4.1.1 Differences between 2-D and 3-D interactions

The unsteady dynamics of spanwise-homogeneous, 2-D STBLIs has been thoroughly investigated for both the impinging-oblique-shock (Dupont et al. Reference Dupont, Haddad and Debiève2006; Touber & Sandham Reference Touber and Sandham2009; Agostini et al. Reference Agostini, Larcheveque, Dupont, Denieve and Dussauge2012; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013; Adler & Gaitonde Reference Adler and Gaitonde2018a) and compression-ramp (Ganapathisubramani et al. Reference Ganapathisubramani, Clemens and Dolling2009; Priebe & Martín Reference Priebe and Martín2012) configurations. Relevant time scales $(\unicode[STIX]{x1D70F}=1/St_{\unicode[STIX]{x1D6FF}})$ for prominent mid-frequency and low-frequency unsteadiness in these spanwise-homogeneous STBLI range from $\unicode[STIX]{x1D70F}\sim O(10)$ for Kelvin–Helmholtz phenomena induced within the separated shear layer, to $\unicode[STIX]{x1D70F}\sim O(300)$ for low-frequency bubble breathing, while the dominant frequencies of the incoming turbulent boundary layer are of higher frequency and coincide with the boundary-layer characteristic time, $\unicode[STIX]{x1D70F}\gtrsim O(1)$ (Agostini et al. Reference Agostini, Larcheveque, Dupont, Denieve and Dussauge2012; Adler & Gaitonde Reference Adler and Gaitonde2018a). The low-frequency unsteadiness appears to scale well with the size of the separation (Dupont et al. Reference Dupont, Haddad and Debiève2006; Dussauge et al. Reference Dussauge, Dupont and Debiève2006) and generally collapses with Strouhal number ($St_{L}=fL/U_{\infty }$) based on the nominal length of the separation bubble, $L$.

The unsteady dynamics of spanwise-non-homogeneous, swept, 3-D, interactions is distinctly different. In these configurations the separation length, $L$, increases across the span with displacement from the VCO, and indeed, we observe $St_{\unicode[STIX]{x1D6FF}}$, not $St_{L}$, to be the most appropriate primary non-dimensionalization for unsteadiness in these interactions, which conspicuously lack the prominent low-frequency ($St_{L}\sim 0.03$, $St_{\unicode[STIX]{x1D6FF}}\sim 0.006$) band of unsteadiness (two decades below the prominent boundary-layer frequencies) found in spanwise-homogeneous interactions. The unsteady features of the SF STBLIs have many similarities to those of the SCR STBLIs in the high-, mid- and low-frequency bands, indicating that the dynamics of 3-D interactions is distinctly different than 2-D interactions, and suggesting that 3-D interactions exhibit commonality of unsteady features for a wide variety of configurations.

Figure 5.

Comparison of premultiplied $(St\times PSD(St))$, wall-pressure PSD between (a,d) 2-D unswept-compression-ramp (sweep $0^{\circ }$, compression $22.5^{\circ }$) and (b,c,e,f) 3-D SCR (sweep $30^{\circ }$, compression $22.5^{\circ }$). Spatial location (horizontal axis) represented by Cartesian ($z/\unicode[STIX]{x1D6FF}=10$; $x/\unicode[STIX]{x1D6FF}$ free: a,b,d,e) and spherical ($r/\unicode[STIX]{x1D6FF}=10$; $\unicode[STIX]{x1D703}$ free: c,f) coordinates. PSD are normalized locally in (a–c) and globally in (d–f). Non-dimensional frequency (vertical axis) formed using boundary-layer thickness ($St_{\unicode[STIX]{x1D6FF}}$: left vertical) and local separation length ($St_{L}$: right vertical). Annotations demarcate the high-frequency integral-scale band ($\cdots \cdots$), mid-frequency shear-layer-coherence band (– – – –, violet), low-mid-frequency bubble-oscillation band (– – –, blue), and low-frequency interaction-breathing band (–   –). Notably, the prominent low-frequency unsteadiness in the 2-D, spanwise-homogeneous interaction is suppressed in the 3-D, swept interaction.

Figure 6.

SCR: premultiplied and normalized fluctuating wall-pressure PSD as a function of azimuthal angle (horizontal). Frequency is described by boundary-layer Strouhal number ($St_{\unicode[STIX]{x1D6FF}}$: left vertical) and local separation arc-length Strouhal number ($St_{L}$: right vertical). The PSD matrix is organized via increasing compression strength left-to-right across the rows and increasing radial displacement from the VCO top-to-bottom down the columns to demonstrate the spatial collapse of the spectra in the conical frame (parallel blue lines) and the lowering of the mid-frequency band of unsteadiness with increasing interaction strength (downward sloping red lines).

