A second order scheme on staggered grids for detonation in gases

Transcription

1 A second order scheme on staggered grids for detonation in gases Chady Zaza IRSN PSN-RES/SA2I/LIE August 1, 2016 HYP 2016, RWTH Aachen Joint work with Raphaèle Herbin, Jean-Claude Latché and Nicolas Therme

2 Introduction 1/23 Fukushima I Nuclear Power Plant, Unit 3, 14 March Hydrogen-air explosion severely damaging the reactor building and leaking radioactive materials in the atmosphere.

3 2/23 Introduction Previous studies at CEA/IRSN in the 80 s concluded that J.-M. Evrard. General description of hydrogen related risks in pressurized water reactor containments. Tech. Rep. SAER/84/392, CEA (1984): If direct initiation of a detonation appears highly improbable, the flame acceleration mechanism, on the hand, is liable to lead to its formation. (...) The turbulences induced by the obstacles in front of the flame cause (...) acceleration of the flame front and may lead to a deflagration detonation transition. A. Forestier and J. Brochard. Tenue d une enceinte de REP 1300 en cas de détonation hydrogène. Tech. Rep. DEMT/87/276, CEA (1987): The chosen scenario leads very likely to a very serious damage of the containment building. Nowadays, the ruin (of the dome in particular) cannot be excluded. Simulating hydrogen-oxygen explosions in ISIS/P2REMICS code: Mixing Low-Mach Low speed Ignition Deflagration Subsonic High speed DDT Detonation Supersonic High speed Projection scheme Explicit scheme Common Staggered Finite-Volume Mesh

4 3/23 Introduction The Zeldovich-von Neumann-Döring (ZND) model for detonation: Steady 1D combustion wave: inert shock wave + chemical reaction (1) with finite rate Self-sustained: shock wave brings reactants to ignition while expansion of products (2) drives shock forward p State B p M S 1 State C State A products (2) (1) reactants ZND states on pressure profile x B C A y = 1 M y = 0 S τ ZND states on Hugoniot curves

5 Introduction The ZND model is a good approximation but detonations are a genuinely multi-dimensional phenomena, driven by transverse instabilities: R T 1 S M R S T 2 I R T 3 S M R S T 4 detonation front at t 1 R S T 1 R S R S M R T 4 detonation S front at t 2 I T 2 T 3 I Front structure: (I)ncident shock (R)eflected shock (M)ach stem (S)lipline (T)riple point Left: soot tracks of a planar detonation in a channel (Strehlow, 1968). Right: open-shutter photograph of a converging cylindrical detonation in a narrow channel (Knystautas Lee, 1971). 4/23

6 Physical Model 5/23 Unimolecular reaction of a perfect gas with reaction rate defined by either Heaviside law or Arrhenius law: ( ) R P EA κ(t ) = κ γ R = γ P = γ 0 exp RT Equation of state: e = p/ρ γ 1 + q 0y Reactive Euler equations: t ρ + div(ρu) = 0 t (ρu) + div(ρu u) + p = 0 t (ρe) + div(ρeu) + div(pu) = 0 t (ρy) + div(ρyu) = ρyκ(t ) Controlling positivity of e : solve an internal energy balance instead?

7 Space Discretization 6/23 Staggered Finite Volume meshes: K P K D σ ε=d σ D σ D σ L u y D σ u y D σ M L u x D σ M Rannacher-Turek/Crouzeix-Raviart grids MAC grid with unknowns: Primal grid ρ K, p K, e K, y K K M Dual grid u Dσ,i σ E S,i, 1 i d

8 Numerical Scheme: Time Algorithm (1) Hydrodynamic step (continuity, mass fraction, internal energy) ρ n+1 K = ρ n K δt div K (ρ n u n ) ȳ n+1 K = 1 ρ n+1 K e n+1 K = 1 ρ n+1 K (2) Chemistry step [ρ n K y n K δt div K (ρ n u n y n )] [ ρ n K en K δt(div K (ρ n e n u n ) + pk n div K (u n ) + 1 ] K Sn K ) y n+1 K = ȳ n+1 K exp( κ(t n K )δt) with T n K = Mpn K Rρ n K (3) Equation of state p n+1 K = (γ 1)ρ n+1 K (en+1 K q 0 y n+1 K ) (4) Hydrodynamic step (momentum) u n+1 σ,i = 1 [ρ n ρ n+1 Dσ uσ,i n δt ( div Dσ (ρ n u n ui n ) + ( ) ] pn+1 ) σ,i D σ 7/23

