The dissolution of high molecular weight polymers and surfactants to wall-bounded shear flows of Newtonian liquids significantly modifies their stability characteristics. The critical Reynolds number Rec first decreases with increasing flow elasticity E until a critical value E=E* is reached and increases back again for E>E*. We explore the mechanisms that cause this behavior in the viscoelastic plane Poiseuille flow of an Oldroyd-B liquid. The minimum in the Rec−E curve arises from two competing contributions to the perturbation vorticity transport: The contribution from the viscoelastic shear stress perturbations that becomes more dissipative with increasing E and that from the viscoelastic normal stress perturbations that becomes more destabilizing with increasing E. Similar behavior is also exhibited by the contributions of the normal and the shear stress perturbations to the kinetic-energy budget. When a Deborah number based on the time scale of the critical disturbance becomes O(1) (≈1.6±0.1, irrespective of the solvent to total viscosity ratio), the dissipative influence of the shear stress perturbations becomes dominant. The elasticity value EC at which this occurs is approximately equal to E*. Moreover, both E* and EC exhibit similar asymptotic dependence on the solvent to total viscosity ratio. Furthermore, E* and EC are of the same order of magnitude as the elasticity values for which the onset of polymer-induced drag reduction is predicted by direct numerical simulations. Finally, we show that the perturbation velocity vector aligns progressively closer with the base flow velocity as E is increased for E<E*, contributing to the initial destabilization.
A modified second order viscoelastic constitutive equation is used to derive a kl type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by P w , for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori P w could be either positive or negative. When P w is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, P w <0 leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the k-l formulation rather than from the closure for P w . An alternative closure for P w , mimicking the viscoelastic stress work predicted by DNS using the Finitely Extensible Nonlinear Elastic-Peterlin fluid model, which is mostly characterized by P w >0 but has also a small region of negative P w in the buffer layer, was also successfully tested. This second model for P w leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained. Nomenclature A parameter of polymer work model, see (33) b ij tensor defined in (28) C D parameter of turbulence model, see (23) C k parameter of k-l turbulence model C ij time-average component of elastic stress tensor defined in (12) D k Turbulent and molecular diffusion of turbulent kinetic energy, defined in (20b) De Deborah number based on bulk velocity of the flow De l Deborah number of turbulence, De l ¼ u1 =l De t friction Deborah number, De C ¼ u C 1 =H H channel half-height k turbulent kinetic energy l length scale in turbulence in order of magnitude analysis and Prandtl's mixing length in turbulence model p pressure P k Production of turbulent kinetic energy, defined in (20a) P w Polymeric work, defined in (20d) t time T ij designates a time-average tensor u scale of velocity fluctuations in order of magnitude analysis and streamwise velocity component in turbulence model equations u i fluctuating velocity vector u iinstantaneous velocity vector u* streamwise velocity normalized by the friction velocity, u þ ¼ u=u C U i time-average velocity vector u t friction velocity in fully-developed channel flow uv Reynolds shear stress Re Reynolds number based on bulk velocity o...
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