Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-19T09:40:17.406Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aluie, H. 2013 Scale decomposition in compressible turbulence. Physica D 247, 54–65.CrossRefGoogle Scholar
Alves-Portela, F., Papadakis, G. & Vassilicos, J.C. 2020 The role of coherent structures and inhomogeneity in near-field interscale turbulent energy transfers. J. Fluid Mech. 896, A16.CrossRefGoogle Scholar
Arun, S., Sameen, A., Srinivasan, B. & Girimaji, S.S. 2019 Topology-based characterization of compressibility effects in mixing layers. J. Fluid Mech. 874, 38–75.CrossRefGoogle Scholar
Casciola, C.M., Gualtieri, P., Benzi, R. & Piva, R. 2003 Scale-by-scale budget and similarity laws for shear turbulence. J. Fluid Mech. 476 (476), 105–114.CrossRefGoogle Scholar
Cimarelli, A. & De Angelis, E. 2012 Anisotropic dynamics and sub-grid energy transfer in wall-turbulence. Phys. Fluids 24, 015102.CrossRefGoogle Scholar
Clark, T.T. & Spitz, P.B. 1995 Two-point correlation equations for variable density turbulence. Tech. Rep. LA-12671-MS. Los Alamos National Lab.CrossRefGoogle Scholar
Clark, T.T. 2020 Two-point evolution equations for incompressible variable density turbulence, arXiv:2011.03345.Google Scholar
Danaila, L., Anselmet, F., Zhou, T. & Antonia, R.A. 2001 Turbulent energy scale budget equations in a fully developed channel flow. J. Fluid Mech. 430, 87–109.CrossRefGoogle Scholar
Davidson, P.A. & Pearson, B.R. 2005 Identifying turbulent energy distributions in real, rather than fourier, space. Phys. Rev. Lett. 95 (21), 1–4.CrossRefGoogle ScholarPubMed
Gatski, T.B. & Bonnet, J.-P. 2009 Compressibility, Turbulence and High Speed Flow. Elsevier.Google Scholar
Gomes-Fernandes, R., Ganapathisubramani, B. & Vassilicos, J.C. 2015 The energy cascade in near-field non-homogeneous non-isotropic turbulence. J. Fluid Mech. 771, 676–705.CrossRefGoogle Scholar
Hamba, F. 2015 Turbulent energy density and its transport equation in scale space. Phys. Fluids 27, 085108.CrossRefGoogle Scholar
Hamba, F. 2018 Turbulent energy density in scale space for inhomogeneous turbulence. J. Fluid Mech. 842 (8), 532–553.CrossRefGoogle Scholar
Hill, R.J. 2002 Exact second-order structure-function relationships. J. Fluid Mech. 468, 317–326.CrossRefGoogle Scholar
von Kármán, T. & Howarth, L. 1938 On the statistical theory of isotropic turbulence. Proc. R. Soc. Lond. A 164 (917), 192–215.CrossRefGoogle Scholar
Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk. SSSR 30, 299–303.Google Scholar
Kolmogorov, A.N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13 (1), 82–85.CrossRefGoogle Scholar
Kumar, G., Girimaji, S.S. & Kerimo, J. 2013 WENO-enhanced gas-kinetic scheme for direct simulations of compressible transition and turbulence. J. Comput. Phys. 234, 499–523.CrossRefGoogle Scholar
Lai, C.C.K., Charonko, J.J. & Prestridge, K. 2018 A Kármán-Howrath-Monin equation for variable-density turbulence. J. Fluid Mech. 843, 382–418.CrossRefGoogle Scholar
Mittal, A. & Girimaji, S.S. 2019 Mathematical framework for analysis of internal energy dynamics and spectral distribution in compressible turbulent flows. Phys. Rev. Fluids 4, 042601.CrossRefGoogle Scholar
Mollicone, J.P., Battista, F., Gualtieri, P. & Casciola, C.M. 2018 Turbulence dynamics in separated flows: the generalised Kolmogorov equation for inhomogeneous anisotropic conditions. J. Fluid Mech. 841, 1012–1039.CrossRefGoogle Scholar
Monin, A.S. 1959 The theory of locally isotropic turbulence. Sov. Phys. Dokl. 4, 271.Google Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Praturi, D.S. & Girimaji, S.S. 2019 Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence. Phys. Fluids 31, 055114.CrossRefGoogle Scholar
Watanabe, T., da Silva, C.B. & Nagata, K. 2020 Scale-by-scale kinetic energy budget near the turbulent/nonturbulent interface. Phys. Rev. Fluids 5, 124610.CrossRefGoogle Scholar
Xu, K., Mao, M. & Tang, L. 2005 A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow. J. Comput. Phys. 203 (2), 405–421.CrossRefGoogle Scholar
Zhou, Y. & Vassilicos, J.C. 2020 Energy cascade at the turbulent/nonturbulent interface. Phys. Rev. Fluids 5, 064604.CrossRefGoogle Scholar