Non-equilibrium Effects in Viscous Reacting Gas Flows

Transcription

1 Issues in Solving the Boltzmann Equation for Aerospace ICERM, Brown University, Providence June 3 7, 2013 Non-equilibrium Effects in Viscous Reacting Gas Flows Elena Kustova Saint Petersburg State University

2 The Boltzmann equation (1872) More than 140 years of studying the Boltzmann equation f t + u f r + F f u = collf Still unsolved Still plenty of surprises and contradictions Still inspires new studies

3 Outline Introduction Reduced-order non-equilibrium fluid dynamic models derived from the Boltzmann equation General idea State-to-state model Multi-temperature models One-temperature models Limitations of models commonly used in CFD Reaction rates and normal mean stress in one-temperature viscous flows Vibrationally non-equilibrium flows. Rate of vibrational relaxation On different contributions to the heat transfer Some features of transport in gases with electronic excitation Conclusions

4 Introduction. Methods for solving the Boltzmann equation D. Hilbert, S. Chapman, D. Enskog, L. Waldmann, H. Grad, G. Bird, M. Kogan, C. Cercignani, S. Vallander, R. Brun, V. Zhdanov, E. Nagnibeda, and many others Linearized Boltzmann equation Model equations (BGK, ES and other modifications) Integral form of the Boltzmann equation Using of variational principle Moment methods (Grad s method and its generalization) Discrete velocities method Asymptotic methods (Hilbert, Chapman Enskog and its generalizations) Numerical solution of the Boltzmann equation Direct simulations Monte Carlo

5 Introduction. Gases with internal degrees of freedom Different ways of description: Classical: both translational and internal degrees of freedom are described classically (Taxman, Kagan) Quantum mechanical: both translational and internal degrees of freedom are quantized (Waldmann, Snider). Quasi-classical: while translational degrees of freedom are treated classically, the internal modes are quantized (Wang Chang, Uhlenbeck)

6 Introduction. Gases with internal degrees of freedom Let f cij ( r, u, t) be a distribution function of c particles over velocity u c, vibrational and rotational energies ε c i, εci j. The Wang Chang Uhlenbeck equation (1951): J (γ) cij f cij + u c f cij + F t r f cij = J cij = J (γ) cij, u c γ is specified by the cross section of a microscopic process γ Dimensionless form (in the absence of mass forces): f cij t + u c f cij r = γ 1 ε γ J γ cij, ε γ τ γ θ 1.

7 Reduced-order models for fluid dynamics Weak and strong non-equilibrium flows Weakly non-equilibrium flows τ γ θ f cij t + u c f cij r = 1 ε Jtotal cij, ε τ fp θ 1. Strongly non-equilibrium flows γ : τ γ θ f cij t + u c f cij r = 1 ε Jrap cij + J sl cij, ε τ rap τ sl 1.

8 Modified Chapman Enskog method Distribution functions depend on r and t only through macroscopic parameters and their gradients: f cij ( r, u, t) = f cij ( u c, ρ λ ( r, t), ρ λ ( r, t),...), Characteristic times of physical chemical processes differ essentially, some of them proceed on the gas-dynamic time scale θ. The basis of the method is to establish the hierarchy of characteristic times τ rap τ sl θ Collision operators are divided into two groups: operators of rapid and slow processes: J rap cij, Jcij, sl

9 Modified Chapman Enskog method. Collision invariants Collision invariants ψ cij + ψ dkl = ψ c i j + ψ d k l Collision invariants for all processes ψ (λ) cij, λ = 1,..., 5 : m c, m c u c, m c u 2 c 2 + ε c ij Additional collision invariants for the most frequent collisions. ψ (µ) cij, µ = 1,..., M Number of additional invariants depend on the deviation from equilibrium.

10 Modified Chapman Enskog method Fluid dynamic variables corresponding to the collision invariants of all processes ρ λ = ψ (λ) f cij d u c, λ = 1,..., 5 cij λ = 1: density ρ λ = 2, 3, 4: velocity v λ = 5: specific energy U Macroscopic variables corresponding to additional invariants of rapid processes ρ µ = ψ (µ) cij f cij d u c, µ = 1,..., M cij Can be different depending on flow conditions

11 Modified Chapman Enskog method Governing equations Conservation equations correspond to the invariants of all processes ρ (λ) t + cij ψ (λ) cij u c f cij r d u c = 0, λ = 1,..., 5 M relaxation equations ρ (µ) t + cij ψ (µ) cij u c f cij r d u c = cij ψ (µ) cij J sl cijd u c, µ = 1,..., M Production term in the right-hand side is specified by slow processes.

