fully coupled maxwell/navier-stokes simulation of...

Fully Coupled Maxwell/Navier-Stokes Simulation ofElectromagnetic Hypersonics Including Accurate Transport

Models

D.D’Ambrosio⋆, D. Giordano¶ and D. Bruno‡

⋆Politecnico di Torino - Dipartimento di Ingegneria Aeronautica e Spaziale

Corso Duca degli Abruzzi, 24 - 10129 Torino - Italy¶ESA/ESTEC - Aerothermodynamics Section

Keplerlaan 1, 2200 AG Noordwijk, The Netherlands‡CNR - IMIP

Via Orabona, 4 - 70126 Bari - Italy

A Navier-Stokes solver for non-equilibrium aerothermodynamics is coupled with a fully implicit solver ofthe Maxwell equations. Transport phenomena are modelled using a high accurate Chapman-Enskog methodthat accounts for the effects of concentration, pressure and temperature gradients and provides anisotropicdiffusion velocity, heat flux and viscosity in the presence of a magnetic field. Gas neutrality is not enforced sothat the electric charge density can be locally different from zero. Numerical tests are carried out on a flat-faced cylinder to test the capabilities of the implicit Maxwell solver and of the coupled Maxwell/Navier-Stokessystem.

I. Introduction

The effects of the interaction between electromagnetic fields and partially ionized gases are interesting for atmo-spheric reentry applications because they can potentiallyreduce the heat flux to a space vehicle surface and allowfor aerodynamic control. Electromagneto-fluid dynamics isgoverned by the coupled set of the the Maxwell andNavier-Stokes equations. Starting from this complete model, different simplifications based on order-of-magnitudeconsiderations are frequently made to obtain a less complexset of governing equations. In the last years, D’Ambrosioand Giordano considered the possibility of solving the fullset of the Maxwell and Navier-Stokes equations in a cou-pled fashion. This resulted in a series of papers that demonstrated that solving the coupled system is feasible with areasonable expense of computer processing time and memory.1–5 More recently, the same approach was embraced byMacCormack, who also solved the coupled Maxwell and Navier-Stokes equations in paper presented at the last AIAAPlasmadynamics Conference.6 The full set of electromagneto-fluid dynamics equations is an extremely powerful in-vestigation tool that accounts for effects that are considered negligiblea priori by the approximated models. One ofits merits is that it can be used to verify when and where the simplifying assumptions on which the reduced modelsare based are valid, and when and where they are not.

Despite the completeness of the electromagnetic model, an important element was missing in the studies presentedin Refs. 1–6, as it is in most numerical simulations of electromagneto-fluid dynamics interactions. The transportmodel and the definitions of electrical conductivity and conduction current density were not consistent with each otherand the plasma was forced to be neutral. In particular, in Refs. 1–5, an arbitrarily constant electrical conductivitywas used in connection with the generalized Ohm law to test the numerical solution techniques and the effect ofthe presence of a magnetic field was not included in the evaluation of transport coefficients. However, as pointedout by Giordano in Ref. 7, the definition of the conduction current density directly descends from the definition ofthe mass diffusive fluxesJmi

of electrically charged species, to that the direct application of the generalized Ohmlaw is not necessary. In addition, since in the first-order Chapman-Enskog theory the components mass diffusiveflux depends not only on the generalized electric fieldE + v × B, but on partial pressure gradients and temperaturegradients also, there are contributions to electric currents due to pressure and thermal diffusion of charged species also.

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American Institute of Aeronautics and Astronautics

Note that an explicit computation of the electrical conductivity is not needed for computational purposes. If desired,presso-electrical, thermoelectrical and electrical conductivities tensors could be post-processed starting from diffusioncoefficients. Finally, the presence of a magnetic field influences transport phenomena, which become anisotropic, asreported by Bruno et alii in Refs. 8, 9. Components mass diffusive fluxes and heat fluxes arise in directions paralleland normal to the magnetic field, and they are not necessarilyaligned with diffusion driving forces and temperaturegradients. The viscosity coefficient is a five-components vector. Part of these phenomena are usually incorporated inthe so-called Hall effect, which is frequently accounted for in approximated magneto-fluid dynamics models in theform of a tensorial electrical conductivity. In this paper,the Hall effect will be consistently linked with the componentsmass diffusive fluxes. Last, but not the least, the plasma will not be forced to be neutral, so that an electric chargedensityρc and the related convection current densityρcv may be present in the flowfield.

The possible presence of a convection current and the anisotropy of transport coefficients in the presence of amagnetic field has a strong impact on the resulting flowfields.At a microscopic level, all originates from the dynamicsof particles carrying an electric chargeq moving in a magnetic fieldB with speedv and thus subject to the Lorentzforceqv ×B. At the macroscopic levels, this results in forces that are due to convection currentsρcv and conductioncurrentsJQ. Consider for example an axially symmetric configuration, as those that are frequently adopted in plasmawind tunnels to carry out experiments on the electromagneto-fluid dynamics interaction. One consequence of theplasma non-neutrality and, possibly to a much larger extent, of transport coefficients anisotropy is that such flows,which are axisymmetric in absence of a magnetic field, will develop a swirling motion in the presence of an appliedaxisymmetric magnetic field. In addition, an induced magnetic field component will appear in the azimuthal directionand electric fields and electric currents will have components in the meridian plane also.

