4.1. General flow features

Base flows are obtained for a hollow cylinder/flare with a series of cylinder radii and flare deflection angles and its two-dimensional counterpart with different ramp angles. For convenience, the critical deflection angles for incipient and secondary separation are denoted by α ₁ and α ₂, respectively.

The skin friction coefficient C_f and surface pressure coefficient C_p are defined as

(4.1a,b)\begin{equation}{C_f} = \frac{{{\tau _w}}}{{0.5{\rho _\infty }u_\infty ^2}},\quad {C_p} = \frac{{{p_w}}}{{0.5{\rho _\infty }u_\infty ^2}},\end{equation}

where τ_w and p_w are the skin shear stress and surface pressure, respectively. The locations of the separation and reattachment points are identified when C_f crosses zero upstream and downstream, respectively. The size of the separation region L_sep is measured from the separation point to the reattachment point along the x-axis. A grid independence study (see Appendix A) shows that 600 × 350 grid points are sufficient for both CFD simulations and GSA. The triple-deck calculations are performed for the axisymmetric and planar flows with 1000 × 200 and 800 × 200 grid points, respectively. A good agreement is observed between the obtained triple-deck solutions and data in the literature (see Appendix B).

Figure 2 presents the skin friction curves at various deflection angles for different cylinder radii R/L = 0.2, 0.5, 1.0 and ∞. The open symbols represent the separation and reattachment points. The enlarged views near the juncture are also displayed. Following an initial decrease upstream of separation due to the development of the laminar boundary layer, C_f experiences a rapid decay near the separation point to the first local minimum and then a gradual growth along the cylinder. Downstream of the juncture, a downward trend of C_f to the second local minimum is observed in the separation region. After the noticeable second local minimum, the skin friction progressively increases around the reattachment point. The existence of two local minima in C_f inside the separation zone indicates a fully separated flow. As α is increased, the extent of the separation zone accordingly grows. The magnitudes of both the second minimum and the local peak near the juncture also increase with increasing α. These features are similar to the findings in compression corner flows (Smith & Khorrami Reference Smith and Khorrami1991; Korolev, Gajjar & Ruban Reference Korolev, Gajjar and Ruban2002; Shvedchenko Reference Shvedchenko2009; Gai & Khraibut Reference Gai and Khraibut2019). Beyond a threshold deflection angle α ₂, secondary separation occurs, which is characterized by a positive local peak near the juncture. The secondary bubble emerges at 15° ≤ α ₂ ≤ 16° for R/L = 0.2. When the cylinder radius is increased to R/L = 0.5 and 1.0, α ₂ decreases to 14°–15° and 13°–14°, respectively. It is observed that the emergence of secondary separation is delayed to a higher deflection angle when the cylinder radius is decreased. With respect to the compression corner flow (R/L = ∞), the critical ramp angle for secondary separation is 11°–12°, which is lower than those for the axisymmetric flows. The above results can be interpreted by the triple-deck theory through the scaled deflection angle α* to determine the critical angles for incipient separation and secondary separation. For R/L = ∞ (R* = ∞), secondary separation occurs at α* = 4.51–4.92. As the cylinder radius is decreased to R/L = 1.0, 0.5 and 0.2 (R* = 24.74, 12.37 and 4.948), the occurrence of secondary separation is delayed to α* = 5.33–5.74, 5.74–6.15 and 6.15–6.57, respectively. Additional CFD simulations show that the critical angles for incipient separation are α* = 1.64–2.05 for R* = 4.948 and 12.37 and α* = 1.23–1.64 for R* = 24.74 and ∞.

Figure 2.

The skin friction coefficient distributions for (a) R/L = 0.2, (b) R/L = 0.5, (c) R/L = 1.0 and(d) R/L = ∞ at different deflection angles with enlarged views near the juncture (PHAROS). Open symbols: separation and reattachment points. Horizontal dashed line: zero skin friction. Vertical dashed line: juncture.

As seen in figure 2, the length of the separation region and the magnitude of the second minimum in C_f depend on the deflection angle α for a given cylinder radius. Burggraf (Reference Burggraf1975) and Korolev et al. (Reference Korolev, Gajjar and Ruban2002) showed a linear dependence between $L_{sep}^\ast $ and α ^*3/2 for a compression corner based on the triple-deck theory. Here, $L_{sep}^\ast $ is plotted versus α ^*3/2 for varying scaled cylinder radii R* in figure 3(a). The linear dependence still holds for hollow-cylinder/flare flows with the slope decreasing as the cylinder radius is decreased. Korolev et al. (Reference Korolev, Gajjar and Ruban2002) also noted that the magnitude of the wall shear stress minimum ahead of reattachment |τ_{w ,min}| is proportional to α* for a compression corner flow. Figure 3(b) presents the variations in |C_{f ,min}| against α* for the considered cases. It is confirmed that a linear relationship between |C_{f ,min}| and α* is satisfied for both the compression corner and hollow-cylinder/flare flows.

Figure 3.

The (a) scaled length of the separation region as a function of α ^*3/2 and (b) magnitude of the second minimum in C_f as a function of α* for axisymmetric and planar configurations (PHAROS).

In figure 4, the scaled length of the separation region and the magnitude of the second skin friction minimum are plotted against the scaled cylinder radius for varying deflection angles. In figure 4(a), as the cylinder radius is increased, the size of the separation bubble experiences an increase and eventually converges to the planar results denoted by the dashed lines. Similarly, the magnitude of the friction minimum grows with increasing R* and approaches the planar data in figure 4(b).

