Numerical Study on Wake Wave Characteristics Around a Transom Stern Vessel

Abstract

The wake characteristics behind a transom stern vessel play a crucial role in determining its hydrodynamic performance, resistance, and environmental impact. This hydrodynamic phenomenon involves violent wave breaking, posing significant challenges for experimental analysis. In this study, we explore the complex wake dynamics behind a transom stern vessel using high-fidelity three-dimensional numerical simulations. A sharp volume of fluid method is employed to capture the gas–liquid interface, while the immersed boundary method is applied to simulate the ship hull boundaries. A distinct advantage of the present simulation is the capability to conduct quantitative analysis within the turbulent two-phase mixing region characterized by significant air entrainment, which is difficult for traditional experimental and theoretical approaches. The research focuses on the interaction between free surface dynamics, air entrainment and turbulent vortex structures, which collectively shape the wake region. The main flow features of wakes, including wave patterns across various Froude numbers, air entrainment and the evolution of bubbly wakes, are investigated. Furthermore, the correlation between turbulent vortex structures and violent interface breaking is examined.

1. Introduction

In modern naval hydrodynamics, the study of wake flow characteristics is as critical as understanding the resistance of the hull itself. The transom stern configuration, characterized by its abrupt vertical truncation, is a prevalent feature in high-speed displacement vessels such as destroyers and frigates (Oving, 1986) [1]. From a design perspective, the transom stern facilitates clean flow separation at high Froude numbers, which effectively extends the virtual length of the hull and minimizes energy loss to the stern wave system (Doctors et al., 2007) [2]. However, this geometric discontinuity induces a complex, high-energy wake region that significantly influences the vessel’s hydrodynamic signature, acoustic characteristics, and the performance of downstream propulsion systems.

The wake evolution aft of a transom stern is primarily governed by the interaction between inertial forces, gravity, and turbulent diffusion. As the flow detaches from the transom edge, it forms an inherently unstable shear layer, which gives rise to large-scale turbulent vortex structures. These vortices, including spanwise rollers and longitudinal bilge vortices, dominate the near-wake and gradually decay as they move into the far-field (Bhushan et al., 2012) [3]. Understanding the spectral characteristics of these vortices is essential, as their interaction with the free surface can induce periodic pressure fluctuations and structural vibrations (Shen et al. 2002) [4]. In addition, at higher speeds, the convergence of the flanking waves and the centerline flow results in the “rooster tail” phenomenon, a highly non-linear wave structure that serves as a primary source of energy dissipation (Maki, 2006) [5]. Behind the rooster tail, a pair of diverging waves dominates the wave pattern. High-fidelity studies further indicate that a dry-transom wake can be organized into physically distinct regions, including a converging-corner-wave (hollow) region, a rooster-tail region, and a downstream diverging-wave region (Hendrickson et al., 2019; Gong et al., 2023) [6,7]. Over a wide range of draft-based Froude numbers, the onset and intensity of ventilation, air entrainment, and wake topology exhibit systematic trends that are now being quantified with modern simulations (Yang et al., 2023) [8].

A distinguishing feature of high-speed transom wakes is the intense air entrainment occurring at the air–water interface. As the stern wave breaks or plunges into the wake, large volumes of air are captured and fragmented into a spectrum of bubble sizes (Deane & Stokes, 2002; Hendrickson et al., 2019) [6,9]. This mixed-phase layer alters effective density and buoyancy gradients and modulates turbulent kinetic energy near the surface, which complicates both prediction and interpretation of wake signatures (Hendrickson et al., 2019; Yang et al., 2023) [8,10]. Recent progress in fundamental free-surface turbulence has also clarified how Froude and Weber numbers regulate surface deformation and near-surface turbulence transfer, including regimes where air entrainment can be sustained in statistically steady conditions (Mostert et al., 2022; Calado & Balaras, 2025) [11,12]. In parallel, modeling efforts are advancing toward mechanistic entrainment closures that connect turbulence–interface interactions to entrainment rates, which is essential for practical-scale wake prediction (Yuan et al., 2024) [13].

Historically, the study of these phenomena has relied heavily on Experimental Fluid Dynamics (EFD), including towing tanks with Laser Doppler Anemometry (LDA) and Particle Image Velocimetry (PIV). While EFD provides indispensable validation data, resolving the immediate transom region at high speed remains difficult because of the violent multiphase mixing, intermittent ventilation, and strong turbulence production. Optical-probe measurements around ship models have highlighted the challenge of capturing bubble size and velocity distributions in breaking and sweep-down conditions (Khaddaj-Mallat et al., 2022) [14]. Moreover, “scale effects” remain a central concern, since model-scale experiments and simulations may not fully preserve the coupled turbulence–breaking–bubble physics that controls small-scale wake structure (Drazen et al., 2014; Wyatt et al., 2014) [15,16]. Recent numerical analyses of scale effects in ship wave breaking (for benchmark hulls) further emphasize that breaking topology and entrained-air dynamics can change with scale even at the same nominal Froude number (Wang et al., 2023) [17].

