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TECHNICAL PAPERS

# On the Mechanism Responsible for Turbulent Drag Reduction by Dilute Addition of High Polymers: Theory, Experiments, Simulations, and Predictions

[+] Author and Article Information
J. Jovanović, M. Pashtrapanska, B. Frohnapfel, F. Durst

Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, Cauerstrasse 4, 91058 Erlangen, Germany

J. Koskinen

Neste Jacobs Oy, POB 310, 06101 Porvoo, Finland

K. Koskinen

Outokumpu Research Oy, POB 60, 28101 Pori, Finland

In 1,2,3,4$Pij=−uiuk¯∂U¯j∕∂xk−ujuk¯∂U¯i∕∂xk$ represents the production of the turbulent stresses by mean motion, $x2$ measures the distance from the wall, $ϵh$ is the homogeneous part of the turbulent dissipation rate defined by 10, and $A$, $C$, and $F$ are scalar functions that depend on the anisotropy invariants and the turbulent Reynolds number.

J. Fluids Eng 128(1), 118-130 (Aug 02, 2005) (13 pages) doi:10.1115/1.2073227 History: Received April 09, 2004; Revised August 02, 2005

## Abstract

Turbulent drag reduction by dilute addition of high polymers is studied by considering local stretching of the molecular structure of a polymer by small-scale turbulent motions in the region very close to the wall. The stretching process is assumed to restructure turbulence at small scales by forcing these to satisfy local axisymmetry with invariance under rotation about the axis aligned with the main flow. It can be shown analytically that kinematic constraints imposed by local axisymmetry force turbulence near the wall to tend towards the one-component state and when turbulence reaches this limiting state it must be entirely suppressed across the viscous sublayer. For the limiting state of wall turbulence, the statistical dynamics of the turbulent stresses, constructed by combining the two-point correlation technique and invariant theory, suggest that turbulent drag reduction by homogeneously distributed high polymers, cast into the functional space which emphasizes the anisotropy of turbulence, resembles the process of reverse transition from the turbulent state towards the laminar flow state. These findings are supported by results of direct numerical simulations of wall-bounded turbulent flows of Newtonian and non-Newtonian fluids and by experiments carried out, under well-controlled laboratory conditions, in a refractive index-matched pipe flow facility using state-of-the art laser-Doppler anemometry. Theoretical considerations based on the elastic behavior of a polymer and spatial intermittency of turbulence at small scales enabled quantitative estimates to be made for the relaxation time of a polymer and its concentration that ensure maximum drag reduction in turbulent pipe flows, and it is shown that predictions based on these are in very good agreement with available experimental data.

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## Figures

Figure 1

Behavior of a polymer in solution at equilibrium (top) and its response to stretching by turbulent motions at small scales very close to the wall (bottom). Here RN and RF are hydrodynamical and Flory radius, respectively.

Figure 2

Anisotropy-invariant map of the tensor aij=uiuj¯∕q2−1∕3δij and the limiting values of scalar invariants IIa=aijaji and IIIa=aijajkaki for the different states of the turbulence, after Lumley and Newmann (26). Here uiuj¯ is the Reynolds stress tensor and q2 is its trace q2=usus¯. According to Lumley (27), all realistic turbulence must exist within the area delineated by the map.

Figure 3

Distribution of the turbulent dissipation rate ϵ versus distance from the wall, normalized on the inner variables uτ and ν, in a plane channel flow (DR=0) from direct numerical simulations of Kim (28), sketched ϵ profiles for nonvanishing DR, and the limiting state at the wall for maximum DR

Figure 4

Anisotropy invariant mapping of turbulence in a channel flow reproduced from 28-33. Data which correspond to low Reynolds number (based on the channel half-width and the wall friction velocity) show the trend as Re→(Re)crit towards the theoretical solution valid for small, neutrally stable, statistically stationary axisysmmetric disturbances (34). The shading indicates the area occupied by the stable disturbances: for such disturbances it is expected that the laminar regime in a flat plate boundary layer will persist up to very high Reynolds numbers.

Figure 5

Anisotropy-invariant mapping of turbulence in a fully developed channel flow with drag reduction from direct numerical simulations of Dimitropulos (20). The trend in the data at the wall (x2=0) strongly supports the conclusion that DR increases as turbulence approaches the one-component limit.

Figure 6

Drag reduction in pipes of different diameters versus a polymer time scale normalized by a viscous time scale ν∕uτ2 in a fully developed turbulent pipe flow (from Durst (45)) and the predicted value of a Deborah number for the maximum drag reduction effect

Figure 7

Measured drag reduction effects at different PAA concentrations in aqueous solutions as a function of the Reynolds number; from Tilli (53). The solid line represents prediction of the optimum concentration of a polymer for the maximum drag reduction effect.

Figure 8

Closed-loop pipe flow test section

Figure 9

Layout of the LDA optical system

Figure 10

Measurements at Reτ=810 in comparison with experimental results of Durst (43) and DNS results of Mansour (55) and Eggels (56)

Figure 11

Illustration of the molecular structure of a FORTUM polymer sample after Koskinen (58): M≈25×106gmol−1,Mdodecane=168.4320gmol−1,Moctane=112.2880gmol−1,l=2×1.54Å,rd∕o=C12H24∕C8H16=1∕3,Nmonomer=Mpolymer∕[rd∕oMoctane+(1−rd∕o)Mdodecane]

Figure 12

DNS data from Mansour (55) for x2+<5 plotted on expanded scale in the anisotropy invariant map

Figure 13

Velocity profile and turbulent intensity profile in the region very close to the wall

Figure 14

Behavior of the mean velocity with time for degrading polymer solutions

Figure 15

Behavior of the axial turbulent intensity component in degrading polymer solutions

Figure 16

Behavior of the tangential turbulent intensity component in degrading polymer solutions. The measurement resolution decreases in time since the dimensionless size of the measuring volume increases; the plotted results are therefore limited to that time period when the measuring volume did not extend out of the viscous sublayer

Figure 17

Drag reduction for different concentrations of a polymer

Figure 18

Measurement results plotted on the anisotropy invariant map (together with those shown in previous figure) demonstrate that with decreasing DR the data points move away from the one-component limit

Figure 19

Illustration of the mechanism responsible for polymer drag reduction utilizing the results of direct numerical simulations of Dimitropulos (20)

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