Propagation speed of turbulent fronts in pipe flow at high Reynolds numbers

We investigated the propagation of turbulent fronts in pipe flow at high Reynolds numbers by direct numerical simulation. used a technique combining moving frame reference and an artificial damping to isolate short periodic pipes, which enables us explore bulk number up Re = $10^5$ with affordable computation power. measured speed downstream front observed that fit $1.971-(Re/1925)^{-0.825}$ (in unit speed) well captures this above $Re\simeq 5000$. The increases monotonically as increases, stark contrast decreasing trend 10000$ reported Wygnanski & Champagne (1973). upstream overall agrees former studies $0.024 + (Re/1936)^{-0.528}$ fits our data those from literature. Based on analysis dynamics, we proposed both speeds would keep their respective monotonic trends further. show that, numbers, local transition tip is via high-azimuthal-wavenumber structures high-shear region near wall, whereas low-azimuthal-wavenumber low-shear center. This difference possibly responsible for asymmetric scalings between fronts.