On the Co-existence of Transonic Buffet and Separation-Bubble Modes for the OALT25 Laminar-Flow Wing Section

For the present work, numerical simulations are performed using the in-house code, SBLI (Yao et al. 2009), which has been well-validated and used for studies on shock-wave/boundary-layer interaction (Touber and Sandham 2009; Sansica 2015), subsonic wing sections (Jones et al. 2008; De Tullio and Sandham 2019), as well as transonic buffet (Zauner 2019; Zauner and Sandham 2020; Moise et al. 2022b). This code has shown good performance for massively-parallelised simulations using structured multi-block grids on several high-performance computer architectures. The compressible three-dimensional Navier–Stokes equations are normalised by the airfoil chord c, the freestream density \(\rho _{\infty }\), streamwise velocity \(u_{\infty }\), and temperature \(T_{\infty }\) and solved in a non-dimensional form using fourth-order finite difference schemes (central at interior and the Carpenter scheme (Carpenter et al. 1999) at boundaries) for spatial discretisation. The pressure is normalised by \(\rho _{\infty } u_{\infty }^2\). A low-storage third-order Runge–Kutta scheme is used for temporal discretisation, where one convective time unit is defined as the ratio of the airfoil chord over the freestream velocity. A total variation diminishing (TVD) scheme is employed to capture features of shock waves (Sansica 2015). The TVD scheme is deactivated around the leading edge to avoid the formation of spurious structures, as discussed in Zauner et al. (2019). Zonal characteristic boundary conditions (Sandberg and Sandham 2006) are enforced at the outlet, whereas integral characteristic boundary conditions (Sandhu and Sandham 1994) are applied at the remaining outer boundaries.

We will use for the present contribution a Spectral-Error based Implicit Large-Eddy Simulation (SE-ILES) scheme. For this method, spectral error indicators are computed locally every \(N_E\) time steps according to Jacobs et al. (2018) and are used to control a sixth-order filter (Visbal and Gaitonde 2002) in order to remove scales, which are not sufficiently resolved by the grid. Further away from the airfoil, the sixth-order filter is permanently activated to maintain numerical stability in regions of increased grid spacing. Details of this approach can be found in Zauner and Sandham (2019b), where this method has been validated against DNS for a transonic buffet test case and successfully applied in multiple studies (Zauner and Sandham 2020; Moise et al. 2022a, b).

The grids used in the present study have been designed and generated using an open-source parametric grid-generation tool for high-fidelity grids developed at the University of Southampton (Zauner and Sandham 2018a, b). High-order polynomials are used to define the geometry of grid lines and corresponding grid-point distributions. Such body-fitted grids are not necessarily limited to a single geometry, as grid lines are designed with respect to their position (coordinates) and local curvature with respect to grid points along the airfoil contour. Therefore, it is possible to use grids with very similar characteristics and resolution for different airfoil geometries. Here, grids previously used for simulations of Dassault Aviation’s V2C airfoil and already assessed in terms of spatial resolution (Zauner and Sandham 2019b, 2018b; Zauner et al. 2018), were adapted to the OALT25 geometry.

In the present work we will use two different CH-type grids for the OALT25 profile, referred to as ‘standard’ and ‘refined’ grids. The topology and resolution of the standard grid is very similar to grids used in Zauner and Sandham (2020) and Moise et al. (2022a) for the V2C airfoil and have been shown adequate to capture the main phenomena of interest. For both simulations, the C-block of the grid has a radius of 7.5c, while the two H-blocks capture 5.5c of the wake behind the airfoil. The spanwise extent of the domain is limited to 0.05c, which has been shown sufficient to capture transonic buffet, when comparing with test cases up to \(L_z=1.0c\). Frequencies match very well between simulations of different aspect ratios and experiments. However, buffet amplitudes can be underestimated in narrow-domain cases, oscillations can be more irregular, and conditions of on- and off-set need to be treated with care. However, we consider the underlying mechanism well captured by much cheaper narrow-domain simulations, which allow us more extensive parametric studies. The refined grid contains additional grid points within the region close to the airfoil surface and is only used for simulations at high Reynolds numbers. Close-ups are shown for grids in Fig. 1, plotting only every 15th grid point.

The standard grid in Fig. 1a, consists of approximately 90 million points (one block of \(1495\times 550\times 50\) points and two blocks of \(799\times 564\times 50\)), where the blunt trailing edge of \(0.5\%c\) contains 30 grid points. The grid clustering is relatively denser close to the aerofoil surface, in its wake, and in the region where the shock wave is expected. The wall-normal and wall-tangential spacings at the airfoil surface vary between \(1\times 10^{-4}\) to \(1.7\times 10^{-4}\) and \(4\times 10^{-4}\) to \(2\times 10^{-3}\), respectively.

