1. Introduction
Shock wave–turbulent boundary-layer interactions (STBLIs) occur in a broad range of high-speed devices, including supersonic intakes, over-expanded rocket nozzles and transonic airfoils. The presence of low-frequency unsteadiness and large-scale flow structures in STBLIs has been extensively reported in previous studies (Dussauge, Dupont & Debiève Reference Dussauge, Dupont and Debiève2006; Clemens & Narayanaswamy Reference Clemens and Narayanaswamy2014).
Significant efforts have been devoted to exploring the physical origin of low-frequency unsteadiness in STBLIs. Two main mechanisms have been proposed to explain the unsteadiness: upstream and downstream mechanisms (Souverein et al. Reference Souverein, Dupont, Debieve, Dussauge, Van, Bas and Scarano2010; Clemens & Narayanaswamy Reference Clemens and Narayanaswamy2014). The upstream mechanism suggests that the shock motions are influenced by the upstream turbulent boundary layer (TBL). Beresh, Clemens & Dolling (Reference Beresh, Clemens and Dolling2002) identified a relationship between the upstream TBL and the shock motions: a fuller velocity profile causes the shock to move downstream (and vice versa). This relationship was verified by Hou, Clemens & Dolling (Reference Hou, Clemens and Dolling2003) using particle image velocimetry (PIV) experimentally. Later, Ganapathisubramani et al. (Reference Ganapathisubramani, Clemens and Dolling2007, Reference Ganapathisubramani, Clemens and Dolling2009) discovered superstructures in the upstream TBL with streamwise lengths of
$(40{-}50)\delta _{0}$
using PIV and Taylor’s hypothesis, where
$ \delta _{0}$
is the boundary-layer thickness. These superstructures were believed to be vital in driving the low-frequency unsteadiness. Regarding the downstream mechanism, much attention has been paid to connecting the dynamics of the separation bubble with the unsteady separation shock movement. Pirozzoli & Grasso (Reference Pirozzoli and Grasso2006) performed a short-time direct numerical simulation in a Mach 2.25 impinging shock interaction. They modelled the low-frequency unsteadiness as acoustic resonance similar to Rossiter modes (Rossiter Reference Rossiter1964) in cavity flows. Piponniau et al. (Reference Piponniau, Dussauge, Debieve and Dupont2009) investigated the relation between the shock position and the turbulent separation bubble (TSB) size based on conditionally averaged PIV velocity fields of an impinging shock interaction. They found that the shock moves upstream when the TSB size is large and downstream when the TSB size is small. They proposed a self-sustained model in which mass loss due to fluid entrainment is recharged by the flapping of the TSB. Subsequently, Touber & Sandham (Reference Touber and Sandham2011) developed a mathematical model to illustrate the low-frequency motions, similar to the Plotkin model (Plotkin Reference Plotkin1975). They argued that the low-frequency unsteadiness is an intrinsic property but needs to be excited by extrinsic low-frequency disturbances. Grilli et al. (Reference Grilli, Schmid, Hickel and Adams2012) and Pasquariello, Hickel & Adams (Reference Pasquariello, Hickel and Adams2017) also numerically verified the intrinsic mode in a compression ramp and impinging shock interaction, respectively. Recently, Hao (Reference Hao2023) drew a similar conclusion based on Reynolds-averaged Navier–Stokes (RANS) equations and linear stability analysis. Clemens & Narayanaswamy (Reference Clemens and Narayanaswamy2014) summarised that both upstream and downstream mechanisms may work in shock-induced turbulent separated flows, whereas the dominant mechanism may depend on the interaction strengths.
Although much work has been done to understand the nominally two-dimensional (2-D) dynamics of STBLIs, three-dimensional (3-D) effects associated with TSBs have received less attention, particularly concerning large-scale spanwise structures. These spanwise structures may lead to streaks of low and high heat flux distributions. Two types of large-scale spanwise structures have been reported: Görtler-like vortices downstream of the interaction region and large-scale spanwise motions within the interaction region. Generally, the scale of Görtler-like vortices is comparable to the boundary-layer thickness, while the latter’s size is comparable to that of the separation region.
Streamwise vortices downstream of the interaction region have been widely reported in turbulent flows by numerical simulations (Loginov, Adams & Zheltovodov Reference Loginov, Adams and Zheltovodov2006; Priebe et al. Reference Priebe, Tu, Rowley and Martín2016; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017) and experiments (Schuelein & Trofimov Reference Schuelein and Trofimov2011; Li et al. Reference Li, Zhang, Yu, Lin, Tan and Sun2022). These streamwise vortices, often referred to as Görtler-like vortices, exhibit wavelengths of approximately
$ 2\delta _0$
and are primarily caused by centrifugal effects. Floryan (Reference Floryan1991) suggested that the Görtler-like vortices are influenced by disturbances in the upstream TBL and classified them as either steady or unsteady. The steady pattern means these vortices are visible in the time-averaged flow, typically represented by the contour of the mean skin-friction coefficient. For turbulent separated flows, both steady (Loginov et al. Reference Loginov, Adams and Zheltovodov2006; Grilli, Hickel & Adams Reference Grilli, Hickel and Adams2013; Tong et al. Reference Tong, Yu, Tang and Li2017; Helm & Martín Reference Helm and Martín2021) and unsteady (Priebe et al. Reference Priebe, Tu, Rowley and Martín2016; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017; Li et al. Reference Li, Zhang, Yu, Lin, Tan and Sun2022) patterns have been reported. However, there is no comprehensive research to elucidate the essential reasons for these different patterns, to the best of the authors’ knowledge.
For the large-scale spanwise structures within the interaction region, Jenquin, Johnson & Narayanaswamy (Reference Jenquin, Johnson and Narayanaswamy2023) observed large-scale spanwise pressure undulations within the intermittent region at Mach 2.5 using 2-D surface pressure field imaging. They found that the shear layer events induced the upstream-propagating pressure perturbations in the vicinity of the reattachment region, which influenced the streamwise motions of the separation shock. However, the authors clearly noted that the sources of spanwise rippling in the separation shock could not be identified due to limitations in the signal-to-noise ratio. Ceci et al. (Reference Ceci, Palumbo, Larsson and Pirozzoli2023, Reference Ceci, Palumbo, Larsson and Pirozzoli2024) conducted a series of non-swept and swept impinging shock interactions in wide computational domains. Their results revealed that spanwise structures near the mean separation line are proportional to the separation bubble size
$ {L}_{\textit{sep}}$
, with a characteristic spanwise wavelength of
$ {\lambda }_{z}\approx 2{L}_{\textit{sep}}$
. They proposed that these spanwise structures were responsible for the rippling of the separation line, behaving as large-scale spanwise corrugations. For non-swept STBLIs, they regarded these structures as the signatures of 2-D breathing motions. Liu et al. (Reference Liu, Chen, Zhang, Tan, Liu and Peng2024) investigated multiscale spanwise unsteadiness in a sidewall-confined
$ {25}^{\circ }$
compression ramp interaction. Their modal analysis revealed that low-frequency unsteadiness in the intermittent region consists of quasi-2-D streamwise oscillations and spanwise unsteadiness. While the streamwise oscillations correspond to the 2-D shock motions, the origins of the large-scale spanwise unsteadiness remained unknown. Similar large-scale structures associated with TSBs have also been reported in incompressible flows by Dau et al. (Reference Dau, Borgmann, Little and Weiss2023) and Borgmann et al. (Reference Borgmann, Cura, Weiss and Little2024). They observed spanwise structures with a wavelength of
$ 0.8 {L}_{\textit{sep}}$
using spectral proper orthogonal decomposition (SPOD) of PIV data. Despite the differences in flow conditions and reported spanwise wavelengths, these studies consistently indicate the potential existence of large-scale spanwise structures associated with TSBs. These structures may provide a basis for understanding and modelling the 3-D motions of TSBs in nominally 2-D STBLIs.