These considerations are now discussed quantitatively, with various figures describing the wall-pressure power-spectral-density (PSD) dependence on streamwise coordinate $(X/\unicode[STIX]{x1D6FF})$ or azimuthal angle $(\unicode[STIX]{x1D703})$ for each of the geometries described in table 2. The azimuthal angle (see schematic in figure 2) describes the angle (in the plane of the incoming boundary-layer wall and with vertex at the VCO) between the streamwise direction and the point of interest. Results for the unswept compression ramp are shown in figure 5(a,d), for the SCR in figures 5(b,c,e,f), 6 and 7 and for the SF in figures 8 and 9. Each will be discussed in context below. The ramp corner is located at $\unicode[STIX]{x1D703}=60^{\circ }$ for the SCR interactions. For both swept configuration types, large azimuthal angle corresponds to upstream displacement, whereas small azimuthal angle corresponds to downstream displacement. Except for figure 5(d–f), the PSD at each spatial location has been normalized such that the integral of premultiplied power over all resolvable frequencies is unity. In all cases, the PSD is obtained using Welch’s windowed overlapping segments estimator method by splitting the signal into 8 segments with 50 % overlap using Hamming windows. The PSD is premultiplied by the corresponding Strouhal number based on the mean incoming boundary-layer thickness to represent the power distribution effectively on a logarithmic scale. Each simulation is integrated in time in excess of $1500\unicode[STIX]{x1D6FF}/U_{\infty }$, with temporal data resolution of $0.05\unicode[STIX]{x1D6FF}/U_{\infty }$. This facilitates the reproduction of the known low-frequency band in 2-D interactions, with convergence of the spectra over more than 10 low-frequency cycles. In figures 6–9, spectra obtained for different interaction strengths and at different radii from the VCO are presented in a matrix, with annotative lines to illustrate various scaling laws for the fluctuation spectra.

Figure 7.

SCR: premultiplied and normalized fluctuating wall-pressure PSD as a function of azimuthal angle (horizontal). Frequency is described by boundary-layer Strouhal number ($St_{\unicode[STIX]{x1D6FF}}$: left vertical) and local separation arc-length Strouhal number ($St_{L}$: right vertical). The PSD matrix is organized via increasing radial displacement from the VCO left to right across the rows and increasing compression strength top to bottom down the columns to demonstrate the competing 2-D scaling (horizontal red lines) and 3-D scaling (downward sloping read lines) for the mid-frequency unsteadiness. Additionally, the increase of interaction spreading rate with interaction strength is demonstrated by the non-parallel blue lines.

Figure 8.

SF: premultiplied and normalized fluctuating wall-pressure PSD as a function of azimuthal angle (horizontal). Frequency is described by boundary-layer Strouhal number ($St_{\unicode[STIX]{x1D6FF}}$: left vertical) and local separation arc-length Strouhal number ($St_{L}$: right vertical). The PSD matrix is organized via increasing compression strength left to right across the rows and increasing radial displacement from the VCO top to bottom down the columns to demonstrate the spatial collapse of the spectra in the conical frame (parallel blue lines) and the lowering of the mid-frequency band of unsteadiness with increasing interaction strength (downward sloping red lines).

Figure 9.

SF: premultiplied and normalized fluctuating wall-pressure PSD as a function of azimuthal angle (horizontal). Frequency is described by boundary-layer Strouhal number ($St_{\unicode[STIX]{x1D6FF}}$: left vertical) and local separation arc-length Strouhal number ($St_{L}$: right vertical). The PSD matrix is organized via increasing radial displacement from the VCO left to right across the rows and increasing compression strength top to bottom down the columns to demonstrate the competing 2-D scaling (horizontal red lines) and 3-D scaling (downward sloping read lines) for the mid-frequency unsteadiness. Additionally, the increase of interaction spreading rate with interaction strength is demonstrated by the non-parallel blue lines.

Spectral comparison between the 2-D, unswept compression ramp and 3-D, SCR is presented in figure 5, along with annotation of the frequency bands relevant to 2-D STBLIs as described by Adler & Gaitonde (Reference Adler and Gaitonde2018a). Prominent low-frequency unsteadiness (two decades lower than the frequencies associated with the incoming turbulent boundary layer) is absent from the spectra of this SCR and all, swept, 3-D interactions of the present database. Notably, the comparison between the 2-D compression-ramp interaction (a) and the 3-D SCR interaction (b,c) differs only as a result of the introduction of sweep. The upstream flow conditions, numerical schemes and duration of temporal integration are maintained; this demonstrates that the 3-D interaction exhibits a suppressed low-frequency band relative to the 2-D interaction. This finding is consistent with the experimental observations of Gibson & Dolling (Reference Gibson and Dolling1991) as well as Schmisseur & Dolling (Reference Schmisseur and Dolling1992, Reference Schmisseur and Dolling1994), for the SF, along with Erengil & Dolling (Reference Erengil and Dolling1993), for the SCR. These studies identify the lowest significant unsteadiness in the swept interactions in a band only one decade below the characteristic boundary-layer frequencies. We classify this unsteadiness as mid-frequency unsteadiness rather than low-frequency unsteadiness, because it is largely associated with convective coherent structures formed in the separated shear layer (see § 4.2), not stationary large-scale (‘breathing’) unsteadiness of the separated region as a whole; necessarily, its mechanism is more similar to mid-frequency unsteadiness in comparable spanwise-homogeneous interactions, rather than the low-frequency unsteadiness in such interactions. Recent experiments also confirm this observation that the prominence of the mid-frequency fluctuation band is increased relative to the that of the low-frequency fluctuation band in swept interactions (Arora, Mears & Alvi Reference Arora, Mears and Alvi2019; Vanstone & Clemens Reference Vanstone and Clemens2019). We note that, by an absolute measure, the mid-frequency content in swept interactions may comprise lower or higher frequencies than the corresponding content in 2-D interactions, because the actual frequency depends on the local length and velocity scales. Relative to the mid-frequency band, these results indicate that swept interactions suppress a mechanism responsible for a still lower frequency band that would be present in the absence of sweep.

