  Nonmodal amplification of stochastic disturbances in elasticity-dominated
channel flows of Oldroyd-B fluids is analyzed in this work. For
streamwise-constant flows with high elasticity numbers $\mu$ and finite
Weissenberg numbers $We$, we show that the linearized dynamics can be
decomposed into slow and fast subsystems, and establish analytically that the
steady-state variances of velocity and polymer stress fluctuations scale as $O
(We^2)$ and $O (We^4)$, respectively. This demonstrates that large velocity
variance can be sustained even in weakly inertial stochastically driven channel
flows of viscoelastic fluids. We further show that the wall-normal and spanwise
forces have the strongest impact on the flow fluctuations, and that the
influence of these forces is largest on the fluctuations in streamwise velocity
and the streamwise component of the polymer stress tensor. The underlying
physical mechanism involves polymer stretching that introduces a lift-up of
flow fluctuations similar to vortex tilting in inertia-dominated flows. The
validity of our analytical results is confirmed in stochastic simulations. The
phenomenon examined here provides a possible route for the early stages of a
bypass transition to elastic turbulence and might be exploited to enhance
mixing in microfluidic devices.
