Perspectives on Boltzmann Equation Analysis in Hypersonic Flows - - PowerPoint PPT Presentation

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

1

Presented at the ICERM Workshop on “Issues in Solving the Boltzmann Equation for Aerospace Systems”, June 3 – 7, 2013

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 – Predict local rarefaction effects

• n vehicle control, aerodynamics

– Design new propulsion system

4

• Problems may span continuum and

rarefied regimes

• Unconventional design space
• Experiments difficult and expensive

Two-step uncoupled approach for high altitude plume simulation

5

• Continuum breakdown surface (green) and rarefied inflow boundary

(blue), used for two-step CFD/DSMC simulation of generic rocket exhaust plume at high altitude

VanGilder, D. B., Chartrand, C. C., Papp, J., Wilmoth, R., and Sinha, N., “Computational Modeling of Nearfield to Farfield Plume Expansion,” AIAA Paper 2007-5704, 2007.

• Nearfield continuum regions

assigned to CFD, farfield assigned to DSMC

• Smoothing operation used to

reduce surface area of DSMC inflow boundary

Flow Characterization in Terms of Knudsen Number

• Knudsen number(Kn) characterizes departure from equilibrium distribution of

molecular velocities in dilute gas flows

• 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

/ Kn L  

Mathematical Models for Simulation of Continuum and Rarefied Flight Regimes

Transition to Rarefied Kn > 0.1, Both Macro- and Microscopic approaches

• Higher Order

Continuum equations,

• viz. DSMC, Gas-

Kinetic BGK schemes, Moment methods, etc. Rarefied regime Kn = 1-10, Microscopic approach

• Particle simulation

(DSMC method)

• Boltzmann equation

Continuum regime, Kn < 0.1, Macroscopic approach

• Euler Equations
• Navier-Stokes Eqns
• Coupled to Internal

Energy Mode Relaxation Eqns

Boltzmann Eqn.

Navier-Stokes Euler Particle or Molecular Model: Continuum Model: Continuum model does not apply

Local Knudsen Number 0.01 0.1 1 10 100 Inviscid Limit Free Molecular Limit Slip Flow Transition Continuum Rarefied

Nonequilibrium in Translational and Internal Energy Modes

Internuclear Separation Energy

v=0 v=1 v=2 v=n0 Continuum v Recombination V-T V-V

D0 De

Dissociation

t0~tT<tR<<tV~td tV-V<<tV-T

• 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

11

) (ξ x ξ I f t f       

x ξd d f

gives number of molecules in phase space element

x and velocity ξ at time t

L.H.S represents the streaming operator R.H.S. denotes collision term leading to discontinuous jumps in velocity space If f is known the macroscopic variables of mass, momentum, energy and stress can be

• btained

x ξd d

at position

* * ' * 2 0 0 '

) ( ) ( ξ ξ d dbd gb ff f f I

b





   

  

ξ

*

ξ

*

ξ ξ

Integrate over the velocity space considering impact parameter b, deflection angle b





Discretized Boltzmann Equation

Discrete Ordinate Representation

12

) ( ) ( ) , ( ) , , ( ) ( ) ( ) (

1 * * ' * 2 0 0 '        

  ξ x ξ ξ ξ x x ξ ξ ξ ξ x ξ I f t f t f t f d dbd gb ff f f I I f t f

N

b

                

  

   

Suitable for monatomic gases

Velocity Grid Discretization Shock Structure

• Schematic of Velocity

Grid

• Determine span of

velocity space based on upstream and downstream velocity

• Distribute evenly by

number of velocity nodes

• Peak of distribution

function is on node

13

Mach No Grid Radius Velocity Sphere Width 1.2 30x15x15 6.639 0.344 5 20x10x10 14.34 1.43 10 30x15x15 25.82 1.72

Efficiency considerations

Mach 5

14

Coarse 14x7x7 Fine 20x10x10

Mach 5

Density, Streamwise Heat Flux

15

Velocity Grid Resolution

16

Mach No Grid Points Upstream peak Ratio: Max velocity spread Up- Downstream Radius Velocity Sphere Width 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

MULTISPECIES BOLTZMANN EQUATION

Internal energy states treated as multiple species for flows with translational and internal energy nonequilibrium

