Coherent Structures in Localized and Global Pipe Turbulence

The recent discovery of unstable traveling waves (TWs) in pipe flow has been hailed as a significant breakthrough with the hope that they populate the turbulent attractor. We confirm the existence of coherent states with internal fast and slow streaks commensurate in both structure and energy with known TWs using numerical simulations in a long pipe. These only occur, however, within less energetic regions of (localized) “puff” turbulence at low Reynolds numbers ($\mathrm{Re}=2000–2400$), and not at all in (homogeneous) “slug” turbulence at $\mathrm{Re}=2800$. This strongly suggests that all currently known TWs sit in an intermediate region of phase space between the laminar and turbulent states rather than being embedded within the turbulent attractor itself. New coherent fast streak states with strongly decelerated cores appear to populate the turbulent attractor instead.

Figure 1

Skin friction coefficient $C_f=-⟨{\partial }_{z}p{⟩}_{r\theta zt}D/2\rho U^2$ for a $50D$ pipe where $\rho$ is the fluid density. Triangles—laminar flow; circles—puffs; solid squares—spatially inhomogeneous turbulence and open squares—homogeneous slug turbulence (the transition is gradual). The lower straight line corresponds to laminar flow and the upper to the Blasius friction law which is initially overshot by the turbulent flow [23]. At $\mathrm{Re}=2300$ the flow alternates between a 1 puff state (circle) and 2 or 3 puff states (square with error bars).

Figure 2

Axial component of vorticity in ($r$, $z$)-plane, $25D$ shown of $50D$ computational domain; (a) slug turbulence at $\mathrm{Re}=2800$; (b) inhomogeneous turbulence at $\mathrm{Re}=2400$ (c) puff turbulence at $\mathrm{Re}=2000$. Cross sections in ($r$, $\theta$) show the axial flow relative to the laminar profile with fast streaks (light/white) and slow streaks (dark/red), contour lines each $0.2U$; (d) $m=4$ and $m=3$ structures seen upstream and downstream of the trailing edge $z_{\mathrm{TE}}$ (flow is left to right); (e) sections from a puff where no clear structures are observed; (f) energetic section at $\mathrm{Re}=2800$, but resembling $m=5$ structure; (g) exact solution with threefold rotational symmetry.

Figure 3

Instantaneous correlations for the puff snapshot of Fig. 2c at positions relative to the trailing edge $z^{\prime }=z-z_{\mathrm{TE}}$. In the upper plot $\theta$ goes from 0 to $2\pi$ vertically; contour intervals of 0.1. The marks below indicate the positions of the cross sections of Fig. 2d.

Figure 4

Probability at different parts of a puff of a good correlation $C_m(z^{\prime })\gtrsim 0.1$ (see Fig. 3) at $\mathrm{Re}=2000$ (left). Probability of a correlation $C_m>C$ at any given point in the flow for $\mathrm{Re}=2400$ (middle) and 2800 (right).

Figure 5

Roll and streak energies at different parts of the puff of Fig. 2c ($\mathrm{Re}=2000$). Expanding in Fourier modes $m$ in $\theta$, the total streak and roll energies are defined as $E_{\mathrm{streak}}(z^{\prime })≔{\mathrm{Re}}^2\pi \sum _{m\ne 0}\int |u_{mz}^{\prime }{|}^{2}rdr$ and $E_{\mathrm{roll}}(z^{\prime })≔{\mathrm{Re}}^2\pi \sum _{m\ne 0}\int (|u_{mr}^{\prime }{|}^2+|u_{m\theta }^{\prime }{|}^2)rdr$ in units $\rho {\nu }^2$. Here ${\mathit{u}}^{\prime }=(u_r^{\prime },u_{\theta }^{\prime },u_z^{\prime })$ is the deviation from the laminar profile, for easier comparison with the TWs. Energies for two-, three- and fourfold rotationally symmetric TWs [11] are shown using lower thick, middle thin and upper thick red solid lines, respectively (the closed loops are produced by the finite continuum of TW axial wave numbers which can exist at $\mathrm{Re}=2000$). The pink short-dotted line “cloud” in the top right hand corner corresponds to a slug at $\mathrm{Re}=2800$.

Figure 6

Correlations in inhomogeneous turbulence ($\mathrm{Re}=2400$) at time intervals of $4D/U$, in a fixed window of $25D$ of the domain. The middle snapshot corresponds to that of Fig. 2b. Same scales as Fig. 3. The black lines mark positions of waves at the different times indicating the translation of the TWs. The phase speeds appear $\approx (1\pm 0.1)U$.