Moments Of Distribution Function Research Articles

Central-moment discrete unified gas-kinetic scheme for incompressible two-phase flows with large density ratio

In this paper, we proposed a central moment discrete unified gas-kinetic scheme (DUGKS) for multiphase flows with large density ratio and high Reynolds number. Two sets of kinetic equations with central-moment-based multiple relaxation time collision operator are employed to approximate the incompressible Navier-Stokes equations and a conservative phase field equation for interface-capturing, respectively. In the framework of DUGKS, the first moment of the distribution function for the hydrodynamic equations is defined as velocity instead of momentum. Meanwhile, the zeroth moments of the distribution function and external force are carefully defined such that a artificial pressure evolution equation can be recovered. Moreover, the Strang splitting technique for time integration is employed to avoid the calculation of spatial derivatives in the force term at cell faces. For the interface-capturing equation, two equivalent DUGKS methods that deal with the diffusion term differently using a source term as well as a modified moments of the equilibrium distribution function are presented. Several benchmark tests that cover a wide a range of density ratios (up to 1000) and Reynolds numbers (up to 105) are subsequently carried out to demonstrate the capabilities of the proposed scheme. Numerical results are in good agreement with the theoretical and experimental data.

Multiple-distribution-function lattice Boltzmann method for convection-diffusion-system-based incompressible Navier-Stokes equations.

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equationswhich are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equationscan be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity. Finally, we also conduct some simulations to test the MDF-LBM and find that the numerical results not only agree with some available analytical and numerical solutions but also have a second-order convergence rate in space.

Conservative multilevel discrete unified gas kinetic scheme for modeling multiphase flows with large density ratios

A novel multilevel discrete unified gas kinetic scheme (MDUGKS) is proposed to efficiently model multiphase flows with large density ratios. By constructing the kinetic flux with a mutable time interval depending on the local mesh spacing, the MDUGKS overcomes the stability problems encountered by the standard DUGKS when operating with the multilevel mesh. With the interpolation of macroscopic variables and mesoscopic distributions handled separately, the moments of distribution functions are maintained consistent with the conservative flow variables. Two kinetic equations corresponding to the conservative Allen–Cahn equation and the hydrodynamic equation are individually solved by the MDUGKS, and six benchmark problems have been conducted to evaluate its performance. Numerical solutions in steady cases produced by the MDUGKS are in well accordance with the theoretical predictions. A limiting density ratio of 109 is achieved in the quiescent droplet. The dynamic processes in unsteady cases anticipated by the MDUGKS agree well with the reference predictions. Comparative results also demonstrate that the MDUGKS behaves consistently with different types of meshes. With the employment of the adaptive multilevel mesh, 80% improvement in computational efficiency could be achieved compared with the uniform mesh. Considering the kinetic nature and the high efficiency, the MDUGKS offers a powerful tool for presenting meaningful insight into understanding the realistic multiphase systems at the mesoscopic scale.

Variant of gas kinetic flux solver for flows beyond Navier-Stokes level.

In this paper, a variant of gas kinetic flux solver (GKFS) is presented for simulation of flows beyond the Navier-Stokes (NS) level. The method retains the framework of GKFS and reconstructs the numerical fluxes by the moments of distribution function at the cell interface, which is given from the local solution of the Boltzmann equation. In the conventional GKFS, the first-order Chapman-Enskog (CE) expansion is utilized to approximate the initial distribution function. By using the differential chain rule, it was found that the CE expansion form could be linked to the stress tensor and the heat flux. For flows in the NS level, the stress tensor and heat flux can be simply calculated from the linearized constitutive relationship and Fourier's law, respectively. However, for flows beyond the NS level, due to the strong nonequilibrium effect, the linearized constitutive relationship and Fourier's law are insufficient to predict the stress tensor and the heat flux. To overcome this difficulty, this paper introduces correction terms to the stress tensor and heat flux in the initial distribution function. These correction terms will take effect in the strong nonequilibrium region for flows beyond the NS level. To avoid finding complex expressions or solving complicated partial differential equations for the correction terms, a simple and iterative procedure is proposed to update the correction terms based on the framework of GKFS. The proposed method is validated by three benchmark cases which cover the flow from the continuum regime to the transition regime. Numerical results show that the present solver can provide accurate solution in the continuum regime. It is indeed the correction terms that take effect in the strong nonequilibrium region for flows beyond the NS level, which enables the present solver to capture the nonequilibrium phenomenon with reasonable accuracy for rarefied flows at moderate Knudsen number.

