that the method underestimated the levels of broadband noise in the frequency range of 50–112 Hz, where the contribution of TVC was significant. The maximum under-prediction in this range was about 28 dB at 72 Hz, and at frequencies above 200 Hz, the broadband noise became more and more under-predicted with increasing frequency. Lidtke et al. [26] performed scale-resolving CFD simulations to predict cavitating propeller noise in off-design conditions. They found that accurately modeling a wide range of frequencies, including tonal and broadband components, was still challenging for scale-resolving simulations. This is due to the need for high spatial resolution when resolving the small length scales involved in high-frequency noise generation mechanisms.

Klapwijk et al. [27] investigated the effect of synthetic inflow turbulence on error estimates under steady inflow conditions using a wing in wet and cavitating conditions. An Eulerian–Lagrangian one-way coupled approach was used by Ku et al. [28] to study the inception of TVCs and the noise produced by underwater submarine propellers. The model's validity was verified by comparing the predicted acoustic pressure spectrum with the measured one. Lidtke et al. [29] simulated sheet cavitation and its noise on a NACA0009 hydrofoil using the large eddy simulation (LES) method with the Schnerr-Sauer cavitation model. As a result, they found that the radiated noise correlated with the second temporal derivative of the total cavitation volume and fluctuations in load.

Moreover, it has been demonstrated that it is impossible to predict high-frequency noise without considering small-scale bubble structures. Additionally, Zhang et al. [30] found that the noise predictions made by the LES and FW-H hybrid methods and the LES and Powell vortex sound theory were similar. The numerical research regarding cavitation noise has demonstrated the possibility of accurately simulating the noise induced by resolvable structures. However, high-frequency noise generated by small-scale bubble structures cannot be accurately predicted.

Research has demonstrated that a multiscale approach incorporating Lagrangian particle tracking and interface-resolving methods [31], [32], [33], [34] can effectively resolve large- and small-scale phase structures. Ling et al. [34] proposed a multiscale model in which the large-scale interfaces were resolved using the volume of fluid (VOF) method to investigate droplet formation in atomization. The small droplets were tracked using a Lagrangian point-particle model. Li et al. [31] achieved a discrete-continuum transition algorithm for simulating the large and small bubbles and their transformation in a gas-stirring process, where small bubbles were tracked using the Lagrangian method and large bubbles were resolved by VOF. The multiscale bubble distribution was well characterized.

To improve the prediction of hydrodynamic cavitation, Xu et al. [35] and Cheng et al. [36] modified the Schnerr-Sauer cavitation model to enhance the TVC prediction by taking the effect of non-condensable gas into account, the gas bubbles were added in the flow field and tracked using the Lagrangian approach to calculate the local total bubble pressure for calculating the local mass transportation. Li et al. [33], [37] proposed a multiscale simulation algorithm to reproduce the complex phase structure of tip-leakage cavitating flows and analyzed the impact of model parameters using the coupled Eulerian-Lagrangian method with considering the transformation between large cavities and small bubbles. Furthermore, Yakubov et al. [38] proposed and validated a hybrid MPI/OpenMP algorithm for parallelising the Eulerian-Lagrangian approach to simulate cavitating flows and improve the computation efficiency. Based on the collapse dynamics of Lagrangian bubbles, Peter and Moctar [39] examined cavitation-induced erosion by applying a multiscale Eulerian-Lagrangian model. An investigation of the flow around a sharp-edged bluff body was conducted by Ghahramani et al. [40] using a hybrid cavitation model, which combined a mixture model with a Lagrangian bubble model. They have pointed out that the correct estimation of small-scale cavities can be as important as that of large-scale structures, because the cavitation noise, erosion and strong vibrations occur at small time and length scales.

Specifically, this work was focused on developing a multiscale model, to replicate the cavitation microstructure, and further to overcome the macroscale model's limitations on predicting cavitation noise. Our work explores the multiscale cavitation phenomenon and its associated noise around a hydrofoil that is part of the NACA66 (National Advisory Committee for Aeronautics). The Eulerian-Lagrangian multiscale cavitation model and the LES approach were employed for the transient cavitating flow, and the noise was predicted using the FW-H acoustic analogy equation. By comparing the experiments with the simulations, we observed that the multiscale approach performs better than the VOF method in predicting multiscale features. Lastly, the relationship between the evolution of the multiscale cavitation structure and the noise signal was examined.

