4.1. Coordinate system and flow angles in the tail region

In this study, streamwise refers to the local flow direction, defined as the tangent to the local streamline. In the cylindrical mid-body section, the radial and wall-normal directions coincide, as do the axial and wall-parallel directions. Hence, the axial velocity ( $ U_{{b}}$ ) is equivalent to both the streamwise and wall-parallel velocity, where upper-case symbols denote mean velocities. In the tail region, however, the surface curvature causes the streamline orientation to vary with wall-normal position. To account for this, a local wall-aligned coordinate system is adopted, with the wall-parallel mean velocity denoted by $ U$ and the wall-normal mean velocity by $ W$ . For simplicity, the wall-parallel velocity is used herein throughout this analysis in place of the streamwise velocity. This choice is consistent with conventional practice in wall-bounded TBLs, where the wall-parallel (streamwise) component captures the dominant mean shear and turbulent transport relative to the surface. Figure 2(c) illustrates the coordinate system in the tail region. The body-axis system ( $ x_{{b}}, z_{{b}}$ ) defines the axial and radial directions and a local surface angle of $ \alpha _{x}$ . Superimposed is the local wall-aligned coordinate system ( $ x, z$ ), representing wall-parallel and wall-normal directions, shown in blue. A point along a local streamline, located at $ (x_{{s}}, z_{{s}})$ , is highlighted to illustrate the local flow angle $ \psi$ , and the total velocity magnitude $ U_{\textit{UW}}$ , which is decomposed into its respective in-plane velocity components, and is shown in both the body-axis ( $ U_{{b}}, W_{{b}}$ ) and wall-aligned axis ( $ U_{}, W_{}$ ) coordinates.

Figure 2.

Contour plots of the local flow angle ( $\psi$ ) with streamlines (solid red lines with arrows), showing the measurement field of view and extracted lines at $x_{{b}}/L = 0.80$ (a) and $x_{{b}}/L\,=\,0.90$ (b). Extracted lines include the radial direction (dotted light green) and wall-normal direction (solid light green). In (b), the wall-normal line is truncated by the limited domain and extended along the downstream outer boundary (light green dash-dot). The wall boundary is shown as a thick solid black line, and the boundary-layer thickness ( $\delta$ ) by a dotted black line. (c) Shows the coordinate system at $x_{{b}}/L = 0.90$ , including the local surface angle ( $\alpha _{x}$ = −17.5 $^\circ$ ), the $x_{{b}}$ – $z_{{b}}$ body axes showing axial and radial directions (black dash-dot lines) and the local wall-parallel and wall-normal axes ( $x{-}z$ , blue). A point along a local streamline at $(x_{{s}}, z_{{s}})$ is highlighted, showing local flow angle ( $\psi$ ,) the in-plane velocity magnitude $U_{\textit{UW}}$ and its velocity components in both body-axis ( $U_{{b}}, W_{{b}}$ ) and wall-aligned coordinates ( $U, W$ ).

To account for flow curvature in the tail, the measured velocity components were transformed from the body-axis to the wall-aligned frame using the local surface angle $ \alpha _{x}$ . The surface angles are $ \alpha _{x} = -8.6^\circ$ and $ -17.5^\circ$ for $ x_{{b}}/L = 0.80$ and 0.90, respectively. The flow angle $ \psi = \tan ^{-1}(W/U)$ , derived from the TR-PIV data, is shown in figure 2(a,b), overlaid with 2-D streamlines. Higher flow angles are observed near the wall, consistent with the surface inclination, which decreases gradually into the free stream. Dotted light green lines indicate the radial traverses, while solid light green lines represent wall-normal traverses. At $ x_{{b}}/L = 0.90$ , the limited axial extent of the measurement domain truncated the wall-normal line before it reached the boundary-layer edge, so it was extended along the right outer boundary (hereafter referred to as the extended wall-normal line), and is illustrated by the dash-dot light green line in figure 2(b).

For a consistent comparison between TR-PIV and hot-wire results, equivalent measurement quantities must be extracted. As the hot-wire was mounted on a vertical traverse, it restricted measurements to radial ( $ z_{{b}}$ ) traverses only. Consequently, the hot-wire profiles at $ x_{{b}}/L = 0.80$ and $ 0.90$ are taken along the radial direction and do not align with the true wall-normal direction. Furthermore, the single-normal hot-wire probe is sensitive to both the instantaneous axial ( $u_{{ib}}$ ) and radial ( $w_{{ib}}$ ) velocity components, effectively measuring the instantaneous velocity magnitude, $u_{{iuw}_{{b}}} = \sqrt {u_{{ib}}^2 + w_{{ib}}^2}$ . In the cylindrical mid-body section, where the mean radial velocity is negligible ( $W_{{b}} \approx 0$ ), the mean velocity magnitude can be approximated by the wall-parallel mean velocity, i.e. $U_{\textit{UW}} \approx U$ . However, in the tail region, a significant downward component, up to 25 % of the axial velocity, makes this approximation invalid (see also Manovski et al. Reference Manovski, Jones, Henbest, Xue, Giacobello and de Silva2020). Additionally, due to the strong APG, the hot-wire may be affected by intermittent separation or reverse flow events near the wall, such as reported in Willert et al. (Reference Willert2018). These limitations are addressed by the TR-PIV measurements, which provide independent measurements of both wall-parallel and wall-normal velocity components and capture reverse flow. A comparison between the hot-wire and TR-PIV measurements, including the different velocity components, radial versus wall-normal profiles and extended wall-normal line profiles is provided in Appendix. This confirms the validity of the velocity component separation and extended wall-normal line approach.

4.2. Mean velocity and Reynolds stress

Figure 3 shows the mean velocity profiles and the wall-parallel Reynolds stresses ( $ \langle uu \rangle = \overline {uu}$ ). For brevity, the Reynolds stress quantity is hereafter denoted simply as $ u^2$ . The data are presented in both inner ( $ U_{\tau }, \nu$ ) and outer ( $ U_{{e}}, \delta$ ) scaling, where $ U_{\tau }$ is determined from independent OFI measurements. Due to limitations in spatial resolution and interference from surface reflections, the TR-PIV measurements begin near the onset of the logarithmic layer. Using inner scaling, figure 3(a) shows that the mean velocity profiles closely follow the logarithmic law of the wall, using the classical constants $\kappa = 0.41$ and $A = 5$ . This behaviour is consistent with observations over streamlined curved surfaces, as previously reported by Meroney & Bradshaw (Reference Meroney and Bradshaw1975). As the downstream distance increases, the boundary layer thickens, the wake region becomes more pronounced and the extent of the logarithmic region decreases. The wall-parallel Reynolds stress, shown in inner scaling, exhibits a relatively flat distribution across the inner region, with similar peak values at middle stations. Turbulence levels decrease slightly through the FPG stations ( $x_{{b}}/L$ = 0.65 and 0.70) and then increase again under the APG $^+$ condition (0.80). More notable deviations include reduced turbulence intensity at $x_{{b}}/L = 0.25$ (due to the low $\textit{Re}$ ), and significantly elevated levels at $x_{{b}}/L$ = 0.90 (under APG $^{++}$ conditions), which can be attributed to the lower local friction velocity.

