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Turbulent spots and waves in a model for plane Poiseuille flow
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Image of FIG. 1.
FIG. 1.

(Color online) Profiles of the basis functions used in the derivation of the model. (a) Polynomials for the in-plane velocity. with dash-dotted (red) line, in dotted (magenta), in dashed (black), and the base flow in solid (blue). (b) Polynomials for the wall-normal velocity. in dashed (red) and in solid (blue).

Image of FIG. 2.
FIG. 2.

Evolution of a turbulent spot depicted by the streamwise velocity . . From top to bottom and left to right: , 112, 208, 272, 448, and 560 (enhanced online). [URL: http://dx.doi.org/10.1063/1.2821912.1]doi: 10.1063/1.2821912.1.

Image of FIG. 3.
FIG. 3.

(Color online) (a) Spatial distribution of . In the turbulent region limited by the (red) box, is mainly positive. (b) The mean profile of the total streamwise velocity (in solid blue). The laminar parabolic profile is shown with the dashed (red) line.

Image of FIG. 4.
FIG. 4.

(Color online) (a) The distribution of the spanwise velocity . The spot is split into small (arbitrary) regions limited by (red) boxes. There are six boxes for and six others for . (b) The mean value of the total spanwise velocity in each bottom box : From left to right: box number (1) with a dotted (black) line, (2) with a thick solid (red) line, (3) with a thin solid (blue) line, (4) with a thin dashed (magenta) line, (5) with a thick dashed (green) line, and (6) with a thin dash-dotted (red) line.

Image of FIG. 5.
FIG. 5.

(Color online) (a) The mean value of the total spanwise velocity in each upper box , shown in Fig. 4 . Same color convention as in Fig. 4(b) . (b) The spatial distribution of . At the channel centerline, the total wall-normal velocity is equal to . Oblique waves are present at the wing tips of the spot (enhanced online). [URL: http://dx.doi.org/10.1063/1.2821912.2]doi: 10.1063/1.2821912.2.

Image of FIG. 6.
FIG. 6.

The spatial distributions of the spanwise velocity (a) and of the streamwise velocity [URL: http://dx.doi.org/10.1063/1.2821912.3] (b) associated with show waves at the wing tips of the spot. Not all the points are represented (enhanced online). [URL: http://dx.doi.org/10.1063/1.2821912.4]doi: 10.1063/1.2821912.3.

doi: 10.1063/1.2821912.4.

Image of FIG. 7.
FIG. 7.

(Color online) The profiles of (in dashed red), (in thin blue), (in thick black) along a streamwise line for (a) and (b). The wave packets are clearly observed for (a) and (b).

Image of FIG. 8.
FIG. 8.

The spatial distribution of (a) represents an inflow toward the spot, whereas (b) represents an outflow from the spot. Outside the spot, the in-plane flow is quadrupolar, as shown in Fig. 9 .

Image of FIG. 9.
FIG. 9.

(a) The flow field exhibits a quadrupolar structure (not all the points are represented). (b) Spatial distribution of the stream function . Outside the spot, it has four lobes with alternating sign. Each lobe corresponds to a region of large recirculation.

Image of FIG. 10.
FIG. 10.

Spatial distribution of (a) and the flow field (b). This spatial distribution represents a crescent vortex in each half-space and . A 3D representation is given in Fig. 11 . The two legs of the crescent vortex are elongated in the streamwise direction and form the streamwise vortices.

Image of FIG. 11.
FIG. 11.

(Color online) 3D reconstruction of the crescent vortex, depicted by and in Fig. 10 . The distribution of has a crescent shape in the shaded plane for some . The two circles represent the cross section of the streamwise vortices. The polynomial is plotted by a solid blue line. For clarity, only the two counter-rotating streamwise vortices spanning the half-space are represented.

Image of FIG. 12.
FIG. 12.

(a) The flow field depicted by vectors. Two dipoles are clearly observed. (b) The Reynolds stress . There are three -elongated patches with positive value.



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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Turbulent spots and waves in a model for plane Poiseuille flow