The refined grid is used for a validation case at increased Reynolds numbers of \(Re=3{,}000{,}000\) only and contains approximately 200 million points (one block of \(1935\times 550\times 100\) points and two blocks of \(799\times 564\times 100\)). For comparison, the grid of a validated simulation by Dandois et al. (2018), targeting the same Reynolds number, contained about 400 million grid points, for a two times wider domain with a larger wall-normal extent of 100c. Around \(x\approx 0.3\), it is notable that the grid is relatively denser in the wall-tangential direction. This would allow a future use of the same grid for studies with boundary-layer tripping applied, as done in Moise et al. (2022b).

The present grids contain roughly 3 times larger cells in streamwise and spanwise direction, compared to DNS resolutions in Zauner et al. (2019). Considering the test case at \(Re=500{,}000\) and \(M=0.735\), the wall-normal spacing does not exceed \(\Delta y^+_w=1.2\) in the transition region \(0.2<x<0.5\). Streamwise and spanwise wall spacings remain below \(\Delta x^+_w=10.66\) and \(\Delta z^+_w=6.76\), where the latter is important to capture streamwise-oriented vortices promoting rapid turbulent breakdown observed in DNS (Zauner et al. 2019). In the turbulent region the normalised spacing peaks at \(x=0.74\) with \(\Delta y^+_w=1.83\), \(\Delta x^+_w=10.85\), and \(\Delta z^+_w=10.37\). Raising the Reynolds number for the same Mach number to \(Re=3{,}000{,}000\), the \(\Delta y^+\) at the wall increases even to \(\Delta y^+_w=5.5\) at \(x \approx 0.8\). Streamwise and spanwise wall spacings reach at that point \(\Delta x^+_w=33.22\) and \(\Delta z^+_w=17.61\), respectively. Even though these values do not strictly follow the rule of \(\Delta y^+<1\) in the fully turbulent aft section of the airfoil, it should be noted that the boundary layers are far from equilibrium and simulation results of interest agree well with reference literature, as shown in Sect. 3. In practice, the grid was iteratively improved to provide good resolution of shear layers further away from the wall, for which a simple \(y^+_w\) criterion is not sufficient. For example, at a wall distance of \(\Delta n \approx 0.01c\) (\(y+\approx 400\)), the cell spacing is only twice the size of the wall cell (\(\Delta y^+ \approx 11\)). The confirmation of adequate grid resolution for the standard grid to effectively capture the viscous sub-layer at a Reynolds number of 500, 000 is provided in appendix A, which includes a study on grid refinement at the wall. Furthermore, the results obtained from simulations at a Reynolds number of 3, 000, 000 will be verified and validated in Sect. 3.

We apply Fourier transformation techniques in order to analyse one-dimensional signals such as aerodynamic histories, after subtracting mean quantities. A typical simulation run time covers about 160 convective time units. Neglecting an initial transient of approximately 25 time units, results in a total signal length of 140 time units. Assuming a signal is split into \(50\%\) overlapping bins containing about 35 time units, each bin contains a sufficient number of cycles at intermediate frequencies (\(>10\) cycles for \(St=0.3\)). However, with the same bin size low-frequency cycles are poorly captured. To overcome this limitation using Hanning windows for Welch’s method (Welch 1967), we gradually decrease the number of bins (leading to an increase in bin size) as frequency is reduced. The resulting ‘composed’ spectra retain the low-frequency content, while reducing the noise in the high-frequency content.

To illustrate the robustness of this strategy, an example of such a composed Fourier spectrum is shown in Fig. 2 considering a representative \(C_L\) signal at \(M=0.71\). The red solid curve denotes the Welch spectrum using 35 time units per bin. For Strouhal numbers \(St>0.1\), modes are selected only from the red spectrum. The low-frequency part of the spectrum (\(St<0.1\)) is expanded by modes from spectra using larger bins containing 47 (blue curve), 70 (cyan curve), and 140 time units (green curve). This leads to the final composed spectrum, marked by black symbols.

Coherent features in the flow field were extracted using spectral proper orthogonal decomposition (Lumley 1970; Glauser et al. 1987; Towne et al. 2018). This decomposition finds an orthonormal basis that ideally represents a given ensemble of realisations of a stochastic process. It can be shown that the ideal basis which is spatio-temporally coherent consists of the eigenfunctions, \(\varvec{\psi }\), of the cross-spectral density tensor, \(\varvec{S}\) (Towne et al. 2018), which satisfyHere, \(\varvec{x}, \varvec{x}' \in \Omega\) are any two points in the domain, \(\Omega\), St is the frequency/Strouhal number, \(\varvec{W}\) is a weight function related to the relevant inner product, \(\lambda\) is the eigenvalue and the subscript i denotes the i-th eigenvalue/eigenfunction. The index i is such that \(\lambda _i\) are sorted in descending order, implying that \(i=1\), represents the most-energetic mode. Thus, the eigenfunction \(\varvec{\psi }_i(\varvec{x},St_0)\) represents the spatial structure of an SPOD mode that oscillates in time with a frequency, \(St_0\). The spatio-temporal variation of this mode is given bywhere \(\Re (\cdot )\) denotes real part and t refers to time.

