Mach-disk formation and shock-structure transitions in underexpanded coflowing jets

Abstract: The near-field shock structures of underexpanded sonic jets exiting into a subsonic coflow are investigated over a range of nozzle pressure ratio (NPR) and coflow-to-nozzle-exit velocity ratio ($U_c$), representative of a propulsive nozzle in subsonic flight. Time-averaged statistics from fully-resolved axisymmetric simulations and inviscid method-of-characteristics (MOC) analysis are used to understand how coflow alters the shock-cell structures, in particular the Mach-disk formation. It is well established that increasing NPR transitions the centerline reflection from regular (characterized by oblique shocks) to Mach reflection (characterized by a near-normal Mach disk). We find that coflow has the opposite influence: a strong coflow shrinks the Mach disk until it vanishes, reverting Mach reflection to regular reflection, so the NPR for this transition increases with $U_c$. This effect has previously been attributed to a reduction in the jet-boundary inclination at the nozzle lip, which confines the lip Prandtl-Meyer fan to a smaller angle and weakens the embedded shock. We show instead that this inclination is determined by the non-uniform pressure the coflow imposes along the jet boundary, which is the primary driver of the shock-structure transitions in coflowing jets. The non-uniform pressure weakens the boundary-reflected compression waves and orients them at shallower angles, so the embedded shock reflects regularly or fails to form. A simulation-informed MOC analysis with this non-uniform pressure boundary condition reproduces the transition behavior with increasing coflow. Coflow also lengthens the first shock cell linearly, which is accurately estimated by a simple correction to Prandtl classical shock-cell length scaling.

• The paper maps Mach-disk formation across NPR 2.84–9.47 and coflow ratios up to 0.8, showing that increasing coflow shrinks or eliminates the disk and shifts Mach reflection to regular reflection.
• A coflow-extended logarithmic model predicts Mach-disk diameter with R² = 0.99 and RMSE of 0.018De, while regime-specific correlations capture coflow-driven shock-cell lengthening.
• The paper identifies non-uniform pressure along the jet boundary—not reduced lip inclination—as the dominant mechanism weakening reflected compression waves and preventing their coalescence into embedded shocks.

Overview and motivation

This paper investigates how a subsonic coflow modifies the near-field shock structures of underexpanded sonic jets issuing from a convergent choked nozzle, with particular attention to Mach-disk formation. The configuration is representative of a propulsive nozzle operating at off-design conditions in subsonic flight, where the jet exhausts into a moving ambient stream rather than a quiescent medium. While it is well established that increasing the nozzle pressure ratio (NPR) drives a transition from regular reflection (oblique shocks) to Mach reflection (a near-normal shock segment, the Mach disk) at the jet centerline, the effect of coflow on this transition had not been systematically mapped. Prior literature is inconsistent: dual coaxial-jet studies report both suppression and enlargement of the Mach disk depending on geometry and annular-stream orientation, while simulated-flight experiments focused on shock-cell spacing and acoustics rather than on the reflection regime itself.

The study combines fully resolved axisymmetric Navier–Stokes simulations with an inviscid method-of-characteristics (MOC) analysis over a two-parameter space: NPR from 2.84 to 9.47 (pressure ratios PR = 1.5–5.0) and coflow-to-nozzle-exit velocity ratio $U_c$ from 0 to 0.8, all combinations considered. The solver uses a sixth-order WENO scheme for inviscid fluxes, fourth-order centered viscous fluxes, RK4 time integration, SAT inflow boundary conditions with a thin inlet shear layer (momentum thickness 2% of the exit radius), NSCBC outflow conditions with sponge zones, and Re = 50,000. Validation against experiments (Panda & Seasholtz; Sugawara et al.) and prior DNS shows excellent agreement in centerline density, Mach-disk location, and density jump across the disk; downstream-of-Mach-disk discrepancies are acknowledged and attributed to loss of axisymmetry, which does not affect the first-shock-cell focus of the work.

