MATHEMATICAL MODELING OF THE LONG TIME EVOLUTION OF THE PULSATING DETONATION WAVE IN THE SHOCK ATTACHED FRAME

Transcription

1 International Conference Quasilinear equations, Inverse problems and their applications MATHEMATICAL MODELING OF THE LONG TIME EVOLUTION OF THE PULSATING DETONATION WAVE IN THE SHOCK ATTACHED FRAME Utkin P.S., Lopato A.I. Institute for Computer Aided Design Russian Academy of Science Moscow Institute of Physics and Technology The reported study was funded by RFBR according to the research project mol_а. Moscow Institute of Physics and Technology, Dolgoprudny, September 2016

2 What is detonation? Detonation is a hydrodynamic wave process of the supersonic propagation of an exothermic reaction through a substance. The dt detonation ti wave (DW) is a self sustained ti shock wave (SW) discontinuity it behind bhid the front of which a chemical reaction is continuously initiated due to heating caused by adiabatic compression. The detonation wave velocities in gaseous mixtures under normal conditions are about 1 3 km/s, front pressures atmospheres. Profile produced by gas motion behind the ideal detonation started at the closed end of the tube Sheperd J.E. // Proc. Comb. Inst Zeldovich von Neumann Doering (ZND) solution for the steady state detonation 2

3 Detonation theory: state of the art DW propagation is characterized by a complicated nonlinear oscillatory process 1D. Pulsations of parameters behind the DW front. 2D. Transverse compression waves that interact with the DW in two dimensional computations and experiments on the DWs propagation ation in narrow gaps. Detonation cells. 3D. Transverse wave propagating in a spiral spin. Leung C. et al. // Physics of Fluids Taylor B.D. et al. // Proc. Comb. Inst Levin V.A. et al. // Comp. Math. and Math. Phys

4 Detonation analogies and simplified models Hydraulic jump and traffic jam are analogous to self sustained DW (ZND like front structures) Kasimov A.R. // J. FluidMech Flynn M.R. et al. // Phys. Rev. E D asymptotic equations: Faria L.M. et al. // J. Fluid. Mech

5 Some problems in detonation calculations Possibility of DW failure in the simulations of the DW long time propagation Higgins i AJ A.J. Approaching dt detonation ti dynamics as an ensemble of interacting ti waves // Proc. 25th ICDERS. 2nd 7th August Leeds, UK. Paper PL3. He L., Lee J.H.S. The dynamical limit of one dimensional detonations // Physics of Fluids Semenov I. et al. Mathematical modeling of detonation initiation via flow cumulation effects // Progress in Propulsion Physics. 8. Proc. EUCASS July Munich, Germany. Why does the detonation wave fail in the numerical computations? Assumptions: Mistakes Mathematical models Computational method 5

6 Aims The aim of the work numerical investigation of weakly unstable and irregular regimes of pulsating DW propagation in two statements the modeling of DW in the laboratory frame () with detonation initiation near the closed end of the channel and modeling in the shock attached frame (),and quantitative i comparison of results using Fourier analysis of the pulsations. 6

7 Governing system of equations in the laboratory frame () Z mass fraction of the reacting mixture component Q heat release A pre exponent exponent factor E activation energy 1D Euler equations + one stage chemical reaction model 7

8 Governing system of equations in the shock attached frame () Z mass fraction of the reacting mixture component Q heat release A pre exponent factor E activation i energy 1D Euler equations + one stage chemical reaction model + shock velocity evolution equation 8

9 Brief review (shock attached frame) Kasimov A.R., Stewart D.S. On the dynamics of the self sustained onedimensional detonations: A numerical study in the shock attached frame // Physics of Fluids (10): Numerical scheme: first approximation order scheme. Shock speed evolution equation: integration of the governing equation along the C + characteristic nearthe shock. Applicability: test with a shock overtaking another shock demonstrates the stability of the algorithm to detonation waves numerical calculations. Results: main regimes of detonation wave propagation p are obtained. The shock dynamics is shown to be determined entirely by the finite region between the shock and the sonic locus. Henrick A.K., Aslam T.D., Powers J.M. Simulations of pulsating one dimensional detonations with true fifth order accuracy // Journal of Computational Physics V P Numerical scheme: fifth approximation order WENO scheme + fifth order Runge Kutta scheme. Shock speed evolution equation: connecting with the momentum flux gradient. Applicability: restrictions strongly unstable detonation waves. Results: the approximation order is shown to be equal to five (steady detonation). For an unstable regime a stable periodic limit cycle is obtained. The phase portrait confirms the unstable regime. The bifurcation diagram of different activation energies (different regimes) is constructed. 9

