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Relative error of the decay rate with the varying ratio of bulk viscosity to shear viscosity. Two cases are tested: τ 21 = 0.5005 and τ 21 = 0.6. In both cases γ = 1.3, Pr = 2.1, and τ 1 = τ 3 . To be seen is that the relative error is generally below 1% over a large range of ν b /ν.
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We point out that the minimal components of the tensorial moments of the distribution that can be independently relaxed in collision without violating rotational symmetry are its irreducible representation (irrep) of SO(3), and a generic multiple-relaxation-time collision model can be constructed by independently relaxing these components. As the s...
Citations
... Each of the Hermite polynomials is further decomposed into components corresponding to an irreducible representation of SO(3) [31] which are the minimum groups closed under spatial rotation. The irreducible tensors are then relaxed with independent relaxation times without violating rotation symmetry [32][33][34][35][36][37]. ...
... By converting via binomial transform the raw moments to the central moments in local frame before relaxing them, a Galilean invariant multiple-relaxation-time LB collision model in Hermite spectral space (SMRT) was obtained [33,34]. Further realizing that the minimum sets of components of the tensorial moments that can be relaxed together without violating rotation symmetry correspond to the irreducible tensors [35], we arrived at the SMRT model [37] which is defined via the local Hermite expansion [37] ( , , ) = ( ) ...
Prediction of non-equilibrium flows is critical to space flight. In the present work we demonstrate that the recently developed spectral multiple-relaxation-time (SMRT) lattice Boltzmann (LB) model is theoretically equivalent to Grad's eigen-system [Grad, H., In Thermodynamics of Gases (1958)] where the eigen-functions obtained by tensor decomposition of the Hermite polynomials are also those of the linearized Boltzmann equation. Numerical results of shock structure simulation using Maxwell molecular model agree very well with those of a high-resolution fast spectral method code up to Mach 7 provided that the relaxation times of the irreducible tensor components match their theoretical values. If a reduced set of relaxation times are used such as in the Shakhov model and lumped-sum relaxation of Hermite modes, non-negligible discrepancies starts to occur as Mach number is raised, indicating the necessity of the fine-grained relaxation model. Together with the proven advantages of LB, the LB-SMRT scheme offers a competitive alternative for non-equilibrium flow simulation. Nomenclature (Nomenclature entries should have the units identified) () = Hermite expansion coefficient of distribution = Collision kernel = peculiar velocity, − () = local Hermite expansion coefficient of distribution = velocity distribution function = Maxwell-Boltzmann distribution = Gaussian distribution of ES-BGK model = target distribution of Shakhov model * Ph. D. Student, Department of Mechanics and Aerospace Engineering. † Chair Professor, IAS = scaled distribution, = 3/2 / = shifted constant integer vector Kn = Knudsen number = linearized Boltzmann collision operator = Gross-Jackson model = Associated Laguerre polynomial = 3rd-order rank-3 traceless moment, ∫ H (3,0) () Ma = Mach number = Hydro-static pressure = pressure tensor, ∫ 2 (,) = spherical moment = Legendre polynomial Pr = Prandtl number = heat flux = 3rd-order moment, ∫ 3 = 4th-order moment, ∫ 4 +1/2 = Sonine polynomial = time, = macroscopic velocity = temperature scaled peculiar velocity, / √ = weight of discrete velocity = space position, = Spherical harmonic = Specific heat ratio = Kronecker delta = Azimuthal angle = temperature = Eigenvalue of linearized BCO = Eigenvalue of Grad's linearized BCO = Kinematic viscosity = molecular velocity = relative velocity 2 = density = Stress tensor = Relaxation time = Polar angle = Perturbed distribution = Eigen-function of Linearized BCO Ω = collision operator = weight function of Hermite polynomials H () = Hermite polynomial Y () = Solid homogeneous spherical harmonic in Cartesian coordinate Subscripts ′ = post-collision values
... With the BGK collision model, the description of the collision as a uniform relaxation process of the distribution function towards its equilibrium is in many cases simplistic. In a previous series of papers [20][21][22], the SMRT collision model was developed where the irreducible components of the Hermite coefficients are relaxed separately in the reference frame moving with the fluid. These components are the minimum tensor components that can be separately relaxed without violating rotation symmetry. ...
... In particular, we have θ (D+1)/2 d (1) F = ρg. Furthermore, if the contributions from the nonequilibrium beyond second order are neglected, we have θ (D+2)/2 d (2) F = −Fg, θ (D+3)/2 d (3) F = 3guF, and θ (D+4)/2 d (4) F = −6gu 1 F. To allow maximum flexibility while preserving rotational symmetry [20][21][22], each H (n) (v) is further decomposed into its traceless components, S (n,k) (v). Let the distribution function have the following expansions: ...
