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Electromagnetic Fluid Dynamics for Aerospace

Applications. Part II: Numerical Simulations Using

Different Physical Models

Domenic D’Ambrosio∗ and Maurizio Pandolfi†

Politecnico di Torino - DIASP, 10129 Torino (TO), Italy

Domenico Giordano‡

ESTEC - Aerothermodynamics Section, 2200 AG Noordwijk, The Netherlands

In this paper, numerical methods for one-dimensional magneto-fluid dynamics are dis-

cussed. Two mathematical models are considered: the full magneto-fluid dynamics equa-

tions, that is the system where the Maxwell and the fluid dynamics balance equations are

coupled, and the simplified magneto-fluid dynamics equations, which are obtained from the

former applying the so-called magnetohydrodynamic approximation. Numerical methods

for the solution of both models are proposed and applied to an idealized magneto-fluid dy-

namics shock-tube problem in one dimension. Comparisons between the results obtained

using the two different methods are carried out to verify their limits of application.

Nomenclature

B magnetic field densityE electric fieldF fluxesI identity matrixJU internal energy diffusive fluxn normal unit vectorv velocity vectorW conservative variablesBi magnetic field componentc speed of light in vacuum, ∼ 2.998 · 108 m/sEi electric field componentem matter energyEt total energyL characteristic length scaleM Mach numberp pressureRe Reynolds numberRem magnetic Reynolds numberS magnetic force number

t timeu, v, w velocity vector componentsV∞ free-stream velocityx, y, z cartesian coordinate directionsS generic volumeV generic surface

Subscripts

0 characteristic reference condition∞ free-stream conditionµ dynamic viscosityε0 permittivity of vacuum, ∼ 8.854 · 10−12 F/mLbc boundary condition at the left sideN cell point indexRbc boundary condition at the right side

Symbols

γ specific heats ratio∆ = (·)N+ 1

2

− (·)N− 1

2

(difference in space)

∆ = (·)K+1 − (·)K (difference in time)λe scalar electrical conductivityλe electrical conductivity tensor

∗Assistant Professor, Dipartimento di Ingegneria Aeronautica e Spaziale, Corso Duca degli Abruzzi 24, Torino, Italy, AIAAMember, ([email protected]).

†Professor, Dipartimento di Ingegneria Aeronautica e Spaziale, Corso Duca degli Abruzzi 24, Torino, Italy, AIAA Fellow,([email protected]).

‡Research engineer, Aerothermodynamics Section, Keplerlaan 1, Noordwijk, The Netherlands, AIAA Member,([email protected]).

Copyright c© 2004 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. withpermission.

1 of 23

American Institute of Aeronautics and Astronautics Paper 2004-2362

τ stress tensorρ densityρc electric charge densityΩ source terms

Superscripts

f fluid dynamicsK time step indexm electromagnetisms simplified MFD

I. Introduction

The recent interest in magneto-fluid dynamics as a mean to control the flow during atmospheric re-entryor sustained hypersonic flight has brought the attention of many members of both the computational

fluid dynamics (CFD) and the computational electromagnetics (CEM) communities towards the field ofcomputational magneto-fluid dynamics (CMFD). The shared experience of these scientists has provided arich heritage of knowledge for CMFD, as noted in Ref.1.

In magneto-fluid dynamics for aerospace applications, it is presently common practice to adopt a set ofgoverning equations that results from the application of simplifying hypothesis to the most general formula-tion, which is composed of the Maxwell and the Navier-Stokes equations. The so-called MHD approximationtransforms the full magneto-fluid dynamics (FMFD) equations into the simplified magneto-fluid dynamics(SMFD) equations. The magnetohydrodynamic approximation is reasonable in most engineering applica-tions, included those related to hypersonic motion. However, as noted in Refs.2 and 3, there could be regionsof the flow where those effects neglected using the simplified MFD equations could be important. In addition,the FMFD system is more flexible than the SMFD one to adopt constitutive laws for the electric currentsthat differ from the generalized Ohm law.

Attempts to numerically compute the FMFD equations are very scarce in the literature, if not com-pletely absent. The nearest application, to the authors’ knowledge, is related to the so-called two-fluidsplasma model, which treats the plasma as a combination of electron and ion fluids coupled through theelectromagnetic field.4, 5 Conversely, most numerical methods for computational magneto-fluid dynamics arebased on the SMFD model and have been developed following the guidelines coming from CFD methods, asit is demonstrated by the large number of publications where numerical procedures for CMFD are borrowedfrom different families of CFD methods.6–18

In this paper, we present two different numerical methods. The one conceived to solve the full MFDequations is presented in Section II, while the other, which concerns the SMFD system, is described in SectionIII. By now, we restrict our attention to one-dimensional problems, as our present aim is to verify the limitof applicability of the two numerical procedures, especially in relation to different magnetic Reynolds numberregimes. In Section IV, we show results obtained applying the two models to the problem of an idealizedmagneto-fluid dynamics shock tube.

II. Numerical simulation of the full magneto-fluid dynamics equations

A. Governing equations

The full magneto-fluid dynamics equations (FMFD) are made up of the Navier-Stokes equations and theMaxwell equations, which are coupled one to each other through source terms that describe the mutualinteractions between electromagnetism and fluid dynamics. The mathematical form of the full magneto-fluiddynamics equations (FMFD), which we briefly remind in the following, is discussed in detail in a companionpaper.3

The non-dimensional form of the full magneto-fluid dynamics equations reads like:

∂ρc

∂t+ ∇ · (ρcv) +

c2

V 2∞

√γ M Rem ∇ · JQ = 0 (1a)

∂ρ

∂t+ ∇ · (ρv) = 0 (1b)

∂ρv

∂t+ ∇ · (ρvv + pI) −

√γM

Re∇ · τ =

V 2∞

c2S ρcE +

V 2∞

c2S ρcv ×B +

√γM S Rem JQ ×B (1c)

∂ (ρem)

∂t+ ∇ · [(ρem + p)v] +

√γM

Re∇ · (JU − τ · v) =

V 2∞

c2S ρcv ·E +

√γM S Rem JQ · E (1d)

