At non-negligible values of the Knudsen number Kn (defined as the ratio between the mean free pat... more At non-negligible values of the Knudsen number Kn (defined as the ratio between the mean free path of the fluid particles in a gas and the characteristic length of the domain), the Navier-Stokes equations lose applicability [1, 2]. Such rarefied gas flows can be approached within the framework of the Boltzmann equation [3-5]. This equation describes the six-dimensional phase-space evolution of the distribution function f , where f (t, x, p)d 3 xd 3 p gives the number of particles at time t which are contained in an infinitesimal volume d 3 x centred on x, having momenta in an infinitesimal range d 3 p about p. Because of its complexity, the Boltzmann equation can be solved analytically only in a very limited number of cases. Alternatively, numerous well-established approaches to the numerical solutions of the Boltzmann equation are now currently used for academic or engineering purposes, of which we only mention the direct simulation Monte Carlo (DSMC) technique [6], the discrete velocity models (DVMs) [7-9], the discrete unified gas-kinetic scheme (DUGKS) [10-12] and the lattice Boltzmann (LB) models [13-20]. The LB models are a particular type of DVMs and are derived from the Boltzmann equation using a simplified version of the collision operator, as well as an appropriate discretisation of the momentum space, which ensure the recovery of the moments of the distribution function f up to a certain order N. Originally derived nearly 30 years ago from the lattice gas automata [17, 19, 20], the LB
We discuss general features of thermal lattice Boltzmann models based on half-range Gauss quadrat... more We discuss general features of thermal lattice Boltzmann models based on half-range Gauss quadratures, specialising to the half-range Gauss-Hermite and Gauss-Laguerre cases. The main focus of the paper is on the construction of high order half-range Hermite lattice Boltzmann (HHLB) models. The performance of the HHLB models is compared with that of Laguerre lattice Boltzmann (LLB) and full-range Hermite lattice Boltzmann (HLB) models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density, macroscopic velocity, temperature and heat fluxes in the two-dimensional Couette flow. The Bhatnagar-Gross-Krook (BGK) collision term is used throughout the paper. To reduce the computational costs of the numerical simulations, we use mixed lattice Boltzmann models, constructed using different quadrature methods on each Cartesian axis. For Kn ? 0.01 , the HLB models require the least number of velocities to satisfy our convergence test. When Kn ? 0.05 , the HLB models are outperformed in terms of number of velocities employed by both the LLB and the HHLB models. Moreover, we find that the HHLB models require less quadrature points than the LLB models at all tested values of Kn, which we attribute to the Maxwellian form of the weight function for the half-range Hermite polynomials.
Deposition of inertial solid particles transported by turbulent flows is modelled in a framework ... more Deposition of inertial solid particles transported by turbulent flows is modelled in a framework of a statistical approach based on the particle velocity Probability Density Function (PDF). The particle-turbulence interaction term is closed in the kinetic equation by a model widely inspired from the famous BGK model of the kinetic theory of rarefied gases. A Gauss-Hermite Lattice Boltzmann model is used to solve the closed kinetic equation involving the turbulence effect. The Lattice Boltzmann model is used for the case of the deposition of inertial particles transported by a homogeneous isotropic turbulent flows. Even if the carrier phase is homogeneous and isotropic, the presence of the wall coupled with particle-turbulence interactions leads to inhomogeneous particle distribution and non-equilibrium particle fluctuating motion. Despite these complexities the predictions of the Lattice Boltzmann model are in very good accordance with random-walk simulations. More specifically the mean particle velocity, the r.m.s. particle velocity and the deposition rate are all well predicted by the proposed Lattice Boltzmann model.
In this paper we apply a finite difference lattice Boltzmann model to study the phase separation ... more In this paper we apply a finite difference lattice Boltzmann model to study the phase separation in a two-dimensional liquid-vapor system. Spurious numerical effects in macroscopic equations are discussed and an appropriate numerical scheme involving flux limiter techniques is proposed to minimize them and guarantee a better numerical stability at very low viscosity. The phase separation kinetics is investigated and we find evidence of two different growth regimes depending on the value of the fluid viscosity as well as on the liquid-vapor ratio.
