3.1. Flow patterns and air entrainment

To provide an intuitive overview of the simulated cases, this section first classifies the flow patterns and presents the corresponding instantaneous flow snapshots. Previous studies have characterized the flow patterns associated with a free-surface flow past a horizontally submerged cylinder. Bouscasse et al. (Reference Bouscasse, Colagrossi, Marrone and Souto-Iglesias2017) identified three key aspects defining the flow regimes: (i) the near-cylinder flow state, classified as KV, Coandă flow without shedding, metastable or barrier regime; (ii) the free-surface response, including flat surface, spilling breaker and plunging breaker; and (iii) the interaction between the separated shear layers and the free surface, categorized as no interaction, interaction, recirculation or large vortex formation. Zhao et al. (Reference Zhao, Wang, Zhu, Cao, Bao, Zhou and Han2022) identified five distinct patterns based directly on the wake topology: a jet, parallel shear layers, one-sided vortex shedding, free-surface modulated-KV and KV. Building on insights from these previous classifications, in this section, we adopt a classification that organizes the flow patterns based on the free-surface deformation and the behaviour of the cylinder’s separated shear layers.

Figure 3.

Classification of free-surface patterns by breakup dynamics and streamwise location relative to the cylinder. Each flow pattern is extracted from the instantaneous flow fields of representative cases. The $x$ – $y$ planes are coloured by the spanwise vorticity ( $\omega _z$ ), and the colour legend is identical to that shown in figure 4. The free surface is represented by the isosurface of the VOF function value $\psi = 0.5$ .

Figure 3. Long description

A diagram showing different free-surface patterns around a circular cylinder in a two-phase air-water system. Panel A: Upstream wave breaking. The image shows a circular cylinder with a wave breaking upstream, indicated by a red dashed ellipse. Panel B: Upstream non-breaking. The image shows a circular cylinder with a non-breaking wave upstream, indicated by a red dashed ellipse. Panel C: Downstream spilling breaking. The image shows a circular cylinder with a spilling breaking wave downstream, indicated by a red dashed ellipse. Panel D: Downstream plunging breaking. The image shows a circular cylinder with a plunging breaking wave downstream, indicated by a red dashed ellipse.

Figure 3 summarizes the classification results based on the free-surface morphology. We report, for the first time, the occurrence of free-surface breaking in the region upstream of the cylinder. To exclude numerical artefacts or spurious effects arising from an insufficient upstream domain length, the grid was kept unchanged while the cylinder was shifted an additional $6D$ downstream. Wave breaking persisted under this modified configuration. Downstream of the cylinder, significant surface breaking is observed in all cases due to the relatively small submergence depths and relatively small surface tension considered in this study. Our results confirm the presence of two downstream breaking patterns, namely, spilling breaking and plunging breaking. This is consistent with the findings of Bouscasse et al. (Reference Bouscasse, Colagrossi, Marrone and Souto-Iglesias2017), Guo et al. (Reference Guo, Ji, Han, Liu, Wu and Kuai2023) and Moradi & Mojra (Reference Moradi and Mojra2024). Notably, the breaking behaviours of the upstream and downstream free surfaces appear to be independent. Consequently, the free-surface morphology can be classified separately based on the upstream and downstream conditions:

1. (i) upstream surface – wave breaking (WB) and non-breaking (NB);

2. (ii) downstream surface – spilling breaking (SB) and plunging breaking (PB).

Figure 4 presents the instantaneous flow snapshots for various Froude numbers ( $\textit{Fr}$ ) at a fixed gap ratio $G = 0.2$ . Figures 5, 6 and 7 display the instantaneous flow snapshots for different gap ratios at $\textit{Fr} = 0.8$ , 1.1 and 1.4, respectively. These visualizations illustrate the instantaneous spanwise vorticity ( $\omega _z$ ) in an $x$ – $y$ plane and the free-surface geometry.

Figure 4.

Instantaneous free surface and spanwise vorticity contours ( $\omega _z$ ) illustrating the influence of the Froude number $\textit{Fr}$ : (a) $\textit{Fr} = 0.8$ (case 3a), (b) $\textit{Fr} = 1.0$ (case 3b), (c) $\textit{Fr} = 1.1$ (case 3c), (d) $\textit{Fr} = 1.2$ (case 3d), (e) $\textit{Fr} = 1.3$ (case 3e), (f) $\textit{Fr} = 1.4$ (case 3f), (g) $\textit{Fr} = 1.6$ (case 3g), (h) $\textit{Fr} = 1.8$ (case 3h) and (i) $\textit{Fr} = 2.0$ (case 3i). The gap ratio is fixed at $G = 0.2$ for all simulations, and case labels correspond to table 1. The free surface is represented by the isosurface of the VOF function value $\psi = 0.5$ .

Figure 5.

Instantaneous free surface and spanwise vorticity contours ( $\omega _z$ ) illustrating the influence of the gap ratio $G$ at a fixed Froude number $\textit{Fr} = 0.8$ : (a) $G = 0.025$ (case 1a), (b) $G = 0.1$ (case 2a), (c) $G = 0.2$ (case 3a) and (d) $G = 0.4$ (case 4a). Case labels correspond to table 1. The colour legend is identical to that shown in figure 4. The free surface is represented by the isosurface of the VOF function value $\psi = 0.5$ .

Figure 5. Long description

The image contains four separate images showing flow past a circular cylinder with varying gap ratios at a fixed Froude number. Each image illustrates the instantaneous free surface and spanwise vorticity contours. Panel A: The image shows the flow with a gap ratio of 0.025. The free surface is represented by the isosurface of the VOF function value. Panel B: The image shows the flow with a gap ratio of 0.1. The free surface is represented by the isosurface of the VOF function value. Panel C: The image shows the flow with a gap ratio of 0.2. The free surface is represented by the isosurface of the VOF function value. Panel D: The image shows the flow with a gap ratio of 0.4. The free surface is represented by the isosurface of the VOF function value.

Figure 6.

Instantaneous free surface and spanwise vorticity contours ( $\omega _z$ ) illustrating the influence of the gap ratio $G$ at a fixed Froude number $\textit{Fr} = 1.1$ : (a) $G = 0.025$ (case 1c), (b) $G = 0.1$ (case 2c), (c) $G = 0.2$ (case 3c) and (d) $G = 0.4$ (case 4c). Case labels correspond to table 1. The colour legend is identical to that shown in figure 4. The free surface is represented by the isosurface of the VOF function value $\psi = 0.5$ .

Figure 7.

Instantaneous free surface and spanwise vorticity contours ( $\omega _z$ ) illustrating the influence of the gap ratio $G$ at a fixed Froude number $\textit{Fr} = 1.4$ : (a) $G = 0.025$ (case 1f), (b) $G = 0.1$ (case 2f), (c) $G = 0.2$ (case 3f) and (d) $G = 0.4$ (case 4f). Case labels correspond to table 1. The colour legend is identical to that shown in figure 4. The free surface is represented by the isosurface of the VOF function value $\psi = 0.5$ .

Figure 7. Long description

The image contains four separate visualizations labeled Panel A, Panel B, Panel C, and Panel D. Each panel shows the instantaneous free surface and spanwise vorticity contours. Panel A shows the case with a gap ratio of 0.025. Panel B shows the case with a gap ratio of 0.1. Panel C shows the case with a gap ratio of 0.2. Panel D shows the case with a gap ratio of 0.4. Each panel includes a circular cylinder positioned beneath a deformable free surface. The free surface is represented by the isosurface of the VOF function value. The color legend is identical to that shown in figure 4.

Figure 8 shows the free-surface breaking patterns for all cases in the parameter space. In figure 8(a), upstream breaking is illustrated, where diamonds indicate WB and triangles indicate NB. Figure 8(b) presents the downstream breaking, where pentagrams correspond to SB and diamonds correspond to PB. Specifically, at $G = 0.2$ , as $\textit{Fr}$ increases from 1.0 to 1.1, the upstream surface transitions from NB to WB, as shown in figures 4(b) and 4(c). As $\textit{Fr}$ continues to increase to 1.3, the upstream surface remains in the WB pattern (figure 4 c–e). However, as $\textit{Fr}$ increases from 1.3 to 1.4, the upstream surface reverts from WB to NB (figures 4 e and 4 f). Subsequently, for all cases with $\textit{Fr} \gt 1.4$ , the surface maintains the NB pattern (figure 4 g–i). At $\textit{Fr} = 1.4$ , increasing $G$ from 0.1 to 0.2 causes the upstream surface to transition from WB to NB, as illustrated in figures 7(b) and 7(c). This indicates that the breaking behaviour of the upstream free surface is governed by the combined effects of $\textit{Fr}$ and $G$ .

Figure 8.

Regime map of the free-surface breaking patterns identified in figure 3, shown as a function of the Froude number $\textit{Fr}$ and gap ratio $G$ . (a) Upstream breaking: diamonds denote WB, and triangles denote NB. The black dash–dotted line marks the boundary between the two regimes. (b) Downstream breaking: pentagrams denote SB, and diamonds denote PB. The black dashed line indicates the boundary between these regimes. The case $\textit{Fr}=1$ , $G=0.5$ is taken from Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025).