Figure 3.

The TR-PIV wall-parallel velocity ( $U$ ) mean profiles and wall-parallel Reynolds stress ( $u^2$ ) with (a) inner and (b) outer normalisations. Vertical dashed lines indicate the location of extracted spectra and will be discussed in § 4.4.

Figure 3. Long description

The image contains two line graphs labeled as Panel A and Panel B. Panel A: This line graph shows the inner normalized mean profiles of wall-parallel velocity and wall-parallel Reynolds stress. The x-axis is labeled z+ and ranges from 10^2 to 10^3. The y-axes are labeled U+ and u2+ with U+ ranging from 0 to 36 and u2+ ranging from 0 to 12. The graph includes a dashed line representing the log law. Multiple lines in varying shades of green and yellow represent different datasets. Panel B: This line graph shows the outer normalized mean profiles of wall-parallel velocity and wall-parallel Reynolds stress. The x-axis is labeled z/δ and ranges from 0 to 1.25. The y-axes are labeled U/Ue and u2/Ue^2 with U/Ue ranging from 0 to 1 and u2/Ue^2 ranging from 0 to 0.01. Vertical dashed lines indicate specific locations of extracted spectra. Multiple lines in varying shades of green and yellow represent different datasets, with the legend indicating different xb/L values ranging from 0.25 to 0.90.

Figure 3(b) shows the same quantities using outer scaling with a linear horizontal axis. The mean velocity profiles collapse well in the outer region. The outer-scaled wall-parallel Reynolds stress exhibits peak turbulence levels at $x_{{b}}/L = 0.45$ , with a decrease observed with downstream distance across the inner region of the layer. For the outermost portion of the layer ( $z/\delta \gt 0.80$ ), the Reynolds stress profiles collapse well for all stations beyond $x_{{b}}/L = 0.25$ .

As previously discussed, DNS results are used as a basis for comparison to evaluate the TBL characteristics along the body of revolution. A flat-plate ZPG DNS dataset (Sillero et al. Reference Sillero, Jiménez and Moser2013) is available at momentum-based Reynolds numbers ( $\textit{Re}_{\theta }$ ) comparable to those at $x_{{b}}/L = 0.45$ , 0.65 and 0.80 in the current study. The key DNS conditions are summarised in table 4, showing an increase in $\textit{Re}_{\theta }$ of approximately 1000 across the selected stations, comparable to the increase observed in our measurements. Matching $ \textit{Re}_{\theta }$ is preferred for comparing boundary layers under varying pressure gradients, as it reflects the global development of the boundary layer and is less sensitive to local flow effects (Österlund Reference Österlund2000). In contrast, $ \textit{Re}_{\tau }$ is more influenced by the near-wall dynamics and local pressure gradients. Figure 4 presents wall-parallel Reynolds stress profiles from both DNS (a) and experimental measurements using wall-normal TR-PIV and radial hot-wire (b), normalised using outer scaling with a logarithmic horizontal axis. For the hot-wire results, the radial coordinate $z_{{wb}}$ is defined relative to the wall, with the origin located at the surface. In the DNS results, increasing $\textit{Re}_{\theta }$ is associated with a progressive reduction in turbulence intensity across the inner region of the boundary layer ( $z/\delta \lt 0.5$ ). A similar trend is observed in the radial hot-wire profiles from our study (figure 4 b): turbulence levels decrease from ZPG to FPG, although the reduction is more pronounced, likely due to the influence of the FPG. At $x_{{b}}/L = 0.80$ (APG $^{+}$ ), a more substantial decrease in turbulence is evident. The wall-normal TR-PIV turbulence profiles exhibit the expected attenuation of levels due to limited spatial resolution but follow the same overall trend as the hot-wire measurements, further supporting the observed reduction in turbulence levels. There is a subtle difference in the TR-PIV results, which exhibit a shallow plateau in the profiles. In contrast, both the DNS and hot-wire data display a more steady increase in turbulence intensity as they approach the wall. This plateau is likely an artefact of limited spatial resolution, as observed in other studies, often with a more pronounced outer peak, in ZPG layers (Hutchins et al. Reference Hutchins, Nickels, Marusic and Chong2009). While an outer peak is more commonly observed in APG layers (Harun et al. Reference Harun, Monty, Mathis and Marusic2013; Sanmiguel Vila et al. Reference Sanmiguel Vila, Vinuesa, Discetti, Ianiro, Schlatter and Örlü2020), inadequate spatial resolution can also affect the shape of the turbulence profile in the outer region (Dróżdż et al. Reference Dróżdż, Örlü, Sokolenko, Schlatter, Elsner and Niegodajew2024). Nevertheless, the attenuation and shallow plateau observed in the TR-PIV results do not compromise the validity of the analysis or the conclusions of this study, since the focus is on the larger scales (lower frequencies) in the outer portion of the flow.

Figure 4.

Wall-parallel Reynolds stress profiles with outer scaling: (a) flat-plate ZPG DNS data from Sillero et al. (Reference Sillero, Jiménez and Moser2013); and (b) present experimental results from wall-normal TR-PIV and radial hot-wire measurements, based on the instantaneous velocity magnitude ( $u_{{iuw}_{{b}}}$ ). Vertical dashed lines mark the locations where spectra were extracted, as discussed in § 4.4.

Table 4.

Boundary-layer parameters of the DNS dataset extracted from Sillero et al. (Reference Sillero, Jiménez and Moser2013). The coloured line in each column denotes the colour used for that DNS case in subsequent figures.

Table 4 Long description

A table comparing boundary-layer parameters of DNS datasets. The table has four rows and three columns. The columns are labeled DNS_a, DNS_b, and DNS_c. The rows are labeled Colour used in figures, Pressure gradient, Re_τ, and Re_θ. Row 1: DNS_a, black line; DNS_b, gray line; DNS_c, light gray line. Row 2: DNS_a, ZPG; DNS_b, ZPG; DNS_c, ZPG. Row 3: DNS_a, 1040; DNS_b, 1307; DNS_c, 1612. Row 4: DNS_a, 3000; DNS_b, 4000; DNS_c, 5000.

