Abstract
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.
Highlights
Different flow regimes can be classified based on the Knudsen number (Kn), which is defined as the ratio of the microscopic mean free path of the particles λ and hydrodynamic length scale l
We are in the process of optimizing the discrete unified gas kinetic scheme (DUGKS) code including decreasing the number of the discrete particle velocities, but such efforts are beyond the scope of the current paper, and they will be reported in the future
We have proposed a new kinetic model with both an adjustable Prandtl number and a tunable ratio of bulk to shear viscosity and applied the model to simulate compressible decaying homogeneous isotropic turbulence (CDHIT) under the DUGKS approach
Summary
Different flow regimes can be classified based on the Knudsen number (Kn), which is defined as the ratio of the microscopic mean free path of the particles λ and hydrodynamic length scale l. Frapolli et al. overcame the low Mach-number limit of the traditional LBM and proposed an entropic lattice Boltzmann model (ELBM) to simulate compressible transonic and supersonic flows using a D3Q73 lattice velocity model with temperature-dependent weights. Liao et al. applied the GKS to simulate CDHIT at a Taylor microscale Reynolds number of Reλ0 = 72.0 and the turbulent Mach number of Mat0 = 0.1–0.6 They concluded that the GKS is adequate for the DNS of moderately compressible homogeneous isotropic turbulence as far as the low-order turbulence statistics are concerned. In Appendixes A–E, we include the details on Hermite polynomials and our newly designed novel E39,77 discrete particle velocity model (Appendix A), Hermite expansion of the equilibrium, the implementation of the fifth-order WENO scheme (Appendix B), the Chapman–Enskog analysis of our redesigned model (Appendix C), explicit expressions of two reduced distribution functions (Appendix D). The evolution equation of mean square velocity divergence is derived in detail (Appendix E)
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