12 Modified Chapman Enskog method Zero- and first-order solutions The solution is sought in the form Zero-order solution f cij = n=0 ɛ n f (n) cij ( J rap cij f (0), f (0)) = 0 Zero-order distribution function is not local equilibrium First-order solution f (1) cij = f (0) cij (1 + ϕ cij ) The first order correction is found from the integral equations I rap cij (ϕ) = J sl(0) cij Df (0) cij I rap cij is the linearized operator of rapid processes, Df (0) cij is the streaming operator

13 State-to-state model Time hierarchy Macroscopic variables vibrational state populations velocity temperature Macroscopic equations τ tr < τ rot τ vibr < τ react θ ρ dα ci dt = J mci + r ρ dv dt = P ρ du = q + P : v dt ξ r ν r,ci M c, c = 1,.., L, i = 0,..., L c

14 Multi-temperature models Time relation τ tr < τ rot < τ VV τ VT < τ react θ Macroscopic variables chemical species mass fractions velocity temperature vibrational temperatures Macroscopic equations ρ dα c dt = J mc + r ξ r ν r,c M c, c = 1,.., L, ρ dα ce v,c dt = q v + E v,c ρ dv dt = P ρ du = q + P : v dt

15 One-temperature model Time relation τ tr < τ int τ react θ Macroscopic variables chemical species mass fractions velocity temperature Macroscopic equations ρ dα c dt = J mc + r ρ dv dt = P ρ du = q + P : v dt ξ r ν r,c M c, c = 1,.., L,

16 Comparison of models. Compressive flows SHOCK HEATED GAS T 0 = 293 K, p 0 = 100 Pa, M 0 = 15 N 2 flow behind a shock wave, M 0 = 15, T 0 = 293 K, p 0 = 100 Pa n i / n Vibrational distributions Heat flux q, kw/m ' 1 3' 2' i x, cm Figure : Vibrational populations (a) and heat flux (b) behind the shock front. FIG. lines: 1. x Reduced = 0.03 mm; level dashed populations lines: of x = N2 0.8 behind mm. a 1, shock 1 : state-to-state wave. Solid approach; 2, 2 : two-temperature approach; 3, 3 : one-temperature approach. FIG. 3. Heat flux behind a shock wave as a function of x. 1: state-to-state approach; 2: two-temperature approach; 3: one-temperature approach.

17 Comparison of models. Expanding flows NOZZLE FLOW Conic nozzle, 21, T = 7000 K, p = 100 atm. N 2 flow in a conic nozzle, T = 7000 K, p = 1 atm n i / n 1 4 Vibrational distributions i q, W/m 2 Heat flux FIG. 4. Vibrational distributions, x/r = 50. 1: STS model; 2: 2-temperature, anharmonic oscillator; 3: 2-temperature, harmonic oscillalator; 3: 2-temperature, harmonic oscillator; 4: 1-temperature. FIG. 7. Heat flux. 1: STS model; 2: 2-temperature, anharmonic osciltor; 4: 1-temperature. Figure : Vibrational populations (a) and heat flux (b) along the nozzle axis x/r

18 Limitations of commonly used models CFD, common practice: Using the Law of Mass Action (LMA) in viscous flow solvers Using the Landau-Teller expression for the rate of vibrational relaxation Neglecting the bulk viscosity and non-equilibrium reaction contributions in the normal mean stress Neglecting thermal diffusion in heat and mass transfer Neglecting electronic excitation

19 Normal stress and reaction rates in one-temperature viscous flows Integral operators of rapid and slow processes J rap ci = J tr ci + J int ci, J sl ci = J react ci, ɛ = τ tr τ react Governing equations ρ dα c + J mc = ξ r ν rc M c, c = 1,.., L, dt r ρ dv + P = 0, dt ρ du + q + P : v = 0. dt

20 Transport and production terms Mass diffusive flux J mc (r, t) = m c c c f ci (r, u, t) du c i Pressure tensor P(r, t) = ci m c c c c c f ci (r, u, t) du c Energy flux q(r, t) = ci ( mc c 2 c 2 + ε c i ) c c f ci (r, u, t) du c Production term ξ r ν rc M c = m c r i J sl ci du c, Jci sl = Jci Jci 2 3

21 Integral operators for slow processes Exchange reactions J 2 2 ci = ( fc i m 3 dd c c s i c kk i f d k md 3 f ci s k d mcs 3 c i f dk m 3 d sd k ) Dissociation reactions m 3 c m3 d sc i sd i d k k gσc cidk dωd u d J 2 3 ci = d kk [ f dk f c f f h3 s c i ( mc m c m f ) 3 f cif dk] σ c i d k cidk, σci, diss d gσ diss ci, d du ddu c du f du d, are the reaction cross sections

22 Production terms Reaction rates ξ r = k f, r L c=1 ( ) (r) ν ρc M c rc k b, r L c=1 ( ρc Reaction rate coefficients fci f dk k f, r = N A g σ f, r d 2 Ω du d du c, n c n d iki k M c ) ν (p) rc r = ex, di, fc k b, ex = N i f d k A g σ b, ex d 2 Ω du d du c, n c n d iki k k b, di = NA 2 fc f f f dk σ b, di du c du d du c du f du n c n f n d. d ik σ f, r is the cross section of rth reaction

