Simplifying the complexity of pipe flow

Transitional pipe flow is modeled as a one-dimensional excitable and bistable medium. Models are presented in two variables, turbulence intensity and mean shear, that evolve according to established properties of transitional turbulence. A continuous model captures the essence of the puff-slug transition as a change from excitability to bistability. A discrete model, which additionally incorporates turbulence locally as a chaotic repeller, reproduces almost all large-scale features of transitional pipe flow. In particular, it captures metastable localized puffs, puff splitting, slugs, localized edge states, a continuous transition to sustained turbulence via spatiotemporal intermittency (directed percolation), and a subsequent increase in turbulence fraction toward uniform, featureless turbulence.

Figure 1

Regimes of transitional pipe flow. Left column is from full DNS with $4\times {10}^7$ degrees of freedom in a periodic pipe 200D long. Flow is from left to right. Shown are instantaneous values of turbulence intensity $q$ and axial velocity $u$ along the pipe axis. (a) Equilibrium puff at $\mathrm{Re}=2000$. (b) Puff splitting at $\mathrm{Re}=2275$. The downstream (right) puff split from the upstream one at an earlier time. (c) Expanding slug flow at $\mathrm{Re}=3200$. Right column shows corresponding states from the simple one-dimensional model (3, 4, 5, 6) (d) $R=2000$, (e) $R=2100$, and (f) $R=3200$.

Figure 2

The distinction between puffs and slugs seen as the difference between excitability and bistablilty in Eqs. (1) and (2). Phase planes show nullclines at (a) $r=0.7$ and (b) $r=1$. The fixed point $(1,0)$ corresponds to parabolic flow. In (b) the additional stable fixed point corresponds to stable turbulence. Solution snapshots show (c) a puff at $r=0.7$ and (d) a slug at $r=1$. These solutions are plotted in the phase planes with arrows indicating increasing $x$.

Figure 3

Illustration of the discrete model. (a) Local dynamics in the $u$-$q$ phase plane. Within a wedge-shaped region $q$ undergoes transient chaos, while outside it decays monotonically to $q=0$. The region varies with $R$ as indicated. (b) Map used to produce transient chaos. Parameter $\alpha$ (which depends on $u$ and $R$), is the lower boundary separating monotonic and chaotic dynamics.

Figure 4

Three regimes of pipe flow from simulations of the discrete model (3, 4, 5, 6). Space-time diagrams (time upward) illustrate (a) decaying puff at $R=1900$, (b) puff splitting at $R=2200$, and (c) slug formation from an edge state at $R=3000$. For ease of comparison with published work on pipe flow, solutions are shown in a frame comoving with structures. Turbulence intensity $q$ is plotted with $q=1.8$ in white. In (a) and (b) the scale is linear with $q=0$ in black, while in (c) the scale is logarithmic with $q⩽{10}^{-3}$ in black. Dimension bars indicate space and time scales. The top space scale applies also to (b).

Figure 5

Main figure is a bifurcation diagram for model turbulence in the thermodynamic limit. The turbulence fraction $F_t$ is plotted throughout the transitional regime. The onset of sustained turbulence, via spatiotemporal intermittency, occurs continuously at $R_c\simeq 2046.2$. $F_t$ increases with $R$ and saturates near $R=2800$. Asymptotic regimes (laminar, intermittent, featureless) are labeled, along with corresponding transient dynamics (decaying puffs, puff splitting, slugs). The onset of featureless turbulence is not sharp, as indicated by gray shading. Space-time plots illustrate the dynamics near the ends of the transitional regime ($R=2058$, $R=2720$, $R=2880$) with $q$ plotted in frames comoving with structures (color map indicated, dark is laminar). (a) Mean lifetimes for decaying (circles) and splitting (squares) puffs crossing at $R_{\times }\simeq 2040$. (b) log-log plot of $F_t$ versus $R-R_c$. Best fit to the solid (red) points determines $R_c$ and the slope.

Figure 6

Parameters ${\epsilon }_1$ and ${\epsilon }_2$ chosen to match a typical puff from DNS at $\mathrm{Re}=2000$. In the left plot ${\epsilon }_2=0.2$, while ${\epsilon }_1$ has values 0.02 (red), 0.04 (blue), and 0.06 (green). In the right plot ${\epsilon }_1=0.04$ and ${\epsilon }_2$ has values 0.1 (red), 0.2 (blue), and 0.4 (green). $r=0.7$. DNS is the irregular black curve. The unlabeled (blue) curves in the two plots correspond to the values of ${\epsilon }_1$ and ${\epsilon }_2$ used in the paper.

Figure 7

Effect of parameter $k$. Top row, $k=1$; bottom row, $k=2$. In each case snapshots are plotted in the $u$-$q$ phase plane. The left two plots show randomly chosen snapshots of solutions with three closely spaced puffs. The rightmost plots are of solutions following a quench from high $R$ and contain a large number of puffs. Top row is at $R=1760$ and the bottom row is at $R=2000$. Both values of $R$ are close to the transition to sustained turbulence for the corresponding value of $k$.

Figure 8

Effect of parameter $\beta$. Top row, $\beta =-0.4$; bottom row, $\beta =0$. In each case snapshots of solutions are shown on either side on the transition from localized puffs to expanding turbulence. (Top left) $R=1750$; (top right) $R=1780$; (bottom left) $R=1760$; (bottom right) $R=1800$.

Figure 9

Further effect of parameter $\beta$. (Left) $\beta =0.3$ ($R=1980$); (right) $\beta =0.5$ ($R=2180$). In each case snapshots of solutions are shown just after a puff split. There is little to distinguish the cases.

Figure 10

Effect of parameter $\gamma$. (Left) $\gamma =0.80$, with $R=2400$; (middle) $\gamma =0.95$, with $R=2000$, and is the same plot as bottom middle of Fig. 7. Arrows indicate the relevant region of the $u$-$q$ phase plane. (Right) DNS of a puff at $\mathrm{Re}=2000$, exactly the same as in Fig. 6.

Figure 11

Effect of parameters $c$ and $d$. In each case snapshots of solutions are shown just after puff splitting. Splitting is not very sensitive to $c$ and $d$ around values used in the paper. (Top left) $c=0.45$, $d=0.1$, $R=2480$; (top right) $c=0.45$, $d=0.2025$, $R=1960$; (bottom left) $c=0.35$, $d=0.15$, $R=2120$; (bottom right) $c=0.55$, $d=0.15$, $R=2080$.

Figure 12

Lifetime statistics for decaying and splitting model puffs. (a) Time series of total energy $E$ for three puffs illustrating abrupt decay at unpredictable times ($R=1800$). Bottom row shows exponential (memoryless) probabilities $P$ for (b) decaying puffs ($R=1800,1850,1900,1950,2000$) and (c) splitting puffs ($R=2060,2090,2120,2150,2180$).