2.1. Model

The test article is a 7 $^\circ$ half-angle circular cone and 20 $^\circ$ half-angle flare (figure 1). It consists of three main components: three interchangeable nose tips, a frustum and four flare petals. The frustum and flare petals have surface-normal holes for pressure sensors. The noses were fabricated from stainless steel, with nose radii of $r_{{n}} = 0.01$ , 0.99 and 5.00 mm. The flare petals and frustum were fabricated from aluminium. Due to the various nose radii tested, the cone length $L$ is not constant and depends on the nose radius $r_{ {n}}$ . The $x$ -coordinate is defined from the tip the respective nose used in the experiment.

Figure 1.

Illustration of the cone–flare test article (dimensions are in millimetres).

2.2. Facility

Experiments were conducted in the Air Force Office of Scientific Research–Notre Dame Large Mach-6 Quiet Tunnel (ANDLM6QT). The ANDLM6QT has a 60 m-long driver tube and 63 cm nozzle exit diameter (Lakebrink, Borg & Kimmel Reference Lakebrink, Borg and Kimmel2018). Heating blankets on the driver tube permit stagnation temperatures from 435 to 590 K. A pneumatically actuated fast-acting shutter valve at the contraction exit opens to start a run, which lasts $\approx 1$ s. During a test run, the variation in the free-stream unit Reynolds number is less than 10 %. The ANDLM6QT is operational with conventional-noise levels of 1.6 %–2.3 %, depending on free-stream Reynolds number, and a free-stream Mach number of 5.7 (Lawson Reference Lawson2025), typical for a conventional-noise tunnel (Schneider Reference Schneider2000, Reference Schneider2001; Duan et al. Reference Duan2018).

2.3. Test conditions

Five free-stream unit Reynolds numbers were tested for each nose radius. Table 1 summarises the flow conditions with corresponding uncertainties. Sutherland’s law was used to compute viscosity (Sutherland Reference Sutherland1893). The free-stream quantities were computed using isentropic relations with the measured time-varying stagnation pressure, initial stagnation temperature and measured Mach number from Hoberg & Juliano (Reference Hoberg and Juliano2023).

Table 1.

Flow conditions.

3.1. PCB piezotronics pressure sensors

Four PCB Piezotronics 132B38 pressure sensors were used to measure pressure fluctuations on the model’s conical forebody, primarily for alignment with the flow (§ 4). Their measurement range is 345 kPa, sensitivity is 20.3 mV/kPa and resolution is 7 Pa. Their low-frequency limit is 11 kHz, and high-frequency response limit is 1 MHz. The conversion from measured voltage to pressure fluctuation was performed using the PCB factory calibration. The sensors were sampled at 2 MHz using a HBM Genesis 17ta data-acquisition system.

3.2. Infrared thermography

The model surface temperature was measured with a 14-bit InfraTech 8300 hp IR camera with a spectral range of 2.0–5.5 $\unicode{x03BC}$ m (mid-wave IR). The camera resolution is $640\times 512$ pixels with a frame rate set to 355 fps. A 25 mm focal length lens was used with its factory calibration. The spatial resolution of the measurements was 0.20 mm. The overall relative uncertainty of the IR camera temperature measurement is $\pm 0.4$ %, in which the systematic uncertainty and resolution are $\pm 1$ K and 0.02 K, respectively (InfraTec 2016).

Aluminium’s low surface emissivity was addressed by applying a matte-black self-adhesive film (3M $^{\textrm {TM}}$ Wrap Film Series 1080) to a 180 $^\circ$ sector of the frustum and flare for improved IR thermography (Running et al. Reference Running, Rataczak, Zaccara, Cardone and Juliano2022). The vinyl wrap footprint was evaluated using a computer-aided design sheet-metal feature, and fiducial marks (0.5 mm radius, 2 thermograph pixels) enabled accurate image mapping. The wrap was applied at the frustum/tip interface, 102 mm downstream of the sharp nose ( $x/L = 0.25$ ), allowing the boundary layer to develop before the forward-facing step; at this location, the computed boundary-layer thickness at the highest Reynolds number, $ \textit{Re}_{\infty }$ is 0.55 mm, five times the film thickness. Thermograph temperatures were mapped using a perspective transformation matrix (PTM) (Haralick Reference Haralick1980). As uncertainty propagation through the PTM inversion is impractical, a Monte Carlo simulation was used, giving a mapped-pixel uncertainty of $\pm 0.30$ mm. Viewing angles were computed based on camera and model geometry, and a directional emissivity correction was applied.

Heat flux was calculated by solving the transient 1-D heat equation through space and time (Zaccara, Edelman & Cardone Reference Zaccara, Edelman and Cardone2020). Lateral heat conduction was neglected due to the short test durations, consistent with prior studies showing minimal through-thickness thermal penetration under similar conditions (Rataczak, Running & Juliano Reference Rataczak, Running and Juliano2021; Running et al. Reference Running, Rataczak, Zaccara, Cardone and Juliano2022) The FORTRAN QCALC subroutine was translated to ${\textrm {MATLAB}}^{\circledR }$ for this purpose (Juliano, Adamczak & Kimmel Reference Juliano, Adamczak and Kimmel2015). QCALC assumes 1-D heat transfer and uses a second-order Euler-explicit finite-difference approximation to solve for the temperature distribution through the wall; heat flux is obtained from a second-order approximation to the derivative of the temperature profile at the outer surface (Boyd & Howell Reference Boyd and Howell1994). In this work, a cylindrical coordinate system was used with the local radius of curvature, thickness, and material properties of the vinyl wrap and underlying aluminium wall. The initial condition is based on the assumption of a uniform through-wall temperature distribution, equal to the pre-run surface temperature at each pixel. The measured surface temperature was applied as the front-face boundary condition, and an adiabatic back-face boundary condition was imposed. Radiation was neglected due to relatively low temperatures. The full temperature history, starting from the assumedly isothermal pre-run, was used to solve transient heat conduction and resolve surface heat flux. A 2 h interval between runs ensured thermal equilibrium and consistent initial conditions, preventing residual thermal gradients from affecting the results.

The heat flux was converted to the non-dimensional heat flux, the Stanton number. Throughout this work, a modified Stanton number is used

(3.1) \begin{align} St = \frac {{\dot {q''}_{w\textit{all}}}} {c_p \rho _{\infty } u_{\infty } \left (T_0 - T_{w\textit{all}} \right )}, \end{align}

where $\dot {q''}_{w\textit{all}}$ , $c_p$ , $\rho _{\infty }$ , $u_{\infty }$ , $T_0$ and $T_{w\textit{all}}$ are the heat flux, specific heat capacity of air at constant pressure, free-stream density, free-stream velocity, stagnation temperature and wall temperature, respectively. The standard heat capacity of air at constant pressure, $1004$ J kg^–1 K^–1, was assumed. The uncertainty in $ \textit{St}$ is computed by analysing and propagating the uncertainty associated with each of the parameters in (3.1). The total uncertainty is calculated by taking the square root of the sum of the squares of all uncertainties. The uncertainty range on the Stanton number is $+10/{-}8$ %. For each test condition, the Stanton number was evaluated during the first steady-flow period after startup transients subsided; it varied minimally despite slight heat-flux changes because the driving temperature difference also decreased.

3.3. Background-oriented schlieren

Background-oriented schlieren was used to measure off-wall density-gradient and density fields. The accuracy and precision of the measurement technique for high-speed flows were validated in a previous study (Davami et al. Reference Davami, Juliano, Moreto and Liu2025b ). The system included high-resolution and high-speed cameras, a background, and two LED arrays. A Canon EOS RP with 24–105 mm lens recorded $3840\times 2160$ pixels at 24 fps to resolve the mean flow, while a Phantom v1840 with a 90 mm Tamron lens recorded $1000\times 100$ pixels at 30 000 fps to capture the unsteady dynamics. Depth of field was maximised by using an aperture of $f/22$ (Molnar et al. Reference Molnar, LaLonde, Combs, Léon, Donjat and Grauer2024), and measurements were performed simultaneously, with the high-speed camera slightly off-axis.

Two $210\times 297$ mm speckled sheets, each with 200 000 randomly distributed 0.25 mm dots, covered the 0.41 m acrylic window and were back illuminated by two LED arrays. With a video resolution of $3840\times 2160$ pixels, the dots corresponded to 2.3 pixels in diameter, yielding a density of 321 particles cm $^2$ . Images were analysed in PIVlab (Thielicke & Sonntag Reference Thielicke and Sonntag2017) by cross-correlating wind-off and wind-on photographs. The maximum displacement between two images was approximately three pixels, prompting the use of a minimum interrogation window size of $8\times 8$ pixels as long as sufficiently high correlation values associated with high dot-pattern density allow. A four-pass refinement from $64\times 64$ to $8\times 8$ pixels with 50 % overlap yielded a $480\times 480$ displacement grid at 0.3 mm resolution. To suppress random noise, consecutive displacement fields separated by 0.04 s were averaged.

Optical parameters from the set-up were used to convert from image to physical space, with an uncertainty of $\pm 0.4$ mm for a given pixel. Assuming axisymmetry, displacement fields were processed using the filtered back projection technique (FBPT) to reconstruct centreplane gradient fields, from which density was obtained by solving the 2-D Poisson equation (Venkatakrishnan & Meier Reference Venkatakrishnan and Meier2004; Davami et al. Reference Davami, Juliano, Moreto and Liu2025b ). Uncertainty quantification followed Rajendran et al. (Reference Rajendran, Zhang, Bhattacharya, Bane and Vlachos2019), with maximum displacement uncertainty of $\pm 0.026$ pixels ( $\pm 0.15$ mm) corresponding to a density uncertainty of $\pm 0.01$ kg m^– $^3$ near shocks. Figure 2(a) presents a representative line-of-site-integrated vertical displacement field ( $\Delta y$ ), while figure 2(b) displays the corresponding FBPT reconstruction on the (axisymmetric) model centreplane. The streamwise and centreline-normal coordinates, $x$ and $y$ , are normalised by the cone length, placing the cone–flare junction at $x/L = 1$ . Shock waves appear as large negative $\Delta y$ , while the shear layer shows positive $\Delta y$ .

Figure 2.

The BOS-measured displacement fields. (a) Line-of-sight integrated. (b) Centreplane.

