2.1. Experimental apparatus

The experiments were conducted in a stratified towing tank located at Shanghai Jiao Tong University. The main tank, constructed with a steel frame and transparent glass panels, measures 6 m in width, 6 m in length, and 1 m in depth, which connected via pipelines to a 3.0 $\textrm {m}^{3}$ freshwater tank, a 5.0 $\textrm {m}^{3}$ saltwater mixing tank, three storage tanks and an array of mushroom-shaped horizontal injection nozzles mounted on a liftable frame, as illustrated in figure 1. The main tank’s substantial lateral cross-sectional area was specifically designed to reduce lateral confinement effects, and porous wave-absorbing material was installed along the tank boundaries. Previous studies have shown that reflection effects can significantly distort internal-wave structures in narrower tanks (Robey Reference Robey1997; Wang et al. Reference Wang, Chen and You2017). In the present study, an effective observation time window free from reflected internal waves was identified in order to capture the uncontaminated direct wake response; a detailed justification is given in Appendix A.

Figure 1.

Schematic diagram of the stratified tank system.

The stratified fluid environment was established using a modified approach adapted from the two-tank method of Oster (Reference Oster1965). This approach, referred to as the pycnocline injection technique, relies on shear-layer mixing and was specifically redesigned to maintain a sharp stratification in large-volume systems. Unlike conventional bottom-filling methods, where gravitational intrusions and limited control over pycnocline thickness often compromise the quality of the stratification, the present configuration provides enhanced stability and greater precision in forming the desired density profile. Specifically, freshwater was introduced gently atop a saltwater base through a $3\times 3$ array of mushroom-shaped horizontal nozzles mounted on a vertically adjustable frame. By controlling both the vertical speed of the nozzle array and the flow rate, a stable and well-defined stratification was achieved. The spatial uniformity and temporal stability of the resulting stratified field are further validated in Appendix B. The density field established in the present study ranged from 998 to $1025 \ \mathrm{kg\,m^{-3}}$ , forming a pycnocline approximately $0.2 \ \mathrm{m}$ below the free surface. The corresponding buoyancy-frequency distribution, shown in figure 2, reached a maximum value of $2.4 \ \mathrm{rad\,s^{-1}}$ . To ensure consistency of the stratified field, the background stratification was checked before and after each experiment. The relative deviation of the characteristic buoyancy frequency was defined as $\varepsilon = |N_{\textit{after}}-N_{\textit{before}}|/N_{\textit{before}}$ . When $\varepsilon \lt 2\,\%$ , the stratification was regarded as acceptable, and the updated buoyancy frequency was used to make a slight adjustment to the Froude number for the subsequent run. Otherwise, the stratification was regenerated using the same top-injection procedure under identical nozzle flow rate and vertical translation speed.

Figure 2.

Vertical distribution of density and buoyancy frequency.

A computer-controlled towing mechanism was installed along the longitudinal axis of the tank. This system employed a continuous stainless-steel wire (0.5 mm diameter) looped around pulleys mounted on the sidewalls, enabling precise constant-speed towing. The towed body was maintained at a fixed submergence depth of 0.26 m by tensioned stainless-steel wires threaded through the centre. This mounting approach minimised transverse oscillations caused by asymmetric vortex shedding. The wake generated by the support wires, whose Reynolds number was approximately two orders of magnitude smaller than that of the sphere, was confirmed to be much weaker and smaller in scale than the main wake of the sphere (Meunier et al. Reference Meunier2018). Additional control runs without the towed body further confirmed that the wires alone did not generate detectable disturbances above the background noise level.

The experimental model consisted of spheres fabricated via 3-D printing using a polyamide composite reinforced with 40 % glass beads. Neutral buoyancy was achieved by embedding lead weights into the spheres. Two sphere diameters, 5 and 7.5 cm, were employed to facilitate investigation across a broad range of Reynolds and Froude numbers. The depth-to-diameter ratio, $H/D$ , was sufficiently large for both diameters, effectively suppressing wall and bottom effects, as recommended by Spedding (Reference Spedding1997). The Reynolds number ranged from $2.38 \times 10^3$ to $4.76 \times 10^4$ , encompassing the near-wake regimes described by Chomaz et al. (Reference Chomaz, Bonneton and Hopfinger1993). For each case with identical motion parameters, three repeated experiments were conducted to ensure the reliability and repeatability of the results. Previous studies (Spedding et al. Reference Spedding, Browand and Fincham1996b ; Spedding Reference Spedding1997) have demonstrated that late-time stratified wake evolution is only weakly dependent on $\textit{Re}$ within this range. Consequently, the Froude number was adopted as the primary control parameter in the present study. For fixed $N$ and $D$ , $\textit{Fr}$ varies linearly with $\textit{Re}$ in this experimental configuration, ranging from 0.42 to 8.33.

2.2. Measurement techniques

Internal density fluctuations were measured using high-precision dual-electrode conductivity microprobes with a tip width of 5 mm. The measured conductivity was converted to density using pre-established calibration curves based on laboratory reference solutions. For wave-height measurement, the background vertical density profile near the pycnocline was first obtained by vertically traversing the probe, thereby reconstructing the initial isopycnal distribution. When the pycnocline was disturbed, the instantaneous density measured at each probe was referenced to this background profile to determine the corresponding vertical displacement from equilibrium, which was taken as the local wave height. To enable wide-domain, near-instantaneous capture of internal wave structures, a custom-designed multichannel data acquisition system was developed. Each acquisition cycle lasted 0.2 s, significantly shorter than the dominant internal wave period $2\pi /N\approx 2.6$ s, thereby ensuring that the measured isopycnal displacements represent quasi-instantaneous snapshots of the wave field. This high temporal resolution was essential for scanning across a large array of probes.

The system accommodated up to 300 conductivity probes arranged in a $10\times 30$ grid. Probes were mounted on an adjustable and portable steel frame, and remained fixed in the fluid during the measurement rather than moving with the flow; each probe extended approximately 5 cm beyond the frame to minimise structural interference near the pycnocline. This design ensured that disturbances from the support system were negligible compared with those induced by the turbulent wake, thus preserving the fidelity of the internal wave measurements.

The primary probe array covered a region measuring 0.9 m in the streamwise direction and 3.8 m laterally, located sufficiently downstream of the towed sphere such that the internal wave field could be considered quasi-steady. The layout of the planar array is shown in figure 3, including both front (a) and side (b) views. All probes were positioned within the pycnocline layer to maximise sensitivity to isopycnal fluctuations. Probe spacing ranged from 0.1 to 0.2 m, depending on the specific test configuration and resolution requirements. Comparison tests using single- and multiprobe arrangements showed that, within this spacing range, the disturbance introduced by the probes to the flow field remained small; see Appendix C for details. Compared with the more commonly used Schlieren and particle image velocimetry (PIV) methods, the finite size of the probes limits the spatial resolution of the measured field. However, this method can directly provide more accurate wave-height information, rather than relying on conversion from visual observations. A right-handed Cartesian coordinate system was adopted, with the origin defined as the vertical projection of the sphere’s centre onto the free surface at the moment towing ceased. The positive $y$ -axis points laterally to the right of the sphere’s direction of motion, and the $z$ -axis is positive downwards. For organisational clarity, probes aligned at identical streamwise positions were grouped into rows. These rows were numbered sequentially from 1 to 10, corresponding to the order in which the sphere passed by. It should be noted that the measured signal may include local vertical displacements rather than propagating waves, mainly during the early near-wake stage. With the subsequent buoyancy-controlled adjustment of the wake, however, this displacement is progressively converted into internal-wave motion, so that the later signal is more appropriately interpreted in terms of wave propagation.

