As ultra-high bypass turbofan engines continue to be adopted and developed further, there is a greater need for new acoustic liner designs to mitigate low-frequency (<1000 Hz) noise. Conventional perforate over honeycomb acoustic liners are impractical for low-frequencies due to the weight and volume penalties they impose, whereas foams are generally only effective at higher frequencies. Recent studies on folded-core and membrane-embedded liners have shown promising results in addressing low-frequencies. Given the diversity in the design space that these new approaches open up, the development of quantitative tunable performance metrics (TPMs) is desirable to facilitate fast and efficient automation of design iterations. Beginning with a generic low-frequency performance (LFP) metric, tunable versions that emphasize various spectral characteristics such as max absorption peak, lower bound, and overall absorption within a bandwidth of interest are defined. These TPMs are applied to various low-frequency design studies to demonstrate the ability to discriminate between candidate configurations tailored to address specific spectral characteristics. Utilizing the framework of TPMs, flexible wall liner configurations were studied to ascertain their low-frequency absorption characteristics. COMSOL Multiphysics simulations were correlated with experiments done using the normal-incidence impedance tube. The response of the membrane was measured in-situ using laser vibrometry. In addition to the high-frequency absorption peak expected from the cavity resonance, the flexible wall introduces a tunable low-frequency peak that correlates well with the modal response of the membrane. A numerical parametric study was also conducted to assess the influence of parameters such as membrane properties and placement as well as the relative location of flexible wall cavities within multi-cavity liner configurations. It is found that non-locally reacting flexible wall liner configurations could significantly widen the bandwidth of appreciable absorption in the low-frequency regime. Further development of flexible wall liners using TPMs to automate the design process could provide practical solutions to address low-frequency acoustic noise mitigation.