Conservative Discrete Ordinate Method Research Articles

An implicit parallel UGKS solver for flows covering various regimes

This paper presents an engineering-oriented UGKS solver package developed in China Aerodynamics Research and Development Center (CARDC). The solver is programmed in Fortran language and uses structured body-fitted mesh, aiming for predicting aerodynamic and aerothermodynamics characteristics in flows covering various regimes on complex three-dimensional configurations. The conservative discrete ordinate method and implicit implementation are incorporated. Meanwhile, a local mesh refinement technique in the velocity space is developed. The parallel strategies include MPI and OpenMP. Test cases include a wedge, a cylinder, a 2D blunt cone, a sphere, and a X38-like vehicle. Good agreements with experimental or DSMC results have been achieved.

High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation

AbstractHigh-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.

A conservative discrete ordinate method for solving semiclassical Boltzmann-BGK equation with Maxwell type wall boundary condition

A robust and efficient numerical method in phase space combining the conservative discrete ordinate method and high order spatial discretization is proposed for the semiclassical Boltzmann equation with Bhatnagar–Gross–Krook (BGK) relaxation time approximation. A general Maxwell type partially diffuse and partially specular reflection solid wall boundary condition for modeling the gas–surface interactions for semiclassical rarefied gas flows is devised. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to evolve the solution in physical space and time. The conservation property of the BGK collision integral is ensured to be fulfilled at the discrete quadrature level approximation. Both specular and diffuse reflection boundary conditions are implemented. Extensive computations of one- and two-dimensional semiclassical rarefied gas dynamical flows covering wide range of flow regimes for three statistics are presented to illustrate the method and boundary conditions treatment. The effect of accommodation coefficient on the rarefied flow field is also examined.

Internal energy relaxation in shock wave structure

The Wang Chang-Uhlenbeck (WCU) equation is numerically integrated to characterize the internal structure of Mach 3 and Mach 5 shock waves in a gas with excitation in the internal energy states for the treatment of inelastic collisions. Elastic collisions are modeled with the hard sphere collision model and the transition rates for the inelastic collisions modified appropriately using probabilities based on relative velocities of the colliding particles. The collision integral is evaluated by the conservative discrete ordinate method [F. Tcheremissine, “Solution of the Boltzmann kinetic equation for high-speed flows,” Comput. Math. Math. Phys. 46, 315–329 (2006); F. Cheremisin, “Solution of the Wang Chang-Uhlenbeck equation,” Dokl. Phys. 47, 487–490 (2002)] developed for the Boltzmann equation. For the treatment of the diatomic molecules, the internal energy modes in the Boltzmann equation are described quantum mechanically given by the WCU equation. As a first step in the treatment of the inelastic collisions by the WCU equation, a two- and three-quantum system is considered to study the effect of the varying of (1) the inelastic cross section and (2) the energy gap between the quantum energy states. An alternative method, the direct simulation Monte Carlo method, is used for the Mach 3 shock wave to ensure the consistency of implementation in the two methods and there is an excellent agreement between the two methods. The results from the WCU implementation showed consistent trends for the Mach 3 and Mach5 standing shock waves simulations. Inelastic contributions change the downstream equilibrium state and allow the flow to transition to the equilibrium state further upstream.

Kinetic solution of the structure of a shock wave in a nonreactive gas mixture

The multispecies Boltzmann equation is numerically integrated to characterize the internal structure of a Mach 3 shock wave in a hard sphere gas. The collision integral is evaluated by the conservative discrete ordinate method [F. G. Tcheremissine, Comput. Math. Math. Phys. 46, 315 (2006)]. There was excellent agreement of macroscopic variables [Kosuge et al.., Eur. J. Mech. B/Fluids 20, 87 (2001)]. The effect of species concentration and mass ratio on the behavior of macroscopic variables and distribution functions in the structure of the shock wave is considered for both two- and three-species gas mixtures. In a binary mixture of gases with different masses and varying concentrations, the temperature overshoot of the parallel component of temperature near the center of the shock wave is highest for the heavy component when the concentration of the heavy component is the smallest. The rise in the parallel component of temperature is revealed by the behavior of the distribution function.

A conservative discrete ordinate method for model Boltzmann equations

A class of conservative discrete ordinate method (C-DOM) for the Bhatnagar–Gross–Krook (BGK) model Boltzmann equation is presented. The C-DOM is the extension of the discrete ordinate method in my previous study (Yang and Huang, 1995 [3]). In the C-DOM, a conservative molecular collision process is employed and thus the conservation properties of the collision integral are maintained at the molecular level. For a broad range of Knudsen number, several test problems, including unsteady shock-tube problem and supersonic/hypersonic flows over circular cylinder, are utilized to demonstrate the performance and validity of the DOM and C-DOM. Results show that the C-DOM can greatly reduce the computer time and memory requirements in hypersonic rarefied gas flow computations.

Conservative numerical methods for model kinetic equations

A new conservative discrete ordinate method for nonlinear model kinetic equations is proposed. The conservation property with respect to the collision integral is achieved by satisfying at the discrete level approximation conditions used in deriving the model collision integrals. Additionally to the conservation property, the method ensures the correct approximation of the heat fluxes. Numerical examples of flows with large gradients are provided for the Shakhov and Rykov model kinetic equations.

Study of a shock wave structure in gas mixtures on the basis of the Boltzmann equation

The structure of a shock wave for a binary gas mixture is studied on the basis of numerical solution of the complete kinetic Boltzmann equation for the model of hard sphere molecules. For the evaluation of the collision integral we apply a generalization of the conservative discrete ordinate method (the kernel method for a single gas was developed by Tcheremissine) for binary gas mixtures and for the case of cylindrical symmetry. The transition from up-stream to downstream uniform state is presented by macroscopic values and by distribution functions.