Roney L. Thompson

A new constitutive equation and its performance in contraction flows

Abstract A new constitutive equation for incompressible materials is obtained by assuming that the stress tensor is an isotropic function of two kinematic quantities, namely, the rate-of-strain tensor and the relative-rate-of-rotation tensor. A representation theorem is employed to obtain the most general symmetric form of this function. The arising coefficients are assumed to be functions of the second invariants of the two tensors only. Because the second invariant of the relative-rate-of-rotation tensor is an indicator of the flow strength for several flows of engineering interest, the equation is thus sensitive to it. Forms of these functions are proposed, which lend to the constitutive equation the capability of fitting closely and independently data for shear viscosity, first normal stress coefficient, second normal stress coefficient, and extensional viscosity. This constitutive equation is used in conjunction with the equations of mass and momentum conservation to obtain the partial differential equations that govern the axisymmetric flow through a 4xa0:xa01 abrupt contraction. These differential equations are integrated using the finite volume method to obtain velocity, stress and flow-type fields. The effect on flow pattern of parameters related to normal stresses and extensional viscosity is investigated. It is observed that the vortex size increases when the level of extensional viscosity is increased, while it mildly decreases when the parameter related to normal stress coefficients is increased. Moreover, the stress power is highly sensitive to the normal stress parameter.

A general transformation procedure for differential viscoelastic models

A general transformation procedure (GTP) for viscoelastic differential constitutive equations is proposed. This procedure is devised to transform a differential model (e.g. Oldroyd-B, Whitex96Metzner (WM) and Phan-Thienx96Tanner (PTT)) into another model which has just one differential equation to be solved. The resulting model is suitable for three-dimensional and time-dependent flows. A new generalized objective time derivative is employed in the procedure, lending important features to the transformed model. For example, for the case of the Oldroyd-B model, the transformed model is capable of predicting non-zero second normal stress differences. Calculations of the stress field have been performed with an Oldroyd-B based GTP model (the GTP[OldB] model) for a 4:1 contraction plane flow. The results obtained have been analyzed with the aid of a flow-type classifier, and were found to be in good agreement with the ones obtained with the original model.

Elliptical, parabolic, and hyperbolic exchanges of energy in drag reducing plane Couette flows

In the present paper, we investigate the polymer–turbulence interaction by discriminating between the mechanical responses of this system to three different subdomains: elliptical, parabolic, and hyperbolic, corresponding to regions where the magnitude of vorticity is greater than, equal to, or less than the magnitude of the rate of strain, respectively, in accordance with the Q-criterion. Recently, it was recognized that hyperbolic structures play a crucial role in the drag reduction phenomenon of viscoelastic turbulent flows, thanks to the observation that hyperbolic structures, as well as vortical ones, are weakened by the action of polymers in turbulent flows in a process that can be referred to as flow parabolization. We employ direct numerical simulations of a viscoelastic finite extensible nonlinear elastic model with the Peterlin approximation to examine the transient evolution and statistically steady regimes of a plane Couette flow that has been perturbed from a laminar flow at an initial time a...

Analysis of uncertainties and convergence of the statistical quantities in turbulent wall-bounded flows by means of a physically based criterion

The present paper provides an analysis of the statistical uncertainties associated with direct numerical simulation (DNS) results and experimental data for turbulent channel and pipe flows, showing a new physically based quantification of these errors, to improve the determination of the statistical deviations between DNSs and experiments. The analysis is carried out using a recently proposed criterion by Thompson et al. [“A methodology to evaluate statistical errors in DNS data of plane channel flows,” Comput. Fluids 130, 1–7 (2016)] for fully turbulent plane channel flows, where the mean velocity error is estimated by considering the Reynolds stress tensor, and using the balance of the mean force equation. It also presents how the residual error evolves in time for a DNS of a plane channel flow, and the influence of the Reynolds number on its convergence rate. The root mean square of the residual error is shown in order to capture a single quantitative value of the error associated with the dimensionles...