2. Computational approaches

2.1. Numerical set-up

The present wall-resolved LES is performed by solving the spatially filtered Navier–Stokes (N–S) equations for incompressible Newtonian flow

(2.1) \begin{align} \left . \begin{array}{c} \dfrac {{\partial {{\tilde u}_i}}}{{\partial t}} + \dfrac {{\partial ( {{{\tilde u}_i}{{\tilde u}_j}} )}}{{\partial {x_j}}} = - \dfrac {{\partial \tilde p}}{{\partial {x_i}}} + \dfrac {1}{\textit{Re}}\dfrac {{{\partial ^2}{{\tilde u}_i}}}{{\partial {x_j}\partial {x_j}}} - \dfrac {{\partial {\tau _{\textit{ij}}}}}{{\partial {x_j}}},\\ \dfrac {{\partial {{\tilde u}_i}}}{{\partial {x_i}}} = 0, \end{array} \right \} \end{align}

where $u_i$ is the velocity, $p$ is the pressure and $\textit{Re} = {{{U_\infty }L} / \nu }$ is the Reynolds number, with $U_\infty$ denoting the free-stream velocity, $L$ the length of SUBOFF model and $\nu$ the kinematic viscosity. The summation convention is applied with $i,j=1,2,3$ . Note that (2.1) are solved based on the Cartesian coordinate system. The symbol $\widetilde { ( \boldsymbol{\cdot })}$ denotes the spatial filtering, and for the sake of simplicity, this symbol will not appear in subsequent parameters. The sub-grid stress $\tau _{\textit{ij}} = \widetilde {{u_i}{u_j}} - {\tilde u_i}{\tilde u_j}$ is parameterised by the wall-adapting local eddy-viscosity model (Nicoud & Ducros Reference Nicoud and Ducros1999). Based on the Pressure implicit with splitting of operators algorithm in OpenFOAM, the governing equations are solved using the finite-volume method. The backward scheme is adopted for time integration (Jasak Reference Jasak1996), i.e.

(2.2) \begin{equation} \frac {{\partial {u_i}}}{{\partial t}} = \frac {1}{{\Delta t}}\left ( {\frac {3}{2}{{\left . {{u_i}} \right |}_{t + 1}} - 2{{\left . {{u_i}} \right |}_t} + \frac {1}{2}{{\left . {{u_i}} \right |}_{t - 1}}} \right )\!. \end{equation}

Spatial derivatives are discretised by the second-order central-difference scheme. Note that for the discretisation of the convective term, the improved Total Variation Diminishing scheme is applied to eliminate numerical oscillations.

A 3-D axisymmetric domain is used for simulation of flow over a SUBOFF bow, as shown in figure 1(a). For convenience, three coordinate systems are used in the following. One is the Cartesian coordinate system ( $x,y,z$ ), or equivalently ( $x_1, x_2, x_3$ ). The second one is the cylindrical coordinate system ( $x,r,\theta$ ), representing the coordinates in the streamwise, radial and spanwise directions, respectively. The third one is the axisymmetric bow-profile coordinate system ( $s,n,\theta$ ), representing the arc/tangential, wall-normal and spanwise directions, respectively. The axisymmetric bow-profile coordinate system is orthogonal, thus the equations in ( $s,n,\theta$ ) can be simplified from a general curvilinear coordinate system. More details about the axisymmetric bow-profile coordinate system are introduced in § 2.3. The origin of the coordinate systems is located at the nose of the bow. The distance from the inlet to the nose of the bow is 3.6R, where R is the maximum radius of the bow. By plotting the normal lines from the bow surface with an equal height of 3.6R, the inlet surface can be obtained. Here, a 1/120-scale SUBOFF model with R = 0.0508 and R = 0.8712 m is used, which is the same as that in the experiment of Jimenez et al. (Reference Jiménez, Hultmark and Smits2010b ). The bow radius (r) can be expressed as follows (Groves, Huang & Chang Reference Groves, Huang and Chang1989):

(2.3) \begin{equation} r = \begin{cases} R\left \{ {\dfrac {{1.126395101x}}{{0.06096}}{{\left ( {\dfrac {{0.3x}}{{0.06096}} - 1} \right )}^4}} \right . \\{\text{ }} + 0.442874707{\left ( {\dfrac {x}{{0.06096}}} \right )^2}{\left ( {\dfrac {{0.3x}}{{0.06096}} - 1} \right )^3} \\{\text{ }} + {\left . {1 - \left ( {\dfrac {{1.2x}}{{0.06096}} + 1} \right ){{\left ( {\dfrac {{0.3x}}{{0.06096}} - 1} \right )}^4}} \right \}^{{1 / {2.1}}}},x \lt 0.2032{m} \\R,x \geqslant 0.2032{m}. \end{cases} \end{equation}

The relatively small domain may bring the confinement effect. Thus, to avoid it, spatially variant velocities are employed along the inflow surface, as plotted in figure 1(b), rather than a uniform free-stream velocity $U_{\infty }$ . The profiles are obtained from the LES of 2-D axisymmetric flow over the bow in a large domain. In this 2-D simulation, a uniform free-stream velocity $U_{\infty }$ is applied along the inflow surface, and the domain is two orders of magnitude larger than that of Kumar & Mahesh (Reference Kumar and Mahesh2018). Then the inflow condition is given as $U_x = {U_{x,2D}}$ , $U_y = {U_{r,2D}}\cos \theta$ and $U_z = {U_{r,2D}}\sin \theta$ . No-slip condition is applied at the bow surface and constant-pressure condition is used at the outlet.

Figure 1.

Computational settings: (a) schematic of computational domain and boundary conditions; (b) mean velocity profiles along the inflow surface; (c) schematic of grid subdivision for the bow surface and the x–y plane.

To deal with the singularity of the cylindrical coordinate system when the radius tends to zero (O’Sullivan & Breuer Reference O’sullivan and Breuer1994), an O-block mesh (shown in figure 1 c) is constructed in the present simulations, instead of a rotationally symmetric one. Simulations of flow over the SUBOFF bow are preformed at a Reynolds number of $\textit{Re}{_L} = 1.1 \times {10^6}$ . The grid sizes at the bow surface satisfy $\Delta {s^ + } \lt 24$ and $r\Delta {\theta ^ + } \lt 12$ , and the first wall-normal grid resolution satisfies $\Delta n_w^ + \lt 1$ . In total 570–870 million grids are used. The computational time step is $\Delta t \times {{{U_\infty }} / L} = 2.7 \times {10^{ - 4}}$ , ensuring that the maximum local Courant number is less than 1. After reaching the statistically steady state, the data over fifteen flow-through-body times are then used to compute the statistics.

Details for the numerical validation can be found in Appendices A and B, and verification of the inlet/outlet computational settings can be found in Appendix C.

2.2. Generation of turbulent inflow

To induce flow transition in the boundary layer over the bow, a turbulent flow field is artificially pre-generated and serves as the inflow condition (figure 1 a). The Kraichnan technique (Saad et al. Reference Saad, Cline, Stoll and Sutherland2017) is adopted to generate the isotropic turbulent flow field, which is supposed to satisfy the von Karman–Pao spectrum (Bailly & Juve Reference Bailly and Juve1999), i.e.

(2.4) \begin{equation} E(\kappa ) = a \frac {{u_{\textit{rms}}^{\prime 2}}}{{{\kappa _e}}}\frac {{{{\left ( {\kappa /{\kappa _e}} \right )}^4}}}{{{{\left [ {1 + \left ( {\kappa /{\kappa _e}} \right )} \right ]}^{17/6}}}}\exp \! \left [ { - 2{{\left ( {\frac {\kappa }{{{\kappa _\eta }}}} \right )}^2}} \right ]\!,a \approx 1.453, \end{equation}

where $u'_{\textit{rms}}$ is the root mean square of the velocity fluctuation, $\kappa$ is the wavenumber, $\kappa _e$ is related to the integral length scale $L \approx {{0.747} / {{\kappa _e}}}$ and $\kappa _\eta$ is the corresponding wavenumber for the Kolmogorov scale. Note that the turbulence intensity $I=u'_{\textit{rms}}/U_{\infty }$ also serves as an input parameter in (2.4). Velocities are reconstructed based on the spectrum model

(2.5) \begin{equation} {\boldsymbol{u}}({\boldsymbol{x}}) = 2\sum \limits _{m = 1}^M {{q_m}} \cos \! \left ( {{\kappa _m}{{\widehat {\boldsymbol{k}}}_m} \boldsymbol{\cdot }{\boldsymbol{x}} + {\psi _m}} \right ){\widehat {{\sigma }}_m},{q_m} = \sqrt {E\left ( {{\kappa _m}} \right )\Delta \kappa }, \end{equation}

where $M$ is the mode number, $\kappa _m$ is the wavenumber, $\widehat {\boldsymbol{k}}_m$ is the unit direction vector related with $\kappa _m$ , $\boldsymbol{x}$ is the coordinate of discrete points, $\psi _m$ is the random phase angle and $\widehat {{\sigma }}_m$ is the unit direction vector used to compulsively satisfy the divergence-free condition in the entire flow field. As a result, the energy spectrum of the generated turbulent flow field agrees well with the theoretical expression (2.4), as shown in figure 2(a). The contours of streamwise velocity are displayed in figure 2(b). The Kraichnan method has been widely applied in generating the initial flow field (Chung, Mishra & Ihme Reference Chung, Mishra and Ihme2022; Chen et al. Reference Chen, Li, Wang, Hu and Xiao2024; Idelsohn, Gimenez & Onate Reference Idelsohn, Gimenez, Larreteguy, Nigro, Sívori and Oñate2024) and turbulent inflow (Markesteijn & Karabasov Reference Markesteijn and Karabasov2019; Markesteijn et al. Reference Markesteijn, Gryazev, Karabasov, Ayupov, Benderskiy and Lyubimov2020), where good agreements with experimental data can be found.

