Moment realizability and the validity of the Navier–Stokes equations for rarefied gas dynamics

1998 ◽  
Vol 10 (12) ◽  
pp. 3214-3226 ◽  
Author(s):  
C. David Levermore ◽  
William J. Morokoff ◽  
B. T. Nadiga
Keyword(s):  
Gas Dynamics ◽  
Stokes Equations ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Moment Realizability

Status of the Navier–Stokes Equations in Gas Dynamics. A Review

2018 ◽  
Vol 53 (1) ◽  
pp. 152-168
Author(s):  
V. S. Galkin ◽  
S. V. Rusakov
Keyword(s):  
Gas Dynamics ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

An analytically predictive model for moderately rarefied gas flow

2012 ◽  
Vol 698 ◽  
pp. 406-422 ◽  
Author(s):  
Thomas Veltzke ◽  
Jorg Thöming
Keyword(s):  
Mass Flow ◽  
Flow Rate ◽  
Stokes Equations ◽  
Rarefied Gas ◽  
Knudsen Diffusion ◽  
Convective Transport ◽  
Surface Topology ◽  
Fickian Diffusion ◽  
Navier Stokes ◽  
Navier Stokes Equations

AbstractIn microducts deviation from continuum flow behaviour of a gas increases with rarefaction. When using Navier–Stokes equations to calculate a flow under slightly and moderately rarefied conditions, slip boundary conditions are used which in turn refer to the tangential momentum accommodation coefficient (TMAC). Here we demonstrate that, in the so-called slip and transition regime, the flow in microducts can be reliably described by a consistently non-empirical model without considering the TMAC. We obtain this equation by superposition of convective transport and Fickian diffusion using two-dimensional solutions of Navier–Stokes equations and a description for the Knudsen diffusion coefficient as derived from kinetic theory respectively. For a wide variety of measurement series found in the literature the calculation predicts the data accurately. Surprisingly only size of the duct, temperature, gas properties and inlet and outlet pressure are necessary to calculate the resulting mass flow by means of a single algebraic equation. From this, and taking the discrepancies of the TMAC concerning surface roughness and nature of the gases into account, we could conclude that neither the diffusive proportions nor the total mass flow rates are influenced by surface topology and chemistry at Knudsen numbers below unity. Compared to the tube geometry, the model slightly underestimates the flow rate in rectangular channels when rarefaction increases. Likewise, the dimensionless mass flow rate and the diffusive proportion of the total flow are distinctly higher in a tube. Thus the cross-sectional geometry has a significant influence on the transport mechanisms under rarefied conditions.

Rarefied Gas Flow Analysis of a Suborbital Shuttle With the Academic CFD Code Galatea

2017 ◽  
Author(s):  
Angelos G. Klothakis ◽  
Georgios N. Lygidakis ◽  
Ioannis K. Nikolos
Keyword(s):  
Gas Flow ◽  
Microelectromechanical Systems ◽  
Stokes Equations ◽  
Rarefied Gas ◽  
Flow Analysis ◽  
Velocity Slip ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Rarefied Gas Flows ◽  
Wide Range

During the past decade considerable efforts have been exerted for the simulation of rarefied gas flows in a wide range of applications, like the flow over suborbital vehicles, in microelectromechanical systems, etc. Such flows appear to be significantly different from those at the continuum regime, making the Navier-Stokes equations to fail without further amendment. In this study an in-house academic CFD solver, named Galatea, is modified appropriately to account for rarefied gases. The no-slip condition on solid walls is no longer valid, hence, velocity slip and temperature jump boundary conditions are applied instead. Additionally, a second-order accurate slip model has been incorporated, namely, this of Beskok and Karniadakis, increasing the accuracy in the same area but avoiding simultaneously the numerical difficulties, entailed by the computation of the second derivative of slip velocity when complex geometries and unstructured grids are coupled. The proposed solver is validated against rarefied laminar flow over a suborbital shuttle, designed by the Azim’UTBM team. The obtained results are compared with those extracted with the parallel open-source kernel SPARTA, which is based on the DSMC method. A satisfactory agreement is reported between the two methodologies, demonstrating the potential of the modified solver to simulate effectively such flows.