In the present work, the force term is first derived in the spectral multiple-relaxation-time high-order lattice Boltzmann model. The force term in the Boltzmann equation is expanded in the Hermite temperature rescaled central moment space (RCM), instead of the Hermite raw moment space (RM). The contribution of nonequilibrium RCM moments beyond second order are neglected. For the collision operator in the RCM space, each order of the force term can be incorporated directly. Through the transformation between the RCM space and the RM space, the force term for practical numerical implementation in the RM space can be derived. It can be demonstrated that the present force scheme is self-consistent for the isothermal flow and compressible thermal flow with adjustable Prandtl number via the numerical experiments.
... With the BGK collision model, the description of the colli-sion as a uniform relaxation process of the distribution function towards its equilibrium is in many cases simplistic. In a previous series of papers [18,21,22], the SMRT collision model was developed where the irreducible components of the Hermite coefficients are relaxed separately in the reference frame moving with the fluid. These components are the minimum tensor components that can be separately relaxed without violating rotation symmetry. ...
... In particular, we have d d d F is not null. To allow maximum flexibility while preserving rotational symmetry [18,21,22], each H (n) (v v v) is further decomposed into its traceless components, S (n,k) (v v v). Let the distribution function has the following expansions: ...
We present an a priori derivation of the force scheme for lattice Boltzmann method based on kinetic theoretical formulation. We show that the discrete lattice effect, previously eliminated a posteriori in BGK collision model, is due to first-order space-time discretization and can be eliminated generically for a wide range of collision models with second-order space-time discretization. Particularly, the force scheme for the recently developed spectral multiple-relaxation-time (SMRT) collision model is obtained and numerically verified.
... 为Z. Guo等 [76] 2002年推导的力项格式(具体离散在 后文中阐述)；式(53-2)即为X. He等 [75] 1998年设计 ...
Numerical stability issue has always been the bottleneck that plagues the wide application of lattice Boltzmann methods in the flow scenarios of high Reynolds number and high Mach number. To improve the numerical stability and accuracy of the collision models in the field of lattice Boltzmann methods has been a research hotspot and difficulty in the past three decades. As an important branch of many collision models, regularized collision model has made massive theoretical progress in recent years, and has been widely applied in the fields of high Reynolds number turbulent flow, high Mach number compressible flow and aeroacoustics flow. The present paper systematically reviews the development history of the regularized collision models, and establishes a systematic theoretical framework to illustrate the underlying theoretical connections among different regularized collision models. Meanwhile, the present work reveals the theoretical essence of the regularized multi-relaxation collision model, which is to rapidly relax the higher-order kinetic moments that cannot be expressed by the discrete lattice model in the coordinate-independent and self-similar moment space, so as to avoid the fake modes affecting the hydrodynamic equations.
Body-force modelling in the lattice Boltzmann method (LBM) has been studied extensively in the incompressible limit but rarely discussed for thermal compressible flows. Here we present a systematic approach of incorporating body force in the LBM which is valid for thermal compressible and non-equilibrium flows. In particular, a LBM forcing scheme accurate for the energy equation with second-order time accuracy is given. New and essential in this scheme is the third-moment contribution of the force term. It is shown via Chapman–Enskog analysis that the absence of this contribution causes an erroneous heat flux quadratic in Mach number and linear in temperature variation. The theoretical findings are verified and the necessity of the third-moment contribution is demonstrated by numerical simulations.
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’.
Despite yielding correct hydrodynamics in the continuum limit, the Bhatnagar–Gross–Krook collision model is too simplistic to model the full details of the collision, which becomes increasingly important as the quasi-equilibrium assumption breaks down. In a recent phenomenological collision model, independent relaxation rates are assigned to the components of the tensorial Hermite expansion of the distribution corresponding to the irreducible representations of SO(3), yielding arguably the most general form of multirelaxation without violating rotation symmetry. Here we show that by using the relaxation rates obtained analytically from Boltzmann collision term with Maxwell molecular model, lattice Boltzmann method yields results in good agreement with the accurate fast spectral method in simulation of the spontaneous Rayleigh–Brillouin scattering problem. The hydrodynamically insignificant relaxation rates of the higher moments are found to be significant as the Knudsen number increases. These results suggest that with properly tuned relaxation rates, the collision model could potentially mimic the behavior of arbitrary collision kernels.
