^{1,a)}, Damien Biau

^{1,b)}, Alessandro Bottaro

^{1}and Masato Nagata

^{2}

### Abstract

The flow in a square duct is considered. Finite amplitude approximate traveling wave solutions, obtained using the self-sustaining-process approach introduced by Waleffe [Phys. Fluids9, 883 (1997)], are obtained at low to moderate Reynolds numbers and used as initial conditions in direct numerical simulations. The ensuing dynamics is analyzed in a suitably defined phase space. Only one among the traveling wave solutions found is capable of surviving for a long time, with the flow trajectory forming quasiregular loops in phase space. Eventually, also this trajectory escapes along the manifold of a chaotic saddle and relaminarization ensues.

The financial support of the EU (Program Marie Curie Grant No. EST FLUBIO 20228-2006) and of the Italian Ministry of University and Research (Grant No. PRIN 2005-092015-002) are gratefully acknowledged. H.W. wishes to acknowledge also the support from the Foundation Blanceflor Boncompagni-Ludovisi as well as the Japan Society for the Promotion of Science (JSPS).

I. INTRODUCTION

II. DEFINITIONS AND GOVERNING EQUATIONS

A. The laminar flow

III. THE SELF-SUSTAINING PROCESS

A. The streamwise rolls

B. The streamwise streaks

C. The linear instability analysis

D. The feedback

IV. DIRECT NUMERICAL SIMULATIONS

A. Definitions

B. Results

V. CLOSING REMARKS

### Key Topics

- Laminar flows
- 41.0
- Turbulent flows
- 30.0
- Flow instabilities
- 19.0
- Duct flows
- 18.0
- Rotating flows
- 17.0

## Figures

The rectangular duct flow. The flow is confined by the intervals and . The unit vectors in the , , and directions are , , and , respectively. The schematic parabolic-like velocity profile represents the laminar flow.

The rectangular duct flow. The flow is confined by the intervals and . The unit vectors in the , , and directions are , , and , respectively. The schematic parabolic-like velocity profile represents the laminar flow.

Plot of the two least stable symmetric modes of Eq. (28). The color coding goes from most negative (darkest) to most positive (lightest).

Plot of the two least stable symmetric modes of Eq. (28). The color coding goes from most negative (darkest) to most positive (lightest).

The streamwise rolls (arrows) and streamwise streaks (contour levels) at , . Left plot: The flow at and the contour levels range between the minimum and the maximum of or to 0.0705 in steps of 0.05. Right plot: and the contour levels range between the minimum and the maximum of or to 0.0567 in steps of 0.06. The color coding goes from most negative (darkest) to most positive (lightest).

The streamwise rolls (arrows) and streamwise streaks (contour levels) at , . Left plot: The flow at and the contour levels range between the minimum and the maximum of or to 0.0705 in steps of 0.05. Right plot: and the contour levels range between the minimum and the maximum of or to 0.0567 in steps of 0.06. The color coding goes from most negative (darkest) to most positive (lightest).

The growth rate (solid line) and the phase speed (dashed line) as a function of the streamwise wavenumber at , , , and symmetry I of the wave. The points of neutral stability are indicated by the two bigger circles and serve as the wavy part of the SSP. The squares are obtained with and confirm that a resolution with is adequate.

The growth rate (solid line) and the phase speed (dashed line) as a function of the streamwise wavenumber at , , , and symmetry I of the wave. The points of neutral stability are indicated by the two bigger circles and serve as the wavy part of the SSP. The squares are obtained with and confirm that a resolution with is adequate.

The growth rate (solid line) and the phase speed (dashed line) as a function of the streamwise wavenumber at , , , and symmetry I of the wave. The point of neutral stability is pointed out by the bigger circle and serves as the wavy part of the SSP solution. The squares are solutions with . The curve marked by ◇ shows the unstable mode at .

The growth rate (solid line) and the phase speed (dashed line) as a function of the streamwise wavenumber at , , , and symmetry I of the wave. The point of neutral stability is pointed out by the bigger circle and serves as the wavy part of the SSP solution. The squares are solutions with . The curve marked by ◇ shows the unstable mode at .

The real and the imaginary part of the eigenfunction of the wave at , , , , and symmetry I of the wave. This solution corresponds to the point of neutral stability in Fig. 4. The arrows correspond to the cross-section velocity of the wave or and . Left plot: The contour levels range between the minimum and the maximum of or to 0.0649 in steps of 0.009. Right plot: The contour levels range between the minimum and the maximum of or to 0.0246 in steps of 0.003. The color coding goes from most negative (darkest) to most positive (lightest).

The real and the imaginary part of the eigenfunction of the wave at , , , , and symmetry I of the wave. This solution corresponds to the point of neutral stability in Fig. 4. The arrows correspond to the cross-section velocity of the wave or and . Left plot: The contour levels range between the minimum and the maximum of or to 0.0649 in steps of 0.009. Right plot: The contour levels range between the minimum and the maximum of or to 0.0246 in steps of 0.003. The color coding goes from most negative (darkest) to most positive (lightest).

