scholarly journals Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory

2018 ◽  
Author(s):  
Alexander Felix Beckmann ◽  
Anirudh Singh Rana ◽  
Manuel Torrilhon ◽  
Henning Struchtrup
Keyword(s):  
Kinetic Theory ◽  
Boundary Conditions ◽  
Rarefied Gas ◽  
Continuum Hypothesis ◽  
Transport Equations ◽  
Particle Methods ◽  
Navier Stokes ◽  
Knudsen Numbers ◽  
Constant Coefficients ◽  
Onsager Theory

Due to 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 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 to 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.

scholarly journals Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 680 ◽  
Author(s):  
Alexander Beckmann ◽  
Anirudh Rana ◽  
Manuel Torrilhon ◽  
Henning Struchtrup
Keyword(s):  
Kinetic Theory ◽  
Boundary Conditions ◽  
Rarefied Gas ◽  
Continuum Hypothesis ◽  
Direct Simulation Monte Carlo ◽  
Transport Equations ◽  
Particle Methods ◽  
Navier Stokes ◽  
Knudsen Numbers ◽  
Constant Coefficients

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.

scholarly journals Slip Velocity Distribution in a Parallel Plate Channel on Asymmetric Thermal Boundary Condition

2016 ◽  
Vol 35 ◽  
pp. 113-126
Author(s):  
Md Tajul Islam
Keyword(s):  
Boundary Conditions ◽  
Velocity Distribution ◽  
Parallel Plate ◽  
Control Volume ◽  
Navier Stokes ◽  
Working Fluid ◽  
Slip Boundary Conditions ◽  
Cross Sectional ◽  
Slip Boundary ◽  
Knudsen Numbers

Steady, laminar and fully developed flows in parallel plate microchannel with asymmetric thermal wall conditions are solved by control volume technique. In order to examine the influence of Reynolds number and Knudsen number on the velocity distributions, a series of simulations are performed for different Reynolds and Knudsen numbers. Nitrogen gas is used as working fluid and we assume the fluid as continuum but employ the slip boundary conditions on the walls. The Navier-Stokes and energy equations are solved simultaneously. The results are found in good agreement with those predicted by analytical solutions in 2D continuous flow model employing first order slip boundary conditions. It is concluded that the rarefaction flattens the velocity distribution. If the product of Reynolds numbers and Knudsen numbers is fixed, the cross sectional average velocity is fixed for incompressible flow.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 113-126

scholarly journals On the modelling of isothermal gas flows at the microscale

2008 ◽  
Vol 604 ◽  
pp. 235-261 ◽  
Author(s):  
DUNCAN A. LOCKERBY ◽  
JASON M. REESE
Keyword(s):  
Boundary Conditions ◽  
Constitutive Relations ◽  
Navier Stokes ◽  
Test Case ◽  
Gas Flows ◽  
Hydrodynamic Models ◽  
Slip Boundary ◽  
Knudsen Numbers ◽  
Isothermal Gas ◽  
Non Equilibrium

This paper makes two new propositions regarding the modelling of rarefied (non-equilibrium) isothermal gas flows at the microscale. The first is a new test case for benchmarking high-order, or extended, hydrodynamic models for these flows. This standing time-varying shear-wave problem does not require boundary conditions to be specified at a solid surface, so is useful for assessing whether fluid models can capture rarefaction effects in the bulk flow. We assess a number of different proposed extended hydrodynamic models, and we find the R13 equations perform the best in this case.Our second proposition is a simple technique for introducing non-equilibrium effects caused by the presence of solid surfaces into the computational fluid dynamics framework. By combining a new model for slip boundary conditions with a near-wall scaling of the Navier--Stokes constitutive relations, we obtain a model that is much more accurate at higher Knudsen numbers than the conventional second-order slip model. We show that this provides good results for combined Couette/Poiseuille flow, and that the model can predict the stress/strain-rate inversion that is evident from molecular simulations. The model's generality to non-planar geometries is demonstrated by examining low-speed flow around a micro-sphere. It shows a marked improvement over conventional predictions of the drag on the sphere, although there are some questions regarding its stability at the highest Knudsen numbers.

scholarly journals Numerical Simulation of Rarefied Gas Flows with Specified Heat Flux Boundary Conditions

2015 ◽  
Vol 17 (5) ◽  
pp. 1185-1200 ◽  
Author(s):  
Jianping Meng ◽  
Yonghao Zhang ◽  
Jason M. Reese
Keyword(s):  
Heat Flux ◽  
Boundary Conditions ◽  
High Speed ◽  
Rarefied Gas ◽  
Flow Behavior ◽  
Velocity Slip ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Rarefied Gas Flows ◽  
Flux Boundary Conditions

AbstractWe investigate unidirectional rarefied flows confined between two infinite parallel plates with specified heat flux boundary conditions. Both Couette and force-driven Poiseuille flows are considered. The flow behaviors are analyzed numerically by solving the Shakhov model of the Boltzmann equation. We find that a zero-heat-flux wall can significantly influence the flow behavior, including the velocity slip and temperature jump at the wall, especially for high-speed flows. The predicted bimodal-like temperature profile for force-driven flows cannot even be qualitatively captured by the Navier-Stokes-Fourier equations.