4.3. Boundary-layer thickness

In figure 5(a), the boundary-layer thickness determined from the TR-PIV data is plotted together with the equivalent flat-plate ZPG boundary-layer thickness ( $\delta _{{fp}}$ ). The developing $\delta _{{fp}}$ was determined for equilibrium conditions using an iterative method that integrates the integral-momentum framework with a composite velocity profile, combining the law of the wall and the law of the wake, as established by Coles (Reference Coles1956). The approach uses a constant free-stream velocity ( $U_{{e}} = 29\,\mathrm{m\,s^{-1}}$ ), a constant wake parameter ( $\varPi = 0.55$ ) and a wake function $w_{{f}}(\delta ) =2$ , consistent with typical values reported in Coles (Reference Coles1956). Additional details of the iterative method used to determine the equivalent flat-plate ZPG boundary layer thickness are provided in the supplementary material (S0.1). It should be noted that $\delta _{{fp}}$ begins at the trip dot location, which marks the onset of turbulent development and is based on the axial $x_{{b}}$ development distance. Along the cylindrical mid-body section, the measured thickness agrees closely with the flat-plate solution. Approaching the tail section, the layer encounters an increasing FPG at $ x_{{b}}/L = 0.65$ and $ 0.70$ , resulting in a contraction of the boundary layer. This behaviour was previously reported from high-resolution PIV and Pitot measurements (Manovski et al. Reference Manovski, Jones, Henbest, Xue, Giacobello and de Silva2020), and is consistent with observations in boundary-layer sink flow (Jones et al. Reference Jones, Marusic and Perry2001) and in flows over aerofoils with varying pressure gradients (Vinuesa et al. Reference Vinuesa, Bobke, Örlü and Schlatter2016). Downstream of the FPG region, the flow experiences an increasing APG, leading to a rapid recovery of the boundary-layer thickness to near the flat-plate value by $ x_{{b}}/L = 0.80$ . Beyond this point, the layer continues to grow rapidly, and by $ x_{{b}}/L = 0.90$ , it deviates significantly from the flat-plate prediction due to the conical contraction of the tail and the presence of a strong APG. The persistence of a canonical-like boundary-layer evolution up to $ x_{{b}}/L = 0.80$ is a key feature that motivates the spectral analysis and associated scaling presented in the following section. In figure 5(b), a magnified view of the aft body is shown in body coordinates (with a 1:1 axis aspect ratio), illustrating the growth of $\delta$ relative to $\delta _{{fp}}$ over the curved tail surface. This view indicates that the rapid increase in $\delta$ downstream is influenced by the strong APG and is likely augmented by the curvature of the tail surface, which slopes away from the flow.

Figure 5.

(a) Boundary-layer thickness ( $\delta$ , black dots) along the model length from TR-PIV measurements, compared with the ZPG flat-plate prediction ( $\delta _{{fp}}$ , blue line). (b) Magnified aft-body view in normalised body coordinates, showing boundary-layer thickness relative to the flat-plate prediction over the curved model surface (thick black line); with a 1:1 axis aspect ratio. The legend in (b) applies to (a).

Figure 5. Long description

Panel A: A line graph shows boundary-layer thickness (black dots) along the model length from TR-PIV measurements, compared with the ZPG flat-plate prediction (blue line). The x-axis represents the normalized body length (x_b/L) ranging from 0 to 1, and the y-axis represents the boundary-layer thickness (delta) in millimeters ranging from 0 to 40. The graph includes a label for trip dots at approximately 0.1 on the x-axis. Panel B: A magnified line graph shows boundary-layer thickness relative to the flat-plate prediction over the curved model surface (thick black line) in normalized body coordinates. The x-axis represents the normalized body length (x_b/L) ranging from 0.65 to 0.95, and the y-axis represents the normalized boundary-layer thickness (z_b/L) ranging from 0.02 to 0.08. The legend in Panel B applies to both panels, indicating the flat plate TBL prediction (blue line), current data (black dots), linear interpolation (dotted line), and model surface (thick black line).

4.4. Spectral analysis

Spectral analysis is performed on the wall-parallel velocity fluctuation time series $ u(t)$ , from which the power spectral density (PSD), $ {\varPhi }_{uu}(f)$ , is computed. The PSD characterises the distribution of turbulent kinetic energy across temporal frequencies $ f$ . In this study, $ {\varPhi }_{uu}(f)$ was estimated using Welch’s modified periodogram method, employing a Hamming window with 50 % overlap to reduce spectral variance while maintaining adequate frequency resolution. Spectral estimation for the TR-PIV data employed two discrete fast Fourier transform (FFT) window lengths: $f_{{a}}$ /100 and $2^{12}$ samples. This dual-resolution approach improved convergence at both high and low frequencies. The corresponding frequency resolutions were approximately 100 Hz and $f_{{a}}$ /2 $^{12}$ Hz, respectively. Throughout this study, outer-scaled non-dimensional frequencies ( $f' = f\delta /U_{{e}}$ ) are used to characterise the spectral content. The spectral uncertainty was previously estimated by Manovski et al. (Reference Manovski, Abu Rowin, Ng, Gulotta, Giacobello, de Silva, Hutchins and Marusic2025b ) to be approximately 5 % near the energy peak, with significantly lower uncertainty across the remainder of the spectrum.

In the analysis of broadband turbulence, it is useful to express the wall-parallel velocity Reynolds stress or variance ( $u^2$ ) in terms of the energy spectrum. According to Parseval’s theorem, the variance of a signal is equal to the integral of its spectral energy density over all frequencies. Subsequently, the area under the curve of a premultiplied spectrum plotted against $ \ln (f)$ is directly proportional to the total turbulent energy. Using normalised quantities, we therefore plot $ f{\varPhi }_{uu}/U^2_{{e}}$ versus $ f'$ on a logarithmic axis, enabling interpretation of the relative contributions of different frequencies to the overall energy content.

The spectral analysis is focused on two wall-normal locations. The first, at $z\,\approx \,0.2\delta$ , represents a wall measurement closer to the wall. The $z \approx 0.2\delta$ location generally lies within the logarithmic region for all cases, where small-scale structures contribute moderately to the turbulent energy content. Closer to the wall, the influence of small scales increases further, resulting in greater attenuation due to spatial resolution limitations. The second location, at $z \approx 0.5\delta$ , lies within the outer wake region of the boundary layer for all stations, where the influence of small scales on the measurements is reduced. At greater wall-normal distances, the contribution of small-scale structures to the spectral energy is considered negligible for the present measurements (Manovski et al. Reference Manovski, Abu Rowin, Ng, Gulotta, Giacobello, de Silva, Hutchins and Marusic2025b ). Both wall heights are indicated in the mean and Reynolds stress plots in figures 3 and 4.

