259 research outputs found
Comparing macroscopic continuum models for rarefied gas dynamics : a new test method
We propose a new test method for investigating which macroscopic continuum models, among the many existing models, give the best description of rarefied gas flows over a range of Knudsen numbers. The merits of our method are: no boundary conditions for the continuum models are needed, no coupled governing equations are solved, while the Knudsen layer is still considered. This distinguishes our proposed test method from other existing techniques (such as stability analysis in time and space, computations of sound speed and dispersion, and the shock wave structure problem). Our method relies on accurate, essentially noise-free, solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation; in this paper, the BGK model and the ES-BGK model equations are considered. Our method is applied to test whether one-dimensional stationary Couette flow is accurately described by the following macroscopic transport models: the Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and the regularized 13 moment equations (two types: the original, and that based on an order of magnitude approach). The gas molecular model is Maxwellian. For Knudsen numbers in the transition-continuum regime (Kn less-than-or-equals, slant 0.1), we find that the two types of regularized 13 moment equations give similar results to each other, which are better than Grad's original 13 moment equations, which, in turn, give better results than the Burnett equations. The Navier-Stokes-Fourier equations give the worst results. This is as expected, considering the presumed accuracy of these models. For cases of higher Knudsen numbers, i.e. Kn > 0.1, all macroscopic continuum equations tested fail to describe the flows accurately. We also show that the above conclusions from our tests are general, and independent of the kinetic model used
What does an ideal wall look like?
This paper deals with the interface between a solid and an ideal gas. The surface of the solid is considered to be an ideal wall, if the flux of entropy is continuous, i.e., if the interaction between wall and gas is non-dissipative. The concept of an ideal wall is discussed within the framework of kinetic theory. In particular it is shown that a non-dissipative wall must be adiabatic and does not exerts shear stresses to the gas, if the interaction of a gas atom with the wall is not influenced by the presence of other gas atoms. It follows that temperature jumps and slip will be observed at virtually all walls, although they will be negligibly small in the hydrodynamic regime (i.e., for small Knudsen numbers
Inflating a Rubber Balloon
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.A spherical balloon has a non-monotonic pressure-radius characteristic. This fact leads to interesting stability properties when two balloons of different radii are interconnected, see [1, 2, 3]. Here, however, we investigate what happens when a single balloon is inflated, say, by mouth. We simulate that process and show how the maximum of the pressure-radius characteristic is overcome by the pressure in the lungs and how the downward sloping part of the characteristic is ‘bridged’ while the lung pressure relaxes
Inconsistency of a dissipative contribution to the mass flux in hydrodynamics
The possibility of dissipative contributions to the mass flux is considered
in detail. A general, thermodynamically consistent framework is developed to
obtain such terms, the compatibility of which with general principles is then
checked--including Galilean invariance, the possibility of steady rigid
rotation and uniform center-of-mass motion, the existence of a locally
conserved angular momentum, and material objectivity. All previously discussed
scenarios of dissipative mass fluxes are found to be ruled out by some
combinations of these principles, but not a new one that includes a smoothed
velocity field v-bar. However, this field v-bar is nonlocal and leads to
unacceptable consequences in specific situations. Hence we can state with
confidence that a dissipative contribution to the mass flux is not possible.Comment: 18 pages; submitted to Phys. Rev.
Coupled constitutive relations: a second law based higher order closure for hydrodynamics
In the classical framework, the Navier-Stokes-Fourier equations are obtained
through the linear uncoupled thermodynamic force-flux relations which guarantee
the non-negativity of the entropy production. However, the conventional
thermodynamic description is only valid when the Knudsen number is sufficiently
small. Here, it is shown that the range of validity of the
Navier-Stokes-Fourier equations can be extended by incorporating the nonlinear
coupling among the thermodynamic forces and fluxes. The resulting system of
conservation laws closed with the coupled constitutive relations is able to
describe many interesting rarefaction effects, such as Knudsen paradox,
transpiration flows, thermal stress, heat flux without temperature gradients,
etc., which can not be predicted by the classical Navier-Stokes-Fourier
equations. For this system of equations, a set of phenomenological boundary
conditions, which respect the second law of thermodynamics, is also derived.
