Early Transient Period in the Evolution of the Area of a Passive Front Propagating in a Strong Turbulence

TL;DR

This paper analytically investigates the early evolution of a passive front in strong turbulence, revealing how turbulence influences front area growth and fluid consumption rate, with key dependence on large-scale turbulent velocities.

Contribution

It provides an analytical description of the initial stage of passive front evolution in high Reynolds number turbulence, highlighting the role of large-scale eddies and the physical mechanism limiting front growth.

Findings

Mean front consumption velocity is proportional to rms turbulent velocity.

Exponential increase in front area due to small-scale eddies at early times.

Front area growth is limited by collisions of front elements after initial exponential increase.

Abstract

Influence of statistically stationary, homogeneous, and isotropic turbulence on the mean area of a passive self-propagating front and, hence, on the rate of fluid consumption by the front is analysed in the case of asymptotically high turbulent Reynolds number $Re_L$ and asymptotically high ratio of the Kolmogorov velocity to a constant speed $u_0$ of the front. By considering an early stage of the front evolution, the mean (over the studied early stage) front area and consumption velocity $\bar{u}_T$ are analytically determined. The analysis shows that the mean $\bar{u}_T$ is proportional to the rms turbulent velocity, which characterizes large-scale turbulent eddies, even if the instantaneous rate of an increase in the front area is mainly controlled by the smallest Kolmogorov eddies. A straightforward dependence of the mean area or $\bar{u}_T$ on the Kolmogorov velocity, length, and time scales vanishes due to the following physical mechanism. At high $Re_L$, turbulent stretching created by small-scale eddies increases the front area exponentially with time, whereas a volume bounded by the leading and trailing edges of the front grows significantly slower. Therefore, after a short time, the volume is tightly filled by the front and the mean distance between opposed front elements becomes small with respect to Kolmogorov length scale. Subsequently, such front elements collide, thus, reducing the front area and limiting the mean $\bar{u}_T$.