To investigate the underlying mechanism of the occurrence of free-surface breaking in the upstream region of the cylinder, we extracted the streamwise gradient of the mean pressure, $\partial \langle p \rangle _{zt} / \partial x$ , along the streamwise direction at a vertical height of $y/D = 0$ upstream of the cylinder. The variation of this gradient with respect to the $x$ -coordinate is presented in figure 9. To avoid visual clutter, only representative cases close to the transition status are presented, with cases yielding similar results omitted from the plot. A combination analysis with the upstream free-surface geometry (figure 8 a) and the mean pressure gradients (figure 9) reveals that all cases exhibiting upstream breaking correspond to a distinct pressure gradient hill, as indicated by the arrows in figure 9. Physically, this hill signifies a strong local adverse pressure gradient upstream of the cylinder induced by the blockage effect of the gap above the cylinder. This gradient imposes a significant deceleration on the water and leads to fluid accumulation upstream, which in turn induces a rise in the free surface. The occurrence of upstream breaking within a limited Froude number range can be interpreted as the result of a competition between blockage-induced pressure-gradient deceleration and gravitational restoring effects. At relatively low Froude numbers, gravitational effects remain dominant and suppress vertical interface deformation. The flow inertia is insufficient to generate a pronounced blockage effect or sustain a recirculating backflow against gravity. Consequently, the adverse pressure gradient is weak and the free surface does not develop a significant elevation. As the Froude number increases, inertial effects become comparable to gravity, allowing the upstream deceleration and associated backflow to overcome gravitational restraint. This promotes the formation of a pronounced pressure-gradient hill and leads to substantial fluid accumulation and surface elevation. In this intermediate regime ( $1.1 \leqslant \textit{Fr} \leqslant 1.4$ ), the balance between inertia-driven blockage and gravity enables the free surface to exceed a critical elevation, thereby triggering breaking. At higher Froude numbers, although the inertia is stronger, the relative influence of gravity is diminished and the interface becomes highly deformable. Consequently, the flow adopts a more streamlined shape and can more readily adjust to pass through the gap above the cylinder by deflecting the interface away from the body, rather than building up upstream. This reduces the effective blockage and suppresses the development of a strong adverse pressure gradient and associated back pressure. The residence time for upstream accumulation is therefore shortened, and the free surface does not reach a sufficiently elevated state to trigger breaking.

Figure 9.

Variation of the mean pressure gradient $\partial \langle p \rangle _{zt} / \partial x$ along the $x$ -coordinate at the vertical height $y/D = 0$ upstream of the cylinder. To avoid excessive clutter in the plot, results from some cases with similar outcomes are omitted. The black arrows indicate the pressure gradient hill. Upstream WB cases are shown using lines without symbols, while NB cases are shown using lines with filled symbols.

To quantitatively characterize and predict the onset of free-surface breaking upstream of the cylinder, as sketched in figure 10, we define a reference upstream location, $x_{\!f}$ , and the cylinder centre location, $x_c$ , as $x_{\!f} / D = -2$ and $x_c / D = 0$ , respectively. The depth-averaged streamwise velocity components at these two locations are calculated as follows:

(3.1) \begin{align} u_{avg}|_{x = x_{\!f}} &= \frac {\int _{y_b|_{x=x_{\!f}}}^{y_{\textit{surf}}|_{x=x_{\!f}}}\langle u\rangle _{zt}\mathop {\text{d}y} }{y_{\textit{surf}}|_{x = x_{\!f}}-y_b|_{x = x_{\!f}}} , \end{align}

(3.2) \begin{align} u_{avg}|_{x = x_c} &= \frac {\int _{y_b|_{x=x_c}}^{y_{\textit{surf}}|_{x=x_c}}\langle u\rangle _{zt}\mathop {\text{d}y} }{y_{\textit{surf}}|_{x = x_c}-y_b|_{x = x_c}} , \end{align}

where $y_{\textit{surf}}|_x$ and $y_{b}|_x$ denote the vertical coordinates of the mean free surface and the lower integration limits, respectively, at a given streamwise location $x$ . The deceleration effect of the adverse pressure gradient can be quantified by the loss of kinetic energy per unit volume associated with the streamwise velocity component,

(3.3) \begin{equation} \Delta K = \rho _w \left (U_0 - u_{avg}|_{x = x_{\!f}}\right )^2 - \rho _w\left (U_0 - u_{avg}|_{x = x_c}\right )^2 . \end{equation}

Furthermore, we define a dimensionless parameter to represent the magnitude of this kinetic energy loss relative to the gravitational potential energy per unit height $P_g = \rho _w g$ :

(3.4) \begin{equation} C_{b} = \frac {\Delta K}{P_g}. \end{equation}

Figure 10.

Schematic diagram illustrating the definition of physical quantities used for the upstream surface breaking criterion. The flow field data are selected from case 1c ( $\textit{Fr}=1.1$ , $G=0.025$ ), coloured by the mean streamwise velocity ( $\langle u \rangle _{zt}$ ).

Figure 11.

Heat map of the actual values for the upstream surface breaking criterion $C_b$ at different upstream reference locations. Here (a) $x_{\!f} / D = -2$ , (b) $x_{\!f} / D = -2.2$ and (c) $x_{\!f} / D = -1.5$ . The colour scale indicates the magnitude. Cases where upstream surface breaking occurred are encircled by dashed lines and marked with a hatched shading.

Figure 11. Long description

Panel A: A heat map showing the distribution of values for the upstream surface breaking criterion at an upstream reference location of xf/D = -2. The x-axis represents the Froude number (Fr) ranging from 0.8 to 2.0, and the y-axis represents the parameter G ranging from 0.025 to 0.400. The color scale indicates the magnitude of the values, with darker reds representing higher values. Cases where upstream surface breaking occurred are encircled by dashed lines and marked with a hatched shading. Notable values include 0.53 at Fr = 1.1 and G = 0.025, and 0.45 at Fr = 1.2 and G = 0.100. Panel B: A heat map showing the distribution of values for the upstream surface breaking criterion at an upstream reference location of xf/D = -2.2. The x-axis represents the Froude number (Fr) ranging from 0.8 to 2.0, and the y-axis represents the parameter G ranging from 0.025 to 0.400. The color scale indicates the magnitude of the values, with darker reds representing higher values. Cases where upstream surface breaking occurred are encircled by dashed lines and marked with a hatched shading. Notable values include 0.45 at Fr = 1.1 and G = 0.100, and 0.45 at Fr = 1.2 and G = 0.200. Panel C: A heat map showing the distribution of values for the upstream surface breaking criterion at an upstream reference location of xf/D = -1.5. The x-axis represents the Froude number (Fr) ranging from 0.8 to 2.0, and the y-axis represents the parameter G ranging from 0.025 to 0.400. The color scale indicates the magnitude of the values, with darker reds representing higher values. Cases where upstream surface breaking occurred are encircled by dashed lines and marked with a hatched shading. Notable values include 0.69 at Fr = 1.3 and G = 0.100, and 0.65 at Fr = 1.3 and G = 0.200.

The results calculated using (3.4) are presented in figure 11(a). The heatmap displays the computed $C_b$ values for each case, with the colour scale indicating the magnitude. The dashed box in the figure outlines the pattern where upstream surface breaking occurs. It is evident that the $C_b$ criterion effectively distinguishes between breaking and NB conditions, with the critical threshold for breaking identified as $C_b \geqslant 0.26$ . To assess the sensitivity to the reference location, the classification criterion was evaluated at several streamwise positions. As shown in figure 11(a–c) present the heatmaps of $C_b$ at $x/D=-2.0$ , $-2.2$ and $-1.5$ , respectively. The corresponding critical values are found to vary (0.26, 0.23 and 0.45, respectively), indicating that the threshold is not universal and depends on the measurement location. Nevertheless, a consistent separation between flow regimes is obtained, demonstrating the robustness of the proposed indicator. Therefore, the threshold values identified in the present study are specific to the present configuration and should not be regarded as universal. It should be noted that $C_b$ is a correlation-based indicator that is a posteriori, as the kinetic energy loss $\Delta K$ includes dissipation associated with the breaking process itself (Duncan Reference Duncan2001; Drazen et al. Reference Drazen, Melville and Lenain2008; Iafrati Reference Iafrati2011). Therefore, it is used here to characterize upstream breaking behaviour rather than to predict its onset.

As illustrated in figure 8(b), the SB and PB breaking patterns of the free surface downstream of the cylinder are represented by pentagrams and diamonds, respectively, with the black dashed line demarcating the boundary between them. Specifically, as shown in figures 4(e) and 4(f), at a fixed gap ratio $G = 0.2$ , an increase in the $\textit{Fr}$ from 1.3 to 1.4 triggers a transition in the downstream breaking mode from SB to PB. Similarly, at a fixed Froude number $\textit{Fr} = 1.4$ , increasing $G$ from 0.1 to 0.2 results in a transition from the SB pattern to the PB pattern, as depicted in figures 7(b) and 7(c). It is noteworthy that although we classify the downstream surface breaking patterns at $\textit{Fr} = 1.4$ under different gap ratios into two distinct flow patterns here, the transition between them is gradual. In other words, the boundary at $G = 0.2$ is not strictly defined. The downstream breaking behaviours for the remaining cases can be identified by cross-referencing figure 8(b) with the instantaneous interface topology and flow field in figures 4, 5, 6 and 7. Collectively, these observations demonstrate that, similar to the upstream surface, the breaking pattern of the downstream free surface is governed by the combined influence of $\textit{Fr}$ and $G$ .

We subsequently elucidate the physical mechanisms underlying the different downstream breaking patterns. As water flows past the submerged cylinder, it accelerates through the gap above the cylinder, forming a jet-like flow that induces a local pressure reduction and a corresponding depression of the free surface immediately downstream. For relatively small Froude numbers and small gap ratios in the present study, the free surface is comparatively stiff and located close to the cylinder. Under these conditions, the jet-like flow remains weak (see the analysis of the mean streamwise velocity $\langle u \rangle _{zt}$ in § 3.3), gravity dominates the dynamics, and the interface is only gently displaced. The water therefore spills forward gradually without significant overturning, giving rise to SB (see figures 5 a–d and 6 a–d). As the Froude number and gap ratio increase, gravitational effects weaken relative to inertia and the jet-like flow intensifies (see again the behaviour of $\langle u \rangle _{zt}$ in § 3.3), leading to a larger downstream flux and stronger wake-induced suction. In this regime, the free surface is both farther from the cylinder and more deformable, allowing it to be drawn into the wake region. This suction promotes the formation of a pronounced surface crest downstream of the cylinder, which subsequently grows in elevation as the flow advects downstream. Once the kinetic energy of the jet exceeds a critical threshold, a distinct hydraulic jump forms downstream of the cylinder. The lowest point of the free surface immediately downstream corresponds to the hydraulic jump toe, indicated by arrow 1 in figure 7(c). Farther downstream, the free surface elevation increases continuously and reaches a maximum within the surface roller region, marked by the red dashed circle. In this region, a persistent jet lip impinges on the free surface (arrow 2), triggering reversed plunging motion and ultimately resulting in PB. In essence, the transition from spilling to plunging breaking occurs when the flow inertia (primarily governed by a sufficiently large Froude number in combination with an appropriate gap ratio) surpasses the restoring effect of gravity. Under these conditions, the interface becomes soft enough and sufficiently separated from the cylinder to be entrained into the wake, allowing a crest to form and overturn. This interpretation is consistent with the trends observed in the present three-dimensional simulations and with earlier observations from two-dimensional studies (Bouscasse et al. Reference Bouscasse, Colagrossi, Marrone and Souto-Iglesias2017; Moradi & Mojra Reference Moradi and Mojra2024; Patel et al. Reference Patel, Sun, Yang and Zhu2025).

