Anomalous transport driven by ion temperature gradient instability in an anisotropic deuterium-tritium plasma

In the nowadays and future fusion devices such as ITER and CFETR, as the use of various heating schemes, the parallel and perpendicular temperature of plasmas can be different; this temperature anisotropy may have significant effects on the turbulence. In this work, the anomalous transport driven by the ion temperature gradient instability is investigated in an anisotropic deuterium-tritium (D-T) plasma. The anisotropic factor $\alpha$, defined as the ratio of perpendicular temperature to parallel temperature, is introduced to describe the temperature anisotropy in the equilibrium distribution function of D. The linear dispersion relation in local kinetic limit is derived, and then numerically evaluated to study the dependence of mode frequency on the anisotropic factor $\alpha$ and the proportion for T particle $\vareT$ by choosing three sets of typical parameters, denoted as the cyclone base case (CBC), ITER and CFETR cases. Based on the linear results, the mixing length model approximation is adopted to analyze the quasi-linear particle and energy fluxes for D and T. It is found that choosing small $\alpha$ and large $\vareT$ is beneficial for the confinement of particle and energy for D and T. This work may be helpful for the estimation of turbulent transport level in the ITER and CFETR devices.

On hydrodynamics in the presense of strong external potential field

On the base of the Boltzmann kinetic equation, hydrodynamics of a dilute gas in the presence of the strong external potential field is investigated. First of all, a gravitational field is meant, because the consistent development of hydrodynamics in this environment is of great practical importance. In the present paper it is assumed that it is possible to neglect the influence of the field on the particle collisions. The study is based on the Chapman–Enskog method in a Bogolyubov’s formulation, which uses the idea of the functional hypothesis. Consideration is limited to steady gas states, which are subjected to a simpler experimental study. Chemical potential μ0 of the gas at the point where the external field has zero value and its temperature T are selected as the reduced description parameters of the system. In equilibrium, in the presence of the field, these values do not depend on the coordinates. It is assumed that in thehydrodynamic states T and μ0 are weakly dependent on the coordinates and therefore their gradients, considered on the scale of the free path length of the gas, are small. The kinetic equation, accounting for the functional hypothesis, gives an integro-differential equation for a gas distribution function at the hydrodynamic stage of evolution. This equation is solved in perturbation theory in gradients of T and μ0. The main approximation is analyzed for possibility of the system to be in a local equilibrium by means of comparing it with an equilibrium distribution function. Next, the distribution function is calculated in the first approximation in gradients and it is expressed in terms of solutions Ap , Bp of some first kind integral Fredholm equations. An approach to the approximate solution of these equations is discussed. The found distribution function is used to calculate the fluxes of the number of gas particles and their energy in the first order in gradients T and μ0 . Kinetic coefficients, which describe the structure of these fluxes, are introduced. Matrix elements of the operator of the linearized collision integral (integral brackets) are used for their research. It is a question of validity of the principle of symmetry of kinetic coefficients and definition of their signs.

Molecular dynamics lattice gas equilibrium distribution function for Lennard–Jones particles

The molecular dynamics lattice gas (MDLG) method maps a molecular dynamics (MD) simulation onto a lattice gas using a coarse-graining procedure. This is a novel fundamental approach to derive the lattice Boltzmann method (LBM) by taking a Boltzmann average over the MDLG. A key property of the LBM is the equilibrium distribution function, which was originally derived by assuming that the particle displacements in the MD simulation are Boltzmann distributed. However, we recently discovered that a single Gaussian distribution function is not sufficient to describe the particle displacements in a broad transition regime between free particles and particles undergoing many collisions in one time step. In a recent publication, we proposed a Poisson weighted sum of Gaussians which shows better agreement with the MD data. We derive a lattice Boltzmann equilibrium distribution function from the Poisson weighted sum of Gaussians model and compare it to a measured equilibrium distribution function from MD data and to an analytical approximation of the equilibrium distribution function from a single Gaussian probability distribution function. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

Revisiting the Lattice Boltzmann Method Through a Nonequilibrium Thermodynamics Perspective

In this paper, we incorporate a nonequilibrium thermodynamics perspective that is consistent with the Onsager reciprocity principle into the lattice Boltzmann framework to propose a novel regularized lattice Boltzmann formulation for modeling the Navier–Stokes–Fourier equations. The new method is applied to one-dimensional (1D) isothermal situations wherein the advantages of incorporating such a nonequilibrium perspective can be explicitly appreciated. In such situations, the nonequilibrium contribution of the lattice populations obtained by the new method completely vanishes, and the lattice update is entirely reduced to evaluating the equilibrium distribution function. Such a counterintuitive 1D mesoscopic description is not obtained in any other existing lattice Boltzmann scheme. We therefore numerically test the proposed formulation on two complex problems, namely, shockwave and nonlinear wave propagation, and compare results with analytical results along with six existing lattice Boltzmann schemes; it is found that the new method indeed yields results that are more stable and accurate. These results highlight the potency of the nonequilibrium thermodynamics-based approach for obtaining accurate and stable lattice Boltzmann computations, and provide new insights into established lattice Boltzmann simulation methods.

