2.1. Governing equations

Large-eddy simulations (LES) are incapable of resolving vortices of all scales in the flow field. By applying a filter to the incompressible Navier–Stokes equations, the governing equations for LES can be obtained as

(2.1) \begin{align} \frac {\partial \tilde {u}_i}{\partial t}+\frac {\partial \tilde {u}_i\tilde {u}_j}{\partial x_j} =-\frac 1\rho \frac {\partial \tilde {p}}{\partial x_i}+\nu \frac {\partial ^2\tilde {u}_i}{\partial x_j\partial x_j}-\frac {\partial \tau _{\it{{ij}}}^\it{sgs}}{\partial x_j},\\[-29pt] \nonumber \end{align}

(2.2) \begin{align} \frac {\partial \tilde {u}_i}{\partial x_i} =0, \\[2pt] \nonumber \end{align}

where $\tilde {u}_i$ ( $i = 1, 2, 3$ ) is the filtered velocity component in the $x_i$ direction, $\tilde {p}$ is the filtered pressure, $\nu$ is the kinematic viscosity of the fluid, and $\tau _{\it{ij}}^{\it{sgs}}$ is the subgrid-scale (SGS) stress term that is given by

(2.3) \begin{equation} \tau _{\mathit{ij}}^\mathit{sgs}=2\nu _{\mathit{sgs}}\tilde {S}_{\mathit{ij}}+\tfrac 13\tau _{\mathit{kk}}^\mathit{sgs}\delta _{\mathit{ij}}, \end{equation}

with $\tilde {S}_{\mathit{ij}}$ the resolved strain rate tensor, $\delta _{\mathit{ij}}$ the Kronecker delta, and $\nu _{\mathit{sgs}}$ the SGS eddy viscosity. In this study, the wall-adapting local eddy viscosity (WALE) model (Nicoud & Ducros Reference Nicoud and Ducros1999) is applied to compute the eddy viscosity

(2.4) \begin{equation} \nu _{\mathit{sgs}}=\left(C_w\unicode{x1D6E5}\right )^2 \! \frac {\left(S_{\mathit{ij}}^dS_{\mathit{ij}}^d\right)^{3/2}}{\left(\tilde {S}_{\mathit{ij}}\tilde {S}_{\mathit{ij}}\right)^{5/2}+\left(S_{\mathit{ij}}^dS_{\mathit{{ij}}}^d\right)^{5/4}}, \end{equation}

where $C_w=0.325$ is the WALE coefficient, $\unicode{x1D6E5}$ is the cube root of the local cell volume, and $S_{\mathit{ij}}^d$ is the traceless symmetric part of the square of the velocity gradient tensor.

The present simulation is performed using the open-source CFD platform OpenFOAM. The spatial discretisation employs a second-order central differencing scheme for both convective and diffusive terms to ensure low numerical dissipation, which is critical for resolving turbulence structures in WRLES. The temporal term is discretised using a second-order implicit backward scheme. The PISO algorithm (Issa Reference Issa1986) is employed to handle the pressure–velocity coupling, with three pressure corrections and one extra loop for non-orthogonality at each time step. The momentum equation is solved using the bi-conjugate gradient stabilised (PBiCGStab) solver with the diagonal-based incomplete LU (DILU) preconditioner. For the pressure Poisson equation, we use the conjugate gradient solver with the diagonal-based incomplete Cholesky preconditioner. The tolerance is set to $10^{-7}$ . The present numerical algorithm has been validated for a variety of problems over a range of Reynolds numbers in our previous study (Fan et al. Reference Fan, Liu, Zhao and Wan2024a ; He et al. Reference He, Zhou, Zhao and Wan2024b ; He, Zhao & Wan Reference He, Zhao and Wan2025), demonstrating good agreement with experimental and direct numerical simulations (DNS) data in terms of mean velocity profiles, Reynolds stresses and wall pressure statistics.

2.2. Computational domain and mesh

The computational domain and mesh are described as follows. Figure 2 shows the computational domain adopted in the present study, which is larger than domains used in Posa & Balaras (Reference Posa and Balaras2016) and Kumar & Mahesh (Reference Kumar and Mahesh2018). Taking the stagnation point at the bow of the hull as the origin, the $x$ -axis is aligned with incoming flow direction. The computational domain extends over length $40D$ , and width and height $20D$ , where $D$ represents the diameter at the parallel mid-body of the hull. The hull length is $L = 8.6D$ , and the domain spans $-10D \lt x \lt 30D$ , $-10D \lt y \lt 10D$ and $-10D \lt z \lt 10D$ . An unstructured mesh is used in this study, with graded refinement applied near the hull. Our previous study (He et al. Reference He, Zhou, Zhao and Wan2024b ) on turbulent channel flow demonstrated that for WRLES, in addition to ensuring that the first wall-normal grid layer satisfies $\Delta y_{w}^{+}\lt 1$ , the spanwise (or azimuthal) grid resolution should also be kept below 15 to avoid log-law mismatch in the near-wall velocity profile. This constraint arises because near-wall velocity streaks, which contribute to wall friction, require relatively fine spanwise grid resolution. Furthermore, the streamwise grid resolution $\Delta x^{+}$ plays a crucial role in accurately capturing high-frequency wall pressure fluctuations, similar to the effect of sensor size on high-frequency measurements in experimental studies. As the streamwise grid is refined, the high-frequency content of the wall pressure spectrum increases and eventually converges to DNS results. In present study, given the primary focus on wall pressure fluctuations, the wake region is not further refined. The near-wall streamwise grid resolution at the mid-body is $\Delta x^{+}=24$ , which is finer than resolutions used in previous studies (Kumar & Mahesh Reference Kumar and Mahesh2018; Morse & Mahesh Reference Morse and Mahesh2021). The azimuthal grid resolution is $r^+\,\Delta \theta =12$ , and the height of the first grid layer is $\Delta y_{w}^{+}=0.92$ . The total number of grids is approximately 540 million. The time step used in the simulation is $\Delta t^*=\Delta t\,U_\infty /L=8.69\times 10^{-5}$ , ensuring that the Courant–Friedrichs–Lewy number is satisfied with $\it{CFL} \lt 1$ . The unsteady simulations are run for six flow-past times, with the first 2.5 flow-past times used for flow development, and the remaining 3.5 flow-past times used for statistical analysis, which is longer than in the WRLES study by Kumar & Mahesh (Reference Kumar and Mahesh2018) (two flow-past times for flow development, and two flow-past times used for data sampling). In Appendix A, we further examine the effect of the sampling time period to ensure that the flow is fully developed and the sampling duration is sufficient.

Figure 2.

Computational domain.

To improve numerical stability and accelerate convergence in the WRLES, a steady-state Reynolds-averaged Navier–Stokes (RANS) simulation is first performed on the same computational mesh. The resulting steady-state pressure and velocity fields from the precursor RANS solution are then used as the initial conditions for the WRLES computation. For the boundary conditions, a uniform inflow is prescribed at the inlet to achieve a hull-length Reynolds number $\it{Re}_{\mathit{{L}}}=1.1\times 10^{6}$ . At the outlet, a zero-gradient condition is imposed on the velocity field, while the pressure is set to zero. The SUBOFF model hull is assigned a no-slip velocity condition and a zero-gradient pressure condition. The remaining domain boundaries are treated as symmetry planes. At this Reynolds number, the flow remains predominantly laminar over a significant portion of the hull. To prompt rapid transition to turbulence, a numerical trip is introduced at the bow ( $x/D=0.75$ ) by applying a steady wall-normal velocity. The trip location is the same as that used in the experiments by Jiménez et al. (Reference Jiménez, Hultmark and Smits2010b ) and the simulations by Kumar & Mahesh (Reference Kumar and Mahesh2018) and Morse & Mahesh (Reference Morse and Mahesh2021). Following the perturbation, the flow quickly transitions from laminar to turbulent.