Figure 3.

Schematic of the planar array probe layout: (a) front view; (b) side view.

Figure 4.

Distribution of experimental cases among the four near-wake regimes.

3.1. SLW regime

The minimum Froude number investigated in the experiments, $\textit{Fr} = 0.42$ , lies within the SLW regime. Under this condition, flow separation behind the sphere is negligible, and the flow field is dominated entirely by lee waves. Figure 5 presents the spatial distribution of internal wave amplitudes at different dimensionless times. The instant at which the sphere exits the upstream boundary of the observation domain is defined as ${\textit{Nt}} = 0$ , as shown in figure 5(a). At ${\textit{Nt}} = 0$ , the internal wave wake consists of both divergent and transverse components. The outer divergent waves complete one full oscillation cycle within the observation area, while the inner transverse waves exhibit approximately two complete cycles. As time progresses, with the observation domain acting as a fixed spatial window, the entire internal wave wake translates downstream following the sphere. Its overall morphology remains essentially unchanged, which is a defining characteristic of lee waves.

Figure 5.

Spatiotemporal distribution of internal wave wake at $\textit{Fr} = 0.42$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 10$ ; (c) ${\textit{Nt}} = 20$ ; (d) ${\textit{Nt}} = 40$ ; (e) ${\textit{Nt}} = 60$ ; (f) ${\textit{Nt}} = 80$ .

Figure 6.

Spatiotemporal distribution of internal wave wake at $\textit{Fr} = 0.83$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 10$ ; (c) ${\textit{Nt}} = 20$ ; (d) ${\textit{Nt}} = 40$ ; (e) ${\textit{Nt}} = 60$ ; (f) ${\textit{Nt}} = 80$ .

When the Froude number is increased to $\textit{Fr} = 0.83$ , the corresponding spatiotemporal distributions of internal wave amplitudes are shown in figure 6. According to the classification of Chomaz et al. (Reference Chomaz, Bonneton and Hopfinger1993), this condition falls just within the T regime, where wake separation is expected to initiate, and wake-effect internal waves may emerge. However, the wave amplitudes observed in figure 6 are larger than those at $\textit{Fr} = 0.42$ . This indicates that the amplitude of lee waves continues to increase with increasing Froude and Reynolds numbers. Within the observation domain, the wave field remains composed exclusively of the divergent and transverse components characteristic of lee waves. As the sphere moves farther downstream, the divergent waves spread laterally, and no clear wake-effect internal waves are observed in the region between the two divergent lobes.

Notably, the present amplitude variation agrees more closely with the thin-layer results of Robey (Reference Robey1997) and Wang et al. (Reference Wang, Chen and You2017) than with the continuous-stratification measurements of Chomaz et al. (Reference Chomaz, Bonneton and Hopfinger1993). As discussed by Meunier et al. (Reference Meunier2018), this difference is likely related to the distinct dynamics introduced by thin-layer stratification. The present observation that lee wave amplitudes at $\textit{Fr} = 0.83$ exceed those at $\textit{Fr} = 0.42$ is therefore more consistent with the latter studies. These features indicate that, under this condition, the near-wake of the sphere remains effectively within the SLW regime, and the internal wave wake is still dominated by lee waves.

Figure 7 further presents the time-series waveform extracted by the probes in the first row using Robey’s line array configuration. Because lee waves are steady in the reference frame of the moving body, the temporal evolution of wave amplitude is consistent with the spatial distributions discussed previously. Under steady towing conditions, a linear correspondence exists between spatial and temporal representations, allowing planar array measurements and line array measurements to be directly related. This result confirms that the line array technique employed in earlier studies remains effective for capturing lee wave characteristics.

Figure 7.

Time-series waveform of the internal wave wake in the SLW regime: (a) $\textit{Fr} = 0.42$ ; (b) $\textit{Fr} = 0.83$ .

3.2. T regime

When the towing speed is increased to $\textit{Fr} = 1.67$ , the spatiotemporal distribution of the internal wave wake in the T regime is shown in figure 8. Compared with figure 6, the lee wave system in figure 8 is characterised solely by divergent waves with a reduced opening angle, accompanied by a clear decrease in wave amplitude. As the flow enters the transition regime, additional wake-related unsteady motions begin to emerge in the wake. Correspondingly, wake-generated waves appear and modify the amplitude distribution of the wave field within the angular sector of lee wave divergence.

In the central region of the lee wave wake, $y \in [-0.6,\,0.6] \ \mathrm{m}$ , the transverse wave component is replaced by complex and irregular internal waves possibly induced by small-scale vortex structures. During the NEQ stage, these turbulent structures are progressively constrained by stratification, while internal waves predominantly exhibitsemicircular or circular concentric spatial patterns, as illustrated in the blue box in figure 8. Their wavelengths and amplitudes are significantly smaller than those of the outer lee waves, and their wavefronts appear as arc-shaped patterns concentrated in the central wake region. These observational differences suggest that they should not be regarded simply as a direct continuation of the body-generated ‘V’-shaped wave system; instead, they are more consistently interpreted as a distinct wake-related wave component associated with local moving structures in the central wake.

Figure 8.

Spatiotemporal distribution of internal wave wake at $\textit{Fr} = 1.67$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 10$ ; (c) ${\textit{Nt}} = 20$ ; (d) ${\textit{Nt}} = 40$ ; (e) ${\textit{Nt}} = 60$ ; (f) ${\textit{Nt}} = 80$ .

Figure 9.

High-resolution spatiotemporal distribution of wake-effect internal waves at $\textit{Fr} = 1.67$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 20$ ; (c) ${\textit{Nt}} = 40$ ; (d) ${\textit{Nt}} = 60$ .

To further resolve the detailed characteristics of wake-effect internal waves, the probe density in the region $y \in [-0.6,\,0.6] \ \mathrm{m}$ was increased to a spacing of 5 cm in the $y$ -direction, while keeping the spatial arrangement of the probe array unchanged in both position and orientation. Although a higher probe density may enhance probe-interference effects, the assessment in Appendix C shows that the disturbance at this spacing remains within an acceptable range. High-resolution spatial distributions of internal wave amplitude at selected times are presented in figure 9. Figure 9(a) corresponds to the NW stage, during which only lee waves are observed and vortex-induced disturbances have a negligible influence on the wave field. At ${\textit{Nt}} = 20$ , figure 9(b) reveals a sequence of semicircular or circular internal waves aligned in the downstream direction. These wake-effect internal waves, which are distinctly different from the ‘V’-shaped lee waves and constitute the dominant component during the NEQ stage, are referred to herein as collapse-stage internal waves. Such waves have repeatedly been observed during the wake-collapse stage under the stabilising influence of background stratification (Bonnier et al. Reference Bonnier, Eiff and Bonneton2000; Brucker & Sarkar Reference Brucker and Sarkar2010). In an equivalent-source sense, the downstream translation of wake structures is more closely related to the forced internal-wave component, whereas the semicircular and circular waves in the central region are more plausibly interpreted as the response to a vertically oscillating source during wake collapse. As ${\textit{Nt}}$ increases, the lateral extent of wake influence gradually expands. Within the finite observation window, the wake-effect internal waves appear as arcs with increasing radii, as shown in figures 9(c) and 9(d). This behaviour is consistent with the coherent circular internal waves observed by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) using fluorescent dye visualisation at ${\textit{Nt}} \approx 50$ . The larger-radius coherent internal waves observed during the later NEQ stage may be attributed to an increase in the effective source scale associated with wake collapse, together with a gradual weakening of the collapse intensity.

