Kinetic relaxation models for reacting gas mixtures

Kinetic relaxation models for reacting gas mixtures M. Groppi Department of Mathematics and Computer Science University of Parma - ITALY Main collaborators: Giampiero Spiga, Giuseppe Stracquadanio, Univ. of Parma Giovanni Russo, Univ. of Catania Novi Sad, 26 September 2014 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 1 / 32

Outline 1 BGK Models for reacting gas mixtures M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 2 / 32

Outline 1 BGK Models for reacting gas mixtures 2 Numerical Approximation: High Order Methods for BGK Models M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 2 / 32

Outline 1 BGK Models for reacting gas mixtures 2 Numerical Approximation: High Order Methods for BGK Models 3 ES-BGK model for a binary mixture M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 2 / 32

Introduction BGK Models for reacting gas mixtures The nonlinear Boltzmann equation is an optimal tool of investigation for many gas dynamic regimes M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 3 / 32

Introduction BGK Models for reacting gas mixtures The nonlinear Boltzmann equation is an optimal tool of investigation for many gas dynamic regimes Relaxation models of BGK type are flexible and reliable kinetic approximations: based on much simpler collision operators, they retain the essential features of the Boltzmann equation (conservation laws, collision equilibria, H-theorem) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 3 / 32

BGK Models for reacting gas mixtures Introduction The nonlinear Boltzmann equation is an optimal tool of investigation for many gas dynamic regimes Relaxation models of BGK type are flexible and reliable kinetic approximations: based on much simpler collision operators, they retain the essential features of the Boltzmann equation (conservation laws, collision equilibria, H-theorem) In spite of severe basic difficulties, standard BGK approaches can be extended to polyatomic molecules, gas mixtures, reactive flows (Andries Aoki Perthame 2002, G. Spiga 2004, Bisi G. Spiga 2009, Bisi Caceres 2014), including the simple bimolecular reaction A 1 + A 2 A 3 + A 4 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 3 / 32

BGK Models for reacting gas mixtures Reactive Boltzmann equations f i t + v f i = Qi ME + Qi CH Q i i = 1,..., 4. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 4 / 32

BGK Models for reacting gas mixtures Reactive Boltzmann equations f i t + v f i = Qi ME + Qi CH Q i i = 1,..., 4. Seven conservation laws (Q i + Q j ) dv = 0, (i, j) = (1, 3), (1, 4), (2, 4), 4 m i vq i dv = 0, i=1 4 i=1 ( 1 2 m iv 2 + E i ) Q i dv = 0. Seven parameter family of collision equilibria ( ) 3/2 mi ( M i (v) = n i exp m i 2πKT 2KT (v u)2) i = 1,..., 4 with number densities related by the mass action law ( ) n 1 n 2 m1 m 3/2 ( ) 2 E = exp, E = E 3 + E 4 E 1 E 2 n 3 n 4 m 3 m 4 KT M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 4 / 32

BGK Models for reacting gas mixtures Reactive Boltzmann equations f i t + v f i = Qi ME + Qi CH Q i i = 1,..., 4. Seven conservation laws (Q i + Q j ) dv = 0, (i, j) = (1, 3), (1, 4), (2, 4), 4 m i vq i dv = 0, i=1 4 i=1 ( 1 2 m iv 2 + E i ) Q i dv = 0. H theorem in terms of the reactive entropy functional 4 ( ) H = f i log f i /mi 3 dv. i=1 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 4 / 32

BGK Models for reacting gas mixtures Relaxation time approximation - Reactive BGK equations I From an idea by Andries, Aoki, Perthame (2002) for inert gases: only one global collision operator for each species Positivity of density and temperature fields, entropy inequality, and indifferentiability principle are fulfilled f i t + v f i = Q i = ν i (M i f i ) i = 1,..., 4 M i (v) = ñ i ( mi 2πK T i ) 3/2 ( exp m ) i 2K T (v ũ i ) 2 i Auxiliary parameters ñ i, ũ i, T i are determined by imposing that reactive Boltzmann and BGK equations prescribe the same exchange rates for species densities, mass velocities and temperatures (chemical exchange rates can be made explicit only under simplifying assumptions: Maxwell molecules, slow reaction) (G-Spiga 2004) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 5 / 32

