Macroscopic and kinetic modelling of rarefied polyatomic gases

2016 ◽  
Vol 806 ◽  
pp. 437-505 ◽  
Author(s):  
Behnam Rahimi ◽  
Henning Struchtrup
Keyword(s):  
Kinetic Model ◽  
Degrees Of Freedom ◽  
Kinetic Modelling ◽  
Higher Order ◽  
Macroscopic Model ◽  
Order Effects ◽  
Moment Equations ◽  
Polyatomic Gases ◽  
First Order ◽  
Order Set

A kinetic model and corresponding high-order macroscopic model for the accurate description of rarefied polyatomic gas flows are introduced. The different energy exchange processes are accounted for with a two term collision model. The proposed kinetic model, which is an extension of the S-model, predicts correct relaxation of higher moments and delivers the accurate Prandtl ($Pr$) number. Also, the model has a proven linear H-theorem. The order of magnitude method is applied to the primary moment equations to acquire the optimized moment definitions and the final scaled set of Grad’s 36 moment equations for polyatomic gases. At the first order, a modification of the Navier–Stokes–Fourier (NSF) equations is obtained. At third order of accuracy, a set of 19 regularized partial differential equations (R19) is obtained. Furthermore, the terms associated with the internal degrees of freedom yield various intermediate orders of accuracy, a total of 13 different orders. Thereafter, boundary conditions for the proposed macroscopic model are introduced. The unsteady heat conduction of a gas at rest is studied numerically and analytically as an example of a boundary value problem. The results for different gases are given and effects of Knudsen numbers, degrees of freedom, accommodation coefficients and temperature-dependent properties are investigated. For some cases, the higher-order effects are very dominant and the widely used first-order set of the NSF equations fails to accurately capture the gas behaviour and should be replaced by the proposed higher-order set of equations.

scholarly journals Reputations for Resolve and Higher-Order Beliefs in Crisis Bargaining

2021 ◽  
pp. 002200272199554
Author(s):  
Allan Dafoe ◽  
Remco Zwetsloot ◽  
Matthew Cebul
Keyword(s):  
Belief Change ◽  
Higher Order ◽  
Future Research ◽  
Order Effects ◽  
Survey Experiment ◽  
Experimental Setting ◽  
Crisis Bargaining ◽  
First Order ◽  
Qualitative Evidence ◽  
Higher Order Effects

Reputations for resolve are said to be one of the few things worth fighting for, yet they remain inadequately understood. Discussions of reputation focus almost exclusively on first-order belief change— A stands firm, B updates its beliefs about A’s resolve. Such first-order reputational effects are important, but they are not the whole story. Higher-order beliefs—what A believes about B’s beliefs, and so on—matter a great deal as well. When A comes to believe that B is more resolved, this may decrease A’s resolve, and this in turn may increase B’s resolve, and so on. In other words, resolve is interdependent. We offer a framework for estimating higher-order effects, and find evidence of such reasoning in a survey experiment on quasi-elites. Our findings indicate both that states and leaders can develop potent reputations for resolve, and that higher-order beliefs are often responsible for a large proportion of these effects (40 percent to 70 percent in our experimental setting). We conclude by complementing the survey with qualitative evidence and laying the groundwork for future research.

Evaluation of Grad's Second Problem Using Different Higher Order Continuum Theories

2020 ◽  
Vol 143 (1) ◽  
Author(s):  
Ravi Sudam Jadhav ◽  
Amit Agrawal
Keyword(s):  
Kinetic Model ◽  
Hard Sphere ◽  
Pressure Profile ◽  
Higher Order ◽  
Collision Integral ◽  
Simple Problem ◽  
Burnett Equations ◽  
Moment Equations ◽  
Maxwell Molecules ◽  
Continuum Theories