A thermal lattice Boltzmann model for a van der Waals fluid is proposed. In the continuum, the mo... more A thermal lattice Boltzmann model for a van der Waals fluid is proposed. In the continuum, the model reproduces at second order of a Chapman-Enskog expansion, the theory recently introduced by A. Onuki ͓Phys. Rev. Lett. 94, 054501 ͑2005͔͒. Phase separation has been studied in a system quenched by contact with external walls. Pressure waves favor the thermalization of the system at initial times and the temperature, soon with respect to typical times of phase separation, becomes homogeneous in the bulk. Alternate layers of liquid and vapor form on the walls and disappear at late times.
The time evolution of the morphology of homogeneous phases during spinodal decomposition is descr... more The time evolution of the morphology of homogeneous phases during spinodal decomposition is described using a family of morphological measures known as Minkowski functionals. They provide the characteristic length scale L of patterns in a convenient, statistically robust, and computationally inexpensive way. They also allow one to study the scaling behavior of the content, shape, and connectivity of spatial structures and to define the crossover from the early stage decomposition to the late stage domain growth. We observe the scaling behavior L ∼ t α with α = 2/3, α = 1/2, and α = 1/3 depending on the viscosity of the fluid. When approaching the spinodal density ρsp, we recover the prediction L ∼ (ρ − ρsp) −1/2 for the early time spinodal decomposition.
A Lattice Boltzmann model is developed to account for the competition between surface tension and... more A Lattice Boltzmann model is developed to account for the competition between surface tension and dipolar interaction in magnetic fluids. The description of the interactions was kept as simple as possible to identify and isolate the nature of the essential interactions in the examined situations. The model is used thereafter in order to simulate the deformation of a magnetic fluid drop under the action of an external magnetic field, as well as the onset of the normal field instability in magnetic fluids. The success of the model, and easily identified deviations, demonstrate that the model is a powerful and versatile tool for the study of magnetic and nonmagnetic fluid systems with interfaces.
The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to c... more The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to construct a family of lattice Boltzmann models. When all quadrature orders Q x , Q y , Q z equal N + 1, the Laguerre lattice Boltzmann model LLB(Q x ,Q y ,Q z) exactly recovers all moments up to order N of the Maxwell-Boltzmann equilibrium distribution function f (eq) , calculated over any Cartesian octant of the three-dimensional momentum space. Results of Couette flow simulations at Kn = 0.1, 0.5, 1.0 and in the ballistic regime are reported. Specific microfluidic effects (velocity slip, temperature jump, longitudinal heat flux) are well captured up to Kn = 0.5, as demonstrated by comparison to direct simulation Monte Carlo results. Excellent agreement with analytic results is obtained in the ballistic regime.
In this paper, we consider the capabilities of the Boltzmann equation with the Shakhov or ellipso... more In this paper, we consider the capabilities of the Boltzmann equation with the Shakhov or ellipsoidal models for the collision term to capture the characteristics of rarefied gas flows. The benchmark is performed by comparing the results obtained using these kinetic model equations with direct simulation Monte Carlo (DSMC) results for particles interacting via ab initio potentials. The analysis is restricted to channel flows between parallel plates and we consider three flow problems, namely: the heat transfer between stationary plates, the Couette flow and the heat transfer under shear. The simulations are performed in the nonlinear regime for the 3 He, 4 He, and Ne gases. The reference temperature ranges between 1 K and 3000 K for 3 He and 4 He and between 20 K and 5000 K for Ne. While good agreement is seen up to the transition regime for the direct phenomena (shear stress, heat flux driven by temperature gradient), the relative errors in the cross phenomena (heat flux perpendicular to the temperature gradient) exceed 10% even in the slip-flow regime. The kinetic model equations are solved using the finite difference lattice Boltzmann algorithm based on half
Abstract. In this paper introduce a finite difference lattice Boltzmann model capable to describe... more Abstract. In this paper introduce a finite difference lattice Boltzmann model capable to describe the hydrodynamics and the phase separation of a two-dimensional liquid-vapor system. A numerical scheme involving flux limiter techniques is proposed to minimize spurious numerical effects. We report some simulation results of the spinodal decomposition process and we find evidence of two different growth regimes depending on the system viscosity.
Phase separation of a two-dimensional van derWaals fluid subject to a gravitational force is stud... more Phase separation of a two-dimensional van derWaals fluid subject to a gravitational force is studied by numerical simulations based on lattice Boltzmannmethods implemented with a finite difference scheme. A growth exponent α=1 is measured in the direction of the external force.
