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.

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.

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.

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.

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.

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.

Boosting the convergence of low-variance DSMC by GSIS

The low-variance direct simulation Monte Carlo(LVDSMC)is a powerful method to simulate low-speed rarefied gas flows.However,in the near-continuum flow regime,due to limitations on the time step and spatial cell size,it takes plenty of time to find the steady-state solution.Here we remove these deficiencies by coupling the LVDSMC with the general synthetic iterative scheme(GSIS)which permits the simulation at the hydrodynamic scale rather than the much smaller kinetic scale.As a proof of concept,we propose the stochastic-deterministic coupling method based on the Bhatnagar-Gross-Krook kinetic model.First,macroscopic synthetic equations are derived exactly from the kinetic equation,which not only contain the Navier-Stokes-Fourier constitutive relation,but also encompass the higher-order terms describing the rarefaction effects.Then,the high-order terms are extracted from LVDSMC and fed into synthetic equations to predict the macroscopic properties which are closer to the steady-state solution than LVDSMC.Finally,the state of simulation particles in LVDSMC is updated to reflect the change of macroscopic properties.As a result,the convergence to steady state is greatly accelerated,and the restrictions on cell size and the time step are removed.We conduct the Fourier stability analysis and simulate several canonical rarefied gas flows to demonstrate the advantages of LVDSMC-GSIS:when the Knudsen number is lower than 0.1,it can use the grid size about 10 times larger than that in traditional DSMC,and it can reduce the computational cost by two orders of magnitude in the flow regime.