scholarly journals A multiple-relaxation-time collision model by Hermite expansion

2021 ◽  
Vol 379 (2208) ◽  
pp. 20200406 ◽  
Author(s):  
Xiaowen Shan ◽  
Xuhui Li ◽  
Yangyang Shi
Keyword(s):  
Relaxation Time ◽  
Relaxation Process ◽  
Rotation Group ◽  
Dynamics Simulation ◽  
Numerical Dissipation ◽  
Numerical Verification ◽  
Relaxation Rates ◽  
Hermite Expansion ◽  
Collision Model ◽  
Irreducible Components

The Bhatnagar–Gross–Krook (BGK) single-relaxation-time collision model for the Boltzmann equation serves as the foundation of the lattice BGK (LBGK) method developed in recent years. The description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is, in many scenarios, simplistic. Based on a previous series of papers, we present a collision model formulated as independent relaxations of the irreducible components of the Hermite coefficients in the reference frame moving with the fluid. These components, corresponding to the irreducible representation of the rotation group, are the minimum tensor components that can be separately relaxed without violating rotation symmetry. For the 2nd, 3rd and 4th moments, respectively, two, two and three independent relaxation rates can exist, giving rise to the shear and bulk viscosity, thermal diffusivity and some high-order relaxation process not explicitly manifested in the Navier–Stokes-Fourier equations. Using the binomial transform, the Hermite coefficients are evaluated in the absolute frame to avoid the numerical dissipation introduced by interpolation. Extensive numerical verification is also provided. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

A GENERAL MULTIPLE-RELAXATION-TIME BOLTZMANN COLLISION MODEL

2007 ◽  
Vol 18 (04) ◽  
pp. 635-643 ◽  
Author(s):  
XIAOWEN SHAN ◽  
HUDONG CHEN
Keyword(s):  
Relaxation Time ◽  
Lattice Boltzmann ◽  
Lattice Structure ◽  
Transport Coefficients ◽  
Hermite Polynomials ◽  
Simple Extension ◽  
Relaxation Rates ◽  
Collision Model ◽  
Multiple Relaxation Time ◽  
Lattice Boltzmann Models

We formulate a simple extension to the Bhatnagar-Gross-Krook collision model by expanding the distribution function in Hermite polynomials and assigning a relaxation time to each hydrodynamic moment. By discretizing the velocity space, multiple-relaxation-time lattice Boltzmann models can be constructed. The transport coefficients are analytically calculated and numerically verified. At the lowest order, allowing different relaxation rates for the second and third Hermite components results in a variable Prandtl number. Comparing with the previously proposed multiple-relaxation-time lattice Boltzmann models, the present formulation is general in the sense that it is independent of the underlying lattice structure and does not require a procedure for transformation of base vectors.

scholarly journals Thermodynamic Analysis of Relaxation Model for Non-Equilibrium Phase Behavior of Hydrocarbon Mixtures

2021 ◽  
Vol 2090 (1) ◽  
pp. 012138
Author(s):  
I M Indrupskiy ◽  
P A Chageeva
Keyword(s):  
Relaxation Time ◽  
Phase Behavior ◽  
Relaxation Process ◽  
Thermodynamic Analysis ◽  
Balance Equation ◽  
Equilibrium Phase ◽  
Characteristic Relaxation Time ◽  
Relaxation Model ◽  
Entropy Balance ◽  
Non Equilibrium

Abstract Mathematical models of phase behavior are widely used to describe multiphase oil and gas-condensate systems during hydrocarbon recovery from natural petroleum reservoirs. Previously a non-equilibrium phase behavior model was proposed as an extension over generally adopted equilibrium models. It is based on relaxation of component chemical potentials difference between phases and provides accurate calculations in some typical situations when non-instantaneous changing of phase fractions and compositions in response to variations of pressure or total composition is to be considered. In this paper we present a thermodynamic analysis of the relaxation model. General equations of non-equilibrium thermodynamics for multiphase flows in porous media are considered, and reduced entropy balance equation for the relaxation process is obtained. Isotropic relaxation process is simulated for a real multicomponent hydrocarbon system with different values of characteristic relaxation time using the non-equilibrium model implemented in the PVT Designer module of the RFD tNavigator simulation software. The results are processed with a special algorithm implemented in Matlab to calculate graphs of the total entropy time derivative and its constituents in the entropy balance equation. It is shown that the constituents have different signs, and the greatest influence on the entropy is associated with the interphase flow of the major component of the mixture and the change of the total system volume in the isotropic process. The characteristic relaxation time affects the rate at which the entropy is approaching its maximum value.