Figure 6 presents the premultiplied energy spectra of wall-parallel velocity fluctuations, normalised as $ f{\varPhi }_{uu}/U_{{e}}$ , plotted against the non-dimensional frequency $ f' = f\delta /U_{{e}}$ . Spectra are shown at $ z \approx 0.2\delta$ and $ z \approx 0.5\delta$ . Results include DNS of a ZPG flat-plate TBL as a reference, alongside radial hot-wire and wall-normal TR-PIV measurements at various axial locations. For the hot-wire measurements, the spectra of the velocity-magnitude fluctuations are referred to, for brevity, as $ {\varPhi }_{u_{\textit{UW}}u_{\textit{UW}}} = {\varPhi }_{u_{\textit{UW}}^2}$ . The normalised Nyquist frequency cutoffs ( $f'_{{c}} = f_{{c}}\delta /U_{{e}}$ ) for the hot-wire and TR-PIV, reported in tables 3 and 2, respectively, are generally beyond the range of the dominant energy content. The DNS spectra in figure 6(a,b) exhibit an energy peak that reduces in magnitude with increasing $ \textit{Re}_{\theta }$ at both wall-normal positions. At high frequencies, the DNS spectra collapse well across Reynolds numbers, whereas the low-frequency range is more difficult to interpret due to truncation caused by the limited streamwise extent of the dataset. The radial hot-wire results in figure 6(c,d) follow similar trends. At the first two measurement stations ( $ x_{{b}}/L = 0.45$ and 0.65), the peak energy is similar and decreases with downstream location. At these locations, both high- and low-frequency components show reasonable collapse at both wall heights. At $x_{{b}}/L$ = 0.80, the high-frequency content collapses consistently, while the reduction in peak energy is more pronounced than in the DNS. The low-frequency energy content is further diminished at the 0.90 location, which also exhibits a slightly greater peak energy than 0.80, shifted to a higher frequency, along with elevated high-frequency energy levels. Given the radial orientation and the hot-wire’s response to total velocity magnitude ( $ u_{{iuw}_{{b}}}$ ), interpretation of spectral features, particularly under strong curvature, may be misleading. Therefore, TR-PIV spectra based on wall-normal profiles are included in figure 6(e, f) for all six axial locations. The small gaps observed in each profile arise from the transition between large and small FFT window lengths used to compute the TR-PIV spectra. Although the spectra are attenuated at small scales, they still exhibit similar trends to those of the DNS and radial hot-wire. High frequencies for the first four measurement locations (0.25, 0.45, 0.65 and 0.70) exhibit a good collapse across both wall heights. The peak energy location remains at a similar frequency for the first five locations, with a progressive decrease in peak levels downstream. The lower frequencies (which are well resolved) show reasonable collapse for $0.45-0.80$ . For the first location, at $x_{{b}}/L = 0.25$ , the boundary layer is still very thin and at a relatively low Reynolds number; its low- and high-frequency content is at a marginally lower energy level with a narrower spectral distribution. The $ x_{{b}}/L = 0.80$ location, further from the wall at 0.5 $\delta$ , exhibits reduced energy levels at low frequencies, and the peak energy has decreased more substantially. As observed in the hot-wire results, the $ x_{{b}}/L = 0.90$ spectrum peak shifts toward higher frequencies, with a slightly elevated peak energy and increased high-frequency content across the spectrum.

Figure 6.

Premultiplied energy spectra of wall-parallel velocity fluctuations, normalised as $f{\varPhi }_{uu}/U^2_{{e}}$ , plotted against the non-dimensional frequency $f' = f\delta /U_{{e}}$ . Spectra are shown at two locations: $z \approx 0.2\delta$ (left) and $z \approx 0.5\delta$ (right). Results are presented for: (a,b) DNS of a ZPG flat plate from Sillero et al. (Reference Sillero, Jiménez and Moser2013), and measured axial locations for (c,d) radial hot-wire and (e, f) wall-normal TR-PIV.

The near-canonical boundary-layer thickness growth observed along most of the model (up to $ x_{{b}}/L = 0.80$ ) motivates rescaling the outer-scaled frequency axis for the spectra using the equivalent flat-plate thickness ( $ \delta _{{fp}}$ ; see table 1) for all stations. This approach enables the evaluation of whether the spectral content remains consistent with an unperturbed ZPG boundary layer or if deviations emerge due to local pressure-gradient effects and curvature. Figure 7 presents the premultiplied energy spectra of wall-parallel velocity fluctuations, normalised as $ f{\varPhi }_{uu}/U_{{e}}$ , plotted against a new non-dimensional frequency $ f'' = f\delta _{{fp}}/U_{{e}}$ . Rescaled spectra are shown for the hot-wire and TR-PIV at the same two wall-normal locations. Only the frequency axis is rescaled by $\delta _{{fp}}$ to assess spectral collapse and highlight upstream history effects, while the spectra are shown at 0.2 $\delta$ and 0.5 $\delta$ based on the local boundary-layer thickness ( $\delta$ ). This approach preserves the physical development of the layer under the influence of pressure gradients and curvature. The rescaling results in improved collapse of the high-frequency content across all axial stations, observed in both the hot-wire and TR-PIV results. Notably, over a large portion of the axisymmetric body, encompassing varying pressure gradients, the high-frequency (small-scale) energy contributions appear largely invariant with Reynolds number when rescaled using the equivalent flat-plate thickness. This behaviour aligns with DNS by Sillero et al. (Reference Sillero, Jiménez and Moser2013) for a canonical flat-plate ZPG TBL over a similar Reynolds-number range (figure 6 a,b). Such invariance is classically attributed to small-scale energy $\textit{Re}$ independence in inner(viscous)-scaled spectra, as shown in flat-plate ZPG experiments, for example, see Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009), Marusic et al. (Reference Marusic, Mathis and Hutchins2010a ) and Samie et al. (Reference Samie, Hutchins and Marusic2018) at much higher $\textit{Re}_{\tau }$ , and Ng et al. (Reference Ng, Monty, Hutchins, Chong and Marusic2011) in turbulent channel flows. Extending this view to partial outer scaling, Samie (Reference Samie2017) observed collapse of inner-scaled spectral energy at high outer-scaled wavenumbers ( $k_x\delta$ ) in flat-plate ZPG layers at $\textit{Re}_{\tau } = 6{\,}000$ – $20{\,}000$ for $z/\delta \approx 0.1$ –0.3, although the highest wavenumbers retained some Reynolds-number dependence, well above the limited $\textit{Re}$ range of the present study.

Figure 7.

Premultiplied energy spectra of wall-parallel velocity fluctuations, normalised as $f{\varPhi }_{uu}/U^2_{{e}}$ , plotted against the non-dimensional frequency $f'' = f\delta _{{fp}}/U_{{e}}$ , where $\delta _{{fp}}$ is the equivalent flat-plate boundary-layer thickness. Spectra are shown at two locations: $z \approx 0.2\delta _{}$ (left) and $z \approx 0.5\delta _{}$ (right). Results correspond to measured axial locations for: (a,b) radial hot-wire and (c,d) wall-normal TR-PIV.

Figure 7. Long description

The image contains four line graphs labeled (a), (b), (c), and (d). Each graph plots premultiplied energy spectra of wall-parallel velocity fluctuations, normalized as fΦ_uu/U_e^2, against the non-dimensional frequency fδ_fp/U_e. Panel A: A line graph shows spectra at z_wb approximately equal to 0.2 delta. The x-axis represents the non-dimensional frequency fδ_fp/U_e, and the y-axis represents the normalized energy spectra fΦ_uu/U_e^2. Different lines represent various x_b/L values: 0.45, 0.65, 0.80, and 0.90. Panel B: Another line graph shows spectra at z_wb approximately equal to 0.5 delta with the same x_b/L values. Panel C: A line graph displays spectra at z approximately equal to 0.2 delta, with x_b/L values of 0.25, 0.45, 0.65, 0.70, 0.80, and 0.90. Panel D: The final line graph shows spectra at z approximately equal to 0.5 delta with the same x_b/L values as in Panel C. Each graph illustrates how the energy spectra vary with frequency and location, highlighting differences in spectral behavior at different measurement points and methods.

