The coordinate system used to project the Navier-Stokes equations.
(a) Streamfunctions Ψ1,a–c (η) and (b) corresponding velocity profiles u 0(η) for Ψ1,a (η) = 0.033(η − 3η 3 + 2η 4) (thin solid), Ψ1,b (η) = 0.7(η − 3η 3 + 2η 4)2 (dashed), and Ψ1,c (η) = 14(η − 3η 3 + 2η 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 − 3η 3 + 2η 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 Ψ6(η, φ) = (η 4 − 2η 5 + η 6) sin(6φ), 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.
(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.
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