The downstream-surface breaking pattern also influences the bubble size. Bubble size distributions play an important role in a range of multiphase flow processes, including gas transfer in geophysical free-surface flows (Asher & Wanninkhof Reference Asher and Wanninkhof1998; Woolf et al. Reference Woolf2007; Kiger & Duncan Reference Kiger and Duncan2012; Deike et al. Reference Deike, Melville and Popinet2016; Di Giorgio et al. Reference Di Giorgio, Pirozzoli and Iafrati2025) and hydrodynamic loading and wake characteristics in naval hydrodynamics (Hendrickson et al. Reference Hendrickson, Weymouth, Yu and Yue2019; Li et al. Reference Li, Liu, Wan and Hu2021; Tong et al. Reference Tong, Li and Yang2026). In canonical free-surface turbulent flows, the statistical characteristics of bubble populations have been extensively documented, including the influence of surface-parallel and surface-normal vortices on air-entrainment volume, as well as the distinction between fragmentation-dominated and degassing-dominated regimes (Yu et al. 2019, Reference Yu, Hendrickson, Campbell and Yue2020; Hendrickson et al. Reference Hendrickson, Yu and Yue2022; Gaylo et al. Reference Gaylo, Hendrickson and Yue2024; Gaylo & Yue Reference Gaylo and Yue2025). While these previous studies have established the fundamental physics of air entrainment and the universal scaling laws of bubble size distributions in canonical turbulent flows, the present work extends these insights to a specific class of flows involving the interaction between a submerged cylinder and a deformable free surface, with particular emphasis on the small-gap-ratio, high-Froude-number limit. Adopting the methodology described in Herrmann (Reference Herrmann2010), the physical properties of the bubbles are calculated. Following Li et al. (Reference Li, Yang and Zhang2024) and Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025), we computed the size spectra of the bubble number density $S_b (R_{\textit{eff}} )$ for all simulated cases as

(3.5) \begin{equation} {S}_b\left (R_{\textit{eff}}\right ) = \left \langle \frac {n(R_{\textit{eff}},t;b)}{V b} \right \rangle _t \!, \end{equation}

where $n(R_{\textit{eff}},t;b)$ is the number of bubbles, whose effective radii fall between $R_{\textit{eff}}$ and $R_{\textit{eff}} + b$ in a given fluid volume $V$ at time $t$ . In the present study, $b = 0.001D$ is selected. The fluid volume $V$ is a cuboid in a subdomain with $x/D \in [0.5, 27.0]$ , $y/D \in [-5.0, 2.0]$ , $z/D \in [0, 3]$ , which contains most air bubbles. The effective spherical radius is defined as

(3.6) \begin{equation} R_{\textit{eff}} = \left (\frac {3 v_e}{4 \pi }\right )^{1/3} , \end{equation}

where $v_e$ represents the volume of an individual bubble.

Figure 12 illustrates the influence of the Froude number on the bubble size spectrum $S_b(R_{\textit{eff}})$ . To ensure numerical rigour, the bubble size spectrum is presented for $R_{\textit{eff}} \geqslant 1.5\varDelta$ (corresponding to $R_{\textit{eff}} = 0.0375D$ ), where $\varDelta = 0.025D$ denotes the grid spacing, following the resolution threshold suggested by Yu et al. (Reference Yu, Hendrickson, Campbell and Yue2019). Within this range, the smallest bubble sizes remain close to the grid scale. Their trends should therefore be interpreted with caution, as they may be indicative rather than fully resolved. As $\textit{Fr}$ increases, the transition from spilling to plunging breaking is driven by a shift from gravity-dominated to inertia-dominated dynamics. To interpret these results, we adopt the population-balance framework of Gaylo et al. (Reference Gaylo, Hendrickson and Yue2024) and Gaylo & Yue (Reference Gaylo and Yue2025). The governing physics is characterized using the degassing flux ratio, defined as the ratio of degassing to entrainment air flux. For Froude numbers up to $\textit{Fr} = 1.6$ , the downstream surface exhibits predominantly spilling behaviour or moderate air entrainment. In the SB cases ( $\textit{Fr} \lt 1.4$ ) and the moderately PB case at $\textit{Fr} = 1.6$ , the degassing flux ratio remains near unity. The bubble population balance is primarily degassing-dominated, showing that the flow is dominated by gravity or buoyancy. Consequently, the large-bubble distribution follows steep power-law scaling of approximately $R_{\textit{eff}}^{-5.83}$ (figure 12 a–c), while the small-bubble distribution follows a $R_{\textit{eff}}^{-4.3}$ power law. These exponents are consistent with the gravity/buoyancy-controlled regimes identified in recent mechanistic analyses (Gaylo et al. Reference Gaylo, Hendrickson and Yue2024; Gaylo & Yue Reference Gaylo and Yue2025). While $\textit{Fr} = 1.6$ is classified as a PB, the turbulent fragmentation at this stage is not yet intense enough to overwhelm the degassing process, leading to the retention of gravity-driven scaling.

Figure 12.

Effect of the Froude number $\textit{Fr}$ on the size spectra of the bubble number density $S_b(R_{\textit{eff}})$ at a fixed gap ratio $G = 0.2$ : (a) $\textit{Fr} = 0.8$ (case 3a), (b) $\textit{Fr} = 1.2$ (case 3d), (c) $\textit{Fr} = 1.6$ (case 3g) and (d) $\textit{Fr} = 2.0$ (case 3i). Case labels correspond to table 1.

A distinct regime shift occurs for the strongly plunging cases at $\textit{Fr} \geqslant 1.8$ (specifically cases 3h and 3i). In this fragmentation-dominated regime, the intensified TKE and jet impact drive a much more vigorous breakup process. As a result, the bubble size distribution across a broader range of scales shifts away from the degassing-controlled slopes and approaches the classical $R_{\textit{eff}}^{-10/3}$ scaling (figure 12 d), consistent with inertia-dominated fragmentation (Garrett et al. Reference Garrett, Li and Farmer2000; Deane & Stokes Reference Deane and Stokes2002).

Within the parameter range investigated, the gap ratio ( $G$ ) exerts a secondary influence on the spectra. At $\textit{Fr} = 0.8$ and $1.1$ , where the flow remains in the spilling mode, variations in $G$ result in minimal changes to the spectral characteristics. At $\textit{Fr} = 1.4$ , although increasing $G$ from 0.1 to 0.2 induces a transition from spilling to plunging, the transformation is gradual. In this transitional state, the fragmentation remains relatively weak, and the bubble size spectrum retains a profile similar to the degassing-dominated behaviour shown in figures 12(a) and 12(b).

Figure 13.

Classification of separated shear-layer patterns based on interactions between the cylinder’s separated shear layer and the jet-like flow above the cylinder. Each flow pattern is extracted from the instantaneous flow fields of representative cases. The $x$ – $y$ planes are coloured by the spanwise vorticity ( $\omega _z$ ), and the colour legend is identical to that shown in figure 4. The free surface is represented by the isosurface of the VOF function value $\psi = 0.5$ .

Previous studies (Zhao et al. Reference Zhao, Wang, Zhu, Ping, Bao, Zhou, Cao and Cui2021; Sun et al. Reference Sun, Lu, Patel, Zhu and Yang2025; Patel et al. Reference Patel, Sun, Yang and Zhu2025) demonstrate that the jet-like flow above the cylinder significantly influences the separated shear layers of the cylinder. The results of the present study reveal four distinct shear layer patterns within the parameter space investigated, as summarized in figure 13. These patterns are defined as follows:

1. (i) SV;

2. (ii) metastable vortex shedding (MV);

3. (iii) Kármán-like vortex shedding with jet (KJV);

4. (iv) KV.

Figure 14.

Regime map showing the separated-shear-layer patterns identified in figure 13, plotted as a function of the Froude number $\textit{Fr}$ and gap ratio $G$ . Marker shape represents the separated-shear-layer pattern: triangles represent SV, pentagrams represent MV, circles represent KJV and diamonds represent KV. The red dashed line schematically indicates the boundaries between different patterns. The case $\textit{Fr}=1$ , $G=0.5$ is taken from Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025).