The numerical code provided in Schmidt and Towne (2019) was used for performing SPOD. It adopts the Welch approach for Fourier transforms and is a streaming algorithm, which computes only the first few of the most-energetic modes. Here, only the first two dominant modes are examined (i.e., \(i = 1\) and 2). Moise et al. (2022a) have shown clear dominance of the first mode, when reconstructing flow features associated with transonic buffet. Therefore, we will limit our modal analysis to the dominating first mode and do not consider higher modes. The domain chosen is the \(z = 0\) plane. Snapshots at different time instants are constructed by arranging the density, velocity components and pressure on this plane into a column vector. The snapshots were sampled at time intervals of 0.064 (sampling frequency \(F_s \approx 15\)). To compute the Fourier transform using the Welch approach, snapshots are divided into blocks of \(T_B \approx 40\). Note that the lowest frequency associated with coherent oscillations occurs for \(St > 0.05\), implying that at least two oscillation cycles are captured in a block. The weighting function, \(\varvec{W}\) is chosen based on the approximate volume associated with each grid point. To compare the spatial structures of SPOD modes obtained for different cases, the phase within an oscillation cycle, \(\phi = 2\pi St_0 \, t\), of the spatio-temporal SPOD mode,must be chosen appropriately. Here, we choose \(\phi = 0\) as occurring when \(\Im (\varvec{\psi }(x_{\textrm{TE}},St_0) \exp (\textrm{i}\phi )) = 0\), where \(\Im (\cdot )\) corresponds to the imaginary part and \(\varvec{x}_{\textrm{TE}}\) represents the point on the upper corner of the trailing edge. Thus, contours shown in Figs. 6 and 10 are at \(\phi = 0\). Movies showing the spatial structure’s variation with phase are provided in the supplementary material. Further details can be found in Moise et al. (2022a and 2022b).

Following on the work of Dandois et al. (2018), we consider ONERA’s OALT25 laminar-flow wing geometry at a fixed angle of attack \(\alpha =4^{\circ }\) for all the present simulations. The fluid is considered to be air, which can be modelled at present conditions as a perfect gas with a specific heat ratio of \(\kappa =1.4\), satisfying Fourier’s law of heat conduction with a Prandtl number of \(Pr=0.72\). It is also assumed to be Newtonian and satisfying Sutherland’s law, with the Sutherland coefficient as \(T_s=110.4\) at a reference temperature \(T_r=268.67\,K\).

Starting from \(M=0.735\) and \(Re=500{,}000\), either Mach or Reynolds numbers are varied in order to study their effect on low- and intermediate-frequency oscillations separately. An overview of the test matrix and information about dominant spectral peaks is provided in Table 1.

Studying the OALT25 profile, Dandois et al. (2018) observed oscillations which were fundamentally different from those corresponding to cases with tripped boundary layers. Test cases subjected to fully-turbulent separation bubbles and shock/boundary-layer interactions typically show large-scale shock motion and lift fluctuations at low Strouhal numbers around \(St<0.1\). For free-transitional test cases, however, shock oscillations were more localised and mainly limited to the shock foot. These lift oscillations occurred at significantly higher frequencies corresponding to \(St \approx 1.1\). These observations were confirmed by wind-tunnel experiments at similar free-transitional conditions of \(M=0.735\) and \(Re=3{,}000{,}000\) (Brion et al. 2019), where an additional peak was observed in spectra of lift-fluctuations at significantly lower frequencies around \(St\approx 0.06\), which is in the range of typical buffet frequencies observed for the OALT25 profile.

