1. 62 nd GEC 10/20/2009 Slide 1 Surrogate Models of Electrical Conductivity in Air* Nicholas Bisek, Mark J. Kushner, Iain Boyd University of Michigan Jonathan Poggie US Air Force Research Laboratory * Work supported by Collaborative Center in Aeronautical Sciences (AFRL and Boeing)

2. 62 nd GEC 10/20/2009 Slide 2 Agenda Plasma-based Control of High Speed Air Vehicles Conductivity Models: Need for generality Surrogate (Design of Experiments) Modeling Base Case Approach Examples Concluding Remarks

3. 62 nd GEC 10/20/2009 Slide 3 Mach 5 20 0 100 Altitude [km] Near-Space Q = 140 W “Supersonic Plasma Flow Control Experiments,” AFRL-VA-WP-TR-2006-3006, Dec. 2005. Net roll Net pitch-up Shock mitigation Radio blackout Virtual Cowl MHD Power Generator PLASMA CONTROL OF HYPERSONIC VEHICLES Motivation/Goals Plasma-based Control Affects boundary layers No moving parts Extremely rapid actuation Minimal aerothermal penalty when non-operational

4. 62 nd GEC 10/20/2009 Slide 4 PLASMA CONTROL OF HYPERSONIC VEHICLES-MODELS  Desire (and need) for general modeling tools that are applicable to predict peformance, optimize design of re-entry vehicles and hypersonic craft.  Wide range of geometries- 3D approach required.  Magnetic field capable  Altitudes, Mach speed  Composition (e.g., Earth vs Venus vs Mars)  High performance computing (massively parallel, many weeks/case)  Rate limiting step is properly representing conductivity in context of vast dynamic range in conditions  Pressures from mTorr to many atm.  Composition  Temperature (ambient to many eV)  Computationally tractable.

5. 62 nd GEC 10/20/2009 Slide 5 Motivation/Goals Unstructured NS solver 2D/axisymmetric/3D grids Parallelized (MPI calls) Thermal non-equilibrium Non-equilibrium chemistry LeMANS (Michigan Aerothermodynamic Navier-Stokes) code Experiment: Nowlan (‘63) Mach 14 Air at 42 km L = 0.2 m U ∞ = 2185 m / s T ∞ = 60 K T w = 300 K

6. 62 nd GEC 10/20/2009 Slide 6 LeMANS-MHD Mesh Input Conditions LeMANS (NS equations) MHD σ model Semi-empiric Boltzmann σ model Semi-empiric Boltzmann Iterate Nonequilibrium Parallelized Hall effect Nonequilibrium Parallelized Hall effect

7. 62 nd GEC 10/20/2009 Slide 7 Several approximate models exist for various ranges. None fully capture the behavior. Several approximate models exist for various ranges. None fully capture the behavior. Electrical Conductivity - Air p = 1 atm

8. 62 nd GEC 10/20/2009 Slide 8 Charge quasineutrality e-e collisions Determine the electrical conductivity from the electron mobility Computationally prohibitive direct coupling Charge quasineutrality e-e collisions Determine the electrical conductivity from the electron mobility Computationally prohibitive direct coupling Mesh Input Conditions LeMANS (NS equations) MHD Iterate Boltzmann Approach Weng, & Kushner, Physical Review A, Vol. 42, No. 10. σ model Semi-empiric Boltzmann σ model Semi-empiric Boltzmann

9. 62 nd GEC 10/20/2009 Slide 9 Surrogate (DOE) Modeling ID Dimensions Surrogates Accuracy CPU-Cost Global Sensitivity Reduced Dimensions ID Dimensions Surrogates Accuracy CPU-Cost Global Sensitivity Reduced Dimensions 1 st order PRS Surrogates Toolbox Felipe Viana – U. of F. Matlab library

10. 62 nd GEC 10/20/2009 Slide 10 Dimension in Surrogate Space E/N, n species Transform species mole fractions dimensions into species angles    Argon: Ar, Ar+ Air:N 2, O 2, NO, N, O, N 2 +, O 2 +, NO +, N +, O + 1D reduction Need a minimum of 2 x 2 n points in DOE

11. 62 nd GEC 10/20/2009 Slide 11 Surrogates Polynomial Response Surface (PRS) Easy to implement Minimal coefficients 1 st Order PRS

12. 62 nd GEC 10/20/2009 Slide 12 Accuracy - Argon Standard error (E) Percent error (PE) Standard error (E) Percent error (PE)

13. 62 nd GEC 10/20/2009 Slide 13 CPU COST - IMPLEMENTABLE PRS models are comparable to semi-empirical models

14. 62 nd GEC 10/20/2009 Slide 14 Global Sensitivity Remove unnecessary dimensions and rerun. Reduced Order Methods (ROM) Ionic species appear more sensitive. Remove unnecessary dimensions and rerun. Reduced Order Methods (ROM) Ionic species appear more sensitive.

15. 62 nd GEC 10/20/2009 Slide 15 Air Surrogate Model E/N, N 2, O 2, NO, N, O, N 2 +, O 2 +, NO +, N +, O + 11D  2 11 sub-domains 4096 learning pts 3072 testing pts

16. 62 nd GEC 10/20/2009 Slide 16 3D Blunt Elliptic Cone Mach 12.6 air at 40 km Dipole magnetic field to reduce heat transfer Mach 12.6 Air at 42 km L = 3 m U ∞ = 4000 m / s T ∞ = 250 K T w = 300 K

17. 62 nd GEC 10/20/2009 Slide 17 3D Blunt Elliptic Cone Mach 12.6 air at 40 km

18. 62 nd GEC 10/20/2009 Slide 18 Concluding Remarks High Performance Computing on massively parallel computers becoming commonplace in aerospace plasma applications. Desire to incorporate fundamental, general techniques to represent plasma transport which are computationally tractable. Surrogate-DOE techniques have captured these goals. Investment up-front to develop surrogate model but can be automated and reused. Applicable to non-terrestrial atmospheres Improvements Real time adjustment of domain to refine surrogate model