Figure 10 further shows the time-series waveform obtained using the line array method. Unlike the SLW regime shown in figure 7, figure 10 clearly reveals wake-effect internal waves within the region $y \in [-0.6,\,0.6] \ \mathrm{m}$ . With increasing time, the characteristic scale of the collapse-stage internal waves grows progressively. In the time domain, these waves appear as distinct ‘ $\cap$ ’- or ‘ $\cup$ ’-shaped waveforms occupying the wedge-shaped region associated with the inner lee waves. The discrepancy between line array and planar array observations arises from the unsteady propagation of collapse-stage internal waves. As semicircular or circular internal waves propagate, their waveforms evolve continuously, resulting in narrower upper portions and wider lower portions when sampled discretely. Consequently, because wake-generated waves propagate at a speed related to the wake drift rather than the body speed, the traditional line-array analysis of wake-effect internal waves proposed by Robey (Reference Robey1997), together with the corresponding theoretical models, may have some limitations in resolving their unsteady spatial structure when interpreted in a body-fixed reference frame.

Figure 10.

Time-series waveform of the internal wave wake in the T regime at $\textit{Fr} = 1.67$ .

3.3. 3-D regime

A further increase in towing speed marks the transition into the 3-D near-wake regime, in which large-scale coherent wake structures emerge as described by Chomaz et al. (Reference Chomaz, Bonneton and Hopfinger1993). At $\textit{Fr} = 3.33$ and $\textit{Re} = 1.9 \times 10^4$ , the near wake of the sphere would be expected to develop pronounced three-dimensional, large-scale oscillatory structures. The corresponding internal wave amplitude field is shown in figure 11. Although outer lee waves remain visible, their amplitudes are markedly smaller than those of the wake-effect internal waves and are also reduced compared with lee waves observed at lower Froude numbers. During the NW stage, as illustrated in figure 11(a), periodic isopycnal undulations induced by large-scale coherent vortex structures can already be identified. This process is commonly associated with the generation of forced internal waves (Bonneton et al. Reference Bonneton, Chomaz and Hopfinger1993), although such wave features are not readily distinguishable in the instantaneous amplitude field. After the near-wake reaches its maximum vertical extent, it undergoes collapse, with the AC process completed at approximately ${\textit{Nt}} = 10$ (Lin et al. Reference Lin, Boyer and Fernando1993; Bonnier & Eiff Reference Bonnier and Eiff2002). During the collapse stage, internal waves emerge near the centres of the turbulent patches in the amplitude field and complete roughly one oscillation cycle, as shown in figures 11(b) and 11(c). Compared with the case at $\textit{Fr} = 1.67$ (figure 8), the internal wave signatures at this higher Froude number are more pronounced and extend over a wider spatial region. This observation suggests that the characteristics of collapse-stage internal waves may be closely related to the scale and spatial organisation of the wake vortex street, both of which are expected to vary with the Froude number. As time progresses, the radii of the collapse-stage internal waves gradually increase. These waves begin to overlap with one another and coexist with randomly distributed internal waves, giving rise to a complex wake-effect internal wave field.

Figure 11.

Spatiotemporal distribution of internal wave wake at $\textit{Fr} = 3.33$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 10$ ; (c) ${\textit{Nt}} = 20$ ; (d) ${\textit{Nt}} = 40$ ; (e) ${\textit{Nt}} = 60$ ; (f) ${\textit{Nt}} = 80$ .

Figure 12.

High-resolution spatiotemporal distribution of wake-effect internal waves at $\textit{Fr} = 3.33$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 20$ ; (c) ${\textit{Nt}} = 40$ ; (d) ${\textit{Nt}} = 60$ .

The high-resolution evolution of wake-effect internal waves is further illustrated in figure 12. The vortex street structure is clearly identifiable during the near-wake stage, as shown in figure 12(a). After stratification begins to influence the wake during the AC stage, figure 12(b) reveals two circular collapse-stage internal waves of comparable radius and wavelength, arranged sequentially in the streamwise direction. Such features are much less evident at the same stage in the T regime. At later times, the circular internal waves continue to expand radially as they propagate downstream, for example at ${\textit{Nt}} = 40$ (figure 12 c). They gradually fragment and spread laterally, eventually merging with the outer internal wave system, while their wavelengths remain nearly constant. Throughout the NEQ stage, no significant additional semicircular or circular internal waves are generated. This behaviour can be attributed to the fact that large-scale wake vortices collapse primarily within the interval $2 \lt Nt \lt 8$ , during which the initial buoyancy effects continuously excite multiple internal waves during the accelerated collapse phase. Once this phase concludes, the generation of new large-amplitude circular internal waves almost ceases.

Figure 13.

Spatiotemporal distribution of internal wave wake at $\textit{Fr} = 6.67$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 10$ ; (c) ${\textit{Nt}} = 20$ ; (d) ${\textit{Nt}} = 40$ ; (e) ${\textit{Nt}} = 60$ ; (f) ${\textit{Nt}} = 80$ .

Figure 14.

High-resolution spatiotemporal distribution of wake-effect internal waves at $\textit{Fr} = 6.67$ : (a) ${\textit{Nt}} = 0$ ; (b) ${\textit{Nt}} = 20$ ; (c) ${\textit{Nt}} = 40$ ; (d) ${\textit{Nt}} = 60$ .

For higher Froude and Reynolds numbers ( $\textit{Fr} \gt 3.3$ , $\textit{Re} \gt 1.43 \times 10^4$ ), representative spatiotemporal distributions of internal waves are shown in figure 13. Increasing $\textit{Fr}$ and $\textit{Re}$ leads to larger scales and spacings of coherent vortex structures in the wake, resulting in more pronounced internal wave oscillations during both the NW and AC stages. In addition to the collapse-stage internal waves, large-wavelength internal waves are observed outside the main collapse region. Unlike lee waves, these waves are concentrated near the wake centreline. At the same time, they retain a distinct ‘V’-shaped planform similar to that of lee waves. These large-wavelength waves are identified as forced internal waves generated by the wake-related displacement effect. Owing to their long wavelengths, these features are more distinctly captured in line array time-series measurements, which are analysed in detail later. Notably, further increases in the Froude number produce only minor changes in the overall waveform morphology and propagation characteristics. To elucidate the detailed structure of wake-effect internal waves, figure 14 demonstrates that, for $\textit{Fr} \gt 3.3$ , these waves exhibit similar structural characteristics and evolutionary processes. The primary differences lie in their wavelength and amplitude. Compared with figure 12, both quantities increase with increasing Froude number.

The line array time series of internal wave amplitudes in the 3-D regime is shown in figure 15. In contrast to the spatially limited planar array measurements, the line array technique captures long-wavelength lee waves in the outermost wake region, highlighting its advantage in resolving the complete structure of long-wavelength wave systems. At this regime, the amplitude of lee waves is smaller than that of wake-effect internal waves, and the wavefront angle in the time-series is larger, confining wake-effect internal waves to a wedge-shaped region near the centreline. Unlike the case in the T regime, long-wavelength forced internal waves also emerge, forming the boundaries of the ‘V’-shaped region occupied by wake-effect internal waves. Within forced internal waves, the wave field is dominated by narrow-top and wide-bottom ‘ $\cap$ ’- or ‘ $\cup$ ’-shaped structures. These signatures arise from semicircular or circular collapse-stage internal waves, which propagate slowly through the probe array while their characteristic radii progressively increase.

Figure 15.

Time-series waveform of the internal wave wake in the 3-D regime: (a) $\textit{Fr} = 3.33$ ; (b) $\textit{Fr} = 6.67$ .