10 Dimensionless procedure Characteristic scales parameters in front of the DW and a half reaction length : l 12 pa paμ ρa pa ua = Ta = ρa ρar 1 uznd ( Z) DCJ = 12AZ exp EρZND Z pznd Z ( ( ) ( )) dz Dimensionless ρ p u = = = ˆ T pˆ = = ˆ E = ˆ Q ˆ ˆ ˆ = ˆ D = ˆ l ρ p u T E Q D l = 2 2 ρ p u T ρˆ μu u u l a a a a a a a 1 variables: ˆ ( ) ˆ ˆ A uznd Z DCJ A= = u l 12 12Zexp Eρ ˆˆ ( Z) pˆ ( Z) ( ) a ZND ZND dz 1/2 Dimensionless ZND profiles: Dˆ ˆ + ˆ CJ γ DCJ 1 DCJ γ uˆ 0( Z) = ( 1 + Z) pˆ 0( Z) = 1 + Z 2 γ+ 1 Dˆ + 1 ˆ CJ γ DCJ + 1 ˆ 2 ˆ 2 ( ) γ DCJ + 1 DCJ γ = ρˆ 0 Z 1 Z Dˆ = + ( ) ˆ + ( 2 ) ˆ 2 CJ γ γ 1 Q γ 1 Q γ + 1 Dˆ ( ˆ + ) CJ 2 2 γ DCJ 1 1 Erpenbeck J.J. Stability of steady state equilibrium detonations // Physics of Fluids V. 5. P Semenko R. et al. Set valued solutions for non ideal detonation // Shock waves V. 26. P

11 Computational algorithm Physical processes splitting technique Finite volume method Second approximation order ENO reconstruction Courant Isaacson Rees numerical flux Second order Runge Kutta explicit scheme for time stepping Euler implicit method for solving equations of energy and chemical reactions Parallelization (MPI) Algorithm of integration ti of the LSW velocity evolution () ( { } ) { } ( ) A i+ 12 { } { } CIR n n 1 n n n f i + = 12 f u i + f u i+ 1 + u + i ui { ( ) } + { ( ) } n = Ω n Λ n Ω n Ω n + 1 Λ n + 1 Ω n A i i i i i i i+ 1 2 Lopato A.I., Utkin P.S. // Comp. Research Model

12 Algorithm for shock speed calculation (1).. dξ = u+ c D dτ dp du + ρc ( γ 1) Qρω = 0 dτ dτ n ξ, a= a ξ, ξ, b= b ξ, ξ, c= c ξ, ξ. n n n n n n ( * * ) ( * * ) ( * * ) n 1 * ξ * 1 1 ( ) ( ) ( ) ( ) n n n n n n n n 2 a ξ*, ξ* τ + b ξ*, ξ* = u* + c* D, n n 1 n 1 n n 1 n 1 n 1 n 1 2 a ξ*, ξ * τ + b ξ*, ξ * = u * + c * D. The system is solved numerically with Newton iterations. 12

13 Algorithm for shock speed calculation (2).. dp du + ρc ( γ 1) Qρω = 0 dτ dτ New LSW speed D n+1 is determined dt dfrom the second equation. Three point one sided approximation of the derivatives: ( 0 * * ) ( 0 * * ) ( ) n+ 1 n n 1 n+ 1 n+ 1 n+ 1 n n 1 n+ 1 n+ 1 αp + λp + δp + ρ c αv + λv + δv γ 1 Qρ ω = 0 n n Δ τ Δ τ α= +, λ= + = 1 1, δ. n n n n n n n 1 Δτ Δ τ +Δτ Δτ Δτ Δ τ +Δτ Calculation of the SW velocity D n+1 = M n+1 c n+1 on (n+1) th time layer asaa solution of nonlinear equation. Here, Rankine Hugoniot conditions are applied n+ 1 n+ 1 ( γ )( M ) ( M ) n+ 1 n+ + ( γ )( M ) 2 2 n+ 1 n n n n+ 1 ( M ) Z p γ γ ρ v =, =, =, = 1. P γ γ R 2 1 C γ M n n 1 n n 1 Variables p*, p*, v*, v* are calculated using quadratic interpolation procedure. Lopato A.I., Utkin P.S. Detailed simulation of the pulsating detonation wave in the shock attached frame // Computational Mathematics and Mathematical Physics

