Numerical MHD of Aircraft Re-entry; Fluid Dynamic, Electromagnetic and Chemical Effects G. Seller Dipartimento di Scienze e Tecnologie Fisiche ed Energetiche, Tor Vergata University, Italy M. Capitelli, S. Longo Chemistry Department, Bari University,

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Numerical MHD of Aircraft Re-entry Fluid
Dynamic, Electromagnetic and Chemical EffectsG.
SellerDipartimento di Scienze e Tecnologie
Fisiche ed Energetiche, Tor Vergata University,
ItalyM. Capitelli, S. LongoChemistry
Department, Bari University, ItalyI.
ArmeniseIMIP-CNR, Bari, Italy

• Magnetohydrodynamics
• Fluid dynamics
• Electromagnetism
• Chemistry
• Physical behavior of Plasma (Generalized Ohm Law
  or other)

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Fluid dynamics for MHD and other
applicationsFinite difference method.
Space-centred explicit scheme of first order with
numerical stiffness

• In implementing a program for MHD upwind methods
  for fluid dynamics are typically used. However,
  there are some advantages using alternative, more
  simple schemes, so that computational time is
  lower and problem resolution are more easy. In
  this work a scheme with the following features is
  used
• Euler equations (no viscosity)
• Cartesian grid
• Finite difference
• Space-centred scheme with numerical stiffness
  (Lax-Fredrichs parametric scheme)

• Finite difference space - centred scheme
  Qi,jtn1 Qi,jn -Dt?(((Fx)i1,jn- (Fx)i-1,jn
  )/Dx ((Fy)i,j1n - (Fy)i,j-1n ) /Dy)
• It is well-known that finite difference space
  - centred scheme, if applied to Euler equations,
  is intrinsically unstable actually, a
  chessboard instability arises the nearest
  cells will have very different variable value,
  with large space and time oscillations that
  increases exponentially.
• Stabilized scheme Lax-Fredrichs
• It is possible to obtain a controlled stiffness
  with the introduction of a parameter, the G
  stiffness coefficient the resultant formula is
  Qi,jtn1 Qi,jn ?(1- Dt/G) - Dt ? (( Fx)i1,jn
  - (Fx)i-1,jn ) / Dx ((Fy)i,j1n -
  (Fy)i,j-1n)/Dy) Dt/4G?(Qi1,jn Qi-1,jn
  Qi,j1n Qi,j-1n )

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Advantages with modified Lax-Fredrichs scheme

• First order scheme, without oscillations near
  discontinuity
• (This is important MHD actually Brio e Wu
  (1981) found a complex structure in MHD shock
  waves (compound waves with compressions and
  expansions). This structure is not identifiable
  in a second order scheme (Lax - Wendroff) because
  of the numerical oscillation)
• The scheme simplicity is computational
  time-saving.
• The scheme is extremely flexible, so that
  implementation of new condition and equation (as
  Maxwell equation for MHD and chemistry) is very
  easy.
• Actually, a simple ad-hoc, computational
  time-saving Maxwell solver has been implemented.

The scheme has been validate by comparison with
results of an upwind scheme and with results of
a particle method.
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Comparison between results of a MHD and a
particle scheme in a simple fluid dynamic,
time-dependent case. Two-dimensional expansion of
a circular blast of high density in a reflective
square box.
Initial radius of the blast 0.10 m Time of
comparison 1.2 ms Density outside the blast
r0 Density inside the blust r0?100
Centre of the blast X0 0.25 m - Y0 0.00
m (on the wall)
Fig.15a
Fig.15a
There is a good agreement between maximum values,
as well as in general shape of waves and time
behaviour (also the velocities show an analogous
good matching). The less good agreement shown by
minimum values is due to fluctuation typical of
particle method, expecially in low density region.
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Fluid Dynamic Results
Comparison between modified Lax-Friedrichs scheme
and an upwind scheme (Mach3 - Blunt
Body) Pressure Diagram Upwind scheme
Lax-Friedrichs scheme Both (Comparison)
Fig.1
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Magnetic fields in MHD
Magnetic fields in MHD

• A MHD scheme include two features, both
  characterised by a parameter
• 1) equation for magnetic field time-evolution
• Parameter Reynolds magnetic number Rem
  ?0?0V0L
• (induced magnetic field fluid dynamics modify
  magnetic field)
• 2) magnetic terms in momentum and energy
  equations
• Parameter Magnetic force number S
  B02/?0?0V02
• (magnetic fields modify fluid dynamics)
• In a full MHD case both number are non
  negligible.
• Several different conditions could be imposed on
  body surface, but all of them are somewhat
  arbitrary. Actually the magnetic field normally
  penetrates in the body so a less arbitrary
  choose is calculating magnetic fields also inside
  the body.
• This can be made by solving the magnetic field
  equation without fluid dynamics
• ?B /?t -rot (rot B/??) inside the body. Only
  at mesh limit a simple continuity condition is
  necessary.
• Conductivity ? is included in the rot operator
  because it could be non-uniform.
• B fields are not defined in mesh vertexes but on
  mesh cell sides this makes calculation of div
  and rot accurate. Calculation of currents (which
  give a better physical comprehension of MHD
  scheme) is easy, too.

