3. What Is A Magnetic Nozzle?
Figure: Schematic of the VASIMR concept
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4. Why This Research?
• Why electric propulsion? High Isp
• Why a magnetic nozzle? High thrust
• Why is more work needed? Plasma detachment
• Why is simulation the right tool? Difficult to test on earth
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7. The Discontinuous Galerkin Numerical Method
Basic definition:
• Linear combination of basis
functions
• Solution discontinuous at edges
Advantages:
• Efficient parallelization
• High order accuracy
Disadvantages:
• Difficulty maintaining positivity Figure: Example of solution
represented with DG
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9. Ideal Magnetohydrodynamics (MHD)
Three key assumptions:
• A single temperature and
pressure well defined
• Electrons mass negligible
• Ion gyrations are small and fast:
λ Ri , τ Ωci
Three key properties:
• Anisotropic wave propagation
• ‘Frozen in condition’:
B · dS = const.
• Generalized Ohm’s law:
E − v × B = 0
k ⊥ B
k ∥ B
Alfvén
Sonic
Slow Magnetosonic
Fast Magnetosonic
Figure: Waves in MHD
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10. Ideal MHD Equation System
• Mass: ∂ρ
∂t + · [ρv] = 0
• momentum: ∂(ρv)
∂t + · ρvv + PI − BB
µ0
+ B2
2µ0
I = 0
• energy: ∂E
∂t + · E + P + B2
2µ0
v − (B·v)
µ0
B = 0
• Flux: ∂B
∂t + · [vB − Bv] = 0
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11. The Role Of Resistivity
Derives from electron-ion collisions within the plasma. Introduces a
first approximation of kinetic effects.
• E + v × B = ηj
• ‘frozen in condition’ violated: B · dS = const.
• η ∝ T
−3/2
e
• Total energy conserved
• ‘Ohmic heating’: magnetic energy converted into thermal energy
• Current calculated from magnetic field: j = µ−1
0 × B
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12. Resistive MHD Equation System
• mass: ∂ρ
∂t + · [ρv] = 0
• momentum: ∂(ρv)
∂t + · ρvv + PI − BB
µ0
+ B2
2µ0
I = 0
• energy: ∂E
∂t + · E + P + B2
2µ0
v − B·v
µ0
B + η
µ0
(j × B) = 0
• flux: ∂B
∂t + · [vB − Bv] + × (ηj) = 0
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13. Controlling Divergence Error
The condition · B = 0 can be violated in numerical simulations.
• Result of discretization error
• Produces non-physical behavior
• Number of approaches for addressing this
Technique chosen for error correction:
• Non-physical variable ψ introduced
• Equation governing it ∂ψ
∂t + c2
h · B = −
c2
h
c2
p
ψ is introduced
• Coupled to magnetic field equations with ψ
• Advects and diffuses away · B error
• ψ vanishes as ∆x, ∆t → 0
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14. resistive MHD with divergence correction
• mass: ∂ρ
∂t + · [ρv] = 0
• momentum: ∂(ρv)
∂t + · ρvv + PI − BB
µ0
+ B2
2µ0
I = 0
• energy: ∂E
∂t + · E + P + B2
2µ0
v − B·v
µ0
B + η
µ0
(j × B) = 0
• flux: ∂B
∂t + · [vB − Bv] + ψ + × (ηj) = 0
• error: ∂ψ
∂t + c2
h · B = −
c2
h
c2
p
ψ
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15. Splitting the Magnetic Field
• This application involves an externally applied, constant magnetic
field
• Natural representation is to split magnetic field into two
components, B0 and B1
• B0 is calculated once at the beginning of the simulation
• B1 is evolved in time, with the applied B0 component being called
at each time.
• not a linearization of B
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16. Types Of Boundary Condition Used
‘Ghost cell’ BC:
• Boundary cells interact with non-physical ‘ghost’
• Value of the solution chosen to create desired boundary condition
• High order accuracy requires extrapolation
‘Flux’ BC:
• No ‘ghost’ cell is used
• The flux across the boundary is chosen to create desired boundary
condition
• High order without extrapolation
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17. Inflow Boundary Condition
• P-total Balance:
P (r) = Pmin +
B2(r) − B2
max
2µ0
(1)
• edge-smoothing:
Γ (r) =
1
2
erfc
r − rin
(2rin)
, e.g. v = Γ vin (2)
• Plasma injected parallel to B-field
v = |v(r)| ·
B
|B|
(3)
• ‘ramp-up’: linear increase over fixed time interval from v = 0 to
v = vin, and ρ = 0 to ρ = ρin.
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18. Far-Field Boundary Condition
Boundary condition needed that will not generate or reflect waves.
Key properties of boundary condition:
• Copy all conserved variables into ghost cell
• Recalculate energy:
•
Eg = Eb +
B2
0 |g
2µ0
−
B2
0 |b
2µ0
(4)
• lower order of accuracy, not a concern since far away from plume.
