Applying the Staggered Mesh Godunov (SMG) Method to Reactive Flows

Transcription

1 New Models and Hydrocodes, Lisbon, Portugal,19-23 May 2008 Applying the Staggered Mesh Godunov (SMG) Method to Reactive Flows Gabi Luttwak 1 and Joseph Falcovitz 2 1 Rafael, P.O. Box 2250, Haifa 31021, Israel 2 Institute of Mathematics, The Hebrew University of Jerusalem, Israel

2 ZND-Detonation Wave Inert Shock (Von Neumann state) Exothermic Reaction Region 0 < λ < 1 degree (fraction) of reaction increases Density & Pressure decrease from VN to CJ point Reaction rate increases with Pressure and/or Temperature y( ) f ( p, T)

3 Detonation stability p and p V positive feedback p as Rayleigh Expansion of reaction products This process may be stable or unstable (*) Large pressure power and/or high activation temperature => instability (*) : k ( 1 ) T (*) Bdzil, Stewart, Sharpe et al. and others p p cj n e a T

4 Stability in numerical simulation of detonation Classical VN pseudo-viscosity: Active only in regions of compression No artificial viscosity in the reaction region Numerical instabilities may arise Adding negative linear pseudo-viscosity in expansion helps. But: Its coefficient is arbitrary Will damp any regions of smooth flow So we restrict its use to the reaction region For large reaction zone width, this can add excessive damping (*) see Vittelo and Souers

5 The Staggered Mesh Godunov (SMG/Q) Scheme SMG schemes for 3D ALE hydrodynamics were presented at Oxford 2005,Prague 2007, APS 2003,2005. Use Riemann problem (RP) solutions to capture shocks. Vertex-centered velocities have jumps on the in-cell corner zone faces. These are simplified, collision RP with continuous p,. They are solved in the normal to shock direction, i.e. along the velocity difference. u L u R

6 The Staggered Mesh Godunov (SMG/Q) Scheme Let p* be the RP solution pressure. We take q p* as a uni-axial tensor pseudo-viscosity acting along the shock direction Its impulse is imparted to the two neighboring vertices. Its work, which must be dissipative, is added to the zone internal energy. p cell Some of these ideas are related to Christensen s split-q, and to the edge viscosity and compatible hydro-scheme by Caramana et al.

7 The Staggered Mesh Godunov (SMG/Q) Scheme The velocities on either side of the face, serving as data for the RP, are evaluated from the vertex velocities using a cell centered velocity gradient, limited to preserve a monotonic velocity distribution. This limiter serves as shock detector. The resulting scheme captures shocks with sharp monotonic profiles. It also has a strong and natural mesh stabilizing effect. u u u L u L Limited slope C u u u R R F Q F Q Original slope Velocity jump for RP

8 SMG in Reactive Flows Captures sharply and monotonically the leading VN shock The RP solution for weak jumps is linear and dissipative in compression and expansion alike Introduces a damping mechanism into the simulation of the reaction zone. Has no arbitrary coefficients The limited second order data for the RP restricts the damping to the non-smooth regions of flow In a well resolved reaction zone such regions may arise due to burn-rate instabilities

9 Detonation flows Run to detonation resolved reaction zone pop plot SDT Ln(run) Ln(p) Run of detonation coarser mesh time burn, DSD, quasi-rr D(κ) Critical diameter Corner turning Significant to: d c kr jet formation in shaped charges with wave-shapers

10 Empirical Reaction Rates let use non-dimensional form! λ mass fraction of reaction products: D J r y( ) y( ) F( p,, T ); r reaction zone width 1 first order depletion, Forest Fire (1 ), 2 / 3, 2 / 3, 2 / 9 2branch IGG ( Tarver) parabola BTC ( Partom) F( p,, T ) n p pj, p pact 0, p pact Ta exp( ) T k J, act, ( ) 0, act 0 1, J ; G in IGG, JWL, Forest ; J fire Arrhenius ; CJ volume burn, IG in IGG

11 Reaction rates (Time Burn, Quasi-RR) at large zones Burn region > Shock width: r C max( r,2x) Combine Time Burn and QRR: D rc max ( t t B ), ( p p act ) max( k p p j, )(1 ) ) p p j n k 0, x 0 0.8, ( x) ; 1, x 0 Heaviside step function

12 Possible refinements Partial energy release: Effect of initial density, temperature/entropy De/re-sensitization - 1 ) ( 0 mx s T r r TMD,, 0 ), ( 2 TMD s t p r