In contrast, the frequency characteristics of the high- and mid-frequency bands translate relatively well from the 2-D to 3-D interactions. The high-frequency band lowers in frequency moving through the interaction with decreasing local $U/\unicode[STIX]{x1D6FF}$. The mid-frequency band becomes prominent after separation and persists throughout the interaction, relatively independent of the azimuthal coordinate.

For the present discussion, there is little need to distinguish between the subtle differences in the 3-D spectra resulting from the use of the Cartesian surface (b,e) or spherical surface (c,f) on which the spatial dependence (horizontal axis) of the surface pressure spectra are plotted; however, use of the spherical surface more appropriately reflects the mean-flow symmetry, and captures the mid-frequency band more consistently, especially for small azimuthal angle (large streamwise coordinate). A notable distinction arises between the amplitude behaviour of the high- and mid-frequency bands of the 2-D and 3-D interactions. For both bands, the amplitude of the downstream (small azimuth) fluctuations decays more rapidly for the swept interaction than the unswept interaction, due to the positive spanwise velocity component downstream of the interaction in the swept configuration.

4.1.2 Spectral trends in 3-D interactions

As discussed in § 3.2, the post-separation mean flow of swept STBLIs exhibits regions of shear with both 2-D and 3-D similarity scaling laws. This section demonstrates the influence of these competing scaling laws on the manner in which high-frequency upstream fluctuations are processed by the interaction, and particularly, the development of mid-frequency fluctuations within the interaction. Specifically, it is observed that the high-frequency fluctuations (integral-scale turbulence, $St_{\unicode[STIX]{x1D6FF}}=0.2\lesssim f\lesssim St_{\unicode[STIX]{x1D6FF}}=3$, with non-dimensional frequency $f$) retain the 2-D scaling from the upstream boundary layer throughout the interaction; i.e. the development of high-frequency fluctuations with 3-D scaling within the interaction does not have adequate time for growth to overtake (in amplitude) the incoming 2-D scaling high-frequency fluctuations. However, the mid-frequency fluctuations (coherent shear-layer turbulence, $St_{L}=0.1\lesssim f\lesssim St_{\unicode[STIX]{x1D6FF}}=0.2$) exhibit aspects of 2-D and 3-D scaling at the upper and lower bounds of the mid-frequency band, respectively; i.e. the developing mid-frequency fluctuations with 3-D scaling within the interaction do indeed display adequate growth characteristics to compare with the amplitude of the developing 2-D-scaling mid-frequency fluctuations. Several quantitative demonstrations of the variability of 2-D versus 3-D mid-frequency scaling as a function of radial coordinate, interaction strength and interaction type follow.

The relationship of the wall-pressure PSD spatial signature to the interaction strength and radial displacement from the VCO is described in figures 6 and 7 for the SCR interactions followed by figures 8 and 9 for the SF interactions. The plots of figures 6 (SCR) and 8 (SF) are arranged in a matrix by increasing interaction strength from left to right across the rows and increasing radius from the VCO from top to bottom down the columns. Several lines crossing different panels are displayed to illustrate the principal observations. The blue lines facilitate comparison of the spatial scaling of the surface fluctuations for different radii. The parallel vertical nature of the blue lines reflects the near perfect (spatial) collapse of the wall-pressure fluctuation spectra in spherical coordinates for each interaction. A similar observation may be made regarding the red lines, which facilitate comparison of the frequency scaling of the surface fluctuations for different interaction strengths. When these are perfectly horizontal, the phenomenon exhibits 2-D frequency scaling with Strouhal number based on the incoming boundary-layer thickness ($St_{\unicode[STIX]{x1D6FF}}$ on the left axis), independent of the interaction strength. The Strouhal number scaling based on the local separation arc length is designated with $St_{L}$ on the right axis: diverging (downward sloping) red lines reflect the thickening of the separated shear layer and stronger 3-D scaling (scaling with $St_{L}$) of mid-frequency fluctuations with increasing interaction strength.

Conversely, the plots of figures 7 (SCR) and 9 (SF) are arranged in a matrix by increasing radius from the VCO from left to right across the rows and increasing interaction strength from top to bottom down the columns. In this case, the blue lines facilitate comparison of the spatial scaling of the surface fluctuations for different interaction strengths. The diverging blue lines reflect the increase in interaction spreading rate with increasing interaction strength; that is, the arc length between separation and attachment grows more rapidly with increasing radius for stronger interactions. In this case, the red lines facilitate comparison of the frequency scaling of the surface fluctuations for different radii. Horizontal alignment reflects 2-D frequency scaling (scaling with $St_{\unicode[STIX]{x1D6FF}}$ on the left axis), which is especially applicable for the high-frequency surface fluctuations. With Strouhal number based on the local separation arc-length ($St_{L}$) on the right axis, the diverging red lines thus reflect the competing 2-D (horizontal with constant $St_{\unicode[STIX]{x1D6FF}}$) and 3-D (downward sloping with constant $St_{L}$) scaling exhibited by the mid-frequency fluctuations.