BGK Models for reacting gas mixtures A different reactive BGK relaxation model (Bisi-G.-Spiga 2010) BGK collision terms Q i = ν i (M i f i ) i = 1,..., 4 where M i (v) = n i ( mi 2πK T ) 3/2 ( exp m i (v ū)2) 2K T Maxwellian attractors are accommodated at (auxiliary) common velocity ū and temperature T. Auxiliary parameters n i, ū, T are determined by imposing that reactive Boltzmann and BGK equations prescribe the same collision invariants (7 conditions) and the mass action law at equilibrium. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 6 / 32

Properties BGK Models for reacting gas mixtures Conservation laws are exactly reproduced by both BGK models M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 7 / 32

Properties BGK Models for reacting gas mixtures Conservation laws are exactly reproduced by both BGK models Reactive Boltzmann and BGK equations share in both cases the correct collision equilibria, including mass action law M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 7 / 32

Properties BGK Models for reacting gas mixtures Conservation laws are exactly reproduced by both BGK models Reactive Boltzmann and BGK equations share in both cases the correct collision equilibria, including mass action law Entropy dissipation: the reactive BGK models have the same Lyapunov functional of the Boltzmann equations for the stability of collision equilibria 4 H = i=1 ( ) f i log f i /mi 3 dv M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 7 / 32

Properties BGK Models for reacting gas mixtures Conservation laws are exactly reproduced by both BGK models Reactive Boltzmann and BGK equations share in both cases the correct collision equilibria, including mass action law Entropy dissipation: the reactive BGK models have the same Lyapunov functional of the Boltzmann equations for the stability of collision equilibria 4 H = i=1 ( ) f i log f i /mi 3 dv Drawback: both models fail in quantitatively reproducing the proper transport coefficients in the asymptotic continuum limit M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 7 / 32

Numerical Approximation: High Order Methods for BGK Models Numerical approximation: Semi-lagrangian methods Idea: follow the evolution along the characteristics M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 8 / 32

Numerical Approximation: High Order Methods for BGK Models Numerical approximation: Semi-lagrangian methods Idea: follow the evolution along the characteristics Semi-lagrangian formulation for the simple case of BGK equation for a monoatomic gas in 1-D df (x, v, t) = 1 ( ) M[f ](x, v, t) f (x, v, t), dt ɛ dx dt = v, x(0) = x, f (x, v, 0) = f 0 (x, v) t 0, x, v R. x becomes a time dependent variable and its equation gives: x(t) = x + vt, t 0, x, v R (characteristic straight lines) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 8 / 32

Numerical Approximation: High Order Methods for BGK Models Numerical approximation: Semi-lagrangian methods Idea: follow the evolution along the characteristics Semi-lagrangian formulation for the simple case of BGK equation for a monoatomic gas in 1-D df (x, v, t) = 1 ( ) M[f ](x, v, t) f (x, v, t), dt ɛ dx dt = v, x(0) = x, f (x, v, 0) = f 0 (x, v) t 0, x, v R. x becomes a time dependent variable and its equation gives: x(t) = x + vt, t 0, x, v R (characteristic straight lines) Semi-lagrangian treatment of the convective part avoids the classical CFL stability restriction. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 8 / 32

Numerical Approximation: High Order Methods for BGK Models Implicit first order Semi-lagrangian scheme Let fij n f (x i, v j, t n ) be the approximate solution. Possible stiffness (small ɛ) implicit formulation. f n+1 ij = f ij n + t ɛ (Mn+1 ij fij n+1 ). (1) Here f ij n = f (t n, x i = x i v j t, v j ) can be calculated by (linear) interpolation from {f n.j }. f n+1 t n+1 ij v j > 0 t n f n ij x i 2 x i 1 x i x i+1 x i M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 9 / 32

Numerical Approximation: High Order Methods for BGK Models Solution of the implicit step Equation (1) is non linear. Indeed M[f ] i,j n+1 depends on fij n+1 through its moments. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 10 / 32