Table 1.

Computational parameters for different cases. Higher turbulence intensity level is given to achieve the target level at the x = 0 plane.

Figure 2.

(a) Energy spectrum of synthetic turbulence and (b) streamwise velocity for the isotropic turbulent field by using Kraichnan’s technique.

Based on the generated isotropic velocity fluctuations, by adding the mean values in figure 1(b), Taylor’s frozen hypothesis is then utilised to transform the streamwise velocity into signals in the temporal direction. The generated flow data are then enforced as the turbulent inflow within the region $r \leqslant 1.4R$ . The Heaviside function (2.6) is adopted to transit from the turbulence region to the free-stream region at $r = 1.4R$ (Chang et al. Reference Chang, Hou, Merriman and Osher1996), i.e.

(2.6) \begin{equation} \phi = {H_{\alpha _d} }(d) = \left \{ \begin{array}{l} 1,d \gt \alpha _d ,\\[4pt]0,d \lt - \alpha _d ,\\[4pt]\dfrac {1}{2} \left(1 + \dfrac {d}{\alpha _d } + \dfrac {1}{\pi }\sin \left(\dfrac {{\pi d}}{\alpha _d }\right)\right)\,{\rm otherwise}, \end{array} \right . \end{equation}

where $d = r - 1.4R$ is the distance from the interface $r = 1.4R$ , and $\alpha _d$ is the width for the transition area. Here, $\alpha _d/R\approx 0.1$ is used. Then, the turbulence intensity is expressed as $I = {I_0}\phi$ at the inflow region, where $I_0$ is the prescribed ITI.

As seen in table 1, four cases with different inflow turbulent intensities ranging from 0 % to 55 % are simulated. Before encountering the bow ( $x/L \lt 0$ ), the flow can be regarded as homogeneous turbulence that decays exponentially downstream. By adjusting the turbulence intensity at the inflow surface, the turbulence intensities within $r/R=0.6{-}1.1$ at the $x = 0$ plane of 0 %, 2 %, 5 % and 8 % are chosen as cases 0–3, respectively, as listed in table 1. The previous studies showed that the incoming flow field of a navigating submarine in a real underwater environment is very complex, with measured turbulence intensities ranging from 0.03 % to 7 % (Osborn Reference Osborn1978; Reference Osborn1980; Witte et al. Reference Witte, Brattegard, Graf and Springer1997; Gargett Reference Gargett1999; Greene, Hendricks & Gregg Reference Greene, Hendricks and Gregg2015). Therefore, this study focuses on the flow around SUBOFF with incoming turbulence intensities ranging from 0 % to 8 % at x = 0. The turbulent integral length scale at the inflow in all cases is set to be 0.06R, which is within the range of integral scales under real sea conditions (Bouruet-Aubertot et al. Reference Bouruet-Aubertot, Cuypers, Ferron, Dausse, Ménage, Atmadipoera and Jaya2018). Moreover, for the flow to develop fully downstream, the bow length in cases 0 and 1 is larger than that in cases 2 and 3.

In figure 3(a), variations of the turbulence intensity in cases 1–3 with the streamwise position are plotted. In each case, the results at different radius positions ( ${r / R} = 0{-}1.1$ ) collapse well with the theoretical one, demonstrating the homogeneity of turbulence in the incoming flow region. According to the classical theory of isotropic turbulence (Saffman Reference Saffman1967), the turbulence intensity $I$ varies as $I \sim { ( {x - {x_{\textit{inlet}}}} )^{ - \alpha }}$ , where $x$ is the distance along the downstream evolution, $x_{\textit{inlet}}$ is the streamwise location of the inlet region and $\alpha$ is the attenuation factor. In cases 1–3, the factor $\alpha$ is within the scope given in Saffman (Reference Saffman1967). In figure 3(b), the pressure spectra in the incoming flow region are presented. Specifically, the scaling law ${\varPhi _{pp}} ( \omega ) \sim {\omega ^{{{ - 7} / 3}}}$ is observed in the mid-frequency region, which is a typical feature of homogeneous turbulence (Gotoh & Fukayama Reference Gotoh and Fukayama2001).

Figure 3.

Characteristics of incoming turbulence evolution: (a) turbulence intensity at different radial positions; (b) pressure spectra at different streamwise positions, scaled by the outer scales. The non-dimensional spectra are represented by ${{{\varPhi _{pp}}D} /{( {p_{\textit{rms}}^{\prime 2}U} )}}$ versus ${\omega D} / U$ , where $D$ is the maximum diameter of the bow, $U$ is the local mean streamwise velocity, $ \omega$ is the angular frequency and $p'_{\textit{rms}}$ is the root mean square of the local pressure fluctuation.

2.3. Expressions in the axisymmetric bow-profile coordinate system

For facilitating analysis in the following sections, the axisymmetric bow-profile coordinate system ( $s,n,\theta$ ) is used. The velocity components ${u_i} ( {i = s,n,\theta } )$ are projected along the curvilinear coordinate lines. The N–S equations in the ( $s,n,\theta$ ) coordinates can be written as follows:

(2.7) \begin{align} & \frac {{\partial {u_i}}}{{\partial t}} + {u_i}\frac {{\partial {u_i}}}{{\partial {\xi _i}}} + {u_j}\frac {{\partial {u_i}}}{{\partial {\xi _j}}} + {u_k}\frac {{\partial {u_i}}}{{\partial {\xi _k}}} + {K_{\textit{ij}}}{u_i}{u_j} - {K_{\textit{ji}}}{u_j}{u_j} + {K_{\textit{ik}}}{u_i}{u_k} - {K_{\textit{ki}}}{u_k}{u_k}\nonumber \\& = - \frac {\partial }{{\partial {\xi _i}}}\left ( {\frac {p}{\rho }} \right ) + \nonumber \\& \nu \left \{ {{{\nabla} ^2}{u_i} + 2{K_{\textit{ij}}}\frac {{\partial {u_j}}}{{\partial {\xi _i}}} + 2{K_{\textit{ik}}}\frac {{\partial {u_k}}}{{\partial {\xi _i}}} - 2{K_{\textit{ji}}}\frac {{\partial {u_j}}}{{\partial {\xi _j}}} - 2{K_{\textit{ki}}}\frac {{\partial {u_k}}}{{\partial {\xi _k}}} + {\alpha _{\textit{ii}}}{u_i} + {\alpha _{\textit{ij}}}{u_j} + {\alpha _{\textit{ik}}}{u_k}} \right \}\nonumber \\& \frac {{\partial {u_i}}}{{\partial {\xi _i}}} + \frac {{\partial {u_j}}}{{\partial {\xi _j}}} + \frac {{\partial {u_k}}}{{\partial {\xi _k}}} + \left ( {{K_{\textit{ji}}} + {K_{\textit{ki}}}} \right ){u_i} + \left ( {{K_{\textit{ij}}} + {K_{\textit{kj}}}} \right ){u_j} + \left ( {{K_{\textit{ik}}} + {K_{\textit{jk}}}} \right ){u_k} = 0 , \end{align}

where $i,j,k=s,n,\theta$ and ${\partial / {\partial {\xi _s}}} = {\partial / { ( {{h_s}\partial s} )}}$ , ${\partial / {\partial {\xi _n}}} = {\partial / { ( {{h_n}\partial n} )}} = {\partial / {\partial n}}$ and ${\partial / {\partial {\xi _\theta }}} = {\partial / { ( {{h_\theta }\partial s} )}}$ . Here, $K_{\textit{ij}}$ and $\alpha _{\textit{ij}}$ are parameters related to the Lame coefficient $h_i$ (Richmond, Chen & Patel Reference Richmond, Chen and Patel1986) as follows:

(2.8) \begin{equation} {K_{\textit{ij}}} = \frac {1}{{{h_i}}}\frac {{\partial {h_i}}}{{\partial {\xi _j}}},{\alpha _{\textit{ij}}} = \frac {{\partial {K_{\textit{ij}}}}}{{\partial {\xi _i}}} - \frac {{\partial {K_{\textit{ji}}}}}{{\partial {\xi _j}}} - {K_{\textit{kj}}}\left ( {{K_{\textit{ki}}} + {K_{\textit{ji}}}} \right ) + {K_{\textit{ki}}}{K_{\textit{ij}}}. \end{equation}

Note that the repeated subscripts in (2.7) and (2.8) do not conduct the summation operation. More details can be found in Appendix D.