Regarding the origins of the two types of large-scale spanwise structures: (i) Görtler-like vortices, which primarily arise from streamline concave curvature effects induced by TSBs; (ii) the recently reported structures within the intermittent region, whose physical explanation remains an open question. As suggested by Liu et al. (Reference Liu, Chen, Zhang, Tan, Liu and Peng2024), stability analysis methods may provide valuable insights to the origins of the latter spanwise structures. Stability analysis methods, including resolvent analysis (McKeon & Sharma Reference McKeon and Sharma2010) and global stability analysis (GSA), have recently gained popularity for uncovering linear mechanisms in turbulent flows. Resolvent analysis has been widely used to investigate dominant coherent structures in turbulent jets and channel flows (McKeon & Sharma Reference McKeon and Sharma2010; Luhar, Sharma & McKeon Reference Luhar, Sharma and McKeon2014; Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Brès2018; Abreu et al. Reference Abreu, Cavalieri, Schlatter, Vinuesa and Henningson2020), while two recent studies by Hao (Reference Hao2023) and Cura et al. (Reference Cura, Hanifi, Cavalieri and Weiss2024) effectively applied it to model low-frequency bubble dynamics. Global stability analysis has also been successfully utilised to analyse low-frequency shock motions in STBLIs (Touber & Sandham Reference Touber and Sandham2009; Pirozzoli et al. Reference Pirozzoli, Larsson, Nichols, Bernardini, Morgan and Lele2010; Nichols et al. Reference Nichols, Larsson, Bernardini and Pirozzoli2017; Hao Reference Hao2023). Notably, a comparison of the resolvent analysis and GSA results reveals a modal resonance mechanism that drives low-frequency bubble breathing in both supersonic and low-speed flows (Hao Reference Hao2023; Cura et al. Reference Cura, Hanifi, Cavalieri and Weiss2024).
The objective of this study is to identify the mechanisms driving large-scale spanwise motions and to characterise the role of these motions in the dynamics of STBLIs. Specifically, this work focuses on three main aims: (i) validate the existence of these large-scale spanwise structures using modal analysis based on the large-eddy simulation (LES) database, as discussed in § 3.2; (ii) examine the origins of these structures through stability analysis, as detailed in § 4.1; (iii) analyse the 3-D motions of the TSB using pressure fluctuations in the interaction region, as discussed in § 4.2.
2. Computational set-up and numerical approach
2.1. Computational set-up
Figure 1. Computational domain and boundary conditions.
Figure 1 depicts the compression ramp configuration with a ramp angle of
$ 25^{\circ }$
. Throughout the paper,
$ x$
,
$ y$
and
$ z$
denote the Cartesian coordinates, with the origin located at the corner. The domain sizes are as follows:
$ L_{x1} = 39.6\delta _{0}$
,
$ L_{x2} = 17.6\delta _{0}$
,
$ L_{y} = 7.3\delta _{0}$
and
$ L_{z}=15\delta _{0}$
, where
$ \delta _{0} = 2.27$
mm is the TBL thickness at the reference position
$ x_{0}$
. Stations
$ x_{0}-x_{5}$
are located at
$ x=$
$ -17.5\delta _{0}$
,
$ -4.9\delta _{0}$
,
$ -2\delta _{0}$
,
$ 0$
,
$ \delta _{0}$
and
$ 3.5\delta _{0}$
, respectively. The computational mesh consists of
$ 1321 \times 136 \times 682$
grid points in the streamwise, wall-normal and spanwise directions. The mesh resolution is
$ \Delta x^{+}\approx 20$
,
$ \Delta z^{+}\approx 10$
, and
$ \Delta {y}_{w}^{+}\approx 1.1$
(at the reference station
$ x_{0}$
), which meets the grid requirement of LES (Choi & Moin Reference Choi and Moin2012).
The free-stream parameters match those of the experiment conducted by Zheltovodov et al. (Reference Zheltovodov, Trofimov, Schuelein and Yakovlev1990), with density
$ \rho _{\infty }=0.314$
kg m−
$^{3}$
, temperature
$ T_{\infty }=108$
K, Mach number
$M_{\infty } =2.95$
and Reynolds number
$Re_{\delta _{0}} =63500$
. Implicit LES are performed using an in-house multi-block parallel finite-volume solver named PHAROS (Hao, Wen & Wang Reference Hao, Wen and Wang2019; Hao & Wen Reference Hao and Wen2020; Hao et al. Reference Hao, Cao, Wen and Olivier2021). The inviscid fluxes are solved by means of the low-dissipative sixth-order kinetic preserving scheme (Pirozzoli Reference Pirozzoli2010) and the Ausm
$ ^{+}$
-up scheme (Liou Reference Liou2006) with the fifth-order weighted essentially non-oscillatory reconstruction (Jiang & Shu Reference Jiang and Shu1996), using a switch based on the Jameson sensor (Jameson, Schmidt & Turkel Reference Jameson, Schmidt and Turkel1981). The viscous terms are computed using the second-order central scheme. The third-order low-storage Runge–Kutta scheme is applied for time integration with a time step of 8 ns. The total simulation time is approximately 12.7 ms. Flow samples are collected for a duration of 11.2 ms (
$3024\delta _{0}/u_{\infty }$
) once the separation zone is fully established, which occurs after approximately 1.5 ms. Here,
$ u_{\infty }$
represents the free-stream velocity. A perfect gas assumption is applied, with a Prandtl number of 0.72, a specific heat ratio of 1.4 and a turbulent Prandtl number of 0.9. The molecular viscosity
$ \mu$
is obtained from Sutherland’s law.
Boundary conditions are also listed in the figure. The extended digital filter technique (Touber & Sandham Reference Touber and Sandham2009; Ceci et al. Reference Ceci, Palumbo, Larsson and Pirozzoli2022) is employed to introduce inflow turbulence. A supersonic outflow boundary condition is applied to the outlet boundary. A sponge zone approximately
$ 2\delta _{0}$
thick is placed at the far-field boundary to eliminate any reflections (Mani Reference Mani2012). An isothermal no-slip boundary condition is implemented at the wall, with a fixed wall temperature
$ T_{w} = 275.4$
K, consistent with Loginov et al. (Reference Loginov, Adams and Zheltovodov2006). Periodic boundary conditions are used in the spanwise directions.
2.2. Global stability analysis
Global stability analysis is based on the assumption of a triple decomposition of unsteady flow into three parts: mean flow, coherent structures and turbulent (or incoherent) fluctuations (Reynolds & Hussain Reference Reynolds and Hussain1972), given as
\begin{equation} \boldsymbol{Q}(x,y,z,t)=\boldsymbol{Q_{2-D} } (x,y)+\boldsymbol{{Q'}}(x,y,z,t)+\boldsymbol{Q'}^{, t}(x,y,z,t), \end{equation}
where
$ \boldsymbol{Q}$
is the vector of conservative variables,
$ \boldsymbol{Q_{2-D} }$
is the 2-D Favre-averaged base flow,
$ \boldsymbol{{Q'}}$
is the 3-D perturbation (coherent) and
$ \boldsymbol{Q'}^{, t}$
is the vector of turbulent fluctuations (incoherent). Substituting (2.1) into the Navier–Stokes (N-S) equations and neglecting higher-order terms leads to
\begin{equation} \frac {\partial \boldsymbol{{Q'}} }{\partial t}=\boldsymbol{\mathcal{A}}(\boldsymbol{Q_{2-D} } ) \boldsymbol{{Q'}}, \end{equation}
where
$ \boldsymbol{\mathcal{A}}$
is the linearised N-S operator. The terms involving the incoherent term
$ \boldsymbol{Q'}^{, t}$
in the operator
$ \boldsymbol{\mathcal{A}}$
are modelled by the eddy viscosity
$ \mu _{t}$
. This modelling approach, however, is valid when a sufficient scale separation exists between coherent structures and turbulent fluctuations, as demonstrated by Reynolds & Hussain (Reference Reynolds and Hussain1972) and Reau & Tumin (Reference Reau and Tumin2002). The perturbation
$ \boldsymbol{{Q'}}$
is written in the following modal form:
\begin{equation} \boldsymbol{{Q'}}(x,y,z,t) = \boldsymbol{\hat {Q}}(x,y)\mathrm{exp}[ i\beta z-i({\omega }_{r}+i{\omega }_{i})t], \end{equation}
where
$ \hat {\boldsymbol{Q}}$
is the 2-D eigenfunction,
$ \beta$
is the spanwise wavenumber,
$ \omega _{r}$
is the angular frequency and
$ \omega _{i}$
is the growth rate. The flow is globally stable for
$ \omega _{i}\lt 0$
and unstable for
$ \omega _{i}\gt 0$
. Substituting (2.3) into (2.2) leads to an eigenvalue problem, which is solved using the implicitly restarted Arnoldi method implemented in ARPACK (Sorensen et al. Reference Sorensen, Lehoucq, Yang and Maschhoff1996) at a given wavenumber
$ \beta$
. The key point in solving the problem is constructing and discretising the operator
$ \boldsymbol{\mathcal{A}}$
, which consists of inviscid and viscous Jacobians. To improve the accuracy, the inviscid fluxes are computed using the modified Steger–Warming scheme near discontinuities and a central scheme in smooth regions, as detected by a modified Ducros sensor (Hendrickson, Kartha & Candler Reference Hendrickson, Kartha and Candler2018). The viscous fluxes are obtained using a second-order central difference scheme. Boundary conditions are consistent with those in figure 1, except for the left boundary, which is set as the far field. Influences of grid resolution and domain size in GSA are verified in Appendix A.
