Based on the stability analysis with no linearization and expansion, it is argued that instability in the lattice BGK model is originated from the linear relaxation hypothesis of collision in the model. The hypothesis...Based on the stability analysis with no linearization and expansion, it is argued that instability in the lattice BGK model is originated from the linear relaxation hypothesis of collision in the model. The hypothesis stands up only when the deviation from the local equilibrium is weak. In this case the computation is absolutely stable for real fluids. But for flows of high Reynolds number, this hypothesis is violated and then instability takes place physically. By performing a transformation a quantified stability criteria is put forward without those approximation. From the criteria a sufficient condition for stability can be obtained and serve as an estimation of the limited Reynolds number as high as possible.展开更多
The Boltzmann-Bhatnagar-Gross-Krook（BGK）model is investigated for its validity regarding the collision term approximation through relaxation evaluation. The evaluation is based on theoretical analysis and numerical ...The Boltzmann-Bhatnagar-Gross-Krook（BGK）model is investigated for its validity regarding the collision term approximation through relaxation evaluation. The evaluation is based on theoretical analysis and numerical comparison between the BGK and direct simulation Monte Carlo（DSMC） results for three specifically designed relaxation problems. In these problems, one or half component of the velocity distribution is characterized by another Maxwellian distribution with a different temperature. It is analyzed that the relaxation time in the BGK model is unequal to the molecular mean collision time. Relaxation of component distribution fails to involve enough contribution from other component distributions, which makes the BGK model unable to capture details of velocity distribution, especially when discontinuity exists in distribution. The BGK model,however, predicts satisfactory results including fluxes during relaxation when the temperature difference is small. Particularly, the model-induced error in the BGK model increases with the temperature difference, thus the model is more reliable for low-speed rarefied flows than for hypersonic flows.展开更多
In this paper,we present a conservative semi-Lagrangian finite-difference scheme for the BGK model.Classical semi-Lagrangian finite difference schemes,coupled with an L-stable treatment of the collision term,allow lar...In this paper,we present a conservative semi-Lagrangian finite-difference scheme for the BGK model.Classical semi-Lagrangian finite difference schemes,coupled with an L-stable treatment of the collision term,allow large time steps,for all the range of Knudsen number[17,27,30].Unfortunately,however,such schemes are not conservative.Lack of conservation is analyzed in detail,and two main sources are identified as its cause.First,when using classical continuous Maxwellian,conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points.However,for a small number of grid points in velocity space such error is not negligible,because the parameters of the Maxwellian do not coincide with the discrete moments.Secondly,the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme.As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number.To treat the first problem and ensure machine precision conservation of mass,momentum and energy with a relatively small number of velocity grid points,we replace the continuous Maxwellian with the discrete Maxwellian introduced in[22].The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in[26].In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving(AP)for the underlying Euler limit,as the Knudsen number vanishes.The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.展开更多
In this paper,a lattice Boltzmann BGK(LBGK)model is proposed for simulating incompressible axisymmetric flows.Unlike other existing axisymmetric lattice Boltzmann models,the present LBGK model can eliminate the compre...In this paper,a lattice Boltzmann BGK(LBGK)model is proposed for simulating incompressible axisymmetric flows.Unlike other existing axisymmetric lattice Boltzmann models,the present LBGK model can eliminate the compressible effects only with the smallMach number limit.Furthermore the source terms of themodel are simple and contain no velocity gradients.Through the Chapman-Enskog expansion,the macroscopic equations for incompressible axisymmetric flows can be exactly recovered fromthe present LBGKmodel.Numerical simulations of theHagen-Poiseuille flow,the pulsatile Womersley flow,the flow over a sphere,and the swirling flow in a closed cylindrical cavity are performed.The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.展开更多
A new discrete Boltzmann model, the discrete ellipsoidal statistical Bhatnagar-Gross-Krook （ES- BGK） model, is proposed to simulate nonequilibrium compressible flows. Compared with the original discrete BGK model, t...A new discrete Boltzmann model, the discrete ellipsoidal statistical Bhatnagar-Gross-Krook （ES- BGK） model, is proposed to simulate nonequilibrium compressible flows. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in the Burnett level, two kinds of discrete velocity model are introduced and the relations between nonequilibrium quantities and the viscous stress and heat flux in the Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, those based on the Navier-Stokes or the Burnett equations.展开更多
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 fiel...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.展开更多
文摘Based on the stability analysis with no linearization and expansion, it is argued that instability in the lattice BGK model is originated from the linear relaxation hypothesis of collision in the model. The hypothesis stands up only when the deviation from the local equilibrium is weak. In this case the computation is absolutely stable for real fluids. But for flows of high Reynolds number, this hypothesis is violated and then instability takes place physically. By performing a transformation a quantified stability criteria is put forward without those approximation. From the criteria a sufficient condition for stability can be obtained and serve as an estimation of the limited Reynolds number as high as possible.