Figure 4.

The (a) scaled length of the separation region and (b) magnitude of the second minimum in C_f as a function of R* for varying deflection angles (PHAROS). Horizontal dashed line: the limiting planar results.

The surface pressure coefficients for various cylinder radii and deflection angles are shown in figure 5. The initial rise in surface pressure upstream of the separation point is governed by the free-interaction process (Chapman, Kuehn & Larson Reference Chapman, Kuehn and Larson1958). Downstream of separation, a pressure plateau forms, and both its size and magnitude increase with increasing α. A small ‘dip’ can be observed near the juncture at the end of the pressure plateau for large α, which becomes progressively noticeable as α is increased. This local drop indicates a local low-pressure region near the juncture, which is consistent with the observations in compression corner and double-cone flows (Smith & Khorrami Reference Smith and Khorrami1991; Korolev et al. Reference Korolev, Gajjar and Ruban2002; Gai & Khraibut Reference Gai and Khraibut2019; Hao et al. Reference Hao, Fan, Cao and Wen2022). Near the reattachment point, C_p experiences a drastic rise, which becomes steeper with increasing α. This pressure climb continues until C_p reaches the peak value, which grows with α.

Figure 5.

The distributions of the surface pressure coefficient for (a) R/L = 0.2, (b) R/L = 0.5, (c) R/L = 1.0 and (d) R/L = ∞ with different deflection angles (PHAROS). Open symbols: separation and reattachment points. Vertical dashed line: juncture.

For R/L = 0.5, figure 6 presents the density gradients in the separation region and the non-dimensional pressure scaled by ${\rho _\infty }u_\infty ^2$ in the juncture region at α = 14° and 15°. At α = 14°, the primary vortex core is located over the flare surface downstream of the juncture, and the secondary separation bubble is not seen. At α = 15°, a secondary vortex emerges beneath the primary bubble close to the juncture, while the primary bubble fragments into two vortices. A smaller secondary bubble also appears on the flare. At α = 14°, there is a low-pressure zone near the vortex core, which may balance the centrifugal force of a fluid element rotating about the core (Jeong & Hussain, Reference Jeong and Hussain1995). The ‘dip’ in C_p is assumed to reflect this low-pressure zone on the flare. As α grows, the low pressure near the primary vortex core remains almost unchanged, while the pressure in the upstream part of the separation zone increases in accordance with the change in the plateau pressure (see figure 5).

Figure 6.

Contours of (a) density gradient magnitude and (b) non-dimensional pressure scaled by ${\rho _\infty }u_\infty ^2$ with streamlines superimposed for R/L = 0.5 at α = 14° (left column) and 15° (right column) (PHAROS).

The axisymmetric effects are then studied by comparing the solutions with different cylinder radii. Figure 7 presents the distributions of C_f and C_p at two deflection angles, α = 5° and 13°. The open symbols represent the separation and reattachment points. A detailed view of the skin friction distributions near the juncture at α = 13° is also shown here. As α = 5° is slightly higher than the incipient separation angle, only a marginal separation region is observed, and a pressure plateau has not yet formed. At α = 13°, the emergence of two local minima in C_f and a constant plateau in C_p suggests a large separation region. As R is decreased, the surface pressure upstream of separation gradually decreases (Kluwick et al. Reference Kluwick, Gittler and Bodonyi1984; Kluwick, Gittler & Bodonyi Reference Kluwick, Gittler and Bodonyi1985), whereas the skin friction shows an opposite trend (White & Majdalani Reference White and Majdalani2006). The length of the separation region and the magnitude of the local minima of C_f both increase with R for a fixed α, which corresponds to figure 4. Furthermore, the plateau and peak pressures both grow with increasing R. The pressure overshoot on the flare surface is due to the strong compression waves induced by flow reattachment. In fact, if the flow is inviscid, the pressure immediately downstream of the juncture obeys the oblique shock relations. Due to the axisymmetric effects on the viscous–inviscid interaction, the oblique shock solution can never be achieved. When R is progressively increased to recover the two-dimensional case, the surface pressure far downstream of reattachment increasingly approaches the value predicted by the oblique shock relations. Conversely, as the cylinder body becomes slenderer, the decay of C_p to a lower limit value determined by the conical shock relations becomes more rapid. Such trends in C_p on the flare are indicative of the axisymmetric effects.

Figure 7.

The distributions of the skin friction coefficient (left column) and surface pressure coefficient (right column) for varying cylinder radii at (a) α = 5° and (b) α = 13° (PHAROS). Horizontal dashed line: zero skin friction for C_f, pressure predicted by the oblique shock relations (upper) and conical shock relations (lower) for C_p. Vertical dashed line: juncture.

Figure 7 demonstrates that for a given α, the axisymmetric effects decay with increasing R, and the separation and reattachment processes are milder in an axisymmetric flow than in its two-dimensional counterpart. The length of the separation region, the peak and plateau pressures and the skin friction minima increase with increasing R and eventually recover to the two-dimensional solutions. Furthermore, the occurrence of both incipient and secondary separation is postponed with decreasing R, which is indicative of more pronounced transverse curvature effects. As R is decreased, the axisymmetric effects lead to a slightly higher C_f upstream of separation and a decreasing pressure overshoot downstream of reattachment, thus reducing the size of the separation bubble and delaying the occurrence of separation.