In this paper, we will show numerical results obtained simulating the flowfield about a blunt-faced cylinder in aplasma consisting of argon atoms, singly charged ions, and electrons. The geometrical, fluid dynamics and magneticconfiguration is similar to the one that was recently set up inthe experiments by Gulhan et alii at DLR-Koeln.10 Thefully coupled Maxwell and Navier-Stokes equations will be solved as in Ref. 5 with the addition of the completetransport model described in Refs. 8, 11. The axisymmetric EMFD equations will include the momentum equationin the azimuthal direction, the azimuthal velocity component and the projection of the Maxwell equations in everycoordinate direction.

II. Full magneto-fluid dynamics equations

II.A. Maxwell equations

The Maxwell equations read like∂B

∂t+ ∇× E = 0 (1)

ε0∂E

∂t− ε0c

2∇× B = −j = −ρcv − JQ (2)

The electric current density,j, represents the flux of electric charge due to convection (convection current) and diffusion(conduction current):

j = ρcv + JQ (3)

Electric charge density and conduction current density canbe obtained directly from the species partial densities andfrom the mass-diffusive fluxes that appear in the governing equations of fluid dynamics:7

ρc = eNA

Nspe∑

i=1

ρi

Mi

σis (4)

JQ = eNA

Nspe∑

i=1

Jmi

Mi

σis (5)

The summation above is extended to all species,σis is +1 for ions, -1 for electrons and 0 for neutral species,e is theelectric charge,Jmi is the mass-diffusive flux of speciesi andρi is the species partial density.

The numerical solution of the Maxwell equations can be improved by an appropriate normalization. Here we reportfour possible forms, all sharing the reference variables Lref for lengths,Vref for velocities andBref for the magnetic fieldcomponents:

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a) tref = Lref/Vref, Eref = VrefBref, JQref = λrefVrefBref

∂B

∂t+ ∇× E = 0 (6a)

∂E

∂t− c2∇× B = −ρcv − c2 (Rem)Vref

JQ (6b)

b) tref = L ref/Vref, Eref = cBref, JQref = λrefcBref

∂B

∂t+ c∇× E = 0 (7a)

∂E

∂t− c∇× B = −ρcv − c2 (Rem)Vref

JQ (7b)

c) tref = L ref/Vref, Eref = cBref, JQref = λrefVrefBref

∂B

∂t+ c∇× E = 0 (8a)

∂E

∂t− c∇× B = −ρcv − c (Rem)Vref

JQ (8b)

d) tref = L ref/c, Eref = cBref, JQref = λrefVrefBref

∂B

∂t+ ∇× E = 0 (9a)

∂E

∂t−∇× B = −1

cρcv − (Rem)Vref

JQ (9b)

where

c =c

Vref

(10a)

(Rem)Vref=λrefVrefL ref

ε0c2magnetic Reynolds number (10b)

Forms b) and c) provide the best numerical results because the elements of the Jacobian of the electromagneticfluxes are of the same order of magnitude. Thus, form c) will beused in the remaining part of this paper.

The two Gauss’ laws for magnetostatics and electrostatics,that directly derive from the Maxwell equations, mustalways be numerically satisfied:

∇ · B = 0 (11)

ε0∇ ·E = ρc (12)

Equation (12) can be used to define the reference electric charge density

ρcref = ε0Eref

L ref

(13)

and, in nondimensional form, it will read as∇ ·E = ρc (14)

In the numerical method that we propose here, we will integrate the Maxwell equations using a finite volumesmethod. To this purpose, the integral form of the equations will be used:

∫

V

∂B

∂tdV + c

∫

S

(n× E) dS = 0 (15)

∫

V

∂E

∂tdV − c

∫

S

(n× B) dS = −c∫

V

(ρcv

c+RemJQ

)

dV (16)

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II.B. Navier-Stokes equations

The Navier-Stokes equations for a non-equilibrium chemically reacting gas are linked to the Maxwell equationsthrough source terms that represent Lorentz force and Jouleheating. In integral and non-dimensional form theyread like:

∂ρ

∂t+ ∇ · (ρv) = 0 (17a)

∂ (ρv)

∂t+ ∇ · (ρvv + pI) −

√γM

Re∇ · τ = S ρcE + S

(ρcv

c+Rem JQ

)

× B (17b)

∂E

∂t+ ∇ · [(E + p)v] +

√γM

Re∇ · (JU − τ · v) = c S

(ρcv

c+Rem JQ

)

·E (17c)

∂ρi

∂t+ ∇ · (ρiv) + ∇ · (Jmi) = Ωi (17d)

The non-dimensional parameterS is called themagnetic force numberand it is defined as:

S =ε0c

2B2ref

ρrefV 2ref

(18)

The productS Rem gives non-dimensional parameterQ, which is called themagnetic interaction parameterand it isdefined as:

Q =λrefB

2refL

ρrefVref

(19)

The total energy per unit volume,E, is given by

E =

n∑

i=1

ρiei + ρ‖v‖2

2(20)

SymbolJU represents the flux of energy due to transport phenomena. As noticed before, the conduction current density,JQ, can be computed as described in Eq. (5) without the need of invoking the generalized Ohm law. To this purpose,transport phenomena in a ionized gas in the presence of an electromagnetic field should be properly modelled.