14 Shock overtaking another shock test τ 14

15 Stable regime E = 25, Q = 50, γ = 1.2, Δx = calc1 N1= 4000, Δ 1= p front pvn.. calc k ln( Δ2 Δ1) pfront pvn 10. ~ Ch k 1.72 N = 2000, Δ = p p ln 2 calc front vn

16 Weakly unstable regime (1) E = 26, Q = 50, γ = 1.2, Δx = Δp ~ 10 4 T = T LIN = T = Lee H.I., HI Stewart t D.S. DS Cl Calculation lti of linear detonation dt ti instability: one dimensional i instability of plane detonation dt ti // Journal of Fluid Mechanics V P

17 Weakly unstable regime (2) E = 26, Q = 50, γ = 1.2, Δx = t τ Long term numerical research 17

18 Weakly unstable regime (3) E = 26, Q = 50, γ = 1.2, Δx = Fourier spectral analysis (5000 units of time) p = Ak cos 2 vn k= 0 p ( πν t ) k A A ν ν 18

19 Weakly unstable regime (4) Fourier spectra of two statements demonstrate similar values of peak frequencies A ν Peak ν() ν ()

20 Irregular regime (1) E = 28, Q = 50, γ = 1.2, Δx = Δp ~ 10 4 pmax 1.65 = pmin 0.76 p p max 1.68 = min

21 Irregular regime (2) E = 28, Q = 50, γ = 1.2, Δx = t t Long term numerical research 21

22 Irregular regime (3) E = 28, Q = 50, γ = 1.2, Δx = Fourier spectral analysis (5000units of time) p = Ak cos ( 2 πνkt ) p vn k= 0 A A ν ν 22

23 Irregular regime (4) Fourier spectra of two statements demonstrate similar values of peak frequencies Peak ν() ν () A ν 23

24 Strongly unstable regime (1) E = 35, Q = 50, γ = 1.2, Δx =

25 Strongly unstable regime (2) E = 35, Q = 50, γ = 1.2, Δx =

26 Conclusion 1. The numerical investigation of pulsating DW propagation using two statements e ts laboratory ato frame and dshock attached frame is carried out. The second statement includes the special numerical algorithm of the integration of shock speed evolution equation with the second approximation order based on the grid characteristic method. 2. The calculation of the weak unstable mode gives a DW that tends to the stable limit cycle. Both statements give similar results that correlate with the known analytical solution. The spectral Furrier analysis demonstrates that main frequency peaks coincide although the regions between the peaks somewhat differ. 3. Numerical investigation of the irregular mode demonstrates the tendency of the front pressure reducing for laboratory statement. The spectral Furrier analysis confirms chaospresence. 4. Further investigations long term calculation of strongly unstable mode for both statements. 26

27 Publications 1. Лопато, А.И., Уткин, П.С. Математическое моделирование пульсирующей волны детонации с использованием ENO схем различных порядков аппроксимации // Компьютерные исследования и моделирование Т. 6, 5. С Лопато, А.И., Уткин, П.С. Особенности расчета детонационной волны с использованием схем различных порядков аппроксимации // Математическое моделирование Т. 27, 7. С Лопато, А.И., Уткин, П.С. О двух подходах к математическому моделированию детонационной волны // Математическое моделирование Т. 28, 2. С Lopato, A.I., Utkin, PS P.S. Two approaches to the mathematical modeling of detonation wave // Mathematical Models and Computer Simulations V. 8, No. 5. P Лопато, А.И., Уткин, П.С. Детальное математическое моделирование пульсирующей детонационной волны в системе координат, связанной с лидирующим скачком // Журнал вычислительной математики и математической физики Т. 56, 5. С Lopato, A.I., Utkin, P.S. Detailed simulation of the pulsating detonation wave in the shockattached frame // Computational Mathematics and Mathematical Physics V. 56, No. 5. P Lopato, A.I., Utkin, P.S. Towards second order algorithm for the pulsating detonation wave modeling in the shock attached frame // Combustion Science and Technology V. 188, No. 11 (to appear). 27