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Scaling laws of compressible turbulence

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Abstract

Spatial scaling laws of velocity kinetic energy spectra for the compressible turbulence flow and the density-weighted counterparts are formulated in terms of the wavenumber, dissipation rate, and Mach number by using a dimensional analysis. We apply the Barenblatt’s incomplete similarity theory to both kinetic and density-weighted energy spectra. It shows that, within the initial subrange, both energy spectra approach the –5/3 and –2 power laws of the wavenumber when the Mach number tends to unity and infinity, respectively.

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Abbreviations

u :

flow velocity

v :

density-weighted velocity

u′i :

fluctuation of flow velocity

L :

length dimension

t :

time dimension

m :

mass dimension

k :

wavenumber

ν :

kinematic viscosity

ε :

dissipation rate

E :

kinetic energy spectrum

E ρ :

density-weighted energy spectrum

Ma :

Mach number

Re :

Reynolds number

c :

speed of sound

γ :

ratio of specific heat

η :

Kolmogorov length

C K :

Kolmogorov constant

C :

coefficient constant

C ρ :

coefficient constant

A :

adjustable parameter

σ :

adjustable parameter

β :

adjustable parameter

β ρ :

adjustable parameter

E c :

compressible part of E ρ

E s :

solenoidal part of E ρ

c c :

compressible part constant

c s :

solenoidal part constant

λ :

constant

d(Ma):

exponent

h(Ma):

exponent

α :

exponent

l :

box size

l ν :

box size of sequence ν

ρ :

mass density

ρ 0 :

local reservoir value of density

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Acknowledgements

This paper commemorates my beloved father, Zhongchuan SUN. Supports from the South Africa National Research Foundation (NRF), Cape Peninsula University of Technology (CPUT), and the State Key Laboratory for Turbulence and Complex Systems at Peking University are gratefully acknowledged. The author would like to express his most sincere thanks to reviewers for their high level academic comments and corrections, and their professionalism inspires me deeply. The author wishes to take this opportunity to appreciate the general discussion on turbulence with Professors Shiyi CHEN, Zhensu SHE, Cunbiao LEE, Yipeng SHI, and Jiancun WANG.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Cape Peninsula University of Technology, Cape Town, 7535, South Africa

    Bohua Sun

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Correspondence to Bohua Sun.

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Project supported by the National Research Foundation of South Africa (No. 93918)

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Sun, B. Scaling laws of compressible turbulence. Appl. Math. Mech.-Engl. Ed. 38, 765–778 (2017). https://doi.org/10.1007/s10483-017-2204-8

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