Abstract In our earlier work (Jadhav, and Agrawal, 2020, “Grad's second problem and its solution within the framework of Burnett hydrodynamics,” ASME J. Heat Transfer, 142(10), p. 102105), we proposed Grad's second problem (examination of steady-state solution for a gas at rest upon application of a one-dimensional heat flux) as a potential benchmark problem for testing the accuracy of different higher order continuum theories and solved the problem within the framework of Burnett hydrodynamics. In this work, we solve this problem within the moment framework and also examine two variants, Bhatnagar–Gross–Krook (BGK)–Burnett and regularized 13 moment equations, for this problem. It is observed that only the conventional form of Burnett equations which are derived retaining the full nonlinear collision integral are able to capture nonuniform pressure profile observed in case of hard-sphere molecules. On the other hand, BGK–Burnett equations derived using BGK-kinetic model predict uniform pressure profile in both the cases. It seems that the variants based on BGK-kinetic model do not distinguish between hard-sphere and Maxwell molecules at least for the problem considered. With respect to moment equations, Grad 13 and regularized 13 moment equations predict consistent results for Maxwell molecules. However, for hard-sphere molecules, since the exact closed form of moment equations is not known, it is difficult to comment upon the results of moment equations for hard-sphere molecules. The present results for this relatively simple problem provide valuable insights into the nature of the equations and important remarks are made in this context.

The Dynamics of a Ligament-Suspended North-Seeking Gyroscope

1979 ◽  
Vol 21 (2) ◽  
pp. 123-131 ◽  
Author(s):  
C. H. J. Fox ◽  
L. Maunder
Keyword(s):  
Approximate Solution ◽  
Steady State ◽  
Free Vibration ◽  
Higher Order ◽  
Experimental Confirmation ◽  
Order Effects ◽  
Linear Vibration ◽  
First Order ◽  
Dynamical Characteristics ◽  
Theoretical Predictions

The dynamical characteristics of a North-seeking gyroscope are ivestigated. Five modes of free vibration are predicted. The response to harmonic horizontal disturbances of the point of suspension contains resonances associated with the natural fequencies that are predictable from first-order linear vibration theory. A higher order approximate solution shows that significant second-order effects occur in the form of ultra-harmonic resonances and a steady-state error in indicated North. Experimental confirmation of the theoretical predictions is reported.

scholarly journals A higher order perfectly matched layer formulation for finite-difference time-domain seismic wave modeling

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T1-T16 ◽  
Author(s):  
David P. Connolly ◽  
Antonios Giannopoulos ◽  
Michael C. Forde
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Staggered Grid ◽  
First Order ◽  
Enhanced Absorption ◽  
Absorption Performance ◽  
Difference Time

We have developed a higher order perfectly matched layer (PML) formulation to improve the absorption performance for finite-difference time-domain seismic modeling. First, we outlined a new unsplit “correction” approach, which allowed for traditional, first-order PMLs to be added directly to existing codes in a straightforward manner. Then, using this framework, we constructed a PML formulation that can be used to construct higher order PMLs of arbitrary order. The greater number of degrees of freedom associated with the higher order PML allow for enhanced flexibility of the PML stretching functions, thus potentially facilitating enhanced absorption performance. We found that the new approach can offer increased elastodynamic absorption, particularly for evanescent waves. We also discovered that the extra degrees of freedom associated with the higher order PML required careful optimization if enhanced absorption was to be achieved. Furthermore, these extra degrees of freedom increased the computational requirements in comparison with first-order schemes. We reached our formulations using one compact equation thus increasing the ease of implementation. Additionally, the formulations are based on a recursive integration approach that reduce PML memory requirements, and do not require special consideration for corner regions. We tested the new formulations to determine their ability to absorb body waves and surface waves. We also tested standard staggered grid stencils and rotated staggered grid stencils.

scholarly journals Molecular Extended Thermodynamics of Rarefied Polyatomic Gases with a New Hierarchy of Moments

