Distinct Organizational States of Fully Developed Turbulent Pipe Flow

Organizational states of turbulence are identified through novel analysis of large scale pipe flow experiments at a Reynolds number of 35 000. The distinct states are revealed by an azimuthal decomposition of the two-point spatial correlation of the streamwise velocity fluctuation. States with dominant azimuthal wave numbers corresponding to $k_{\theta }=2$,3,4,5,6 are discovered and their structure revealed as a series of alternately rotating quasistreamwise vortices. Such organizational states are highly reminiscent of the nonlinear traveling wave solutions previously identified at Reynolds numbers an order of magnitude lower.

Figure 1

Organizational states at $\mathrm{Re}=35000$ corresponding to $k_{\theta }=2,3,4,5$, and 6 (top to bottom), shown by (a) the spatial correlation of the streamwise velocity fluctuation in the radial-azimuthal plane, red contours show $R_{uu}=0.05$ and 0.1, blue contours $R_{uu}=-0.05$ and $-0.1$; the dot marks the center of the correlation at ($r_0$, ${\theta }_0$), (b) instantaneous snapshots of the velocity field, (c) conditionally averaged velocity fields with $u_0<0$, (d) conditionally averaged velocity fields with $u_0>0$. In (b)—(d) the color maps show streamwise velocity fluctuations (red indicates $u>0$, blue indicates $u<0$, and white indicates $u\approx 0$) and vector arrows show the direction of the velocity in the azimuthal-radial plane.

Figure 2

An example of the axial coherence of the wave number states. Top: Variation of azimuthal wave number ($k_{\theta }$) with axial distance ($x$), showing significant axial coherence of the wave number states. Middle: The corresponding correlation map at $r_0/R=0.75$ showing the correlations that lead to the state allocation and also the transitions between wave number states (red indicates $R_{uu}>0$, blue $R_{uu}<0$, white $R_{uu}\approx 0$ and $s$ is the arclength, i.e., $s=r_0\theta$). Bottom: The corresponding instantaneous velocity fluctuations, where red indicates $u>0$, blue $u<0$, and white $u\approx 0$.