To overcome these challenges, Computational Fluid Dynamics (CFD) has become an essential tool for detailed wake analysis. Early numerical studies focused on Reynolds-Averaged Navier–Stokes (RANS) methods to predict ship resistance. While RANS methods are computationally efficient, they often fail to resolve the fine-scale, unsteady features of the wake because they “smooth out” the turbulence. Recently, there has been a shift toward high-fidelity methods like Large Eddy Simulation (LES) and Detached Eddy Simulation (DES). These approaches allow for a much more detailed analysis of the flow field by resolving large-scale turbulent vortices directly. Modern CFD also uses the Volume of Fluid (VOF) (Hirt & Nichols, 1981) [18] method to track the complex air-water interface. This allows researchers to study air entrainment, where air bubbles are pulled into the wake by breaking waves—a phenomenon that is nearly impossible to measure accurately in a lab (Wang et al., 2016) [19].

Despite the growing body of work on ship wakes, systematic numerical studies specifically focusing on the wake characteristics of transom stern vessels remain limited, particularly in the context of coupled free-surface turbulence and multiphase flow phenomena. This study aims to fill this gap by conducting a comprehensive numerical investigation of the wake dynamics around a transom stern vessel under various operational conditions.

In this study, high-fidelity three-dimensional numerical simulations are utilized to analyze the wake characteristics aft of a transom stern vessel. Section 2 details the governing equations, computational setup, numerical methods and mesh convergence study. Section 3 provides a detailed discussion of the wake evolution, flow field analysis, air entrainment, and turbulent vortex dynamics. Concluding remarks are given in Section 4.

2. Numerical Setup and Methods

The numerical simulations in this study are conducted using a robust, in-house high- fidelity CFD solver. The fidelity and reliability of this code have been rigorously established through validation against a wide range of violent free surface deformations and complex multiphase flows. These cases include bow wave breaking (Hu et al., 2021, 2023, 2024; Liu et al., 2024) [20,21,22,23], unsteady hydraulic jumps (Li et al., 2021, 2022) [24,25], and bubble collapse dynamics (Yang et al., 2021, 2023) [26,27]. A distinguishing feature of this solver is its capability to fully resolve the two-phase flow field, ensuring an accurate representation of strong nonlinear interactions such as air entrainment and droplet at the gas–liquid interface. A brief overview of the numerical methods follows, with full details available in the aforementioned references.

The topological evolution of the interface is captured using the Coupled Level-Set and Volume of Fluid (CLSVOF) method, which combines the mass conservation of VOF with the sharp interface tracking of the Level-Set method (Sussman et al., 2007) [28]. To ensure numerical stability in two-phase flows characterized by high density ratios, a mass and momentum consistent advection scheme is adopted (Liu et al., 2022; Li et al., 2022) [25,29]. Additionally, surface tension effects, which are essential for modeling bubble and droplet dynamics, are handled via a Sharp Surface-Force (SSF) formulation (Sussman et al., 2007) [28] based on the Ghost Fluid Method (Liu & Hu, 2017) [30], allowing for a sharp treatment of interface discontinuities.

2.1. Governing Equations

The governing equations for incompressible two-phase flows are the continuity and momentum equations. In a conservative framework, these equations are formulated as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \mathbf{u}\right)=0,$$

(1a)

$${\displaystyle \frac{\partial \left(\rho \mathbf{u}\right)}{\partial t}}+\nabla ·\left(\rho \mathbf{u}\mathbf{u}\right)=-\nabla p+\nabla ·\mathsf{\tau }+\rho \mathbf{g}+{\mathbf{T}}_{\mathsf{\sigma }},$$

(1b)

where $\mathbf{u}$ represents the velocity vector, and $p$ denotes the pressure. The viscous stress tensor $\mathsf{\tau }$, for a Newtonian fluid, is defined as $\mathsf{\tau }=\mu \left[\nabla \mathbf{u}+{\left(\nabla \mathbf{u}\right)}^T\right]$, where $\mu$ and $\rho$ signify the dynamic viscosity and density of the fluid, respectively. These physical properties are determined based on the distribution of the immiscible phases. The term $\mathbf{g}$ represents the acceleration of gravity, while ${\mathbf{T}}_{\mathsf{\sigma }}$ accounts for the surface tension, which is modeled as a singular force distributed exclusively at the gas–liquid interface.

To track the evolution of the interface, the advection equation for the volume fraction $\alpha$ is solved:

$${\displaystyle \frac{\partial \alpha }{\partial t}}+\nabla ·\left(\alpha \mathbf{u}\right)=0,$$

(2)

The local density $\rho$ and dynamic viscosity $\mu$ within the computational domain are updated based on the volume fraction $\alpha$ using a linear interpolation scheme:

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

(3a)

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

(3b)

where the subscripts $l$ and $a$ correspond to the liquid and air phases, respectively.

