Resolved Viscous Detonation in H 2 /O 2 /Ar Using Intrinsic Low - PDF document

Resolved Viscous Detonation in H 2 /O 2 /Ar Using Intrinsic Low Dimensional Manifolds and Wavelet Adaptive Multilevel Representation by Joseph M. Powers Associate Professor Department of Aerospace and Mechanical Engineering University of Notre Dame presented to the University of Illinois at Urbana-Champaign’s Center for Simulation of Advanced Rockets Urbana, Illinois 22 March 2000 Support: LANL, NSF, and AFOSR

Acknowledgments Prof. Samuel Paolucci, Faculty, ND-AME, Mr. Sandeep Singh, Ph.D. Candidate, ND-AME, Mr. Yevgenii Rastigejev, Ph.D. Candidate, ND-AME, Dr. Christopher Bowman, Post-Doc, ND-AME, Dr. John Liau, LANL, Dr. John Lyman, LANL, Dr. Steven Son, LANL.

Outline • Introduction to issues in nitramine (e.g. HMX or RDX) combus- tion • Introduction to issues in viscous detonations • Intrinsic Low Dimensional Manifold (ILDM) technique (Maas & Pope, 1992) • Wavelet Adaptive Multilevel Representation (WAMR) technique (Paolucci & Vasilyev) • Results for one-dimensional viscous H 2 /O 2 /Ar detonation with detailed kinetics • Preliminary results for HMX gas phase combustion • Conclusions

RDX/HMX COMBUSTION • part of ongoing theoretical/experimental LANL study of low pres- sure ∼ 10 atm combustion of explosives (Son, Liau, et al.), • similar to solid propellant combustion, • preheat zone in semi-infinite solid, • two-phase bubbly liquid foam layer, • gaseous flame region, • gas phase reactions greatest computational burden in simulations. T (K) solid multicomponent reacting foam 3000 gas mixture heat flux 300 0 10 x (cm)

Some Important Questions • Do we have resolved, accurate solutions for HMX combustion? • How can ILDM improve the calculation of HMX combustion? • How can ILDM, derived for well-stirred systems, be used ratio- nally in systems in which convection and diffusion are important?

Motivation for ILDM • Detailed finite rate kinetics critical in reactive fluid mechanics • Common detailed kinetic models are computationally expensive. • Expense increases with – number of species and reactions modeled (linear effect), – stiffness –ratio of slow to fast time scales, (geometric effect). • chemical time scales typically more demanding than convection- diffusion • Reduced kinetics necessary given current computational resources.

RDX GAS PHASE COMBUSTION SIMULATION • very similar to HMX • Uses Yetter’s 45 species, 232 reaction detailed kinetics mecha- nism, • Constant pressure • well-stirred • fastest time scales ∼ 10 − 16 s ! • stiffness ratio (fastest time scale/slowest time scale) ∼ 10 11 b) a) 20 12 10 10 condition number 15 10 10 10 λ i (1/s) 10 8 10 10 5 6 10 10 −10 −5 0 −10 −5 0 10 10 10 10 10 10 t (ms) t (ms) d) c) −3 −3 x 10 x 10 4 1.15 3.5 1.1 [CO 2 ] (mole/cc) [N 2 ] (mole/cc) 3 2.5 1.05 2 1 1.5 1 0.95 −10 −5 0 −10 −5 0 10 10 10 10 10 10 t (ms) t (ms)

Compressible Reactive Navier-Stokes Equations ∂ρ ∂t + ∂ ∂x ( ρu ) = 0 , mass ∂t ( ρu ) + ∂ ∂ ρu 2 + P − τ � � = 0 , momentum ∂x � � �� � � � � ∂ e + u 2 + ∂ e + u 2 ρ ρu + u ( P − τ ) + q = 0 , energy ∂t 2 ∂x 2 ∂t ( ρy l ) + ∂ ∂ ∂x ( ρuy l + j l ) = 0 , ( l = 1 , . . ., L − 1) , elements ∂t ( ρY i ) + ∂ ∂ ∂x ( ρuY i + J i ) = ˙ ω i M i , ( i = 1 , . . . , N − L ) , species τ = 4 3 µ∂u ∂x, Newtonian gas with Stokes’ assumption � ˆ � T N � � q = − k ∂T � d ˆ � h o ∂x + J i i + c pi T T , Fourier’s law T o i =1 J i = − ρ D ∂Y i ∂x , ( i = 1 , . . . , N ) , Fick’s law N φ il � y l = m l Y i , ( l = 1 , . . ., L − 1) , element mass fraction M i i =1 N φ il � j l = m l J i , ( l = 1 , . . ., L − 1) , element mass flux M i i =1 N � Y i = 1 , mass fraction constraint i =1 L � y l = 1 , element mass fraction constraint l =1 J � N � ν ′ � − E j � ρY k � � kj a j T β j exp � � ν ′′ ij − ν ′ ω i = ˙ , ( i = 1 , . . . , N − L ) law of mass action ij ℜ T M k j =1 k =1 N Y i � P = ρ ℜ T , thermal equation of state M i i =1 � T N � � T − ℜ T T o c pi ( ˆ T ) d ˆ � h o e = Y i i + . caloric equation of state M i i =1 N species, L elements, J reactions 3 N + L + 6 equations in 3 N + L + 6 unknowns