Numerical Approximation: High Order Methods for BGK Models Solution of the implicit step Equation (1) is non linear. Indeed M[f ] i,j n+1 depends on fij n+1 through its moments. Let φ(v) be the vector φ(v) = (1, v, v 2 ) T. Compute the moments of f n+1 ij : fij n+1 φ = f ij n φ + t ɛ (Mn+1 ij fij n+1 )φ. From the conservation, we have (Mij n+1 fij n+1 )φ = 0 fij n+1 φ = f n ij φ Hence we immediately find the macroscopic variables ρ n+1 i, ui n+1 Ti n+1 corresponding to fij n+1 using f ij n and with these values the approximated Maxwellian is updated. and M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 10 / 32

Numerical Approximation: High Order Methods for BGK Models Higher order: RK and BDF schemes (G.-Russo-Stracquadanio 2014) L- stable Diagonally implicit Runge-Kutta schemes (DIRK) of order 2 and 3 have been considered to obtain high order in time, providing the correct fluid dynamic limit (Russo - Santagati 2011) Drawback: high number of interpolations required BDF (Backward Difference Formula) methods allow same order of accuracy at lower cost. Applying BDF to the Lagrangian formulation of the BGK model we get f n+1 ij = 18 11 f n+1 ij = 4 3 (1) f n ij 9 11 (1) f n ij 1 3 (2) fij n 1 + t ɛ (Mn+1 ij fij n+1 ) (BDF 2) (2) f n 1 ij + 2 11 (3) fij n 2 + t ɛ (Mn+1 ij fij n+1 ) (BDF 3) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 11 / 32

Numerical Approximation: High Order Methods for BGK Models Higher order: RK and BDF schemes (G.-Russo-Stracquadanio 2014) L- stable Diagonally implicit Runge-Kutta schemes (DIRK) of order 2 and 3 have been considered to obtain high order in time, providing the correct fluid dynamic limit (Russo - Santagati 2011) Drawback: high number of interpolations required BDF (Backward Difference Formula) methods allow same order of accuracy at lower cost. Applying BDF to the Lagrangian formulation of the BGK model we get f n+1 ij = 18 11 f n+1 ij = 4 3 (1) f n ij 9 11 (1) f n ij 1 3 (2) fij n 1 + t ɛ (Mn+1 ij fij n+1 ) (BDF 2) (2) f n 1 ij + 2 11 (3) fij n 2 + t ɛ (Mn+1 ij fij n+1 ) (BDF 3) (s) f n ij= f n (x i sv j t, v j ), s = 1, 2, 3, obtained by interpolation. High order in space is obtained by WENO reconstruction (Carlini-Ferretti-Russo 2005). M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 11 / 32

Numerical Approximation: High Order Methods for BGK Models Example: RK2 and BDF2 x i 2 x i 1 f n+1 t n+1 t n + c 1 t t n F (1,1) i 2,j v j > 0 F (1,1) i 1,j f (2,n) ij K (2,1) i,j f (1,n) ij x (2) x (3) x (1) ij F (2,2) ij x i F (1,1) ij 2 char. 1 char. x i 2 x i 1 f n+1 t n+1 ij t n t n 1 f n 1,2 ij v j > 0 f n,1 ij x 2 x 1 x i x i+1 Figure: Left RK2, right BDF2. Interpolation is needed in red circles. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 12 / 32

Numerical Approximation: High Order Methods for BGK Models Remarks These high order semi-lagrangian methods for the classical 1-D BGK equation, based on both DIRK and BDF schemes for time integration, have been successfully extended to 3-D in velocity problems (Chu reduction) 1-D boundary problems (reflective and diffusive boundary conditions) systems of BGK equations modelling reactive gas mixtures Owing to their semi-lagrangian nature, large CFL numbers can be used, allowing for large time step without degrading the accuracy and stability M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 13 / 32

Numerical Approximation: High Order Methods for BGK Models Remarks These high order semi-lagrangian methods for the classical 1-D BGK equation, based on both DIRK and BDF schemes for time integration, have been successfully extended to 3-D in velocity problems (Chu reduction) 1-D boundary problems (reflective and diffusive boundary conditions) systems of BGK equations modelling reactive gas mixtures Owing to their semi-lagrangian nature, large CFL numbers can be used, allowing for large time step without degrading the accuracy and stability The computational cost can be further reduced by developing variants that avoid interpolation and allow larger time steps M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 13 / 32

