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# A Study of the Effect of Polymer Solution in Promoting Friction Reduction in Turbulent Channel Flow

[+] Author and Article Information
F. R. Cunha

Department of Mechanical Engineering,  University of Brasília, Campus Universitário, 70910-900, Brasília-DF, Brazil

M. Andreotti

Laboratoire des Ecoulements Géophysiques et Industriels, Institut National Polytechnique de Grenoble, Ecole Nationale Supérieure d’Hydraulique et de Mécanique de Grenoble, 1025, rue de la Piscine,  Domaine Universitaire, BP 95, 38402 Saint-Martin-d’Hères Cedex, France.

J. Fluids Eng 129(4), 491-505 (Nov 08, 2006) (15 pages) doi:10.1115/1.2436579 History: Received February 14, 2006; Revised November 08, 2006

## Abstract

In this work, turbulent drag reduction in a pipe is investigated by using laser Doppler velocimetry. The effect of decreasing the friction factor of the flow is obtained by addition of high molecular weight polymers. The mechanism of drag reduction is explained in terms of a stress anisotropy that inhibits the transversal transport of momentum by turbulent fluctuations. Semi-theoretical models based on a nonlinear constitutive equation, which takes into account an extra extensional rate of strain in the flow produced by the local additive orientation, are presented. The semi-theoretical models used to predict the friction factor of the flow in the presence of the polymer have successfully described the experimental measurements. The results have revealed a reduction in the friction factor of 65% for a concentration of $350ppm$ in volume of polyacrylamide (PAMA) in an aqueous solution. In addition, the flow statistics, such as the axial and radial velocity fluctuations, the normalized autocorrelation functions as well as the power spectra for both velocity fluctuation components, are examined for the Newtonian flow of pure water and the flow of a $120ppm$ solution of PAMA at the same friction velocity. Next, the results are compared in order to characterize the effect of the additive on the turbulent flow.

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Copyright © 2007 by American Society of Mechanical Engineers
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## Figures

Figure 1

Schematic experimental setup: (1) main reservoir, (2) digital balance, (3) secondary reservoir, (4) switch flow, (5) centrifuge pump, and (6) pressure gages

Figure 2

Schematic representation of a polyacrylamide macromolecule: (a) macromolecule in dynamic equilibrium, (b) macromolecule stretch by the flow, and (c) structural unity of polyacrylamide

Figure 3

Dimensionless effective shear viscosity of the solution as a function of shear rate for several values of particle volume fraction ϕ. The points represent +, ϕ=0; ×, ϕ=2×10−5; ∗, ϕ=4×10−5; ◻, ϕ=8×10−5; ∎, ϕ=1.2×10−4; ▵, ϕ=2×10−4; 엯, ϕ=3×10−4; ●, ϕ=3.5×10−4; and ▴, ϕ=4.5×10−4

Figure 4

Dimensionless effective shear viscosity as a function of the additive volume fraction. The solid line (1) fits the experimental data with the Einstein theory (30), where μ̃=μ(1+1.4×104ϕ); The dashed line (2) refers to Batchelor and Green’s theory (41), μ̃=μ(1+1.4×104ϕ−1.6×107ϕ2).

Figure 5

Dimensionless effective shear viscosity as a function of the experimentation time: ◻, γ̇=132s−1; ∎, γ̇=139s−1; 엯, γ̇=158s−1; ●, γ̇=178s−1; ▴: γ̇=198s−1; ▵: γ̇=211s−1; ×: γ̇=264s−1

Figure 6

Friction factor for dilute aqueous solutions of PAMA in the turbulent regime. Note that Ref1∕2∼Rew. The solid line represents the Kárman-Prandtl logarithmic law for smooth pipes, and the symbols represent: ●, ϕ=0; 엯, ϕ=8×10−5; ∗, ϕ=2×10−4; ◻, ϕ=3.5×10−4; ∎, ϕ=4.5×10−4.

Figure 7

The curve of maximum drag reduction for 350ppm. The solid line represents the Kárman-Prandtl logarithmic law for smooth pipes; the symbols ● denotes the friction factor measurements for pure water, ϕ=0ppm; the symbols ∎ denote the maximum friction factor measured by the present work and the upper dotted line corresponds to the asymptotic Virks (43) maximum drag reduction given by f1∕2=19log10(Ref1∕2)−32.4

Figure 8

The inverse of the square root of the friction factor as a function of the polymer volume fraction. Comparison between experimental data and the semi-theoretical models proposed in the present work for Re=105 and ℓ∕b=107: 엯, experimental data; solid line: constant shear rate additive function; ◻: logarithm additive function; ▵, iterative model of the additive function (numerical integration). The best-fit constant of the semi-empirical models was C=2×10−4.

Figure 9

The mean flow velocity profile as a function of wall distance: 엯, experimental data for pure water (ϕ=0); solid line denotes the Prandtl’s universal logarithmic law; ∎, constant shear rate additive function; ●, logarithmic shear rate additive function; ▴, iterative shear rate additive function (numerical solution); ◻, experimental data for ϕ=120ppm

Figure 10

Nondimensional velocity fluctuations for pure water: (a) axial fluctuations and (b) radial fluctuations

Figure 11

The nondimensional velocity fluctuations for a volume fraction ϕ=1.2×10−4 of the polymer solution: (a) axial fluctuations and (b) radial fluctuations

Figure 12

The mean flow velocity profile as a function of wall distance. ●, pure water (ϕ=0); dashed line denotes Prandtl universal logarithmic law; ◻, experimental data for (ϕ=120ppm); and the solid line denotes the numerical solution for our semi-empirical model. The solid curve of intersection represents the velocity profile of the viscous boundary layer.

Figure 13

Normalized velocity fluctuation autocorrelation function in the axial direction. The plots correspond to the flow of pure water at position y+=50. In the inset, the autocorrelation function has been plotted in a log-log scale in order to show the exponential decay of this function with a correlation time about 1∕250.

Figure 14

Normalized velocity fluctuation autocorrelation function in the radial direction. The plots correspond to the flow of pure water at position y+=50. In the inset, the autocorrelation function has been plotted in a log-log scale in order to show the exponential decay of this function with a correlation time about 1∕200.

Figure 15

Normalized velocity fluctuation autocorrelation function in the axial direction. The plots correspond to the flow of a 120ppm solution of PAMA at position y+=50. In the inset, the autocorrelation function has been plotted in a log-log scale in order to show the exponential decay of this function with a correlation time about 1∕185.

Figure 16

Normalized velocity fluctuation autocorrelation function in the radial direction. The plots correspond to the flow of a 120ppm solution of PAMA at position y+=50. In the inset, the autocorrelation function has been plotted in a log-log scale in order to show the exponential decay of this function with a correlation time about 1∕85.

Figure 17

Dimensionless diffusion coefficient associated with the radial component of the velocity fluctuations at the position y+=50. The solid line denotes the 120ppm solution of PAMA and dashed line, pure water.

Figure 18

Dimensionless power spectra at y+=50 as a function of the frequency for the axial component of the velocity fluctuations for pure water (▴) and the solution of 120ppm of PAMA (∎), measured with LDV

Figure 19

Dimensionless power spectra at y+=50 as a function of the frequency for the axial component of the velocity fluctuations for pure water (▴) and the solution of 120ppm of PAMA (∎), measured with LDV

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