基金supported by the National Natural Science Foundation of China(91116013,11372325,and 11111120080)
文摘The Boltzmann-Bhatnagar-Gross-Krook（BGK）model is investigated for its validity regarding the collision term approximation through relaxation evaluation. The evaluation is based on theoretical analysis and numerical comparison between the BGK and direct simulation Monte Carlo（DSMC） results for three specifically designed relaxation problems. In these problems, one or half component of the velocity distribution is characterized by another Maxwellian distribution with a different temperature. It is analyzed that the relaxation time in the BGK model is unequal to the molecular mean collision time. Relaxation of component distribution fails to involve enough contribution from other component distributions, which makes the BGK model unable to capture details of velocity distribution, especially when discontinuity exists in distribution. The BGK model,however, predicts satisfactory results including fluxes during relaxation when the temperature difference is small. Particularly, the model-induced error in the BGK model increases with the temperature difference, thus the model is more reliable for low-speed rarefied flows than for hypersonic flows.
基金supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1801-02.
文摘In this paper,we present a conservative semi-Lagrangian finite-difference scheme for the BGK model.Classical semi-Lagrangian finite difference schemes,coupled with an L-stable treatment of the collision term,allow large time steps,for all the range of Knudsen number[17,27,30].Unfortunately,however,such schemes are not conservative.Lack of conservation is analyzed in detail,and two main sources are identified as its cause.First,when using classical continuous Maxwellian,conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points.However,for a small number of grid points in velocity space such error is not negligible,because the parameters of the Maxwellian do not coincide with the discrete moments.Secondly,the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme.As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number.To treat the first problem and ensure machine precision conservation of mass,momentum and energy with a relatively small number of velocity grid points,we replace the continuous Maxwellian with the discrete Maxwellian introduced in[22].The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in[26].In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving(AP)for the underlying Euler limit,as the Knudsen number vanishes.The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.
基金One of authors(T.Zhang)is grateful to Liang Wang for useful discussions.This work is financially supported by the National Basic Research Program of China(Grant No.2011CB707305)the National Natural Science Foundation of China(Grant Nos.51006040 and 51006039)the Fundamental Research Funds for the Central Universities,Hust(Grant Nos.2010MS131 and 2010JC005).
文摘In this paper,a lattice Boltzmann BGK(LBGK)model is proposed for simulating incompressible axisymmetric flows.Unlike other existing axisymmetric lattice Boltzmann models,the present LBGK model can eliminate the compressible effects only with the smallMach number limit.Furthermore the source terms of themodel are simple and contain no velocity gradients.Through the Chapman-Enskog expansion,the macroscopic equations for incompressible axisymmetric flows can be exactly recovered fromthe present LBGKmodel.Numerical simulations of theHagen-Poiseuille flow,the pulsatile Womersley flow,the flow over a sphere,and the swirling flow in a closed cylindrical cavity are performed.The results agree well with the analytic solutions and the existing numerical or experimental data reported in some previous studies.
基金The authors would like to acknowledge the support of the National Natural Science Foundation of China （Grant Nos. 11475028, 11772064, 11502117, and U1530261）, and Science Challenge Project （Grant Nos. JCKY2016212A501 and TZ2016002）.
文摘A new discrete Boltzmann model, the discrete ellipsoidal statistical Bhatnagar-Gross-Krook （ES- BGK） model, is proposed to simulate nonequilibrium compressible flows. Compared with the original discrete BGK model, the discrete ES-BGK has a flexible Prandtl number. For the discrete ES-BGK model in the Burnett level, two kinds of discrete velocity model are introduced and the relations between nonequilibrium quantities and the viscous stress and heat flux in the Burnett level are established. The model is verified via four benchmark tests. In addition, a new idea is introduced to recover the actual distribution function through the macroscopic quantities and their space derivatives. The recovery scheme works not only for discrete Boltzmann simulation but also for hydrodynamic ones, for example, those based on the Navier-Stokes or the Burnett equations.
基金The project supported by the National Natural Science Foundation of China (19772059, 19889209)
文摘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.
