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.

Analysis of Rarefied Effects of a Nano-scale Magnetic Head/Disk

When the decrease in the space between magnetic head and disk arrived at 10 nm or less, which is much lower than the mean free path of gas molecules, the gas flow presents distinctive features against the macro features because of the rarefied effects. The modified Reynolds equation considering rarefied gas effect is used to calculate the rarefied region of a negative pressure magnetic head working in the distance of 10 nm. Inverse Knudsen number was adopted to calculating the ratio of the rarefied area. According to the numerical resuks, discussions and analyses are then presented to reveal the rarefied effect on the working performances of a magnetic head. The results show that the magnetic head works in the slip-flow and transition regions and moves to the transition region with the increase in velocity. Furthermore, the maximum rarefied effects occur at the side edges where the flying height is thinner and pressure is lower, rather than in the minimum flying height on the rear. The results also show that with considering the rarefied effects, the load-carrying capacity of the magnetic head and the maximum pressure decrease significantly, but the minimum pressure slightly changes.

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.

Rarefied planar jets into vacuum

This paper presents a fundamental gaskinetic study on high speed rarefied jets expanding into vacuum from a cluster of planar exits.Based on the corresponding exact expressions for one planar jet,this paper straightforwardly derives the combined multiple jet flowfield solutions of density and velocity components,however,for the combined temperature and pressure solutions,extra attention shall be practiced.Several direct simulation Monte Carlo simulation results are provided and they validate these analytical solutions of rarefied planar jet flows.

Effect of Ballistic Bouncing of Gas Particles across a Microchannel on Rarefied Gas Flows

This paper proposes a novel computationally efficient method of modeling rarefied gas flow in microchannels based on the newly discovered and mathematically proven Ballistic Principle of the Property Balance in Space (BPPBS). The mechanism of influence of the effect of rarefication on the gas flow is specifically investigated. Also, a differential form of the momentum balance equation governing gas flow in the channel between two parallel plates due to the pressure gradient along the channel and its exact implicit solution in the form of an integral equation have been derived. The theory does not use the generalized concept of viscosity based on the variable mean free path (MFP) in the Knudsen layer (KL). Comparing the normalized flow rate as a function of the inverse Knudsen number according to the current theory and the experimental data shows good agreement in the range of the inverse Knudsen number from 0.01 to about 40. The correlation factor is found to be about 0.995. The results show that our approach based on the BPPBS offers substantial and practical advantages in modeling and simulation of rarefied gases. The validity of the widely disseminated claim of the geometry-dependent MFP in the KL was analyzed.

Efficient molecular model for squeeze-film damping in rarefied air

Based on the energy transfer model(ETM) proposed by Bao et al.and the Monte Carlo(MC) model proposed by Hutcherson and Ye, this paper proposes an efficient molecular model(MC-S) for squeeze-film damping(SQFD) in rarefied air by releasing the assumption of constant molecular velocity in the gap.Compared with the experiment data, the MC-S model is more efficient than the MC model and more accurate than ETM.Besides, by using the MC-S model, the feasibility of the empirical model proposed by Sumali for SQFD of different plate sizes is discussed.It is proved that, for various plate sizes, the accuracy of the empirical model is relatively high.At last, the SQFD of various vibration frequencies is discussed, and it shows that, for low vibration frequency, the MC-S model is reduced to ETM.

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.

Flows of a Rarefied Gas between Coaxial Circular Cylinders with Nonuniform Surface Properties

Flows of a rarefied gas between coaxial circular cylinders with nonuniform surface properties are studied on the basis of kinetic theory. It is assumed that the outer cylinder is a diffuse reflection boundary and the inner cylinder is a Maxwell-type boundary whose accommodation coefficient varies in the circumferential direction. Three fundamental flows are studied: 1) a flow caused by the rotation of the outer cylinder (Couette flow), 2) a flow induced between the cylinders at rest kept at different temperatures (heat transfer problem), and 3) a flow induced by the circumferential temperature distribution along the cylindrical surfaces (thermal creep flow). The linearized ES-BGK model of the Boltzmann equation is numerically analyzed using a finite difference method. The time-independent behavior of the gas is studied over a wide range of the gas rarefaction degree, the radii ratio, and a parameter characterizing the distribution of the accommodation coefficient. Due to an effect of nonuniform surface properties, a local heat transfer occurs between the gas and the cylindrical surfaces in Couette flow;a local tangential stress arises in the heat transfer problem. However, the total heat transfer between the two cylinders in Couette flow and the total torque acting on the inner cylinder in the heat transfer problem vanish irrespective of the flow parameters. Two nondegenerate reciprocity relations arise due to the effect of nonuniform surface properties. The reciprocity relations among the above-mentioned three flows are numerically confirmed over a wide range of the flow parameters. The force on the inner cylinder, which also arises due to the effect of nonuniform surface properties in Couette flow and the heat transfer problems, is studied.