Multi-Species Boltzmann Equation in Momentum Space

18

l j l j l j j S l b b j A

d d dbd gb f f f f I N B A I f m t f t I t I t f t f C B A species for I f m t f

N N

p p x p p p x x p p p x p x p       

                       

) ~ ~ ( 4 1 ,... 2 , 1 ,.. , ) ( ) , ( ) , , ( ), ( ) , , ( ) , , ( ,... , , x p

1 1 2 1 1 i i

                    

    

      

Suitable for non-reacting multi-species gases

Velocity Distribution Function in Mach 3 Shock Wave for 2- Species Gas Mixture mB/mA=0.25, dB/dA=1, cB=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

• r

Generalized Boltzmann Equation

22

  

        

                        

n m n m j i n m l k j i l l k j i l k j i l k j i l k j i j i j i l k j i l k j i l k j l k j i

q q q p p Section Cross for Model al et Anderson d f f )f f q q q q Q Q f t f

, , ) , ( ) , ( 2 , ) , ( ) , ( 2 k , ) , ( ) , ( , ) , ( ) , ( , , ) , ( ) , ( , ) , ( ) , ( , ) , ( ) , ( , , , , ) , ( ) , (

) || || 2 1 ( ) || || 2 1 ( q ) ( ) ( Ω) , ( . , Ωd ) Ω , ( ) ( ) ( ) ( (

* * * * * * * * * * * * * * *

                              

   

re l re l * * * * * * *

ξ ξ ξ ξ, ξ ξ, ξ ξ, ξ ξ ξ, ξ ξ ξ ξ x ξ

Suitable for reacting multispecies gases with internal energy excitation of diatomic molecules

Influence of Inelastic Cross Section – Mach 3

23

f in (f) approaches that of elastic; has higher average velocity

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

Chapman Enskog Analysis

25

ci ci ci c c i r ci j ci ci Tci rel Tci dk dk cidk ci ci cij dk cij cij cij dk dk cij cij cij cij ci j c c rot ci ci ci j c cij slow cij rapid cij cij react vib rot el cij cij cij

V n kT d D p T q vI S I p p P T D d D V ly respective A B D from

• btained

D ts Coefficien Transport G v F v B d D T A n f f kT kT c m T Z n s kT m f J J J J f t f

   

                                                                                   t t t t 2 5 2 ) ( ln , , , , : ln 2 exp ) ( 2 1 ; ~

) ( ) 1 ( 2 2 / 3 ) ( c

x u

• Continuum governing equations
• Obtain transport coefficients

Continuum Equations – Inviscid

• Mass conservation
• Momentum conservation
• Energy conservation

,... 1 , ) ( ) (        n u t

n n n

    







*

n n n

  ) ( ) (           p u u u t   

] ) / ( ) (        u p e e t    

27

Coupling Flow Equation to Master Equation

n n n

u t         ) .( ) ( 

48 ...., 2 , 1 ,  n

2 2 2 2

) ( ) ( N n N N n N     M N M n N    2 ) (

2

]} ) ( ~ ) ( ) ( ) ( [ { 1

2 N n nD N n nD n VT n n VT n

n Continuum k Continuum n k n n k n n k M                   





Invisicid Equations with master Equation for internal energy excitation Useful to predict aerodyamic forces and moments in low Knudsen regime

28

Equilibrium-Weighted Probability N2-N2 Collisions at 4,000 K

Vibrational quantum number Equilibrium-Weighted Probability

10 20 30 40 50 10

• 10

10

• 8

10

• 6

10

• 4

10

• 2

10 10

2

VT Dissociation V-V Resonant V-V Off Resonant (with ground state)

Various internal energy exchange processes assume importance at different quantum levels

Continuum Equations - Navier-Stokes

29

             

 

) ( ) / ( ) ( Energy Total

• f
• n

Conservati ) ( Energy l Vibrationa

• f
• n

Conservati ~ ) ( Momentum

• f
• n

Conservati Mass

• f
• n

Conservati                                            

 

t         t         v V h q v p e t e specific state non for q q V v e t e vv t v specific state non for V v t specific state for V V v t

c c c vib vib vib trans c vib vib c c c c ci ci c ci ci

     