To understand the generation mechanism of skin friction, the dynamic decomposition of the skin friction coefficient $C_f$ can be derived by double integrations of the Reynolds-averaged form of (2.7). Here, $C_f$ is defined as ${C_f} = {{{\tau _w}} / { ( {0.5\rho U_\infty ^2} )}}$ , where $\tau _w$ is the local wall shear stress at the bow surface, and $U_{\infty }$ is the free-stream velocity, which is used at the inlet for LES of 2-D axisymmetric flow over the bow in a large domain, as mentioned in § 2.1. The dynamic decomposition of $C_f$ is expressed as follows:

(2.9) \begin{align} & {C_f} = \underbrace {\frac {2}{{{\textit{Re}_\delta }}}{{\left . {{U_s}} \right |}_{{\delta _{99}}}}\beta }_{{C_{\nu 1}}}\underbrace { - \frac {4}{\textit{Re}}\left ( {\beta - 1} \right )\int _0^1 {{U_s}{\rm d}n} }_{{C_{\nu 2}}}\underbrace { - 2\int _0^1 {\frac {r}{{{{\left . r \right |}_0}}}\overline {{{u'}_s}{{u'}_n}} {\rm d}n} }_{{C_R}} \nonumber\\& \underbrace { - 2 \! \int _0^1 \!\! {\left ( {1 - n} \right )\!\left (\! {{U_s}\frac {1}{h_s}\frac {{\partial {U_s}}}{{\partial s}} + {U_n}\frac {{\partial {U_s}}}{{\partial n}}} \! \right )\!{\rm d}n} }_{{C_{C1}}}\underbrace { - 2 \!\left ( {\beta - 1} \right ) \!\int _0^1 \!\! {\big ( {1 - {n^2}} \big ) \!\left ( \! {{U_s}\frac {1}{h_s}\frac {{\partial {U_s}}}{{\partial s}} + {U_n}\frac {{\partial {U_s}}}{{\partial n}}} \! \right ){\rm d}n} }_{{C_{C2}}}\nonumber\\& \underbrace { - 2\int _0^1 {\left ( {1 - n} \right )\frac {1}{h_s}\frac {{\partial P}}{{\partial s}}{\rm d}n} }_{{C_{P1}}}\underbrace { - 2\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big)\frac {1}{h_s}\frac {{\partial P}}{{\partial s}}{\rm d}n} }_{{C_{P2}}}\nonumber\\& + \underbrace {2\int _0^1 {\left ( {1 - n} \right )\left ( {\frac {1}{{{\textit{Re}}}}{{\nabla} ^2}{U_s} - \frac {1}{{{\textit{Re}}}}\frac {{{\partial ^2}{U_s}}}{{\partial {n^2}}} + \frac {1}{h_s}\frac {{\partial \overline {{{u'}_s}{{u'}_s}} }}{{\partial s}}} \right ){\rm d}n} }_{{C_{D1}}}\nonumber\\& + \underbrace {2\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big)\left ( {\frac {1}{{{\textit{Re}}}}{{\nabla} ^2}{U_s} - \frac {1}{{{\textit{Re}}}}\frac {{{\partial ^2}{U_s}}}{{\partial {n^2}}} + \frac {1}{h_s}\frac {{\partial \overline {{{u'}_s}{{u'}_s}} }}{{\partial s}}} \right ){\rm d}n} }_{{C_{D2}}} + {C_{f,\textit{geo}}}, \end{align}

where $U_i$ is the mean velocity, $u'_i$ is the velocity fluctuation, $P$ is the mean pressure, ${\mathop {\textit{Re}_\delta }\nolimits } = {{{U_\infty }{\delta _{99}}} / \nu }$ , with $\delta _{99}$ the nominal thickness of boundary layer on the bow surface, and $\beta = {{{{\left . r \right |}_{{\delta _{99}}}}} / {{{\left . r \right |}_0}}}$ denotes the ratio of the radius at the outer edge of the boundary layer to that of the bow surface. The symbol $\left . {} \right |_0$ represents the value at the wall ( $n = 0$ ) and $\overline { ( \boldsymbol{\cdot })}$ denotes the average operation. Moreover, the terms containing the geometric parameters $K_{\textit{ij}}$ as defined in (2.8) are collected in $C_{f,\textit{geo}}$ , i.e.

(2.10) \begin{align} & {C_{f,\textit{geo}}} = - 2\int _0^1 {\left ( {1 - n} \right ){K_{\textit{sn}}}{U_s}{U_n}{\rm d}n} \nonumber \\& - 2\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big){K_{\textit{sn}}}{U_s}{U_n}{\rm d}n} \nonumber \\& - 2\int _0^1 {\left ( {1 - n} \right )\big [ {\left ( {2{K_{\textit{sn}}} + {K_{\theta n}}} \right )\overline {{{u'}_s}{{u'}_n}} + {K_{\theta s}}\overline {{{u'}_s}{{u'}_s}} - {K_{\theta s}}\overline {{{u'}_\theta }{{u'}_\theta }} } \big ]{\rm d}n} \nonumber \\& - 2\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big)\big [ {\left ( {2{K_{\textit{sn}}} + {K_{\theta n}}} \right )\overline {{{u'}_s}{{u'}_n}} + {K_{\theta s}}\overline {{{u'}_s}{{u'}_s}} - {K_{\theta s}}\overline {{{u'}_\theta }{{u'}_\theta }} } \big ]{\rm d}n} \nonumber\\& + \frac {2}{{{\text{Re}}}}\int _0^1 {\left ( {1 - n} \right )\left ( {2{K_{\textit{sn}}}\frac {{\partial {U_n}}}{{\partial s}} + {\alpha _{\textit{nn}}}{U_s} + {\alpha _{\textit{sn}}}{U_n}} \right ){\rm d}n} \nonumber\\& + \frac {2}{{{\text{Re}}}}\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big)\left ( {2{K_{\textit{sn}}}\frac {{\partial {U_n}}}{{\partial s}} + {\alpha _{\textit{nn}}}{U_s} + {\alpha _{\textit{sn}}}{U_n}} \right ){\rm d}n}. \end{align}

Details of the derivation can be found in Appendix E. This decomposition formulation is an extension of that developed by Xia et al. (Reference Xia, Huang, Xu and Cui2015), which is based on the Cartesian coordinate system. Note that this expression is different from the Fukagata-Iwamoto-Kasagi (FIK) formulation (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002), which is obtained by triple integrations in the wall-normal direction, in order to relate $C_f$ to the mean streamwise velocity. Accordingly, the FIK method is appropriate for situations with constant flow rate, while the present formulation is more suitable for cases of a given external flow velocity (Xia et al. Reference Xia, Huang, Xu and Cui2015).

As dictated in (2.9), the skin friction coefficient in the ( $s,n,\theta$ ) system can be decomposed into six parts: contributions from the viscous term, the development of external flow velocity and the flow rate, $C_{\nu 1}$ and $C_{\nu 2}$ ; contribution from the Reynolds shear stress, $C_R$ ; contributions from the mean convection terms, $C_{C1}$ and $C_{C2}$ ; contributions from the mean-pressure-gradient terms, $C_{P1}$ and $C_{P2}$ ; contributions from the derivatives of other terms, $C_{D1}$ and $C_{D2}$ ; contribution from the direct geometric influence related to parameters $K_{\textit{ij}}$ , $C_{f,\textit{geo}}$ . Specifically, for the TBL flow over a flat plate, in term $C_R$ the values of $r$ and ${{\left . r \right |}_0}$ are equal; $\beta =1$ and $C_{\nu 2}$ , $C_{C2}$ , $C_{P2}$ and $C_{D2}$ are equal to zero; the geometric parameters $K_{\textit{ij}}=0$ , and thus $C_{f,\textit{geo}}$ is equal to zero. The detailed analysis of the contribution terms in (2.9) will be presented in § 5.