In all of the spectra, high frequencies typical of integral scale features of the incoming turbulent boundary layer $(St_{\unicode[STIX]{x1D6FF}}\sim 1)$ are prominent upstream of the interaction region. This frequency band results from the coherent structure of the boundary layer and naturally scales with the outer-scale Strouhal number, $St_{\unicode[STIX]{x1D6FF}}$. These frequencies are modulated throughout the interaction, monotonically decreasing through the pre-separation, interaction and post-attachment zones, as the boundary layer grows and slows, responding to decreases in local $U/\unicode[STIX]{x1D6FF}$. This high-frequency content completely collapses (spatially) in the conical (3-D) frame. It is also essentially independent of the separation length; therefore it is not modulated with increasing radial displacement from the VCO (comparing along the columns in figures 6 and 8 or comparing along the rows in figures 7 and 9, horizontal alignment of the red lines delineating the upper and lower bounds of the high-frequency band is observed). This frequency content is relatively independent of the sweep or compression strengths, except for delineation of the pre-separation, interaction and post-attachment zones by the separation and attachment lines which depend on the strength of the interaction.

Mid-frequency content $(0.01\lesssim St_{\unicode[STIX]{x1D6FF}}\lesssim 0.2,0.1\lesssim St_{L}\lesssim 1)$ corresponding to coherent Kelvin–Helmholtz-like (K–H) shear-layer phenomena is present between the separation and attachment locations. This content is also found in spanwise-homogeneous interactions (Agostini et al. Reference Agostini, Larcheveque, Dupont, Denieve and Dussauge2012; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013; Adler & Gaitonde Reference Adler and Gaitonde2018a); however, the three-dimensional nature of the SCR or SF mean flow leads to different spectral properties. In spanwise-homogeneous interactions, the mid-frequency coherent structures convect downstream after attachment and persist in the downstream spectra for highly separated interactions (Agostini et al. Reference Agostini, Larcheveque, Dupont, Denieve and Dussauge2012); however, in these three-dimensional interactions, the coherent fluctuations also prominently convect in the spanwise direction. Both the SCR and SF flows exhibit strong cross-flow underneath the main separated flow (Adler & Gaitonde Reference Adler and Gaitonde2019a), which influences the convection of coherent structures in the mid-frequency range.

In the SCR and SF interactions, mid-frequency unsteadiness demonstrates a linear dependence on the local separation length as demonstrated in figures 6 and 8 (by the downward sloping red line forming the lower bound of the mid-frequency range); i.e. $St_{L}$ (3-D scaling) is more appropriate than $St_{\unicode[STIX]{x1D6FF}}$ (2-D scaling) for describing this lower bound of the mid-frequency content. However, the content scales only weakly with $St_{L}$, because the upper bound is indicated by a horizontal red line and better described by $St_{\unicode[STIX]{x1D6FF}}$. The frequency of mid-frequency content is reduced while the amplitude slightly increases for stronger interaction strength (more evident for the SCR configurations), which reflects stronger 3-D scaling in stronger interactions – the spatial spreading rate is larger for swept interactions with a stronger shock (Adler & Gaitonde Reference Adler and Gaitonde2019a). The frequency of mid-frequency content also reduces linearly with increasing radial displacement from the VCO given a fixed compression strength (evident in all spectra), which reflects that all 3-D interactions exhibit 3-D scaling of the lower bound of the mid-frequency spectra.

These observations of mid-frequency unsteadiness in SCR interactions are consistent with the experimental observations of Erengil & Dolling (Reference Erengil and Dolling1993), who describe the dependence of unsteadiness on ramp sweep angle. They find that the intensity of shock unsteadiness decreases with increasing sweep, which is demonstrated by a reduction in root-mean-square pressure fluctuations as well as a decrease in the size of the intermittent STBLI interaction region with increasing sweep. They also observe that the dominant frequencies are an order of magnitude higher in highly swept cases compared with unswept cases.

We confirm these observations with comparison to previous spanwise-homogeneous simulations (Adler & Gaitonde Reference Adler and Gaitonde2018a), and connect these observations to the absence of low-frequency unsteadiness, which typically manifests as large scale, low-frequency, coherent but stationary, ‘breathing’ displacement of the separation and shock structure. We suggest that these results do not simply describe the transition of a single mechanism of ‘low-frequency’ unsteadiness to higher frequency with increasing interaction sweep; rather, the evidence indicates that a dominant mechanism resulting in low-frequency behaviour in 2-D STBLIs is suppressed when sweep is added, and the apparent ‘low-frequency’ (really mid-frequency) unsteadiness now results from the mid-frequency mechanism associated with the development of coherence in the post-separation shear layers.

The conceptual connection between the competing 2-D and 3-D scalings of the mean-flow shear layers and the realization of competing 2-D and 3-D scalings in the mid-frequency (shear-layer) band of unsteadiness is now summarized utilizing the SCR configuration as an example. Figure 4(a) and (b) shown earlier isolated the outer (main) and inner (cross-flow) shear layers, respectively. The outer shear layer for all swept, 3-D interactions collapses with a spanwise-homogeneous (2-D) similarity scaling (e), whereas the inner shear layer for all simple, 3-D interactions collapses with a conical (3-D) similarity scaling (f). These 2-D and 3-D mean-flow scalings influence the development of mid-frequency unsteadiness in swept, 3-D interactions (figure 7a–c), which is related to convectively coherent fluctuations in the shear layers (see § 4.2). The upper bound of the mid-frequency band ($St_{\unicode[STIX]{x1D6FF}}\sim 0.2$: horizontal red line) reflects a 2-D frequency scaling based on the global, upstream, boundary-layer thickness, $\unicode[STIX]{x1D6FF}$; whereas the lower bound of the mid-frequency band ($St_{L}\sim 0.1$: downward sloping red line) reflects a 3-D frequency scaling based on the local, separation arc length, $L$. A notable question arises as to whether there is an ultimate lower bound to the 3-D scaling as it continues to decrease with increasing VCO displacement, providing greater scale separation from the integral scale, high-frequency turbulence, and how this frequency scaling (and a potential lower bound thereto) is related to the phenomena of secondary separation.