Large-eddy simulation of wall-bounded turbulent flow with high-order discrete unified gas-kinetic scheme

In this paper, we introduce the discrete Maxwellian equilibrium distribution function for incompressible flow and force term into the two-stage third-order Discrete Unified Gas-Kinetic Scheme (DUGKS) for simulating low-speed turbulent flows. The Wall-Adapting Local Eddy-viscosity (WALE) and Vreman sub-grid models for Large-Eddy Simulations (LES) of turbulent flows are coupled within the present framework. Meanwhile, the implicit LES are also presented to verify the effect of LES models. A parallel implementation strategy for the present framework is developed, and three canonical wall-bounded turbulent flow cases are investigated, including the fully developed turbulent channel flow at a friction Reynolds number (Re) about 180, the turbulent plane Couette flow at a friction Re number about 93 and lid-driven cubical cavity flow at a Re number of 12000. The turbulence statistics, including mean velocity, the r.m.s. fluctuations velocity, Reynolds stress, etc. are computed by the present approach. Their predictions match precisely with each other, and they are both in reasonable agreement with the benchmark data of DNS. Especially, the predicted flow physics of three-dimensional lid-driven cavity flow are consistent with the description from abundant literature. The present numerical results verify that the present two-stage third-order DUGKS-based LES method is capable for simulating inhomogeneous wall-bounded turbulent flows and getting reliable results with relatively coarse grids.

Large-eddy Simulation of Wall-bounded Turbulent Flow with High-order Discrete Unified Gas-Kinetic Scheme

In this paper, we introduce the incompressible discrete Maxwellian equilibrium distribution function and external forces into the two-stage third-order Discrete Unified Gas-Kinetic Scheme (DUGKS) for simulating low-speed incompressible turbulent flows with forcing term. The Wall-Adapting Local Eddy-viscosity (WALE) and Vreman sub-grid models for Large-Eddy Simulations (LES) of wall-bounded turbulent flows are coupled within the present framework. In order to simulate the three-dimensional turbulent flows associated with great computational cost, a parallel implementation strategy for the present framework is developed, and is validated by three canonical wall-bounded turbulent flows, viz., the fully developed turbulent channel flow at a friction Reynolds number (Re) about 180, the turbulent plane Couette flow at a friction Re number about 93 and three-dimensional lid-driven cubical cavity flow at a Re number of 12000. The turbulence statistics are computed by the present approach with both WALE and Vreman models, and their predictions match precisely with each other. Especially, the predicted flow physics of three-dimensional lid-driven cavity are consistent with the description from abundant literatures. While, they have small discrepancies in comparison to the Direct Numerical Simulation (DNS) due to the relatively low grid resolution. The present numerical results verify that the present two-stage third-order DUGKS-based LES method is capable for simulating inhomogeneous wall-bounded turbulent flows and getting reliable results with relatively coarse grids.

Compressibility in lattice Boltzmann on standard stencils: effects of deviation from reference temperature

With growing interest in the simulation of compressible flows using the lattice Boltzmann (LB) method, a number of different approaches have been developed. These methods can be classified as pertaining to one of two major categories: (i) solvers relying on high-order stencils recovering the Navier–Stokes–Fourier equations, and (ii) approaches relying on classical first-neighbour stencils for the compressible Navier–Stokes equations coupled to an additional (LB-based or classical) solver for the energy balance equation. In most cases, the latter relies on a thermal Hermite expansion of the continuous equilibrium distribution function (EDF) to allow for compressibility. Even though recovering the correct equation of state at the Euler level, it has been observed that deviations of local flow temperature from the reference can result in instabilities and/or over-dissipation. The aim of the present study is to evaluate the stability domain of different EDFs, different collision models, with and without the correction terms for the third-order moments. The study is first based on a linear von Neumann analysis. The correction term for the space- and time-discretized equations is derived via a Chapman–Enskog analysis and further corroborated through spectral dispersion–dissipation curves. Finally, a number of numerical simulations are performed to illustrate the proposed theoretical study. This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Validity of the molecular-dynamics-lattice-gas global equilibrium distribution function

The molecular-dynamics-lattice-gas (MDLG) method establishes a direct link between a lattice-gas method and the coarse-graining of a molecular dynamics (MD) approach. Due to its connection to MD, the MDLG rigorously recovers the hydrodynamics and allows to validate the behavior of the lattice-gas or lattice-Boltzmann methods directly without using the standard kinetic theory approach. In this paper, we show that the analytical definition of the equilibrium distribution function remains valid even for very high volume fractions.

Lattice Boltzmann study of two-phase hydrocarbon fluids based on Peng–Robinson free energy model

In this paper, a lattice Boltzmann (LB) model is presented to study the two-phase hydrocarbon fluid systems. Based on the Peng–Robinson (P–R) free energy model, a Cahn–Hilliard type equation is derived to describe the interfacial properties. In the corresponding LB method, the gradient contribution of chemical potential is treated as source term, while the homogeneous part is put in the equilibrium distribution function, which guarantees its scale order in the Chapman–Enskog analysis. In the numerical experiments, the realistic hydrocarbon components of propane are numerically studied by the presented LB model in three dimensions. The numerical results show that the predicted surface tension and capillary pressure are in good agreement with laboratory data.