Intense structures of different momentum fluxes in turbulent channels

The effect of different definitions of the momentum flux on the properties of the coherent structures of the logarithmic region of wall-bounded turbulence is investigated by comparing the structures of intense tangential Reynolds stress with those of the alternative flux proposed in [Jiménez, J. Fluid Mech. 809, 585 (2016)]. Despite the fairly different statistical properties of the two flux definitions, it is found that their intense structures show many similarities, such as the dominance of “wall-attached” objects, and geometric self-similarity. The new structures are wider, but not taller, than the classical ones, and include both high- and low-momentum regions within the same object. It is concluded that they represent the same phenomenon as the classical groups of a sweep, an ejection, and a roller, suggesting that these groups should be considered the fundamental coherent structures of the momentum flux. The present results show that the properties of these composite momentum structures are robust with respect to the definition of the fluxes.

Figure 1

Percolation diagram for the identification of (a) Qs of $uv$, adapted from [2], and (b) Ops of ${\varphi }_{xy}$. The solid lines are the ratio of the volume of the largest object to the volume of all identified objects; the dashed ones are the ratio of the number of identified objects to the maximum number of objects. The vertical line is the nominal threshold: $H=1.75$ in (a), and $H=2$ in (b). Symbols denote the two Reynolds numbers, as in Table 1.

Figure 2

Joint PDF of the distance from the closest wall to the bottom $\left(y_{\min }\right)$ and top $\left(y_{\max }\right)$ of the structures. (a) Qs, (b) Ops. Contours include 40%, 90%, and 98.8% of data. ${\text{Re}}_{\tau }=934$.

Figure 3

(a) Joint PDF of the wall-parallel dimensions $({\mathrm{\Delta }}_x,{\mathrm{\Delta }}_z)$ of attached structures. Contours include 50% and 99.8% of the data. The two diagonals are ${\mathrm{\Delta }}_x=1.3{\mathrm{\Delta }}_z$ and ${\mathrm{\Delta }}_x=2.5{\mathrm{\Delta }}_z$. (b) PDF of the wall-parallel aspect ratio ${\mathrm{\Delta }}_x/{\mathrm{\Delta }}_z$ of tall attached objects $(y_{\min }^{+}>100)$. The two vertical lines mark the peak value of the PDF, obtained by parabolic fitting: ${\mathrm{\Delta }}_x/{\mathrm{\Delta }}_z=1.3$ and ${\mathrm{\Delta }}_x/{\mathrm{\Delta }}_z=2.5$. (c) As in (b) for the wall-normal aspect ratio ${\mathrm{\Delta }}_x/{\mathrm{\Delta }}_y$ of tall attached objects. The vertical lines are ${\mathrm{\Delta }}_x/{\mathrm{\Delta }}_y=1.8$ and ${\mathrm{\Delta }}_x/{\mathrm{\Delta }}_y=2.0$. (d) Joint PDF of the volume and height of attached structures. Contours include 50% and 99.8% of data. The diagonal line is $Vol.\propto {\mathrm{\Delta }}_y^{2.5}$. In all panels, solid and dashed lines correspond to Ops and Qs, respectively, and symbols denote the two Reynolds numbers, as in Table 1.

Figure 4

(a) Snapshot of low/high-momentum regions (blue for $u<0$, and red for $u>0)$, and $uv$ above the identification threshold for Qs (translucent contour). Flow is from left to right. (b) As in (a), for Ops. ${\text{Re}}_{\tau }=2003$ at $y/h=0.10$.

Figure 5

(a) PDF of the low-momentum fraction, $F$, as defined in Eq. (6), for tall attached Ops in the logarithmic layer. (b) Conditionally averaged volume of Ops as a function of $F$, where the overline denotes averaging over all the tall attached objects in the logarithmic layer. Symbols denote the two Reynolds numbers, as in Table 1.

Figure 6

Two-dimensional PDF of the relative position of (a) Closest Op relative to another Op. (b) Closest low-momentum Op $\left(F\ge 0.5\right)$ relative to a high-momentum Op $\left(F<0.5\right)$. (c) Closest $Q$ relative to an Op. (d) Closest sweep-ejection $Q$ pair relative to an Op. Bold line contours and darker filled areas contain 10% of the data. Thinner contours and lighter areas contain 50% of the data. Shaded contours are ${\text{Re}}_{\tau }=934$ and lines are ${\text{Re}}_{\tau }=2003$. The dashed ellipses correspond to the mean size of the reference object.

Figure 7

Instantaneous three-dimensional representational of a strong Op (yellow central object), and its closest sweep (red) and ejection (blue). Flow is from lower left to upper right. ${\text{Re}}_{\tau }=2003$.

Figure 8

(a) Cross section at ${\delta }_x=0$ of the conditionally averaged field around tall attached Ops in the logarithmic region, looking in the direction of the flow. The color background is $\tilde{u}$, increasing from blue to red; arrows are the velocity in the cross-plane; the heavier contour is ${\widetilde{{\varphi }_{xy}}}^{+}=-2.1$, and the lighter double one is ${\widetilde{uv}}^{+}=-2.1$. (b) As in (a), for the conditionally averaged field around tall attached sweep-ejection pairs in the logarithmic region. Symbols are as in (a), but the heavier double contour is ${\widetilde{uv}}^{+}=-1.3$, and the lighter one is ${\widetilde{{\varphi }_{xy}}}^{+}=-1.3$. ${\text{Re}}_{\tau }=934$.

Figure 9

(a) Three-dimensional plot of the conditionally averaged field around tall attached Ops in the logarithmic layer. The translucent isosurface is ${\widetilde{{\varphi }_{xy}}}^{+}=-2.1$, colored by $\tilde{u}$; the yellow solid isosurfaces are ${\widetilde{uv}}^{+}=-2.1$; arrows are the conditional cross-flow velocity: gray ones at ${\delta }_x=0$, and black ones at ${\delta }_x=3$. Flow is from lower left to upper right. (b) Enlargement of the smaller sub-box in (a). ${\text{Re}}_{\tau }=934$.