A Well-Conditioned Numerical Method for Resolvent Analysis of Viscoelastic Channel Flows

Linear analyses provide useful information about the potential for transition to nonlinear states. While a modal approach furnishes information about long-time growth or decay of initial conditions, non-modal approaches give insight into the amplification of disturbances in a linearly stable flow. Here, we conduct non-modal analysis of inertialess 2D viscoelastic channel flows. Our analysis reveals large stress gradients in the near-wall region (for plane Poiseuille flow) and in the channel center (for plane Couette flow). These steep stress gradients can only be resolved using recently developed well-conditioned spectral methods, e.g., the ultraspherical and spectral integration methods. Furthermore, even if the discretization method is well-conditioned, computation of frequency-responses can be erroneous if singular values are obtained as the eigenvalues of a cascade connection of the resolvent operator with its adjoint. We address this issue by introducing a feedback interconnected system that avoids matrix inverses and allows reliable frequency-response calculations of viscoelastic channel flows at high Weissenberg numbers ($\sim$500). The steep stress gradients that we identify may play a role in explaining recent experiments concerning transition to elastic turbulence.