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

2021 ◽  
Vol 3 (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.

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

2020 ◽  
Author(s):  
Tao Chen ◽  
Lianping Wang ◽  
Jun Lai ◽  
Shiyi Chen
Keyword(s):  
Boltzmann Equation ◽  
General Framework ◽  
Source Term ◽  
Distribution Functions ◽  
Inverse Design ◽  
Navier Stokes ◽  
Source Terms ◽  
Hermite Expansion ◽  
Mesoscopic Models ◽  
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.

scholarly journals Higher-order dissipation in the theory of homogeneous isotropic turbulence

2016 ◽  
Vol 803 ◽  
pp. 250-274 ◽  
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.

scholarly journals Inertial Frame Independent Forcing for Discrete Velocity Boltzmann Equation: Implications for Filtered Turbulence Simulation

2012 ◽  
Vol 12 (3) ◽  
pp. 732-766 ◽  
Author(s):  
Kannan N. Premnath ◽  
Sanjoy Banerjee
Keyword(s):  
Boltzmann Equation ◽  
Lattice Boltzmann ◽  
Inertial Frame ◽  
Fluid Velocity ◽  
Central Moment ◽  
Distribution Functions ◽  
Inverse Transformation ◽  
Lattice Boltzmann Equation ◽  
Source Terms ◽  
Central Moments

AbstractWe present a systematic derivation of a model based on the central moment lattice Boltzmann equation that rigorously maintains Galilean invariance of forces to simulate inertial frame independent flow fields. In this regard, the central moments, i.e. moments shifted by the local fluid velocity, of the discrete source terms of the lattice Boltzmann equation are obtained by matching those of the continuous full Boltzmann equation of various orders. This results in an exact hierarchical identity between the central moments of the source terms of a given order and the components of the central moments of the distribution functions and sources of lower orders. The corresponding source terms in velocity space are then obtained from an exact inverse transformation due to a suitable choice of orthogonal basis for moments. Furthermore, such a central moment based kinetic model is further extended by incorporating reduced compressibility effects to represent incompressible flow. Moreover, the description and simulation of fluid turbulence for full or any subset of scales or their averaged behavior should remain independent of any inertial frame of reference. Thus, based on the above formulation, a new approach in lattice Boltzmann framework to incorporate turbulence models for simulation of Galilean invariant statistical averaged or filtered turbulent fluid motion is discussed.

Estimation of Source Term Uncertainty in a Severe Accident With Correlated Variables

2014 ◽  
Author(s):  
Xiaoyu Zheng ◽  
Hiroto Itoh ◽  
Hitoshi Tamaki ◽  
Yu Maruyama
Keyword(s):  
Uncertainty Analysis ◽  
Nuclear Power ◽  
Source Term ◽  
Correlation Coefficients ◽  
Sampling Technique ◽  
Latin Hypercube Sampling ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Severe Accident ◽  
Source Terms

The quantitative evaluation of the fission product release to the environment during a severe accident is of great importance. In the present analysis, integral severe accident code MELCOR 1.8.5 has been applied to estimating uncertainty of source term for the accident at Unit 2 of the Fukushima Daiichi nuclear power plant (NPP) as an example and to discussing important models or parameters influential to the source term. Forty-two parameters associated with models for the transportation of radioactive materials were chosen and narrowed down to 18 through a set of screening analysis. These 18 parameters in addition to 9 parameters relevant to in-vessel melt progression obtained by the preceding uncertainty study were input to the subsequent sensitivity analysis by Morris method. This one-factor-at-a-time approach can preliminarily identify inputs which have important effects on an output, and 17 important parameters were selected from the total of 27 parameters through this approach. The selected parameters have been integrated into uncertainty analysis by means of Latin Hypercube Sampling technique and Iman-Conover method, taking into account correlation between parameters. Cumulative distribution functions of representative source terms were obtained through the present uncertainty analysis assuming the failure of suppression chamber. Correlation coefficients between the outputs and uncertain input parameters have been calculated to identify parameters of great influences on source terms, which include parameters related to models on core components failure, models of aerosol dynamic process and pool scrubbing.

scholarly journals Vibrations of Nonlinear Elastic Structure Excited by Compressible Flow

2021 ◽  
Vol 11 (11) ◽  
pp. 4748
Author(s):  
Monika Balázsová ◽  
Miloslav Feistauer ◽  
Jaromír Horáček ◽  
Adam Kosík
Keyword(s):  
Compressible Flow ◽  
Nonlinear Elasticity ◽  
Vocal Tract ◽  
Stokes Equations ◽  
Vocal Folds ◽  
Nonlinear Material ◽  
Navier Stokes ◽  
Arbitrary Lagrangian Eulerian ◽  
Navier Stokes Equations ◽  
Reliable Solution

This study deals with the development of an accurate, efficient and robust method for the numerical solution of the interaction of compressible flow and nonlinear dynamic elasticity. This problem requires the reliable solution of flow in time-dependent domains and the solution of deformations of elastic bodies formed by several materials with complicated geometry depending on time. In this paper, the fluid–structure interaction (FSI) problem is solved numerically by the space-time discontinuous Galerkin method (STDGM). In the case of compressible flow, we use the compressible Navier–Stokes equations formulated by the arbitrary Lagrangian–Eulerian (ALE) method. The elasticity problem uses the non-stationary formulation of the dynamic system using the St. Venant–Kirchhoff and neo-Hookean models. The STDGM for the nonlinear elasticity is tested on the Hron–Turek benchmark. The main novelty of the study is the numerical simulation of the nonlinear vocal fold vibrations excited by the compressible airflow coming from the trachea to the simplified model of the vocal tract. The computations show that the nonlinear elasticity model of the vocal folds is needed in order to obtain substantially higher accuracy of the computed vocal folds deformation than for the linear elasticity model. Moreover, the numerical simulations showed that the differences between the two considered nonlinear material models are very small.

scholarly journals Research of reliability and efficiency of technological processes of mechanical assembly production on the basis of the common semi-Markov model

2018 ◽  
Vol 224 ◽  
pp. 01138
Author(s):  
Yuri Rapatskiy ◽  
Mikhail Zamorenov ◽  
Vadim Kopp ◽  
Yuri Obzherin ◽  
Vladimir Gusev ◽  
...  
Keyword(s):  
Distribution Functions ◽  
Mechanical Assembly ◽  
Renewal Equations ◽  
Technological Processes ◽  
Threaded Connections ◽  
Proposed Model ◽  
The Common ◽  
Semi Markov Processes ◽  
Simulation Parameters ◽  
Assembly Production

In the article a common semi-Markov mathematical model is considered that allows one to investigate the productivity and reliability of various technological processes of mechanical assembly production. The proposed model allows to study, inter alia, technological processes of manufacturing parts with screw and assemblies of threaded connections. Mathematical apparatus of the research is the theory of semi-Markov processes with a common phase space, which operates with a common kind of random variables distribution functions. If the considering process in the system is a subsystem located on a higher level of hierarchy, the hierarchical model for compatibility with each other levels as output simulation parameters required distribution functions. In the proposed model, based on the decision of the Markov renewal equations depend not only on the torque characteristics, but also the distribution function of time per unit of output service according to different kinds of undervalued failures.

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

1979 ◽  
Vol 34 (5) ◽  
pp. 585-593 ◽  
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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