Two new differential equations of turbulent dissipation rate and apparent viscosity for non-newtonian fluids (original) (raw)
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International Communications in Heat and Mass Transfer, 2012
The Flow of inelastic Non-Newtonian fluids is involved in many biological and industrial applications like nanofluids. Despite many years have passed since the beginning of the study of turbulent Non-Newtonian fluids, most of the studies carried out focus the attention on viscoelastic-fluids. In order to make accurate and low-cost prediction on turbulent inelastic Non-Newtonian fluids flow, a RANS Generalized Newtonian Fluid (GNF) turbulence model is required based on exact transport equation of turbulent variables. In a previous paper [52] we achieved the exact transport equations for turbulent kinetic energy and dissipation rate through the introduction of an apparent viscosity transport equation in 2D case for sake of simplicity. The object of this paper is to extend the results given in [52] in 3D case giving the full mathematical demonstration of the exact-equations. The modelling of the unknown terms it is left for a future work.
A Model for the Effect of Turbulence on the Molecular Viscosity of Generalized Newtonian Fluids
2003
A model is derived to account for the effect of turbulence on the molecular viscosity of a purely viscometric fluid obeying a power-law equation. The Generalized Newtonian model is modified to mimic extensional effects and this introduces a second kinematic invariant into the viscosity equation. The dependence of viscosity on the strain rate invariant is also non-linear and consequently affected by turbulene. The molecular viscosity model derived initially is modified to account for this extra dependence.
Evaluating flows of non—Newtonian fluids by the method of equivalent Newtonian viscosity
AIChE Journal, 1975
Many flow problems arising in engineering design of polymer piocessing equipment are approximated as fully developed isothermal pressure flows in variously shaped geometrical conduits such as parallel plates, tubes, etc. . The rheological behavior of the melts in such flows can often be adequately described as pseudoplastic or shear dependent, but time independent and inelastic. Every polymer melt has a different shear rate dependence. In a typical flow curve (that is, shear stress or non-Newtonian viscosity versus shear rate curves), the non-Newtonian viscosity continuously drops with increasing shear rate from an asymptotic value at low shear rates. The most commonly used non-Newtonian constitutive equation, the power law model, although of great practical value, cannot accurately represent the rheological behavior of the melt over a wide range of shear rates. It fails in particular in the low shear rate range, and since the shear rates in pressure flow extend from zero at the center to a maximum value at the wall, by using the power law model to calculate the flow rate, pressure gradient relationship, an unavoidable error, is introduced into the calculation. One obvious way to partly overcome this problem would be to employ another constitutive equation which is free from the power law model deficiency, for example, the Ellis model, which is a 3-parameter model. Such models, however, being nonlinear pose some problems regarding the statistical reliability of determining the best values of the parameters. The inaccuracies in calculating flow rates with the aid of relatively simple constitutive equations can be avoided, of course, by numerical integration of the complete experimentally determined flow curve. This is, however, a tedious or time consuming process and it will be shown below how by defining an Equivalent Newtonian Viscosity (ENV) it can be avoided after evaluating once a set of coefficients. The method will be demonstrated on parallel plate geometry.
Flow of a generalized second grade non-Newtonian fluid with variable viscosity
Continuum Mechanics and Thermodynamics, 2004
A modified constitutive equation for a second grade fluid is proposed so that the model would be suitable for studies where shear-thinning (or shear-thickening) may occur. In addition, the dependence of viscosity on the temperature follows the Reynolds equation. In this paper, we propose a constitutive relation, (18), which has the basic structure of a second grade fluid, where the viscosity is now a function of temperature, shear rate, and concentration. As a special case, we solve the fully developed flow of a non-Newtonian fluid given by (11), where the effects of concentration are neglected.
A Model of Averaged Molecular Viscosity for Turbulent Flow of Non-Newtonian Fluids
2014
A novel turbulence model for flows of viscoplastic fluid is presented. It is based on the Reynolds-Averaged approach. A closed model for the averaged viscosity that takes into account its nonlinear dependence on the fluctuating rate of deformation tensor is proposed. Test calculations were performed for power-law fluid and Herschel–Bulkley fluid flows in a straight round pipe. Numerical data obtained with the use of the proposed model are compared with the results of direct numerical simulations. The proposed model adequately describes the reduction in the turbulent transport of momentum with decreasing power-law index and with increasing yield stress of the fluid.