II.C. Transport model

The transport model that is used in this work is described in Refs. 8,11 and is based on the Chapman-Enskog method.A noticeable consequence of the presence of a magnetic field is that transport properties cease to be isotropic, butthey depend on the orientation of the magnetic field. A new frame of reference with components parallel (‖), per-pendicular (⊥) and tangent (t) to the magnetic field vector must be defined. To move from cartesian components,vc = vx, vy, vz, to rotated components,vr =

v‖, v⊥, vt

, a rotation matrix[R] must be defined:

vr = [R]vc (21)

[R] =

bx by bz

−by bx +b2z

1 + bx− bybz

1 + bx

−bz − bybz1 + bx

bx +b2y

1 + bx

(22)

where

bi =Bi

‖B‖ (23)

Diffusion velocity and heat flux components in the directions parallel, perpendicular and tangential to the magneticfield can be obtained solving algebraic systems. For the parallel component we have:

α [A]

x‖

=

b‖

(24)

and for the normal and tangential components:

(α [A] + ιβ [B])

x⊥ + ιxt

=

b⊥ + ιbt

(25)

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Matrices[A] and [B] contain elements that are related to collisional cross sections. Their rank depends upon thenumber of species and upon the order of approximation of the Chapman-Enskog method. The right-hand side containsthe diffusion driving forces and the temperature gradients:

b‖

=

...

−d‖i−d‖i+i

...5

2xi

(∇T )‖

T5

2xi+1

(∇T )‖

T...

0

0...

;

b⊥ + ιbt

=

...

−(

d⊥i + ιdti

)

−(

d⊥i+1 + ιdti+1

)

...5

2

xi

T

[

(∇T )⊥

+ ι (∇T )t]

5

2

xi+1

T

[

(∇T )⊥

+ ι (∇T )t]

...

0

0...

(26)

with

d‖i = (∇xi)

‖ + (xi − yi) (∇ ln p)‖ − xi

kBT

(

ρci −mi

ρc

ρ

)

[

E‖ + (v × B)‖]

(27a)

d⊥i = (∇xi)⊥

+ (xi − yi) (∇ ln p)⊥ − xi

kBT

(

ρci −mi

ρc

ρ

)

[

E⊥ + (v × B)⊥]

(27b)

dti = (∇xi)

t+ (xi − yi) (∇ ln p)

t − xi

kBT

(

ρci −mi

ρc

ρ

)

[

Et + (v × B)t]

(27c)

The unknowns are the diffusion velocity (Jmi = ρiwi) and the heat flux:

x‖

=

...

xiw‖i

xi+iw‖i+1

...

q∗i‖

q∗i+1

‖

...

h.o.t.

h.o.t....

;

x⊥ + ιxt

=

...

xiw⊥i + ι (xiw

ti)

xi+1w⊥i+1 + ι

(

xi+1wti+1

)

...

q∗i⊥ + ιq∗i

t

q∗i+1

⊥ + ιq∗i+1

t

...

h.o.t.

h.o.t....

(28)

q‖ = −5

2p

n∑

i=1

q∗i‖ ; q⊥ = −5

2p

n∑

i=1

q∗i⊥ ; qt = −5

2p

n∑

i=1

q∗it (29)

II.D. Axis-symmetric finite-volumes form

Here, we will restrict our attention to axisymmetric fields.This means that changes in the axialx-directions and radialr-direction will be allowed, but changes in the azimuthalθ-direction will be assumed to be impossible. However, norestrictions are made on the azimuthal components of vectorfields. In the finite volumes discretization, the equationsystem to be solved for each computational cell is:

∂ρ

∂tAr +

4∑

i=1

[ρ (uxnx + urnr)]i ℓiri = 0 (30)

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∂ (ρux)

∂tAr +

4∑

i=1

[ρux (uxnx + urnr)]i ℓiri +

4∑

i=1

(p nx)i ℓiri −4∑

i=1

(τxxnx + τrxnr)i ℓiri =

= SρcExAr + S (jrBθ − jθBr) Ar

(31)

∂ (ρur)

∂tAr +

4∑

i=1

[ρur (uxnx + urnr)]i ℓiri +4∑

i=1

(p nr)i ℓiri −4∑

i=1

(τxrnx + τrrnr)i ℓiri

+ (−p) A− (−τθθ) A =

= SρcExAr + S (jθBx − jxBθ) Ar

(32)

∂ (ρuθ)

