A Molecular Gas Dynamics Study of Hypersonic Boundary Layer Second Mack Mode Instabilities

A flat-plate laminar boundary layer is simulated at Mach 6 and unit Reynolds number of 1.1e7 using the Direct Simulation Monte Carlo (DSMC) method to capture and analyze spontaneous second-mode instability growth. Power spectral density (PSD) analysis identifies dominant frequencies of 200-400 kHz, in line with linear stability theory (LST) predictions. Near-wall perturbations remain confined within the unstable regions known from linear theory. Dynamic mode decomposition (DMD) of unsteady flowfield snapshots reveals wave packets of spatially coherent modes having wavelengths and phase speeds characteristic of the acoustic second mode; their growth and decay occur exclusively within LST-predicted unstable bounds. Targeted interaction with these flow instabilities is demonstrated for an acoustic vibrating surface (AVS), where forcing at the unstable frequency of 300 kHz results in amplified waves downstream, while at the stable frequency of 500 kHz AVS-induced disturbances are damped. This further emphasizes the ability of the present kinetic simulations to capture and describe linear perturbations at high Reynolds numbers and suggests that DSMC will be a useful tool for understanding theoretically founded control of laminar-turbulent transition in hypersonic boundary layers.

💡 Research Summary

The paper presents a pioneering Direct Simulation Monte Carlo (DSMC) investigation of second‑mode (Mack) instability in a Mach 6, flat‑plate hypersonic boundary layer at a unit Reynolds number of 1.1 × 10⁷ m⁻¹. Traditional studies of this high‑frequency acoustic mode rely on linear stability theory (LST) and continuum‑based DNS, which struggle to capture receptivity, slip, temperature‑jump, and shock‑boundary‑layer coupling effects that become important at high Mach numbers and low‑disturbance environments. By solving the Boltzmann equation with DSMC, the authors naturally include rarefaction, stochastic thermal fluctuations, and non‑equilibrium phenomena without ad‑hoc modeling.

The computational setup uses nitrogen modeled with a Variable Hard Sphere (VHS) model, a major‑frequency collision scheme, and a grid resolution of 4 × 10⁻⁵ m (≈ 60 λ) with a 1 ns time step. A total of 1.5 × 10⁹ simulated particles evolve over 0.6 ms, with the final 0.2 ms sampled for unsteady analysis. Validation against the compressible Blasius solution confirms that the mean flow field (velocity, temperature, boundary‑layer thickness) is accurately reproduced, providing a solid baseline for instability detection.

Time‑resolved temperature and velocity data are collected at 180 probes distributed through the shear layer. Power spectral density (PSD) analysis of these signals reveals dominant peaks in the 200–400 kHz range, matching the unstable frequency band predicted by LST for the given Mach and Reynolds numbers. This demonstrates that DSMC can generate spontaneous second‑mode disturbances solely from intrinsic particle noise, without any external forcing.

Dynamic Mode Decomposition (DMD) is then applied to the full set of primitive variable snapshots. The algorithm extracts a small number of coherent global modes, each characterized by a complex eigenvalue whose real part gives the growth/decay rate and the imaginary part yields the oscillation frequency. The leading DMD mode exhibits a wavelength of roughly 2 mm and a phase speed of about 1.5 U∞, consistent with the acoustic second‑mode. Its growth rate is positive precisely within the LST‑identified unstable region (√Re ≈ 500–1150). A secondary mode with shorter wavelength shows near‑zero growth, representing background acoustic noise. By integrating the growth rates along the streamwise direction, the authors compute an N‑factor that reaches values of 9–10, close to the empirical transition threshold.

To explore active control, an acoustic vibrating surface (AVS) 1 mm wide is placed at x = 60 mm. The AVS is driven at two frequencies: 300 kHz (inside the unstable band) and 500 kHz (outside). At 300 kHz, the surface‑generated acoustic wave synchronizes with the naturally occurring second‑mode packet, leading to downstream amplification; the PSD amplitude rises by more than 30 % and the DMD‑derived growth rate increases by roughly 0.8–1.0. Conversely, at 500 kHz the forced wave is quickly damped, the PSD level drops, and the DMD growth rate becomes negative, confirming frequency‑selective attenuation. These results validate that DSMC can faithfully reproduce both passive receptivity and active forcing mechanisms in a high‑Mach, high‑Re environment.

Overall, the study establishes three key contributions: (1) DSMC is capable of capturing linear second‑mode instability at Reynolds numbers previously unattainable for kinetic simulations; (2) a combined PSD‑DMD workflow provides a data‑driven, quantitative description of the instability’s spectral content, spatial structure, and growth characteristics, complementing traditional LST; and (3) targeted acoustic forcing via an AVS can selectively amplify or suppress the second‑mode, demonstrating a viable pathway for transition control in hypersonic vehicles. The authors suggest future work to extend the approach to three‑dimensional geometries, chemically reacting flows, and experimental validation, thereby broadening the applicability of kinetic methods to realistic re‑entry and high‑speed flight scenarios.