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.

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.

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.

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.

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.

THERMAL CREEP FLOW IN THE RAREFIED GAS

The usual heat flow moves along the direction from high temperature place to the low one,as often observed in the daily life.However,when the gas is very rarefied,the gas may move along a different way,that is,the so-called thermal creep flow moves along the direction from the low temperature place to the high one.In this note,we will survey our recent mathematical works on this topic,mainly based on[27]and[25].

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.

Spacecraft aerodynamics and trajectory simulation during aerobraking

This paper uses a direct simulation Monte Carlo （DSMC） approach to simulate rarefied aerodynamic characteristics during the aerobraking process of the NASA Mars Global Surveyor （MGS） spacecraft. The research focuses on the flowfield and aerodynamic characteristics distribution under various free stream densities. The vari- ation regularity of aerodynamic coefficients is analyzed. The paper also develops an aerodynamics-aeroheating-trajectory integrative simulation model to preliminarily calculate the aerobraking orbit transfer by combining the DSMC technique and the classical kinematics theory. The results show that the effect of the planetary atmospheric density, the spacecraft yaw, and the pitch attitudes on the spacecraft aerodynamics is significant. The numerical results are in good agreement with the existing results reported in the literature. The aerodynamics-aeroheating-trajectory integrative simulation model can simulate the orbit transfer in the complete aerobraking mission. The current results of the spacecraft trajectory show that the aerobraking maneuvers have good performance of attitude control.

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.

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.

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.

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.

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.

A mean free path approach to the micro/nanochannel gas flows

We investigate the gas flows near to solid surfaces in terms of the local spatial variation in the molecular mean free path(MFP).Molecular dynamics(MD)is the appropriate scientific tool for obtaining molecularly-accurate dynamic information in micro and nano-scale gas flows,and has been used to evaluate the molecular mean free path of gases.In the calibration procedure,the viscosity of a gas in the homogeneous case can be recovered in our MD simulations and reach good agreement with the theoretical prediction and data from NIST.In surface-bounded gas flows,if the collisions between gas molecules and walls are counted,a spatially-varying mean free path is presented,and for the first time we have observed that the distribution of the free paths deviates from the exponential one and spikes appear in their distributions at larger Kn,i.e.in the transition flow regime.Based on elementary kinetic theory,the effective viscosity of the gas derived from the mean free path has been incorporated into the framework of the continuum-fluid dynamics equations,and micro-Couette flows are performed to demonstrate this potential application.

Numerical simulation of hypersonic thermochemical nonequilibrium flows using nonlinear coupled constitutive relations

To predict aeroheating performance of hypersonic vehicles accurately in thermochemical nonequilibrium flows accompanied by rarefaction effect,a Nonlinear Coupled Constitutive Relations(NCCR)model coupled with Gupta’s chemical models and Park’s two-temperature model is firstly proposed in this paper.Three typical cases are intensively investigated for further validation,including hypersonic flows over a two-dimensional cylinder,a RAM-C II flight vehicle and a type HTV-2 flight vehicle.The results predicted by NCCR solution,such as heat flux coefficient and electron number densities,are in better agreement with those of direct simulation Monte Carlo or flight data than Navier-Stokes equations,especially in the extremely nonequilibrium regions,which indicates the potential of the newly-developed solution to capture both thermochemical and rarefied nonequilibrium effects.The comparisons between the present solver and NCCR model without a two-temperature model are also conducted to demonstrate the significance of vibrational energy source term in the accurate simulation of high-Mach flows.

Numerical study of conduction and radiation heat losses from vacuum annulus in parabolic trough receivers

Parabolic trough receiver is a key component to convert solar energy into thermal energy in the parabolic trough solar system.The heat loss of the receiver has an important influence on the thermal efficiency and the operating cost of the power station.In this paper,conduction and radiation heat losses are analyzed respectively to identify the heat loss mechanism of the receiver.A 2-D heat transfer model is established by using the direct simulation Monte Carlo method for rarefied gas flow and heat transfer within the annulus of the receiver to predict the conduction heat loss caused by residual gases.The numerical results conform to the experimental results,and show the temperature of the glass envelope and heat loss for various conditions in detail.The effects of annulus pressure,gas species,temperature of heat transfer fluid,and annulus size on the conduction and radiation heat losses are systematically analyzed.Besides,the main factors that cause heat loss are analyzed,providing a theoretical basis for guiding the improvement of receiver,as well as the operation and maintenance strategy to reduce heat loss.

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.

Convergence ofADistributional Monte Carlo Method for the Boltzmann Equation

Direct Simulation Monte Carlo(DSMC)methods for the Boltzmann equation employ a point measure approximation to the distribution function,as simulated particles may possess only a single velocity.This representation limits the method to converge only weakly to the solution of the Boltzmann equation.Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and L∞solutions of the Boltzmann equation.This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC.We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods.Toward this end,we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

Analytical Solution for the Lattice Boltzmann Model Beyond Naviers-Stokes

To understand lattice Boltzmann model capability for capturing nonequilibrium effects,the model with first-order expansion of the equilibrium distribution function is analytically investigated.In particular,the velocity profile of Couette flows is exactly obtained for the D2Q9 model,which shows retaining the first order expansion can capture rarefaction effects in the incompressible limit.Meanwhile,it clearly demonstrates that the D2Q9 model is not able to reflect flow characteristics in the Knudsen layer.

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.