Investigation of the kinetic model equations

Multiscale kinetic inviscid flux extracted from a gas-kinetic scheme for simulating incompressible and compressible flows

A kinetic inviscid flux (KIF) is proposed for simulating incompressible and compressible flows. It is constructed based on the direct modeling of multiscale flow behaviors, which is used in the gas-kinetic scheme (GKS), the unified gas-kinetic scheme (UGKS), the discrete unified gas-kinetic scheme (DUGKS), etc. In KIF, the discontinuities (such as the shock wave) that cannot be well resolved by mesh cells are mainly solved by the kinetic flux vector splitting (KFVS) method representing the free transport mechanism (or microscale mechanism), while in other flow regions that are smooth, the flow behavior is solved mainly by the central-scheme-like totally thermalized transport (TTT). The weights of KFVS and TTT in KIF is automatically determined by those in the theory of direct modeling. Two ways of choosing the weights in KIF are proposed, which are actually the weights adopted in the UGKS and the DUGKS, respectively. By using the test cases of the Sod shock tube, the rarefaction wave, the boundary layer of a flat plate, the cavity flow, hypersonic flow over a circular cylinder, the shock and turbulent boundary iteration, and transonic flow over a three-dimensional M6 wing, the validity and accuracy of the present method are examined. The KIF does not suffer from the carbuncle phenomenon, and does not introduce extra numerical viscosity in smooth regions. Especially in the case of hypersonic cylinder, it gives quite sharp and clear density and temperature contours. The KIF can be viewed as an inviscid-viscous splitting version of the GKS. By doing this splitting, it is easy to be used in traditional computational fluid dynamics frameworks. It can also be classified as a type in the numerical schemes based on the kinetic theory that are represented by the works of Sun et al. [Adv. Appl. Math. Mech. 8, 703 (2016)10.4208/aamm.2015.m1071] and Ohwada et al. [J. Comput. Phys. 362, 131 (2018)JCTPAH0021-999110.1016/j.jcp.2018.02.019], except the weights are determined by the weights of direct modeling.

Conserved discrete unified gas-kinetic scheme with unstructured discrete velocity space

Discrete unified gas-kinetic scheme (DUGKS) is a multiscale numerical method for flows from continuum limit to free molecular limit, and is especially suitable for the simulation of multiscale flows, benefiting from its multiscale property. To reduce integration error of the DUGKS and ensure the conservation property of the collision term in isothermal flow simulations, a conserved-DUGKS (C-DUGKS) is proposed. On the other hand, both DUGKS and C-DUGKS adopt Cartesian-type discrete velocity space, in which Gaussian and Newton-Cotes numerical quadrature are used for calculating the macroscopic physical variables in low-speed and high-speed flows, respectively. However, the Cartesian-type discrete velocity space leads to huge computational cost and memory demand. In this paper, the isothermal C-DUGKS is extended to the nonisothermal case by adopting coupled mass and inertial energy distribution functions. Moreover, since the unstructured mesh, such as the triangular mesh in the two-dimensional case, is more flexible than the structured Cartesian mesh, it is introduced to the discrete velocity space of C-DUGKS, such that more discrete velocity points can be arranged in the velocity regions that enclose a large number of molecules, and only a few discrete velocity points need to be arranged in the velocity regions with a small amount of molecules in it. By using the unstructured discrete velocity space, the computational efficiency of C-DUGKS is significantly increased. A series of numerical tests in a wide range of Knudsen numbers, such as the Couette flow, lid-driven cavity flow, two-dimensional rarefied Riemann problem, and the supersonic cylinder flows, are carried out to examine the validity and efficiency of the present method.

Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory

A conservative discrete velocity method (DVM) is developed for the ellipsoidal Fokker-Planck (ES-FP) equation in prediction of nonequilibrium neutral gas flows in this paper. The ES-FP collision operator is solved in discrete velocity space in a concise and quick finite difference framework. The conservation problem of the discrete ES-FP collision operator is solved by multiplying each term in it by extra conservative coefficients whose values are very close to unity. Their differences to unity are in the same order of the numerical error in approximating the ES-FP operator in discrete velocity space. All the macroscopic conservative variables (mass, momentum, and energy) are conserved in the present modified discrete ES-FP collision operator. Since the conservation property in a discrete element of physical space is very important for the numerical scheme when discontinuity and a large gradient exist in the flow field, a finite volume framework is adopted for the transport term of the ES-FP equation. For nD-3V (n<3) cases, a nD-quasi nV reduction is specifically proposed for the ES-FP equation and the corresponding FP-DVM method, which can greatly reduce the computational cost. The validity and accuracy of both the ES-FP equation and FP-DVM method are examined using a series of 0D-3V homogenous relaxation cases and 1D-3V shock structure cases with different Mach numbers, in which 1D-3V cases are reduced to 1D-quasi 1V cases. Both the predictions of 0D-3V and 1D-3V cases match well with the benchmark results such as the analytical Boltzmann solution, direct full-Boltzmann numerical solution, and DSMC result. Especially, the FP-DVM predictions match well with the DSMC results in the Mach 8.0 shock structure case, which is in high nonequilibrium, and is a challenge case of the model Boltzmann equation and the corresponding numerical methods.

Structure of velocity distributions in shock waves in granular gases with extension to molecular gases

Velocity distributions in normal shock waves obtained in dilute granular flows are studied. These distributions cannot be described by a simple functional shape and are believed to be bimodal. Our results show that these distributions are not strictly bimodal but a trimodal distribution is shown to be sufficient. The usual Mott-Smith bimodal description of these distributions, developed for molecular gases, and based on the coexistence of two subpopulations (a supersonic and a subsonic population) in the shock front, can be modified by adding a third subpopulation. Our experiments show that this additional population results from collisions between the supersonic and subsonic subpopulations. We propose a simple approach incorporating the role of this third intermediate population to model the measured probability distributions and apply it to granular shocks as well as shocks in molecular gases.

Discrete unified gas kinetic scheme for all Knudsen number flows. II. Thermal compressible case

This paper is a continuation of our work on the development of multiscale numerical scheme from low-speed isothermal flow to compressible flows at high Mach numbers. In our earlier work [Z. L. Guo et al., Phys. Rev. E 88, 033305 (2013)], a discrete unified gas kinetic scheme (DUGKS) was developed for low-speed flows in which the Mach number is small so that the flow is nearly incompressible. In the current work, we extend the scheme to compressible flows with the inclusion of thermal effect and shock discontinuity based on the gas kinetic Shakhov model. This method is an explicit finite-volume scheme with the coupling of particle transport and collision in the flux evaluation at a cell interface. As a result, the time step of the method is not limited by the particle collision time. With the variation of the ratio between the time step and particle collision time, the scheme is an asymptotic preserving (AP) method, where both the Chapman-Enskog expansion for the Navier-Stokes solution in the continuum regime and the free transport mechanism in the rarefied limit can be precisely recovered with a second-order accuracy in both space and time. The DUGKS is an idealized multiscale method for all Knudsen number flow simulations. A number of numerical tests, including the shock structure problem, the Sod tube problem in a whole range of degree of rarefaction, and the two-dimensional Riemann problem in both continuum and rarefied regimes, are performed to validate the scheme. Comparisons with the results of direct simulation Monte Carlo (DSMC) and other benchmark data demonstrate that the DUGKS is a reliable and efficient method for multiscale flow problems.