23 Zero-order approximation Maxwell-Boltzmann distribution function ( f (0) mc ) 3/2 nc ci = 2πkT Transport terms: Governing equations: Zc int (T ) sc i exp J mc = 0, q = 0, P = pu ρ dα c dt = r ρ dv + p = 0, dt ρ du + p v = 0. dt ( m ccc 2 ) 2kT εc i kt ξ (0) r ν rc M c, c = 1,.., L,

24 Zero-order production terms Zero-order reaction rates: ξ (0) r = k (0) L f, r c=1 ( ρc M c ) ν (r) rc k (0) b, r Zero-order reaction rate coefficient: f, r = N A k (0) iki k L c=1 ( ρc M c ) ν (p) rc f (0) ci f (0) dk n c n d g σ f, r d 2 Ω du d du c, If the reaction cross sections σ f, r are known the zero order rate coefficients can be easily calculated. Alternatively, the Arrhenius law can be applied for the k (0) f, r calculation, and the equilibrium constant for the backward reaction rate coefficient.

25 Using the detailed balance principle, we obtain: ξ (0) r = ω r k (0) f,r L c=1 ( nc N A ) ν (r) rc = ω r k (0) f,r L c=1 ( ρc M c ) ν (r) rc ω r is the chemical reaction characteristics: ( ) Ar ω r = 1 exp R G T A r is the affinity of a chemical reaction r.

26 Summary on the zero-order (Euler) approximation In the zero-order approximation, the stress tensor takes the diagonal form pu and does not depend on chemical reactions The rate of chemical reaction r depends only on the affinity of the appropriate reaction A r and does not depend on the affinities of other reactions Therefore, no cross effects between different chemical reactions and between normal mean stress and chemical reactions appear in inviscid gas flows

27 Summary on the zero-order (Euler) approximation The law of mass action is valid in the zero-order (inviscid) flow approximation of the Chapman-Enskog method The zero-order chemical-reaction rate coefficient can be calculated by averaging the corresponding cross section over the Maxwell-Boltzmann distribution The main problem in the modeling of inviscid reacting flows is the correct determination of the zero-order rate coefficients of chemical reactions k (0) f,r, k(0) b,r. No other uncertainties occur in the Euler equations Existing CFD models used for one-temperature inviscid flows are more or less self-consistent

28 First-order solution In the first-order approximation, the distribution function takes the form ( ) f ci = f (0) ci 1 + φ (1) ci The first-order correction to the distribution function: φ (1) ci = 1 n A ci ln T 1 D d n ci d d 1 n B ci : v 1 n F ci v 1 Gci r ω r n d functions A ci, D d ci, B ci, F ci and G r ci are found from the linear integral equations. The first-order normal mean stress and chemical reaction rates are determined by the scalar functions F ci, Gci r, velocity divergence v and chemical reaction characteristics ω r r

29 First-order stress tensor The normal mean stress P = πu + 2µ ( v) 0 S (π + p) = R G T r l vr ω r l vv v, coefficients l vr, l vv are determined by the bracket integrals l vr = [F, G r ] N A, l vv = kt [F, F ]. There is a connection between coefficients l vr, l vv, relaxation pressure p rel and bulk viscosity coefficient ζ: R G T r l vr ω r = p rel, l vv = ζ

30 First-order reaction rates The chemical-reaction rate in the first-order approximation takes the form ξ r = ξ r (0) + ξ r (1) The first-order correction ξ (1) r = l rv v + R G T s l rs ω s The kinetic coefficients l rv = [G r, F ] N A, l rs = 1 R G T [G r, G s ] N A

31 First order reaction rate coefficients First order reaction rate coefficients: k f, r = k (0) f, r (T ) k (1) f, r (α1,..., α L, ρ, T ) k (1) f, r (α1,..., α L, ρ, T, v). First order corrections: k (1) f, r = N A n iki k f (0) ci f (0) dk n cn d k (1) f, r = v N (0) A f dk n n iki k cn d k (1) f, r ci f (0) (G ci + G dk ) g σ f, r d 2 Ω du c du d (F ci + F dk ) g σ f, r d 2 Ω du cdu d are due to deviations from Maxwell-Boltzmann distributions k (1) f, r are due to spatial non-homogeneity If there is no internal degrees of freedom, coefficients k (1) f, r = 0