Journal of Non-Newtonian Fluid Mechanics, 2003
Based on a generalised Newtonian fluid (GNF) model, modified to account for strain-thickening of the extensional viscosity, this paper derives transport equations for mass, momentum, Reynolds stresses, turbulent kinetic energy and its rate of dissipation. An analysis of order of magnitude identifies the relevant new terms and suggestions are made to model those terms needed to ensure closure in the perspective of a low Reynolds number k-ε model. Specifically, a closed model for the time-average viscosity is proposed that takes into account its non-linearity and dependence on the second and third invariants of the fluctuating rate of deformation tensor. The turbulence model is qualitatively shown to increase the rate of decay of turbulent kinetic energy in isotropic grid turbulence for certain rheological conditions. The performance of the turbulence model in a pipe flow is assessed in a companion paper by Cruz and Pinho [J. Non-Newtonian Fluid Mech., in press].
A compressible turbulence model for the dissipation rate
Thermal Science
In this work, the ability of a Reynolds stress model to compute turbulent homogeneous shear flow with significant compressibility effects is discussed. Several studies of compressible turbulent flows carried out in the past years have shown that the pressure strain correlation is mainly responsible for the strong changes in the magnitude of the Reynolds stress anisotropies. Two recent compressible models of this term are considered in conjunction with the standard model of the dissipation rate of the turbulent kinetic energy to predict compressible homogeneous flow highly sheared are tested. It is found that deficiencies appear in the calculations even if the pressure strain model is improved by compressibility corrections. Consistent with earlier studies, this deficiency is attributed to the use of the incompressible model for turbulent dissipation. However, a compressibility correction of this equation model uncovers the main focus of the present study. This correction makes the s...
Turbulence modeling based on non-Newtonian constitutive laws
2011
This work revisits the analogy between Newtonian turbulence and non-Newtonian laminar flows. Several direct numerical simulations (DNS) data of a plane channel flow, for a large range of Reynolds numbers (180 ≤ Reτ ≤ 2000) were explored. The profiles of mean velocity and second moment quantities were used to extract viscometric functions in the non-Newtonian modeling framework. The Reynolds stress tensor is expressed in terms of a set of basis kinematic tensors based on a projection of a nonlinear framework. The coefficients of the model are given as functions of the intensity of the mean strain tensor. The apparent eddy turbulent viscosity, the first and second normal stress differences are presented as function of the shear rate. One of the advantages of the new algebraic nonlinear power law constitutive equation derived in the paper, is that is only dependent on the mean velocity gradient and can be integrated up to the wall.
On the connection between Maximum Drag Reduction and Newtonian fluid flow
To date, the most successful turbulence control technique is the dissolution of certain rheology-modifying additives in liquid flows, which results in a universal maximum drag reduction (MDR) asymptote. The MDR asymptote is a well-known phenomenon in the turbulent flow of complex fluids; yet recent direct numerical simulations of Newtonian fluid flow have identified time intervals showing key features of MDR. These intervals have been termed "hibernating turbulence" and are a weak turbulence state which is characterised by low wall-shear stress and weak vortical flow structures. Here, in this experimental investigation, we monitor the instantaneous wall-shear stress in a fully-developed turbulent channel flow of a Newtonian fluid with a hot-film probe whilst simultaneously measuring the streamwise velocity at various distances above the wall with laser Doppler velocimetry. We show, by conditionally sampling the streamwise velocity during low wall-shear stress events, that the MDR velocity profile is approached in an additive-free, Newtonian fluid flow. This result corroborates recent numerical investigations, which suggest that the MDR asymptote in polymer solutions is closely connected to weak, transient Newtonian flow structures.
Non-isotropic dissipation in non-homogeneous turbulence
Journal of Fluid Mechanics
On the basis of the two-point velocity correlation equation a new tensor lengthscale equation and in turn a dissipation rate tensor equation and the pressurestrain correlation are derived by means of asymptotic analysis and frame-invariance considerations. The new dissipation rate tensor equation can account for non-isotropy effects of the dissipation rate and streamline curvature. The entire analysis is valid for incompressible as well as for compressible turbulence in the limit of small Mach numbers. The pressure-strain correlation is expressed as a functional of the two-point correlation, leading to an extended compressible version of the linear formulation of the pressure-strain correlation. In this turbulence modelling approach the only terms which still need ad hoc closure assumptions are the triple correlation of the fluctuating velocities and a tensor relation between the length scale and the dissipation rate tensor. Hence, a consistent formulation of the return term in the pressure-strain correlation and the dissipation tensor equation is achieved. The model has been integrated numerically for several different homogeneous and inhomogeneous test cases and results are compared with DNS, LES and experimental data.