The surface tension force ${\mathbf{T}}_{\sigma }$ in the momentum equation is modeled as a localized force at the interface:

$${\mathbf{T}}_{\sigma }=\sigma \kappa \delta \mathbf{n},$$

(4)

where $\sigma$ is the surface tension coefficient, $\kappa$ represents the local interface curvature, and $\mathbf{n}$ denotes the unit normal vector to the interface. $\delta$ is the Dirac delta function, which is concentrated at the interface. In this study, the curvature $\kappa$ is estimated using the Height Function (HF) method integrated with a curvature smoothing strategy. This approach ensures second-order spatial accuracy and significantly enhances the numerical stability of the surface tension model, even when employing large computational time steps.

2.2. Numerical Methods

To efficiently resolve small-scale features, such as millimeter-range bubbles and droplets, a block structured adaptive mesh refinement (AMR) strategy is implemented (Liu & Hu, 2018) [31]. The finest mesh resolution is selectively applied to critical regions, primarily near the gas–liquid interface. This approach ensures high-fidelity resolution of multiscale flow structures while significantly reducing the overall computational overhead. The entire domain is partitioned into structured blocks, which serve as the fundamental units for data manipulation. This block-based framework facilitates the implementation of high-order schemes with wide stencils, such as the WENO (Weighted Essentially Non-Oscillatory) scheme (Shu, 2003) [32], ensuring both accuracy and numerical stability.

To represent the presence of the submerged hull, an efficient immersed boundary (IB) method is employed to enforce the relevant boundary conditions. In this framework, the effect of the solid body is modeled by introducing an artificial forcing term as a source on the right-hand side of the momentum equation. This forcing term is dynamically reconstructed at each time step based on the velocity boundary conditions and the local velocity field adjacent to the body surface. In the present IB implementation, a signed distance function (SDF) field is utilized to implicitly identify the position of the body interface. To enhance the accuracy of the velocity reconstruction at the forcing points, an improved Moving Least Squares (MLS) interpolation scheme is adopted. This scheme, which also utilizes the signed distance information, operates within a circular support domain. This approach allows for the straightforward construction of high-order interpolations, ensuring both numerical accuracy and geometric flexibility in representing the hull’s boundary.

The turbulence treatment is handled using a Large Eddy Simulation (LES) technique. The large-scale motions are directly resolved, while the effect of the small-scale eddies is accounted for using a Smagorinsky subgrid-scale (SGS) model [33] with a standard Smagorinsky constant.

2.3. Computational Set-Up

The vessel employed in the current numerical simulations is the RV Tangaroa, a representative transom stern vessel, as illustrated in Figure 1. The ship is operated by Earth Sciences New Zealand (formerly NIWA) and has principal dimensions of $70\mathrm{m}$ in length and $13.8\mathrm{m}$ in beam. For the present numerical investigation, a scale factor of $\lambda =70$ is applied. The resulting computational model geometry is presented in Figure 1 (showing front, rear, and top views), with a model length of $L=1.0\mathrm{m}$ and a transom beam of approximately $B\approx 0.2\mathrm{m}$. In the current study, the static transom immersion is $d=0.08\mathrm{m}$.

As shown in Figure 2, the schematic of the computational domain is presented. The primary focus of this simulation is to capture the key features of the wake behind a transom stern vessel. Figure 2b illustrates the positioning of the vessel at the front of the computational domain. The computational domain is sized with dimensions of $L_x=3.0\mathrm{m}$, $L_y=1.0\mathrm{m}$, and $L_z=1.0\mathrm{m}$, corresponding to the streamwise, spanwise, and vertical directions, respectively. This domain size ensures that the wake-breaking region is adequately captured, allowing for a comprehensive analysis of the wake dynamics.

The applied boundary conditions are as follows: a uniform inflow condition is specified at the inlet boundary (the $-x$ plane), representing the incoming flow. At the outlet boundary (the $+x$ plane), a mass-conserving outflow condition is employed to allow the fluid to exit the domain naturally, with a damping zone near the boundary to minimize wave reflection effects. Symmetry conditions are applied on the $-y$ and $+y$ planes, assuming zero gradient extrapolations for both pressure and velocity fields. On the bottom boundary ($-z$ plane), an impenetrable slip condition is imposed, ensuring no penetration of the flow, while on the top boundary ($+z$ plane), a constant pressure condition is applied to model the open atmosphere above the fluid. The hull remains in a fixed position throughout the simulation, with both trim and sinkage fully restrained. In this numerical setup, the ship hull is stationary within the computational domain, and a uniform inflow with the target velocity is prescribed at the inlet. The time histories presented in the study show the evolution of the wake flow, from the initial impulse of the stream to the attainment of a quasi-steady state.