Fluids ◽  
2021 ◽  
Vol 6 (2) ◽  
pp. 62
Author(s):  
Takashi Arima ◽  
Tommaso Ruggeri
Keyword(s):  
Degrees Of Freedom ◽  
Maximum Entropy Principle ◽  
Extended Thermodynamics ◽  
Moment Equations ◽  
Field Equations ◽  
Polyatomic Gases ◽  
Moment System ◽  
Rarefied Polyatomic Gases ◽  
Entropy Principle ◽  
Internal Degrees Of Freedom

The aim of this paper is to construct the molecular extended thermodynamics for classical rarefied polyatomic gases with a new hierarchy, which is absent in the previous procedures of moment equations. The new hierarchy is deduced recently from the classical limit of the relativistic theory of moments associated with the Boltzmann–Chernikov equation. The field equations for 15 moments of the distribution function, in which the internal degrees of freedom of a molecule are taken into account, are closed with the maximum entropy principle. It is shown that the theory contains, as a principal subsystem, the previously polyatomic 14 fields theory, and in the monatomic limit, in which the dynamical pressure vanishes, the differential system converges, instead of to the Grad 13-moment system, to the Kremer 14-moment system.

scholarly journals A Note on Identity and Higher Order Quantification

2009 ◽  
Vol 7 ◽  
Author(s):  
Rafal Urbaniak
Keyword(s):  
Higher Order ◽  
Second Order ◽  
Identity Relation ◽  
First Order ◽  
Order Language ◽  
Order Set

It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not definable. The point is that the definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed. This paper is a follow-up to (Urbaniak 2006), where it has been proven that no actual axiomatization of Leśniewski’s Ontology determines the standard semantics for the epsilon connective.

Higher-Order Stabilized Perturbation for Recursive Eigen-Decomposition Estimation

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Paul Mucchielli ◽  
Basuraj Bhowmik ◽  
Budhaditya Hazra ◽  
Vikram Pakrashi
Keyword(s):  
Real Time ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Mode Shapes ◽  
Accurate Estimation ◽  
Perturbation Approach ◽  
Mode Identification ◽  
First Order ◽  
Ill Conditioned Matrices ◽  
Eigen Decomposition

Abstract Eigen-decomposition remains one of the most invaluable tools for signal processing algorithms. Although traditional algorithms based on QR decomposition, Jacobi rotations and block Lanczos tridiagonalization have been proposed to decompose a matrix into its eigenspace, associated computational expense typically hinders their implementation in a real-time framework. In this paper, we study recursive eigen perturbation (EP) of the symmetric eigenvalue problem of higher order (greater than one). Through a higher order perturbation approach, we improve the recently established first-order eigen perturbation (FOP) technique by creating a stabilization process for adapting to ill-conditioned matrices with close eigenvalues. Six algorithms were investigated in this regard: first-order, second-order, third-order, and their stabilized versions. The developed methods were validated and assessed on multiple structural health monitoring (SHM) problems. These were first tested on a five degrees-of-freedom (DOF) linear building model for accurate estimation of mode shapes in an automated framework. The separation of closely spaced modes was then demonstrated on a 3DOF + tuned mass damper (TMD) problem. Practical utility of the methods was probed on the Phase-I ASCE-SHM benchmark problem. The results obtained for real-time mode identification establishes the robustness of the proposed methods for a range of engineering applications.

Numerical Investigation of the Physics of Higher Order Effects Generated by Wave Paddles

2020 ◽  
Author(s):  
Yun Zhi Law ◽  
Hui Liang ◽  
Harrif Santo ◽  
Kian Yew Lim ◽  
Eng Soon Chan
Keyword(s):  
Free Surface ◽  
Higher Order ◽  
The Other ◽  
Order Effects ◽  
Harmonic Waves ◽  
Regular Wave ◽  
Wave Tank ◽  
First Order ◽  
Higher Harmonic ◽  
Higher Order Effects