For the budget analysis in the curvilinear coordinate system, Richmond et al. (Reference Richmond, Chen and Patel1986) derived a general form of the Reynolds stress transport equations. However, their equations cannot express the impact of geometric curvature on energy transfer in the different budget terms. To separate the geometric and non-geometric influences, the Reynolds stress transport equations are derived from another perspective. Details of the derivation can be found in Appendix F. In a general curvilinear coordinate system, the Reynolds stress transport equations are expressed as follows:

(2.11) \begin{equation} \begin{array}{l} \underbrace {\frac {{\partial \big ( {\overline {{{u}^\prime_i}{{u}^\prime_{\!j}}} } \big )}}{{\partial t}} + {\boldsymbol{U}} \boldsymbol{\cdot }\boldsymbol{\nabla }\big ( {\overline {{{u^\prime}_{\!i}}{{u}^\prime_{\!j}}} } \big )}_{Convection,non - geo}\\= \underbrace { - \overline {{{u}^\prime_{\!j}}{\boldsymbol{u'}}} \boldsymbol{\cdot }\boldsymbol{\nabla }{U_i} - \overline {{{u}^\prime_{\!i}}{\boldsymbol{u'}} \boldsymbol{\cdot }} \boldsymbol{\nabla }{U_{\!j}}}_{P,non - geo}\underbrace { - \overline {{\boldsymbol{u'}} \boldsymbol{\cdot }\boldsymbol{\nabla }\left ( {{{u}^\prime_{\!i}}{{u}^\prime_{\!j}}} \right )} }_{DT,non - geo}\underbrace { - \left [ {{{\boldsymbol{e}}_i} \boldsymbol{\cdot }\frac {{\boldsymbol{\nabla }\big ( {\overline {p'{{u}^\prime_{\!j}}} } \big )}}{\rho }{\textrm { + }}{{\boldsymbol{e}}_{\!j}} \boldsymbol{\cdot }\frac {{\boldsymbol{\nabla }\big ( {\overline {p'{{u}^\prime_{\!i}}} } \big )}}{\rho }} \right ]}_{DP}\\+ \underbrace {\left [ {{{\boldsymbol{e}}_i} \boldsymbol{\cdot }\overline {p'\frac {{\boldsymbol{\nabla }{{u}^\prime_{\!j}}}}{\rho }} + {{\boldsymbol{e}}_{\!j}} \boldsymbol{\cdot }\overline {p'\frac {{\boldsymbol{\nabla }{{u}^\prime_{\!i}}}}{\rho }} } \right ]}_\varPhi + \underbrace {\nu \Delta \big ( {\overline {{{u}^\prime_{\!i}}{{u}^\prime_{\!j}}} } \big )}_{D\nu ,non - geo}\underbrace { - 2\nu \overline {\boldsymbol{\nabla }{{u}^\prime_{\!i}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{u}^\prime_{\!j}}} }_{\varepsilon ,non - geo} + {R_{\textit{geo}}}, \end{array} \end{equation}

where ${\boldsymbol{u}} = {\boldsymbol{U}} + {\boldsymbol{u'}}$ is the velocity vector, $\boldsymbol{U}$ is the mean velocity vector, $\boldsymbol{u'}$ is the velocity fluctuation vector, $p'$ is the pressure fluctuation and ${\boldsymbol{e}}_i$ is the unit vector in the ith direction. Also, $P$ is the production term, $DT$ is the turbulent transport term, $DP$ is the pressure transport term, $\varPhi$ is the pressure-strain or redistribution term, $D\nu$ is the viscous diffusion term and $\varepsilon$ is the dissipation term. The terms with the symbol $non - geo$ do not include the geometric effects, while the geometric terms are represented by $R_{\textit{geo}}$ , i.e.

(2.12) \begin{equation} \begin{array}{l} {R_{\textit{geo}}} = \underbrace {\overline {{{u}^\prime_{\!j}}{\boldsymbol{U}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{\boldsymbol{e}}_i} \boldsymbol{\cdot }{\boldsymbol{u'}}} + \overline {{{u}^\prime_{\!i}}{\boldsymbol{U}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{\boldsymbol{e}}_{\!j}} \boldsymbol{\cdot }{\boldsymbol{u'}}} }_{Convection,geo}\\[4pt] + \underbrace {\overline {{{u}^\prime_{\!j}}{\boldsymbol{u'}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{\boldsymbol{e}}_i} \boldsymbol{\cdot }{\boldsymbol{U}}} + \overline {{{u}^\prime_{\!i}}{\boldsymbol{u'}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{\boldsymbol{e}}_{\!j}} \boldsymbol{\cdot }{\boldsymbol{U}}} }_{P,geo}\\[4pt] + \underbrace {\overline {{{u}^\prime_{\!j}}{\boldsymbol{u'}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{\boldsymbol{e}}_i} \boldsymbol{\cdot }{\boldsymbol{u'}}} + \overline {{{u}^\prime_{\!i}}{\boldsymbol{u'}} \boldsymbol{\cdot }\boldsymbol{\nabla }{{\boldsymbol{e}}_{\!j}} \boldsymbol{\cdot }{\boldsymbol{u'}}} }_{DT,geo}\\[4pt] \underbrace { - \overline {\nu {{u}^\prime_{\!j}}\Delta {{u}^\prime_{\!i}}} + \nu \overline {{{u}^\prime_{\!j}}\Delta {\boldsymbol{u'}} \boldsymbol{\cdot }{{\boldsymbol{e}}_i}} - \overline {\nu {{u}^\prime_{\!i}}\Delta {{u}^\prime_{\!j}}} + \nu \overline {{{u}^\prime_{\!i}}\Delta {\boldsymbol{u'}} \boldsymbol{\cdot }{{\boldsymbol{e}}_{\!j}}} }_{D\nu + \varepsilon ,geo}. \end{array} \end{equation}

The gradient of base vectors appears in $R_{\textit{geo}}$ , indicating the direct impact of geometric curvature on the flow evolution. Note that the geometric terms of the redistribution term and the pressure transport term are opposite, and the resulting impact is exactly offset. (2.11) has been thoroughly validated in the Cartesian, cylindrical and spherical coordinate systems. In the Cartesian coordinate system, the gradient of base vectors disappears, so $R_{\textit{geo}}$ is equal to zero. Detailed results of the budget analysis will be presented in § 6.

3. Flow field and turbulent statistics

The mean wall pressure coefficient $C_p$ of case 2 is plotted in figure 4, which is defined as ${C_p} = {{ ( {p - {p_\infty }} )} / { ( {0.5\rho U_\infty ^2} )}}$ ( $p_\infty$ is the reference pressure taken near the inflow). Basically, the bow surface can be divided into two parts. One is the favourable-pressure-gradient (FPG) region, in which the pressure drops rapidly downstream within a range of ${x / L} = 0{-}0.03$ or ${s / L} = 0{-}0.05$ . The other is the weak-pressure-gradient (WPG) region. In the WPG region, adverse and favourable pressure gradients alternate, being much weaker than that in the FPG region. The flow features of these two regions will be discussed and compared in detail. If not stated specifically, the following results are assumed to be from case 2. Velocities are non-dimensionalised by $U_\infty$ , and pressure is non-dimensionalised by ${\rho U_\infty ^2} / 2$ . The superscript ‘+’ represents that the quantities are non-dimensionalised by the wall viscous unit.

Figure 4.

Distribution of the mean wall pressure coefficient of case 2 in the FPG region and the WPG region. The red line represents the mean wall pressure coefficient, and the black line represents the local radius of the bow.

In figure 5(a,b), contours of instantaneous velocities and pressure are presented in the streamwise–radial ( $x {-} r$ ) plane. The strong turbulence from the inflow decays gradually along the streamwise direction in the upstream region ( $x \lt 0$ ). As the flow encounters the bow ( $x \gt 0$ ), a radial outward velocity ${U_r} \gt 0$ is formed. Using the Q-criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988), where $Q=-0.5({{\partial {u_i}}}/{{\partial {x_j}}})({{\partial {u_j}}}/{{\partial {x_i}}})$ , visualisation of the instantaneous near-wall vortical structures is plotted in figure 5(c). Patterns of vortices in the FPG and WPG regions are quite different. In the FPG region, the vortices from the incoming flow directly collide with the front of the bow, and then form the structures that are mainly stretched along the arc direction due to flow acceleration. These stretched structures can be observed more clearly at the zone between the FPG and WPG regions. In the WPG region, turbulence arises in the boundary layer, and the hairpin vortical structures are generated. Figure 6 also displays the instantaneous flow fields in the TBL around the bow. It can be seen that with the growth of the TBL thickness, the vortical structures evolve and gradually lift up. In the WPG region, low- and high-speed streaks are formed, and small-scale pressure fluctuation is observed over the surface.

Figure 5.

Contours of instantaneous flow fields in case 2: (a) streamwise velocity and radial velocity; (b) spanwise velocity and pressure coefficient; (c) vortical structures, coloured by tangential velocity.

Figure 6.

Instantaneous flow fields in the TBL over the bow surface: (a) vortical structures identified by Q-criterion; (b) WPF; (c) the low- and high-speed streaks in the ${n^ + } \approx 11$ surface; (d) velocity cross-sections at different streamwise positions; (e) TBL along the bow profile. (a,d,e) are coloured by the tangential velocity; (b) is coloured by WPF; (c) is coloured by the tangential velocity fluctuation. Only half of the streamwise length (x/L $\lt$ 0.18) is plotted.