Consistent with studies focusing on transom stern hydrodynamics, the Froude number ($Fr$) in this work is defined as $Fr=U/\sqrt{gd}$, where $U$ denotes the inflow velocity and $d$ is the ship draft, which remains constant. This definition is consistently applied throughout the subsequent analysis of wake patterns and air entrainment. Based on the characteristic length of the ship model and the transom draft, the Reynolds number and Weber number for the $Fr=1.41$ case are $Re=1.25\times {10}^6$ and $We=1.7\times {10}^3$, respectively. The investigated working conditions, covering a wide range of velocities, are listed in

Table 1. Parameters for numerical simulations.

Case	Fr	Inflow Velocity U (m/s)	Draft d (m)
A1	1.41	1.25	0.08
A2	1.69	1.5	0.08
A3	1.98	1.75	0.08
A4	2.14	1.89	0.08
A5	2.26	2	0.08
A6	2.54	2.25	0.08

. The numerical results are presented in non-dimensional units, with both length and time scales normalized by the ship length $L$ (set to 1 m) and the reference time scale $T$ (set to 1 s), respectively.

Figure 3 illustrates the implementation of a block-structured adaptive Cartesian grid used in the numerical simulation. The meshing strategy is designed to optimize computational efficiency by dynamically allocating resources only where necessary. As shown in the initial setup in Figure 3a, the grid is locally refined primarily around the hull structure and the initial free surface interface, ensuring distinct capture of the geometry and the air-water boundary while maintaining a coarser grid in the far-field to reduce computational cost. Figure 3b presents the steady-state free surface topology, while Figure 3c shows the corresponding top-down view of the mesh. It is evident that the mesh refinement is not static; rather, it dynamically evolves to track the wake pattern. The grid density increases specifically in the Kelvin wake region, where wave resolution is critical, while remaining coarse in undisturbed regions.

To further highlight the efficiency of this method, Figure 3d provides a magnified view of the near-wake region corresponding to the red dashed box in Figure 3b. Figure 3d displays the local grid distribution, and Figure 3e shows the corresponding free-surface details. A key feature of this algorithm is that refinement is not applied uniformly across the entire free surface. Instead, a curvature-based criterion is utilized, where high-resolution grids are generated exclusively in regions exhibiting large free-surface curvature (i.e., the wave crests, troughs, and turbulent breaking zones) (Liu et al., 2025) [34]. As seen in Figure 3d, the mesh is coarsened immediately adjacent to the high-curvature wave structures where the flow is relatively flat. Compared to traditional uniform refinement methods, this selective, feature-driven adaptation significantly reduces the total number of grid points and substantially improves computational efficiency without compromising the accuracy of the wave capture.

2.4. Mesh Convergence Study

To assess the numerical reliability and grid independence of the simulation, a mesh convergence analysis was performed by varying the maximum adaptive refinement level on the free surface. It should be noted that the current mesh convergence study primarily focuses on the convergence of the free-surface wave profile. Since the main objective of this research is to investigate the wave-induced wake characteristics and the complex air-water mixing dynamics behind the transom stern, the stabilization of the wave elevation is considered the most direct and critical metric for ensuring the reliability of the following wake analysis. Figure 4 presents a comparison of the steady-state free surface across three distinct grid densities: a coarse configuration (Level 3–6, 65,071 surface points), a medium configuration (Level 3–7, 430,015 surface points), and a fine configuration (Level 3–8, 2,733,334 surface points). A fine simulation (Level 3–8) requires approximately 96 h of wall-clock time on a cluster using 160 CPU cores (Intel Xeon Gold 6248) to achieve a statistically steady wake flow. From a macroscopic perspective, the global wave patterns, including the Kelvin wake angle and the position of the primary divergent waves, exhibit strong consistency across all three resolution levels, indicating that the fundamental hydrodynamic features are captured even at lower grid densities. However, significant differences arise in the resolution of small-scale flow structures. As the refinement level increases, the fidelity of the interface capture improves markedly. While the coarse mesh (Figure 4a) yields a relatively smooth surface that smears out high-frequency disturbances, the fine mesh (Figure 4c) resolves intricate details, such as the sharp crests of breaking waves and the fine-scale turbulence within the wake.

Figure 5 quantitatively validates the grid independence by comparing wave characteristics across three refinement levels. As illustrated in Figure 5a, the wake trajectories of the medium mesh (Level 3–7) and the fine mesh (Level 3–8) demonstrate excellent agreement, while the coarse mesh (Level 3–6) shows a slight deviation. Regarding the longitudinal flow profiles in Figure 5b, all three refinement levels exhibit a consistent overall trend. However, the Level 3–8 mesh displays more pronounced local oscillations compared to the coarser levels. These fluctuations are attributed to the fact that the profiles in Figure 5b are instantaneous snapshots; with the highest grid density, Level 3–8 captures finer-scale transient flow structures and sharper interface gradients that are otherwise smoothed out by the numerical dissipation in Level 3–6 and 3–7. This confirms that the simulation effectively resolves the rich physical details of the turbulent wake. Consequently, an optimized hybrid strategy is adopted: Level 3–7 is utilized for macroscopic wake analysis, while Level 3–8 is reserved for resolving fine-scale turbulent structures.

