In this paper, we develop an efficient Boltzmann-equation-based mesoscopic approach to simulate three-dimensional (3D) compressible turbulence, using reduced Gauss-Hermite quadrature (GHQ) orders by redefining the second distribution in terms of the total energy in the double distribution function approach. This allows the use of two sets of 3D off-lattice discrete particle velocity models, namely, a 27 discrete velocity model of the seventh-order GHQ accuracy (D3V27A7) combined with a 13 discrete velocity model of the fifth-order GHQ accuracy (D3V13A5), to achieve full consistency with the Navier-Stokes-Fourier system. The source terms in the Boltzmann-Bhatnagar-Gross-Krook system are designed to adjust both the Prandtl number and bulk-to-shear viscosity ratio. Compressible decaying homogeneous isotropic turbulence (DHIT) is simulated at low and moderate turbulent Mach numbers to validate our code. It is observed that the simulation results are in good agreement with those in the existing literatures. Furthermore, the terms in the transport equation of turbulent kinetic energy are analyzed in detail, to illustrate four different transient stages from the initial random flow field to the developed DHIT. It is shown that the transient pressure-dilatation transfer happens rapidly, while the small-scale vortical structures take a longer time to establish physically. Compared to the existing literatures, our approach represents the most efficient mesoscopic scheme for compressible turbulence under the double distribution function formulation.

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