^{1,a)}and H. I. Andersson

^{1}

### Abstract

A helical fin in a circular pipe is a means to generate swirling flow and the swirling motion will in turn modulate the turbulence field. This paper reports on a direct numerical simulation aimed to explore such flow phenomena. The full Navier-Stokes equations expressed in a cylindrical coordinate system are integrated numerically in time and the helical fin is embedded in the structured grid by means of an immersed boundary method. The statistically steady three-dimensional flow field exhibits a helical symmetry with an azimuthal mean velocity which amounts to about 50% of the axial mean velocity component. The variation of the mean flow over the cross-sections can be explained by the variations of the nine non-zero components of the Reynolds stresstensor. Particular attention is paid to the contribution of the Reynolds stresses to the skin-friction coefficient. Appreciable levels of fluctuating helicity are observed in the vicinity of the pipe wall and reflect that the swirling motion breaks the structural symmetry in conventional pipe flow.

This work has been supported by the Research Council of Norway through a grant of computing time (Programme for Supercomputing).

I. INTRODUCTION

II. FLOW CONFIGURATION AND NUMERICAL RESOLUTION

III. NUMERICAL PROCEDURE

A. Averaging

B. Verification of the spatial- and the time-domain

IV. RESULTS

A. Mean velocities, pressure, and vorticity

B. Velocityfluctuations

C. Pipe friction

D. Fluctuatingvorticity and helicity

V. CONCLUDING DISCUSSION

### Key Topics

- Turbulent flows
- 27.0
- Rotating flows
- 24.0
- Turbulent pipe flows
- 23.0
- Friction
- 16.0
- Vortex dynamics
- 15.0

## Figures

Pipe with a left-handed helix fin. Only half the axial domain is shown.

Pipe with a left-handed helix fin. Only half the axial domain is shown.

The lines show the locations at which the results will be presented in the (*r, θ*)-section.

The lines show the locations at which the results will be presented in the (*r, θ*)-section.

Frictional velocity *u _{τ} * with respect to azimuthal distance

*θ*from the fin.

_{f}Frictional velocity *u _{τ} * with respect to azimuthal distance

*θ*from the fin.

_{f}Schematic of the fin-implementation in the three orthogonal grid planes: (a) *(θ, z*)-plane; (b) (*r, θ*)-plane; and (c) (*r, z*)-plane.

Schematic of the fin-implementation in the three orthogonal grid planes: (a) *(θ, z*)-plane; (b) (*r, θ*)-plane; and (c) (*r, z*)-plane.

Two-point correlations in the three coordinate directions *i = θ, r,* and *z* at *θ _{f} = π.* (a)

*y/R =*0.1 and (b)

*y/R =*0.75.

Two-point correlations in the three coordinate directions *i = θ, r,* and *z* at *θ _{f} = π.* (a)

*y/R =*0.1 and (b)

*y/R =*0.75.

Mean Reynolds normal stress at *θ _{f} = π.* Line legends as in Figure 5. 200 (symbols) and 280 (lines) time samples.

Mean Reynolds normal stress at *θ _{f} = π.* Line legends as in Figure 5. 200 (symbols) and 280 (lines) time samples.

(Color) Visualization of helical averaged variables in a (*r, θ*)-plane. (a) The colour indicates normalized axial velocities *U _{z}/U_{P} *. The scale is from 0 to 0.7. The vectors show the normalized in-plane velocities scaled from 0 to 0.3. (b) The normalized pressure variations caused by the fin. The scale is from –0.01 to 0.06.

(Color) Visualization of helical averaged variables in a (*r, θ*)-plane. (a) The colour indicates normalized axial velocities *U _{z}/U_{P} *. The scale is from 0 to 0.7. The vectors show the normalized in-plane velocities scaled from 0 to 0.3. (b) The normalized pressure variations caused by the fin. The scale is from –0.01 to 0.06.

Axial mean velocity *U _{z} *. (a) Radial variation. (b) Azimuthal variation.

Axial mean velocity *U _{z} *. (a) Radial variation. (b) Azimuthal variation.

Azimuthal mean velocity *U _{θ} *. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Azimuthal mean velocity *U _{θ} *. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Radial mean velocity *U _{r} *. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Radial mean velocity *U _{r} *. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Axial mean vorticity Ω_{ z }. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Axial mean vorticity Ω_{ z }. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Mean pressure variations caused by the fin. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Mean pressure variations caused by the fin. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Axial normal stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Axial normal stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Azimuthal normal stress . Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Azimuthal normal stress . Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Radial normal stress . Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Radial normal stress . Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

shear-stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

shear-stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

shear-stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

shear-stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

shear-stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

shear-stress. Legends as in Figure 8. (a) Radial variation. (b) Azimuthal variation.

Turbulent contribution to the friction coefficient. Legends as in Figure 8. (a) Weighted contribution. (b) Cumulative contribution defined in Eq. (7).

Turbulent contribution to the friction coefficient. Legends as in Figure 8. (a) Weighted contribution. (b) Cumulative contribution defined in Eq. (7).

(Color online) Instantaneous visualization of fluctuating axial velocities. The black line shows the outline of the fin. (a) *r — θ*-section at z/R = 0. (b) *θ — z*-section at y/R = 0.1 (r = 0.9 R).

(Color online) Instantaneous visualization of fluctuating axial velocities. The black line shows the outline of the fin. (a) *r — θ*-section at z/R = 0. (b) *θ — z*-section at y/R = 0.1 (r = 0.9 R).

Fluctuating vorticity in the near-wall region, *y/R ≤* 0.1. (a) Axial component. (b) Azimuthal component, legends as in (a).

Fluctuating vorticity in the near-wall region, *y/R ≤* 0.1. (a) Axial component. (b) Azimuthal component, legends as in (a).

Fluctuating helicity, legends as in Figure 21. (a) Axial component, . (b) Azimuthal component, .

Fluctuating helicity, legends as in Figure 21. (a) Axial component, . (b) Azimuthal component, .

## Tables

Spatial resolution. The subscripts c and 1 refer to the centreline and the node next to the wall, respectively.

Spatial resolution. The subscripts c and 1 refer to the centreline and the node next to the wall, respectively.

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