Model Boltzmann Equation Research Articles

A conservative discrete ordinate method for model Boltzmann equations

A class of conservative discrete ordinate method (C-DOM) for the Bhatnagar–Gross–Krook (BGK) model Boltzmann equation is presented. The C-DOM is the extension of the discrete ordinate method in my previous study (Yang and Huang, 1995 [3]). In the C-DOM, a conservative molecular collision process is employed and thus the conservation properties of the collision integral are maintained at the molecular level. For a broad range of Knudsen number, several test problems, including unsteady shock-tube problem and supersonic/hypersonic flows over circular cylinder, are utilized to demonstrate the performance and validity of the DOM and C-DOM. Results show that the C-DOM can greatly reduce the computer time and memory requirements in hypersonic rarefied gas flow computations.

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.

A NEW MULTI-DIMENSIONAL GENERAL RELATIVISTIC NEUTRINO HYDRODYNAMIC CODE FOR CORE-COLLAPSE SUPERNOVAE. I. METHOD AND CODE TESTS IN SPHERICAL SYMMETRY

We present a new general relativistic (GR) code for hydrodynamic supernova simulations with neutrino transport in spherical and azimuthal symmetry (1D/2D). The code is a combination of the CoCoNuT hydro module, which is a Riemann-solver based, high-resolution shock-capturing method, and the three-flavor, energy-dependent neutrino transport scheme VERTEX. VERTEX integrates the neutrino moment equations with a variable Eddington factor closure computed from a model Boltzmann equation and uses the ray-by-ray plus approximation in 2D, assuming the neutrino distribution to be axially symmetric around the radial direction, and thus the neutrino flux to be radial. Our spacetime treatment employs the ADM 3+1 formalism with the conformal flatness condition for the spatial three-metric. This approach is exact in 1D and has been shown to yield very accurate results also for rotational stellar collapse. We introduce new formulations of the energy equation to improve total energy conservation in relativistic and Newtonian hydro simulations with Eulerian finite-volume codes. Moreover, a modified version of the VERTEX scheme is developed that simultaneously conserves energy and lepton number with better accuracy and higher numerical stability. To verify our code, we conduct a series of tests, including a detailed comparison with published 1D results for stellar core collapse. Long-time simulations of proto-neutron star cooling over several seconds both demonstrate the robustness of the new CoCoNuT-VERTEX code and show the approximate treatment of GR effects by means of an effective gravitational potential as in PROMETHEUS-VERTEX to be remarkably accurate in 1D. (abridged)

Slow evaporation and condensation on a spherical droplet in the presence of a noncondensable gas

A spherical droplet is placed in a binary mixture composed of the vapor of the droplet and another gas which neither evaporates nor condenses (a noncondensable gas). The mixture is in an equilibrium state at rest at infinity. A slow steady flow of the vapor caused by weak evaporation or condensation, under the influence of the noncondensable gas, is investigated on the basis of a linearized model Boltzmann equation. Numerical analyses by means of a finite-difference method are carried out for a wide range of the Knudsen number (i.e., from a large to small droplet compared to the molecular mean free path). The numerical results, together with analytical solutions for small and large Knudsen numbers, clarify the behavior the mixture, i.e., the mass- and heat-flow rates from or onto the droplet as well as spatial distributions of the macroscopic quantities, in the entire range of gas rarefaction. The solution for the steady heat transfer problem between a solid sphere and a binary gas mixture is also obtained as a byproduct.

Velocity inversion of micro cylindrical Couette flow: A lattice Boltzmann study

In this work the micro gas flow between two concentric cylinders is investigated by a lattice Boltzmann equation (LBE) model with multiple relaxation times. A local kinetic boundary condition is proposed for the LBE to model the gas–wall interaction. Numerical simulations are conducted to examine the tangential velocity distribution under different flow conditions. It is shown that the proposed LBE can capture the velocity inversion phenomenon successfully. Comparisons with the Navier–Stokes solutions and DSMC results are also made and it is shown that the LBE yields better predictions.

Finite-difference-based multiple-relaxation-times lattice Boltzmann model for binary mixtures

In this paper, we propose a finite-difference-based lattice Boltzmann equation (LBE) model with multiple-relaxation times (MRT), in which the distribution functions of individual species evolve on a same regular lattice without any interpolations. Furthermore, the use of the MRT enables the model more flexible so that it can be applied to mixtures of species with different viscosities and adjustable Schmidt number. Some numerical tests are conducted to validate the model, the numerical results are found to agree well with analytical solutions/or other numerical results, and good numerical stability of the proposed LBE model is also observed.