2. Model formulation

2.1 vol. of fluid (VOF) method

A VOF method can be used for simulating the macroscale morphology of cavitation. As a result, the governing equations can be expressed as follows (Eqs. (1), (2)):

$$\frac{\partial {\alpha }_v}{\partial t}+\mathrm{\nabla }·\left({\alpha }_v\overrightarrow{u}\right)=\frac{\dot{m}}{{\rho }_v},$$ (1)

$$\frac{\partial \left({\rho }_m\overrightarrow{u}\right)}{\partial t}+\mathrm{\nabla }·\left({\rho }_m\overrightarrow{u}\overrightarrow{u}\right)=-\mathrm{\nabla }P+\mathrm{\nabla }\left[{\mu }_m\left(\mathrm{\nabla }\overrightarrow{u}+\mathrm{\nabla }{\overrightarrow{u}}^T\right)\right]+{\rho }_m\overrightarrow{g}+{\overrightarrow{F}}_s+{\overrightarrow{F}}_i,$$ (2)

where ${\overrightarrow{F}}_s$ is the surface tension [41] and ${\overrightarrow{F}}_i$ is the resultant force interacting with the microbubbles for the momentum coupling. ${\rho }_m$ and ${\mu }_m$ represent the density and viscosity of the mixture, respectively, which are defined as follows (Eqs. (3), (4)):

$${\rho }_m={\alpha }_v{\rho }_v+\left(1-{\alpha }_v\right){\rho }_l,$$ (3)

$${\mu }_m={\alpha }_v{\mu }_v+\left(1-{\alpha }_v\right){\mu }_l,$$ (4)

The mass transfer rate $\dot{m}$ is given by the Schnerr–Sauer cavitation model as follows (Eq. (5)) [42]:

$$\left\{\begin{array}{c}{\dot{m}}_e=C_e\frac{3{\alpha }_v\left(1-{\alpha }_v\right)}{R_B}\frac{{\rho }_v{\rho }_l}{{\rho }_m}\sqrt{\frac{2}{3}\frac{\left(p_v-p\right)}{{\rho }_l}},p<p_v,\\ {\dot{m}}_c=C_c\frac{3{\alpha }_v\left(1-{\alpha }_v\right)}{R_B}\frac{{\rho }_v{\rho }_l}{{\rho }_m}\sqrt{\frac{2}{3}\frac{\left(p-p_v\right)}{{\rho }_l}},p\ge p_v,\end{array}\right)$$ (5)

where ${\dot{m}}_e$ and ${\dot{m}}_c$ are the evaporation and condensation terms, respectively. $C_e$ and $C_c$ are the coefficients of evaporation and condensation fixed at 1.0 and 0.2, respectively [43], and $p_v$ is the saturated vapour pressure set to 3169 Pa. $R_B$ is the bubble radius represented as follows (Eq. (6)):

$$R_B={\left(\frac{{\alpha }_v}{1-{\alpha }_v}\frac{3}{4\pi }\frac{1}{n_B}\right)}^{1/3}.$$ (6)

where $n_B$ is the bubble number density, which is 1 × 10¹¹ [43], [57].

2.2. Discrete bubble model

Under the current computing capability, the microscale bubbles in cavitating flows cannot be directly resolved by the VOF method due to its dependence on mesh resolution. Microscale bubbles can be tracked using a Lagrangian tracking method called the discrete bubble model (DBM), represented as follows (Eq. (7)):

$$m_B\frac{d{\overrightarrow{u}}_B}{\mathit{dt}}=\frac{m_B\overrightarrow{g}\left({\rho }_v-{\rho }_l\right)}{{\rho }_v}+{\overrightarrow{F}}_{\mathit{Drag}}+{\overrightarrow{F}}_{\mathit{VM}}+{\overrightarrow{F}}_{\mathit{PG}},$$ (7)

where ${\overrightarrow{F}}_{\mathit{Drag}}$, ${\overrightarrow{F}}_{\mathit{VM}}$, and ${\overrightarrow{F}}_{\mathit{PG}}$ are the drag force, virtual-mass force, and pressure gradient force, respectively, represented as follows (Eqs. 8–10):

$${\overrightarrow{F}}_{\mathit{Drag}}=m_B\frac{18}{{\rho }_v{d}_B}\frac{{\rho }_m{C}_D}{24}\left|{\overrightarrow{u}}_B-\overrightarrow{u}\right|\left(\overrightarrow{u}-{\overrightarrow{u}}_B\right),$$ (8)

$${\overrightarrow{F}}_{\mathit{VM}}=C_{\mathit{VM}}{m}_B\frac{{\rho }_m}{{\rho }_v}\left({\overrightarrow{u}}_B\mathrm{\nabla }\overrightarrow{u}-\frac{d{\overrightarrow{u}}_B}{\mathit{dt}}\right),$$ (9)

$${\overrightarrow{F}}_{\mathit{PG}}=\frac{{\rho }_m}{{\rho }_v}{m}_B{\overrightarrow{u}}_B\mathrm{\nabla }\overrightarrow{u},$$ (10)

where $d_B$ represents the discrete bubble diameter, $C_D$ is the drag coefficient [44], and $C_{\mathit{VM}}$ is the coefficient of virtual-mass force (equals 0.5).

