One Equation Model for Turbulent Channel Flow with Second Order Viscoelastic Corrections

Abstract

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...

A FENE-P k–ε turbulence model for low and intermediate regimes of polymer-induced drag reduction

“…of Eq. (18), which needs modelling, multiplied by such coefficient of proportionality. The essential feature of the model is to capture this Weissenberg number dependence away from the wall, and because NLT ij is not that important close to the wall any discrepancy in the near wall region will be of little consequence to the turbulence model predictions.…”

“…However, this constitutive equation is not truly elastic in nature, therefore it is unable to capture important features of elastic fluids, namely the memory effect and its spatio-temporal variation. To illustrate this feature, using a simplified version of the second order fluid equation, Pinho et al [18] derived a simple one-equation k-l turbulence model that was capable of predicting drag reduction by incorporating the time-average elastic shear stress in the momentum equation. This feature especially required a closure for the elastic stress work (interaction between fluctuating elastic stress and rate of strain tensors) appearing in the transport equation for turbulence kinetic energy, provided the elastic stress work was mostly dissipative.…”

A low Reynolds number turbulence closure for viscoelastic fluids

“…9 shows a comparison between this model and the DNS results for the quantity appearing in the viscoelastic turbulent transport of k. The main features of CU iiy are well captured by Eq. (29). To comply with the damping effect of the wall, and to match the predictions of the model with the DNS data, it was necessary to introduce a damping function f CU = 1 − exp (−y + /26.5).…”

“…Pinho [28] pointed out that all these models are based on the eddy viscosity concept and require ad-hoc, flow-dependent, modifications since they were developed without reference to fluid rheology or constitutive modeling. More recently Pinho et al [29] developed a one-equation turbulence model which accounts for second-order viscoelastic corrections by employing closure approximations for the correlations between the fluctuations in the rate of strain and the viscoelastic stress. While such models can predict DR by suitable choice of correlation coefficients that mimic DNS data, they are strictly only applicable to small DR values (low Deborah numbers).…”

A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow

“…While the DNS work cited above is very useful, for instance for lower-resolution model developments (Pinho et al [13], and Thais et al [14]), it appears that it has been limited in the Reynolds number and/or in the channel dimension that could be achieved with the computer power and algorithms available. Earlier DNS computations (Sureshkumar et al [2], and Dimitropoulos et al [3]) started at Re s0 = 125 and Re s0 = 180, where Re s0 = qu s0 h/g 0 is the friction Reynolds number based on the zero-shear friction velocity at the wall u s0 , the channel mid-gap h, the polymer solution density q and zero-shear viscosity g 0 .…”