32 Summary on the first-order (Navier Stockes) approximation In the first-order approximation, we found the cross-coupling effects between chemical reactions and normal mean stress The reaction rates depend on the velocity divergence The rate of the rth reaction is affected by other reactions, therefore the existence of the cross-coupling effects between various chemical reactions in viscous flows becomes evident The law of mass action does not hold in the first-order approximation The ratio k f, r /k b, r is not equal to the equilibrium constant K eq Symmetry of coefficients l vr, l rv, and l rs can be proved based on the symmetry properties of bracket integrals. Therefore, the Onsager Casimir reciprocity relations are valid

33 Summary on the first-order (Navier Stockes) approximation Existing CFD models for viscous flows do not account for these effects and are not completely self-consistence. This can lead to loss of accuracy in strongly non-equilibrium flow simulations The effects can be small but apriori we do not know this. We encourage to check above contributions before neglecting them To implement the non-equilibrium effects we should modify the commonly used Navier Stockes governing equations in order to take into account the bulk viscosity, the contribution of chemical reactions to the normal mean stress, and the first-order corrections to the reaction rates Developing the efficient numerical algorithms for the calculation of the first-order contributions: ζ, p rel, k (1) f, r, k (1) f, r, k (1) b, r, and k (1) b, r is of particular importance

34 Compressive N 2 /N flow behind a shock wave The initial conditions are M 0 = 15, T 0 = 293 K, p 0 = 100 Pa. Approximate study: First, the flow parameters and their derivatives are found in the inviscid flow approximation; then these quantities are used as input parameters for the calculation of the first-order normal mean stress and reaction rate coefficients η, ζ 103, Pa s bulk viscosity coefficient (a) 0 p rel, ζ div v, Pa (b) ζ div v p rel shear viscosity coefficient x, cm x, cm Figure : Shear and bulk viscosity coefficients (a); first-order corrections to the normal mean stress (b) behind the shock front as functions of x.

35 Compressive N 2 /N flow behind a shock wave p, π, atm (a) 0.45 δ, % (b) π p x,cm 0.00 x, cm Figure : Pressure p and normal mean stress π (a); per cent first-order correction to the normal mean stress (b) behind the shock front. the bulk viscosity coefficient exceeds the shear viscosity coefficient p rel and the term ζ v associated to bulk viscosity are of the same order; moreover, the absolute value of p rel is higher in a compressive flow. In order to stay self-consistent, one should take into account simultaneously both effects. The total contribution of the correction terms to the normal mean stress is weak: π is slightly higher compared to p; the correction δ < 0.5%

36 Compressive N 2 /N flow behind a shock wave k f, m 3 /mole/s (a) k f (0) k 6000 b, m 6 /mole 2 /s (b) k f k (1) f (p rel ) k (1) b (div v) 0 k (1) f (div v) x, cm-3000 x, cm k b (0) k b k b (1) (p rel ) Figure : Dissociation (a) and recombination (b) rate coefficients behind the shock front as functions of x. the contribution of the terms effect, is weak k (1) k (1) b (1) (1) k f, k b associated to the bulk viscosity the correction terms f, connected to the relaxation pressure play an important role: in the beginning of the relaxation zone, their values are of the same order (and even higher for k (1) b ) as the zero-order rate coefficients k (0) f, k (0) b

37 Compressive N 2 /N flow behind a shock wave 180 δ, % (a) δ k b 20 δ k f =δ ξ 0 x, cm ξ, mole/m 3 /s (b) ξ ξ (0) Figure : First-order corrections to the reaction rate coefficients (a) and reaction rate (b) behind the shock front. Taking into account the first-order effects leads to a considerable decrease of the total reaction rate coefficient close to the shock front the per cent correction reaches 80% for the dissociation and 170% for the recombination rate coefficients The difference between ξ and ξ (0) is significant; the value of the first-order correction to ξ is approximately equal to that for the dissociation rate coefficient (up to 80%). x, cm

38 Expanding N 2 /N flow in a nozzle A conic nozzle with an angle 21. The low pressure throat conditions are T = 7000 K, p = 1 atm η, ζ 103, Pa s (a) 100 p rel, ζ div v, Pa (b) bulk viscosity coefficient 10 1 ζ div v p rel 0.05 shear viscosity coefficient 0.00 x/r E x/r Figure : Shear and bulk viscosity coefficients (a); first-order corrections to the normal mean stress (b) along the nozzle axis as functions of x/r. T = 7000 K, p = 1 atm.

39 Expanding N 2 /N flow in a nozzle p, π, atm π p (a) 2.0 δ, % (b) x/r x/r Figure : Pressure p and normal mean stress π (a); per cent first-order correction to the normal mean stress (b) along the nozzle axis. shear and bulk viscosity coefficients are of the same order of magnitude the relaxation pressure is one to two orders lower compared to the term ζ v The first-order effects lead to a slight decrease of the normal mean stress; the per cent correction is about 2% close to the throat and decreases rapidly with x/r.

