The influence of flow-induced non-Newtonian fluid properties on turbulent drag reduction

Friction factors and velocity profiles in turbulent drag reduction can be compared to Newtonian fluid turbulence when the shear viscosity at the wall shear rate is used for the Reynolds number and the local shear viscosity is used for the non-dimensional wall distance. On this basis, an apparent maximum drag reduction asymptote is found which is independent of Reynolds number and type of drag reducing additive. However, no shear viscosity is able to account for the difference between the measured Reynolds stress and the Reynolds stress calculated from the mean velocity profile (the Reynolds stress deficit). If the appropriate local viscosity to use with the velocity fluctuation correlations includes an elongational component, the problem can be resolved. Taking the maximum drag reduction asymptote as a non-Newtonian flow, with this effective viscosity, leads to agreement with the concept of an asymptote only when the solvent viscosity is used in the non-dimensional wall distance.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

Laminar,transitional and turbulent annular flow of drag-reducing polymer solutions

Mean and rms axial velocity-profile data obtained using laser Doppler anemometry are presented together with pressure-drop data for the flow through a concentric annulus (radius ratio κ = 0.506) of a Newtonian (a glycerine–water mixture) and non-Newtonian fluids—a semi-rigid shear-thinning polymer (a xanthan gum) and a polymer known to exhibit a yield stress (carbopol). A wider range of Reynolds numbers for the transitional flow regime is observed for the more shear-thinning fluids. In marked contrast to the Newtonian fluid, the higher shear stress on the inner wall compared to the outer wall does not lead to earlier transition for the non-Newtonian fluids where more turbulent activity is observed in the outer wall region. The mean axial velocity profiles show a slight shift (～5%) of the location of the maximum velocity towards the outer pipe wall within the transitional regime only for the Newtonian fluid.     

A numerical study of the motion of a spherical drop rising in shear-thinning fluid systems

The dynamical processes of a Newtonian spherical drop rising freely through shear-thinning fluids expressed by the generalized Cross-Carreau (Carreau–Yasuda) model were considered experimentally and computationally. The local effects of shear-thinning on the drop motion, which are hard to evaluate from an experimental approach, are clearly revealed by the numerical results. The relation between the drop motion and the change in viscosity is clearly indicated and considered in detail for the cases in which the non-Newtonian fluids are highly and weakly pseudoplastic. In addition, the modified Reynolds and Morton numbers are discussed in order to describe the drop or bubble free motion in shear-thinning fluids using a more convenient and practical form.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

A physically based correlation for the effects of power law rheology and inclination on slug bubble rise velocity

A study has been made of the motion of long bubbles in inclined pipes containing viscous Newtonian and non-Newtonian liquids. A semi-theoretical expression for the rise velocity of air bubbles in water is derived on the hypothesis that the dominant factor is the momentum exchange of the bubble underflow, i.e. the bubble nose shape. The correlation calls on empirical inputs from established literature on bubble rise speeds at high Reynolds number. The effects of increasing Newtonian viscosity are analysed with reference to the momentum exchange and it is shown how viscosity reduces the inclination dependence of the bubble Froude number. Results from an experimental survey in seven different non-Newtonian liquids in three different diameter pipes are presented. These data are correlated so as to decouple the effects of surface tension and viscosity. An empirical relation is proposed for the effective shear rate in the fluid travelling around the bubble nose. Our correlation is compared to literature data from a broad range of Reynolds numbers with excellent agreement except at shallow angles.     

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.     

The stratified flow of gas and non-newtonian liquid in horizontal pipes

Predictions of pressure drop and holdup are presented for the stratified flow of gas and non-Newtonian liquid obeying the Ostwald-de Waele power law model. The model of Taitel & Dukler (1976) for gas/Newtonian liquid flow is extended to liquids possessing either shear-thinning or shear-thickening laminar flow behaviour and computed results are given for flow behaviour indices in the range $\text{0.1 ≤ n ≤ 2}$. In particular, conditions are defined for drag reduction of the liquid flow by the presence of the gas. It is concluded that drag reduction occurs over the largest ranges of liquid and gas flow rates at the lowest $\text{n}$ values, provided that liquid flow remains laminar, but that maximum drag reduction may be expected for shear-thickening liquids with $\text{n}$ values of 2 or greater. Ratios of the liquid flow rate in the presence of gas to that for liquid flow alone under a constant pressure gradient are also presented. These ratios frequently exceed unity and are greatest for highly shear-thinning liquids.Although the Taitel & Dukler approach is consistent with experiments on gas/Newtonian liquid flow, and, in addition, appears to be valid for immiscible Newtonian liquid-liquid systems, provided that the viscosity ratio of the two phases is at least five, experiments are required to confirm its applicability for gas/non-Newtonian systems.     