3. Results and Discussion

3.1. Development of Wake Around a Transom Stern Vessel

As depicted in Figure 6, the visualization captures the fully developed free-surface flow field, where the transient startup effects have dissipated. In the fully developed flow regime, the wake extends downstream, forming a broad V-shaped envelope of diverging waves. The simulation effectively resolves fine-scale wave breaking and air entrainment, highlighting the complex interaction between the hull and the surrounding flow.

Figure 7 illustrates the temporal evolution of the wake behind the transom stern vessel, capturing the interaction of the hull on the flow $t=0.05$ to $t=1.25$. The development process can be categorized into three distinct phases: for the initial formation phase ($t=0.05-0.2$), at $t=0.05$ (Figure 7a), the wake is in its nascent stage. Only minor disturbances are visible aft of the stern, marking the onset of flow separation. By $t=0.2$ (Figure 7b), these initial instabilities evolve into more pronounced bubbly structures. The vortices generated at the transom begin to interact, leading to the elongation of the bubbly flow region downstream.

For transitional and expansion phase ($t=0.2-1.0$): As the simulation progresses to $t=0.3$ (Figure 7c), the wake exhibits significant development, characterized by the emergence of larger, coherent trailing vortices. By $t=0.4$ (Figure 7d), a clear recirculation zone establishes at the stern, signaling the transition from initial chaotic turbulence to more organized vortex structures. From $t=0.5-0.75$ (Figure 7e,f), the wake expands laterally. This period is marked by intense turbulent mixing and strong shear layers. The interaction between large-scale vortices and the surrounding potential flow intensifies, causing the breakdown of large coherent structures into finer turbulent scales.

For the fully developed phase ($t=1.0-1.25$): At $t=1.0$ (Figure 7g), the distinct vortical regions become clearly stratified, extending further downstream. Ultimately, at $t=1.25$ (Figure 7h), the flow field achieves a quasi-steady state. In this fully developed regime, transient startup effects have dissipated, and the wake manifests as a broad, V-shaped envelope of diverging waves, extending into the far field.

Figure 8 illustrates the temporal evolution of the bow wave breaking process, from the initial transient stage to a quasi-steady state. As the vessel advances, the stagnation pressure at the stem induces a significant elevation of the free surface, leading to the formation of a steep bow wave. Due to the high Froude number, the wave profile rapidly steepens and becomes unstable, which is clearly captured in the visualization. In the early stages (Figure 8a–c), the water surface ahead of the stem rises sharply due to the impulsive motion of the hull, causing a coherent wave crest to form and steepen. As the flow develops, the steepness of the bow wave increases to the point where it exceeds the stability limit. At the onset of breaking (Figure 8d), the wave crest starts to overturn and plunge both forward and sideways, initiating the air entrainment process. In the later stages (Figure 8e,f), the breaking bow wave stabilizes into a continuous structure. The resulting white water and foam propagate along the hull sides, interacting with the shoulder wave.

Figure 9 provides a magnified view of the near-field wake evolution aft of the transom stern, capturing the transition from initial flow separation to a fully developed turbulent wake state. In the early stage (Figure 9a), the flow separates cleanly from the sharp bottom edge of the transom, with a distinct transom hollow visible immediately behind the stern. By $t=0.35$ (Figure 9b), the converging flow from beneath the hull and the sides meets the centerline, causing a rapid elevation of the free surface, which manifests as the onset of a “rooster tail” structure. As the flow develops (Figure 9c), the rooster tail becomes unstable and breaks violently. This breaking process is marked by intense air entrainment (visualized as white regions), where large volumes of air are captured and fragmented into bubbles. By $t=0.65$ (Figure 9d), the wake expands laterally as spilling breakers develop along the leading edges of the diverging waves. The interaction between these breaking waves and the surrounding flow generates a highly convoluted interface and introduces significant vorticity, creating the complex surface textures and the characteristic V-shaped turbulent wake observed downstream. In the final stages (Figure 9e,f), the wake achieves a quasi-steady state. The chaotic breaking at the rooster tail transitions into a continuous stream of bubbly flow that convects downstream. A prominent V-shaped diverging wave pattern is clearly formed, delineating the turbulent wake region.