Analysis of the ferrofluid microstructure based on the static magnetic measurements

In this work, we applied regularized inversion method [New J. Phys. 19, 073012 (2017)] for microstructure analysis of 10 ferrofluid samples with different ferroparticle concentrations, which were obtained by diluting of the first one. The experimental measurements of the magnetization M(H) of all samples were subjected to numerical inversion using 120 ferroparticle fractions. Since volume concentration of ferrofluid samples does not exceed 3%, the properties of each fraction were described using Langevin theory. The magnetogranulomertic analysis confirmed that all 10 samples have the same magnetic moment distribution function, which has two peaks. The first (small) peak in the magnetic moment distribution function demonstrates the presence of a certain number of aggregates or correlated ferroparticle structures in the samples, whose magnetic moments have been compensated. The second peak corresponds to ordinary single particles distributed in the sample. The main advantage of using numerical inversion method is the absence of an a priori fixed form of the distribution function instead of a standard log-normal or Γ-distribution.

A novel solver for simulation of flows from continuum regime to rarefied regime at moderate Knudsen number

A novel and simple solver for simulation of flows from continuum regime to rarefied regime at moderate Knudsen number is developed in this work. The present solver combines good features of gas kinetic flux solver (GKFS), discrete velocity method (DVM) and the moment method. Like the GKFS, in the present solver, the macroscopic governing equations are discretized by finite volume method and the numerical fluxes at cell interfaces are evaluated by the local solution of Boltzmann equation. To get the local solution of Boltzmann equation, the initial distribution function is reconstructed with the help of Grad's distribution function, which is inspired from the moment method. For the high order moments in Grad's distribution function, they are computed directly by the moments of distribution function, which is inspired from DVM. In principle, the present solver only needs to discretize the physical space and solves the governing equations resulted from conservation laws of mass, momentum and energy. Thus, its governing equations are much simpler than those of the moment method. To validate the proposed solver, some test examples covering continuum regime and rarefied regime are simulated. Numerical results showed that the present solver can give accurate prediction in the continuum regime and reasonable results as the traditional moment method in the rarefied regime at moderate Knudsen number.

Graphical Magnetogranulometry of EMG909

The magnetization curve of the commercially available ferrofluid EMG909 is measured. It can adequately be described by a superposition of four Langevin terms. The effective dipole strength of the magnetic particles in this fluid is subsequently obtained by a graphical rectification of the magnetization curve based on the inverse Langevin function. The method yields the arithmetic and the harmonic mean of the magnetic moment distribution function, and a guess for the geometric mean and the relative standard deviation. It has the advantage that it does not require a prejudiced guess of the distribution function of the poly-disperse suspension of magnetic particles.

An adjoint method for neoclassical stellarator optimization

Stellarators are a promising route to steady-state fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating high-dimensional spaces, often necessitating the use of gradient-based methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring $O(N)$ flux computations, where $N$ is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finite-collisionality neoclassical fluxes can be used within an optimization loop. With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large ( $O(10^{2})$ ), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjoint-based optimization of the field strength to obtain minimal bootstrap current on a surface. With adjoint-based derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Furthermore, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.

Bilinear plasma susceptibility of strongly non-uniform magnetized plasmas

The bilinear plasma susceptibility describing the nonlinear coupling of electromagnetic and two electrostatic waves in strongly non-uniform magnetized plasmas is derived using both the distribution function moments equations and the kinetic approach. The possibility of the drastic enhancement of nonlinear coupling leading to a decrease of the two-plasmon decay instability threshold in strongly inhomogeneous plasmas is demonstrated.

Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

SummaryIn the conventional discrete velocity method (DVM), the local solution of collisionless Boltzmann equation with a piecewise constant distribution for the distribution function is utilized to reconstruct distribution function at the cell interface and then calculate numerical flux of Boltzmann equation for updating the distribution function at cell center. In this process, a numerical dissipation will be introduced into the solution due to neglecting of the collision effect at the cell interface. This numerical dissipation may deteriorate the solution accuracy of conventional DVM in the continuum flow regime, in which the particle collision happens frequently. To overcome this defect, two improved schemes are first presented in this work, in which the local discrete solution of Boltzmann equation with Shakhov model is adopted to evaluate the distribution function at the cell interface, while the equilibrium state of the local solution is computed by different ways. One of the improved schemes evaluates the equilibrium state exactly by the moments of distribution functions according to the compatibility condition, while the other computes the equilibrium state approximately by a simple average at the cell interface. Since the collision effect is incorporated in evaluation of numerical flux, the improved schemes can provide reasonable solutions in all flow regimes. On the other hand, they introduce some extra computational efforts for determining the collision term at the cell interface as compared with the conventional DVM. To assess the performance of different methods for simulation of flows in all flow regimes, a comprehensive study is then carried out in this work.