2.3. Validation of present WRLES

To validate the accuracy of the present numerical method, figure 3 shows the mean pressure and skin friction coefficient on the hull surface, which are defined as

(2.5) \begin{equation} C_p=\frac {p-p_\infty }{0.5\rho U_\infty ^2},\quad C_f=\frac {\tau _w}{0.5\rho U_\infty ^2}, \end{equation}

where $p_{\infty }$ represents the reference pressure, and $\tau _w$ denotes the wall shear stress. Since Jiménez’s experiment (Jiménez et al. Reference Jiménez, Hultmark and Smits2010b ) for the DARPA SUBOFF model did not provide $C_f$ data, and the reference pressure selection also differed from the experiment by Huang et al. (Reference Huang, Liu, Groves, Forlini, Blanton and Gowing1992), the results here are only compared with Huang et al. (Reference Huang, Liu, Groves, Forlini, Blanton and Gowing1992) at Reynolds number $\it{Re}_{\mathit{{L}}}=1.2\times 10^{7}$ . The application of numerical tripping introduces minor fluctuations in the WRLES results within the bow region ( $0.08\leqslant x/L\leqslant 0.1$ ). The pressure coefficient $C_p$ is known to be relatively insensitive to Reynolds number. However, the friction coefficient $C_f$ is significantly affected by it, and the experimental $C_f$ is scaled to the simulation Reynolds number using the scaling law $C_f\sim \it{Re}^{-0.2}$ . This scaling is strictly valid only for TBLs under ZPG conditions. The predicted $C_f$ in the parallel mid-body region shows good agreement with the scaled experimental data, but deviations appear in the stern region where strong APGs are present due to the limitations of the scaling law under non-equilibrium conditions.

Figure 3.

Mean pressure and skin friction coefficients on the hull. The experimental results (Huang et al. Reference Huang, Liu, Groves, Forlini, Blanton and Gowing1992) at $\it{Re}_{\mathit{{L}}}=1.2\times 10^{7}$ are shown for comparison.

Figure 4 presents the mean velocity profile, r.m.s. of velocity fluctuations ( $u_{\mathit{{x,rms}}}^{+}$ , $u_{\mathit{{r,rms}}}^{+}$ , $u_{\mathit{\theta ,rms}}^{+}$ ) and Reynolds stress $\langle u_x u_r\rangle ^+$ profile at the mid-body ( $x/L=0.42$ ). The WRLES results (Kumar & Mahesh Reference Kumar and Mahesh2018) at the same $\it{Re}_L$ , and DNS results (Jiménez et al. Reference Jiménez, Hoyas, Simens and Mizuno2010a ) for a flat plate at a momentum thickness Reynolds number $\it{Re}_{\theta }=1551$ , are shown for comparison. As mentioned in the Introduction, an axisymmetric TBL is characterised by slightly higher wall friction compared to a flat-plate TBL at similar Reynolds numbers. The outer edge velocity of the boundary layer is smaller, and this behaviour is consistent with observations reported in previous studies (Kumar & Mahesh Reference Kumar and Mahesh2018; Morse & Mahesh Reference Morse and Mahesh2021). In terms of the r.m.s. of velocity fluctuations and Reynolds stress, the WRLES predictions for the near-wall inner layer agree well with both the WRLES results from Kumar & Mahesh (Reference Kumar and Mahesh2018) and the DNS results for a flat plate from Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010a ). The axisymmetric BOR exhibits a more rapid decay in wall-normal turbulence intensity compared to the flat plate. In the region where $y^{+}\gt 100$ , the r.m.s. values of streamwise velocity fluctuations are lower than the corresponding flat-plate results, which is also consistent with the findings of Kumar & Mahesh (Reference Kumar and Mahesh2018).

Figure 4.

Mean flow characteristics of the TBL at the mid-body of the hull ( $x/L=0.42$ ): (a) mean streamwise velocity profile; (b) r.m.s. of velocity fluctuations ( $u_{\mathit{{x,rms}}}^{+}$ , $u_{\mathit{{r,rms}}}^{+}$ , $u_{\mathit{\theta ,rms}}^{+}$ ) and Reynolds stress $\langle u_xu_r\rangle ^+$ . The red line represents the present WRLES results, while the blue line corresponds to the WRLES results from Kumar & Mahesh (Reference Kumar and Mahesh2018) at the same $\it{Re}_L$ . The scatter points represent DNS results for a flat plate at $\it{Re}_\theta =1551$ from Jiménez et al. (Reference Jiménez, Hoyas, Simens and Mizuno2010a ).

Figure 5 shows the profiles of pressure coefficients ( $C_p$ ), streamwise velocity ( $U_x$ ), radial velocity ( $U_r$ ) and r.m.s. of velocity fluctuations ( $u_{\mathit{{x,rms}}}$ , $u_{\mathit{{r,rms}}}$ , $u_{\mathit{\theta ,rms}}$ ) at the stern of the hull ( $x/L=0.904$ and 0.978). The WRLES results from Kumar & Mahesh (Reference Kumar and Mahesh2018) at the same $\it{Re}_L$ , and the experimental results from Huang et al. (Reference Huang, Liu, Groves, Forlini, Blanton and Gowing1992) at $\it{Re}_{\mathit{{L}}}=1.2\times 10^{7}$ , are shown for comparison. Here, $r_0$ is the local radius of the hull at the stern, and $R$ is the radius of the mid-body. The velocity profiles in figures 5(c,d) reveal the progressive thickening of TBL in the stern region. Due to flow separation, the peaks of velocity fluctuation r.m.s. in figure 5(f) shift away from the wall. From figures 5(e,f), it can be observed that the predicted r.m.s. of velocity fluctuations within the boundary layer agree well with the data of Kumar & Mahesh (Reference Kumar and Mahesh2018). However, the predicted results outside the boundary layer ( $(\mathit{r}-\mathit{r}_0)/\mathit{R}\gt 0.75$ ) are slightly lower than those of Kumar & Mahesh (Reference Kumar and Mahesh2018). Given that the present study focuses primarily on the flow dynamics within the boundary layer, we did not apply additional grid refinement in the wake region. While the WRLES study by Kumar & Mahesh (Reference Kumar and Mahesh2018) specially targeted the wake. The grid resolution at the stern away from the wall is lower than that in Kumar & Mahesh (Reference Kumar and Mahesh2018), which can result in higher dissipation and lower predicted r.m.s. of velocity fluctuations. Overall, the WRLES predictions in this study agree well with existing WRLES datasets (Kumar & Mahesh Reference Kumar and Mahesh2018) and experimental measurements (Huang et al. Reference Huang, Liu, Groves, Forlini, Blanton and Gowing1992), validating the accuracy of the present numerical simulation.

Figure 5.

