A high-order moment approach for capturing non-equilibrium phenomena in the transition regime

Abstract

The method of moments is employed to extend the validity of continuum-hydrodynamic models into the transition-flow regime. An evaluation of the regularized 13 moment equations for two confined flow problems, planar Couette and Poiseuille flows, indicates some important limitations. For planar Couette flow at a Knudsen number of 0.25, they fail to reproduce the Knudsen-layer velocity profile observed using a direct simulation Monte Carlo approach, and the higher-order moments are not captured particularly well. Moreover, for Poiseuille flow, this system of equations creates a large slip velocity leading to significant overprediction of the mass flow rate for Knudsen numbers above 0.4. To overcome some of these difficulties, the theory of regularized moment equations is extended to 26 moment equations. This new set of equations highlights the importance of both gradient and non-gradient transport mechanisms and is shown to overcome many of the limitations observed in the regularized 13 moment equations. In particular, for planar Couette flow, they can successfully capture the observed Knudsen-layer velocity profile well into the transition regime. Moreover, this new set of equations can correctly predict the Knudsen layer, the velocity profile and the mass flow rate of pressure-driven Poiseuille flow for Knudsen numbers up to 1.0 and captures the bimodal temperature profile in force-driven Poiseuille flow. Above this value, the 26 moment equations are not able to accurately capture the velocity profile in the centre of the channel. However, they are able to capture the basic trends and successfully predict a Knudsen minimum at the correct value of the Knudsen number.

Evaporation-driven vapour microflows: analytical solutions from moment methods

“…For instance, in the 13-moment approximation Ψ A = 1, c i , C i C j and C 2 C i /2, corresponding to the 13 moments (5 conservative variables, 5 components of σ ij and 3 components of q i ). A further extension of the moment equations contain even higher-order moments (see below) and here we follow the same notation as used in Gu & Emerson (2009) for the R26 equations, where ∆, Ω i , R ij , m ijk , ψ ijk and φ ijkl are scalar, first-, second-, third-and fourth-order symmetric trace-free tensors, respectively, defined as…”

“…Besides the Boltzmann equation, rarefaction effects that are beyond the resolution of the NSF system can be predicted by extended macroscopic moment equations (Struchtrup 2005;Gu & Emerson 2009), which were the subject of a recent article in the Annual Review of Fluid Mechanics (Torrilhon 2016). The moment equations form a set of partial differential equations describing the evolution of macroscopic quantities, such as mass density, temperature, velocity, heat flux, stress tensor and so on, defined as moments of the distribution function.…”

“…To extend the applicability of the moment method further into the transition regime, Gu & Emerson (2009) used the regularisation process to obtain the regularised 26-moment (R26) equations. The kinetic wall boundary conditions for the R26 equations were also derived in the same paper using the Maxwell accommodation model.…”

Evaporation boundary conditions for the R13 equations of rarefied gas dynamics

“…12 The method can be extended to higher moment numbers. 21,22 Details of the R13 equations depend on the molecular model, and in the following we consider only the wellestablished equations for Maxwell molecules. 12 Equations for hard sphere molecules, 23 polyatomic gases, 24 and gas mixtures 25 are discussed elsewhere in the literature.…”

Compressibility and rarefaction effects on entropy and entropy generation in micro/nano Couette flow using DSMC

“…The micro/nano-Couette flow has been widely applied in existing literature for different purposes. Some of the sample works can be found in [7][8][9][10][11][12][13][14]. Meanwhile, entropy and entropy generation concepts are wildly used in the literature for different important purposes such as: evaluating a system's irreversibility, optimizing a thermodynamic system or as a means of examining continuum breakdown in rarefied fluids.…”