• Treatment for state-to-state kinetic rates and multi-temperature models
• Note transport coefficients derivable from Boltzmann equation
• Useful for prediction of heat transfer to the aerospace vehicle

Mach 20 Flow with Dissociation

Master Eqn vs Multi-Temp Model Actual Physics in Flowfield

• Degree of nonequilibrium described at every point in the flowfield

Master Eqn Model Multi-Temp Model

CONTINUUM BREAKDOWN AND HYBRID METHODS

Local rarefaction in hypersonic flows

• Hypersonic flows with low global Kn tend to include regions of high local Kn

1. Interior of strong shocks 2. Strong expansion regions 3. Shock layer near sharp leading edge – location of maximum heat transfer 4. High gradient boundary layer regions 5. Shock-shock, shock-boundary layer interaction

32

2 1 5 4 3

Motivation for Boltzmann-NS Hybrid Solver Development

Navier-Stokes Requirement: Accurate, consistent coupling Direct Boltzmann or DSMC

Elegant Coupling at Interfaces of Boltzmann and NS regions Enables hypersonic full trajectory analysis including heat loads Kinetic/Continuum Analysis DSMC Boltzmann

Boltzmann is an attractive option

• Accurate and consistent

Problem:

• Current single regime codes unsuitable for coupling

Solution:

• Develop and implement compatible solution

methodologies for coupling

33

50,000 100,000 150,000 200,000 250,000 300,000 350,000 50 100 150 200 250 300 Time (sec) Altitude (ft) Continuum

• Low altitude
• Long length scales

Rarefied

• High altitude
• Sharp edges

Hypersonic Shock-Boundary Layer Interaction

• Hypersonic SBLI in laminar low-density flow, with shock impinging on flat

plate boundary layer

• Contours of maximum gradient-length local Knudsen number

34

max

max , ,

tra tra

Kn T V T a               

Issues in Hybrid Schemes

• Predicting Continuum Breakdown
• Exchanging Information between Continuum and

Rarefied Domains (more challenging for DSMC hybrid)

• Consistency in Treatment of Modeling Flow Physics in

Both Domains

• Type of Coupling (Weak or Strong, one-way)

ERRORS AND UNCERTAINTIES

Errors and uncertainties

• Discretization errors should be quantified to ensure

accurate simulation

• Model form errors can contribute to uncertainty in

simulation results (e.g. BGK approximation)

• Other errors result from extrapolation of experimental

conditions used in determining model parameter values

• Both model form errors and parametric errors lead to

epistemic uncertainty

Nominal simulation results for example case

• Mach 15.6 flow of N2 over 25/55 double cone, KnD = 8.410-4
• Surface pressure (left) and heat flux(right); comparison of CUBRC

experimental data with results from HAP, DS2V and SMILE DSMC codes

38 geometry geometry

Sensitivity analysis results

• Percent contribution of each input parameter to total uncertainty in output

quantities

COMPUTATIONAL LOAD ISSUES

Comparison of efficiency between Boltzmann, DSMC, BGK

• Goal: Evaluate accuracy and efficiency of various models in

Unified Flow Solver (UFS) for simple high speed flow problems

• Compare UFS with established NS and DSMC codes &

experiments

• Test cases: Argon flow over a cylinder

– Freestream Mach number = 4 – Kn = 0.3

• Kn = 0.3: UFS Boltzmann, BGK;

MONACO DSMC

Simulation method CPU hours

Boltzmann (UFS) 1929.4 BGK (UFS) 18.5 DSMC (MONACO) 3.75

Efficiency comparison for Kn = 0.3

CPU hours with existing solvers

• Particle representation at molecular level
• Particles possess microscopic properties, each

simulated particle represents large collection of atoms/molecules

• Particles move according to assigned velocities
• Collisions handled probabilistically
• Probabilities depend on energy dependent

