In this paper, a new mesoscopic approach with both the adjustable Prandtl number and the ratio of bulk to shear viscosity has been developed to simulate three-dimensional compressible decaying homogeneous isotropic turbulence under the framework of discrete unified gas kinetic scheme (DUGKS). In the new approach, two reduced model Boltzmann equations with newly designed source terms are solved. In the continuum limit, the Navier–Stokes–Fourier system can be recovered by applying the Chapman–Enskog analysis. A three-dimensional DUGKS code has been developed, incorporating the fifth-order weighted essentially non-oscillatory scheme to better reconstruct the particle distribution functions at the cell interfaces. In addition, a new lattice velocity model with 77 discrete particle velocities is applied to ensure that the accuracy of the Gauss–Hermite quadrature is up to the ninth-order, and as such, the heat flux can be accurately evaluated. To validate our code, we simulate two cases with different initial turbulent Mach numbers and Taylor microscale Reynolds numbers. The simulation results converge with the increase in resolution and agree well with the results from the literature. As a direct application of our DUGKS, we briefly study the influence of bulk viscosity on turbulence statistics and flow structures. Our results show that the DUGKS is a reliable tool for simulating compressible decaying isotropic turbulence at low and moderate turbulent Mach numbers. More parametric studies are needed in the future to further explore the full capabilities of this specific mesoscopic method.

In this paper, we implemented the Boltzmann-equation-based mesoscopic model, developed recently by Chen et al. [“Inverse design of mesoscopic models for compressible flow using the Chapman–Enskog analysis,” Adv. Aerodyn. 3, 5 (2021)], to simulate three-dimensional (3D) forced compressible isotropic turbulence. In this model, both the Prandtl number and the ratio of bulk to shear viscosity can be arbitrary prescribed. The statistically stationary turbulent flow is driven by a large-scale momentum forcing in the Fourier space, with the internal heating due to the viscous dissipation at small scales being removed by a thermal cooling function. Under the framework of discrete unified gas kinetic scheme (DUGKS), a 3D direct numerical simulation code has been developed, incorporating a generalized Strang-splitting scheme. The weighted essentially non-oscillatory (WENO) scheme is used to increase local spatial accuracy in the reconstruction of particle distribution functions at the cell interface. A 3D discrete particle velocity model with a ninth-order Gauss–Hermite quadrature accuracy is used to ensure accurate evaluation of viscous stress and heat flux in the continuum regime. We simulate forced compressible isotropic turbulence at both low and high turbulent Mach numbers. A direct comparison is performed with the results obtained from a hybrid compact finite difference-WENO scheme solving directly the Navier–Stokes–Fourier system. The comparison validates our DUGKS code and indicates that DUGKS is a reliable and promising tool for simulating forced compressible isotropic turbulence. The work represents a first study to directly simulate forced compressible turbulence by a mesoscopic method based on the Boltzmann equation.

To investigate how the nonuniform fluid density distribution caused by large temperature variations affects the development of unsteady natural convection, we perform a series of direct numerical simulations of two-dimensional compressible natural convection in an air-filled square cavity. The cavity has a hot wall on the left and a cold wall on the right, and two horizontal walls are adiabatic. The simulations are done using a kinetic approach based on a modeled Boltzmann equation, from which the fully compressible Navier–Stokes–Fourier equations are recovered. No Boussinesq approximation or low Mach number approximation is made. An extra source term is introduced to adjust the fluid Prandtl number. Simulations are performed for a range of Rayleigh numbers (107−109) with a fixed dimensionless temperature difference of ε=0.6 to determine the critical Rayleigh number and study the development of unsteady flow. To illustrate the instability mechanism, instantaneous fluctuation field, time trace of temperature, and velocity at selected monitoring points, the spectrum and other statistics are presented and discussed. As expected, significant differences are observed between the instability of compressible natural convection and the Boussinesq-type natural convection. With a large temperature difference, the transition to unsteady flow is asymmetric for the flows near the hot wall and cold wall. For the Rayleigh number range we studied, the cold wall region is dominated by low-frequency impact instability of the boundary thermal jet at the bottom corner. For the hot wall region, besides the upper corner impact instability, a boundary layer instability featuring high-frequency oscillations is observed.

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