T HE D ISCONTINUOUS G ALERKIN F INITE E LEMENT M ETHOD FOR M ULTI - FLUID P LASMA M ODELING Éder M. Sousa 1 PI: Jean-Luc Cambier 2 ERC I NC . 1 , I N -S PACE P ROPULSION B RANCH 2 , A IR F ORCE R ESEARCH L ABORATORY E DWARDS A IR F ORCE B ASE , CA USA RQ-West 6.1 Research Review, University of California, Los Angeles CA January, 2015 Distribution A: Approved for public release; distribution unlimited AFTC/PA clearance No. 15010, 16 January 2015 D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION É.M. S OUSA (AFRL/RQRS) 1 / 11
O UTLINE 1 T HE M ULTI - FLUID P LASMA M ODEL 2 T HE D ISCONTINUOUS G ALERKIN F INITE E LEMENT M ETHOD 3 P ROBLEMS OF I NTEREST É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 2 / 11
T HERE ARE MULTIPLE PLASMA MODELS . Ensemble average of particles 3-Dimensions + 3-Velocities distribution Distribution function: f s ( x , v ,t) Evolve the particles position and velocity Evolve the distribution function É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 3 / 11
P LASMA CAN BE MODELED AS A FLUID . Take moments of the distribution function, f s ( x , v ,t) � ρ s = m s f s d v � ρ s u s = m s v f s d v p s = ρ s T s = 1 � w 2 f s d v 3 m s h s = 1 � w 2 w f s d v 2 m s Bulk properties of the fluid are obtained Assumes local There can be multi particles thermodynamic equilibrium species in a plasma. and high collisionality É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 4 / 11
F LUID PROPERTIES AND FIELDS ARE COUPLED . F LUID PROPERTIES : ∂ρ s ∂ρ s u s + ∇ · ( ρ s u s u s + p s I ) = ρ s q s ∂ t + ∇ · ( ρ s u s ) = S s , ( E + u s × B ) + R s ∂ t m s γ − 1 + 1 ∂ǫ s ∂ t + ∇ · (( ǫ s + p s ) u s ) = ρ s q s p s 2 ρ s | u s | 2 u s · E + Q s , ǫ s = m s S s , R s , and Q s account for all collisional effects in the plasma E LECTROMAGNETIC F IELDS : ∂ B ∂ t + ∇ × E = 0 , ∇ · B = 0 1 ∂ E q s q s � � ∂ t − ∇ × B = − µ o ρ s u s , ǫ o ∇ · E = ρ s c 2 m s m s s s É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 5 / 11
T EMPORAL MULTISCALES EXIT IN PLASMAS . C HARACTERISTIC SPEEDS : � γ P s v cs = , c ρ s C HARACTERISTIC FREQUENCIES : � n s q 2 s ω ps = , ǫ o m s Example lab plasma (seconds): q s B ω pe = 10 − 14 , L 1 c = 10 − 9 , 1 ω ci = ω cs = m s 10 − 8 , L v ci = 10 − 5 4 √ π e 4 Z 4 n r log Λ ν sr = 3 √ m s T 3 / 2 s É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 6 / 11
A DVANTAGES OF THE M ODEL The Multi-Fluid Plasma Model (MFPM) is less computationally expensive than kinetic models Multi-fluid effects become relevant at small spacial and temporal scales The model includes finite electron mass and speed-of-light There is an inherent assumption of neutrality in MHD but not in the MFPM Neutrality does not allow local charge separation Displacement current effects are resolved in the MFPM savings , CPU LTE Generalization − − − − − − − − − − − − − → ← − − − − − − − − Kinetic MFPM MHD É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 7 / 11
T HE DISCONTINUOUS G ALERKIN (DG) METHOD . ∂ Q ∂ t + ∇ · ← → F = S An ideal numerical method for the MFPM should: resolve high frequency be high-order accurate oscillations capture shocks not impose strict time-steps P OLYNOMIAL EXPANSION : � Q = c i v i i c i coefficient v i Legendre polynomials É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 8 / 11
DG IS EASILY P ARALLELIZABLE . ∂ Q ← → ← → � � � � v p ∂ t dV = L p ( Q ) = v p S dV − v p F · d A + F · ∇ v p dV Ω Ω ∂ Ω Ω Explicit Runge-Kutta time integration (2nd, 3rd or 4th order) CFL = c ∆ t / ∆ x ≤ 1 / ( 2 p − 1 ) , p is the polynomial order É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 9 / 11
S OME APPLICATIONS OF I NTEREST F IELD R EVERSED H YDRODYNAMIC C ONFIGURATION (FRC) I NSTABILITIES Rayleigh-Taylor Kelvin-Helmholtz Formation, translation Additionally: Turbulence, Radiation, Plasma Reactions, Ionization and Recombination, LASER-plasma interactions, ... É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 10 / 11
S UMMARY The Multi-fluid plasma model (MFPM) is implemented less computational expensive than kinetic Model assumes Local Thermodynamic Equilibrium is more general than the MHD model The discontinuous Galerkin (DG) numerical method is chosen higher-order accurate (2nd-16th) resolves high frequency oscillations captures shocks Applications of interest include the FRC, Hydro instabilities, Turbulence, etc ... Thank You. É.M. S OUSA (AFRL/RQRS) D ISTRIBUTION A: A PPROVED FOR PUBLIC RELEASE ; UNLIMITED DISTRIBUTION 11 / 11
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