Figure 3 shows (a) distributions of mean wall-pressure coefficient, \(\overline{C_p}\), the root-mean-square of fluctuating component, \(C'_{p,rms}\), and (b) histories of lift coefficient \(C_L\) for various Reynolds numbers. Statistical values of coefficients are computed according towhere N denotes the total number of data points. For now, we only focus on the blue, magenta, and green curves corresponding to \(Re=3{,}000{,}000\). The blue and magenta curves denote the present simulation results using standard (90 million grid points) and refined (200 million grid points) grids respectively, while the green curve corresponds to LES results from Dandois et al. (2018) using ONERA’s second-order accurate CFD code elsA (400 million grid points, but for a larger domain, see Sect.  2.2). Experimental measurements from Brion et al. (2019) are denoted by red dots and error bars corresponding to \(5\%\) pressure variations of the freestream according to \(\Delta C_p = \epsilon \ p_{\infty } \ 2 / (\rho _{\infty } u_{\infty }^2)\), where \(\epsilon =0.05\). Overall, the \(\overline{C_p}\) distributions at increased Re agree well, despite small differences in \(\overline{C_p}\) levels over the fore part and shock position. The former may be due to the influence of boundary-layer tripping applied on the pressure side for the case of Dandois et al. (2018) and Brion et al. (2019), while the mean shock position appears to be relatively sensitive to the grid resolution. However, the dynamic behaviour of the flow is hardly affected, which is indicated by the good agreement between dashed lines corresponding to \(C'_{p,rms}\) as a function of x. While the \(C'_{p,rms}\) peaks arise from shock-wave oscillations centered around \(x\approx 0.63\), the plateau further upstream (\(0.4<x<0.6\)) results from pressure fluctuation within the laminar separation bubble, as will be discussed later in more detail. Also lift fluctuations agree well in Fig. 3b, where the same colour code is used for different cases. All three cases exhibit clear intermediate-frequency fluctuations (period approximately equal to one time unit), which are slightly modulated by low-frequency undulations (period approximately equal to 12 time units).

To better assess the frequency content of these lift fluctuations, Fig. 4a shows Fourier spectra, with spectral peaks observed in experiments of Brion et al. (2019) indicated by vertical black broken lines. Again, we observe a good agreement between blue, magenta and green curves showing intermediate-frequency peaks in spectra in the range of \(0.9<St<1.1\) (highlighted by square symbols) with similar power-spectral densities and associated harmonics at \(St \approx 2\). The present simulations show a weak peak at low frequencies \(St \approx 0.08\) (circles), which is below the cut-off frequency of the spectrum associated with the work of Dandois et al. (2018). This low-frequency bump is slightly above the Strouhal numbers, \(St \approx 0.06\), observed in wind-tunnel tests of Brion et al. (2019). Boundary-layer tripping on the pressure side is not expected to impact the frequency significantly (Moise et al. 2022b). Even though Zauner and Sandham (2020) showed minor sensitivity of low frequencies to the spanwise domain extent, we cannot rule this effect out. The difference may instead be due to wind-tunnel confinement effects such as the presence of side and top walls, which are not captured in the current simulations. Nevertheless, considering the experimental and numerical challenges when comparing wind-tunnel test and simulation results, the agreement with respect to the low-frequency mode is good.

Although the blue curve in Fig. 4a shows some quantitative differences compared to cases using finer grids (the magenta and green curves), we still capture the same physical flow phenomena with very similar scales and amplitudes. This gives us confidence that the standard grid is well suited for simulations carried out at lower Reynolds numbers. Furthermore, we have some justification for the use of narrow domains and periodic boundary conditions in recovering the main flow characteristics observed in wind-tunnel tests.

After having verified and validated our observations at high Reynolds numbers of \(Re=3{,}000{,}000\) and \(M=0.735\), we consider a set of simulations at the same Mach number, but decreasing Reynolds numbers down to \(Re=500{,}000\) to study Re scaling effects. All these simulations use the standard grid.

Looking again at Figs. 3 and 4, simulations at \(Re=2{,}000{,}000\), 1, 000, 000, and 500, 000 are denoted by red, orange, and black curves, respectively. Looking at Fig. 3b, we can see that low-frequency oscillations strengthen with decreasing Reynolds numbers, indicating established buffet at low Re and a trend towards buffet offset at high Re for a fixed Mach number and angle of attack. In previous studies of the V2C profile, a similar trend was observed: while the flow was fully stalled at \(Re=200{,}000\), buffet was fully developed at \(Re=500{,}000\) before it weakened again towards higher Re (Zauner et al. 2019; Moise et al. 2022a) and eventually dies out (Szubert et al. 2016). An opposite trend is observed for intermediate-frequency oscillations, as they become less pronounced during high-lift phases when decreasing Re. At \(Re=500{,}000\), we eventually observe large-amplitude lift oscillations characteristic of transonic buffet. Averaging of dynamic effects associated with shock-wave motion leads to a rather smooth \(\overline{C_p}\) distribution on the suction side shown in Fig. 3a, where the pressure change associated with the main shock wave is smeared out. Consequently, the \(C'_{p,rms}\) peak turns into a broad bump and increased pressure fluctuations are even observed in the fore part of the airfoil. On the pressure side we observe a plateau in \(\overline{C_p}\) at \(x\approx 0.7\) corresponding to a laminar separation bubble. Looking at spectra shown in Fig. 4a, for decreasing Reynolds numbers we can observe a clear low-frequency peak evolving at a rather constant Strouhal number of \(St \approx 0.08\). The intermediate-frequency peak, however, decreases in magnitude and shifts towards lower Strouhal numbers, reaching \(St\approx 0.4\) at \(Re=500{,}000\). Oscillations at similar intermediate frequencies have been also observed for Dassault Aviation’s V2C profile at the same Reynolds number, but were less pronounced (Zauner and Sandham 2019a).