3.2. Wake Characteristics Across Different Froude Number Conditions

Figure 10 presents a comparative visualization of the steady-state free-surface topology across a range of velocities. As the speed increases, distinct changes in the wake characteristics are observed. At lower velocities (Figure 10a,b), the wake remains relatively calm, with air entrainment confined to the immediate vicinity of the transom. The wake pattern is diffuse, with a wider spreading angle. The diverging waves propagate more broadly, and the V-shaped envelope appears more obtuse. As the velocity increases to $Fr=1.98$ and beyond (Figure 10c–f), there is a marked intensification of interfacial instability. The white aerated regions expand significantly, indicating that higher Froude numbers induce more violent wave breaking and stronger turbulent mixing. The wake envelope shows a clear narrowing trend. The diverging wave arms align more closely with the centerline, effectively reducing the apparent wake angle.

At the highest speeds, the flow enters a specific high-speed regime within the investigated range. The wake is characterized by intense, chaotic wave breaking that extends significantly downstream. The diverging wave crests become sharper and more persistent compared to the lower-speed cases, forming a more acute V-shape. In this study, the “wake spreading angle” is defined as the local orientation of these diverging wave crests relative to the ship’s centerline. This near-field hydrodynamic feature is distinct from the global Kelvin wake envelope angle, as the latter governs the far-field wave boundary which typically forms beyond the current computational domain. As the Froude number increases, the energy of the near-field wake becomes more concentrated along these diverging arms. This transition is a characteristic feature of high-Froude-number flows, where the transverse wave components diminish and the diverging waves dominate, sweeping back more aggressively relative to the vessel’s path.

To further quantify the effect of Froude number on wake topology, the spatial trajectories of the wake envelope are extracted and plotted in Figure 11. Figure 11a clearly illustrates the narrowing trend of the wake envelope as the Froude number increases from $Fr=1.69$ to $Fr=2.54$, consistent with observations in Figure 10. At lower speeds (e.g., $Fr=1.69$, red triangles), the wake exhibits a larger spreading angle, extending significantly in the transverse direction. As the Froude number increases, the wake envelope progressively contracts towards the centerline. This data quantitatively demonstrates the “wake narrowing” phenomenon, where the hydrodynamic energy becomes increasingly confined to a narrower sector behind the vessel at higher speeds.

Figure 11b illustrates the centerline wave elevation ($z/L$) to characterize the development of the generated stern wave. A clear phase shift is observed in the wave profile. As the Froude number increases, the location of the primary wave crest moves significantly downstream (to larger $x/L$ values). At lower Froude numbers, the wave crest is steep and close to the transom. However, as the flow transitions to the highest Froude numbers ($Fr=2.26$ and $2.54$), the downstream migration of the peak slows down, and the wave profile begins to stabilize and flatten. This indicates that the vessel has entered a relatively stable wave regime, in which the stern wave manifests as an elongated, steady streak rather than a steep, breaking crest.

Figure 12 visualizes the three-dimensional topology of the free surface colored by velocity magnitude at four different speeds. For $Fr=1.41$, the primary wave crest forms immediately aft of the transom, indicating a relatively low Froude number regime where the wavelength is short. The high-speed region is confined to a small area near the hull and the first wave crest. The velocity decays rapidly as the wake spreads. As the speed increases, the rooster tail significantly elongates. The peak of the wave crest moves further downstream, effectively increasing the virtual length of the hull. The wave slope becomes more gradual, transitioning into a long, stable high-speed wake.

The Froude number based on the transom draft, $Fr$, plays a critical role in the transition of the flow regime at the stern. As illustrated in Figure 13, a primary observation is the distinct transition in the transom ventilation state. At the lowest velocity ($Fr=1.41$), the flow exhibits a “wetted” regime, where the hydrodynamic forces are insufficient to clear the stern, resulting in a chaotic recirculation zone that keeps the transom face partially immersed. Conversely, as the velocity increases to $Fr=2.26$, the flow transitions to a fully “dry” regime, characterized by clean flow separation from the sharp bottom edge of the transom and complete ventilation of the stern area. In addition, in the low-speed wetted regime, the free surface immediately aft of the hull is highly irregular and turbulent due to the unsteady interaction between the backflow and the main stream. However, in the high-speed dry regime, a stable “transom hollow” develops, where the free surface appears remarkably smooth. This smoothness indicates a coherent flow region prior to the onset of turbulent breaking further downstream.

To further characterize the hydrodynamic response of the free surface, the variations in the wave height immediately aft of the stern and the diverging wave characteristic length are plotted against the Froude number in Figure 14. In the lower Froude number range, the wave height is relatively high, corresponding to the wetted flow regime. At these speeds, the transom is partially immersed, and the recirculation of the flow causes the water level to rise against the stern face. However, as the Froude number increases, a sharp, monotonic decrease in wave height is observed. This rapid decay marks the transition from a wetted to a fully dry (ventilated) regime. As the flow separates cleanly from the bottom edge of the transom, a “transom hollow” is formed, causing the local water level to drop significantly. Figure 14b depicts the variation in the diverging wave characteristic length with respect to the Froude number. A strong inverse correlation is observed, where the characteristic length drops steeply. This significant monotonic reduction serves as quantitative confirmation of the wake narrowing phenomenon described earlier. It indicates that as the ship speed increases, the lateral extent of the diverging waves diminishes rapidly, confining the wake energy to a narrower spatial envelope immediately aft of the stern.