40 Expanding N 2 /N flow in a nozzle k f, m 3 /mole/s (a) 5x10 6 k b, m 6 /mole 2 /s 4x10 6 (0) k 3x10 6 f k f 2x10 6 k b (1) (div v) (b) 0.02 k (1) f (div v) 1x10 6 (0) k b k (1) b (p rel ) k (1) 0.00 f (p rel ) k -1x10 6 b x/r x/r Figure : Dissociation (a) and recombination (b) along the nozzle axis. (1) k f, the contribution of the corrections determined by the relaxation pressure is considerably lower compared to the role of the terms k (1) f, k (1) b governed by the velocity divergence: the term k (1) f is negligible, and k (1) b k (1) b k (1) b

41 Expanding N 2 /N flow in a nozzle δ, % (a) ξ, mole/m 3 /s ξ (b) δ k b =ξ ξ (0) δ k 0 f x/r x/r Figure : First-order corrections to the reaction rate coefficients (a) and reaction rate (b) along the nozzle axis. For the dissociation rate coefficient, the first-order corrections are important close to the throat; their contribution is about 30% near the critical section and decreases to % in the expanding part For the recombination rate coefficient, the first-order effects are significant in the whole flow range For the reaction rate, the first-order effects are found to be important; the difference is determined mainly by recombination reaction; the first-order correction is approximately equal to δk b.

42 Expanding N 2 /N flow in a nozzle The high pressure throat conditions: T = 7000 K, p = 100 atm. Qualitative results are similar: bulk and viscosity coefficients take approximately the same values the contribution of relaxation pressure is much less compared to the term ζ v Quantitatively, the first-order corrections are very low: the correction to the normal mean stress does not exceed 0.05% the correction to the dissociation rate coefficient is less than 0.25% the correction to the recombination rate coefficient as well as to the reaction rate achieves the maximum value 2.5%. Thus one can conclude, that in a high pressure expanding flow, the first-order effects appear to be negligible.

43 Summary on the first order corrections in one-temperature flows Although the bulk and shear viscosity coefficients take close values, the effect of bulk viscosity on the normal mean stress is found to be weak; the same holds for the contribution of p rel to π. The maximum correction to the normal mean stress is about 2% in a low-pressure expanding flow and is much less for other cases. Alternatively, the contribution of the first-order corrections to the reaction rates and rate coefficients is much more important: up to 170% for the recombination rate coefficient and 120% for the reaction rates. Exception is for the high-pressure expanding flow, where all first-order corrections are found to be small.

44 Vibrationally non-equilibrium flows State-to-state model Relaxation equations (equations of detailed vibrational-chemical kinetics for the level populations) ρ dα ci dt = J mci + r ξ r ν r,ci M c, c = 1,.., L, i = 0,..., L c Rate of transitions/reactions ξ r = 1 Jcij r du c, N A j Jcij r r, vibr = Jcij + J r, react cij

45 Vibrationally non-equilibrium flows Multi-temperature models Relaxation equations ρ dα c dt = J mc + r ξ r ν r,c M c, c = 1,.., L, ρ dα ce v,c dt = q v + E v,c, Rate of vibrational energy relaxation E v,c = N A r i ε c i c = 1,..., L mol ξ r ν r,ci For anharmonic oscillators, vibrational energy ε c i is not a collisional invariant, and relaxation equation should be written for the specific numbers of vibrational quanta W c, ρ c W c = i in ci

46 State-to-state model. Zero-order approximation The distribution function f (0) cij is given by Maxwell-Boltzmann distribution over velocity and rotational energy The rate of non-equilibrium process r ( ( )) N A ξ r (0) Ar = 1 exp f (0) cij f (0) kt dkl σ f, r (g)du c du d jj ll r stands for any vibrational energy transition or state-specific chemical reaction

47 State-to-state model. Zero-order approximation A r are generalized affinities of state-specific chemical reactions and vibrational transitions: A r, ex = 3 2 kt ln m cm d m c m d + kt ln Z ci rot + (ε c + ε d ε c ε d ) + A r, dr = 3 m kt ln 2 c 3 m c m f 2 n ci Zdk rot c i Z rot Z rot kt ln n cin dk d k n c i n + d k ( ε c i + εd k εc i ε d k ), rot ln(2πkt ) + 3kT ln h + kt ln Zci kt ln + (ε c + ε f ε c ε c i ). n c n f

48 State-to-state model. Zero-order approximation For the particular case of VV transitions within the same species c, A r takes the simplified form A i + A k = A i + A k A r, VV = kt ln n in k n i n k For VT transitions A i + M = A i + M A r, VT = kt ln n i n i + (ε i + ε k ε i ε k ) + (ε i ε i )

49 State-to-state model. Zero-order approximation If we introduce ( ) Ar ω r = 1 exp kt then the zero-order rate of vibrational transitions and state-specific chemical reactions can be written as a linear function of ω r : k (0) f,r ( ) (r) ν ξ r (0) = ω r k (0) f,r (T ρci )ΠL c=1π Lc r,ci i=0 M c is the zero-order rate coefficient for the r th vibrational transition or chemical reaction. The last expression is the generalized mass action law for coupled chemical reactions and vibrational transitions