Despite a general overestimation of the size of the separation region (Katzer Reference Katzer1989), the triple-deck theory is accurate in anticipating the occurrence of flow separation. Figure 8 presents the distributions of scaled wall shear stress for hollow-cylinder/flare flows and compression corner flows by solving the triple-deck equations at varying α*. Note that, due to the rounded juncture, the minimum shear stress emerges upstream of the juncture when α* is small (Kluwick et al. Reference Kluwick, Gittler and Bodonyi1984). The distributions of wall shear stress for axisymmetric flows are generally consistent with those for two-dimensional flows. The incipient separation emerges at $\alpha _1^\ast= 2.05\unicode{x2013} 2.46$ for R* = 4.948 and 12.37, and 1.64–2.05 for R* = 24.74. For compression corner flows, the critical angle for incipient separation is $1.64 \le \alpha _1^\ast \le 2.05$. The exact calculations give $\alpha _1^\ast \approx 1.85$, which is in reasonable agreement with the values obtained by Cassel, Ruban & Walker (Reference Cassel, Ruban and Walker1995), Grisham, Dennis & Lu (Reference Grisham, Dennis and Lu2018) and Exposito, Gai & Neely (Reference Exposito, Gai and Neely2021). This value is higher than the typical value of 1.57 (Rizzetta et al. Reference Rizzetta, Burggraf and Jenson1978; Ruban Reference Ruban1978) due to a rounded corner. The critical incipient separation angle for an axisymmetric flow is much higher than that for the corresponding planar problem and increases with decreasing cylinder radius, which corresponds well with the findings of Huang & Inger (Reference Huang and Inger1983) and Kluwick et al. (Reference Kluwick, Gittler and Bodonyi1984). Similar trends are observed for the occurrence of secondary separation. For the planar case, secondary separation is observed at $4.10 \le \alpha _2^\ast \le 4.51$, which agrees well with the results of Smith & Khorrami (Reference Smith and Khorrami1991), Korolev et al. (Reference Korolev, Gajjar and Ruban2002), Shvedchenko (Reference Shvedchenko2009) and Gai & Khraibut (Reference Gai and Khraibut2019). For the axisymmetric flow, the critical angle of secondary separation decreases from 6.15–6.57 for R* = 4.948 to 5.33–5.74 for R* = 12.37 and 4.92–5.33 for R* = 24.74. In effect, the triple-deck solutions show that the critical deflection angles of both incipient and secondary separation decrease with increasing cylinder radius and eventually approach the values for the compression corner case. Although triple-deck theory generally provides a slight overprediction of the incipient separation angles and a marginal underprediction of the secondary separation angles, the triple-deck solutions exhibit the same overall trends as the CFD solutions in terms of the axisymmetric effects.

Figure 8.

The scaled wall shear stress distributions obtained by the triple-deck theory for different cylinder radii (a) R* = 4.948, (b) R* = 12.37, (c) R* = 24.74 and (d) R* = ∞ at various deflection angles (triple-deck solutions). Vertical dashed line: juncture. Horizontal dashed line: scaled wall shear stress of zero and one.

It is significant to quantitatively assess the accuracy of the triple-deck theory in terms of estimating the length of the separation region. Despite satisfactory qualitative results, Burggraf et al. (Reference Burggraf, Rizzetta, Werle and Vatsa1979) found that the quantitative accuracy was only obtained at very high Reynolds numbers. Katzer (Reference Katzer1989) indicated an overprediction of the separation region for finite Reynolds numbers. The discrepancy was found to increase with increasing Mach number and decreasing Reynolds number. With respect to the influence of wall temperature, Exposito et al. (Reference Exposito, Gai and Neely2021) revealed an overestimation of the triple-deck theory for T_w/T ₀ ≥ 0.15 and an underestimation for T_w/T ₀ ≤ 0.15. Figure 9 compares the scaled length of the separation region obtained by both triple-deck and CFD solutions. Similar to figure 3, the triple-deck results exhibit a linear trend with α ^*3/2 for both planar and axisymmetric cases in figure 9(a). However, the noticeable discrepancy of the triple-deck and base-flow results in their slopes indicates a general overprediction of the triple-deck theory. The overestimation substantially grows with α because the linearized theory assumed in triple-deck theory no longer holds for higher α. The influence of cylinder radius is taken into consideration in figure 9(b). The discrepancy between the base-flow results and the triple-deck data increases as the cylinder radius is increased. The triple-deck theory in the axisymmetric regime assumes that the scaled body radius satisfies R* ∼ O(1) (Huang & Inger Reference Huang and Inger1983; Kluwick et al. Reference Kluwick, Gittler and Bodonyi1984). Therefore, a better estimation is obtained for a slightly smaller body radius (Kluwick et al. Reference Kluwick, Gittler and Bodonyi1985). As the cylinder radius is increased, the interaction law which relates the pressure and displacement thickness is not appropriate for consideration, thus leading to the overestimation of the length of the separation region. The hot wall (T_w/T ₀ = 1.02) in the present instance also contributes to the overestimation in figure 9 (Exposito et al. Reference Exposito, Gai and Neely2021).

Figure 9.

The scaled length of the separation region as a function of (a) α ^*3/2 for R* = 12.37 and ∞ and (b) R* for a constant deflection angle α* = 3.69 (PHAROS and triple-deck solutions). Open symbols: base-flow data. Solid symbols: triple-deck data. Horizontal dashed line in (b): limiting planar results.

The above discussion reveals that the triple-deck theory provides a satisfactory prediction for the occurrence of incipient and secondary separation. However, this asymptotic theory exhibits limitations in quantitatively estimating the length of the separation region.