2 of 23


∂B

∂t+ ∇×E = 0 (1e)

V 2∞

c2

∂E

∂t− γM2 ∇×B = −V 2

∞

c2ρcv −√

γ M Rem JQ (1f)

with the constitutive relation

em =1

γ − 1

p

ρ+

1

2‖v‖2 (2a)

Equations (1) and (2) are in non-dimensional form. The similarity parameters that appear in them aredefined as:

Re =ρ0V∞L

µ0

Reynolds number (3)

M =V∞

√

γp0/ρ0

Mach number (4)

Rem =λ0V∞L

ε0c2Magnetic Reynods number (5)

S =ε0c

2B20

ρ0V 2∞

Magnetic force number (6)

where the subscript ′0′ refers to characteristic reference conditions and L is a characteristic length scale.Note that velocity is made non-dimensional using V0 =

√

p0/ρ0 as the characteristic speed of the flow, whilethe similarity parameters are defined using the free-stream velocity, V∞. This explains the appearance ofterms such as

√γM in Eqs. (1).

Since the aim of this work is to develop numerical methods capable of coupling the Maxwell and the fluiddynamics equations, we decide to simplify Eqs. (1a)–(1f) using the following approximations:

Re →∞ inviscid flow (7a)

ρc = 0 null electric charge density (7b)

λe ≡λe = const. constant and scalar electrical conductivity (7c)

JQ = λe (E + v ×B) generalized Ohm law (7d)

Thus, Eqs. (1a)–(1f) are changed into:∂ρ

∂t+ ∇ · (ρv) = 0 (8a)

∂ρv

∂t+ ∇ · (ρvv + pI) =

√γM S Rem JQ ×B (8b)

∂ (ρem)

∂t+ ∇ · [(ρem + p)v] =

√γM S Rem JQ · E (8c)

∂B

∂t+ ∇×E = 0 (8d)

V 2∞

c2

∂E

∂t− γM2 ∇×B = −√

γ M Rem JQ (8e)

Note that the condition expressed by Eq. (7b) represents the same approximation made for obtaining thesimplified magneto-fluid dynamics equations. However, since the displacement current term is retainedin Eq. (8e), the mathematical nature of the full magneto-fluid dynamics equations is preserved and thenumerical method presented in the following Subsection can be easily extended to account for the electriccharge density.

As the numerical method presented here adopts a cell-centered finite volume discretization of the com-putational domain, we need to write Eqs. (8) in their integral form:

∫

V

∂ρ

∂tdV +

∫

S

ρv · n dS = 0 (9a)

3 of 23


∫

V

∂ρv

∂tdV +

∫

S

ρv v · n dS +

∫

S

p I · n dS =√

γM S Rem

∫

V

JQ ×B dV (9b)

∫

V

∂ (ρem)

∂tdV +

∫

S

(ρem + p)v · n dS =√

γM S Rem

∫

V

JQ ·E dV (9c)

∫

V

∂B

∂tdV +

∫

S

n×E dS = 0 (9d)

V 2∞

c2

∫

V

∂E

∂tdV − γM2

∫

S

n×B dS = −√γ M Rem

∫

V

JQ dV (9e)

Moreover, since attention is focused on one-dimensional flows, we reduce Eqs. (9) into:

∫ x2

x1

∂W

∂tdx + F2 − F1 =

∫ x2

x1

Ω dx (10)

where

W =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ρ

ρ u

ρ v

ρ w

ρ em

By

Bz

Ey

Ez

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

; F =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ρ u

p + ρ u u

ρ v u

ρ w u

ρ em + p

−Ez

Ey

γM2c2

V 2∞

Bz

−γM2c2

V 2∞

By

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(11)

and

Ω = λe

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0√

γM S Rem

[

Ey Bz − Ez By + v Bx By + w Bx Bz − u(

B2y + B2

z

)]

√γM S Rem

[

Ez Bx − Ex Bz + w By Bz + u By Bx − v(

B2z + B2

x

)]

√γM S Rem

[

Ex By − Ey Bx + u Bz Bx + v Bz By − w(

B2x + B2

y

)]

√γM S Rem

[

‖E‖2 + Ex (vBz − wBy) + Ey (wBx − uBz) + Ez (uBy − vBx)]

0

0√

γMc2

V 2∞

Rem (Ey + wBx − uBz)

√γM

c2

V 2∞

Rem (Ez + uBy − vBx)

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(12)

Note that the first five elements of vectors W, F and Ω refer to the fluid dynamics balance equations, whilethe last four are pertinent to electromagnetism. The coupling between the two phenomenologies occurs onlythrough the source term, Ω, while in W and F the first five and the last four components are completelyuncoupled.

At this stage, the mathematical model has been completely defined and it requires a numerical methodfor its solution.

B. Numerical method

The major problem in the numerical simulation of the full magneto-fluid dynamics equations resides in thefact that the signals propagation speeds typical of fluid dynamics differ orders of magnitude from the signalpropagation speeds typical of electromagnetism. The former are of the order of the speed of sound, or atmost one order of magnitude larger (flying at Mach 20 at high altitude, the fastest propagation speed is about7000 m/s), while the latter are as fast as the speed of light (that is about 3 · 108 m/s). Since there are at

4 of 23


least 5 orders of magnitude separating signal propagation speeds in fluid dynamics and in electromagnetism,the resulting coupled system is a very stiff one, and requires a special treatment.

Here, we want to solve the system of Eqs. (10)–(12) using a finite volume scheme and a time-dependentmethod. Solution methods for fluid dynamics and for electromagnetism, when considered separately, arewell known.1 However, the coupling requires that the two systems are solved simultaneously. Stabilityconsiderations request that, using explicit schemes, the common time step be very small, namely of theorder of ∆x/c. This is definitely a too small time step for efficient fluid dynamics computations. The wayto overcome such an obstacle is to solve the Maxwell equations using implicit schemes, that allow for largecomputational time steps as they are nominally unconditionally stable. The fact that the speed of light isa constant parameter implies that the homogeneous electromagnetic part of the FMFD system has lineareigenvalues, so that the application of implicit schemes to the Maxwell equations allows for very large timesteps (comparatively to the Maxwell system). Thus, one can conceive to march the fluid dynamics systemin time for one iteration using the values of the electromagnetic field variables frozen at the beginning ofthe time step, and then to march the electromagnetics system in time for a couple of iterations until thesame time value of the companion fluid dynamics step is reached. In this second step, one can decide to usethe flow-field values frozen at the beginning of the fluid dynamics time step, or those obtained at the endof it, or an average value. Clearly, such a way of proceeding does not possess the time resolution necessaryto resolve electromagnetic phenomena that evolve with very high frequency, even though, in principle, themethod permits to reduce the electromagnetic step at will.