Mean flow characteristics of the TBL at the stern of the hull, for (a,c,e) $x/L=0.904$ , (b,d, f) $x/L=0.978$ . (a,b) Mean pressure coefficients profile; (c,d) mean streamwise ( $U_x$ ) and radial ( $U_r$ ) velocity profile; (e, f) r.m.s. of velocity fluctuations ( $u_{\mathit{{x,rms}}}$ , $u_{\mathit{{r,rms}}}$ , $u_{\mathit{\theta ,rms}}$ ). The WRLES results from Kumar & Mahesh (Reference Kumar and Mahesh2018) at the same $\it{Re}_L$ , and the experimental results from Huang et al. (Reference Huang, Liu, Groves, Forlini, Blanton and Gowing1992) at $\it{Re}_{\mathit{{L}}}=1.2\times 10^{7}$ , are shown for comparison.

3.1. Overview of the flow fields

Figure 6 shows the instantaneous three-dimensional vortex structures using the Liutex vortex identification method (Liu et al. Reference Liu, Gao, Tian and Dong2018). Iso-surfaces are extracted at threshold $\tilde {\Omega }_R=0.52$ , with the non-dimensional streamwise velocity used for colouring. The vortex structures illustrate that numerical tripping initiates a rapid transition from laminar to turbulent flow, with near-wall turbulence exhibiting multi-scale characteristics. Along the flow direction, the TBL continuously develops, and the vortex structures attached to the hull grow rapidly, leading to an increase in boundary layer thickness. At the stern, APG induces flow separation, generating a wake region densely populated by vortical structures.

Figure 6.

Instantaneous near-wall vortical structures visualised by the Liutex vortex identification method ( $\tilde {\Omega }_R=0.52$ ). The vortical structures are coloured by instantaneous streamwise velocity normalised by the mean inflow velocity $U_{\infty }$ .

Figure 7 presents the instantaneous flow fields around the SUBOFF model, showcasing both the instantaneous streamwise velocity and pressure distributions. The velocity field highlights concentration of fluctuations within the near-wall boundary layer. In the instantaneous pressure field shown in figure 7(b), the stagnation point at the bow corresponds to a high-pressure region, while the stern region is characterised by an APG. The mid-body exhibits a near ZPG, with pronounced pressure fluctuations observed throughout.

Figure 7.

Instantaneous flow field of (a) streamwise velocity and (b) pressure.

3.2. Boundary layer development

Building upon the analysis of instantaneous flow fields, we further investigate the development of the TBL. Figure 8 presents the spatial evolution of TBL parameters along the hull. The complex curvature variations in the stern region render traditional methods, such as the $99\, \%$ freestream velocity criterion, inadequate for reliably quantifying boundary layer thickness under APG. Recent studies have introduced advanced approaches for boundary layer thickness estimation in flows with curved geometries, such as vorticity-based (Spalart & Watmuff Reference Spalart and Watmuff1993; Coleman, Rumsey & Spalart Reference Coleman, Rumsey and Spalart2018) and total-pressure-based (Patel, Nakayama & Damian Reference Patel, Nakayama and Damian1974; Griffin, Fu & Moin Reference Griffin, Fu and Moin2021; He et al. Reference He, Zhou, Zhao and Wan2024a ) methodologies. This work employs the total-pressure-based method, defining the boundary layer thickness as the location where $99\, \%$ of the total pressure $C_{\mathit{{p,total}}}$ is recovered. Figure 8(a) depicts the distribution of key boundary layer integral parameters, thickness $\delta$ , displacement thickness $\delta ^{*}$ , and momentum thickness $\theta$ along the hull. As the boundary layer evolves downstream, all integral thickness parameters increase monotonically, with accelerated growth under the APG in the stern region. A slight dip is observed at $x/L\approx 0.77$ , primarily due to the local FPG. Figure 8(b) presents the distribution of local Reynolds numbers, include the friction Reynolds number $\it{Re}_\tau =\delta u_\tau /\nu$ and momentum thickness Reynolds number $\it{Re}_\theta =\theta U_\infty /\nu$ . In addition, the ratios of boundary layer thicknesses $\delta /\delta ^{*}$ , $\delta /\theta$ and shape factor $H=\delta ^{*}/\theta$ are shown in figures 8(c,d), respectively. All the parameters are highly sensitive to FPG and APG. At the parallel mid-body, $H$ is approximately 1.4, whereas at the stern, $H$ rapidly increases from 1.3 under FPG conditions to nearly 1.7 under APG conditions.

Figure 8.

Evolution of TBL parameters along the hull: (a) boundary layer thickness $\delta$ , displacement thickness $\delta ^{*}$ and momentum thickness $\theta$ ; (b) friction Reynolds number $\it{Re}_\tau$ and momentum thickness Reynolds number $\it{Re}_\theta$ ; (c) thickness ratios $\delta /\delta ^{*}$ , $\delta /\theta$ , and (d) shape factor $H$ .

Figure 9 shows the evolution of Clauser pressure gradient parameter $\beta$ (Clauser Reference Clauser1954) and acceleration parameter $K$ (Kline et al. Reference Kline, Reynolds, Schraub and Runstadler1967) along the hull, defined as

(3.1) \begin{equation} \beta =\frac {\delta ^*}{\tau _w}\frac {{\rm d}P}{{\rm d}x},\quad K=\frac {\nu }{U_e^2}\frac {{\rm d}U_e}{{\rm d}x}, \end{equation}

where the $\beta$ parameter can be defined using either $P_e$ or $P_w$ , corresponding to the local static pressure at the edge of the boundary layer and on the wall, respectively. Figure 9(a) reveals a markedly stronger near-wall pressure gradient compared to that at the boundary layer edge, consistent with the results reported by Morse & Mahesh (Reference Morse and Mahesh2021). The pressure gradient around the SUBOFF model exhibits a complex spatial evolution, with alternating regions of APG and FPG along the hull. Beyond $x/L\gt 0.3$ , the influence of the numerical trip becomes negligible, and a near ZPG is observed over most of the parallel mid-body. Notably, a small FPG region is observed before the APG zone at the stern, approximately within $0.71 \lt x/L\lt 0.78$ . In this region, the flow accelerates, with the acceleration parameter $K\gt 0$ peaking at $x/L\approx 0.75$ . Due to the FPG, the local boundary layer thickness decreases, as shown in figure 8(a). Subsequently, for $0.79 \lt x/L\lt 0.94$ , the flow experiences a strong APG due to the contraction of the body shape. Beyond $x/L\approx 0.94$ , the streamwise concave curvature induces another FPG region.

Figure 9.

Evolution of (a) Clauser pressure gradient parameter $\beta$ and (b) acceleration parameter $K$ along the hull.

Figure 10 shows the evolution of transverse curvature parameters $\delta /r_0$ and $r_0^{+}$ along the hull. As noted in the Introduction, these parameters govern transverse curvature effects. For large local curvature radius $r_0$ , transverse curvature minimally impacts boundary layer dynamics, primarily influencing the outer layer (Piquet & Patel Reference Piquet and Patel1999). For small $r_0$ , particularly when $\delta /r_0\gt 1$ , transverse curvature alters TBL behaviour, elevating wall friction (Alam Reference Alam2020) and attenuating low-frequency wall pressure fluctuations (Neves et al. Reference Neves, Parviz and Moser1994). As the flow develops downstream, $\delta /r_0$ continuously increases. In the FPG region, rising local wall friction drives $r_0^{+}$ upwards. Conversely, in the APG region, $r_0^{+}$ rapidly decreases.

Figure 10.

Evolution of transverse curvature parameters ( $\delta /r_0$ and $r_0^{+}$ ) along the hull.

