Tabular Equations of State for HE Simulations in the VTF

1. Eric O. Morano and Joseph E. Shepherd CIT ASCI-ASAP Site Visit October 22-23, 2001 Pasadena, CA Tabular Equations of Statefor HE Simulations in the VTF

2. Reactive Flow Model – Single Phase • Conservative two-component 2D Euler equations: • Discretization: • Finite Volume discretization with Strang’s time-splitting technique • Roe’s approximate Riemann solver extended by Glaister • Adaptive mesh refinement with GrACE (various criteria) density mass products mass-fraction reaction rate pressure velocity vector momentum internal energy energy

3. Reactants (R): Mie-Grüneisen (MG) Products (P): Jones-Wilkins-Lee (JWL) Equilibrium EoS Model for Mixtures mixture rules mechanical equilibrium thermal equilibrium • Pressure, temperature, sound speed and pressure derivatives are • pre-computed and stored in a look-up table. • All variables subsequently interpolated during flow computation. Shock-wave initiation of heterogeneous reactive solids, Johnson, Tang and Forest, J. Appl. Phys. 57 (9), 1985.

4. HMX_sp: Example of the Temperature Temperature [K] Reactants - MG"dominated" EoS Mixture EoS Mixture EoS l = 0.0 l = 0.2 l = 0.4 e[kJ/g] e[kJ/g] e[kJ/g] Invalid EoS range Invalid EoS range Invalid EoS range v[cc/g] v[cc/g] v[cc/g] Products - JWL "dominated" EoS Mixture EoS l = 0.6 Mixture EoS l = 0.8 l = 1.0 e[kJ/g] e[kJ/g] e[kJ/g] Invalid EoS range v[cc/g] v[cc/g] v[cc/g]

5. means there is a solution means there is no solution Interpolation Technique Interpolation is performed on the 3 planes defined by (e,v,l), there are 4 possible cases 1. All points surrounding the desired value are defined: Trilinear interpolation 3. 3 points in a given plane surrounding the desired value are defined: Pick the closest point 2. 4 points in the same plane surrounding the desired value are defined: Bilinear interpolation 4. No points surrounding the desired value are defined: Failure (recompute a new table)

6. Diffracting Detonation – Set-Up Pressure [GPa] • (60mm x 40mm) = (240x160) mesh - GrACE NO AMR • 10 processors initial ZND profile reflective VN CJ Dirichlet reflective reflective f = 0 HMX_sp reflective • EoS table parameters: • l in [ 0.0, 1.0] with nl = 50 • e in [-4.0,12.0] with ne = 100 • v in [ 0.3, 2.0] with nv = 50

7. States Accessed at Final Time Products the green-to-purple contour corresponds to l = 0.975 ± 0.025 (e,v) states accessed by l = 0.975 ± 0.025 when the front has passed the corner products l = 1.00 reaction zone 0.05 < l < 0.95 shock location CJ ~ (0.41 , 2.02) + HMX_sp l = 0.00 JWL "theoretical" isentrope

8. States Accessed at Final Time l = 0.80 ± 0.05 l = 0.70 ± 0.05 l = 0.90 ± 0.05 l > 0.75 l > 0.85 l > 0.95 invalid EoS range: (e,v) values appear to result in negative squared soundspeed

9. States Accessed at Final Time l = 0.50 ± 0.05 l = 0.60 ± 0.05 l = 0.40 ± 0.05 l > 0.65 l > 0.55 l > 0.45

10. States Accessed at Final Time l = 0.10 ± 0.05 l = 0.30 ± 0.05 l = 0.20 ± 0.05 l > 0.15 l > 0.35 l > 0.25 more invalid EoS range: (e,v) values may result in negative temperatures

11. l = 0.025 ± 0.025 States Accessed at Final Time Reactants Note the close proximity of the VN state wrt. the area of invalid EoS, hence the requirement for a very finely discretized table (e,v) states accessed by l = 0.025 ± 0.025 when the front has passed the corner l > 0.05 invalid EoS range VN ~ (0.33 , 5.71) + close-up shock location HMX_sp l = 0.00 MG "theoretical" Hugoniot

12. Mie-Grüneisen EoS for LX-17 reactants • JWL EoS for both reactants and products available in literature by Tarver et al. • Our mixture EoS is based upon a Mie-Grüneisen EoS for the reactants. • How do we obtain a Mie-Grüneisen form of the reactants EoS? 1. Use the jump conditions to compute the Hugoniot. 2. Use a weighted nonlinear fit algorithm (Mathematica) to compute the necessary constants (i.e. c and s in the Us = c + s up relation) in order to approach the VN point as closely as possible. 3. Refine the value of the CJ state (UCJ, eCJ, vCJ) with the tangency criterion at CJ until acceptable agreement with literature.

13. Mie-Grüneisen EoS for LX-17 reactants Tarver et al.: Mie-Grüneisen fitted expression UCJ = 7.596 km/s PCJ = 27.500 GPa vCJ = 0.394 cc/g PVN = 33.740 GPa vVN = 0.359 cc/g Hugoniot from jump conditions Fitted values: Original Rayleigh line UCJ = 7.606 km/s PCJ = 27.499 GPa vCJ = 0.394 cc/g PVN = 33.619 GPa vVN = 0.365 cc/g Fitted Rayleigh line

14. LX-17 ZND Profile and Reaction Rate • In "Modeling Two-Dimensional Shock Inititiation and Detonation Wave Phenomena in PBX 9404 and LX-17", UCRL-84990 (1981), Tarver and Hallquist propose the "Ignition and Growth" reaction rate: z 2/9 2/3 2/9 4 lt = I (1-l) (rs/r0-1) + G (1-l) l P -1 -z -1 where I = 50 ms , G = 500 ms Mbars and z = 3.0 for LX-17 and where rs is the density of the shocked explosive, i.e.rs = rVN in a ZND profile. • Using the previous mixture EoS and a weighted nonlinear fit (Mathematica) for the reaction rate we obtain a new expression for an "Ignition and Growth" like rate that is not based on rs, since this quantity is difficult to obtain in an Eulerian code: a b z a lt = I (1-l) + G (1-l) l P Note that the constants above have no physical significance, merely the result of a fit, and may take several different values according to the fit performed.

15. LX-17 ZND Profile and Reaction Rate Ignition & Growth Fitted Rate DZND = 1.05 mm

16. LX-17 ZND Diffracting Past a Corner Products Mass Fraction Temperature [K] products l = 1.00 reaction zone 0.01 < l < 0.95 shock location LX-17 l = 0.00 • EoS table parameters: • l in [ 0.0, 1.0] with nl = 40 • e in [-4.0, 4.0] with ne = 400 • v in [ 0.3, 3.3] with nv = 300 • (6mm x 4mm) = (960x640) mesh – GrACE NO AMR • 10 processors