A theoretical model for the collective motion of proteins by means of principal component analysis

Open Physics ◽  
2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Hiqmet Kamberaj
Keyword(s):  
Molecular Dynamics ◽  
Principal Component Analysis ◽  
Relaxation Time ◽  
Friction Coefficient ◽  
Principal Component ◽  
Component Analysis ◽  
Dynamics Simulation ◽  
Coarse Grained ◽  
Dynamical Friction ◽  
Collective Coordinates

AbstractA coarse grained model in the frame work of principal component analysis is presented. We used a bath of harmonic oscillators approach, based on classical mechanics, to derive the generalized Langevin equations of motion for the collective coordinates. The dynamics of the protein collective coordinates derived from molecular dynamics simulations have been studied for the Bovine Pancreatic Trypsin Inhibitor. We analyzed the stability of the method by studying structural fluctuations of the Ca atoms obtained from a 20 ns molecular dynamics simulation. Subsequently, the dynamics of the collective coordinates of protein were characterized by calculating the dynamical friction coefficient and diffusion coefficients along with time-dependent correlation functions of collective coordinates. A dual diffusion behavior was observed with a fast relaxation time of short diffusion regime 0.2–0.4 ps and slow relaxation time of long diffusion about 1–2 ps. In addition, we observed a power law decay of dynamical friction coefficient with exponent for the first five collective coordinates varying from −0.746 to −0.938 for the real part and from −0.528 to −0.665 for its magnitude. It was found that only the first ten collective coordinates are responsible for configuration transitions occurring on time scale longer than 50 ps.

A Multi-Relaxation-Time Finite Volume Discrete Boltzmann Method for Viscous Flows

2019 ◽  
Author(s):  
Leitao Chen ◽  
Hamid Sadat ◽  
Laura Schaefer
Keyword(s):  
Relaxation Time ◽  
Boltzmann Equation ◽  
Finite Volume ◽  
Dynamic Models ◽  
Viscous Flows ◽  
Constitutive Law ◽  
Collision Model ◽  
New Model ◽  
Mesh Resolution ◽  
Boltzmann Method

Abstract Conventional constitutive law-based fluid dynamic models solve the conservation equations of mass and momentum, while kinetic models, such as the well-known lattice Boltzmann method (LBM), solve the propagation and collision processes of the Boltzmann equation-governed particle distribution function (PDF). Such models can provide an a priori modeling platform on a more fundamental level while easily reconstructing macroscopic variables such as velocity and pressure from the PDF. While the LBM requires a rigid and uniform grid for spatial discretization, another similar unique kinetic model known as the finite volume discrete Boltzmann method (FVDBM) has the ability to solve the discrete Boltzmann equation (DBE) on unstructured grids. The FVDBM can easily and accurately capture curved and more complicated fluid flow boundaries (usually solid boundaries), which cannot be satisfactorily realized in the LBM framework. As a result, the FVDBM preserves the physical advantages of the LBM over the constitutive law-based model approach, but also incorporates a better boundary treatment. However, the FVDBM suffers larger diffusion errors compared to the LBM approach. Building on our previous work, the FVDBM is further developed by integrating the multi-relaxation-time (MRT) collision model into the existing framework. Compared to the existing FVDBM approach that uses the Bhatnagar–Gross–Krook (BGK) collision model, which is also known as the single-relaxation-time (SRT) model, the new model can significantly reduce diffusion error or numerical viscosity, which is essential in the simulation of viscous flows. After testing the new model, the MRT-FVDBM, and the old model, the BGK-FVDBM, on Taylor-Green vortex flow, which can quantify the diffusion error of the applied model, it is found that the MRT-FVDBM can reduce the diffusion error at a faster rate as the mesh resolution increases, which renders the MRT-FVDBM a higher-order model than the BGK-FVDBM. At the highest mesh resolution tested in this paper, the reduction of the diffusion error by the MRT-FVDBM can be up to 30%.

