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Unified description of turbulent entrainment

Published online by Cambridge University Press:  03 December 2020

Maarten van Reeuwijk Open the ORCID record for Maarten van Reeuwijk [Opens in a new window] ,
J. Christos Vassilicos Open the ORCID record for J. Christos Vassilicos [Opens in a new window]  and
John Craske Open the ORCID record for John Craske [Opens in a new window]
Maarten van Reeuwijk*
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
J. Christos Vassilicos
Affiliation:
Univ. Lille, CNRS, ONERA, Arts et Metiers ParisTech, Centrale Lille, UMR 9014 – LMFL – Laboratoire de Mécanique des Fluides de Lille – Kampé de Feriet, F-59000 Lille, France
John Craske
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: m.vanreeuwijk@imperial.ac.uk
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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 $\bar {V}_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\bar {V}_g=\bar {u}_\zeta - \bar {\textrm {D}}h_t/\bar {\textrm {D}}t$, where $\bar {\textrm {D}}/\bar {\textrm {D}} t$ is the material derivative of the average flow field and $\bar {u}_\zeta$ 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.

JFM classification

Boundary Layers: Free shear layers Convection: Plumes/thermals Wakes/Jets: Jets

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JFM Papers
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Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A12
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© The Author(s), 2020. Published by Cambridge University Press

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