Based on the Boussinesq approximation,
$ \mu _{t}$
can be calculated using a least-squares method (Lilly Reference Lilly1992), which has been widely applied in turbulent flows (Raiesi, Piomelli & Pollard Reference Raiesi, Piomelli and Pollard2011; Abe et al. Reference Abe, Mizobuchi, Matsuo and Spalart2012; Coleman, Rumsey & Spalart Reference Coleman, Rumsey and Spalart2018; Fan et al. Reference Fan, Uy, Hao and Wen2024). Furthermore, the frozen eddy-viscosity strategy (Carini et al. Reference Carini, Airiau, Debien, Léon and Pralits2017) is employed. The GSA solver for laminar flows (Hao et al. Reference Hao, Cao, Wen and Olivier2021) is used, adopting the effective viscosity
$ {\mu }_{\textit{eff}}={\mu }_{t}+\mu$
.
2.3. Spectral proper orthogonal decomposition
Spectral proper orthogonal decomposition (Lumley Reference Lumley2007) is utilised to identify coherent structures of turbulent flows. Similar to ”standard” POD (Lumley Reference Lumley2007), SPOD aims to find an optimal orthogonal basis for flow data. However, SPOD modes evolve both in time and space. According to Towne, Schmidt & Colonius (Reference Towne, Schmidt and Colonius2018), SPOD modes combine the features of both POD and dynamic mode decomposition, allowing for the identification of optimal coherent structures.
In this study, we follow the procedures outlined by Towne et al. (Reference Towne, Schmidt and Colonius2018) and utilise their code to seek SPOD modes. First, the discrete Fourier transform is applied to the LES database, combining Welch’s method (Welch Reference Welch1967) and a standard Hamming window to minimise spectral leakage. This step transforms the flow data in time space into frequency space
$\hat {(\boldsymbol{\cdot})}$
. Then, the cross-spectral density tensor
$ \boldsymbol{S}$
can be constructed at each frequency, given as
\begin{equation} \boldsymbol{S}=\boldsymbol{\hat {U}}\boldsymbol{\hat {U}}^{*}, \end{equation}
where
$ \boldsymbol{\hat {U}}$
is a data matrix containing flow data at the same frequency, and
$(\boldsymbol{\cdot})^{*}$
is the Hermitian transpose. The next step is to solve an eigenvalue problem to obtain the eigenvalues
$ \boldsymbol{\varLambda }$
and the corresponding SPOD modes
$ \boldsymbol{\hat {\varPsi }}$
at each frequency, given as
\begin{equation} \boldsymbol{S}\boldsymbol{W}\boldsymbol{\hat {\varPsi }} =\boldsymbol{\hat {\varPsi }}\boldsymbol{\varLambda }, \end{equation}
where
$ \boldsymbol{W}$
is the weight matrix. For compressible flows, the compressible energy form defined by Chu (Reference Chu1965) is applied to
$ \boldsymbol{W}$
. The SPOD modes are sorted by decreasing energy, i.e.
$ {\lambda }_{{fk}}^{1}\geqq {\lambda }_{{fk}}^{2}\geqq \ldots \geqq {\lambda }_{{fk}}^{{n}_{blk}}$
. The subscript
$ fk$
represents the
$ k$
th frequency, while the superscript
$ {n}_{blk}$
indicates the mode number. The first SPOD mode is referred to as the optimal mode because it is optimal in terms of energy. The second and subsequent modes are referred to as suboptimal modes.
Table 1. Summary of mean-flow parameters for the TBL at the reference position
$ x_{0}$
.
$ \delta _{0}$
, the nominal thickness of the TBL, based on 0.99
$ u_{\infty }$
;
${\delta }^{*}$
, the displacement thickness of the TBL;
$\theta$
, the momentum thickness of the TBL;
${C}_{f}$
, the skin-friction coefficient.
3. Large-eddy simulation results
3.1. Instantaneous and mean flow features
Table 1 presents key properties of the TBL at the reference station
$ x_{0}$
. Our results align well with previous experimental data (Zheltovodov et al. Reference Zheltovodov, Trofimov, Schuelein and Yakovlev1990) and LES results (Loginov et al. Reference Loginov, Adams and Zheltovodov2006). Figure 2(
$ a$
) shows the Van Driest transformed mean velocity profile (Van Driest Reference Van and Edward1951) at
$ x_{0}$
, where
\begin{equation} y^{+}=\frac {\bar {\rho }_{w} u_{\tau } y}{\bar {\mu }_{w}}, \quad U_{V D}^{+}=\frac {1}{{u}_{\tau }}\int _{0}^{\bar {u}}\sqrt {\frac {\bar {\rho } }{\bar {{\rho }}_{w}}}{\rm d}\bar {u} . \end{equation}
In these expressions,
$ \bar {\rho }_{w}$
is the mean wall density,
$ \bar {\mu }_{w}$
is the mean wall viscosity,
$ {u}_{\tau }$
is the friction velocity and
$ \bar {\rho }$
and
$ \bar {u}$
represent the mean density and streamwise velocity, respectively. Good agreement is achieved between the transformed velocity profile with the classical solutions
$ U_{V D}^{+}=y^{+}$
and
$ U_{V D}^{+}=(1/0.41)$
ln
$y^{+}+5.1$
, as well as with the experimental data (Zheltovodov et al. Reference Zheltovodov, Trofimov, Schuelein and Yakovlev1990). Figure 2(
$ b$
) presents a comparison of the distributions of mean density
$ \bar {\rho }$
, velocity
$ \bar {u}$
and temperature
$\bar {T}$
with the experimental results, normalised by the free-stream parameters
$ \rho _{\infty }$
,
$u_{\infty }$
and
$T_{\infty }$
. The computed density and velocity profiles show good agreement with the experimental data, while minor discrepancies are noted in the temperature profile.
Figure 2. (
$ a$
) Van Driest transformed mean velocity profile and (
$ b$
) distributions of mean qualities of the TBL at the reference station
$ x_{0}$
.
The density-scaled root-mean-square (r.m.s.) intensities
$ \sqrt {\bar {\rho }/\bar {\rho }_{w}}(\boldsymbol{\cdot })^{\prime}_{\textit{rms}}/{u}_{\tau }$
(where
$ (\boldsymbol{\cdot })$
denotes streamwise velocity
$ u$
, vertical velocity
$ v$
and spanwise velocity
$ w$
) of our results are compared with direct numerical simulations (DNS) results of incompressible and compressible TBLs (Wu & Moin Reference Wu and Moin2009; Bernardini & Pirozzoli Reference Bernardini and Pirozzoli2011) in figure 3. In the inner layer, the density-scaled r.m.s. values fall within two DNS databases. In the outer layer, the decreasing trends are also similar, with the velocity fluctuations approaching zero at
$ y=1.4\delta _{0}$
.
Figure 3. Density-scaled r.m.s.
$ \sqrt {\bar {\rho }/\bar {\rho }_{w}}(\boldsymbol{\cdot })^{\prime}_{rms}/{u}_{\tau }$
(where
$ (\boldsymbol{\cdot })$
denotes streamwise velocity
$ u$
, vertical velocity
$ v$
and spanwise velocity
$ w$
) at the reference station
$ x_{0}$
in inner layer (
$ a$
) and outer layer (
$ b$
). Wu & Moin (Reference Wu and Moin2009), incompressible DNS data; Bernardini & Pirozzoli (Reference Bernardini and Pirozzoli2011), compressible DNS data at
$ Re_{\theta }=4300$
.
Figure 4. Distributions of (
$ a$
) the skin-friction coefficient
$ C_{f}$
and (
$ b$
) the pressure coefficient
$ C_{p}$
. Reported quantities are averaged over time and spanwise direction.
