Unified description of turbulent entrainment

TL;DR

This paper develops a unified mathematical framework for turbulent entrainment applicable to various free shear flows, linking local and global descriptions, and analyzing effects of unsteadiness on entrainment.

Contribution

It introduces a comprehensive mathematical description of turbulent entrainment that applies to unsteady flows and connects small-scale and large-scale flow features.

Findings

The global entrainment velocity is expressed in terms of flow variables.

The model reproduces known results for classical flows like jets and wakes.

Unsteady conditions affect the entrainment coefficient, incorporating time-dependency effects.

Abstract

We present a mathematical description of turbulent entrainment that is applicable to free shear problems that evolve in space, time or both. Defining the global entrainment velocity $\overline V_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\overline V_g=\overline u_\zeta - \overline{D}h_t/\overline{D}t$, where $\overline D/\overline D t$ is the material derivative of the average flowfield and $\overline u_\zeta$ is the average velocity perpendicular to the flow direction across the interface located at $\zeta=h_t$. The description is shown to reproduce well-known results for the axisymmetric jet, the planar wake and the temporal jet, and provides a clear link between the local (small-scale) and global (integral) descriptions of turbulent entrainment. Application to unsteady jets/plumes demonstrates that, under unsteady conditions, the entrainment coefficient $\alpha$ no longer only captures entrainment of ambient fluid, but also time-dependency effects due to the loss of self-similarity.