Second-order Transport Equation Research Articles

Charge–velocity correlation transport equations in gas–solid flow with triboelectric effects

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

Convergence behavior of second-order transport equations in near-void problems

In this paper, a newly developed second-order least-squares form of the SN equations is tested in 3-D with both thin-slab problems and the PECOS vehicle re-entry problem. Its performance in voids and near-voids is then compared to that of the SPN, even-parity SN, and self-adjoint angular flux (SAAF) methods through convergence tests in these two problems. A rebalance scheme is also proposed for the least-squares SN to regain particle balance for this generally non-conservative form. Our results indicate that the least-squares SN method is capable of producing accurate results in these kinds of problems where the SPN, even-parity SN fail to produce trustworthy results. While SAAF can produce results comparable with least-squares SN, it can be ill-posed in some extreme cases.

Relativistic second order dissipative hydrodynamics from an effective fugacity quasiparticle model

In this work, a second-order relativistic viscous hydrodynamic model has been presented based on the effective fugacity quasiparticle model (EQPM). The hydrodynamic model has been derived from the effective relativistic second-order transport equation under the EQPM for a multiparticle (two-component) system and solving it in Grad's 14-moment method. The EQPM describes the strongly interacting thermal system of QCD interactions through its fugacity parameters extracted from updated lattice equations of state. The proper time evolution of temperature and pressure anisotropy is observed to be affected significantly due to the inclusion of the EQPM compared to an ideal system.

Modeling and analysis of a two-phase thin film model with insoluble surfactant

In this paper we consider a two-phase thin film consisting of two immiscible viscous fluids endowed with a layer of insoluble surfactant on the surface of the upper fluid. The governing equations for the two film heights and the surfactant concentration are derived using a lubrication approximation. Taking gravitational forces into account but neglecting capillary effects, the resulting system of evolution equations is parabolic, strongly coupled, of second order and degenerated in the equations for the two film heights. Incorporating on the contrary capillary forces and neglecting the effects of gravitation, the system of evolution equations is parabolic, degenerated and of fourth-order for the film heights, strongly coupled to a second-order transport equation for the surfactant concentration. Local well-posedness and asymptotic stability are shown for both systems.

An examination of the high-order statistics of developing jets, lazy and forced plumes at various axial distances from their source

In this study, direct numerical simulations of a turbulent free jet (Re = 2000), a lazy plume (), and a forced plume (Re = 1684, Ri = 0.025) are compared. The evolution of the various fluxes and the so-called source parameter, Γ, are examined as a function of distance from the source. The first-, second-, and third-order statistics of the flows are calculated and discussed. The radial profiles of such statistics, as well as that of the turbulent kinetic energy balance and other second-order transport equations are examined at two axial distances, one axial distance before the flows have adjusted to their similarity solution, and the other beyond the similarity adjustment length scale. Vortical structures are visualised and discussed along with entrainment. The source term Γ was not found to monotonically decrease with axial distance from the source as predicted by past researchers. While the mean flow and turbulent velocity statistics of the simulated lazy and forced plumes took on similar behaviour far from the sources turbulent statistics which involve buoyancy did not.

Wiener chaos and uniqueness for stochastic transport equation

We prove a uniqueness result for the stochastic transport linear equation (STLE), without any W 1 , 1 or BV hypothesis on the coefficient, which is needed for the corresponding deterministic equation. We use Wiener chaos decomposition to pass from the STLE to a deterministic second-order transport equation with uniqueness property.

Much ado about nothing: response matrices for void regions

Weak form Galerkin approaches are examined for obtaining response matrices for void regions – that is regions where nothing is present. The need arises in spherical harmonics methods based on second-order forms of the neutron transport equation; the methods fail in voids because the cross-section appearing in the equation’s denominator then vanishes. The diffusion approximation, being the lowest-order spherical harmonic method, is first employed as a vehicle for examining response matrices derived from both primal and dual weak forms of the mixed-first-order and second-order transport equations. Those for which discretization results in singular matrix equations as the cross-section goes to zero are rejected. First-order- mixed formulations with modified natural boundary conditions are shown to lead to nonsingular response matrices for voids. The primal method is chosen as the better of the two candidates for generalization from the diffusion to the transport equations, and the transport formulation is presented.

Tunneling splittings in vibrational spectra of non-rigid molecules: IX. Malonaldehyde and its isotopomers as a test case for fully coupled multidimensional tunneling dynamics

Twenty one dimensional potential energy surfaces (PES) and the tunneling coordinate dependent kinematic matrices of malonaldehyde and of several of its isotopomers (D, 13C) are constructed in the low-energy region (<3000 cm −1) using quantum-chemical data. Even though the barrier heights calculated with different methods differ strongly (from 2.8 to 10.3 kcal mol −1), all PES become near identical after scaling, fitting only one semiclassical parameter γ, which defines the scales of potential, energy and action. The results of dynamical calculations carried out in dimensionless variables are therefore quite insensitive to the choice of the quantum-chemical method used for the construction of the Hamiltonian. Choosing the value of γ such that the calculated tunneling splitting in the ground state coincides with the experimental value, the corresponding barrier height is determined as 4.30 kcal mol −1 and the vibrational spectrum of the transition state is obtained. The perturbative instanton approach developed in the previous papers of this series is used to solve the dynamical problem without reducing the number of degrees of freedom. The role of all 20 transverse vibrations in proton tunneling is characterized. The tunneling path and globally uniform semiclassical wave functions are evaluated from the fourth-order Hamilton–Jacobi equation and the second-order transport equation. Tunneling splittings in the ground and low-lying excited states are calculated and isotope effects of H/D and 13C/12C substitutions are predicted.

A connection between the Lagrangian stochastic–convective and cumulant expansion approaches for describing solute transport in heterogeneous porous media

The equation describing the ensemble-average solute concentration in a heterogeneous porous media can be developed from the Lagrangian (stochastic–convective) approach and from a method that uses a renormalized cumulant expansion. These two approaches are compared for the case of steady flow, and it is shown that they are related. The cumulant expansion approach can be interpreted as a series expansion of the convolution path integral that defines the ensemble-average concentration in the Lagrangian approach. The two methods can be used independently to develop the classical form for the convection–dispersion equation, and are shown to lead to identical transport equations under certain simplifying assumptions. In the development of such transport equations, the cumulant expansion does not require a priori the assumption of any particular distribution for the Lagrangian displacements or velocity field, and does not require one to approximate trajectories with their ensemble-average. In order to obtain a second-order equation, the cumulant expansion method does require truncation of a series, but this truncation is done rationally by the development of a constraint in terms of parameters of the transport field. This constraint is less demanding than requiring that the distribution for the Lagrangian displacements be strictly Gaussian, and it indicates under what velocity field conditions a second-order transport equation is a reasonable approximation.

Investigation of heat transfer Prandtl number dependence by [formula omitted] and [formula omitted] solutions

On the basis of second-order transport equations of velocity and temperature fluctuations, we present a numerical analysis of axial and radial turbulent heat flux for pipe flows with different Prandtl numbers. Distributions of different additives in the balance equations of axial and radial heat fluxes were analyzed under the effects of Re and Pr. Predicted parameter distributions were in a good agreement with experimental data of different authors over wide ranges of the operation parameters.