Another delicate point arises when source terms become large. In fact, it is well known that the numericalsolution of hyperbolic equations becomes very hard when stiff source terms are present. Such a problem istypical, for instance, of the numerical simulation of chemically reacting flows. In the full MFD equations,source terms become stiff when the magnetic Reynolds number is large. In order to increase the value ofRem above which the numerical method fails, the classical strategy of treating the source terms implicitlycan be used.

Putting into practice what we just anticipated results in the development of a two-steps numerical pro-cedure. In the first step, the fluid dynamic part of the equations is solved using a second order explicitintegration scheme for the homogeneous part and an implicit procedure for the source term. During thisstep, the electromagnetic variables are considered as frozen. Indicating with Wf , Ff and Ωf that parts ofW, F and Ω pertinent to the fluid dynamics equations, we write:

∆WfN

∆t∆x +

(

Ff

N+ 1

2

)K

−(

Ff

N−1

2

)K

=(

ΩfN

)K+ 1

2 ∆x =

=(

ΩfN

)K∆x +

[

(

∂ΩfN

∂WfN

)K∆Wf

N

∆t

∆t

2

]

∆x

(13)

where

∂ΩfN

∂WfN

=√

γM S Rem λe

0 0 0 0 0

− (v ×B)x

ρ−

B2y + B2

z

ρ

BxBy

ρ

BxBz

ρ0

−(v ×B)y

ρ

ByBx

ρ−B2

x + B2z

ρ

ByBz

ρ0

− (v ×B)z

ρ

BzBx

ρ

BzBy

ρ−

B2x + B2

y

ρ0

− (v ×B) · Eρ

− (E×B)x

ρ−

(E×B)y

ρ− (E×B)z

ρ0

(14)

In this paper, any Jacobian matrix will be meant to be computed at time step K. Therefore, from now on,we will omit the superscript K when writing such terms.

It is clear from Eqs. (13)–(14) that the mass balance equation, which has no source term, can be solvedseparately from the others:

ρK+1N = ρK

N − ∆F ρ ∆t

∆x(15)

Conversely, the momentum and energy balance equations are coupled through the source terms, whose

5 of 23


implicit integration results in the system:

∆Wf

N

∆t∆x +

(

Ff

N+ 1

2

)K

−(

Ff

N−1

2

)K

=(

Ωf

N

)K

∆x +

[

∂Ωf

N

∂ρN

∆ρN

∆t

∆t

2

]

∆x +

[

∂Ωf

N

∂Wf

N

∆Wf

N

∆t

∆t

2

]

∆x (16)

The signˆabove vectors Wf , Ff and Ωf indicates that the mass balance equations is excluded from Eq. (16).The solution of Eq. (16) requires the inversion of a (4x4) matrix for each computational cell:

[A]N ∆Wf

N = −(

∆Ff

N

)K ∆t

∆x+

(

Ωf

N

)K

∆t +∂Ωf

N

∂ρN

∆ρN

∆t

2(17)

where

[A]N =

1 + CfB2

y + B2z

ρ

∆t

2−Cf BxBy

ρ

∆t

2−Cf BxBz

ρ

∆t

20

−Cf ByBx

ρ

∆t

21 + Cf B2

x + B2z

ρ

∆t

2−Cf ByBz

ρ

∆t

20

−Cf BzBx

ρ

∆t

2−Cf BzBy

ρ

∆t

21 + Cf

B2x + B2

y

ρ

∆t

20

Cf(E ×B)x

ρ

∆t

2Cf

(E×B)y

ρ

∆t

2C

(E ×B)z

ρ

∆t

21

(18)

withCf =

√γM S Rem λe (19)

The elements of [A]N must be computed using the local values of B, E and v at each N cell. Note that, since

the time-variation of density, ∆ρ, is already known from Eq. (15), the derivatives with respect to densitythat are present in the source term components are treated explicitly in Eq. (16), with

∂Ωf

N

∂ρN

= Cf

[

− (v ×B)x

ρ, −

(v ×B)y

ρ, − (v ×B)z

ρ, − (v ×B) · E

ρ

]T

(20)

At this point, the only missing information concerns the way in which fluxes Ff are computed. In thiswork, such an operation is carried out using an upwind flux-difference splitting method widely used by thefirst two authors for computing high speed flows.19, 20

Once fluid dynamics variables have been computed as described above, the second step consists in solvingthe electromagnetic part of the equations system. In this case, the integration scheme is fully implicit, insuch a way that the same time-integration interval of fluid dynamics, ∆t, can be used without running intostability problems. Fluid dynamics variables are treated as if they were frozen at the values obtained afterthe first step of the procedure. Therefore, the integration scheme for the electromagnetic part of the FMFDequations is:

∆WmN

∆t∆x +

(

Fm

N+ 1

2

)K+1

−(

Fm

N− 1

2

)K+1

= (ΩmN )

K+ 1

2 ∆x (21)

Assuming that the fluxes at each cell interface depend on the two neighboring cells only, Eq. (21) can beexpanded as:

∆WmN

∆t∆x+

(

Fm

N+ 1

2

)K

+∂Fm

N+ 1

2

∂WmN+1

∆WmN+1

∆t

∆t

2+

∂FmN+ 1

2

∂WmN

∆WmN

∆t

∆t

2−

−(

Fm

N−1

2

)K

−∂Fm

N−1

2

∂WmN

∆WmN

∆t

∆t

2−

∂FmN−

1

2

∂WmN−1

∆WmN−1

∆t

∆t

2=

=(ΩmN )