Focus on element conservation • L − 1 explicit element conservation equations formed along with N − L species evolution equations, instead of the typical N − 1 species equations, • facilitates a proper use of ILDM in upcoming operator splitting, • In general element mass fractions change due to mass diffusion ρdy l dt = − ∂j l ∂x. • With simple Fick’s Law with no preferential diffusion ρdy l dt = D ∂  ρ∂y l    . ∂x ∂x • In uniformly premixed problem with no boundary influences then, all element concentrations are constant for all time: dy l dt = 0 .

Operator Splitting Technique • Equations are of form ∂t q ( x, t ) + ∂ ∂ q , f , g ∈ ℜ N +2 . ∂x f ( q ( x, t )) = g ( q ( x, t )) , where T    e + u 2    , ρy l , ρY i q =  ρ, ρu, ρ .  2 • f models convection and diffusion • g models reaction source terms • Splitting 1. Inert convection diffusion step: ∂t q ( x, t ) + ∂ ∂ ∂x f ( q ( x, t )) = 0 , d dt q i ( t ) = − ∆ x f ( q i ( t )) . ∆ x is any spatial discretization operator, here a wavelet operator. 2. Reaction source term step: ∂ ∂t q ( x, t ) = g ( q ( x, t )) , d dt q i ( t ) = g ( q i ( t )) . • Operator splitting with implicit stiff source solution can induce non- physical wave speeds! (LeVeque and Yee, JCP 1990)

ILDM Implementation in Operator Splitting • Form of equations in source term step:     ρ 0             ρu 0                 d     e + u 2 � �     ρ = 0 .     2     dt             ρy l 0                      ρY i   ω i M i ˙  l = 1 , . . . , L − 1 , i = 1 , . . . , N − L. • Equations reduce to ρ = ρ o , u = u o , e = e o , y l = y lo , dY i dt = ˙ ω i M i , i = 1 , . . . , N − L ρ o • ˙ ω i has dependency on ρ , e , y l , and Y i • ODEs for Y i are stiff, usually solved with implicit methods. • ODEs for Y i can be attacked with manifold methods to remove stiffness with ILDM with ρ , e , y l , . . . , y L − 1 parameterization.

Intrinsic Low-Dimensional Manifold Method (ILDM) • Uses a dynamical systems approach, • Does not require imposition of ad hoc partial equilibrium or steady state assumptions, • Fast time scale phenomena are systematically equilibrated, • Slow time scale phenomena are resolved in time, • Computation time reduced by factor of ∼ 10 for non-trivial com- bustion problems; manifold gives much better roadmap to find solution relative to general implicit solution techniques (Norris, 1998)

Necessary Dimension of ILDM • Spatial discretization of PDEs results in a set of adiabatic, iso- choric well-stirred reactors, • N species with L elements at constant e and ρ gives rise to a ( N − L )-dimensional phase space (same as composition space), • To resolve M slow time scales, we identify M -dimensional sub- spaces (manifolds), M < ( N − L ), embedded within the ( N − L )- dimensional phase space on which the M slow time scale events evolve, – Fast time scale events rapidly move to the manifold, – Slow time scale events move on the manifold, – Because of convection-diffusion, e , ρ , y l vary, requiring a K = M + L + 1-dimensional manifold. – If y l conserved (premixed with no preferential diffusion), di- mension of manifold is reduced by L − 1. – e.g., for M = 1 in premixed H 2 /O 2 /Ar with no preferential diffusion, we need K = 3. – e.g., for M = 1 in non-premixed HMX (with H , O , C , and N ) in Ar , we need K = 7. For isobaric, K = 6.

Implementation of ILDMs with Convection-Diffusion • To minimize phase error, must integrate full equations until suf- ficiently close to ILDM • When near ILDM, M slow equations are integrated, other vari- ables found by table lookup • Convection-diffusion step applied to all variables perturbs sys- tem from ILDM • In next reaction step, project to ILDM at different value of ρ , e , y 1 , . . . , y N − 1 . Projection onto Y manifold at a different state B Perturbation off the manifold due to convection and ILDM ( ρ , e , y ,..., y ) diffusion 2 2 1, 2 L-1,2 ILDM ( ρ , e , y ..., y ) 1 1 1,1 L-1,1 Y A