Figure 14 illustrates the distribution of these shear layer patterns within the parameter space defined by $\textit{Fr}$ and $G$ . Distinct marker shapes are employed to differentiate the flow patterns: triangles represent SV; pentagrams denote the MV; circles indicate KJV, in which the vortex shedding is strongly modulated by the jet-like flow above the cylinder; and diamonds represent KV. The boundaries between these patterns are demarcated by red dashed lines. As evident from the figure, the SV state is observed exclusively at the lowest Froude number $\textit{Fr} = 0.8$ and the smallest gap ratio $G = 0.025$ (case 1a) under investigation. As mentioned in Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025), the jet-like flow above the cylinder exerts a stabilizing effect on the flow. At $\textit{Fr} = 1.0$ and $G = 0.5$ , vortex shedding was suppressed at $\textit{Re} = 400$ ; however, as $\textit{Re}$ increased to 500, the enhanced flow instability overcame the stabilizing effect, leading to the onset of vortex shedding (Sun et al. Reference Sun, Lu, Patel, Zhu and Yang2025). In the present study, $\textit{Re} = 2000$ is significantly higher than 400, resulting in more pronounced instability. Consequently, a stronger stabilizing effect from the jet-like flow is required to suppress vortex shedding. This explains why the SV pattern is observed only at the specific combination of $\textit{Fr} = 0.8$ and $G = 0.025$ (case 1a). This case corresponds to the minimum gap height ( $h^*$ ) above the cylinder, where the distance between the cylinder’s separated shear layers and the free surface is minimized (see figure 5 a). The overlying jet-like flow exerts a potent stabilizing influence on the cylinder’s shear layers, thereby completely suppressing vortex shedding. As shown in figure 14, a transition from the MV to KJV occurs when $G$ increases from 0.2 to 0.4 at $\textit{Fr} = 0.8$ and 1.1, and when $G$ increases from 0.1 to 0.2 at $\textit{Fr} = 1.4$ (refer to figures 5 c and 5 d, 6 c and 6 d, 7 a and 7 b). On the other hand, at a fixed gap ratio $G = 0.2$ , increasing $\textit{Fr}$ from 1.3 to 1.4 triggers a similar transition from MV to KJV (see figures 4 e and 4 f). Subsequently, as $\textit{Fr}$ further increases from 1.6 to 1.8, a second transition occurs, shifting the pattern from KJV to KV, as shown in figures 4(g) and 4(h). Within the parameter range considered in the present study, a small $G$ and low $\textit{Fr}$ lead to a strong stabilizing effect of the jet-like flow, manifested as SV or MV. As the gap ratio and Froude number increase, the flow becomes more symmetric, the stabilizing effect of the jet-like flow weakens, and its impact on vortex shedding gradually diminishes.

Figure 15.

Isosurfaces of $Q / U_0^{2} D^{-2} = 10$ coloured by streamwise velocity ( $u$ ) for cases at (a) $\textit{Fr} = 0.8$ , $G = 0.025$ (case 1a), (b) $\textit{Fr} = 1.1$ , $G = 0.2$ (case 3c), (c) $\textit{Fr} = 1.4$ , $G = 0.2$ (case 3f) and (d) $\textit{Fr} = 2.0$ , $G = 0.2$ (case 3i). Each case is shown in (i) a perspective view and (ii) a down–top view. Background contours represent spanwise vorticity ( $\omega _z$ ) in the $x$ – $y$ and $x$ – $z$ planes, respectively.

Figure 15. Long description

Panel A: The first set of graphs shows isosurfaces colored by streamwise velocity for case 1a. The perspective view (i) displays a 3D representation with a color gradient indicating velocity. The downtop view (ii) provides a top-down perspective of the same data. Background contours represent spanwise vorticity in the x-z plane. Panel B: The second set of graphs illustrates isosurfaces colored by streamwise velocity for case 3c. The perspective view (i) and downtop view (ii) are shown with background contours representing spanwise vorticity in the x-z plane. Panel C: The third set of graphs presents isosurfaces colored by streamwise velocity for case 3f. The perspective view (i) and downtop view (ii) are displayed with background contours representing spanwise vorticity in the x-z plane. Panel D: The fourth set of graphs shows isosurfaces colored by streamwise velocity for case 3i. The perspective view (i) and downtop view (ii) are presented with background contours representing spanwise vorticity in the x-z plane.

The results also demonstrate that variations in $\textit{Fr}$ and $G$ significantly influence the instantaneous vortex structures of the flow, primarily by modulating the states of the separated shear layers of the cylinder. Figure 15 presents the instantaneous vortex structures for representative cases of each separated-shear-layer pattern, visualized using Q-criterion isosurfaces (Hunt et al. Reference Hunt, Wray and Moin1988).

Figure 15(a) depicts the instantaneous vortex structures for case 1a ( $\textit{Fr} = 0.8$ , $G = 0.025$ ), corresponding to the SV pattern. In this case, the jet-like flow is extremely narrow (figure 5 a). The two-phase shear flow above the jet-like flow is situated in close proximity to the cylinder’s upper separated shear layer, resulting in a strong interaction that generates numerous small-scale vortex structures near the trailing edge of the shear layer. The jet-like flow adheres closely to the cylinder and tilts downward significantly from the streamwise direction. Farther downstream, this deflection promotes vorticity annihilation between the two-phase shear flow and the separated shear layers in the deeper water region (Zhao et al. Reference Zhao, Wang, Zhu, Ping, Bao, Zhou, Cao and Cui2021). Consequently, vortex structures are sparse near the free surface initially and only gradually rise towards the surface as they diffuse downstream.

Figure 15(b) displays the vortex structures for case 3c ( $\textit{Fr} = 1.1$ , $G = 0.2$ ), which falls under the MV pattern. Here, the vertical thickness of the jet-like flow has increased. At the trailing edge, the two-phase shear flow and the cylinder’s separated shear layers are separated by a jet-like flow with nearly negligible vorticity, preventing direct mixing in its vicinity (Zhao et al. Reference Zhao, Wang, Zhu, Ping, Bao, Zhou, Cao and Cui2021; Sun et al. Reference Sun, Lu, Patel, Zhu and Yang2025). Notably, regular spanwise planar vortex structures emerge at the trailing edge of the separated shear layers (indicated by the red dashed circle). This contrasts with the fragmented small-scale structures typically observed in single-phase cylinder flow at $\textit{Re} = 2000$ , indicating that the jet-like flow exerts a suppressing effect on the wake instability. Near the initially shed vortex cores, the two-phase shear flow mixes with the wake developing from the cylinder’s shear layers, leading to the formation of smaller-scale vortex structures, consistent with the findings of Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025).

Figure 15(c) illustrates the vortex structures for case 3f ( $\textit{Fr} = 1.4$ , $G = 0.2$ ), characterized by a KJV pattern. In this pattern, the vertical thickness of the jet-like flow expands further, and large-scale, planar vortex structures form in the near-wake region (indicated by the red dashed circle). Since the downstream free surface has transitioned to the PB pattern, the two-phase shear flow associated with the SB pattern has disappeared. Consequently, the two-phase flow region shifts upward to the surface roller region near the free surface. The interaction between the vorticity from the cylinder’s separated shear layers and the two-phase flow region is reduced, resulting in small-scale vortex structures being concentrated primarily within the surface roller region.

Figure 15(d) presents the instantaneous vortex structures for case 3i ( $\textit{Fr} = 2.0$ , $G = 0.2$ ), which corresponds to KV. At this stage, the vertical thickness of the jet-like flow is significantly increased. The jet continues to exert a suppressing effect on the cylinder’s separated shear layers, inhibiting vorticity in the wake and resulting in large-scale planar vortex structures (indicated by the red dashed circle). Small-scale vortex structures are found to be almost entirely confined within the two-phase flow region, concentrated in the surface roller region.

The analysis above indicates that the stabilizing effect of the jet-like flow continuously influences the vortex structures in the near-wake region. As $\textit{Fr}$ and $G$ increase, the flow becomes more symmetric, and the jet-like flow strengthens in terms of flow rate (see the related analysis of the mean streamwise velocity $\langle u \rangle _{zt}$ in § 3.3), allowing it to flush farther downstream. Consequently, the planar-vortex region extends farther downstream.

3.2. Hydrodynamic forces and vortex shedding

This section begins by analysing the lift and drag coefficients of the cylinder. The lift and drag coefficients of the cylinder are calculated, respectively, as

(3.7) \begin{align} C_l &= \frac {F_l}{0.5 \rho _w U_0^2 A}, \end{align}

(3.8) \begin{align} C_d &= \frac {F_d}{0.5 \rho _w U_0^2 A}, \end{align}

where $F_l$ and $F_d$ are the lift and drag forces, respectively, and $A = L_z D$ is the projected area. In all cases under investigation, the cylinder is fully submerged, and therefore, the buoyancy remains constant. To focus on the hydrodynamic force, $F_l$ is defined with the buoyancy excluded as

(3.9) \begin{equation} F_l = F_{\!y} - F_b , \end{equation}

where $F_{\!y}$ denotes the total vertical force exerted on the cylinder, and $F_b$ is the buoyancy force, defined as

(3.10) \begin{equation} F_b = \frac {\rho _w g \pi {D}^2 L_z}{4} . \end{equation}

Figure 16.

Variation of (a) the mean lift coefficient ( $\langle C_l \rangle _t$ ) and (b) mean drag coefficient ( $\langle C_d \rangle _t$ ) with Froude number $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ . The error bars denote the corresponding r.m.s. and are omitted where smaller than the marker size.

Figure 17.

Variation of the mean gap height $\langle h^* \rangle _{t}$ above the cylinder with Froude number $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ .

Figure 16 presents the variation of the mean lift coefficient $\langle C_l \rangle _t$ and drag coefficient $\langle C_d \rangle _t$ as $\textit{Fr}$ increases from 0.8 to 2.0, with a fixed gap ratio $G = 0.2$ . The error bars denote the corresponding r.m.s., and are omitted where smaller than the marker size. Figure 16(a) shows that the lift coefficient is negative. This is consistent with the downward thrust of the jet-like flow mentioned in studies such as Bouscasse et al. (Reference Bouscasse, Colagrossi, Marrone and Souto-Iglesias2017), Colagrossi et al. (Reference Colagrossi, Nikolov, Durante, Marrone and Souto-Iglesias2019), Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025) and Patel et al. (Reference Patel, Sun, Yang and Zhu2025). As $\textit{Fr}$ increases, the absolute value of $\langle C_l \rangle _t$ gradually decreases and approaches zero, consistent with the findings of Patel et al. (Reference Patel, Sun, Yang and Zhu2025). This change can be attributed to the increase in the mean gap height $\langle h^* \rangle _{t}$ above the cylinder, as shown in figure 17, reducing the asymmetry of the flow field. In figure 17, the error bars denote the corresponding r.m.s., and the error bars are omitted as they are smaller than the marker size. On the other hand, increasing $\textit{Fr}$ causes the weakening of gravitational effects, thereby diminishing the downward thrust. Regarding the mean drag coefficient $\langle C_d \rangle _t$ , as shown in figure 16(b), its magnitude also exhibits a monotonic decreasing trend with increasing $\textit{Fr}$ . This similarly stems from the combined effects of flow symmetry and weakened gravitational effects, leading to a reduced vortex shedding frequency and consequently a smaller pressure difference between the front and rear of the cylinder. It is noteworthy that when $\textit{Fr}$ increases from 1.3 to 1.4, the decrease in the mean lift and drag coefficients is significantly greater than in other $\textit{Fr}$ intervals. Combined with figures 4(e) and 4(f), it can be observed that within this interval, the free surface morphology downstream of the cylinder transitions from SB to PB. This abrupt change in flow pattern significantly alters the pressure distribution in the flow field, resulting in substantial changes in the lift and drag coefficients.