9 Numerical Scheme: Convection Operators Convection operators on the primal grid discretized as: div K (ρuz) = 1 K σ E(K) F K,σ z σ with F K,σ = σ ρ σ ( d i=1 Convection operators on the dual grid discretized as: div Dσ (ρuz) = 1 F σ,ε z ε D σ ε E(D σ) u σ,i n K,σ,i ) with F σ,ε calculated such that (Ansanay-Alex Babik Latché Vola, 2010): D σ ( ρ n+1 D δt σ ρ n ) D σ + Fσ,ε n = 0 ε E(D σ) Convected quantities: upwinding with respect to the material velocity u n. Improved accuracy at contact waves : MUSCL-like interpolation of ρ σ, e σ with a limitation procedure such that p and u are well preserved at contacts (Therme, 2015) p σ = βp K + (1 β)p L, β [0, 1] 8/23

10 Numerical Scheme: Discrete Regularization Term (1) Hydrodynamic step (continuity, mass fraction, internal energy) ρ n+1 K = ρ n K δt div K (ρ n u n ) ȳ n+1 K = 1 ρ n+1 K e n+1 K = 1 ρ n+1 K (2) Chemistry step [ρ n K y n K δt div K (ρ n u n y n )] [ ρ n K en K δt(div K (e n ρ n u n ) + pk n div K (u n ) + 1 ] K Sn K ) y n+1 K = ȳ n+1 K exp( κ(t n K )δt) with T n K = Mpn K Rρ n K (3) Equation of state p n+1 K = (γ 1)ρ n+1 K (en+1 K q 0 y n+1 K ) (4) Hydrodynamic step (momentum) u n+1 σ,i = 1 [ρ n ρ n+1 Dσ uσ,i n δt ( div Dσ (ρ n u n u n i ) + ( p n+1 ) ] ) σ,i D σ Regularization term S n K compensates upwind diffusion in time and space generated in the momentum balance: S µ x,t u 2 It can be proved: S > 0 e > 0 Under CFL, S > 0 S n K is the counterpart of residual term R n+1 σ from the kinetic energy balance. It is found by deriving a discrete kinetic energy balance from the momentum equation. Then S n K can be designed accordingly. 9/23

11 Numerical Scheme: Discrete Regularization Term (1) Hydrodynamic step (continuity, mass fraction, internal energy) ρ n+1 K = ρ n K δt div K (ρ n u n ) ȳ n+1 K = 1 ρ n+1 K e n+1 K = 1 ρ n+1 K (2) Chemistry step [ρ n K y n K δt div K (ρ n u n y n )] [ ρ n K en K δt(div K (e n ρ n u n ) + pk n div K (u n ) + 1 ] K Sn K ) y n+1 K = ȳ n+1 K exp( κ(t n K )δt) with T n K = Mpn K Rρ n K (3) Equation of state p n+1 K = (γ 1)ρ n+1 K (en+1 K q 0 y n+1 K ) (4) Hydrodynamic step (momentum) u n+1 σ,i = 1 [ρ n ρ n+1 Dσ uσ,i n δt ( div Dσ (ρ n u n u n i ) + ( p n+1 ) ] ) σ,i D σ Regularization term S n K compensates upwind diffusion in time and space generated in the momentum balance: S µ x,t u 2 It can be proved: S > 0 e > 0 Under CFL, S > 0 S n K is the counterpart of residual term R n+1 σ from the kinetic energy balance. It is found by deriving a discrete kinetic energy balance from the momentum equation. Then S n K can be designed accordingly. 9/23

12 Numerical Scheme: Discrete Regularization Term (1) Hydrodynamic step (continuity, mass fraction, internal energy) ρ n+1 K = ρ n K δt div K (ρ n u n ) ȳ n+1 K = 1 ρ n+1 K e n+1 K = 1 ρ n+1 K (2) Chemistry step [ρ n K y n K δt div K (ρ n u n y n )] [ ρ n K en K δt(div K (e n ρ n u n ) + pk n div K (u n ) + 1 ] K Sn K ) y n+1 K = ȳ n+1 K exp( κ(t n K )δt) with T n K = Mpn K Rρ n K (3) Equation of state p n+1 K = (γ 1)ρ n+1 K (en+1 K q 0 y n+1 K ) (4) Hydrodynamic step (momentum) u n+1 σ,i = 1 [ρ n ρ n+1 Dσ uσ,i n δt ( div Dσ (ρ n u n u n i ) + ( p n+1 ) ] ) σ,i D σ Regularization term S n K compensates upwind diffusion in time and space generated in the momentum balance: S µ x,t u 2 It can be proved: S > 0 e > 0 Under CFL, S > 0 S n K is the counterpart of residual term R n+1 σ from the kinetic energy balance. It is found by deriving a discrete kinetic energy balance from the momentum equation. Then S n K can be designed accordingly. 9/23