Thermal Electromagnetic Radiation of Rarefied Gas

The Boltzmann kinetic equation for rarefied radiating gas is found. It is shown, that process of radiation is defined by excitation of atoms at their collision, and also spontaneous radiation of quantums at transition of electrons to the basic power level and the compelled radiation of quantums at collision of the excited atoms. It is shown, that distributions on velocities of the excited and not excited atoms submit to various laws. Distinctions in laws of distribution of the excited and not excited atoms define power parameters of radiating gas, and also a share of radiating molecules in gas.

3D DSMC Simulation of Rarefied Gas Flows around a Space Crew Capsule Using OpenFOAM

An open source Direct Simulation Monte Carlo (DSMC) code, called as dsmcFoam in OpenFOAM, is used to study a blunt body with the shape of a space crew capsule return vehicle. The rarefied gas has the Knudsen number with 0.03. The flow with a Mach number 4.35 over the capsule was simulated by DSMC. The distributions of velocity field and temperature around the capsule were calculated. This study may provide some useful information for the reentry of the return vehicle.

A Closed-Form Solution of a Kinetic Integral Equation for Rarefied Gas Flow in a Cylindrical Duct

A spectral method based on Hermite cubic splines expansions combined with a collocation scheme is used to develop a solution for the vector form integral S-model kinetic equation describing rarefied gas flows in cylindrical geometry. Some manipulations are made to facilitate the computational treatment of the singularities inherent to the kernel. Numerical results for the simulation of flows generated by pressure and thermal gradients, Poiseuille and thermal-creep problems, are presented.

Variation character of stagnation point heat flux for hypersonic pointed bodies from continuum to rarefied flow states and its bridge function study

This paper is a research on the variation character of stagnation point heat flux for hypersonic pointed bodies from continuum to rarefied flow states by using theoretical analysis and numerical simulation methods. The newly developed near space hypersonic cruise vehicles have sharp noses and wingtips,which desires exact and relatively simple methods to estimate the stagnation point heat flux. With the decrease of the curvature radius of the leading edge,the flow becomes rarefied gradually,and viscous interaction effects and rarefied gas effects come forth successively,which results in that the classical Fay-Riddell equation under continuum hypothesis will become invalid and the variation of stagnation point heat flux is characterized by a new trend. The heat flux approaches the free molecular flow limit instead of an infinite value when the curvature radius of the leading edge tends to 0. The physical mechanism behind this phenomenon remains in need of theoretical study. Firstly,due to the fact that the whole flow regime can be described by Boltzmann equation,the continuum and rarefied flow are analyzed under a uniform framework. A relationship is established between the molecular collision insufficiency in rarefied flow and the failure of Fourier's heat conduction law along with the increasing significance of the nonlinear heat flux. Then based on an inspiration drew from Burnett approximation,control factors are grasped and a specific heat flux expression containing the nonlinear term is designed in the stagnation region of hypersonic leading edge. Together with flow pattern analysis,the ratio of nonlinear to linear heat flux Wr is theoretically obtained as a parameter which reflects the influence of nonlinear factors,i.e. a criterion to classify the hypersonic rarefied flows. Ultimately,based on the characteristic parameter Wr,a bridge function with physical background is constructed,which predicts comparative reasonable results in coincidence well with DSMC and experimental data in the whole flow regime.

