Axisymmetric Cone-Cylinder-Flare Configurations

• Axisymmetric cone-cylinder-flare configurations are piecewise-axisymmetric geometries that integrate conical, cylindrical, and flared sections to study diverse instability and transition mechanisms.
• They enable detailed analysis of hypersonic stability, boundary-layer evolution, and shock-induced separation using frameworks like LST, PSE, and global resolvent methods.
• Both external and internal configurations illustrate how variations in flare angle and nose bluntness critically affect shock structures, separation, and heat transfer dynamics.

Axisymmetric cone-cylinder-flare configurations are piecewise-axisymmetric geometries comprising a conical section, a cylindrical section, and a downstream flare or diffuser. In current research, the term covers both external hypersonic bodies designed to combine forebody boundary-layer development with aft recompression or separation, and internal axisymmetric expansions connecting cylindrical ducts through conical or smoothly flared walls (Caillaud et al., 2023, González et al., 2012). Their common technical significance is that they place canonical cone physics, near-cylindrical development, and flare-induced pressure-gradient effects within a single axisymmetric setting, making them useful for stability analysis, transition studies, and reduced inviscid modeling (Esquieu et al., 12 Sep 2025).

1. Geometric archetype and canonical variants

For external hypersonic studies, the canonical modern family consists of slender axisymmetric bodies with a nearly sharp nose, a $5^\circ$ cone, a cylindrical mid-body, and a downstream flare whose angle determines whether the aft interaction remains attached or separates. The review literature identifies two primary configurations: CCF3-5, designed to remain attached under recompression, and CCF10, designed to produce a laminar separation bubble; both share the same nose, cone, and cylinder and differ only in flare angle (Esquieu et al., 12 Sep 2025). The shared geometry is specified by a nose radius $R_n = 0.1\ \text{mm}$, half-cone angle $5^\circ$, a cylinder section of $0^\circ$ local angle, an overall length constraint up to about $800\ \text{mm}$, and a maximum diameter constraint up to about $115\ \text{mm}$; the main junction coordinates are $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$ at the nose-cone junction, $(398.15,\ 34.93)\ \text{mm}$ at the cone-cylinder junction, and $(525.67,\ 34.93)\ \text{mm}$ at the cylinder-flare junction (Esquieu et al., 12 Sep 2025).

Configuration	Shared forebody	Flare role
CCF3-5	$R_n=0.1\ \text{mm}$, $R_n = 0.1\ \text{mm}$0 cone, common cylinder	$R_n = 0.1\ \text{mm}$1 flare, attached recompression
CCF10	Same as CCF3-5	$R_n = 0.1\ \text{mm}$2 flare, laminar separation bubble
CCF12	Outgrowth of CCF3-5/CCF10	$R_n = 0.1\ \text{mm}$3 flare, stronger separated-flow study

The global-stability study of the hypersonic CCF10 uses the same basic archetype and names the model by its flare angle, $R_n = 0.1\ \text{mm}$4, with a $R_n = 0.1\ \text{mm}$5 cone and two nose options: an effectively sharp case with $R_n = 0.1\ \text{mm}$6 mm and a blunt case with $R_n = 0.1\ \text{mm}$7 mm (Caillaud et al., 2023). In that literature, the cone establishes the forebody instability environment, the cylinder continues or modifies the boundary-layer development, and the flare imposes adverse pressure gradient and possible shock-induced separation.

A second, internally focused usage appears in the inviscid diffuser literature, where an axisymmetric expansion joins an upstream cylinder of radius $R_n = 0.1\ \text{mm}$8 to a downstream cylinder of radius $R_n = 0.1\ \text{mm}$9 over $5^\circ$0 with wall $5^\circ$1. Two explicit profiles are considered: a conical profile,

$5^\circ$2

and a curved profile,

$5^\circ$3

where $5^\circ$4. The first is exactly a cone or diffuser, and the second is a smooth flare-like transition (González et al., 2012). This internal formulation is directly relevant because it isolates how conical and smoothly flared axisymmetric transitions behave when they connect two cylindrical sections.