∂tAr +

4∑

i=1

[ρuθ (uxnx + urnr)]i ℓiri −4∑

i=1

(τxθnx + τrθnr)i ℓiri =

= SρcEθ Ar + S (jxBr − jrBx) Ar

(33)

∂E

∂tAr +

4∑

i=1

[(E + p) (uxnx + urnr)]i ℓiri +4∑

i=1

(JUxnx + JUr

nr)i ℓiri

−4∑

i=1

[(τxxux + τrxur + τθxuθ)nx + (τxrux + τrrur + τθruθ)nr]i ℓiri =

= cS (jxEx + jrEr + jθEθ) Ar

(34)

∂ρj

∂tAr +

4∑

i=1

[ρj (uxnx + urnr)]i ℓiri +

4∑

i=1

[(

Jmj

)

xnx +

(

Jmj

)

rnr

)

iℓiri = Ωj Ar j = 1, Nspe (35)

∂Bx

∂tAr + c

[

4∑

i=1

(Eθnr)i ℓiri + (Eθ) A

]

= 0 (36)

∂Br

∂tAr + c

4∑

i=1

(−Eθnx)i ℓiri = 0 (37)

∂Bθ

∂tAr + c

[

4∑

i=1

(Ernx − Exnr)i ℓiri + (−Ex) A

]

= 0 (38)

∂Ex

∂tAr − c

[

4∑

i=1

(Bθnr)i ℓiri + (Bθ) A

]

= −c jxAr (39)

∂Er

∂tAr − c

4∑

i=1

(−Bθnx)i ℓiri = −c jr Ar (40)

∂Eθ

∂tAr − c

[

4∑

i=1

(Brnx −Bxnr)i ℓiri + (−Bx) A

]

= −c jθ Ar (41)

whereA, r, ℓi andri are the projection of a cell in the meridian plane, its radialdistance from the axis, the projectionof thei− th cell surface on the meridian plane and its radial distance from the axis, respectively.

In the absence of an applied magnetic field in the meridian plane (Bappx = Bapp

r = 0) and of an initial azimuthalvelocity component, electric currents will be allowed in the meridian plane only (jx 6= 0, r 6= 0, jθ = 0). ExaminingEqs. (30)–(41) we can notice that the consequence of this is that the only induced electromagnetic field componentsdifferent from zero will beEx, Er andBθ and that no azimuthal velocity component will be activated.Conversely,in the presence of an applied magnetic field in the meridian plane, an azimuthal velocity component different fromzero will create a swirling motion in the flowfield. Moreover,diffusion, and thus electric currents also, will be allowedto flow in the azimuthal direction with the consequence of inducing an azimuthal electric field,Eθ and an inducedmagnetic field in the meridian plane.

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III. Numerical solution of the Maxwell equations

III.A. Full implicit integration of the Maxwell equations

In the work described in this paper the Maxwell equations areintegrated using a fully implicit scheme:

JM∆WBE = −FM − SM Ar (42)

whereWBE = Bx, Br, Bθ, Ex, Er, EθT (43)

FM =

c[

∑4

i=1(Eθnr)i ℓiri + (Eθ) A

]

c∑4

i=1(−Eθnx)i ℓiri

c[

∑4

i=1(Ernx − Exnr)i ℓiri + (−Ex) A

]

−c[

∑4

i=1(Bθnr)i ℓiri + (Bθ) A

]

−c∑4

i=1(−Bθnx)i ℓiri = −c jr Ar

−c[

∑4

i=1(Brnx −Bxnr)i ℓiri + (−Bx) A

]

(44)

SM = 0, 0, 0, jx, jr, jθT (45)

and

JM =Ar

∆tI +

4∑

i=1

(

∂FM

∂WBE

)

i

+∂SM

∂WBE

Ar (46)

The Jacobian elements∂SM

∂WBE

are obtained through numerical differentiation of the electric current density compo-

nents with respect to the electromagnetic field components.The Jacobian elements∂FM

∂WBE

are computed analytically.

Their form depends on the numerical schemes that is used to evaluate electromagnetic fluxes at cells interfaces, asubject that will be addressed to in the next subsection.

III.B. A flux-difference splitting method for the Maxwell eq uations

The integration in time of the Maxwell equations using a finite volumes technique requires the evaluation of theintegrals

∫

S(n × E) dS and

∫

S(n × B) dS. This means that one has to compute the tangential components of E

andB at each computational cell surface. To obtain a stable explicit numerical method, a flux-difference splittingtechnique is used. The technique is very similar to the one that we use for the flow solver12 and consists in solvinga one dimensional Riemann problem normal to each cell surface. We define the locally one-dimensional Riemannproblem and its solution as follows.

