Dye visualization of bypass transition in the experiments. Two different streak breakdown sequences developing on a low-speed streak are shown. Left column: Sinuous antisymmetric breakdown. Right column: Varicose symmetric breakdown. From J. Mans, Ph.D. thesis, Eindhoven University of Technology (2007); courtesy of Mans and de Lange.
Three-dimensional visualization of the numerical simulations of bypass transition pertaining to two different types of structures prior to the formation of a turbulent spot. (a) Sinuous breakdown. (b) Varicose breakdown. Isocontours of positive and negative streamwise disturbance velocities in medium (red) and dark gray (blue) isocontours, respectively, and light gray (green) surfaces of the vortex identification criterion (Ref. 17) .
Wall-normal profile of the streamwise velocity disturbance for the optimal spatial transient growth in comparison with the experiments by Westin et al. (Ref. 24). The wall-normal coordinate is scaled with . Adapted from Andersson et al. (Ref. 26).
Nonlinear development of the optimal streaks. The dashed lines indicate linearly stable streaks, whereas the thick solid line represents the amplitude of the streaky flow used for the simulation of the nonlinear impulse response in Sec. III C.
Development of the linear impulse response on a parallel streak in a wall-parallel plane for three different times , , . The background color represents the streamwise velocity of the base flow, with high to lower velocity ranging from dark (red) to light (yellow). The isolines indicate the wall-normal perturbation velocity; dotted lines denote negative values. From L. Brandt, Ph.D. thesis, KTH Stockholm (2003) (Ref. 30).
Temporal growth rate vs streamwise wavenumber determined from the linear impulse response for various Reynolds numbers (: ; —: ; : ; - - - -: ) and the inviscid limit (—). The streak amplitude is . From L. Brandt, Ph.D. thesis, KTH Stockholm (2003) (Ref. 30).
Top view of the flow development for the nonlinear impulse response on a spatially evolving streak at times (a) , (b) , and (c) . Low-speed streaks are indicated by dark gray (blue) isocontours; medium gray (red) isocontours correspond to high-speed streaks .
Top view of the early flow development for the nonlinear impulse response. Medium gray (red) and dark gray (blue) isocontours correspond to high- and low-speed streaks (see Fig. 7); white and black isocontours are positive and negative wall-normal velocity at times (a) and (b) . (c) Light gray (green) isocontours correspond to the criterion at time .
Spanwise fluctuation energy for the nonlinear impulse response. (a) Early linear evolution with interval and (b) later nonlinear evolution, with interval .
Temporal energy growth for the nonlinear impulse response. The streamwise maximum of the perturbation energy integrated over cross-stream planes is displayed vs time.
Typical image sequence obtained with the combined PIV-LIF technique. The vector plot represents the disturbance field, while the gray scale indicated the dye intensity, here black states high dye intensity and white low intensity. . Real aspect ratio. From J. Mans, Ph.D. thesis, Eindhoven University of Technology, 2007; courtesy of Mans and de Lange.
Three-dimensional visualizations of sinuous streak instability prior to the formation of a turbulent spot. Isocontours of positive and negative streamwise disturbance velocities in medium (red) and dark gray (blue), respectively, and surfaces of the vortex identification criterion in light gray (green).
Complex eigenfunctions of the wall-normal velocity component pertaining to modes A and B used together with the Zaki–Durbin model problem, at the inlet . (—) Real part; (- - - -) imaginary part. (a) Penetrating low-frequency mode A. (b) Sheltered high-frequency mode B.
Streamwise velocity fluctuation in a -plane for the penetrating mode A and nonpenetrating mode B. (- - - -) boundary-layer thickness. Contour spacing and starting level for mode A: 0.025; for the nonpenetrating mode B: 0.0005.
Wall-normal fluctuation at for () penetrating mode A, (- - - -) nonpenetrating mode B, and (—) both modes A and B together.
Skin-friction coefficient for various cases. (—) Temporally and spatially averaged simulation results; (—) instantaneous for combination of modes A and B. (- - - -) and () only penetrating (a) and nonpenetrating (b) modes, respectively. () Laminar [Blasius, ] and turbulent correlations. Note that the laminar correlation virtually collapses with of mode B.
Top view of instantaneous wall-normal velocity at fixed wall-normal distance , ranging from (black) to 0.02 (white). Plotted range , , at relative time .
Three-dimensional top view of the flow structures pertaining to the Zaki–Durbin model problem. Isocontours of (medium and dark gray, red, and blue) and (light gray, green). Relative times , , and . True aspect ratio, spanwise range .
Wall-normal cuts displaying contours of the streamwise disturbance velocity for the Zaki–Durbin model problem at relative times , , and . Spanwise position , contour spacing of 0.025, negative values are denoted by dashed lines.
Parameters for the various numerical simulations presented. The Reynolds number denotes the location of the inflow of the computational domain, which is chosen larger for the cases involving the study of the instability only.
Parameters of the low-frequency (A) and high-frequency (B) modes of the continuous spectrum of the Orr–Sommerfeld/Squire equation. Note that modes A and B each contains both contributions. The spanwise width of the domain is .
Comparison of breakdown characteristics for sinuous streak breakdown. The wavelength is in units of the local displacement boundary-layer thickness , the velocities of the free-stream velocity , and the growth rate of . The three values for the propagation velocity relate to the velocity of the tail, center, and leading edge of the instability, respectively.
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