Simulations of Contrail Optical Properties and Radiative Forcing for Various Crystal Shapes

The aim of this study is to investigate the sensitivity of radiative-forcing computations to various contrail crystal shape models. Contrail optical properties in the shortwave and longwave ranges are derived using a ray-tracing geometric method and the discrete dipole approximation method, respectively. Both methods present good correspondence of the single-scattering albedo and the asymmetry parameter in a transition range (3–8 µm). There are substantial differences in single-scattering properties among 10 crystal models investigated here (e.g., hexagonal columns and plates with different aspect ratios, and spherical particles). The single-scattering albedo and the asymmetry parameter both vary by up to 0.1 among various crystal shapes. The computed single-scattering properties are incorporated in the moderate-resolution atmospheric radiance and transmittance model(MODTRAN) radiative transfer code to simulate solar and infrared fluxes at the top of the atmosphere. Particle shapes have a strong impact on the contrail radiative forcing in both the shortwave and longwave ranges. The differences in the net radiative forcing among optical models reach 50% with respect to the mean model value. The hexagonal-column and hexagonal-plate particles show the smallest net radiative forcing, and the largest forcing is obtained for the spheres. The balance between the shortwave forcing and longwave forcing is highly sensitive with respect to the assumed crystal shape and may even change the sign of the net forcing. The optical depth at which the mean diurnal radiative forcing changes sign from positive to negative varies from 4.5 to 10 for a surface albedo of 0.2 and from 2 to 6.5 for a surface albedo of 0.05. Contrails are probably never that optically thick (except for some aged contrail cirrus), however, and so will not have a cooling effect on climate

Three-dimensional central-moments-based lattice Boltzmann method with external forcing: A consistent, concise and universal formulation

The cascaded or central-moments-based lattice Boltzmann method (CM-LBM) is a robust alternative to the more conventional BGK-LBM for the simulation of high-Reynolds number flows. Unfortunately, its original formulation makes its extension to a broader range of physics quite difficult. To tackle this issue, a recent work [A. De Rosis, Phys. Rev. E 95, 013310 (2017)] proposed a more generic way to derive concise and efficient three-dimensional CM-LBMs. Knowing the original model also relies on central moments that are derived in an adhoc manner, i.e., by mimicking those of the Maxwell-Boltzmann distribution to ensure their Galilean invariance a posteriori, a very recent effort [A. De Rosis and K. H. Luo, Phys. Rev. E 99, 013301 (2019)] was proposed to further generalize their derivation. The latter has shown that one could derive Galilean invariant CMs in a systematic and a priori manner by taking into account high-order Hermite polynomials in the derivation of the discrete equilibrium state. Combining these two approaches, a compact and mathematically sound formulation of the CM-LBM with external forcing is proposed. More specifically, the proposed formalism fully takes advantage of the D3Q27 discretization by relying on the corresponding set of 27 Hermite polynomials (up to the sixth order) for the derivation of both the discrete equilibrium state and the forcing term. The present methodology is more consistent than previous approaches, as it properly explains how to derive Galilean invariant CMs of the forcing term in an a priori manner. Furthermore, while keeping the numerical properties of the original CM-LBM, the present work leads to a compact and simple algorithm, representing a universal methodology based on CMs and external forcing within the lattice Boltzmann framework.Comment: Published in Phys. Fluids as Editor's Pic

Lattice Boltzmann Model for The Volume-Averaged Navier-Stokes Equations

A numerical method, based on the discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.Comment: 9 pages, 2 figure

Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data

In this paper, a backward Euler method is discussed for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in L2-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition

Numerical solution of the time-fractional Fokker-Planck equation with general forcing

We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^\alpha)$ for a uniform time step $k$, where $\alpha\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.Comment: 3 Figure

Incorporating Forcing Terms in Cascaded Lattice-Boltzmann Approach by Method of Central Moments

Cascaded lattice-Boltzmann method (Cascaded-LBM) employs a new class of collision operators aiming to improve numerical stability. It achieves this and distinguishes from other collision operators, such as in the standard single or multiple relaxation time approaches, by performing relaxation process due to collisions in terms of moments shifted by the local hydrodynamic fluid velocity, i.e. central moments, in an ascending order-by-order at different relaxation rates. In this paper, we propose and derive source terms in the Cascaded-LBM to represent the effect of external or internal forces on the dynamics of fluid motion. This is essentially achieved by matching the continuous form of the central moments of the source or forcing terms with its discrete version. Different forms of continuous central moments of sources, including one that is obtained from a local Maxwellian, are considered in this regard. As a result, the forcing terms obtained in this new formulation are Galilean invariant by construction. The method of central moments along with the associated orthogonal properties of the moment basis completely determines the expressions for the source terms as a function of the force and macroscopic velocity fields. In contrast to the existing forcing schemes, it is found that they involve higher order terms in velocity space. It is shown that the proposed approach implies "generalization" of both local equilibrium and source terms in the usual lattice frame of reference, which depend on the ratio of the relaxation times of moments of different orders. An analysis by means of the Chapman-Enskog multiscale expansion shows that the Cascaded-LBM with forcing terms is consistent with the Navier-Stokes equations. Computational experiments with canonical problems involving different types of forces demonstrate its accuracy.Comment: 55 pages, 4 figure