K∆x +

[

∂ΩmN

∂WmN

∆WmN

∆t

∆t

2

]

∆x

(22)

As the homogenous part of the Maxwell equations does not provide for the presence of discontinuities, acentered scheme can be used to evaluate the elements of Fm at each cell interface:

Fm

N± 1

2

=1

2(Fm

N + FmN±1) (23)

6 of 23


The solution of Eq. (22) requires the inversion of a block tridiagonal matrix, [D], across the whole compu-tational domain. The linear system to be solved is:

[D]

∆Wm

= −∆Fm ∆t

∆x+ Ωm ∆t (24)

As anticipated, matrix [D] is a block tridiagonal matrix. If Eq. (23) is used to compute Fm, then thesubdiagonal blocks [L], the superdiagonal blocks [R] and diagonal blocks [C] have the form:

[L] =

0 0 01

2

∆t

∆x

0 0 −1

2

∆t

∆x0

0 −K

2

∆t

∆x0 0

K

2

∆t

∆x0 0 0

; [R] =

0 0 0 −1

2

∆t

∆x

0 01

2

∆t

∆x0

0K

2

∆t

∆x0 0

−K

2

∆t

∆x0 0 0

[C] =

1 0 0 0

0 1 0 0

0 −CmuN

∆t

21 + Cm ∆t

20

CMuN

∆t

20 0 1 + Cm ∆t

2

(25)

where

K = γM2 c2

V 2∞

(26)

Cm =√

γMc2

V 2∞

Remλe (27)

At the boundaries of the computational domain, the form of [L], [R] and [C] will change according tothe enforced boundary conditions.

1. Accuracy

The fluid dynamic part of the FMFD equations is computed reconstructing the convective fluxes with asecond order scheme of the ENO family,21 as described in Ref.20. In addition, source terms are computedat the intermediate time step K + 1/2 with an implicit scheme. Thus, the numerical method for the fluiddynamic step has a nominal second order accuracy both in space and in time.

The Maxwell equations are solved using central differences for computing the fluxes, which, therefore, aresecond order accurate in space. The integration in time is performed with an implicit evaluation of the fluxesat the electromagnetic time step K +1, while source terms are evaluated at the intermediate electromagnetictime step K + 1/2. Though this is not requested by stability constraints, three electromagnetic time stepsof amplitude ∆t/3 are provided within each fluid dynamic time interval, ∆t.

III. Numerical simulation of the simplified magneto-fluid dynamics equations

A. Governing equations

If the MHD approximation is applied, the magneto-fluid dynamics equations are reduced to a simplifiedsystem where the electric field E becomes a dependent variable that can be reconstructed from the values ofB and v. The SMFD model is widely discussed in Ref.3.

In integral form, the SMFD equations are given by:

∫

V

∂Ws

∂tdV +

∫

S

Fs · n dS =

∫

V

Ωs dV (28)

7 of 23


where

Ws =

∣

∣

∣

∣

∣

∣

∣

∣

∣

ρ

ρv

B

Et

∣

∣

∣

∣

∣

∣

∣

∣

∣

; Fs =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ρv

ρvv +

(

p + γM2SB2

2

)

I− γM2S BB

vB −Bv(

Et + p + γM2SB2

2

)

v − γM2S (v · B)B

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(29)

and

Ωs =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0

0

−√

γM

Rem

∇×(∇×B

λe

)

√γM

Rem

γM2S1

λe

[

(∇×B)2 + B · ∇2B]

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(30)

with

Et =

(

ρem + γM2SB2

2

)

(31)

Equations (28)–(31) are written in non-dimensional form using the same characteristic reference values ofthe FMFD model.

If we restrict our model to one-dimension, we remain with the following system of equations:∫ x2

x1

∂Ws

∂tdx + Fs

2 − Fs1 =

∫ x2

x1

Ωs dx (32)

where

Ws =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ρ

ρ u

ρ v

ρ w

By

Bz

Et

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

; Fs =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

ρ u

ρuu + p + γM2SB2

2− γM2S Bx Bx

ρvu − γM2S By Bx

ρwu − γM2S Bz Bx

By u − v Bx

Bz u − w Bx(

Et + p + γM2SB2

2

)

u − γM2S (uBx + vBy + wBz) Bx

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(33)

and

Ωs =1

λe

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0

0

0

0√γM

Rem

(

∂2By

∂x2

)

√γM

Rem

(

∂2Bz

∂x2

)

√γM

Rem

γM2S

[

(

∂By

∂x

)2

+

(

∂Bz

∂x

)2

+ By

∂2By

∂x2+ Bz

∂2Bz

∂x2

]

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(34)

As previously anticipated, the electric field E can be computed from the relation:

E =

√γM

Rem

1

λe

∇×B − v ×B (35)

The electric charge density, ρc does not appear in Eqs. (32)–(34), but, in general, it can be reconstructedfrom the Gauss law for electricity:

ρc = ε0∇ ·E (36)

8 of 23


Enforcing the magnetohydrodynamic approximation completely changes the mathematical nature of theMFD equations. The propagation of electromagnetic signals with the speed of light is completed removedand the homogenous SMFD system (that is in the limit of infinite viscous and magnetic Reynolds numbers)is characterized by eigenvalues and eigenvectors that correspond to seven different signals with the respectivepropagation speeds. Details on the subject are given in Ref.22 and are widely reported in the literature.23–25

B. Numerical method

Numerical methods for the simplified MFD equations can be very different depending on the magneticReynolds number of the application. In fact, the system contained in Eqs. (28)–(30) presents source termsthat become large for small values of Rem. When such a situation occurs, the SMFD equations are furthermodified and the so-called Low-Rem SMFD equations are obtained.3, 26 In this case, considerable simplifica-tion is introduced by noting that the induced magnetic field is negligible and the Ampere’s law is discarded.Conversely, in the general SMFD formulation, no assumption is made on the magnetic Reynolds number. Inthis case, the magnetic induction equation is explicitly solved to obtain the induced magnetic field.