4.2.1 Spatio-temporal visualization

Figure 10.

SCR: band-pass-filtered pressure fluctuations for $22.5^{\circ }$ compression, $30.0^{\circ }$ sweep interaction. Spatial coordinates normalized by incoming boundary-layer thickness. Convective shear-layer coherence in the mid-frequency band is evident in (a–f). Convective but less coherent high-frequency unsteadiness is evident in (g,h). Annotations track the evolution of high (red) and low (blue) extrema pressures through consecutive panels.

Figure 11.

SF: band-pass-filtered pressure fluctuations for $15.0^{\circ }$ sweep interaction. Spatial coordinates normalized by incoming boundary-layer thickness. Convective shear-layer coherence in the mid-frequency band is evident in (a–f). Convective but less coherent high-frequency unsteadiness is evident in (g,h). Annotations track the evolution of high (red) and low (blue) extrema pressures through consecutive panels.

In this section, to illustrate the spatio-temporal nature of the fluctuations associated with the prominent frequency bands observed in the spectral analysis of § 4.1, a band-isolation analysis of the unsteady flow is performed for the SCR and SF interactions. For this purpose, a series of low-pass, finite-impulse-response, window-based filters are applied in sequence to obtain a complete band-isolated basis for the instantaneous flow (Adler & Gaitonde Reference Adler and Gaitonde2018a). Figures 10 and 11 describe the structure of flow features observed via pressure fluctuations in the high-frequency $(St_{\unicode[STIX]{x1D6FF}}>0.2)$ and mid-frequency $(0.01<St_{\unicode[STIX]{x1D6FF}}<0.1)$ bands for the reference interactions: the ($22.5^{\circ }$ compression, $30.0^{\circ }$ sweep) SCR and the ($15.0^{\circ }$ attack) SF, respectively. No significant content is observed in the low-frequency $(St_{\unicode[STIX]{x1D6FF}}<0.01)$ band for any of the swept, 3-D interactions.

The mid-frequency bands exhibit spatio-temporally coherent fluctuations that form after separation and travel through the separated shear layer, with a surface pressure footprint that largely follows the streamlines of the outer (main) shear layer. These structures are convective in nature, as demonstrated by the temporally sequential panels (a,c,e and b,d,f) of figures 10 and 11, and they continue downstream after attachment, like in spanwise-homogeneous interactions (Agostini et al. Reference Agostini, Larcheveque, Dupont, Denieve and Dussauge2012; Aubard et al. Reference Aubard, Gloerfelt and Robinet2013; Adler & Gaitonde Reference Adler and Gaitonde2018a), as demonstrated in figures 10(b,d,f) and 11(b,d,f). However, unlike spanwise-homogeneous STBLIs, these three-dimensional interactions also result in convection of the structures associated with this frequency in the spanwise direction, through the separated flow, as demonstrated in figures 10(a,c,e) and 11(a,c,e). The competing downstream and spanwise convection directions in this band interfere both constructively and destructively, leading to intermittent in-phase and anti-phase switching of pressure fluctuations in the $x{-}z$ plane; these characteristic directions correspond, roughly, to the mean flow directions of the outer and inner shear layers, respectively. Similar to spanwise-homogeneous interactions (Agostini et al. Reference Agostini, Larcheveque, Dupont, Denieve and Dussauge2012), these convecting coherent fluctuations influence the shock through disturbance propagation to the shock wave along the local characteristic direction, resulting in mid-frequency shock unsteadiness which is correlated with the passage of these convective structures (figures 10b,d,f and 11b,d,f). The coherent size of mid-frequency content is larger for stronger interactions and for larger radius, likely because the thicker shear-layer results in larger instability wavelengths, facilitating more coherent disruption of the shock. Additionally, the convective speed $({\approx}0.3U_{\infty })$ of the mid-frequency coherent structures in the main separated flow near the main shock is similar between all of the configurations and is comparable with observations of mid-frequency, coherent, vortical content in spanwise-homogeneous interactions (Aubard et al. Reference Aubard, Gloerfelt and Robinet2013). However, the convective speed of the mid-frequency content in the cross-flow (underneath the main separated flow) depends on the interaction strength. For stronger interactions, in which the cross-flow is robust and less closely aligned with the main separated flow, the mid-frequency pressure fluctuations travelling in the cross-flow direction appear less coherent and disrupted by the pressure footprint from the structures in the separated flow above.

The high-frequency band contains contributions from integral- and fine-scale turbulence and is not as coherently organized as the mid-frequency band. High-frequency pressure fluctuations are shown for a representative snapshot of both the SCR and SF interactions in figures 10(g,h) and 11(g,h). In both interactions, the amplitude of this band increases locally across the separation and attachment lines as the flow is compressed through the separation and attachment shocks. After attachment, the amplitude of these fluctuations decreases; this observation is connected to the post-interaction reduction in turbulent kinetic energy. Unlike the mid-frequency fluctuations, the coherent size of these fluctuations appears relatively independent of interaction strength and radial location; this is consistent with the observation that they are associated with integral- and fine-scale turbulence seeded from the incoming turbulent boundary-layer (2-D similarity) and not a result of developing mid-scale coherent dynamics (3-D similarity).