Abstract When free-surface waves are generated using wave paddles to produce the desired waves, higher order effects might be inevitable for some cases. These can be due to the mismatch in the wave paddle displacement and non-linear free-surface wave kinematics, as well as the moving boundary of wave paddles. Such higher order effects are often manifested as higher harmonic waves, which can propagate independently (or free waves). The presence of such waves will contaminate the quality of the tank test, and together with effects due to scaling and finite size of tank, it is important to reduce or mitigate such effects as much as possible in a wave tank in order to simulate a more realistic scenario. This study investigates the above problem in a systematic manner by using a fully-nonlinear numerical wave tank based on the three-dimensional time-domain Harmonic Polynomial Cell (HPC) method. Wave is generated by flap-type wave paddles on one end of the tank, and is damped on the other end. The paddle boundary conditions are satisfied on the instantaneous paddles surfaces, and the free surface is tracked by the generalized semi-Lagrangian scheme. In this study, first order paddle signal is used to generate regular waves, and the focus is on characterising the behaviour of the generated free higher harmonic waves. We first look into a rectangular wave tank where the paddles are distributed at one side of the tank. Upon the generation of an oblique regular wave (primary wave), it is observed that the generated free waves propagate at a different angle/direction. An explicit analytical expression is derived for the direction of the free waves, which agrees with the numerical observation. Besides propagating at a different direction, the free waves also interact with the primary waves resulting in additional bound waves of the first and third harmonics. Next, we consider a circular wave tank, where paddles along half of the circumference are used to generate planar regular wave, while paddles at the other half are assumed to be able to fully absorb the wave. The generated free waves are observed to focus at a particular region in the tank due to constructive interference. To eliminate or at least mitigate such undesired waves, correction to first order paddle signal is required. Second order correction scheme based on Schaffer (1996) is implemented for such purpose. Preliminary results seem to suggest that second order correction to the paddle signal can only mitigate but cannot completely eliminate the existence of free higher harmonic waves.

First- and higher-order effects of curvature and torsion on the flow in a helical rectangular duct

1996 ◽  
Vol 314 ◽  
pp. 113-138 ◽  
Author(s):  
C. Jonas Bolinder
Keyword(s):  
Secondary Flow ◽  
Stable Solution ◽  
Higher Order ◽  
Convective Transport ◽  
Order Effects ◽  
First Order ◽  
Order Solution ◽  
Aspect Ratios ◽  
Curvature And Torsion ◽  
Square Ducts

A series expansion method is employed to determine the first-order terms in curvature ε and torsion η of fully developed laminar flow in helical square ducts and in helical rectangular ducts of aspect ratio two. The first-order solutions are compared to solutions of the full governing equations. For toroidal square ducts with zero pitch, the first-order solution is fairly accurate for Dean numbers, De = Re ε1/2, up to about 20, and for straight twisted square ducts the first-order solution is accurate for Germano numbers, Gn = η Re, up to at least 50 where Re is the Reynolds number. Important conclusions are that the flow in a helical duct with a finite pitch or torsion to the first order (i.e. with higher-order terms in ε and η neglected) is obtained as a superposition of the flow in a toroidal duct with zero pitch and a straight twisted duct; that the secondary flow in helical non-circular ducts for sufficiently small Re is dominated by torsion effects; and that for increasing Re, the secondary flow eventually is dominated by effects due to curvature. Torsion has a stronger impact on the flow for aspect ratios greater than one. A characteristic combined higher-order effect of curvature and torsion is an enlargement of the lower vortex of the secondary flow at the expense of the upper vortex, and also a shift of the maximum axial flow towards the upper wall. For higher Reynolds numbers, bifurcation phenomena appear. The extent of a few solution branches for helical ducts with finite pitch or torsion is determined. For ducts with small torsion it is found that the extent of the stable solution branches is affected little by torsion. Physical velocity components are employed to describe the flow. The contravariant components are found useful when describing the convective transport in the duct.

Sign in / Sign up

Export Citation Format

Share Document