Lattice Boltzmann equation for axisymmetric thermal flows

A simple lattice Boltzmann equation (LBE) model for axisymmetric thermal flow is proposed in this paper. The flow field is solved by a quasi-two-dimensional nine-speed (D2Q9) LBE, while the temperature field is solved by another four-speed (D2Q4) LBE. The model is validated by a thermal flow in a pipe and some nontrivial thermal buoyancy-driven flows in vertical cylinders, including Rayleigh–Bénard convection, natural convection, and heat transfer of swirling flows. It is found that the numerical results agree excellently with analytical solution or other numerical results.

Computation of heat transfer and fluid flow in an obstructed channel using lattice Boltzmann method

PurposeThe purpose of this paper is to investigate the laminar flow and heat transfer characteristics in a two‐dimensional horizontal channel with two square blocks placed side‐by‐side using a numerical scheme based on a coupling between the lattice Boltzmann method and the finite difference method.Design/methodology/approachThe multiple‐relaxation‐time (MRT) lattice Boltzmann equation model coupled with the finite difference method are used to predict numerically the velocity and the temperature fields.FindingsA complex structure of the fluid flow was observed for various dimensionless block separation distance (G). An unsteady flow was found when the two blocks are placed side by side (G = 0). For G < 1.5, the presence of each block develops the street of Van Karman which generates complex binary vortex street. In the opposite case (G > 1.5), the effect of this parameter (G) on the fluid is reduced, whereas, the distance between the blocks and the nearest walls have a great influence on the fluid flow and the heat transfer. When the obstacles are posed on the walls (G = 3), an important heat exchange between the blocks and the nearest walls is noted.Originality/valueThis study offers more knowledge on natural convection in an obstructed channel. Furthermore, this work shows the effectiveness of the MRT lattice Boltzmann equation model for this kind of geometry.

Entropy and chaos in the Kac model

We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochasticmodel of Boltzmann dynamics, and its relation to the entropy function forsolutions of Kac's one dimensional nonlinear model Boltzmann equation. We proveresults that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved hereshow that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open and the upper bounds obtained here show that this problem is somewhat subtle.

Acoustic Scattering by a Localized Thermal Disturbance

Direct aeroacoustic simulation, which simulates acoustic radiation associated with unsteady fluid flows, involves capturing waves in both aerodynamic and acoustic scales. The presence of a thermal field further complicates this simulation. There are few analytical thermal-acoustic problems that could serve as benchmark to validate computational aeroacoustics schemes. This paper attempts to seek a theoretical solution of a simple problem involving scattering of acoustic plane waves by a localized zero-heat-gain/loss thermal disturbance and use it as a benchmark to verify the validity and extent of a numerical gas-kinetic scheme based on the modeled Boltzmann equation for the study of aeroacoustics resulting from thermal-flow/acoustic interaction. Two limiting cases with different acoustic wavelengths are attempted. Theoretical solutions are established for both cases, with the center of an axisymmetric thermal disturbance located at the origin of the physical domain. The incident acoustic wave has a dimensionless amplitude given by 1 x 10 -4 and is propagated from left to right of the coordinate system. The simulation of the gas-kinetic scheme is carried out using the same computational setting as for the theoretical solutions. Riemann invariants are stipulated at the open boundaries. Numerical results are compared with the theoretical solutions and good agreement is obtained for the short-wavelength case only. The discrepancy for the long-wavelength case could be partially attributed to the inappropriateness of the Riemann invariants for oblique waves at the boundaries.

Lattice Boltzmann 3D flow simulations on a metallic foam

In this paper a lattice Boltzmann method (LBM) is used to simulate isothermal incompressible flow in a RCM-NCX-1116 metallic foam. The computational technique is a multiple relaxation time (MRT) lattice Boltzmann equation model. Computer aided X-ray micro-tomography is used to obtain 3D images of the metallic foam, providing the geometry and information required for LB simulations of a single phase flow. Pressure drops are computed and successfully compared to experimental measures and correlated with Ergun’s equation. Invariance of Ergun’s parameters A and B with the sampling rate of the images is observed.