At $ z_{{b}} \approx 0.2\delta _{}$ , figure 7(a), the hot-wire low-frequency contents now show a good collapse for the 0.80 and 0.90 locations; however, these results should be interpreted cautiously due to the radial orientation and $u_{{iuw}_{{b}}}$ measurement. The TR-PIV at $ z \approx 0.2\delta _{}$ in figure 7(c) shows a similar result but with a better spectral collapse across all six stations. At both wall-normal heights, the TR-PIV data show a reduction in peak energy from the nominal maximum at $ x_{{b}}/L = 0.45$ . That is, except at $ x_{{b}}/L = 0.90$ , where the low- and high-frequency ranges at $0.2\delta$ collapse reasonably well, while the peak energy remains elevated, likely due to the combined effects of APG $^{++}$ and increased curvature. At the outer location ( $\approx 0.5\delta$ ), for both the 0.80 and 0.90 tail stations, a reduction in low-frequency energy is again observed. However, unlike with $\delta$ scaling, the two profiles now almost fully overlap, with the 0.90 station showing only a slightly elevated peak. The spectra in figures 6 and 7 were also normalised using the displacement thickness, $\delta ^*$ (not shown); this alternative outer scaling produced qualitatively similar collapse, indicating that the observed trends are not sensitive to the choice between $\delta$ and $\delta ^*$ .

In this analysis, we intentionally use normalised frequencies rather than converting the time series into wavelength or wavenumber spectra, as such conversions require either Taylor’s hypothesis or an estimated frequency-dependent convection velocity (determined in § 5), both of which introduce additional assumptions. Furthermore, in the tail section, where the boundary layer evolves rapidly under increasing APG strength and curvature, projecting the time series onto spatial wavelengths is not strictly valid, particularly for long wavelengths. When the two conversion methods were applied to estimate the wall-parallel wavelength ( $\lambda _x$ ), they produced broadly similar spectral results. Rescaling with $\delta _{{fp}}$ provided a comparable degree of spectral collapse across a range of length scales, although it remained less effective than using the outer-scaled frequency $f''$ .

Taking the 1-D spectra at each wall-normal location, figure 8 presents normalised premultiplied energy spectrograms of the wall-parallel velocity fluctuations from TR-PIV for four axial positions, $ x_{{b}}/L = 0.45, 0.65, 0.80$ and $ 0.90$ , plotted as a function of wall-normal distance against normalised frequency ( $ f''$ ) and viscous time scale ( $t^+$ ). The results for $ x_{{b}}/L = 0.25$ and $ 0.70$ are omitted for conciseness, as the $ 0.25$ case (APG) closely resembles the $ 0.45$ case (ZPG), while the $ 0.70$ case (FPG $^{+}$ ) is almost identical to the $ 0.65$ case (FPG). The acquisition Nyquist frequency ( $f'_{{c}}$ ) and the theoretical spatial resolution cutoff frequency based on twice the $\textit{IW}$ size ( $f'_{{2 \times IW}}$ ) are shown and are located beyond the primary energy-containing region of the spectrum. The $\textit{IW}$ cutoff frequencies substantially underestimate the actual cutoffs, as they do not account for the full 3-D spatial filtering effects inherent in TR-PIV. The red and magenta lines indicate the $-1$ and $-3$ dB cutoff frequencies, respectively, which properly account for the full 3-D filtering. These thresholds were determined from comparison with DNS at $x_{{b}}/L = 0.45$ and with hot-wire spectra at $x_{{b}}/L = 0.65$ ; see Manovski et al. (Reference Manovski, Abu Rowin, Ng, Gulotta, Giacobello, de Silva, Hutchins and Marusic2025b ) for details. For the $x_{{b}}/L = 0.80$ and $0.90$ locations, the thresholds were obtained by referencing the hot-wire spectra against the corresponding TR-PIV results, using the radial direction and $u_{{iuw_b}}$ . These thresholds define the frequencies beyond which TR-PIV measurements experience significant spectral attenuation due to spatial resolution limits. Figure 8 shows that the overall extent and shape of the spectrograms remain similar across wall-normal heights and measurement stations. The results confirm that the energetic frequencies scale more closely with the upstream development when normalised by the equivalent flat-plate boundary-layer thickness ( $\delta _{{fp}}$ ), while the wall-normal extent of the spectra adjusts to the local boundary-layer thickness ( $\delta$ ). This behaviour is evident from the alignment of the spectral contours along both the normalised frequency and wall-normal axes across the measurement stations. In all cases, a broad spectral peak is centred at a common normalised frequency ( $f'' \approx 0.1-1$ ). The peak energy decreases at $ x_{{b}}/L = 0.65$ and $ 0.80$ compared with the maximum observed at $ 0.45$ , whereas at $ 0.90$ , a renewed increase is evident and is consistent with the trends observed in the 1-D profiles shown in figure 7. Further insight into the evolving dynamics can be gained by examining the frequency-dependent convection velocities at each measurement location, as explored in the following section.

Figure 8.

At axial locations $x_{{b}}/L = 0.45$ (a), 0.65 (b), 0.80 (c) and 0.90 (d): normalised premultiplied 1-D energy spectrograms ( $f{\varPhi }_{uu}/U_{{e}}^2$ ) of wall-parallel velocity fluctuations from TR-PIV, plotted as a function of wall-normal distance for varying normalised frequencies ( $f'' = f\delta _{{fp}}/U_{{e}}$ ) and viscous time scales ( $t^+$ ). The TR-PIV cutoff frequencies are indicated for the $-1$ dB (red) and $-3$ dB (magenta) thresholds. The acquisition Nyquist frequency ( $f'_{{c}}$ ) and the theoretical spatial resolution cutoff frequency ( $f'_{{2 \times IW}}$ ) are also shown. Vertical dashed lines indicate the wall-normal locations at which 1-D spectra are extracted. High-, intermediate- and low-frequency regions are indicated with arrows in (a) for reference and are discussed in § 4.5.

The spectral peak at $z/\delta \approx 0.2-0.3$ observed in the spectrogram at $ x_{{b}}/L = 0.45$ (figure 8 a), coinciding with the plateau in the Reynolds stress profile, is likely an artefact of limited spatial resolution, as previously discussed. In contrast, at $ x_{{b}}/L = 0.90$ , a narrower spectral peak at $z/\delta \approx 0.2$ appears further from the wall in dimensional units (9.2 mm vs. 3.2 mm), where the TR-PIV is more reliable and the reduced spectral energy is better resolved; yet in normalised wall-normal coordinates ( $ z/\delta$ ), its position is similar to that at $ x_{{b}}/L = 0.45$ . The peak at $ x_{{b}}/L = 0.90$ therefore appears more physically plausible and aligns with previous observations in APG flat-plate studies, where it is commonly identified as an outer boundary-layer peak (Harun et al. Reference Harun, Monty, Mathis and Marusic2013). At this location, the spectral features also closely resemble those reported for an axisymmetric body by Balantrapu et al. (Reference Balantrapu, Hickling, Alexander and Devenport2021), where a distinct outer peak was also observed. As discussed in the Introduction, Balantrapu et al. (Reference Balantrapu, Hickling, Alexander and Devenport2021) proposed ESL scaling to characterise turbulence structure in the tail of an axisymmetric model. In the present study, however, no velocity inflection point is observed at the two tail measurement locations, precluding direct application of ESL scaling. It is important to note that the geometry used by Balantrapu et al. (Reference Balantrapu, Hickling, Alexander and Devenport2021) included a 2 : 1 elliptical nose, followed by a constant-diameter cylindrical mid-body that transitions sharply into a steep tail cone. Although Reynolds numbers in the tail region of both studies are comparable, the sharp geometric discontinuity and the stronger APG in their set-up may have had a more pronounced effect on boundary-layer development and outer-layer turbulence structure.