4.1. Evolution of wall pressure fluctuations

Figure 12 illustrates the streamwise evolution of the wall pressure r.m.s. normalised by reference dynamic pressure $q_{\infty }$ , local dynamic pressure $q_e$ , local wall shear stress $\tau _w$ , and local maximum magnitude of Reynolds shear stress $\tau _{max }$ . Here, $q_{\infty }=0.5\rho U_{\infty }^{2}$ and $q_e=0.5\rho U_e^{2}$ , and $\tau _{max }$ is defined as

(4.1) \begin{equation} \tau _{max }=-\rho \! \left \langle u_su_n\right \rangle _{max } \!, \end{equation}

where $u_s$ and $u_n$ are obtained from axial velocity $u_x$ and radial velocity $u_r$ through the coordinate transformation

(4.2) \begin{equation} u_s=u_x\cos \alpha -u_r\sin \alpha ,\quad u_n=u_x\sin \alpha +u_r\cos \alpha , \end{equation}

in which $s$ and $n$ represent the tangential and normal directions relative to the wall, respectively, and $\alpha$ is the angle between the wall tangential direction and the axial direction of SUBOFF.

Figure 12.

Wall pressure r.m.s. values normalised by: (a) reference dynamic pressure $q_{\infty }$ and local dynamic pressure $q_e$ ; (b) local wall shear stress $\tau _w$ and local maximum magnitude of Reynolds shear stress $\tau _{max }$ .

As the flow moves downstream, the r.m.s. of the wall pressure fluctuations follows a non-monotonic trend, characterised by an initial decrease, followed by an increase, and a subsequent decrease. When normalised by $q_e$ , the r.m.s. values diminish more sharply in FPG regions but amplify more rapidly in APG regions. This behaviour stems from competing mechanisms: FPG suppresses wall pressure fluctuations while increasing outer edge velocity $U_e$ , whereas APG intensifies wall pressure fluctuations while reducing $U_e$ . Consequently, $q_e$ proves suboptimal for scaling wall pressure fluctuations under spatially varying pressure gradients.

In contrast, the wall shear stress $\tau _w$ appears to be a more effective scaling parameter for wall pressure fluctuations. For a flat-plate TBL, the ratio $\mathit{p}_{w,\mathit{rms}}/\mathit{\tau }_w$ increases monotonically with $\it{Re}_\tau$ . Prior studies (Farabee & Casarella Reference Farabee and Casarella1991; Abe, Matsuo & Kawamura Reference Abe, Matsuo and Kawamura2005; Cohen & Gloerfelt Reference Cohen and Gloerfelt2018) established an empirical relationship for ZPG flows:

(4.3) \begin{equation} \left (p_{w,rms}/\tau _{w}\right )^{2}=6.5+1.86\ln(\it{Re}_{\tau }/333), \end{equation}

suggesting Reynolds number dependence. However, in the aft section of the parallel mid-body ( $0.5\lt \mathit{x}/\mathit{L}\lt 0.7$ ), the ratio $\mathit{p}_{w,\mathit{rms}}/\mathit{\tau }_w$ does not exhibit a slight increase with $\it{Re}_\tau$ . Instead, it remains approximately constant at 2.1. This behaviour can be attributed to the influence of a mild FPG. As shown in figure 9, the Clauser pressure gradient parameter $\mathit{\beta }$ slightly decreases in the range $0.5\lt \mathit{x}/\mathit{L}\lt 0.7$ , while the acceleration parameter $\mathit{K}$ gradually increases. Additionally, it is noteworthy that the value $\mathit{p}_{w,\mathit{rms}}/\mathit{\tau }_w \approx 2.1$ over the parallel mid-body is slightly lower than experimental observations of ZPG TBL in the corresponding Reynolds number range (Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2013), which can be attributed to the influence of transverse curvature. As discussed in the Introduction, transverse curvature effects can lead to increased wall shear stress and a slight suppression of wall pressure fluctuations.

In the range $0.86\lt x/L\lt 0.9$ , $p_{w,rms}$ and $\tau _w$ decline concurrently, stabilising at a plateau $p_{w,rms}/{\tau _w}\approx 6$ . As shown in figure 9, this plateau corresponds to the APG-dominant regime, where $\beta _e$ peaks before decreasing. The streamwise curvature is approximately zero, and the pressure gradient remains relatively stable in this region. A similar trend was observed experimentally by Balantrapu et al. (Reference Balantrapu, Alexander and Devenport2023) for flow over BOR under APG. In their study, the stern region of the BOR experienced a nearly uniform APG without streamwise curvature, resulting in $p_{w,rms}/{\tau _w}\approx 7$ . They concluded that wall-friction-induced motion remains a dominant contributor to pressure fluctuations in attached flows, even under strong APG. The present numerical results align with this observation, though the simulated ratio $p_{w,rms}/{\tau _w}\approx 6$ is slightly lower than the experimental value. Two factors can explain this discrepancy: the Reynolds number effects and the pressure gradient strength. The Reynolds number $\it{Re}_L=1.1\times 10^6$ in the study is lower than the experimental $\it{Re}_L=1.8\times 10^6$ . The local friction Reynolds number $\it{Re}_\tau$ is also smaller. Moreover, the APG in this study ( $\beta _e= 6{-}8$ ) is weaker than in the experiments ( $\beta _e=11{-}16$ ). For axisymmetric APG flows, $p_{w,rms}/\tau _w$ can correlate with the local pressure gradient parameters (such as $\beta _e$ ), reflecting the boundary layer’s proximity to equilibrium. As noted by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018), equilibrium boundary layers require a near-constant streamwise pressure gradient to achieve self-similarity, enabling scaling via $\tau _w$ . However, the relationship $p_{w,rms}/{\tau _w}\approx 6$ breaks down when streamwise curvatures and strong streamwise pressure gradient variations present. In non-equilibrium flows (e.g. $0.79\lt x/L\lt 0.85$ , where $\beta _e$ increases rapidly), $p_{w,rms}/\tau _w$ rises sharply from 1.5 to 6. Beyond $x/L\gt 0.9$ , transverse and concave streamwise curvature effects cause $\beta _e$ to decrease abruptly, reducing $p_{w,rms}/\tau _w$ .

Among the three pressure scaling parameters, the local maximum magnitude of the Reynolds shear stress $\tau _{max }$ appears to be the most effective scaling parameter for wall pressure fluctuations. As discussed in the Introduction, prior studies (Simpson et al. Reference Simpson, Ghodbane and Mcgrath1987; Na & Moin Reference Na and Moin1998; Abe Reference Abe2017; Schatzman & Thomas Reference Schatzman and Thomas2017) have highlighted the significant role of the turbulent shear term in governing wall pressure fluctuations within APG TBLs. These works proposed scaling wall pressure fluctuations using the maximum turbulent shear stress under APG conditions. However, the flow in the present configuration demonstrates greater complexity due to the continuous variation in streamwise curvature along the stern part, which includes both convex and concave curvatures. This curvature variation is further coupled with an evolving transverse curvature. In the APG region without streamwise curvature ( $0.86\lt \mathit{x}/\mathit{L}\lt 0.9$ ), $\tau _w$ is demonstrated to be an effective scaling parameter. However, across the entire APG region ( $0.79\lt \mathit{x}/\mathit{L}\lt 0.94$ ), the variation in $\mathit{p}_{w,\mathit{rms}}/\mathit{\tau }_{max }$ (ranging from 1.7 to 2.7) is significantly smaller compared to $\mathit{p}_{w,\mathit{rms}}/\mathit{q}_e$ (ranging from 0.0053 to 0.0124) and $\mathit{p}_{w,\mathit{rms}}/\mathit{\tau }_w$ (ranging from 1.5 to 6).

