A priori DNS development of a closure for the nonlinear term of the evolution equation of the conformation tensor for FENE-P fluids (original) (raw)

Polymer stress tensor in turbulent shear flows

Physical Review E, 2005

The interaction of polymers with turbulent shear flows is examined. We focus on the structure of the elastic stress tensor, which is proportional to the polymer conformation tensor. We examine this object in turbulent flows of increasing complexity. First is isotropic turbulence, then anisotropic (but homogenous) shear turbulence and finally wall bounded turbulence. The main result of this paper is that for all these flows the polymer stress tensor attains a universal structure in the limit of large Deborah number De ≫ 1. We present analytic results for the suppression of the coil-stretch transition at large Deborah numbers. Above the transition the turbulent velocity fluctuations are strongly correlated with the polymer's elongation: there appear high-quality "hydro-elastic" waves in which turbulent kinetic energy turns into polymer potential energy and vice versa. These waves determine the trace of the elastic stress tensor but practically do not modify its universal structure. We demonstrate that the influence of the polymers on the balance of energy and momentum can be accurately described by an effective polymer viscosity that is proportional to to the cross-stream component of the elastic stress tensor. This component is smaller than the stream-wise component by a factor proportional to De 2 . Finally we tie our results to wall bounded turbulence and clarify some puzzling facts observed in the problem of drag reduction by polymers.

Symmetric factorization of the conformation tensor in viscoelastic fluid models

The positive-definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability. (C.R. Doering).

Application of the Log-Conformation Tensor Approach to Three Dimensional Viscoelastic Flows

2007

A finite-volume method is applied to the numerical simulation of creeping flows of viscoelastic fluids through a 4:1 square-square three dimensional abrupt contraction. The calculation of the polymer stress contribution is carried out with the log-conformation methodology . The log-conformation scheme allowed converged solutions at higher Deborah numbers for the Oldroyd-B fluid than the standard method when imposing steady flow conditions, but not for the UCM model. This does not mean that the log-conformation technique is less effective for the UCM fluid, but suggests that this flow is most probably time-dependent above a critical Deborah number. In any case, this work confirms the advantages of the log-conformation approach vis-a-vis the standard procedure.

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

Journal of Non-Newtonian Fluid Mechanics, 2011

a b s t r a c t A new low-Reynolds-number k-e turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k-e turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89-108].

Polymer conformations and hysteretic stresses in nonstationary flows of polymer solutions

EPL (Europhysics Letters), 2009

The low Reynolds number flow of a polymer solution around a cylinder engenders a nonlinear drag force vs. the flow velocity. A velocity quench of such a flow gives rise to a long time relaxation and hysteresis of the stress due to history-dependent elastic effects. Our results suggest that such hysteretic behavior has its origin in the long time relaxation dynamics and hysteresis of the polymer conformations.

Free surface flows of polymer solutions with models based on the conformation tensor

Journal of non-newtonian fluid mechanics, 2002

A computational method is presented for analyzing free surface flows of polymer solutions with the conformation tensor. It combines methods of computing Newtonian free surface flows [215]. Modifications are introduced to compute a traceless velocity gradient, to impose inflow boundary conditions on the conformation tensor that are independent of the specific model adopted, and to include traction boundary conditions at free surfaces and open boundaries.

Report on the IUTAM symposium on viscoelastic fluid mechanics: effects of molecular modeling

Journal of Non-newtonian Fluid Mechanics, 1999

On 21±25 June 1998, 47 experts from 14 countries met at Stanford to examine in a workshop setting the current state and future impact of molecular modeling on viscoelastic fluid mechanics. The meeting was particularly timely, since research in the field is becoming increasingly molecular in focus. The reasons behind this were clearly discussed throughout the symposium, and include: The development and increasing use of molecular-based simulations to understand the dynamics and stress in complex and`strong' flows of polymer solutions and melts. These simulations include thè micro±macro' simulation where coarse-grained models are immediately incorporated into fluid mechanical field calculations. Associated and complementary to these coarse-grained simulations are more detailed Brownian dynamics simulations where mechanical models for polymers are developed and simulated including detail down to scales of the Kuhn step or persistence length of the chain. The rapid development of experimental tools capable of examining the molecular configurations of polymers during flow and, with some inference, the associated polymeric stresses. These include a variety of optical polarimetry methods, scattering techniques (light and neutron primarily), as well as strong optical microscopy and optical`tweezer' methods. The increasing interest in the rheology of polymer solutions and melts far from equilibrium. In these complex flow situations, molecular models serve as an invaluable guide to understanding the complicated physics of the processes. These include polymeric dynamics in strong extensional flows, melt flows in the`fast flow' regime where the reptation model is altered significantly by flowinduced molecular stretch, and polymeric flows near interfaces or in highly confined channels and pores where the flow dimension is comparable to the molecular dimension. The importance of developing new models for polymeric fluids with increasing material complexity such as polymeric liquid crystals or highly branched polymer melts. The meeting was divided into seven sessions, with a single session encompassing an entire morning or afternoon. Each session included a plenary talk followed by four to five topical presentations

Property preserving reformulation of constitutive laws for the conformation tensor

Theoretical and Computational Fluid Dynamics, 2018

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg numbers. This study presents a new approach to preserve the symmetric positive definiteness of the conformation tensor and to bound the magnitude of its eigenvalues. The idea behind this transformation is lies with the matrix logarithm formulation. Under the logarithmic transformation, the eigenvalue spectrum of the new conformation tensor varies from infinite positive to infinite negative. But, reconstruction the classical formulation from unbounded eigenvalues doesn't achieve meaningful results. This enhanced formulation, hyperbolic tangent, prevails the previous numerical failure by bounding the magnitude of eigenvalues in a manner that positive definite is always satisfied. In order to evaluate the capability of the hyperbolic tangent formulation we performed a numerical simulation of FENE-P fluids in a rectangular channel in the context of the finite element method. Under this new transformation, the maximum attainable Weissenberg number increases 21.4% and 112.5% comparing the standard log-conformation and classical constitutive equation respectively.