4.5. Linear coherence spectra and frequency-dependent inclination angle

The linear coherence spectrum (LCS) (Bendat & Piersol Reference Bendat and Piersol2010) is used to quantify the wall-normal coupling between two velocity signals across different frequencies. For two synchronously measured velocity time series between two wall-normal distances, $u(z)$ and $u(z_{{r}})$ , the LCS is defined as

(4.1) \begin{equation} \gamma ^2 (f; z, z_{{r}}) \equiv \frac {\left | \langle \hat {u}(f; z), {\hat {u}^*}(f; z_{{r}}) \rangle \right |^2} {\langle |\hat {u}(f; z)|^2 \rangle \, \langle |\hat {u}(f; z_{{r}})|^2 \rangle } = \frac {\left | {\varPsi }_{uu}(f; z, z_{{r}}) \right |^2} {{\varPhi }_{uu}(f; z) \, {\varPhi }_{uu}(f; z_{{r}})}, \end{equation}

where $z_{{r}}$ denotes the reference wall height, here taken to be 0.2 $\delta$ or 0.5 $\delta$ , $\hat {u}(f; z)$ denotes the temporal Fourier transform of $u(z)$ , the superscript $^*$ represents the complex conjugate, $\langle \boldsymbol{\cdot }\rangle$ indicates ensemble averaging and $|\boldsymbol{\cdot }|$ denotes the modulus. Since $\gamma ^2$ corresponds to the squared magnitude of the cross-spectrum, ${\varPsi }_{uu}(f; z, z_{{r}})$ , normalised by the respective energy spectra of $u(z)$ and $u(z_{{r}})$ , it provides a normalised measure of linear coherence bounded between 0 and 1 ( $0 \leqslant \gamma ^2 \leqslant 1$ ).

Analysis of the linear coherence spectrograms (blue lines in figure 9) reveals distinct trends across the frequency spectrum, offering insight into the scale-dependent organisation of turbulent motions. The blue iso-lines denote levels of linear coherence ( $\gamma ^2$ ), where higher coherence corresponds to contours closer to the reference probe location (shown by vertical dot-dashed line), and the wall-normal extent of these contours indicates the spatial reach of correlated turbulent motions. For the ZPG case (figure 9 a,b), coherence computed with reference heights $z_{{r}} = 0.2\delta$ and $0.5\delta$ remains high across the entire boundary layer at low frequencies ( $f'' \lesssim 0.1$ ) and then progressively flattens with decreasing frequency, indicating near frequency independence. This behaviour is consistent with the observations of Baars & Marusic (Reference Baars and Marusic2020), where the LCS was presented using the wall-parallel wavelength ( $\lambda _x$ ) as the scale variable. At very low frequencies, the LCS estimates become increasingly sensitive to ensemble limitations and noise; consequently, results obtained using the larger FFT window size ( $2^{12}$ ) are not shown in the figure. At intermediate frequencies ( $0.1 \lesssim f'' \lesssim 1.0$ ), corresponding to the most energetic region, coherence across the layer decreases with increasing frequency, consistent with a hierarchical scale organisation (Baars & Marusic Reference Baars and Marusic2020). At higher frequencies ( $f'' \gtrsim 1.0$ ), a narrow neck of nearly constant coherence at both wall heights is observed, reflecting the confined spatial extent of small-scale motions. Beyond the theoretical resolution limit ( $f'' \gtrsim f'_{2\times {IW}}$ ), the TR-PIV measurements reach their effective bandwidth limit, and both spectral and coherence results in this range are considered unreliable. With downstream evolution, the LCS exhibits similar behaviour at the FPG locations (not shown). At the mild APG location ( $x_{{b}}/L = 0.80$ ; figure 9 c,d), the results remain fairly consistent with those of the ZPG case, providing further evidence that the turbulent characteristics of the boundary layer largely retain ZPG behaviour. The delineation of low-, intermediate- and high-frequency ranges is kept constant for all stations, as these correspond approximately to the same frequency organisation observed under ZPG conditions, and is therefore used consistently throughout this paper. At the strong APG location ( $x_{{b}}/L = 0.90$ ; figure 9 e, f), a narrowing of coherence is observed across all frequencies, indicating a reduced spatial extent of energetic coupling.

Figure 9.

Colour contours of frequency-dependent inclination angle ( $\chi$ ) at axial locations $x_{{b}}/L = 0.45$ (a,b), 0.80 (c,d) and 0.90 (e, f), shown for reference wall heights $z_r/\delta = 0.2$ (left) and $z_r/\delta = 0.5$ (right). Inclination angle is plotted as a function of wall-normal distance for varying normalised frequencies ( $f'' = f\delta _{{fp}}/U_{{e}}$ ) and viscous time scales ( $t^+$ ). Blue iso-lines denote the linear coherence spectrogram ( $\gamma ^2$ ) with iso-levels 0.1:0.1:0.7, and black iso-lines denote premultiplied energy spectrograms ( $f\varPhi _{uu}/U_{{e}}^2$ ), reproduced from figure 8. The TR-PIV cutoff frequencies corresponding to $-1$ dB (red) and $-3$ dB (magenta) thresholds are indicated, along with the theoretical spatial resolution limit ( $f'_{2\times {IW}}$ ). High-, intermediate- and low-frequency regions are marked with arrows in (a) for reference.

Figure 9. Long description

Panel A: A contour plot shows the frequency-dependent inclination angle at an axial location of 0.45 and a reference wall height of 0.2. The inclination angle is plotted as a function of wall-normal distance for varying normalized frequencies and viscous time scales. Blue iso-lines denote the linear coherence spectrogram with iso-levels ranging from 0.1 to 0.7, and black iso-lines denote premultiplied energy spectrograms. The TR-PIV cutoff frequencies corresponding to -3 dB and -1 dB thresholds are indicated, along with the theoretical spatial resolution limit. High-, intermediate-, and low-frequency regions are marked with arrows for reference. Panel B: A similar contour plot shows the frequency-dependent inclination angle at an axial location of 0.45 and a reference wall height of 0.5. Panel C: A contour plot shows the frequency-dependent inclination angle at an axial location of 0.80 and a reference wall height of 0.2. Panel D: A contour plot shows the frequency-dependent inclination angle at an axial location of 0.80 and a reference wall height of 0.5. Panel E: A contour plot shows the frequency-dependent inclination angle at an axial location of 0.90 and a reference wall height of 0.2. Panel F: A contour plot shows the frequency-dependent inclination angle at an axial location of 0.90 and a reference wall height of 0.5.

