A popular technique: hybrid method Limitations Numerical rather than physical Artifacts Time coupling MD NS Dynamic scale changes International Journal for Multiscale Computational Engine Athough hybrid methods provide signifi- Hadjiconstantinou cant savings by limiting molecular solutions IntJ Multiscale Comput Eng 3 189-202, 2004 only to the regions where they are needed, so- lution of time-evolving problems, which span a large range of timescales, is still not possible Discussion of Hybrid if the molecular domain, however small needs Atomistic-Continuum method to be integrated for the total time of interest for Multiscale hydrodynamics Hybrid method is inappropriate for problems with dynamic scale changes
A popular technique: hybrid method MD NS Limitations Artifacts Time coupling Numericalrather than physical Dynamic scale changes Hadjiconstantinou Int J Multiscale Comput Eng 3 189-202, 2004 Hybrid method is inappropriate for problems with dynamic scale changes
Efforts based on kinetic description of flows Discrete Ordinate Method(DOM)(1 2 Time-splitting scheme for kinetic equations (similar with DSMc dt(time step<(collision time) dx cell size)< (mean-free-path numerical dissipation dt Works well for highly non-equilibrium flows, but encounters difficult fo continuum flows Asymptotic preserving (AP)scheme 3,4) Consistent with the chapman -Enskog representation in the continuum limit (Kn→0) dt (time step) is not restricted by(collision time at least 2nd-order accuracy to reduce numerical dissipation [51 Aims to solve continuum flows, but may encounter difficulties for free molecular flows [1]J. Y. Yang and J. C Huang, J. Comput. Phys. 120, 323(1995 [2]A. N Kudryavtsev and A. A Shershnev, J. Sci. Comput. 57, 42(2013) [3]S Pieraccini and G. Puppo, J. Sci. Comput. 32, 1(2007) [4]M. Bennoune, M. Lemo, and L Mieussens, J Comput. Phys. 227, 3781(2008) [5]K Xu and J -C Huang, J. Comput. Phys. 229, 7747(2010)
Efforts based on kinetic description of flows # Discrete Ordinate Method (DOM) [1,2] : • Time-splitting scheme for kinetic equations (similar with DSMC) • dt (time step) < (collision time) • dx (cell size) < (mean-free-path) • numerical dissipation dt # Asymptotic preserving (AP) scheme [3,4] : Works well for highly non-equilibrium flows, but encounters difficult for continuum flows Aimsto solve continuum flows, but may encounter difficultiesfor free molecular flows • Consistent with the Chapman-Enskog representation in the continuum limit (Kn → 0) • dt (time step) is not restricted by (collision time) • at least 2 nd -order accuracy to reduce numerical dissipation [5] [1] J. Y. Yang and J. C. Huang, J. Comput. Phys. 120, 323 (1995) [2] A. N. Kudryavtsev and A. A. Shershnev, J. Sci. Comput. 57, 42 (2013). [3] S. Pieraccini and G. Puppo, J. Sci. Comput. 32, 1 (2007). [4] M. Bennoune, M. Lemo, and L. Mieussens, J. Comput. Phys. 227, 3781 (2008). [5] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010)
Efforts based on kinetic description of flows Unified Gas-Kinetic Scheme(UGKS)(; Coupling of collision and transport in the evolution Dynamicly changes from collision-less to continuum according to the local flow The nice ap property a dynamic multi-scale scheme, efficient for multi-regime flows In this report, we will present an alternative kinetic scheme (Discrete Unified Gas-Kinetic Scheme), sharing many advantages of the UGKS method but having some special features [1]K Xu and J -C Huang, J Comput. Phys. 229, 7747(2010)
# Unified Gas-Kinetic Scheme (UGKS) [1] : Efforts based on kinetic description of flows A dynamic multi-scale scheme, efficientfor multi-regime flows • Coupling of collision and transport in the evolution • Dynamicly changes from collision-less to continuum according to the local flow • The nice AP property [1] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010) In this report, we will present an alternative kinetic scheme (Discrete Unified Gas-Kinetic Scheme), sharing many advantages of the UGKS method, but having some special features
Outline ● Motivation Formulation and properties Numerical results ● Summary
Outline • Motivation • Formulation and properties • Numerical results • Summary
Kinetic model ( bGK-type ?x籽f=W?籐fq Distribution function f= f(x, x, t) Particel velocity Equilibrium: eq- fx,W(x,t),J(x,1),…J Conserved variables Flux Example eq = fu 2pRT)3+Ky2 exp Maxwell (standard BGK) ERT (1-Pr) c xq cc Shakhoy model 5pRT Rt ES eXpc語? ES model det(2pL) LE E RT
# Kinetic model (BGK-type) 1 eq t f f f f t ? 籽 = W ? - 轾 x 臌 f f t = ( , , ) x x Distribution function Particel velocity [ , ( , ), ( , ),...] eq eq Equilibrium: f f t t = x W x J x Conserved variables Flux Example: 2 (3 )/ 2 exp (2 ) 2 eq M K c f f R T R T r p + 轾 = = -犏犏臌 Maxwell (standard BGK) 2 1 (1 P r) 5 5 eq S M c f f f pR T R T 殒 鳄 犏 × ç ÷ = = + - - ? 犏 çç桫 ÷ 臌 c q Shakhov model ES model 1 exp det (2 ) 2 eq ES f f r p 轾 = = -犏 譒 ? L 犏臌 c c 1 1 P r ij ij ij R T d s 骣ç ÷ L = + - ? çç桫 ÷