13 EOS + Closure JWL for products (, e) Linear U s c 0 su Hugoniot + Mie-Gruneisen EOS to go off - Hugoniot for un-reacted HE: p s (, e) Closure try simple mix-rule: p F ( ) G Chemical energy released only into the products Pdv compression work shared by both: p(, e) G jwl p ( ) Q jwl F ( ) G e s jwl s (, e Q) jwl (1 ) p s jwl e (, e Q)

14 Calculations Use an HMX based HE 1-D sustained shock initiation: 1D mesh 2D Saltmann-problem like mesh

15 1D sustained shock initiation 1D 1000x1x1 zones mesh a slab of 10x0.01x0.01 mm On a 2D skewed mesh(*) to test the mesh dependence 500x10x1 zones over a slab of 10x0.2x0.02 mm Composed of 5 blocks of 100x10x1 as shown: Wall BC on the sides, constant velocity piston (900m/s) on the left side (*) as in the Saltzman test case

16 1D initiation by a 900m/s piston pressure time histories

17 1D initiation by a 900m/s piston degree of reaction time histories

18 1D initiation on a 2D-skewed mesh isobars near the leading front T=0.2,0.7 T=0.2 T=0.7

19 1D initiation on a 2D-skewed mesh isobars near the leading front at T=1.2,1.4 T = 1.0 T = 1.2

20 1D initiation on a 2D-skewed mesh isobars near the leading front at T=1.4,1.5 T=1.4 T=1.5

21 1D initiation on a 2D-skewed mesh Legend for the above isobars

22 1D initiation on a 2D-skewed mesh pressure and lambda history

23 Conclusions The SMG scheme for reactive flow: Introduces a damping mechanism in the reaction zone Improves the propagation of detonation in distorted meshes.

24 References 1. Luttwak G., Falcovitz J., "Staggered Mesh Godunov (SMG) Schemes for ALE Hydrodynamics", Workshop On Numerical Methods For Multi-Material Flows, Oxford, UK, Sept. 2005, 2. Luttwak, G. Staggered Mesh Godunov (SMG) Schemes for Lagrangian Hydrodynamics, p , Shock Compression of Condensed Matter-2005, Furnish M. D. et al Eds, (2006), AIP, CP Luttwak G., Sliding and Multifluid Velocities in Staggered Mesh (MMALE) Codes ", Conference/Workshop On Numerical Methods For Multi-Material Flows, Prague, Sept. 2007, www-troja.fjfi.cvut.cz/~multimat07 4. Luttwak, G., "Comparing Lagrangian Godunov and Pseudo-Viscosity Schemes for Multi-Dimensional Impact Simulations, p , Shock Compression of Condensed Matter-2001, Furnish M. D. et al Eds, (2002), AIP, CP620.

25 References 5. Sharpe J.D., Proc. Roy. Soc. London, A453,p2603,(1997) 6. Watt S.D., Sharpe J.D. J. Fluid Mech. 522,p329,(2005) 7. Short M., Aslam T.D., Bdzil J.B., Henrick A, Quirk J.J., Hydrodynamic Stability Properties of Condensed Phase Detonations., 13 th Int. Symp. Detonation, (2006) 8. Vitello P., Souers P.C., Stability Effects of Artificial Viscosity in Detonation Modeling, 12 th Int. Symp. Detonation, (2002) 9. Christensen R.B., "Godunov Methods on a Staggered Mesh. An Improved Artificial Viscosity", L.L.N.L report UCRL-JC , (1990). 10. Caramana E.J., Shaskov M.J., Whalen P.P., J. Comp. Phys. 144, p70, (1998).

26 More References 11. Mader C.L., Numerical Modeling of Detonations, U.C. Press,Berkeley,(1979) 12. Lee E.L., Tarver C.M., Phys. Fluids 23(12),p2362,(1980) 13. Partom Y., Hydro-reactive computations, Shock Compr. Cond. Matter- 2001,p460, Furnish M. D. et al eds, (2002), AIP, CP Murphy M.J., et al., Modeling Shock Initiation in Comp. B, 10 th Int.Det. Symp.,(1993) 15. Souers P.C. et al., Prop., JWL+., Prop. Expl., Pyro. 25,p54(2000) 16. Lambert D.E., et al. J. Fluid Mech., 546, p227, (2006) 17. Luttwak G., Hayek M., Ginzburg A., " On the transient effects of the detonation propagation in an HMX based explosive, presented at the 9 th Int. Det.,(1989), unpublished