The linear coherence analysis provides a measure of the strength of coupling between motions at different wall-normal positions, while the associated phase information offers insight into their relative orientation. The frequency-dependent inclination angle, $\chi (f; z, z_{{r}})$ , is derived from the phase of the cross-spectrum ${\varPsi }_{uu}(f; z, z_{{r}})$ and quantifies the phase shift between velocity fluctuations as a function of frequency, thereby revealing how the inclination of these motions changes with temporal frequency. It is defined as

(4.2) \begin{equation} \chi (f; z, z_{{r}}) = \tan ^{-1} \left ( \frac {\text{Im} [{\varPsi }_{uu}(f; z, z_{{r}})]}{\text{Re} [{\varPsi }_{uu}(f; z, z_{{r}})]} \right )\!, \end{equation}

where $\text{Re} [\boldsymbol{\cdot }]$ and $\text{Im} [\boldsymbol{\cdot }]$ denote the real and imaginary parts, respectively. A positive value of $\chi$ indicates motions that are inclined downstream with increasing wall-normal distance.

The resulting frequency-dependent inclination angles are shown in figure 9 and were derived from the unwrapped phase of the cross-spectrum. The calculation was performed only where the LCS exceeded a 5 % threshold, thereby excluding incoherent regions. For the ZPG case (figure 9 a,b), the inclination angle remains low at the lowest frequencies, with near-wall values of approximately $15^{\circ }$ . This is consistent with the results of Marusic & Heuer (Reference Marusic and Heuer2007), who reported similar angles from simultaneous wall-shear-stress and velocity measurements in the log layer of TBLs across a wide range of Reynolds numbers. At intermediate and high frequencies, the inclination angle increases with frequency, approaching $90^{\circ }$ at the highest frequencies. This behaviour is consistent with reduced coherence and the near-isotropic nature of small-scale turbulence, where the phase relationship weakens and the inclination angle inferred from the cross-spectrum tends towards $90^{\circ }$ . With downstream development, the inclination angle exhibits a similar distribution for the FPG cases (not shown) and the APG $^{+}$ case (figure 9 c,d). At the APG $^{++}$ location (figure 9 e, f), a slight increase in the inclination angles is observed at low frequencies.

For all measurement stations, figure 10 shows the extracted local inclination angles at two reference wall heights, $ z_{{r}} \approx 0.2\delta$ and $ 0.5\delta$ , for an intermediate frequency (a,b) at $ f'' \approx 0.3$ , corresponding approximately to the location of peak spectral energy, and for the low frequencies (c,d) at $ f'' \approx 0.1$ . Since the frequency-dependent inclination angle is computed between wall-normal points but defined relative to the local surface, the local flow angle obtained from the TR-PIV field was added ( $+\psi$ , circles with black outline) to express the inclination relative to the body-axis coordinate system at the downstream stations ( $ x_{{b}}/L = 0.80$ and 0.90). It should be noted that at $z = z_r$ , the inclination angle is zero; the small non-zero values observed near the vertical lines arise from finite spatial resolution and scaling effects, such that the evaluation points do not coincide exactly with the reference heights ( $z_r = 0.2$ and $0.5$ ).

Figure 10.

Frequency-dependent inclination angles ( $\chi$ ) at two reference wall heights, $ z_{{r}} \approx 0.2\delta$ (left) and $ 0.5\delta$ (right), for intermediate frequencies (a,b; $ f'' \approx 0.3$ ), and for low frequencies (c,d; $ f'' \approx 0.1$ ). Circles with black outlines denote values corrected by adding the local flow angle ( $+\psi$ ) to express inclination relative to the body-axis coordinate system.

At intermediate frequencies (figure 10 a,b), inclination angles are larger ( $20-50^{\circ }$ ) but show a more pronounced downstream reduction. After correction for the local flow angle, this decrease becomes more evident, particularly at $ x_{{b}}/L = 0.90$ , where the angles fall below those at 0.80. This behaviour suggests that intermediate-scale motions are more responsive to local curvature changes, adjusting more readily in regions of gradual and steep curvature, whereas the larger scales retain their upstream organisation over longer distances.

For the low frequencies (figure 10 c,d), smaller angles appear near the wall compared with those at mid-layer heights, consistent with stronger wall-shear influence close to the surface. Across the $0.25-0.80$ stations, a consistent reduction in inclination angle with downstream development is evident. This trend reflects the combined effects of increasing Reynolds number and boundary-layer thickening, both of which are known to reduce the average inclination of coherent motions as the outer region expands and large-scale structures become more elongated and less steeply inclined (e.g. Head & Bandyopadhyay Reference Head and Bandyopadhyay1981; Hutchins & Marusic Reference Hutchins and Marusic2007). At the 0.90 ( $ \text{APG}^{++}$ ) station, the inclination angle increases significantly when measured in local wall coordinates; however, after correcting for the local flow angle, the body-axis inclination decreases and approaches values similar to those of the 0.80 case for both wall heights. Between $ x_{{b}}/L = 0.70$ and $0.80$ , the flow-angle-corrected inclination decreases by roughly $10^{\circ }$ , which may be attributed to the gradual increase in local curvature along the axis. In contrast, between $0.80$ and $0.90$ , the curvature increases more rapidly; however, the flow-angle-corrected inclination angle remains approximately constant. This suggests that the large-scale structures retain a similar global orientation despite the sharper geometric curvature, implying that the available wall-parallel distance is insufficient for the structures to respond further.

4.6. Two-point spatial correlations and inclination angle

The two-point spatial correlation of the wall-parallel velocity in the TBL over the axisymmetric body of revolution is used to examine the spatial extent and organisation of coherent structures within the TBL. It is defined as

(4.3) \begin{equation} R_{uu}(x_{{r}}, z_{{r}}; x_{{r}}+\Delta x, z_{{r}}+\Delta z) = \frac {\left \langle u(x_{{r}}, z_{{r}})\, u(x_{{r}}+\Delta x,\, z_{{r}}+\Delta z) \right \rangle } {\sqrt {\left \langle u^2(x_{{r}}, z_{{r}}) \right \rangle \left \langle u^2(x_{{r}}+\Delta x,\, z_{{r}}+\Delta z) \right \rangle }}, \end{equation}

where $(x_{{r}}, z_{{r}})$ denotes the reference location about which the correlation is computed by averaging instantaneous velocity fields over time, and $(\Delta x, \Delta z)$ are the wall-parallel and wall-normal spatial separations, respectively.

Rather than inferring spatial correlations from temporal data using Taylor’s hypothesis, this analysis employs high-resolution snapshot PIV measurements from Manovski et al. (Reference Manovski, Jones, Henbest, Xue, Giacobello and de Silva2020) under near identical test conditions, providing a well-resolved, wide-field view of the instantaneous velocity field. The experimental parameters for that dataset are described in the aforementioned reference. The snapshot PIV provides direct spatial information in the axial and radial directions, enabling the investigation of the spatial extent of coherent structures within the TBL. For the two tail stations, the local surface angle ( $\alpha _x$ ) was applied to transform the spatial correlation fields into the wall-aligned coordinate system, ensuring that the computed structure orientation and scales are expressed relative to the local surface.