4.1. Transition regimes

A fundamental distinction between lee waves and wake-effect internal waves lies in their propagation speed relative to the moving underwater body. Robey (Reference Robey1997) demonstrated that, as the towing speed of the body increases, the propagation speed of wake-effect internal waves approaches an asymptotic value lower than the body’s towing speed and thereafter remains essentially unchanged. This property provides an effective criterion for distinguishing dominant wave systems in different parameter regimes by analysing the propagation speed of the prevailing waves within the observation region. On this basis, the transition regime of the wake can be defined.

Consider two observation points located at the same transverse position (identical $y$ coordinate) and separated by a streamwise distance $\Delta x$ . The temporal evolution of the internal wave signals recorded at these two points can be analysed using correlation techniques. The time lag $\Delta t$ corresponding to the maximum correlation coefficient is then used to estimate the velocity of the dominant internal wave component in the wake:

(4.1) \begin{equation} U_{{iw}} = \frac {\Delta x}{\Delta t}. \end{equation}

According to the definition of the Froude number, the internal wave correlation Froude number is defined as

(4.2) \begin{equation} \textit{Fr}_{{iw}} = \frac {U_{{iw}}}{\textit{ND}}. \end{equation}

The correlation Froude number quantifies the streamwise velocity component of the dominant internal wave group, thereby characterising the propagation properties of the excited internal waves.

However, previous studies have typically estimated the correlation Froude number using a limited number of observation points. For example, Wang et al. (Reference Wang, Chen and You2017) employed four pairs of probes arranged symmetrically about the longitudinal plane of a sphere and reported a discrete distribution of correlation Froude numbers in the range 0.5–1.2. Similarly, Robey (Reference Robey1997) placed four probes in the streamwise direction at $y = 40 \ \mathrm{cm}$ and obtained an average correlation Froude number of 0.78. Such estimates are reasonably accurate when steady lee waves dominate the wake. In contrast, wake-effect internal waves exhibit pronounced spatiotemporal randomness and non-stationarity, making accurate estimation of the correlation velocity difficult when relying on a limited number of fixed-point measurements.

To overcome this limitation, the present study computes a large ensemble of internal wave correlation Froude number samples using all observation points sharing the same $y$ coordinate within a broad spatial domain of $3.8 \ \mathrm{m} \times 0.9 \ \mathrm{m}$ . This enables estimation of the overall probability density distribution $f_p$ of the internal wave correlation Froude number:

(4.3) \begin{equation} f_p(\textit{Fr}_{{iw}}) = \frac {1}{\textit{Qh}} \sum _{q=1}^\textit{Q}K \left ( \frac {\textit{Fr}_{{iw}} - \textit{Fr}_{{iw}, q}}{h} \right )\!, \end{equation}

where $ Q$ denotes the total number of samples ( $ Q = 1350$ in this study), and $ K$ is a Gaussian kernel function,

(4.4) \begin{equation} K(x) = \frac {1}{\sqrt {2\pi }} \mathrm{e}^{-x^2 / 2}. \end{equation}

The kernel function satisfies normalisation and symmetry, ensuring that the calculated mean equals the sample mean. The optimal bandwidth $ h$ is estimated using the standard deviation $ \sigma$ as follows:

(4.5) \begin{equation} h = \left ( \frac {4\sigma ^5}{3Q} \right )^{1/5}. \end{equation}

Observations in § 3 indicate that wake-effect internal waves are primarily confined to a narrow transverse range for ${\textit{Nt}} \lt 50$ . Accordingly, the probe array is divided into an inner region ( $y \in [-0.4 \ \mathrm{m},\,0.4 \ \mathrm{m}]$ ) and an outer region ( $y \in [-1.9 \ \mathrm{m},\,-0.4 \ \mathrm{m}] \cup [0.4 \ \mathrm{m},\,1.9 \ \mathrm{m}]$ ) when computing probability density distributions. Although both regions contain contributions from lee waves and wake-effect internal waves, their dominant components differ, making region-wise analysis more effective for identifying spatially transition characteristics.

Figures 16–18 present the probability density distributions of the correlation Froude number under different towing Froude numbers. In these figures, the light grey area outlined by a solid line and the dark grey area outlined by a dashed line correspond to the inner and outer regions, respectively. In figure 16, the peaks of both distributions are closely aligned with the towing Froude number, indicating that the internal wave field is dominated by lee waves and propagates primarily with the moving body. This observation is consistent with the spatiotemporal characteristics in figures 5 and 6. Although Chomaz et al. (Reference Chomaz, Bonneton and Hopfinger1993) suggested that the near wake enters the T regime at $\textit{Fr} = 0.83$ , from the perspective of internal wave excitation, the wake at this Froude number remains lee wave dominated and is therefore identified here as a pretransition regime.

Figure 16.

Probability density distribution of correlation Froude number in the SLW region: (a) $\textit{Fr} = 0.42$ ; (b) $\textit{Fr} = 0.83$ . Light grey denotes the inner region; dark grey denotes the outer region.

Figure 17.

Probability density distribution of correlation Froude number in the T region: (a) $\textit{Fr} = 1.67$ ; (b) $\textit{Fr} = 2.50$ . Light grey denotes the inner region; dark grey denotes the outer region.

With increasing towing Froude number, the probability density distributions in both the inner and outer regions develop a bimodal structure, as shown in figure 17(a). A secondary peak appears at Froude numbers lower than the towing value, most prominently in the inner region, indicating the emergence of wake-effect internal waves propagating at speeds smaller than that of the towed body. At this stage, however, the probability density peak associated with these wake-effect internal waves remains weaker than that of the lee-wave peak in both regions, implying that lee waves still dominate the wake statistics. As the towing Froude number is further increased, in figure 17(b), the dominant peak in the inner region shifts towards $\textit{Fr}_{{iw}} \approx 0.4$ . Despite this shift, the probability density distribution in the inner region remains relatively broad, while the outer region continues to be entirely dominated by lee waves. This confirms that, at $\textit{Fr} = 2.50$ , the wake is still within the transition regime, as wake-effect internal waves have not yet become dominant over an extended spatial range. It is worth noting that, during the transition from lee waves to wake-effect internal waves, the outer region peak in figure 17(b) does not necessarily provide an exact estimate of the pure lee-wave Froude number, because partial overlap with the newly emerged wake-effect component may introduce a slight bias in the statistical peak location.

Figure 18.

Probability density distribution of correlation Froude number in the 3-D regime: (a) Fr = 3.33; (b) Fr = 4.17; (c) Fr = 5.00; (d) Fr = 5.83; (e) Fr = 6.67; (f) Fr = 8.33. Light grey denotes the inner region; dark grey denotes the outer region.

Beyond the transition regime, the probability density distribution of the correlation Froude number converges toward a stable form, as shown in figure 18. In the inner region, Froude numbers concentrated in the range $0.3$ – $0.4$ become dominant, corresponding to the semicircular and circular internal waves observed during the NEQ stage. In the outer region, correlation Froude numbers in the range 2–2.8 are also observed, corresponding to the ‘V’-shaped forced internal waves identified in the time-series waveform of the wake. This lack of clear dependence on the towing Froude number may be attributed to the gradual deceleration of the wake source during downstream propagation, such that the largest-amplitude internal waves are excited when the source speed falls within a relatively narrow range. These features mark the completion of the transition process, indicating that the wake has fully entered a regime dominated by wake-effect internal waves. Under certain probe configurations, residual lee wave signatures persist, manifesting as a secondary peak in the probability density near the towing Froude number in this regime.

Overall, the combined methodology of spatial partitioning and statistical analysis for internal wave systems yields more reliable estimates of the correlation Froude number pertaining to the dominant collapse-stage internal waves, while also enabling the identification of secondary forced waves. As a result, the transition from a lee-wave-dominated regime to a wake-effect-internal-wave-dominated regime can be quantitatively identified using this approach, with the diagnostic criteria based on the emergence of bimodality in the probability density distributions and on comparisons of the relative peak magnitudes of wake-effect internal waves and lee waves in both the inner and outer regions.

