Exact Relation for Correlation Functions in Compressible Isothermal Turbulence

Abstract

Compressible isothermal turbulence is analyzed under the assumption of homogeneity and in the asymptotic limit of a high Reynolds number. An exact relation is derived for some two-point correlation functions which reveals a fundamental difference with the incompressible case. The main difference resides in the presence of a new type of term which acts on the inertial range similarly as a source or a sink for the mean energy transfer rate. When isotropy is assumed, compressible turbulence may be described by the relation -2/3ε(eff)r = F(r)(r), where F(r) is the radial component of the two-point correlation functions and ε(eff) is an effective mean total energy injection rate. By dimensional arguments, we predict that a spectrum in k(-5/3) may still be preserved at small scales if the density-weighted fluid velocity ρ(1/3)u is used.

Exact relations for energy transfer in self-gravitating isothermal turbulence

“…This form looks nearly similar to the previously derived exact relations for compressible turbulence [50][51][52]. In order to obtain the concerned form, we combine the above expressions and obtain for R H (and similarly for R ′ H )…”

“…With our old expression of thermodynamic energy correlator i.e. at n = 1 [50], we get at acoustic limit, R U (1; r) = c 2 s ρ0 ρρ ′ − ρ 2 /2 . After Fourier transform, it gives thermodynamic energy spectral density which is two times the required energy spectral density.…”

“…The derivation of the exact relation relates to general compressible turbulence whereas the principal physical motivation lies in understanding the role of gravity in cold star-forming clouds where the turbulence is usually supersonic. The derivation follows the methodology previously outlined by Galtier and Banerjee [50], Banerjee and Galtier [51,52]. However, this work revisits the definition of two-point correlation functions which renders the final exact relation simpler and entirely expressible in terms of two-point fluctuations.…”

“…The same form has also been used in Refs. [50,57,58]. Another possibility is to express the kinetic energy density as the square of the variable w ≡ √ ρu and to define the correlator as w · w ′ /2 in order to guarantee the positive definiteness of kinetic energy density in the spectral space [55,56].…”

Supersonic turbulence in 3D isothermal flow collision

“…Typical ways of forcing are continuous energy input in each grid cell or occasional energy input at randomly selected grid cells, to mimic energy input into the interstellar medium by, for example, collimated or spherical outflows or supernova explosions (Li & Nakamura 2006;de Avillez & Breitschwerdt 2007;Wang et al 2010;Moraghan et al 2013). Only rather recently have analytical results been presented that demonstrate the existence of an intermediate scaling range and derive scaling relations for compressible turbulence (Aluie 2011;Galtier & Banerjee 2011;Banerjee & Galtier 2013).…”