Multiscale modeling of wake-induced propeller cavity bursting

When a marine propeller exceeds its critical speed, cavitation typically initiates at the blade tips. Under the influence of the nonuniform wake generated by the hull, tip vortex cavities may undergo so-called bursting.¹ During this process, these cavities rapidly expand, leading to a sharp increase in vortex core pressure and subsequent violent cavity collapse. The triggered cavity collapse can cause a significant increase in broadband hull pressure fluctuations and noise,² which is undesirable in marine engineering. To obtain more accurate predictions of cavity bursting and the noise that it induces, high-precision numerical simulations of propeller cavitation are essential.

The complex process of tip vortex cavity bursting presents substantial challenges for numerical simulation. Previous studies have identified several factors that may contribute to cavity bursting. Kuiper³ attributed this phenomenon to abrupt variations in blade loading caused by the hull wake, highlighting the need for accurate wake prediction. Given limitations on computational resources, a common strategy is to specify inlet boundary conditions to represent the nonuniform wake.^4,5 However, this approach may be inadequate for the current problem, since it fails to accurately capture the turbulent vortices generated by the hull. To address this, we will employ a wake generator to produce a more realistic and dynamic wake.

Additionally, Ge et al.⁶ suggested that interactions between sheet/cloud cavitation and tip vortex cavitation play a crucial role in bursting. Sheet/cloud cavities consist of numerous microscale bubble clusters,^7,8 and bursting tip vortex cavities also exhibit complex multiscale behavior, with phase interfaces transitioning from continuous cavities to dispersed bubble clusters.⁹ Thus, the bursting phenomenon is characterized by highly unsteady and multiscale dynamics. Conventional mass transport models are constrained by grid resolution and are therefore inadequate for simulating such multiscale processes. Recently, multiscale cavitation models based on an Euler–Lagrange (E–L) framework have been developed.^10–13 These models employ mass transport models for grid-resolved scale cavitation and particle tracking techniques for sub-grid-scale cavitation, ensuring smooth transitions across scales. This approach offers significant advantages in simulating multiscale cavitation phenomena. However, its application has been largely confined to stationary hydrofoils, with no reported cases involving rotating machinery.

This study aims to investigate the feasibility of applying an E–L hybrid model to simulate wake-induced propeller cavity bursting. We focus on the cavitating flows around a INSEAN E779A propeller,¹⁴ a well-established benchmark test case for simulations of propeller cavitation. The remainder of this paper is structured as follows. Section II introduces the novel E–L hybrid model. Section III details the numerical setup for the test case of an INSEAN E779A propeller. Section IV A evaluates the ability of the novel model to simulate cavity bursting and the associated pressure fluctuations, with calibration against available measured data. Section IV B further explores the microscale bubble characteristics during the process of multiscale cavity bursting. Finally, Sec. V summarizes the findings and presents the conclusions of this study.

We simulate the cavitating flow around an INSEAN E779A propeller positioned behind wake-generating plates. This is a skewed four-blade model propeller, with a diameter D of 227.27 mm. The corresponding experiments were conducted at the Italian Navy’s cavitation tunnel facility (C.E.I.M.M.).¹⁴ 

As shown in Fig. 2(a), the computational domain is constructed on the basis of the setup used in the workshop of the Fourth International Symposium on Marine Propulsors.²³ The dimensions of this tunnel are 0.6 × 0.6 × 2.2 m³, with a corner radius of 0.1 m in the cross-section. The coordinate origin is located at the propeller plane center, positioned 4D away from the outlet center. The wake-generating plates are modeled following the descriptions in the literature,¹⁴ with the downstream side of the plates located 0.445D from the propeller plane.

An unstructured mesh strategy is employed, as shown in Fig. 2(b). We use a prism layer with ten nodes on the no-slip walls to accurately predict boundary layer flow. The height of the first layer is set to ensure a dimensional distance y⁺ of around 1, meeting the requirements of WMLES. A Cartesian cut-cell mesh is applied to the remaining regions, generating a base mesh with a total cell number of 17.4 × 10⁶ (coarse mesh). Furthermore, the present prediction accuracy of propeller cavitation noise is limited to only the first few blade-passing frequencies. Ge et al.⁶ have pointed out that tip vortex cavitation generates propeller cavitation noise with higher blade-passing frequencies. Therefore, to accurately capture high-frequency propeller cavitation noise, special refinement of the tip vortex region is necessary. With additional refinement in areas where cavitation may occur, two refined meshes are generated, with total cell numbers of 29.9 × 10⁶ (medium mesh) and 97.6 × 10⁶ (fine mesh).

