Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21], [22], [23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.     

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.     

Numerical study of the transverse supersonic flow of a diatomic rarefied gas past a plate

The two-dimensional supersonic rarefied gas flow past an infinite plate placed normally to the flow is analyzed. The gas possesses rotational degrees of freedom. The problem is stated for a model kinetic equation and is solved by applying a second-order accurate implicit conservative finite-difference method. The gas parameters correspond to nitrogen. The results are compared with those obtained for a monatomic gas. The influence exerted by the rotational degrees of freedom and the boundary conditions at the plate’s surface on the aerodynamic characteristics of the plate and the flow pattern is illustrated.     

A model of incomplete phase transitions of gas hydrates in a porous medium

The kinetics of incomplete phase transitions of gas hydrates in a porous medium is considered. In such transformations, the solid hydrate and its decomposition products coexist in extended regions. The conservation laws that take into account mass, momentum, and energy transfer between the components of the medium are formulated. The concept of phase transition dissipation is introduced. A general form of the constitutive relations, which is necessary and sufficient for the entropy inequality to be satisfied in any processes involving a change of state of the medium, is found. A potential of the skeleton that takes into account the surface energy, the latent energy of the phase transition and the temperature dependence is proposed. A thermodynamically consistent kinetic equation is formulated. The conditions under which the phase transition of a hydrate begins and is completed are found. The problem of isothermal dissociation of a hydrate when the pressure on part of the boundary of the body is reduced is examined. The influence of several parameters of the model on the seepage of the decomposition products is investigated.     

Kinetic model of the Boltzmann equation for a power-law intermolecular interaction potential

A model kinetic equation approximating the Boltzmann equation with a linearized collision integral is constructed to describe rarefied gas flows at moderate and low Knudsen numbers. The kinetic model describes gas flows with a power-law intermolecular interaction potential and involves five relaxation parameters. The structure of a shock wave is computed, and the results are compared with an experiment for argon.     

Nonequilibrium jet boundary layer in a polyatomic gas

The two-dimensional nonequilibrium hypersonic free jet boundary layer gas flow in the near wake of a body is studied using a closed system of macroscopic equations obtained (as a thin-layer version) from moment equations of kinetic origin for a polyatomic single-component gas with internal degrees of freedom. (This model is can be used to study flows with strong violations of equilibrium with respect to translational and internal degrees of freedom.) The solution of the problem under study (i.e., the kinetic model of a nonequilibrium homogeneous polyatomic gas flow in a free jet boundary layer) is shown to be related to the known solution of the well-studied simpler problem of a Navier-Stokes free jet boundary layer, and a method based on this relation is proposed for solving the former problem. It is established that the gas flow velocity distribution along the separating streamline in the kinetic problem of a free jet boundary layer coincides with the distribution obtained by solving the Navier-Stokes version of the problem. It is found that allowance for the nonequilibrium nature of the flow with respect to the internal and translational degrees of freedom of a single-component polyatomic gas in a hypersonic free jet boundary layer has no effect on the base pressure and the wake angle.     

Modified method of splitting with respect to physical processes for solving radiation gas dynamics equations

An approach based on a modified splitting method is proposed for solving the radiation gas dynamics equations in the multigroup kinetic approximation. The idea of the approach is that the original system of equations is split using the thermal radiation transfer equation rather than the energy equation. As a result, analytical methods can be used to solve integrodifferential equations and problems can be computed in the multigroup kinetic approximation without iteration with respect to the collision integral or matrix inversion. Moreover, the approach can naturally be extended to multidimensional problems. A high-order accurate difference scheme is constructed using an approximate Godunov solver for the Riemann problem in two-temperature gas dynamics.     

Multiscale approach to computation of three-dimensional gas mixture flows in engineering microchannels

A multiscale approach to computing real gas flows in engineering microchannels on high-performance computer systems in a wide range of Knudsen numbers is described. The numerical implementation of the approach combines the solution of quasigasdynamic equations and the molecular dynamics method. Following the approach, the parameters of the real gas equation of state are found at the molecular level, the kinetic gas properties are calculated, and the form of boundary conditions on the microchannel walls are determined. The technique is verified by computing several test problems. The results agree well with available theoretical and experimental data.     

