^{1,a)}, C. Chin

^{1}, H. M. Blackburn

^{2}, A. Ooi

^{1}and I. Marusic

^{1}

### Abstract

Direct numerical simulations of fully developed turbulent pipe flow that span the Reynolds number range 90 ≲ δ^{+} ≲ 1000 are used to investigate the evolution of the mean momentum field in and beyond the transitional regime. It is estimated that the four layer regime for pipe flow is nominally established for δ^{+} ⩾ 180, which is also close to the value found for channel flow. Primary attention is paid to the magnitude ordering and scaling behaviors of the terms in the mean momentum equation. Once the ordering underlying the existence of four distinct balance layers is attained, this ordering is sustained for all subsequent increases in Reynolds number. Comparisons indicate that pipe flow develops toward the four layer regime in a manner similar to that for channel flow, but distinct from that for the boundary layer. Small but discernible differences are observed in the mean momentum field development in pipes and channels. These are tentatively attributed to variations in the manner by which the outer region mean vorticity field develops in these two flows.

The authors gratefully acknowledge the financial support of the Australian Research Council and the computational resources of the Australian Partnership for Advanced Computing (MAS Grant Nos. p46, m45, and d77), as well as the Victorian Partnership for Advanced Computing (Project Nos. pMelb0037 and pMelb0061). Thanks are also extended to Dr. X. Wu and Dr. P. Moin, and Dr. H. Kawamura, Dr. H. Abe, and Dr. K. Shingai for making their simulation data bases available on the internet.

I. INTRODUCTION

A. The four layer regime

B. The underlying layer hierarchy

C. Objectives

II. NUMERICAL METHODS AND DATA SETS

III. RESULTS

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Reynolds stress modeling
- 37.0
- Channel flows
- 33.0
- Viscosity
- 12.0
- Vortex dynamics
- 11.0
- Boundary value problems
- 9.0

## Figures

Sketch of the ratio of the mean viscous force (VF) to mean effect of turbulent inertia (TI). This ratio reveals the four layer force balance structure of turbulent wall-bounded flows.^{8} Note that this sketch is for a fixed Reynolds number, as the layer boundaries depend on δ^{+}. Note also that layer I in the zero pressure gradient turbulent boundary layer (dotted line in I) differs from that of channel or pipe flow. In the boundary layer all of the terms in the mean momentum equation approach zero as *y* → 0.

Sketch of the ratio of the mean viscous force (VF) to mean effect of turbulent inertia (TI). This ratio reveals the four layer force balance structure of turbulent wall-bounded flows.^{8} Note that this sketch is for a fixed Reynolds number, as the layer boundaries depend on δ^{+}. Note also that layer I in the zero pressure gradient turbulent boundary layer (dotted line in I) differs from that of channel or pipe flow. In the boundary layer all of the terms in the mean momentum equation approach zero as *y* → 0.

Ratio of the mean viscous force to the mean effect turbulent inertia in pipe flow for 90 ≲ δ^{+} ≲ 1000.

Ratio of the mean viscous force to the mean effect turbulent inertia in pipe flow for 90 ≲ δ^{+} ≲ 1000.

Inner-normalized mean viscous force and time-averaged turbulent inertia profiles in pipe flow for 90 ≲ δ^{+} ≲ 1000.

Inner-normalized mean viscous force and time-averaged turbulent inertia profiles in pipe flow for 90 ≲ δ^{+} ≲ 1000.

Positions of the inner and outer maxima of the *dT* ^{+}/*dy* ^{+} profiles of Fig. 3. Note that the inner peak approaches a fixed *y* ^{+} value (), while the outer peak approaches a fixed *y*/δ value (*y*/δ ≃ 0.5). The vertical line (δ^{+} ≃ 180) indicates the present estimate for the onset of the four layer regime.

Positions of the inner and outer maxima of the *dT* ^{+}/*dy* ^{+} profiles of Fig. 3. Note that the inner peak approaches a fixed *y* ^{+} value (), while the outer peak approaches a fixed *y*/δ value (*y*/δ ≃ 0.5). The vertical line (δ^{+} ≃ 180) indicates the present estimate for the onset of the four layer regime.

Profiles of *T* ^{+} and *WdT* ^{+}/*dy* ^{+} at δ^{+} = 171.

Profiles of *T* ^{+} and *WdT* ^{+}/*dy* ^{+} at δ^{+} = 171.

Actual and predicted values of versus δ^{+}: actual values, •; values predicted from Eq. (10), ■.

Actual and predicted values of versus δ^{+}: actual values, •; values predicted from Eq. (10), ■.

Layer width distribution of the *L* _{β} hierarchy for transitional and four layer regime pipe flow; (a) linear axes and (b) logarithmic axes. Note that the Reynolds number of each profile is given by the end point position *y* ^{+} = δ^{+} in (b). The curve-fit of the *W*(*y* ^{+}) distribution at δ^{+} = 1002 is over the range and given by 7.52 + 0.644*y* ^{+}. The dashed profile is that of Wu and Moin^{18} at δ^{+} = 1142.

Layer width distribution of the *L* _{β} hierarchy for transitional and four layer regime pipe flow; (a) linear axes and (b) logarithmic axes. Note that the Reynolds number of each profile is given by the end point position *y* ^{+} = δ^{+} in (b). The curve-fit of the *W*(*y* ^{+}) distribution at δ^{+} = 1002 is over the range and given by 7.52 + 0.644*y* ^{+}. The dashed profile is that of Wu and Moin^{18} at δ^{+} = 1142.

Comparison of the linear portion of the *W*(*y* ^{+}) profiles from channel and pipe flow at δ^{+} ≃ 1000. Data are from the present simulation, δ^{+} = 1002, the pipe flow simulation of Wu and Moin,^{18} δ^{+} = 1142, and the channel flow simulation of Kawamura *et al.*,^{21} δ^{+} = 1016.

Comparison of the linear portion of the *W*(*y* ^{+}) profiles from channel and pipe flow at δ^{+} ≃ 1000. Data are from the present simulation, δ^{+} = 1002, the pipe flow simulation of Wu and Moin,^{18} δ^{+} = 1142, and the channel flow simulation of Kawamura *et al.*,^{21} δ^{+} = 1016.

## Tables

Scaling behaviors of the layer thicknesses and velocity increments associated with the mean momentum equation in turbulent wall-flows in the four-layer regime, see Fig. 1. Note that the layer IV properties are asymptotically attained as δ^{+} → ∞, see Ref. 5.

Scaling behaviors of the layer thicknesses and velocity increments associated with the mean momentum equation in turbulent wall-flows in the four-layer regime, see Fig. 1. Note that the layer IV properties are asymptotically attained as δ^{+} → ∞, see Ref. 5.

Summary of computational parameters.

Summary of computational parameters.

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