Due to the coexistence of convex and concave streamwise curvatures, the flow in the stern region features alternating zones of FPGs and APGs. To facilitate systematic analysis, based on the pressure gradient distribution in figure 9 and the r.m.s. of the wall pressure fluctuations in figure 12, the stern boundary layer is further divided into six distinct regions, as shown in figure 13.

1. (i) FPG-1 ( $0.71\lt x/L\lt 0.75$ ): the flow gradually transitions from ZPG to FPG, reaching the peak of the FPG at $x/L\approx 0.75$ .

2. (ii) FPG-2 ( $0.75\lt x/L\lt 0.79$ ): the FPG gradually decreases and returns to the ZPG at $x/L\approx 0.79$ .

3. (iii) APG-1 ( $0.79\lt x/L\lt 0.85$ ): this region is characterised by convex streamwise curvature, where the flow transitions from ZPG to APG.

4. (iv) APG-2 ( $0.85\lt x/L\lt 0.90$ ): the streamwise curvature is unchanged, with the wall pressure r.m.s. and wall shear stress exhibiting a linear relationship, i.e. $p_{w,rms}/\tau _{w}\approx 6$ .

5. (v) APG-3 ( $0.90\lt x/L\lt 0.94$ ): the flow is approaching a concave streamwise curvature, with APG gradually decreasing and eventually returning to ZPG.

6. (vi) FPG-3 ( $0.94\lt x/L\lt 0.99$ ): due to the change in streamwise curvature, the flow transitions back to FPG again, with flow separation occurring very close to the stern.

Figure 13.

Schematic diagram of the flow region division at the stern.

Figure 14 illustrates the evolution of wall pressure fluctuations frequency spectra at the stern, normalised by reference pressure $p_{\mathit{{ref}}}=20 \ \unicode{x03BC}\text{Pa}$ , with the fluid density specified as $1.225\, \text{kg}\, \text{m}^{-3}$ . The inset displays the corresponding pre-multiplied spectra $\omega\, \phi _{\mathit{{pp}}}(\omega )$ , where the $x$ -axis employs a logarithmic scale, and the $y$ -axis uses a linear scale. This ensures that the area under the pre-multiplied curve corresponds to the total energy of wall pressure fluctuations. Figure 15 further visualises the spatially continuous pre-multiplied power spectra as contour plots, highlighting their temporal and spatial evolution. In figure 14(a), a mild strengthening of the FPG correlates with a modest reduction in spectral levels at low-to-mid frequencies ( $\omega \lt 1000\, \rm { rad\, s^{-1}}$ ), and a slight elevation at high frequencies ( $\omega \gt 1000\, \rm { rad\, s^{-1}}$ ). This trend suggests that stronger FPG suppresses large-scale turbulent structures while amplifying small-scale fluctuations. In figure 14(b), as the FPG weakens and transitions toward a ZPG in region FPG-2, spectral levels decrease uniformly across all frequencies.

Figure 14.

Evolution of wall pressure fluctuation frequency spectra at the stern: (a) FPG-1, (b) FPG-2, (c) APG-1, (d) APG-2, (e) APG-3, (f) FPG-3. The inset shows corresponding pre-multiplied form $\omega\, \phi _{\mathit{{pp}}}(\omega )$ . The SUBOFF model and probe locations are displayed at the top.

Figure 15.

Contour of wall pressure fluctuations pre-multiplied energy frequency spectra at the stern.

Subsequently, the TBL encounters a strong APG induced by the convex streamwise curvature. While prior studies (Na & Moin Reference Na and Moin1998; Rozenberg et al. Reference Rozenberg, Robert and Moreau2012; Cohen & Gloerfelt Reference Cohen and Gloerfelt2018; Lee Reference Lee2018) have examined the isolated effects of APG on wall pressure fluctuations, conclusively demonstrating that APG amplifies low-frequency energy and attenuates high-frequency components, the dynamic evolution of APG (emerges, intensifies and decays) in this work reveals a more nuanced response. As shown in figures 14(c) and 14(d), the spectra exhibit a distinct two-stage evolution: a sharp rise in low- and mid-frequency energy during APG onset, followed by progressive attenuation of mid- and high-frequency energy as the APG stabilises. In the APG-1 region, the parameter $\beta _e$ increases rapidly from 1.05 to 5.2. Correspondingly, figure 14(c) reveals a sharp escalation in the low- and mid-frequency spectral energy ( $\omega \lt 1000\, \rm { rad\, s^{-1}}$ ), while the attenuation of high-frequency components ( $\omega \gt 1000\, \rm { rad\, s^{-1}}$ ) remains relatively subdued at this stage. However, as $\beta _e$ reaches its peak and gradually begins to decline, a rapid reduction in spectral energy emerges across the mid- and high-frequency range ( $\omega \gt 150\, \rm { rad\, s^{-1}}$ ), consistent with experimental observations (Balantrapu et al. Reference Balantrapu, Alexander and Devenport2023).

In the APG-3 region, as shown in figure 14(e), $\beta _e$ gradually decreases due to the introduction of concave streamwise curvature. A decay in wall pressure fluctuations is observed across the entire frequency range. Notably, the reduction of APG does not lead to a replenishment of high-frequency spectral levels, which can be attributed to the shifting of turbulent structure away from the wall. This phenomenon will be further discussed later. Downstream in the FPG-3 region, shown in figure 14(f), the concave streamwise curvature locally enhances the FPG, accelerating the flow and amplifying high-frequency wall pressure fluctuations through increased small-scale turbulence production. Eventually, flow separation occurs beyond $x/L\gt 0.99$ as the streamwise curvature transitions from concave to convex.

4.2. Scaling laws of wall pressure fluctuations

To identify the dominant contributors to wall pressure fluctuations across different frequency bands, it is critical to examine the scaling law governing these fluctuations in axisymmetric boundary layers with varying pressure gradient. Over the past few decades, numerous scaling approaches (Schloemer Reference Schloemer1966; Choi & Moin Reference Choi and Moin1990; Farabee & Casarella Reference Farabee and Casarella1991; Keith et al. Reference Keith, Hurdis and Abraham1992; Na & Moin Reference Na and Moin1998) have been proposed. However, owing to the multi-scale nature of turbulence, no universal scaling law has been established that achieves consistent collapse across all frequency ranges.

Figure 16 illustrates the non-dimensional PSD of wall pressure fluctuations under different scaling methods in the stern region. Six common scaling methods have been examined: outer scaling ( $q_e$ , $\delta$ and $U_e$ ), inner scaling ( $\tau _w$ , $\delta _v$ and $u_\tau$ ), mixed scaling ( $\tau _w$ , $\delta$ and $U_e$ ), Zagarola–Smits scaling ( $\tau _w$ , $\delta ^2/\delta ^*$ and $U_e$ ), Reynolds shear stress scaling ( $\tau _{max }$ , $\delta$ and $U_e$ ), and displacement thickness scaling ( $\tau _{max }$ , $\delta ^{*}$ and $U_e$ ). As shown in figure 14, the wall pressure spectra exhibit a generally monotonic trend within each flow region. Therefore, we select the start and end points of each region for comparison, which better highlights the variation of wall pressure fluctuations across different stern regions under various scaling laws. Figure 16 clearly illustrates that when both varying streamwise and transverse curvatures are present, conventional inner or mixed scaling methods completely fail. Among all the scaling approaches, normalisation using the local maximum Reynolds shear stress yields the best collapse, consistent with the trend observed in figure 12. Across the entire stern region, the variation of $\mathit{p}_{w,\mathit{rms}}/\mathit{\tau }_{max }$ is minimal. This underscores the pivotal role of $\tau _{max }$ in governing wall pressure fluctuations, particularly in non-equilibrium boundary layers. Notably, the Zagarola–Smits scaling performs poorly in the stern region of the SUBOFF model, primarily because $\delta$ and $\delta ^*$ are not equivalent length scales, as their ratio varies with the pressure gradient parameter $\beta _e$ (figure 8 c). Compared to the displacement thickness $\delta ^{*}$ scaling (figure 16 f), the boundary layer thickness $\delta$ (figure 16 e) is clearly a more effective scaling parameter.