Model Kinetic Equations and the Description of Gas Flows at Various Relaxation Stages

Various relaxation stages in high-velocity and high-temperature gases with physicochemical processes are considered on the basis of model kinetic equations. Macroscopic equations are derived in the zero approximation of the modified Chapman–Enskog method and expressions for the flow members of gas-dynamic equations in terms of intensive and extensive parameters are deduced. A formula for the velocity of sound (as the velocity of propagation of small perturbations) is derived using the parameter æ, which is not a constant under the considered conditions.     

Rarefied gas flow through a pipe of variable square cross section into vacuum

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results.     

Phase boundary solutions to model kinetic equations via the Conley index theory. Part I

In this paper, we consider phase boundary solutions to a four-velocity kinetic model of a kinetic equation governing the motion of van der Waals fluids. These solutions connect such equilibrium states, which are saddle critical points of the suitable dynamic system. Solutions of this type can be interpreted as dynamic phase transition. The mathematical apparatus is that of the Coney index theory.     

A fully meshless method for ‘gas – evaporating droplet’ flow modelling

A meshless method for modelling two-phase flows with phase transition is described. The method is based on consideration of three systems: viscous-vortex blobs, thermal-blobs and droplets; and can be applied for numerical simulation of 2D non-isothermal flows of ‘gas-evaporating droplets’ in the framework of the one-way coupled two-fluid approach. The carrier phase is viscous incompressible gas. The dispersed phase is presented by a cloud of identical spherical droplets, and, due to evaporation, the radius and mass of droplets are time dependent. The carrier phase parameters are calculated using the viscous-vortex and thermal-blob method; the dispersed phase parameters are calculated using the Lagrangian approach. Two applications have been considered: (i) a standard benchmark – Lamb vortex; (ii) a cold spray injected into a hot quiescent gas. In the latter problem three cases corresponding to three droplet sizes were investigated. The smallest droplets (of the three cases considered) are more readily entrained by the carrier phase and form ring-like structures; the flow shows better mixing. Larger droplets evaporate less intensively. The medium sized droplets collect into two narrow bands stretched along the jet axis. The largest droplets form a two-phase jet, which remains close to the jet axis. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A modified DNA genetic algorithm for parameter estimation of the 2-Chlorophenol oxidation in supercritical water

Based on the mechanism of biological DNA genetic information and evolution, a modified DNA genetic algorithm (MDNA-GA) is proposed to estimate the kinetic parameters of the 2-Chlorophenol oxidation in supercritical water. In this approach, DNA encoding method, choose crossover operator and frame-shift mutation operator inspired by the biological DNA are developed for improving the global searching ability. Besides, an adaptive mutation probability which can be adjusted automatically according to the diversity of population is also adopted. A local search method is used to explore the search space to accelerate the convergence towards global optimum. The performance of MDNA-GA in typical benchmark functions and kinetic parameter estimation is studied and compared with RNA-GA. The experimental results demonstrate that the proposed algorithm can overcome premature convergence and yield the global optimum with high efficiency.     

Determination of the analytical parametric relationship for output spectrum of Volterra systems based on its parametric characteristics

The output frequency response function (OFRF) of Volterra systems can be described as a polynomial function of model parameters. However, the analytical determination of the OFRF is very computationally intensive, especially for higher order OFRF. To circumvent this problem, a numerical method can be adopted, provided that a series of simulation or experimental data for this polynomial function are given. In this study, it is theoretically shown that the analytical parametric relationship of OFRF up to any order can be determined accurately by using a simple Least Square method and every specific component of the output spectrum can also be determined explicitly, based on the OFRF's parametric characteristics. Practical techniques to obtain a unique and accurate solution for the Least Square method are discussed. This study provides a fundamental result for the determination of the analytical parametric relationship for this kind of system polynomial functions by using numerical methods.     

