Nautilus: Robust and accurate code for modeling chemically reacting plasma …doeplasma.eecs.umich.edu › files › Hakim_Ammar_2011_03_25.pdf · 2011-04-12 · Nautilus: Robust

Nautilus: Robust and accurate code for modelingchemically reacting plasma fluid flows

Ammar Hakim, John Loverich, Peter Stoltz

1Tech-X Corporation5621, Arapahoe Avenue Suite A

Boulder, CO. [email protected]

March 25, 2011

Ammar Hakim ([email protected]) Nautilus: Numerics and Applications March 25, 2011 1 / 33

Tech-X Corporation is located in Boulder

Corporate website: http://www.txcorp.com

I We have 70 employees, 2/3 Ph.Ds,spread over 3 states, 2 countries.

I Working on computational fusion,beam and accelerator physics, GPUprogramming and computer sciencetopics.

I Products: VORPAL (EM/PIC),GPULib, OOPIC Pro.

I Lead institution on FACETSSciDAC and partner in several otherSciDACs (COMPASS, SWIM, RF).

Figure: Ultra-intense laser propagationthrough under-dense plasma


http://www.txcorp.com

Nautilus team and funding sources

Nautilus is a new commercial plasma fluid mechanics code (1.0 releaseplanned for summer 2011)

I Nautilus team: Ammar Hakim,John Loverich, Peter Stoltz, SeanZhou, Sudhakar Mahalingam andMahmood Miah.

I Funding sources: Department ofEnergy SBIR Phase I and Phase IIgrants, Department of DefenceSBIR/STTR Phase I grants.


Aim: Present Nautilus theory, numerics and showapplications in plasma physics

I Overview and features of Nautilus

I Definition, examples and properties of hyperbolic balance laws

I Overview of numerical schemes implemented in Nautilus

I Applications: jet propagation and merging in vacuum for highenergy-density physics application, Field reversed configurationformation and hypersonic flow over blunt-bodies.


Nautilus is a multi-physics code for hyperbolic balancelaws coupled to plasma chemistry

Hyperbolic balance laws are building blocks for complex physicalphenomena.

I Nautilus can solve multiple equation systems: Maxwell equations,Euler/Navier-Stokes equations, MHD and two-fluid equations.

I Arbitrary set of chemical reactions can be included. Rates can beentered as Arrhenius form or as tabulated data.

I Nautilus solvers run on body-fitted grids. Porting algorithms tounstructured grids is almost complete (part of 1.0 release).

I Solvers can be combined in arbitrary manner without recompilation.

I An extensive set of synthetic diagnostics can be inserted insimulations.


Example: Euler equations for neutral fluid flow

Neutral inviscid flow is described by Euler equations

∂n

∂t+ n∇ · u + u · ∇n = 0

∂u

∂t+ u · ∇u +

1

mn∇p =

q

m(E + u× B)

∂p

∂t+ u · ∇p + γp∇ · u = 0.

This is weakly hyperbolic, isotropic system with eigenvalues{u ± cs , u, u, u}, where cs =

√γp/mn.

This system is interesting in itself, and is also a building block forNavier-Stokes equations, two-fluid equations, and MHD equations.


Hyperbolic balance laws have number of properties thatare important for schemes to satisfy

Creating numerical schemes that provably satisfy mathematical propertiesof the equations will greatly help in verification process.

I Hyperbolic balance laws allow for discontinuous solutions. I.e. shocks,rarefactions and contact discontinuities can develop even fromsmooth initial conditions. Schemes must be able to handle this, i.e.be shock capturing.

I Even if true shocks do not form (due to diffusion), small scalefluctuations and sharp gradients need to be captured.

I If a hyperbolic balance law is isotropic, so must be the numericalscheme, i.e. be grid and coordinate independent.

I Schemes must preserve invariant domains. For example, n ≥ 0, p ≥ 0and Pij is semi-positive definite.


Three broad classes of methods are implemented inNautilus to solve hyperbolic balance laws

1. Use Taylor series expansion and replace derivatives using the balancelaw. I.e. U(x , t + ∆t) = U(x , t) + ∆t∂U/∂t + . . .. Replace ∂U/∂tusing the PDE. Leads to fully discrete schemes. Example, Wave

Propagation Scheme of Randy LeVeque (see, for example, LeVeque 2006).

2. Assume we know average solution in each cell. Then, construct ahigh-order polynomial representation of the solution in each cell usingneighbor averages. Leads to semi-discrete schemes which are thenintegrated using a ODE solver. Example, MUSCL schemes (see

Kulikovskii, 2001).

3. Represent solution in each cell using set of basis functions. Leads todiscontinuous Galerkin schemes.


Applications of Nautilus: Plasma jets, FRC formation andhypersonic reacting flows

I Merging plasma jets for creation of high-energy density states inlaboratory. (DoE Phase II SBIR).

I Field-Reversed Configuration (FRC) formation for use in magnetisedtarget fusion applications. (DoD Phase I SBIR).

I Hypersonic reacting flow over blunt bodies. (DoD Phase I STTR).


Merging plasma jets can be used to create high-energydensity states

Figure: Two dimensional jet implosion on a target. On the upper right we see jetsmerging prior to imploding the target. At this point some ram energy is lost toshock heating of the converging jet. Bottom right shows merged jets forming aplasma liner just before compressing the magnetized target. Image taken fromHsu, J. Fusion Energy, 2008


The Plasma Liner Experiment (PLX) is being built atLos-Alamos National Laboratory


PLX construction has started and chamber is being setupat LANL


Figure: Merging of 26 jets in 3D geometry. In each hemisphere there 13 jetsplaced along latitude slices. Merging chamber geometry at LANL constraintslocation and number of jets that can be used in the experiment


Jet merging can be used to create an “accretion disk” inthe laboratory. Experiment proposed by D. Ryutov.

