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Scale-space energy density function transport equation for compressible inhomogeneous turbulent flows

Published online by Cambridge University Press:  11 June 2021

S. Arun Open the ORCID record for S. Arun [Opens in a new window] ,
A. Sameen Open the ORCID record for A. Sameen [Opens in a new window] ,
Balaji Srinivasan  and
Sharath S. Girimaji
S. Arun
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
A. Sameen*
Affiliation:
Department of Aerospace Engineering, IIT Madras, Chennai600036, India
Balaji Srinivasan
Affiliation:
Department of Mechanical Engineering, IIT Madras, Chennai600036, India
Sharath S. Girimaji
Affiliation:
Department of Ocean Engineering, Texas A&M University, College Station, TX, USA
*
†Email address for correspondence: sameen@ae.iitm.ac.in
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Abstract

The scale-space energy density function $E(\boldsymbol {x},\boldsymbol {r})$ is defined as the derivative of the two-point velocity correlation $Q_{ii}(\boldsymbol {x},\boldsymbol {r})$ as $E(\boldsymbol {x},r_\alpha ) = -(\partial Q_{ii}(\boldsymbol {x},\boldsymbol {r})/\partial r_\alpha )/{2}$, where $\boldsymbol {x}$ is the spatial coordinate of interest and $\boldsymbol {r}$ is the separation vector. The function $E$ describes the turbulent kinetic energy density of scale $|\boldsymbol {r}|$ at a location $\boldsymbol {x}$ and can be considered as the generalization of the spectral energy density function concept to inhomogeneous flows. In this work, we derive the scale-space energy density function transport equation for compressible flows to develop a better understanding of scale-to-scale energy transfer and the degree of non-locality of the energy interactions. Specifically, the effects of variable-density and dilatation on an energy cascade are identified. It is expected that these findings will yield deeper insight into compressibility effects on canonical energy cascades, which will lead to improved models (at all levels of closure) for mass flux, density variance, pressure-dilatation, pressure–strain correlation and dilatational dissipation processes. Direct numerical simulation (DNS) data of mixing layers at different Mach numbers are used to characterize the scale-space behaviour of different turbulence processes. The scaling of the energy density function that leads to self-similar evolution at the two Mach numbers is identified. The scale-space (non-local) behaviour of the production and pressure dilatation at the centre-plane is investigated. It is established that production is influenced by long-distance (order of vorticity thickness) interactions, whereas the pressure dilatation effects are more localized (fraction of momentum thickness) in scale space. The analysis of DNS data demonstrates the utility of the energy density function and its transport equation to account for the relevance of various physical mechanisms at different scales.

JFM classification

Turbulent Flows: Compressible turbulence Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A31
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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