III.B.1. Locally one-dimensional Riemann problem

Riemann invariants and propagation speeds:

dR−T-E = dEz + dBt ; λ = −c (47)

dR+T-E = dEz − dBt ; λ = +c (48)

dR−T-M = dEt − dBz ; λ = −c (49)

dR+T-M = dEt + dBz ; λ = +c (50)

Indicating with ’l’ and ’r’ left and right states respectively, and with ’*’ the intermediate state, after integration ofthe Riemann invariants, the Riemann problem solution is:

(

R−T-E

)r= E∗

z +B∗t (51)

(

R+T-E

)l= E∗

z −B∗t (52)

(

R−T-M

)r= E∗

t −B∗z (53)

(

R+T-M

)l= E∗

t +B∗z (54)

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and thus

B∗t =

(R−T-E)

r − (R+T-E)

l

2=

Blt +Br

t

2− El

z − Erz

2(55)

E∗t =

(R+T-M)

l+ (R−

T-M)r

2=

Elt + Er

t

2+Bl

z −Brz

2(56)

B∗z =

(R+T-M)

l − (R−T-M)

r

2=

Blz +Br

z

2+El

t − Ert

2(57)

E∗z =

(R+T-E)

l+ (R−

T-E)r

2=

Elz + Er

z

2− Bl

t −Brt

2(58)

SubscriptsT-EandT-M stand fortransverse electricandtransverse magnetic, the two propagation modes in which thesignal can be decomposed. In transverse electric propagation mode, componentsEx, Ey andBz are perturbed, whilein the transverse magnetic mode the signal is a combination of theBz, Ex andEy components.

III.B.2. The half Riemann problem at the boundaries

In case only the magnetic or the electric field are specified atthe boundaries, Riemann invariants can be used toevaluate what is missing. In particular, we have:

a) Bt is known,Ez is unknown:

Ebcz = Ein

z −(

Bint −Bbc

t

)

(59)

b) Ez is known,Bt is unknown:

Bbct = Bin

t −(

Einz − Ebc

z

)

(60)

c) Et is known,Bz is unknown:

Bbcz = Bin

z +(

Eint − Ebc

t

)

(61)

d) Bz is known,Et is unknown:

Ebct = Ein

t +(

Binz −Bbc

z

)

(62)

(63)

III.B.3. Maxwell fluxes Jacobian

Using the Riemann solver described above, the electromagnetic fluxes at cells interfaces are given as follows:

FM = c

nyE∗z

−nxE∗z

E∗t

−nyB∗z

nxB∗z

−B∗t

= c

ny

(

Elz + Er

z

2−

−nyBlx + nxB

ly + nyB

rx − nxB

ry

2

)

−nx

(

Elz + Er

z

2−

−nyBlx + nxB

ly + nyB

rx − nxB

ry

2

)

−nyElx + nxE

ly − nyE

rx + nxE

ry

2+Bl

z −Brz

2

−ny

(

Blz +Br

z

2+

−nyElx + nxE

ly + nyE

rx − nxE

ry

2

)

nx

(

Blz + Br

z

2+

−nyElx + nxE

ly + nyE

rx − nxE

ry

2

)

−(

−nyBlx + nxB

ly − nyB

rx + nxB

ry

2− El

z − Erz

2

)

(64)

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Therefore, the dependence of the electromagnetic fluxes with respect to’left’ and’right’ cells is easily obtained:

(

∂FM

∂WBEl

)

i

=1

2c

n2y −nxny 0 0 0 ny

−nxny n2x 0 0 0 −nx

0 0 1 −ny nx 0

0 0 −ny n2y −nxny 0

0 0 nx −nxny n2x 0

ny −nx 0 0 0 1

ℓiri (65)

(

∂FM

∂WBEr

)

i

=1

2c

−n2y nxny 0 0 0 ny

nxny −n2x 0 0 0 −nx

0 0 −1 −ny nx 0

0 0 −ny −n2y nxny 0

0 0 nx nxny −n2x 0

ny −nx 0 0 0 −1

ℓiri (66)

These elements of the Jacobian matrix can be computed just once at the beginning of the numerical simulation.Conversely, the complete Jacobian must be computed at each time step because of the non-linear dependence of theelectric current density from the electromagnetic field components and because the integration time step may changefrom step to step.

III.C. Enforcing ∇ · B = 0 and∇ ·E = ρc/ε0

The Gauss’ laws for magnetostatics and electrostatics defined in Eqs. (11) and (12) may not be statisfied by the numer-ical solution of the Maxwell equations due to discretization errors. Especially when the magnitude of the computedelectromagnetic field is large, it is necessary to enforce the two conditions independently from the integration of theMaxwell equations. To this purpose, we use a projection method similar to the one described in Ref. 13. Thenumer-ical solutions of the electromagnetic field,Bn andEn, are corrected by a quantity that can be defined as the gradientof two scalar functions,ψB andψE, respectively.

B = Bn + ∇nψB (67a)

E = En + ∇nψE (67b)

The needed correction can be computed by applying the divergence operator to equations Eqs. (67a) and (67b) andusing Eqs. (11) and (14):

∇ ·Bn + ∇ · ∇nψB = 0 (68)

∇ · En + ∇ · ∇nψE = ρc (69)

In practice, a Poisson equation has to be solved for each Gauss’ law. Ignoring the numerical error in evaluatingρc, thetwo equations will read like:

∇n · ∇nψB = −∇n ·Bn (70)

∇n · ∇nψE = ρnc −∇n ·En (71)

The projection method will also be affected by the spatial accuracy of the Poisson solver.