An investigation of non-newtonian flow past a sphere

Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finite-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60.Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV_∞P/η₀, where η₀ is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines.     

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

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–ε), 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.     

Numerical simulation of the non-Newtonian blood flow through a mechanical aortic valve

This work focuses on the comparison between Newtonian and non-Newtonian blood flows through a bileaflet mechanical heart valve in the aortic root. The blood, in fact, is a concentrated suspension of cells, mainly red blood cells, in a Newtonian matrix, the plasma, and consequently its overall behavior is that of a non-Newtonian fluid owing to the action of the cells’ membrane on the fluid part. The common practice, however, assumes the blood in large vessels as a Newtonian fluid since the shear rate is generally high and the effective viscosity becomes independent of the former. In this paper, we show that this is not always the case even in the aorta, the largest artery of the systemic circulation, owing to the pulsatile and transitional nature of the flow. Unexpectedly, for most of the pulsating cycle and in a large part of the fluid volume, the shear rate is smaller than the threshold level for the blood to display a constant effective viscosity and its shear thinning character might affect the system dynamics. A direct inspection of the various flow features has shown that the valve dynamics, the transvalvular pressure drop and the large-scale features of the flow are very similar for the Newtonian and non-Newtonian fluid models. On the other hand, the mechanical damage of the red blood cells (hemolysis), induced by the altered stress values in the flow, is larger for the non-Newtonian fluid model than for the Newtonian one.     

A unified theory of turbulent flow

Summary A general theory of turbulent flow is applied to incompressible flow in a circular pipe. The theoretical mean velocity distribution is found to be in good agreement with experiment, but there is some discrepancy in the normal stress distribution. The available pressure drop data are used to estimate the value of the apparent wall velocity as a function of Reynolds number and roughness. It is found that the results can be represented by simple expressions which in turn imply simple expressions for the pressure drop as a function of Reynolds number and roughness. However, it has not been possible to derive these results from fundamental considerations. The basis of Reynolds analogy and the application of the theory to channel flow are also discussed.     

Studies on two-phase co-current air/non-Newtonian shear-thinning fluid flows in inclined smooth pipes

In this work, co-current flow characteristics of air/non-Newtonian liquid systems in inclined smooth pipes are studied experimentally and theoretically using transparent tubes of 20, 40 and 60 mm in diameter. Each tube includes two 10 m long pipe branches connected by a U-bend that is capable of being inclined to any angle, from a completely horizontal to a fully vertical position. The flow rate of each phase is varied over a wide range. The studied flow phenomena are bubbly flow, stratified flow, plug flow, slug flow, churn flow and annular flow. These are observed and recorded by a high-speed camera over a wide range of operating conditions. The effects of the liquid phase properties, the inclination angle and the pipe diameter on two-phase flow characteristics are systematically studied. The Heywood–Charles model for horizontal flow was modified to accommodate stratified flow in inclined pipes, taking into account the average void fraction and pressure drop of the mixture flow of a gas/non-Newtonian liquid. The pressure drop gradient model of Taitel and Barnea for a gas/Newtonian liquid slug flow was extended to include liquids possessing shear-thinning flow behaviour in inclined pipes. The comparison of the predicted values with the experimental data shows that the models presented here provide a reasonable estimate of the average void fraction and the corresponding pressure drop for the mixture flow of a gas/non-Newtonian liquid.     

Evaluation of flush mounted hot-film sensors for skin friction reduction measurements in viscoelastic polymer solutions

The purpose of this investigation was to evaluate the performance of flush mounted hot-film sensors for mean wall shear stress measurement in turbulent flows of dilute drag reducing polymer solution. A series of pipe flow expriments were conducted over a range of Reynolds numbers and polymer solution concentrations to compare the level of skin friction drag reduction measured by hot-film sensors with values calculated from pipe pressure drop. It is shown that water calibrated hot-film sensors consistently underestimate the wall shear stress suggesting that Reynolds analogy is not valid in dilute polymer solutions. The Newtonian form of the relationship between the wall shear stress and the heat transfer remains valid in dilute polymer solutions. However, multiplicative and additive factors in the relationship are shown to increase linearly with the logarithm of the polymer concentration.     

