Resolvent analysis to inform viscoelastic coatings for turbulent drag reduction

Abstract

Viscoelastic compliant coatings offer a passive route to modify wall-bounded turbulence; however, their effectiveness for drag reduction remains unresolved. We perform resolvent analysis of turbulent boundary layers over linear viscoelastic continuum, and apply it to incompressible hydrodynamic and compressible aerodynamic zero-pressure-gradient turbulent boundary layers, using both standard and eddy viscosity resolvent formulations. Across a wide range of storage modulus E and coating thickness H, viscoelastic surfaces amplify near-wall-cycle-type modes while also attenuating the resolvent gain of very large scale motions (VLSMs) by up to 50%, which together result in a reduction of Reynolds stress. For density-matched coatings representative of aqueous incompressible flows, however, these favorable bands lie entirely within the regime where the effective coatings are linearly unstable to traveling wave flutter, rendering them practically unrealizable. Optimizing material damping does not eliminate this but provides a pathway to use weaker sub-optimal interactions. In supersonic flow, the large solid-to-fluid density ratio (O(1000)) shifts the favorable interaction to substantially higher moduli, weakening the achievable reduction in turbulence production to a few percent. However, the strongest interaction band occurs in the linearly stable regime. These results suggest that compliant wall drag reduction via coupling with high gain modes is fundamentally constrained by flow-induced structural instabilities in incompressible applications, whereas the high density ratios of supersonic flow offer a much narrower but stable window for practical coatings.