5.1. Laminar and turbulent base flows

The presented results were computed for the cone–flare test article for all three nose radii, using a grid of $2201 \times 826$ points along the streamwise and wall-normal directions, respectively. Figure 5 is a representative grid, for $r_{ {n}}=0.99$ mm and adapted to $ \textit{Re}_\infty =5.8\times 10^6$ m^–1. The nose region is shown with every other streamwise line and every twentieth wall-normal point; the full geometry is shown with every tenth line in both directions. The laminar flow solutions were computed with a second-order accurate algorithm as implemented in the finite-volume compressible Navier–Stokes flow solver VULCAN-CFD (visit http://vulcan-cfd.larc.nasa.gov for further information about the VULCAN-CFD solver) (Baurle & Edwards Reference Baurle and Edwards2020). The VULCAN-CFD solution is based on the full Navier–Stokes equations and uses the solver’s built-in capability to iteratively adapt the computational grid to the bow shock and boundary layer (Scholten et al. Reference Scholten, Paredes, Li, White, Baurle and Choudhari2022; Scholten Reference Scholten2024). Adaptation ensures enough points are clustered next to the model surface to resolve the thickness of the boundary layer within the separation region. For this process, the boundary-layer edge is defined as the wall-normal position where $h_0/h_{0,\infty }=0.99$ , with $h_0$ denoting the total enthalpy, i.e. $h_0\equiv h+ ({1}/{2})(\bar {u}^2+\bar {v}^2+\bar {w}^2)$ , where $h=c_p \bar {T}$ is the static enthalpy. An offset is also applied and ensures the chosen number of cells in the wall-normal direction will properly resolve the entropy layer as well. Sutherland’s law for air is used to calculate the dynamic viscosity as a function of temperature. An isothermal wall with $T_w=290$ K is used and free-stream conditions were selected to replicate most of the experimental runs (table 2).

Figure 6 presents Mach number contours of two representative laminar basic-state solutions for nose radius $r_{ {n}} = 0.1$ mm, illustrating the effect of $ \textit{Re}_\infty$ on the topology of the SWBLI. The upper Mach number contour is for $ \textit{Re}_\infty = 5.8 \times 10^6$ m^–1, while the lower is $ \textit{Re}_\infty = 10.1 \times 10^6$ m^–1. The separation and reattachment locations are denoted by ‘ $\circ$ ’. As $ \textit{Re}_\infty$ increases, the laminar boundary layer on the conical forebody becomes less resistant to separation, resulting in earlier separation for $ \textit{Re}_\infty = 10.1 \times 10^6$ m^–1 than for $ \textit{Re}_\infty = 5.8 \times 10^6$ m^–1. Despite different separation locations, the reattachment location is nearly identical for both $ \textit{Re}_\infty$ .

Table 2.

Computational flow conditions.

Figure 5.

An $r_{ {n}}=0.99$ mm grid adapted to $ \textit{Re}_\infty =5.8\times 10^6$ m^–1. (a) Nose region, (b) Full geometry.

Figure 6.

Effect of $ \textit{Re}_{\infty }$ on the laminar SWBLI ( $r_{ {n}} = 0.01$ mm).

Figure 7 presents the effect of nose radius on the laminar SWBLI for $ \textit{Re}_\infty =5.8\times 10^6$ m^–1. The Mach number contour for the upper domain is for $r_{ {n}} = 0.1$ mm, while the lower is for $r_{ {n}} = 5.0$ mm. The differences in $x_s/L$ and $x_r/L$ are slightly exaggerated between the upper and lower contours due to the shorter cone length for $r_{ {n}} = 5.0$ mm, which is 36 mm shorter. Nevertheless, as the nose radius increases, the boundary layer on the conical forebody is thicker, and the edge Mach number decreases. This leads to earlier separation, later reattachment, and increased separation bubble length. Additionally, a blunter nose results in a thicker entropy layer on the cone, influencing the development of instabilities (Paredes et al. Reference Paredes, Choudhari, Li, Jewell and Kimmel2019b ; Kennedy et al. Reference Kennedy, Jewell, Paredes and Laurence2022). The separation bubble properties are tabulated in table 3 for all laminar conditions computed.

Table 3.

Separation bubble properties for all nose radii and Reynolds numbers.

Figure 7.

Effect of $r_{ {n}}$ on the laminar SWBLI ( $ \textit{Re}_\infty =5.8\times 10^6$ m^–1).

In addition to the laminar solutions, fully turbulent solutions were computed at the same free-stream Reynolds numbers using the Menter-Shear stress transport (SST) Reynolds-averaged Navier–Stokes (RANS) turbulence model (Menter Reference Menter1994) to compare with experimental results. Additional details of the implementation of the model in the VULCAN-CFD solver are found in Baurle & Edwards (Reference Baurle and Edwards2020).

5.2. Stability analysis

The methodologies used for the analysis of disturbance amplification are equivalent to those used by Paredes et al. (Reference Paredes, Scholten, Choudhari, Li, Benitez and Jewell2022). The evolution of convective instabilities was calculated with the linear PSE and the HLNSE frameworks. Global stability analysis of the laminar flow solution was also performed.

The Cartesian coordinates are represented by $(x,y,z)$ . The computational coordinates are defined as an orthogonal body-fitted coordinate system along the cone, with $(\xi ,\eta ,\zeta )$ denoting the wall-parallel (‘streamwise’), wall-normal and azimuthal coordinates, respectively, and $(u,v,w)$ representing their corresponding velocity components. The same orientation of the coordinate system and velocities is maintained along the flare with a non-orthogonal transformation of the two-dimensional grid. The density and temperature are denoted by $\rho$ and $T$ , respectively. The vector of basic-state variables is $\bar {\boldsymbol{q}}(\xi ,\eta ,\zeta )=({\bar {\rho }},{\bar {u}},\bar {v},\bar {w},{\bar {T}})^\top$ and the vector of perturbation variables is denoted by ${\tilde {\boldsymbol{q}}}(\xi ,\eta ,\zeta ,t)=(\tilde {\rho },\tilde {u},\tilde {v},{\tilde {w}},\tilde {T})^\top$ . For axisymmetric geometries at zero degrees angle of attack, the basic-state variables are independent of the azimuthal coordinate and the linear perturbations can be assumed to be harmonic in time and in the azimuthal direction, which leads to the following expression for the perturbations

(5.1) \begin{align} {\tilde {\boldsymbol{q}}}(\xi ,\eta ,\zeta ,t)=\breve {\boldsymbol{q}}(\xi ,\eta ) \exp \left [\textrm {i} \left (m\zeta -\omega t\right )\right ]+\textrm {c.c.}, \end{align}

where the vector of disturbance functions is $\breve {\boldsymbol{q}}(\xi ,\eta )=(\breve {\rho },\breve {u},\breve {v},\breve {w},\breve {T})^\top$ , $m$ is the azimuthal wavenumber, $\omega$ is the angular frequency and $\textrm {c.c.}$ refers to the complex conjugate.

The disturbance functions $\breve {\boldsymbol{q}}(\xi ,\eta ,\zeta )$ satisfy the HLNSE (Paredes et al. Reference Paredes, Choudhari, Li, Jewell, Kimmel, Marineau and Grossir2019a ), which involve coefficient functions that depend on the basic-state variables and parameters, and on the angular frequency and azimuthal wavenumber of the perturbation

(5.2) \begin{align} \boldsymbol{L}_{\textit{HLNSE}}\,\breve {\boldsymbol{q}}(\xi ,\eta )=\breve {\boldsymbol{f}}, \end{align}

where $\breve {\boldsymbol{f}}$ represents a potential forcing term.

The PSE approximation to the HLNSE is based on isolating the rapid phase variations in the streamwise direction by introducing the disturbance ansatz

(5.3) \begin{align} \breve {\boldsymbol{q}}(\xi ,\eta ,\zeta )=\hat {\boldsymbol{q}}(\xi ,\eta ,\zeta ) \exp \left [ \textrm {i} \int _{\xi _0}^{\xi }\alpha (\xi ')\,\textrm {d}\xi '\right ]\!, \end{align}

where the unknown, streamwise-varying wavenumber $\alpha (\xi )$ is determined in the course of the solution by imposing an additional constraint

(5.4) \begin{align} \int _\eta \hat {\boldsymbol{q}}^* \frac {\partial \hat {\boldsymbol{q}}}{\partial \xi }\,h_\xi \,h_\zeta \,\textrm {d}\eta \,\textrm {d}\zeta =0, \end{align}

and requiring the amplitude functions $\hat {\boldsymbol{q}}(\xi ,\eta ,\zeta )=(\hat {\rho },{\hat {u}},\hat {v},\hat {w},\hat {T})^\top$ to vary slowly in the streamwise direction in comparison with the phase term $\exp [ \textrm {i} \int _{\xi _0}^{\xi }\alpha (\xi ')\,\textrm {d}\xi ' ]$ . Substituting (5.3) into the HLNSE and invoking scale separation between the streamwise coordinate and the other two directions to neglect the viscous terms with streamwise derivatives, the PSE are obtained in the form

(5.5) \begin{align} \left (\boldsymbol{L}_{\textit{PSE}}+\boldsymbol{M}_{\textit{PSE}}\frac {1}{h_\xi }\frac {\partial }{\partial \xi }\right )\hat {\boldsymbol{q}}(\xi ,\eta )=0. \end{align}

The onset of laminar-turbulent transition is estimated by using the logarithmic amplification ratio, the so-called $N$ -factor, relative to the location $\xi _{I}$ where the disturbance first becomes unstable

(5.6) \begin{align} N_\phi =-\int _{\xi _{I}}^{\xi }\alpha _i(\xi ')\,\mathrm{d}\xi '+\ln \big [\hat {\phi }(\xi )/\hat {\phi }(\xi _{I})\big ]. \end{align}

Here, $\hat {\phi }$ denotes an amplitude norm of $\hat {\boldsymbol{q}}$ at a given $\xi$ , e.g. wall-pressure disturbance or total disturbance energy $E$ (Chu Reference Chu1956; Mack Reference Mack1969).

The evolution of convective boundary-layer instabilities is analysed with a hybrid methodology comprising PSE and HLNSE solutions across overlapping streamwise domains. The linear amplification of planar and oblique, first and second Mack mode disturbances along the cone, is computed with PSE until just upstream of the cone–flare corner. The HLNSE is used to calculate the development of the instability waves through the remaining length of the geometry. Figure 8 shows the comparison of the $N$ -factor envelopes based on the total disturbance energy for the planar and oblique waves with different azimuthal wavenumbers shown by the legend, for the lowest and highest free-stream Reynolds numbers. The effect of a lower Reynolds number is to reduce the amplification of the disturbances. At the separation location around $x/L \approx 0.75$ ( $x\approx 0.31$ m), an $N$ -factor of ${\approx} 4.88$ is predicted for $ \textit{Re}_\infty =7.6\times 10^6$ $\textrm {m}^{-1}$ , while a higher $N_E\approx 7.46$ is found for $ \textit{Re}_\infty =10.1\times 10^6$ $\textrm {m}^{-1}$ . Experimental measurements indicate the boundary layer is laminar upstream of separation, thus these amplification factors are not sufficient to lead to transition onset.