IV. Numerical solution of the Navier-Stokes equations

The Navier-Stokes equations are discretized according to afinite volumes method based on the same grid wherethe Maxwell equations are computed. Convective fluxes are evaluated using an upwind flux-difference splitting tech-nique12 and diffusive fluxes through a centered scheme. Second orderaccuracy in space and in time is achieved usingan Essentially Non-Oscillatory scheme. Chemical source terms are treated implicitly to avoid instabilities due to fastchemical reactions. The volumetric forcej × B and heatingj · E due to the electromagnetic field are also treatedimplicitly, at least as far as fluid dynamics variables are concerned. To this purpose, the Jacobian of the electromag-netic source terms is computed through numerical differentiation and then summed to the Jacobian of chemical sourceterms, so that a tight coupling between fluid dynamics and chemical and electromagnetic source terms is enforced.

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V. Coupling technique for Maxwell and Navier-Stokes equations

Maxwell and Navier-Stokes equations are loosely coupled here. For each time step, we solve first the Navier-Stokes equations while keeping the electromagnetic field frozen. Then, we integrate the Maxwell equations using thesame time step of fluid dynamics or using a series of sub-stepsuntil the same simulation time of fluid dynamics isreached. During the Maxwell time step, fluid dynamics variables are assumed to be frozen. As an attempt to mitigatethe unstable behavior that will be discussed later, a predictor-corrector scheme was also attempted. It consists inmaking one full fluid dynamics step, then one Maxwell step using the new fluid dynamics variables, and finally inrecovering the initial fluid dynamics variables and in marching the Navier-Stokes equations forward in time againwith the updated electromagnetic field values. Indicating with F andM the Navier-Stokes and Maxwell operators,respectively, this can by summarized as

WK+1,⋆F = F (WK

F ,WKBE) explicit N.-S. (72a)

WK+1BE = M (WK+1,⋆

F ,WKBE,W

K+1BE ) implicit Maxwell (72b)

WK+1,⋆F = F (WK

F ,WK+1BE ) explicit N.-S. (72c)

VI. Numerical results

VI.A. Geometrical configuration and computational grid

Numerical experiments have been carried out about a flat-faced cylinder that contains a coil arranged as in Fig. 1.Such a model is very similar to configuration C3 described in Ref. 10.

CoordinateX

Co

ord

ina

teY

0 0.02 0.04 0.06 0.08 0.1

0

0.01

0.02

0.03

0.04

0.05

current carrying wires

Figure 1. Meridian section of the flat-faced cylindrical model including current carrying wires.

The computational domain for both Maxwell and Navier-Stokes equations have been discretized using the meshshown in Fig. 2. The Maxwell equations are solved everywhere, including the interior of the model. The computationaldomain must be extended far from the model with respect to typical CFD simulations because the electromagnetic fieldwill affect regions outside the shock layer.

CoordinateX

Co

ord

ina

teY

-0.5 0 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CoordinateX

Co

ord

ina

teY

0 0.02 0.04 0.06 0.08

0

0.01

0.02

0.03

Figure 2. Computational mesh (left). Enlargement in the vicinity of the model (right).

VI.B. Applied magnetic field reconstruction using the implicit Maxwell solver

As a first numerical experiment, we demonstrate the capability of the implicit Maxwell solver to compute theappliedmagnetic field. A similar results was shown in Ref. 5 using a different numerical technique and on a slightly different

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configuration. The capability of autonomously computing the applied magnetic field is interesting because it avoidsus the necessity of interpolating an externally computed magnetic field in our computational domain. In order tocompute the applied magnetic field using the Maxwell equations, we need to specifyB andE of the boundaries ofthe computational domain. In this case, the boundaries are the freestream, the wires surfaces and the symmetry axis.At the freestream, we assume the magnetic field to be zero and we evaluate the electric field using a half Riemannproblem. At the the wires surfaces, we specify the magnetic field that we obtain by integrating the generalized Biot-Savart-Laplace law defined in Eq. (73)

B (r) =µ0

4π

∫∫∫

V

j (r′) × (r − r′)

|r − r′|3dV (73)

For axially symmetric configurations, there is an analytical solution to Eq. (73) that contains elliptic integrals, whichare evaluated numerically.

The solution obtained solving the Maxwell equations with the implicit scheme described above is compared inFig. 3 with the solution of the generalized Biot-Savart-Laplace law (Eq. (73)). The agreement is quite satisfactory.Enforcing the∇ ·B = 0 constraint was mandatory to obtain such a good result.