4.2. Evolution stages

Internal waves generated by turbulent wakes are often embedded in random background fluctuations, which makes their direct and unified characterisation challenging. For such signals, spectral analysis provides an effective and robust analytical framework. By neglecting nonlinear excitation effects, the propagation processes of both wake-effect internal waves and lee waves can be regarded as linear (Gilreath & Brandt Reference Gilreath and Brandt1985). Under this assumption, a two-dimensional Fourier transform can be applied to decompose the internal wave amplitude field at a given time into elementary wave components distributed over different frequency bands and propagation directions. This enables the extraction of dominant wave characteristics from a spectral perspective.

Because the amplitude of internal waves varies slowly compared with their propagation, a two-dimensional Fourier transform can be directly applied to the instantaneous spatial amplitude field to obtain the normalised wavenumber–amplitude spectrum $\hat {\eta }(k_x,k_y)$ . Owing to the conjugate symmetry of the Fourier spectrum, dimensionality reduction can be performed in polar wavenumber space, yielding the one-dimensional wavenumber–amplitude spectrum $\eta _{k_h}$ and the directional (phase)–amplitude spectrum $\eta _{\theta }$ :

(4.6) \begin{equation} \eta _{k_h}(k_h) = \int _{-\pi }^{\pi } \hat {\eta }(k_h, \theta )\, k_h \,\mathrm{d}\theta , \quad \eta _\theta (\theta ) = \int _{0}^{\infty } \hat {\eta }(k_h, \theta )\, k_h \,\mathrm{d}k_h, \end{equation}

where $ k_h = \sqrt {k_x^2 + k_y^2}$ denotes the horizontal wavenumber and $\theta =\arctan (k_x/k_y)$ represents the propagation direction. In these one-dimensional spectra, the horizontal axis corresponds to the wavenumber or propagation direction of the elementary waves, while the vertical axis represents the magnitude of their Fourier coefficients.

Although the use of a two-dimensional discrete Fourier transform introduces a certain degree of numerical dissipation, this method effectively extracts key information, such as amplitude, wavenumber and propagation direction, of the target wave systems from complex composite wave fields. This facilitates subsequent quantitative statistical analysis of the amplitude characteristics of the random wave spectrum at any given time. The significant wave height is defined from the cumulative one-third wave energy in the spectrum as

(4.7) \begin{equation} h_s = 4 \left ( \int _0^{k_{h,1/3}} \eta _{k_h}(k_h)\,\mathrm{d}k_h \right )^{1/2}, \end{equation}

where $ k_{h,1/3}$ represents the wavenumber corresponding to one-third of the total wave energy, i.e.

(4.8) \begin{equation} \int _0^{k_{h,1/3}} \eta _{k_h}(k_h)\,\mathrm{d}k_h = \frac {1}{3}\int _0^{\infty } \eta _{k_h}(k_h)\,\mathrm{d}k_h. \end{equation}

Using this approach, the significant wave height over the observation region is calculated at 0.1 s intervals, allowing a quantitative description of the wake evolution. In the lee-wave-dominated regime prior to transition, the temporal evolution of the significant wave height is shown in figure 19. During this regime, the significant wave height reaches its maximum value over the entire dimensionless time range at ${\textit{Nt}}\lt 2$ . Thereafter, the significant wave height decays continuously at an approximately constant rate. By ${\textit{Nt}}=60$ , it decreases to about half of its peak value. Subsequently, owing to dissipation and the absence of additional excitation sources, the overall amplitude of the internal wave field continues to diminish. The peak significant wave height in figure 19(b) is larger than that in figure 19(a), and the significant wave height remains consistently higher throughout the wake evolution. This observation further confirms that the sphere wake at $\textit{Fr}=0.83$ remains within the SLW regime and has not undergone transition, in agreement with the transition classification before.

Figure 19.

Temporal evolution of significant wave height in the lee-wave-dominated regime: (a) $\textit{Fr} = 0.42$ ; (b) $\textit{Fr} = 0.83$ .

Figure 20.

Temporal evolution of significant wave height after the onset of wake-effect internal waves: (a) $\textit{Fr} = 1.67$ ; (b) $\textit{Fr} = 3.33$ .

As the wake enters the T regime, the significant wave heights during the NW stage (figure 20 a) are smaller than those observed in the lee-wave-dominated case (figure 19 b), reflecting the progressive attenuation of lee-wave contributions with increasing Froude number. As collapse-stage internal waves begin to emerge, the significant wave height increases accordingly. With the diminishing influence of lee waves and the growing dominance of collapse-stage internal waves, the maximum significant wave height shifts to approximately ${\textit{Nt}} \approx 5$ . This identifies the interval $2 \lt Nt \lt 5$ as an initial accelerated collapse phase within the T regime. Overall, the significant wave height in this regime remains below that observed at $\textit{Fr} = 0.83$ , indicating that the wake energy is still insufficient to sustain high-amplitude wake-effect internal waves.

Beyond the T regime, the wake energy is expected to increase further, accompanied by increasingly complex unsteady motions. Prior to the wake reaching its maximum vertical extent, pronounced oscillations are observed, after which wake-generated internal waves are continuously radiated into the surrounding stratified fluid during the wake-collapse stage. As shown in figure 20(b), the significant wave height increases steadily over the interval $2 \lt Nt \lt 8$ , attaining a maximum near ${\textit{Nt}} \approx 8$ . This time window coincides with the large-scale vortex collapse time reported by Lin et al. (Reference Lin, Boyer and Fernando1993) and the AC stage identified by Bonnier & Eiff (Reference Bonnier and Eiff2002). This also indirectly suggests that strong initial buoyancy effects play a central role in the excitation of the high-amplitude semicircular or circular internal waves observed during this stage. Comparison with the evolution over $2 \lt Nt \lt 5$ in figure 20(a) further suggests that increasing Froude and Reynolds numbers extend the duration over which buoyancy-driven excitation remains effective. Subsequently, viscous dissipation becomes dominant and the significant wave height begins to decay. Nevertheless, a weak secondary increase is observed around ${\textit{Nt}} \approx 20$ , reflecting the continued radiation of energy in the form of internal waves during the NEQ stage. Consequently, for ${\textit{Nt}} \gt 10$ , the significant wave height no longer exhibits the monotonic decay characteristic of the lee-wave-dominated regime.

5.1. Waveform

Because collapse-stage internal waves arise from unsteady turbulent wake dynamics and coexist with multiple interacting wave systems, their signatures are not readily discernible in the raw wave fields. This complexity necessitates a dedicated filtering strategy to isolate the dominant component during the NEQ stage.

The filtering procedure consists of four steps.

(1) The raw wave height signal is convolved with a Gaussian kernel function $ K$ to achieve Gaussian filtering. This smooths the wave height field data, reducing local noise and minor fluctuations:

(5.1) \begin{equation} \eta * K = \sum _{i=-\infty }^\infty \sum _{j=-\infty }^\infty \eta (x-i, y-j) K(i, j). \end{equation}

(2) For each point within the observation domain, a set of nested circular contours is constructed by identifying radii at which the wave amplitude changes sign inside and outside a circular neighbourhood centred at that point. These contours, with radii $r_0,r_1,\ldots ,r_n$ in an arithmetic progression, represent candidate crest and trough boundaries of internal waves at a given time. A circularity criterion is then applied to reject signals that do not conform to the semicircular or circular geometry typical of collapse-stage internal waves. To accommodate deviations from perfect circularity (e.g. due to nonlinear effects and wake unsteadiness), an edge-thickness parameter is introduced to tolerate non-ideal boundaries.