Figure 18.

Variation of (a) the mean lift coefficient $\langle C_l \rangle _t$ and (b) mean drag coefficient $\langle C_d \rangle _t$ with the gap ratio $G$ at $\textit{Fr} = 0.8$ , 1.1 and 1.4. The error bars denote the corresponding r.m.s. and are omitted where smaller than the marker size.

Figure 19.

Variation of the mean gap height $\langle h^* \rangle _t$ above the cylinder with the gap ratio $G$ at $\textit{Fr} = 0.8$ , 1.1 and 1.4.

Figure 18, on the other hand, shows the variation of the mean lift and drag coefficients of the cylinder with $G$ for different $\textit{Fr}$ . The error bars denote the corresponding r.m.s., evaluated as described for figure 16, and are omitted where smaller than the marker size. As shown in figure 18(a), for all three Froude numbers, the absolute value of $\langle C_l \rangle _t$ decreases with increasing $G$ . This is because an increase in $G$ leads to an increase in the mean gap height $\langle h^* \rangle _t$ above the cylinder (as shown in figure 19), which reduces the asymmetry of the flow field near the cylinder, thereby gradually weakening the downward thrust. However, the variation of the mean drag coefficient $\langle C_d \rangle _t$ with $G$ shows distinct differences for the three Froude numbers (figure 18 b). For $\textit{Fr} = 0.8$ and 1.1, $\langle C_d \rangle _t$ generally exhibits an increasing trend with increasing $G$ . Under these conditions, gravitational effects are relatively significant, and the free surface morphology downstream of the cylinder is SB (see figures 5, 6 and 8 b). As $G$ increases, $\langle h^* \rangle _t$ increases, and the increased jet-like flow flux (see the related analysis of the mean streamwise velocity $\langle u \rangle _{zt}$ in § 3.3.2) enhances the adverse pressure gradient in the wake region, thus increasing the form drag, manifested as an overall increase in $\langle C_d \rangle _t$ . In contrast, for $\textit{Fr} = 1.4$ , $\langle C_d \rangle _t$ first decreases and then increases with increasing $G$ . Here, flow inertia dominates, and the free surface morphology downstream of the cylinder is in a transition status from SB to PB (see figures 7 and 8 b). When $G$ is small ( $G = 0.025$ and 0.1), the free surface disturbance is intense, with coexisting spilling and plunging breaking, resulting in higher drag. As $G$ increases to a moderate value (approximately 0.2), the reduction in free surface disturbance leads to a decrease in drag. When $G$ further increases to 0.4, the increased jet-like flow flux (see the related analysis of the mean streamwise velocity $\langle u \rangle _{zt}$ in § 3.3.2) significantly enhances momentum transport and leads to pronounced plunging breaking. The large-scale hydraulic jump structure introduces wave drag, causing the drag to rise again.

Figure 20.

Frequency spectra of lift and drag coefficients for different cases: (a) $\textit{Fr} = 1.0$ , $G = 0.2$ (case 3b), (b) $\textit{Fr} = 1.4$ , $G = 0.2$ (case 3f), (c) $\textit{Fr} = 1.6$ , $G = 0.2$ (case 3g) and (d) $\textit{Fr} = 2.0$ , $G = 0.2$ (case 3i).

Figure 20. Long description

Four line graphs depict frequency spectra of lift and drag coefficients for different cases. Panel A: A line graph shows the frequency spectra of lift and drag coefficients for case 3b. The x-axis represents the dimensionless frequency fD/U0, and the y-axis represents the amplitude. The orange line represents the lift coefficient, and the blue line represents the drag coefficient. The graph shows multiple peaks and variations in amplitude across the frequency range. Panel B: A line graph shows the frequency spectra of lift and drag coefficients for case 3f. The x-axis represents the dimensionless frequency fD/U0, and the y-axis represents the amplitude. The orange line represents the lift coefficient, and the blue line represents the drag coefficient. The graph shows a prominent peak at a higher frequency. Panel C: A line graph shows the frequency spectra of lift and drag coefficients for case 3g. The x-axis represents the dimensionless frequency fD/U0, and the y-axis represents the amplitude. The orange line represents the lift coefficient, and the blue line represents the drag coefficient. The graph shows a prominent peak at a higher frequency. Panel D: A line graph shows the frequency spectra of lift and drag coefficients for case 3i. The x-axis represents the dimensionless frequency fD/U0, and the y-axis represents the amplitude. The orange line represents the lift coefficient, and the blue line represents the drag coefficient. The graph shows a prominent peak at a higher frequency and an additional notable feature indicated by an arrow.

To further investigate the variation characteristics of the cylinder’s lift and drag signals, the frequency spectra of lift coefficient $\varPhi _l$ and drag coefficient $\varPhi _d$ for all cases are calculated. Figure 20 presents the results for representative cases. In most cases, a unimodal high-frequency signal $f_h$ and multimodal low-frequency signals $f_l$ coexist, as shown in figures 20(b) and 20(c), with their magnitudes differing by approximately one order of magnitude, i.e. $f_h / f_l = O(10)$ . This again validates the findings of Bouscasse et al. (Reference Bouscasse, Colagrossi, Marrone and Souto-Iglesias2017), Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025) and Patel et al. (Reference Patel, Sun, Yang and Zhu2025). As shown in figure 20(a), for case 1a ( $\textit{Fr} = 0.8$ , $G = 0.025$ ), since the vortex shedding is suppressed, the spectrum does not include a major frequency peak corresponding to vortex shedding. The spectrum shows a broad frequency band with small amplitudes.

Our previous study (Sun et al. Reference Sun, Lu, Patel, Zhu and Yang2025) pointed out that for single-phase flow past a cylinder, the dominant frequency of the drag coefficient, induced by vortex shedding, is twice that of the lift coefficient. However, under two-phase inflow conditions, when the jet-like flow effect is strong, the low-pressure zone persistently resides on the upper side of the cylinder near the jet-like flow. Periodic variations in the position and intensity of this low-pressure zone simultaneously cause fluctuations in both the streamwise and vertical pressure differences, leading to equal high-frequency components in the lift and drag coefficients. Within the parameter range under investigation, the jet-like flow influence is weak, leading to the coexistence of two high-frequency peaks of $\varPhi _d$ : one with a frequency twice that of the high-frequency peak of the $\varPhi _l$ , and the other with a frequency equal to that of the lift coefficient’s high-frequency peak. This phenomenon is evident from figure 20(d), for $\textit{Fr} = 2.0$ , $G=0.2$ (case 3i). As indicated by the arrow in the figure, the frequency spectrum of the drag coefficient $\varPhi _d$ exhibits not only a signal at $f_h$ , corresponding to the high-frequency component of $\varPhi _l$ , but also an additional signal at $2 f_{h}$ . This phenomenon reflects a transition from two-phase flow with significant influence from the jet-like flow to single-phase flow without such influence. As flow symmetry gradually increases, the low-pressure zone is no longer confined solely to the upper side (jet-like flow side) of the cylinder, but alternates between both sides of the cylinder. This transition causes the frequency of the streamwise pressure difference variation to evolve from solely corresponding to the lift coefficient’s high frequency $f_h$ to simultaneously containing both $f_h$ and $2f_h$ , eventually manifesting entirely as $2f_h$ in single-phase flow.

Figure 21.

Regime map of vortex-shedding frequency patterns. Symbols denote the dominant frequency peaks: blue squares represent no shedding (NS), orange triangles indicate single-harmonic (SH) behaviour and green diamonds represent double-harmonic (DH) behaviour. The red dashed line indicates the regime boundaries. The case $\textit{Fr}=1$ , $G=0.5$ is taken from Sun et al. (Reference Sun, Lu, Patel, Zhu and Yang2025).

Figure 22.

Variation of the shear-layer separation angles on the cylinder surface with Froude number $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ . Here $\theta _{st}$ and $\theta _{sb}$ denote the upper and lower surface separation angles, respectively.

The spectral regimes are classified into three categories: NS, SH and DH, corresponding, respectively, to the absence of a dominant vortex-shedding frequency, the presence of only $f_h$ , and the coexistence of $f_h$ and its second harmonic $2f_h$ . Figure 21 shows the regime in the $(\textit{Fr},G)$ parameter space. The majority of cases fall within the SH regime. The DH regime is observed primarily at high Froude numbers combined with relatively large gap ratios, where the increased mean gap height weakens the flow asymmetry. In contrast, the NS regime is only identified in a single case ( $\textit{Fr}=0.8$ , $G=0.025$ ) within the present ( $\textit{Fr},G$ ) parameter space.