After having assessed the effect of low- and intermediate-frequency phenomena on aerodynamic wing characteristics, we would now like to study the spatial structure of associated phenomena. Figure 4b shows leading eigenvalues \(\lambda _1\) of SPOD of two-dimensional snapshots at different Reynolds numbers. These spectra look qualitatively very similar to those in Fig. 4a. The SPOD eigenvalue spectra are less smooth compared to the Fourier spectra of \(C_L\), due to the fixed number of bins used in the SPOD, but the trends in the low and intermediate frequency modes are the same. For the lower Re case, harmonics of the buffet mode coexist in the intermediate-frequency range, which makes the separation of both phenomena very difficult using SPOD. In contrast to the lift spectra, SPOD contains more information about the wake behind the airfoil and we can observe an additional broadband bump at \(St \approx 2\), where distinct modes appear at higher Reynolds numbers very close to harmonics of intermediate-frequency phenomena. These high-frequency phenomena are associated with a von-Karman vortex street and have also been observed and discussed by Moise et al. (2022a) for the V2C airfoil.

Before discussing the shape of dominant coherent structures extracted by SPOD, we first consider the contours of local time- and span-averaged Mach number in Fig. 5 for representative Reynolds numbers of (a) \(Re=3{,}000{,}000\) and (b) \(Re=500{,}000\) and instantaneous fields in (c) and (d) at high and low lift conditions respectively for the lower Re case. Based on previous work on V2C wing sections with different domain sizes at similar conditions (Zauner and Sandham 2020; Moise et al. 2022a) that showed minor variations of coherent features in the spanwise direction, it is sufficient to focus the present work on two-dimensional cross sections (instantaneous at \(z=0\) as well as span-averaged), which are obtained by 3D simulations. For the high Re case, we observe a large supersonic region, delineated by the green sonic curve with a distinct shock wave at \(x \approx 0.6\). A region of high velocity occurs just outside the boundary layer near the mid-chord location, leading to a small bump in the sonic line extending the supersonic region slightly in the downstream direction. Similar features have been observed in experiments by Pearcey et al. (1968) and are often referred to as “supersonic tongue” Shock (2011). For the \(Re=500{,}000\) case, this bump is more pronounced and extends further away from the surface, which is an indicator for significant flow separation (this can be confirmed, looking at \(C_f\) distribution in Fig. 17 in the appendix). Compared to the high Re case, the time-averaged supersonic region appears smaller and smoother due to large-scale unsteadiness. The global flow field at higher Re is not subjected to such significant variations (see \(C_L\) histories) and therefore no instantaneous snapshots are shown for brevity. The instantaneous snapshot of the \(Re=500{,}000\) case at the high-lift phase Fig. 5c demonstrates the high-frequency small-scale structures. Behind the shock foot the boundary layer thickens and undergoes rapid transition to turbulence, which is again confirmed in \(C_f\) distributions of Fig. 17 in the appendix. Then, in the wake region behind the airfoil we see a clear von-Karman vortex street forming. During low lift phases, we observe in Fig. 5d flow features that are reminiscent of stall. We should mention at this point that all instantaneous quantities and snapshots of this work are taken at \(z=0\), while statistical values are averaged in the spanwise direction. As already noted by Zauner and Sandham (2020), low-frequency oscillations at the present flow conditions show strong spanwise coherence. This is confirmed in the present simulations even for intermediate-frequency oscillations associated with bubble instabilities, as shown in Appendix C (Fig. 18).

Looking now at SPOD modes showing density-fluctuation contours in Fig. 6, which correspond to dominant flow oscillations around the mean flow, we can see low- (left column) and intermediate-frequency modes (right column) for representative Reynolds numbers of \(Re=3{,}000{,}000\) (top), \(Re=2{,}000{,}000\) (middle), and \(Re=500{,}000\) (bottom). For low-frequency phenomena at high Reynolds numbers, we can confirm rather localised oscillations of the shock wave accompanied by fluctuations of the trailing-edge region, which are \(180^{\circ }\) out of phase. This phase relation has been examined and confirmed by Moise et al. (2022b) for multiple airfoil geometries and reference literature. Structures in the wake correspond to periodic up- and downward deflection (‘flapping’). When reducing the Reynolds number to \(Re=2{,}000{,}000\), we observe a \(\lambda\)-structure occurring within the supersonic region with the leading leg associated with boundary-layer separation. Although less pronounced, this lambda structure also exists at \(Re=3{,}000{,}000\). The mode shape for the \(Re=3{,}000{,}000\) case has a striking resemblance to the globally unstable mode reported in Sartor et al. (2018) (see their Fig. 12a) for transonic buffet on an OAT15A aerofoil under fully-turbulent conditions. This serves as the justification for referring to the low-frequency modes observed here for free-transition conditions as transonic buffet modes.