Some of the benchmark problems in fluid mechanics are studied to show the
applicability of the derived equations and boundary conditions.Comment: 20 pages, 6 figures, Proceedings of the Royal Society A (Open access
article
Evaporation boundary conditions for the linear R13 equations based on the onsager theory
Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed
Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic Matlab program
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic
model for many applications in thermodynamics, econophysics and sociodynamics.
Despite recent hardware improvements, the solution of the Boltzmann equation
remains extremely challenging from the computational point of view, in
particular by deterministic methods (free of stochastic noise). This work aims
to improve a deterministic direct method recently proposed [V.V. Aristov,
Kluwer Academic Publishers, 2001] for solving the HIBE with a generic
collisional kernel and, in particular, for taking care of the late dynamics of
the relaxation towards the equilibrium. Essentially (a) the original problem is
reformulated in terms of particle kinetic energy (exact particle number and
energy conservation during microscopic collisions) and (b) the computation of
the relaxation rates is improved by the DVM-like correction, where DVM stands
for Discrete Velocity Model (ensuring that the macroscopic conservation laws
are exactly satisfied). Both these corrections make possible to derive very
accurate reference solutions for this test case. Moreover this work aims to
distribute an open-source program (called HOMISBOLTZ), which can be
redistributed and/or modified for dealing with different applications, under
the terms of the GNU General Public License. The program has been purposely
designed in order to be minimal, not only with regards to the reduced number of
lines (less than 1,000), but also with regards to the coding style (as simple
as possible).Comment: 35 pages, 4 figures, it describes the code HOMISBOLTZ to be
distributed with the pape
Optimal prediction for moment models: Crescendo diffusion and reordered equations
A direct numerical solution of the radiative transfer equation or any kinetic
equation is typically expensive, since the radiative intensity depends on time,
space and direction. An expansion in the direction variables yields an
equivalent system of infinitely many moments. A fundamental problem is how to
truncate the system. Various closures have been presented in the literature. We
want to study moment closure generally within the framework of optimal
prediction, a strategy to approximate the mean solution of a large system by a
smaller system, for radiation moment systems. We apply this strategy to
radiative transfer and show that several closures can be re-derived within this
framework, e.g. , diffusion, and diffusion correction closures. In
addition, the formalism gives rise to new parabolic systems, the reordered
equations, that are similar to the simplified equations.
Furthermore, we propose a modification to existing closures. Although simple
and with no extra cost, this newly derived crescendo diffusion yields better
approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment
Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor
correction
Inertial Frame Independent Forcing for Discrete Velocity Boltzmann Equation: Implications for Filtered Turbulence Simulation
We present a systematic derivation of a model based on the central moment
lattice Boltzmann equation that rigorously maintains Galilean invariance of
forces to simulate inertial frame independent flow fields. In this regard, the
central moments, i.e. moments shifted by the local fluid velocity, of the
discrete source terms of the lattice Boltzmann equation are obtained by
matching those of the continuous full Boltzmann equation of various orders.
This results in an exact hierarchical identity between the central moments of
the source terms of a given order and the components of the central moments of
the distribution functions and sources of lower orders. The corresponding
source terms in velocity space are then obtained from an exact inverse
transformation due to a suitable choice of orthogonal basis for moments.
Furthermore, such a central moment based kinetic model is further extended by
incorporating reduced compressibility effects to represent incompressible flow.
Moreover, the description and simulation of fluid turbulence for full or any
subset of scales or their averaged behavior should remain independent of any
inertial frame of reference. Thus, based on the above formulation, a new
approach in lattice Boltzmann framework to incorporate turbulence models for
simulation of Galilean invariant statistical averaged or filtered turbulent
fluid motion is discussed.Comment: 37 pages, 1 figur
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