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 μ 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(We2) and O(We4), 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 fluctuations in the 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.
Bibliographical noteFunding Information:
M.R.J. would like to thank Prof. Petar V. Kokotović for stimulating discussions and insightful comments. This work was supported in part by the National Science Foundation under CAREER Award CMMI-06-44793 (to M.R.J.), by the Department of Energy under Award DE-FG02-07ER46415 (to S.K.), and by the University of Minnesota Digital Technology Center ’s 2010 Digital Technology Initiative Seed Grant (to M.R.J. and S.K.).
- Elastic turbulence
- Inertialess flows
- Polymer stretching
- Singular perturbations
- Variance amplification
- Viscoelastic fluids