2. Sectionwise flow physics on hypersonic cone-cylinder-flare bodies

The design philosophy of the hypersonic CCF family is explicitly sectional. The cone is intended to reproduce a canonical sharp-cone boundary layer with strong second-mode growth; the cone-cylinder junction generates an expansion and favorable pressure gradient; the cylinder provides an expanded, relatively stabilizing region; and the flare introduces recompression, which can either re-amplify disturbances while remaining attached or generate separated shock/boundary-layer interaction (Esquieu et al., 12 Sep 2025).

On the cone, the review identifies a quasi-constant conical pressure field, a thin boundary layer, edge Mach number about $5^\circ$5, and strong favoring of second-mode instability. At the cone-cylinder junction, the geometric expansion causes a favorable pressure gradient, boundary-layer acceleration, a rapid increase in boundary-layer thickness, and strong damping or detuning of second-mode waves that were amplified on the cone. The cylinder therefore acts as a stabilizing section even though the edge Mach number rises to about $5^\circ$6 (Esquieu et al., 12 Sep 2025).

The flare reverses that trend. For CCF3-5, the incoming boundary layer withstands the adverse pressure gradient: the flow remains attached, skin friction approaches zero at the junction but does not become negative, the boundary layer initially thins on the flare, and a narrow frequency band of second-mode waves is strongly re-amplified downstream. For CCF10, the adverse pressure gradient is too strong: the boundary layer separates near the cylinder-flare junction, a large laminar separation bubble forms, an induced separation shock appears near separation, pressure rises into a plateau, skin friction becomes negative over the separated region, a recompression or reattachment shock forms downstream, and wall heating rises strongly near reattachment where the boundary layer is thinnest (Esquieu et al., 12 Sep 2025).

The Mach-6 global-stability study resolves that separated base state in more detail. At $5^\circ$7, $5^\circ$8, and $5^\circ$9, the laminar base flow over CCF10 includes a detached shock, expansion waves, separation and reattachment shocks, and a recirculation region at the cylinder-flare junction; relative to the sharp nose, the blunt nose produces a much thicker entropy layer from the detached bow shock, a longer separation bubble, reduced edge Mach number $0^\circ$0, and a thicker boundary layer upstream of the flare (Caillaud et al., 2023).

A persistent misconception is that a CCF body can be treated as a cone with an aft appendage. The literature instead treats the geometry as a deliberately coupled sequence of instability environments. A plausible implication is that downstream flare behavior cannot be inferred from cone-only stability results without accounting for the cone-cylinder expansion, the cylinder’s stabilizing role, and the flare-induced pressure rise.

3. Stability, receptivity, and transition analysis

Two distinct linear frameworks dominate the external CCF literature. For attached-flow configurations such as CCF3-5, the review employs Linear Stability Theory (LST) and linear Parabolized Stability Equations (PSE). LST is used under the parallel-flow assumption to obtain local spatial growth rates, modal frequencies, wave-angle dependence, and starting conditions for PSE; PSE then retains weak streamwise evolution and computes disturbance evolution, integrated growth, and $0^\circ$1-factors (Esquieu et al., 12 Sep 2025). The review gives the PSE amplification measure as

$0^\circ$2

and correlates transition with the integrated growth of linear instability waves through the $0^\circ$3 framework (Esquieu et al., 12 Sep 2025).

For CCF3-5 at Mach 6, zero angle of attack, high edge Mach number, and cold wall, the dominant instability is the second mode of Mack, most amplified for two-dimensional waves $0^\circ$4, while the first mode is lower frequency, oblique, and less amplified over the full geometry (Esquieu et al., 12 Sep 2025). On the cone, measured and computed second-mode peaks agree well: in conventional noise, measured peaks are around $0^\circ$5–$0^\circ$6 at $0^\circ$7 and $0^\circ$8–$0^\circ$9 at $800\ \text{mm}$0; in quiet flow at $800\ \text{mm}$1, measured cone peaks are around $800\ \text{mm}$2–$800\ \text{mm}$3, and computed $800\ \text{mm}$4 values reach about $800\ \text{mm}$5 and $800\ \text{mm}$6 at the two cone sensor stations (Esquieu et al., 12 Sep 2025). On the attached flare, measured peaks near $800\ \text{mm}$7 agree with predicted amplification in the $800\ \text{mm}$8–$800\ \text{mm}$9 band, with $115\ \text{mm}$0-factors about $115\ \text{mm}$1 and $115\ \text{mm}$2 at the two flare sensor locations and approaching $115\ \text{mm}$3 near the end of the model (Esquieu et al., 12 Sep 2025).