4.2.2 Temporal autocorrelations of mid-frequency fluctuations

Figure 12.

Temporal autocorrelations of low-pass-filtered pressure fluctuations $(St_{\unicode[STIX]{x1D6FF}}\lesssim 0.05)$ in the SCR (a) and SF (b) interactions demonstrate competing 2-D and 3-D scaling laws in the mid-frequency fluctuation band. In regions of significant 3-D mean-flow symmetry (solid lines), correlations indicate a reduced rate of amplitude decay with increasing radius. However, for regions of significant 2-D mean-flow symmetry (dashed lines), the change in correlation decay rate with increasing radius is insignificant. 2-D correlations in (b) are rescaled for improved clarity.

Additional quantitative evidence to distinguish 2-D from 3-D similarity scaling laws for mid-frequency fluctuations in swept, 3-D STBLIs may be gathered from the comparison of various temporal autocorrelations of pressure fluctuations. Figure 12 presents autocorrelations of low-pass-filtered $(St_{\unicode[STIX]{x1D6FF}}\lesssim 0.05)$ pressure fluctuations for representative SCR (a) and SF (b) interactions at three different radial displacements from the VCO; the solid lines correspond to a region of the flow that exhibits 3-D mean-flow scaling in the inner (cross-flow) shear layer, whereas the dashed lines correspond to a region of the flow that exhibits 2-D mean-flow scaling in the outer (main) shear layer, just downstream of separation (Adler & Gaitonde Reference Adler and Gaitonde2019a). The low-pass filtering operation isolates pressure fluctuations associated with the lower portion of the mid-frequency band. For both interaction types, the autocorrelations associated with the flow regions of 2-D scaling are independent of the radius (spanwise homogeneous), whereas the autocorrelations associated with the flow regions of 3-D scaling are dependent on radius, with successive correlation peaks increasing in amplitude and spacing with increasing radius. Both the amplitude increase and period decrease of the 3-D autocorrelation peaks with increasing radius are consistent with the decrease in the frequency of mid-frequency unsteadiness with increasing radius (figures 6–9). Essentially, the periodic behaviour of the lower range of the mid-frequency band increases in coherence with increasing radius. Necessarily, because of the similarity of the flow velocity at each of the radial stations, these trends are also related to the spatial enlargement of mid-frequency coherence with increasing radius (figures 10 and 11).

4.3.1 Overview of method

In addition to the phenomenological differences in unsteadiness between 2-D and 3-D STBLIs described above, there are significant differences in the underlying linear dynamics of these flows. Using the recently described method of Adler & Gaitonde (Reference Adler and Gaitonde2018a), we now compare the dynamic linear response of a representative, swept, 3-D STBLI (the SCR with $30.0^{\circ }$ sweep and $37.5^{\circ }$ compression) to the results of the published 2-D impinging shock (IS) interaction (Adler & Gaitonde Reference Adler and Gaitonde2018a). The dynamic linear response results from the implicit, constrained linearization of the Navier–Stokes equations about an unsteady (LES or direct numerical simulation) base flow, subject to some perturbation of initial conditions or forcing. The implicit linearization is achieved in practice through the synchronization of two simulations, which also accounts for the linearization of sub-grid models, boundary conditions, and other implicit aspects of the numerical method. This method of stability analysis may be classified as fully three-dimensional in space and non-modal (Schmid Reference Schmid2007; Theofilis Reference Theofilis2011). The analysis is fully 3-D, in that three non-homogeneous spatial dimensions are assumed for the base flow and perturbations Additionally, the base flow is assumed to vary in time, which is facilitated by the non-modal approach, through which the base flow and perturbation solutions are calculated via coupled initial value problems.

In essence, for the discretized Navier–Stokes equations $(\boldsymbol{F})$ acting on the conserved state vector $(\unicode[STIX]{x1D731})$,

(4.1)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{i}}{\unicode[STIX]{x2202}t}=F_{i}[\unicode[STIX]{x1D731}],\end{eqnarray}$$

the evolution of the linear perturbation to the state vector $(\unicode[STIX]{x1D731}^{P})$ of an unsteady base flow $(\unicode[STIX]{x1D731}^{B})$ may be described,

(4.2)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{i}^{P}}{\unicode[STIX]{x2202}t}=\frac{\unicode[STIX]{x2202}F_{i}[\unicode[STIX]{x1D731}^{B}]}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{j}^{B}}\unicode[STIX]{x1D6F7}_{j}^{P}+S_{i}[\unicode[STIX]{x1D731}^{B},\unicode[STIX]{x1D731}^{P}],\end{eqnarray}$$

with forcing term $(\boldsymbol{S})$. In practice, the linearization is performed implicitly via the synchronized evolution of two nonlinear simulations, and the perturbations are constrained to the linear regime. The evolution of the constrained linear perturbations (denoted $\unicode[STIX]{x1D731}^{P_{c}}$) may be described as

