Super resonance: Breaking the bandwidth limit of resonant modes and its application to flow control

Abstract

We report the discovery of super resonance--a new regime of resonant behavior in which a mode's out-of-phase response persists far beyond its classical bandwidth. This effect emerges from a coiled phononic structure composed of a locally resonant elastic metamaterial and architected to support multiple internal energy pathways. These pathways converge at a single structural location, enabling extended modal dominance and significantly broadening the frequency range over which a resonant phase is sustained. We demonstrate by direct numerical simulations the implications of this mechanism in the context of flow instability control, where current approaches are inherently constrained by the characteristically narrow spectral bandwidth of conventional resonances. Using a super-resonant phononic subsurface structure interfacing with a channel flow, we show passive simultaneous suppression of four unstable flow perturbations across a frequency range more than five times wider than that is achievable with a standard resonance in an equivalent uncoiled structure. By enabling broadband, passive control of flow instabilities, super resonance overcomes a longstanding limitation in laminar flow control strategies. More broadly, it introduces a powerful new tool for phase-engineered wave-matter interaction. The ability to preserve out-of-phase modal response across wide spectral ranges establishes a fundamental advance in the physics of resonance, with far-reaching implications for suppressing fully developed turbulent flows and beyond.

Figures (9)

• Figure 1: Super-resonant coiled PSub design: A schematic of the concept of a coiled phononic subsurface with multiple structural connectivity. The physical model, shown in (a), features a central voided beam that is rotationally locked at the turning junctions, leading to a full preservation of the phonon band structure willey2022coiled. The beam "trunk" has an array of branching resonators in the form of small cantilevered beams$-$this configuration in its entirety is amenable to fabrication. The dimensions of the PSub unit cell are as follows based on the parameters shown in (a): $L_{\text{UC}} = 20$ mm, $w_{\text{c}} = 1.44$ mm, $h_{\text{c}} = 4.63$ mm, $l_{\text{h}} = 1.17$ mm, $w_{\text{h}} = 1.12$ mm, $l_1 = 18.23$ mm, $w_1 = 2.06$ mm, $l_2 = 12.93$ mm, $w_2 = 4.68$ mm, $w_{\text{f}} = 0.4$ mm. The central beam component is composed of ABS polymer with density $\rho_{\text{ABS}}=1200$ kg/m$^3$ and Young's modulus $E_{\text{ABS}}=1$ GPa. The resonant branches are made from aluminium with density $\rho_{\text{Al}} = 2700$ kg/m$^3$ and Young's modulus $E_{\text{Al}} = 68.8$ GPa. In our computational investigation, this 3D structure is modeled in the form of a 1D rod, as shown in (b), with effective Young's modulus $E_{\text{eff}} = 0.2$ GPa and density $\rho_{\text{eff}} = 450$ kg/m$^3$, containing an array of spring-mass resonators attached based on the following parameters: $L_{\text{UC}} = 20$ mm, $k_{\text{res},1} = 2.6 \times 10^{10}$ N/m, $k_{\text{res},2} = 7.11 \times 10^{11}$ N/m, $m_{\text{res},1} = 1.62$ g, $m_{\text{res},2} = 4.2$ g. Material damping is introduced to the rod component in the form of viscous proportional damping with constants $q_1=0$ and $q_2=6\times 10^{-8}$kianfar2023phononicNJP. Given that our interest is in only the longitudinal motion, the rod model provides a good approximation of the coiled PSub response, owing to the rotational locking applied at the junctions. The selected simple model parameters deviate less than 2% from the actual unit-cell equivalent configuration. With one coiling cycle, we have two structural points that are simultaneously interfacing with the flow, labeled Junctions 1 and 3. With two coiling cycles, Junctions 1, 3, and 5 are in contact with the flow, and so on in an odd-number progression. We find that with three coiling cycles, comprising Junctions 1, 3, 5, 7, super-resonant behavior emerges (see Fig. \ref{['fig:F2']}). While the coiled PSub design is confined to a specific finite dimension along the spanwise direction, its response can still be considered one-dimensional when normalized by the surface area interacting with the flow, marked in red and denoted by $A$. The excitation of the structure is marked by $F$ and the corresponding input into the flow is marked by $\Sigma\eta_{i}$. In this work, we consider only a single coiled PSub covering the full span of the channel. However, application to longer spanwise distances (e.g., for wider channels or boundary-layer flows) may be realized by installation of a periodic arrangement of the coiled PSub along that direction (side view). Additional rows may also be added along the streamwise direction to form a PSub lattice setup hussein2025scatterless.