Figure 8.

Evolution of $N$ -factor envelopes based on total disturbance energy ( $N_{E,\textit{en}v}$ ). Legend reflects the different azimuthal wavenumbers; (a) $ \textit{Re}_\infty =5.8\times 10^6$ $\textrm {m}^{-1}$ , (b) $ \textit{Re}_\infty =12.1\times 10^6$ $\textrm {m}^{-1}$ .

The GSA is based on the HLNSE, with the real-valued angular frequency $\omega$ from (5.1) replaced by a complex value $\varOmega =\omega +\textrm {i}\sigma$ , where $\sigma$ is the temporal growth rate of the disturbance. After setting $\breve {\boldsymbol{f}}=0$ and defining $\hat {\boldsymbol{q}}\equiv \breve {\boldsymbol{q}}$ , (5.2) can be written as the generalised eigenvalue problem

(5.7) \begin{align} \boldsymbol{A} \hat {\boldsymbol{q}} = \varOmega\!\boldsymbol{B} \hat {\boldsymbol{q}}, \end{align}

where the leading eigenvalues $\varOmega$ and eigenvectors $\hat {\boldsymbol{q}}$ are calculated with the Arnoldi algorithm (Saad Reference Saad1980).

Grid sensitivity was analysed for $ \textit{Re}_\infty =12.1\times 10^6$ $\textrm {m}^{-1}$ at all nose radii using different grid resolution for the computation of the laminar basic state. For each grid, the solution was used to compute the azimuthal wavenumber and growth rate associated with the maximum global instability (table 4). The relative differences between the azimuthal wavenumbers associated with the maximum growth rate between the $2001 \times 751$ and $2401 \times 901$ grids are under $10\,\%$ , and the lower $ \textit{Re}_\infty$ cases have lower relative differences. Therefore, this worst case justifies the $2201 \times 826$ grid be used for the computation of all basic states.

Table 4.

Wavenumber associated with the most unstable global disturbance using different basic-state grid resolutions at $ \textit{Re}_\infty =12.1\times 10^6$ $\textrm {m}^{-1}$ .

The leading global instability growth rate at each of the selected conditions given in table 2 is shown in figure 9. For all Reynolds numbers computed, the flow was globally unstable. The leading unstable mode is a short-wavelength stationary disturbance with wavenumber increasing from $m=50$ to $m=71$ as the Reynolds number increases. The eigenspectra computed for the lowest Reynolds number is given in figure 10 for all three nose radii. Oscillatory mode families exist but remain less amplified than the dominant, stationary, mode. The eigenspectrum remains relatively similar between $r_n=0.1$ and $r_n=0.99$ mm. For the largest nose radius, additional oscillatory modes become unstable.

Figure 9.

Leading instability growth rate from GSA as function of wavenumber, $ \textit{Re}_{\infty }$ , and $r_{ {n}}$ .

Figure 10.

Eigenspectra obtained by GSA for $ \textit{Re}_\infty =5.8\times 10^6$ $\textrm {m}^{-1}$ . The grey area indicates stable eigenvalues. The wavenumber at the peaks is given by the number near each one; (a) $r_n=0.1$ mm, (b) $r_n=0.99$ mm, (c) $r_n=5$ mm.

Figure 11.

Real part of streamwise-velocity perturbation corresponding to the leading disturbance as computed by GSA ( $r_{ {n}} = 0.99$ mm): (a) $ \textit{Re}_\infty =5.8\times 10^6$ $\textrm {m}^{-1}$ and $m = 50$ ; (b) $ \textit{Re}_\infty =12.1\times 10^6$ $\textrm {m}^{-1}$ and $m = 70$ .

The real part of the azimuthal velocity component of the corresponding global mode eigenfunctions for the lowest and highest Reynolds numbers for the $1$ mm tip at peak growth rates are presented in figure 11. The boundary-layer edge is displayed for each mode shape as a solid black line and, unlike in the grid adaptation process, is defined more precisely by $h_0/h_{0,\infty }=0.995$ . In addition, the first and second separation regions are shown by dashed black lines at $u = 0$ . This short-wavelength mode is concentrated at the reattachment location with no significant presence at the corner. It straddles the separation line of the first, larger, separated region.

5.3. Direct numerical simulations

Direct numerical simulations were performed for $r_{ {n}} = 0.99$ mm at $ \textit{Re}_\infty =5.8\times 10^6$ $\textrm {m}^{-1}$ and $7.6\times 10^6$ $\textrm {m}^{-1}$ to compare with experiment and GSA, and to characterise the nonlinear behaviour of the finite-amplitude global modes and their effect on the formation of thermal striations along the flare.

To simulate the boundary-layer flow by DNS, the full, 3-D, compressible Navier–Stokes equations in conservation form were solved numerically in cylindrical coordinates. The working fluid is air and falls within the perfect gas regime. The usual constitutive relations for a Newtonian fluid were used: the viscous stress tensor is linearly related to the rate-of-strain tensor, and the heat-flux vector is linearly related to the temperature gradient through Fourier’s law. The coefficient of viscosity $\mu$ is computed from Sutherland’s law, and the coefficient of thermal conductivity $\kappa$ is computed from $\kappa =\mu C_p/\textit{Pr}$ , with a molecular Prandtl number of $ \textit{Pr} = 0.72$ . The inviscid fluxes of the governing equations were computed using a seventh-order weighted essentially non-oscillatory (WENO) scheme (Jiang & Shu Reference Jiang and Shu1996). The viscous fluxes were discretised using a fourth-order central-difference scheme, and time integration was performed using a third-order, low-storage, Runge–Kutta scheme (Williamson Reference Williamson1980). A similar WENO-based numerical solver has been successfully applied to study hypersonic transitional flows including transitional SWBLIs over a compression ramp at Mach 7.7 (Cao et al. Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021) and a cone–cylinder–flare model at Mach 6 (Li et al. Reference Li, Choudhari, Paredes and Scholten2024).

To set up the DNS, the unperturbed boundary-layer flow and axisymmetric separation bubble computed with the VULCAN-CFD flow solver were used to obtain a shock-adapted solution for the entire cone–flare model. Then a subset of the computational domain of $x/L = [0.367, 1.258]$ that covers the separation and reattachment regions was used for the DNS with the same grid distribution. Figure 12(a) visualises the DNS subdomain along with the full-domain VULCAN-CFD solution along a meridional plane for the DNS case at $ \textit{Re}_{\infty } =5.8\times 10^6$ m^–1. Here, the inflow of the DNS domain is placed significantly upstream of the cone–flare junction to allow for inflow adjustment and guarantee adequate decay of any instabilities from the inflow boundary so that the global instabilities from the separation bubble can be isolated. The DNS domain extends in the wall-normal direction from the cone surface to just below the bow shock. The inflow and the upper boundary conditions for the DNS were obtained by using the VULCAN-CFD solution. An unsteady, non-reflecting boundary condition based on Thompson (Reference Thompson1987) is used at the outflow boundary. For the no-slip wall, isothermal conditions are specified with the wall temperature of $T_{w} = 290$ K.

Figure 12.

The DNS subdomain along with full-domain basic-state Mach number contours computed by VULCAN-CFD ( $r_{ {n}} = 0.99$ mm and $ \textit{Re}_{\infty } =5.8\times 10^6$ m^–1); (a) 2-D DNS, (b) 3-D DNS $m=4$ .

For each Reynolds number, a steady simulation with a 2-D subdomain of $x/L = [0.367, 1.258]$ , referred to as 2-D DNS, was first performed and validated against the full-domain VULCAN-CFD solution. Next, the 2-D domain of 2-D DNS was extruded along the azimuthal direction with a spanwise extent of $\phi$ , and a 3-D unsteady simulation with periodic boundary conditions in the spanwise direction was performed, referred to as 3-D DNS (figure 12 b). Both the 2-D and 3-D DNS runs use a total number of grid points of $N_x = 740$ and $N_r = 540$ in the streamwise and radial directions, respectively. The spanwise extent of the primary 3-D DNS was selected to be $\phi =90^\circ$ (corresponding to an azimuthal wavenumber $m = 4$ ), wide enough to accommodate multiple thermal streaks within the domain. A total of $N_\phi =800$ points were used.

For each case, the 3-D DNS was initialised with no disturbances inside the domain or at the boundaries. As demonstrated in Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021) and Li et al. (Reference Li, Choudhari, Paredes and Scholten2024), the growth of instability waves can arise from the extremely low-level perturbations provided by numerical round-off error, which enables the examination of intrinsic instabilities in the fluid-dynamic system. The 3-D DNS was run with a constant time step of $\Delta t = 1.2$ ns for at least 5.7 ms. A fully developed flow field was established within $t \approx 1.40$ ms for $ \textit{Re}_{\infty } =5.8\times 10^6$ m^–1 and $t \approx 1.0$ ms for $ \textit{Re}_{\infty } =7.6\times 10^6$ m^–1. These values were determined by monitoring the exponential temporal growth and saturation of the spanwise-velocity fluctuations within the separation bubble, which is shown later.

Table 5 summarises the DNS cases along with the spatial and temporal parameters. $N_x$ , $N_\phi$ and $N_r$ are the number of grid points in the streamwise, azimuthal and radial directions, respectively. Here, $\Delta x_s$ is the streamwise wall-parallel grid spacing, $ (r\Delta \phi )^+_w$ is the azimuthal grid spacing at the wall and $\Delta y_{n,w}$ is the wall-normal grid spacing at wall. The superscript ‘+’ denotes normalisation by the viscous length near the end of the DNS domain at $x/L = 1.13$ . The selection of grid and other aspects of the numerical solution were based on extensive experience with a similar class of flows, including independent verification of the DNS tools against multiple solvers from different computational groups for the scaled ROTEX-T cone–flare geometry (Davami et al. Reference Davami2025c ). Additional assessment of the grid resolution for the DNS is included in Appendix A.

Table 5.

Summary of DNS cases along with spatial and temporal parameters.

Table 6 compares the properties of the separation bubble between the 2-D DNS baseflow, 3-D DNS and VULCAN-CFD solutions. Excellent agreement is achieved between 2-D DNS and VULCAN-CFD, indicating the validity of the laminar base flow computed by the DNS code. Very small differences are seen in separation and reattachment locations and the size of the separation bubble between the laminar base flow and the averaged mean flow of the 3-D DNS, indicating that the flare boundary layer remains laminar. Nevertheless, the finite-amplitude global modes cause an appreciable deviation in the normalised maximum reverse velocity from the laminar base flow.

Table 6.

