Towards an immersed-boundary resolvent analysis framework for wall-bounded flows with various surface configurations

New flow control paradigms for wall-bounded flows are leveraging increasingly sophisticated actuation strategies targeted near the wall surface including riblets, anisotropic porous and textured surfaces, and passively adaptive metamaterials. This new direction brings a growing need for predictive computational tools that can accurately and efficiently account for these complex surface effects, potentially involving fully coupled fluid-structure interactions (FSI). The resolvent analysis framework is a natural choice for these aims, as it provides a fast computational model rooted in an analysis of the true linearized operator for the wall-bounded flows of interest. Yet, while there are a number of resolvent-based strategies designed to incorporate surface effects, most are formulated for a specific configuration and cannot be readily extended to different setups. We present developments towards a more general formalism rooted in an immersed boundary (IB) treatment of the surface that we call IB resolvent. The proposed method inherits the benefits of versatile flow-surface treatment that high-fidelity IB methods have enjoyed for decades, formally incorporated into a spanwise-streamwise-temporal homogeneous resolvent framework that has become a workhorse for wall-bounded flow analysis and control design. We present the mathematical formulation for the case of turbulent channel flow with a static sinusoidal wall boundary condition and examine its performance. We also demonstrate that when the wall is equivalent to a flat surface, the IB resolvent results match those of the conventional resolvent analysis. While the results are focused on this canonical setting, the derivation is constructed to facilitate straightforward extensions to more complex configurations.