Most numerical methods for solving the general formulation of the SMFD equations in compressible flowconditions adopt upwind schemes originally developed for the ideal magnetohydrodynamics (MHD) equations,that is the homogenous and inviscid part of the SMFD equations. The approach replicates the one followedby numerical methods for compressible flows: first upwind schemes were developed for the Euler equationsin one-dimension and then extended to two and three-dimensions and to Navier-Stokes. Upwind methodsdeveloped to solve the ideal MHD equations belong to the Roe family,8–10 to the flux-vector splitting family,11

to the HLLE family,12, 13 to the gas-kinetics theory based flux-splitting family,14 to the PPM family,15 tothe Lax-Friedrichs family16 and to the Osher family.22 With the exception of the gas-kinetic FVS method,all the above cited upwind methods follow the Godunov approach and thus necessitate knowledge of theeigenvectors and eigenvalues system that characterize the one-dimensional ideal MHD equations. Othernumerical methods for MHD adopt the TVD scheme17 or a combination of compact-difference for spacederivatives and the classical Runge-Kutta method for time derivatives.18

When such methods are extended to more than one dimension, special procedures must be adopted inorder to enforce the ∇ · B = 0 constraint in the integration of the discretized equations.11, 27–31

Here, we present a numerical method conceived for the general formulation of the SMFD equations, whichis based on the upwind solver presented in Ref.22. The integration of the equations system is carried out inan explicit fashion as far as the convective part is concerned. In addition, source terms are computed withan implicit scheme to overcome stability problems that may occur for small values of the magnetic Reynoldsnumber, when the source term becomes very large.

We start our description of the method with discretizing the one-dimensional SMFD equations using acell-centered finite volume approximation:

∆Ws

∆t+ FsK

N+ 1

2

− FsKN− 1

2

= ΩsK+ 1

2

N (37)

The first four equations contained in the system of Eq. (37), namely the mass and momentum balanceequations, have no source term and they can be solved using a standard explicit scheme:

∆W iN = − ∆t

∆x∆F i

N (38)

Conversely, the magnetic induction equations and the total energy balance equation are characterizedby the presence of source terms, whose implicit integration provides that they be computed at time stepK +1/2. In this work, the space derivatives of the magnetic field contained in the source terms are computedaccording to the formulas:

∂Bi

∂x≈

[

d1 (Bi)N+1 + d2 (Bi)N + d3 (Bi)N−1

]

∆x(39a)

∂2Bi

∂x2≈

[

l1 (Bi)N+1 + l2 (Bi)N + l3 (Bi)N−1

]

∆x2(39b)

9 of 23


where, for internal points, we use

d1 =1

2d2 = 0 d3 = −1

2(40a)

l1 = 1 l2 = −2 l3 = 1 (40b)

while other coefficients have to be adopted for any particular boundary condition. Thus, the source termsdepend not only on the values of B at the local mesh point, but also at the two neighboring cells:

ΩsK+ 1

2

N = ΩsKN +

[

∂ΩsN

∂WsN−1

∆WsN−1

∆t+

∂ΩsN

∂WsN

∆WsN

∆t+

∂ΩsN

∂WsN+1

∆WsN+1

∆t

]

∆t

2(41)

Note that the source terms in the two magnetic induction equations are uncoupled, so that those equationscan be solved separately. In general, for each component of the magnetic induction equation, it will benecessary to solve a system that requires the inversion of a tridiagonal matrix whose rank is equal to thenumber of cells:

[T]

∆Bi

= −

F Bi

∆t

∆x+

ΩBi

∆t (42)

The elements of the tridiagonal matrix, [T], are:

a = −∂

ΩBi

N

∂ BiN−1

∆t

2= −

√γM

Rem

1

λ

l3∆x2

∆t

2subdiagonal term (43a)

b = 1 −∂

ΩBi

N

∂ BiN

∆t

2= 1−

√γM

Rem

1

λ

l2∆x2

∆t

2diagonal term (43b)

c = −∂

ΩBi

N

∂ BiN+1

∆t

2= −

√γM

Rem

1

λ

l1∆x2

∆t

2superdiagonal term (43c)

In addition, the half-implicit discretization of the total energy balance equation doesn’t require theinversion of a matrix, because the source term component ΩEt only depends from the space derivatives of

B. Thus, we can solve for

∆Et

using the equation:

∆EtN + ∆F Et

N

∆t

∆x=

(

ΩEt

N

)K

∆t+

+

∑

i=y,z

∂ΩEt

N

∂BiN−1

∆BiN−1 +∑

i=y,z

∂ΩEt

N

∂BiN

∆BiN +∑

i=y,z

∂ΩEt

N

∂BiN+1

∆BiN+1

∆t

2

(44)

with

∂ΩEt

N

∂BiN−1

=

√γM

Rem

γM2S1

λ

[

2

(

∂Bi

∂x

)

N

d3

∆x+ Bi

l3∆x2

]

(45a)

∂ΩEt

N

∂BiN

=

√γM

Rem

γM2S1

λ

[

2

(

∂Bi

∂x

)

N

d2

∆x+ Bi

l2∆x2

+

(

∂2Bi

∂x2

)

N

]

(45b)

∂ΩEt

N

∂BiN+1

=

√γM

Rem

γM2S1

λ

[

2

(

∂Bi

∂x

)

N

d1

∆x+ Bi

l1∆x2

]

(45c)

In Eq. (44), all the increments

∆By

and

∆Bz

are known values previously obtained from the solution

of Eq. (42).

1. Accuracy

An ENO reconstruction21 and the use of the MINMOD limiter provide a reconstruction of the initial datafor the computation of the convective fluxes that leads to a second order of accuracy in space. Secondorder of accuracy in time is achieved by adding a further correction to these values to account for the timedependency of the solution. In addition, source terms are computed at the intermediate step K + 1/2 andthe partial derivatives appearing in them are evaluated using finite difference formulas that are also secondorder accurate. Thus, globally, the numerical method for the SMFD equations has a nominal second orderaccuracy both in space and in time.

