A general framework for the inverse design of mesoscopic models based on the Boltzmann equation

Abstract In this paper, we present a general framework for the inverse-design of mesoscopic models based on the Boltzmann equation. Starting from the single-relaxation-time Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using Navier-Stokes-Fourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leading-order Chapman-Enskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the Navier-Stokes-Fourier system in the continuum limit. These constraints allow for adjustable bulk-to-shear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three well-known mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the Chapman-Enskog expansion at the NSF order. Finally, based on the higher-order Chapman-Enskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored under this framework.

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Inverse design of mesoscopic models for compressible flow using the Chapman-Enskog analysis

Advances in Aerodynamics ◽

10.1186/s42774-020-00059-2 ◽

2021 ◽

Vol 3 (1) ◽

Cited By ~ 1

Author(s):

Tao Chen ◽

Lian-Ping Wang ◽

Jun Lai ◽

Shiyi Chen

Keyword(s):

Boltzmann Equation ◽

Compressible Flow ◽

Source Term ◽

Distribution Functions ◽

Inverse Design ◽

Navier Stokes ◽

Source Terms ◽

Hermite Expansion ◽

Mesoscopic Models ◽

Proposed Model

AbstractIn this paper, based on simplified Boltzmann equation, we explore the inverse-design of mesoscopic models for compressible flow using the Chapman-Enskog analysis. Starting from the single-relaxation-time Boltzmann equation with an additional source term, two model Boltzmann equations for two reduced distribution functions are obtained, each then also having an additional undetermined source term. Under this general framework and using Navier-Stokes-Fourier (NSF) equations as constraints, the structures of the distribution functions are obtained by the leading-order Chapman-Enskog analysis. Next, five basic constraints for the design of the two source terms are obtained in order to recover the NSF system in the continuum limit. These constraints allow for adjustable bulk-to-shear viscosity ratio, Prandtl number as well as a thermal energy source. The specific forms of the two source terms can be determined through proper physical considerations and numerical implementation requirements. By employing the truncated Hermite expansion, one design for the two source terms is proposed. Moreover, three well-known mesoscopic models in the literature are shown to be compatible with these five constraints. In addition, the consistent implementation of boundary conditions is also explored by using the Chapman-Enskog expansion at the NSF order. Finally, based on the higher-order Chapman-Enskog expansion of the distribution functions, we derive the complete analytical expressions for the viscous stress tensor and the heat flux. Some underlying physics can be further explored using the DNS simulation data based on the proposed model.

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Electron Energy Distribution Functions of Hydrogen: The Effect of Superelastic Vibrational Collisions and of the Dissociation Process

Zeitschrift für Naturforschung A ◽

10.1515/zna-1979-0511 ◽

1979 ◽

Vol 34 (5) ◽

pp. 585-593 ◽

Cited By ~ 15

Author(s):

M. Capitelli ◽

M. Dilonardo

Keyword(s):

Boltzmann Equation ◽

Energy Distribution ◽

Electron Energy ◽

Distribution Functions ◽

Electron Energy Distribution ◽

Hydrogen Atoms ◽

Dissociation Process ◽

Excitation Rate ◽

The Boltzmann Equation ◽

Energy Distribution Functions

Abstract Electron energy distribution functions (EDF) of molecular H2 have been calculated by numerically solving the Boltzmann equation including all the inelastic processes with the addition of superelastic vibrational collisions and of the hydrogen atoms coming from the dissociation process. The population densities of the vibrational levels have been obtained both by assuming a Boltz-mann population at a vibrational temperature different from the translational one and by solving a system of vibrational master equations coupled to the Boltzmann equation. The results, which have been compared with those corresponding to a vibrationally cold molecular gas, show that the inclusion of superelastic collisions and of the parent atoms affects the EDF tails without strongly modifying the EDF bulk. As a consequence the quantities affected by the EDF bulk, such as average and characteristic energies, drift velocity, 0-1 vibrational excitation rate are not too much affected by the inclusion of superelastic vibrational collisions and of parent atoms, while a strong influence is observed on the dissociation and ionization rate coefficients which depend on the EDF tail. Calculated dissociation rates, obtained by EDF's which take into account both the presence of vibrationally excited molecules and hydrogen atoms, are in satisfactory agreement with experimental results.

