Abstract:
The kinetic theory of gases has suggested the idea of viscosity to model the effect of thermal fluctuations on the resolved flow. Supported by the assumed analogy between molecules and the eddies in a turbulent flows, the idea of an eddy viscosity has been put forward in the pioneering work by Lord Kelvin and Osborne Reynolds. In over hundred years of turbulence modeling, the numerical schemes adopted to simulate turbulent flow - with the exception of the Lattice Boltzmann methods - have never exploited this analogy in any other way. In this work, a gas-kinetic scheme is modified to simulate turbulent flow; the turbulent relaxation time is deduced from assumed turbulent quantities. The new scheme does not adopt an eddy viscosity, yet it relies even more strongly on the analogy between thermal and turbulent fluctuations, as turbulence dynamics is mathematically modeled by the Boltzmann equation. In the gas-kinetic scheme, a measure of the degree of rarefaction is introduced, as the ratio between unresolved and resolved time scales of motion. At low rarefaction, the turbulent gas-kinetic scheme deviates negligibly from a conventional Navier-Stokes scheme. However, as the degree of rarefaction increases, the kinetic effects become evident. This phenomenon is evident in the mathematical description of the turbulent stress tensor and also in numerical experiments. This study does not propose an innovative turbulence model or technique. It addresses the fact that the traditional coupling numerical scheme and turbulence modeling might improve the physical consistence of numerical simulations.

Abstract:
Numerical schemes derived from gas-kinetic theory can be applied to simulations in the hydrodynamics limit, in laminar and also turbulent regimes. In the latter case, the underlying Boltzmann equation describes a distribution of eddies, in line with the concept of eddy viscosity developed by Lord Kelvin and Osborne Reynolds at the end of the nineteenth century. These schemes are physically more consistent than schemes derived from the Navier-Stokes equations, which invariably assume infinite collisions between gas particles (or interactions between eddies) in the calculation of advective fluxes. In fact, in continuum regime too, the local Knudsen number can exceed the value 0.001 in shock layers, where gas-kinetic schemes outperform Navier-Stokes schemes, as is well known. Simulation of turbulent flows benefit from the application of gas-kinetic schemes, as the turbulent Knudsen number (the ratio between the eddies' mean free path and the mean flow scale) can locally reach values well in excess of 0.001, not only in shock layers. This study has investigated a few cases of shock - boundary layer interaction comparing a gas-kinetic scheme and a Navier-Stokes one, both with a standard k-\omega turbulence model. Whereas the results obtained from the Navier-Stokes scheme are affected by the limitations of eddy viscosity two-equation models, the gas-kinetic scheme has performed much better without making any further assumption on the turbulent structures.

Abstract:
The multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields [J. Comput. Phys. 226 (2007) 2003-2027] is extended to resistive magnetic flows. The non-magnetic part of the magnetohydrodynamics equations is calculated by a BGK solver modified due to magnetic field. The magnetic part is treated by the flux splitting method based gas-kinetic theory [J. Comput. Phys. 153 (1999) 334-352 ], using a particle distribution function constructed in the BGK solver. To include Lorentz force effects into gas evolution stage is very important to improve the accuracy of the scheme. For some multidimensional problems, the deviations tangential to the cell interface from equilibrium distribution are essential to keep the scheme robust and accurate. Besides implementation of a TVD time discretization scheme, enhancing the dynamic dissipation a little bit is a simply and efficient way to stabilize the calculation. One-dimensional and two-dimensional shock waves tests are calculated to validate this new scheme. A three-dimensional turbulent magneto-convection simulation is used to show the applicability of current scheme to complicated astrophysical flows.

Abstract:
Recently, the discrete unified gas-kinetic scheme (DUGKS) [Z. L. Guo \emph{et al}., Phys. Rev. E ${\bf 88}$, 033305 (2013)] based on the Boltzmann equation is developed as a new multiscale kinetic method for isothermal flows. In this paper, a thermal and coupled discrete unified gas-kinetic scheme is derived for the Boussinesq flows, where the velocity and temperature fields are described independently. Kinetic boundary conditions for both velocity and temperature fields are also proposed. The proposed model is validated by simulating several canonical test cases, including the porous plate problem, the Rayleigh-b\'{e}nard convection, and the natural convection with Rayleigh number up to $10^{10}$ in a square cavity. The results show that the coupled DUGKS is of second order accuracy in space and can well describe the convection phenomena from laminar to turbulent flows. Particularly, it is found that this new scheme has better numerical stability in simulating high Rayleigh number flows compared with the previous kinetic models.

Abstract:
The unified gas kinetic scheme (UGKS) of K. Xu et al. [K. Xu and J.-C. Huang, J. Comput. Phys., 229, pp. 7747--7764, 2010], originally developed for multiscale gas dynamics problems, is applied in this paper to a linear kinetic model of radiative transfer theory. While such problems exhibit purely diffusive behavior in the optically thick (or small Knudsen) regime, we prove that UGKS is still asymptotic preserving (AP) in this regime, but for the free transport regime as well. Moreover, this scheme is modified to include a time implicit discretization of the limit diffusion equation, and to correctly capture the solution in case of boundary layers. Contrary to many AP schemes, this method is based on a standard finite volume approach, it does neither use any decomposition of the solution, nor staggered grids. Several numerical tests demonstrate the properties of the scheme.