Numerical Approximation: High Order Methods for BGK Models Remarks These high order semi-lagrangian methods for the classical 1-D BGK equation, based on both DIRK and BDF schemes for time integration, have been successfully extended to 3-D in velocity problems (Chu reduction) 1-D boundary problems (reflective and diffusive boundary conditions) systems of BGK equations modelling reactive gas mixtures Owing to their semi-lagrangian nature, large CFL numbers can be used, allowing for large time step without degrading the accuracy and stability The computational cost can be further reduced by developing variants that avoid interpolation and allow larger time steps These schemes are asymptotic preserving, namely able to capture the fluid dynamic limit M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 13 / 32

Numerical Approximation: High Order Methods for BGK Models Numerical Test - Reactive BGK equations I We considered a mixture of four gases with the following values of the molecular masses: m 1 = 58.5, m 2 = 18, m 3 = 40, m 4 = 36.5. As initial data we have chosen Maxwellian reproducing the following moments (Riemann problem): (ρ 0, u 0, p 0 ) = (ρ 01, ρ 02, ρ 03, ρ 04 ) = { (1, 0, 5/3), x < 0.5, (1/8, 0, 1/6), x > 0.5, { (1/10, 2/10, 3/10, 4/10), x < 0.5, (1/80, 2/80, 3/80, 4/80), x > 0.5,, u 0i = 0, i = 1,..., 4. Results obtained with the BDF2 scheme. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 14 / 32

Numerical Approximation: High Order Methods for BGK Models Evolution of densities, ɛ 10 2 0.5 0.45 0.4 0.35 Initial gas densities, E=500 gas 1 gas 2 gas 3 gas 4 0.5 0.45 0.4 0.35 Evolution of densities, E=500, t fin =2 gas 1 gas 2 gas 3 gas 4 Density 0.3 0.25 0.2 Density 0.3 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 6 4 2 0 2 4 6 x 0 6 4 2 0 2 4 6 x M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 15 / 32

Numerical Approximation: High Order Methods for BGK Models Evolution of velocities, ɛ 10 2 0.2 0.15 0.1 Initial gas velocities, E=500 gas 1 gas 2 gas 3 gas 4 2 1.8 1.6 1.4 Evolution of velocities, E=500, t fin =2 gas 1 gas 2 gas 3 gas 4 Velocity 0.05 0 0.05 Velocity 1.2 1 0.8 0.6 0.1 0.4 0.15 0.2 0.2 6 4 2 0 2 4 6 x 0 6 4 2 0 2 4 6 x M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 16 / 32

Numerical Approximation: High Order Methods for BGK Models Evolution of temperatures, ɛ 10 2 60 58 56 54 Initial gas temperatures, E=500 gas 1 gas 2 gas 3 gas 4 130 120 110 Evolution of temperatures, E=500, t fin =2 gas 1 gas 2 gas 3 gas 4 Temperature 52 50 48 Temperature 100 90 46 80 44 70 42 40 6 4 2 0 2 4 6 x 60 6 4 2 0 2 4 6 x M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 17 / 32

ES-BGK models ES-BGK model for a binary mixture The ellipsoidal BGK (ES-BGK) models (Holway 1966, Andries et al. 2000, Brull Schneider 2008) are devised to correctly reproduce transport coefficients: fundamental moments are bound to relax to equilibrium at a faster rate than the distribution function, allowing to fit transport coefficients from Chapman Enskog expansion in terms of the additional relaxation parameters M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 18 / 32

ES-BGK model for a binary mixture ES-BGK models The ellipsoidal BGK (ES-BGK) models (Holway 1966, Andries et al. 2000, Brull Schneider 2008) are devised to correctly reproduce transport coefficients: fundamental moments are bound to relax to equilibrium at a faster rate than the distribution function, allowing to fit transport coefficients from Chapman Enskog expansion in terms of the additional relaxation parameters In case of mixtures Fick s law for diffusion velocities and Soret and Dufour effects show up in addition to Newton and Fourier constitutive equations Recent substantial breakthrough in the extension of ES BGK models to inert gas mixtures by Brull, Pavan, Schneider 2012: correct N N Fick diffusion matrix, preserving positivity and consistency constraints, obtained by requiring equalization of species velocities (N 1 additional relaxation parameters) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 18 / 32