In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implemen... more In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implementation of boundary conditions at the kinetic level. Such boundary conditions induce a discontinuity in the distribution function with respect to the component of the momentum which is normal to the boundary. Expanding the distribution function with respect to half-range polynomials allows this discontinuity to be captured. The implementation of this concept has been reported in the literature only for force-free flows. In the case of general forces which can have non-zero components in the direction perpendicular to the walls, the implementation of the force term requires taking the momentum space gradient of a discontinuous function. Our proposed method deals with this difficulty by employing the theory of distributions. We validate our procedure by considering the simple one-dimensional flow between diffuse-reflective walls of equal or different temperatures driven by the constant grav...
We present here a comparison between collision-streaming and finite-difference lattice Boltzmann ... more We present here a comparison between collision-streaming and finite-difference lattice Boltzmann (LB) models. This study provides a derivation of useful formulae which help one to properly compare the simulation results obtained with both LB models. We consider three physical problems: the shock wave propagation, the damping of shear waves, and the decay of Taylor–Green vortices, often used as benchmark tests. Despite the different mathematical and computational complexity of the two methods, we show how the physical results can be related to obtain relevant quantities.
Abstract We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffus... more Abstract We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffuse reflection boundary conditions apply. We present a systematic procedure for the construction of the force term in lattice Boltzmann models based on mixed Cartesian quadratures, where the quadrature on each axis is selected independently. We find that, at non-negligible value of the Knudsen number, half-range quadratures outperform the full-range Gauss–Hermite quadratures for the direction perpendicular to the diffuse-reflecting plates, while the quadrature on the periodic direction along the flow is the full-range Gauss–Hermite quadrature. Our results are validated against numerical results available in the literature.
A previous theoretical model is enlarged to elucidate some features of the optogalvanic signals o... more A previous theoretical model is enlarged to elucidate some features of the optogalvanic signals originating from the metastable states in hollow cathode neon discharges. Numerical experiments were performed on the basis of this model to obtain ...
Public reporting burden for this collection of information is estimated to average 1 hour per res... more Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.
At non-negligible values of the Knudsen number Kn (defined as the ratio between the mean free pat... more At non-negligible values of the Knudsen number Kn (defined as the ratio between the mean free path of the fluid particles in a gas and the characteristic length of the domain), the Navier-Stokes equations lose applicability [1, 2]. Such rarefied gas flows can be approached within the framework of the Boltzmann equation [3-5]. This equation describes the six-dimensional phase-space evolution of the distribution function f , where f (t, x, p)d 3 xd 3 p gives the number of particles at time t which are contained in an infinitesimal volume d 3 x centred on x, having momenta in an infinitesimal range d 3 p about p. Because of its complexity, the Boltzmann equation can be solved analytically only in a very limited number of cases. Alternatively, numerous well-established approaches to the numerical solutions of the Boltzmann equation are now currently used for academic or engineering purposes, of which we only mention the direct simulation Monte Carlo (DSMC) technique [6], the discrete velocity models (DVMs) [7-9], the discrete unified gas-kinetic scheme (DUGKS) [10-12] and the lattice Boltzmann (LB) models [13-20]. The LB models are a particular type of DVMs and are derived from the Boltzmann equation using a simplified version of the collision operator, as well as an appropriate discretisation of the momentum space, which ensure the recovery of the moments of the distribution function f up to a certain order N. Originally derived nearly 30 years ago from the lattice gas automata [17, 19, 20], the LB
We discuss general features of thermal lattice Boltzmann models based on half-range Gauss quadrat... more We discuss general features of thermal lattice Boltzmann models based on half-range Gauss quadratures, specialising to the half-range Gauss-Hermite and Gauss-Laguerre cases. The main focus of the paper is on the construction of high order half-range Hermite lattice Boltzmann (HHLB) models. The performance of the HHLB models is compared with that of Laguerre lattice Boltzmann (LLB) and full-range Hermite lattice Boltzmann (HLB) models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density, macroscopic velocity, temperature and heat fluxes in the two-dimensional Couette flow. The Bhatnagar-Gross-Krook (BGK) collision term is used throughout the paper. To reduce the computational costs of the numerical simulations, we use mixed lattice Boltzmann models, constructed using different quadrature methods on each Cartesian axis. For Kn ? 0.01 , the HLB models require the least number of velocities to satisfy our convergence test. When Kn ? 0.05 , the HLB models are outperformed in terms of number of velocities employed by both the LLB and the HHLB models. Moreover, we find that the HHLB models require less quadrature points than the LLB models at all tested values of Kn, which we attribute to the Maxwellian form of the weight function for the half-range Hermite polynomials.