To model the nucleation process, cavitation cores are randomly assigned in a region where the pressure is below the nucleation pressure to simulate the growth and collapse of discrete bubbles. Zwart-Gerber-Belamri's cavitation model [45] assumes an initial bubble radius of 1 × 10^-6 m. Furthermore, using the simplified Rayleigh-Plesset equation, we can simulate the size evolution of discrete bubbles as follows (Eq. (11)):

$$\frac{d{m}_B}{\mathit{dt}}=\left\{\begin{array}{c}{C}_{\mathit{evp}}4\pi {\rho }_v{R}_B^{\prime 2}\sqrt{\frac{2}{3}\left(\frac{\left|C_{\mathit{sp}}{p}_v-p\right|}{{\rho }_l}\right)},p<C_{\mathit{sp}}{p}_v,\\ C_{\mathit{cond}}4\pi {\rho }_v{R}_B^{\prime 2}\sqrt{\frac{2}{3}\left(\frac{\left|p-C_{\mathit{sp}}{p}_v\right|}{{\rho }_l}\right)},p\ge C_{\mathit{sp}}{p}_v,\end{array}\right)$$ (11)

where $C_{\mathit{sp}}$, $C_{\mathit{evp}}$ and $C_{\mathit{cond}}$ are the coefficients of saturated pressure, evaporation and condensation in the discrete bubble model, and have values of 1.25, 0.5 and 0.01, respectively [37].

The volume fraction of discrete bubbles is evaluated by integrating the volume of all bubbles in each cell and the total volume fraction including both macroscale and microscale parts is calculated as (Eq. (12)):

$${\alpha }_v={\alpha }_v\text{+}\frac{{\sum }_{B_i\in cell}{V}_{B_i}}{V_{\mathit{cell}}}$$ (12)

Then the total volume of cavity $V$ is calculated by volume integrating in the flow field as (Eq. (13)):

$$V=\sum \left({\alpha }_v{V}_{\mathit{cell}}\text{+}\sum _{B_i\in cell}{V}_{B_i}\right)$$ (13)

where $V_{\mathit{cell}}$ represents the cell volume.

2.3. Large eddy simulation (LES)

LES can simulate large-scale vortices as well as model small-scale vortices directly. An LES model can reproduce the detailed information of unsteady flows compared to a RANS model. The mass and momentum conservation equations for vapour/two-phase liquid flows are shown below (Eqs. (14), (15)).

$$\frac{\partial {\rho }_m}{\partial t}+\frac{\partial }{\partial x_i}\left({\rho }_m{\overline{u}}_i\right)=0,$$ (14)

$$\frac{\partial }{\partial x_i}\left({\rho }_m{\overline{u}}_i\right)+\frac{\partial }{\partial x_j}\left({\rho }_m{\overline{u}}_i{\overline{u}}_j\right)=\frac{\partial }{\partial x_i}\left({\mu }_m\frac{\partial {\overline{u}}_i}{\partial x_j}\right)-\frac{\partial \overline{p}}{\partial x_i}-\frac{\partial {\tau }_{\mathit{ij}}}{\partial x_j}+F$$ (15)

where ${\overline{u}}_i$ and ${\overline{u}}_j$ are the filtered velocities in the i and j directions, respectively, $F$ is the force acting on the fluid including the interacting force from discrete bubbles, and ${\tau }_{\mathit{ij}}$ is the sub-grid scale (SGS) stress term, which is defined as follows (Eq. (16)):

$${\tau }_{\mathit{ij}}={\rho }_m{\overline{u}}_i{\overline{u}}_j-{\rho }_m\overline{u_i{u}_j},$$ (16)

The eddy-viscosity model is expressed as follows (Eq. (17)):

$${\tau }_{\mathit{ij}}-\frac{1}{3}{\tau }_{\mathit{kk}}{\delta }_{\mathit{ij}}=-2{\mu }_t{\overline{S}}_{\mathit{ij}},$$ (17)

In previous studies [46], the wall-adapting local eddy-viscosity (WALE) model was applied, and the SGS stress was expressed as follows (Eq. (18)):

$${\mu }_t={\rho }_m{\left(C_w\mathrm{\Delta }\right)}^2\frac{{\left(S_{\mathit{ij}}^d{S}_{\mathit{ij}}^d\right)}^{3/2}}{{\left({\overline{S}}_{\mathit{ij}}{\overline{S}}_{\mathit{ij}}\right)}^{5/2}+{\left(S_{\mathit{ij}}^d{S}_{\mathit{ij}}^d\right)}^{5/4}},$$ (18)

where $C_w$ is the empirical coefficient of the WALE model equal to 0.325, and the $S_{\mathit{ij}}^d$ and ${\overline{S}}_{\mathit{ij}}$ are defined as follows (Eqs. (19), (20)):

$$S_{\mathit{ij}}^d=\frac{1}{2}\left({\overline{g}}_{\mathit{ij}}^2+{\overline{g}}_{\mathit{ji}}^2\right)-\frac{1}{3}{\delta }_{\mathit{ij}}{\overline{g}}_{\mathit{kk}}^2,$$ (19)

$${\overline{S}}_{\mathit{ij}}=\frac{1}{2}\left(\frac{\partial {\overline{u}}_i}{\partial x_j}+\frac{\partial {\overline{u}}_j}{\partial x_i}\right).$$ (20)

2.4. Hydroacoustic model

An acoustic analogy is widely used for engineering simulations and applications based on the FW-H equation, which is extended from Lighthill's acoustic analogy [47] and developed by Ffowcs-Williams and Hawkings [19] to incorporate the influence of surfaces in arbitrary motion. Using integrals of source field quantities, the sound at any arbitrary location can be determined. Di Francescantonio et al. [48] presented a new boundary integral formulation to evaluate the radiated noise, which permits the selection of permeable surfaces within the fluid domain to integrate. By integrating the permeable surfaces, noise sources in the volume enclosed by the surfaces are accounted for without performing volume integration [49]. This method has been proven to be effective in hydroacoustic analysis for cavitating flows [9], [50], and is also employed in the present work.