An analytical study on nonequilibrium dissociating gas flow behind a strong bow shockwave under rarefied conditions

In this paper, an explicitly analytical shock mapping relation is approximately deduced based on the theoretical modeling of the chemical nonequilibrium stagnation flow towards a slightly blunted nose. Based on the relation, the complex reacting stagnation flow problem can be discussed under the framework of the simplest normal shockwave flow. Therefore, a quantita- tively meaningful criterion for dissociation nonequilibrium flow, that is a specific Damk hler number Da d , is naturally intro- duced as the ratio of the mapping length of the stagnation streamline and the characteristic nonequilibrium scale. Da d is found to be dependent on the flow's rarefaction criterion W r , that is a specific Knudsen number. Then, based on Da d , a normalized analytical formulation is obtained to quantitatively predict the actual degrees of dissociation at the outer edge of the stagnation point boundary layer (SPBL). At last, the direct simulation Monte Carlo (DSMC) method is employed to validate the analytical results, and the related flow mechanism is discussed. The present study not only shows nonequilibrium features of the flow problem, but also provides an indispensable basis for the following study on the nonequilibrium SPBL heat transfer.

Analytical method of nonlinear coupled constitutive relations for rarefied non-equilibrium flows

It is well known that Navier-Stokes equations are not valid for those high-Knudsen and high-Mach flows, in which the local thermodynamically non-equilibrium effects are dominant. To extend the non-equilibrium describing the ability of macroscopic equations, Nonlinear Coupled Constitutive Relation(NCCR) model was developed from Eu’s generalized hydrodynamic equations to substitute linear Newton’s law of viscosity and Fourier’s law of heat conduction in conservation laws. In the NCCR model, how to solve the decomposed constitutive equations with reasonable computational cost is a key ingredient of this scheme. In this paper, an analytic method is proposed firstly. Compared to the iterative procedure in the conventional NCCR model, the analytic method not only obtains exact roots of the decomposed constitutive polynomials, but also preserves the nonlinear constitutive relations in the original framework of NCCR methods. Numerical tests to assess the efficiency and accuracy of the proposed method are conducted for argon shock structures, Couette flows, two-dimensional hypersonic flows over a cylinder and threedimensional supersonic flows over a three-dimensional sphere. These superior advantages of the current method are expected to render itself a powerful tool for simulating the hypersonic rarefied flows and microscale flows of high Knudsen number for engineering applications.

Rarefied gas effect in hypersonic shear flows

Recently,as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude,the numerical simulation based on the Navier-Stokes(NS)equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime.As first noted by Zhou and Zhang(Science in China,2015),the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows.In this paper,we present the scope,physical concept,mathematical model of this shear non-equilibrium effect in hypersonic flows,as well as the way of considering this effect in conventional computational fluid mechanics(CFD)for engineering applications.Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model,named direct simulation Monte Carlo(DSMC)data-improved Navier-Stokes(DiNS)model.

Theoretical modeling of chemical nonequilibrium stagnation point boundary layer heat transfer under rarefied conditions

This paper theoretically studies the recombination-dominated nonequilibrium reacting flow inside the stagnation point bound- ary layer （SPBL） and the heat transfer characteristics under rarefied conditions. A general model is intuitively proposed to de- scribe the energy transfer and conversion along the stagnation streamline towards a slightly blunted nose with non-catalytic wall surface. It is found that the atoms recombination effects inside the SPBL could be equivalent to a modification on the de- gree of dissociation in the external flow. As a result, a recombination nonequilibrium criterion Dar, that is a specific DamktSh- let number, is introduced to characterize the nonequilibrium degree of the reacting flow in the SPBL, and then, based on the general model and Dar, a bridging function indicating the nonequilibrium chemical effects on the SPBL heat transfer is estab- lished. By using the explicitly analytical bridging function, the flow and heat transfer mechanisms, including the real gas flow similarity law and the nonequilibrium flow regimes classification, are discussed. In addition, the direct simulation Monte Carlo （DSMC） method has also been employed to systematically validate the analytical results.

