^{1}and Beverley J. McKeon

^{1}

### Abstract

A streamwise-constant model is presented to investigate the basic mechanisms responsible for the change in mean flow occuring during pipe flow transition. The model is subject to two different types of forcing: a simple forcing of the axial momentum equation via a deterministic form for the streamfunction and a stochastic forcing of the streamfunction equation. Using a single forced momentum balance equation, we show that the shape of the velocity profile is robust to changes in the forcing profile and that both linear non-normal and nonlinear effects are required to capture the change in mean flow associated with transition to turbulence. The particularly simple form of the model allows for the study of the momentum transfer directly by inspection of the equations. The distribution of the high- and low-speed streaks over the cross-section of the pipe produced by our model is remarkably similar to one observed in the velocity field near the trailing edge of the puff structures present in pipe flow transition. Under stochastic forcing, the model exhibits a quasi-periodic self-sustaining cycle characterized by the creation and subsequent decay of “streamwise-constant puffs,” so-called due to the good agreement between the temporal evolution of their velocity field and the projection of the velocity field associated with three-dimensional puffs in a frame of reference moving at the bulk velocity. We establish that the flowdynamics are relatively insensitive to the regeneration mechanisms invoked to produce near-wall streamwise vortices, such that using small, unstructured background disturbances to regenerate the streamwise vortices in place of the natural feedback from the flow is sufficient to capture the formation of the high- and low-speed streaks and their segregation leading to the blunting of the velocity profile characteristic of turbulent pipe flow. We propose a “quasi self-sustaining process” to describe these mechanisms.

The authors gratefully acknowledge the support of the AFOSR Grant No. FA 9550-09-1-0701 (program manager John Schmisseur).

I. INTRODUCTION

II. DESCRIPTION OF THE MODEL AND NUMERICAL METHODS

III. SIMPLIFIED 2D/3C MODEL WITH DETERMINISTIC FORCING

IV. STOCHASTIC FORCING OF THE 2D/3C MODEL

V. CONCLUSIONS

### Key Topics

- Rotating flows
- 32.0
- Turbulent flows
- 27.0
- Laminar flows
- 26.0
- Turbulence simulations
- 21.0
- Turbulent pipe flows
- 19.0

## Figures

The coordinate system used to project the Navier-Stokes equations.

The coordinate system used to project the Navier-Stokes equations.

(a) Streamfunctions Ψ_{1,a–c }(*η*) and (b) corresponding velocity profiles *u* _{0}(*η*) for Ψ_{1,a }(*η*) = 0.033(*η* − 3*η* ^{3} + 2*η* ^{4}) (thin solid), Ψ_{1,b }(*η*) = 0.7(*η* − 3*η* ^{3} + 2*η* ^{4})^{2} (dashed), and Ψ_{1,c }(*η*) = 14(*η* − 3*η* ^{3} + 2*η* ^{4})^{3} (dash-dot) and experimental velocity profile of Ref. 38 at Re = 24 600 (thick solid).

(a) Streamfunctions Ψ_{1,a–c }(*η*) and (b) corresponding velocity profiles *u* _{0}(*η*) for Ψ_{1,a }(*η*) = 0.033(*η* − 3*η* ^{3} + 2*η* ^{4}) (thin solid), Ψ_{1,b }(*η*) = 0.7(*η* − 3*η* ^{3} + 2*η* ^{4})^{2} (dashed), and Ψ_{1,c }(*η*) = 14(*η* − 3*η* ^{3} + 2*η* ^{4})^{3} (dash-dot) and experimental velocity profile of Ref. 38 at Re = 24 600 (thick solid).

Model output for deterministic forcing: (a) contours of the streamfunction Ψ = 0.033 (*η* − 3*η* ^{3} + 2*η* ^{4}) sin *φ*, (b) vector plot of the corresponding in-plane velocities, and (c) contours of the resulting axial velocity field.

Model output for deterministic forcing: (a) contours of the streamfunction Ψ = 0.033 (*η* − 3*η* ^{3} + 2*η* ^{4}) sin *φ*, (b) vector plot of the corresponding in-plane velocities, and (c) contours of the resulting axial velocity field.

Contours of the axial velocity induced by the streamfunction Ψ_{6}(*η*, *φ*) = (*η* ^{4} − 2*η* ^{5} + *η* ^{6}) sin(6*φ*), the light and dark filled contours correspond to regions of the flow, respectively, faster and slower than laminar.

Contours of the axial velocity induced by the streamfunction Ψ_{6}(*η*, *φ*) = (*η* ^{4} − 2*η* ^{5} + *η* ^{6}) sin(6*φ*), the light and dark filled contours correspond to regions of the flow, respectively, faster and slower than laminar.

Time traces of the centerline velocity from three different simulations, respectively, at *Re* = 2200 with 0.0005 and 0.002 rms noise levels (a) and (c) and at *Re* = 10 000 with 0.002 rms noise level (b). The resolution in the radial direction is *N* = 48. (d) Zoom on the time interval during which the samples of Figure 6 are taken. The vertical lines indicate the sampling instants.

Time traces of the centerline velocity from three different simulations, respectively, at *Re* = 2200 with 0.0005 and 0.002 rms noise levels (a) and (c) and at *Re* = 10 000 with 0.002 rms noise level (b). The resolution in the radial direction is *N* = 48. (d) Zoom on the time interval during which the samples of Figure 6 are taken. The vertical lines indicate the sampling instants.

(Color online) Contours of the axial velocity, subfigures (a) to (c), and of the swirling strength for the in-plane velocities, subfigures (d) to (f), computed, respectively, at *t* = 1620, *t* = 1700, and *t* = 1740 dimensionless time units.

(Color online) Contours of the axial velocity, subfigures (a) to (c), and of the swirling strength for the in-plane velocities, subfigures (d) to (f), computed, respectively, at *t* = 1620, *t* = 1700, and *t* = 1740 dimensionless time units.

Diagram detailing the different stages of the QSSP. The dashed lines represent unmodeled effects.

Diagram detailing the different stages of the QSSP. The dashed lines represent unmodeled effects.

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