The blue regions around shock and separation waves become thicker for decreasing Re, and eventually spread over the entire supersonic region for \(Re=500{,}000\). The spatial organisation of SPOD modes at intermediate frequencies are very similar for the high-Re test cases. For \(Re=500{,}000\), however, we observe a larger wavelength since the Strouhal numbers have decreased (recalling the spectra in Fig. 4). At higher Re, modal features within the supersonic region are limited to oscillations around the separation wave, whereas for \(Re = 500{,}000\), the mode structure extends beyond the supersonic region.

While modal structures and frequencies in Figs. 4b and 6, respectively, are consistent for both modes at low and high frequencies across the considered range of Re, particularly the low-frequency buffet mode changes significantly in amplitude. This can be explained by changing onset conditions towards increased Reynolds numbers. The general trend has been shown in experimental studies by McDevitt et al. (1976) and Schauerte and Schreyer (2023) for free and forced transition cases, respectively. Also numerical studies by Szubert et al. (2016) and Moise et al. (2022a) showed that buffet onset occurs at larger angles of attack at increased Re for the V2C profile at similar conditions. Even though it has not been confirmed yet for the OALT25 profile, we assume also for the present test case increased buffet amplitudes at higher angles of attack at \(Re\approx 3\) million.

Despite some quantitative variations in wave lengths, Strouhal numbers and extents of regions covered by the mode shape, we observe similar mode structures at various Re. Therefore, by studying the moderate Re case in more detail, we expect to be able to project our conclusions at least to Re typical of wind-tunnel experiments.

Firstly we consider the frequency scaling of the mode. Table 2 summarises relevant quantities for four different configurations of the OALT25 (simulation data from Dandois et al. (2018) is added to present results at \(Re=3,000,000\)) and one V2C test case from Zauner and Sandham (2019a). The various dimensionless frequencies are \(St=fc/U_\infty\) as obtained from the simulations, \(St_L=fL_{sep}/U_\infty\) based on the interaction length \(L_{sep}\), defined as the distance from separation to reattachment, and \(St_{R}=fL_{sep}|U_R |/U_\infty ^2=St_L |U_R|/U_\infty\), where \(U_R\) is the maximum velocity in the reverse-flow region (approximated by the minimum x-velocity component). We see that \(St_L\) is not constant over the different cases and is an order of magnitude higher than the values of this parameter seen in shock-induced separation bubbles. In that case the bubble response is known to be a form of growth and shrinkage known as bubble breathing. Although the precise mechanism is still debated, one line of argument (e.g. Touber and Sandham 2009) is that it arises from the response of the separating boundary layer and the associated separation shock to stochastic forcing. Here the separation shock is rather weak in comparison with the shock impinging cases and the Mach number is not far into the supersonic regime. As another difference to cases with shock generators, the shock wave and impingement point for the current case are not fixed and the separation bubble is observed to periodically move back and forth, as it will be shown later in more detail. Correcting \(St_L\) by the factor \(|U_R |/U_\infty\) leads to a Strouhal number, where present cases collapse much better, showing \(St_{R} \approx 0.022\), even though the onset of low-frequency buffet makes the comparison more difficult for \(M>0.7\) at \(Re = 500,000\). The proposed scaling suggests a role for the reverse-flow vortex in the mechanism for the intermediate mode, but some caution is required, considering the limited number of data points.

To examine the intermediate-frequency separation bubble mode in detail we consider the \(M=0.7\) case with \(Re=500,000\), where this mode containing multiple shock waves is well established. Before discussing the dynamics of the wave structures, we introduce some terminology. Figure 11 extracts the main features that will be useful in the discussion. As mentioned previously, a prominent feature is a \(\lambda\)-shock structure that extends over the front part of the airfoil. The front leg of the \(\lambda\)-shock is the compression wave due to boundary-layer separation from the airfoil surface, which can be termed a separation wave. The upper part of the \(\lambda\)-shock terminates the supersonic region well above the airfoil, resulting in the first lobe structure seen in the average Mach number contours in Fig. 9a. Even though the flow closer to the surface remains supersonic, the rear leg of the \(\lambda\)-shock connects down to the top of the separation bubble on the airfoil surface. This shock reflects from the sonic line at the apex of the separated flow region as an expansion fan (coloured blue in Fig. 11), following a structure that is well known from impinging shock wave studies. The turning of the flow due to the expansion wave supports reattachment of the boundary layer (Lindsey et al. 1947). The impact of the shock on the separation is to stop the separation region from growing, so its maximum height is at the impingement location, forming a ‘ridge’ feature. We will use the terminology ‘ridge wave’ as a shorthand for the V-shaped shock-expansion wave pattern formed as the flow turns over the top of the separation bubble.