(4.3)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{i}^{P_{c}}}{\unicode[STIX]{x2202}t}=F_{i}[\unicode[STIX]{x1D731}^{B}+\unicode[STIX]{x1D731}^{P_{c}}]-F_{i}[\unicode[STIX]{x1D731}^{B}]-\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6F7}_{i}^{P_{c}}+S_{i}[\unicode[STIX]{x1D731}^{B},\unicode[STIX]{x1D731}^{P_{c}}]+O(\Vert \unicode[STIX]{x1D731}^{P_{c}}\Vert ^{2}),\end{eqnarray}$$

with constraining term $(\unicode[STIX]{x1D6FC})$ and forcing term $(\boldsymbol{S})$; a posteriori verification is performed to ensure that the constraining strategy satisfactorily minimizes the nonlinear error term $(O(\Vert \unicode[STIX]{x1D731}^{P}\Vert ^{2}))$. Concerning the perturbation response to impulse, the unconstrained linear perturbations may be related to the constrained linear perturbations,

(4.4)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{i}^{P}(t)=\unicode[STIX]{x1D6F7}_{i}^{P_{c}}(t)\int _{t_{0}}^{t}\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F},\end{eqnarray}$$

with initial condition,

(4.5)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{i}^{P}(t_{0})=\unicode[STIX]{x1D6F7}_{i}^{P_{c}}(t_{0}).\end{eqnarray}$$

For time-dependent base flows, we generalize the classification of instabilities based on the Green’s function mapping of impulse $(\unicode[STIX]{x1D731}^{P}(t_{0}))$ to response $(\unicode[STIX]{x1D731}^{P}(t))$ of the unconstrained perturbations

(4.6)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{i}^{P}(t)={\mathcal{G}}_{ij}(t,t_{0})\unicode[STIX]{x1D6F7}_{j}^{P}(t_{0}).\end{eqnarray}$$

In this fully 3-D non-modal framework, a flow is classified as unstable if the impulse gain (the maximum norm of the Green’s function) is unbounded,

(4.7)$$\begin{eqnarray}\limsup _{t\rightarrow \infty }\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{\infty }\rightarrow \infty \quad \text{for some}:t_{0}.\end{eqnarray}$$

An unstable flow is additionally classified as absolutely unstable if

(4.8)$$\begin{eqnarray}\limsup _{t\rightarrow \infty }\frac{\Vert {\mathcal{G}}_{i_{0}j_{0}}(t,t_{0})\Vert _{\infty }}{\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{\infty }}\in (0,1]\quad \text{for some, finite, non-boundary set of}:i_{0},\,j_{0},\,t_{0},\end{eqnarray}$$

indicating that the maximum gain is sustained within a finite (spatially localized), time-independent set of degrees of freedom $(i_{0},j_{0})$. Alternatively, an unstable flow is additionally classified as convectively unstable if

(4.9)$$\begin{eqnarray}\limsup _{t\rightarrow \infty }\frac{\Vert {\mathcal{G}}_{i_{0}j_{0}}(t,t_{0})\Vert _{\infty }}{\Vert \boldsymbol{{\mathcal{G}}}(t,t_{0})\Vert _{\infty }}=0\quad \text{for all, finite, non-boundary sets of}:i_{0},\,j_{0},\,t_{0},\end{eqnarray}$$

indicating that the maximum gain is not sustained within a finite (spatially localized), time-independent set of degrees of freedom; rather, the maximum gain is convective in character. Although the concepts of absolute and convective instability are often described in the context of a quasi-parallel flow, the aforementioned generalized definitions allow for analogous concepts to be discussed in the context of a 3-D unsteady flow. Similarities between the quasi-parallel and 3-D descriptions of absolute and convective instability exist in both the Green’s function definitions as well as in the qualitative behaviour of perturbation evolution. Similar to the classical qualitative descriptions of these phenomena in quasi-parallel flow, the dynamic linear impulse response of an absolutely unstable 3-D flow exhibits spatially bounded, self-sustaining perturbations, seemingly driven by an internal (oscillator-like) dynamics, whereas the convectively unstable 3-D flow exhibits convective perturbations, which are necessarily fed by an external source for sustenance, and are seemingly driven by the flow amplifying or filtering the externally introduced perturbations.

As discussed in detail in by Adler & Gaitonde (Reference Adler and Gaitonde2018a), the constraining term in (4.3) is chosen such that

(4.10)$$\begin{eqnarray}\Vert \unicode[STIX]{x1D731}^{P_{c}}(t)\Vert _{\infty }=\Vert \unicode[STIX]{x1D731}^{P_{c}}(t_{0})\Vert _{\infty }\;\Longrightarrow \;\Vert \boldsymbol{{\mathcal{G}}}^{c}(t,t_{0})\Vert _{\infty }\sim O(1)\quad \text{for all}:t,\,t_{0},\end{eqnarray}$$

where $\Vert \boldsymbol{{\mathcal{G}}}^{c}(t,t_{0})\Vert _{\infty }$ represents the gain of the constrained system,

(4.11)$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{i}^{P_{c}}(t)={\mathcal{G}}_{ij}^{c}(t,t_{0})\unicode[STIX]{x1D6F7}_{j}^{P_{c}}(t_{0}).\end{eqnarray}$$

Thus, an absolute instability in the constrained system is identified by a self-sustaining (finite perturbation magnitude persisting in time) spatially localized perturbation field for some initial condition. Conversely, the absence of an absolute instability in the constrained system is identified by a locally decaying constrained perturbation field at all non-boundary locations.

4.3.2 Differences between 2-D and 3-D interactions

Figure 13.