The effect of the expansion ratio on a turbulent non-Newtonian recirculating flow

 Measurements of the mean and turbulent flow characteristics of shear-thinning moderately elastic 0.1% and 0.2% xanthan gum aqueous solutions were carried out in a sudden expansion having a diameter ratio of 2. The inlet flow was turbulent and fully developed, and the results were compared with data for water in the same geometry and with previous published Newtonian and non-Newtonian data in a smaller expansion of diameter ratio equal to 1.538. An increase in expansion ratio led to an increase in the recirculation length and in the axial normal Reynolds stress at identical normalised locations, but the difference between Newtonian and non-Newtonian characteristics was less intense than in the smaller expansion. An extensive comparison of mean and turbulent flow characteristics was carried out in order to understand the variation of flow features. Received: 31 July 2000 / Accepted: 27 August 2001     

Anomalous predictions of pressure drop and heat transfer in ducts of arbitrary cross-section with modified power law fluids

 The apparent viscosities of purely viscous non-Newtonian fluids are shear rate dependent. At low shear rates, many of such fluids exhibit Newtonian behaviour while at higher shear rates non-Newtonian, power law characteristics exist. Between these two ranges, the fluid's viscous properties are neither Newtonian or power law. Utilizing an apparent viscosity constitutive equation called the “Modified Power Law” which accounts for the above behavior, solutions have been obtained for forced convection flows. A shear rate similarity parameter is identified which specifies both the shear rate range for a given fluid and set of operating conditions and the appropriate solution for that range. The results of numerical solutions for the friction factor–Reynolds number product and for the Nusselt number as a function of a dimensionless shear rate parameter have been presented for forced fully developed laminer duct flows of different cross-sections with modified power law fluids. Experimental data is also presented showing the suitability of the “Modified Power Law” constitutive equation to represent the apparent viscosity of various polymer solutions. Received on 21 August 2000     

Laminar length of a non-Newtonian fluid jet

The laminar length of a submerged jet of a non-Newtonian fluid is measured employing a flow visualization technique, which makes use of the birefringent property of the fluid and a circular polariscope. Fluids of two different concentrations are studied. The results indicate that the concentration, hence the non-Newtonian nature of the fluid, has no influence on the laminar length of the jet for 600 〈 Re 〈 1100. In the Reynolds number range of 50 to 200 the laminar length is affected by the viscous properties of the fluid. The results also indicate that there is not much difference in the laminar length of a Newtonian and non-Newtonian jet for 600 〈 Re 〈 1100.The experimental facility is also used as a “falling head” capillary viscometer. The laminar length data and the viscosity data are taken simultaneously. The viscosity obtained from the present test facility is found to agree with that obtained using a standard rotary viscometer in trend only.     

A correlation for yield stress fluid flow through packed beds

Relatively few correlations are available for non-Newtonian fluid flows through packed beds, even though such fluids are frequently used in industry. In this paper, a correlation is presented for yield stress fluid flow through packed beds. The correlation is developed by introducing the yield stress model in place of the Newtonian model used in deriving Erguns equation. The resulting model has three parameters that are functions of the geometry and roughness of the particle surfaces. Two of the parameters can be deduced in the limit as the yield stress becomes negligible and the model reduces to Erguns equation for Newtonian fluids. The third model parameter is determined from experimental data. The correlation relates a defined friction factor to the dimensionless Reynolds and Hedstrom numbers and can be used to predict pressure drop for flow of a yield stress fluid through a packed bed of spherical particles. Conditions for flow or no-flow are also determined in the correlation. Comparison of model calculations, between a Newtonian and a yield stress fluid for flow penetration into a packed bed of spheres, shows the yield stress fluid initially performs similar to the Newtonian fluid, at large Reynolds numbers. At lower Reynolds numbers the yield stress effect becomes important and the flow rate significantly decreases when compared to the Newtonian fluid.     

Computational Modelling of Non-Newtonian Effects on Flow in Channels with Moving Wall Indentations

Non-Newtonian effects in a channel with moving wall indentations are assessed numerically by a finite volume method for solving the unsteady incompressible Navier-Stokes equations and using a power-law model exhibiting shear thinning viscosity and Casson's model as the constitutive equations for the non-Newtonian fluid. The computations show that for a non-Newtonian fluid, there are differences in the velocity profiles and in the structure and size of the reversed flow regions as compared with the corresponding Newtonian fluid. The comparison of non-Newtonian and Newtonian wall shear stress reveals a slight decrease in the magnitude on the average for the non-Newtonian case, eventually resulting in the strength of the “wave train” being slightly weaker than those corresponding to a Newtonian fluid.