Figure 4 compares the distributions of the skin-friction coefficient
$ C_{f}$
and the pressure coefficient
$ C_{p}$
of our results with the experimental and LES results (Loginov et al. Reference Loginov, Adams and Zheltovodov2006). The values of
$ C_{f}$
and
$ C_{p}$
are defined by
\begin{equation} {C}_{f}=\frac {2{\tau }_{w}}{{\rho }_{\infty }{u}_{\infty }^{2}}, {C}_{p}=\frac {2\bar {p}_{w}}{{\rho }_{\infty }{u}_{\infty }^{2}}, \end{equation}
where
$ {\tau }_{w}$
and
$ \bar {p}_{w}$
are the averaged wall shear stress and wall pressure, respectively. On the flat plate, the two LES results for
$ {C}_{f}$
are nearly identical but slightly higher than the experimental data. Near the separation point, the skin friction decreases suddenly to negative values and recovers after reattachment. The separation and reattachment points of the previous LES results (Loginov et al. Reference Loginov, Adams and Zheltovodov2006) are located further downstream, while the separation lengths
$ L_{\textit{sep}}$
are both
$ 7\delta _{0}$
. As noted by Loginov et al. (Reference Loginov, Adams and Zheltovodov2006), the minor discrepancy between the LES and experimental results in
$ {C}_{f}$
distributions is influenced by two main factors: (i) experimental uncertainty (
$6\,\% {-} 10\,\%$
) in skin-friction measurements (Borisov et al. Reference Borisov, Vorontsov, Zheltovodov, Pavlov and Shpak1993, Reference Borisov, Zheltovodov, Maksimov, Fedorova and Shpak1999), and (ii) localised discrepancies stemming from the unspecified spatial alignment of experimental data relative to convergence–divergence-line pairs. Furthermore, 3-D flow relieving effects in experiments tend to decrease the separation length. Regarding
$ {C}_{p}$
, all three distributions exhibit similar increases near the separation and reattachment points. Overall, the present LES results agree well with previous LES results and experimental data. The current grid resolutions are adequate.
Instantaneous 3-D vortical structures identified using the
$ Q$
criterion are shown in figure 5, along with the density contour in the
$ x$
-
$ y$
plane. Vortices are primarily generated near the shock foot and propagate downstream over the TSB. These vortices are in the form of streamwise vortices with a spanwise wavelength of approximately
$ 2\delta _{0}$
, referred to as Görtler-like vortices. Zhuang et al. (Reference Zhuang, Tan, Liu, Zhang and Ling2017) also observed these Görtler-like vortices immediately downstream of the separation shock using a Rayleigh scattering technique. No apparent large-scale spanwise structures (generally larger than the Görtler-like vortices) are observed.
Figure 5. Instantaneous 3-D flow structures extracted using the
$ Q$
criterion, coloured by streamwise velocity
$ {u}/{{u}_{\infty }}$
from
$ -0.4$
to
$ 0.4$
. The iso-surface value of
$ Q$
is set to
$ 5\,\%$
of its maximum. The contour in the
$ x$
-
$ y$
plane represents the density distribution.
Spectral analysis is performed on wall-pressure signals. Welch’s method (Welch Reference Welch1967) is employed for spectral estimation, using a standard Hamming window to weight the data. The signals are divided into three segments with a
$ 50\,\%$
overlap and the length of each segment is approximately
$1500\delta _{0}/u_{\infty }$
(
$ 214L_{\textit{sep}}/u_{\infty }$
). Figure 6 shows the spanwise-averaged spectrum of wall pressure as a function of Strouhal number
$ St=fL_{\textit{sep}}/u_{\infty }$
along the streamwise direction. The mean separation line, corner and reattachment line are marked in the figure.
Upstream of the interaction region, the contour presents a broadband bump centred around
$ St =5-7$
, with no significant low-frequency content. This broadband bump corresponds to energetic scales in the undisturbed TBL. In the intermittent region, the energy is primarily concentrated in the low-frequency region, with a peak frequency of
$ St= 0.042$
and an intermittent length of approximately
$ 2\delta _{0}$
. The peak frequency falls within the range of
$ St=0.02{-}0.05$
(Dussauge et al. Reference Dussauge, Dupont and Debiève2006). Downstream of the interaction region, the energetic scales shift back to the high-frequency zone, although some low-frequency structures persist. Interestingly, Jenquin et al. (Reference Jenquin, Johnson and Narayanaswamy2023) identified a prominent peak in the frequency range
$ St=0.2{-}0.4$
near the mean reattachment line, they concluded that it is the signature of Görtler-like structures. However, this peak is not evident in our spectrum and in other simulations (Touber & Sandham Reference Touber and Sandham2009; Grilli et al. Reference Grilli, Schmid, Hickel and Adams2012; Priebe & Martín Reference Priebe and Martín2012; Pasquariello et al. Reference Pasquariello, Hickel and Adams2017; Ceci et al. Reference Ceci, Palumbo, Larsson and Pirozzoli2023), which needs further investigations in future studies.
Figure 6. Contour of weighted power spectral density (WPSD) of wall-pressure signals. The dashed line indicates the mean separation point
$ x_{s}$
, the dashed dotted line denotes the mean reattachment point
$ x_{r}$
and the solid line marks the corner.
3.2. Evidence of large-scale spanwise motions
3.2.1. Proper orthogonal decomposition results
Space-only POD is performed on wall pressure to examine potential spanwise structures surrounding the mean separation line. Figure 7(
$ a{,}b$
) shows the leading POD mode and the mode energy distribution of the first 100 modes. Spanwise structures near the mean separation line can be observed in the mode, characterised by a large spanwise wavelength of the order of
$ O(15\delta _{0})$
. According to Ceci et al. (Reference Ceci, Palumbo, Larsson and Pirozzoli2023, Reference Ceci, Palumbo, Larsson and Pirozzoli2024), the spanwise structures are signatures of the rippling of the separation line. Interestingly,
$ O(15\delta _{0})$
-scale structures with opposing signs emerge near the mean reattachment line. The correlation between the structures near the mean separation and reattachment lines indicates these structures may be associated with motions of the whole TSB. Most of the energy is captured by numerous high-order modes, with the leading mode occupying only approximately 1.2
$\%$
of the total energy. The low energy ratio may be attributed to extra acoustic disturbances arising at the inlet (Ceci et al. Reference Ceci, Palumbo, Larsson and Pirozzoli2022). The cumulative energy of the first 100 POD modes accounts for 30
$\%$
of the total energy. Consequently, the motions surrounding the mean separation line are multimodally coupled and cannot be simply reconstructed using only the first few energetic modes.
A low-pass filter is commonly used to investigate the low-frequency dynamics in STBLIs, as noted by Priebe & Martín (Reference Priebe and Martín2012) and Tong et al. (Reference Tong, Yu, Tang and Li2017). Therefore, we apply a low-pass filter with a cutoff frequency
$St_{cut}=0.05$
to the raw wall pressure to isolate low-frequency fluctuations. This cutoff frequency is around the characteristic frequency band of the shock motions. Figure 7(
$ c$
) displays the leading POD mode derived from the filtered wall pressure. Compared with the unfiltered leading POD mode, the spanwise structures near the mean separation and reattachment lines are nearly identical in shape and position, which indicates that the large-scale spanwise structures are primarily associated with the low-frequency dynamics. Figure 7(
$ d$
) shows the corresponding energy distribution for the first 100 modes and the cumulative energy. The leading mode captures 25
$\%$
of the total energy, exhibiting a low-rank feature. The first 40 POD modes collectively account for nearly 100
$\%$
of the total energy. It is clear that the low-pass filter effectively removes high-frequency components while preserving low-frequency large-scale structures. The leading POD modes shown in figure 7(
$ a{,} c$
) resemble those observed in the non-swept case noted by Ceci et al. (Reference Ceci, Palumbo, Larsson and Pirozzoli2023, Reference Ceci, Palumbo, Larsson and Pirozzoli2024).
Figure 7. (
$ a$
) Leading POD mode and (
$ b$
) corresponding energy distribution of the first 100 POD modes from the raw wall-pressure signals; (
$ c$
) filtered (low-pass filter,
$ St_{cut}=0.05$
) leading POD mode and (
$ d$
) its associated energy distribution of the first 100 POD modes. The dashed lines in (
$ a{,}c$
) indicate the mean separation point
$ x_{s}$
, while the dash dot lines represent the mean reattachment point
$ x_{r}$
.