Figure 16.

Non-dimensional PSD of wall pressure fluctuations under different scaling methods: (a) outer scaling ( $q_e$ , $\delta$ and $U_e$ ); (b) inner scaling ( $\tau _w$ , $\delta _v$ and $u_\tau$ ); (c) mixed scaling ( $\tau _w$ , $\delta$ and $U_e$ ); (d) Zagarola–Smits scaling ( $\tau _w$ , $\delta ^{2}/\delta ^{*}$ and $U_e$ ); (e) Reynolds shear stress scaling ( $\tau _{max }$ , $\delta$ and $U_e$ ); (f) displacement thickness scaling ( $\tau _{max }$ , $\delta ^{*}$ and $U_e$ ).

Figure 17 illustrates the non-dimensional PSD of wall pressure fluctuations under different scaling methods in the APG-2 region ( $0.85\lt x/L\lt 0.90$ ). In this region, the streamwise curvature is approximately zero, and the r.m.s. wall pressure exhibits a linear relationship with the wall shear stress, specifically $p_{w,rms}/\tau _w\approx 6$ . This relationship closely resembles the experimental results studied by Balantrapu et al. (Reference Balantrapu, Alexander and Devenport2023) in their BOR experiment. Among the selected scaling methods, the mixed scaling ( $\tau _w$ , $\delta$ and $U_e$ ) achieves an excellent collapse of the wall pressure fluctuation spectra across the entire frequency range. This finding is in complete agreement with the experimental conclusions of Balantrapu et al. (Reference Balantrapu, Alexander and Devenport2023), confirming $\tau _w$ , $\delta$ and $U_e$ as the optimal scaling parameters in this flow regime. Notably, the scaling retains its effectiveness within the viscous region (figure 17 b), consistent with experimental observations. The viscous time scale $\nu /u_\tau ^2$ exhibits dependence on the outer time scale $\delta /U_e$ , suggesting coupling between inner viscous dynamics and outer flow scales. Additionally, scaling using the local maximum magnitude of the Reynolds shear stress $\tau _{max }$ (figure 17 e) also collapses the spectra across the entire frequency range.

Figure 17.

Non-dimensional PSD of wall pressure fluctuations under different scaling methods in the APG-2 region ( $0.85\lt x/L\lt 0.90$ ): (a) outer scaling ( $q_e$ , $\delta$ and $U_e$ ); (b) inner scaling ( $\tau _w$ , $\delta _v$ and $u_\tau$ ); (c) mixed scaling ( $\tau _w$ , $\delta$ and $U_e$ ); (d) Zagarola–Smits scaling ( $\tau _w$ , $\delta ^{2}/\delta ^{*}$ and $U_e$ ); (e) Reynolds shear stress scaling ( $\tau _{max }$ , $\delta$ and $U_e$ ); (f) displacement thickness scaling ( $\tau _{max }$ , $\delta ^{*}$ and $U_e$ ).

Figures 14 and 15 demonstrate that the strong APG at the stern drives abrupt changes in the wall pressure fluctuation spectra within localised regions. In the APG-1 region shown in figure 14(c), the low-frequency component of wall pressure fluctuations increases sharply by approximately 10 dB, while the APG-3 region exhibits a broadband reduction in spectral energy levels across all frequencies. For the two APG regions, we also conducted additional and detailed comparisons of these scaling methods in Appendix B. The results further confirm the significant advantage of scaling based on the Reynolds shear stress.

4.3. Space–time characteristics of wall pressure fluctuations

Since turbulence is a spatiotemporal coupled problem, performing a one-dimensional spectral analysis of wall pressure fluctuations solely in the time domain inevitably results in the loss of spatial information. In this subsection, we investigate the space–time characteristics of wall pressure fluctuations by analysing their wavenumber–frequency spectra and space–time correlation functions. Analysing wavenumber–frequency spectra typically requires the assumption of streamwise uniformity. However, the flow over the SUBOFF stern exhibits significant streamwise pressure gradient variations. The APG boundary layers, in particular, are strongly influenced by upstream flow history (Bobke et al. Reference Bobke, Vinuesa, Örlü and Schlatter2017) and the specific manner of gradient application, complicating direct comparisons with datasets derived from simpler configurations. To mitigate this, our wavenumber–frequency spectral analysis is restricted to the parallel mid-body section, where the conventional assumption of negligible streamwise boundary layer thickening remains valid.

Figure 18 presents the space–time contour (spanning half the flow-past time) of wall pressure fluctuations along the streamwise direction in the mid-body region ( $x/L=0.3{-}0.7$ ) of the hull. The fluctuations are normalised by the wall shear stress $\tau _w$ . The contours reveal alternating high- and low-pressure regions, clearly illustrating the convection and distortion of vortical structures. According to Taylor’s frozen-flow hypothesis (Taylor Reference Taylor1938), the ‘world lines’ of these pressure fluctuations should form perfectly straight trajectories. However, significant deviations from this idealised behaviour are evident, arising from random sweeping effects (Kraichnan Reference Kraichnan1964; Tennekes Reference Tennekes1975) caused by large-scale structures and nonlinear interactions among small-scale turbulent motions.

Figure 18.

Space–time contours of wall pressure fluctuations along the streamwise direction in the mid-body region ( $x/L=0.3{-}0.7$ ) of the hull.

To compute the wavenumber–frequency spectra of wall pressure fluctuations, a standard methodology involves deploying a streamwise-aligned array of pressure probes to capture space–time fluctuating pressure signals. These signals are stored in a two-dimensional matrix $p(x=m\Delta X, t=n\Delta T)$ , where $m=1,2,3,\ldots, M$ and $n=1,2,3,\ldots, N$ . Here, $\Delta X$ denotes the spatial interval between adjacent probes, and $\Delta T$ represents the temporal sampling interval. In this study, the probe spacing $\Delta X$ matches the computational grid resolution, while $\Delta T$ corresponds to the simulation time step. The acquired pressure data (25 600 time steps) are segmented into 19 overlapping blocks with 50 % overlap to enhance statistical convergence. To mitigate spectral leakage, each segment is multiplied by a Hanning window. Four independent probe arrays, spaced uniformly along the streamwise direction, are circumferentially averaged to improve signal-to-noise ratios. The windowed space–time signals are then transformed into the wavenumber–frequency domain via a two-dimensional discrete Fourier transform:

(4.4) \begin{equation} \hat {p} \!\left (k_x,\omega \right )=\frac {\sqrt {8/3}}{MN}\sum _{m=1}^M\sum _{n=1}^Nw \!\left (n\Delta T\right )\,p \!\left (m\Delta X,n\Delta T\right )\times \exp^{{-}{\rm i} \left (m\Delta X\,k_x-n\Delta T\,\omega \right )}, \end{equation}

where the factor $\sqrt {8/3}$ compensates for the amplitude attenuation induced by the Hanning window $w(t)$ , preserving the r.m.s. value of $p$ . The wavenumber–frequency spectrum $\phi _{\mathit{{pp}}}(k_x,\omega )$ is obtained by ensemble-averaging the squared Fourier coefficients:

(4.5) \begin{equation} \phi _{\mathit{{pp}}}\!\left (k_x,\omega \right )=\frac {\overline {\hat {p} \!\left (k_x,\omega \right )\,\hat {p}^* \!\left (k_x,\omega \right )}}{\Delta k_x \Delta \omega }, \end{equation}

where $\Delta k_x=2\unicode{x03C0} /{(M\!\Delta\kern-1pt X)}$ and $\Delta \omega =2\pi /(N\!\Delta\kern-1pt T)$ define the streamwise wavenumber and frequency resolutions, respectively. This procedure fully maps the fluctuating pressure field from the space–time domain to the wavenumber–frequency domain.