10 of 23


IV. Numerical results

A. The shock-tube problem

The shock-tube problem can be seen as a particular Riemann problem with stationary initial conditions. Ina Riemann problem, the collapse of the initial discontinuity generates a pattern of waves depending on thegiven initial left and right states. In fluid dynamics, three waves are generated: a contact discontinuity andtwo acoustic waves, that can be shocks or expansion fans (in general, there can be one expansion fan and oneshock, or two shocks, or two expansion fans). In ideal magneto-fluid dynamics (or magnetohydrodynamics,MHD), the presence of seven characteristic waves (one contact discontinuity, two fast magnetoacoustic waves,two slow magnetoacoustic waves and two Alfven waves) allows for various possible configurations and thedebate is still open about which are admissible and which are not.22, 25, 32–34

If we restrict out scope to the particular case of the shock-tube problem, where the gas is initiallystationary, then, in fluid dynamics, the two acoustic waves are certainly an expansion fan and a shock wave.In MHD, if we add the further conditions that the problem be planar, that is with magnetic field and velocitycomponents lying on the same plane, and that the initial magnetic field be constant, we will certainly obtaina solution consisting of two magnetoacoustic expansion fans (a fast one and a slow one), two magnetoacousticshock waves (a fast one and a slow one) and a contact discontinuity. The Alfven waves are not present inplanar problems.

B. Initial conditions

We consider here a planar one-dimensional shock-tube problem, whose left- and right-side initial states aregiven in Table 1. With respect to the usual fluid dynamics test case, an initially constant magnetic field is

Table 1. Left and right states of the initial discontinuity for the considered Riemann problem.

ρ [kg/m3] p [Pa] u [m/s] v [m/s] w [m/s] Bx [T] By [T] Bz [T]

Left side 12.25 1 · 106 0.0 0.0 0.0 0.75 1.0 0.0

Right side 1.225 1 · 105 0.0 0.0 0.0 0.75 1.0 0.0

also applied and the gas is assumed to be electrically conductive. The electric field is initially equal to zero.The computational domain ranges from x/L = 0 to x/L = 1.0.

The similarity parameters that characterize our test case are given in Table 2. They were obtained using

Table 2. Similarity parameters pertinent to the reference values of Table 3.

S Rem/λe [ohm·m] Q/λe [ohm·m] M γM2S√

γM/(Rem/λe) [mho/m] V∞/c

0.714 4.24·10−4 3.03·10−4 1.0 1.0 2789.01 1.127·10−6

the reference values listed in Table 3. In particular, V0 and B0 have been chosen is such a way that M = 1and γM2S = 1.

Table 3. Characteristic reference values.

ρ0 [kg/m3] p0 [Pa] V0 [m/s] V∞ [m/s] B0 [T] L

1.225 1 · 105 0.285.71428 338.06169 0.35424929 1.0

We note that, in the planar one-dimensional shock-tube problem, the electric field component Ex cannotbe generated. Thus, according to Eq. (36), the electric charge density must be null, unless a gradient ofEx is given as an initial condition. Here, we set the initial conditions such that Ex = Ey = Ez = 0 and,therefore, the fact of neglecting ρc in the full magneto-fluid dynamics equations is an exact condition.

11 of 23


C. Boundary conditions

Normally, simple-wave boundary conditions can be safely used in the numerical solution of classical shock-tube problems, both in fluid dynamics and in MHD. Nevertheless, when source terms are present in theequations, such boundary conditions cannot be used, in particular when source terms are effective rightat the boundaries of the computational domain. Thus, one is forced to choose for a different boundarycondition. In the present case, we noted that, for intermediate values of λe, a dispersive wave reaches theright border of the domain producing a sub-magneto-sonic outflow velocity. Therefore, in the SMFD code,we decided to impose, at the right boundary, an exit pressure and an exit magnetic field component By withvalues equal to those of the initial right state, that is

(p)Rbc =1 · 105 Pa (46a)

(By)Rbc

=1.0 T (46b)

Conversely, we maintained the simple wave boundary condition at the left border, which is not affected bysignificant disturbances.

Concerning the FMFD code, one has to make a distinction between the fluid dynamic and electromagneticsteps. For the first one, the same boundary conditions of the SMFD code were applied as far as the fluiddynamic variables are concerned (in particular, condition Eq. (46a)). For the electromagnetic step, a valuefor the magnetic field component By has been imposed at both boundaries, that is

(By)Lbc

=1.0 T (47a)

(By)Rbc

=1.0 T (47b)

D. Results

We present numerical simulations of the shock-tube problem described in Section IV.B that were obtainedsolving the full magneto-fluid dynamics (FMFD) equations as described in Section II and the simplifiedmagneto-fluid dynamics (SMGD) equations as discussed in Section III.

In our numerical experiments, the electric conductivity was considered as constant and different testswere carried out using different values of it. The basic computational grid is composed of 1000 points. InFigs. 1–6, we show the results that are pertinent to electrical conductivities λe equal to 1, 102, 103, 104,105, 106 S/m. Each figure shows the behavior of a different magnetogasdynamic variable. The borderlinecases of null electrical conductivity, that is Euler flow, and of infinite electrical conductivity, that is MHD,are represented in the plots as a dash-dotted line with a ’square’ symbol and as dotted line with a ’delta’symbol, respectively. The intermediate cases of finite λe are plotted as a solid line when the solution wasobtained solving the SMFD equations and as a dashed line when the FMFD equations were adopted.

As already anticipated, the pure fluid dynamic solution is characterized by a shock wave and a contactsurface that move rightwards and by an expansion fan whose outermost characteristics go leftwards. TheMHD solution contains a weak fast shock that moves rightwards, followed by a stronger slow shock and bya contact surface; a fast expansion fan moves leftwards, while a slow-sonic slow expansion opens in bothdirections. Where the slow-sonic transition of the latter occurs, there’s a change in sign of the velocitycomponent, v.

For small values of the electrical conductivity, the flow-field is very close to the Euler one. The inducedmagnetic field is very small and it initially scales linearly with the electrical conductivity, as shown in Fig. 7.As λe grows stronger, however, the three wave structure starts to get deformed and it finally takes the shapeof the MHD solution for large values of λe. The magnitude of the induced magnetic field becomes important.