50 Limit transitions for generalized affinities Multi-temperature model, harmonic oscillators Boltzmann vibrational distribution ( n ci = (T v,c ) exp Generalized affinity A r, ex = 3 2 kt ln m cm d m c m d n c Zc vibr + kt ln Z c rot Zc rot Z rot d Z rot d εc i kt v,c vibr Zc Zc vibr ) (T v,c )Zd vibr (T v,d ) )Z vibr d (T v,d ) (T v,c kt ln n cn d + (ε c + ε d ε c ε d ) + (ε c i n c n + εd k εc i ε d k d ( ) kt ε c i kt v,c + εd k kt v,d εc i kt v,c εd k kt v,d )

51 Limit transitions for generalized affinities Multi-temperature model, harmonic oscillators VT relaxation in a single-component gas ) ) A r, VT = (ε i ε i ) (1 TTv = ε ii (1 TTv For harmonic oscillators, only single-quantum jumps are allowed. Therefore, for VT relaxation in a single-component system there are only two types of reactions ω 1 = 1 exp ( hν kt (1 TTv )) ; ω 2 = 1 exp Zero-order rate of vibrational relaxation E v (0) E v (0) = nhν n i 2N A i r=1,2 ω r k (0) f,r ( hν )) (1 TTv kt

52 Limit transitions for generalized affinities Multi-temperature model, anharmonic oscillators Treanor vibrational distribution n ci = Z vibr c Generalized affinity n c A r, ex = 3 2 kt ln m cm d m c m d +iε c 1 ( ) T 1 +kε d 1 T 1,c (T, T 1,c ) exp +kt ln Z c rot Zc rot ( εc i iε c ) 1 iεc 1 kt kt 1,c Z rot d Z rot d vibr Zc Zc vibr kt ln n cn d + (ε c + ε d ε c ε d ) + n c n d ( ) ( T T 1 i ε c 1 T 1,d T 1,c (T, T 1,c )Zd vibr (T, T 1,d ) (T, T 1,c )Z vibr d (T, T 1,d ) ) ( ) 1 k ε d T 1 1 T 1,d

53 Limit transitions for generalized affinities Multi-temperature model, anharmonic oscillators VT relaxation in a single-component gas A r, VT = ( i i ) ε 1 ( 1 T T 1 ) = iε 1 ( 1 T T 1 ) Multi-quantum (M) transitions are allowed: 1-quantum : A 11 = ε 1 ( 1 T T 1 ) ; A 12 = ε 1 ( 1 T T 1 ), 2-quantum : A 21 = 2ε 1 ( 1 T T 1 ) ; A 22 = 2ε 1 ( 1 T T 1 ),... M-quantum : A M1 = Mε 1 ( 1 T T 1 ) ; A M2 = Mε 1 ( 1 T T 1 ) Then the number of possible VT reactions is 2M. Zero-order rate of vibrational relaxation W (0) is obtained in the similar form as a linear function of ω r but it depends on i rather than ε i and summation is taken over 2M reactions

54 Limit transitions for generalized affinities One-temperature model Vibrational distribution Generalized affinity A r, ex = 3 2 kt ln mcm d m c m d n ci = n c Zc vibr +kt ln Z c int (T ) exp Z int d Z int c Z int d ( ) εc i kt kt ln ncn d +(ε c + ε d ε c ε d ) n c n d A r coincides with the classical definition of the affinity of exchange reaction If only vibrational transitions take place in the mixture (and no chemical reactions) then the A r are identically zero, and we have the case of complete thermodynamic equilibrium

55 State-to-state model. First-order solution First-order distribution function ( f (1) cij = f (0) cij 1 n A cij ln T 1 n dk 1 n F cij v 1 ) G r n cijω r r D dk cij d dk 1 n B cij : v Scalar fluxes are specified by the terms F cij v and r G r cij ω r This representation of the last term becomes possible because the zero order rates of transitions are expressed as linear functions of the scalar force ω r.

56 Cross-coupling effects for the state-to-state model Stress tensor P = πu + 2µ( v) s o µ is the shear viscosity, π is the normal mean stress π = p kt r l vr ω r +l vv v, l vr = [F, G r ], l vv = kt [F, F ] First-order rate of non-equilibrium vibrational transitions and chemical reactions (0) (1) ξ r = ξ r + ξ r N ξ(1) A r = kt l rs ω s l rv v, l rv = [G r, F ], l rs = [G r, G s ] kt s Normal mean stress and rates of transitions-reactions are the linear functions of the same scalar forces v and ω r and are strongly coupled. The kinetic coefficients are symmetric l vr = l rv, l rs = l sr due to symmetry properties of bracket integrals. Therefore the Onsager-Casimir reciprocity relations are verified for the case of strong vibrational-chemical coupling.