4.2. Occurrence of global instability

To identify the global stability of azimuthally periodic perturbations imposed on the base flows, GSA is performed over a wide range of azimuthal wavenumbers with a series of cylinder radii and flare deflection angles.

The global instability of the appropriate compression corner flow is considered first. Figure 10 shows the growth rates of the most unstable modes as a function of the spanwise wavenumber for various deflection angles. At the lowest deflection angle α = 9°, only stable modes are observed. At α = 10°, the compression corner flow becomes globally unstable with the presence of a stationary unstable mode whose largest growth rate occurs at βL = 16.9 (λ/L = 0.37). When α is increased to 11°, the unstable mode is shifted to a smaller wavenumber of βL = 14.1 (λ/L = 0.45). As α is further increased, more unstable modes gradually emerge. At α = 12°, three unstable modes including two stationary unstable modes (modes 1 and 3) and an oscillatory unstable mode (mode 2) are captured by the GSA. The mode shown in figure 10 corresponds to mode 1, which has the largest growth rate among the three modes. At α = 13°, four unstable modes exist, including two stationary modes and two oscillatory modes. The strongest mode reaches its maximum growth rate at βL = 67.9 (λ/L = 0.093).

Figure 10.

Variations in the growth rates of the most unstable modes as a function of spanwise wavenumber at different flare deflection angles for R/L = ∞ (GSA). Vertical dashed line: the most unstable wavenumber. Horizontal dashed line: zero growth rate.

The growth rates and angular frequencies of the three unstable modes captured at α = 12° are plotted as a function of the spanwise wavenumber in figure 11.Modes 1 and 2 reach their maximum growth rates at βL = 60.5 (λ/L = 0.104) and βL = 66.7 (λ/L = 0.094), respectively. The weaker stationary mode (mode 3) with the growth rates peaking at approximately βL = 11.2 (λ/L = 0.56) corresponds to the stationary modes observed at α = 10° and 11°. Figure 12 presents the contours of the real part of the pressure perturbation p′ and spanwise velocity perturbation w′ for modes 1 and 2 at their respective most unstable spanwise wavenumbers. The features of the pressure and spanwise velocity perturbations are consistent with previous findings in compression corner flows (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021; Hao et al. Reference Hao, Cao, Wen and Olivier2021). The pressure perturbations mainly exist in the downstream half of the separation zone, with the positive and negative parts periodically coupled to each other. The wave-like structures further stretch downstream of the reattachment.

Figure 11.

Variations in (a) growth rates and (b) frequencies of the unstable modes as a function of spanwise wavenumber for R/L = ∞ at α = 12° (GSA). Vertical dashed line: the most unstable wavenumber. Horizontal dashed line: zero growth rate and angular frequency.

Figure 12.

Real parts of (a) the spanwise velocity perturbation and (b) pressure perturbation for mode 1 (left column) with βL = 60.5 and mode 2 (right column) with βL = 66.7 at α = 12° for the two-dimensional problem (GSA). The contour levels are uniformly distributed between ±0.1 of the maximum |w′| and |p′|, respectively.

For an intermediate radius R/L = 0.5, figure 13 presents the growth rates of the captured dominant unstable modes as a function of azimuthal wavenumber at various deflection angles. Note that these dominant unstable modes shown in figure 13 are all stationary. At α = 11°, the flow is globally stable. A globally unstable mode appears at α = 12°. At α = 13°, the peak growth rate further increases and shifts to a slightly smaller wavenumber, which corresponds to the trend shown in figure 10. Two stationary unstable modes are observed at α = 14°. One belongs to the same family as those at lower angles (not shown), while the other becomes dominant with the most unstable azimuthal wavenumber of m = 36. A further slight increase in the flare angle has profound consequences on the global instability. At α = 15°, four unstable modes are captured. The maximum growth rate of the most unstable mode occurs at m = 40. Recall that the secondary separation angle is 14°–15° for R/L = 0.5. The growth rate of the most unstable mode increases with α. At α = 12° and 13°, there is only one unstable stationary mode with a relatively small wavenumber. As α approaches and surpasses α ₂, the flow is strongly destabilized and dominated by a new stationary mode with a much larger wavenumber, which is consistent with the compression corner flow.

Figure 13.

Variations in the growth rates of the most unstable modes as a function of azimuthal wavenumber at different flare deflection angles for R/L = 0.5 (GSA). Vertical dashed line: the most unstable wavenumber. Horizontal dashed line: zero growth rate.

For R/L = 0.5, the flow is strongly destabilized at α = 15°. The growth rates and angular frequencies of the four unstable modes against the azimuthal wavenumber are plotted in figure 14. Note that mode 1 is the mode shown in figure 13 at α = 15°. The distributions of angular frequencies suggest that modes 1 and 3 are stationary, whereas modes 2 and 4 are oscillatory. The growth rate of mode 1 peaks at m = 40, whereas mode 2 reaches its maximum growth rate at a slightly larger wavenumber m = 42. The dominant frequency of mode 2 at m = 42 is approximately f = ω_r/2${\rm \pi}$ = 3.0 kHz ( f L/u_∞ = 0.56). Modes 3 and 4 reach their maximum growth rates at considerably lower azimuthal wavenumbers m = 8 and 14, respectively. Note that mode 3 is associated with the stationary unstable modes captured at α = 12°, 13° and 14°, as shown in figure 13.

Figure 14.

Variations in (a) growth rates and (b) angular frequencies of the most unstable modes as a function of azimuthal wavenumber at α = 15° for R/L = 0.5 (GSA). Vertical dashed line: the most unstable wavenumber. Horizontal dashed line: zero growth rate and angular frequency.

