Intermittency patterns in the chaotic transition of the planar flow past a circular cylinder

The present paper investigates the planar flow past a circular cylinder for Reynolds numbers between 1000 and 10 000. The flow is studied as a dynamical system, so the present investigation is motivated by the presence of complex patterns, as the Reynolds number increases, in the force-time signals when the system goes from the periodic to the chaotic regime. The 2D numerical simulations were performed with a Lagrangian particle method called diffused vortex hydrodynamics (DVH). This computational approach allows high spatial resolutions with an accurate description of different vortical scales shed in the flow field. Flow analysis was executed with the typical tools used in the study of dynamical systems (i.e., Fourier spectra, Poincaré sections, and phase space maps) and is supported by discussion of the near and far wake topologies. During the transition of the system from a regular to a chaotic regime, the lift time signal shows intermittent irregular patterns.

Figure 1

Left: wake shed by a ship model of length $2\text{m}$ and advancing in drift at $0.2\text{m/s}$. Colours are representative of the vertical component of the vorticity. Right: wake pattern of the tanker Argo Merchant inclined about ${45}^{{}^{\circ }}$ to the current and leaking crude oil (NASA photograph) (see Ref. [11]).

Figure 2

Cylinder vorticity wake for Re = 1000 case at maximum lift, see Supplemental Material [44].

Figure 3

Lift time signal at regime conditions and corresponding Fourier analysis for Re = 1000 case. The dominant and higher harmonics are highlighted with red circles.

Figure 4

Autocorrelation function for Re = 1000 case and corresponding PSD, where $\tau$ indicates the nondimensional time lag of the signal.

Figure 5

Phase map (top-left) and Poincaré section (bottom-right) for Re = 1000 case. The 3D plot of the phase map with the time axis is depicted top-right. The phase space is drawn bottom-left. The phase map lines, as well as the Poincaré section dots, are contoured darker blue with time.

Figure 6

Cylinder vorticity wake for at maximum lift. Top Re = 2000, bottom Re = 3000, see Supplemental Material [44].

Figure 7

Cylinder vorticity wake modulation at Re = 2000.

Figure 8

Lift time signal at regime conditions and corresponding Fourier analysis for Re = 2000 (top) and Re = 3000 (bottom) cases. The low frequency modulation is superimposed with a dash line.

Figure 9

Autocorrelation functions for Re = 2000 (top) and Re = 3000 (bottom) cases and corresponding PSD. With $\tau$ the nondimensional time lag of the signal is indicated.

Figure 10

Phase map (left), 3D view of the map (center) and Poincaré section (right) for Re = 2000 (top) and Re = 3000 (bottom) cases. The phase map lines, as well as the Poincaré section dots, are contoured darker blue with time.

Figure 11

Different shedding mechanisms observed at Re = 3100 referenced with the corresponding number in vorticity fields. Top: vorticity fields at different nondimensional times. Bottom: lift time signal and corresponding Fourier spectrum; with red dots the shedding time instants depicted on top frames and indicated with the corresponding number.

Figure 12

Example of the shedding pause at Re = 3100. Top: the vorticity fields; bottom: the lift (black solid) and drag (blue solid) time signals.

Figure 13

Phase maps for different time ranges at Re = 3100. Top: the lift time signal; the time ranges considered (delimited with solid lines) are highlighted. Bottom: the corresponding maps. For the sake of readability, the frames are numbered and the same numbers are indicated at the corresponding time range. The maps are contoured darker with time (the contour ranges are adapted to the corresponding interval).

Figure 14

Autocorrelation function (on the left frame) and power spectral density (on the right frame) for the case Re = 3100. With $\tau$ the nondimensional time lag of the signal is indicated.

Figure 15

Poincaré sections for different time ranges at Re = 3100. Small green diamonds: the full set of maxima; black circles: the maxima of the periodic time ranges; red triangles: the maxima of the irregular ones. For the sake of readability, the frames are numbered and the same numbers are indicated in the corresponding time range.

Figure 16

Vorticity far field for time instants falling in ranges 2, 3, and 5 of Fig. 13.

Figure 17

From top to bottom: lift time signals and Fourier spectra for Reynolds numbers from 3200 (top) to 3500 (bottom). Different time ranges where specific behaviours are found are separated with vertical solid lines. The irregular time ranges are highlighted in red.

Figure 18

From top to bottom: lift time signals and Fourier spectra for Reynolds numbers from 3600 (top) to 4000 (bottom). Different time ranges where specific behaviours are found are separated with vertical solid lines. The irregular time ranges are highlighted in red.

Figure 19

Poincaré sections for Reynolds numbers spanning from 3200 to 4000. Maxima related to regular time ranges are represented with circles, irregular time ranges with red triangles.

Figure 20

Ratio between irregular number of maxima over the total maxima for Reynolds numbers ranging from 3200 to 4000.

Figure 21

From top to bottom: lift time signals and Fourier spectra for different Reynolds numbers. From (top): 5000, 6000, 8000, 10 000 (bottom). The cases Re = 7000 and Re = 9000 were investigated but are not shown for the sake of readability. The irregular time ranges are highlighted in red.

Figure 22

Poincaré sections for Reynolds number spanning from 5000 to 10 000. Maxima related to regular time ranges are represented with circles, irregular time ranges with red triangles.

Figure 23

Vorticity field for Re = 10 000. The left frame depicts the far field at maximum lift, while the right frame is a magnification of the near wake, see Supplemental Material [44].

Figure 24

Maximum Lyapunov exponents distribution for all the Reynolds numbers investigated. The exponents are normalised with respect to the maximum value.

Figure 25

Time averaged drag coefficient for the Reynolds numbers investigated. With error bars, the variance associated to every time signal.

Figure 26

Autocorrelation function (on the left frames) and power spectral density (on the right frames) for cases from Re = 3200 (top) to Re = 3900 (bottom). With $\tau$ the nondimensional time lag of the signal is indicated.

Figure 27

Autocorrelation function (on the left frames) and power spectral density (on the right frames) for cases from Re = 4000 (top) to Re = 10 000 (bottom). With $\tau$ the nondimensional time lag of the signal is indicated.

Figure 28

Computational multiblock grid used for numerical simulations with FVM algorithm.

Figure 29

Flow past a circular cylinder at Re = 2000: lift force comparison between DVH (top) and FVM (bottom) together with the corresponding Fourier transforms (right frames).

Figure 30

Flow past a circular cylinder at Re = 2000: vorticity fields comparison between DVH (left) and FVM (right).

Figure 31

Flow past a circular cylinder at Re = 10 000: lift force comparison between DVH (top) and FVM (bottom).

Figure 32

Flow past a circular cylinder at Re = 10 000: vorticity fields comparison between DVH (left) and FVM (right).

Figure 33

Continuous wavelet transform with Morse wavelet for the lift signal at Re = 1000 (top), Re = 3100 (center), Re = 3200 (bottom). The contour lines correspond to $95\%$ of the maximum wavelet amplitude.