scholarly journals On the accuracy of macroscopic equations for linearized rarefied gas flows

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Lei Wu ◽  
Xiao-Jun Gu
Keyword(s):  
Boltzmann Equation ◽  
Volume Diffusion ◽  
Rarefied Gas ◽  
Hermite Polynomials ◽  
Velocity Distribution Function ◽  
Gas Flows ◽  
Burnett Equations ◽  
Moment Equations ◽  
Rarefied Gas Flows ◽  
Macroscopic Equations

AbstractMany macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and “thermo-mechanically consistent" Burnett equations based on the argument of “volume diffusion”. This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on “volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

Towards the development of a multiscale, multiphysics method for the simulation of rarefied gas flows

2010 ◽  
Vol 661 ◽  
pp. 262-293 ◽  
Author(s):  
DAVID A. KESSLER ◽  
ELAINE S. ORAN ◽  
CAROLYN R. KAPLAN
Keyword(s):  
Heat Flux ◽  
Boltzmann Equation ◽  
Conservation Laws ◽  
Rarefied Gas ◽  
Multiscale Methods ◽  
Time Interval ◽  
Gas Flows ◽  
Transition Regime ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation

We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continuum field, and the cycle is repeated. The target applications for this type of method are transition-regime gas flows for which standard continuum models (e.g. Navier–Stokes equations) cannot be used, but solution of Boltzmann's equation is prohibitively expensive. The use of molecular-level data to close the conservation laws significantly extends the range of applicability of the continuum conservation laws. In this study, the CM3 is used to perform two proof-of-principle calculations: a low-speed Rayleigh flow and a thermal Fourier flow. Velocity, temperature, shear-stress and heat-flux profiles compare well with direct-simulation Monte Carlo solutions for various Knudsen numbers ranging from the near-continuum regime to the transition regime. We discuss algorithmic problems and the solutions necessary to implement the CM3, building upon the conceptual framework of the heterogeneous multiscale methods.

scholarly journals The Half-Range Moment Method in Harmonically Oscillating Rarefied Gas Flows

Fluids ◽  
2021 ◽  
Vol 6 (1) ◽  
pp. 17
Author(s):  
Giorgos Tatsios ◽  
Alexandros Tsimpoukis ◽  
Dimitris Valougeorgis
Keyword(s):  
Boundary Conditions ◽  
Stokes Flow ◽  
Moment Method ◽  
Rarefied Gas ◽  
Gas Flows ◽  
Moment Equations ◽  
Rarefied Gas Flows ◽  
Flow Parameters ◽  
Discrete Velocity Method ◽  
Wide Range

The formulation of the half-range moment method (HRMM), well defined in steady rarefied gas flows, is extended to linear oscillatory rarefied gas flows, driven by oscillating boundaries. The oscillatory Stokes (also known as Stokes second problem) and the oscillatory Couette flows, as representative ones for harmonically oscillating half-space and finite-medium flow setups respectively, are solved. The moment equations are derived from the linearized time-dependent BGK kinetic equation, operating accordingly over the positive and negative halves of the molecular velocity space. Moreover, the boundary conditions of the “positive” and “negative” moment equations are accordingly constructed from the half-range moments of the boundary conditions of the outgoing distribution function, assuming purely diffuse reflection. The oscillatory Stokes flow is characterized by the oscillation parameter, while the oscillatory Couette flow by the oscillation and rarefaction parameters. HRMM results for the amplitude and phase of the velocity and shear stress in a wide range of the flow parameters are presented and compared with corresponding results, obtained by the discrete velocity method (DVM). In the oscillatory Stokes flow the so-called penetration depth is also computed. When the oscillation frequency is lower than the collision frequency excellent agreement is observed, while when it is about the same or larger some differences are present. Overall, it is demonstrated that the HRMM can be applied to linear oscillatory rarefied gas flows, providing accurate results in a very wide range of the involved flow parameters. Since the computational effort is negligible, it is worthwhile to consider the efficient implementation of the HRMM to stationary and transient multidimensional rarefied gas flows.

scholarly journals Simulation of 2D rarefied gas flows based on the numerical solution of the Boltzmann equation