Solution of Thermally Developing Zone in Short Micro-/Nanoscale Channels

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
Masoud Darbandi ◽  
Shidvash Vakilipour
Keyword(s):  
Boundary Conditions ◽  
Gas Flow ◽  
Stokes Equations ◽  
Rarefied Gas ◽  
Critical Role ◽  
Navier Stokes ◽  
Channel Inlet ◽  
Constant Wall Temperature ◽  
The Real ◽  
Structure Of Flow

We numerically solve the Navier–Stokes equations to study the rarefied gas flow in short micro- and nanoscale channels. The inlet boundary conditions play a critical role in the structure of flow in short channels. Contrary to the classical inlet boundary conditions, which apply uniform velocity and temperature profiles right at the real channel inlet, we apply the same inlet boundary conditions, but at a fictitious position far upstream of the real channel inlet. A constant wall temperature incorporated with suitable temperature jump is applied at the channel walls. Our solutions for both the classical and extended inlet boundary conditions are compared with the results of other available Navier–Stokes and lattice Boltzmann solvers. It is shown that the current extended inlet boundary conditions can effectively improve the thermofluid flow solutions in short micro- and nanoscale channels.

scholarly journals Thermophoresis of a spherical particle: modelling through moment-based, macroscopic transport equations

2019 ◽  
Vol 862 ◽  
pp. 312-347 ◽  
Author(s):  
Juan C. Padrino ◽  
James E. Sprittles ◽  
Duncan A. Lockerby
Keyword(s):  
Boltzmann Equation ◽  
Kinetic Theory ◽  
Gas Dynamics ◽  
Transport Equations ◽  
The Body ◽  
Closed Form Expression ◽  
Moment Equations ◽  
Thermophoretic Force ◽  
Knudsen Numbers ◽  
The Boltzmann Equation

We consider the linearized form of the regularized 13-moment equations (R13) to model the slow, steady gas dynamics surrounding a rigid, heat-conducting sphere when a uniform temperature gradient is imposed far from the sphere and the gas is in a state of rarefaction. Under these conditions, the phenomenon of thermophoresis, characterized by forces on the solid surfaces, occurs. The R13 equations, derived from the Boltzmann equation using the moment method, provide closure to the mass, momentum and energy conservation laws in the form of constitutive, transport equations for the stress and heat flux that extend the Navier–Stokes–Fourier model to include non-equilibrium effects. We obtain analytical solutions for the field variables that characterize the gas dynamics and a closed-form expression for the thermophoretic force on the sphere. We also consider the slow, streaming flow of gas past a sphere using the same model resulting in a drag force on the body. The thermophoretic velocity of the sphere is then determined from the balance between thermophoretic force and drag. The thermophoretic force is compared with predictions from other theories, including Grad’s 13-moment equations (G13), variants of the Boltzmann equation commonly used in kinetic theory, and with recently published experimental data. The new results from R13 agree well with results from kinetic theory up to a Knudsen number (based on the sphere’s radius) of approximately 0.1 for the values of solid-to-gas heat conductivity ratios considered. However, in this range of Knudsen numbers, where for a very high thermal conductivity of the solid the experiments show reversed thermophoretic forces, the R13 solution, which does result in a reversal of the force, as well as the other theories predict significantly smaller forces than the experimental values. For Knudsen numbers between 0.1 and 1 approximately, the R13 model of thermophoretic force qualitatively shows the trend exhibited by the measurements and, among the various models considered, results in the least discrepancy.

An Investigation of Heat Transfer in a Cavity Flow in the Noncontinuum Regime

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Chariton Christou ◽  
S. Kokou Dadzie
Keyword(s):  
Heat Transfer ◽  
Continuum Model ◽  
Volume Diffusion ◽  
Rarefied Gas ◽  
Transfer Problem ◽  
Diffusion Flux ◽  
Navier Stokes ◽  
Transition Flow ◽  
Flow Equations ◽  
Knudsen Numbers

Volume diffusion (or bi-velocity) continuum model offers an alternative modification to the standard Navier–Stokes for simulating rarefied gas flows. According to this continuum model, at higher Knudsen numbers the contribution of molecular spatial stochasticity increases. In this paper, we study a microcavity heat transfer problem as it provides an excellent test for new continuum flow equations. Simulations are carried out for Knudsen numbers within the slip and higher transition flow regimes where nonlocal-equilibrium and rarefaction effects dominate. We contrast the predictions by a Navier–Stokes model corrected by volume diffusion flux in its constitutive equations to that of the direct simulation Monte Carlo (DSMC) method and the standard Navier–Stokes model. The results show improvement in the Navier–Stokes prediction for the high Knudsen numbers. The new model exhibits proper Knudsen boundary layer in the temperature and velocity fields.

Numerical Methods for the Kinetic Theory of Fluids

2018 ◽  
Author(s):  
Sauro Succi
Keyword(s):  
Molecular Dynamics ◽  
Monte Carlo ◽  
Numerical Methods ◽  
Kinetic Theory ◽  
Computational Efficiency ◽  
Lattice Boltzmann ◽  
Direct Simulation Monte Carlo ◽  
Particle Methods ◽  
Direct Simulation ◽  
Simulation Monte Carlo

This chapter provides a bird’s eye view of the main numerical particle methods used in the kinetic theory of fluids, the main purpose being of locating Lattice Boltzmann in the broader context of computational kinetic theory. The leading numerical methods for dense and rarified fluids are Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC), respectively. These methods date of the mid 50s and 60s, respectively, and, ever since, they have undergone a series of impressive developments and refinements which have turned them in major tools of investigation, discovery and design. However, they are both very demanding on computational grounds, which motivates a ceaseless demand for new and improved variants aimed at enhancing their computational efficiency without losing physical fidelity and vice versa, enhance their physical fidelity without compromising computational viability.

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