Discrete unified gas-kinetic scheme for compressible flows

Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and related Problems, May 21-27, 2014, Beijing, china Discrete unified gas-kinetic scheme for compressible flows haoli guo (Huazhong University of Science and Technology Wuhan, China Joint work with Kun Xu and Ruijie Wang(Hong Kong University of Science and Technology

Discrete unified gas-kinetic scheme for compressible flows Zhaoli Guo (Huazhong University of Science and Technology, Wuhan, China) Joint work with Kun Xu and Ruijie Wang (Hong Kong University of Science and Technology) Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Outline ● Motivation Formulation and properties ● Numerical resu|ts Summary

Outline • Motivation • Formulation and properties • Numerical results • Summary

Motivation Non-equilibrium flows covering different flow regimes Re-Entry vehicle Chips Inhalable particles 103 10 101 10 10

Motivation Non-equilibrium flows covering different flow regimes Re-Entry Vehicle Chips Inhalable particles 10 10 -3 10-1 100 10-2

Challenges in numerical simulations Modern cfd: Based on Navier-Stokes equations Efficient for continuum flows does not work for other regimes Particle Methods: (MD, DSMC.) · noise · Small time and cell size Difficult for continuum flows /low-speed non-equilibrium flows Method based on extended hydrodynamic models: Theoretical foundations Numerical difficulties( Stability boundary conditions Limited to weak-nonequilibrium flows

Challenges in numerical simulations • Based on Navier-Stokes equations • Efficient for continuum flows • does not work for other regimes • Noise • Small time and cell size • Difficult for continuumflows / low-speed non-equilibrium flows Modern CFD: Particle Methods: (MD, DSMC… ) • Theoretical foundations • Numerical difficulties (Stability, boundary conditions,……) • Limited to weak-nonequilibrium flows Method based on extended hydrodynamic models :

Lockerby's test (2005, Phys. Fluid) the most common high-order continuum equation sets(Grad's 13 th moment, Burnett, and super-Burnett equations)cannot capture the Knudsen Layer, Variants of these equation families have however, been proposed and some of them can qualitatively describe the Knudsen layer structure. the quantitative agreement with kinetic theory and dSmc data is only slight result from kinetic theory. We find that, for a benchmark case, the most common higher-order continuum equation sets(Grads 13 moment, Burnett, and super-Burnett equations) cannot capture the Knudsen layer. Variants of these equation families have, however, been proposed and some of sm (Pe sul a suin IMt et Nar stles lan ll wats them can qualitatively describe the Knudsen layer structure. To make quantitative comparisons, we f-h artaud lume t- b NGk Biwa i-l Ikale ef the ILN obtain additional boundary conditions(needed for unique solutions to the higher-order equations) from kinetic theory. However, we find the quantitative agreement with kinetic theory and DSMc ata is only slight. o 2005 American Institute of Physics. [ DOI: 10.1063/1.]

Lockerby’s test (2005, Phys. Fluid)  = const the most common high-order continuum equation sets (Grad’s 13 moment, Burnett, and super-Burnett equations ) cannot capture the Knudsen Layer, Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure … the quantitative agreement with kinetic theory and DSMC data is only slight

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 ÷