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mode
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e
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Range
of
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Mode characteristics
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Perturbation
animation
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Flow
animation
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1
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0.01
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0≤l≤2.87
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Antisymmetry-breaking
Dean vortices oscillate in counter phase
Zero pseudo streamline is strongly perturbed
An increase of through flow intensifies the
Dean vortices, which slow down the through flow and consequently weaken by
themselves
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Perturbation_e=0p01_l=0.avi
Perturbation_e=0p01_l=2.avi
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Flow_e=0p01_l=0.avi
Flow_e=0p01_l=2.avi
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2
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0.01
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2.87≤l≤5
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Perturbations
are located inside
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Perturbation_e=0p01_l=4.avi
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Flow_e=0p01_l=4.avi
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0.03
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1.1≤l≤5
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a weaker vortex and are advected
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Perturbation_e=0p05_l=4.avi
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Flow_e=0p05_l=4.avi
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0.05
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1≤l≤5
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along the streamlines
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Perturbation_e=0p1_l=3p5.avi
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Flow_e=0p1_l=3p5.avi
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0.075
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1.9≤l≤5
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Zero pseudo streamline is strongly
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0.1
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perturbed
Instability of boundary layer is assumed
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3
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0.03
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0≤l≤1.1
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Antisymmetry-preserving
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Perturbation_e=0p03_l=0.avi
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Flow_e=0p03_l=0.avi
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0.05
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0≤l≤1
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Zero pseudo streamline is not perturbed
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Perturbation_e=0p03_l=1.avi
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Flow_e=0p03_l=1.avi
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0.075
0.1
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0≤l≤0.9
0≤l≤0.6
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Dean vortices oscillate in phase and do
not interact
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Perturbation_e=0p05_l=0.avi
Perturbation_e=0p075_l=0.avi
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Flow_e=0p05_l=0.avi
Flow_e=0p075_l=0.avi
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Instability of boundary layer is assumed
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Perturbation_e=0p1_l=0.avi
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Flow_e=0p1_l=0.avi
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4
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0.075
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0.9≤l≤1.9
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Small
circumferential wavenumber
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0.1
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0.6≤l≤2.8
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downstream cross-flow wave located inside
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Perturbation_e=0p1_l=1p5.avi
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Flow_e=0p1_l=1p5.avi
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0.2
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0.4≤l≤2.55
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a weaker vortex
Zero pseudo streamline is strongly
perturbed
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Perturbation_e=0p2_l=1p5.avi
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Flow_e=0p2_l=1p5.avi
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5
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0.1
0.2
0.3
0.4
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4.2≤l≤5
2.55≤l≤5
2.6≤l≤2.88
2.2≤l≤2.35
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Large
circumferential wavenumber
upstream cross-flow wave
Zero pseudo streamline, if exists, is
strongly perturbed
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Perturbation_e=0p1_l=4p5.avi
Perturbation_e=0p2_l=3.avi
Perturbation_e=0p2_l=4.avi
Perturbation_e=0p2_l=5.avi
Perturbation_e=0p3_l=2p7.avi
Perturbation_e=0p4_l=2p3.avi
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Flow_e=0p1_l=4p5.avi
Flow_e=0p2_l=3.avi
Flow_e=0p2_l=4.avi
Flow_e=0p2_l=5.avi
Flow_e=0p3_l=2p7.avi
Flow_e=0p4_l=2p3.avi
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6
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0.2
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0≤l≤0.4
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Antisymmetry-breaking
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Perturbation_e=0p2_l=0.avi
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Flow_e=0p2_l=0.avi
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0.3
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0≤l≤1.37
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Dean vortices do not interact
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Perturbation_e=0p3_l=0.avi
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Flow_e=0p3_l=0.avi
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0.4
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0≤l≤0.7
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Zero pseudo streamline is weakly
perturbed
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Perturbation_e=0p3_l=1.avi
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Flow_e=0p3_l=1.avi
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0.5
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0≤l≤0.81
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At larger perturbations are located
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Perturbation_e=0p4_l=0.avi
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Flow_e=0p4_l=0.avi
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0.6
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0≤l≤0.3