ES-BGK model for a binary mixture ES-BGK model for a binary inert mixture (G.-Monica-Spiga 2011) Further constraint (in addition to conservations): equalization of drift velocities of the two species in the evolution additional relaxation parameter η M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 19 / 32

ES-BGK model for a binary mixture ES-BGK model for a binary inert mixture (G.-Monica-Spiga 2011) Further constraint (in addition to conservations): equalization of drift velocities of the two species in the evolution additional relaxation parameter η ES-BGK collision operator: ˆQi = ν(g i f i ) ( ) 3 mi 2 G i [f 1, f 2 ](v) = n i exp [ m ] i ( v u ) 2 2πT 2T i, i = 1, 2 Theorem The pair (G 1, G 2 ) is the minimizer of the entropy 2 H[f 1, f 2 ] = (f i log f i f i ) d 3 v R 3 i=1 in the class of distribution functions for which the ES-BGK operator ˆQ i fulfils the conservation laws, drift velocities equalize with inverse relaxation times η, and species temperatures are equal. M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 19 / 32

ES-BGK model for a binary mixture Properties of the ES-BGK model Auxiliary fields u i, T determined in terms of the actual velocities u i and gas temperature T : u (G) = u, T (G) = T ν [ (u 1 u 1 ) + (u 2 u 2 ) ] = η(u 1 u 2 ) from which u i = ( 1 η ) u i + η ν ν u nkt = 2 n i KT i + η ( 2 η ) 2 ρ i (u i u) 2 3ν ν i=1 ν η 2ν for positivity of temperatures i=1 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 20 / 32

ES-BGK model for a binary mixture Properties of the ES-BGK model Auxiliary fields u i, T determined in terms of the actual velocities u i and gas temperature T : u (G) = u, T (G) = T ν [ (u 1 u 1 ) + (u 2 u 2 ) ] = η(u 1 u 2 ) from which u i = ( 1 η ) u i + η ν ν u nkt = 2 n i KT i + η ( 2 η ) 2 ρ i (u i u) 2 3ν ν i=1 ν η 2ν for positivity of temperatures The ES-BGK model satisfies an H-theorem in terms of the Boltzmann H functional H[f 1, f 2 ] Collision equilibria are Maxwellian distribution functions at common mean velocity and temperature i=1 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 20 / 32

ES-BGK model for a binary mixture Asymptotic Chapman Enskog analysis In the hydrodynamic limit w.r.t. the Knudsen number ε, two transport coefficients can be quantitatively fitted at leading order O(ε), allowing to correctly reproduce the Fick diffusion law 2 u i u = D ij n j D 11 = ε η KT ρ j=1 ρ 2 ρ 1, D 12 = D 21, 2 D ij ρ j = 0 j=1 and the Newton law P = nkt I µ ( u + u T 2 ) 3 u I, µ = ε ν nkt but only qualitatively the Fourier conduction law (and also Soret and Dufour effects) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 21 / 32

ES-BGK model for a binary mixture Reactive ES-BGK model (G.-Spiga 2014) A 2 excited state of species A 1, with energy of chemical bond E; common mass m Irreversible binary slow reaction (de-excitation process) A 2 + A i A 1 + A i, i = 1, 2 Scaled reactive ES-BGK equations (dimensionless) f i t + v f i = ν ( ) Gi [f 1, f 2 ] f i + Ji [f 1, f 2 ], i = 1, 2 ε J i chemical collision operator (Boltzmann-type, five-fold integral) M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 22 / 32

ES-BGK model for a binary mixture Weak form for any pair of smooth test functions ϕ i (v) 2 ϕ i (v)j i [f 1, f 2 ](v)d 3 v = i=1 = g σ21(g, 11 χ) [ ϕ 1 (v21)+ϕ 11 1 (w21) ϕ 11 2 (v) ϕ 1 (w) ] f 2 (v)f 1 (w) dvdwdˆn + g σ22(g, 12 χ) [ ϕ 1 (v22)+ϕ 12 2 (w22) ϕ 12 2 (v) ϕ 2 (w) ] f 2 (v)f 1 (w) dvdwdˆn M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 23 / 32

