Overall Assessment of Heat Transfer for a Rarefied Flow in a Microchannel with Obstacles Using Lattice Boltzmann Method

The objective of this investigation is to assess the effect of obstacles on numerical heat transfer and fluid flow momentum in a rectangular microchannel(MC).Two distinct configurations were studied:one without obstacles and the other with alternating obstacles placed on the upper and lower walls.The research utilized the thermal lattice Boltzmann method(LBM),which solves the energy and momentum equations of fluids with the BGK approximation,implemented in a Python coding environment.Temperature jump and slip velocity conditions were utilized in the simulation for the MC and extended to all obstacle boundaries.The study aims to analyze the rarefaction effect,with Knudsen numbers(Kn)of 0.012,0.02,and 0.05.The outcomes indicate that rarefaction has a significant impact on the velocity and temperature distribution.The presence of nine obstacles led to slower fluid movement inside the microchannel MC,resulting in faster cooling at the outlet.In MCs with obstacles,the rarefaction effect plays a crucial role in decreasing the Nusselt number(Nu)and skin friction coefficient(Cf).Furthermore,the study demonstrated that the obstacles played a crucial role in boosting fluid flow and heat transfer in the MC.The findings suggest that the examined configurations could have potential applications as cooling technologies in micro-electro-mechanical systems and microdevice applications.

An Empirical Method for Prediction of Hypersonic Rarefied Flow-Field Structure

Numerical simulations are presented about the effects of gas rarefaction on hypersonic flow field.Due to the extremely difficult experiment,limited wind-tunnel conditions and high cost,most problems in rarefied flow regime are investigated through numerical methods,in which the direct simulation Monte-Carlo(DSMC)method is widely adopted.And the unstructured DSMC method is employed here.Flows around a vertical plate at a given velocity 7 500 m/s are simulated.For gas rarefaction is judged by the free-stream Knudsen number(Kn),two vital factors are considered:molecular number density and the plate′s length.Cases in which Kn varies from 0.035 to13.36 are simulated.Flow characters in the whole rarefied regime are described,and flow-field structure affected by Knis analyzed.Then,the dimensionless position D*of a certain velocity in the stagnation line is chosen as the marker of flow field to measure its variation.Through flow-field tracing and least-square numerical method analyzing,it is proved that hypersonic rarefied flow field expands outward linearly with the increase of 1/2Kn.An empirical method is proposed,which can be used for the prediction of the hypersonic flow-field structure at a given inflow velocity,especially the shock wave position.

The effect of surface roughness on rarefied gas flows by lattice Boltzmann method

This paper studies the roughness effect combining with effects of rarefaction and compressibility by a lattice Boltzmann model for rarefied gas flows at high Knudsen numbers. By discussing the effect of the tangential momentum accommodation coefficient on the rough boundary condition, the lattice Boltzmann simulations of nitrogen and helium flows are performed in a two-dimensional microchannel with rough boundaries. The surface roughness effects in the microchannel on the velocity field, the mass flow rate and the friction coefficient are studied and analysed. Numerical results for the two gases in micro scale show different characteristics from macroscopic flows and demonstrate the feasibility of the lattice Boltzmann model in rarefied gas dynamics.

General Synthetic Iterative Scheme for Unsteady Rarefied Gas Flows

In rarefied gas flows,the spatial grid size could vary by several orders of magnitude in a single flow configuration(e.g.,inside the Knudsen layer it is at the order of mean free path of gas molecules,while in the bulk region it is at a much larger hydrodynamic scale).Therefore,efficient implicit numerical method is urgently needed for time-dependent problems.However,the integro-differential nature of gas kinetic equations poses a grand challenge,as the gain part of the collision operator is non-invertible.Hence an iterative solver is required in each time step,which usually takes a lot of iterations in the(near)continuum flow regime where the Knudsen number is small;worse still,the solution does not asymptotically preserve the fluid dynamic limit when the spatial cell size is not refined enough.Based on the general synthetic iteration scheme for steady-state solution of the Boltzmann equation,we propose two numerical schemes to push the multiscale simulation of unsteady rarefied gas flows to a new boundary,that is,the numerical solution not only converges within dozens of iterations in each time step,but also asymptotically preserves the Navier-Stokes-Fourier limit in the continuum flow regime,when the spatial grid is coarse,and the time step is large(e.g.,in simulating the extreme slow decay of two-dimensional Taylor vortex,the time step is even at the order of vortex decay time).The properties of fast convergence and asymptotic preserving of the proposed schemes are not only rigorously proven by the Fourier stability analysis for simplified gas kinetic models,but also demonstrated by several numerical examples for the gas kinetic models and the Boltzmann equation.