The porous FW-H equation can be derived from the Navier-Stokes equations and continuity equation as follows (Eq. (21)):

$$\frac{1}{c^2}\frac{{\partial }^2{p}^{\prime }}{\partial t^2}-\frac{{\partial }^2{p}^{\prime }}{\partial x_i^2}=\frac{{\partial }^2{T}_{\mathit{ij}}}{\partial x_i\partial x_j}-\frac{\partial }{\partial x_i}\left[P_{\mathit{ij}}\delta \left(f\right)\frac{\partial f}{\partial x_j}\right]+\frac{\partial }{\partial t}\left[{\rho }_0{u}_i\delta \left(f\right)\frac{\partial f}{\partial x_i}\right],$$ (21)

where $p^{\prime }$ is defined as $p^{\prime }=p-p_0$, which is the sound pressure in the far field. $\delta \left(f\right)$ represents the Dirac-$\delta$ function. $P_{\text{ij}}$ and $T_{\text{ij}}$ represent the compressive stress tensor and Lighthill’s stress tensor, respectively. $c$ represents the sound speed and ${\rho }_0$ is the ambient density.

A solution to Eq. (21) can be expressed as the sum of the three terms at any arbitrary receiver $\mathbf{x}$ as follows (Eq. (22)):

$$p^{\prime }\left(\mathbf{x},t\right)=p_T^{\prime }\left(\mathbf{x},t\right)+p_L^{\prime }\left(\mathbf{x},t\right)+p_Q^{\prime }\left(\mathbf{x},t\right),$$ (22)

The three terms on the right-hand side of Eq. (22) are represented as follows: $p_T^{\prime }\left(\mathbf{x},t\right)$ is the thickness of the noise source (monopole). Cavitation development and collapse are responsible for the thickness acoustic source. Alternatively, it is referred to as cavitation noise. $p_L^{\prime }\left(\mathbf{x},t\right)$ is the loading (dipole) noise source. In addition to low-frequency noise, it is an unsteady pulse force generated by turbulent and inhomogeneous flow fields on the hydrofoil surface. $p_Q^{\prime }\left(\mathbf{x},t\right)$ describes the quadrupole noise resulting from the hydrofoil's boundary layer and vortex shedding, which is a high-frequency noise.

Following Eqs. (23), (24), (25), (26) consider the intermediate acoustic variables.

$$U_i=\frac{\rho }{{\rho }_0}{u}_i,$$ (23)

$$L_i=\left(P_{\mathit{ij}}+\rho u_i{u}_j\right){\widehat{n}}_j,$$ (24)

$$U_n=U_i{\widehat{n}}_i,$$ (25)

$$L_r=L_i{\widehat{r}}_i,$$ (26)

For a stationary source surface S, $p_T^{\prime }\left(\mathbf{x},t\right)$ and $p_L^{\prime }\left(\mathbf{x},t\right)$ are expressed as follows (Eqs. (27), (28)):

$$4\pi p_T^{\prime }\left(\mathbf{x},t\right)={\int }_S{\left[\frac{{\rho }_0{\dot{U}}_n}{r}\right]}_{\mathit{ret}}\text{d}S,$$ (27)

$$4\pi p_L^{\prime }\left(\mathbf{x},t\right)=\frac{1}{c}{\int }_S{\left[\frac{{\dot{L}}_r}{r}\right]}_{\mathit{ret}}\text{d}S+{\int }_S{\left[\frac{L_r}{r^2}\right]}_{\mathit{ret}}\text{d}S,$$ (28)

where ${\left[...\right]}_{\mathit{ret}}$ denotes terms evaluated at the retarded time. Since the receiver locations are far from the source surface S, $p_T^{\prime }\left(\mathbf{x},t\right)$ can be expressed as follows (Eq. (29)):

$$p_T^{\prime }\left(\mathbf{x},t\right)={\left[\frac{{\rho }_0\overline{V}}{4\pi \overline{r}}\right]}_{\overline{\mathit{ret}}},$$ (29)

where $\overline{\mathit{ret}}$ and $\overline{r}$ represent the average retarded time and the average distance of all noise sources on the surface S, respectively, and V represents the cavity volume in the source surface S [51]. Thus, $p_T^{\prime }\left(\mathbf{x},t\right)$ is proportional to the second derivative of the cavity volume.

3. Configurations for simulation

3.1. Domains and meshes of the computation

Fig. 1 illustrates the computational domain and boundary conditions. Dimensions of the domain are 850 mm × 75 mm × 250 mm. This hydrofoil (NACA66) has a chord length (C) of 100 mm, an angle of attack of 8°, and a span length of 0.75C. Hydrofoils are positioned at the center in the direction of height. From the leading edge, the inlet and outlet of the calculation domain are situated at 3.5C and 5C, respectively. In this case, the inlet velocity is 10 m/s, and the outlet pressure is based on the cavitation number. Validation is provided by the experiment conducted by Wu et al. [52], whereas the simulation is conducted with the condition of σ = 1.5. To ignore the wall effect, the hydrofoil surfaces are designed as no-slip walls, and the side walls are designed as free-sliding walls.

