Scaling of energy amplification in viscoelastic channel and Couette flow

TL;DR

This paper analytically investigates how energy amplification in viscoelastic flows scales with Reynolds and Weissenberg numbers, revealing significant sensitivity to parameters and potential for flow control via polymer stress forcing.

Contribution

It derives analytical scaling laws for energy amplification in viscoelastic flows in strong and weak elasticity limits, highlighting the role of polymer stress forcing.

Findings

Weak elasticity scaling similar to O(Re^3) with elastic correction

Strong elasticity scaling as O(Wi^3) with viscous correction

Polymer stress forcing can greatly enhance energy amplification

Abstract

The linear amplification of disturbances is critical in setting up transition scenarios in viscoelastic channel and Couette flow, and may also play an important role when such flows are fully turbulent. As such, it is of interest to assess how this amplification, defined as the steady-state variance maintained under Gaussian white noise forcing, scales with the main nondimensional parameters: the Reynolds ($Re$) and Weissenberg ($Wi$) numbers. This scaling is derived analytically in the two limits of strong and weak elasticity for when the forcing is streamwise-constant. The latter is the relevant forcing for capturing the overall behaviour because it was previously shown to have the dominant contribution to amplification. The final expressions show that for weak elasticity the scaling retains a form similar to the well-known O($Re^3$) relationship with an added elastic correction. For strong elasticity, however, the scaling is O($Wi^3$) with a viscous correction. The key factor leading to such a mirroring in the scaling is the introduction of forcing in the polymer stress. The results demonstrate that energy amplification in a viscoelastic flow can be very sensitive to the model parameters even at low $Re$. They also suggest that energy amplification can be significantly increased by forcing the polymer stress, thereby opening up possibilities such as flow control using systematically designed polymer stress perturbations.