DSMC modeling of rarefied ionization reactions and applications to hypervelocity spacecraft reentry flows

The DSMC modeling is developed to simulate three-dimensional(3D)rarefied ionization flows and numerically forecast the communication blackout around spacecraft during hypervelocity reentry.A new weighting factor scheme for rare species is introduced,whose key point is to modify the corresponding chemical reaction coefficients involving electrons,meanwhile reproduce the rare species in resultants and preserve/delete common species in reactants according to the weighting factors.The resulting DSMC method is highly efficient in simulating weakly inhomogeneous flows including the Couette shear flow and controlling statistical fluctuation with high resolution.The accurate reliability of the present DSMC modeling is also validated by the comparison with a series of experimental measurements of the Shenzhou reentry capsule tested in a low-density wind tunnel from the HAI of CARDC.The obtained electron number density distribution for the RAM-C II vehicle agrees well with the flight experiment data,while the electron density contours for the Stardust hypervelocity reentry match the reference data completely.In addition,the present 3D DSMC algorithm can capture distribution of the electron,N+and O+number densities better than the axis-symmetric DSMC model.The introduction of rare species weighting factor scheme can significantly improve the smoothness of the number density contours of rare species,especially for that of electron in weak ionization case,while it has negligible effect on the macroscopic flow parameters.The ionization characteristics of the Chinese lunar capsule reentry process are numerically analyzed and forecasted in the rarefied transitional flow regime at the flying altitudes between 80 and 97 km,and the simulations predict communication blackout altitudes which are in good agreement with the actual reentry flight data.For the spacecraft reentry with hypervelocity larger than the second cosmic speed,it is forecasted and verified by the present DSMC modeling that ionization reactions will cover the windward capsule surface,leading to reentry communication blackout,and the communication interruption must be considered in the communication design during reentry in rarefied flow regimes.

Data-driven nonlinear constitutive relations for rarefied flow computations

To overcome the defects of traditional rarefied numerical methods such as the Direct Simulation Monte Carlo(DSMC)method and unified Boltzmann equation schemes and extend the covering range of macroscopic equations in high Knudsen number flows,data-driven nonlinear constitutive relations(DNCR)are proposed first through the machine learning method.Based on the training data from both Navier-Stokes(NS)solver and unified gas kinetic scheme(UGKS)solver,the map between responses of stress tensors and heat flux and feature vectors is established after the training phase.Through the obtained off-line training model,new test cases excluded from training data set could be predicated rapidly and accurately by solving conventional equations with modified stress tensor and heat flux.Finally,conventional one-dimensional shock wave cases and two-dimensional hypersonic flows around a blunt circular cylinder are presented to assess the capability of the developed method through various comparisons between DNCR,NS,UGKS,DSMC and experimental results.The improvement of the predictive capability of the coarsegraining model could make the DNCR method to be an effective tool in the rarefied gas community,especially for hypersonic engineering applications.

On the accuracy of macroscopic equations for linearized rarefied gas flows

Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level,either from the mesoscopic Boltzmann equation or some physical arguments,including(i)Burnett,Woods,super-Burnett,augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation,(ii)Grad 13,regularized 13/26 moment equations,rational extended thermodynamics equations,and generalized hydrodynamic equations,where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials,and(iii)bi-velocity equations and“thermo-mechanically consistent"Burnett equations based on the argument of“volume diffusion”.This paper is dedicated to assess the accuracy of these macroscopic equations.We first consider the RayleighBrillouin scattering,where light is scattered by the density fluctuation in gas.In this specific problem macroscopic equations can be linearized and solutions can always be obtained,no matter whether they are stable or not.Moreover,the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem.Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation.We find that(i)the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion,(ii)for the moment method,the more moments are included,the more accurate the results are,and(iii)macroscopic equations based on“volume diffusion"do not work well even when the Knudsen number is very small.Therefore,among about a dozen tested equations,the regularized 26 moment equations are the most accurate.However,for moderate and highly rarefied gas flows,huge number of moments should be included,as the convergence to true solutions is rather slow.The same conclusion is drawn from the problem of sound propagation between the transducer and receiver.This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function.This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

Structure of low-head two-phase flow in flat nozzlesduring outflow of the liquid into rarefied medium

Outcomes of experimental researches of the low-pressure adiabatic flow of the boiling liquid through two-dimensional Laval nozzles in a vacuum atmosphere were adduced.Requirements of critical conditions of flow were determined.Structural forms of a stream were investigated and their connection with crisis of flow was shown.It was established periodic non-stationary macrostructures of a stream which was stipulated by the rotational gear of origin of a vapor phase.