The surfaces parallel to the bow wall are extracted at ${n^ + } \approx 20$ and ${n^ + } \approx 100$ , which are estimated by the friction velocity $u_\tau$ at the location ${s / L} = 0.32$ , as shown in figure 7. An enlarged view of the WPG region is presented in figure 7(e–h). In the FPG region, disturbance from the inflow is suppressed significantly by the strong FPG. The flow structures are produced mainly by the incoming flow, and thus the characteristic sizes of both the pressure and tangential velocity fluctuations are similar to those in the inflow, which are significantly larger than those in the WPG region. The pressure fluctuation structures are stretched along the spanwise direction in the WPG region, as seen in figure 7(a,b). The streaky structures are observed at ${n^ + } \approx 20$ in the buffer layer (figure 7 c), while such structures are not apparent at ${n^ + } \approx 100$ in the logarithmic layer (figure 7 d). Specifically, the average spacing between the neighbouring low-speed streaks is approximately $\Delta {d^ + } \approx 120$ in the $n^ +\approx 20$ surface at s/L = 0.32, which is slightly larger than $\Delta {d^ + } \approx 100$ observed in turbulent flow over a flat plate (Smith et al. Reference Smith and Metzler1983; Rajaee, Karlsson & Sirovich Reference Rajaee, Karlsson and Sirovich1995; Lagraa, Labraga & Mazouz Reference Lagraa, Labraga and Mazouz2004).

Figure 7.

Contours of pressure fluctuation (a,b,e,f) and velocity streaks (c,d,g,h). Panels (e–h) are the enlarged views of (a–d) marked by the square: (a,c,e,g) ${n^ + } \approx 20$ ; (b,d,f,h) ${n^ + } \approx 100$ .

Figure 8.

Spectra of streamwise velocity fluctuation (a,b), pressure fluctuation (c,d) and Reynolds shear stress (e,f) at different positions over the bow: (a,c,e) ${s / L} = 0.03$ ; (b,d,f) ${s / L} = 0.32$ . Colour contours represent the present data of case 2 and solid lines represent the turbulent channel flow data at $\textit{Re}{_\tau } = 360$ .

To further quantitatively provide the distribution of fluctuations at different scales, the spanwise pre-multiplied spectra of streamwise velocity fluctuation, pressure fluctuation and Reynolds shear stress are shown in figure 8. Two typical streamwise locations are selected: one is at ${s / L} = 0.03$ in the FPG region, and the local friction Reynolds number is $\textit{Re}_\tau = {{{u_\tau }{\delta _{99}}} / \nu } = 35$ , where $\delta _{99}$ is the nominal thickness of the boundary layer; the other is at ${s / L} = 0.32$ in the WPG region, where $\textit{Re}_\tau = 300$ . The turbulent channel flow data at $\textit{Re}_\tau = 360$ are also plotted, which are comparable to the spectra at ${s / L} = 0.32$ . In the FPG region, there is apparently energy scattered outside the boundary layer, while nearly all the energy is distributed within the boundary layer in the WPG region. The spanwise characteristic scales for velocity and pressure fluctuations are approximately 2500 viscosity units in the FPG region, which are an order of magnitude higher than those in the WPG region. Moreover, as shown in figure 8(e), different from the traditional channel flow, there exist two peaks in the FPG region for the spectrum of Reynolds shear stress. The negative peak indicates the negative Reynolds shear stress within the boundary layer, while the positive peak appears beyond the boundary layer, which reflects a typical feature of the bow flow.

Figure 9.

Boundary layer statistics for different cases: (a) nominal thickness; (b) shape factor; (c) momentum-thickness Reynolds number; (d) friction Reynolds number. In (a,c,d), brown dashed line: flat-plate theory 1, i.e. (A2) in Appendix A; blue dashed line: flat-plate theory 2, i.e. (A3) in Appendix A. In (b), black dashed line represents the streamwise pressure gradient $\text{d}{C_p}/\text{d}x$ , and its values correspond to the left $y$ -axis.

Figure 9 shows the evolution of the nominal thickness $\delta _{99}$ , the shape factor $H$ , the momentum-thickness Reynolds number $\textit{Re}{_\theta }$ and the friction Reynolds number $\textit{Re}_\tau $ along the bow surface. The theoretical results for the zero-pressure-gradient turbulent boundary layer (ZPGTBL) are also included for comparison. As shown in figure 9(a,c,d), $\delta _{99}$ , $\textit{Re}_\theta $ and $\textit{Re}_\tau $ increase monotonically along the bow surface for all the cases. However, in figure 9(b), the variation trend of $H$ is affected remarkably by both the pressure gradient and the boundary layer flow state. The streamwise pressure gradient $\text{d}{C_p}/\text{d}x$ for case 0 is included to show its relationship to the shape factor. When the ITI is relatively low, i.e. cases 0 and 1, the flow state remains laminar until downstream locations, and the variation of $H$ is mainly influenced by the streamwise pressure gradient; it roughly correlates with the pressure gradient ( $\text{d}{C_p}/\text{d}x$ ), i.e. $H$ increases (decreases) as $\text{d}{C_p}/\text{d}x$ increases (decreases). It should be mentioned that, owing to the effect of streamwise curvature of the bow surface, the variation trend of $H$ is different from that of a flat-plate boundary layer, where $H$ correlates with the pressure coefficient ( $C_p$ ) (Spalart & Watmuff Reference Spalart and Watmuff1993). When the ITI is high, i.e. cases 2 and 3, the development of the flow state plays a major role, and thus $H$ decreases monotonically downstream.

The ITI has a significant impact on the evolution of the boundary layer along the bow surface. For cases 0–2, the higher the turbulence intensity of the inflow, the earlier the turbulence occurs in the boundary layer. Therefore, with the increase of ITI, $\delta _{99}$ , $\textit{Re}_\theta $ and $\textit{Re}_\tau $ increase monotonically, and the shape factor decreases monotonically. According to the small difference between case 2 and case 3 in the FPG region, as seen in figure 9, larger ITI will not cause the location of turbulence to advance to the FPG region further. The large inflow disturbance does not excite the turbulence to arise in the boundary layer, owing to the stabilising effect of the strong FPG in the FPG region (Finnicum & Hanratty Reference Finnicum and Hanratty1988; Balin & Jansen Reference Balin and Jansen2021). That is why the tripping devices are not placed close to the stagnation point $x = 0$ , both in the previous simulations and experiments (Jiménez et al. Reference Jiménez, Hultmark and Smits2010b , Reference Jiménez, Reynolds and Smits2010c ; Erm, Jones & Henbest Reference Erm, Jones and Henbest2012; Posa & Balaras Reference Posa and Balaras2016; Saeidinezhad, Dehghan & Dehghan Manshadi Reference Saeidinezhad, Dehghan and Dehghan Manshadi2020; Morse & Mahesh Reference Morse and Mahesh2023).

The profiles of mean tangential velocity at different streamwise locations are plotted in figure 10. The locations are at ${s / L} = 0.006$ , 0.01 and 0.03 in the FPG region, and ${s / L} = 0.09$ , 0.21 and 0.32 in the WPG region. In the FPG region, the profiles collapse well for the different cases, due to the fact that there is no turbulence arising in this region. The acceleration of flow causes the upward shift of velocity profiles. In the WPG region, the log layer is observed first in case 3, which has the largest ITI. With the development of TBL, the velocity profiles in cases 2 and 3 converge gradually as the turbulence becomes fully developed. Considering the later occurrence of turbulence in cases 0 and 1, the log layer does not appear before ${s / L} = 0.32$ .

Figure 10.

Mean tangential velocity profiles at different streamwise locations in (a,b,c) the FPG region and (d,e,f) the WPG region. The inset shows the locations for different points: p1 ( ${s / L} = 0.006$ ), p2 ( ${s / L} = 0.01$ ), p3 ( ${s / L} = 0.03$ ), p4 ( ${s / L} = 0.09$ ), p5 ( ${s / L} = 0.21$ ), p6 ( ${s / L} = 0.32$ ); the dashed red line represents the bow profile.

Figure 11 shows the profiles of velocity fluctuations and Reynolds shear stress at different streamwise locations for case 2. In the FPG region, the velocity fluctuations decrease downstream along the bow due to the FPG. In the WPG region, the pressure gradient is relatively small, and the velocity fluctuations increase along the streamwise direction within the boundary layer, but the opposite trend is observed outside the boundary layer. The amplitude of Reynolds shear stress is relatively small in the FPG region, and increases continuously in the WPG region. In addition, in the FPG region and the early WPG region, the positive value of Reynolds shear stress occurs at the outer region or beyond the boundary layer, which can be also found in figure 8(e).

Figure 11.

Profiles of (a) the root mean square (r.m.s.) of tangential velocity fluctuation, (b) r.m.s. of normal velocity fluctuation, (c) r.m.s. of spanwise velocity fluctuation and (d) Reynolds shear stress at different streamwise locations for case 2.

