A New Gas-Kinetic Scheme for Multifluid Flows

MNMCFF2014 A New Gas-Kinetic Scheme for Multifluid Flows Qibing Li Department of Engineering Mechanics, Tsinghua University, Beijing 100084 lqb@tsinghua.edu.cn 2014.5 Beijing

Content • Introduction • Gas-kinetic scheme • Gas-kinetic scheme for multifluid flow • Numerical tests • Conclusions

Introduction • Compressible multifluid flow is important for many engineering applications, thus attracts many numerical researches • Inertial-confinement-fusion (ICF) • Underwater gas jet flow • Challenges to CFD • Pressure oscillation at material interface • Conservation • Robustness

Existing numerical methods based on macro variables • Solving each component separately, with the help of interface capture/tracking technique • Level set, VOF, GFM, … • Flow mixing ? • Directly computing the multifluid mixture • Sharp interface ? • Source term for interfacial physics ? • Challenges/open questions of FVM • Entropy satisfaction ? • Consistent numerical dissipation ? • Genuinelymultidimensional scheme ?

FLUID MODELING Continuum Models Molecular Models Euler Deterministic Navier-Stokes Statistical Burnett Liouville MD Chapman-Enskog DSMC Boltzmann BGK

Based on kinetic theory for micro particles, many new CFD methods have been constructed and show good performance • LBM, KFVS, DSMC, GKS (Gas-kinetic scheme), … • Gas-kinetic scheme (K. Xu, JCP2001) • Satisfaction of entropy condition inherently • Coupling of free movement and collision of particles • Easy to develop a genuinely multidimensional scheme • 2nd-order accuracy both in spatial and temporal directions • Wide applied fields • Hypersonic viscous flow, chemical reactive flow, rarefied flow, MHD, shallow water flow, …

Gas-kinetic scheme for multimaterial flows • Perfect gases mixture: with a scalar to represent mass fraction of different gas (Lian & Xu 2000; Jiang & Ni 2004; Li, Fu & Xu 2005) • Two mass species (Xu 1997; Kotelnikov & Montgomery 1997) • General EOS for KFVS (Chen & Jiang 2011) • Baer-Nunziato two-phase flow (Pan, Zhao, Tian & Wang 2012; Pan, Zhao & Wang 2013) • Gas/water with stiffened EOS(Li & Fu 2011) • Numerical mixing at cell interface, resulting in numerical oscillation when cell size is much larger than interface width • Objective of the present study • To construct a new gas-kinetic scheme for multimaterial flow with stiffened EOS

y F x o Gas-Kinetic Scheme • Brief description BGK equation

BGK equation • The distribution function f is the possibility of a particle with velocity u and internal freedom ξ at physical space x and time t . g is the Maxwellian distribution: • τ is the collision time • Macro quantities can be obtained through integration

Integral solution of BGK equation • Key steps to solve BGK equation Q1D/DS

GKS for Multifluid Flow • Considering gas and water, equation of state • Perfect gas model • Stiffened water model • Equilibrium assumption for a mixture model • The gas and water achieve equilibrium state with equal temperature, pressure and velocity inside a computational cell during a time step • Volume fraction of gas is denoted by α • Phase transition is not considered

Kinetic method for gas/water flow • Gas and water are described by distribution functions separately • Conservative variables

Flux at cell interface: Stratified model (JCP 1984, Stewart & Wendroff; JCP 2003, Abgrall & Saurel; JCP 2007, Chang & Liou) • Flux comes from three different interactions: gas-gas, water-water and gas-water interaction • Gas-water model ? • Riemann solver • Non-mixing kinetic model Water Water

Non-mixing kinetic model • Incident particles bounce back from the material interface (e.g. specular reflection) • Velocity of interface can be determined by the constraint for pressure equilibrium, which is computed with the incident distribution function • Densities at both sides can also be determined • Flux for gas-water can be calculated by pressure, velocity and densities at the interface Uinterface

Advantages for the present model • Non-oscillation of pressure at interface • Conservation • Easy to extend to high-order accuracy or multidimensional flow • Easy to consider viscous effect inside each component

Numerical Tests • Shock tube for gas and water • Interaction of shock and cylindrical water column • Interaction of a cylindrical converging shock and SF6 bubble

Water-air shock tube (Chang & Liou 2007) C1 t=0.0023s

C2 t=0.0027s

Interaction of shock and cylindrical water columns Numerical Schlieren （t=0.02ms、0.04ms、0.06ms、0.0115ms） 19

Interaction of a cylindrical converging shock and SF6 bubble Initial density Volume fraction（t=0.015） Density（t=0.015）

Volume fraction （t=0.01） Density（t=0.01）

Conclusions • A non-mixing kinetic model for gas-water interface is proposed including a stiffened EOS; • A new gas-kinetic scheme for gas-water flow is developed based on a mixture model; • The applications in several typical flow problems show its good performance. • Further validation and improvement are under study • 3D • Efficiency (AMR) ?

Thanks!

Remarks • With the increasing of time, the flux function shows a transition from particle free transport to equilibrium state, like upwind to central manner, thus a smart time-dependent dissipation is included • The above-mentioned scheme is a genuinely multidimensional scheme. If remove the coefficients related to tangential slopes, one can obtain the DS or Q1D scheme • Boundary conditions: usual B.Cs, or kinetic B.Cs • The solution is a combination of Maxwellian distributions, thus the macro variables and flux can be calculated easily,which means the computational cost is comparable with traditional scheme based on macro variables • The present scheme is a one-step method, with 2nd accuracy both in space and time. Only ONE integral point (the center of the cell interface) is required • The BGK scheme inherits good parallel performance 24

Chapman-Enskog expansion at NS eqs. Level: 25 25

2nd-order GKS-NS 1st-order Taylor expansion H[x]: Heaviside function 26 26

A New Gas-Kinetic Scheme for Multifluid Flows