It is noted that a sufficiently strong rear leg (compression wave) of the \(\lambda\)-shock can turn the flow subsonic (as seen for example in Zauner et al. (2019) for a different airfoil geometry). However, for the present test case at \(M=0.7\), the flow near the surface remains supersonic until reaching the normal shock. Similar flow structures have also been observed in experiments by Spee (1969). For some cases, a train of several weak reflected waves can be observed in Schlieren visualisations, before a final shock wave terminates the supersonic region.

Passing the ridge wave, the local freestream is still supersonic and this continues up to the terminating shock wave sketched in Fig. 11. The terminating shock determines the rear lobe of the supersonic region in Fig. 9a. Boundary-layer transition occurs in the vicinity of the reattachment position, which is close to the shock-foot location and, at this Mach number, the boundary layer remains attached up to the trailing edge. The wake mode is sketched as a von-Karman vortex street, developing behind the airfoil.

One other prominent feature, that leads us into a discussion of the dynamics, is the presence of upstream propagating waves in the subsonic region downstream of the \(\lambda\)-shock and above and behind the terminating shock. Many acoustic waves are present in the flow, some originating at the trailing edge, which are widely known as ‘Kutta waves’ (Tijdeman 1975), and some from the transition region. Such upstream-propagating wave structures have been observed in simulations (e.g. Garnier and Deck 2010) as well as experiments (e.g. Jacquin et al. 2009). Above the terminating shock, these waves take on a more coherent form and strengthen into upstream-propagating shock waves, with a supersonic flow (relative to their propagation speed) ahead of them. It is interesting to note that very similar flow features have been observed in experiments of Börner and Niehuis (2021) over transonic turbine blades in a cascade configuration. In that application, the top bound of the supersonic flow region is restricted by the wake of a neighbouring blade and the upstream-propagation and potential influence of acoustic waves appears significantly restricted. Nevertheless they also observe two distinct phenomena at low and intermediate frequencies, which appear to be related with buffet and separation-bubble instabilities.

Having established a terminology and identified the principal physical features, we can now look at the flow development during a cycle of the intermediate mode. Figure 12 shows five representative snapshots of the streamwise pressure gradient \(\partial p / \partial x\) during one cycle of the mode evolution, together with an x/t diagram in part (f) showing the location of the snapshots with horizontal black lines. The x/t diagram contains contours of \(\partial p / \partial x\) in a plane located 0.05c above the airfoil surface. The method of plotting shows compression regions (e.g. shock waves) in red and expansion (e.g. expansion fan) in blue. A corresponding movie can be found in supplementary material and online (https://​youtu.​be/​_​9TwbAgkAU4). It can be seen in Fig. 12f that the intermediate mode follows a regular oscillation and frame (e) at \(t=80.8\) has a very similar structure as frame (a) at \(t=78.5\). The period of \(\tau \approx 2.3\) corresponds to the value of \(St=0.42\) given in Table 2. Although Fig. 12f only covers a small portion of a low-frequency cycle, relatively regular intermediate-frequency oscillations are observed throughout the simulation, as shown in Fig. 7b. The first point to note as we move from frame (a) to (b) is the bifurcation of the terminating shock, the upper part of which moves upstream, merging with one of the upstream propagating waves above the terminating shock. The bifurcation can be seen in (b) at a distance 0.08c from the surface. The rear part of the bifurcated shock remains almost stationary, while the front part moves upstream and the bifurcation point moves towards the surface. Eventually, between (d) and (e) the front bifurcated wave merges with the \(\lambda\)-shock. The top part of the \(\lambda\)-shock weakens out as it moves into a region with reducing Mach number and eventually ends up as acoustic radiation into the region upstream of the airfoil. Similar bifurcations of shock waves have been reported by Spee (1969).

A final point to note from Fig. 12f is that all the wave features participate in the intermediate mode oscillation. At \(x=0.2\) we have oscillations on the separation compression wave. In the region \(0.4<x<0.45\) we see the ridge wave structure (shock and expansion waves) moving forward and backwards and at \(x=0.5\) we see oscillations in the terminating shock wave. With reference to the ridge wave, the x location of the separation wave lags by roughly \(90^\circ\), indicating that the bubble is neither in a pure ‘breathing’ mode, which would require the waves to be out of phase, nor in a simple up- and down-stream motion which would require the waves to be in phase. The movement of the ridge wave is close to being out of phase with the terminating shock location, given by the green sonic line in Fig. 12f, such that the ridge wave is furthest upstream when the terminating wave is furthest downstream.