Fig. 1.

Computational domain and boundary conditions.

In the current model, the computational domain is discretised into structured hexahedron cells, as shown in Fig. 2. During the refinement procedure, the mesh at the cavitation areas is ensured to have acceptable orthogonal quality and improve the simulation accuracy. Approximately 100 × 120 meshes are located adjacent to the hydrofoil walls. The first layer's cell height is 0.012 mm, and its growth ratio is 1. A refined mesh is used near the hydrofoil's leading edge and tail to capture the cavitation process's intricate details.

Fig. 2.

The distribution of meshes in the computational domain for (a) side walls and (b) hydrofoils.

3.2. Numerical schemes and noise setups

In the present work, the momentum equation is solved using the bounded central differencing spatial discretisation algorithm, and the bounded second-order implicit time formulation is utilised for the LES method. To enhance the simulation stability, the implicit scheme is used for solving the equation of the volume fraction in the VOF method, which is discretised as (Eq. (30)):

$$\frac{{\alpha }_v^{n+1}{\rho }_v^{n+1}-{\alpha }_v^n{\rho }_v^n}{\mathrm{\Delta }t}{V}_{\mathit{cell}}+\sum _f\left({\rho }_v^{n+1}{U}_f^{n+1}{\alpha }_{v,f}^{n+1}\right)=\dot{m}{V}_{\mathit{cell}}$$ (30)

where n is the index for the previous time step, U is the volume flux through the face and the subscript f represents the face value.

For pressure–velocity coupling, the pressure-implicit with splitting of operators (PISO) scheme, which is suitable for transient simulations, is utilised. In this scheme, the momentum equation using the operator H for the finite-difference representation is first solved using the pressure field $P^n$ from the old time step as (Eq. (31)):

$$\frac{1}{\mathrm{\Delta }t}\left[{\left(\rho u\right)}_i^{\ast }-{\left(\rho u\right)}_i^n\right]=H\left(u_i^{\ast }\right)-{\mathrm{\Delta }}_i{P}^n+S_i$$ (31)

where ${\left(\rho u\right)}_i^{\ast }$ is the flux calculated by the intermediate velocity field $u_i^{\ast }$ which may not satisfy the zero-divergence condition. So a corrector step is performed using a new velocity field $u_i^{\ast \ast }$ and a corresponding new pressure field $P^{\ast }$ obtained by the pressure equation seeking the zero-divergence condition, which is taken as (Eqs. (32), (33)):

$$\frac{1}{\mathrm{\Delta }t}\left[{\left(\rho u\right)}_i^{\ast \ast }-{\left(\rho u\right)}_i^n\right]=H\left(u_i^{\ast }\right)-{\mathrm{\Delta }}_i{P}^{\ast }+S_i$$ (32)

$${\mathrm{\Delta }}_i^2{P}^{\ast }={\mathrm{\Delta }}_{i}H\left(u_i^{\ast }\right)+{\mathrm{\Delta }}_i{S}_i+\frac{1}{\mathrm{\Delta }t}{\mathrm{\Delta }}_i{\left(\rho u\right)}_i^n\text{+}\frac{1}{\mathrm{\Delta }{t}^2}\left({\rho }^{\ast }-{\rho }^n\right)$$ (33)

A further corrector step can be introduced to obtain $u_i^{\ast \ast \ast }$ and $P^{\ast \ast }$. By conducting corrector steps until a pre-defined tolerance is met, the new velocity can be consistent with the new pressure.

Moreover, two values of time step (1 × 10^-5 s and 5 × 10^-6 s) are set to test the time step independency, the convergence criteria for all variables are set as 1 × 10^-4. The computational code for VOF-DBM coupled simulations is developed using user defined functions (UDF) embed into the FLUENT software. The VOF and DBM equations are solved for continuous vapour phase and discrete bubbles, respectively. In the region with vapour volume fraction larger than 0.6 (as the maximum volume fraction of discrete phase is about 0.6), the discrete bubbles are removed and the mass transfer rate is calculated using the source term in Eq. (5). Otherwise, the growing and collapsing terms of discrete bubbles in the dispersed region are modelled using Eq. (11). The algorithm for transformation between VOF and DBM [37] is also implemented for bubbles to grow into continuous ones and large cavities to collapse into discrete bubbles. Thus, the coupling between VOF and DBM is achieved.

For the FW-H approach to calculate the far-field sound pressure signal, noise integration surfaces must be established for tracking velocity and pressure variations. The setup location of the noise integration surface is shown in Fig. 3. To ensure that the cavitation area does not extend beyond the integration surface when the cavity drops off, it is necessary to set the noise integration surface as large as possible to cover the whole cavitation area. Further, the locations of the noise receivers are indicated in Fig. 3 as R1-5. Several noise receiving points are set up around the hydrofoil at varying distances to measure sound pressure distribution in the same direction. For compact sound sources, the receivers and integration surfaces must satisfy the Minimum Frequency Parameter (MFP) criteria [53].