Figure 15 quantitatively depicts the longitudinal migration of the primary wave crest (rooster tail peak) as a function of the Froude number. A distinct monotonic increasing trend is observed: as the Froude number rises from $1.41$ to $2.54$, the non-dimensional position of the wave crest ($x/L$) shifts significantly downstream, from approximately $1.05$ to $1.75.$ This downstream shift is intrinsically governed by the dispersive nature of surface gravity waves. According to linear wave theory, the wavelength of the steady ship wake scales with the square of the velocity ($\lambda \propto V^2$). Consequently, as the vessel speed increases, the generated stern wave system elongates. The fluid momentum carries the free-surface disturbance further downstream before gravity can restore it to form the first peak, resulting in the observed lag in the crest position.

Figure 16 quantifies the relationship between the Froude number and the wake spreading angle. In the speed range ($Fr<2.0$), the angle declines sharply from approximately ${55}^{°}$ to ${30}^{°}$, providing quantitative confirmation of the rapid lateral contraction of the wake envelope as velocity increases. As the flow transitions to the high-speed regime $Fr>2.0$, this narrowing trend persists, although the rate of decrease becomes more gradual. Unlike the fixed angle predicted for low-speed Kelvin wakes, the observed spreading angle continues to diminish with increasing $Fr$. This behavior is consistent with linear wave theory, which suggests that for high-speed vessels, the effective wave envelope angle asymptotically approaches zero in the limit of an infinite Froude number, reflecting the high concentration of wave energy within an increasingly narrow sector aft of the hull.

3.3. Air Entrainment

Figure 17 presents the temporal evolution of the underwater bubble distribution ($t=0.25$ to $t=1.25$), clearly delineating the primary sources and transport mechanisms of air entrainment around the hull. The visualization reveals that bubble generation is predominantly localized in two distinct hydrodynamic zones: the bow region and the stern wake. At the fore of the vessel, a continuous stream of bubbles originates near the stem. This phenomenon is directly attributed to the plunging breaking of the bow wave, where the overturning wave crest impacts the undisturbed free surface. The entrapped air is fragmented into polydisperse bubbles, which are subsequently captured by the flow and advected downstream along the hull flanks, forming elongated bubble streamers that mark the ship’s track.

In contrast to the streamlined flow at the bow, the stern region exhibits a much more voluminous and turbulent bubble structure. A massive cloud of bubbles is generated immediately aft of the transom, resulting from the complex interaction between the high-speed flow separation and the violent breaking of the rooster tail. As the simulation progresses to a fully developed state ($t=1.25$), turbulent diffusion dominates the wake dynamics. The stern bubble cloud expands significantly in both lateral and vertical directions, ultimately forming a dense, V-shaped bubbly wake. This wake structure persists far downstream, merging with the advected bow streamers to constitute the comprehensive underwater acoustic and hydrodynamic footprint of the vessel.

Figure 18 plots the time history of the total underwater bubble number density. The underwater bubble number density is defined as the total number of discrete air bubbles identified by the interface-capturing method divided by the total volume of the computational domain. This density provides a macroscopic measure of the bubble population within the vessel’s wake region. The evolution is characterized by an initial phase of rapid accumulation ($t<1.0$) driven by the onset of wave breaking and air entrainment. Following this, the growth rate decays, and the curve reaches a stable plateau, oscillating around a constant mean value. This asymptotic behavior indicates that a dynamic equilibrium between bubble generation and dissipation has been established.

Figure 19 presents the temporal evolution of the normalized total entrained air volume. In contrast to the stable convergence of the bubble number density, the volume history exhibits pronounced high-frequency oscillations throughout the simulation. These fluctuations are physically attributed to the highly unsteady dynamics of large-scale air cavities; the continuous cycle of large void formation, rapid fragmentation into smaller bubbles, and sudden degassing at the free surface causes significant instantaneous variations in the total integral volume, preventing the curve from settling into a smooth steady state.