13 Numerical Scheme: Discrete Regularization Term S K defined to compensate the source term appearing in the following discrete kinetic energy balance (Nguyen, 2012) (Therme, 2015) δt ( ( ρ n+1 D σ u n+1 σ,i ) 2 ρ n Dσ ( u n σ,i ) 2 ) div D σ (ρu u 2 ) + ( p n+1 ) σ,i uσ,i n = 1 D σ Rn+1 σ,i with for all test function φ, when h, δt 0: ( ) d δt SK n φ n+1 K R n+1 σ,i φ n+1 σ 0 σ E S,i n 0 K M i=1 Moreover if the following discrete duality property holds, K M K p K div K uφ K + d D σ u σ,i ( p) σ,i φ σ σ E S,i i=1 Ω pu φ We recover a global total energy balance at the limit. 10/23

14 Numerical Scheme: Discrete Regularization Term 10/23 S K defined to compensate the source term appearing in the following discrete kinetic energy balance (Nguyen, 2012) (Therme, 2015) δt ( ( ρ n+1 D σ u n+1 σ,i ) 2 ρ n Dσ ( u n σ,i ) 2 ) div D σ (ρu u 2 ) + ( p n+1 ) σ,i uσ,i n = 1 D σ Rn+1 σ,i with for all test function φ, when h, δt 0: ( ) d δt SK n φ n+1 K R n+1 σ,i φ n+1 σ 0 σ E S,i n 0 K M The need for the discrete duality property constrains the choice for the discretization of the pressure gradient (centered): K M K p K div K u + d i=1 i=1 σ E S,i D σ u σ,i ( p) σ,i = 0

15 Numerical Scheme: Conserving Entropy Why solve ρ/e/p/u (1) instead of ρ/u/e/p (2)? Formally, the entropy balance can be recovered from the mass and internal energy balances as follows: s = s(ρ, e) = ln(ρ) 1 γ 1 ln e ρ s [ t ρ + div(ρu)] + es ρ [ t(ρe) + div(ρue) + p div u] ( = t s + u s ) p div u γ 1 ρe }{{} =0 With choice (1) we have time-level consistency between div(ρ n u n ) (from mass balance) and p n div u n (from internal energy). On the other hand the entropy balance cannot be recovered with choice (2) : u n+1 σ,i = 1 [ ρ n ρ n+1 Dσ uσ,i n δt (div Dσ (ρ n u n u n i ) + ( p n ) σ,i ) ] D σ e n+1 K = 1 ρ n+1 K [ ρ n K ek n δt(div K (e n ρ n u n ) + pk n div K (u n+1 ) + 1 ] K Sn K) 11/23

16 12/23 Numerical Scheme: Conserving Entropy Choice (1) vs choice (2) : non-conservation of entropy in rarefactions. Both yield a weak solution, but only the one from choice (2) is entropic. 1,000 1, pressure 500 pressure position position ρ u e p ρ e p u

17 Application: CJ Detonation 13/23 Exact ZND solution (q 0 = 25, Arrhenius law with κ 0 = 200, E A = 25) A N A 1/2 y A 1 A 1/2 30 A N A 0 p A x H C R A τ p A 1/2 A 1 A N A x

18 Application: CJ Detonation 13/23 Numerical solution, cells (400 cells/l 1/2 ) A N A 1/2 y A 1 A 1/2 30 A N A 0 p A x 15 C 40 A 1/2 A N 10 5 H R A τ p 20 A 1 A x

19 Application: CJ Detonation 13/23 Numerical solution, 300 cells (4 cells/l 1/2 ). 40 A N 1 A 1 A N 35 y A 0 p A x 15 C 40 A 1 A N 10 R p 20 5 A 0 H τ A x

20 Application: Detonation Diffraction 14/23 Detonation diffraction around a corner (Xu Aslam Stewart, 1997). u CJ unburnt state pressure horizontal axis Kinetic law: Arrhenius κ(t ) = κ 0 exp ( E ) A RT κ 0 = E A /M = 25 Initial data: Given unburnt state Matching ZND solution (CJ)

21 Application: Detonation Diffraction 15/23 Comparison of numerical results on a grid. 120 Isovalues of density at t = 14 (Xu Aslam Stewart, 1997). 120 Isovalues of density at t = 14 with our scheme (15 values)

22 Application: Detonation Diffraction 16/23 Comparison of numerical results on a grid. 120 Shock front at different times (solid line) : our results (red) vs (Xu Aslam Stewart, 1997) (black). 120 Isovalues y {0.1, 0.5, 0.9} (dashed) and y = 1 (solid line) for our results (red) vs (Xu Aslam Stewart, 1997) (black)