Shock-structure transitions across the NPR–coflow parameter space

The central finding is that coflow acts oppositely to NPR: increasing $U_c$ shrinks the Mach disk until it vanishes, reverting centerline Mach reflection to regular reflection. Equivalently, the transition NPR for regular-to-Mach reflection increases with coflow. For the representative PR = 2.25 (NPR = 4.26) jet, a Mach disk of diameter $0.129\,D_e$ at $x/D_e = 1.105$ exists in quiescent ambient; at $U_c = 0.4$ the disk has shrunk to $0.08\,D_e$ and moved to $x/D_e = 1.193$; at $U_c = 0.6$ the embedded shock reflects regularly as an oblique shock; and at $U_c = 0.8$ no embedded shock forms at all before the centerline. A weak bow-shaped shock appears near $x/D_e \simeq 1.9$ at the highest coflow, confirmed by a three-dimensional DNS not to be an axisymmetric artifact.

Mach-disk diameters over the full parameter space collapse onto a coflow-extended Antsupov-type logarithmic fit,

$U_c$0

with $U_c$1 and RMSE of 0.018 ($U_c$2 of $U_c$3). Coflow thus steepens the logarithmic slope linearly while lowering the intercept quadratically; negative values indicate absence of a Mach disk. This correlation recovers eight of nine "no Mach disk" cases in the parameter map. In quiescent ambient, extrapolation of the present data yields a transition NPR of 3.57, within the unsettled literature range of roughly 3.1–3.9 — itself evidence that the zero-coflow transition value remains unresolved.

Mechanism: non-uniform boundary pressure versus lip inclination

The paper tests, and ultimately revises, the explanation offered in earlier work — that coflow suppresses the Mach disk by reducing the jet-boundary inclination $U_c$4 at the nozzle lip, thereby confining the Prandtl–Meyer fan and weakening the embedded shock. Measured inclinations do decrease nearly monotonically with $U_c$5, independent of NPR. However, an MOC calculation in which coflow is imposed solely through this inclination fails to reproduce the observed transition: compression waves still coalesce into an embedded shock at all coflow levels, contradicting the Navier–Stokes solutions. Prescribing $U_c$6 also implicitly prescribes a uniform boundary pressure above ambient ($U_c$7 between 1.17 and 1.30), reducing the effective pressure ratio but leaving the free-jet constant-pressure condition intact.

The correct mechanism, established by examining the boundary streamline pressure in the simulations, is that the subsonic coflow imposes a non-uniform static pressure along the entire jet boundary. The ambient stream accelerates over the diverging portion of the jet (favorable pressure gradient) up to the maximum-diameter point, then decelerates (adverse gradient) beyond it. Along the favorable-gradient section, reflected compression waves are weakened — because the falling boundary pressure requires less pressure recovery than the incident expansion drop — and emitted at progressively shallower Mach angles, spreading them apart so they fail to coalesce into an embedded shock. Domain-independence checks confirm this pressure variation is physical rather than a numerical artifact.

When the simulation-derived pressure profile is imposed as the MOC boundary condition, the inviscid analysis reproduces the full phenomenology: coalescence onset shifting downstream with increasing $U_c$8, complete absence of coalescence at $U_c$9 and 0.8, and recombination of characteristics near $0.129\,D_e$0 at $0.129\,D_e$1 where the adverse pressure gradient steepens — matching the bow-shaped structure seen in the Navier–Stokes solution. The reduced lip inclination reported previously is thereby shown to be a local consequence of the imposed pressure distribution, not an independent cause. This mechanism also offers an explanation for the unexplained shock-cell weakening measured by André et al. in flight-simulation experiments, and subsumes the expansion-fan-narrowing account of Ahmad et al. One residual discrepancy remains: for $0.129\,D_e$2, MOC predicts similar embedded-shock formation whereas the simulated Mach-disk diameters differ substantially ($0.129\,D_e$3 versus 0.19), tentatively attributed to viscous displacement effects of the thicker low-coflow shear layer; the authors state plainly that this explanation has not been independently verified.

