Abstract
It has been shown recently that the Fokker-Planck kinetic model can be employed as an approximation of the Boltzmann equation for rarefied gas flow simulations [4, 5, 10]. Similar to the direct simulation Monte-Carlo (DSMC), the Fokker-Planck solution algorithm is based on the particle Monte-Carlo representation of the distribution function. Yet opposed to DSMC, here the particles evolve along independent stochastic paths where no collisions need to be resolved. This leads to significant computational advantages over DSMC, considering small Knudsen numbers [10]. The original Fokker-Planck model (FP) for rarefied gas flow simulations was devised according to the Maxwell type pseudo-molecules [4, 5]. In this paper a consistent Fokker-Planck equation is derived based on the Boltzmann collision integrals and maximum entropy distribution. Therefore the resulting model fulfills the H-theorem and leads to correct relaxation of velocity moments up to heat fluxes consistent with hard sphere interactions. For assessment of the model, simulations are performed for Mach 5 flow around a vertical plate using both Fokker-Planck and DSMC simulations. Compared to the original FP model, significant improvements are achieved at high Mach flows.
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Abstract
The Boltzmann equation is often used to model rarefied gas flow in the transition or kinetic regime for moderate to large Knudsen numbers. However, standard moment methods like Grad’s approach lack hyperbolicity of the equations. We point out the failure of Grad’s method and overcome the deficiencies with the help of the new hyperbolic moment models called QBME and HME, derived by an operator projection framework. The new model equations are in partially-conservative form meaning that a subset of the equations cannot be written in conservative form due to some changes in these equations. This leads to additional numerical difficulties. The influence of the partially-conservative terms on the solution is analyzed and we present a numerical scheme for the solution of the partially-conservative PDE systems, namely the PRICE-C scheme by Canestrelli. Furthermore, a shock structure test case is used to compare the accuracy of the different hyperbolic moment models to a discrete velocity reference solution. The results show that the new hyperbolic models achieve higher accuracy than the standard Grad model despite the fact that the model equations cannot be fully written in conservative form.
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Abstract
For any study of microflows it is crucial to understand the collision dynamics of the molecules involved. In the present work we will discuss the collision dynamics of chemically reacting hard spheres(CRHS). The inability of the classical smooth inelastic hard spheres, which have been extensively used in the past to study granular gases, to describe the collision dynamics of chemically reacting hard spheres has been discussed. Using the model of rough inelastic hard spheres as a motivation, a new model has been proposed for chemically reacting hard spheres which has been further used to derive certain useful velocity transformations. A methodology to compute the moments of the Boltzmann’s collision operator arising from chemical reactions, using Grad’s distribution function, has been discussed in detail. Finally explicit expressions for the rates of the reaction have been obtained which contain contributions from higher order moment and thus can be used for non-equilibrium chemically reacting flows.
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Abstract
The applicability of the order of magnitude method [H. Struchtrup, “Stable transport equations for rarefied gases at high orders in the Knudsen number,” Phys. Fluids 16, 3921–3934 (2004)] is extended to binary gas mixtures in order to derive various sets of equations—having minimum number of moments at a given order of accuracy in the Knudsen number—for binary mixtures of monatomic-inert-ideal gases interacting with the Maxwell interaction potential. For simplicity, the equations are derived in the linear regime up to third order accuracy in the Knudsen number. At zeroth order, the method produces the Euler equations; at first order, it results into the Fick, Navier–Stokes, and Fourier equations; at second order, it yields a set of 17 moment equations; and at third order, it leads to the regularized 17-moment equations. The transport coefficients in the Fick, Navier–Stokes, and Fourier equations obtained through order of magnitude method are compared with those obtained through the classical Chapman–Enskog expansion method. It is established that the different temperatures of different constituents do not play a role up to second order accurate theories in the Knudsen number, whereas they do contribute to third order accurate theory in the Knudsen number. Furthermore, it is found empirically that the zeroth, first, and second order accurate equations are linearly stable for all binary gas mixtures; however, although the third order accurate regularized 17-moment equations are linearly stable for most of the mixtures, they are linearly unstable for mixtures having extreme difference in molecular masses.
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Abstract
This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number—the ratio between the mean free path of the particles and a macroscopic length scale—becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann’s equation in kinetic gas theory. With moment approximations, extended fluid dynamic equations can be derived, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low-Mach number setting for which both evolution equations and boundary conditions are well established. Conversely, nonlinear, high-speed processes require special closures that are still under development. Current approaches are examined, along with the challenge of computing shock wave profiles based on continuum equations.
Reference
Annu. Rev. Fluid Mech., 48, Annual Reviews, Palo Alto, CA, 2016
Abstract
By a further study of the mechanism of the hyperbolic regularization of the moment system for the Boltzmann equation proposed in Cai et al. (Commun Math Sci 11(2):547–571, 2013), we point out that the key point is treating the time and space derivative in the same way. Based on this understanding, a uniform framework to derive globally hyperbolic moment systems from kinetic equations using an operator projection method is proposed. The framework is so concise and clear that it can be treated as an algorithm with four inputs to derive hyperbolic moment systems by routine calculations. Almost all existing globally hyperbolic moment systems can be included in the framework, as well as some new moment systems including globally hyperbolic regularized versions of Grad’s ordered moment systems and a multi-dimensional extension of the quadrature-based moment system.
Reference
J. Stat. Phys. 162 (2016), no. 2, 457–486
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Abstract
Because of stability constraints, most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar. This problem emerges with the $M_1$ system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the $M_1$ system is non-linear, and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability. In this paper, we propose a numerical method that overcomes the stability constraint and preserves the realizability property. For this purpose, we relax the $M_1$ system to obtain a linear flux term. Then we extend the stencil of the difference quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation example.
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Abstract
We derive Lattice Boltzmann (LBM) schemes to solve the Linearized Euler Equations in 1D, 2D, and 3D with the future goal of coupling them to an LBM scheme for Navier Stokes Equations and a Finite Volume scheme for Linearized Euler Equations. The derivation uses the analytical Maxwellian in a BGK model. In this way, we are able to obtain second-order schemes. In addition, we perform an $L^2$-stability analysis. Numerical results validate the approach.
Reference
Comput. Math. Appl. 72 (2016), no. 2, 311–327
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Abstract
The transport of photons and electrons is studied in the field of radiotherapy to compute the dose, that is, the quantity of energy transferred to the medium by a beam of particles at each position. A kinetic model is proposed, and to decrease the computation times, it is reduced through a moment extraction. Entropy-based angular moment models of order up to two ($M_1$ and $M_2$ models) are shown to provide accurate results compared to a reference code with much lower computational costs.