The FMFD and the SMFD solutions shown in Figs. 1–6 are almost perfectly one on top of the other. Itshould be noted that the source terms for the two models behave in a diametrically opposed way. In fact, inFMFD, small values of the magnetic Reynolds number mean small source terms, while the opposite it truein SMDF. Thus, having a small Rem is an easy situation for solving the FMFD equations, but it is a stiffproblem for SMFD. Conversely, a large Rem doesn’t create difficulties for computing the SMFD model, butit makes the solutions of the FMFD equations a difficult task.

In our tests, we experienced that the implicit treatment of source terms is largely beneficial to ourcomputational methods. In fact, in the conditions of the present test case, we are able to run our codes upto λe = 1 · 106 S/m using the FMFD model and, on the other hand, down to λe = 1 · 10−2 S/m with theSMFD model. These results are particularly stunning, as the wave structure of the homogenous part of the

12 of 23


equations, which provides for a ’fluid dynamic’ three waves structure in FMFD and for a five waves structurein SMFD, is completely disrupted by the effect of the source terms, which produce five waves or three wavesstructures, respectively. In particular, the sharp numerical capture of fluid dynamic shocks with the SMFDcode used at very low values of λe was unexpected.

Numerical difficulties are encountered by the FMFD solver only for λe = 1·106 S/m and in correspondenceof the fast magneto-acoustic shock wave, where a kink is clearly visible. Such a problem, however, completelydisappears increasing the number of grid points by a factor 10 (from 1000 points to 10000 points), as shownin the grid convergence analysis reported in Figs. 8(a)–8(b). However, one should keep in mind that thesevalues of λe represent the borderline for the numerical application of the FMFD solver.

On the other hand, the SMFD solver has a problem, for values of λe smaller than 104 S/m, right at thelocation of the fluid dynamic shock wave. There, one can clearly see that the plots of the velocity componentv display a kink (Fig. 4), which doesn’t exist in the numerical solutions of the FMFD equations. When thegrid is refined (Fig. 9), the disturbance remains, though it reduces its extension. The kink apex is locatedexactly where the plot of the magnetic field component By makes an angle (Fig. 5). This aspect requiresfurther investigation. The electric field component Ez (Fig. 6), which, within the SMFD model, is computedfrom the primitive variables using Eq. (35), manifests a similar problem, which is due to the fact that v isused to compute Ez . In addition, a second kink is present where the slow expansion is sonic, at x/L = 0.5.In this case, the disturbance is due to a very slight oscillation in the slope of By that occurs at the slow-sonicpoint, as it typically happens in all upwind schemes. This fact produces an oscillation in the derivative(∂By/∂x) that must be computed to evaluate Ez . Such an effect is more or less amplified according to themagnitude of the magnetic Reynolds number, which divides ∇×B in Eq. (35). The smaller is the magneticReynolds number, the more the oscillation is magnified.

Apart from these two problems, the numerical solutions obtained using the FMFD and SMFD modelsare satisfactory. No differences can be perceived between the results related to the two approaches, exceptthose mentioned above. The numerical solution of the FMFD equations has been proved feasible, with aburden equal to a factor 3 in computing time.

V. Conclusion

Two numerical methods, one for the full magneto-fluid dynamic equations and another for the simplifiedmagneto-fluid dynamic equations have been developed. In the considered test case, they should theoreticallygive exactly the same results, as it happens, in fact, in our numerical experiments. While the physical modelbased upon the SMFD equations is widely used in the scientific community, the FMFD model is rarelyadopted. Here, we have shown that its numerical solution is feasible without an excessive burden in terms ofcomputing time. The advantage of the FMFD model is that it could be used also in regions of the flow whereterms containing the electric charge density, which are neglected in the SMFD model, become important. Inaddition, the FMFD model is not indissolubly linked to the generalized Ohm law, in the sense that it allowsfor using any other constitutive law that relates the electric current with the other electromagnetic and fluiddynamic variables.

Finally, both numerical methods have proved to be very robust, as it was possible to use each of them inborderline situations in terms of magnitude of the magnetic Reynolds number. In the next future, it is theintention of the authors to extend the methods presented here to more than one dimension and to situationswhere taking into account or not the electric charge density could make a difference between FMFD andSMFD.

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x/L

p/p 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(a) λe = 1 · 100 S/m

x/L

p/p 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(b) λe = 1 · 102 S/m

x/L

p/p 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(c) λe = 1 · 103 S/m

x/L

p/p 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

(d) λe = 1 · 104 S/m

x/L

p/p 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 105 mho/m

Rem = 4.24 x 101

S = 0.714

Q = 3.03 x 101

λ

(e) λe = 1 · 105 S/m

x/L

p/p 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λ

(f) λe = 1 · 106 S/m

Figure 1. Pressure plots obtained using different values of λe in the considered shock-tube problem.15 of 23


x/L

ρ/ρ 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(a) λe = 1 · 100 S/m

x/L

ρ/ρ 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(b) λe = 1 · 102 S/m

x/L

ρ/ρ 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(c) λe = 1 · 103 S/m

x/L

ρ/ρ 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

(d) λe = 1 · 104 S/m

x/L

ρ/ρ 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 105 mho/m

Rem = 4.24 x 101

S = 0.714

Q = 3.03 x 101

λ

(e) λe = 1 · 105 S/m

x/L

ρ/ρ 0

0 0.25 0.5 0.75 10

1

2

3

4

5

6

7

8

9

10

11SMFDFMFDMHDEuler

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λ

(f) λe = 1 · 106 S/m

Figure 2. Density plots obtained using different values of λe in the considered shock-tube problem.16 of 23


x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFDFMFDMHDEuler

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(a) λe = 1 · 100 S/m

x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFDFMFDMHDEuler

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(b) λe = 1 · 102 S/m

x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFDFMFDMHDEuler

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(c) λe = 1 · 103 S/m

x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFDFMFDMHDEuler

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

(d) λe = 1 · 104 S/m

x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFDFMFDMHDEuler

e = 105 mho/m

Rem = 4.24 x 101

S = 0.714

Q = 3.03 x 101

λ

(e) λe = 1 · 105 S/m

x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFDFMFDMHDEuler

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λ

zoom

(f) λe = 1 · 106 S/m

Figure 3. Velocity u plots obtained using different values of λe in the considered shock-tube problem.17 of 23


