Fokker-Planck model for computational studies of monatomic rarefied gas flows

In this study, we propose a non-linear continuous stochastic velocity process for simulations of monatomic gas flows. The model equation is derived from a Fokker-Planck approximation of the Boltzmann equation. By introducing a cubic non-linear drift term, the model leads to the correct Prandtl number of 2/3 for monatomic gas, which is crucial to study heat transport phenomena. Moreover, a highly accurate scheme to evolve the computational particles in velocity- and physical space is devised. An important property of this integration scheme is that it ensures energy conservation and honours the tortuosity of particle trajectories. Especially in situations with small to moderate Knudsen numbers, this allows to proceed with much larger time steps than with direct simulation Monte Carlo (DSMC), i.e. the mean collision time not necessarily has to be resolved, and thus leads to more efficient simulations. Another computational advantage is that no direct collisions have to be calculated in the proposed algorithm. For validation, different micro-channel flow test cases in the near continuum and transitional regimes were considered. Detailed comparisons with DSMC for Knudsen numbers between 0.07 and 2 reveal that the new solution algorithm based on the Fokker-Planck approximation for the collision operator can accurately predict molecular stresses and heat flux and thus also gas velocity and temperature profiles. Moreover, for the Knudsen Paradox, it is shown that good agreement with DSMC is achieved up to a Knudsen number of about

Simulation of gas micro flows based on finite element and finite volume method

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.The Regularized 13-Moment Equations state a model for describing rarefied gas flows. The equations are based on a moment approximation of the Boltzmann equation and aim at an accurate prediction up to Knudsen numbers of 0.5. This paper is concerned with the numerical treatment of the PDE system, with special focus on slow flows. Finite elements and finite volumes are applied and the results of both approaches are discussed and the pros and cons are outlined

Modeling of electric arcs: A study of the non-convective case with strong coupling

In this paper, we investigate a mathematical model for electric arcs. The model is based on the equations of magnetohydrodynamics, where the flow and heat transfer in a plasma is coupled to electrodynamics. Our approach neglects convection and yields a reaction-diffusion model that includes only the core phenomena of electric arcs: Ohmic heating and nonlinear electric conductivity. The equations exhibit interesting mathematical properties like non-unique steady states and instabilities that can be linked to electric arc properties. Additionally, a 3D axisymmetric simulation of the creation and extinction of an electric arc is presented based on a strongly coupled numerical algorithm for the non-convective model. The approach is especially suited for high-current arcs where strong coupling becomes necessar

Grad’s Moment Equations for Binary Hard Sphere Gas-Mixtures

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.The derivation of non-linear Grad’s 2×26-moment (2×G26) equations for a binary gasmixture of monatomic-inert-ideal hard sphere gases is sketched, although—for conciseness—only the linear 2×G26 equations are illustrated and analysed. The linear stability analysis is performed on 2×G26 equations by studying the dispersion relation and by considering the plane wave solution, it is shown that the 2×G26 equations for binary hard sphere gas-mixture are linearly stable

Quantum Hydrodynamic Model by Moment Closure of Wigner Equation

In this paper, we derive the quantum hydrodynamics models based on the moment closure of the Wigner equation. The moment expansion adopted is of the Grad type firstly proposed in \cite{Grad}. The Grad's moment method was originally developed for the Boltzmann equation. In \cite{Fan_new}, a regularization method for the Grad's moment system of the Boltzmann equation was proposed to achieve the globally hyperbolicity so that the local well-posedness of the moment system is attained. With the moment expansion of the Wigner function, the drift term in the Wigner equation has exactly the same moment representation as in the Boltzmann equation, thus the regularization in \cite{Fan_new} applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation is turned to be a linear source term, which can only induce very mild growth of the solution. As the result, the local well-posedness of the regularized moment system for the Wigner equation remains as for the Boltzmann equation

Higher order finite difference schemes for the magnetic induction equations

We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. The finite difference schemes are based on Summation by Parts (SBP) operators for spatial derivatives and a Simultaneous Approximation Term (SAT) technique for imposing boundary conditions. We present various numerical experiments that demonstrate both the stability as well as high order of accuracy of the schemes.Comment: 20 page

Moment Closure - A Brief Review

Moment closure methods appear in myriad scientific disciplines in the modelling of complex systems. The goal is to achieve a closed form of a large, usually even infinite, set of coupled differential (or difference) equations. Each equation describes the evolution of one "moment", a suitable coarse-grained quantity computable from the full state space. If the system is too large for analytical and/or numerical methods, then one aims to reduce it by finding a moment closure relation expressing "higher-order moments" in terms of "lower-order moments". In this brief review, we focus on highlighting how moment closure methods occur in different contexts. We also conjecture via a geometric explanation why it has been difficult to rigorously justify many moment closure approximations although they work very well in practice.Comment: short survey paper (max 20 pages) for a broad audience in mathematics, physics, chemistry and quantitative biolog