Planar collisionless jet impingement on a specular reflective plate

This paper presents a fundamental gas-kinetic study on a high speed planar rarefied jet impinging on a flat plate of specular reflections. Based on previous collisionless planar free jet results, it is straightforward to obtain jet impingement flowfield solutions, and jet impingement for specular reflective plate surface properties. Several direct simulation Monte Carlo simulation results are provided and they validate these analytical solutions of rarefied planar jet flows. The results can find applications in many disciplines, such as materials processing, molecular beams, and space engineering.

DIRECT NUMERICAL TEST OF THE B-G-K MODEL EQUATION BY THE DSMC METHOD

In the present paper the rarefied gas how caused by the sudden change of the wall temperature and the Rayleigh problem are simulated by the DSMC method which has been validated by experiments both in global flour field and velocity distribution function level. The comparison of the simulated results with the accurate numerical solutions of the B-G-K model equation shows that near equilibrium the BG-K equation with corrected collision frequency can give accurate result but as farther away from equilibrium the B-G-K equation is not accurate. This is for the first time that the error caused by the B-G-K model equation has been revealed.

Semiclassical Lattice Boltzmann Simulations of Rarefied Circular Pipe Flows

Computations of microscopic circular pipe flow in a rarefied quantum gas are presented using a semiclassical axisymmetric lattice Boltzmann method.The method is first derived by directly projecting the Uehling-Uhlenbeck Boltzmann-BGK equations in two-dimensional rectangular coordinates onto the tensor Hermite polynomials using moment expansion method and then the forcing strategy of Halliday et al.[Phys.Rev.E.,64(2001),011208]is adopted by adding forcing terms into the resulting microdynamic evolution equation.The determination of the forcing terms is dictated by yielding the emergent macroscopic equations toward a particular target form.The correct macroscopic equations of the incompressible axisymmetric viscous flows are recovered through the Chapman-Enskog expansion.The velocity profiles and the mass flow rates of pipe flows with several Knudsen numbers covering different flow regimes are presented.It is found the Knudsen minimum can be captured in all three statistics studied.The results also indicate distinct characteristics of the effects of quantum statistics.

An Efficient Nonlinear Multigrid Solver for the Simulation of Rarefied Gas Cavity Flow

We study efficient simulation of steady state for multi-dimensional rarefied gas flow,which is modeled by the Boltzmann equation with BGK-type collision term.A nonlinear multigrid solver is proposed to resolve the efficiency issue by the following approaches.The unified framework of numerical regularized moment method is first adopted to derive the high-quality discretization of the underlying problem.A fast sweeping iteration is introduced to solve the derived discrete problem more efficiently than the usual time-integration scheme on a single level grid.Taking it as the smoother,the nonlinear multigrid solver is then established to significantly improve the convergence rate.The OpenMP-based parallelization is applied in the implementation to further accelerate the computation.Numerical experiments for two lid-driven cavity flows and a bottom-heated cavity flow are carried out to investigate the performance of the resulting nonlinear multigrid solver.All results show the wonderful efficiency and robustness of the solver for both first-and second-order spatial discretization.

Rotational Slip Flow in Coaxial Cylinders by the Finite-Difference Lattice Boltzmann Methods

Recent studies on applications of the lattice Boltzmann method(LBM)and the finite-difference lattice Boltzmann method(FDLBM)to velocity slip simulations are mostly on one-dimensional(1D)problems such as a shear flow between parallel plates.Applications to a 2D problem may raise new issues.The author performed numerical simulations of rotational slip flow in coaxial cylinders as an example of 2D problem.Two types of 2D models were used.The first were multi-speed FDLBM models proposed by the author.The second was a standard LBM,the D2Q9 model.The simulations were performed applying a finite difference scheme to both the models.The study had two objectives.The first was to investigate the accuracies of LBM and FDLBM on applications to rotational slip flow.The second was to obtain an experience on application of the cylindrical coordinate system.The FDLBM model with 8 directions and the D2Q9 model showed an anisotropic flow pattern when the relaxation time constant or the Knudsen number was large.The FDLBM model with 24 directions showed accurate results even at large Knudsen numbers.

