A coupled discrete unified gas-kinetic scheme for Boussinesq flows
Introduction
In recent years, kinetic methods have drawn particular attention as newly-developing alternative computational fluid dynamics (CFD) technology. Unlike conventional CFD methods based on direct discretizations of the Navier–Stokes equations, kinetic methods are based on the kinetic theory or the micro particle dynamics, which provides the theoretical connection between hydrodynamics and the underlying microscopic physics, and thus provides efficient tools for multiscale flows. Up to date, a variety of kinetic methods have been proposed, such as the lattice gas cellular automata (LGCA) [1], the lattice Boltzmann equation (LBE) [2], [3], the gas-kinetic scheme (GKS) [4], [5], [6], [7], and the smoothed particle hydrodynamics(SPH) [8], among which the GKS and LBE are specifically designed for CFD.
Both GKS and LBE are compressible schemes for hydrodynamic equations based on gas-kinetic models, but the GKS is a finite-volume (FV) scheme originally designed for compressible flows, while LBE is a finite-difference scheme originally designed for nearly incompressible isothermal flows with low Mach number [9], [10]. Later both schemes are extended to low speed thermal flows [11], [12], [13], [14], [15], [16]. Generally, thermal effects in nearly incompressible flows can lead to large compressibility errors for a compressible scheme [12], and in order to reduce such difficulty, the mass and momentum equations are decoupled from the energy equation. Such strategy has been adopted in both GKS and LBE methods [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].
Recently, starting from the Boltzmann equation, a discrete unified gas-kinetic scheme (DUGKS) was proposed for isothermal flows in all Knudsen regimes [6]. The DUGKS is a FV method, which combines the advantages of the GKS in its flux modeling and the LBE methods in its expanded Maxwellian distribution function and discrete conservative collision operator. In addition, the DUGKS has the asymptotic preserving (AP) property in capturing both rarefied and Navier–Stokes solutions in the corresponding flow regimes [5].
Particularly, although sharing the common kinetic origin, some distinctive features also exist between DUGKS and LBE methods. First, both LBE and DUGKS methods evolve in discrete phase space (physical and particle velocity space) and discrete time, in the LBE the phase space and time step are coupled due to the particle motion from one node to another within a time step, whereas the DUGKS does not suffer from this restriction and the time step is fully determined by the Courant–Friedrichs–Lewy (CFL) condition. Second, the streaming process coupling between the discrete velocity and the underlying regular lattice in standard LBE makes it quite difficult to be extended to non-uniform mesh, while for the DUGKS the non-uniform mesh can be easily employed without additional efforts. More importantly, there also exist modeling differences between standard LBE and DUGKS in the particle evolution process. The standard LBE separates the particle streaming and collision process in its algorithm development. But, the particle transport and collision are fully coupled in DUGKS. This dynamic difference, as well as different discretization errors in boundary condition treatment, will lead to solution deviation in their flow simulations. Consequently, it has been demonstrated that the DUGKS can achieve identical accurate results for the incompressible flows in comparison with the LBE methods, but is more robust and stable [22]. We also notice a non-classical LBE model was developed recently in which the particle streaming and collision are not separated [23], [24].
Although the DUGKS has such distinctive features, the original DUGKS is only designed for isothermal flows which limits its applications [6]. The motivation of this work is to develop a DUGKS for near incompressible thermal flows under the decoupling strategy, where the velocity and temperature fields are described by two respective DUGKS models which are coupled under the Boussinesq assumption. Kinetic boundary conditions are also proposed for both the velocity and temperature fields. To validate the performance of the coupled DUGKS, two-dimensional (2D) porous plate problem, the Rayleigh–Bénard problem and the natural convection in a square cavity at Rayleigh number from 103 up to 1010 are simulated.
The rest of this paper is organized as follows. In Section 2, the coupled DUGKS and the kinetic boundary conditions for velocity and temperature fields are developed, some numerical tests are made in Section 3 to validate the performance of the new scheme, and a brief summary is presented in Section 4.
Section snippets
Coupled discrete unified gas-kinetic scheme
In this section, we first introduce the gas-kinetic model for the Boussinesq flows. Then, the DUGKS based on the model will be derived for velocity and temperature fields, respectively. The two evolution equations are coupled based on the Boussinesq assumption. The kinetic boundary conditions and algorithm for velocity and temperature fields are introduced finally.
Numerical results
In this section, several numerical simulations are conducted to validate the proposed model, including the porous plate problem, the Rayleigh–Bénard convection, and the natural convection in a square cavity. In our simulations, are taken for feq and geq although they can be different in theory, and the three-point Gauss–Hermite quadrature is used to evaluate the moments, which yields the following discrete velocities and associated weights, For the
Conclusions
In this paper, a coupled discrete unified gas-kinetic scheme is developed for the Boussinesq flows. The velocity field and temperature field are separately described by two distributions, and the DUGKS with an external force term is presented in the DUGKS algorithm. The simulation results demonstrate that the coupled DUGKS is of second order accuracy, and can accurately describe the laminar and turbulent thermal convection. Particularly, in comparison with the LBE methods, the coupled DUGKS can
Acknowledgment
This study is financially supported by the National Natural Science Foundation of China (grant no. 51125024) and the Fundamental Research Funds for the Central Universities (grant no. 2014TS119).
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