Two-dimensional numerical methods in electromagnetic ...

Two-dimensional numerical methods in electromagnetic hypersonics including fully coupled Maxwell equations D.D’Ambrosio?∗and D. Giordano¶† ?

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

We describe a numerical technique for solving the coupled Maxwell and Navier-Stokes equations. The method is based on altering the magnitude of the speed of light in the Maxwell equations in order to make the characteristic time scale of electromagnetism comparable with the one of fluid dynamics during the transient to a steady state solution. The method is not time accurate, but provides interesting results at steady state. It can also be used to numerically evaluate the applied magnetic field provided that the distribution of the latter is known on a surface that contains the magnetic field source (solenoid or permanent magnet). To test the technique, we show the results obtained on an axial symmetric configuration.

I.

Introduction

The mutual interaction of electrically conducting flows and electromagnetic fields has been subject of interest for the aerospace community in the last years due to the envisaged possibility of using the magneto-fluid dynamics interaction for controlling hypersonic flows.1–12 In this framework, scientists belonging mainly, but not only, to the CFD community have begun to perform computational magneto-fluid dynamics simulations by adapting numerical methods for compressible flows to physical models based on the so-called magnetohydrodynamic approximation.10, 11, 13–18 In a recent paper, the authors presented a critical review of the physical models which are normally used in magneto-fluid dynamics simulations for aerospace applications.12 In synthesis, the most general model is represented by the full magneto-fluid dynamics equations (FMFD), that is, by the coupled system of the Maxwell and the Navier-Stokes equations. Other simplified magneto-fluid dynamics (SMFD) models descend from the FMFD system after enforcing the magnetohydrodynamics approximation which is based on the assumption of negligible displacement current, negligible convection current and applicability of the generalized Ohm law. The resulting model leads to what is sometimes called the resistive MHD equations, which describe a diffusive-convective magnetic regime where the equations of fluid dynamics and of electrodynamics are integrated into a single set of equations, composed of a modified form of the Navier-Stokes equations and of the magnetic induction equation, which directly descends from the Faraday equation. In the borderline case of very low magnetic Reynolds number, the induced magnetic field is assumed to be negligible with respect to the applied field, so that the magnetic induction equation becomes unnecessary and the Amp`ere equation can be substituted by the condition that the electric-current field be solenoidal. The resulting model is characterized by a loose coupling between fluid-dynamics and electrodynamics. In Ref.12, the authors investigated the possibility of using the full magneto-fluid dynamics equations (FMFD) for simulating compressible flows with electromagnetic interaction. The study was restricted to computational methods for one-dimensional flows, but it demonstrated that the numerical solution of the fully coupled equations of electromagnetism and fluid dynamics is feasible with a burden equal to a factor 3 in computing time. The adoption of the full magneto-fluid dynamics (FMFD) model is interesting for different reasons. One is to prove the validity of the magnetohydrodynamic approximation through extensive comparison of numerical results obtained using both full and simplified models in a wide range of conditions. Another one is related to the independence of the full MFD model ∗ Assistant † Research

Professor, [email protected], Member AIAA. Engineer, [email protected], Member AIAA.

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from the generalized Ohm law, which makes it possible to use different approximations for the conduction current in case it were necessary. A third reason is that a numerical method for the FMFD equations can be used without encountering numerical difficulties starting from conditions of zero electrical conductivity up to medium/high values of the latter, while the SMFD models are separately exploitable either for very small values of the electrical conductivity (the low-Rem approximation) or for intermediate values only (the resistive MHD model). In this paper, we will describe one possible numerical method for solving in two dimensions the full magneto-fluid dynamics (FMFD) equations. The attention will be focused on the numerical issues which characterize the computational solution of such a model. Since the characteristic evolution time of the Maxwell equations is definitely faster than the characteristic time of fluid dynamics, the coupled electromagnetic and fluid dynamics system is mathematically very stiff. The design of any numerical technique for their solution has to be capable of circumventing this problem. The proposed numerical method is based on the artificial modification of the Maxwell equation and aims at modifying the characteristic time scale of the latter. Such a technique is an alternative to the fully implicit integration of the Maxwell equations.

II. II.A.

Full magneto-fluid dynamics equations

Maxwell equations

The Maxwell equations, which read like ∂B +∇×E=0 ∂t ∂E ε0 − ε0 c2 ∇ × B = −j ∂t are made non-dimensional using the reference variables Eref , Bref , Vref , λref and take the form ∂B + K∇ × E = 0 ∂t ∂E − c˜2 ∇ × B = −˜ c2 (Rem )Vref j ∂t The non-dimensional parameters appearing in Eqs.(3) and (4) are defined as K

c Vref λref Vref Lref (Rem )Vref = ε0 c2 Eref K= Vref Bref c˜ =

(1) (2)

(3) (4)

(5a) magnetic Reynolds number loading factor

(5b) (5c)

Since we will integrate the Maxwell equations using a finite volumes method, the integral form is the most adequate: Z Z ∂B dV + K (n × E) dS = 0 (6) S V ∂t Z Z Z ∂E 2 2 K dV − c˜ (n × B) dS = −˜ c (Rem )Vref j dV (7) V ∂t S V II.B.