To analyse the vortex shedding behaviour, this section first calculates and examines the changes of the separation angles on the upper and lower surface shear layers of the cylinder. In this study, the leading edge of the cylinder is defined as $\theta = {0}^\circ$ . The angular coordinate increases clockwise along the upper surface and anticlockwise along the lower surface, with the trailing edge corresponding to $\theta = {180}^\circ$ . Figure 22 presents the variation of the upper surface shear layer separation angle $\theta _{st}$ and the lower surface shear layer separation angle $\theta _{sb}$ with the Froude number, with a fixed gap ratio $G = 0.2$ . The error bars denote $\pm 1$ standard deviation of $\theta _{st}$ and $\theta _{sb}$ evaluated on a cycle-by-cycle basis, i.e. from the distribution of $\theta _{st}$ and $\theta _{sb}$ computed for individual vortex shedding cycles. The error bars are omitted since they are smaller than the marker size. The figure shows a tendency for the upper and lower surface separation angles to converge towards an intermediate value, reflecting the enhancement of flow symmetry. As $\textit{Fr}$ increases, the upper surface separation angle $\theta _{st}$ continuously decreases. This is attributed to the increase of the mean gap height $\langle h^* \rangle _t$ above the cylinder (see figure 17), which weakens the constraining effect imposed by the jet-like flow, causing the separation point to move upstream. Meanwhile, the lower surface separation angle $\theta _{sb}$ also continuously increases due to the overall enhancement of flow symmetry, but the magnitude of change is relatively small. This is because the flow domain below the cylinder is nearly semi-infinite, making it less susceptible to changes in flow symmetry on the upper surface. A similar trend is observed in the variation of separation angles with $G$ . Figure 23 shows that for $\textit{Fr} = 0.8$ , 1.1 and 1.4, as $G$ increases, the upper surface separation angle $\theta _{st}$ continuously decreases (see figure 23 a), while the lower surface separation angle $\theta _{sb}$ continuously increases (see figure 23 b). This phenomenon primarily stems from the increase of $\langle h^* \rangle _t$ (see figure 19) caused by the increase in $G$ .

Figure 23.

Variation of the shear layer separation angles on the cylinder surfaces with gap ratio $G$ for Froude numbers $\textit{Fr} = 0.8$ , 1.1 and 1.4. (a) Upper surface separation angle $\theta _{st}$ and (b) lower surface separation angle $\theta _{sb}$ . The error bars denote $\pm 1$ standard deviation based on cycle-to-cycle variations in $\theta _{st}$ and $\theta _{sb}$ , and are omitted where smaller than the marker size.

Figure 24.

Variation of the Strouhal number $\textit{St}$ with the Froude number $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ . The green dash–dotted line and the blue dashed line represent the fitted variation laws between $\textit{St}$ and $\textit{Fr}$ for $\textit{Fr} \leqslant 1.3$ and $\textit{Fr} \geqslant 1.4$ , respectively. The error bars denote $\pm 1$ standard deviation based on cycle-to-cycle variations in $\textit{St}$ and are omitted where smaller than the marker size.

Figure 25.

Variation of the Strouhal number $\textit{St}$ with gap ratio $G$ for Froude numbers $\textit{Fr} = 0.8$ , 1.1 and 1.4. For the case with $\textit{Fr}=0.8$ , $G=0.025$ (case 1a), vortex shedding is suppressed. Therefore, the value of $\textit{St}$ is not provided in the figure. The error bars denote $\pm 1$ standard deviation based on cycle-to-cycle variations in $\textit{St}$ and are omitted where smaller than the marker size.

The characteristics of the separated shear layers directly influence the vortex shedding behaviour. To quantify vortex shedding, the Strouhal number is calculated as

(3.11) \begin{equation} \textit{St} = \frac {\textit{fD}}{U_0} , \end{equation}

where $f$ represents the vortex shedding frequency, which is also the dominant high frequency of the cylinder’s lift coefficient. Figure 24 illustrates the variation of $\textit{St}$ with $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ . The error bars denote $\pm 1$ standard deviation of $\textit{St}$ evaluated on a cycle-by-cycle basis, i.e. from the distribution of $\textit{St}$ computed for individual vortex shedding cycles, and are omitted where smaller than the marker size. It can be observed that as $\textit{Fr}$ increases, the value of $\textit{St}$ decreases monotonically. Figure 25 shows the variation of $\textit{St}$ with $G$ for $\textit{Fr} = 0.8$ , 1.1 and 1.4. The error bars denote $\pm 1$ standard deviation, evaluated as described for figure 24, and are omitted where smaller than the marker size. It can be seen that as $G$ increases, the value of $\textit{St}$ decreases monotonically for the three Froude numbers. This variation primarily stems from the reduction in the constraining effect of the free surface on the separated shear layers due to the increased gap height ( $\langle h^* \rangle _t$ ). For the case $\textit{Fr} = 0.8$ , $G = 0.025$ (case 1a), the mean gap height ( $\langle h^* \rangle _t$ ) above the cylinder is extremely small. The influence of the jet-like flow is very strong, which strongly suppresses the shear layers and prevents vorticity accumulation. As a result, vortex shedding is completely inhibited, and as such, the value of $\textit{St}$ for this case is not provided in figure 25. Increases in either $\textit{Fr}$ or $G$ lead to an increase of $\langle h^* \rangle _t$ , reducing the flow asymmetry near the cylinder. This weakening of the constraint imposed by the two-phase shear flow and the jet-like flow on the vortex structures at the trailing edge of the cylinder’s separated shear layers allows the vortex roll-up process to draw a greater vorticity feed from the shear layers. As vorticity accumulates, the vortex structures eventually shed, leading to the observed continuous decreasing trend of $\textit{St}$ .

Notably, from figure 24, it can be observed that at $G = 0.2$ , the decrease of $\textit{St}$ is more pronounced as $\textit{Fr}$ increases from 1.3 to 1.4. Furthermore, as indicated by the green dash–dotted line and the blue dashed line in the figure, $\textit{St}$ follows different variation laws for $\textit{Fr}\leqslant 1.3$ and $\textit{Fr} \geqslant 1.4$ :

(3.12) \begin{equation} \textit{St} = \begin{cases} -0.14\textit{Fr} + 0.73, & 0.8 \leqslant \textit{Fr} \leqslant 1.3 , \\ -0.29\textit{Fr} + 0.85, & 1.4 \leqslant \textit{Fr} \leqslant 2.0 . \end{cases} \end{equation}

This phenomenon is closely related to the transition in the free surface breaking pattern downstream of the cylinder: for $\textit{Fr} \leqslant 1.3$ , the downstream free surface exhibits SB, while for $\textit{Fr} \geqslant 1.4$ , it exhibits PB (see figures 4 and 8 b). The difference in breaking patterns significantly alters the flow characteristics, including the pressure field, which in turn affects the vorticity accumulation and shedding process in the cylinder’s separated shear layers, resulting in the different variation law of $\textit{St}$ with respect to $\textit{Fr}$ .

3.3. Turbulence statistics

This section discusses the statistical properties of the flow field, including the mean streamwise velocity and TKE, $k$ , which is defined as

(3.13) \begin{equation} k = \frac {1}{2}\left (\langle {\rho u^\prime u^\prime }\rangle _{zt} + \langle {\rho v^\prime v^\prime }\rangle _{zt} + \langle {\rho w^\prime w^\prime }\rangle _{zt} \right )\!. \end{equation}

3.3.1. Effect of froude number

Figure 26.

Contours of the mean streamwise velocity ( $\langle u \rangle _{zt}$ ) for different Froude numbers $\textit{Fr}$ at a fixed gap ratio $G=0.2$ : (a) $\textit{Fr} = 0.8$ (case 3a), (b) $\textit{Fr} = 1.3$ (case 3e), (c) $\textit{Fr} = 1.4$ (case 3f) and (d) $\textit{Fr} = 2.0$ (case 3i). Case labels correspond to table 1. The green dashed line indicates the mean free surface position, represented by the $\langle \psi \rangle _{zt} = 0.5$ isopleth.

Figure 26 presents the contours of the mean streamwise velocity $\langle u \rangle _{zt}$ for different Froude numbers at a fixed gap ratio $G = 0.2$ . Significant differences are observed between $\textit{Fr} \leqslant 1.3$ and $\textit{Fr} \geqslant 1.4$ . Cases with $0.8 \lt \textit{Fr} \lt 1.3$ are similar to those with $\textit{Fr} = 0.8$ and 1.3, and cases with $1.4 \lt \textit{Fr} \lt 2.0$ are similar to those with $\textit{Fr} = 1.4$ and 2.0, and as such, those cases are omitted. For $\textit{Fr} \leqslant 1.3$ , where the free surface downstream of the cylinder exhibits SB (see figures 4 a–e and 8 b), the two-phase shear flow above the jet-like flow continuously entrains air into the water. The entrained bubbles further break up, some rise to the surface due to buoyancy, while others are advected into the downstream wake region (Sun et al. Reference Sun, Lu, Patel, Zhu and Yang2025). Two distinct recirculation zones are present: one is the wake recirculation zone near the cylinder trailing edge, and the other is the two-phase flow recirculation zone located near the free surface above the jet-like flow (Sun et al. Reference Sun, Lu, Patel, Zhu and Yang2025). It can be observed that as $\textit{Fr}$ increases, the downward tilt angle of the jet-like flow decreases, leading to a shorter recirculation zone length near the free surface. When $\textit{Fr}$ reaches 1.4, the free surface morphology downstream of the cylinder transitions to PB. At this point, the flow symmetry is significantly enhanced, and the characteristic features of the jet-like flow induced by the spilling breaker almost disappear. It is replaced by a jet-like flow corresponding to a plunging breaker with a larger vertical thickness extending downstream. Concurrently, the two-phase shear flow originating from the free surface above the jet-like flow also vanishes (see figure 4 f–i), resulting in a substantial weakening of the constraining effect on the cylinder’s separated shear layers. For $\textit{Fr} \geqslant 1.4$ , instantaneous flow fields shown in figure 4(f–i) indicate that the plunging breaking zone is located within the surface roller region of the hydraulic jump. As $\textit{Fr}$ increases further, the flow inertia continues to strengthen, causing the position of the plunging breaking zone to gradually shift downstream (figures 4 f–i, 26 c and 26 d).

Figure 27.

Vertical profiles of the mean streamwise velocity $\langle u \rangle _{zt}$ at different streamwise locations (a) $x/D = 1$ and (b) $x/D = 3$ for cases with different Froude numbers $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ .

Figure 27. Long description

The image contains two sets of line graphs labeled as (a) and (b). Each set includes six individual line graphs. Panel A shows data for x/D = 1, and Panel B shows data for x/D = 3. Each line graph plots the vertical profiles of the mean streamwise velocity at different streamwise locations. The x-axis represents the normalized mean streamwise velocity (u_z)/U_0, ranging from -0.5 to 0.5. The y-axis represents the normalized vertical distance y/D, ranging from -4.0 to 1.5. Each graph within the panels is labeled with different Froude numbers (Fr): 0.8, 1.1, 1.3, 1.4, 1.8, and 2.0. The lines in the graphs show the variation in mean streamwise velocity profiles for these different Froude numbers. The trends indicate how the velocity profiles change with varying Froude numbers at the specified streamwise locations.