57 Cross-coupling effects for the state-to-state model In the linearized case ω r A r kt Scalar fluxes become linear functions of generalized affinities, which is consistent with the results of linear irreversible thermodynamics The generalized mass action law does not work in a viscous flow since ξ r depends on the velocity divergence and affinities of all transitions/reactions For self-consistent CFD simulations of viscous compressible flows, it is necessary to include all the first-order correction terms. Including only the bulk viscosity in the fluid-dynamics equations is not self-consistent if the corresponding term l rv v in the reaction-rate expressions is neglected.

58 Cross-coupling effects for multi-temperature models Single-component gas Distribution function ( f (1) ij =f (0) ij 1 n A ij ln T 1 n A(1) ij ln T v 1 n B ij : v 1 n F ij v 1 n r=1,2 For anharmonic oscillators, we have T 1 instead of T, the summation is taken over r = 1, 2,..., 2M, and ω r are calculated differently Normal mean stress is obtained in a similar form The rate of vibrational energy relaxation for harmonic oscillators E v = E v (0) + E (1) v, E v (1) = r=1,2 ε r kt E v,r (1), ( ) E v,r (1) = kt l rs ω s l rv v, l rv = [G r, F ], l rs = [G r, G s ] s=1,2 ε 1 = hν, ε 2 = hν. For anharmonic oscillators, the summation is taken from 1 to 2M, and ε r should be replaced by mhν, where m is the number of quanta transferred in the collision. G r ij ω r )

59 Cross-coupling effects for multi-temperature models Cross-coupling terms are written in the form similar to that in the state-to-state model The kinetic coefficients are symmetric l vr = l rv, l rs = l sr Therefore the reciprocity Onsager Casimir relations are valid for the multi-temperature case The use of Landau-Teller expression for the rate of vibrational relaxation E v = ρ E v (T ) E v (T v ) τ vibr is not justified for viscous flows because of cross-coupling terms

60 Numerical example 100 δ, % (a) 100 δ, % (b) N 2 T=5000 K p=100 Pa, div v=1000s -1 p=1000 Pa, div v=1000s -1 p=1000 Pa, div v=2000s -1 O 2 T=5000 K p=100 Pa, div v=1000s -1 p=100 Pa, div v=2000s -1 p=1000 Pa, div v=1000s -1 p=1000 Pa, div v=2000s T v, K T v, K Contribution of the first-order correction to the total rate δ as a function of T v for N 2 (a) and O 2 (b). For low pressure and large velocity divergence E v (1) may be of the same order as E v (0). For O 2 the effect is weaker, the mean contribution of v is within 1-2%. E (1)

61 Numerical example 400 u v (1), J/m 3 /c (a) 2 δ, % (b) N 2 O T=5000 K, p=100 Pa, div v=1000 s T v, K -12 x/r N 2, p 0 =1atm, T 0 =7000 K O 2, p 0 =1atm, T 0 =4000 K First-order correction E v (1) as a function of T v (a) and contribution of the first-order correction to the total rate δ as a function of x/r in a nozzle. Close to the throat (particularly for nitrogen), the first-order effects can influence noticeably the rate of vibrational relaxation, whereas with rising x/r (R is the throat radius), the contribution of the first-order correction decreases.

62 Summary on vibrational non-equilibrium flows A self-consistent kinetic model relating the rates of non-equilibrium processes and the normal mean stress to the velocity divergence and chemical reaction/transition affinities is proposed. In the inviscid approximation, cross effects between reaction rates and diagonal elements of the viscous stress tensor do not appear. Cross effects between reaction/transition rates and diagonal elements of the viscous stress tensor exist in viscous gas flows; the rate of each reaction is affected by other reactions and flow compressibility; the law of mass action is violated for viscous flows; the Landau-Teller expression for the rate of vibrational relaxation does not hold.

63 State-to-state model. Heat and mass transfer features State-specific diffusion velocity V ci = dk D cidk d dk D Tc ln T = V DVE ci + V MD c + V TD c is the contribution of vibrational energy diffusion, characteristic feature of the state-to-state approach V DVE ci Vc MD Vc TD is the mass diffusion is the thermal diffusion In CFD, basically the Fick s law is used V c = D c x c either with the effective species diffusion coefficients or constant Schmidt number. Thermal diffusion is systematically neglected.