Figure: Nine jets in equatorial plane fired slightly offset from the center. Jets willmerge creating a spinning “accretion disk”. Axial outflow is then compressedusing cusp-field produced by coils places on either side. Figure taken from D.D.Ryutov, Astrophys. Space Sci, 2010.


Initial set of experiments will use six merging jets to formspinning disk of plasma

Figure: Six merging in-plane jets. Density and magnetic field in the planeperpendicular to jets is shown.


Late in time magnetic field compression is seen withsignificant axial momentum

Figure: Six merging in-plane jets. Density and magnetic field in the planeperpendicular to jets is shown.


Initial cusp magnetic field is created using set of coils

Figure: Initial magnetic field is axisymmetric and created with a set of coils oneither side of merging disk.


Magnetic field is trapped due to merging jets and getstwisted

Figure: Magnetic field gets trapped and twist due to spinning disk of plasma.


Magnetic field is trapped due to merging jets and getstwisted

Figure: Top view of twisted magnetic field.


Field Reversed Configurations are being explored for use inpotential fusion machine

Figure: Schematic diagram of Tri-Alpha Energy, Inc. C-2 FRC experiment. Twoθ-pinch preformed energetic compact toroids are merged to form a stable FRC inthe confinement region. The C-2 FRC has ports for neutral beam heating,rotating magnetic fields and is highly stable. Figure taken from M.W.Binderbauer et. al, Physical Review Letters, 2010.


Two-Fluid FRC formation with theta-pinch coils

Figure: Schematic diagram of a Field Reversed Configuration. The device iscylindrical and a r − z plane slice through the axis is shown. Figure taken fromRPPL website (http://depts.washington.edu/rppl/)


Initial magnetic fields from theta-pinch and mirror coils

Figure: Equilibrium magnetic field stream lines produced by a set of theta-pinchand mirror coils. The domain is a cylinder tube 50 cms long by 10 cm radius. Thecentral theta-pinch coil is 36 cm long and was placed at a radius of 7 cm. Themirror coils were 10 cm long each, with a 2 cm gap between the theta-pinch coilsand the mirror coils. The peak bias axial field is about -0.4 Tesla. After reversalthe peak axial field is about 3.0 Tesla.


Theta-pinch coil currents are reversed to triggerreconnection and FRC formation

Figure: Poloidal magnetic field structure after FRC formation. This plot showsthe expected closed field-line topology of the FRC poloidal field, indicatingformation was successful. When the FRC is fully formed the current in thetheta-pinch coils are reversed with respect to the initial bias field.


Plasma density shows same order-of-magnitude as in AFRLexperiment

Figure: Plasma number density after FRC formation. The FRC has undergoneradial and axial contraction and has achieved a peak number density of about5× 1022 m−3. This occurs about 2 µsec after the current reverses in thetheta-pinch coils.


Electrons develop anti-symmetric vortexes in the poloidalplane generating toroidal magnetic fields

Figure: Toroidal magnetic field at 4 µsec. In ideal-MHD theory toroidal field iszero. However, such fields are observed in experiments. As the formation processis symmetric about the FRC mid-plane, the net toroidal flux is zero. Toroidalfields decay and electron momentum is transferred to ion energy.


Electron and ion temperatures show significant differences

Figure: Electron (left) and ion (right) temperatures at 4 µsec. The color schemeis selected to show 225 eV for the brightest color for the electrons and 400 eV forthe ions. In this simulation the ions are hotter than the electrons, clearly showingtwo-temperature effects, which can not be captured in a single temperature MHDmodel.


In atmosphere hypersonic flight is being actively pursuedby the Air-Force for both manned and unmanned missions

Figure: The experimental X-51A Waverider is an unmanned, autonomoussupersonic combustion ramjet-powered hypersonic flight test demonstrator for theU.S. Air Force.


At hypersonic speeds fluid physics becomes very complex

I Complex shock structures can form and shock-boundary layerinteraction can lead to separated turbulent flow.

I High temperatures can cause dissociation of air (oxygen, nitrogen)and chemical reactions are initiated.

I Ionization of nitrous oxide NO into NO+ and electrons can create athin plasma layer around aircraft causing radio blackout.

We have a DoD STTR in partnership with George Washington Universityto simulate such reacting hypersonic flows with Nautilus.


Initial simulations were performed for sphere-cone bluntbody assembly

Figure: Steady-state temperature over blunt body for 7600 m/s flow at 61 kmusing a 7-species air model that includes 18 different chemical reactions.


Density for each species is tracked to better understand airchemistry

Figure: Number density of N2, O2, NO, N, O, NO+ and electrons at steady-statefor 7600 m/s flow at 61 km. Note the thin plasma layer formed close to the noseof the assembly.


Figure: Number density of electrons as a function of normalized distance fromnose. The electron density is primary factor determining radio-black out.


Conclusions: Overview of Nautilus was presented andseries of applications shown

I Nautilus is a powerful, multi-physics code that can handle a widevariety of fluid equations as well as electromagnetism and chemicalreactions.

I Simulations of merging plasma jets show formation of accretion diskthat traps and twists magnetic field. Formation of axial jets is beingstudied at present.

I FRC formation with the two-fluid model was shown. Unique featuresnot observable with restive or ideal MHD simulations (toroidal fields,electron vortex currents) are seen.

I A seven species model of air chemistry was used to simulate flow overa sphere-cone assembly. Simulations are being improved to comparewith experimental test-flight and laboratory data.


Nautilus: Robust and accurate code for modeling chemically reacting plasma …doeplasma.eecs.umich.edu › files › Hakim_Ammar_2011_03_25.pdf · 2011-04-12 · Nautilus: Robust

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