Figure 11 presents two-point spatial correlations of the wall-parallel velocity fluctuations, $R_{uu}$ , obtained from snapshot PIV (Manovski et al. Reference Manovski, Jones, Henbest, Xue, Giacobello and de Silva2020). The plots compare reference heights of $z_r/\delta = 0.2$ (left) and $0.5$ (right) across axial reference locations $x_{{r}}/L$ = $x_{{b}}/L = 0.45,\ 0.65,\ 0.80$ and $0.90$ . From $x_{{b}}/L = 0.45$ to $0.65$ , the correlation lobes exhibit a slight increase in size, whereas toward the tail ( $x_{{b}}/L = 0.80$ and $0.90$ ) their extent decreases notably. To quantify their geometry, the correlation contours are analysed using a threshold-based principal component procedure. The correlation contours are not strictly elliptical, particularly near the wall. A threshold of $R_{uu}=0.3$ is therefore used to isolate the dominant coherent structure, for which the contour is approximately elliptical. The major and minor axes are determined using a weighted principal component analysis of the region enclosed by the threshold contour. The eigenvector associated with the largest eigenvalue defines the major-axis direction, from which the mean inclination angle, $\chi _{{m}}$ , is calculated relative to the wall-parallel direction. The projected extents of the contour along the principal directions define the aspect ratio, $A\mkern-7mu R$ , which quantifies the degree of wall-parallel elongation of these structures. Table 5 summarises the extracted values of $\chi _{{m}}$ with an estimated uncertainty of $\pm 2^{\circ }$ and $A\mkern-7mu R$ for all stations and for both reference wall heights using a threshold of $R_{uu}$ = 0.3. Over the parallel mid-body section, the mean inclination angle remains similar ( $11-13^{\circ }$ ) with axial distance for both heights. In the tail, there is a notable increase in the inclination angles, which is consistent with the increasing axial curvature and the flow reorienting with the local streamline direction. At the near-wall reference ( $z_{{r}} \approx 0.2\delta$ ), the mean inclination angles are smaller and aspect ratios larger than at the mid-layer reference ( $z_{{r}} \approx 0.5\delta$ ), indicating stronger wall-parallel elongation of coherent motions in regions of higher wall shear. With increasing axial distance, the inclination angles increase and the aspect ratio decreases, suggesting that coherent motions become more inclined and more compact. The final two rows in figure 11 present the same two-point correlations expressed in the body-axis coordinate system, allowing the structure orientation to be assessed relative to the global axial direction rather than the local wall-aligned frame. In this global axis system it is noted that structural inclination remains relatively unaltered compared with upstream conditions. In table 5, the corresponding body-axis values are also listed. In body-axis coordinates, the mean inclination angle remains largely consistent, with only a slight reduction near the wall. The aspect ratio in the body-axis frame indicates that, at the higher reference height ( $z_r \approx 0.5\delta$ ), the structures become progressively more elongated along the model length, whereas at the near-wall reference height ( $z_r \approx 0.2\delta$ ) the aspect ratio remains approximately unchanged with increasing distance along the model. Together, these trends indicate axial persistence of the flow structures as the wall curves away. These aspects are explored further in § 6, along with body-aligned spatial correlation lobes shown at an $R_{uu}=0.3$ threshold.

Figure 11.

Two-point spatial correlations of the wall-parallel velocity fluctuations, $R_{uu}$ , shown as a function of wall-parallel and wall-normal separation normalised by the local boundary-layer thickness, $\delta$ . Results are obtained from snapshot high-resolution PIV (Manovski et al. Reference Manovski, Jones, Henbest, Xue, Giacobello and de Silva2020). Columns compare reference heights of $z_{{r}}/\delta = 0.2$ (left) and $0.5$ (right) with the each correlation centred at axial stations $x_{{b}}/L = 0.45$ (a,b), $0.65$ (c,d), $0.80$ (e, f) and $0.90$ (g,h). The corresponding correlations expressed in the body-axis coordinate system are shown for $x_{{b}}/L$ = 0.80 (i, j) and $0.90$ (k,l).

Figure 11. Long description

Panel A: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at an axial location 0.45 a reference height of 0.2δ, with wall-parallel and wall-normal separation normalized by the local boundary-layer thickness. The horizontal axis represents the wall-parallel separation, and the vertical axis represents the wall-normal separation. Panel B: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at a reference height of 0.5δ, with the same axes as Panel A. Panel C: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at an axial location 0.65 and a reference height of 0.2δ, with the same axes as Panel A. Panel D: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at a reference height of 0.5δ, with the same axes as Panel A. Panel E: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at an axial location 0.80 and a reference height of 0.2δ, with the same axes as Panel A. Panel F: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at a reference height of 0.5δ, with the same axes as Panel A. Panel G: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at an axial location 0.90 and a reference height of 0.2δ, with the same axes as Panel A. Panel H: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at a reference height of 0.5δ, with the same axes as Panel A. Panel I: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at an axial location 0.80 and a reference height of 0.2δ in the body-axis coordinate system, with the horizontal axis representing the wall-parallel separation and the vertical axis representing the wall-normal separation. Panel J: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at a reference height of 0.5δ in the body-axis coordinate system, with the same axes as Panel I. Panel K: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at an axial location 0.90 and a reference height of 0.2δ in the body-axis coordinate system, with the same axes as Panel I. Panel L: A contour plot showing spatial correlations of wall-parallel velocity fluctuations at a reference height of 0.5δ in the body-axis coordinate system, with the same axes as Panel I.

Table 5.

Mean inclination angle ( $\chi _{{m}}$ ) and aspect ratio ( $A\mkern-7mu R$ ) extracted from two-point spatial correlations using a threshold of $R_{uu}$ = 0.3 at two reference wall heights, $z_r/\delta = 0.5$ and $0.2$ , for different axial stations $x_{{b}}/L$ . The corresponding body-axis values ( $\chi _{{m{b}}}$ and $A\mkern-7mu R_{{b}}$ ) are also included. The estimated uncertainty in the mean inclination angle is $\pm 2^{\circ }$ .

Table 5. Long description

The table presents data on mean inclination angles and aspect ratios extracted from two-point spatial correlations using a threshold of 0.3 at two reference wall heights, 0.5 delta and 0.2 delta, for different axial stations. The table has 10 rows and 6 columns. The columns are labeled as follows: xb/L, chim (degrees), chim b (degrees), AR, ARb. The rows are grouped into two sections based on the reference wall heights. The first section corresponds to the reference height of approximately 0.5 delta, and the second section corresponds to the reference height of approximately 0.2 delta. Each section contains values for different axial stations. The values in the table include mean inclination angles (chim and chim b) in degrees, and aspect ratios (AR and ARb). The table also includes corresponding body-axis values (chim b and ARb). The data shows variations in mean inclination angles and aspect ratios at different axial stations and reference wall heights.