Figure 27 compares the vertical profiles of $\langle u \rangle _{zt}$ at different streamwise locations for various Froude numbers at a fixed gap ratio $G = 0.2$ . To avoid excessive clutter in the plot, only representative results are presented, with cases yielding similar results omitted from the plot. At $x/D = 1$ (figure 27 a), in the region near $y/D = 0$ , all cases exhibit a prominent positive peak, corresponding to the jet-like flow above the cylinder. For $\textit{Fr} \leqslant 1.3$ , the peak velocities of the jet-like flow are relatively close, whereas for $\textit{Fr} \geqslant 1.4$ , the peak increases monotonically with $\textit{Fr}$ . Since the mean gap height $\langle h^* \rangle _t$ above the cylinder also gradually increases with $\textit{Fr}$ (see figure 17), the flow flux rate through the gap increases with $\textit{Fr}$ . Near $y/D = -0.4$ , negative velocities appear for $\textit{Fr} = 0.8$ (case 3a) and $\textit{Fr} = 1.0$ (case 3b, omitted for clarity), corresponding to the recirculation zone near the free surface (see figure 26 a). The recirculation intensity in other cases is weaker, and no negative velocities are observed at this location. Near $y/D = -1.4$ , a significant negative region corresponds to the recirculation zone near the cylinder’s trailing edge (see figure 26). For cases with $\textit{Fr} \leqslant 1.3$ , the peak magnitudes in this recirculation zone are similar, reflecting the minor influence of $\textit{Fr}$ variation on the recirculation zone under the SB state downstream. In contrast, for $\textit{Fr} \geqslant 1.4$ , the PB leads to enhanced symmetry and vorticity accumulation at the trailing edge of the separating shear layers, elongating the recirculation zone in the wake. Consequently, the magnitude of the reverse flow velocity at $x/D = 1$ decreases as $\textit{Fr}$ increases. In the region $y/D \leqslant -1.8$ , $\langle u \rangle _{zt}$ decreases monotonically with increasing $\textit{Fr}$ and gradually approaches the inflow velocity $U_0$ . Comparing the mean streamwise velocity distributions above and below the cylinder, it is evident that as $\textit{Fr}$ increases, more water tends to flow above the cylinder rather than underneath it.

Figure 27(b) compares the vertical profiles of $\langle u \rangle _{zt}$ at $x/D = 3$ , a location already outside the core influence region of the jet-like flow. Near $y/D = -0.5$ , cases with $\textit{Fr} \leqslant 1.3$ still exhibit significant negative values, corresponding to the recirculation zone near the free surface, and its intensity weakens as $\textit{Fr}$ increases, consistent with the shrinking trend of the recirculation zone shown in figures 26(a) and 26(b). This recirculation zone essentially disappears for $\textit{Fr} \geqslant 1.4$ . For $-2 \leqslant y/D \leqslant -1$ , for cases with $\textit{Fr} \leqslant 1.3$ , due to their similar flow structures (see figures 26 a and 26 b), $\langle u \rangle _{zt}$ increases monotonically with $\textit{Fr}$ . In contrast, for cases with $\textit{Fr} \geqslant 1.4$ , the hydraulic jump toe shifts farther downstream as $\textit{Fr}$ increases (figures 4 f–i, 26 c and 26 d), resulting in a decrease in $\langle u \rangle _{zt}$ at $x/D = 3$ . In the deep-water region ( $y/D \leqslant -2$ ), similar to the trend at $x/D = 1$ , $\langle u \rangle _{zt}$ decreases with increasing $\textit{Fr}$ and approaches the inflow velocity $U_0$ , further supporting the previous conclusion of enhanced symmetry with an increasing Froude number.

Figure 28.

Contours of the TKE, $k$ , for different Froude numbers $\textit{Fr}$ at a fixed gap ratio $G = 0.2$ : (a) $\textit{Fr} = 0.8$ (case 3a), (b) $\textit{Fr} = 1.3$ (case 3e), (c) $\textit{Fr} = 1.4$ (case 3f) and (d) $\textit{Fr} = 2.0$ (case 3i). Case labels correspond to table 1. The green dashed line indicates the mean free surface position, represented by the $\langle \psi \rangle _{zt} = 0.5$ isopleth.

Figure 28 presents the contours of the TKE ( $k$ ) for different Froude numbers at a fixed gap ratio $G = 0.2$ . All cases exhibit two high-TKE zones: one located in the near-wake region close to the cylinder trailing edge, and the other in the two-phase flow region. Again, distinct differences in the distribution of TKE characteristics exist between $\textit{Fr} \leqslant 1.3$ and $\textit{Fr} \geqslant 1.4$ , attributed to the different free surface breaking patterns downstream of the cylinder. For $\textit{Fr} \leqslant 1.3$ , the two-phase flow region is concentrated in the shear layer above the jet-like flow. This region, characterized by intense air entrainment and mixing, forms a prominent core of TKE (figures 28 a and 28 b). As $\textit{Fr}$ increases, the high-TKE zone in the two-phase flow region moves closer to the free surface in the vertical direction since the downward tilt angle of the jet-like flow gradually decreases, while its streamwise position remains almost unchanged. The position of the high-TKE zone in the cylinder near-wake region remains relatively stable. For $\textit{Fr} \geqslant 1.4$ , where the downstream free surface exhibits PB (see figures 4 f–i and 8 b), the high-TKE zone in the two-phase flow region shifts to the surface roller region of the hydraulic jump. As $\textit{Fr}$ increases, the high-TKE zones in both the two-phase flow region and near-wake region move downstream. Additionally, the near-wake high-TKE zone shifts upward due to a decrease in the downward tilt angle of the jet-like flow (figures 28 c and 28 d).

Figure 29.

Variation of the peak value of TKE ( $k$ ) in the near-wake region $k_{mw}$ and two-phase region $k_{mt}$ with increasing Froude ( $\textit{Fr}$ ) number at a fixed gap ratio $G=0.2$ . The error bars denote the r.m.s. of $k$ .

To quantitatively compare the influence of $\textit{Fr}$ on $k$ , the variation of the peak values of TKE in the near-wake region $k_{mw}$ and two-phase flow region $k_{mt}$ with $\textit{Fr}$ is analysed, as shown in figure 29. Error bars in this figure indicate the r.m.s. of $k$ , defined as

(3.14) \begin{equation} k_{\textit{rms}} = \sqrt {\left\langle \left ( \frac {1}{2}\left (\rho u^{\prime }u^{\prime } + \rho v^{\prime }v^{\prime } + \rho w^{\prime }w^{\prime }\right ) - k\right )^2 \right\rangle _{zt}} . \end{equation}

As $\textit{Fr}$ increases from 0.8 to 1.1, both $k_{mt}$ and $k_{mw}$ show an increasing trend. This stems from the reduction of the suppression effect of jet-like flow on the vortex shedding. For $1.1 \leqslant \textit{Fr} \leqslant 1.3$ , both values enter a plateau phase. At this stage, the elevation difference of the free surface between the downstream drop and subsequent rise increases, leading to a greater consumption of gravitational potential energy. This increased energy expenditure ultimately offsets the benefits of improved flow mixing. For $\textit{Fr} \geqslant 1.4$ , the peak value of TKE in the near-wake region $k_{mw}$ generally continues to increase with $\textit{Fr}$ . This reflects the continued promotion of wake mixing. In contrast, the peak value of TKE in the two-phase region $k_{mt}$ exhibits a decreasing trend with increasing $\textit{Fr}$ . On the one hand, as $\textit{Fr}$ increases, inertia strengthens and gravitational effects weaken. With increasing $\textit{Fr}$ , the height of the surface roller in the hydraulic jump gradually increases (see figures 4 f–i and 28 c and 28 d). Since the two-phase flow region is located within this surface roller region, the energy required to overcome the gravity for the flow to travel from the hydraulic jump toe to the surface roller region increases. On the other hand, the increase in $\textit{Fr}$ leads to an increase in the mean gap height $\langle h^* \rangle _t$ above the cylinder, resulting in enhanced flow symmetry. This reduces the constraint on the flow, enhances mixing and consequently increases the kinetic energy dissipated. The combined effect of these two factors causes $k_{mt}$ to exhibit a decreasing trend.

3.3.2. Effect of gap ratio

Figure 30.

Contours of mean streamwise velocity $\langle u \rangle _{zt}$ for different gap ratios ( $G$ ) at a fixed Froude number $\textit{Fr} = 1.1$ : (a) $G = 0.025$ (case 1c), (b) $G = 0.1$ (case 2c), (c) $G = 0.2$ (case 3c) and (d) $G = 0.4$ (case 4c). Case labels correspond to table 1. The green dashed line indicates the mean free surface position, represented by the $\langle \psi \rangle _{zt} = 0.5$ isopleth.

Figure 30. Long description

Panel A: A heat map showing mean streamwise velocity contours for a gap ratio of 0.025 at a fixed Froude number of 1.1. The x-axis is labeled x/D and ranges from -4 to 16. The y-axis is labeled y/D and ranges from -4 to 2. The color scale at the bottom ranges from -0.2 to 1.2, indicating the normalized streamwise velocity. The green dashed line represents the mean free surface position. A black circle is visible near the origin. Panel B: A heat map showing mean streamwise velocity contours for a gap ratio of 0.1 at a fixed Froude number of 1.1. The axes and color scale are the same as in Panel A. The green dashed line and black circle are also present. Panel C: A heat map showing mean streamwise velocity contours for a gap ratio of 0.2 at a fixed Froude number of 1.1. The axes and color scale are the same as in Panel A. The green dashed line and black circle are also present. Panel D: A heat map showing mean streamwise velocity contours for a gap ratio of 0.4 at a fixed Froude number of 1.1. The axes and color scale are the same as in Panel A. The green dashed line and black circle are also present.