4. Wall pressure fluctuation

Wall pressure fluctuation over the bow surface in cases 1, 2 and 3 is shown in figure 12. The disturbance from the incoming flow produces the pressure fluctuation over the bow surface, and it decays rapidly due to the strong FPG in the FPG region. In case 1, the WPF decreases to almost zero due to small ITI, and thus the turbulence generation only occurs in the posterior area of the WPG region, where a turbulent spot can be recognised in figure 12(a). In cases 2 and 3, the WPF after the FPG region is still large enough, leading to the turbulence generation in the boundary layer. The characteristic scale of the pressure fluctuation in the FPG region is much larger than that in the WPG region, which can be also observed in figure 7.

Figure 12.

Contours of WPF over the bow surface in different cases: (a) case 1; (b) case 2; (c) case 3. Supplementary movie 1 is available online.

The mean pressure coefficient $C_p$ is plotted in figure 13(a), where the previous experiment (Huang & Liu Reference Huang and Liu1994) and numerical simulation (Gross, Kremheller & Fasel Reference Gross, Kremheller and Fasel2011) are included for comparison. The results collapse well and show a good agreement. Note that the experiment of Huang & Liu (Reference Huang and Liu1994) was conducted at ${{\mathop {\textit{Re}}\nolimits } _L} = 1.2 \times {10^7}$ , and the Reynolds number of flow simulation in Gross et al. (Reference Gross, Kremheller and Fasel2011) is ${{\mathop {\textit{Re}}\nolimits } _L} = 1.4 \times {10^7}$ . The mean pressure coefficient is not affected by the ITI and the Reynolds number. The radius of streamwise curvature $R_{\textit{cur}v\textit{e}}$ is also plotted, appearing to roughly correlate with $C_p$ , both of which increase in the regions of $x/L=0.04{-}0.11$ and $x/L=0.18{-}0.23$ , and decrease in the region of $x/L=0.11{-}0.18$ . Therefore, the mean pressure coefficient is mainly related to the geometric curvature if there is no trip wire at the wall. Root-mean-square values of WPF are presented in figure 13(b) for cases 0–3. Higher ITI leads to larger pressure fluctuation. In the FPG region, the WPF falls rapidly, and the value becomes nearly zero in case 1. For cases 2 and 3, the changing trend of pressure fluctuation is sensitive to several aspects. Firstly, an important factor is the ITI, as mentioned above. Secondly, the pressure fluctuation changes with the pressure gradient. The adverse pressure gradient causes the fluctuation to increase along the bow. In other words, the pressure fluctuation is affected by the geometric curvature of the bow. Finally, approaching to turbulent equilibrium state will also affect the distribution of the pressure fluctuations. In the rear position of the WPG region, the flow is close to a fully developed turbulent state, resulting in collapse of the curves for cases 2 and 3.

Figure 13.

Mean value (a) and r.m.s. (b) of pressure coefficient over the bow surface. The dashed red line represents the radius of streamwise curvature of the bow.

Figure 14 shows the spectra of WPF for different inflow turbulence intensities and different streamwise locations. Three scaling schemes are applied to characterise the spectra to show the differences between the FPG and WPG regions. Note that, for case 1, turbulence in the boundary layer occurs after ${x / L} = 0.37$ , and thus the WPF signals in the WPG region are selected at the turbulence generation region, different from cases 2 and 3. The scaling methods of WPF spectra with outer/inner variables have been used in previous studies, such as the pressure spectrum scaling in turbulent flow over a flat plate (Kim Reference Kim1989; Farabee & Casarella Reference Farabee and Casarella1991; Hu et al. Reference Hu, Morfey and Sandham2006), a rough wall (Blake Reference Blake2017), an axisymmetric body (Hong et al. Reference Hong, Shin, Jeon and Kim2008; Hong Reference Hong2015) and other curved walls (Goody & Simpson Reference Goody and Simpson2000; Lee & Sung Reference Lee and Sung2001). Note that the outer-layer variable scaling scheme 2 in figure 14(c,d) shows some difference from the previous studies, due to the influence of ITI in the present study. As shown in figure 14(a,b), higher turbulence level leads to larger pressure fluctuation, causing the upward shift of spectra from case 1 to case 3. In the FPG region, the pressure spectra shift downward along the arc direction due to the suppression effect of the FPG. In the WPG region, turbulence arises, causing the growth of amplitude of the pressure spectra, especially in the high-frequency region.

In the high-frequency region, the scaling law ${\varPhi _{pp}} ( \omega ) \sim {\omega ^{ - 5}}$ is found in figure 14, which is the typical characteristic of ZPGTBL (Goody & Simpson Reference Goody and Simpson2000; Aupoix Reference Aupoix2015). In the FPG region, slight offset to the law is recognised owing to the strong pressure gradient. Since small-scale turbulence structures do not appear in the FPG region, as seen in figure 8, it is reasonable to scale the pressure spectra by the outer-layer variable scheme 1 in figure 14(a) (Hong et al. Reference Hong, Shin, Jeon and Kim2008). However, the collapse of spectra for cases 1–3 at different positions is not observed with this scaling scheme. Considering that both different inflow turbulent intensities and continuously changing the pressure gradient along the bow lead to variation of the pressure fluctuation, it is better to introduce the r.m.s. of WPF as the characteristic quantity (scheme 2), as shown in figure 14(c). This scaling scheme results in a more satisfactory collapse of the pressure spectra in the FPG region. In the WPG region, the low-frequency wall pressure spectra are scaled fairly well by the outer-layer variable schemes 1 and 2, although there is a considerable discrepancy for the high-frequency spectra. According to the previous experimental and numerical studies (Farabee & Casarella Reference Farabee and Casarella1991; Lee, Blake & Farabee Reference Lee, Blake and Farabee2005), the inner-layer variable is more appropriate to scale the spectra in the high-frequency region. Due to the small-scale structures, it is helpful to fit all the spectra in the WPG region for cases 1–3 by introducing the inner-layer variable, as seen in figure 14(f).

Figure 14.

The WPF spectra over the bow surface scaled by different variables in (a,c,e) the FPG region (solid line: ${s / L} = 0.006$ , dash line: ${s / L} = 0.01$ , dash dot line: ${s / L} = 0.03$ ) and (b,d,f) the WPG region (solid line: ${s / L} = 0.09$ for case 2 and 3, ${s / L} = 0.39$ for case 1; dash line: ${s / L} = 0.21$ for case 2 and 3, ${s / L} = 0.50$ for case 1; dash dot line: ${s / L} = 0.32$ for case 2 and 3, ${s / L} = 0.62$ for case 1). Different scaling schemes are applied: (a,b) the outer-layer variable scaling scheme 1, represented by ${{\varPhi _{pp}}} / {( {{\delta _{99}}{\rho ^2}{U^3}})}$ versus ${\omega {\delta _{99}}} / U$ ; (c,d) the outer-layer variable scaling scheme 2, represented by ${{\varPhi _{pp}}U} / {( {{\delta _{99}}p\prime _{\textit{rms}}^2} )}$ versus ${\omega {\delta _{99}}} / U$ ; (e, f) the inner-layer variable scaling, represented by ${{\varPhi _{pp}}} / {( {\nu {\rho ^2}u_\tau ^2})}$ versus ${\omega \nu } / {u_\tau ^2}$ , where $U$ is the mean tangential velocity at the edge of the boundary layer and $p{'_{\textit{rms}}}$ is the r.m.s. of WPF.

The spatial distributions of WPF spectra in cases 1 and 2 are presented in figures 15 and 16, respectively. Moreover, the spectra averaged along the circumferential direction are also presented in figure 17. For case 1, which has low ITI (figures 15 and 17 a), the WPF spectrum drops significantly in the FPG region at low frequency (St = 2.3). As the frequency becomes higher (St $\gt$ 29.6), the WPF spectrum shows the same variation trend as the streamwise curvature radius in figure 13(a). At around x/L = 0.3, a peak of the WPF spectrum is observed at the low-frequency (St $\lt$ 29.6) area and a valley exists as the frequency becomes higher (St $\gt$ 29.6), which shows that the natural transition occurs in this region.

Figure 15.

The WPF spectra over the bow surface at different frequencies for case 1. The non-dimensional frequency is denoted by the Strouhal number $St = {{fL} / U_\infty }$ , and the spectrum $\varPhi _{pp}$ is scaled by ${p\prime _{\textit{rms}}^2L} / {{U_\infty }}$ .

Figure 16.

The WPF spectra over the bow surface at different frequencies for case 2. The frequency and spectrum are scaled by the same variables as in figure 15.

Figure 17.

Variations of the WPF spectra along the streamwise direction at different frequencies for (a) case 1 and (b) case 2, which are obtained by averaging over the circumferential direction.

