Perspectives on Boltzmann Equation Analysis in Hypersonic Flows Eswar Josyula Air Force Research Laboratory, Ohio Alexander Alekseenko California State University, Northridge, California Jonathan Burt Universal Technology Corp, Dayton, Ohio Presented at the ICERM Workshop on “ Issues in Solving the Boltzmann Equation for Aerospace Systems ”, June 3 – 7, 2013 1
Outline • Motivation Numerical Simulation of Boltzmann Eqn Modeling Internal Energy in Molecules Breakdown of the Continuum Assumption Uncertainties and Sensitivities in Predictions • Closing Thoughts
Critical Challenges in Hypersonic Aerosciences • Aerodynamic force and moment prediction • Aerothermal heating prediction • Rocket/spacecraft plume characterization • Boundary layer transition prediction • Measurements: BL transition, heat transfer, kinetic rates • Wind tunnel testing
Flow Examples Impact on Aerospace Design • Continuum breakdown may occur for: – Hypersonic atmospheric flows • Atmospheric entry (Above Mach18) • Hypersonic cruise vehicles (Mach 5-7) – Small scale internal/external flows • Micro/nano devices – Low density flows • Materials processing • Spacecraft thruster plumes • Design objectives: – Reduce heating • Problems may span continuum and – Predict local rarefaction effects rarefied regimes on vehicle control, aerodynamics • Unconventional design space – Design new propulsion system • Experiments difficult and expensive 4
Two-step uncoupled approach for high altitude plume simulation • Continuum breakdown surface (green) and rarefied inflow boundary (blue), used for two-step CFD/DSMC simulation of generic rocket exhaust plume at high altitude • Nearfield continuum regions assigned to CFD, farfield assigned to DSMC • Smoothing operation used to reduce surface area of DSMC inflow boundary VanGilder, D. B., Chartrand, C. C., Papp, J., Wilmoth, R., and Sinha, N., “Computational Modeling of Nearfield to 5 Farfield Plume Expansion,” AIAA Paper 2007 -5704, 2007.
Flow Characterization in Terms of Knudsen Number • Knudsen number(Kn) characterizes departure from equilibrium distribution of molecular velocities in dilute gas flows / Kn L • Length scale L calculated from vehicle dimensions, characteristic gradient lengths or other flow quantities • Continuum flow assumed if Kn << 1 • Continuum assumptions: – Near-equilibrium distribution of molecular velocities – Diffusive fluxes proportional to macroscopic gradients – No-slip boundary conditions 6
Mathematical Models for Simulation of Continuum and Rarefied Flight Regimes Local Knudsen Number 0.01 0.1 1 10 100 Inviscid Limit Slip Flow Transition Free Molecular Limit Continuum Rarefied Particle or Molecular Boltzmann Eqn. Model: Continuum Model: Euler Navier-Stokes Continuum model does not apply Transition to Rarefied Continuum regime, Kn > 0.1, Both Macro- Rarefied regime Kn < 0.1, Macroscopic and Microscopic Kn = 1-10, Microscopic approach approaches approach Euler Equations Higher Order Particle simulation Navier-Stokes Eqns Continuum equations, (DSMC method) Coupled to Internal Boltzmann equation viz. DSMC, Gas- Energy Mode Kinetic BGK schemes, Relaxation Eqns Moment methods, etc.