scholarly journals Molecular Association of Tetramethylurea and Chlorobenzene Molecules in Microwave Frequency Range

2010 ◽  
Vol 65 (10) ◽  
pp. 854-858
Author(s):  
Vimal Sharma ◽  
Nagesh Thakur
Keyword(s):  
Relaxation Time ◽  
Dielectric Relaxation ◽  
Viscous Flow ◽  
Relaxation Process ◽  
Binary Mixtures ◽  
Variation Method ◽  
Dielectric Relaxation Time ◽  
Flow Process ◽  
Dielectric Relaxation Process ◽  
Energy Parameters

The dielectric constant ε´ and dielectric loss ε´´ of the binary mixtures of tetramethylurea (TMU) and chlorobenzene (CB) have been calculated at 9.883 GHz by using standard standing microwave techniques. Gopalakrishna’s single frequency concentration variation method has been used to calculate dipole moment μ and dielectric relaxation time τ for different mole fractions of TMU in the binary mixture at different temperatures of 25 °C, 30 °C, 35 °C, and 40 °C. The variation of dielectric relaxation time with the mole fraction of TMU in the whole concentration range of the binary mixtures was found to be non-monotonic. The solute-solute and solute-solvent type of molecular associations may be proposed based upon above observations. Using Eyring rate equations the energy parameters ΔH, ΔF, and ΔS for the dielectric relaxation process and the viscous flow process have been calculated at the given temperatures. It is found from the comparison of energy parameters that, just like the viscous flow process, the dielectric relaxation process can also be treated as a rate process.

The effect of compaction of a porous material confiner on detonation propagation

2017 ◽  
Vol 834 ◽  
pp. 434-463 ◽  
Author(s):  
Mark Short ◽  
James J. Quirk
Keyword(s):  
Relaxation Time ◽  
Porous Material ◽  
Time Scale ◽  
Sound Speed ◽  
Volume Fraction ◽  
Relaxation Rates ◽  
Slab Geometry ◽  
Local Flow ◽  
Detonation Propagation ◽  
Detonation Speed

The fluid mechanics of the interaction between a porous material confiner and a steady propagating high explosive (HE) detonation in a two-dimensional slab geometry is investigated through analytical oblique wave polar analysis and multi-material numerical simulation. Two HE models are considered, broadly representing the properties of either a high- or low-detonation-speed HE, which permits studies of detonation propagating at speeds faster or slower than the confiner sound speed. The HE detonation is responsible for driving the compaction front in the confiner, while, in turn, the high material density generated in the confiner as a result of the compaction process can provide a strong confinement effect on the HE detonation structure. Polar solutions that describe the local flow interaction of the oblique HE detonation shock and equilibrium state behind an oblique compaction wave with rapid compaction relaxation rates are studied for varying initial solid volume fractions of the porous confiner. Multi-material numerical simulations are conducted to study the effect of detonation wave driven compaction in the porous confiner on both the detonation propagation speed and detonation driving zone structure. We perform a parametric study to establish how detonation confinement is influenced both by the initial solid volume fraction of the porous confiner and by the time scale of the dynamic compaction relaxation process relative to the detonation reaction time scale, for both the high- and low-detonation-speed HE models. The compaction relaxation time scale is found to have a significant influence on the confinement dynamics, with slower compaction relaxation time scales resulting in more strongly confined detonations and increased detonation speeds. The dynamics of detonation confinement by porous materials when the detonation is propagating either faster or slower than the confiner sound speed is found to be significantly different from that with solid material confiners.

Sign in / Sign up

Export Citation Format

Share Document