Computed separation bubble properties for $r_{ {n}} = 0.99$ mm.

To monitor the initial growth of instability waves arising from the extremely low-level perturbations provided by numerical round-off error and the subsequent saturation and establishment of 3-D flow for each 3-D DNS case, we follow the approach by Cao et al. (Reference Cao, Hao, Klioutchnikov, Olivier and Wen2021) and consider the temporal evolution of the quadratic mean of the spanwise velocity at a streamwise position, which is defined as

(5.8) \begin{align} A_w \equiv \sqrt {\frac {1}{N_r N_\phi }\sum _{j=1}^{N_\phi }\sum _{k=1}^{N_r} \left ( w/U_\infty \right )_{j,k}^2 } , \end{align}

where $w$ is the azimuthal velocity, $U_\infty$ is the free-stream velocity and $N_\phi$ and $N_r$ are the number of grid points in the azimuthal and radial directions, respectively. Figure 13 shows the time evolution of $A_w$ inside the separation bubble near reattachment ( $x/L = 1.05$ ) for both 3-D DNSs. The round-off errors of the order of $A_w = 10^{-7}$ at the very beginning start to grow exponentially at $t \approx 1.0$ and $0.7$ ms, respectively, for $ \textit{Re}_{\infty } =5.8\times 10^6$ and $ \textit{Re}_{\infty } =7.6\times 10^6$ m^–1, until reaching the asymptotic level of $A_w = 10^{-2}$ at $t \approx 1.4$ and $1.0$ ms at each Reynolds number, which marks the start of the ‘saturated’ regime. The black dashed and dash-dotted lines represent the maximum growth rate predicted by GSA, $13.1$ ms^–1 for $ \textit{Re}_{\infty } =5.8\times 10^6$ m^–1 and $20.1$ ms^–1 for $ \textit{Re}_{\infty } =7.6\times 10^6$ m^–1. The comparison of exponential instability growth from DNS data matches the maximum growth rates predicted by the GSA extremely well, suggesting that the global instability of the recirculation bubble is the primary cause of 3-D flow structures such as reattachment streaks.

Figure 13.

The DNS instability growth measured by $A_w$ (5.8) near reattachment ( $x/L = 1.05$ ). (a) Full time series. (b) Linear growth regime.

To assess the flow state in the DNS, the temporal evolution of spanwise-averaged surface Stanton number ( $ \textit{St}$ ) is presented in figure 14 for $ \textit{Re}_{\infty } = 5.8\times 10^6$ and $7.6\times 10^6$ m^–1, with $r_{ { n}} = 0.99$ mm. The temporal evolution of spanwise-averaged separation (leftmost black line) and reattachment (rightmost black line) locations are also provided, determined as the location where $\tau _{ {w}} = 0$ , while neglecting the few locations within the separation bubble where $\tau _w$ reaches zero but does not indicate reattachment. The standard deviations of $x_s(t)$ and $x_r(t)$ about their mean, indicating the extent of the intermittent region, are $\sigma _s = 2.8$ mm, $\sigma _r = 1.3$ mm at $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1, and $\sigma _s = 2.5$ mm, $\sigma _r = 1.5$ mm at $ \textit{Re}_{\infty } = 7.6\times 10^6$ m^–1. At both Reynolds numbers, saturation of the global instabilities causes a slight upstream shift of the separation location over time, whereas the average reattachment location remains essentially stationary, leading to an overall enlargement of the separation bubble. Large intermittent spurts of $ \textit{St}$ near reattachment are apparent once the flow saturates (horizontal red dashed line) – $ \textit{St}$ almost doubles compared with the linear growth regime. Although the reattachment location is nearly identical for both $ \textit{Re}_\infty$ (0.04 % difference), the magnitude of heating spurts increases significantly. A movie of the temporal evolution of surface heating is provided in the supplementary movie is available at https://doi.org/10.1017/jfm.2025.10905.

Figure 14.

The DNS results of the temporal evolution of Stanton number ( $r_{ {n}} = 0.99$ mm). (a) $\textit{Re}_{\infty} = 5.8\times 10^6\,\textrm{m}^{-1}$ , (b) $\textit{Re}_{\infty} = 7.6\times 10^6\,\textrm{m}^{-1}$ .

The DNS flow state is further assessed in figure 15, which presents streamwise Stanton number profiles from the 3-D DNS (time/spanwise averaged) alongside laminar and turbulent VULCAN-CFD predictions for $ \textit{Re}_{\infty } = 5.8\times 10^6$ and $7.6\times 10^6$ m^–1 for $r_{ {n}} = 0.99$ mm. Over the flare, the 3-D DNS exhibits $ \textit{St}$ values higher than the laminar base flow, yet substantially lower than those predicted by the turbulent RANS solution. Nonlinear saturation of the unstable global modes within the recirculation bubble leads to increased peak heating on the flare (Davami et al. Reference Davami2025c ). Given that Menter’s SST model provides a reasonable estimate of turbulent heating on the flare, these comparisons indicate that the 3-D DNS remains laminar along the conical forebody but reaches the onset of transition at reattachment, with intermittent turbulent bursts – some exceeding $ \textit{St} = 3.5\times 10^{-3}$ , comparable to the RANS-predicted turbulent levels – as evidenced in figure 14(b).

Figure 15.

Comparison of base-flow and time-/spanwise-averaged mean-flow solutions ( $r_{ {n}} = 0.99$ mm); (a) $ \textit{Re}_\infty =5.8\times 10^6$ $\textrm {m}^{-1}$ , (b) $ \textit{Re}_\infty =7.6\times 10^6$ $\textrm {m}^{-1}$ .

Figure 16 presents time-averaged skin-friction coefficient ( $C_f \equiv \tau _w/(({1}/{2})\rho _\infty u_\infty ^2)$ ) distributions from DNS. Separation is identified by $C_f$ changing from positive to negative, and reattachment by the reverse, from negative to positive. Time averages were computed over snapshots spanning the saturated portion of each case, as given in table 5. For $ \textit{Re}_{\infty } =5.8\times 10^6$ m^–1, separation occurs at $x_{ {s}}/L = 0.735$ while reattachment ranges from $x_r/L = 1.120$ to 1.132 during the saturated regime. At $ \textit{Re}_{\infty } =7.6\times 10^6$ m^–1, separation shifts upstream to $x_{ {s}}/L= 0.711$ while reattachment remains nearly unchanged. The Reynolds-number dependence of $x_{ {s}}$ and $x_{ {r}}$ agrees with VULCAN-CFD base-flow results (figure 6). Although the experimental model ends at $x/L = 1.13$ , computations extend the domain, capturing the full evolution of the reattached boundary layer.

Figure 16.

The DNS-computed time-averaged skin-friction contours ( $r_{ {n}} = 0.99$ mm). (a) $\textit{Re}_{\infty} = 5.8\times 10^6\,\textrm{m}^{-1}$ , (b) $\textit{Re}_{\infty} = 7.6\times 10^6\,\textrm{m}^{-1}$

6.1. Global surface heating

Infrared thermography provided high-spatial-resolution, non-intrusive, quantitative measurements of the surface temperature distribution, from which surface heat flux and Stanton number were calculated. Figure 17 presents global experimental $ \textit{St}$ distributions temporally filtered over 0.11 s (40 frames) at $ \textit{Re}_{\infty } = 7.6\times 10^6$ and $10.0\times 10^6$ m^–1 for $r_{ {n}} = 0.99$ mm. The streamwise and centreline-normal coordinates, $x$ and $y$ , are normalised by the cone length for $r_{ {n}} = 0.99$ mm ( $L=406.6$ mm) such that the cone–flare junction is located at $x/L = 1$ . The distinct decrease in $ \textit{St}$ upstream of the cone–flare junction indicates laminar boundary-layer separation. This low-heating region is commonly referred to as a ‘ $ \textit{St}$ bucket’. As will be shown in § 6.3, the point at which $ \textit{St}$ begins to decrement aligns excellently with separation. The abrupt increase in $ \textit{St}$ along the flare correlates closely with the reattachment location.

Figure 17.

Time-averaged $ \textit{St}$ distributions ( $r_{ {n}} = 0.99$ mm). (a) $\textit{Re}_{\infty} = 7.6\times 10^6\,\textrm{m}^{-1}$ , (b) $\textit{Re}_{\infty} = 10.0\times 10^6\,\textrm{m}^{-1}$

Streamwise $ \textit{St}$ profiles facilitate a more in-depth comparison of separation, reattachment, and the overall state of the flow. Figure 18 presents spanwise-averaged, experimental streamwise $ \textit{St}$ profiles for the five primary $ \textit{Re}_{\infty }$ conditions for $r_{ {n}} = 0.99$ mm. The profiles on the cone (figure 18 a) and flare (figure 18 b) are plotted separately to independently optimise their scales. Time-averaged DNS-computed $ \textit{St}$ profiles for $ \textit{Re}_{\infty } = 5.8\times 10^6$ and $7.6\times 10^6$ m^–1 are provided for comparison. Laminar and turbulent VULCAN-CFD computed $ \textit{St}$ profiles at the minimum and maximum $ \textit{Re}_{\infty }$ are also included to assess the boundary-layer state. Off-wall separation and reattachment locations (see § 6.2) are denoted by ‘ $\circ$ ’ for experiment, ‘ $\triangle$ ’ for DNS and ‘ $\square$ ’ for VULCAN-CFD, with colours corresponding to $ \textit{Re}_{\infty }$ . Note that computed reattachment locations for laminar VULCAN-CFD and DNS are outside of the experimental domain (refer to table 6 and figure 12 b). Turbulent RANS simulations indicate that the SWBLI strength is insufficient to induce flow separation.

Figure 18.

Value of $ \textit{St}(x/L)$ ( $r_{ {n}} = 0.99$ mm). Separation and reattachment locations are denoted by ‘ $\circ$ ’ for experiment, ‘ $\triangle$ ’ for DNS and ‘ $\square$ ’ for VULCAN-CFD. (a) Cone, (b) Flare.

