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Remarkable drag reduction in non-affine viscoelastic turbulent flows
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10.1063/1.4774239
/content/aip/journal/pof2/25/1/10.1063/1.4774239
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/1/10.1063/1.4774239

Figures

Image of FIG. 1.
FIG. 1.

Three-dimensional rendering of the isosurfaces of vortex sheet identified by [ ] , shown using gray shade and vortex tube identified by shown using heavier gray (red online) from HIT. (a) Newtonian, (b) α = 0.0, and (c) α = 1.0. The whole domain is shown.

Image of FIG. 2.
FIG. 2.

Distribution of PDF for the alignment between the torque force vector due to the polymer stress and the vortex-stretching vector. (a) α = 0.0 (dashed line (red)) and (b) α = 1.0 (solid line (blue)).

Image of FIG. 3.
FIG. 3.

Three-dimensional rendering of the isosurfaces of vortex tube (, gray (red)) and the eigenvectors of [ ] along the vortex tube shown using the white arrows in HIT. (a) , (b) , and (c) .

Image of FIG. 4.
FIG. 4.

Isocontours of the elongation viscosity μ and the second NSD, (τ − τ), plotted on the isosurfaces of vortex tubes () from α = 0.0 case in HIT. (a) μ and (b) (τ − τ).

Image of FIG. 5.
FIG. 5.

Distribution of PDF of the eigenvalues of the polymer stress in HIT. (a) α = 0.0, [τ] (dotted line/black), [τ] (dashed line/red), [τ] (solid line/blue); (b) α = 1.0, [τ] (dotted line/black), [τ] (solid line/red), and [τ] (dashed line/blue).

Image of FIG. 6.
FIG. 6.

Three-dimensional rendering of isosurfaces of vortex tube (, heavy gray (red)), isosurfaces of vortex sheet ([ ] , gray) and the polymer force vectors (gray arrows/blue), and the vortex-stretching vectors (white arrows), from the α = 0.0 case in HIT. (a) and on tube, (b) on tube, and (c) and on sheet.

Image of FIG. 7.
FIG. 7.

Three-dimensional rendering of isosurfaces of vortex sheet ([ ] , gray) and the eigenvectors of [ ] along the vortex sheet (gray arrows/blue) in HIT. (a) , (b) , and (c) .

Image of FIG. 8.
FIG. 8.

Isocontours of NSDs plotted on the isosurfaces of vortex sheets ([ ] ) from α = 1.0 case in HIT. (a) first NSD (τ − τ) and (b) second NSD (τ − τ).

Image of FIG. 9.
FIG. 9.

Three-dimensional rendering of the isosurfaces of [ ] (gray) and the polymer force vectors from the case with α = 1.0 in HIT. (a) (gray arrows) and (b) (gray arrows/blue).

Image of FIG. 10.
FIG. 10.

Mean normalized velocity profiles as a function of the normalized distance from the wall during drag reduction obtained in pipe flow. The light gray straight dashed line shows the Newtonian log-law, the light gray straight dotted line the Virk's maximum DR limit. (a) the profiles in the cases of Newtonian flow (solid line (black)), α = 0.0 (long dashed line (black)), 0.1 (two-point dashed line (green)), 0.5 (heavier gray dashed line (red)), 0.9 (one-point dashed line (blue)) and (b) the profiles in the cases of α = 0.0 (dashed line (blue)), 0.5 (long-dashed line (red)), 1.0 (solid line (black)). The one-point dashed line shows the viscous sublayer profile.

Image of FIG. 11.
FIG. 11.

Shear stress contributions as a function of the distance from the wall in pipe flow obtained in the case of = 25.0, β = 0.9. : Reynolds stress (dashed line (black)), ⟨τ⟩: polymer stress (solid line (red)), Viscous denotes viscous stress (one-point dashed line (blue)), Total denotes the total stress (long-dashed line (green)). (a) α = 0.0, (b) α = 0.5, and (c) α = 1.0.

Image of FIG. 12.
FIG. 12.

Three-dimensional rendering of the isosurfaces of vortex sheet ([ ] , gray) and tube (, heavier gray/red) in pipe flow. (a) Newtonian, (b) α = 0.0, (c) α = 0.1, (d) α = 0.5, (e) α = 0.9, and (f) α = 1.0. The whole computational domain is shown.

Image of FIG. 13.
FIG. 13.

Isocontours of NSDs plotted on the isosurfaces of vortex sheets ([ ] ) from α = 1.0 case in pipe flow. (a) first NSD (τ − τ) (gray/red) and (b) second NSD (τ − τ) (gray/blue).

Image of FIG. 14.
FIG. 14.

Isocontours of first NSD and second NSD plotted on the isosurfaces of vortex tubes () from α = 0.0 case in pipe flow. (a) (τ − τ) (gray/blue) and (b) (τ − τ) (gray/red).

Image of FIG. 15.
FIG. 15.

Distribution of PDF of the eigenvalues of the polymer stress τ, [τ] (solid line/black), [τ] (one-point dashed line/red), [τ] (dashed line/blue) in pipe flow. (a) α = 0.0 and (b) α = 1.0.

Image of FIG. 16.
FIG. 16.

(a) Three-dimensional rendering of isosurface of vortex tube (, heavy gray/red) and the polymer force vectors (gray arrows/blue) from α = 0.0 case in pipe flow; (b) Isosurface of vortex sheet ([ ] , gray) and vectors (gray arrows/green) in α = 1.0 case.

Image of FIG. 17.
FIG. 17.

Schematics of arrangement of the dominant polymer force vectors on vortex tube and vortex sheet. The gray arrows (blue) denote the polymer force vectors, the black arrows the polymer connector vectors, the white arrow the vortex stretching vector. (a) Arrangement of force vectors along vortex tube (gray/red) and (b) the force vectors along vortex sheet (gray).

Image of FIG. 18.
FIG. 18.

Schematics of a parallelogram defined by the two material line elements and and material surface element represented by area vector . The gray arrow (green) denoted as shows vector, the gray arrow (red) denoted as shows vector, and the gray arrow (blue) denoted as shows vector. The surface drawn using the heavy gray shade represents the vortex sheet and the plane drawn using lighter gray shows the hyperplane.

Image of FIG. 19.
FIG. 19.

Isosurfaces of the polymer energy (heavy gray/sky blue) and the vortex sheets ([ ] , gray) obtained from α = 1.0 case in HIT.

Tables

Generic image for table
Table I.

Parameters for computed cases in HIT: grid resolution criterion ( is the maximum wave number (=64)); Taylor microscale Reynolds number ; average kinetic energy of the solvent ; average dissipation rate ɛ; integral length scale ; average Taylor microscale ; average Kolmogorov length ( = (ν/ɛ)); eddy turnover time according to , ; average turbulent time scale according to the Taylor micro scale, τ(= / ); average of production term of the polymer energy ; average of the polymer energy (= ± τ/2); average rate of energy addition ; average production term of the solvent energy due to the polymer stress .

Generic image for table
Table II.

Computed cases and the drag reduction rate % obtained in each case from pipe flow. The results shown in the column “2nd-order” are obtained using the 2nd-order upwind method for the Oldroyd derivatives in the JS model. Other results are obtained using the MINMOD method.

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/content/aip/journal/pof2/25/1/10.1063/1.4774239
2013-01-17
2014-04-17
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Remarkable drag reduction in non-affine viscoelastic turbulent flows
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/1/10.1063/1.4774239
10.1063/1.4774239
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