823 research outputs found
Numerical simulation of rarefied supersonic flows using a fourth-order maximum-entropy moment method with interpolative closure
Maximum-entropy moment methods allow for the modelling of gases from the
continuum regime to strongly rarefied conditions. The development of
approximated solutions to the entropy maximization problem has made these
methods computationally affordable. In this work, we apply a fourth-order
maximum-entropy moment method to the study of supersonic rarefied flows. For
such conditions, we compare the maximum-entropy solutions to results obtained
from the kinetic theory of gases at different Knudsen numbers. The analysis is
performed for both a simplified model of a gas with a single translational
degree of freedom (5-moment system) and for a typical gas with three degrees of
freedom (14-moment system). The maximum-entropy method is applied to the study
of the Sod shock-tube problem at various rarefaction levels, and to the
simulation of two-dimensional low-collisional crossed supersonic jets. We show
that, in rarefied supersonic conditions, it is important to employ accurate
estimates of the wave speeds. Since analytical expressions are not presently
available, we propose an approximation, valid for the 14-moment system. In
these conditions, the solution of the maximum-entropy system is shown to
realize large degrees of non-equilibrium and to approach the Junk subspace, yet
provides a good overall accuracy and agreement with the kinetic theory.
Numerical procedures for reaching second-order accurate discretizations are
discussed, as well as the implementation of the 14-moment solver on Graphics
Processing Units (GPUs)
Simulating fluid flows in micro and nano devices : the challenge of non-equilibrium behaviour
We review some recent developments in the modelling of non-equilibrium (rarefied) gas flows at the micro- and nano-scale, concentrating on two different but promising approaches: extended hydrodynamic models, and lattice Boltzmann methods. Following a brief exposition of the challenges that non-equilibrium poses in micro- and nano-scale gas flows, we turn first to extended hydrodynamics, outlining the effective abandonment of Burnett-type models in favour of high-order regularised moment equations. We show that the latter models, with properly-constituted boundary conditions, can capture critical non-equilibrium flow phenomena quite well. We then review the boundary conditions required if the conventional Navier-Stokes-Fourier (NSF) fluid dynamic model is applied at the micro scale, describing how 2nd-order Maxwell-type conditions can be used to compensate for some of the non-equilibrium flow behaviour near solid surfaces. While extended hydrodynamics is not yet widely-used for real flow problems because of its inherent complexity, we finish this section with an outline of recent 'phenomenological extended hydrodynamics' (PEH) techniques-essentially the NSF equations scaled to incorporate non-equilibrium behaviour close to solid surfaces-which offer promise as engineering models. Understanding non-equilibrium within lattice Boltzmann (LB) framework is not as advanced as in the hydrodynamic framework, although LB can borrow some of the techniques which are being developed in the latter-in particular, the near-wall scaling of certain fluid properties that has proven effective in PEH. We describe how, with this modification, the standard 2nd-order LB method is showing promise in predicting some rarefaction phenomena, indicating that instead of developing higher-order off-lattice LB methods with a large number of discrete velocities, a simplified high-order LB method with near-wall scaling may prove to be just as effective as a simulation tool
A kinetic perspective on k-epsilon turbulence model and corresponding entropy production
In this paper, we present an alternative derivation of the entropy production in turbulent flows, based on a formal analogy with the kinetic theory of rarefied gas. This analogy allows proving that the celebrated k-epsilon model for turbulent flows is nothing more than a set of coupled BGK-like equations with a proper forcing. This opens a novel perspective on this model, which may help in sorting out the heuristic assumptions essential for its derivation, such as the balance between turbulent kinetic energy production and dissipation. The entropy production is an essential condition for the design and optimization of devices where turbulent flows are involved
Nonlinear Effects in Squeeze Film Gas Damping on Microbeams
We consider squeeze film gas damping during microbeam motion away and toward a substrate as occurs during opening and closing of RF switches and other MEMS devices. The numerical solution of the gas damping problem in two-dimensional geometries is obtained based on the Boltzmann-ESBGK equation. The difference in damping force between downward and upward moving beams is shown to vary from as little from as 5% for low beam velocities of 0.1m/s to more than 200% at 2.4m/s. For a constant velocity magnitude of 0.8m/s, this difference increases from 60% to almost 90% when the pressure is reduced by an order of magnitude. The numerical simulations are consistent with earlier observations of a significantly higher damping force during the closing of a capacitive RF MEMS switch reported by Steeneken et al. (JMM, 15, 176-184, 2005). The physical mechanism leading to this non-linear dependence of the damping force on velocity has been attributed to the differences in the flow rarefaction regime for the gas in the microgap
Exponential Runge-Kutta methods for stiff kinetic equations
We introduce a class of exponential Runge-Kutta integration methods for
kinetic equations. The methods are based on a decomposition of the collision
operator into an equilibrium and a non equilibrium part and are exact for
relaxation operators of BGK type. For Boltzmann type kinetic equations they
work uniformly for a wide range of relaxation times and avoid the solution of
nonlinear systems of equations even in stiff regimes. We give sufficient
conditions in order that such methods are unconditionally asymptotically stable
and asymptotic preserving. Such stability properties are essential to guarantee
the correct asymptotic behavior for small relaxation times. The methods also
offer favorable properties such as nonnegativity of the solution and entropy
inequality. For this reason, as we will show, the methods are suitable both for
deterministic as well as probabilistic numerical techniques
A Multiscale Kinetic-Fluid Solver with Dynamic Localization of Kinetic Effects
This paper collects the efforts done in our previous works [P. Degond, S.