Efficient method for computing rarefied gas flow in a long finite plane channel

The linearized kinetic S-model is used to study the nonisothermal steady rarefied gas flow driven by differences in pressure and temperature in a plane channel between long finite parallel plates joining two tanks of infinite volume. An efficient composite (asymptotic) method is developed: a one-dimensional asymptotic solution corresponding to an infinitely long channel is constructed in the middle part of the computational domain, while a solution of the two-dimensional kinetic equation matched with the middle-part asymptotic solution is constructed near the ends of the channel. The latter solution is found numerically by a high-order accurate conservative method. The basic quantity to be computed is the gas flow rate through the channel. Characteristic flow features are also investigated. The resulting solutions are compared with previously known results.     

Numerical study of unsteady rarefied diatomic gas flows in a plane microchannel

The numerical solution of a kinetic equation for a diatomic gas (nitrogen) is used to study two-dimensional unsteady gas flows in a plane microchannel caused by discontinuous in the initial distributions of macroscopic gas parameters. The plane discontinuity fronts are perpendicular to the walls of the channel. The arising flows are model ones for gas flows in a shock tube and a microchannel. The reflection of an incident shock wave from a flat end face is studied. It is found that the gas piles up at the cold wall, which slows down the shock wave detachment. The numerical results are in qualitative agreement with experimental data.     

A direct search method for determination of DAEM kinetic parameters from nonisothermal TGA data (note)

In this study, a simple direct search method to be used for the determination of distributed activation energy model (DAEM) kinetic parameters from the nonisothermal thermogravimetric analysis (TGA) data of coals has been introduced. Process steps of direct search method that depends on the grid technique have been given. The method has been applied to the nonisothermal TGA data of one Turkish coal and one imported coal, and DAEM kinetic parameters of these coal samples have been determined. Calculated model results from determined kinetic parameters have been compared with nonisothermal TGA data of the coals.     

Control of time-periodic systems via symbolic computation with application to chaos control

A general method for the control of linear time-periodic systems employing symbolic computation of Floquet transition matrix is considered in this work. It is shown that this method is applicable to chaos control. Nonlinear chaotic systems can be driven to a desired periodic motion by designing a combination of a feedforward controller and a feedback controller. The design of the feedback controller is achieved through the symbolic computation of fundamental solution matrix of linear time-periodic systems in terms of unknown control gains. Then, the Floquet transition matrix (state transition matrix evaluated at the end of the principal period) can determine the stability of the system owing to classical techniques such as pole placement, Routh–Hurwitz criteria, etc. Thus it is possible to place the Floquet multipliers in the desired locations to determine the control gains. This method can be applied to systems without small parameters. The Duffing’s oscillator, the Rössler system and the nonautonomous parametrically forced Lorenz equations are chosen as illustrative examples to demonstrate the application.     

Kinetic model of the Boltzmann equation for a diatomic gas with rotational degrees of freedom

A system of model kinetic equations is proposed to describe flows of a diatomic rarefied gas (nitrogen). A conservative numerical method is developed for its solution. A shock wave structure in nitrogen is computed, and the results are compared with experimental data in a wide range of Mach numbers. The system of model kinetic equations is intended to compute complex-geometry three-dimensional flows of a diatomic gas with rotational degrees of freedom.     

Numerical modelling of rarefied gas flow through a slit at arbitrary pressure ratio based on the kinetic equation

A rarefied gas flow through a thin slit at an arbitrary gas pressure ratio is calculated on the basis of the kinetic model equations (BGK and S-model) applying the discrete velocity method. The calculations are carried out for the whole range of the gas rarefaction from the free-molecular regime to the hydrodynamic one. Numerical data on the flow rate and distributions of density, bulk velocity and temperature are reported. Comparisons of the present results with those based on the direct simulation Monte Carlo method and on the linearized BGK kinetic equation are performed. The conditions of applicability of the linearized theory are discussed.