3.4. Turbulent Vortex Structures

Figure 20 illustrates the spatiotemporal evolution of coherent turbulent structures around the hull, identified using the Q-criterion isosurfaces (shown in yellow). This visualization provides a fundamental hydrodynamic explanation for the interface breaking and air entrainment phenomena observed in the previous sections. The distribution of these vortex structures exhibits a strong spatial correlation with the regions of intense free-surface deformation, confirming that the generation of underwater bubble flows is intrinsically driven by high-vorticity dynamics. At the bow region, a cluster of intense vortex structures is generated immediately upon the ship’s impact with the water. As the bow creates a plunging breaker, the overturning free surface generates strong shear layers that roll up to form primary spanwise vortices. These structures subsequently break down into complex three-dimensional streamwise vortices due to instability and are advected downstream along the hull flanks. This mechanism directly accounts for the continuous “bubble streamers” observed in Figure 17, as the persistence of these high-vorticity structures maintains the air entrainment process along the ship’s sides.

In the stern region, the visualization reveals a more voluminous and chaotic accumulation of vortex structures, which serves as the primary engine for the massive wake air entrainment. The sharp edge of the transom stern induces a massive flow separation, resulting in the shedding of shear layers that rapidly evolve into large-scale turbulent eddies. Simultaneously, the breaking of the rooster tail wave injects high turbulent kinetic energy into the flow. The yellow isosurfaces in Figure 20c clearly depict this complex interaction, where the wake is dominated by hairpin-like vortices and turbulent coherent structures. The spilling breakers occurring along the diverging wave fronts are primarily responsible for the physical disintegration of the free surface. These breaking processes introduce intense vorticity and turbulence into the flow, where the resulting hydrodynamic forces overcome surface tension, leading to the entrapment and entrainment of large volumes of air into the wake. The temporal sequence from $t=0.3$ to $t=2.0$ highlights the downstream convection and lateral expansion of these structures, eventually establishing a fully developed, V-shaped turbulent wake that sustains the dispersion of microbubbles far behind the vessel.

Figure 21 provides a magnified view of the coherent vortex structures critical to air entrainment. In the bow region (Figure 21a), the plunging breaking of the bow wave generates a high-density cluster of turbulent structures. These vortices do not dissipate immediately but adhere to the hull surface and are advected downstream, forming the hydrodynamic basis for the bubble streamers. In the stern region (Figure 21b), the flow is dominated by massive separation from the sharp transom edge. This creates a thick, chaotic shear layer composed of dense, small-scale eddies. These structures constitute the turbulent core of the wake, driving the violent interface fragmentation and sustaining the volumetric air entrainment aft of the vessel.

4. Conclusions

In this study, high-fidelity three-dimensional numerical simulations were conducted to investigate the complex wake dynamics, air entrainment characteristics, and turbulent vortex evolution around a transom stern vessel. The simulation effectively captured the full temporal evolution of the flow field, from the transient startup to the fully developed quasi-steady state. The results indicate that the wake development process undergoes three distinct phases: the initial formation, the transitional expansion, and the fully developed phase. In the final quasi-steady regime, the wake forms a broad, V-shaped envelope of diverging waves, where the formation and violent breaking of the “rooster tail” are the primary mechanisms for energy dissipation and air entrainment at the stern.

A critical hydrodynamic transition was identified regarding the effect of the Froude number on wake topology. As the vessel speed increases, the flow regime shifts from a “wetted” state, characterized by chaotic recirculation and partial transom immersion, to a “dry” or fully ventilated state, marked by clean flow separation and the formation of a stable transom hollow. Simultaneously, a “wake narrowing” phenomenon was quantitatively observed; as the Froude number rises, the diverging wave characteristic length decreases, and the wake spreading angle contracts significantly. Additionally, the primary stern wave crest was observed to shift downstream.

With regard to air entrainment, the study identified two distinct generation mechanisms: the plunging breaking of the bow wave, which creates hull-flank bubble streamers, and the massive flow separation combined with rooster tail collapse at the stern, which forms a dense, V-shaped bubbly wake. Statistical analysis of the multiphase flow revealed that, while the underwater bubble number density reaches dynamic equilibrium in the fully developed stage, the total entrained air volume continues to exhibit high-frequency oscillations. These fluctuations are attributed to the highly unsteady dynamics of large-scale air cavity formation, fragmentation, and degassing.

Moreover, the analysis of coherent structures using the Q-criterion highlighted a strong spatial correlation between turbulent vortices and interfacial instability. In the bow region, shear layers from plunging breakers roll up into streamwise vortices, which transport bubbles downstream. In the stern region, the sharp transom edge induces massive separation, generating a thick shear layer. These high-vorticity structures are identified as the primary drivers that overcome surface tension, disintegrating the free surface and sustaining the volumetric air entrainment observed in the wake.

In summary, this work offers a comprehensive understanding of the hydrodynamic interplay between free-surface deformation, multiphase flow, and turbulent structures, providing valuable insights for the design and acoustic signature analysis of high-speed transom stern vessels. However, some limitations remain. The current study primarily focuses on wake topology and air entrainment, without a detailed quantitative analysis of the ship’s resistance components. Future research will explore the coupling effects of dynamic ship motions on the wake signature and provide a more holistic assessment of hydrodynamic performance.