The pressure-based SIMPLEC algorithm is adopted for pressure–velocity coupling. A bounded second-order implicit scheme is applied to the transient term, and a bounded central differencing scheme is used for the convection term. The time-step size is chosen as 1° of propeller rotation, which falls within the range of 0.5°–2° recommended by the International Towing Tank Conference guidelines.²⁴ The residual target for each time step is set to 1 × 10⁻⁴, with a maximum of 30 iterations allowed per time step.

This subsection assesses the effects of mesh refinement on the prediction of tip vortex bursting. Figure 3 presents the predicted flow structures at a blade angle θ of 140°. The coarse mesh fails to resolve a continuous tip vortex, owing to significant numerical dissipation caused by its low resolution. Refining the mesh improves the prediction, with the medium mesh successfully capturing a more continuous tip vortex and its bursting pattern. The fine mesh, while resolving additional secondary vortices shed from the trailing edge, does not significantly change the predicted bursting behavior of the tip vortex. Since the primary focus of this study is on vortex bursting, and the secondary vortices are expected to have only a small impact on this phenomenon, the medium mesh is deemed sufficient and is therefore adopted for further investigation.

In this subsection, we discuss the potential of the novel E–L hybrid model for predicting cavity bursting dynamics and the associated noise. The predicted cavity patterns are first validated against available experimental data.¹⁴  Figure 4 depicts the cavity patterns at three blade angles. For θ = 0°, the E–L hybrid model accurately predicts the sheet cavity patterns, performing similarly to traditional homogeneous mixture models reported in the literature.⁵ However, the E–L hybrid model demonstrates significant advantages at larger blade angles, where the continuous phase interface breaks up, leading to complex multiscale features, such as the bubbly tip vortex cavity at θ = 30° and the cavity bursting at θ = 60°.

It is important to note that the patterns of cavity bursting are not consistently well reproduced across all cavitation numbers. For higher cavitation numbers (σ = 7.5 and 5.5), only the part of the tip vortex cavity inside the wake is captured, whereas for the lowest cavitation number (σ = 3.5), nearly the entire tip vortex cavity is captured, including the sharply enlarged cavity due to bursting. It is difficult to maintain a pressure in the tip vortex core lower than the saturated one, despite refined mesh resolution. This suggests that further modifications to the current model, particularly to account for physical hysteresis effects,^25,26 are necessary to accurately simulate the tip vortex cavity at large blade angles.

Figure 5 further details the bursting process for σ = 3.5. Capturing the detachment of bubble clusters from the leading edge is crucial, since the interaction between sheet and tip vortex cavities is believed to be a key mechanism of bursting.⁶ The E–L hybrid model successfully captures this feature at θ = 35°. Additionally, two-way transitions between resolved cavities and discrete bubbles can be observed. Between θ = 35° and 40°, discrete bubbles near the blade tip merge into the tip vortex cavity, while between θ = 45° and 50°, the tip vortex cavity breaks up into discrete bubbles. The predicted evolutions of cavity pattern during this process agree well with the measured results.

Figure 6 compares the measured and predicted pressure spectra. Tip vortex cavity bursting typically elevates broadband humps between the 3rd and 13th harmonics of the blade passing frequency.⁶ This characteristic is also reflected in the measured spectra. Owing to its accurate prediction of bursting dynamics, the E–L hybrid model successfully reproduces the range and intensity of these broadband humps for σ = 7.5 and 3.5, although some discrepancies are observed for σ = 5.5. Notably, the measured broadband hump intensity for σ = 5.5 is weaker than that for σ = 7.5, which is in contrast with the common understanding that noise intensity increases as cavitation number decreases. This anomaly suggests that more experimental evidence is needed to clarify the relationship between bursting intensity and cavitation number.

Overall, the agreement between the measured and predicted results demonstrates that the E–L hybrid model is capable of accurately predicting the complex dynamics of tip vortex cavity bursting and the associated noise.