ES-BGK model for a binary mixture Weak form for any pair of smooth test functions ϕ i (v) 2 ϕ i (v)j i [f 1, f 2 ](v)d 3 v = i=1 = g σ21(g, 11 χ) [ ϕ 1 (v21)+ϕ 11 1 (w21) ϕ 11 2 (v) ϕ 1 (w) ] f 2 (v)f 1 (w) dvdwdˆn + g σ22(g, 12 χ) [ ϕ 1 (v22)+ϕ 12 2 (w22) ϕ 12 2 (v) ϕ 2 (w) ] f 2 (v)f 1 (w) dvdwdˆn post-collisional velocities vij hk = 1 ( v + w + gij hk ˆn ) wij hk = 1 ( v + w gij hk ˆn ) 2 2 ( g21 11 = g22 12 = g + = g 2 + 4E ) 1 2 m M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 23 / 32

ES-BGK model for a binary mixture Collision invariants for the whole ES-BGK collision operator ˆQ i + J i ( 1 (1, 1) (mv, mv) 2 mv 2, 1 ) 2 mv 2 + E M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 24 / 32

ES-BGK model for a binary mixture Collision invariants for the whole ES-BGK collision operator ˆQ i + J i ( 1 (1, 1) (mv, mv) 2 mv 2, 1 ) 2 mv 2 + E Collision equilibrium ( ) 3 f eq m 1 (v) = n 2 exp [ m ] ( ) 2 v u, f eq 2 2πT 2T (v) = 0 Five exact but not closed macroscopic conservation equations for number density n, momentum ρu and total energy 1 2 ρu2 + 3 2 nt + n 2E M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 24 / 32

ES-BGK model for a binary mixture Macroscopic equations The hydrodynamic limit for ε 0 by a Chapman-Enskog asymptotic expansion up to the Navier-Stokes level requires invariants of the dominant operator ˆQ alone (all species densities, momentum, and kinetic energy), whose equilibria are Maxwellians with free densities and temperature. Macroscopic conservation equations n t + (nu) = 0 (ρu) + (ρu u) + (nt ) + Π = 0 t t ( 1 2 ρu2 + 3 [( 1 2) 2 nt + En + 2 ρu2 + 5 2) 2 nt + En u ] +Π u + q + En 2 (u 2 u) = 0 n 2 t + (n 2u) = [n 2 (u 2 u)] 2 S 2j j=1 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 25 / 32

ES-BGK model for a binary mixture Asymptotic expansion Presence of inhomogeneous terms S 2j (reactive rates), contributed by the slow process and vanishing at chemical equilibrium Hydrodynamic closure requires constitutive equations for diffusion velocity u 2 u, viscous stress Π = P nkt I, heat flux q, and reactive sources S 2j Asymptotic expansion f i fi ε = f (0) i + εf (1) i i = 1, 2 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 26 / 32

ES-BGK model for a binary mixture Asymptotic expansion Presence of inhomogeneous terms S 2j (reactive rates), contributed by the slow process and vanishing at chemical equilibrium Hydrodynamic closure requires constitutive equations for diffusion velocity u 2 u, viscous stress Π = P nkt I, heat flux q, and reactive sources S 2j Asymptotic expansion f i fi ε = f (0) i + εf (1) i i = 1, 2 with n 1, n 2, u, T unexpanded, and with u i = u + εu (1) i, T i = T + εt (1) i, S 2j = S (0) 2j + εs (1) 2j subject to constraints 2 ρ i u (1) i = 0, i=1 2 i=1 n i T (1) i = 0 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 26 / 32

Leading term ES-BGK model for a binary mixture f (0) i ( ) 3/2 = fi M mi ( (v) = n i exp m 2πT 2T (v u)2) i = 1, 2 M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 27 / 32

Leading term ES-BGK model for a binary mixture f (0) i ( = fi M mi (v) = n i 2πT ( S (0) m 2j =n 2 n j 2πT with σ2j ch (g) angle integrated cross sections. ) 3/2 exp ( m 2T (v u)2) i = 1, 2 ) 3 [ gσ2j ch (g) exp m ] 2T (v 2 + w 2 ) dvdw First order correction f ε i = G i [f ε 1, f ε 2 ] ε ν ( f M i t + v f M i ) + ε ν J i[f M 1, f M 2 ] + O(ε 2 ) with J i [f1 M, f 2 M] = JCH(0) i explicit (but complicated) functions of the hydrodynamic variables, isotropic in c = v u M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 27 / 32