After having identified the behaviour of distinct flow structures, we now want to further analyse their connections during a cycle of the intermediate frequency oscillation. To assist with this, we will use two more x/t diagrams, shown on Fig. 13, based on (a) pressure fluctuations at a distance 0.05c above the airfoil surface and (b) skin friction contours on the surface. Some propagating features are identified on the figures and will be discussed in the following paragraphs. Note that, in interpreting Fig. 13a, we need to be careful making connections to shock and expansion waves, as the centre of a shock wave, for example, is located by a rapid change in colour from blue to red, not by the centre of the red region.

From Fig. 13a we can readily identify the main features. Linear instabilities develop in the separation bubble, where velocity profiles contain an inflexion point and form Kelvin-Helmholtz roll-ups with strong spanwise coherence. Such structures have been predicted for similar flows using linear stability analysis (Zauner et al. 2017; Zauner and Sandham 2019a) and typical phase speeds agree well with present convection speeds of \(U_c \approx 0.2\), corresponding to the slope of the dotted line labelled I in Fig. 13. Even after break-down to turbulence, these vortex structures maintain spanwise coherence, leaving traces over the aft-section of the airfoil at \(St>10\). Further details are provided by Zauner and Sandham (2020), where Q-criteria visualisations of these phenomena for a different airfoil are shown in their Fig. 11. The pressure oscillation caused by movement of the separation compression wave is seen at \(x=0.2\). We also note the bifurcation of the terminating shock, with the leading wave moving upstream along the lines labelled II, eventually merging with the ridge wave at \(x=0.4\). Near the separation wave, we again observe features in the contours that appear to align with the extension of line II. A third feature that appears is the line labelled III, which can be associated with traces of acoustic waves circumventing the supersonic region at a speed corresponding to the slope of the magenta line of approximately \(U_a \approx U_{\infty }-U_{\infty }/M=0.43\) (\(U_{\infty }\) denotes the freestream velocity). The magenta line matches the contour gradient between \(0.25<x<0.4\) particularly well. Within the same x range, traces of acoustic structures can be also seen at different time instances (e.g. blue stripe at \(t=75,5\)), even though they appear relatively weak in the contour plots. These waves pass through the supersonic regions at significantly higher frequencies compared to those corresponding to the bubble mode. However, the acoustic field may well be moderated by unsteadiness at lower frequencies, leading to changes in contour patterns of Fig. 13 corresponding to acoustic speeds. The velocity associated with line II (\(U_{II} \approx 0.087\)) is about 20 times slower than typical acoustic waves, but of the same order as the maximum reverse-flow velocity (\(U_R=-0.147\)). The mechanism leading to the slope of features aligned with II is not clear, but one could speculate about some near-wall structures being responsible.

More insights are possible from the skin friction plot in Fig. 13b, where sketched lines are identical with (a). Here the separation point is shown by the white line at \(x \approx 0.15\). The reattachment is obscured by the unsteadiness, but can be traced as the colour change from deep blue to red at \(x \approx 0.525\). During the upstream motion of the separation wave (e.g. \(76.6<t<77.8\)) the \(C_f\) is dropping, which favours the development of the linear instabilities mentioned before. As a consequence, the reattachment region starts spreading upstream. This observation aligns with local linear stability results of Zauner and Sandham (2019a), where laminar boundary layers become unstable further upstream during low-lift phases (i.e. when the separation wave is close to its most upstream position). The region just upstream of reattachment, where transition is starting, forms a roughly triangular-shaped region, bounded by line I and a slope corresponding to velocities similar to maximum reverse-flow velocities of \(U_{r,max} \approx -0.147\) which may indicate an upstream convection of disturbances helping to sustain the process of transition to turbulence.

The above discussion has concerned only the \(M=0.7\) \(Re=500,000\) case. From analysis of SPOD mode shapes, some changes in this mode can be observed as Re increases, the most obvious being that the separation bubble reduces in size and the upper portion and trailing shock of the \(\lambda\)-shock merge with the termination shock, as can be seen in the SPOD modes on Fig. 6. This suggests that the shock bifurcation is not a crucial part of the mechanism, which instead involves a connection between fluctuations inside the separated boundary layer and upstream-propagating waves travelling up the main shock and around the top of the supersonic flow region. According to the Strouhal number arguments above, the period is set by the separation length and the strength of the reverse flow vortex. It would be interesting to investigate whether such modes can be detected in shock-induced separation bubbles under simplified flow conditions considering free- as well as forced transition upstream of the separation point. Furthermore, it would be also interesting to explore the possibility of such bubble modes existing at incompressible conditions. Eventually, global stability and/or resolvent analysis are needed to gain better understanding of this intermediate-frequency separation-bubble mode.