For case 2 (figures 16 and 17 b), the WPF spectra show a similarly changing trend in the FPG region. Owing to the turbulence generation in the WPG region, both the body shape and the ITI act together in the mid- and high-frequency regions (St = 18.2–293.7). The WPF spectrum is larger at the straight section (x/L $\gt$ 0.23) in the mid-frequency region (St = 18.2–29.6), where the turbulence energy is higher. By comparing the variations of the WPF spectra in figure 17(b) and the radius of streamwise curvature of the bow in figure 13(a), when the frequency is much higher (St $\gt$ 473.5), there is a strong peak caused by the body shape of the bow.

5. Skin friction

Instantaneous distributions of the skin friction fluctuation over the bow surface in cases 1–3 are presented in figure 18. The skin friction coefficient $C_f$ is defined as ${C_f} = {{{\tau _w}} / { ( {0.5\rho U_\infty ^2} )}}$ , where $\tau _w$ is the wall shear stress at the bow surface. The turbulence intensity in case 1 is relatively low, leading to the late generation of turbulence, and a turbulent spot during the transition process is observed, similar to that in Wu (Reference Wu2023). For the situation with higher inflow turbulence intensities in cases 2 and 3, the streaky structures arise at the front of the WPG region, showing evidence of turbulence generation. The length scale of velocity fluctuation in the FPG region is much larger than that in the WPG region, due to the impact of incoming flow.

Figure 18.

Contours of skin friction fluctuation over the bow surface in different cases: (a) case 1; (b) case 2; (c) case 3. Supplementary movie 2 is available online.

The mean skin friction coefficient $C_{\!f}$ along the bow surface is presented in figure 19(a). Different from the pressure coefficient, $C_{\!f}$ is sensitive to the Reynolds number and the ITI, and no self-similarity can be observed in any of the cases. In the WPG region, higher turbulence intensity results in larger skin friction. The differences between case 0 and the other cases are ${C_{\!f,0}} = {C_{\!f,Casei}} - {C_{f,Case{\textrm {0}}}}$ , where i = 1, 2, 3, are given in figure 19(c). For case 1, the position of turbulence generation is approximately at ${x / L} = 0.37$ . After that, turbulence develops and $C_{f,0}$ grows continuously. For cases 2 and 3, in the FPG region, there exists very slight discrepancy with case 0, and thus $C_{f,0}$ is almost zero. Subsequently, thanks to the small adverse pressure gradient at the front part of the WPG region, turbulence arises immediately, and $C_{f,0}$ grows rapidly. At the straight section, the skin friction coefficients in cases 2 and 3 collapse as the turbulence equilibrium state is reached. Root-mean-square values of skin friction fluctuation $C^{\prime }_{f,\textit{rms}}$ are plotted in figure 19(b). It is seen that the higher the ITI, the larger the skin friction fluctuation along the bow surface. The skin friction fluctuation along the whole bow surface is zero in case 0. For case 1, $C^{\prime }_{f,\textit{rms}}$ decays continuously until the growth of turbulence fluctuation. For cases 2 and 3, $C^{\prime }_{f,\textit{rms}}$ decreases over the front of the bow. In the region with small adverse pressure gradient, the active disturbance leads to turbulence generation rapidly. At the straight section of the bow, the skin friction fluctuation in cases 2 and 3 is almost the same.

Figure 19.

Mean value (a), r.m.s. (b) and net mean value (c) of skin friction coefficient over the bow surface.

Furthermore, based on the dynamic decomposition of $C_{\!f}$ in the curvilinear coordinates ( $s,n,\theta$ ), as dictated in (2.9), the generation mechanism of the skin friction can be clarified. According to our calculation of (2.9) by using the flow field data, the results show that the terms related to the geometric parameters $K_{\textit{ij}}$ are negligible, as well as $C_{\nu 2}$ and $C_{D2}$ , and that ${{\nabla} ^2}{U_s} \approx {{{\partial ^2}{U_s}} / {\partial {n^2}}}$ . Therefore, (2.9) can be simplified as

(5.1) \begin{equation} \begin{array}{l} {C_{\!f}} \approx \underbrace {\frac {2}{{{\textit{Re}}}}{{\left . {{U_s}} \right |}_{{\delta _{99}}}}\beta }_{{C_\nu }}\underbrace { - 2\int _0^1 {\frac {r}{{{{\left . r \right |}_0}}}\overline {{{u'}_{\!s}}{{u'}_{\!n}}} {\rm d}n} }_{{C_R}}\underbrace { - 2\int _0^1 {\left ( {1 - n} \right )\frac {1}{h_s}\frac {{\partial \overline {{{u'}_{\!s}}{{u'}_{\!s}}} }}{{\partial s}}{\rm d}n} }_{{C_D}}\\ \underbrace { - 2\int _0^1 {\left ( {1 - n} \right )\left ( {{U_s}\frac {1}{h_s}\frac {{\partial {U_s}}}{{\partial s}} + {U_n}\frac {{\partial {U_s}}}{{\partial n}}} \right ){\rm d}n} }_{{C_{C1}}}\\ \underbrace { - 2\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big)\left ( {{U_s}\frac {1}{h_s}\frac {{\partial {U_s}}}{{\partial s}} + {U_n}\frac {{\partial {U_s}}}{{\partial n}}} \right ){\rm d}n} }_{{C_{C2}}}\\ \underbrace { - 2\int _0^1 {\left ( {1 - n} \right )\frac {1}{h_s}\frac {{\partial P}}{{\partial s}}{\rm d}n} }_{{C_{P1}}}\underbrace { - 2\left ( {\beta - 1} \right )\int _0^1 {\big( {1 - {n^2}} \big)\frac {1}{h_s}\frac {{\partial P}}{{\partial s}}{\rm d}n} }_{{C_{P2}}}. \end{array} \end{equation}

Note that the geometric effect is not explicitly reflected in the above equation. Instead, indirect influences of geometric curvature play a role by altering the mean flow field and modulating the turbulence development along the bow surface, which subsequently impacts the terms of (5.1).

Figure 20.

Dynamic decomposition of the skin friction coefficient: (a) comparison with the value calculated by definition; (b–d) contribution ratios of different terms.

The skin friction coefficient $C_{f,d}$ calculated by (5.1) is compared with that obtained directly by its definition ${C_{\!f}} = {{ ( {{{\nu \partial {U_s}} / {\partial n}}} )} / { ( {0.5U_\infty ^2} )}}$ in figure 20(a), which shows a good agreement. Ratios of the terms in (5.1) to local mean skin friction coefficient are shown in figure 20(b–d). The variations of all the terms are sensitive to the streamwise curvature of the bow. In other words, the bluff body induces variations of the mean velocity and pressure gradient, which then contribute to the other terms. As a consequence of the flow acceleration in the FPG region, the mean tangential velocity of external flow $\left . {{U_s}} \right |_{{\delta _{99}}}$ grows rapidly along the arc direction from the stagnation point. Despite the decrease of the Reynolds number ${\mathop {\textit{Re}}\nolimits } = {{{U_\infty }{\delta _{99}}} / \nu }$ , the fast increase of $\left . {{U_s}} \right |_{{\delta _{99}}}$ causes the sustainable growth of $C_\nu$ , which cannot be ignored in the FPG region and the front part of the WPG region. The Reynolds shear stress term $C_R$ is nearly zero in the front part of the bow owing to the turbulence suppression effect. In the WPG region, turbulence grows gradually, and the ratio of $C_R$ keeps the value of approximately 40 % as the turbulence reaches its equilibrium state. Here, $C_D$ is relatively small over the whole bow surface, and is zero in the ZPGTBL flow. Its two growth zones correspond to the area of increasing curvature of the bow, which is also the region where the adverse pressure gradient exists (seen in figure 13 a). This reflects the influence of geometric shape on the development of Reynolds stress. The values of $C_{C1}$ , $C_{C2}$ , $C_{P1}$ and $C_{P2}$ also show a similarly changing trend. In figure 20(c), it can be seen that the mean convection and pressure-gradient terms, $C_{C1}$ and $C_{P1}$ , contribute to the local skin friction the most overall. The most obvious inflection point of $C_{C1}$ and $C_{P1}$ appears at the junction of the curved wall and the straight sections, at approximately ${x / L} = 0.23$ . Note that $C_{P1}$ reflects the influence of not only the pressure gradient, but also the boundary layer thickness. Therefore, considering their combined influences, the maximum value appears at ${x / L} = 0.23$ , rather than at the stagnation point intuitively, where the largest pressure gradient exists. In figure 20(d), it can be considered that ${C_{C2}} \approx 0.1{C_{C1}}$ and ${C_{P2}} \approx 0.1{C_{P1}}$ , owing to the fact that the boundary layer thickness is an order of magnitude smaller than the local radius of the bow. Nonetheless, the contribution of $C_{C2}$ and $C_{P2}$ to the skin friction cannot be ignored. It should be mentioned that the mean-pressure-gradient terms $C_{P1}$ and $C_{P2}$ also reflect the effect of the streamwise curvature, since the radius of streamwise curvature $R_{\textit{cur}v\textit{e}}$ plotted in figure 13(a) appears to roughly correlate with the mean pressure coefficient $C_{p}$ .