On the cone, upstream of the $ \textit{St}$ buckets, the experimentally measured and DNS-computed $ \textit{St}$ profiles align closely with one another and are within the range of the laminar VULCAN-CFD solutions for the highest and lowest $ \textit{Re}_{\infty }$ . The presence of $ \textit{St}$ buckets, coupled with the strong agreement with the laminar computed solutions, substantiates that the boundary layer remained laminar prior to separation at all $ \textit{Re}_{\infty }$ for both the experimental and DNS results. For $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1, the DNS predicts separation the earliest, followed by VULCAN-CFD, and the experimental results occur farthest downstream. In the experiment, as $ \textit{Re}_{\infty }$ increases, the separation location shifts downstream. Davami et al. (Reference Davami2025c ) showed that the angular deflection of the flow at separation is largely insensitive to $ \textit{Re}_{\infty }$ ; therefore, as $ \textit{Re}_{\infty }$ increases, reattachment concomitantly shifts upstream, and the separation bubble shrinks. The trend of a progressively smaller separation bubble with increasing $ \textit{Re}_{\infty }$ is indicative of a transitional SWBLI, indicating that the flow is transitioning along the shear layer or just downstream of reattachment (Becker Reference Becker1956; Babinsky & Harvey Reference Babinsky and Harvey2011; Cao et al. Reference Cao, Hao and Guo2024). The opposite trend is predicted by VULCAN-CFD, as expected, since the flow is entirely laminar throughout the computational domain (i.e. a fully laminar SWBLI). The DNS results predict an upstream shift in separation as $ \textit{Re}_{\infty }$ increases.

Figure 19.

Effect of nose radius on $ \textit{St}$ for $ \textit{Re}_{\infty } = 12\times 10^6$ m^–1. Markers denote separation and reattachment locations. (a) Cone, (b) Flare.

On the flare, the flow state is assessed experimentally by analysing the streamwise behaviour of Stanton number. Davami et al. (Reference Davami2025c ) succinctly summarised that streamwise $ \textit{St}$ can display zero, one or two local maxima, contingent on the flow state on the flare. For all $ \textit{Re}_{\infty }$ except the highest, $ \textit{St}$ increases consistently until the end of the flare, with no discernible maxima. This lack of a peak suggests either a transitional reattaching boundary layer or the absence of reattachment. However, since off-wall measurements confirm reattachment, it indicates that a transitional boundary layer reattaches. This is corroborated by the experimental $ \textit{St}$ being between the laminar and turbulent values computed by VULCAN-CFD. For the highest $ \textit{Re}_{\infty }$ , the heating profile exhibits two peaks. The first peak, caused by compression, suggests that the boundary layer is likely turbulent at reattachment, especially since the heating magnitude closely matches the corresponding RANS computation. The second peak occurs farther downstream of reattachment at $x/L = 1.09$ , and is due to separation happening close to (but upstream of) the corner, where the separation and reattachment shocks interact (Running et al. Reference Running, Juliano, Borg, Jewell and Kimmel2019). This interaction, an Edney type VI (Edney Reference Edney1968), generates an expansion wave that impinges on the flare downstream of reattachment, reducing both density and heating. Discrepancies in reattachment heating and topological differences of the SWBLI between experiment and computation are discussed and analysed in § 6.4.

Nose radius affected the SWBLI through the modulated flow on the conical forebody. Figure 19 presents $ \textit{St}(x/L)$ for $ \textit{Re}_{\infty } = 12.0 \times 10^6$ m^–1 and nose radii of 5.00, 0.99 and 0.01 mm, along with laminar and turbulent VULCAN–CFD $ \textit{St}$ profiles for $r_{ {n}} = 0.99$ mm. The flow separated earliest for the 5.00 mm radius nose, thus delaying reattachment. This behaviour was consistent across all $ \textit{Re}_{\infty }$ , as higher nose bluntness stabilises the flow on the conical forebody (Brinich Reference Brinich1957). Bibin et al. (Reference Bibin, Srikanth, Ganesh and Vinayak2016) showed numerically that while leading-edge bluntness reduces separation in 2-D SWBLIs by producing a thicker entropy layer and stronger favourable pressure gradient, in axisymmetric flows the thinner entropy layer resulting from 3-D relieving effects increases separation length.

Figure 20.

Experimental density fields ( $r_{ {n}} = 0.99$ mm); (a) $ \textit{Re}_{\infty } =5.8\times 10^6$ m^–1, (b) $ \textit{Re}_{\infty } = 7.6\times 10^6$ m^–1, (c) $ \textit{Re}_{\infty } =10.0\times 10^6$ m^–1, (d) $ \textit{Re}_{\infty } = 12.0\times 10^6$ m^–1.

For all nose radii in figure 19, surface heating and off-wall measurements indicate laminar flow at separation and transitional flow at reattachment. For the sharp nose, $ \textit{St}$ rises upstream of the cone–flare junction, and since it remains below the turbulent computation, the boundary layer is inferred to be transitioning prior to the SWBLI. The stability analysis shows that the $N$ -factor exceeds 8 at the cone–flare junction for $ \textit{Re}_{\infty }=12.1\times 10^6\,\text{m}^{-1}$ (figure 8 b), above critical levels ( $N\approx 7$ ) typically associated with transition onset in conventional hypersonic facilities (Horvath et al. Reference Horvath, Berry, Hollis, Singer and Chang2002). For the sharp nose tip at $ \textit{Re}_{\infty } = 12.0\times 10^6$ m^–1, the BOS-identified separation location coincides with the streamwise $ \textit{St}$ plateau at $x/L = 0.98$ . A slight plateau followed by an abrupt $ \textit{St}$ increase is therefore taken as the surface-heating signature of transitional separation. A distinct heating peak at reattachment indicates that the flow is nearing the end of transition, as further supported by the close agreement with the turbulent RANS VULCAN–CFD heating magnitude.

6.2. Global off-wall density

Concurrent with IR thermography, BOS measured off-wall density and density-gradient fields provided high-spatial-resolution, non-intrusive, quantitative assessment of separation and reattachment locations (Davami et al. Reference Davami, Juliano, Moreto and Liu2025b , Reference Davamic ). Figure 20 shows normalised density fields for $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 to $12.0\times 10^6$ m^–1 for $r_{ {n}} = 0.99$ mm (see table 1). At each $ \textit{Re}_{\infty }$ , the bow shock enters at $y/L = 0.175$ , with $\rho /\rho _{\infty } \gt 1$ downstream. The separation shock is indicated by a rapid near-surface density rise ( $\rho /\rho _{\infty }\approx 2.0$ ), while the reattachment shock occurs near the flare surface ( $\rho /\rho _{\infty }\approx 3.0$ ), enclosing a low-density recirculation bubble. As $ \textit{Re}_{\infty }$ increases, the boundary layer on the conical forebody thins, bringing the shear layer closer to the surface, while the separation point shifts downstream, the reattachment point moves upstream, and the recirculation bubble shrinks, consistent with IR results.

Figure 21 presents normalised density fields computed from VULCAN-CFD for $r_{ {n}} = 0.99$ mm; figure 21(a) is a laminar base flow at $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and figure 21(b) is a turbulent RANS solution at $ \textit{Re}_{\infty } = 12.1\times 10^6$ m^–1. For $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1, the computed and experimental density fields agree excellently. Although the separation and reattachment locations differ slightly – because the experimental SWBLI is transitional, whereas the computation is fully laminar – the density magnitudes match well throughout the domain except very near the wall, due to unknown boundary conditions along the surface (Liu & Katz Reference Liu and Katz2006; Liu & Moreto Reference Liu and Moreto2021; Abassi, Wang & Liu Reference Abassi, Wang and Liu2025).

Figure 21.

VULCAN-CFD density fields ( $r_{ {n}} = 0.99$ mm). (a) Laminar, $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1. (b) Turbulent (RANS), $ \textit{Re}_{\infty } = 12.1\times 10^6$ m^–1.

As was discussed in § 6.1, for a fully laminar SWBLI, the separation bubble increases in size as $ \textit{Re}_{\infty }$ increases. Consequently, comparisons between laminar base flows and experimental density fields at higher $ \textit{Re}_{\infty }$ demonstrate worse agreement regarding separation. The turbulent VULCAN-CFD results reveal that, regardless of $ \textit{Re}_{\infty }$ , the adverse pressure gradient is insufficient to cause separation and the normalised density fields are virtually identical. As $ \textit{Re}_{\infty }$ increases in the experiment, the transition front shifts upstream of the cone–flare junction. Thus, the experimental density field progressively approaches the fully turbulent VULCAN-CFD solution.

6.3. Global comparison of surface and off-wall results

Surface and off-wall results are presented in figure 22, for time-averaged experimental measurements, laminar VULCAN-CFD base flow and DNS time averaged over the linear and saturated growth regimes, for $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. Stanton number distributions are the centre contour, with normalised density shown above and density-gradient magnitude below. The experimental peak $ \textit{St}$ exceeds the DNS predictions by more than a factor of two (note the different scales for the $ \textit{St}$ contours). The discrepancies in flare heating are expected to be due to fundamental differences in disturbance environments between experiments and simulations. This is consistent with previous SWBLI experiments, which show that reattachment heating under noisy conditions can exceed that in quiet flow by more than a factor of two (Schneider Reference Schneider2004; Simeonides Reference Simeonides2008; Chynoweth et al. Reference Chynoweth, Edelman, Gray., McKiernan and Schneider2017; Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2021; Leidy et al. Reference Leidy, Paredes, Scholten, Juliano and Davami2025). Discrepancies in separation length between experiment and simulation are also evident. Previous studies have demonstrated that free-stream noise strongly influences the separation and reattachment locations for transitional flare-induced SWBLIs (Threadgill et al. Reference Threadgill, Hader, Singh, Tsakagiannis, Fasel, Little, Lugrin, Bur, Chiapparino and Stemmer2024; Davami et al. Reference Davami2025c ; Leidy et al. Reference Leidy, Paredes, Scholten, Juliano and Davami2025).

Figure 22.

Surface and off-wall results ( $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm). (a) Experiment, (b) laminar base flow, (c) DNS (linear growth regime), (d) DNS (saturated growth regime).

Comparison of the global off-wall and surface results in figure 22 reveals good agreement between the separation shock foot location and the decrease in $ \textit{St}$ on the conical forebody. At reattachment, the shock foot impinges upstream of peak $ \textit{St}$ , with the maximum heating rates observed near the end of the flare. Thermal streaks are present on the flare, extending from within the separation bubble downstream to the reattachment shock and continuing to the end of the flare in both experiment and DNS.

Measured and computed normalised separation length $l/L$ (where $l\equiv x_{ {r}}-x_{ {s}}$ ) are shown in figure 23(a) as functions of $ \textit{Re}_{\infty }$ for all nose radii. Experimentally, a separation bubble was present for all $ \textit{Re}_{\infty }$ , with its mean length decreasing as $ \textit{Re}_{\infty }$ increased due to the downstream shift of the separation point. This trend indicated a transitional interaction, where the flow was laminar before separation and became transitional or turbulent upon reattachment at higher $ \textit{Re}_{\infty }$ . In contrast, the VULCAN-CFD computations were run with a fully laminar separation bubble, leading to an increase in separation length with increasing $ \textit{Re}_{\infty }$ as well as a larger separated region for a given $ \textit{Re}_{\infty }$ compared with the experiment. For both the experiment and computations, a blunter nose results in a larger separation bubble. Figure 23(b) presents the per cent difference between measured and computed separation lengths as a function of $ \textit{Re}_{\infty }$ for each nose radius.

