  This paper extends the resolvent formulation proposed by McKeon & Sharma
(2010) to consider turbulence-compliant wall interactions. Under this
formulation, the turbulent velocity field is expressed as a linear
superposition of propagating modes, identified via a gain-based decomposition
of the Navier-Stokes equations. Compliant surfaces, modeled as a complex
wall-admittance linking pressure and velocity, affect the gain and structure of
these modes. With minimal computation, this framework accurately predicts the
emergence of the quasi-2D propagating waves observed in recent direct numerical
simulations. Further, the analysis also enables the rational design of
compliant surfaces, with properties optimized to suppress flow structures
energetic in wall turbulence. It is shown that walls with unphysical negative
damping are required to interact favorably with modes resembling the energetic
near-wall cycle, which could explain why previous studies have met with limited
success. Positive-damping walls are effective for modes resembling the
so-called very large-scale motions (VLSMs), indicating that compliant surfaces
may be better suited for application at higher Reynolds number. Unfortunately,
walls that suppress structures energetic in natural turbulence are also
predicted to have detrimental effects elsewhere in spectral space. Consistent
with previous experiments and simulations, slow-moving spanwise-constant
structures are particularly susceptible to further amplification. Mitigating
these adverse effects will be central to the development of compliant coatings
that have a net positive influence on the flow.