23 Application: Detonation in a Channel (Heaviside) Ozone decomposition detonation in a channel (Bao-Jin, 2000) Frame of reference : laboratory frame INFLOW burnt mixture ρ b, p b, u b, y b unburnt mixture ρ u, p u, u u, y u OUTFLOW Kinetic law: Heaviside κ(t ) = κ 0 H(T ) { 0 if T < Tign H(T ) = 1 otherwise T ign = 500 K κ 0 = s 1 Initial data: Given unburnt state Matching burnt state for a CJ detonation but with u b 2u bcj Symmetric deformation of the front into the unburnt state 17/23

24 Application: Detonation in a Channel (Heaviside) 18/23 Transverse displacement of the triple points ( cells) : density contours near the detonation front between t = 0 and t = (Bao Jin, 2000) Our results

25 Application: Detonation in a Channel (Heaviside) 18/23 Transverse displacement of the triple points ( cells) : density contours near the detonation front between t = 0 and t = (Bao Jin, 2000) Our results

26 Application: Detonation in a Channel (Arrhenius) Same problem, with the following changes Frame of reference : attached to shock (u s 1140 m s 1 ) OUTFLOW burnt/reacting mixture ρ b, p b, u b, y b unburnt mixture ρ u, p u, u u, y u INFLOW Kinetic law: Arrhenius κ(t ) = κ 0 H(T ) exp ( E ) A RT T ign = 500 K κ 0 = 10 8 s 1 R = J kg 1 K 1 E A = J kg 1 Initial data: Given unburnt state Matching ZND solution (CJ) Density: random perturbation r [0, 1] in small region after shock as ρ = ( r)ρ 19/23

27 20/23 Application: Detonation in a Channel (Arrhenius) Cellular structure: procedure for tracking triple points (max t0<t<t 1 p(x, t) in the laboratory frame) in parallel. Each rank r performs the following operations: 1 Define the domain of interest D at time t [t 0, t 1 ] y L y Pressure field in shock frame at time t > t 0 D = [x 0 σ(t), L x ] [0, L y ] σ(t) = ε 0 + (t t 0 )u s 0 σ(t) D x 0 L x x 2 Compute partitions of D over self and neighboring ranks y Pressure field in shock frame at time t > t 0 Ω = 0 i nrank Ω i for i = 0,..., n rank : Π i = D Ω i 0 D Π 3 Π 4 Π 5 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 x

28 Application: Detonation in a Channel (Arrhenius) Cellular structure: procedure for tracking triple points (max t0<t<t 1 p(x, t) in the laboratory frame) in parallel. Each rank r performs the following operations: 3 Determine sub-regions of Π r (local frame) to be sent and to which ranks Π i = Π i shifted by (x 0 σ(t), 0) for i = 0,..., n rank : Ω sendi = Π r Ω i 4 Determine from which ranks to receive Π r and the matching sub-regions (lab frame) for i = 0,..., n rank : Ω recvi = Π i Ω r y 0 Pressure field in laboratory frame at time t > t 0 Π 3 Π4 Π5 x 0 σ(t) Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 Parallel communications at time t 5 Π5 Ω 3 3 Π3 Ω 1 Π4 Ω 3 Π4 Ω1 1 4 Π4 Ω 2 2 x 20/23

29 20/23 Application: Detonation in a Channel (Arrhenius) Cellular structure: procedure for tracking triple points (max t0<t<t 1 p(x, t) in the laboratory frame) in parallel. Each rank r performs the following operations: 5 Non-blocking Send/Recv and update of max t>t0 p(x, t) in the lab frame y Pressure field in shock frame at time t > t 0 σ(t) for i = 0,..., n rank : D if Ω sendi { } : Send p Ωsendi to rank i 0 x 0 x for i = 0,..., n rank : y max t>t0 p(x, t) (laboratory frame) if Ω recvi { } : Receive from rank i update max t 0<t<t 1 p(x, t) on Ω recvi 0 x 0 σ(t) Π 3 Π4 Π5 Ω 1 Ω 2 Ω 3 Ω 4 Ω 5 x

30 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = 0 Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

31 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = t 0 = 19700δt = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

32 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

33 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

34 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

35 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

36 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

37 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

38 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

39 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

40 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = Pressure Mass fraction max p(x, t) t 0 <t<t 1 21/23

41 Application: Detonation in a Channel (Arrhenius) Numerical solution, cells, t = t 1 = 45063δt = Pressure Mass fraction max p(x, t) t 0 <t<t 1 L C = /23

42 Application: Detonation in a Channel (Arrhenius) 22/23 Detonation front structure: (I)ncident shock (weak shock), (R)eflected shock (transverse wave), (M)ach stem and (T)riple point

43 Perspectives 23/23 1 Detailed chemistry 2 Interaction with more complex obstacles 3 3D unstable detonation 4 Coupling with codes for deflagration and low Mach flows 5 Higher order in space and time