For R/L = 0.5 and α = 15°, figure 15 presents the contours of the real parts of pressure perturbation p′ and azimuthal velocity perturbation w′ for modes 1 and 2 at m = 40 and 42, respectively. Modes 1 and 2 are structurally similar to the dominant stationary and oscillatory unstable modes shown in figure 12 and the unstable modes observed in supersonic double-wedge flows, compression corner flows, double-cone flows and hollow-cylinder/flare flows (Sidharth et al. Reference Sidharth, Dwivedi, Candler and Nichols2018; Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021; Hao et al. Reference Hao, Cao, Wen and Olivier2021, Reference Hao, Fan, Cao and Wen2022; Lugrin et al. Reference Lugrin, Beneddine, Garnier and Bur2021a). The unstable modes are attributed to the separated flow, and they are identical in nature. Appendix C presents the distributions of streamwise and radial velocity perturbations, density perturbations and temperature perturbations for modes 1 and 2.

Figure 15.

Real parts of (a) the pressure perturbations and (b) azimuthal velocity perturbations for mode 1 (left column) at m = 40 and mode 2 at m = 42 (right column) for R/L = 0.5 at α = 15° (GSA). The contour levels are uniformly distributed between ±0.1 of the maximum |w′| and |p′|, respectively.

The growth rates of the most unstable modes as a function of azimuthal wavenumber at different deflection angles are shown in figure 16 for R/L = 0.2 and 1.0. For R/L = 0.2, the separated flow becomes unstable at α = 13°, which is higher than that for R/L = 0.5. As α is increased to 14° and 15°, the stationary unstable mode is observed at m = 5. When α is further increased to 16°, the maximum growth rate of the least stable mode occurs at m = 19. For R/L = 1.0, global instability occurs at α = 11°, which is lower than that for R/L = 0.5. As α is increased to 12°, an unstable mode is observed at m = 18. At α = 13° and 14°, the most unstable modes are shifted to considerably higher wavenumbers m = 66 and 72, respectively. It is observed that the variations of the maximum ω_i and corresponding m in different α for R/L = 0.2 and 1.0 are similar to those unveiled for R/L = 0.5. However, the emergence of global instability is suppressed as R is decreased. For R/L = 0.2 and 1.0, a series of stationary and oscillatory unstable modes emerge at higher deflection angles. These obtained modes are similar to the unstable modes presented in figure 15 and thus are not shown here for brevity.

Figure 16.

Variations in the growth rates of the most unstable modes as a function of azimuthal wavenumber for (a) R/L = 0.2 and (b) R/L = 1.0 with increasing deflection angles (GSA). Vertical dashed line: the most unstable azimuthal wavenumber. Horizontal dashed line: zero growth rate.

Similarly, the GSA results can be scaled using the triple-deck theory to determine the critical deflection angles where the flow becomes globally unstable. For the planar case R* = ∞, α* = 3.69–4.10, which agrees well with the stability boundary for a compression corner flow by Hao et al. (Reference Hao, Cao, Wen and Olivier2021). For R* = 24.74, 12.37 and 4.948, α* = 4.10–4.51, 4.51–4.92 and 4.92–5.33, respectively. A decrease in cylinder radius postpones the occurrence of global instability. When the deflection angle gradually approaches and exceeds the critical secondary separation angle α ₂, the flow is strongly destabilized with the emergence of more unstable modes and a pronounced increase in the growth rate of the most unstable mode.

4.3. Direct numerical simulations

Direct numerical simulations are performed for the canonical case R/L = 0.5 and its planar counterpart R/L = ∞ at α = 15° to verify the foregoing GSA results and illustrate the temporal evolution of three-dimensional structures.

The GSA is similarly performed for the compression corner flow at α = 15°. Since the planar flow is much more unstable than its axisymmetric equivalent, a finer mesh (600 × 450) is used for the base-flow simulations and GSA to ensure grid convergence. The GSA results of the planar case at α = 15° are shown in Appendix D. Note that the three most unstable modes (modes 1, 2 and 3) reach their maximum growth rates at βL = 74, 60 and 72, respectively.

Direct numerical simulations are performed for both axisymmetric and planar flows at α = 15° to verify the GSA results. For the axisymmetric case, the medium mesh (600 × 350) is rotated around the flow axis over an azimuthal angle of 36° with 200 grid cells in the azimuthal (ϕ) direction. This azimuthal angle is chosen because the growth rates of mode 1 and mode 2 peak at m = 40 and 42, respectively. The initial flow field is generated through a duplication of the base flow along the azimuthal direction. Similarly, for the planar case, the mesh with 600 × 450 grid points is extended along the spanwise (z) direction for 42 mm, which corresponds to approximately five wavelengths of mode 1 at R/L = ∞. The 200 grid points are equally spaced in the spanwise direction. The three-dimensional flow field is generated by extending the base-flow solution in the spanwise direction. The inlet and upper boundary conditions are given by the free-stream profiles. Periodic boundary conditions are enforced on the azimuthal/spanwise boundaries. Without introducing any external or internal perturbations, the numerical round-off error is expected to induce global instability. The three-dimensional simulations are conducted using PHAROS with a second-order implicit method (Peterson Reference Peterson2011) for time integration. The physical time step is set to 20 ns for both axisymmetric and planar simulations. To ensure a sufficiently developed flow, the simulated time is set to t = 20 ms (tu_∞/L = 108) for R/L = 0.5 and t = 8 ms (tu_∞/L = 43) for R/L = ∞.

