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Coherent flow states in a square duct
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10.1063/1.2978357
/content/aip/journal/pof2/20/9/10.1063/1.2978357
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/9/10.1063/1.2978357

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

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).

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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 .

Image of FIG. 6.
FIG. 6.

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).

Image of FIG. 7.
FIG. 7.

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).

Image of FIG. 8.
FIG. 8.

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).

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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).

Image of FIG. 12.
FIG. 12.

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.

Image of FIG. 13.
FIG. 13.

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

Image of FIG. 14.
FIG. 14.

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).

Image of FIG. 15.
FIG. 15.

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

Generic image for table
Table I.

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

Generic image for table
Table II.

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

Generic image for table
Table III.

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

Generic image for table
Table IV.

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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/content/aip/journal/pof2/20/9/10.1063/1.2978357
2008-09-25
2014-04-25
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Coherent flow states in a square duct
http://aip.metastore.ingenta.com/content/aip/journal/pof2/20/9/10.1063/1.2978357
10.1063/1.2978357
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