Figure 31.

Contours of mean streamwise velocity $\langle u \rangle _{zt}$ for different gap ratios ( $G$ ) at a fixed Froude number $\textit{Fr} = 1.4$ : (a) $G = 0.025$ (case 1f), (b) $G = 0.1$ (case 2f), (c) $G = 0.2$ (case 3f) and (d) $G = 0.4$ (case 4f). Case labels correspond to table 1. The green dashed line indicates the mean free surface position, represented by the $\langle \psi \rangle _{zt} = 0.5$ isopleth.

This section proceeds to analyse the influence of the gap ratio $G$ on the distribution of the mean streamwise velocity $\langle u \rangle _{zt}$ at $\textit{Fr} = 0.8$ , 1.1 and 1.4. Figures 30 and 31 present the contours of $\langle u \rangle _{zt}$ for different $G$ at $\textit{Fr} = 1.1$ and 1.4, respectively. Cases with $\textit{Fr} = 0.8$ are similar to those with $\textit{Fr} = 1.1$ , and as such, the plots are omitted. For $\textit{Fr} = 1.1$ (figure 30), the free surface downstream of the cylinder exhibits SB in all cases. As $G$ increases, the mean gap height $\langle h^* \rangle _t$ above the cylinder increases (figure 17), enhancing flow symmetry and weakening the constraining effect of the jet-like flow. The downward tilt angle of the jet-like flow gradually decreases, and the near-surface recirculation zone shortens. For $\textit{Fr} = 1.4$ (figure 31), the influence of $G$ is more pronounced. For $G \leqslant 0.1$ , the downstream free surface exhibits SB (figures 7 a, 7 b and 8 b), and a recirculation zone persists near the surface (figures 31 a and 31 b). As $G$ increases to 0.2 and above, the breaking pattern gradually transitions to PB (figures 7 c, 7 d and 8 b), leading to the shortening and eventual disappearance of the near-surface recirculation zone (figures 31 c and 31 d).

Figure 32.

Vertical profiles of the mean streamwise velocity $\langle u \rangle _{zt}$ at $x/D = 1$ for cases with different gap ratio $G$ at (a) $\textit{Fr} = 0.8$ , (b) $\textit{Fr} = 1.1$ and (c) $\textit{Fr} = 1.4$ .

Figure 32. Long description

The image contains three sets of line graphs labeled (a), (b), and (c), each showing vertical profiles of the mean streamwise velocity for different gap ratios at specific Froude numbers. Panel A: For Fr = 0.8, four line graphs are shown with gap ratios G = 0.025, G = 0.1, G = 0.2, and G = 0.4. The y-axis represents y/D, and the x-axis represents the normalized mean streamwise velocity (u_z)/U0. Each line graph shows distinct trends and patterns for different gap ratios. Panel B: For Fr = 1.1, similar line graphs are presented with the same gap ratios. The y-axis and x-axis labels remain consistent with Panel A, and each line graph exhibits unique trends. Panel C: For Fr = 1.4, the line graphs follow the same format with gap ratios G = 0.025, G = 0.1, G = 0.2, and G = 0.4. The y-axis represents y/D, and the x-axis represents the normalized mean streamwise velocity (u_z)/U0. Each line graph shows different patterns and trends for the given gap ratios.

Figure 32 provides a further comparison of the vertical profiles of $\langle u \rangle _{zt}$ at the streamwise location $x/D = 1$ for the corresponding cases. Observation reveals that a prominent positive peak of $\langle u \rangle _{zt}$ exists near $y/D = -0.6$ , corresponding to the jet-like flow above the cylinder. For all three Froude numbers, the peak value of $\langle u \rangle _{zt}$ increases monotonically with increasing gap ratio. A significant negative peak of $\langle u \rangle _{zt}$ is located near $y/D = -1.2$ , corresponding to the recirculation zone near the cylinder’s trailing edge. The influence of the gap ratio on the magnitude of this reverse flow velocity is relatively noticeable for $\textit{Fr} = 0.8$ (see figure 32 a), but shows little variation for the other Froude numbers (see figures 32 b and 32 c). In the deep-water region ( $y/D \leqslant -1.8$ ), where the influence of flow asymmetry diminishes, the velocity distributions of different cases converge.

Figure 33.

Contours of TKE ( $k$ ) for different gap ratios ( $G$ ) at a fixed Froude number $\textit{Fr} = 1.1$ : (a) $G = 0.025$ (case 1c), (b) $G = 0.1$ (case 2c), (c) $G = 0.2$ (case 3c) and (d) $G = 0.4$ (case 4c). Case labels correspond to table 1. The green dashed line indicates the mean free surface position, represented by the $\langle \psi \rangle _{zt} = 0.5$ isopleth.

Figure 34.

Contours of TKE ( $k$ ) for different gap ratios ( $G$ ) at a fixed Froude number $\textit{Fr} = 1.4$ : (a) $G = 0.025$ (case 1f), (b) $G = 0.1$ (case 2f), (c) $G = 0.2$ (case 3f) and (d) $G = 0.4$ (case 4f). Case labels correspond to table 1. The green dashed line indicates the mean free surface position, represented by the $\langle \psi \rangle _{zt} = 0.5$ isopleth.

Figure 34. Long description

The image contains four contour plots labeled (a), (b), (c), and (d), each representing the distribution of turbulent kinetic energy (TKE) for different gap ratios (G) at a fixed Froude number (Fr = 1.4). Panel A: The contour plot for G = 0.025 shows the TKE distribution with the x-axis labeled as x/D and the y-axis labeled as y/D. The green dashed line indicates the mean free surface position. Panel B: The contour plot for G = 0.1 shows the TKE distribution with the same axis labels and the green dashed line indicating the mean free surface position. Panel C: The contour plot for G = 0.2 shows the TKE distribution with the same axis labels and the green dashed line indicating the mean free surface position. Panel D: The contour plot for G = 0.4 shows the TKE distribution with the same axis labels and the green dashed line indicating the mean free surface position. Each plot includes a color bar indicating the TKE values ranging from 0 to 0.20.

Figures 33 and 34 present the contours of TKE ( $k$ ) for different gap ratios at $\textit{Fr} = 1.1$ and 1.4, respectively. The results for $\textit{Fr} = 0.8$ are similar to $\textit{Fr} = 1.1$ , and as such, the plots are omitted. Similarly, all cases exhibit two high-TKE zones. For cases with $\textit{Fr} = 1.1$ , as $G$ increases, the downward tilt angle of the jet-like flow decreases. Consequently, the high-TKE zone in the two-phase flow region moves upward with respect to the cylinder, while its streamwise position remains almost unchanged. The position of the high-TKE zone in the near-wake region remains almost unchanged. For $\textit{Fr} = 1.4$ (figure 34), the downstream free surface is in a transitional state between SB and PB (see figures 7 and 8 b). The two-phase flow region shifts to the surface roller region of the hydraulic jump. As $G$ increases, the inertia of the jet-like flow strengthens, and its downward tilt angle decreases. Consequently, both the high-TKE zones in the two-phase flow region and the near-wake region move downstream, with the near-wake one also shifting upward.

Figure 35.

Variation of the peak value of TKE ( $k$ ) in the (a) near-wake region $k_{mw}$ and (b) two-phase region $k_{mt}$ with gap ratio ( $G$ ) at $\textit{Fr} = 0.8$ , 1.1 and 1.4. The error bars denote the r.m.s. of $k$ .

Figure 35 further presents the variation of the peak values of TKE in the near-wake region $k_{mw}$ and two-phase flow region $k_{mt}$ with $G$ . Error bars in this figure indicate the r.m.s. of $k$ , evaluated as described for figure 29. In the near-wake region (see figure 35 a), $k_{mw}$ generally shows an increasing trend with increasing $G$ for all three Froude numbers. Note that $k_{mw}$ for the case with $\textit{Fr}=0.8$ , $G=0.025$ (case 1a) is higher than that for $\textit{Fr}=1.1$ , $G=0.025$ (case 1c). This is because, in the former case, the mean gap height $\langle h^* \rangle _t$ above the cylinder is extremely small, leading to intense mixing between the jet-like flow and the separated shear layers in the near-wake region (figure 15 a), which elevates the TKE. For $G=0.4$ , $k_{mw}$ for $\textit{Fr}=1.1$ and 1.4 are similar, indicating a plateau phase. This occurs because the mean gap height ( $\langle h^* \rangle _{t}$ ) is sufficiently large that further increases in $\textit{Fr}$ have only a weak influence on the flow, representing a marginal effect. In the two-phase flow region (figure 35 b), the variation of $k_{mt}$ with $G$ exhibits three distinct patterns depending on $\textit{Fr}$ : it first increases and then decreases for $\textit{Fr}=0.8$ ; increases initially and then stabilizes for $\textit{Fr}=1.1$ ; and decreases monotonically for $\textit{Fr}=1.4$ . For $\textit{Fr} \leqslant 1.1$ , where the downstream free surface exhibits SB, there exists a critical gap ratio value: below this value, an increase in $G$ weakens the blockage effect of the jet-like flow, promoting flow mixing in the two-phase flow region and thus increasing $k_{mt}$ ; above this value, a further increase in $G$ enhances symmetry, reducing air entrainment and causing $k_{mt}$ to decrease or remain constant. For $\textit{Fr}=1.4$ , where the downstream free surface is in a transitional state from SB to PB, an increase in $G$ enhances flow symmetry. Consequently, more system energy is consumed by wake mixing and by the work done against gravity during surface recovery, leading to a continuous decrease in $k_{mt}$ . This trend is similar to the effect of increasing $\textit{Fr}$ at a fixed gap ratio $G=0.2$ (figure 29). The above analysis again indicates that in the problem of two-phase flow past a submerged cylinder, the influences of $\textit{Fr}$ and $G$ are highly coupled: an increase in $\textit{Fr}$ enhances both inertial effects and flow symmetry, while an increase in the $G$ value primarily enhances flow symmetry. Their combined action on the flow topology results in a complex nonlinear response in the peak values of $k$ .