Deposition of inertial solid particles transported by turbulent flows is modelled in a framework ... more Deposition of inertial solid particles transported by turbulent flows is modelled in a framework of a statistical approach based on the particle velocity Probability Density Function (PDF). The particle-turbulence interaction term is closed in the kinetic equation by a model widely inspired from the famous BGK model of the kinetic theory of rarefied gases. A Gauss-Hermite Lattice Boltzmann model is used to solve the closed kinetic equation involving the turbulence effect. The Lattice Boltzmann model is used for the case of the deposition of inertial particles transported by a homogeneous isotropic turbulent flows. Even if the carrier phase is homogeneous and isotropic, the presence of the wall coupled with particle-turbulence interactions leads to inhomogeneous particle distribution and non-equilibrium particle fluctuating motion. Despite these complexities the predictions of the Lattice Boltzmann model are in very good accordance with random-walk simulations. More specifically the mean particle velocity, the r.m.s. particle velocity and the deposition rate are all well predicted by the proposed Lattice Boltzmann model.
In this paper we apply a finite difference lattice Boltzmann model to study the phase separation ... more In this paper we apply a finite difference lattice Boltzmann model to study the phase separation in a two-dimensional liquid-vapor system. Spurious numerical effects in macroscopic equations are discussed and an appropriate numerical scheme involving flux limiter techniques is proposed to minimize them and guarantee a better numerical stability at very low viscosity. The phase separation kinetics is investigated and we find evidence of two different growth regimes depending on the value of the fluid viscosity as well as on the liquid-vapor ratio.
A thermal lattice Boltzmann model for a van der Waals fluid is proposed. In the continuum, the mo... more A thermal lattice Boltzmann model for a van der Waals fluid is proposed. In the continuum, the model reproduces at second order of a Chapman-Enskog expansion, the theory recently introduced by A. Onuki ͓Phys. Rev. Lett. 94, 054501 ͑2005͔͒. Phase separation has been studied in a system quenched by contact with external walls. Pressure waves favor the thermalization of the system at initial times and the temperature, soon with respect to typical times of phase separation, becomes homogeneous in the bulk. Alternate layers of liquid and vapor form on the walls and disappear at late times.
The time evolution of the morphology of homogeneous phases during spinodal decomposition is descr... more The time evolution of the morphology of homogeneous phases during spinodal decomposition is described using a family of morphological measures known as Minkowski functionals. They provide the characteristic length scale L of patterns in a convenient, statistically robust, and computationally inexpensive way. They also allow one to study the scaling behavior of the content, shape, and connectivity of spatial structures and to define the crossover from the early stage decomposition to the late stage domain growth. We observe the scaling behavior L ∼ t α with α = 2/3, α = 1/2, and α = 1/3 depending on the viscosity of the fluid. When approaching the spinodal density ρsp, we recover the prediction L ∼ (ρ − ρsp) −1/2 for the early time spinodal decomposition.
A Lattice Boltzmann model is developed to account for the competition between surface tension and... more A Lattice Boltzmann model is developed to account for the competition between surface tension and dipolar interaction in magnetic fluids. The description of the interactions was kept as simple as possible to identify and isolate the nature of the essential interactions in the examined situations. The model is used thereafter in order to simulate the deformation of a magnetic fluid drop under the action of an external magnetic field, as well as the onset of the normal field instability in magnetic fluids. The success of the model, and easily identified deviations, demonstrate that the model is a powerful and versatile tool for the study of magnetic and nonmagnetic fluid systems with interfaces.
The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to c... more The Gauss-Laguerre quadrature method is used on the Cartesian semiaxes in the momentum space to construct a family of lattice Boltzmann models. When all quadrature orders Q x , Q y , Q z equal N + 1, the Laguerre lattice Boltzmann model LLB(Q x ,Q y ,Q z) exactly recovers all moments up to order N of the Maxwell-Boltzmann equilibrium distribution function f (eq) , calculated over any Cartesian octant of the three-dimensional momentum space. Results of Couette flow simulations at Kn = 0.1, 0.5, 1.0 and in the ballistic regime are reported. Specific microfluidic effects (velocity slip, temperature jump, longitudinal heat flux) are well captured up to Kn = 0.5, as demonstrated by comparison to direct simulation Monte Carlo results. Excellent agreement with analytic results is obtained in the ballistic regime.