For separated-flow CCF10, local LST is explicitly judged unsuitable because the recirculation bubble, strong streamwise gradients, reversed flow, shear-layer growth, and shock interaction make the flow highly non-parallel (Esquieu et al., 12 Sep 2025). The corresponding Mach-6 study therefore adopts a global linear stability and resolvent framework. The steady laminar base flow is a fixed point of the compressible Navier-Stokes equations computed with the BROADCAST toolbox through a coarse non-shock-aligned solution, shock extraction, a fine shock-aligned mesh, interpolation, and pseudo-transient continuation or pseudo-Newton iterations until

$115\ \text{mm}$4

(Caillaud et al., 2023). Perturbations are decomposed as

$115\ \text{mm}$5

leading to the global eigenproblem

$115\ \text{mm}$6

and the resolvent operator

$115\ \text{mm}$7

with Chu compressible disturbance energy used as the response norm (Caillaud et al., 2023).

That analysis shows two instability classes on CCF10: convective disturbances along the cone and downstream boundary layer, including first mode, second mode, and steady streaks; and global instabilities associated with the separated flow at the cylinder-flare junction, including steady and weakly unsteady three-dimensional bubble modes (Caillaud et al., 2023). For the sharp nose, the strongest full-geometry resolvent peaks occur near $115\ \text{mm}$8 for the first mode, $115\ \text{mm}$9 for the second mode, and $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$0 for steady streaks; on the cone-only domain, the second mode is strongest, but over the full geometry the first mode becomes the dominant resonance peak (Caillaud et al., 2023). The paper also states that by the end of the flare in the sharp case, the first-mode and second-mode waves have comparable energetic content, so both should be considered in realistic transition scenarios (Caillaud et al., 2023).

Bluntness changes that landscape. In the blunt $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$1 mm case, the sharp first-mode and second-mode peaks vanish; only a strong steady-streak peak remains, with maximum gain

$(x,y) = (0.0920,\ 0.0997)\ \text{mm}$2

at

$(x,y) = (0.0920,\ 0.0997)\ \text{mm}$3

and frequencies above about $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$4 kHz show only weak broadband non-normal amplification (Caillaud et al., 2023). The authors relate this to transition reversal on blunt hypersonic cones and identify an entropy-layer-supported forcing and response structure rather than a standard first- or second-mode boundary-layer mechanism (Caillaud et al., 2023).

4. Loss of axial symmetry and separated-flow unsteadiness

Geometric axisymmetry does not imply flow axisymmetry. This point appears in both CCF-specific studies and in related cone and conical-compression analyses. In the Mach-6 CCF10 global study, the unstable global modes all have $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$5, so the dominant intrinsic instabilities of the cylinder-flare recirculation bubble are non-axisymmetric and three-dimensional even though the base flow is axisymmetric and the angle of attack is zero (Caillaud et al., 2023).

A stronger warning comes from the Mach-16 study of single and double cones. There, the strongest amplification occurs for the non-axisymmetric azimuthal wavenumber $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$6, close to the cone tip, because the conical shock lies close to the viscous shear layer; in the separated double-cone interaction, the axisymmetric and fully three-dimensional DSMC solutions differ in almost all of the main flow structures, with the three-dimensional solution giving a smaller separation bubble, weaker shocks, and differences in heat transfer and streamwise skin friction on the order of $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$7 for instantaneous time snapshots (Karpuzcu et al., 2024). This supports the CCF inference that a nominally axisymmetric flare interaction may inherit upstream non-axisymmetric content from the forecone, not merely generate asymmetry locally.

