Laser Doppler velocimetry (LDV) and video flow visualization are used to investigate the creeping motion of a highly elastic, constant-viscosity fluid flowing past a cylinder mounted centrally in a rectangular channel. A sequence of viscoelastic flow transitions are documented as the volumetric flow rate past the cylinder is increased and elastic effects in the fluid become increasingly important. Velocity profiles clearly show that elasticity has almost no effect on the kinematics upstream of the cylinder, but that the streamlines in the wake of the cylinder are gradually shifted further downstream . Finite element calculations with a nonlinear constitutive model closely reproduce the evolution of the steady two-dimensional velocity field. However, at a well defined set of flow conditions the steady planar stagnation ow in the downstream wake is experimentally observed to become unstable to a steady, three-dimensional cellular structure. The Reynolds number at the onset of the flow instability is less than 0.05 and inertia plays little role in the flow transition, LDV measurements in the wake close to the cylinder reveal large spatially periodic fluctuations of the streamwise velocity that extend along the length of the cylinder and more than five cylinder radii downstream of the cylinder. Fourier analysis shows that the characteristic spatial wavelength of these flow perturbations scales closely with the cylinder radius R . Flow visualization combined with LDV measurements also indicates that the perturbations in the velocity field are confined to the narrow region of strongly extensional flow near the downstream stagnation point. A second flow transition is observed at higher flow rates that leads to steady translation of the cellular structure along the length of the cylinder and time-dependent velocity oscillations in the wake. Measurements of the flow instability are presented for a range of cylinder sizes, and a stability diagram is constructed which shows that the onset point of the wake instability depends on both the extensional deformation of the fluid in the stagnation flow and the shearing flow between the cylinder and the channel.

The wake instability in viscoelastic flow past confined circular cylinders

Publisher Summary This chapter focuses on heat transfer behavior of viscoelastic fluid in turbulent pipe flow Although the asymptotic values of the heat transfer and friction factor can be calculated, there exist no firm criteria for determining whether asymptotic conditions exist Predictions of the intermediate values of the friction and heat transfer are not yet possible, even if the rheology and the thermal properties of the aqueous polymer solution are known To deals with the problems, the Weissenberg or Deborah number has to be taken into account The behavior of viscoelastic fluids flowing turbulently in noncircular channels or over external surfaces represents a relatively unexplored area of fluid mechanics Open channel flow of viscoelastic fluid is another interesting field currently being investigated The chapter concludes that to approach turbulent heat transfer behavior analytically, the usual and simplest method is to solve the uncoupled energy equation using the empirically determined velocity profile Hence, it is essential to understand the fluid mechanics of non-Newtonian fluids as well as the rheology

Non-Newtonian Fluids in Circular Pipe Flow

A low Reynolds number k–e turbulence model is developed for predicting turbulent wall flows of viscoelastic fluids. The model uses a non-linear molecular viscosity that is affected by the turbulent fluctuations and a new damping function is introduced to account for near-wall effects. This new function was made equal to the eddy viscosity damping function which was derived taking into account viscometric and elastic effects. Flow predictions compare favourably with results from experiments with several viscoelastic fluids, especially the friction factor and the mean velocity. Comparisons of the turbulence kinetic energy are less good, but the model is able to capture the shift of the peak turbulence kinetic energy and rate of dissipation away from the wall, the decrease in those peak values and of the production of k, and the Reynolds shear stress deficit across the pipe. An advantage of the present single-point turbulence closure relative to previous attempts at modelling polymer drag reduction, is the fact that here the input is only the mean velocity and fluid properties and no other modifications of the turbulence model are required to deal with different fluids. Further developments of the turbulence model are suggested at the end.

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Turbulent pipe flow predictions with a low Reynolds number k–ε model for drag reducing fluids

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–e 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]. © 2003 Elsevier B.V. All rights reserved.

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A GNF framework for turbulent flow models of drag reducing fluids and proposal for a k–ε type closure

Remarkable power saving in a fluid transport system is possible if the surfactant drag reduction technology is used Application of surfactant drag reduction to district heating and cooling systems has been investigated in the past The establishment of the scale-up law in drag-reducing pipe flows is one of the most important problems in this application Main purpose of this study is aimed to develop a reliable scale-up law in surfactant drag-reducing flows As the basic data of surfactant solutions, both non-Newtonian viscosity and viscoelasticity were experimentally determined A turbulent eddy diffusivity model based on the Maxwell model was employed to estimate the drag reduction of surfactant solutions The predictions by the turbulence model developed in this study with proper rheological characteristics of surfactant solutions has resulted in a reliable estimation of the pipe diameter effect in surfactant drag-reducing flows over the pipe diameter range from 11 to 150mm

On pipe diameter effects in surfactant drag-reducing pipe flows

Literature Cited 1) Astakhov, V.A., M. M. Dubinin and P. G. Romankov: Theor. Osn. Khim. Techn., 3, 292 (1969) 2) Dubinin, M. M.: Chem. Rev., 60, 235 (1960) 3) Dubinin, M. M. : \"Chemistry and Physics of Carbon\", Vol. 2 (P. L. Walker ed.) p. 51, Marcel Dekker (1966) 4) Dubinin, M. M.: \"Adsorption-Desorption Phenomena\" (Ricca ed.), p. 3, Academic Press (1972) 5) Dubinin, M. M. and V. A. Astakhov: \"Molecular Sieve Zeolites II\", Adv. Ser. 102, p 69 (1971) 6) Eguchi, Y.: Kagaku Kojo, 13, No. 9, 45 (1969) 7) Eguchi, Y.: /. Japan Petrol Inst., 13, 106 (1970) 8) Kawazoe, K., Y. A. Astakhov, T. Kawai and Y. Eguchi: Kagaku Kogaku, 35, 1006 (1971) 9) Kawazoe, K. and T. Kawai: Seisan Kenkyu, 25, 513 (1973)

Turbulent pipe flow of dilute polymer solutions

The dynamic equation with the generalized Maxwell model is solved numerically in moderate Reynolds number range. The results of calculation reveal the effect of elasticity in cross flow around a circular cylinder. The electrochemical method is used for the measurement of Sherwood number in cross flow of dilute polymer solutions. Comparison of calculated values with the experimental results shows good agreement.

Hiromoto Usui

Papers

Turbulent pipe flow of dilute polymer solutions

Transport phenomena of viscoelastic fluid in cross flow around a circular cylinder