Fig. 3.

A diagrammatic representation of the FW-H integration surface and the locations of the noise receivers.

4. Results and discussion

4.1. Model validation and independence study

An investigation of the effect of mesh resolution on simulation results is conducted by comparing three meshes: coarse, medium, and fine. Fig. 4 shows the distribution of y⁺ values on the hydrofoil surface using different meshes. Based on the y⁺ values (0 ∼ 3) at the surface of the hydrofoil, it is evident that the LES requirements for simulations are generally met. An overview of the parameters used for mesh generation and the comparison of average drag and lift coefficients are presented in Table 1. Observations show that the coarse mesh simulations of C_l and C_d have large errors. In contrast, those of medium and fine meshes are closer to each other. Further simulations use the medium mesh to balance computing demands and simulation accuracy.

Fig. 4.

Y⁺ values for (a) coarse, (b) medium, and (c) fine meshes.

Table 1.

Mesh resolutions and the results of the mesh independence test.

Mesh	Cells on hydrofoil	Cl	Cd
Coarse	70 × 100	1.159	0.096
Medium	80 × 120	1.125	0.092
Fine	90 × 140	1.125	0.092

Fig. 5 illustrates the temporal evolution of the lift coefficient C_l and drag coefficient C_d for a hydrofoil simulated under the fine and medium meshes. Lift and drag coefficients fluctuate periodically as a result of cavitation shedding. Fig. 6 shows the power spectrum density (PSD) of C_d under the medium and fine meshes. In this case, we find that the PSD of C_d follows the −7/3 power law. As shown in Fig. 6, the main frequency of the fluctuation is 45.58 Hz, while the corresponding period is 21.93 ms. According to the experimental data [52], the period of cavitation shedding is 22.29 ms. The discrepancy between the simulation and the experiment may result from the 1 mm gap between the hydrofoil and the side wall, as well as from the roughness of the side wall (not considered in the simulation). The difference between the experimental and simulation cavitation shedding periods is approximately 1.6%, indicating that the simulation is accurate.

Fig. 5.

Temporal evolution of (a) lift coefficient C_l and (b) drag coefficient C_d for hydrofoils with fine and medium meshes.

Fig. 6.

Power spectral density (PSD) of C_d.

To investigate the sensitivity of the time step, Fig. 7 shows the results of sensitivity analysis of the time step performed in the present work. From both the curves of lift coefficients and the cavitation features in a typical cycle, it can be found that the two results using 0.01 ms and 0.005 ms are close to each other. The results indicate that the time step used in the present simulation is adequate as the implicit scheme for VOF allows a relatively large time step.

Fig. 7.

Evolution of (a) C_l and (b) cavitation features using different time steps.

4.2. Unsteady and multiscale features of cavitation

Fig. 8 compares cavitation characteristics observed experimentally and predicted by numerical simulation using the macroscale (VOF) and multiscale (VOF-DBM) approaches, with a cavitation number of 1.5. The surface of hydrofoil experiences periodic cycles of cavitation development, shedding, and collapse. It is evident from the evolution of the cavity volume that cavitation is shed periodically. During each cycle of cavity evolution, four stages can be distinguished: the development of sheet cavitation, the cutoff and collapse of sheet cavitation, the shedding of cloud cavitation, and the collapse of cloud cavitation. As shown in Fig. 8, the sheet cavitation develops from the leading edge to the trailing edge and grows to a location where X/C = 0.8 from 0/10 T to 6/10 T. Cavitation generates re-entrant jets at the tail of the cavity, which travels from the sides of the trailing edge toward the center of the leading edge. As illustrated at 9/10 T in Fig. 8, when the re-entrant jets converge at the hydrofoil's midsection, the cavity's tail cuts off and falls off on the downstream side. A re-entrant jet caused by an adverse pressure gradient in the downstream flow field is responsible for causing the cavity shedding process. VOF-DBM results in comparison to VOF results reveal numerous discrete bubbles around the trailing edge of the hydrofoil. Experimental observations support the location of discrete bubbles. Accordingly, the VOF-DBM method can accurately predict the transition between large cavities and microscale bubbles in the cavitating flows. Compared to VOF results, these results are more consistent with the experiment.

Fig. 8.

Comparison of cavitation characteristics between (a) experimental results [52] and simulations predicted by (b) macroscale (VOF) and (c) multiscale (VOF-DBM) (σ = 1.5, α = 8°, U_∞ = 10 m/s).

Fig. 9 illustrates the evolution of cavity volume $V_{\mathit{vap}}$ over several typical cycles calculated using VOF and VOF-DBM methods. The results show that the cavity volume predicted by the VOF-DBM method is larger than that predicted by the VOF method; however, the variation period of the cavity volume remains unchanged. In particular, the cavity volume calculated by the VOF-DBM method is approximately 20% larger than that calculated by the VOF method when discrete bubbles are considered. Accordingly, it can be concluded that the macroscale VOF method cannot capture the distribution and transport of discrete microbubbles at the current mesh resolution. Multiscale models, on the other hand, are capable of acquiring these microscale structures.