Numerical Simulation and Modeling of Laminar Developing Flow in Channels and Tubes With Slip

2013 ◽  
Vol 135 (10) ◽  
Author(s):  
Y. S. Muzychka ◽  
R. Enright
Keyword(s):  
Stokes Equations ◽  
Rarefied Gas ◽  
Slip Boundary Condition ◽  
Momentum Equation ◽  
Navier Stokes ◽  
Cfd Simulations ◽  
Navier Stokes Equations ◽  
Navier Slip ◽  
Slip Boundary ◽  
Rarefied Gas Flows

Analytical solutions for slip flows in the hydrodynamic entrance region of tubes and channels are examined. These solutions employ a linearized axial momentum equation using Targ's method. The momentum equation is subjected to a first order Navier slip boundary condition. The accuracy of these solutions is examined using computational fluid dynamics (CFD) simulations. CFD simulations utilized the full Navier–Stokes equations, so that the implications of the approximate linearized axial momentum equation could be fully assessed. Results are presented in terms of the dimensionless mean wall shear stress, τ⋆, as a function of local dimensionless axial coordinate, ξ, and relative slip parameter, β. These solutions can be applied to either rarefied gas flows when compressibility effects are small or apparent liquid slip over hydrophobic and superhydrophobic surfaces. It has been found that, under slip conditions, the minimum Reynolds number should be ReDh>100 in order for the approximate linearized solution to remain valid.

scholarly journals On the apparent permeability of porous media in rarefied gas flows

2017 ◽  
Vol 822 ◽  
pp. 398-417 ◽  
Author(s):  
Lei Wu ◽  
Minh Tuan Ho ◽  
Lefki Germanou ◽  
Xiao-Jun Gu ◽  
Chang Liu ◽  
...  
Keyword(s):  
Porous Medium ◽  
Correction Factor ◽  
Knudsen Number ◽  
Stokes Equations ◽  
Rarefied Gas ◽  
Gas Permeability ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Apparent Gas Permeability ◽  
Rarefied Gas Flows

The apparent gas permeability of a porous medium is an important parameter in the prediction of unconventional gas production, which was first investigated systematically by Klinkenberg in 1941 and found to increase with the reciprocal mean gas pressure (or equivalently, the Knudsen number). Although the underlying rarefaction effects are well known, the reason that the correction factor in Klinkenberg’s famous equation decreases when the Knudsen number increases has not been fully understood. Most of the studies idealize the porous medium as a bundle of straight cylindrical tubes; however, according to the gas kinetic theory, this only results in an increase of the correction factor with the Knudsen number, which clearly contradicts Klinkenberg’s experimental observations. Here, by solving the Bhatnagar–Gross–Krook equation in simplified (but not simple) porous media, we identify, for the first time, two key factors that can explain Klinkenberg’s experimental results: the tortuous flow path and the non-unitary tangential momentum accommodation coefficient for the gas–surface interaction. Moreover, we find that Klinkenberg’s results can only be observed when the ratio between the apparent and intrinsic permeabilities is ${\lesssim}30$; at large ratios (or Knudsen numbers) the correction factor increases with the Knudsen number. Our numerical results could also serve as benchmarking cases to assess the accuracy of macroscopic models and/or numerical schemes for the modelling/simulation of rarefied gas flows in complex geometries over a wide range of gas rarefaction. Specifically, we point out that the Navier–Stokes equations with the first-order velocity-slip boundary condition are often misused to predict the apparent gas permeability of the porous medium; that is, any nonlinear dependence of the apparent gas permeability with the Knudsen number, predicted from the Navier–Stokes equations, is not reliable. Worse still, for some types of gas–surface interactions, even the ‘filtered’ linear dependence of the apparent gas permeability with the Knudsen number is of no practical use since, compared to the numerical solution of the Bhatnagar–Gross–Krook equation, it is only accurate when the ratio between the apparent and intrinsic permeabilities is ${\lesssim}1.5$.

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