(3) The wave-elevation strength per unit area in the annulus enclosed by adjacent contours is evaluated. A threshold $\varGamma$ is imposed to retain only those contour pairs that exhibit sufficient signal strength over one wave period, thereby reducing errors caused by weak signals near the wave edges:

(5.2) \begin{equation} \left | \frac {\eta _{r_1, r_2}}{\pi (r_1^2 - r_2^2)} \right | \geqslant \varGamma \quad \cup \quad \left | \frac {\eta _{r_2, r_3}}{\pi (r_2^2 - r_3^2)} \right | \geqslant \varGamma . \end{equation}

(4) The filtered field is binarised according to the sign of $\eta$ , yielding the final extraction result.

This filtering procedure is not intended to reconstruct all small-scale features in the raw wake field, but to extract the dominant coherent wavefronts associated with collapse-stage internal waves. Its reliability is assessed using both physical and visual consistency criteria. The retained structures satisfy the semicircular or circular wavefront geometry, the alternating signs of adjacent crest–trough contours, and a sufficient signal strength over approximately one wave period. The filtered results are further compared with the original elevation fields to confirm that the main wavefront locations, shapes and temporal evolution are preserved, while weak, fragmented and non-circular disturbances are removed.

Using the above procedure, internal waves at ${\textit{Nt}} = 15$ and ${\textit{Nt}} = 45$ for $\textit{Fr} = 3.33$ are extracted, as shown in figure 21. Comparison of the wave fields before and after filtering demonstrates that the proposed method effectively isolates the semicircular or circular concentric waveforms associated with collapse-stage internal waves, while preserving the overall structure of the raw data. In the contour representation, the extracted signals appear as relatively complete and well-defined periodic waveforms. In the original elevation field at ${\textit{Nt}} = 15$ , approximately three successive wavefronts of collapse-stage internal wave can be visually identified, whereas the filtering procedure retains two complete waveforms. Owing to mutual interference, these wavefronts are not perfectly regular, and their centres exhibit slight offsets from the wake centreline as a result of wake oscillations. At later times during the NEQ stage, the characteristic scale of the collapse-stage waves increases and multiple semicircular or circular wavefronts begin to overlap, as shown in figure 21(d–f). After filtering based on geometric coherence and signal-strength criteria, the resulting wave structures become significantly clearer, with characteristic radii on the order of $0.2$ – $0.3 \ \mathrm{m}$ .

Figure 21.

Extraction of semicircular or circular internal waves at $\textit{Fr} = 3.33$ : (a) ${\textit{Nt}} = 15$ , before filtering; (b) ${\textit{Nt}} = 15$ , after filtering; (c) ${\textit{Nt}} = 15$ , contour lines; (d) ${\textit{Nt}} = 45$ , before filtering; (e) ${\textit{Nt}} = 45$ , after filtering; (f) ${\textit{Nt}} = 45$ , contour lines.

It is worth noting that, in rare cases, multiple semicircular or circular internal waves are excited nearly simultaneously during the collapse stage. For example, at $\textit{Fr} = 6.67$ and ${\textit{Nt}} = 15$ , two laterally connected circular waveforms are observed above the primary wavefront located near $x = 0.6 \ \mathrm{m}$ (figure 22). These additional wavefronts, possibly arising from energy redistribution during the collapse process, typically exhibit smaller characteristic radii and shorter wavelengths.

Figure 22.

Extraction of semicircular or circular internal waves at $\textit{Fr} = 6.67$ and ${\textit{Nt}} = 15$ : (a) before filtering; (b) after filtering; (c) contour lines.

Figure 23.

Phase lines of internal waves excited by an oscillating source: (a) $ \omega _o / N = 0.3$ ; (b) $ \omega _o / N = 0.6$ .

Because semicircular or circular internal waves are observed primarily during the accelerated collapse of coherent wake vortices, their radiation may be interpreted as analogous to that produced by a decaying unsteady forcing. In particular, previous studies suggest that, during this stage, temporal modulations of the momentum and buoyancy fields may act as an effective oscillatory forcing on the surrounding stratified fluid from the perspective of internal-wave generation. This physical picture is closely related to the oscillating-source framework of Voisin (Reference Voisin1991), within which a dimensionless frequency $\xi$ is introduced to account for the Doppler shift induced by source motion:

(5.3) \begin{equation} \xi _\pm = \frac {\omega _\pm }{N} = \frac {\omega _o - U_o k_x}{N}, \end{equation}

where $\omega _o$ denotes the oscillation frequency of the source, $k_x$ is the streamwise wavenumber and $U_o$ is the source velocity. The corresponding phase-line relations can then be expressed in spherical coordinates as

(5.4) \begin{equation} x_* = R_1 \cos \theta _1 = \frac {N\omega _o \sin ^2\phi _1 - N^2\xi _\pm ^3} {(\omega _o - N\xi _\pm )^2 \sin ^2\phi _1} \, \mathrm{sgn}\!\left (\frac {\omega _o}{N} - \xi _\pm \right )\!, \end{equation}

(5.5) \begin{equation} R_{h*} = \sqrt {y_*^2 + z_*^2} = R_1 \sin \theta _1 = \frac {N^2\xi _\pm ^2 \sqrt {\sin ^2\phi _1 - \xi _\pm ^2}} {(\omega _o - N\xi _\pm )^2 \sin ^2\phi _1}, \end{equation}

where $(R_1,\theta _1,\phi _1)$ denote the spherical coordinates, with $\theta _1$ measured from the $x$ -axis and $\phi _1$ representing the azimuthal angle in the transverse plane.

Figure 23 plots the wave-crest phase lines in the $\tilde {x}oy$ plane for $z_* = 0.1,\,0.2,\,0.5, 1,\,1.5,\,2$ and $3$ . Voisin (Reference Voisin1991) noted that when $\omega _o/N\lt 1$ , internal waves generated by an oscillating source consist of a difference wave and sum waves propagating upstream and downstream, respectively; these components are indicated by the blue and black curves in the figure. For $\omega _o/N=0.3$ , the difference wave attains an appreciable amplitude only in a very limited region close to the source, and the wake is dominated by the sum wave components. The solid and dashed black curves represent transverse and dispersive sum waves, respectively. The transverse sum wave propagates downstream with a ‘ $\cup$ ’-shaped front and appears relatively far from the source (around $x_*\approx 8$ ). The dispersive sum wave forms an upstream-propagating ‘ $\cap$ ’-shaped front, which intersects the transverse branch at a caustic point; this intersection gives rise to the closed, circular features characteristic of collapse-stage internal waves. When $\omega _o/N$ increases to 0.6, the wake remains dominated by sum waves, but the dispersive branch appears only beyond a finite downstream distance and its wavefront narrows as $\omega _o/N$ increases. In contrast, the transverse branch appears earlier (around $x_*\approx 3$ ). These trends indicate that the waveform characteristics of internal waves are strongly influenced by the effective collapse velocity ( $\omega _o$ ) of the excitation source and by the stratification strength ( $N$ ).

