^{1,a)}, Kazuma Matsumoto

^{1}and Kosuke Fujiwara

^{1}

### Abstract

We carry out a direct numerical simulation (DNS) study which aims to reveal the mechanism of turbulence drag reduction (DR) in polymer diluted flows. The polymer chains are modeled as elastic dumbbells. This paper focuses on elucidation of effect of introduction of non-affinity to describe the motions of the dumbbells on DR. We consider the cases in which the motions do not precisely correspond to macroscopically-imposed deformation. The Johnson-Segalman (JS) model is adopted to express the polymer stress. Assessment is done in forced homogeneous isotropic turbulence and pipe flow. In both flows, DR exhibits non-monotonous dependence on the strength of non-affinity. DR is maximal when non-affinity is either minimum (slip parameter α = 0.0) or maximum (α = 1.0) and almost no DR is obtained when α = 0.5. Remarkable enhancement of DR is achieved when α = 1.0 in both flows. In pipe flow, the mean velocity profile surpasses the Virk's maximum DR limit and nearly complete relaminarization occurs. This marked DR is not established when α ≠ 1.0. Mechanism of DR applied commonly to both flows is identified. A method to evaluate the normal-stress difference (NSD) and elongation viscosity is proposed using new eigenvector basis which span the isosurface of vortex tube and sheet. It is shown that the first NSD is predominantly positive, while the second NSD is negative along the sheets and tubes in both α = 0.0 and 1.0, implying that the polymer molecules exhibit alignment in a preferential direction in both cases. Mechanism in α = 1.0, however, is distinctively different from that in α = 0.0. When α = 0.0, the connector vector of dumbbell is convected as a contravariant vector representing material line element and elasticity is incurred primarily on filament-like element or the vortex tube. As shown in previous studies, the force exerted by the polymer stress such as the torque force reduces the vortex strength by opposing the vortical motions. When α = 1.0, the connector vector is convected as a covariant vector representing material surface element, and directs outward perpendicularly on the vortex sheet and exert an extra tension on the sheet. Creation of tubes due to rolling-up of the sheet is attenuated by this tensile force and energy cascade is annihilated. In high-DR cases, the elongation viscosity increases and stretching of the sheet and tube is hindered. Consistency of the results obtained in the DNS data with those predicted using an explicit expression of the polymer stress in the JS model is shown. Analogy of DR in α = 1.0 with DR occurring in the fluid diluted with high-concentration cationic surfactant and the fibers is presented. Limitation of the JS model in the intermediate range of 0.0 < α < 1.0 is discussed.

We are grateful to M. Adati for assistance in the development of the DNS codes and S. Takeu for valuable discussions. This work was partially supported by Grants-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan (Grant No. 23560188), and HPCI Systems Research Programs (Grant No. hp120256). Main computations are performed at Cybermedia Centre, Osaka University.

I. INTRODUCTION

II. CONSTITUTIVE EQUATION FOR POLYMER STRESS

III. CONVECTIVE MOTIONS OF DUMBBELLS IN THE JS MODEL

IV. EXPLICIT EXPRESSION OF POLYMER STRESS IN THE JS MODEL

V. RESULTS OF DNS IN HOMOGENEOUS ISOTROPIC TURBULENCE

A. Details of DNS data

B. Identification of vortical structures

C. Mechanism of DR in α = 0.0 in HIT

1. Torque force and normal stress difference

2. Estimate of NSD and elastic energy production using explicit expression of the polymer stress

3. Principal polymer stress

D. Mechanism of DR in α = 1.0 in HIT

VI. RESULTS OF DNS IN PIPE FLOW

A. Details and statistics of DNS data

B. Vortical structures, NSD and polymer force in pipe flow

VII. SUMMARY OF DR MECHANISM

VIII. CONCLUSIONS

### Key Topics

- Polymers
- 153.0
- Rotating flows
- 94.0
- Elasticity
- 35.0
- Viscosity
- 31.0
- Eigenvalues
- 29.0

## Figures

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

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

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)).

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)).

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

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

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

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

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

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

Three-dimensional rendering of isosurfaces of vortex tube (Q, heavy gray (red)), isosurfaces of vortex sheet ([A ij ]+ , 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.

Three-dimensional rendering of isosurfaces of vortex tube (Q, heavy gray (red)), isosurfaces of vortex sheet ([A ij ]+ , 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.

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

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

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

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

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

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

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.

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.

Shear stress contributions as a function of the distance from the wall in pipe flow obtained in the case of We τ0 = 25.0, β = 0.9. : Reynolds stress (dashed line (black)), ⟨τ rθ⟩: 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.

Shear stress contributions as a function of the distance from the wall in pipe flow obtained in the case of We τ0 = 25.0, β = 0.9. : Reynolds stress (dashed line (black)), ⟨τ rθ⟩: 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.

Three-dimensional rendering of the isosurfaces of vortex sheet ([A ij ]+ , gray) and tube (Q, 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.

Three-dimensional rendering of the isosurfaces of vortex sheet ([A ij ]+ , gray) and tube (Q, 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.

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

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

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

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

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

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

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

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

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).

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).

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 R shows R vector. The surface drawn using the heavy gray shade represents the vortex sheet and the plane drawn using lighter gray shows the hyperplane.

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 R shows R vector. The surface drawn using the heavy gray shade represents the vortex sheet and the plane drawn using lighter gray shows the hyperplane.

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

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

## Tables

Parameters for computed cases in HIT: grid resolution criterion (k max is the maximum wave number (=64)); Taylor microscale Reynolds number R λ; average kinetic energy of the solvent ; average dissipation rate ɛ; integral length scale L; average Taylor microscale L T ; average Kolmogorov length L K (L K = (ν3/ɛ)1/4); eddy turnover time according to L, ; average turbulent time scale according to the Taylor micro scale, τ T (= L T /u ′); average of production term of the polymer energy P e ; average of the polymer energy k p (= ± τ ii /2); average rate of energy addition u i f i ; average production term of the solvent energy due to the polymer stress P S .

Parameters for computed cases in HIT: grid resolution criterion (k max is the maximum wave number (=64)); Taylor microscale Reynolds number R λ; average kinetic energy of the solvent ; average dissipation rate ɛ; integral length scale L; average Taylor microscale L T ; average Kolmogorov length L K (L K = (ν3/ɛ)1/4); eddy turnover time according to L, ; average turbulent time scale according to the Taylor micro scale, τ T (= L T /u ′); average of production term of the polymer energy P e ; average of the polymer energy k p (= ± τ ii /2); average rate of energy addition u i f i ; average production term of the solvent energy due to the polymer stress P S .

Computed cases and the drag reduction rate %DR 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.

Computed cases and the drag reduction rate %DR 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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