A Lattice Boltzmann Method Based Numerical Scheme for Microchannel Flows

Conventional lattice Boltzmann method (LBM) is hyperbolic and can be solved locally, explicitly, and efficiently on parallel computers. The LBM has been applied to different types of complex flows with varying degrees of success, and with increased attention focusing on microscale flows now. Due to its small scale, microchannel flows exhibit many interesting phenomena that are not observed in their macroscale counterpart. It is known that the Navier–Stokes equations can still be used to treat microchannel flows if a slip-wall boundary condition is assumed. The setting of boundary conditions in the conventional LBM has been a difficult task, and reliable boundary setting methods are limited. This paper reports on the development of a finite difference LBM (FDLBM) based numerical scheme suitable for microchannel flows to solve the modeled Boltzmann equation using a splitting technique that allows convenient application of a slip-wall boundary condition. Moreover, the fluid viscosity is accounted for as an additional term in the equilibrium particle distribution function, which offers the ability to simulate both Newtonian and non-Newtonian fluids. A two-dimensional nine-velocity lattice model is developed for the numerical simulation. Validation of the FDLBM is carried out against microchannel and microtube flows, a driven cavity flow, and a two-dimensional sudden expansion flow. Excellent agreement is obtained between numerical calculations and analytical solutions of these flows.

The BGK Model with External Confining Potential: Existence, Long-Time Behaviour and Time-Periodic Maxwellian Equilibria

We study global existence and long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential. For an initial datum f 0≥0 with bounded mass, entropy and total energy we prove existence and strong convergence in L 1 to a Maxwellian equilibrium state, by compactness arguments and multipliers techniques. Of particular interest is the case with an isotropic harmonic potential, in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states. This behaviour is shared with the Boltzmann equation and other kinetic models. For all these systems we study the multistability of the time-periodic Maxwellians and provide necessary conditions on f 0 to identify the equilibrium state, both in L 1 and in Lyapunov sense. Under further assumptions on f, these conditions become also sufficient for the identification of the equilibrium in L 1.

Gas-kinetic numerical schemes for one- and two-dimensional inner flows

Several kinds of explicit and implicit finite-difference schemes directly solving the discretized velocity distribution functions are designed with precision of different orders by analyzing the inner characteristics of the gas-kinetic numerical algorithm for Boltzmann model equation. The peculiar flow phenomena and mechanism from various flow regimes are revealed in the numerical simulations of the unsteady Sod shock-tube problems and the two-dimensional channel flows with different Knudsen numbers. The numerical remainder-effects of the difference schemes are investigated and analyzed based on the computed results. The ways of improving the computational efficiency of the gas-kinetic numerical method and the computing principles of difference discretization are discussed.

DELAYED NEUTRINO-DRIVEN SUPERNOVA EXPLOSIONS AIDED BY THE STANDING ACCRETION-SHOCK INSTABILITY

We present results of 2D hydrodynamic simulations of stellar core collapse, which confirm that the neutrino-heating mechanism remains viable for the explosion of a wider mass range of supernova progenitors with iron cores. We used an energy-dependent treatment of the neutrino transport based on the ray-by-ray plus approximation, in which the number, energy, and momentum equations are closed with a variable Eddington factor obtained by iteratively solving a model Boltzmann equation. We focus on the evolution of a 15 Msun progenitor and show that shock revival and the explosion are initiated at about 600 ms post bounce, powered by neutrino energy deposition. Similar to previous findings for an 11.2 Msun star, but significantly later, the onset of the explosion is fostered by the standing accretion shock instability (SASI). This instability exhibits highest growth rates for the dipole and quadrupole modes, which lead to large-amplitude bipolar shock oscillations and push the shock to larger radii, thus increasing the time accreted matter is exposed to neutrino heating in the gain layer. Therefore also convective overturn behind the shock is strengthened. A soft nuclear equation of state that causes a rapid contraction and a smaller radius of the forming neutron star and thus a fast release of gravitational binding energy, seems to be more favorable for an explosion. Rotation has the opposite effect because it leads to a more extended and cooler neutron star and thus lower neutrino luminosities and mean energies and overall less neutrino heating. Neutron star g-mode oscillations and the acoustic mechanism play no important role in our simulations. (abridged)

Lattice Boltzmann equation for microscale gas flows of binary mixtures

Modeling and simulating gas flows in and around microdevices are a challenging task in both science and engineering. In practical applications, a gas is usually a mixture made of different components. In this paper we propose a lattice Boltzmann equation (LBE) model for microscale flows of a binary mixture based on a recently developed LBE model for continuum mixtures [P. Asinari and L.-S. Luo, J. Comput. Phys. 227, 3878 (2008)]. A consistent boundary condition for gas-solid interactions is proposed and analyzed. The LBE is validated and compared with theoretical results or other reported data. The results show that the model can serve as a potential method for flows of binary mixture in the microscale.