The bursting of a tip vortex cavity exhibits highly multiscale features, prompting further investigation into the microscale bubble characteristics. Figure 7(a) illustrates the evolution of bubble numbers during one bursting process, showing a typical trend of increase, decrease, and rebound. The moment corresponding to the maximum bubble number is examined in detail, because it marks the onset of bursting and is expected to have a significant impact on pressure fluctuations. The corresponding blade angles are 15° for σ = 7.5, 25° for σ = 5.5, and 20° for σ = 3.5.

Figure 7(b) presents the bubble size distributions (BSDs) at these selected blade angles. Previous studies of BSDs, primarily conducted on stationary hydrofoils,^8,27–29 have identified two distinct scaling laws. Our predicted BSDs for the propeller also exhibit similar characteristics, suggesting that the mechanisms governing BSDs may be analogous for both propellers and hydrofoils.

To further explore these mechanisms, we analyze bubble velocity characteristics, since the turbulence cascade is known to dominate the BSDs around hydrofoils. Figures 8(a)–8(c) display probability distributions of the bubble velocity, which conform to lognormal functions, as reported in the literature.⁸ In addition to the main peak around 9 m/s, a secondary peak around 18 m/s is observed, indicating that the BSDs may be influenced by two distinct mechanisms.

To elucidate the relationship between bubble velocity and bubble size, Figs. 8(d)–8(f) show joint probability density distributions. These figures also reveal two distinct groups, and the velocity threshold is observed to be around 15 m/s. Further, Figs. 8(g)–8(i) visualize the spatial distributions of these groups, with higher-velocity bubbles concentrating near the blade wall and lower-velocity bubbles being located beyond the boundary layer. The re-entrant jet, a primary mechanism governing near-wall cavity dynamics, has a characteristic velocity given by the inviscid relationship $Vin1+σ$⁠.³⁰ In the present cases, the characteristic velocity ranges from 13.2 to 18.1 m/s, suggesting that the higher-velocity bubble group may be influenced by the re-entrant jet.

According to Liu et al.,⁸ anisotropic turbulence near the wall can result in scaling laws deviating from the theoretical laws, specifically a −4/3 law for small-scale and a −10/3 law for large-scale BSDs. To demonstrate the wall turbulence effect on the bubble size distributions, Table I summarizes the fitted scaling laws for bubble velocities below 15 m/s. The filtered results are much closer to the theoretical scaling laws, indicating that near-wall and freestream turbulence can generate different BSDs characteristics.

In summary, the bubble characteristics predicted around the propeller show similarities to those observed around hydrofoils. The BSDs follow two distinct scaling laws, with anisotropic turbulence near the wall contributing to deviations from theoretical scaling laws.

This study has investigated the potential of a novel Euler–Lagrange hybrid model for simulating wake-induced propeller cavity bursting, using the INSEAN E779A propeller as a test case. Unlike previous studies that specified inlet boundary conditions to define the nonuniform wake, we have utilized a wake generator to produce a more realistic and unsteady wake.

The Euler–Lagrange hybrid model effectively captures the two-way transition between resolved cavities and subgrid bubbles, accurately reproducing the multiscale phase interfaces observed in experiments. Consequently, the model provides a reliable prediction of both cavity bursting dynamics and the associated broadband pressure fluctuations. Compared with traditional homogeneous mixture models, this hybrid model demonstrates superior capability in predicting full-spectrum noise.

Given the significant multiscale features of cavity bursting, further analysis needs to be directed toward the microscale bubble characteristics. Although there has been limited research on bubble size distributions around propellers, comparisons with available data from stationary hydrofoils reveal certain similarities. The distributions around both configurations satisfy two distinct scaling laws, with anisotropic turbulence near the wall causing deviations from the theoretical laws.

The authors would like to thank Dr. Francesco Salvatore and Dr. Francisco Alves Pereira for providing the opening geometry of the INSEAN E779A marine propeller and the experimental data. This work was financially supported by the National Natural Science Foundation of China (Project Nos. 52479085 and 123B2032).

The authors have no conflicts to disclose.

Xincheng Wang: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Validation (lead); Writing – original draft (lead). Yongshuai Wang: Methodology (equal); Software (equal). Huaiyu Cheng: Writing – review & editing (supporting). Bin Ji: Conceptualization (lead); Funding acquisition (lead); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.