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Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

Journal of Computational Physics ◽

10.1016/j.jcp.2007.11.032 ◽

2008 ◽

Vol 227 (8) ◽

pp. 3781-3803 ◽

Cited By ~ 95

Author(s):

Mounir Bennoune ◽

Mohammed Lemou ◽

Luc Mieussens

Keyword(s):

Boltzmann Equation ◽

Navier Stokes ◽

Numerical Schemes ◽

The Boltzmann Equation ◽

Uniformly Stable

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Higher-order dissipation in the theory of homogeneous isotropic turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.489 ◽

2016 ◽

Vol 803 ◽

pp. 250-274 ◽

Cited By ~ 8

Author(s):

Norbert Peters ◽

Jonas Boschung ◽

Michael Gauding ◽

Jens Henrik Goebbert ◽

Reginald J. Hill ◽

...

Keyword(s):

Source Term ◽

Isotropic Turbulence ◽

Stokes Equations ◽

Higher Order ◽

Distribution Functions ◽

Order Structure ◽

Source Terms ◽

Homogeneous Isotropic Turbulence ◽

Higher Order Moments ◽

Even Order

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier–Stokes equations. Noticeably, we also find the volume-averaged dissipation $\unicode[STIX]{x1D700}_{r}$ used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) in the resulting system of equations, because it is related to dissipation correlations.

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Incompressible hydrodynamic approximation with viscous heating to the Boltzmann equation

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202517500440 ◽

2017 ◽

Vol 27 (12) ◽

pp. 2261-2296 ◽

Cited By ~ 1

Author(s):

Yan Guo ◽

Shuangqian Liu

Keyword(s):

Boltzmann Equation ◽

Initial Data ◽

Kinetic Equations ◽

Navier Stokes ◽

Viscous Heating ◽

Text Setting ◽

Hydrodynamic Approximation ◽

Energy Conversation ◽

The Boltzmann Equation

The incompressible Navier–Stokes–Fourier (INSF) system with viscous heating was first derived from the Boltzmann equation in the form of the diffusive scaling by Bardos–Levermore–Ukai–Yang [Kinetic equations: Fluid dynamical limits and viscous heating, Bull. Inst. Math. Acad. Sin.[Formula: see text] 3 (2008) 1–49]. The purpose of this paper is to justify such an incompressible hydrodynamic approximation to the Boltzmann equation in [Formula: see text] setting in a periodic box. Based on an odd–even expansion of the solution with respect to the microscopic velocity, the diffusive coefficients are determined by the INSF system with viscous heating and the super-Burnett functions. More importantly, the remainder of the expansion is proven to decay exponentially in time via an [Formula: see text] approach on the condition that the initial data satisfies the mass, momentum and energy conversation laws.

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THE BOLTZMANN EQUATION IN THE SPACE $L^2\cap L^\infty_\beta$: GLOBAL AND TIME-PERIODIC SOLUTIONS

Analysis and Applications ◽

10.1142/s0219530506000784 ◽

2006 ◽

Vol 04 (03) ◽

pp. 263-310 ◽

Cited By ~ 41

Author(s):

SEIJI UKAI ◽

TONG YANG

Keyword(s):

Boltzmann Equation ◽

Source Term ◽

Decay Rates ◽

Regularity Conditions ◽

Unique Existence ◽

The Boltzmann Equation ◽

Optimal Decay Rates ◽

Optimal Decay ◽

Derivatives Of ◽

Time Periodic

We present a function space in which the Cauchy problem for the Boltzmann equation is well-posed globally in time near an absolute Maxwellian in a mild sense without any regularity conditions. The asymptotic stability of the absolute Maxwellian is also established in this space and, moreover, it is shown that the higher order spatial derivatives of the solutions vanish in time faster than the lower order derivatives. No smallness assumptions are imposed on the derivatives of the initial data, and the optimal decay rates are derived. Furthermore, the Boltzmann equation with a time-periodic source term is solved in the same space on the unique existence and stability of a time-periodic solution which has the same period as the source term. The proof is based on the spectral analysis of the linearized Boltzmann operator.

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