A qualitative assessment of the role of a viscosity depending on the third invariant of the rate-of-deformation tensor upon turbulent non-Newtonian flow

Abstract

The numerical simulation of some non-Newtonian effects in wall and wall-free turbulent flows, such as drag reduction in pipe flows or the decrease in transverse normal Reynolds stresses, has been attempted in the past with a limited degree of success on the basis of modified wall functions applied to traditional turbulence models (k -m), rather than through more realistic rheological constitutive equations. In this work, it is qualitatively shown that if the viscosity function of a generalised Newtonian fluid is assumed to depend on the third invariant of the rate of deformation tensor, there is an increase of the viscous diffusion terms, but especially, of the dissipation of turbulence kinetic energy by a factor equal to the Trouton ratio of the fluid, divided by the Trouton ratio of the solvent, thus indicating a possible way to improve rheological-turbulence modelling.

Two new differential equations of turbulent dissipation rate and apparent viscosity for non-newtonian fluids

“…The k-ε equations are derived in [10][11] for power-law and Herschel-Bulkley fluid using the apparent viscosity of a non-Newtonian fluid in the RANS equations of a Newtonian fluid, but the agreement is not good enough. The introduction of the third invariant of the rate of deformation tensor in the viscosity contributes to an increase of viscous diffusion and dissipation rate in the turbulent kinetic energy confirming the dependence of the viscosity on the second invariant of the rate of deformation tensor in a 2D flow [12]. The Generalized Newtonian Fluid (GNF) constitutive equation is applied to a Bird-Carreau fluid in order to derive a k-ε model for the equations of Reynolds stresses tensor, turbulent kinetic energy and dissipation rate [13].…”

“…Viscosity is on the shear-rate, as shown in [12] in a 2D flow and done in all the papers found in the literature, without an explicit statement of the relation with the shear-rate. A 2D theory of turbulence has been developed since the 70's in [16][17][18][19][20] despite the differences with the real 3D flow and the absence of vortex-stretching terms.…”

“…Due to the difficulty of deriving the governing equations directly in a 3D flow, it is considered a guide to develop first a 2D turbulent model. The importance of studying a 2D model, which can be generalized in 3D, is also remarked in [12], where the authors developed an order of magnitude analysis for a 2D turbulent flow, without loss of generality.…”

Turbulent pipe flow of thixotropic fluids

“…Considering the differences in rig size and flow velocity, the relevant, energetic large scale rates of deformation (s) are larger in our rig than in theirs and this conclusion is reached assuming for simplicity that the fluids are Newtonian and following the arguments in Section 4.2.2 of Oliveira and Pinho (1998). For this estimate s can be either an area-average value of the rate of deformation or its maximum value near the wall.…”

“…For this estimate s can be either an area-average value of the rate of deformation or its maximum value near the wall. In either case Oliveira and Pinho (1998) arrived at expressions of sD=U ¼ gðf ; ReÞ, so that for the same Reynolds number one obtains the same friction factor and consequently s EP ¼ s=8. Alternatively, we could also conclude that s is larger in our rig than in Escudier and Presti's by equating the inviscid estimate of the rate of dissipation of turbulent kinetic energy (e ¼ u 03 =D) with the exact expression for the area-average rate of energy dissipation (hei)…”