3.2.2. Spectral proper orthogonal decomposition results
Spectral proper orthogonal decomposition is then applied to extract coherent structures based on the LES data. A segment length of
$1500\delta _{0}/u_{\infty }$
with a 90
$ \%$
overlap is used, leading to 11 realisations. A standard Hamming window is applied to minimise spectral leakage. It is noted that the results are nearly identical when using a Hann window or a relatively short segment of
$1300\delta _{0}/u_{\infty }$
. Figure 8(
$ a$
) presents the normalised SPOD eigenvalues at station
$ x_{3}$
(
$ x=0$
, at the corner). The energy decreases with increasing frequency across different modes. The dominant frequency for mode 1 is the lowest resolvable frequency,
$ St=0.0046$
. This frequency is almost an order of magnitude lower than the quasi-2-D shock motion frequency
$ St=0.02 -0.05$
(Dussauge et al. Reference Dussauge, Dupont and Debiève2006). The normalised eigenvalues of the SPOD modes at
$ St=0.0046$
are depicted in figure 8(
$ b$
). The optimal mode is significantly more energetic than the other, capturing
$ 61\,\%$
of the total energy. This indicates the presence of low-rank features in the dynamical system in the spanwise direction at this frequency.
Figure 8. At station
$ x_{3}$
: (
$ a$
) SPOD eigenvalues as a function of frequency, normalised by the total flow energy; (
$ b$
) SPOD eigenvalues at the lowest frequency
$ St = 0.0046$
, normalised by the total flow energy at this frequency.
Figure 9. Real parts of (
$ a$
)
$ \hat {u}$
, (
$ b$
)
$ \hat {v}$
and (
$ c$
)
$ \hat {w}$
of the leading SPOD mode at
$ St=0.0046$
at station
$ x_{3}$
. The black dashed lines indicate the separation shock locations and
$ y_{n}$
is the wall-normal distance.
Figure 9 depicts three velocity components of the leading SPOD mode at
$ St=0.0046$
at station
$ x_{3}$
. From the streamwise velocity perturbation
$ \hat {u}$
, two types of coherent structures located at different wall-normal locations are observed: large-scale spanwise structures of
$ O(15\delta _{0})$
surrounding the separation shock (referred to as the shock component) and structures near the wall (referred to as the near-wall component). The near-wall component seems to contain structures with spanwise wavelengths of both
$ O(15\delta _{0})$
and
$ O(2\delta _{0})$
, while the large-scale structures are modulated by the Görtler-like vortices immediately downstream of separation point, as shown in figure 5. Similar modulation effects between structures of different scales inside TSBs have also been documented by Borgmann et al. (Reference Borgmann, Cura, Weiss and Little2024). For the vertical velocity perturbation
$ \hat {v}$
, most fluctuations are attributed to the shock component, while the near-wall component can be barely seen. The near-wall structures remain evident in the spanwise velocity perturbation
$ \hat {w}$
, whereas the shock component is not present around the separation shock but is located beneath it.
Figure 10. Real parts of (
$ a{,}b$
)
$ \hat {u}$
, (
$ c{,}d$
)
$ \hat {v}$
and (
$ e{,}f$
)
$ \hat {w}$
of the leading SPOD modes at
$ St=0.0046$
. (
$ a{,}c{,}e$
) Results for the
$ y$
-
$ z$
planes at stations
$ x_{1}-x_{5}$
; (
$ b{,}d{,}f$
) results for the mid-span plane. The mean streamwise velocity contours on the
$ x$
-
$ y$
plane through
$ z/\delta _{0} = 0$
are shown in (
$ a{,}c{,}e$
), along with black dashed lines indicating the iso-lines of
$ \bar {u}/{u}_{\infty }=0.99$
. The streamlines in (
$ b{,}d{,}f$
) pass through the point (
$ x_{s} , 0.01\delta _{0}$
).
Figure 10
$ (a{,}c{,}e)$
presents the leading SPOD modes at stations
$ x_{1}-x_{5}$
at
$ St=0.0046$
, along with the iso-lines of
$ \bar {u}/{u}_{\infty }=0.99$
that mark the separation shock. At station
$ x_{1}$
(within the intermittent region), the two components merge into a common large-scale spanwise structure. Downstream of
$ x_{1}$
, the coherent structures at different stations resemble those in figure 9 and maintain the two components. The shock components remain of the order of
$ 15\delta _{0}$
in the spanwise direction, suggesting the presence of persistent large-scale structures. The near-wall components remain modulated by Görtler-like structure up to
$ x_{5}$
, where modulation ceases and Görtler-like vortices become dominant. These near-wall structures at
$ x_{5}$
in figure 10
$ (a)$
resemble the turbulent Görtler vortices reported by Zhang, Hao & Uy (Reference Zhang, Hao and Uy2025), which are characterised by counter-rotating pairs.
A clear streamwise evolution of the two types of structures is shown in figure 10(
$ b{,}d{,}f$
), which presents the leading SPOD mode in the mid-span plane at
$ St=0.0046$
. The perturbation
$ \hat {u}$
is excited near the shock foot and develops into the shock and near-wall components. The two components correspond to the two components in the spanwise direction shown in figure 10(
$ a{,}c{,}e$
). The near-wall component continues along both the separation bubble and the reattached boundary layer. In contrast, the shock component persists solely along the shock. Furthermore,
$ \hat {u}$
keeps nearly the same sign throughout the computational domain. Most of
$ \hat {v}$
is concentrated along the separation shock, with only slight fluctuations occurring close to the wall. The sign of
$ \hat {w}$
changes inside the separation bubble.
Figure 11 presents the real part of the leading SPOD mode of wall pressure at
$ St=0.0046$
. As expected, the large-scale spanwise structures are observed surrounding the mean separation line, consistent with the leading POD modes in figures 7(
$ a{,}c$
). Furthermore, similar large-scale spanwise structures with opposite signs also appear near the mean reattachment point
$ x_{r}$
, though these are less organised than those near
$ x_{s}$
. This irregularity is likely caused by the near-wall Görtler-like vortices, which modulate the large-scale spanwise structures near
$ x_{r}$
. The corresponding relationship between structures at
$ x_{s}$
and
$ x_{r}$
indicates that the spanwise rippling may be associated with the entire TSB motions. Combined with the observation of shock rippling at various streamwise locations, this leads to the following hypothesis: the TSB motions induce the rippling of the separation line, which, in turn, causes the rippling of the separation shock. The dynamics of this system is elaborated later.
Figure 11. Real part of the leading SPOD mode of wall pressure at
$ St=0.0046$
. The dashed line indicates the mean separation point
$ x_{s}$
, the dashed dot line denotes the mean reattachment point
$ x_{r}$
.
All in all, the SPOD method successfully extracts the low-frequency coherent structures in the spanwise direction which are excited near the shock foot and persist along the separation shock. These structures manifest as spanwise rippling, with wavelengths of the order of
$ O(15\delta _{0})$
.
3.3. Influence of spanwise width
To confirm that the captured
$ 15\delta _{0}$
(
$ 2L_{\textit{sep}}$
) structures are not artefacts arising from spanwise domain constraints, a larger domain width of
$ L_{z}=30\delta _{0}$
(
$ \approx 4L_{\textit{sep}}$
) is adopted. A coarser mesh is used in the
$ x$
and
$ z$
directions to save the computational cost, with resolutions of
$ \Delta x^{+}\approx 30$
, and
$ \Delta z^{+}\approx 15$
. The total physical time is reduced to
$1300\delta _{0}/u_{\infty }$
.
The SPOD analysis is performed on
$ y$
-
$ z$
planes at different streamwise stations and on wall-pressure signals. The segment length is
$650\delta _{0}/u_{\infty }$
, with a
$ 90\, \%$
overlap. Figure 12 shows the leading SPOD modes at the lowest frequency
$ St=0.0096$
at stations
$ x_{2},x_{3},x_{4}$
and
$x_{5}$
. These modes consist of a near-wall component and a large-scale shock component, resembling the leading modes in figure 10
$ (a{,}c)$
. Two pairs of large-scale spanwise structures surrounding the separation shock are observed at different stations, with a wavelength approximately
$ \lambda _{z}= 15\delta _{0}$
(
$\approx 2L_{\textit{sep}}$
). The wavelength and mode shape align well with those captured in the relatively narrow computational domain (
$ 15\delta _{0}$
), confirming domain-size independence.
Furthermore, figure 13 presents the leading SPOD mode at
$ St=0.0096$
for wall-pressure fluctuations. Consistent large-scale spanwise structures are observed around the mean separation line, accompanied by opposing-sign structures near the reattachment line. The spatial coherence between the leading SPOD modes across streamwise stations and wall-pressure data confirms that these structures are physical features rather than numerical artefacts.
