Compressible Linear Stability Analysis for Passive Transition Delay Using Phononic Materials

Abstract: Phononic materials (PMs) have been proposed as a passive flow control method capable of mitigating Tollmien-Schlichting (TS) wave growth and delaying the onset of boundary layer transition. Their tunable spectral properties allow for the matching of a PM’s response to a narrow-band fluid instability. Recent works have begun to characterize the required response for transition delay, but how these requirements change as a function of increasing freestream flow scales and compressibility influences remains unclear. This work addresses this question through the development of a compressible linear stability method capable of modeling the influence of a PM on TS wave growth in the compressible, subsonic regime. The development of the tool is described, then linear stability analysis is performed, and results are used to establish requirements for a PM capable of TS wave attenuation. The analysis method is then applied to determine how the requirements for TS wave stabilization vary with frequency, Reynolds number, and Mach number. Across all Mach numbers, Reynolds numbers, and frequencies, achieving positive phasing between the wall motion and pressure fluctuations is required for stabilization. However, the exact value of the optimal phase angle varies from 60° to 120°. The changes in the minimum required amplitude follow the trends seen in TS wave growth rates for a rigid wall, wherein a larger magnitude of PM response is required when the rigid wall TS wave growth rate is increased. Finally, an analysis of energy fluxes is performed to further understand the physical drivers behind these trends. The key driver that results in a decrease in TS wave growth rate is a decrease in the Reynolds stress production due to the positive phasing from the PM.

Citation: M. Brotnow, C. Barnes, T. Vincent, V. Ramakrishnan, K. H. Matlack, P. Ansell, "Compressible Linear Stability Analysis for Passive Transition Delay Using Phononic Materials", Under Review, 2026
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