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inside a stronger vortex
An inviscid instability of through flow
is assumed
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Perturbation_e=0p5_l=0.avi
Perturbation_e=0p6_l=0.avi
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Flow_e=0p5_l=0.avi
Flow_e=0p6_l=0.avi
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7
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0.3
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2.9≤l≤5
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The disturbance
propagates upstream the mean through flow
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Perturbation_e=0p3_l=4.avi
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Flow_e=0p3_l=4.avi
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0.4
0.5
0.6
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2.45≤l≤5
2.05≤l≤5
1.75≤l≤5
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Large circumferential wavenumber downstream cross-flow wave developing
in a single Dean vortex flow
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Perturbation_e=0p4_l=4.avi
Perturbation_e=0p5_l=4.avi
Perturbation_e=0p6_l=4.avi
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Flow_e=0p4_l=4.avi
Flow_e=0p5_l=4.avi
Flow_e=0p6_l=4.avi
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8
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0.3
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1.37≤l≤2
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Oscillations in the bulk of
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Perturbation_e=0p3_l=1p55.avi
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Flow_e=0p3_l=1p55.avi
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0.4
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1.33≤l≤1.7
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the counter clockwise vortex that cause
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Perturbation_e=0p4_l=1p4.avi
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Flow_e=0p4_l=1p4.avi
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oscillations in the whole flow.
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Instability of a locally developing
mixing layer is assumed
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9
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0.4
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1.17≤l≤1.34
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Small
circumferential wavenumber
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Perturbation_e=0p4_l=1p1.avi
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Flow_e=0p4_l=1p1.avi
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0.5
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0.81≤l≤1.24
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downstream cross-flow wave
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Perturbation_e=0p5_l=1.avi
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Flow_e=0p5_l=1.avi
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0.6
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0.57≤l≤1.2
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Zero pseudo streamline is strongly
perturbed
Instability of a locally developing
mixing layer is assumed
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Perturbation_e=0p6_l=0p6.avi
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Flow_e=0p6_l=0p6.avi
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10
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0.3
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2≤l≤2.6
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The disturbance
propagates upstream the mean through flow
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Perturbation_e=0p3_l=2p4.avi
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Flow_e=0p3_l=2p4.avi
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0.4
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1.7≤l≤2.2
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Large circumferential wavenumber downstream cross-flow wave
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Perturbation_e=0p4_l=1p8.avi
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Flow_e=0p4_l=1p8.avi
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0.5
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1.24≤l≤1.88
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An inviscid instability of through flow
is
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Perturbation_e=0p5_l=1p5.avi
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Flow_e=0p5_l=1p5.avi
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0.6
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1.2≤l≤1.6
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assumed
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Perturbation_e=0p6_l=1p3.avi
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Flow_e=0p6_l=1p3.avi
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11
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0.4
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2.27≤l≤2.45
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s – independent (k=0)
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0.5
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1.88≤l≤2.05
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large circumferential wavenumber
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Perturbation_e=0p4_l=2p3_k=0.avi
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Flow_ e=0p4_l=2p3_k=0.avi
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0.6
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1.6≤l≤1.75
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downstream cross-flow wave
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Perturbation_e=0p6_l=1p7_k=0.avi
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Flow_ e=0p6_l=1p7_k=0.avi
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Zero pseudo streamline is slightly
perturbed
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12
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0.4
0.6
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0.7≤l≤1.16
0.47≤l≤0.55
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large circumferential wavenumber upstream wave
Instability of a locally developing
mixing layer is assumed
Zero pseudo streamline is slightly
perturbed
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Perturbation_e=0p4_l=1p1avi
Perturbation_e=0p6_l=0p5avi
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Flow_ e=0p4_l=1p1.avi
Flow_ e=0p6_l=0p5.avi
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13
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0.6
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0.3≤l≤0.47
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Small
circumferential wavenumber
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Perturbation_e=0p6_l=0p4avi
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Flow_
e=0p6_l=0p4.avi
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downstream waves developing
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in both vortices
Instability of a locally developing mixing
layer is assumed
Zero pseudo streamline is strongly perturbed
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