One equation model for turbulent channel flow with second order viscoelastic corrections. (English) Zbl 1257.76032
Summary: A modified second order viscoelastic constitutive equation is used to derive a \(k-l\) 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.
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