Programming and modelling environment for studies of gas flows in micro- and nanostructures based on solving the Boltzmann equation

Abstract

A set of programs has been developed for modelling gas flows in micro-and nanostructures. The programs are based on a method for solving the kinetic equation by a finite-difference technique on a fixed space-velocity grid. A projection method is used for calculating the Boltzmann collision integral which ensures that the conservation of mass, momentum, and energy is rigorously satisfied and that the collision integral goes to zero under thermodynamic equilibrium conditions. An explicit flux conservative scheme is used for approximating the differential part. The solution of the resulting system of difference equations is found by splitting into stages of collisional relaxation and free molecular flow. The computational algorithm is realized on a multiprocessor system using MPI technology. A graphical shell has been developed for visualizing the results during the computations and for convenient variation of the physical parameters of the flow under study. Some calculations of flows through a periodic system of square holes and a periodic system of slots whose transverse dimension is on the order of the mean free path of the gas molecules, as well as some model calculations of a micropump design, are given as examples.The development of micro-and nanotechnologies, the construction of microscopic vacuum pumps, filters, gas analyzers, manipulators, and other instruments, and research on techniques for the safe storage of hydrogen, as well as of toxic and radioactive gases, in nanostructures have stimulated interest in studies of gas flows in devices whose characteristic sizes are comparable to the molecular mean free path. The practical importance of these studies is explained by the need to evaluate the working characteristics of the planned devices over a wide range of physical parameters, which determine the properties of the gaseous medium and materials from which they are made. The equations of continuum hydrodynamics cannot be used for numerical modelling of these flows, so that it is necessary to use methods from the theory of rarefied gas flows [1].Computational techniques for the theory of rarefied gases were developed rapidly during the second half of the last century for aerospace applications. Two main approaches were developed: direct statistical modelling and finite-difference solution of the Boltzmann kinetic equation. In order to reduce the volume of calculations, model kinetic equations that were not rigorously justified but were simpler to solve were often used instead of the Boltzmann equation. In the first approach [2], the evolution of a large number of three dimensional vectors depicting the gas molecules is followed in physical space,

A multi-level parallel solver for rarefied gas flows in porous media

“…Numerical solutions of this nonlinear integro-differential equation for realistic problems are tremendously expensive, in terms of computational cost and memory requirement, which is due to the complex five-fold integral I as well as multidimensional phase space (3D spatial domain Ω x , 3D velocity domain Ω v , and 1D temporal domain Ω t for unsteady flows). The full Boltzmann solvers can be classified into stochastic approach, such as the direct simulation Monte Carlo (DSMC) method, and deterministic approach, such as discrete velocity method (DVM) and fast spectral method (FSM) [8][9][10][11][12][13][14]. In order to obtain reasonable approximation of the Boltzmann equation, the full collision integral I is commonly simplified by a relaxation-time collision model, e.g.…”

Immersed boundary method for unsteady kinetic model equations

“…In such cases, the deterministic solution of Boltzmann kinetic model equation provides an efficient and accurate framework. In this framework, the non‐linear collision integral of the Boltzmann equation is replaced by a collision operator such as in the Bhatnagar‐Gross‐Krook (BGK) or the ellipsoidal‐statistical (ES)‐BGK models and has been applied to a wide variety of steady and unsteady problems in rarefied as well as continuum regime in 2‐D and 3‐D .…”