Demonstration of the dynamic linear response to impulse forcing for both a simple, 3-D STBLI and spanwise-homogeneous 2-D STBLI. The impulse response of the simple, 3-D interaction is not self-sustaining, instead convecting toward the domain boundaries (a–f), indicating the absence of an absolute instability, whereas the impulse response of the 2-D interaction is self-sustaining (g–l), indicating the presence of an absolute instability. Representative perturbation quantities are rendered over sequential time steps to demonstrate these contrasting responses to impulse, in which the green annotations provide a static reference to gauge shock location.

The impulse response of the 2-D impinging-shock interaction (Adler & Gaitonde Reference Adler and Gaitonde2018a) and 3-D SCR interaction are compared in figure 13. A complete, 3-D rendering is employed to visualize the SCR interaction, whereas the centreline, spanwise-normal plane is employed to visualize the IS simulation; both examples are rendered from a complete, 3-D, dynamic linear response analysis of the unsteady base flow. The impulse is introduced in both interactions through short-time, broadband, random forcing of the full, perturbation, state vector, in a spanwise-normal line upstream of the interaction. A representative perturbation quantity (vertical velocity for the SCR interaction and streamwise velocity for the IS interaction) is presented at sequential instances of time throughout the panels of figure 13, with green annotations to provide a reference shock location for both configurations along with other properties (pressure, sonic line, and reverse flow line) of the IS base flow.

The SCR interaction is perturbed in the upstream boundary layer (a), and the perturbations grow convectively in the upstream boundary layer (b), before entering the interaction region (c), where strong receptivity (large perturbation growth rate) is observed. The perturbations are processed by the interaction (d); however, no self-sustaining mechanism is observed, rather the perturbations continue to convect toward the simulation boundaries (e) and eventually start to decay in magnitude (f). This lack of self-sustaining perturbations in the simple, 3-D STBLI, is consistent with the discussion of convective instability (lack of absolute instability: equations (4.7)–(4.9)).

The IS interaction is perturbed just upstream of the separated shear layer and reflected shock (g), and the perturbations travel convectively into to interaction region (h), where strong receptivity is demonstrated (i). The perturbations are processed by the interaction (j), in which large perturbation magnitudes remain in the interaction region, relative to downstream, reflecting the strong, absolute, perturbation growth rates around the recirculation region, which ultimately result in a self-sustaining perturbation field, anchored to the interaction region, and reflective of the underlying absolute instability (4.8). The self-sustaining perturbations demonstrate a correlation with the low-frequency motion of the reflected shock wave, exhibiting high (k) and low (l) pressures for downstream and upstream shock displacement, respectively, linking the character of the absolute instability to linear restoring tendencies driving the low-frequency shock unsteadiness (Adler & Gaitonde Reference Adler and Gaitonde2018a).

The impulse responses of equilibrium turbulent boundary layers and swept, 3-D STBLIs exhibit behaviour consistent with the lack of an absolute instability, instead exhibiting only features of convective instability. Considering the mechanistic ramifications for STBLI unsteadiness, the absolutely unstable 2-D STBLI exhibits prominent low-frequency unsteadiness, whereas the convectively unstable 3-D STBLI (lacking the absolute instability) does not exhibit this prominent low-frequency unsteadiness. This raises the question as to whether the absolute instability is necessary to facilitate the low-frequency STBLI unsteadiness (two decades lower than the characteristic integral-scale boundary-layer unsteadiness and exhibiting a clear separation of scales therefrom). Our recent mechanistic description of 2-D STBLIs (Adler & Gaitonde Reference Adler and Gaitonde2018a), which describes the relationship of the linear perturbation dynamics to the evolution of the nonlinear unsteady base flow in several prominent frequency bands, suggests that the linear dynamics associated with the absolute instability is related to (and could be a primary mechanism for) the unsteady motion of the base flow in the low-frequency band. However, this does not exclude the possibility that other mechanisms also contribute to STBLI unsteadiness in that band. Concerning swept, 3-D STBLIs, the low-frequency band (comparable to the prominent low-frequency unsteadiness ubiquitously observed in 2-D STBLIs with otherwise similar flow conditions) is not observed, nor is the absolute instability, supporting the connection between the absolute instability and low-frequency STBLI unsteadiness. Further analysis will be needed to assess precisely how the convectively unstable linear dynamics contributes to the nonlinear mid-frequency unsteadiness in swept 3-D interactions.

Unfortunately, the analysis of the dynamic linear response of a convectively unstable flow (lacking an absolute instability) is not as straightforward as the analysis of an absolutely unstable flow, because the convective instability always retains a signature of the external forcing, whereas the absolute instability is ergodic (forgetting the initial forcing in the long time limit). Hence, many different responses to forcing must be accounted for in the ensemble analysis of convective instabilities, whereas ergodic ensemble averaging in time was sufficient for the absolute instability (the task of ensemble averaging for a convectively unstable flow becomes more computationally intensive). Due to the additional complexity, these analyses, apart from the simple (non-ensemble) analysis of the response to impulse, have not yet been performed. Nonetheless, this difference in the dynamic linear impulse response distinguishes the representative 2-D spanwise-homogeneous interaction from the representative 3-D swept interaction and suggests a strong connection between the exhibition of absolute instability and the manifestation of the corresponding low-frequency shock unsteadiness. This observation represents a significant advance and opens many additional avenues of research in developing a complete understanding of the interaction of the linear dynamics with the nonlinear base flow and shock unsteadiness.