The non-equilibrium attractor for kinetic theory in relaxation time approximation

I demonstrate that the concept of a non-equilibrium attractor can be extended beyond the lowest-order moments typically considered in hydrodynamic treatments. Using a previously obtained exact solution to the relaxation-time approximation Boltzmann equation for a transversally homogeneous and boost-invariant system subject to Bjorken flow, I derive an equation obeyed by all moments of the one-particle distribution function. Using numerical solutions, I show that, similar to the pressure anisotropy, all moments of the distribution function exhibit attractor-like behavior wherein all initial conditions converge to a universal solution after a short time with the exception of moments which are sensitive to modes with zero longitudinal momentum and high transverse momentum. In addition, I compute the exact solution for the distribution function itself on very fine lattices in momentum space and demonstrate that (a) an attractor for the full distribution function exists and (b) solutions with generic initial conditions relax to this solution, first at low momentum and later at high momentum.

DETERMINING OF LANE CAPACITY OF INTERCITY ROADS AND URBAN STREETS

The capacity of intercity roads and urban streets is an important basic input required for planning, analysis and operation of roadway systems. It is used when choosing rational options in organizing traffic and for solving other issues concerning intercity roads and urban streets development planning. There are many mathematical models for calculating lane capacity but they are not an actual assessment of transport facilities that can be used in relevant traffic calculations. Goal. It is required to develop a new effective method which is based on objective characteristics of the traffic flow with obvious assumptions and provides an opportunity for enough accurate and indisputable forecasting of capacity on the benchmark lane. Methodology. The goal can be achieved by abandoning attempts to obtain the exact point value in favor of an interval assessment of lane capacity. At the same time, the basis should be drivers’ decision-making regarding the car speed and the distance from the vehicle ahead in dense traffic flow. Then the distance from the vehicle ahead is random while the car speed can be viewed as an argument of distribution function moments of random distance. Results. The received interval assessment of the lane capacity has wide possibilities for tuning to real parameters of the traffic and can take into account peculiarities of an operation of each separate road lane and its role in the transport system. Originality. The new method of the interval assessment of road lane capacity which is based on random distances between vehicles in the traffic is developed. Practical value. The developed method creates opportunities for determining not only the interval, but also an accurate estimation of the lane capacity when specifying the parameters of the standard traffic.

Laws of Random Walks Derived by A.N. Kolmogorov in 1934

In the first half of the 1930s A.N. Kolmogorov was developing analytical methods for the probability theory and presented the solution of the Fokker–Planck type equation. This solution contains scales for the distribution function moments of the mean squares for velocities and relative displacements of the analyzed objects and for the mixed moments of velocities and coordinates. The exclusion of time from these moments leads to the 2/3 law for the velocity structure function and to the Richardson–Obukhov law for the eddy diffusion. The analysis of the fetch laws for wind waves demonstrates that the Kolmogorov laws are manifested in the growth of wave amplitudes and in the form of elevation spectra. These laws also work in the statistics of the planetary surface relief, in the size distribution of the lithospheric plates, in the energy spectra of cosmic rays, and in other processes. In the equation deduced in 1934, probability distribution functions are derived only under the condition of homogeneity of these functions and thereby allow describing a broad range of phenomena and processes.

Onset of hydrodynamics for a quark-gluon plasma from the evolution of moments of distribution functions

The pre-equilibrium evolution of a quark-gluon plasma produced in a heavy-ion collision is studied in the framework of kinetic theory. We discuss the approach to local thermal equilibrium, and the onset of hydrodynamics, in terms of a particular set of moments of the distribution function. These moments quantify the momentum anisotropies to a finer degree than the commonly used ratio of longitudinal to transverse pressures. They are found to be in direct correspondence with viscous corrections of hydrodynamics, and provide therefore an alternative measure of these corrections in terms of the distortion of the momentum distribution. As an application, we study the evolution of these moments by solving the Boltzmann equation for a boost invariant expanding system, first analytically in the relaxation time approximation, and then numerically for a quark-gluon plasma with a collision kernel given by leading order 2 ↔ 2 QCD matrix elements in the small angle approximation.

Exact collisional moments for plasma fluid theories

The velocity-space moments of the often troublesome nonlinear Landau collision operator are expressed exactly in terms of multi-index Hermite-polynomial moments of distribution functions. The collisional moments are shown to be generated by derivatives of two well-known functions, namely, the Rosenbluth-MacDonald-Judd-Trubnikov potentials for a Gaussian distribution. The resulting formula has a nonlinear dependency on the relative mean flow of the colliding species normalised to the root-mean-square of the corresponding thermal velocities and a bilinear dependency on densities and higher-order velocity moments of the distribution functions, with no restriction on temperature, flow, or mass ratio of the species. The result can be applied to both the classic transport theory of plasmas that relies on the Chapman-Enskog method, as well as to derive collisional fluid equations that follow Grad's moment approach. As an illustrative example, we provide the collisional ten-moment equations with exact conservation laws for momentum- and energy-transfer rates.