Figure 12. Real parts of (
$ a$
)
$ \hat {u}$
and (
$ b$
)
$ \hat {v}$
of the leading SPOD modes at
$ St=0.0096$
at stations
$ x_{2}, x_{3}, x_{4}, x_{5}$
for the
$ 30\delta _{0}$
case and the mean streamwise velocity contour on an
$ x$
-
$ y$
plane through
$ z/\delta _{0} = 0$
. The black dashed line on the
$ x$
-
$ y$
plane indicates the iso-line of
$ \bar {u}/{u}_{\infty }=0.99$
.
Figure 13. Real part of the leading SPOD mode of wall pressure at
$ St=0.0096$
from the
$ 30\delta _{0}$
case. The dashed line indicates the mean separation point
$ x_{s}$
, the dashed dot line denotes the mean reattachment point
$ x_{r}$
.
4. Discussion
4.1. Origins of large-scale spanwise motions
Figure 14. The most unstable modes at different wavenumbers
$ \beta \delta _{0}$
. The black dashed line indicates zero growth rate.
A GSA is conducted to investigate potential global modes of the dynamical system associated with the TSB. Figure 14 shows the most unstable modes at different wavenumbers
$ \beta \delta _{0}$
. As the wavenumber increases, the growth rates rise to the peak at
$ \beta \delta _{0}=0.43$
and subsequently decline, becoming stable when
$ \beta \delta _{0}$
exceeds 1.1. The wavelength of the peak roughly corresponds to
$ \lambda _{z}\approx 15\delta _{0}$
, which justifies the choice of
$ L_{z}$
. Furthermore, all these modes are stationary with
$ \omega _{r}=0$
. Zero-frequency globally unstable modes indicate that small perturbations will grow exponentially without oscillations until nonlinear saturation.
Note that the GSA modes are normalised using the
$ L_{2}$
norm. Consequently, only the relative spatial distributions within each mode are physically meaningful. The eigenfunctions of the 2-D mode (referred to as the shock mode) is shown in figure 15(
$ a$
). The streamwise velocity perturbation
$ \hat {u}$
is predominantly concentrated around the shock foot and the separation shock. Downstream of the shock foot,
$ \hat {u}$
diminishes and becomes barely visible within the separation bubble. Nichols et al. (Reference Nichols, Larsson, Bernardini and Pirozzoli2017) and Hao (Reference Hao2023) captured similar 2-D shock modes that were thought to contribute to low-frequency shock motions, but this mode is not the focus of our study.
Figure 15. Real part of (
$ a$
)
$ \hat {u}$
of the shock mode (
$ \beta \delta _{0}$
= 0.0), (
$ b{-}c$
)
$ \hat {u}$
and
$ \hat {w}$
of the bubble mode (
$ \beta \delta _{0}$
= 0.43). The streamlines pass through the point (
$ x_{s} , 0.01\delta _{0}$
).
Of particular interest is the most unstable 3-D mode (referred to as the bubble mode) shown in figure 15(
$ b{-}c$
). Two-component structures appear in this mode: one along the separation shock and another near the wall. Unlike the shock mode,
$ \hat {u}$
remains at a high level within the separation bubble and downstream of the corner. Additionally,
$ \hat {u}$
maintains the same sign throughout the computational domain, while
$ \hat {w}$
exhibits sign reversal beneath the shear layer. The characteristics of both the shock mode and the bubble mode align well with previous RANS-based GSA results (Hao Reference Hao2023), which indicate that the linear dynamics of TSBs are largely insensitive to turbulence modelling.
Figure 16. The projection coefficient between the leading SPOD mode of the mid-span plane at
$ St=0.0046$
and the global modes over various spanwise wavenumbers. The red dashed line indicates the local maximum
$ \beta \delta _{0}=0.96$
.
Qualitatively, the eigenfunction of the bubble mode resembles the leading SPOD mode of the mid-span plane presented in figure 10(
$ b{,}f$
), exhibiting a similar streamwise evolution of the two components. The quantitative comparison of the two modes is made using their projection, given by
\begin{equation} \gamma =\frac {\left | {\parallel \boldsymbol{{\hat {\varPsi }}}_\boldsymbol{SPOD}(x,y), \boldsymbol{{\hat {Q}}}_\boldsymbol{GSA}(x,y)\parallel }_{E} \right |}{ {\parallel \boldsymbol{{\hat {\varPsi }}_\boldsymbol{SPOD}}(x,y)\parallel }_{E} \times {\parallel \boldsymbol{{\hat {Q}}_\boldsymbol{GSA}}(x,y)\parallel }_{E} }, \end{equation}
where
$ \boldsymbol{{\hat {\varPsi }}_\boldsymbol{SPOD}}(x,y)$
represents the leading SPOD mode, while
$ \boldsymbol{{\hat {Q}}_\boldsymbol{GSA}}(x,y)$
denote the GSA modes over several spanwise wavenumbers. The notation
$ {\parallel \boldsymbol{\cdot }\parallel }_{E}$
denotes the Chu norm (Chu Reference Chu1965). A value of
$ \gamma = 0$
indicates orthogonality between the two modes, while
$ \gamma = 1$
signifies perfect alignment (Abreu et al. Reference Abreu, Cavalieri, Schlatter, Vinuesa and Henningson2020). It should be noted that
$ \boldsymbol{{\hat {\varPsi }}_\boldsymbol{SPOD}}(x,y)$
aggregates structures at different wavenumbers implicitly, whereas
$ \boldsymbol{{\hat {Q}}_\boldsymbol{GSA}}(x,y)$
represents structures at a specific wavenumber. Consequently, the projection can illustrate the similarities between the two types of modes and identify the wavenumber of dominant structures through the peak
$ \gamma (\beta \theta _{0})$
. Cura et al. (Reference Cura, Hanifi, Cavalieri and Weiss2024) applied a similar projection strategy to compare the optimal SPOD mode (derived from planar PIV data) with resolvent modes at different wavenumbers.
Figure 16 depicts the projection coefficient as a function of spanwise wavenumbers. The projection coefficient increases to a peak at
$ \beta \delta _{0}=0.96$
and then decreases as
$ \beta \delta _{0}$
continues to increase. The peak value is
$ \gamma = 0.88$
, which indicates a strong alignment between the two modes. The small discrepancy in the peak wavenumber primarily arises from the different wavelengths of the near-wall component from GSA and SPOD. The GSA models perturbations as
$ 15\delta _{0}$
coherent structures for both shock and near-wall components, while the near-wall Görtler-like vortices are approximately
$ 2\delta _{0}$
in SPOD. The difference in wavelength leads to a decrease in the peak value of
$ \gamma$
and an increase in the wavenumber of the peak
$ \gamma$
.
Figure 17. Comparisons of the leading SPOD modes (
$ a{,}c{,}e$
) at
$ St=0.0046$
and reconstructed 3-D perturbations (
$ b{,}d{,}f$
) using the bubble mode at stations
$ x_{1}-x_{5}$
.
The spanwise structures of the two components from SPOD and GSA are then examined. For ease of comparison, figure 17 presents a comparison of the leading SPOD modes shown in figure 10 and the reconstructed 3-D perturbations using the bubble mode at the same stations
$ x_{1}-x_{5}$
. Qualitatively, the shock components of the bubble mode resemble the coherent structures surrounding the separation shock in the SPOD modes, especially in terms of origin and development path. However, the near-wall component of the perturbations from the bubble mode is not fully reflected in the leading SPOD modes. The centrifugal effects dominate over the global instability near the wall, leading to the modulating effects of the Görtler-like structures on the near-wall spanwise mode of
$ O(15\delta _{0})$
. The effects of global instability are concentrated mainly near the shock foot and along the separation shock. It should be noted that the biased shock components observed in SPOD may be caused by nonlinear effects. Moreover, the bubble mode is stationary, indicating an extremely low characteristic frequency, which is also a key feature of the leading SPOD modes, as previously mentioned.
Figure 18. Wall-normal distributions of the spanwise-averaged Chu energy density from the leading SPOD modes at
$ St=0.0046$
and the bubble mode at stations
$ x_{2}$
,
$ x_{3}$
and
$ x_{4}$
normalised by their respective maximum values near the separation shock.
Figure 19. Reconstructed perturbed flow field using the bubble mode from (2.1), superimposed with wall-pressure fluctuations. The
$ y$
-
$ z$
slices are located at stations
$ x_{2}$
,
$ x_{3}$
and
$ x_{4}$
, with black lines denoting
$ u/u_{\infty }=0$
.