ES-BGK model for a binary mixture First order correction - II Time derivatives of Maxwellians may be eliminated through conservation equations fi M ( ) ( ) Df + v fi M M = i Df M + i (2) t Dt Dt ME CH Mechanical contribution is orthogonal to any isotropic function of c Chemical contribution is isotropic w.r.t. c Ellipsoidal terms G i [f ε 1, f ε 2 ] = fi M (v) + εfi M (v) m T First order distribution functions f (1) i (v) = f M i (v)ψ i (v) = fi M (v) ( 1 η ) u (1) i c + O(ε 2 ) ν [ ψ ME i ] (v) + ψi CH (v) with again mechanical contribution orthogonal to isotropic functions of c and chemical contribution isotropic in c M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 28 / 32

ES-BGK model for a binary mixture First order correction - III Re-computation of u (1) i by integration leads to Fick law u (1) 1 = 1 T n 2 n 1 + 1 T η ρ n 1 η ρ n 2 u (1) 2 = 1 T η ρ n 1 1 T n 1 n 2 η ρ n 2 No thermal diffusion (Soret effect) because of the equal masses M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 29 / 32

ES-BGK model for a binary mixture First order correction - III Re-computation of u (1) i by integration leads to Fick law u (1) 1 = 1 T n 2 n 1 + 1 T η ρ n 1 η ρ n 2 u (1) 2 = 1 T η ρ n 1 1 T n 1 n 2 η ρ n 2 No thermal diffusion (Soret effect) because of the equal masses Diffusion velocities do not depend on the chemical reaction, due to isotropy arguments Moreover, chemical contributions do not affect viscous stress and heat flux (O(ε) terms), that remain the same as for the inert case Conversely, elastic collisions do not affect the chemical terms at Navier-Stokes level M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 29 / 32

ES-BGK model for a binary mixture Navier-Stokes equations n t + (nu) = 0 (ρu) + (ρu u) + (nt ) = ε Π (1) t ( 1 t 2 ρu2 + 3 ) [ (1 2 nt + En 2 + 2 ρu2 + 5 2 nt + En 2 ( = ε Π (1) u + q (1) + En 2 u (1) ) 2 n 2 t + (n 2u) + 2 S (0) 2j = ε (n 2 u (1) ) 2 2 ε j=1 j=1 S (1) 2j ) ] u M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 30 / 32

ES-BGK model for a binary mixture Navier-Stokes equations Newton law Π (1) = 1 ( ν nt u + ( u ) ) T 2 3 ui viscosity ε ν nt Fourier law q (1) = 1 5 nt ν 2 m T thermal conductivity ε 5 nt ν 2 m no Dufour effect since masses are equal Elastic collisions do not affect the chemical source terms S (1) 2j reactive Navier-Stokes equations of the M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 31 / 32

Concluding Remarks ES-BGK model for a binary mixture Same mechanical constitutive equations of the corresponding inert gas mixture, since the evolution is driven by elastic collisions Chemical source terms are O(1) and are independent from the mechanical parameters The relaxation parameter η may be used to fit exactly the Fick diffusion matrix The parameter ν may be used to fit either viscosity or thermal conductivity, but not both coefficients simultaneously M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 32 / 32

Concluding Remarks ES-BGK model for a binary mixture Same mechanical constitutive equations of the corresponding inert gas mixture, since the evolution is driven by elastic collisions Chemical source terms are O(1) and are independent from the mechanical parameters The relaxation parameter η may be used to fit exactly the Fick diffusion matrix The parameter ν may be used to fit either viscosity or thermal conductivity, but not both coefficients simultaneously Future works: introduction of a further relaxation parameter to control the Prandtl number extension of the high order semi-lagrangian methods to ES-BGK models M. Groppi (DMI UniPR) Kinetic relaxation models for reacting gas Novi Sad, 26 September 2014 32 / 32