CoordinateX

Co

ord

inate

Y

-0.05 0 0.05 0.1 0.150

0.02

0.04

0.06

0.08

0.1

Bx0.040.02

0

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12

-0.14

-0.16

-0.18

-0.2

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-0.3

-0.32

CoordinateX

Co

ord

inate

Y

-0.05 0 0.05 0.1 0.150

0.02

0.04

0.06

0.08

0.1

Bx0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

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-0.12

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-0.3

CoordinateX

Co

ord

inate

Y

-0.05 0 0.05 0.1 0.150

0.02

0.04

0.06

0.08

0.1

By0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

-0.02

-0.04

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CoordinateX

Co

ord

inate

Y

-0.05 0 0.05 0.1 0.150

0.02

0.04

0.06

0.08

0.1

By0.140.12

0.1

0.08

0.06

0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12

-0.14

Figure 3. Magnetic field axial component (left) and radial component (right). Comparison between solution of the generalized Biot-Savart-Laplace law (top)and of the Maxwell equations (bottom).

VI.C. Electrically non-neutral hypersonic ionizing Argon flow without applied magnetic field

In order to test the coupling between the Maxwell and Navier-Stokes solver, we run a test case on the flat-facedcylindrical model described above in absence of an applied magnetic field. The gas mixture is pre-ionized Argon. Thefreestream conditions are listed in Table 1 and they are similar to those measured in a recent experiment by Guelan etalii.10 The wall is assumed to be non-catalytic and radiation-adiabatic, with emissivity coefficient equal to 0.85. TheMaxwell equations are solved in the flowfield and in the solid body.

V∞ [m/s] T∞ [K] p∞ [Pa] (xe)∞ (xAr+)∞2350 80 10 6.98 · 10−5 6.98 · 10−5

Table 1. Freestream conditions

The challenge in this simulation is that electrical neutrality is not enforced. The combined action of transportand electromagnetic field will distribute charged and neutral particles according to the physical laws embedded in theMaxwell and Navier-Stokes equations. Since no magnetic field is applied in the meridian plane, transport propertieswill remain isotropic and we expect to see an electric field inthe meridian plane and magnetic field in the azimuthal

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CoordinateX

ρ c[C

/m3 ]

Ex

[V/m

]

-0.04 -0.03 -0.02 -0.01 0

-0.05

0

0.05

-150

-100

-50

0

Ex

ρc

CoordinateX

mo

lefr

actio

n

-0.04 -0.03 -0.02 -0.01 0

6E-05

7E-05

8E-05

9E-05

0.0001

0.00011

xAr+

xe

Figure 4. Electric charge density and electric field axial component along the stagnation line (left). Electrons and ions mole fractions (right).

plane. The initial solution is the flowfield obtained in the same conditions, but assuming electric neutrality and am-bipolar diffusion. The initial electric field is the one thatcomes out assuming that conduction currents are zero in theflowfield.

We tried to run our computation using different time steps. If the fluid dynamics time step is used from thebeginning (CFLCFD = 8 · 10−1, CFLMaxw = 1.39 · 105), the CFD solution blows up immediately and the Maxwell stepcannot even start. To obtain a solution that remains stable for a reasonable number of steps, we have to reduce theCFL of 1000 times at least. Using a CFLCFD = 8 · 10−4, that is CFLMaxw = 139, the number of iterations is sufficientlylarge to provide a meaningful solution. In particular, it isclearly possible to see a loss of neutrality across the shockwave and at the wall. The electric charge density and the axial component of the electric field along the stagnation lineare shown in Fig. 4. As we expect from the electrostatic Gausslaw,Ex decreases when the electric charge density isnegative and increases whenρc is positive. The non-neutrality is due to an accumulation ofelectrons before the shockwave and a reduction of electrons after the shock. We interpret this fact noting that for light particles it is harder tocross the strong adverse pressure gradient with respect to heavy particles. A smaller deviation from neutrality, in this

CoordinateX

elec

tric

curr

entd

ensi

ty[A

/m2 ]

-0.04 -0.03 -0.02 -0.01 0

-2000

-1000

0

1000

2000

ρcu

JQ xjx

Figure 5. Electric current densities along the stagnation line.

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case with an excess of positively charged particles, can also be seen at the wall.Electric current densities distributions along the stagnation line are shown in Fig. 5. The conduction current,JQ,

is the major component of the total electric current densityj.Despite these results are quite encouraging and unique, thenumerical solution does not remain stable for a long

time. Instabilities start to occur in various spots across the shock wave and we were not able to keep them under controlup to now. The situation is depicted in Fig. 6, where electriccharge density and electric field magnitude contours areplotted and electric field lines are superimposed. The spotsthat are located near the shock are regions of exceedinglyhigh electric charge density and electric field magnitude that cannot be taken under control and continue to increaseuntil the numerical simulation blows up. The same behavior occurs using smaller time steps also and the instabilityappears to arise roughly at the same physical time regardless of the used time step.

CoordinateX

Coo

rdin

ateY

-0.04 -0.02 0 0.02 0.04 0.06 0.08

0

0.02

0.04

0.06

0.08

0.1

0.12

rhoc [C/m3]

0.160.140.120.10.080.060.040.020

-0.02-0.04-0.06-0.08-0.1-0.12-0.14-0.16-0.18

CoordinateXC

oord

inat

eY-0.04 -0.02 0 0.02 0.04 0.06 0.08

0

0.02

0.04

0.06

0.08

0.1

|E| [V/m]

90085080075070065060055050045040035030025020015010050

Figure 6. Electric charge density contours and electric field lines (left). Magnitude of the electric field and electric field lines (right).