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19. Choosing Boundary Condition for ψ
• Authors of method found choice of boundary condition to be
arbitrary
• For this application this was found to not be the case
• Different choices of boundary conditions were investigated
• optimal boundary condition found to be ψ = 0
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20. · B at Z-min: no correction vs. ψ = 0
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25. Scaling Study
Parameters Scaled:
• Rm : 101 → 103
• βk: 10−3 → 10−1
Characteristics Compared:
• Magnetic field perturbation
• Radial expansion of plume
• Evolution of energy in beam
Definition of energy metrics:
• ke = (KE
M )/kein
• te = (TE
M )/tein
where:
• M = ρ dV
• KE = 1/2ρv2 dV
• TE = P/ρ(γ − 1) dV
• kein = 1/2v2
in
• tein = Pin/ρin(γ − 1)
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26. Scaling with Rm: B-Field
(a) Magnetic field lines and density
for low Rm
(b) Magnetic field lines and density
for high Rm
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27. Scaling With Rm: Radial Expansion
0.00 0.25 0.50 0.75 1.00 1.25
R/Rin
0.00
0.25
0.50
0.75
1.00
1.25
Z/Rin
Rm =1.2×101 Rm =1.2×103
Figure: Comparison of radial expansion for
different magnetic Reynolds numbers.
• Slightly more
expansion for
higher Rm
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28. Scaling With Rm: Integrated Quantities
0
1
2
3
4
5
6
ke
Rm =1.2×101 Rm =1.2×103
0.0 0.2 0.4 0.6 0.8 1.0
z/rin
0.0
0.5
1.0
1.5
2.0
2.5
3.0
te
Figure: Specific thermal and kinetic energy over
the course of plume propagation. Scaling with
magnetic Reynolds number.
• Deceleration after
z ≈ 0.1rin
• less kinetic energy
gain for lower Rm
• evolution same
till z ≈ 0.1rin
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29. Scaling with βk: B-Field
(a) Magnetic field lines and density
for low βk
(b) Magnetic field lines and density
for high βk
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30. Scaling With βk: Radial Expansion
0.00 0.25 0.50 0.75 1.00 1.25
R/Rin
0.00
0.25
0.50
0.75
1.00
1.25
Z/Rin
k =5.1×10-3
Rm =3.8×102
k =5.1×10-1
Rm =3.8×101
Figure: Comparison of radial expansion for
different values of kinetic plasma beta.
• contraction for
lower βk
• expansion for
higher βk
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31. Scaling With βk: Integrated Quantities
0
5
10
15
20
25
30
35
40
45
ke
k =5.1×10-3
Rm =3.8×102
k =5.1×10-1
Rm =3.8×101
0.0 0.2 0.4 0.6 0.8 1.0
z/rin
0.0
0.5
1.0
1.5
2.0
2.5
3.0
te
Figure: Specific thermal and kinetic energy over
the course of plume propagation. Scaling with
kinetic plasma beta.
• Large difference
due to
normalization
• Deceleration after
z ≈ 0.1rin
• Apparent
difference in ramp
up, may be due
to background
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32. Scaling with Rmax: B-field
(a) Magnetic field lines and density
for larger Rmax .
(b) Magnetic field lines and density
for smaller Rmax .
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33. Scaling With Rmax: Radial Expansion
0.00 0.25 0.50 0.75 1.00 1.25
R/Rin
0.00
0.25
0.50
0.75
1.00
1.25
Z/Rin
rmax =3rin rmax =3
2 rin
Figure: Comparison of radial expansion for
different radial domain sizes.
• No contraction
develops
• Large increase in
expansion
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36. Scaling With Rmax: Integrated Quantities
0
2
4
6
8
10
12
ke
Rmax =3
2 rin Rmax =3rin
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
z/rin
0.0
0.5
1.0
1.5
2.0
2.5
3.0
te
Figure: Specific thermal and kinetic energy over
the course of plume propagation. Scaling with
radial domain size.
• No deceleration
for smaller Rmax
• larger increase in
ke
• larger drop in te
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38. Work Completed
Model Development
• Implemented resistive-MHD in WARPX
• Implemented divergence error correction scheme
• Implemented split B-field
• Determined appropriate ψ boundary condition for this work
• Developed inflow boundary condition
• Developed far-field boundary condition
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39. Work Completed
Preliminary Investigation
• Proof of concept
• performed 81 simulations over range of parameters
• Analyzed scaling of nozzle performance with βk
• Analyzed scaling with Rm
• Analyzed and explained effect of Rmax
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40. Future Work
Repeat the analysis and simulations discussed here with corrected
Rmax , submit findings to AIAA journal. Proposed goals:
• Improve simulation robustness
• Improve confidence of analysis
Proposed topics:
• Positivity preserving scheme
• Two-fluid physics
• Incorporate knowledge from experiment and theory to develop
more sophisticated inflow.
• Quantify effect of background blasma on beam propagation.
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