Figure 18 compares the distributions of the spanwise-averaged Chu energy density along the wall-normal direction for the leading SPOD modes at
$ St=0.0046$
and the bubble mode at stations
$ x_{2}$
,
$ x_{3}$
and
$ x_{4}$
. The Chu energy density is used to represent fluctuating energy distribution along the wall-normal direction, defined as
\begin{equation} \mathrm{Chu \ energy \ density}= \bar {\rho }| \mathbf{u} ' | ^{2} +\frac {\bar {T} }{\bar {\rho }\gamma {M_{\infty }}^{2} }(\rho ')^{2}+\frac {\bar {\rho }}{(\gamma -1)\gamma {M_{\infty }}^{2} \bar {T }}(T')^{2} , \end{equation}
where
$ \mathbf{u}= [ u,v,w ]^T$
. The trends of Chu energy density at different stations are similar, with a first peak close to the wall and a second peak near the shock. The shock components from SPOD and GSA are almost identical, while the near-wall components show significant differences in both strength and position, particularly at station
$ x_{4}$
. These discrepancies are attributed to the modulation effects of the Görtler-like structures. Despite the near-wall differences, the similarities near the separation shock indicate that large-scale spanwise structures captured by SPOD are consistent with those detected by GSA.
Qualitative and quantitative comparisons are made between GSA and SPOD results to demonstrate that the spanwise rippling captured by SPOD primarily arise from global instability. Recall the hypothesis given in § 3.2.2 that the spanwise undulations of the TSB are suspected to be the direct cause of the rippling. To model the spanwise motions of the TSB influenced by global instability, we reconstruct a perturbation field
$ \boldsymbol{{Q'}}$
superimposed with the 2-D mean flow
$ \boldsymbol{Q_{2-D} }$
, as illustrated in figure 19. The perturbation field
$ \boldsymbol{{Q'}}$
is reconstructed using (2.3). The time factor is simplified as a constant by setting
$ | \hat {w} |_{\textit{max}} /u_{\infty }=0.05$
, which is appropriate for the TSB shape.
At different stations, the TSB size varies in the spanwise direction, exhibiting peaks and valleys. These spanwise variations are attributed to the TSB undulations, consistent with the reconstruction by Borgmann et al. (Reference Borgmann, Cura, Weiss and Little2024) using a low-order model. The dynamical system associated with the TSB is summarised as follows. As the TSB oscillates in the streamwise direction, it also experiences spanwise undulations due to global instability. These undulations affect the separation shock, causing its motion to become spanwise-dependent rather than purely two-dimensional. As a result, the shock undergoes similar spanwise undulations, which correspond to the large-scale spanwise structures identified using SPOD in figure 10. The rippling serves as an intuitive representation of the spanwise undulations of the TSB. Furthermore, the opposite-sign large-scale wall-pressure fluctuations are consistent with the POD modes in figure 7(
$ a{,}c$
) and leading SPOD mode in figure 11.
Flow conditions for the present case and prior studies reporting large-scale spanwise structures are summarised in table 2. The free-stream parameters and interaction types vary across these cases, spanning subsonic to hypersonic regimes. While spanwise undulations of TSBs have been observed in reported cases, other works report no such modes. Three potential factors may explain this discrepancy. First, many prior STBLI investigations employed narrow spanwise domains (typically
$ 4\delta _{0}{-}6\delta _{0}$
), which may inadequately resolve large-scale spanwise motions. Second, the emergence of these motions may critically depend on free-stream parameters such as Mach number, temperature ratio and Reynolds number. Third, the separation scales (or the interaction strengths) may influence the manifestation of such spanwise modes. For example, the linear bubble dynamics may dominate in strongly separated flows, whereas other convective linear dynamics may prevail in weakly separated flows. Our present study focuses solely on one case; therefore, we do not assert generality. Future work will systematically investigate the influence of parameters, including the Mach number, Reynolds number, temperature ratio and interaction strength, to determine whether such bubble modes are intrinsic.
Table 2. Summary of studies reporting large-scale spanwise structures. The symbol – indicates the observations of large-scale spanwise pressure fluctuations near the mean separation line in these studies; however, quantitative characterisation of the most energetic spanwise structures (e.g. dominant scales) caused by TSB motions is not given explicitly.
4.2. Coupling of spanwise unsteadiness and quasi-2-D streamwise motions
In previous STBLI studies focusing on the low-frequency unsteadiness in 2-D interactions (e.g. compression ramp and shock impingement), the shock motions are usually modelled as nominally 2-D structures, as summarised by Clemens & Narayanaswamy (Reference Clemens and Narayanaswamy2014). The standard methodology for analysing the 2-D dynamics is to average flow variables in the spanwise direction, as implemented by Grilli et al. (Reference Grilli, Schmid, Hickel and Adams2012), Priebe & Martín (Reference Priebe and Martín2012), Priebe et al. (Reference Priebe, Tu, Rowley and Martín2016) and Pasquariello et al. (Reference Pasquariello, Hickel and Adams2017). However, this quasi-2-D framework inherently excludes low-frequency spanwise unsteadiness, which remains unaccounted for in such simplified models. Here, the spanwise unsteadiness and streamwise oscillations are investigated to characterise their interplay, providing insights into 3-D bubble motions.
Figure 20. (
$ a$
)Four wall-pressure signals at
$ x_{s}$
: spanwise averaged (unfiltered), frequency filtered and spanwise averaged (
$ St_{cut}=0.05$
), frequency-filtered mid-span (
$ St_{cut}=0.05$
) and frequency- and wavelength-filtered mid-span (
$ St_{cut}=0.05$
,
$ \lambda _{z}=15\delta _{0}$
). The black arrows indicate some opposite motions between the filtered spanwise-averaged and filtered mid-span signals. (
$ b$
) Normalised r.m.s. of the frequency-filtered spanwise-averaged pressure signal (blue) and three frequency-filtered wall-pressure signals within the intermittent region at
$ z/\delta _{0}=3.75$
(dash dot), 7.5 (red solid) and 11.25 (dash dot dot). The black dashed line in (
$ a$
) denotes the mean value
$ p_{w}/p_{\infty } =1.3$
at
$ x_{s}$
, and in (
$ b$
) indicates
$ x_{s}$
.
Given that the characteristic frequency of quasi-2-D shock motions lies around
$ St= 0.02{-}0.05$
(Dussauge et al. Reference Dussauge, Dupont and Debiève2006), and the spanwise undulations of TSBs are characterised by a very low-frequency dynamics, the range
$ St= 0.0{-}0.05$
can encompass both types of oscillations. To capture these dynamics, a low-pass filter with a cutoff frequency
$ St_{cut}=0.05$
is employed to the raw and spanwise-averaged wall-pressure signals. Since the streamwise oscillations of the shock foot result in sharp increases or decreases in wall pressure near
$ x_{s}$
, the instantaneous pressure fluctuations can effectively represent the shock motions. Importantly, these shock motions are indicative of the TSB motions. Therefore, the instantaneous pressure fluctuations also serve as a marker for the TSB motions.
Figure 20(
$ a$
) compares four wall-pressure signals at the location
$ x_{s}$
: the spanwise-averaged signal, the frequency-filtered spanwise-averaged signal (
$ St_{cut}=0.05$
), the frequency-filtered mid-span signal (
$ St_{cut}=0.05$
) and the frequency- and wavelength-filtered mid-span signal (
$ St_{cut}=0.05$
,
$ \lambda _{z}=15\delta _{0}$
). Wavelength filtering is achieved by performing a discrete Fourier transform on the frequency-filtered spanwise pressure distribution to isolate the specific
$ \lambda _{z}=15\delta _{0}$
component. The mid-span signal is then reconstructed from this wavelength-specific mode. The frequency-filtered spanwise-averaged signal primarily reflects quasi-2-D oscillations, while the frequency-filtered mid-span signal captures a combination of both spanwise and streamwise motions. The trends of both frequency-filtered signals are similar most of the time, which indicates that the streamwise shock motions remain dominant. However, at certain moments, the two pressure signals diverge and appear on opposite sides of the line
$ p_{w}/p_{\infty } =1.3$
, as marked in the black arrows in the figure. The frequency- and spanwise-filtered mid-span signal almost overlaps with the frequency-filtered signal, which indicates the dominant contribution of the
$ 15\delta _{0}$
spanwise undulations to the marked opposing pressure fluctuations. Figure 20(
$ b$
) shows a comparison between the r.m.s. values of the frequency-filtered spanwise-averaged signal and three frequency-filtered signals at
$ z/\delta _{0}=3.75$
, 7.5 and 11.25. The peak r.m.s. value from the filtered spanwise-averaged signal is lower than those from the three filtered signals, accounting for approximately 66
$ \%$
, 74
$ \%$
and 58
$ \%$
of their respective peak r.m.s. values. The mean ratio of the r.m.s. at
$ x_{s}$
in the spanwise direction is approximately 65
$ \%$
. This high percentage highlights the dominant role of streamwise motions in the 3-D shock dynamics.