64 State-to-state model. Heat and mass transfer features Heat flux in the state-to-state model q = λ T p D Tci + ( ) 5 n ci V ci 2 kt + <εci > rot +ε c v + ε c ci ci Taking into account state-specific diffusion velocity: q = q HC + q MD + q TD + q DVE q HC is the contribution due to heat conduction (Fourier flux) Thermal diffusion and diffusion of vibrational energy is usually neglected

65 Contributions to the heat flux Different contributions to the heat flux in compressive and Contribution expandingofflows various processes to the heat flux behind a SHOCK WAVE Contribution of various processes to the heat flux in a NOZZLE q, kw/m q α /q x, cm FIG. 8. Contribution of various processes to the total heat flux. 1: total energy flux q; 2: q HE ; 3: q T D ; 4: q MD ; 5: q DV E. Shock wave. 1: total heat flux, 2: contribution of heat conduction, 3: thermal diffusion, 4: mass diffusion, 5: diffusion of vibrational energy x/r FIG. 9. Contribution of various processes to the total heat flux. 1: q HC /q; 2: q MD /q; 3: q DV E /q; 4: q T D /q. Nozzle. 1: contribution of heat conduction, 2: mass diffusion, 3: diffusion of vibrational energy, 4: thermal diffusion

66 Contributions to the heat flux in a boundary layer q (W m -2 ) q (W m -2 ) (a) Mixture 25%CO 2, 25%CO, 25%(O 2+O), 25%C, T w = 500 K, T e = 3000 K, p e = 1000 Pa, β = 500 s 1. Non-catalytic surface (b) Mixture 10.4% CO 2, 57.6% CO, 31.7% (O 2+O), 0.3% C, T w = 1500 K, T e = 6000 K, p e = 888 Pa, β = 2708 s 1. Non-catalytic surface e f q F 2 - q TD q F 2 - q TD q MD q DVE 5 - q q MD q DVE 5 - q Fig.6: Contribution to the heat flux as functions of. Different test cases: case MC1- C1, VT2 processes included (a); case MC1-C2, VT2 processes included (b); case

67 Contributions to the diffusion velocity in a boundary layer - V TD-CO2(0,0,0) and - V MD,DVE-CO2(0,0,0) (m s -1 ) - V TD-O and - V MD,DVE-O (m s -1 ) Mixture 10.4% CO 2, 57.6% CO, 31.7% (O 2+O), 0.3% C, T w = 1500 K, T e = 6000 K, p e = 888 Pa, β = 2708 s 1. Non-catalytic surface a V TD- CO2(0,0,0) 2- - V MD,DVE-CO2(0,0,0) b V TD-O V MD,DVE-O Fig.10

68 Contributions to the heat flux in a boundary layer q (W m -2 ) q (W m -2 ) 78.58% (N 2+N), 21.38% (O 2+O), 0.04% NO, T w = 1000 K, T e = 7000 K, p e = 1000 Pa, β = 5000 s 1. (a) Non-catalytic surface (b) Partially catalytic surface a q F 2 - q TD 3 - q MD q F 2 - q TD 3 - q MD q DVE 5 - q q DVE 5 - q Fig.14: Contributions to the heat flux as functions of MA1-A1 for the catalytic wall, case

69 Summary on heat flux contributions For a non-catalytic surface, the main contribution to the heat transfer is given by thermal conductivity and thermal diffusion; the contribution of the vibrational energy diffusion varies depending on the deviation of the flow from thermal equilibrium; the mass diffusion process is negligible. The mass flux near the surface is specified mainly by thermal diffusion. For a catalytic surface, mass diffusion is the main process responsible for the heat transfer; the contribution of thermal diffusion is found to be small; diffusion of vibrational energy can be important close to the wall. For shock waves and nozzle flows, the contribution of thermal diffusion is small. Diffusion of vibrational energy is of importance close to the shock front

70 Some features of transport in gases with electronic excitation For temperatures lower than K electronic excitation practically does not contribute to the heat capacities and transport coefficients. Therefore it is usually neglected At higher temperatures the role of electronic excitation occurs important for both molecules and atoms Capitelli, first works in 1970-s, then interrupted Galkin, Zhdanov, some private discussions Capitelli (Bari), 2000-s, equilibrium plasma of argon and hydrogen atoms with electronic excitation SPbSU, 2000-s, non-equilibrium mixtures of atoms and molecules with electronic excitation

71 Contribution of electronic excitation Contribution of translational and internal degrees of freedom to the heat conductivity coefficient % T, K N N 2 λ tr,n λ int,n λ tr,n2 λ rv,n2 λ int,n

72 Contribution of electronic excitation New effects: Bulk viscosity in mixtures of atoms/ions/electrons Internal heat conductivity coefficients in atomic plasmas Prandtl number Prandtl number 0,72 0,71 0,70 0,69 N N 2 O O 2 0,68 0,67 Temperature [K] 0, Prandtl number for atomic and molecular species with electronic excitation.

73 Conclusions Many non-equilibrium effects are systematically neglected in the CFD, among them bulk viscosity and relaxation pressure viscous contributions to the reaction/transition rates electronic excitation thermal diffusion We encourage to estimate these non-equilibrium effects before taking decision that they are negligible