Nonequilibrium in Translational and Internal Energy Modes t 0 ~ t T < t R << t V ~ t d Dissociation Recombination Continuum t V-V << t V-T v=n 0 Energy D e v V-T D 0 v=2 V-V v=1 v=0 Internuclear Separation Continuous Distribution of Translational Energy Discrete Distribution of Internal Energy
NUMERICAL SIMULATION OF BOLTZMANN EQUATION
Kinetic Description of Gases In one dimension, each particle associated with velocity (u) and position (x), known as phase space In 3-dimensions, state of gas described using molecular velocity distribution f(t, x , u ) The normalized f ( t, x , u ) is the probability density of finding a particle at the velocity space point u at the configuration space location x
Governing Equation f f ξ ( ξ ) I x t ξ d x ξ d x f d d gives number of molecules in phase space element x and velocity ξ at time t at position L.H.S represents the streaming operator R.H.S. denotes collision term leading to discontinuous jumps in velocity space ξ ξ 2 b ξ ξ ' ' ( ) ( ) I f f ff gb dbd d b * * * ξ ξ 0 0 * * Integrate over the velocity space considering impact parameter b , deflection angle If f is known the macroscopic variables of mass, momentum, energy and stress can be obtained 11
Discretized Boltzmann Equation Discrete Ordinate Representation f f ξ ξ ( ) I x t 2 b ξ ξ ' ' ( ) ( ) I f f ff gb dbd d * * * 0 0 N 0 ξ x x ξ ξ ( , , ) ( , ) ( ) f t f t 1 f f ξ ξ ( ) I x t Suitable for monatomic gases 12
Velocity Grid Discretization Shock Structure • Schematic of Velocity • Determine span of Grid velocity space based on upstream and downstream velocity • Distribute evenly by number of velocity nodes Mach Grid Radius Width No Velocity • Peak of distribution Sphere function is on node 1.2 30x15x15 6.639 0.344 5 20x10x10 14.34 1.43 Efficiency considerations 10 30x15x15 25.82 1.72 13
Mach 5 Coarse 14x7x7 Fine 20x10x10 14
Mach 5 Density, Streamwise Heat Flux 15
Velocity Grid Resolution Mach Grid Points Ratio: Radius Width No Upstream Max velocity Velocity peak spread Up- Sphere Downstream 1.2 30x15x15 13 0.915 6.639 0.344 5 20x10x10 5 0.339 14.34 1.43 10 30x15x15 5 0.176 25.82 1.72 16
Internal energy states treated as multiple species for flows with translational and internal energy nonequilibrium MULTISPECIES BOLTZMANN EQUATION
Multi-Species Boltzmann Equation in Momentum Space p f f i , , ,... I for species A B C A x t m i N N 0 0 p x p x p p p x x p p ( , , ) ( , , ) ( ), ( , , ) ( , ) ( ) f t f t I t I t 1 1 p f f , ,.. 1 , 2 ,... I A B N x 0 t m 2 b S ~ ~ 1 p p j j l j l j l ( ) I f f f f gb dbd d d 4 1 1 j l 0 0 b Suitable for non-reacting multi-species gases 18
Velocity Distribution Function in Mach 3 Shock Wave for 2- Species Gas Mixture m B /m A =0.25, d B /d A =1, c B =0.5 Species A (Heavier) 19
Momentum Grid Resolution for Upstream and Downstream Boundaries in Mach 3 Shock Wave 20
INTERNAL ENERGY EXCITATION
Wang-Chang Uhlenbeck Equation or Generalized Boltzmann Equation f f ξ * , Q x ( , ) ( , ) i j k l t * j , k , l , * q q ξ ξ * ξ ξ * ξ, ξ * ξ * * * * * , i j , Ω Ωd ( ( ) ( ) ( ) ( , ) Q f )f f f d ( i , j ) ( k , l ) i j i j ( i , j ) ( k , l ) * q q k l , . Anderson et al Model for Cross Section ξ, ξ * ξ, ξ * * * * , Ω) , , ( , ( ) p ( , ) ( , ) 0 ( , ) ( , ) i j k l i j k l 1 ξ * * 2 , q ( || || ) q re l k ( , ) ( , ) l i j k l 2 ξ, ξ * * , ( ) p ( , ) ( , ) i j k l 1 ξ * 2 , ( || || ) q q re l ( , ) ( , ) m n i j m n 2 , m n Suitable for reacting multispecies gases with internal energy excitation of diatomic molecules 22
Influence of Inelastic Cross Section – Mach 3 f in (f) approaches that of elastic; has higher average velocity 23
Research Areas for Boltzmann Problem • Efficiency considerations – Grid adaption in velocity space – Use of multiple velocity grids for multiple species in gas mixtures – A priori error analysis to determine sensitivity of velocity grid sizes and population of internal energy states affecting macroscopic quantities • Methods that extend continuum NS equations – Use of BGK approx, Moment methods, others – Quantify discrepancy from benchmark full Boltzmann collisions
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