Fig. 9.

Typical cycles of vapor volume evolution.

Using the VOF and VOF-DBM methods, Fig. 10 illustrates the pressure distribution on the surface of the hydrofoil as well as the cavity shapes at two representative instants. In Fig. 10(b), the black dotted circle indicates that VOF-DBM predicts greater low-pressure areas near the trailing edge than VOF. The multiscale model also captures microscale bubbles; negative pressure should accompany bubble transport. Furthermore, the multiscale model can capture the pressure surge caused by microbubble collapse. Based on this study, the VOF-DBM method can capture more detailed characteristics of the transient cavitating flow, including the microscale structures and their effects on the flow field.

Fig. 10.

Comparison of the pressure distribution on the hydrofoil surface and cavitation patterns in (a) macroscale (VOF) and (b) multiscale (VOF-DBM) simulations.

4.3. Sound field characteristics

Acoustic signals generated by the flow around a hydrofoil, including cavitation noise, can be analysed using an acoustic integration surface around the hydrofoil. Fig. 11 illustrates the noise pressure signal $p^{\prime }$ obtained from the monitoring point at R3 over several cavitation periods, where $p^{\prime }$ represents the total noise according to Eq. (21). The total noise is composed of cavitation noise, dipole noise and quadrupole noise, as shown in Eq. (22). The maximum values of total noise calculated by both the VOF and VOF-DBM methods are approximately 200–300 Pa. Due to the inclusion of the sound pressure signal induced by microbubbles in the VOF-DBM method, the acoustic signal calculated by this method has a greater frequency and amplitude.

Fig. 11.

Total sound pressure signal at R3 calculated using two methods: (a) VOF and (b) VOF-DBM.

Fig. 12 illustrates the Sound Pressure Level (SPL) of $p^{\prime }$ R3 based on the frequency spectrum obtained by both methods. In this study, 45.58 Hz was the dominant frequency of the total noise, consistent with the cavitation shedding frequency. This study illustrates that cavitation is the main noise source, while the attached cavity shedding dominates the maximum SPL. It is evident that the SPL calculated by the VOF method and that calculated by the VOF-DBM method are similar in the low-frequency band. Nevertheless, the SPL calculated by the VOF-DBM method is higher than that obtained by the VOF method in the high-frequency band, especially in the frequency band around 10 kHz, which illustrates that the frequency of cavitation noise caused by microbubbles is about 10 kHz.

Fig. 12.

Calculation of SPL for R3 using the (a) VOF and (b) VOF-DBM methods.

The SPL measurements of the monitoring points at R1-5 located above the hydrofoil were plotted, as shown in Fig. 13(a), to examine the differences in noise between different receiving points. In general, the SPL decreases as the distance from the hydrofoil increases. The dominant frequencies are the same in all monitoring points, and the dominant Strouhal number (fC/U_∞) is 0.456. Moreover, we observed that the decaying slope of the sound signal spectrum, monitored at all points, falls between f ^-1 ∼ f ^-2, which is consistent with that described in the experimental investigation [54]. Fig. 13(b) shows the evolution of overall SPL along with the logarithmic distance in the y direction. It illustrates that the SPL decreases linearly with the logarithmic distance. Furthermore, it is found that the noise is a continuous spectrum with signals ranging from low frequencies to high frequencies and the SPL at low frequencies is relatively high.

Fig. 13.

(a) Vertical SPL for receivers at R1-5, (b) evolution of overall SPL at R1-5, along with logarithmic distance.

As shown in Fig. 14, the cavitation (thickness) noise signal at receiver R3 is represented by the VOF and the VOF-DBM methods, where $p_T^{\prime }$ represents the thickness noise or cavitation noise, as shown in Eq. (22). As compared to the total noise $p^{\prime }$ in Fig. 11, the cavitation noise signal is approximately a third of the total noise pressure signal. A cavitation noise signal exhibits strong pulses in every cycle when its $p_T^{\prime }$ value is around the negative peak. In contrast, the strong pulses in the total noise signal when the $p^{\prime }$ value is around the positive peak are not present in the cavitation noise signal, indicating that the strong pulses are not the result of thickness noise caused by the growth and collapse of a cavity. The results from VOF and VOF-DBM indicate that the VOF-DBM method can predict a greater proportion of high-frequency signals, indicating that microbubbles play a significant role in producing high-frequency noise. Unfortunately, the traditional VOF method does not consider microbubbles and other macroscale methods. Additionally, the fundamental frequency of the cavitation noise is the same as the cavitation shedding frequency. Investigating the relationship between the cavitation noise and the unsteady cavity features would be useful.

Fig. 14.

Cavitation noise pressure signals at R3 calculated using two methods (a) VOF and (b) VOF-DBM.