An Efficient Hybrid DSMC/MD Algorithm for Accurate Modeling of Micro Gas Flows

Aiming at simulating micro gas flows with accurate boundary conditions,an efficient hybrid algorithm is developed by combining the molecular dynamics(MD)method with the direct simulation Monte Carlo(DSMC)method.The efficiency comes from the fact that the MD method is applied only within the gas-wall interaction layer,characterized by the cut-off distance of the gas-solid interaction potential,to resolve accurately the gas-wall interaction process,while the DSMC method is employed in the remaining portion of the flow field to efficiently simulate rarefied gas transport outside the gas-wall interaction layer.A unique feature about the present scheme is that the coupling between the two methods is realized by matching the molecular velocity distribution function at the DSMC/MD interface,hence there is no need for one-toone mapping between a MD gas molecule and a DSMC simulation particle.Further improvement in efficiency is achieved by taking advantage of gas rarefaction inside the gas-wall interaction layer and by employing the“smart-wall model”proposed by Barisik et al.The developed hybrid algorithm is validated on two classical benchmarks namely 1-D Fourier thermal problem and Couette shear flow problem.Both the accuracy and efficiency of the hybrid algorithm are discussed.As an application,the hybrid algorithm is employed to simulate thermal transpiration coefficient in the free-molecule regime for a system with atomically smooth surface.Result is utilized to validate the coefficients calculated from the pure DSMC simulation with Maxwell and Cercignani-Lampis gas-wall interaction models.

Smart Wall Model for Molecular Dynamics Simulations of Nanoscale Gas Flows

Three-dimensional molecular dynamics (MD) simulations of gas flows con-fined within nano-scale channels are investigated by introduction of a smart wall modelthat drastically reduces the memory requirements of MD simulations for gas flows.The smart wall molecular dynamics (SWMD) represents three-dimensional FCC wallsusing only 74 wall molecules. This structure is kept in the memory and utilized foreach gas molecule surface collision. Linear Couette flow of argon at Knudsen number10 is investigated using the SWMD utilizing Lennard-Jones potential interactions. Effects of the domain size on the periodicity boundary conditions are investigated usingthree-dimensional simulations. Domain sizes that are one mean-free-path long in theperiodic dimensions are sufficient to obtain domain-size independent MD solutions ofnano-scale confined gas flows. Comparisons between the two- and three-dimensionalsimulations show the inadequacy of two-dimensional MD results. Three-dimensionalSWMD simulations have shown significant deviations of the velocity profile and gasdensity from the kinetic theory based predictions within the force penetration regionof the walls.

SHOCK AND BOUNDARY STRUCTURE FORMATION BY SPECTRAL-LAGRANGIAN METHODS FOR THE INHOMOGENEOUS BOLTZMANN TRANSPORT EQUATION

The numerical approximation of the Spectral-Lagrangian scheme developed by the authors in [30] for a wide range of homogeneous non-linear Boltzmann type equations is extended to the space inhomogeneous case and several shock problems are benchmark. Recognizing that the Boltzmann equation is an important tool in the analysis of formation of shock and boundary layer structures, we present the computational algorithm in Section 3.3 and perform a numerical study case in shock tube geometries well modeled in for ID in x times 3D in v in Section 4. The classic Riemann problem is numerically analyzed for Knudsen numbers close to continuum. The shock tube problem of Aoki et al [2], where the wall temperature is suddenly increased or decreased, is also studied. We consider the problem of heat transfer between two parallel plates with diffusive boundary conditions for a range of Knudsen numbers from close to continuum to a highly rarefied state. Finally, the classical infinite shock tube problem that generates a non-moving shock wave is studied. The point worth noting in this example is that the flow in the final case turns from a supersonic flow to a subsonic flow across the shock.