Coflow-corrected models for cell geometry

The classical Prandtl–Pack scaling $0.129\,D_e$4 is extended to coflowing jets with regime-dependent corrections:

Quantity	Correlation	Regime	$0.129\,D_e$5
Lip inclination $0.129\,D_e$6	$0.129\,D_e$7	both	0.77
Cell length $0.129\,D_e$8	$0.129\,D_e$9	Mach reflection	0.85
Cell length $x/D_e = 1.105$0	$x/D_e = 1.105$1	Regular reflection	0.79

Both prefactors lie close to Pack's classical value of 1.22, and the fits reduce identically to the quiescent scaling at $x/D_e = 1.105$2. Two features carry implications. First, the coflow sensitivity of the cell length roughly doubles upon transition from Mach to regular reflection, which explains why single-coefficient flight corrections calibrated at moderate speeds (e.g., Tam's factor 0.625, valid only for $x/D_e = 1.105$3) underestimate lengthening at higher speeds precisely where they break down. Second, the cell length exhibits a discontinuous fall at the (PR, $x/D_e = 1.105$4) combinations marking the regular-to-Mach transition, attributed to truncation of the cell when the reflected shock launches from the off-axis triple point rather than the centerline. The inclination prefactor of 0.85 conceals a systematic PR dependence (from about 0.76 at PR = 1.5 to 0.88 at PR = 5), acknowledged as producing average deviations up to 3%.

Viscous effects

Viscous mixing is quantified through the vorticity thickness $x/D_e = 1.105$5 of the boundary shear layer, evaluated as a median over $x/D_e = 1.105$6. It decreases monotonically with $x/D_e = 1.105$7 and collapses onto $x/D_e = 1.105$8 against the velocity-ratio parameter $x/D_e = 1.105$9, with $U_c = 0.4$0 — consistent with canonical free-shear-layer scaling. The authors caution that the enforced axisymmetry excludes three-dimensional azimuthal instability and turbulent breakdown, so these comparisons indicate qualitative consistency with shear-layer behavior rather than a quantitative test of turbulent mixing theory. Since the structural transitions occur at higher coflow velocities where the shear layer is thin, viscosity is concluded not to be the primary driver of the shock-structure changes, though it plausibly matters at low coflow (the unexplained diameter discrepancy noted above).

Limitations and open questions

Several limitations bound the results. All production simulations are axisymmetric, justified by agreement with 3-D DNS only up to the first Mach disk; flow downstream of the Mach disk, where axisymmetry is lost, lies outside the validated scope. The 3-D confirmation runs use a reduced Reynolds number of 5,300 versus 50,000 in the axisymmetric production runs, defended on the grounds that the compared quantities are inviscid-core-governed. The MOC analysis is fold-back type: it neither captures shocks explicitly nor treats centerline reflections, so Mach-disk location and diameter cannot be predicted directly, only inferred from compression-wave coalescence. The tentative link between shear-layer displacement thickness and the effective boundary pressure at low coflow is left unverified. Finally, the mechanism is demonstrated for sonic jets from convergent nozzles with matched total temperature; its extension to convergent–divergent nozzles, heated jets, or supersonic coflows remains open, as does a predictive (rather than simulation-informed) model for the boundary-pressure distribution itself.

Conclusion

This work provides the first systematic mapping of subsonic-coflow effects on Mach-disk formation across the NPR–$U_c = 0.4$1 space and identifies a single sufficient mechanism: the coflow-imposed non-uniform boundary pressure weakens and reorients the boundary-reflected compression waves, suppressing embedded-shock formation and reverting Mach reflection to regular reflection. The reduced lip inclination cited in prior studies is demoted to a local consequence of this pressure distribution. An inviscid MOC analysis with simulation-informed pressure boundary conditions reproduces the transition behavior, establishing the inviscid character of the dominant dynamics, while compact correlations extend Prandtl's cell-length scaling to coflowing jets with explicit dependence on the reflection regime.