Related conical-shock / axisymmetric-boundary-layer work at $(x,y) = (0.0920,\ 0.0997)\ \text{mm}$8 reaches a complementary conclusion. In a cone-in-tube configuration, a windward-plane-matched axisymmetric surrogate overpredicts local separation severity relative to the fully three-dimensional case because the 3D interaction supports side overflow and circumferential pressure relief. The study identifies a Mach reflection-like event and a Mach stem-like structure above the front of the separation bubble, and its empirical fit for the circumferentially unseparated region,

$(x,y) = (0.0920,\ 0.0997)\ \text{mm}$9

implies a critical angle of attack of approximately $(398.15,\ 34.93)\ \text{mm}$0 at which that length tends to zero (Weng et al., 2023). Although this is an internal shock-generator problem rather than an external CCF body, it directly informs the interpretation of axisymmetric conical compression, separation topology, and azimuthal nonuniformity.

Low-frequency global unsteadiness appears in an even more extreme limiting geometry. The Mach-6 spike-cylinder study, treated as the $(398.15,\ 34.93)\ \text{mm}$1 limit of an axisymmetric double cone, interprets pulsation as a self-sustained compression/expansion cycle of the separated-flow system rather than continuous mass feeding by the Edney jet. The pulsation Strouhal number

$(398.15,\ 34.93)\ \text{mm}$2

increases monotonically with $(398.15,\ 34.93)\ \text{mm}$3, and for $(398.15,\ 34.93)\ \text{mm}$4 the measured value is $(398.15,\ 34.93)\ \text{mm}$5; the maximum instantaneous pressure is nearly geometry-invariant over the tested range, with $(398.15,\ 34.93)\ \text{mm}$6 as the mean across six cases (Das et al., 1 Jun 2026). Because the spike-cylinder is presented as a limiting cone-cylinder-shoulder configuration, this suggests that sufficiently strong flare or shoulder interactions on axisymmetric bodies may support geometry-controlled global breathing modes in addition to convective instability waves.

A common misconception is therefore that the primary transition problem on a CCF is always an axisymmetric second-mode problem. The literature instead shows at least three distinct routes: canonical cone-mode amplification, non-axisymmetric global bubble modes at the cylinder-flare junction, and low-frequency separated-flow pulsation in strongly separated limiting geometries.

5. Internal diffuser and expansion formulations

In internal-flow usage, the cone-cylinder-flare concept appears as an axisymmetric diffuser joining two cylindrical ducts. The flow is assumed steady, inviscid, and axisymmetric, governed by the steady Euler equations recast as the Bragg-Hawthorne equation. The inlet is prescribed as solid-body rotation plus uniform axial flow,

$(398.15,\ 34.93)\ \text{mm}$7

with streamfunction

$(398.15,\ 34.93)\ \text{mm}$8

and Rossby number

$(398.15,\ 34.93)\ \text{mm}$9

(González et al., 2012).

Using the inlet invariants

$(525.67,\ 34.93)\ \text{mm}$0

the Bragg-Hawthorne equation is solved under a quasi-cylindrical approximation, meaning

$(525.67,\ 34.93)\ \text{mm}$1

relative to the radial and source terms (González et al., 2012). This generalizes the Batchelor-type cylindrical solution by writing

$(525.67,\ 34.93)\ \text{mm}$2

with

$(525.67,\ 34.93)\ \text{mm}$3

In nondimensional variables, the resulting velocity field is

$(525.67,\ 34.93)\ \text{mm}$4

$(525.67,\ 34.93)\ \text{mm}$5

$(525.67,\ 34.93)\ \text{mm}$6

(González et al., 2012).

The main structural result is that the transition-region solution has Beltrami flow structure: $(525.67,\ 34.93)\ \text{mm}$7 or nondimensionally

$(525.67,\ 34.93)\ \text{mm}$8

(González et al., 2012). The flow is thus the superposition of uniform axial translation, solid-body rotation, and a Beltrami field induced by the axisymmetric expansion.

