FLUID In the literature: partially resolved

Direct measurement of directed-percolation exponents in pipe flow

In plain words

Near the onset of turbulence in a pipe, turbulent patches called puffs either decay or split, much like sick individuals recovering or infecting others. Pipe flow has now been placed in the class of this epidemic-type transition (directed percolation), but its critical exponents have not been measured directly in a pipe.

Precise statement

Hagen-Poiseuille flow with $\mathrm{Re} = U D / \nu$ (U bulk velocity, D diameter) has its critical point $\mathrm{Re}_c \sim 2040$, where puff splitting balances decay. Measure directly the turbulent-fraction exponent $\beta$ in $F_t \sim \varepsilon^{\beta}$, $\varepsilon = (\mathrm{Re} - \mathrm{Re}_c)/\mathrm{Re}_c$, and the correlation exponents $\nu_{\mathrm{perp}}$ and $\nu_{\mathrm{par}}$, to test the ($1+1$)-dimensional directed-percolation values $\beta \sim 0.276, \nu_{\mathrm{perp}} \sim 1.097, \nu_{\mathrm{par}} \sim 1.734$, and determine the range of $\varepsilon$ over which this scaling holds before puff jamming sets in. Answer: measured exponents with error bars and the width of the critical range.

What would settle it

Turbulent-fraction and correlation measurements in pipes many thousands of diameters long, over times long compared with puff splitting and decay times, close enough to $\mathrm{Re}_c$ to fit exponents.

Status in the literature

Unverified note

Experiments on pairwise puff interactions combined with simulations place the pipe transition in the directed-percolation class and reveal a jammed puff phase above $\mathrm{Re}_c$ (Lemoult et al., Nature Physics 2024); direct fits of $\beta$, $\nu_{\mathrm{perp}}$ and $\nu_{\mathrm{par}}$ in a pipe are still lacking.