Study on the unified algorithm for three-dimensional complex problems covering various flow regimes using Boltzmann model equation

The Boltzmann simplified velocity distribution function equation describing the gas transfer phenomena from various flow regimes will be explored and solved numerically in this study. The discrete velocity ordinate method of the gas kinetic theory is studied and applied to simulate the complex multi-scale flows. Based on the uncoupling technique on molecular movement and colliding in the DSMC method, the gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by extending and applying the unsteady time-splitting method from computational fluid dynamics. The Gauss-type discrete velocity numerical quadrature technique for different Mach number flows is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established to study the three-dimensional complex flows from rarefied transition to continuum regimes. The parallel strategy adapted to the gas-kinetic numerical algorithm is investigated by analyzing the inner parallel degree of the algorithm, and then the HPF parallel processing program is developed. To test the reliability of the present gas-kinetic numerical method, the three-dimensional complex flows around sphere and spacecraft shape with various Knudsen numbers are simulated by HPF parallel computing. The computational results are found in high resolution of the flow fields and good agreement with the theoretical and experimental data. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view of solving the Boltzmann model equation.

A Discrete Boltzmann Equation Based on a Cub-Octahedron in $\mathbb{R}^3$

In this article we introduce a discrete Boltzmann equation in $\mathbb{R}^3$ velocity space based on a cub-octahedron. We present a layer-wise construction of a regular collision model in a bounded grid of the sphere packing problem. The paper contains the analytical results about the basic kinetic features of the regular collision model. We analyze the computational complexity, report the efficiency of the model, and perform numerical experiments based on the discrete model Boltzmann equation for some standard test problem.

Modeled Boltzmann Equation and Its Application to Shock-Capturing Simulation

A modified equilibrium distribution function for the Bhatnagar-Gross-Krook-type modeled Boltzmann equation has recently been proposed. The function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized Euler equations. It is a combination of a Maxwellian distribution plus three other terms that are moments of particle velocity. The lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the Euler equations; therefore, there is no Mach number limitation in the entire approach. This lattice counterpart was able to replicate aeroacoustics problems involving vorticity-acoustic and entropy-acoustic interactions correctly, and the simulations were carried out using a finite difference lattice Boltzmann method employing only a two-dimensional, nine-velocity lattice. Thus formulated, the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis. This paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one- and two-dimensional Riemann problems. The simulations are carried out using the same two-dimensional, nine-velocity lattice, and identical set of constants and relaxation time; they are compared with theoretical results and those obtained by solving the Euler equations directly using Harten's first-order numerical scheme. Good agreement is obtained for all test cases. However, the modified equilibrium distribution function is not suitable for shock structure simulation; for that, an exact recovery of the Navier-Stokes equations is required.

Gas-kinetic numerical studies of three-dimensional complex flows on spacecraft re-entry

The gas-kinetic numerical algorithm solving the Boltzmann model equation is extended and developed to study the three-dimensional hypersonic flows of spacecraft re-entry into the atmosphere in perfect gas. In this study, the simplified velocity distribution function equation for various flow regimes is presented on the basis of the kinetic Boltzmann–Shakhov model. The discrete velocity ordinate technique and numerical quadrature methods, such as the Gauss quadrature formulas with the weight function 2/ π 1/2exp(− V 2) and the Gauss–Legendre numerical quadrature rule, are studied to resolve the barrier in simulating complex flows from low Mach numbers to hypersonic problems. Specially, the gas-kinetic finite-difference scheme is constructed for the computation of three-dimensional flow problems, which directly captures the time evolution of the molecular velocity distribution function. The gas-kinetic boundary conditions and numerical procedures are studied and implemented by directly acting on the velocity distribution function. The HPF (high performance fortran) parallel implementation technique for the gas-kinetic numerical method is developed and applied to study the hypersonic flows around three-dimensional complex bodies. The main purpose of the current research is to provide a way to extend the gas-kinetic numerical algorithm to the flow computation of three-dimensional complex hypersonic problems with high Mach numbers. To verify the current method and simulate gas transport phenomena covering various flow regimes, the three-dimensional hypersonic flows around sphere and spacecraft shape with different Knudsen numbers and Mach numbers are studied by HPF parallel computing. Excellent results have been obtained for all examples computed.