Figure 23.

Measured and computed separation bubble comparison. (a) Separation length. (b) Per cent difference.

Davami et al. (Reference Davami2025c ) compared flight test data from the ROcket Technology Experiment on Transition (ROTEX-T) cone–flare vehicle with experiments from multiple conventional hypersonic facilities and computations using various computational fluid dynamics (CFD) codes. When the Reynolds number based on cone length was matched, both ground tests and flight exhibited consistent trends: increasing Reynolds number shifted the separation and reattachment points toward the cone–flare junction, shortening the separation region. Heating profiles also followed the experimental trends shown in figure 18. Leidy et al. (Reference Leidy, Paredes, Scholten, Juliano and Davami2025) and Davami et al. (Reference Davami, Leidy, Scholten, Juliano and Paredes2025d ) investigated ROTEX-T under quiet-flow conditions and observed similar trends. For $N$ -factors below the onset of transition, the separation bubble closely matches the laminar base flow predicted by computations. At the onset of transition, the experimentally measured topology of the SWBLI diverges appreciably from the computed laminar base flow.

The sensitivity of the mean flow – observed across quiet and noisy experimental facilities as well as in flight – results in substantial discrepancies between experimental observations and computational predictions. This emphasises the critical need for transitional SWBLI simulation models that extend beyond replicating fully laminar or turbulent states to accurately capture the intermediate transitional stages. These limitations in current computational approaches directly impact design and control strategies for high-speed vehicles equipped with control surfaces (Maughmer, Long & Pagano Reference Maughmer, Long and Pagano1992; Berry et al. Reference Berry, Horvath, Hollis, Thompson and Hamilton II2001; Horvath et al. Reference Horvath, Berry and Merski2004; Bur & Chanetz Reference Bur and Chanetz2009; Coleman & Faruqi Reference Coleman and Faruqi2009; Lee et al. Reference Lee, Zhao, Luo, Zou, Zhang, Zheng and Zhang2024).

6.4. Effect of inflow forcing

The experiments analysed in this work were conducted under conventional-noise conditions, with elevated free-stream noise introducing inflow and free-stream forcing. In contrast, the DNS were performed without external disturbances at the inflow or in the free stream, with instabilities developing solely from minimal numerical round-off perturbations. To examine the observed differences in heating rates along the flare and separation bubble topology, additional numerical simulations with inflow forcing disturbances were performed.

Here, we use the broadband inflow forcing approach by Cao et al. (Reference Cao, Hao and Guo2024) as a simple model to trigger convective instabilities in the laminar boundary layer, which (at least qualitatively) accounts for the effect of ambient disturbances in the experiment. The induced disturbances have been shown to have a broadband spectrum in both frequency and wavelength and therefore allow ‘naturally’ selected preferred modes (Cao et al. Reference Cao, Hao and Guo2024). The method uses the FORTRAN pseudo-random number generator to impose random spanwise-velocity perturbation ( $w'$ ) into the flow at a user-specified domain as

(6.1) \begin{align} w'_{i,j,k,n}/U_\infty =A(2r-1), \end{align}

where $r=\text{random}\_{\text{number()}}$ , $i$ , $j$ and $k$ are the grid points in the streamwise, azimuthal and wall-normal directions, respectively, $n$ represents the time step and $A$ is the specified amplitude of the perturbations and is referred to in percentage of the free-stream velocity $U_\infty$ in this paper. For the results presented here, spanwise-velocity disturbances are applied over a singular streamwise plane of $i=6$ (which is 6 grid points downstream from the inlet), and the perturbation is updated every $50$ time steps to ensure a sufficient time for the convection of disturbances through grid points. The amplitude $A$ is systematically altered from $0.625\,\%$ and $5\,\%$ to show the effect of forcing amplitude on boundary-layer instability growth $A_w$ (5.8) and surface heating $ \textit{St}$ . The simulations were performed using the same grid resolution as in the baseline case, but with a narrower span of $\phi =7.2^\circ$ (corresponding to $m=50$ ), thus $N_x \times N_\phi \times N_r = 740 \times 64 \times 540$ .

Figures 24 and 25 present $A_w$ and $ \textit{St}$ as a function of inflow forcing amplitude $A$ at $ \textit{Re}_{\infty } =5.8\times 10^6$ and $7.6\times 10^6$ m^–1, respectively. As expected, the addition of inflow forcing reduces the time it takes for boundary-layer instabilities to become saturated. However, the linear growth rate and post-saturation disturbance amplitude remain the same as in the zero-forcing case. These results show that by varying the amplitude of the broadband inflow forcing from $0.625\,\%$ to $5\,\%$ significantly increases the peak heating rate in the post-reattachment region, leading to a closer match with the experimental data. The DNS results reproduce experimental flare-heating trends, demonstrating that reattachment heating on the flare is highly sensitive to inflow disturbances. Accounting for inflow forcing in the simulations shows that inflow disturbance spectra directly determine the thermal loads, explaining the persistently elevated heating rates observed in prior studies under noisy flow conditions compared with quiet environments.

Figure 24.

Effect of inflow forcing on DNS for $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 with $r_{ {n}} = 0.99$ mm; (a) $A_w$ near reattachment ( $x/L = 1.05$ )., (b) $ \textit{St}$ .

Figure 25.

Effect of inflow forcing on DNS for $ \textit{Re}_{\infty } = 7.6\times 10^6$ m^–1 with $r_{ {n}} = 0.99$ mm; (a) $A_w$ near reattachment ( $x/L = 1.05$ ), (b) $ \textit{St}$ .

Note that for the non-zero-forcing cases, the overall characteristics of the laminar separation bubble such as the bubble size and the locations of separation and reattachment remain similar to those of the zero-forcing case. Previous studies have demonstrated that axisymmetric transitional separation bubbles contract with increasing $ \textit{Re}$ and that free-stream noise further contracts the bubble for a given $ \textit{Re}$ compared with quiet environments (Leidy et al. Reference Leidy, Paredes, Scholten, Juliano and Davami2025). Tsakagiannis et al. (Reference Davami, Juliano, Little, Duan, Ostoich, Blades, Riley, Benitez, Borg and Spottswood2025a ) captured this effect in DNS by performing a suite of controlled forcing simulations. By progressively increasing the amplitude of forcing on the dominant convective instability upstream of separation, the bubble eventually contracts in length. The different bubble behaviours suggest the importance of forcing characteristics in bubble topology. Another possible factor in the incomplete agreement between DNS and experiment is that the as-computed flare was extended to reduce the influence of the base wake on reattachment (Davami et al. Reference Davami, Leidy, Scholten, Juliano and Paredes2025d ; Leidy et al. Reference Leidy, Paredes, Scholten, Juliano and Davami2025).

7.1. Streak instability

Streamwise striations were experimentally observed on the flare for all $ \textit{Re}_{\infty }$ , indicating the presence of azimuthally periodic perturbations. Figure 26 presents instantaneous $ \textit{St}$ distributions, with the non-dimensionalised streamwise coordinate on the abscissa and the azimuthal angle ( $\phi$ ) on the ordinate. Transitional separation bubbles are highly sensitive to angle of attack (Davami et al. Reference Davami, Juliano, Scholten and Paredes2023b ); even $0.15^\circ$ can cause significant changes in spanwise heating at reattachment (Benitez et al. Reference Benitez, Esquieu, Jewell and Schneider2021; Leidy et al. Reference Leidy, Paredes, Scholten, Juliano and Davami2025). The spanwise variation in $ \textit{St}$ is minimal for all $ \textit{Re}_{\infty }$ in figure 26, indicating that the model was well aligned with respect to the free stream. For the cone–flare configuration, the streaks have smaller wavelengths and amplitudes than 2-D SWBLIs (Running et al. Reference Running, Juliano, Borg and Kimmel2020). Particularly for small turning angles ( $\lesssim 15$ °) and depending on the nose radius, the streaks may not be detectable by thermography due to a low signal-to-noise ratio (Benitez et al. Reference Benitez, Borg and Hill2024a ; Leidy et al. Reference Leidy, Paredes, Scholten, Juliano and Davami2025). Although lateral heat conduction may contribute to slight attenuation of the thermal striations, its influence is limited due to the short test durations (Running et al. Reference Running, Juliano, Borg and Kimmel2020). The streaks wavelength decreases as $ \textit{Re}_{\infty }$ increases, which is consistent with the GSA and DNS. The amplitude and downstream extent of the streaks also decrease as $ \textit{Re}_{\infty }$ increases because of the accelerated transition to turbulence. Recall that at $ \textit{Re}_{\infty } = 12.0\times 10^6$ m^–1, the flow was nearly turbulent at reattachment.

Figure 26.

Experimental $ \textit{St}$ distributions on the flare ( $r_{ {n}} = 0.99$ mm); (a) $5.8\times 10^6$ m^–1, (b) $7.5\times 10^6$ m^–1, (c) $10.0\times 10^6$ m^–1, (d) $12.0\times 10^6$ m^–1.

As a starting point, the case with $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm is examined for comparisons between experiment, DNS and GSA. This case exhibits the closest mean-flow agreement between experiment and computations among all conditions tested, with the separation bubble length differing by 26 % (recall figure 23 b). Quiet-flow experiments on this configuration indicated that, despite reattachment occurring approximately 23 % closer to the cone–flare junction than predicted by laminar base flows, the dominant instability frequencies and mode shapes matched closely provided that nonlinear activity at reattachment was not excessive (i.e. triadic interactions were confined to relatively low frequencies) (Davami et al. Reference Davami, Leidy, Scholten, Juliano and Paredes2025d ).

Figure 27 presents the corresponding heat streaks along the flare for experiment and time-averaged DNS for the linear and saturated growth regimes. For improved visualisation (here only), the streaks are enhanced by subtracting the $ \textit{St}$ distribution from an earlier steady-flow period, with contour levels adjusted to increase their visibility. As the DNS simulation evolves from the linear growth phase to saturation, global unstable modes induce spanwise modulation near reattachment, causing the flow to break down from axisymmetric to three-dimensional on the flare.

Figure 27.

Thermal streaks ( $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm). (a) Experiment, (b) DNS (linear), (c) DNS (saturated).