The real and the imaginary part of the eigenfunction of the wave at , , , , and symmetry I of the wave. This solution corresponds to the point of neutral stability in Fig. 4. The arrows correspond to the cross-section velocity of the wave . Left plot: The contour levels range between the minimum and the maximum of or to 0.0138 in steps of 0.0017. Right plot: The contour levels range between the minimum and the maximum of or to 0.0054 in steps of . The color coding goes from most negative (darkest) to most positive (lightest).

The real and the imaginary part of the eigenfunction of the wave at , , , , and symmetry I of the wave. This solution corresponds to the point of neutral stability in Fig. 4. The arrows correspond to the cross-section velocity of the wave . Left plot: The contour levels range between the minimum and the maximum of or to 0.0138 in steps of 0.0017. Right plot: The contour levels range between the minimum and the maximum of or to 0.0054 in steps of . The color coding goes from most negative (darkest) to most positive (lightest).

The real and the imaginary parts of the eigenfunction of the wave at , , , , and symmetry I of the wave. This solution corresponds to the point of neutral stability in Fig. 5. The arrows correspond to the cross-section velocity of the wave . Left plot: The contour levels range between the minimum and the maximum of or to 0.011 in steps of 0.002. Right plot: The contour levels range between the minimum and the maximum of or to 0.0077 in steps of 0.0015. The color coding goes from most negative (darkest) to most positive (lightest).

The real and the imaginary parts of the eigenfunction of the wave at , , , , and symmetry I of the wave. This solution corresponds to the point of neutral stability in Fig. 5. The arrows correspond to the cross-section velocity of the wave . Left plot: The contour levels range between the minimum and the maximum of or to 0.011 in steps of 0.002. Right plot: The contour levels range between the minimum and the maximum of or to 0.0077 in steps of 0.0015. The color coding goes from most negative (darkest) to most positive (lightest).

The feedback of the stream function at six stations in at , , , , and symmetry I of the wave. The dashed line corresponds to the initial stream function and the solid line to the . The value of is indicated at the top of each individual figure.

The feedback of the stream function at six stations in at , , , , and symmetry I of the wave. The dashed line corresponds to the initial stream function and the solid line to the . The value of is indicated at the top of each individual figure.

The feedback of the stream function at six stations in at , , , , and symmetry I of the wave. The dashed line corresponds to the initial stream function and the solid line to the . The value of is indicated at the top of each individual figure.

Energy of the fluctuations vs time. We use as initial conditions the self-sustaining solutions represented by (dashed line) and (solid line) at and at (dash-dotted line).

Energy of the fluctuations vs time. We use as initial conditions the self-sustaining solutions represented by (dashed line) and (solid line) at and at (dash-dotted line).

The energy of the mean flow vs time. We use as an initial condition the self-sustaining solutions represented by at . The time-averaged eight-vortex state is also shown (top right); it is not symmetric with respect to the diagonals of the cross section.

The energy of the mean flow vs time. We use as an initial condition the self-sustaining solutions represented by at . The time-averaged eight-vortex state is also shown (top right); it is not symmetric with respect to the diagonals of the cross section.

Energy of the fluctuations vs time. We use as an initial condition the self-sustaining solution represented by at .

Energy of the fluctuations vs time. We use as an initial condition the self-sustaining solution represented by at .

The skin friction vs time. We use as an initial condition the self-sustaining solution represented by at . The mean skin friction value, averaged from to , equals 0.0424. The skin friction for the laminar flow equals 0.02 (dashed line).

The skin friction vs time. We use as an initial condition the self-sustaining solution represented by at . The mean skin friction value, averaged from to , equals 0.0424. The skin friction for the laminar flow equals 0.02 (dashed line).

The initial condition (circle) corresponds to the self-sustaining solution represented by (left figure) and (right figure) at . Both initial conditions correspond to and . The arrows indicate the direction in which the time increases. The ideal laminar flow corresponds to and and is indicated by a square.

The initial condition (circle) corresponds to the self-sustaining solution represented by (left figure) and (right figure) at . Both initial conditions correspond to and . The arrows indicate the direction in which the time increases. The ideal laminar flow corresponds to and and is indicated by a square.

## Tables

The five least stable eigenvalues of the symmetric stream function in a square duct.

The five least stable eigenvalues of the symmetric stream function in a square duct.

The four possible symmetries for the linear waves, even and odd .

The four possible symmetries for the linear waves, even and odd .

Properties of the discovered SSP solutions. The measure of the amplitude of the rolls is defined as .

Properties of the discovered SSP solutions. The measure of the amplitude of the rolls is defined as .

Parameters for the three cases studied; is the streamwise wavenumber of the traveling wave and is the initial energy of the fluctuations.

Parameters for the three cases studied; is the streamwise wavenumber of the traveling wave and is the initial energy of the fluctuations.

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