The volume evolution of discrete bubbles is shown in Fig. 15(a), it is found that the evolution period is also 21.93 ms, consistent with the cavitation's primary shedding period. As shown in Fig. 15(b), the Power Spectral Density (PSD) of the cavitation noise induced by discrete microbubbles is primarily characterised by high PSD values at high frequencies. In the process of collapse, the cavity transforms into microscale bubbles that are transported downstream, and the change in pressure around the discrete bubbles results in their collapse. Due to the high frequency of collapse caused by many microbubbles, the cavitation noise caused by microbubbles is also in the high-frequency band and cannot be ignored. The highest PSD of the cavitation noise generated by discrete microbubbles is approximately 10 kHz.

Fig. 15.

(a) Discrete bubble volume evolution over four cycles and (b) PSD of noise by discrete bubbles.

4.4. Relationship between cavitation noise and multiscale features

As shown in Fig. 16(a), the cavitation noise signal and the cavity volume evolution simulated using the VOF-DBM method are plotted to examine the relationship between the cavitation noise and the multiscale cavitation features. As shown in Fig. 16(b), the multiscale simulation and the experiment were compared at three different instants in the evolution curve plotting. Comparing the evolutions of the cavitation noise signal and the cavity volume, the maximum and minimum cavities correspond to the negative and positive peaks of the cavitation noise pressure signal, respectively. Cavitation noise consists of strong pulses when the cavity volume is large and begins to decrease. From this, the strong pulse of noise may have arisen due to the collapse of a large-scale cavity during the transport of a detached cavity.

Fig. 16.

(a) Cavitation noise signal at R3 and volume evolution of the cavity over four cycles, and (b) the three marked instants from both simulation and experiment.

Radiated noise from cavitation is closely related to cavity structures, particularly for strong pulses resulting from cavity collapse. In order to study the details of cavitation features when the strong pulse occurs, the cavitation noise signal and the cavity volume evolution in one cycle are plotted together, and the cavitation features around the two corresponding instants are shown in Fig. 17. There are two signature cavitation noise pulses which are designated as I and II. There is a direct correlation between the intensity of cavitation noise and the acceleration of cavity volume. It can be observed from the curves that the positive cavitation noise pressure is associated with an increase in cavity growth rate. In contrast, the negative cavitation noise pressure is primarily associated with a decrease in cavity growth rate or an increase in cavity collapse rate, except for pulses. According to Fig. 16, the maximum cavity volume occurs when the attached cavity reaches its maximum size and the detached cavity transforms into a cloud of cavities. As seen in Fig. 17, the noise pulse arises after the moment of maximum cavity volume, which is because of the first collapse of the large cloud cavity, as seen in Fig. 17(b) where the cavitation structure evolution around instant I is shown for revealing the noise pulse mechanism. According to the cavitation structure evolution at instant II, some cavity clouds are found to grow and collapse again, resulting in a second strong noise pulse during the cloud cavity transportation. Compared to the first strong noise pulse, the second one is weaker. Furthermore, in addition to the two main noise pulses, other pulses are also found during other cavitation cycles, as the cloud cavity generates and collapses throughout the entire cavity evolution process.

Fig. 17.

(a) Cavitation noise at R3 during a typical cycle and (b) simulated cavity structures at the same time.

5. Conclusions

This study uses the VOF-DBM coupled multiscale cavitation model to investigate the cavitation structures over a wide range of time and space scales and the corresponding cavitation noise characteristics on a NACA66 hydrofoil. The LES approach and FW-H equations are also employed to predict the noise pressure signals in the cavitating flow. VOF and VOF-DBM methods are compared in their ability to predict cavitation structures and noise characteristics. The noise characteristics induced by discrete microbubbles are taken into account, and the relationship between cavitation noise characteristics and multiscale cavity features is investigated. Several key conclusions can be summarised as follows:

1. VOF-DBM can provide a more accurate indication of the transition between large cavities and microscale bubbles in cavitating flows compared to VOF-VOF. VOF-DBM simulation results are more consistent with the experiment than VOF results, as VOF-DBM can more accurately reproduce the bubble cloud transport in the wake region.

2. Main cavity shedding dominates the maximum SPL of the noise. In the low-frequency band, the VOF method provides similar results to the VOF-DBM method. In the high-frequency band, the SPL calculated by the VOF-DBM method is greater than that calculated by the VOF method. Induced by microbubbles, cavitation noise is concentrated around 10 kHz in frequency. Based on the simulation results in the high-frequency band, the multiscale model is more accurate at predicting the noise caused by cavitating flows.

3. According to the cavitation noise pressure signals, the maximum cavity volume corresponds to the negative peak, while the minimum cavity volume corresponds to the positive peak. During the collapse of the cloud cavity, a typical noise characteristic is observed, including two intense pulses. In most cases, the first pulse occurs when a large detached cavity collapses, which happens shortly after the attached cavity grows to its largest proportions. As a result of some cavity clouds growing and then collapsing again, the second pulse occurs.

CRediT authorship contribution statement

Linmin Li: Investigation, Resources, Methodology, Writing – original draft. Yabiao Niu: Investigation, Visualization, Writing – original draft. Guolai Wei: Writing – review & editing. Sivakumar Manickam: Writing – review & editing. Xun Sun: Conceptualization, Writing – review & editing. Zuchao Zhu: Conceptualization, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.