The evolution of global instability is described by the root mean square of the azimuthal/spanwise velocity (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021; Hao et al. Reference Hao, Fan, Cao and Wen2022) at a streamwise location, which is defined as

(4.2)\begin{equation}{\sigma _w} = \sqrt{\frac{1}{{{N_r}{N_\phi }}}\sum\limits_{j = 1}^{{N_r}} {\sum\limits_{k = 1}^{{N_\phi }} {{{\left( {\frac{w}{{{u_\infty }}}} \right)}^2}} } } \end{equation}

for the axisymmetric flow and

(4.3)\begin{equation}{\sigma _w} = \sqrt {\frac{1}{{{N_y}{N_z}}}\sum\limits_{j = 1}^{{N_y}} {\sum\limits_{k = 1}^{{N_z}} {{{\left( {\frac{w}{{{u_\infty }}}} \right)}^2}} } }\end{equation}

for the planar flow, where N_r, N_ϕ, N_y and N_z represent the numbers of grid cells in the radial, azimuthal, normal and spanwise directions, respectively.

Figure 17 presents the temporal evolution of σ_w for the axisymmetric and planar cases at x/L = 1.25, respectively. For R/L = 0.5, the initialization is followed by an exponential increase from tu_∞/L ≈ 6 to 20 and a successive nonlinear saturation. The variation in σ_w enters a quasi-steady stage from tu_∞/L ≈ 50. The growth rate of σ_w in the linear exponential stage agrees well with that of mode 1 at m = 40 predicted by the GSA and denoted by the red dashed line. For R/L = ∞, after the initial adaption, σ_w similarly experiences a rapid increase from tu_∞/L ≈ 5 to 10. A satisfactory agreement is also observed between the DNS data and the GSA prediction in the planar regime. Due to the stronger global instability of the dominant mode, the required time period for the exponential growth is shorter for the planar flow than for its axisymmetric counterpart. After the rapid growth, the compression corner flow similarly enters a saturated stage at approximately tu_∞/L = 12, from which sustained fluctuations of σ_w occur. The saturated values for the axisymmetric and planar flows are of the same order.

Figure 17.

Temporal evolution of (a) the averaged azimuthal velocity for R/L = 0.5 and (b) the averaged spanwise velocity for R/L = ∞ at x/L = 1.25 (DNS).

Figure 18(a) presents the azimuthal velocity distributions of the hollow-cylinder/flare flow in an azimuthal plane (ϕ = 18°) at tu_∞/L = 14.88 in the mid-linear stage. Perturbations mainly exist in the downstream part of the separated region (0.40 < x/L < 1.48) and stretch further downstream through the reattaching shear layer, which is similar to the findings in compression corner flows (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021). The general structure is consistent with that of mode 1 found by GSA, indicating that mode 1 plays a predominant role in the linear growth of the three-dimensional perturbations. Figure 18(b) presents the azimuthal velocity distributions of a wall-normal slice at x/L = 1.21 which is located in the separation bubble. The azimuthal velocity variations unveil a periodicity of m = 40, which corresponds well with the GSA results.

Figure 18.

The azimuthal velocity distributions for R/L = 0.5 and α = 15° at tu_∞/L = 14.88: (a) the x–r plane with ϕ = 18°; (b) the wall-normal slice extracted at x/L = 1.21 (DNS).

Figure 19 shows the spanwise velocity distributions of the compression corner flow in the x–y plane with z/L = 0.5 and the z–y plane with x/L = 1.25 at tu_∞/L = 8.63 which belongs to the exponential stage. Similar to the findings of Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021), figure 19(a) reveals the existence of perturbations in the downstream part of the separation zone (0.17 < x/L < 1.75). Moreover, the perturbations of spanwise velocity are consistent with that of mode 1 captured by GSA. The consistency manifests that mode 1 captured by GSA governs the exponential growth of the three-dimensional perturbations. Figure 19(b) shows the periodic distributions of spanwise velocity. The wavelength of the spanwise velocity variations agrees well with the GSA data, namely λ/L = 0.085 of mode 1.

Figure 19.

The spanwise velocity distributions for R/L = ∞ and α = 15° at tu_∞/L = 8.63: (a) the x–y plane with z/L = 0.5; (b) the wall-normal slice extracted at x/L = 1.25 (DNS).

A fast Fourier transformation is performed for the temporal perturbation of the azimuthal/spanwise velocity near flow reattachment in the quasi-steady stage with a sampling frequency of f_s = 1.0 MHz. Figure 20 presents the power spectral density (PSD) of the azimuthal velocity obtained near the surface at ϕ = 18° and x/L = 1.55 from tu_∞/L = 54 to 97 for the axisymmetric case and the spanwise velocity obtained at x/L = 1.71 and z/L = 0.5 from tu_∞/L = 16 to 43 for the planar case. The power spectra are computed using Welch's method (Welch Reference Welch1967) with eight segments and a 50% overlap. A Hamming window is used for the Fourier transform. For the axisymmetric flow, with a dominant frequency at f L/u_∞ = 0.23, the azimuthal velocity signal shows a broadband low-frequency feature, which agrees well with the dominant frequencies of modes 2 and 4 by the GSA at f L/u_∞ = 0.56 and 0.077, respectively. In figure 20(b), the spanwise velocity signal shows a similar broadband spectrum which covers the dominant frequency of mode 3 ( f L/u_∞ = 0.72) by the GSA. The frequency broadening phenomenon, which was also observed in compression corner flows (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021) and double-cone flows (Hao et al. Reference Hao, Fan, Cao and Wen2022), is associated with the interactions of the critical multiple perturbation modes arising after the linear growth stage (Fan, Hao & Wen Reference Fan, Hao and Wen2022).

