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A Boltzmann model equation (kinetic model) involving the chemical reaction of a multicomponent gaseous mixture is derived based on Groppi's work ["A Bhatnagar–Gross–Krook-type approach for chemically reacting gas mixtures," Phys. Fluids 16, 4273 (2004)], in which the relaxation parameters of elastic collision frequency for rigid elastic spheres are obtained based on the collision term, and the pivotal collision frequency of the chemical reaction is deduced from the chemical reaction rate that is determined by the direct simulation Monte Carlo (DSMC) method. This kinetic model is shown to be conservative, and the H theorem for an endothermic reaction is proven. In the framework of the gas-kinetic unified algorithm, the discrete velocity method, finite volume method, and implicit scheme are applied to solve the proposed kinetic model by introducing a suitable boundary condition at the wall surface. For hypersonic flows around a cylinder, the proposed kinetic model and the corresponding numerical methods are verified for both endothermic and exothermic reactions by comparison of the model's results with results from the DSMC method. The different influences of endothermic and exothermic reactions are also given. Finally, the proposed kinetic model is also used to simulate an exothermic reaction-driven flow in a square cavity. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved gas-kinetic unified algorithm (IGKUA) for high rarefied transition to continuum flows by computable modeling of Boltzmann equation. Compared with the original algorithm, the new method utilizes less needed discrete velocity ordinate points to obtain accurate results and removes the dependency of flow regime on computational time step, which can speed up the convergence in continuum flows. One of the novel strategies adopted in IGKUA is to develop a type of gas-kinetic quadrature rule that can exactly preserve conservation constraint of the model by adjusting the integral weights, increasing efficiency, and reducing nonphysical sources. Another key innovation is to introduce the analytical solutions of colliding-relaxation equation by considering the evolutions of associated macroscopic flow variables first, leading to no limit on the permissible time step. Numerical explicit and implicit schemes for unsteady flows are constructed to solve the physical convective equation, and Fourier spectral method is applied for the molecular-velocity convective movement equation analytically when the flows are under external-force fields. The IGKUA is tested using some numerical examples, including the shock-tube problems, Rayleigh flow, Couette flow, lid-driven cavity, external force-driven Poiseuille flow, and hypersonic flow past an infinite flat plate. Simulation results are in high resolution of the flow fields and match well with the results of the analytical, direct simulation Monte Carlo, Navier–Stokes solvers, and other reference methods. In addition, the new algorithm is better than the original one in the aspects of computational amount and time, which are more obvious when simulating the continuum flows. [ABSTRACT FROM AUTHOR]

Based on the previous researches of the Gas-Kinetic Unified Algorithm (GKUA) for flows from highly rarefied free-molecule transition to continuum, a new implicit scheme of cell-centered finite volume method is presented for directly solving the unified Boltzmann model equation covering various flow regimes. In view of the difficulty in generating the single-block grid system with high quality for complex irregular bodies, a multi-block docking grid generation method is designed on the basis of data transmission between blocks, and the data structure is constructed for processing arbitrary connection relations between blocks with high efficiency and reliability. As a result, the gas-kinetic unified algorithm with the implicit scheme and multi-block docking grid has been firstly established and used to solve the reentry flow problems around the multi-bodies covering all flow regimes with the whole range of Knudsen numbers from 10 to 3.7E−6. The implicit and explicit schemes are applied to computing and analyzing the supersonic flows in near-continuum and continuum regimes around a circular cylinder with careful comparison each other. It is shown that the present algorithm and modelling possess much higher computational efficiency and faster converging properties. The flow problems including two and three side-by-side cylinders are simulated from highly rarefied to near-continuum flow regimes, and the present computed results are found in good agreement with the related DSMC simulation and theoretical analysis solutions, which verify the good accuracy and reliability of the present method. It is observed that the spacing of the multi-body is smaller, the cylindrical throat obstruction is greater with the flow field of single-body asymmetrical more obviously and the normal force coefficient bigger. While in the near-continuum transitional flow regime of near-space flying surroundings, the spacing of the multi-body increases to six times of the diameter of the single-body, the interference effects of the multi-bodies tend to be negligible. The computing practice has confirmed that it is feasible for the present method to compute the aerodynamics and reveal flow mechanism around complex multi-body vehicles covering all flow regimes from the gas-kinetic point of view of solving the unified Boltzmann model velocity distribution function equation. [ABSTRACT FROM AUTHOR]

This article reviews rarefied gas flow computations based on nonlinear model Boltzmann equations using deterministic high-order gas-kinetic unified algorithms (GKUA) in phase space. The nonlinear Boltzmann model equations considered include the BGK model, the Shakhov model, the Ellipsoidal Statistical model and the Morse model. Several high-order gas-kinetic unified algorithms, which combine the discrete velocity ordinate method in velocity space and the compact high-order finite-difference schemes in physical space, are developed. The parallel strategies implemented with the accompanying algorithms are of equal importance. Accurate computations of rarefied gas flow problems using various kinetic models over wide ranges of Mach numbers 1.2–20 and Knudsen numbers 0.0001–5 are reported. The effects of different high resolution schemes on the flow resolution under the same discrete velocity ordinate method are studied. A conservative discrete velocity ordinate method to ensure the kinetic compatibility condition is also implemented. The present algorithms are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the steady normal shock wave structures for different Mach numbers, the two-dimensional flows past a circular cylinder and a NACA 0012 airfoil to verify the present methodology and to simulate gas transport phenomena covering various flow regimes. Illustrations of large scale parallel computations of three-dimensional hypersonic rarefied flows over the reusable sphere–cone satellite and the re-entry spacecraft using almost the largest computer systems available in China are also reported. The present computed results are compared with the theoretical prediction from gas dynamics, related DSMC results, slip N–S solutions and experimental data, and good agreement can be found. The numerical experience indicates that although the direct model Boltzmann equation solver in phase space can be computationally expensive, nevertheless, the present GKUAs for kinetic model Boltzmann equations in conjunction with current available high-performance parallel computer power can provide a vital engineering tool for analyzing rarefied gas flows covering the whole range of flow regimes in aerospace engineering applications. [ABSTRACT FROM AUTHOR]