x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFDFMFDMHDEuler

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(a) λe = 1 · 100 S/m

x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFDFMFDMHDEuler

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(b) λe = 1 · 102 S/m

x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFDFMFDMHDEuler

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(c) λe = 1 · 103 S/m

x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFDFMFDMHDEuler

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λzoom

(d) λe = 1 · 104 S/m

x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFDFMFDMHDEuler

e = 105 mho/m

Rem = 4.24 x 101

S = 0.714

Q = 3.03 x 101

λ

(e) λe = 1 · 105 S/m

x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFDFMFDMHDEuler

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λ

(f) λe = 1 · 106 S/m

Figure 4. Velocity v plots obtained using different values of λe in the considered shock-tube problem.18 of 23


x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

SMFDFMFDMHD

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(a) λe = 1 · 100 S/m

x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

SMFDFMFDMHD

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(b) λe = 1 · 102 S/m

x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

SMFDFMFDMHD

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(c) λe = 1 · 103 S/m

x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

SMFDFMFDMHD

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

(d) λe = 1 · 104 S/m

x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

SMFDFMFDMHD

e = 105 mho/m

Rem = 4.24 x 101

S = 0.714

Q = 3.03 x 101

λ

(e) λe = 1 · 105 S/m

x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

SMFDFMFDMHD

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λ

(f) λe = 1 · 106 S/m

Figure 5. Magnetic field By plots obtained using different values of λe in the considered shock-tube problem.19 of 23


x/L

Ez/(

V0

B0)

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

SMFDFMFDMHD

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(a) λe = 1 · 100 S/m

x/L

Ez/(

V0

B0)

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

SMFDFMFDMHD

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(b) λe = 1 · 102 S/m

x/L

Ez/(

V0

B0)

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

SMFDFMFDMHD

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(c) λe = 1 · 103 S/m

x/L

Ez/(

V0

B0)

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

SMFDFMFDMHD

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

zoom

(d) λe = 1 · 104 S/m

x/L

Ez/(

V0

B0)

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

SMFDFMFDMHD

e = 105 mho/m

Rem = 4.24 x 101

S = 0.714

Q = 3.03 x 101

λ

(e) λe = 1 · 105 S/m

x/L

Ez/(

V0

B0)

0 0.25 0.5 0.75 1-2

-1.75

-1.5

-1.25

-1

-0.75

-0.5

-0.25

0

0.25

SMFDFMFDMHD

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λ

(f) λe = 1 · 106 S/m

Figure 6. Electric field Ez plots obtained using different values of λe in the considered shock-tube problem.20 of 23


x/L

By/B

0

0 0.25 0.5 0.75 11.75

2

2.25

2.5

2.75

3

3.25

3.5

3.75

SMFDFMFDMHD

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

(a) λe = 1 · 104 S/m

x/L

By/B

0

0 0.25 0.5 0.75 1

2.725

2.75

2.775

2.8

2.825

2.85

2.875

2.9 SMFDFMFDMHD

e = 103 mho/m

Rem = 4.24 x 10-1

S = 0.714

Q = 3.03 x 10-1

λ

(b) λe = 1 · 103 S/m

x/L

By/B

0

0 0.25 0.5 0.75 12.8125

2.815

2.8175

2.82

2.8225

2.825

2.8275

2.83 SMFDFMFDMHD

e = 102 mho/m

Rem = 4.24 x 10-2

S = 0.714

Q = 3.03 x 10-2

λ

(c) λe = 1 · 102 S/m

x/L

By/B

0

0 0.25 0.5 0.75 1

2.82278

2.8228

2.82283

2.82285

2.82288

2.8229

2.82293

2.82295SMFDFMFDMHD

e = 100 mho/m

Rem = 4.24 x 10-4

S = 0.714

Q = 3.03 x 10-4

λ

(d) λe = 1 · 100 S/m

Figure 7. Scaling of the induced magnetic field with λe. The vertical range of each frame is scaled in proportionto the value of the electrical conductivity.

21 of 23


x/L

u/V

0

0 0.25 0.5 0.75 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1SMFD -1000 pts.SMFD -10000 pts.FMFD -1000 pts.FMFD -10000 pts.

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λenlargementin next picture

(a) Global picture

x/L

u/V

0

0.86 0.88 0.9 0.92 0.94

0.4

e = 106 mho/m

Rem = 4.24 x 102

S = 0.714

Q = 3.03 x 102

λFMFD -1000 pts.

FMFD -10000 pts.

SMFD -1000 pts.

SMFD -10000 pts.

(b) Enlargement of the square box in Fig. 8(a)

Figure 8. Grid convergence study for λe = 1 · 106 S/m. Velocity component u computed both with SMFD andFMFD and grids of 1000 and 10000 points.

22 of 23


x/L

v/V

0

0 0.25 0.5 0.75 1-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6SMFD -1000 pts.SMFD -10000 pts.FMFD -1000 pts.FMFD -10000 pts.

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

enlargementsin next pictures

(a) Global picture

x/L

v/V

0

0.55 0.6 0.65

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λSMFD -1000 pts.

FMFD -1000 pts.

SMFD -10000 pts.FMFD -10000 pts.

(b) Enlargement of the top square box in Fig. 9(a)

x/L

v/V

0

0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74-0.23

-0.22

-0.21

-0.2

-0.19

-0.18

-0.17

-0.16

-0.15

e = 104 mho/m

Rem = 4.24 x 100

S = 0.714

Q = 3.03 x 100

λ

SMFD -1000 pts.

FMFD -1000 pts.

SMFD -10000 pts.FMFD -10000 pts.

(c) Enlargement of the bottom square box in Fig. 9(a)

Figure 9. Grid convergence study for λe = 1 · 104 S/m. Velocity component v computed both with SMFD andFMFD and grids of 1000 and 10000 points.

23 of 23


AIAA Plasmadynamics and Lasers Conference, 28 June - 1 ... · 35th AIAA Plasmadynamics and Lasers...

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Transcript of AIAA Plasmadynamics and Lasers Conference, 28 June - 1 ... · 35th AIAA Plasmadynamics and Lasers...