2017 ◽  
Author(s):  
Sergey O. Poleshkin ◽  
Ewgenij A. Malkov ◽  
Alexey N. Kudryavtsev ◽  
Anton A. Shershnev ◽  
Yevgeniy A. Bondar ◽  
...  
Keyword(s):  
Boltzmann Equation ◽  
Numerical Solution ◽  
Rarefied Gas ◽  
Gas Flows ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation

scholarly journals Kinetic Methods for Solving Unsteady Problems with Jet Flows

2019 ◽  
pp. 34-51
Author(s):  
A. A. Frolova ◽  
V. A. Titarev
Keyword(s):  
Boltzmann Equation ◽  
Model Equation ◽  
Rarefied Gas ◽  
Strong Dependence ◽  
Gas Flows ◽  
Original Text ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation ◽  
Uniform Grids ◽  
Model Equations

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order  to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model,  Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space.  The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model)  and makes it possible to calculate on non-structured non uniform grids in velocity and  physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

scholarly journals Kinetic Methods for Solving Non-stationary Jet Flow Problems

2018 ◽  
pp. 27-44
Author(s):  
A. A. Frolova ◽  
V. A. Titarev
Keyword(s):  
Boltzmann Equation ◽  
Rarefied Gas ◽  
Velocity Space ◽  
Gas Flows ◽  
Study Objective ◽  
Software Complex ◽  
Rarefied Gas Flows ◽  
The Boltzmann Equation ◽  
Model Equations ◽  
3D Software

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

Derivation of stable Burnett equations for rarefied gas flows

2017 ◽  
Vol 96 (1) ◽  
Author(s):  
Narendra Singh ◽  
Ravi Sudam Jadhav ◽  
Amit Agrawal
Keyword(s):  
Rarefied Gas ◽  
Gas Flows ◽  
Burnett Equations ◽  
Rarefied Gas Flows

scholarly journals Entropic boundary conditions for 13 moment equations in rarefied gas flows

2019 ◽  
Vol 31 (2) ◽  
pp. 021215 ◽  
Author(s):  
Carl Philipp Zinner ◽  
Hans Christian Öttinger
Keyword(s):  
Boundary Conditions ◽  
Rarefied Gas ◽  
Gas Flows ◽  
Moment Equations ◽  
Rarefied Gas Flows

scholarly journals Analysis of the Moment Method and the Discrete Velocity Method in Modeling Non-Equilibrium Rarefied Gas Flows: A Comparative Study

2019 ◽  
Vol 9 (13) ◽  
pp. 2733
Author(s):  
Weiqi Yang ◽  
Shuo Tang ◽  
Hui Yang
Keyword(s):  
Moment Method ◽  
Rarefied Gas ◽  
Hermite Polynomials ◽  
Distribution Functions ◽  
Gas Flows ◽  
Thermally Induced ◽  
Rarefied Gas Flows ◽  
Discrete Velocity Method ◽  
The Moment ◽  
Non Equilibrium

In the present study, the performance of the moment method, in terms of accuracy and computational efficiency, was evaluated at both the macro- and microscopic levels. Three different types of non-equilibrium gas flows, including the force-driven Poiseuille flow, lid-driven and thermally induced cavity flows, were simulated in the slip and transition regimes. Choosing the flow fields obtained from the Boltzmann model equation as the benchmark, the accuracy and validation of Navier–Stokes–Fourier (NSF), regularized 13 (R13) and regularized 26 (R26) equations were explored at the macroscopic level. Meanwhile, we reconstructed the velocity distribution functions (VDFs) using the Hermite polynomials with different-order of molecular velocity moments, and compared them with the Boltzmann solutions at the microscopic level. Moreover, we developed a kinetic criterion to indirectly assess the errors of the reconstructed VDFs. The results have shown that the R13 and R26 moment methods can be faithfully used for non-equilibrium rarefied gas flows in the slip and transition regimes. However, as indicated from the thermally induced case, all of the reconstructed VDFs are still very close to the equilibrium state, and none of them can reproduce the accurate VDF profile when the Knudsen number is above 0.5.

A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics

2013 ◽  
Vol 14 (1) ◽  
pp. 242-264 ◽  
Author(s):  
Jaw-Yen Yang ◽  
Bagus Putra Muljadi ◽  
Zhi-Hui Li ◽  
Han-Xin Zhang
Keyword(s):  
Distribution Function ◽  
Initial Value Problems ◽  
Rarefied Gas ◽  
Relaxation Times ◽  
Velocity Distribution Function ◽  
Gas Flows ◽  
Good Resolution ◽  
Initial Value ◽  
Rarefied Gas Flows ◽  
Wide Range

AbstractAn accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. 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 capture the time evolution of the discretized velocity distribution function in physical space and time. The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved. Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method. The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

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