Comparative study of discrete velocity method and high-order lattice Boltzmann method for simulation of rarefied flows

Abstract A comparative study of discrete velocity method (DVM) and high-order lattice Boltzmann method (HLBM) is made in this work. The key difference between the DVM and HLBM lies in the use of equilibrium distribution function and discrete velocity. In the DVM, the complete Maxwellian distribution function or the Shakhov distribution function is used and the discrete velocity is to represent the continuous velocity domain in the particle velocity space. As a result, the integral forms for the moments of distribution function are approximated by numerical integration. In order to make the quadrature error to be small enough, a large number of discrete velocity points are required for the DVM in general case. In contrast, in the HLBM, the lattice Boltzmann model is utilized as the equilibrium distribution function and discrete velocity. The construction of lattice Boltzmann model is intended to seek a minimal set of velocities in the particle velocity space to represent the objective hydrodynamics. With lattice properties of the lattice Boltzmann model, the weighted sums of distribution functions at those velocity points can exactly satisfy the physical conservation laws. However, the choice of discrete velocity points may have considerable effect on the accuracy of HLBM for fluid flows with large Knudsen number as reported in literatures [26, 29–30, 34–36]. From this study, it was found that HLBM provides reasonable results for fluid flows at moderate Knudsen number and low temperature difference with less computational cost than that of DVM. For fluid flows with large Knudsen number and/or high temperature difference, the DVM is preferred although more computational cost is required.

Kinetics theory of shock-induced structural heterogenization

In considering the shock wave propagation in solid, instead of traditional constitutive equation, the lockingof balance equations is conducted by using the mesoparticle velocity distribution function. In this approach, the dynamically deformed material is considered to be the stochastic medium and the particle velocity and velocity dispersion are determined as the first and second statistical moments of distribution function. Meso-macro- energy exchange is found to be realized through change of the particle velocity dispersion – the second moment of distribution function. The criterion for transition from evolutional regime of meso-macro energy exchange to heterogenization is found to be determined by the ratio of rate of change the velocity dispersion and particle velocity acceleration.

Electrostatic thermal noise in a weakly ionized collisional plasma

AbstractQuasi‐thermal noise (QTN) spectroscopy is a plasma diagnostic technique which enables precise measurements of local electron velocity distribution function moments. This technique is based on measurements and analysis of voltage fluctuations at the antenna terminals, induced by thermal motion of charged particles. In this work, we accommodate, for the first time, this technique to weakly ionized collisional plasmas. It turns out that the QTN spectrum is modified both at low frequencies, increasing the level of power spectrum, and around the plasma frequency, where collisions damp the plasma oscillations and therefore broaden and reduce the amplitude of so called “plasma peak,” while the spectrum at high frequencies is nearly unmodified compared to the collisionless case. Based on these results, we show that QTN spectroscopy enables independent measurements of the collision frequency, electron density, and temperature, provided the ratio of collision frequency to plasma frequency is ν/ωp∼0.1. The method presented here can be used for precise estimation of plasma parameters in laboratory devices and unmagnetized ionospheres, while application in the ionosphere of Earth is possible but limited to small, low‐frequency range due to magnetic field influence.

Asymmetry of velocity distribution function and inhomogeneity-induced flow associated with neutral depletion structure in an ECR plasma

A neutral depletion structure with strong inhomogeneity in the radial direction has been observed in an electron cyclotron resonance plasma. We have measured the velocity distribution function of neutrals with a high resolution laser-induced fluorescence system and examined the relationship between asymmetry of distribution function and flow induced by inhomogeneity. It has been revealed that the third order moment of distribution function, that is, skewness, is proportional to the inhomogeneity-induced flow, and a simple relation between the skewness and the normalized flow velocity has been obtained and confirmed in the experiment.

Magnetic properties of heat treated bacterial ferrihydrite nanoparticles

The magnetic properties of ferrihydrite nanoparticles, which are products of vital functions of Klebsiella oxitoca bacteria, have been studied. The initial powder containing the nanoparticles in an organic shell was subjected to low-temperature (T=160°C) heat treatment for up to 240h. The bacterial ferrihydrite particles exhibit a superparamagnetic behavior. Their characteristic blocking temperature increases from 26 to 80K with the heat treatment. Analysis of the magnetization curves with regard to the magnetic moment distribution function and antiferromagnetic contribution shows that the low-temperature heat treatment enhances the average magnetic moment of a particle; i.e., the nanoparticles coarsen, probably due to their partial agglomeration during heat treatment. It was established that the blocking temperature nonlinearly depends on the particle volume. Therefore, a model was proposed that takes into account both the bulk and surface magnetic anisotropy. Using this model, the bulk and surface magnetic anisotropy constants KV≈1.7×105erg/cm3 and KS≈0.055erg/cm2 have been determined. The effect of the surface magnetic anisotropy of ferrihydrite nanoparticles on the observed magnetic hysteresis loops is discussed.