5.2. Propagation speed and wavelength

Following the above filtering procedure, the propagation speed of collapse-stage internal wave can be evaluated from the measured travel distance and elapsed time, providing an independent validation of the probability density results of the correlation Froude number. Taking $\textit{Fr} = 6.67$ as a representative case, a semicircular internal wave is observed within the measurement window at $t = 13.6 \ \mathrm{s}$ , corresponding to ${\textit{Nt}} \approx 28$ , as shown in figure 24(a). At this instant, the wave crest is located on the outer side, and the leading edge of the wave has just reached the boundary of the observation window at $x=0$ . After 3.2 s, at $t = 16.8 \ \mathrm{s}$ ( ${\textit{Nt}} \approx 34$ ), the wave has propagated approximately half a wavelength in the $x$ direction, with the leading edge of the trough arriving at $x=0$ (figure 24 b). By $t = 20.2 \ \mathrm{s}$ ( ${\textit{Nt}} \approx 41$ ), the trough contour has fully crossed the boundary of the observation window, as shown in figure 24(c). The time interval $\Delta t$ between the arrival of the leading edge of the crest and the passage of the trailing edge of the trough at $x=0$ is recorded. Combining this interval with the measured wavelength of the collapse-stage internal wave yields a propagation speed in the negative $x$ direction of approximately $0.04 \ \mathrm{m\,s^{-1}}$ . This corresponds to a Froude number of about 0.39, which agrees closely with the dominant peak near 0.4 in the probability density distribution of the internal wave correlation Froude number in the inner wake region (see figure 18). This consistency suggests that the dominant contribution to the inner-wake probability density is associated with this semicircular or circular internal waves, with $\textit{Fr}_{{pr}}$ here representing the propagation Froude number of the dominant wave component inferred from two-probe correlation, rather than a direct measure of wake motion.

Figure 24.

Contours of collapsing internal wave propagation at $\textit{Fr} = 6.67$ : (a) $ t = 13.6 \, \mathrm{s}$ ; (b) $ t = 16.8 \, \mathrm{s}$ ; (c) $ t = 20.2 \, \mathrm{s}$ .

Further insight into the propagation characteristics of collapse-stage internal waves is provided by the time-series wave-elevation records from the first row of probes, shown in figure 25. When the leading edge of a collapse-stage internal wave first reaches this probe line, no distinct internal-wave signature is apparent in the time series (figure 25 a). At $t = 16.8 \ \mathrm{s}$ , a pronounced ‘ $\cap$ ’-shaped crest emerges in the central region ( $y \in [-0.2,\,0.2] \ \mathrm{m}$ ), as illustrated in figure 25(b). By $t = 20.2 \ \mathrm{s}$ , a complete wave period has traversed the probe line, and a corresponding ‘ $\cap$ ’-shaped trough appears beneath the crest (figure 25 c). Throughout this interval, the transverse components of the sum-wave system do not intersect the probe line. As a result, once the wave system has fully passed, the time-series record exhibits nested waveforms characteristic of dispersive wave propagation. Moreover, even after one wave period, the internal-wave signal remains intermittently excited and manifests as a two-lobed structure rather than a complete ‘ $\cap$ ’ shape (figure 25 d). This behaviour reflects the combined effects of the relatively low propagation speed of collapse-stage internal waves, their lateral spreading, and the gradual weakening of the wake source in both velocity and energy. Consequently, the spatial structure of collapse-stage internal waves may be significantly distorted when inferred solely from line-array conductivity measurements of the type employed by Robey (Reference Robey1997).

Figure 25.

Time-series waveform of collapse-stage internal waves at $x = 0$ : (a) $ t = 13.6 \, \mathrm{s}$ ; (b) $ t = 16.8 \, \mathrm{s}$ ; (c) $ t = 20.2 \, \mathrm{s}$ ; (d) $ t = 27.4 \, \mathrm{s}$ .

Based on the raw crest and trough data and the contour areas extracted from the filtered fields, the wavelength $\lambda$ of collapse-stage internal waves can be estimated more accurately by computing the mean radial separation between adjacent contours:

(5.6) \begin{equation} \lambda = \frac {2}{n} \sum _{i=1}^n (r_{i+1} - r_i). \end{equation}

The dependence of the dimensionless wavelength of collapse-stage internal waves on the Froude number in the post-transition regime is shown in figure 26. Circular and square symbols denote measurements obtained at ${\textit{Nt}}=10$ and ${\textit{Nt}}=15$ , respectively. With increasing Froude number, the wavelength increases monotonically and gradually approaches a plateau, stabilising at $\lambda \approx 5$ for $\textit{Fr} \gtrsim 7$ . For a given Froude number, the wavelengths measured at ${\textit{Nt}}=10$ and ${\textit{Nt}}=15$ are nearly identical, indicating only a weak temporal dependence during this stage of wake evolution. Also shown in figure 26 are the dimensionless vortex spacings ( $c/D$ ) measured by Lin et al. (Reference Lin, Boyer and Fernando1993) using schlieren visualisation (black symbols with error bars). The vortex spacing exhibits a similar dependence on the Froude number, increasing with $\textit{Fr}$ and approaching an asymptotic value of approximately $5.5$ for $\textit{Fr}\gt 10$ , consistent with the Strouhal scaling observed in uniform flows. Under the present experimental conditions, the wavelength of collapse-stage internal waves after transition is therefore found to be comparable to the characteristic spacing of large-scale wake vortices reported by Lin et al. (Reference Lin, Boyer and Fernando1993). Because Lin et al. (Reference Lin, Boyer and Fernando1993) provided a representative relationship between internal-wave wavelength and wake-vortex spacing, which has since been broadly supported by subsequent studies, we use their result as the principal reference for this comparison.

The ratio between the internal-wave wavelength $\lambda$ and the vortex spacing $c$ is shown in figure 27. Across the range of Froude numbers examined, this ratio remains close to unity. This systematic correspondence provides quantitative evidence of a close association between the large-scale coherent wake vortices and the collapse-stage internal waves. In particular, the comparable horizontal length scales, together with their similar variation with the Froude number, suggest that the observed waves are linked to the spatial organisation of the vortical wake rather than arising independently of it. During the AC stage, the adjustment of coherent vortex structures may therefore act as a localised source of internal-wave radiation, with the vortex spacing providing a representative horizontal length scale for the observed wavelengths. The near equality between wavelength and vortex spacing may also help account for the frequent spatial overlap observed between adjacent wavefronts, particularly as the radii of the semicircular or circular internal waves increase with time during their outward propagation. Although, in some cases, a single large-scale wake structure may be associated with multiple smaller-scale collapse-stage waves, the associated energy may be distributed among these components. As a result, the smaller amplitudes and less frequent occurrence of these secondary waves suggest that their influence on the measured wavelength statistics is limited.

Figure 26.

The wavelength of collapsing internal waves at different towed Froude number: circles, ${\textit{Nt}}=10$ ; squares, ${\textit{Nt}}=15$ ; black symbols with error bars, vortex spacing $(c/D)$ from Lin et al. (Reference Lin, Boyer and Fernando1993). The error bars indicate the uncertainty in determining $c/D$ from side-view shadowgraphs.

Figure 27.

The ratio of collapsing internal wave wavelength to vortex spacing at different towed Froude number: circles, ${\textit{Nt}}=10$ ; squares, ${\textit{Nt}}=15$ .

Based on the observed propagation speeds and wavelengths, the streamwise propagation velocities of collapse-stage internal waves generated by the towed sphere after transition are evaluated over a range of towing Froude numbers. The corresponding mean propagation Froude numbers are summarised in table 1. The resulting values, denoted by $\textit{Fr}_{{pr}}$ , lie within the interval $0.26$ – $0.46$ , in close agreement with the dominant peaks identified in the probability density distributions of the internal wave correlation Froude number in the inner wake region (figure 18). This quantitative consistency further suggests that collapse-stage internal waves constitute the dominant component in the central wake and propagate with comparable propagation Froude numbers across a broad range of towing conditions, as inferred from the wave signal.

Table 1.

Propagation Froude number of collapse-stage internal waves.