Boundary condition for magnetic fields
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Full MHD ResultsConstant and Homogeneous
Electrical ConductivityJoule effect has not been
kept into account for simplicity
All values non-dimensional ? ?/?? ?U ?U/??U?
?V?V/??U? ?e?e???/(??U?)2 pp???/(?U?
)2 TT/T?B B?(??/?)1/2/??U? B/B0?S1/2
? ????/(?0??U?)?L Rem
Blunt Body in hypersonic motion along
x-axis Initial uniform magnetic field along
y-axis Mach7.35 By0.2 S.04
Rem3.6 sPLASMAsBODY
Fig.6
min
max
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Current behaviour in plasma
Comparison among magnetic fields in different
conditions

• Strong currents take place in the portion of
  plasma near the stagnation point.
• Between the body and the shock wave there are
  only weak currents
• Out of the shock wave currents take place in
  opposite direction they have a tail that
  penetrate in undisturbed plasma the tail has
  more penetration when conductivity is lower.

Rem7.2 sBodysPlasma Rem7.2
sBodysPlasma/4 Rem1.8 sBodysPlasma
Fig. 9
Imposed Magnetic Field by coil Blunt Body -
Mach7.35 By0.08 S.02 Rem7.2
sBODYsPLASMA/2
min
max
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Comparison between shock waves with different
magnetic and conductivity conditions
Magnetic field and currents in the
starting-by-coil case
Fig.13
Fig.12
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Non-homogeneous Electrical Conductivity

• In MHD, magnetic field evolution equation is
• ?B/?t rot(V?B) - rot(rot (B/m)/s)
• If s e m are homogeneous, this equation can be
  simplified as
• ?Bk/?t rotk(V?B) (?2Bk/?x2 ?2Bk/?y2
  ?2Bk/?z2)/ms
• (with k?x,y,z)
• If s is not homogeneous this is no longer
  possible. In numerical solution, div B ?0
  generating terms could rise up.
• Used equation
• ?B/?t -rot E rot(V?B - rot (B/m)/s)
• Calculation of B as rot E in a finite-difference
  first-order scheme, in some conditions, has no
  div B ?0 generating terms.

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Low and Non-homogeneous Electrical Conductivity

• In scheme in which electrical conductivity is
  calculated by chemical reactions, usually the
  conductivity is initially zero, then firstly it
  rise up to low values, eventually it reaches
  significant values.
• In MHD, if electrical conductivity becomes zero,
  the rot (B/m)/s term in magnetic field evolution
  equation diverges.
• To avoid these problems a very low artificial,
  homogeneous conductivity is imposed over all the
  scheme.

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Low and Non-homogeneous Electrical Conductivity

• Problems
• - Artificial conductivity has to be low enough
  to not affect fluid dynamic scheme.
• This is verified by comparison with a frozen
  magnetic field case.
• - In case of very low electrical conductivity,
  time step has to be small enough to reach
  stability in the scheme ? High calculation time.
• Solution
• A) Very simple, high speed ad-hoc Maxwell
  solver implementation.
• B) Double-time step implementation
• ?t Fluid Dynamics 10 100 ?t
  Electromagnetics

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Chemistry (N2)

• In the classical MHD method, chemical effects are
  kept in account by means of electrical
  conductivity entering generalised Ohm law.
• Chemical frozen or equilibrium state is
  typically considered
• However, fluid dynamics, magnetic fields and
  chemical kinetics can influence each other then
  they have to be calculated together.
• A first comparison could be made between a
  chemical kinetics scheme and a pure fluid
  dynamics one.
• Reactions N2N2NNN2 N2NNNN
  NNN2e-

Mach?18.91 T?300 K r?1.48 g/m3
p?132 Pa Blunt body diameter2.656
m (Frozen chemistry)
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Chemistry
Chemical Kinetics (flux along x-axis) Blunt body
diameter2.656 m
Frozen Chemistry
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• Fluid Dynamics is high influenced by chemical
  reactions
• - Chemical Reaction absorbs energy (Temperature
  goes down)
• Reactions velocity is influenced by body
  dimensions
• - Small body
• -Low nitrogen dissociation
• -High temperature
• -High ionization.

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Chemistry
Chemical Kinetics (flux along x-axis) Blunt body
diameter2.656 m
Chemical Kinetics (flux along x-axis) Blunt body
diameter26.56 m
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Electrical Conductivity Calculations Chemistry
and Transport Phoenomena Simulation

• Calculation of Electrical Conductivity
• -Locally Boltzmann equation resolution by
  Chapman Enskog method.
• Collision Integrals Calculations
• With Liboff method for collisions between charged
  species (e--e-, e--N2, N2-N2)
• By data fit of Capitelli, Gorse, Longo and
  Giordano (2000) for collisions between electrons
  and neutre species
• (e--N2, e--N)

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Full MHD with chemical derived electrical
conductivity
Chemical Kinetics (flux along x-axis) Blunt body
diameter2.656 m
Initial Homogeneous Magnetic Field along y-axis
- Full MHD (induced magnetic fields)
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Full MHD with chemical derived electrical
conductivity
Initial Homogeneous Magnetic Field along y-axis
- Full MHD (induced magnetic fields)
Chemical Kinetics (flux along x-axis) Blunt body
diameter26.56 m
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Comparison between magnetic field frozen case
and full MHD case
Profile comparison of total density (a) and
pressure (b) along symmetry central line. Fluid
dynamic chemistry case data are plotted in
solid line Full MHD case data are plotted in
dashed line

• Comparison of density ratio and pressure contour
  plots
• Fluid dynamic chemistry (lower side)
• With Full MHD (upper side).

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Conclusions

• Fluid Dynamics, electromagnetism and chemical
  reactions have high influence each others
• Particularly, chemical reaction influence has
  been found.
• In other conditions this influence could be also
  higher.

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Next Developements

• Influence of other transport parameters
• Thermical conductivity,Viscosity, Diffusion
• Tensorial Electric Conductivity
• Other Chemical Species (Oxygen) Complete air
  chemistry.
• Larger Mach number
• Non-Homogeneous initial Magnetic Field (by coils)