The streaks are initially characterised by quantifying the time-varying azimuthal wavenumber spectrum locally. Discrete Fourier transforms (DFTs) of $ \textit{St}$ along the azimuth locations were computed, yielding amplitudes as functions of wavenumber. The input to the DFT was detrended $ \textit{St}$ profiles extracted along $\phi = [-45^\circ , 45^\circ ]$ and averaged in the streamwise direction over approximately $2$ mm. For the experiment, this was performed without temporal averaging for each of 150 snapshots, a 0.42 s duration. The uncertainty quantification in wavenumber followed Running et al. (Reference Running, Juliano, Borg and Kimmel2020). For the DNS, the linear (300 snapshots, 0.27 ms) and saturated (4392 snapshots, 4.8 ms) growth regimes were analysed separately. The DFTs were calculated upstream and downstream of reattachment to investigate the evolution of streaks through the strong compression at reattachment.

Spectral maps of azimuthal wavenumber $m$ at $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 with $r_{ {n}} = 0.99$ mm are shown upstream (figure 28) and downstream (figure 29) of reattachment. Below each spectral map, the time-averaged amplitude is plotted as a function of the azimuthal wavenumber. For this case, GSA showed that the stationary global mode with $m = 50$ exhibited the largest growth rate. The locations analysed for the experiment are shifted upstream to maintain the same relative distance from reattachment as in the computations.

Figure 28.

Normalised spectral maps of azimuthal wavenumber (top) and amplitudes (bottom) upstream of reattachment; $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. (a) Experiment ( $x/L = 1.04$ ), (b) DNS linear ( $x/L = 1.05$ ), (c) DNS saturated ( $x/L = 1.05$ ).

Upstream of reattachment, the experiment displays a broad wavenumber spectrum spanning $m = 30$ –130, with a time-averaged dominant wavenumber of $m = 50 \pm 3$ (figure 28 a). The DNS confirms the dominance of $m = 50$ during the linear growth period (figure 28 b). The stationary global mode is markedly more energetic than the oscillatory modes identified by GSA, which reflects its large exponential growth rate. During the saturated regime, energy spreads across a wider wavenumber range with $m = 50$ being most energetic on average, consistent with the experiment (figure 28 c).

Downstream of reattachment, the experiment reveals two distinct steady wavenumbers: $m = 50 \pm 3$ and its harmonic $m=100 \pm 3$ (figure 29 a). In the DNS, the stationary global mode continues to dominate during linear growth (figure 29 b), even with the local analysis being conducted 41 mm downstream of its central location. As the flow in the DNS approaches saturation (figure 29 c), nonlinear interactions redistribute energy from the dominant wavenumber to its harmonic ( $m = 100$ ) between $t = 1.2$ and 1.5 ms. Nonlinear saturation of the global modes distributes energy across the spectrum $m = 30$ –150, with the time-averaged dominant wavenumber corresponding to both the fundamental and its harmonic, as observed in the experiment. The DNS, however, indicates that this far downstream of the global mode, the harmonic gradually begins to dominate over the fundamental. Although other wavenumbers exhibit some content in the experiment and saturated regime of the DNS, none are as dominant or persistent as the primary wavenumber of $m = 50$ or its harmonic.

Figure 29.

Normalised spectral maps of azimuthal wavenumber (top) and amplitudes (bottom) downstream of reattachment; $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. (a) Experiment ( $x/L = 1.11$ ), (b) DNS linear ( $x/L = 1.15$ ), (c) DNS saturated ( $x/L = 1.15$ ).

Figure 30.

Spectra of normalised POD energy from experiment and DNS and normalised growth rate from GSA ( $ \textit{Re}_{\infty } = 5.8\times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm).

To better understand the streamwise evolution of streaks originating from the stationary global unstable mode, the thermal striations on the flare were also characterised globally using Fourier decomposition coupled with proper orthogonal decomposition (POD), similar to the analysis by Lugrin, Beneddine & Garnier (Reference Lugrin, Beneddine and Garnier2021). The decomposition distinguishes correlated physical information from uncorrelated noise while capturing both spatial and temporal variations in wavenumber. Its input is the entire streamwise extent of the flare, unlike the spectral map analysis, which focused on a specific streamwise location. An azimuthally reconstructed mode shape of a wavenumber thereby highlights where that wavenumber is dominant. At each streamwise station, a DFT of detrended spanwise Stanton number was computed. For the experimental measurements, this was performed for approximately 105 frames, covering 0.3 s, and combined into a matrix of DFT amplitudes as functions of time, wavenumber, and the streamwise spatial coordinate. The POD of this matrix obtains modes for a given wavenumber. Since the decomposition is performed on the DFT output, the energy of a wavenumber is indicated by its highest-energy POD mode.

Figure 30 presents the POD energy ( $\lambda$ ) spectrum with a wavenumber resolution of 4. Each spectrum is normalised by its maximum energy to highlight the most dominant wavenumbers. For comparison, the growth rate ( $\sigma$ ) spectrum from GSA is also included and normalised the same way. The most energetic wavenumbers correspond closely to those previously identified in the spectral maps downstream of reattachment (figure 29) for both the experiment and DNS. The GSA-predicted dominant wavenumber ( $m=50$ ) aligns with the highest energy content in the linear growth regime of the DNS. The experiment and saturated DNS match closely: the fundamental and first harmonic dominate the flow globally, while other wavenumbers are moderately amplified. The greater energy at wavenumbers away from the fundamental in the saturated than linear DNS results suggests that saturation of the global modes drives much of the streak amplification. In the experiment, some energy may also arise from convective mechanisms, but under these conditions most of the streak amplification appears to result from global instabilities.

Figure 31 presents the azimuthally reconstructed $m = 50$ mode shape for $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. From GSA, the mode shape is represented by the azimuthally reconstructed real part of the streamwise heat-flux perturbation of the leading stationary disturbance (figure 31 b) provides the azimuthally reconstructed real part of the streamwise heat-flux perturbation of the leading stationary disturbance computed by GSA for the same condition. For each, a dashed black line marks the mean reattachment location. The mode shapes from the experiment, GSA and DNS are similar, each exhibiting neutral lines both upstream of and at reattachment, across which the heat-flux oscillations change sign. The first band of elevated amplitude is of much shorter streamwise extent than the second. The third (post-reattachment) band reaches peak amplitude only at the downstream end of the flare; a longer flare may experience continued amplification.

Figure 31.

Value of $||\boldsymbol{\nabla }\rho ||$ and surface-heating mode shape. Here, $m = 50$ , $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. (a) Experiment, (b) GSA and VULCAN-CFD, (c) DNS (linear), (d) DNS (saturated).

The magnitude of the density gradient is shown along the $x$ – $z$ plane in figure 31 to provide off-wall context – these streaks originate within the recirculation bubble and amplify downstream of the reattachment shock. Recall the GSA-computed off-wall mode shape for this case was presented in figure 11(a), which showed its centre is at $x/L = 1.05$ –1.06, matching the neutral line in the mode shape. That the neutral line is observed in the experimentally measured mode shape at $x/L = 1.035$ may suggest that the global instability is centred there – farther upstream than GSA predictions due to the smaller separation bubble in the experiment.

Like the present GSA and DNS, the high-fidelity computational studies of Hildebrand et al. (Reference Hildebrand, Dwivedi, Nichols, Jovanović and Candler2018), Sidharth & Candler (Reference Sidharth and Candler2018) and Lugrin et al. (Reference Lugrin, Beneddine and Garnier2022a ), show that global unstable modes in high-speed recirculation bubbles induce sign changes in the streamwise heat-flux instability modes. The good agreement between the mode shapes, along with the similarity in the most amplified wavenumbers predicted by GSA and DNS, provides evidence that global instability of the separation bubble exists and plays a key role in the formation of the reattachment streaks for this flow.

As shown in figure 30, the $m=100$ harmonic also exhibits considerable energy content in the POD energy spectra for both the experiment and saturated regime of DNS. Figure 32 displays its azimuthally reconstructed POD mode shape. The harmonic exhibits two dissimilarities from the fundamental: the harmonic does not have a neutral line and sign change at reattachment, and its oscillations have low amplitude until much closer to reattachment.

Figure 32.

Value of $||\boldsymbol{\nabla }\rho ||$ and surface-heating mode shape. Here, $m = 100$ , $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. (a) Experiment, (b) DNS (saturated).

To determine whether the global instability of the separation bubble dominates at higher free-stream unit Reynolds numbers and nose radii, Fourier decomposition coupled with POD was performed on the thermographs for various $ \textit{Re}_{\infty }$ . Figure 33 presents the energy spectrum as a function of wavenumber for $r_{ {n}} = 0.99$ mm. As $ \textit{Re}_{\infty }$ increases, the most energetic azimuthal wavenumber also increases, consistent with GSA and DNS predictions. At low $ \textit{Re}_{\infty }$ (figure 33 a), energy is concentrated in a few wavenumbers, whereas at high $ \textit{Re}_{\infty }$ (figure 33 b), it is distributed across many more. Convective mechanisms excite a broad range of wavenumbers (Dwivedi et al. Reference Dwivedi, Sidharth, Nichols, Candler and Jovanovic2019; Lugrin et al. Reference Lugrin, Nicolas, Severac, Tobeli, Beneddine, Garnier, Esquieu and Bur2022b ; Davami et al. Reference Davami, Leidy, Scholten, Juliano and Paredes2025d ). At higher $ \textit{Re}_{\infty }$ , turbulence near reattachment and nonlinear interactions across frequencies and scales broaden the energetic wavenumber distribution.

Figure 33.

Normalised spectral-POD wavenumber energy from experiment ( $r_{ {n}} = 0.99$ mm); (a) $ \textit{Re}_{\infty } = 7.6 \times 10^6$ to $ 8.5 \times 10^6$ m^–1, (b) $ \textit{Re}_{\infty } = 10.0 \times 10^6$ to $ 12.0 \times 10^6$ m^–1.

Figure 34 compares dominant azimuthal wavenumbers from experiment and the linear growth regime of DNS with the wavenumbers with the largest growth rate computed from GSA for varying $ \textit{Re}_{\infty }$ and nose radii. For both experiments and GSA, as $ \textit{Re}_{\infty }$ increases, the dominant azimuthal wavenumber increases. Furthermore, at a given $ \textit{Re}_{\infty }$ , a blunt nose yields a smaller dominant wavenumber than the sharper noses. This is likely due to a blunter nose resulting in a more stable flow on the conical forebody, leading to a larger separation bubble, and delayed transition in the reattachment region. This explains why past studies of flare-induced SWBLIs with relatively sharp noses have reported striations that are challenging to detect: the earlier onset of turbulence and higher-wavenumber striations necessitate high-spatial-resolution measurements.

Figure 34.

Dominant azimuthal wavenumbers from GSA, DNSs and experiment.