• Figure 2: Evidence of super resonance: Coiled PSub dispersion and vibration characteristics: (a) Dispersion, (b-c) harmonic frequency response, and (d) performance metric for the coiled PSub with multiple structural connectivity. The dispersion is unchanged with the number of coiling cycles since the design of the unit cell is the same in all cases and band structure preservation is maintained due to the rotational lockings willey2022coiled. The amplitude and phase frequency dependency for the uncoiled (0 coils) and 3-cycle coiled configurations are shown in (b-c). Positive (red) and negative (green) regions of $P$ represent destabilization and stabilization, respectively, upon employment of the PSub for flow control. Zero coiling cycles represent a PSub with 1 point-of-contact with the flow, while 3 coiling cycles represents the same PSub but with 4 points-of-contact with the flow, as demonstraed in Fig. \ref{['fig:F1']}. As the number of coils is increased from 1 to 3, the destabilization and stabilization regions across the frequency domain are narrowed and broadened, respectively. With 3 coiling cycles, super resonance emerges yielding a perfectly contiguous stabilization band that is nearly two times larger than that of the uncoiled reference case. A quasi-super resonance regime also emerges, one that is effectively contiguous and nearly four times larger than that of the reference configuration.
• Figure 3: Super-resonant coiled PSub performance by DNS: (a) Stability map in the frequency-Reynolds number domain obtained by solving the Orr-Sommerfeld equation Orr1907Sommerfeld1909 for the channel. (b) Zoom-in of the performance metric plot for the super-resonant 3-cycle coiled PSub, with frequencies of the four unstable TS waves marked. (c) Time-averaged perturbation kinetic energy in the flow as a function of streamwise position in the channel. The results of five separate simulations are shown: four in which a distinct TS instability wave is controlled (left axis), and one where all four TS instability waves are controlled simultaneously (right axis). In all cases, the 3-cycle coiled PSub successfully causes local stabilization in the flow, with stabilization strength consistent with the corresponding value of the performance metric. The all-modes case demonstrates broadband stabilization spanning the entire unstable region for the given value of $Re$ [shown in (a)] with an intensity that matches the "additive" effect of the $P$ values over the four instability frequencies [shown in (b)]. (d) $K_{\text{p}}$ plot for the 700-Hz instability case showing contrast in performance when the 3-cycle coiled Psub is installed versus using the nominal uncoiled PSub. The results confirm the transformation from destabilization to stabilization due to the coiling.
• Figure 4: Super-resonant coiled PSub behavior by FIK integral analysis: Evaluation of various key flow parameters when the super-resonant 3-cycle coiled PSub is installed versus the rigid-wall case. (a) Streamwise distribution of the averaged skin-friction coefficient $C_f$, (b) streamwise evolution of the "turbulent enhancement" to $C_f$ computed by the FIK equation, (c) streamwise variation of the wall-normal integral of the production rate $P_{\text{r}}$, and (d) wall-normal variation of $P_{\text{r}}$ at streamwise positions corresponding to the leading (Station 1) and trailing (Station 2) edges of the control surface.
• Figure A1: PSub coiling: PSub periodic unit cell with different degrees of coiling: (a) uncoiled case, (b) partially coiled case, (c) fully coiled case. The periodic unit-cell size for the dispersion analysis scaling willey2022coiled is taken as $a = 2 L_\text{UC}\cos{\phi}$.