F.T. Pinho

Numerical simulation of non-linear elastic flows with a general collocated finite-volume method

This paper reports the development and application of a finite-volume based methodology for the calculation of the flow of fluids which follow differential viscoelastic constitutive models. The novelty of the method lies on the use of the non-staggered grid arrangement, in which all dependent variables are located at the center of the control volumes, thus greatly simplifying the adoption of general curvilinear coordinates. The pressure‐velocity‐stress decoupling was removed by the development of a new interpolation technique inspired on that of Rhie and Chow, AIAA 82 (1982) 998. The differencing schemes are second order accurate and the resulting algebraic equations for each variable are solved in a segregated way (decoupled scheme). The numerical formulation especially designed for the interpolation of the stress field was found to work well and is shown to be indispensable for accurate results. Calculations have been carried out for two problems: the entry flow problem of Eggleton et al., J. Non-Newtonian Fluid Mech. 64 (1996) 269, with orthogonal and non-orthogonal meshes; and the bounded and unbounded flows around a circular cylinder. The results of the simulations compare favourably with those in the literature and iterative convergence has been attained for Deborah and Reynolds numbers similar to, or higher than, those reported for identical flow problems using other numerical methods. The application of the method with non-orthogonal coordinates is demonstrated. The entry flow problem is studied in more detail and for this case differences between Newtonian and viscoelastic fluids are identified and discussed. Viscoelasticity is shown to be responsible for the development of very intense normal stresses, which are tensile in the wall region. As a consequence, the viscoelastic fluid is more intensely decelerated in the wall region than the Newtonian fluid, thus reducing locally the shear rates and the role of viscosity in redeveloping the flow. A layer of high stress-gradients is formed at the wall leading edge and is convected below and away from the wall; its effect is to intensify the aforementioned deviation of elastic fluid from the wall.

Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions

The paper presents very accurate numerical results for the vortex size, the vortex intensity and the Couette correction, in planar contraction flows of Oldroyd-B and PTT fluids ( e = 0.25) with both the linear and the exponential stress function, and with a solvent viscosity ratio equal to 1/9. The accuracy of these results is quantified, being generally below 1% (0.3% for most results), and the finest mesh employed had over 1 million degrees of freedom. Such degree of mesh fineness is shown to be required for accurate results with the Oldroyd-B fluid, especially at high Deborah numbers, but the shear-thinning PTT fluid in general does not require the finest meshes. In terms of level of elasticity, steady results for the PTT fluid could be obtained for values of the Deborah number in excess of 100 (linear PTT) and 10,000 (exponential PTT).

Flow of non-Newtonian fluids in a pipe

Abstract Measurements of mean axial velocity and of the three normal stresses have been obtained in fully developed pipe-flow with four concentrations of a polymer (sodium carboxymethyl cellulose) in aqueous solution and with water and viscous Newtonian fluids encompassing a range of Reynolds numbers from 240 to 111,000. The results quantify the delay in transition from laminar to turbulent flow caused by shear-thinning, the suppression of turbulent fluctuations particularly in the radial and tangential components of normal stress, and the drag reduction at the higher Reynolds numbers. They also confirm that the maximum drag reduction asymptote is appropriate to these shear-thinning solutions.

Fully developed laminar flow of purely viscous non-Newtonian liquids through annuli, including the effects of eccentricity and inner-cylinder rotation

The results are presented of extensive numerical calculations, carried out using a highly accurate finite-volume method, for the fully developed laminar flow of an inelastic shear-thinning power-law fluid through an eccentric annulus with inner cylinder rotation. Additional calculations are reported for more complex rheological models, including Cross, Carreau and Herschel–Bulkley, which we relate systematically to the power-law model. Comparisons are made with the results of other recent numerical studies. An extensive bibliography is appended of 100 papers additional to those specifically referenced and concerned with theoretical and numerical investigations of laminar flow of non-Newtonian fluids through annular channels. 2002 Elsevier Science Inc. All rights reserved.

Analytical solution for fully developed channel and Pipe Flow of Phan-Thien/Tanner Fluids

Analytical expressions are derived for the velocity vector, the stress components and the viscosity function in fully developed channel and pipe flow of Phan-Thien–Tanner (PTT) fluids; both the linearized and the exponential forms of the PTT equation are considered. The solution shows that the wall shear stress of a PTT fluid is substantially smaller than the corresponding value for a Newtonian or upper-convected Maxwell fluid, with implications for comparing predicted and measured values in a non-dimensional form.