Abstract:
A Cartesian grid method combined with a simplified gas kinetic scheme is presented for subsonic and supersonic viscous flow simulation on complex geometries. Under the Cartesian mesh, the computational grid points are classified into four different categories, the fluid point, the solid point, the drop point, and the interpolation point. The boundaries are represented by a set of direction-oriented boundary points. A constrained weighted least square method is employed to evaluate the physical quantities at the interpolation points. Different boundary conditions, including isothermal boundary, adiabatic boundary, and Euler slip boundary, are presented by different interpolation strategies. We also propose a simplified gas kinetic scheme as the flux solver for both subsonic and supersonic flow computations. The methodology of constructing a simplified kinetic flux function can be extended to other flow systems. A few numerical examples are used to validate the Cartesian grid method and the simplified flux function. The reconstruction scheme for recovering the boundary conditions of compressible viscous and heat conducting flow with a Cartesian mesh can provide a smooth distribution of physical quantities at solid boundary, and present an accurate solution for the flow study with complex geometry.

Abstract:
The recently proposed discrete unified gas kinetic scheme (DUGKS) is a finite volume method for deterministic solution of the Boltzmann model equation with asymptotic preserving property. In DUGKS, the numerical flux of the distribution function is determined from a local numerical solution of the Boltzmann model equation using an unsplitting approach. The time step and mesh resolution are not restricted by the molecular collision time and mean free path. To demonstrate the capacity of DUGKS in practical problems, this paper extends the DUGKS to arbitrary unstructured meshes. Several tests of both internal and external flows are performed, which include the cavity flow ranging from continuum to free molecular regimes, a multiscale flow between two connected cavities with a pressure ratio of 10000, and a high speed flow over a cylinder in slip and transitional regimes. The numerical results demonstrate the effectiveness of the DUGKS in simulating multiscale flow problems.

Abstract:
The discrete unified gas kinetic scheme (DUGKS) is a finite-volume scheme with discretization of particle velocity space, which combines the advantages of both lattice Boltzmann equation (LBE) method and unified gas kinetic scheme (UGKS) method, such as the simplified flux evaluation scheme, flexible mesh adaption and the asymptotic preserving properties. However, DUGKS is proposed for near incompressible fluid flows, the existing compressible effect may cause some serious errors in simulating incompressible problems. To diminish the compressible effect, in this paper a novel DUGKS model with external force is developed for incompressible fluid flows by modifying the approximation of Maxwellian distribution. Meanwhile, due to the pressure boundary scheme, which is wildly used in many applications, has not been constructed for DUGKS, the non-equilibrium extrapolation (NEQ) scheme for both velocity and pressure boundary conditions is introduced. To illustrate the potential of the proposed model, numerical simulations of steady and unsteady flows are performed. The results indicate that the proposed model can reduce the compressible effect efficiently against the original DUGKS model, and the NEQ scheme fits well with our model as they are both of second-order accuracy. We also implement the proposed model in simulating the three dimensional problem: cubical lid-driven flow. The comparisons of numerical solutions and benchmarks are presented in terms of data and topology. And the motion pattern of the fluid particles in a specific area is characterized for the steady-state cubical lid-driven flows.

Abstract:
In this paper, a semi-Lagrangian gas-kinetic scheme is developed for smooth flows based on the Bhatnagar-Gross-Krook (BGK) equation. As a finite-volume scheme, the evolution of the average flow variables in a control volume is under the Eulerian framework, whereas the construction of the numerical flux across the cell interface comes from the Lagrangian perspective. The adoption of the Lagrangian aspect makes the collision and the transport mechanisms intrinsically coupled together in the flux evaluation. As a result, the time step is independent of the particle collision time and solely determined by the Courant-Friedrichs-Lewy (CFL) conditions. A set of simulations are carried out to validate the performance of the new scheme. The results show that with second-order spatial accuracy, the scheme exhibits low numerical dissipation, and can accurately capture the Navier-Stokers solutions for the smooth flows with viscous heat dissipation from the low-speed incompressible to hypersonic compressible regimes.

Abstract:
Asymptotic preserving (AP) schemes are targeting to simulate both continuum and rarefied flows. Many AP schemes have been developed and are capable of capturing the Euler limit in the continuum regime. However, to get accurate Navier-Stokes solutions is still challenging for many AP schemes. In order to distinguish the numerical effects of different AP schemes on the simulation results in the continuum flow limit, an implicit-explicit (IMEX) AP scheme and the unified gas kinetic scheme (UGKS) based on Bhatnagar-Gross-Krook (BGk) kinetic equation will be applied in the flow simulation in both transition and continuum flow regimes. As a benchmark test case, the lid-driven cavity flow is used for the comparison of these two AP schemes. The numerical results show that the UGKS captures the viscous solution accurately. The velocity profiles are very close to the classical benchmark solutions. However, the IMEX AP scheme seems have difficulty to get these solutions. Based on the analysis and the numerical experiments, it is realized that the dissipation of AP schemes in continuum limit is closely related to the numerical treatment of collision and transport of the kinetic equation. Numerically it becomes necessary to couple the convection and collision terms in both flux evaluation at a cell interface and the collision source term treatment inside each control volume.