The dominant wavenumbers from experiment agree excellently with those from GSA – no more than 7 % different, which is within the measurement precision – except for sharper noses at $ \textit{Re}_{\infty } \gt 10.0 \times 10^6$ m^–1. There are two reasons that the concordance between GSA and experiment would be worse for this case. First, recalling figure 9, GSA predicted rather uniform growth rates for wavenumbers near the maximum growth for sharp noses at $ \textit{Re}_{\infty }=12.1\times 10^6$ m^–1. Growth rates differed by only 2.5 % for wavenumbers from $m=60$ to 85, a wide range of wavenumbers with growth rates very close to the maximum. Second, although all $ \textit{Re}_{\infty }$ examined in this study resulted in a separation bubble, at higher $ \textit{Re}_{\infty }$ and for sharper nose radii, the separation bubble becomes significantly smaller, indicating the transition front is only slightly downstream of the cone–flare junction. Under these conditions, the flow enters a nonlinear regime where linear GSA is no longer valid. However, the excellent agreement among GSA, DNS, and experiment for the lower $ \textit{Re}_\infty$ cases is yet more evidence strongly indicating that the global instability within the separation bubble causes the heating streaks at reattachment for this configuration.

7.2. Off-wall dynamics

The dynamics of the separation bubble was analysed globally by applying spectral-POD (SPOD) to the high-speed BOS measurements and DNS. It is a data-driven method that extracts coherent structures from flow field measurements by considering both spatial and temporal modes, unlike traditional POD, which only analyses spatial modes (Schmidt & Shepherd Reference Schmidt and Shepherd2020). Using a space–time inner product, SPOD decomposes flow snapshots into energetically optimal modes at distinct frequencies, with leading modes generally corresponding to coherent spatial structures (Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018).

Figure 35 presents the normalised SPOD mode energy spectrum for $r_{ {n}} = 0.99$ mm and various $ \textit{Re}_{\infty }$ . The decomposition was applied to the BOS-measured density gradient, computed from 31 blocks of 128 snapshots with 50 % overlap, resolving frequencies up to 15 kHz (Nyquist limit). The spectrum reveals the distribution of energy across spatio-temporal modes, emphasising the dominant frequencies and spatial patterns that drive the dynamics of the system. As explained by Schmidt & Shepherd (Reference Schmidt and Shepherd2020), a significant separation between the first and second SPOD modes indicates the dominance of a particular mechanism. For this flow, the primary mechanism is unsteadiness along the shear layer and at reattachment. The first and second modes in the spectrum are separated in the frequency range of 1 to 5 kHz, with the gap widening as $ \textit{Re}_{\infty }$ increases.

Although the stationary global unstable mode was detected both experimentally and in simulations by leveraging spatial spectral analysis of the surface heat flux, extracting it from off-wall temporal measurements is profoundly challenging, because it appears at zero frequency and is thus dominated by the mean flow and experimental noise. Oscillatory global unstable modes should be detectable experimentally; however, they are primarily centred within the separation bubble, where BOS signal levels are low. Consequently, their unsteady influence is mainly near reattachment, where density-gradient fluctuations are stronger. Note also that in the experiment, the influences of convective amplification and global instability mechanisms are difficult to isolate (Davami et al. Reference Davami, Leidy, Scholten, Juliano and Paredes2025d ).

Figure 35.

Normalised SPOD energy spectra of high-speed BOS measurements ( $r_{ {n}} = 0.99$ mm); (a) $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1, (b) $ \textit{Re}_{\infty } = 8.5 \times 10^6$ m^–1, (c) $ \textit{Re}_{\infty } = 12.0 \times 10^6$ m^–1.

Figure 36.

Real part of the leading SPOD mode from experiment and DNS, and the stationary leading and secondary oscillatory global unstable modes from GSA ( $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm); (a) GSA: $m = 50$ and $f = 0$ kHz, (b) GSA: $m = 53$ and $f = 4.8$ kHz, (c) DNS: $f = 1$ –5 kHz (SPOD), (d) experiment: $f = 1$ –5 kHz (SPOD).

Figure 36 displays the real part of the leading SPOD mode from BOS measurements and DNS, which were both observed consistently from 1 to 5 kHz, and the real part of the density-gradient perturbation associated with the leading stationary and secondary oscillatory global unstable mode identified through GSA for $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. The ordinate is the normalised wall-normal coordinate, and the white line indicates the boundary-layer edge for attached flow or the shear-layer location for separated flow. This location is determined as the wall-normal height corresponding to the peak vertical density gradient, a criterion validated in Davami et al. (Reference Davami2025c ).

The four results in figure 36 exhibit a similar spatial structure, with anti-correlated lobes at the reattachment region. The agreement in frequency and spatial structure suggests that the oscillatory global mode was detected experimentally. The stationary global mode ( $f=0$ kHz) reflects the spatial structure of the reattachment region and indicates a persistent flow pattern on which the oscillatory instabilities are superimposed. The other global oscillatory unstable modes from GSA (not shown) display nearly identical perturbation modes localised at reattachment. Note that the line-of-sight integration inherent to BOS, combined with the finite aperture and depth of field, distorts the experimentally measured mode shape.

Similar low-frequency modes – previously thought to arise from mainly convective mechanisms near reattachment – have been reported for this geometry in multiple high-speed facilities, both noisy and quiet (Davami & Juliano Reference Davami and Juliano2024; Benitez et al. Reference Benitez, Borg, Hill, Davami and Juliano2024b ; Davami et al. Reference Davami, Juliano, Little, Duan, Ostoich, Blades, Riley, Benitez, Borg and Spottswood2025a , Reference Davami, Leidy, Scholten, Juliano and Paredesd ). Additionally, for this geometry and flow conditions, off-wall convective mechanisms at higher frequencies (25–40 kHz), including shear-layer instabilities and shock-layer waves, have been measured in the separation region and at reattachment (Benitez et al. Reference Benitez, Borg, Scholten, Paredes and Jewell2023; Davami et al. Reference Davami and Juliano2024a , Reference Davami2024b ; Davami et al. Reference Davami, Juliano, Little, Duan, Ostoich, Blades, Riley, Benitez, Borg and Spottswood2025a , Reference Davamic , Reference Davami, Leidy, Scholten, Juliano and Paredesd ), along with significantly higher-frequency second-mode waves (200–290 kHz) (Butler & Laurence Reference Butler and Laurence2021, Reference Butler and Laurence2022; Sousa & Laurence Reference Sousa and Laurence2025).

7.3. Comparison of surface and off-wall dynamics

Figure 37 presents unsteady, simultaneously measured, surface and off-wall results for $ \textit{Re}_{\infty } = 5.8 \times 10^6$ to $12.0 \times 10^6$ m^–1 and $r_{ {n}} = 0.99$ mm. The centre contour shows the azimuthally reconstructed POD mode shape of the most energetic azimuthal wavenumber (POD of $\mathcal{F}(St')$ ). Above it, the high-resolution BOS-measured magnitude of the density gradient ( $||\boldsymbol{\nabla }\rho ||$ ) is shown for off-wall context of the mean flow, while the bottom contour displays the leading SPOD mode shape from high-speed BOS measurements of the density gradient (SPOD $\boldsymbol{\nabla }\rho$ ). Included for reference in the off-wall mode shapes are the shear-layer edge (grey dashed line) and the separation shock (grey solid line).

Figure 37.

Simultaneous, global experimental measurements of the unsteady-flow dynamics ( $r_{ {n}} = 0.99$ mm): (a) $5.8\times 10^6$ m^–1, $m = 50$ and $f_{\textit{SPOD} \boldsymbol{\nabla }\rho } = 1-5$ kHz; (b) $7.6\times 10^6$ m^–1, $m = 58$ and $f_{\textit{SPOD} \boldsymbol{\nabla }\rho } = 1-5$ kHz; (c) $8.5\times 10^6$ m^–1, $m = 63$ and $f_{\textit{SPOD} \boldsymbol{\nabla }\rho } = 0.5-6$ kHz; (d) $10.0\times 10^6$ m^–1, $m = 71$ and $f_{\textit{SPOD}\boldsymbol{\nabla }\rho } = 0.5-6$ kHz; (e) $12.0\times 10^6$ m^–1, $m = 82$ and $f_{\textit{SPOD}\boldsymbol{\nabla }\rho } = 0.5{-}7$ kHz.

The experimentally measured streamwise heat-flux perturbations in figure 37 extend from within the recirculation bubble and amplify downstream of reattachment. Furthermore, near reattachment, they exhibit a sign change, and the neutral line shifts upstream with reattachment as $ \textit{Re}_{\infty }$ increases. At $ \textit{Re}_{\infty } = 7.6 \times 10^6$ m⁻¹, the azimuthally reconstructed real part of the streamwise heat-flux perturbation of the leading stationary disturbance computed by GSA, resembles that observed at $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1 (figure 31 b). However, the first band of elevated amplitude, which is induced by the global instability, spans a much shorter streamwise distance – approximately one-third the length. The low heat-flux magnitude upstream of reattachment, combined with the diminished extent of this band, may explain why POD for the higher $ \textit{Re}_{\infty }$ cases does not capture the sign change in the heat-flux perturbations associated with the global instability in that region.

Like for $ \textit{Re}_{\infty } = 5.8 \times 10^6$ m^–1, at $ \textit{Re}_{\infty } = 7.6 \times 10^6$ and $8.5 \times 10^6$ m^–1, the harmonics of the dominant surface-heating wavenumbers also contain elevated energy in the POD energy spectrum. Also like at lower $ \textit{Re}_{\infty }$ (figure 32 a), other higher harmonics amplify near the end of the flare. At $ \textit{Re}_{\infty } = 10.0 \times 10^6$ and $12.0 \times 10^6$ m^–1, the streaks experience less amplification because transition to turbulence occurs farther upstream. At the highest $ \textit{Re}_{\infty }$ , the dominant wavenumber has high amplitude only near reattachment (figure 37 e), while its harmonic and others with significant energy amplify farther downstream (not shown).

The SPOD modes of the off-wall density gradient (i.e. SPOD( $\boldsymbol{\nabla }\rho$ ), lower contour) exhibit an alternating fluctuation pair straddling the shear layer and centred near reattachment. The frequency range over which these modes appear broadens with increasing $ \textit{Re}_{\infty }$ , reflecting elevated unsteadiness along the shear layer and around the reattachment region as the flow transitions. For each $ \textit{Re}_{\infty }$ , the sign changes in the off-wall oscillations correspond closely to those of the streamwise heat-flux instability modes. Since heat flux scales with density (Running et al. Reference Running, Juliano, Borg and Kimmel2020; Davami et al. Reference Davami2025c ), these coupled oscillations shift upstream in tandem with the reattachment location as $ \textit{Re}_{\infty }$ increases.