Critical Behavior in the Relaminarization of Localized Turbulence in Pipe Flow

The statistics of the relaminarization of localized turbulence in a pipe are examined by direct numerical simulation. As in recent experimental data [J. Peixinho and T. Mullin, Phys. Rev. Lett. 96, 094501 (2006)], the half-life for the decaying turbulence is consistent with the scaling $({\mathrm{Re}}_c-\mathrm{Re}{)}^{-1}$, indicating a boundary crisis of the localized turbulent state familiar in low-dimensional dynamical systems. The crisis Reynolds number is estimated as ${\mathrm{Re}}_c=1870$, a value within 7% of the experimental value 1750. We argue that the frequently asked question, of which initial disturbances at a given Re trigger sustained turbulence in a pipe, is really two separate questions: the “local phase space” question (local to the laminar state) of what threshold disturbance at a given Re is needed to initially trigger turbulence, followed by the “global phase space” question of whether Re exceeds ${\mathrm{Re}}_c$ at which point the turbulent state becomes an attractor.

Figure 1

Numerical “puff” at $\mathrm{Re}=1900$. $(r,z)$ section of $(\mathbf{\nabla }\times \mathit{u}{)}_z$. Only $20D$ shown of the $50D$ computational domain.

Figure 2

Probability of relaminarization after time $T$ at $\mathrm{Re}=1740$ is the same for increased resolutions (dt: data with time step halved; dr: data with 60 radial points; dz: data with axial resolution of $\pm 576$), pipe length (L is $100D$ data), and different disturbance (F is data obtained with the puff generated by an initial period of body forcing—the data are shifted in the last case to account for a longer transient period). All data sets effectively overlay the default data WK. Inset: Numerical puff spectrum at $\mathrm{Re}=1900$, $A_n=\underset{km}{max}[\max (u_{nmk}^2,v_{nmk}^2,w_{nmk}^2)]$, where ($u_{nmk}$, $v_{nmk}$, $w_{nmk}$) are the radial, azimuthal, and axial velocity spectral amplitudes with the indices $n$, $m$, and $k$ referring to the (transformed) Chebyshev, Fourier, and Fourier expansions in $(r,\theta ,z)$, respectively (similarly for $A_k$ and $A_m$).

Figure 3

The probability of turbulent lifetime $\ge T$, $P(T)$, for several Re in a periodic pipe of length $16\pi D$.

Figure 4

The reciprocal of the puff half-life $\tau$ plotted against Re. Data plotted: WK—$50D$ data (each data point is the result of 40–60 simulations); PM—experimental data from Ref. 8; FE—reinterpreted $5D$ data [11]; H—experimental data from Ref. 12. Inset: log-plot of $1/\tau$ vs Re.

Figure 5

Trace of perturbation energy versus additional pressure fraction required to maintain fixed mass flux, $1+\beta =⟨{\partial }_{z}p⟩/{\mathrm{d}}_z{p}_{\mathrm{lam}}$ (the origin represents laminar flow), for the three cases of a sustained puff at $\mathrm{Re}=1900$ (solid line), a metastable puff at $\mathrm{Re}=1860$ with sudden relaminarization (dotted line), and the immediate decay of a perturbation (dashed line). The inset shows that the energy trace for the metastable puff essentially overlaps that of the sustained puff before it suddenly laminarizes.

Figure 6

Sketch of the two (independent) thresholds associated with transition: one is amplitude-dependent (and highly form-dependent), indicating when a turbulent episode is triggered, and the other is a global Re-dependent threshold, indicating when the turbulence will be sustained.