Non-equilibrium Effects in Viscous Reacting Gas Flows Elena Kustova - PowerPoint PPT Presentation

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

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

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

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

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)

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 , ε ci � j . The Wang Chang–Uhlenbeck equation (1951): ∂ f cij u c · ∂ f cij F · ∂ f cij J ( γ ) r + � � ∂ t + � = J cij = cij , ∂� ∂� u c γ J ( γ ) cij is specified by the cross section of a microscopic process γ Dimensionless form (in the absence of mass forces): ∂ f cij u c · ∂ f cij 1 ε γ ∼ τ γ � ∂ t + � = J γ cij , θ ≪ 1 . ∂� r ε γ γ

Reduced-order models for fluid dynamics Weak and strong non-equilibrium flows Weakly non-equilibrium flows τ γ ≪ θ ∂ f cij u c · ∂ f cij = 1 ε ∼ τ fp ε J total ∂ t + � , θ ≪ 1 . cij ∂� r Strongly non-equilibrium flows ∃ γ : τ γ ∼ θ ∂ f cij u c · ∂ f cij = 1 ε ∼ τ rap ε J rap cij + J sl ∂ t + � cij , ≪ 1 . ∂� r τ sl

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 J sl cij , cij ,

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 m c u 2 ψ ( λ ) c + ε c cij , λ = 1 , ..., 5 : m c , m c � u c , ij 2 Additional collision invariants for the most frequent collisions. ψ ( µ ) ˜ cij , µ = 1 , ..., M Number of additional invariants depend on the deviation from equilibrium.

Modified Chapman–Enskog method Fluid dynamic variables corresponding to the collision invariants of all processes � � ψ ( λ ) f cij d � ρ λ = 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 � � ψ ( µ ) ˜ ρ µ = ˜ cij f cij d � u c , µ = 1 , ..., M cij Can be different depending on flow conditions

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

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

State-to-state model Time hierarchy τ tr < τ rot ≪ τ vibr < τ react ∼ θ Macroscopic variables vibrational state populations velocity temperature Macroscopic equations ρ d α ci � ˙ = −∇ · J m ci + ξ r ν r , ci M c , c = 1 , .., L , i = 0 , ..., L c dt r ρ d v dt = ∇· P ρ du dt = −∇· q + P : ∇ v

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 d α c � ˙ = −∇ · J m c + ξ r ν r , c M c , c = 1 , .., L , dt r ρ d α c E v , c = −∇· q v + ˙ E v , c dt ρ d v dt = ∇· P ρ du dt = −∇· q + P : ∇ v

One-temperature model Time relation τ tr < τ int ≪ τ react ∼ θ Macroscopic variables chemical species mass fractions velocity temperature Macroscopic equations ρ d α c � ˙ = −∇ · J m c + ξ r ν r , c M c , c = 1 , .., L , dt r ρ d v dt = ∇· P ρ du dt = −∇· q + P : ∇ v

Comparison of models. Compressive flows N 2 flow behind a shock wave, M 0 = 15, T 0 = 293 K, p 0 = 100 Pa n i / n q , kW/m 2 10 - 1 0 3 10 - 2 - 100 3 2 10 - 3 1 1' - 200 3' 10 - 4 2' 1 2 - 300 10 - 5 - 400 10 - 6 10 - 7 - 500 0 5 10 15 20 x , cm i 0 0.5 1.0 1.5 2.0 Figure : Vibrational populations (a) and heat flux (b) behind the shock front.

Comparison of models. Expanding flows N 2 flow in a conic nozzle, T ∗ = 7000 K, p ∗ = 1 atm n i / n 2 q , W/m 1 10 6 10 - 1 10 - 2 10 - 3 2 10 5 10 - 4 10 - 5 1 10 4 1 10 - 6 3 10 - 7 4 4 10 - 8 2 10 3 3 10 - 9 0 10 20 30 40 50 0 10 20 30 40 x/R i Figure : Vibrational populations (a) and heat flux (b) along the nozzle axis.

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

Normal stress and reaction rates in one-temperature viscous flows Integral operators of rapid and slow processes τ tr J rap = J tr ci + J int J sl ci = J react ci , , ǫ = ci ci τ react Governing equations ρ d α c � ˙ dt + ∇ · J m c = c = 1 , .., L , ξ r ν rc M c , r ρ d v dt + ∇· P = 0 , ρ du dt + ∇· q + P : ∇ v = 0 .

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

Integral operators for slow processes Exchange reactions ��� � f c ′ i ′ � f d ′ k ′ f ci f dk � � J 2 ⇋ 2 = − × ci c s c m 3 c ′ s c ′ m 3 d ′ s d ′ m 3 m 3 d s d i i ′ k ′ k dd ′ c ′ kk ′ i ′ d ′ s c ′ i ′ s d ′ k ′ g σ c ′ i ′ d ′ k ′ m 3 c ′ m 3 d Ω d � u d cidk Dissociation reactions ��� � � 3 � � m c J 2 ⇋ 3 � � dk ′ f c ′ f f ′ h 3 s c = f ′ − f ci f dk × ci i m c ′ m f ′ d kk ′ × g σ diss ci , d d u d d u c ′ d u f ′ d u ′ d , σ c ′ i ′ d ′ k ′ , σ diss ci , d are the reaction cross sections cidk