Jin, L. Mieussens, A Smooth Transition Between Kinetic and Hydrodynamic
Equations, J. Comp. Phys., 209 (2005) 665--694.],[P.Degond, G. Dimarco, L.
Mieussens, A Moving Interface Method for Dynamic Kinetic-fluid Coupling, J.
Comp. Phys., Vol. 227, pp. 1176-1208, (2007).],[P. Degond, J.G. Liu, L.
Mieussens, Macroscopic Fluid Model with Localized Kinetic Upscaling Effects,
SIAM Multi. Model. Sim. 5(3), 940--979 (2006)] to build a robust multiscale
kinetic-fluid solver. Our scope is to efficiently solve fluid dynamic problems
which present non equilibrium localized regions that can move, merge, appear or
disappear in time. The main ingredients of the present work are the followings
ones: a fluid model is solved in the whole domain together with a localized
kinetic upscaling term that corrects the fluid model wherever it is necessary;
this multiscale description of the flow is obtained by using a micro-macro
decomposition of the distribution function [P. Degond, J.G. Liu, L. Mieussens,
Macroscopic Fluid Model with Localized Kinetic Upscaling Effects, SIAM Multi.
Model. Sim. 5(3), 940--979 (2006)]; the dynamic transition between fluid and
kinetic descriptions is obtained by using a time and space dependent transition
function; to efficiently define the breakdown conditions of fluid models we
propose a new criterion based on the distribution function itself. Several
numerical examples are presented to validate the method and measure its
computational efficiency.Comment: 34 page
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
Towards an ultra efficient kinetic scheme. Part I: basics on the BGK equation
In this paper we present a new ultra efficient numerical method for solving
kinetic equations. In this preliminary work, we present the scheme in the case
of the BGK relaxation operator. The scheme, being based on a splitting
technique between transport and collision, can be easily extended to other
collisional operators as the Boltzmann collision integral or to other kinetic
equations such as the Vlasov equation. The key idea, on which the method
relies, is to solve the collision part on a grid and then to solve exactly the
transport linear part by following the characteristics backward in time. The
main difference between the method proposed and semi-Lagrangian methods is that
here we do not need to reconstruct the distribution function at each time step.
This allows to tremendously reduce the computational cost of the method and it
permits for the first time, to the author's knowledge, to compute solutions of
full six dimensional kinetic equations on a single processor laptop machine.
Numerical examples, up to the full three dimensional case, are presented which
validate the method and assess its efficiency in 1D, 2D and 3D
Entropic Multi-Relaxation Models for Simulation of Fluid Turbulence
A recently introduced family of lattice Boltzmann (LB) models (Karlin,
B\"osch, Chikatamarla, Phys. Rev. E, 2014) is studied in detail for
incompressible two-dimensional flows. A framework for developing LB models
based on entropy considerations is laid out extensively. Second order rate of
convergence is numerically confirmed and it is demonstrated that these entropy
based models recover the Navier-Stokes solution in the hydrodynamic limit.
Comparison with the standard Bhatnagar-Gross-Krook (LBGK) and the entropic
lattice Boltzmann method (ELBM) demonstrates the superior stability and
accuracy for several benchmark flows and a range of grid resolutions and
Reynolds numbers. High Reynolds number regimes are investigated through the
simulation of two-dimensional turbulence, particularly for under-resolved
cases. Compared to resolved LBGK simulations, the presented class of LB models
demonstrate excellent performance and capture the turbulence statistics with
good accuracy.Comment: To be published in Proceedings of Discrete Simulation of Fluid
Dynamics DSFD 201
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