scholarly journals From Boltzmann to incompressible Navier–Stokes in Sobolev spaces with polynomial weight

2018 ◽  
Vol 17 (01) ◽  
pp. 85-116 ◽  
Author(s):  
Marc Briant ◽  
Sara Merino-Aceituno ◽  
Clément Mouhot
Keyword(s):  
Boltzmann Equation ◽  
Knudsen Number ◽  
Exponential Decay ◽  
Sobolev Spaces ◽  
Stokes Equations ◽  
Linear Part ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Cauchy Problem ◽  
Global Equilibrium

We study the Boltzmann equation on the [Formula: see text]-dimensional torus in a perturbative setting around a global equilibrium under the Navier–Stokes linearization. We use a recent functional analysis breakthrough to prove that the linear part of the equation generates a [Formula: see text]-semigroup with exponential decay in Lebesgue and Sobolev spaces with polynomial weight, independently of the Knudsen number. Finally, we prove well-posedness of the Cauchy problem for the nonlinear Boltzmann equation in perturbative setting and an exponential decay for the perturbed Boltzmann equation, uniformly in the Knudsen number, in Sobolev spaces with polynomial weight. The polynomial weight is almost optimal. Furthermore, this result only requires derivatives in the space variable and allows to connect solutions to the incompressible Navier–Stokes equations in these spaces.

scholarly journals The Random Gas of Hard Spheres

J ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 162-205 ◽  
Author(s):  
Rafail Abramov
Keyword(s):  
Boltzmann Equation ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Mass Density ◽  
Navier Stokes ◽  
Enskog Equation ◽  
Hard Sphere Model ◽  
Collision Dynamics ◽  
Rigid Spheres ◽  
Navier Stokes Equations

The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out by Loschmidt. To avoid Loschmidt’s objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback–Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a non-increasing function of time. Unlike the conventional hard sphere model, the proposed random gas system results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog–Euler and Enskog–Navier–Stokes equations acquire additional effects in both the advective and viscous terms.

Steady-state analytical solutions for the lattice boltzmann equation

Open Physics ◽  
2003 ◽  
Vol 1 (3) ◽  
Author(s):  
Gábor Házi
Keyword(s):  
Steady State ◽  
Boltzmann Equation ◽  
Analytical Solutions ◽  
Poiseuille Flow ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Lattice Boltzmann Equation ◽  
Two Dimensional ◽  
Navier Stokes Equations

AbstractA general class of analytical solutions of the lattice Boltzmann equation is derived for two-dimensional, steady-state unidirectional flows. A subset of the solutions that verifies the corresponding Navier-Stokes equations is given. It is pointed out that this class includes, e.g., the Couette and the Poiseuille flow but not, e.g., the basic Kolmogorov flow. For steady-state non-unidirectional flows, first and second order solutions of the lattice Boltzmann equation are derived. Practical consequences of the analysis are mentioned. Differences between the technique applied here and those used in some earlier works are emphasized.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

Sign in / Sign up

Export Citation Format

Share Document