In this paper, we consider the capabilities of the Boltzmann equation with the Shakhov or ellipso... more In this paper, we consider the capabilities of the Boltzmann equation with the Shakhov or ellipsoidal models for the collision term to capture the characteristics of rarefied gas flows. The benchmark is performed by comparing the results obtained using these kinetic model equations with direct simulation Monte Carlo (DSMC) results for particles interacting via ab initio potentials. The analysis is restricted to channel flows between parallel plates and we consider three flow problems, namely: the heat transfer between stationary plates, the Couette flow and the heat transfer under shear. The simulations are performed in the nonlinear regime for the 3 He, 4 He, and Ne gases. The reference temperature ranges between 1 K and 3000 K for 3 He and 4 He and between 20 K and 5000 K for Ne. While good agreement is seen up to the transition regime for the direct phenomena (shear stress, heat flux driven by temperature gradient), the relative errors in the cross phenomena (heat flux perpendicular to the temperature gradient) exceed 10% even in the slip-flow regime. The kinetic model equations are solved using the finite difference lattice Boltzmann algorithm based on half
Abstract. In this paper introduce a finite difference lattice Boltzmann model capable to describe... more Abstract. In this paper introduce a finite difference lattice Boltzmann model capable to describe the hydrodynamics and the phase separation of a two-dimensional liquid-vapor system. A numerical scheme involving flux limiter techniques is proposed to minimize spurious numerical effects. We report some simulation results of the spinodal decomposition process and we find evidence of two different growth regimes depending on the system viscosity.
Phase separation of a two-dimensional van derWaals fluid subject to a gravitational force is stud... more Phase separation of a two-dimensional van derWaals fluid subject to a gravitational force is studied by numerical simulations based on lattice Boltzmannmethods implemented with a finite difference scheme. A growth exponent α=1 is measured in the direction of the external force.
In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implemen... more In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implementation of boundary conditions at the kinetic level. Such boundary conditions induce a discontinuity in the distribution function with respect to the component of the momentum which is normal to the boundary. Expanding the distribution function with respect to half-range polynomials allows this discontinuity to be captured. The implementation of this concept has been reported in the literature only for force-free flows. In the case of general forces which can have non-zero components in the direction perpendicular to the walls, the implementation of the force term requires taking the momentum space gradient of a discontinuous function. Our proposed method deals with this difficulty by employing the theory of distributions. We validate our procedure by considering the simple one-dimensional flow between diffuse-reflective walls of equal or different temperatures driven by the constant grav...
We present here a comparison between collision-streaming and finite-difference lattice Boltzmann ... more We present here a comparison between collision-streaming and finite-difference lattice Boltzmann (LB) models. This study provides a derivation of useful formulae which help one to properly compare the simulation results obtained with both LB models. We consider three physical problems: the shock wave propagation, the damping of shear waves, and the decay of Taylor–Green vortices, often used as benchmark tests. Despite the different mathematical and computational complexity of the two methods, we show how the physical results can be related to obtain relevant quantities.
Abstract We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffus... more Abstract We consider the 2D force-driven Poiseuille flow between parallel plates, on which diffuse reflection boundary conditions apply. We present a systematic procedure for the construction of the force term in lattice Boltzmann models based on mixed Cartesian quadratures, where the quadrature on each axis is selected independently. We find that, at non-negligible value of the Knudsen number, half-range quadratures outperform the full-range Gauss–Hermite quadratures for the direction perpendicular to the diffuse-reflecting plates, while the quadrature on the periodic direction along the flow is the full-range Gauss–Hermite quadrature. Our results are validated against numerical results available in the literature.
A previous theoretical model is enlarged to elucidate some features of the optogalvanic signals o... more A previous theoretical model is enlarged to elucidate some features of the optogalvanic signals originating from the metastable states in hollow cathode neon discharges. Numerical experiments were performed on the basis of this model to obtain ...
Public reporting burden for this collection of information is estimated to average 1 hour per res... more Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing the burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.
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