cross-section

• Approximations assumed accurate if

mean collision separation (MCS) << , t << tcoll

{

m, x, u erot, evib

Direct Simulation Monte Carlo Method

42

Widely used in Aerospace design

DSMC simulations of atmospheric flow, plume flow around Mir space station

43

• Surface pressure on Mir space station

from Space Shuttle RCS exhaust during docking maneuver

• Complex plume impingement pattern

due to interaction of multiple plumes and surface shadowing effects

Markelov, G. N., Kashkovsky, A. V., and Ivanov, M. S., “Space Station Mir Aerodynamics Along the Descent Trajectory,” Journal of Spacecraft and Rockets, Vol. 38, No. 1, 2001, pp. 43-50. LeBeau, G. J., and Lumpkin, F. E., “Application Highlights of the DSMC Analysis Code (DAC) Software for Simulating Rarefied Flows,” Computer Methods in Applied Mechanics and Engineering, Vol. 191, 2001, pp. 595-609.

• Streamlines from DSMC calculation
• f flow around Mir space station

during controlled descent, at 110 km

• DSMC used to help determine
• ptimal trajectory and orientation

Major Challenge for the Boltzmann Community

Why Boltzmann for Aerospace?

• Errors from

discretization in physical space

• Errors due to numerical

time step

• Errors due to number of

simulated particles

• Errors from

discretization in physical space

• Errors due to numerical

time step

• Errors due to

discretization of velocity space

DSMC Boltzmann

• Fundamental limitations of DSMC include:
• statistical noise, particularly for subsonic flows
• simulated particle representation does not represent

real velocity distribution

• Boltzmann PDE amenable to mathematical treatment

Computational Load

• n one spatial node/cell
• Computational load = No of velocity nodes (cells) X

integrating over impact parameter, scattering angle, and the velocity where n=number of velocity nodes in 1 direction

• No of operations for velocity nodes = n3
• For all impact parameters, scattering angles, and the velocities , no of
• perations is n5
• For the above classical form, total number of operations is n3 x n5 = n8

* * ' * 2 0 0 '

) ( ) (

3

ξ ξ d dbd gb ff f f I

R b





   

n3 n n operations

ξ

*

ξ

*

ξ

ξ

Computational load for n=21 Deterministic solution Boltzmann equation with full collision integral

• Time taken for n8 operations on a single processor of speed 2.3 GHz

is 134 s for 1 time step

• For a 1D problem with 100 spatial nodes, 100 x 134 s = 13, 400s =

3.72 hr

• For 3D problem with 100 spatial nodes, 1003 x 134 s = 37,222 hr
• For 1D problem with 2 internal energy levels, 100 x 134 x 22 = 14.8

hr

• For 1D problem with 10 internal energy levels, 100 x 134 x 202 =

1488 hr

The number of operations Classical form Galerkin (generic) Discontinuous Galerkin Fourier- Galerkin Stochastic evaluation

Computational Load Issues Numerical Solution of the Boltzmann Equation

One time step on single processor, O(n^8) n=9 n=15 n=21 n=33 n=61 <1 sec 9 sec 134 sec 1.6 hrs 232 hrs (est) 0.2 Mb 2.75Mb 14 Mb 138 Mb 2.83 Gb

) (

9

n O ) (

8

n O ) (

8

n O ) (

3

M O n ) (

6

k O

Example: Time and memory requirements in DG velocity method

Fourier-Galerkin and Stochastic methods show promise

Scalability of Collision Integral

One time step using MPI parallelization, n=61, 64 cores 128 cores 256 cores 512 cores 1024 cores 9,432 s 4,801 s 2,499 s 1,259 s 751 s

• 0.97 order of

speed-up 0.94 order of speed-up 0.98 order of speed-up 0.74 order of speed-up n = 61 For n = 21 on 128 cores, the projected CPU seconds per time step is 1.4 s For 3D problem, 1003 spatial nodes, 1003 x 134 s = 37,222 hr on 1 core for 1 time step For 3D problem 1003 spatial nodes, 1003 x 1.4 s = 389 hr on 128 cores for 1 time step

Closing Thoughts

• Boltzmann equation analysis limited by large computational

loads associated with velocity grids. However, 2D benchmark solutions of important aerospace problems achievable

• Boltzmann equation analysis for the complete aerospace

problem includes internal energy excitation, hybrid Boltzmann-NS, and error quantification

• Major challenge in numerical efficiency improvements

consists of reducing the redundant computations in any Boltzmann-like problem