Figure 20.

The PSD of (a) the azimuthal velocity at x/L = 1.55 and r/L = 0.68 for ϕ = 18° and (b) the spanwise velocity at x/L = 1.71 and y/L = 0.71 for z/L = 0.5 (DNS).

Figure 21 presents the evolution of the skin friction coefficient contours in the $x{-}\varPhi$ plane to illustrate the development of three-dimensionality for the hollow-cylinder/flare flow. The isolines represent that C_f = 0, while the solid points denote the separation and reattachment points obtained from base-flow solutions. The initial contour at tu_∞/L = 0 is identical to the axisymmetric base-flow solution. At tu_∞/L = 14.88, the reattachment lines of the primary bubble and the secondary bubble are slightly corrugated, while the separation lines remain nearly unchanged. Meanwhile, azimuthal oscillations and several small bubbles gradually emerge. At tu_∞/L = 28.37, the primary separation line begins to move upstream, while the reattachment line zigzags, indicating the influence of three-dimensionality. Subsequently, the size of the separation region slightly grows, and the reattachment line noticeably meanders. At tu_∞/L = 55.34, 76.93 and 98.51, distinct skin friction streaks are observed along the azimuthal direction downstream of reattachment, which is consistent with the experimental observations (Benay et al. Reference Benay, Chanetz, Mangin, Vandomme and Perraud2006; Lugrin et al. Reference Lugrin, Nicolas, Severac, Tobeli, Beneddine, Garnier, Esquieu and Bur2022). The discrepancy of skin friction coefficient distributions at these instants indicates that the flow is unsteady.

Figure 21.

Contours of the skin friction coefficient for R/L = 0.5 and α = 15° at (a) tu_∞/L = 0, (b) tu_∞/L = 14.88, (c) tu_∞/L = 28.37, (d) tu_∞/L = 55.34, (e) tu_∞/L = 76.93 and (f) tu_∞/L = 98.51 (DNS). Solid lines: isolines of C_f = 0. Solid points: separation and reattachment points of base-flow solutions.

For the planar flow, figure 22 provides the evolution of skin friction coefficient contours at six instants in the x–z plane. The isolines of C_f = 0 are also displayed to identify the separation zones. The separation and reattachment points are marked by solid points. At tu_∞/L = 0, the distribution of C_f is indistinguishable from the two-dimensional converged solutions. As is shown in figure 17, the temporal development of three-dimensionality is considerably faster for the compression corner flow than its axisymmetric equivalent. Consequently, at tu_∞/L = 8.63 in the exponential stage, the reattachment lines of the primary bubble and the secondary bubble become corrugated and several small vortices occur inside the separation region. The rapid evolution of three-dimensionality leads to the occurrence of streamwise streaks which are non-uniformly distributed downstream of reattachment at tu_∞/L = 10.79. The streaks in the spanwise direction correspond to previous experimental observations (Simeonides & Haase Reference Simeonides and Haase1995; Chuvakhov et al. Reference Chuvakhov, Borovoy, Egorov, Radchenko, Olivier and Roghelia2017; Roghelia et al. Reference Roghelia, Chuvakhov, Olivier and Egorov2017a,Reference Roghelia, Olivier, Egorov and Chuvakhovb). The unsteadiness subsequently develops with meandering reattachment lines at tu_∞/L = 19.42, 30.21 and 41.00.

Figure 22.

Contours of the skin friction coefficient for R/L = ∞ and α = 15° at (a) tu_∞/L = 0, (b) tu_∞/L = 8.63, (c) tu_∞/L = 10.79, (d) tu_∞/L = 19.42, (e) tu_∞/L = 30.21 and (f) tu_∞/L = 41.00 (DNS). Solid lines: isolines of C_f = 0. Solid points: separation and reattachment points of base-flow solutions.

Figure 23 provides the distributions of skin friction coefficient by DNS at tu_∞/L = 98.51 for the axisymmetric case and tu_∞/L = 41.00 for the planar case. The shaded area represents the azimuthal/spanwise variations of DNS data. The averaged DNS data correspond to the base-flow results. However, the variations of C_f in azimuthal/spanwise directions are significant downstream of flow reattachment. The length of the separation region for the compression corner flow is significantly larger than its axisymmetric counterpart for both base-flow and DNS results, which manifests a stronger shock-wave/boundary-layer interaction of the planar flow.

Figure 23.

A comparison of the skin friction coefficient distributions between the base-flow and DNS results: (a) the azimuthally averaged C_f for R/L = 0.5 and α = 15° at tu_∞/L = 98.51; (b) the spanwise-averaged C_f for R/L = ∞ and α = 15° at tu_∞/L = 41.00 (DNS). The shaded area denotes the azimuthal/spanwise variations.

