VISCOELASTIC NONLINEAR TRAVELING WAVES AND DRAG REDUCTION IN PLANE POISEUILLE FLOW

摘要

Nonlinear traveling wave solutions to the Navier-Stokes equations in the plane Poiseuille geometry (Waleffe, F. (2003), Phys. Fluids 15, 1517-1534) come into existence through a saddle-node bifurcation at a Reynolds number of 977, very close to the experimentally observed Reynolds number of ～ 1000 for transition to turbulence in this geometry. These traveling waves are comprised of staggered counter-rotating streamwise vortices with a spanwise wavelength of 106 wall units, in good agreement with the experimentally observed value of ～ 100 wall units for buffer layer structures. In the present work, the effect of viscoelasticity on these states is examined, using the FENE-P constitutive model of polymer solutions. The changes to the velocity field for the viscoelastic traveling waves mirror those experimentally observed in fully turbulent flows of polymer solutions near the onset of turbulent drag reduction: drag is reduced, streamwise velocity fluctuations increase and wall-normal fluctuations decrease. The mechanism underlying theses changes is elucidated through an examination of the forces exerted by the polymer molecules on the fluid. The onset Weis-senberg number (shear rate times polymer relaxation time) for drag reduction is insensitive to polymer extensibility or concentration. Above the onset Weis-senberg number, there is a dramatic increase in the critical wall-normal length scale at which the nonlinear traveling waves can exist. This sharp increase in length scale directly correlates with the extensibility and concentration of the polymer molecules and is consistent with the observed shift to higher Reynolds numbers of the transition to turbulence in polymer solutions. The balance of turbulent kinetic energy for the nonlinear traveling waves shows the same qual-ititative changes as are found in full turbulence, as do the overall kinematics of stretching and rotation in the flow. These observations suggest that the mechanism of near-onset drag reduction in flow over smooth walls is captured by the effect of viscoelasticity on these nonlinear traveling waves.