Dynamical landscape of transitional pipe flow

The transition to turbulence in pipes is characterized by a coexistence of laminar and turbulent states. At the lower end of the transition, localized turbulent pulses, called puffs, can be excited. Puffs can decay when rare fluctuations drive them close to an edge state lying at the phase-space boundary with laminar flow. At higher Reynolds numbers, homogeneous turbulence can be sustained, and dominates over laminar flow. Here we complete this landscape of localized states, placing it within a unified bifurcation picture. We demonstrate our claims within the Barkley model, and motivate them generally. Specifically, we suggest the existence of an antipuff and a gap-edge—states which mirror the puff and related edge state. Previously observed laminar gaps forming within homogeneous turbulence are then naturally identified as antipuffs nucleating and decaying through the gap edge. We also discuss alternatives to the suggested bifurcation diagram, which could be relevant for wall-bounded flows other than straight pipes.

Figure 1

Sketch of the bifurcation diagram for transitional pipe flow. Attracting states are solid lines, unstable edge states are dotted. In the deterministic Barkley model used here, $r_{\text{turb}}=0.667,r_{\text{slug}}=0.726,r_{\text{gap}}=0.736$

Figure 2

Spatial profiles of turbulence level $q$ and mean velocity $u$ in the Barkley model: Puff and decay edge at $r<r_{\text{turb}}$ (left) and antipuff and gap edge at $r_{\text{slug}}<r<r_{\text{gap}}$ (right)

Figure 3

Explicitly computed bifurcation diagram for the Barkley model. The stable states for each $r$ (laminar, turbulent, puff, antipuff) are computed by relaxation of the dynamics. The unstable edge states (gap edge, decay edge) are computed by the edge-tracking algorithm described in Sec. 3b2. Note the jump in the $y$ axis from 0.2 to 0.8 at the solid horizontal grid line.

Figure 4

(a) The spatial structure of the gap edge $q\left(x\right)$ corresponds to a homoclinic trajectory of a particle moving in the potential $V_r\left(q\right)$, with ${\partial }_q{V}_r\left(q\right)=f(q,u)$ and $x$ playing the role of time. The particle starts at $q_t$, reaches $q_g$, and returns. (b) When $r=r_{\text{gap}}$, this homoclinic trajectory goes all the way to $q_g=0$.

Figure 5

Matching between upstream and downstream speeds for antipuffs in the asymptotic Barkley model. (a) Below $r=0.75$, there is no solution for $u_{\text{ap}}$. (b) At $r\approx 0.7555$, there is a single solution, $u_{\text{ap}}\approx 1.8$ (c) For $0.7555\lesssim r\lesssim 0.7613$, there are two solutions. (d) At $r\approx 0.7613$, the higher velocity solution occurs at $u_{\text{ap}}=U_0=2$, corresponding to the identification between the downstream front of a puff and that of the unstable antipuff.

Figure 6

Alternative bifurcation diagrams for transitional flow. (a) Existence of two antipuff solutions, with the unstable (U antipuff) and stable antipuff (S antipuff) being created out of a saddle-node bifurcation at ${\text{Re}}_{\text{ap}}$. Note the coexistence of puffs and antipuffs in this regime marked in blue. The intermittent turbulence regime is marked in red. (b) Disappearance of the gap edge before the transition to slugs: ${\text{Re}}_{\text{gap}}<{\text{Re}}_{\text{slug}}$. Like in (a), there are two antipuff solutions and a coexistence region between antipuffs and puffs. The intermittent turbulence regime is absent.

Figure 7

For $r=0.748$, $\sigma =0.22$, the antipuff is long lived, but noise is strong enough to drive a rare transition from turbulent flow into the antipuff (left) and from an antipuff back into the fully turbulent flow (right).

Figure 8

Spatial structure (left) and $q\text{−}u$ plot (right) of the antipuff at $r=0.748$, $\sigma =0.22$. Many realizations of the stochastic antipuff are aligned at the downstream front to obtain its average form. In the $q\text{−}u$ plane, we superimpose a density plot of all considered realizations of the antipuff.