Figure 19 presents the two-dimensional wavenumber–frequency spectra of wall pressure fluctuations in the mid-body region ( $x/L=0.3{-}0.7$ ) of the hull. The results are compared with semi-empirical wavenumber–frequency spectral models for flat-plate TBLs (Corcos Reference Corcos1964; Chase Reference Chase1980; Witting Reference Witting1986). The details of these models are provided in Appendix C. Similar to one-dimensional spectral models, wavenumber–frequency spectral models also require mean boundary layer parameters as inputs. Notably, the Corcos model (Corcos Reference Corcos1964) additionally depends on a self-spectral model. In this study, the Goody model (Goody Reference Goody2004) is selected as the input for the Corcos model. Due to the central symmetry of the wavenumber–frequency spectrum, only the first quadrant ( $\omega \gt 0$ , $k_x\gt 0$ ) is shown. Figure 19(a) reveals that spectral energy at any frequency concentrates near the convection wavenumber $k_c$ , forming a distinct ‘convective ridge’. Contrary to Taylor’s frozen-flow hypothesis (Taylor Reference Taylor1938), the convective ridge broadens with increasing frequency and wavenumber – a phenomenon termed Doppler broadening. The spectral widening reflects the distortion of turbulent eddies, driven by the random sweeping effects of large-scale structures on small-scale eddies and nonlinear interactions among the small-scale eddies. Additionally, the highest spectral energy resides in the low-frequency, low-wavenumber regime, decaying continuously with increasing $\omega$ and $k_x$ , in agreement with experimental (Abraham & Keith Reference Abraham and Keith1998), DNS (Choi & Moin Reference Choi and Moin1990; Yang & Yang Reference Yang and Yang2022), LES (Francis et al. Reference Francis, Ebenezer, Bhattacharyya and Sharma2023; Fan et al. Reference Fan, Zhu, Zhao and Wan2024b ) and semi-empirical (Corcos Reference Corcos1964; Chase Reference Chase1980; Witting Reference Witting1986) results. A notable feature is the contraction of spectral contours towards $(k_x,\omega)=(0,0)$ as $k_x$ approaches zero. This behaviour, linked to the rapid source term in the pressure Poisson equation (Yang & Yang Reference Yang and Yang2022), indicates that the wall pressure spectrum initially rises and then declines with increasing $k_x$ – a trend consistent with the one-dimensional spectra in figure 11.

Figure 19.

(a) Wavenumber–frequency spectra of wall pressure fluctuations at mid-body ( $x/L=0.3{-}0.7$ ) of SUBOFF, for present WRLES. The result is compared with semi-empirical models for the flat-plate TBL: (b) Corcos model; (c) Chase I model; (d) Witting model.

The comparison in figure 19 demonstrates that the high-frequency, high-wavenumber regions of the wavenumber–frequency spectrum from the present WRLES exhibit lower spectral levels than those predicted by semi-empirical models. As noted earlier, this discrepancy stems primarily from streamwise grid resolution limitations and the dissipative effects of the SGS model. Among the three models, the Corcos model (Corcos Reference Corcos1964) significantly overestimates spectral energy away from the convective ridge, likely due to its simplified treatment of spatiotemporal decorrelation mechanisms. Conversely, the Witting model (Witting Reference Witting1986) underestimates the spectral width, failing to adequately capture the random sweeping effects and distortion dynamics of small-scale eddies. In contrast, the Chase I model (Chase Reference Chase1980) shows the closest agreement with the WRLES results, accurately reproducing both the convective characteristics and spectral broadening trends of the wall pressure fluctuations.

To enable quantitative comparison of the wavenumber–frequency spectra, figure 20 shows wavenumber-dependent spectral slices at selected frequencies $\omega \delta /(u_\tau =100,150,200,250)$ in wavenumber–frequency spectra. The comparison reveals that all three semi-empirical models accurately predict the convective peak locations across these frequency bands. Among them, the Chase I model (Chase Reference Chase1980) shows the closest agreement with the present WRLES results, particularly in the low-wavenumber regime. Notably, the convective ridge in the WRLES spectra exhibits asymmetry about the peak – a feature consistent with prior observations in flat-plate TBLs and turbulent channel flows (Abraham & Keith Reference Abraham and Keith1998; Viazzo, Dejoan & Schiestel Reference Viazzo, Dejoan and Schiestel2001; Francis et al. Reference Francis, Ebenezer, Bhattacharyya and Sharma2023; Fan et al. Reference Fan, Zhu, Zhao and Wan2024b ). In summary, transverse curvature effects exert minimal influence on wall pressure fluctuations in the parallel mid-body region compared to flat-plate configurations. Consequently, the Chase I model remains applicable for rapid prediction of wavenumber–frequency spectra in this region, offering practical utility for engineering applications.

Figure 20.

Wavenumber-dependent spectra of wall pressure fluctuations at various frequencies: (a) $\omega \delta /u_{\tau }=100$ , (b) $\omega \delta /u_{\tau }=150$ , (c) $\omega \delta /u_{\tau }=200$ , (d) $\omega \delta /u_{\tau }=250$ .

Another point worth noting is the convection velocity $U_c$ of wall pressure fluctuations, which is the slope of the convective ridge in the wavenumber–frequency spectra, defined as (Wills Reference Wills1971)

(4.6) \begin{equation} \mathit{U_c}(\omega ) = \omega /\mathit{k_c}(\omega ), \end{equation}

where $k_c$ is the convection wavenumber, and

(4.7) \begin{equation} \left (\frac {\partial \mathit{\phi _{pp}}(\mathit{k_x},\omega )}{\partial \mathit{k_x}}\right )_{\mathit{k_x}=\mathit{k_c}(\omega )} = 0. \end{equation}

Figure 21 shows the frequency-dependent convection velocity of wall pressure fluctuation. The results are compared with experiment measurement (Abraham & Keith Reference Abraham and Keith1998) and the Smol’yakov model (Smol’yakov Reference Smol’yakov2006) for a flat-plate TBL. The model expression is given as

(4.8) \begin{equation} \frac {\mathit{U}_c}{\mathit{U_\infty }} = \frac {1.6(\omega \mathit{\delta }^*/\mathit{U_\infty })}{1+16(\omega \mathit{\delta }^*/\mathit{U_\infty })^2}+0.6. \end{equation}

The convection velocity of present WRLES reaches its peak at $\omega \delta ^*/U_\infty = 0.3$ . Then, as frequency increases, the convection velocity decreases from approximately $0.8U_\infty$ at low frequencies to $0.7U_\infty$ at high frequencies, consistent with flat-plate TBL studies (Abraham & Keith Reference Abraham and Keith1998; Smol’yakov Reference Smol’yakov2006).