The source of instability doesn’t seem to be linked to the solution of the Maxwell equations, but rather to thedrift term that is embedded in the transport model implemented in the Navier-Stokes equations. The solution to thisproblem will be the subject of future research.

VII. Conclusions

In this paper we discussed four elements of our research workthat can be considered as new in the framework of thenumerical modelling and simulation of magneto-fluid dynamics. First, we described a fully implicit numerical methodfor the Maxwell equations and we successfully used it to evaluate the magnetostatic field generated by a coil in flat-faced cylinder model. Second, we successfully implementedaccurate transport models based on the Chapman-Enskogmethod in our magneto-fluid dynamics solver. Third, we coupled our implicit Maxwell solver with our explicit non-equilibrium Navier-Stokes solver. Fourth, we did not enforce gas neutrality, thus allowing the electric charge densityto varying in the flowfield according to the governing equations. The latter point resulted to be critical, because stronginstabilities arise after a certain amount of time in those regions where the electric charge density differ from zero. Wesuspect that such a problem is linked with the drift term in the transport model, but we have not found a method tokeep the instability under control yet.

References1D’Ambrosio, D. and Giordano, D., “Electromagnetic Fluid Dynamics for Aerospace Applications. Part I: Classification and Critical Review

of Physical Models,” AIAA Paper 2004-2165,35th AIAA Plasmadynamics and Lasers Conference, Portland, OR, June 2004.

2D’Ambrosio, D., Pandolfi, M., and Giordano, D., “Electromagnetic Fluid Dynamics for Aerospace Applications. Part II: Numerical Sim-ulations Using Different Physical Models,” AIAA Paper 2004–2362,35th AIAA Plasmadynamics and Lasers Conference, Portland, OR, June2004.

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3D’Ambrosio, D. and Giordano, D., “A Numerical Method for Two- Dimensional Hypersonic Fully Coupled Electromagnetic Fluid Dynam-ics,” AIAA Paper 2005–5374,36th AIAA Plasmadynamics and Lasers Conference, Toronto, Canada, June 2005.

4D’Ambrosio, D. and Giordano, D., “Electromagnetic Fluid Dynamics for Aerospace Applications.”Journal of Thermophysics and HeatTranfer, Vol. 21, No. 2, 2007.

5D’Ambrosio, D. and Giordano, D., “Two-Dimensional Numerical Methods in Electromagnetic Hypersonics Including FullyCoupledMaxwell Equations,” AIAA Paper 2008–4013,39

th AIAA Plasmadynamics and Lasers Conference, Seattle, WA, USA, June 2008.6MacCormack, R., “Numerical Simulation of Aerodynamic FlowIncluding Induced Magnetic and Electric Fields,” AIAA 2008-4010,39th

AIAA Plasmadynamics and Lasers Conference, Seattle, WA, June 2008.7Giordano, D., “Hypersonic-Flow Governing Equations with Electromagnetic Fields,” AIAA Paper 2002-2165, May 2002.8Bruno., D., Laricchiuta, A., Capitelli, M., Catalfamo, C.,and Giordano, D., “Transport Properties of Partially Ionized Argon in a Magnetic

Field,” Journal of Thermophysics and Heat Transfer, Vol. 22, No. 3, Jul.-Sep. 2008, pp. 424–433.9Bruno., D., Laricchiuta, A., Capitelli, M., Catalfamo, C.,and Giordano, D., “Transport Properties of High Temperature Air Plasma in the

Presence of Magnetic Field,” AIAA 2008-4015,39th AIAA Plasmadynamics and Lasers Conference, Seattle, WA, June 2008.

10A. Gulhan, B. E., Koch, U., Siebe, F., Riehmer, J., Giordano, D., and Konigorski, D., “Experimental Verification of Heat-Flux Mitigation byElectromagnetic Fields in Partially Ionized-Argon Flows,” Journal of Spacecraft and Rockets, Vol. 46, No. 2, Mar.-Apr. 2009, pp. 274–283.

11Bruno, D., Catalfamo, C., Laricchiuta, A., Giordano, D., and Capitelli, M., “Convergence of ChapmanEnskog Calculation of TransportCoefficients of Magnetized Argon Plasma,”Physics of Plasmas, Vol. 13, No. 7, 2006, pp. 072307, doi:10.1063/1.2221675.

12Pandolfi, M., “A Contribution to the Numerical Prediction ofUnsteady Flows,”AIAA Journal, Vol. 22, No. 5, 1983, pp. 602–610.13Toth, G., “The∇ · B = 0 Constraint in Shock-Capturing Magnetohydrodynamics Codes,” Journal of Computational Physics, Vol. 161,

2000, pp. 605–652.

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