Table 1. Long description

A table with three columns and twelve rows. The columns are labeled D (cm), Fr, and Fr_pr. The table presents data on propagation Froude numbers and towing Froude numbers. Row 1: D (cm), 5.0; Fr, 1.67; Fr_pr, 0.26. Row 2: D (cm), 7.5; Fr, 2.22; Fr_pr, 0.37. Row 3: D (cm), 5.0; Fr, 2.50; Fr_pr, 0.31. Row 4: D (cm), 7.5; Fr, 2.78; Fr_pr, 0.30. Row 5: D (cm), 5.0; Fr, 3.33; Fr_pr, 0.29. Row 6: D (cm), 7.5; Fr, 3.89; Fr_pr, 0.35. Row 7: D (cm), 5.0; Fr, 4.17; Fr_pr, 0.44. Row 8: D (cm), 7.5; Fr, 4.45; Fr_pr, 0.32. Row 9: D (cm), 5.0; Fr, 5.00; Fr_pr, 0.27. Row 10: D (cm), 7.5; Fr, 5.56; Fr_pr, 0.39. Row 11: D (cm), 5.0; Fr, 5.83; Fr_pr, 0.36. Row 12: D (cm), 5.0; Fr, 6.67; Fr_pr, 0.39. Row 13: D (cm), 5.0; Fr, 8.33; Fr_pr, 0.46.

Figure 28.

Dimensionless amplitude of collapse-stage internal waves at different towed Froude numbers for ${\textit{Nt}} \in [10,15]$ : circles, maximum amplitude; solid line, linear fit; dashed line, field-averaged significant amplitude over the same interval.

5.3. Amplitude

The maximum wave amplitude associated with collapse-stage internal waves is extracted for each individual wave period enclosed by the corresponding contours. Figure 28 presents the dimensionless maximum wave amplitude, averaged over the interval ${\textit{Nt}} \in [10,15]$ following the onset of transition, as a function of the towing Froude number (circular symbols). The error bars indicate one standard deviation. For reference, the dashed curve shows the dimensionless significant wave amplitude averaged over the entire internal-wave field over the same time interval.

At low Froude numbers prior to transition, the internal wave wake is dominated by lee waves. Around $\textit{Fr} \approx 0.8$ , the significant wave amplitude attains its maximum, corresponding to the resonant peak of lee-wave excitation. For $\textit{Fr}\lt 2$ , the dimensionless wave amplitude approximately follows the decay law $\eta /D \sim Fr^{-1}$ , consistent with the low- $\textit{Fr}$ lee-wave scaling discussed by Meunier et al. (Reference Meunier2018). As the Froude number increases further into the range $2 \lt Fr \lt 3$ , the significant wave amplitude decreases and remains at a comparatively low level. In this regime, the amplitude of the newly excited semicircular or circular internal waves remains slightly smaller than the significant wave amplitude of the full field, indicating that lee waves continue to govern the wake dynamics. At higher Froude numbers, the amplitudes of collapse-stage internal waves exhibit increasingly large variability, as reflected by the enlarged error bars, owing to stronger interference from coexisting wave components within the wake. Despite this variability, the mean amplitude consistently exceeds the significant wave amplitude averaged over the entire observation region. When interpreted together with the wake-evolution diagrams, these results suggest that, within the 3-D wake regime, the semicircular and circular internal waves observed during the AC stage and associated with the evolution of large-scale coherent wake structures make an increasingly important contribution to the high-amplitude wave field, particularly in the central wake. Accordingly, the increase in the overall significant wave amplitude broadly follows the trend exhibited by the collapse-stage internal waves. These observations are qualitatively consistent with previous studies. Given the differences in amplitude definitions, sampling locations, analysis time windows and background stratification among the available experiments, the comparison is restricted to these qualitative trends.

To interpret the observed scaling of wave amplitude, an equivalent oscillating-source model is constructed based on the collapse characteristics of coherent wake structures. In this formulation, the diameter of the oscillating source is taken to be equal to the instantaneous wake width, the source translation speed is prescribed as the correlation velocity of the collapse-stage internal waves, and the oscillation period is assumed to be twice the duration of the accelerated collapse phase. With these assumptions, the collapse process is idealised as a moving oscillatory forcing acting on the surrounding stratified fluid. Following the simplified ray-tracing formulation proposed by Broutman & Rottman (Reference Broutman and Rottman2004), the internal-wave amplitude field in wavenumber space can be written as

(5.7) \begin{equation} \hat {\eta }_o(k_x,k_y,z) = 2\pi \mathrm{i} \frac {\left (\hat {f}\,\partial k_z/\partial B\right )_{z=z_s}} {\sqrt {M_s/M}} \exp \!\left (\mathrm{i}\int _{z_s}^{z} k_z(k_x,k_y,\xi )\,\mathrm{d}\xi \right ), \end{equation}

where

(5.8) \begin{equation} B = \omega ^2 k_z^2 - \big(N^2-\omega ^2 \big) \big(k_x^2+k_y^2 \big), \quad M(k_x,k_y,z)=\frac {N^2 c_{gz}}{\omega }, \end{equation}

and the subscript $s$ denotes evaluation at the source depth $z_s$ . The internal-wave dispersion relation provides the vertical wavenumber $k_z$ in terms of the horizontal wavenumber $k_h=\sqrt {k_x^2+k_y^2}$ as

(5.9) \begin{equation} k_z = \pm \frac {1}{\omega }\sqrt {k_h^2\left (N^2-\omega ^2\right )}. \end{equation}

Voisin (Reference Voisin1991) showed that, when $\omega _o/N\lt 1$ , sum waves with frequencies in the range $0\lt \xi \lt \omega _o/N$ give rise to transverse waves propagating downstream, whereas sum waves with $\omega _o/N\lt \xi \lt |\sin \phi _1|$ correspond to dispersive waves propagating upstream from the two lateral edges of the source. By applying an inverse Fourier transform, the wave-amplitude fields associated with the transverse and dispersive sum-wave components are separated for $\omega _o/N=0.3$ and $\phi _1=\pi /4$ , as shown in figure 29. In figure 29(a), the dispersive component displays an inverted ‘ $\cap$ ’-shaped structure and propagates in the negative $x$ direction with a relatively long wavelength. In figure 29(b), the transverse component propagates downstream of the source, with amplitudes and wavelengths comparable to those of the dispersive component, forming a ‘ $\cup$ ’-shaped waveform. Together, these results demonstrate that the internal-wave field generated by an equivalent oscillating source captures the essential features of the experimentally observed collapse-stage internal waves, including their semicircular or circular geometry, low characteristic frequency and large amplitude.

Figure 29.

Wave amplitude distributions of sum wave components generated by an oscillating source: (a) dispersive waves and (b) transverse waves.

Figure 30.

Spatiotemporal distributions of wave height measured by four probes in the near-boundary region: (a) $\textit{Fr}=5.0$ ; (b) $\textit{Fr}=8.33$ .

Using the relationships between dimensionless wake parameters and the Froude number reported by Meunier & Spedding (Reference Meunier and Spedding2006), the theoretical amplitude of collapse-stage internal waves predicted by the equivalent oscillating-source model is evaluated following the onset of transition. The resulting prediction is shown by the black solid curve in figure 28. As the Froude number increases, the wake kinetic energy, as well as the wake height and width, all increase, leading to corresponding increases in both the effective diameter and the forcing strength of the equivalent oscillating source. As a result, the theoretical wave amplitude grows with increasing Froude number and follows an approximately linear trend, with a slope that closely matches the experimental measurements. These findings are in line with the interpretation proposed by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) and suggest that an oscillating-source framework provides a more suitable description of internal-wave radiation associated with wake collapse than steady-forcing models.