Linear energy analysis of the elastic disturbances in viscoelastic channel flows: a new perspective from the geometric decomposition

We perform a comprehensive linear non-modal stability analysis of plane Poiseuille flow in Oldroyd-B fluids, employing the recently proposed geometric decomposition of the polymer conformation tensor (Hameduddin et al. , J. Fluid Mech ., 2018, vol. 842, 395–427; J. Fluid Mech ., 2019, vol. 858, 377–406), where perturbation elastic energy has an appropriate definition. In the presence of weak shear flow, e.g. Reynolds number Re = 1, the spanwise-uniform disturbance with a large wavenumber exhibits the largest transient growth, and perturbation elastic energy is the most amplified, proportional to W 2 (where W denotes the Weissenberg number), which is mainly attributed to the combination of different modes in the continuous spectrum (CS). Elastic energy analysis suggests that this growth is physically due to the energy transfer from the perturbation wall-normal velocity to the normal components of the perturbation polymer conformation tensor. As Re increases to Re = 1000, transient growth has a significant increase, and its maximum appears in the oblique disturbance with perturbation kinetic energy dominating amplification instead of elastic energy, which is still induced by the spanwise-uniform components of the input perturbation conformation tensor, just as in the Re = 1 case. Mathematically, this energy growth is caused by the combination of CS and the remaining discrete modes, and it is remarkably larger than that merely caused by CS. Kinetic energy analysis suggests that this significant kinetic energy growth is physically due to the fact that elastic effects enhance energy transfer from the base flow field and the perturbed conformation tensor field to the perturbed hydrodynamic field.