Figure 21. Time sequence of filtered wall-pressure fluctuations, normalised by the free-stream pressure, over a duration of 1.92 ms: (
$ a$
)
$ t=0.128$
ms, (
$ b$
)
$ t=0.768$
ms, (
$ c$
)
$ t=1.408$
ms and (
$ d$
)
$ t=2.048$
ms. The black lines denote the filtered iso-lines of
$ C_{f}=0$
, while the green dashed lines represent the filtered spanwise-averaged iso-lines of
$ C_{f}=0$
. The cutoff frequency is
$ St_{cut}=0.05$
.
Figure 21 presents the time sequence of filtered wall-pressure fluctuations over a duration of 1.92 ms (long-time evolution is shown in the Supplementary movie). The filtered iso-lines of
$ C_{f}=0$
(marked in black solid lines) and the filtered spanwise-averaged iso-lines of
$ C_{f}=0$
(marked in green dashed lines) indicate 3-D large-scale spanwise undulations and quasi-2-D motions of the TSB, respectively. At different time instants, the black lines contract and expand accompanying the green lines simultaneously in the streamwise direction, which implies that both quasi-2-D breathing and spanwise rippling are active. Furthermore, the black lines surround the green lines, with the spanwise rippling corresponding to significant large-scale spanwise pressure fluctuations of approximately
$ 15\delta _{0}$
. These fluctuations indicate that, although the TSB oscillates integrally in the streamwise direction, the spanwise rippling still causes locally opposing pressure fluctuations, as shown in figure 20(
$ a$
). The maximum streamwise distance between the two lines at
$ t=2.048$
ms is approximately 0.5
$ \delta _{0}$
. This value is appreciable and occupies approximately 25
$ \%$
of the intermittent region. These large-scale pressure fluctuations are consistent with previous experimental studies (Jenquin et al. Reference Jenquin, Johnson and Narayanaswamy2023; Liu et al. Reference Liu, Chen, Zhang, Tan, Liu and Peng2024) and are responsible for the modes in figure 7(
$ a{,}c$
) and figure 11.
Similarly, dominant large-scale spanwise pressure fluctuations persist around the 2-D reattachment lines. These fluctuations exhibit opposite signs relative to those near the separation lines, consistent with both the optimal SPOD mode shown in figure 11 and the GSA bubble mode in figure 19. The irregularities of the filtered reattachment lines and the small-scale streamwise streaks of pressure fluctuations are attributed to Görtler-like structures.
Our results demonstrate that streamwise oscillations and spanwise undulations of the TSB coexist, with streamwise motions dominating over spanwise motions. However, the observed 65
$ \%$
dominance ratio may not hold universally. Future studies will therefore investigate additional configurations to further quantify the roles of the two motions.
One may argue that the large-scale spanwise structures of the TSB are caused by the Görtler-like vortices. However, our results demonstrated that the GSA bubble mode primarily drives the large-scale spanwise oscillations of the TSB. The influence of the Görtler-like vortices on the large-scale spanwise structures near the reattachment lines is limited at
$ St\lt 0.05$
, manifesting only as slight spanwise irregularities. In contrast, such irregularities are absent in the structures near the separation line. Moreover, perturbations originating from the Görtler-like structures can propagate both upstream and downstream, potentially influencing the shock system. However, at different streamwise locations, the shock components exhibit similar structures and spanwise wavelengths, suggesting that the Görtler-like structures have minimal impact on the large-scale spanwise structures. Overall, there is no direct evidence to suggest that the spanwise motions of the TSB are driven by Görtler-like vortices.
5. Conclusions
Large-scale spanwise structures in STBLIs over a
$ {25}^{\circ }$
compression ramp at Mach 2.95 are investigated using LES. The leading POD mode of wall pressure within the intermittent region exhibits a distinct spanwise corrugation centred on the mean separation line, with a wavelength of
$ O(15\delta _{0})$
or O(2
$ L_{\textit{sep}}$
). When a low-pass filter with
$ St_{cut} = 0.05$
is applied, the leading POD mode retains a similar structure and wavelength, which indicates that it is primarily associated with low-frequency motions. The SPOD method is employed to extract coherent structures in the spanwise direction from the LES database. At different streamwise stations, the leading SPOD modes of the
$ y$
-
$ z$
planes are characterised by low-frequency features. These modes consist of two components: the shock components with large-scale spanwise structures and the near-wall components. The shock components are excited near the shock foot and sustain along the separation shock, while the near-wall structures seem to be modulated by the Görtler-like vortices.
The GSA identifies a 3-D stationary bubble mode at a spanwise wavelength of
$ 15\delta _{0}$
. Qualitative and quantitative comparisons between the leading SPOD modes and the GSA bubble mode are conducted. Qualitatively, the streamwise evolution and spanwise scales of the shock components observed in the SPOD modes closely resemble those predicted by the bubble mode. Quantitatively, the peak projection coefficient between the leading SPOD mode in the mid-span plane and the bubble mode is as high as 0.88, which strongly indicates a high degree of alignment between the two modes. Furthermore, the Chu energy density distribution trends at different streamwise stations derived from the leading SPOD modes align well with those from the bubble mode, particularly near the separation shock. The qualitative and quantitative evidence significantly suggest that global instability is primarily responsible for the observed large-scale spanwise coherent structures. The reconstructed TSB using the 3-D bubble mode exhibits spanwise undulations, which directly cause the rippling of the shock. The dynamical system of the TSB can be summarised as follows: global instability induces spanwise undulations of the TSB, which in turn excite large-scale spanwise structures along the separation shock.
The coupling of shock motions in the spanwise and streamwise directions is also examined using filtered wall-pressure signals with a cutoff frequency
$ St_{cut}=0.05$
. Filtered signals from the three spanwise stations highlight the dominant role of streamwise oscillations. The peak r.m.s. value of the filtered spanwise-averaged signals accounts for 65
$ \%$
of the peak r.m.s. value in the full spanwise-direction-filtered signals. The contours of pressure fluctuations superimposed with the iso-lines of
$ C_{f}=0$
at different time instants reveal the TSB simultaneously undergoing streamwise oscillations and spanwise undulations. Overall, the present study provides new insights into 3-D effects of TSBs in nominally 2-D STBLIs.
However, our study focuses solely on one single case and cannot indicate the generality of the captured structures. Hence, the 65
$ \%$
dominance ratio only holds in the current case and may vary in other cases under different parameters or interaction types. Future studies will quantify the contributions of these two types of motions to the overall TSBs’ dynamics in different cases.
Supplementary movie
Supplementary movie is available at https://doi.org/10.1017/jfm.2025.10496.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (no. 12472239) and the Hong Kong Research Grants Council (no. 15204322).
Declaration of interests
The authors report no conflict of interest.
Appendix A. Verification of GSA
Figure 22 examines the effects of domain size on GSA mode growth rates. A relatively small domain serves as the new base flow for GSA, as illustrated in figure 22(
$ a$
). The growth rates overlap between both domains and share a common peak at
$ \beta \delta _{0}=0.43$
. These results confirm that the current domain size is sufficiently large for stability analysis.
Figure 22. (
$ a$
) Schematic of two computational domains in GSA and (
$ b$
) growth rates as a function of
$ \beta \delta _{0}$
.
Figure 23. (
$ a$
) Comparison of eigenvalues at
$ \beta \delta _{0} = 0.43$
of two grids. Real parts of (
$ b$
)
$ \hat {u}$
and (
$ c$
)
$ \hat {w}$
at
$ \beta \delta _{0} = 0.43$
from the fine grid.
The influence of grid resolution on GSA is then investigated. Results from the original grid (
$ 1321\times 136$
) and the fine grid (
$1821\times 173$
) share a common peak at
$ \beta \delta _{0}=0.43$
. Figure 23(
$ a$
) compares the eigenvalues of the two meshes at
$ \beta \delta _{0} = 0.43$
. The eigenvalues from the fine mesh are slightly larger than those from the original mesh. Nonetheless, the obtained eigenfunctions shown in figure 23(
$ b{-}c$
) exhibit the same features as those in figure 15 at
$ \beta \delta _{0} = 0.43$
. These similarities demonstrate that both meshes effectively capture the key dynamical features of the TSB.
Open access