4.4. Criterion for triple-deck scaling

The triple-deck theory is an effective tool to correlate the theoretical analyses, experimental data and numerical results for supersonic planar and axisymmetric flows (Gai & Khraibut Reference Gai and Khraibut2019; Exposito et al. Reference Exposito, Gai and Neely2021; Hao et al. Reference Hao, Cao, Wen and Olivier2021, Reference Hao, Fan, Cao and Wen2022). Figure 24 exhibits the variations in the scaled critical flare deflection angle as a function of cylinder radius for the emergence of incipient separation ($\alpha _1^\ast $) and secondary separation ($\alpha _2^\ast $) by CFD and triple-deck solutions and for the occurrence of global instability ($\alpha _{GSA}^\ast $) by GSA. Extensive CFD and GSA simulations are further performed for a series of cylinder radii. According to (3.5), the error bars represent an uncertainty of approximately 0.41 for α* due to a minimum increment of 1° in physical deflection angle α. The horizontal dashed lines represent the limit solutions obtained from the equivalent two-dimensional circumstance. Furthermore, the figure consists of theoretical, numerical and experimental data taken from previous research for supersonic and hypersonic hollow-cylinder/flare flows with a wide range of free-stream Mach numbers from 3.0 to 7.3 and unit Reynolds numbers from 2.8 × 10⁵ m⁻¹ to 1.1 × 10⁷ m⁻¹. The geometric parameters and flow conditions are listed in table 1. For the CFD data, the formation of three-dimensional streamwise streaks in the absence of any external disturbance essentially indicates a globally unstable flow. For the experimental data, owing to the presence of upstream disturbances in the experiments, the global instability is further examined by the current GSA.

Figure 24.

The scaled critical deflection angles as a function of the scaled cylinder radii for the incipient separation $\alpha _1^\ast $ and secondary separation $\alpha _2^\ast $ by CFD and triple-deck theory, and global instability $\alpha _{GSA}^\ast $ by GSA. Horizontal dashed lines: limiting solutions obtained from the equivalent two-dimensional compression corner flow. Red symbols: both presence of incipient separation and absence of secondary separation. Blue symbols: emergence of secondary separation. Open symbols: globally stable. Solid symbols: globally unstable.

Table 1.

Flow conditions and geometric parameters of the collected theoretical, numerical and experimental data shown in figure 24.

In figure 24, it is evident that the critical angles for incipient separation, secondary separation and global instability simultaneously decrease with the cylinder radius and eventually recover to the planar solutions. In accordance with supersonic two-dimensional compression corner flows (Hao et al. Reference Hao, Cao, Wen and Olivier2021) and axisymmetric double-cone flows (Hao et al. Reference Hao, Fan, Cao and Wen2022), global instability emerges at a deflection angle smaller than the secondary separation angle for hollow-cylinder/flare flows. The data taken from the literature denoted by symbols are in reasonable agreement with the separation and global stability boundaries denoted by the solid lines. Therefore, the criterion of the stability boundary with regard to critical deflection angles established for supersonic compression corner and double-wedge flows (Hao et al. Reference Hao, Cao, Wen and Olivier2021) is extended to hollow-cylinder/flare flows.

In figure 25, a series of hollow-cylinder/flare flows taken from the literature are considered to further confirm the separation and stability boundaries. The free-stream conditions are shown in table 2. Note that additional CFD and GSA simulations are carried out for these cases since the separation stage and global stability were not entirely known in the literature. It is seen that the present criterion corresponds well with the considered cases.

Figure 25.

The scaled critical deflection angles as a function of the scaled cylinder radii for the incipient separation $\alpha _1^\ast $ and secondary separation $\alpha _2^\ast $ by CFD and triple-deck theory, and global instability $\alpha _{GSA}^\ast $ by GSA. Horizontal dashed lines: limiting solutions obtained from the equivalent two-dimensional compression corner flow. Red symbols: both presence of incipient separation and absence of secondary separation. Open symbols: globally stable.

Table 2.

Flow conditions and geometric parameters of the collected theoretical, numerical and experimental data presented in figure 25.

It is important to note that the cold-wall effects are not considered in this study. For triple-deck theory, the wall cooling effects on separation for a compression corner flow were studied in subcritical, transcritical and supercritical regimes (Brown, Cheng & Lee Reference Brown, Cheng and Lee1990; Kerimbekov, Ruban & Walker Reference Kerimbekov, Ruban and Walker1994; Cassel, Ruban & Walker Reference Cassel, Ruban and Walker1996; Exposito et al. Reference Exposito, Gai and Neely2021). It was found that wall cooling could postpone the emergence of incipient separation for compression corner flows and diminish the magnitude of the second minimum of wall shear stress. Recently, Hao et al. (Reference Hao, Cao, Wen and Olivier2021) and Hong et al. (Reference Hong, Hao, Uy, Wen and Sun2022) have revealed that low wall temperatures can promote the appearance of secondary separation and global instability for compression corner flows. Therefore, it is likely that the cold-wall effects are also significant to the emergence of flow separation and global instability for axisymmetric hollow-cylinder/flare flows, e.g. the high-enthalpy experiments by Holden et al. (Reference Holden, Wadhams, MacLean and Dufrene2013). Besides, Hong et al. (Reference Hong, Hao, Uy, Wen and Sun2022) found that the thermochemical non-equilibrium effects slightly stabilized double-wedge flows through changing the base flow and the perturbations were thermochemically frozen. The stabilization caused by thermochemical non-equilibrium effects may be similarly observed for the hollow-cylinder/flare flow. In terms of the influence of Mach number, the triple-deck scaling in (3.5) indicates that, as the free-stream Mach number is increased, R* undergoes an increase and α* experiences a decrease.Since $\alpha _1^\ast $, $\alpha _2^\ast $ and $\alpha _{GSA}^\ast $ also decrease with increasing R*, it is not straightforward to determine the influence of Mach number in the axisymmetric regime, which merits further investigation.