The critical condition is centerline stagnation at the outlet: $(525.67,\ 34.93)\ \text{mm}$9 which yields

$R_n=0.1\ \text{mm}$0

and defines the critical Rossby number by

$R_n=0.1\ \text{mm}$1

(González et al., 2012). The reported values are $R_n=0.1\ \text{mm}$2 for $R_n=0.1\ \text{mm}$3, $R_n=0.1\ \text{mm}$4 for $R_n=0.1\ \text{mm}$5, and $R_n=0.1\ \text{mm}$6 for $R_n=0.1\ \text{mm}$7, so $R_n=0.1\ \text{mm}$8 increases with expansion ratio $R_n=0.1\ \text{mm}$9 (González et al., 2012).

An important point, directly relevant to cone-cylinder-flare transitions, is that within this quasi-cylindrical approximation the key results do not depend on the chosen wall profile: the conical and smooth curved profiles give the same critical stagnation condition because the outlet condition depends only on $R_n = 0.1\ \text{mm}$00 (González et al., 2012). This does not establish contour-independence in viscous, separated, or compressible flows; it states profile-insensitivity only within a steady inviscid slow-variation model. The same paper also emphasizes that the solutions do not branch off in this whole-pipe solid-body-rotation case, in contrast with earlier Rankine-core problems where folds occurred (González et al., 2012).

6. Complementary analytical frameworks and limiting assumptions

Several adjacent theories supply local or rigorous structure without constituting a full CCF model. For cylindrical sections, incompressible axisymmetric boundary-layer integral analysis shows explicitly how transverse curvature modifies displacement and momentum thickness definitions and raises skin friction above planar estimates. In zero pressure gradient,

$R_n = 0.1\ \text{mm}$01

while under pressure gradient the paper derives curvature-corrected relations for $R_n = 0.1\ \text{mm}$02, $R_n = 0.1\ \text{mm}$03, and separation-limit behavior (Kumar et al., 2018). This is directly applicable to the constant-radius cylinder section and only qualitatively informative for a true flare, because the derivation assumes fixed radius $R_n = 0.1\ \text{mm}$04.

For cone sections, incompressible global stability analysis of the circular-cone boundary layer shows that the least stable mode is the helical mode $R_n = 0.1\ \text{mm}$05, that all studied modes are temporally stable but convectively unstable, and that increasing semi-cone angle makes the modes more temporally stable while increasing downstream spatial amplification (Vinod et al., 2016). This suggests that even before a cylinder or flare is encountered, the cone can pre-amplify low-order helical content that later enters the downstream interaction region.

On the mathematical side, rigorous compressible and incompressible axisymmetric theories exist for partial subproblems but not for the full external hypersonic CCF body. In a straight cylinder with swirl and vorticity, a transonic shock can be treated as a free boundary in the steady full Euler equations using a Helmholtz decomposition,

$R_n = 0.1\ \text{mm}$06

with transport of entropy $R_n = 0.1\ \text{mm}$07 and angular momentum density $R_n = 0.1\ \text{mm}$08 along streamlines (Park et al., 2019). For a cone-only incompressible Navier-Stokes domain with Navier-Hodge-Lions slip boundary condition, global bounded strong solutions are obtained under even-odd-odd symmetry and the partial smallness condition

$R_n = 0.1\ \text{mm}$09

with no size restriction on the meridional components (Li et al., 2022). These results are not CCF theories; they show that axisymmetric cone or cylinder segments admit geometry-adapted analytical treatments, whereas a full cone-cylinder-flare body introduces junctions, changing wall angle, and mixed attached/separated regimes that such theories do not yet unify.

The dominant approximations in the present literature are therefore sectional and regime-specific. External attached-flow work relies on LST and PSE; separated-flow CCF analysis uses global eigenmodes and resolvent methods about a laminar base flow; internal expansion work uses steady inviscid Bragg-Hawthorne reductions and quasi-cylindrical approximations; and mathematically rigorous results are available mainly for isolated cone or cylinder subdomains (Esquieu et al., 12 Sep 2025, Caillaud et al., 2023, González et al., 2012). This suggests that “axisymmetric cone-cylinder-flare configuration” is best understood not as a single closed theory, but as a research class of geometries whose cone, cylinder, and flare sections are intentionally combined to expose the interaction between convective instability, pressure-gradient modulation, separation, and three-dimensionalization.