The flow of viscoelastic fluids past a cylinder: finite-volume high-resolution methods

Accurate solutions are obtained with the numerical method of Oliveira et al. [J. Non-Newtonian Fluid Mech. 79 (1998) 1] for the inertialess plane flow around a confined cylinder. This numerical procedure is based on the finite-volume method in non-orthogonal block-structured meshes with a collocated arrangement of the dependent variables, and makes use of a special interpolation practice to avoid stress‐velocity decoupling. Two high-resolution schemes (MINMOD and SMART) are implemented to represent the convective terms in the constitutive equations for the upper convected Maxwell and Oldroyd-B fluids, and the resulting predictions of the drag coefficient on the cylinder are shown to be as accurate as existing finite-element method predictions based on the supposedly very accurate h-p refinement technique. Numerical uncertainties are quantified with help of Richardson’s extrapolation technique and the orders of convergence of the differencing schemes are established and shown to be second-order accurate. Calculations performed with a wake-refined mesh predicted the variation of the maximum longitudinal normal stress in the wake as De 3 and De 5 depending on Deborah number.

Effect of a high-resolution differencing scheme on finite-volume predictions of viscoelastic flows

Improved accuracy and enhanced convergence rate are achieved when a finite-volume method (FVM) is used in conjunction with a high-resolution scheme (MINMOD) to represent the stress derivatives in the constitutive equation, because it avoids oscillations of the solution field near sharp stress gradients. Calculations for the benchmark flow of an upper-convected Maxwell fluid through a 4:1 plane contraction were carried out at a constant Reynolds number of 0.01 and varying Deborah numbers in four consistently refined meshes, the finest of which had a normalised cell size of 0.005 in the vicinity of the re-entrant corner. The MINMOD scheme was able to provide converged solutions up to Deborah numbers well beyond those attained by other second-order accurate schemes. The asymptotic behaviour of velocity and stresses near the re-entrant corner was accurately predicted as compared with Hinch’s theory [1]. The simulations improved previous results for the same flow conditions obtained with less accurate schemes, and the present results can be used as benchmark values up to a Deborah value of 3 with quantified numerical uncertainties.

Analysis of forced convection in pipes and channels with the simplified Phan-Thien–Tanner fluid

Abstract Analytical solutions are derived for the temperature distribution and heat transfer coefficient in forced convection of a viscoelastic fluid obeying the simplified Phan-Thien–Tanner constitutive equation in laminar pipe and plane channel flows. The results are valid for fully developed thermal and hydrodynamic flow conditions with a constant heat flux imposed at the wall and include the investigation of the effects of viscous dissipation. A nonvanishing value of the extensional parameter of the fluid model is shown to be essential for the solution to differ significantly from that for a Newtonian, or an elastic fluid without extensibility. Elasticity, only when combined with extensibility, is shown to increase the heat transfer and to reduce the range of temperatures present inside a duct. These beneficial effects of fluid elasticity are enhanced by viscous dissipation.

Study of steady pipe and channel flows of a single-mode Phan-Thien–Tanner fluid

Analytical solutions are derived for the steady state channel and pipe flows of viscoelastic fluids obeying the complete single-mode Phan-Thien–Tanner (PTT) constitutive equation with a linear stress coefficient in the absence of a solvent viscosity contribution. The results include the profiles of all relevant stresses, the axial velocity and the viscosity across the gap. The three material functions of the single-mode PTT model in steady Couette flow are also given and it is shown that the conditions of the maximum point in the shear stress versus shear rate curve are related to the conditions for existence of steady state solutions in the channel and pipe flows. The range of model parameters for which a classical steady solution exists is established.

On the reproducibility of the rheology of shear-thinning liquids

The independent analysis of flow measurements is frequently hampered by incomplete characterisation of the working fluid. This problem is particularly acute in situations which require working fluids with identical properties, such as the development of scaling laws for the turbulent flow of drag-reducing liquids. In this paper, we demonstrate that the viscometric viscosity, loss and storage moduli for two of the most common polymers used for flow experiments, carboxymethylcellulose (CMC) and xanthan gum (XG), are practically insensitive to the chemistry of the tap water used as a solvent, to the method of mixing, and to the biocide added. However, the properties of CMC from two different manufacturers were found to be significantly different, whereas there was no difference between XG solutions prepared from different batches from the same manufacturer. Our conclusion is that for a given concentration in water, the properties of certain non-Newtonian liquids, such as CMC and XG, are essentially fixed and reproducible. Although the situation is less than ideal, comparisons of fluid-flow data from entirely independent laboratories can thus be made even in the absence of direct rheological measurements.