Figure 21.

Frequency-dependent convection velocity of wall pressure fluctuations.

Figure 22 presents contour plots of the space–time correlation functions of wall pressure fluctuations at key streamwise locations. Spatial separation is normalised by the freestream velocity $U_\infty$ and boundary layer thickness $\delta _{\mathit{{mid}}}$ at the parallel mid-body ( $x/L=0.5$ ). The space–time correlation function is defined as

(4.9) \begin{equation} R_{\mathit{{pp}}}(s,t,\Delta s,\Delta t)=\frac {\overline {\left \langle p^{\prime }(s,t)\,p^{\prime }(s+\Delta s,t+\Delta t)\right \rangle }}{\sqrt {\left \langle p^{\prime 2}(s,t)\right \rangle }\sqrt {\left \langle p^{\prime 2}(s+\Delta s,t+\Delta t)\right \rangle }}, \end{equation}

where $\Delta s$ and $\Delta t$ are the streamwise spatial and temporal separations, respectively. The convection velocity of wall pressure fluctuations is derived from the slope ${\rm d}(\Delta s)/{\rm d}(\Delta t)$ . Figures 22(a–f) correspond to the ZPG, the FPG, the onset of the APG region, the peak of the APG region, the decline of the APG region, and the end of the APG region, respectively. The APG markedly reduces convection velocities, while the FPG enhances them. Streamwise coherence exhibits strong sensitivity to pressure gradient magnitude. In the APG onset region (figure 22 c), the streamwise correlation scale decreases rapidly downstream, with further reductions under strengthening APG. Conversely, FPG significantly amplifies streamwise coherence, particularly at large separations (evident in low-level contours), aligning with prior studies (Na & Moin Reference Na and Moin1998; Caiazzo et al. Reference Caiazzo, Pargal, Wu, Sanjosé, Yuan and Moreau2023). As the APG weakens (figures 22 d–f), the streamwise correlation scale recovers, accompanied by contour broadening. This recovery reflects an increase in the correlation time scale of wall pressure fluctuations. Notably, at $x/L=0.85$ , where the APG is strongest, negative contours are observed on both sides of the convective ridge (figure 22 d). This phenomenon will be discussed further in figure 24.

Figure 22.

Contours of two-point correlation of wall pressure fluctuations as functions of streamwise spatial and temporal separations: (a) $x/L=0.50$ , (b) $x/L=0.76$ , (c) $x/L=0.80$ , (d) $x/L=0.85$ , (e) $x/L=0.88$ , (f) $x/L=0.95$ . The solid lines represent contours ranging from 0.2 to 0.95 with increments of 0.15, while the dashed line denotes the contour at $-0.1$ .

Figure 23.

Contours of two-point correlation of wall pressure fluctuations as a function of transverse spatial and temporal separations: (a) $x/L=0.50$ , (b) $x/L=0.76$ , (c) $x/L=0.80$ , (d) $x/L=0.85$ , (e) $x/L=0.88$ , (f) $x/L=0.95$ . Contour levels are from 0.2 to 0.9 with increments of 0.1.

Figure 24.

Two-point correlation coefficients of wall pressure fluctuations as a function of (a,b) streamwise separation and (c,d) transverse separation in regions (a,c) APG-1, (b,d) APG-2.

Figure 23 presents two-point correlation contours of wall pressure fluctuations as functions of transverse spatial and temporal separations. Unlike streamwise coherence, transverse coherence exhibits weaker sensitivity to pressure gradients. Under mild FPG (figure 23 b), the contours remain nearly unchanged compared to the ZPG (figure 23 a). However, with increasing APG (figures 23 c,d), the transverse correlation scale grows as contours elongate spatially – a trend consistent with prior flat-plate and aerofoil studies (Na & Moin Reference Na and Moin1998; Caiazzo et al. Reference Caiazzo, Pargal, Wu, Sanjosé, Yuan and Moreau2023). Notably, as the APG diminishes downstream (figure 23 d–f), the transverse integral scale continues to increase rather than decrease. This behaviour aligns with observations on another LES study of BOR (Zhou et al. Reference Zhou, Wang and Wang2020), but contrasts with flat-plate conclusions (Na & Moin Reference Na and Moin1998; Caiazzo et al. Reference Caiazzo, Pargal, Wu, Sanjosé, Yuan and Moreau2023), which can be attributed to the contraction of geometry at the stern. As the turbulent structures and shear layer motions shift away from the wall, larger outer-layer structures, characterised by longer correlation lengths, dominate the coherence of wall pressure fluctuations. This aligns with the simultaneous growth of streamwise and transverse correlation scales (figures 22(d–f) and 23(d–f)) and the shift of spectral energy to lower frequencies (figure 14 d,e).

In terms of the effect of transverse curvature, APG and the contraction of the SUBOFF stern’s geometry drive rapid boundary layer thickening, amplifying the curvature ratio $\delta /r_0$ to $\sim6:1$ at $x/L=0.95$ (figure 10). Although transverse curvature is generally considered to have a minor effect on wall pressure fluctuations in previous studies (Willmarth & Yang Reference Willmarth and Yang1970; Neves et al. Reference Neves, Parviz and Moser1994; Piquet & Patel Reference Piquet and Patel1999), its influence becomes non-negligible when $\delta /r_0$ is relatively large. Interestingly, the observed increase in correlation scales with increasing transverse curvature at the stern contradicts the findings of Willmarth & Yang (Reference Willmarth and Yang1970) regarding transverse curvature effects. Their study concluded that transverse curvature leads to smaller streamwise and transverse integral scales, suggesting that pressure-generating motions occur closer to the wall compared to a flat plate. However, the experiments by Willmarth & Yang (Reference Willmarth and Yang1970) were conducted on a straight cylinder, with the flow approximately in a ZPG state. For the SUBOFF geometry, the flow history near the stern shows great difference due to the distinct geometric configuration. The influence of streamwise curvature variations seems to be more significant than that of transverse curvature. To isolate the influence of transverse curvature alone, additional simulation cases would be required.

Figure 24 presents two-point correlation coefficients of wall pressure fluctuations as functions of streamwise separation and transverse separation in the APG-1 and APG-2 regions. Under APG conditions, the streamwise correlation curves (figures 24 a,b) exhibit distinct negative lobes flanking the central positive peak at $\Delta s/\delta _{\mathit{{mid}}}=\pm 0.2$ , which is consistent with the observation in figure 22(d). Additionally, the streamwise integral length scales of wall pressure fluctuations are smaller than the transverse counterparts, which can also be observed in the instantaneous non-dimensional wall pressure fluctuation field. As shown in figure 25, the amplitude of wall pressure fluctuations increases rapidly within the stern APG region, accompanied by a sharp reduction in the streamwise integral scale. As the APG weakens, the streamwise integral scale gradually recovers. Beyond $x/L\gt 0.9$ , it is evident that wall pressure fluctuations exhibit larger streamwise and transverse scales. This observation is consistent with the analysis above, primarily attributed to turbulent structure and the local maximum Reynolds shear stress shifting away from the wall. The structure of wall pressure fluctuations in this region tends to more isotropic.

Figure 25.

Instantaneous non-dimensional wall pressure fluctuation field plotted versus $r\theta$ and $x$ .