6. Budget analysis

For the budget analysis in the curvilinear coordinate system, the Reynolds stress transport equations are given in (2.11), including the production term $P$ , the turbulent transport term $DT$ , the pressure transport term $DP$ , the viscous diffusion term $D\nu$ , the pressure-strain or redistribution term $\varPhi$ , the dissipation term $\varepsilon$ and the geometric term $R_{\textit{geo}}$ as given by (2.12). In the curvilinear coordinate ( $s,n,\theta$ ) system, two streamwise locations are selected to present the budget analysis for case 2. One is at ${s / L} = 0.01$ in the FPG region, where the friction Reynolds number is $\textit{Re}_\tau = 20$ , and the other is at ${s / L} = 0.32$ in the WPG region with $\textit{Re}_\tau = 300$ . The budget terms for the tangential, normal and spanwise Reynolds normal stresses and Reynolds shear stress are shown in figure 21. Note that every term in figure 21 is the combination of geometric and non-geometric terms. In the rear position of the WPG region, the budget relationships for the different components are quite similar to those in a turbulent channel flow (Lee & Moser Reference Lee and Moser2018). For the areas beyond the boundary layer ( ${n^ + } \gt 300$ ), the budget terms fall to zero. On the contrary, the budget relationships in the FGP region exist not only within the boundary layer. The turbulent transport term, viscous diffusion term and dissipation term drop to zero beyond the boundary layer, while the other non-zero terms in (2.11) extend outward in the normal direction, much further away from the wall than the boundary layer thickness.

Figure 21.

Budget terms in case 2: (a,c,e,g) ${s/ L} = 0.01$ ; (b,d,f,h) ${s/ L} = 0.32$ . (a,b) Tangential, (c,d) normal and (e,f) spanwise components of Reynolds normal stresses; (g,h) Reynolds shear stress. The dash-dotted black lines represent locations of the outer edge of the boundary layer.

Figure 22.

Budget terms for Reynolds shear stress at ${s / L} = 0.01$ in case 2, with summation of $DP$ and $\varPhi$ .

Figure 23.

Key budget terms at different streamwise positions for case 2: (a,c,e,g) the production term; (b,d,f,h) the redistribution term. (a,b) Tangential, (c,d) normal and (e,f) spanwise components of Reynolds normal stresses; (g,h) Reynolds shear stress.

To clarify the differences between the FPG and WPG regions more clearly, the budget relationships in the FPG region are presented in more detail. As shown in figure 21(a) for $\overline {{{u'}_s}{{u'}_s}}$ , the pressure-strain term plays a major role in the budget. A negative production term is observed, which reflects the suppression effect on turbulence generation in this region. For the normal component (figure 21 c), only the influences of pressure fluctuation, $DP$ , and $\varPhi$ dominate within the boundary layer, while outside of the boundary layer a positive production term becomes the main term, which is zero throughout the boundary layer in turbulent channel flow. For the spanwise component (figure 21 e), a non-zero production term is balanced with $\varPhi$ , especially outside of the boundary layer. For the Reynolds shear stress (figure 21 g), due to the large pressure fluctuation caused by the disturbance from the incoming flow, $DP$ and $\varPhi$ are dominant across the entire region. Since the values of $DP$ and $\varPhi$ are nearly opposite, to show the distribution of the other terms, they are combined and are compared with the other terms in figure 22. It is seen that their summation balances with the production term in the FPG region. Similar to figure 21(a,e), the production term is negative, indicating the suppression effect on turbulence generation.

Since the production and redistribution terms are the key terms in the budget of the Reynolds stresses, they are plotted at different streamwise positions for case 2 in figure 23. Due to the suppression effect of FPG, the key terms drop rapidly in the FPG region. In the WPG region, the turbulence develops in the boundary layer. The same changing trends are observed as those of velocity fluctuations and Reynolds shear stress, as shown in figure 11.

Now the geometric effects are discussed according to (2.12). Due to the fact that the large streamwise curvature exists in the FPG region, the geometric effects are investigated only in this region. The geometric effects of the turbulent transport term and viscous term are small, compared with that of the production term. Therefore, only the geometric and non-geometric parts of the production term are considered, which are expressed as

(6.1) \begin{align} \begin{array}{l} \textit{For}\,\overline {{{u'}_s}{{u'}_s}} ,\left \{ \begin{array}{l} {P_{\textit{non}} - {\textit{geo}}} = - 2\overline {{{u'}_s}{{u'}_s}} \dfrac {1}{{{h_s}}}\dfrac {{\partial {U_s}}}{{\partial s}} - 2\overline {{{u'}_s}{{u'}_n}} \dfrac {{\partial {U_s}}}{{\partial n}},\\{P_{\textit{geo}}} = - 2\overline {{{u'}_s}{{u'}_s}} {K_{\textit{sn}}}{U_n}. \end{array} \right .\\[16pt]\textit{For}\,\overline {{{u'}_n}{{u'}_n}} ,\left \{ \begin{array}{l} {P_{\textit{non}} - {\textit{geo}}} = - 2\overline {{{u'}_s}{{u'}_n}} \dfrac {1}{{{h_s}}}\dfrac {{\partial {U_n}}}{{\partial s}} - 2\overline {{{u'}_n}{{u'}_n}} \dfrac {{\partial {U_n}}}{{\partial n}},\\{P_{\textit{geo}}} = 2\overline {{{u'}_s}{{u'}_n}} {K_{\textit{sn}}}{U_s}. \end{array} \right .\\[16pt]\textit{For}\,\overline {{{u'}_\theta }{{u'}_\theta }} ,\left \{ \begin{array}{l} {P_{\textit{non}} - {\textit{geo}}} = 0,\\{P_{\textit{geo}}} = - 2\overline {{{u'}_\theta }{{u'}_\theta }} \left ( {{K_{\theta s}}{U_s} + {K_{\theta n}}{U_n}} \right )\! .\end{array} \right .\\[12pt]\textit{For} - \overline {{{u'}_s}{{u'}_n}} ,\left \{ \begin{array}{l} {P_{\textit{non}} - {\textit{geo}}} = 2\overline {{{u'}_n}{{u'}_n}} \dfrac {{\partial {U_s}}}{{\partial n}} + \overline {{{u'}_s}{{u'}_s}} \dfrac {1}{{{h_s}}}\dfrac {{\partial {U_n}}}{{\partial s}} + \overline {{{u'}_s}{{u'}_n}} \left ( {\dfrac {1}{{{h_s}}}\dfrac {{\partial {U_s}}}{{\partial s}} + \dfrac {{\partial {U_n}}}{{\partial n}}} \right )\!,\\{P_{\textit{geo}}} = - \overline {{{u'}_s}{{u'}_s}} {K_{\textit{sn}}}{U_s} + \overline {{{u'}_s}{{u'}_n}} {K_{\textit{sn}}}{U_n}, \end{array} \right . \end{array} \end{align}

and are shown in figure 24 for different components of the Reynolds stresses. In the FPG region, the Reynolds shear stress is much smaller than the Reynolds normal stresses, and the terms with $- \overline {{{u'}_s}{{u'}_n}}$ in (6.1) can be ignored. Thus, as seen in figure 24, for the production term of $\overline {{{u'}_n}{{u'}_n}}$ , the non-geometric term is dominant and the geometric effect is negligible. Flow acceleration in the FPG region leads to significant ${\partial {U_s}} / {\partial s}$ , explaining the fact that the non-geometric term plays a major role for the production term of $\overline {{{u'}_s}{{u'}_s}}$ . For the production term of $\overline {{{u'}_\theta }{{u'}_\theta }}$ , the non-geometric term is zero, and the geometric effect contributes to the whole production term. For the production term of $- \overline {{{u'}_s}{{u'}_n}}$ , the geometric and non-geometric effects have similar magnitudes, as seen in figure 24(d).

Figure 24.

Geometric (solid lines) and non-geometric (dashed lines) parts of the production term for case 2: (a) tangential, (b) normal and (c) spanwise components of Reynolds normal stresses; (d) Reynolds shear stress.

According to the above budget analysis, the paths of energy transfer in the FPG region are summarised in figure 25. The FPG suppresses the development of turbulence, and causes the decreases of absolute values of budget terms. As a result, the intensities of Reynolds normal stresses and Reynolds shear stress are decreased rapidly along the streamwise direction. Here, $P$ and $\varPhi$ play the dominant roles in the budget of Reynolds normal stresses outside of the boundary layer, while only $DP$ and $\varPhi$ are dominant for the Reynolds shear stress. The geometric effects are reflected in the production term, especially for the spanwise Reynolds normal stress and Reynolds shear stress.

Figure 25.

Schematic of paths of energy transfer in the FPG region. Red arrows denote the changing of absolute values; black arrows denote the directions of paths of energy transfer; green arrows denote the dominant budget relationships.