Non-modal energy amplification in channel flows of viscoelastic fluids

Energy amplification in channel flows of Oldroyd-B fluids is studied from an input-output point of view by analyzing the responses of the velocity components to spatio-temporal body forces. These inputs into the governing linearized equations are assumed to be harmonic in the streamwise and spanwise directions and stochastic in the wall-normal direction and in time. Such inputs enable the use of powerful tools from linear systems theory that have recently been applied to analyze Newtonian fluid flows. It is found that the energy amplification increases with a decrease in viscosity ratio and increase in Reynolds number and elasticity number. In most of the cases, streamwise constant perturbations are most amplified and the location of maximum energy amplification shifts to higher spanwise wavenumbers with an increase in Reynolds number and elasticity number and decrease in viscosity ratio. For streamwise constant perturbations, an explicit Reynolds number scaling of energy amplification from different forcing to different velocity components is developed, showing the same $Re$-dependence as in Newtonian fluids. At low Reynolds numbers, the energy amplification decreases monotonically when the elasticity number is sufficiently small, but shows a maximum when the elasticity number becomes sufficiently large, suggesting that elasticity can amplify disturbances even when inertial effects are weak.