Navier-Stokes equations

The Navier-Stokes equations for a non-equilibrium chemically reacting gas are linked to the Maxwell equations through source terms that represent Lorentz force and Joule heating. In integral and non-dimensional form they read like: ∂ρ + ∇ · (ρv) = 0 (8a) ∂t √ γM ∂ρv K2 Q + ∇ · (ρvv + pI) − ∇·τ = 2 ρc E + Q j × B (8b) ∂t Re c˜ Rem

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√ γM ∂ (ρe) + ∇ · [(ρe + p) v] + ∇ · (JU − τ · v) = K Q j · E ∂t Re ∂ρi + ∇ · (ρi v) + ∇ · (Jmi ) = Ωi ∂t The non-dimensional parameter Q is called the magnetic interaction parameter and it is defined as: Q=

2 λref Bref L ρref Vref

(8c) (8d)

(9)

Symbol JU represents the flux of energy due to transport phenomena and Jmi is the mass flux of species ’i’ due to transport phenomena. The total current density, j, is the sum of the convection current (jconv = ρc v) and the conduction current density (JQ ). To evaluate JQ , there no need to invoke the generalized Ohm law. In fact, it can be obtained directly from the flowfield solution, which provides electric charge density, velocity and the electric charge flux due to transport phenomena (i.e. JQ ).3 In fact, the electric current density is given by j=v

X Jmi NA X ρi N A σis + e σis e Mi Mi i i

(10)

where the summation is extended to all species, σis is +1 for ions, -1 for electrons and 0 for neutral species, e is the electric charge and ρi is the species density. To this purpose, transport phenomena in a ionized gas in the presence of an electromagnetic field should be properly modelled.

III.

Numerical solution of the Full Magneto-Fluid Dynamics equations

. The characteristic evolution time of the Maxwell equations is definitely faster than the characteristic time of fluid dynamics. For this reason, the coupled electromagnetic and fluid dynamics system is mathematically very stiff and the design of any numerical technique for their solution has to be capable of circumventing this problem. III.A. III.A.1.

Numerical solution of the Maxwell equations Altering the speed of light to accelerate time convergence

In this paper, we describe a technique that is based on the artificial modification of the Maxwell equation aimed at modifying their characteristic time scale. We call such a technique ”the method of fake speed of light”. The technique is based on the modification of the Amp`ere equation (2) and makes it possible to integrate the Maxwell equations using explicit schemes. In fact, the explicit integration of the original Maxwell equations requires that the time-step used be of the order of ∆l axw ∆tM = (11) expl c where ∆l is the minimum cell size. Such a time step is orders of magnitude smaller than the time step required for the integration of the fluid dynamics equations and cannot be realistically used in a coupled simulation. A trick that can be used to obtain a fast convergence of the Maxwell equations consists in changing the original Amp`ere equation (Eq.(2)) into ∂E a2 0 − 0 a2 ∇ × B = − 2 j (12) ∂t c where a is a ”fake speed of light” that can be arbitrarily chosen to increase the characteristic time scale of electromagnetic phenomena. Since Eqs. (2) and (12) coincide at steady state, the numerical technique is based on the relaxation in time of the ”fake” Maxwell equations, at each fluid dynamics iteration, until time convergence is achieved to a certain tolerance. Of course, the results obtained during the transient are non physical and the technique can be used only when a steady-state solution is sought for. III.A.2.

A flux-difference splitting method for the Maxwell equations

The integration in time ofRthe Maxwell equations using a finite volumes technique requires the evaluation of the inteR grals S (n × E) dS and S (n × B) dS. This means that one has to compute the tangential components of E and B at each computational cell surface. To obtain a stable explicit numerical method, a flux-difference splitting technique

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is used. The technique is very similar to the one that we use for the flow solver19 and consists in solving a one dimensional Riemann problem normal to each cell surface. In two-dimensional or axial symmetric cases, depending on the particular electromagnetic configuration, we define the Riemann problem and its solution as follows.

III.A.3.

Riemann problem in T-E mode: Bx , By , Ez

Riemann invariants and propagation speeds: dR− dR+

= K dEz + c˜ dBt = K dEz − c˜ dBt

; ;

λ = −˜ c λ = +˜ c

(13) (14)

Indicating with ’l’ and ’r’ left and right states respectively, and with ’*’ the intermediate state, after integration of the Riemann invariants, the RP solution is:
r R− = K Ezr + c˜ Btr (15)
+ l l l = K Ez − c˜ Bt R (16) r

III.A.4.

Bt∗

=

Ez∗

=

l

(R− ) − (R+ ) 2˜ c r + l (R ) + (R− ) 2K

(17) (18)

Riemann problem in T-M mode: Bz , Ex , Ey

Riemann invariants and propagation speed: dR− dR+

= K dEt − c˜ dBz = K dEt + c˜ dBz

; ;

λ = −˜ c λ = +˜ c

(19) (20)

Indicating with ’l’ and ’r’ left and right states respectively, and with ’*’ the intermediate state, after integration of the Riemann invariants, the RP solution is:
r R− = K Etr − c˜ Bzr (21)
+ l l l R = K Et + c˜ Bz (22) l

Bz∗

=

Et∗

=

r

(R+ ) − (R− ) 2˜ c r + l (R ) + (R− ) 2K

(23) (24)

In case only the magnetic or the electric field are specified at the boundaries, Riemann invariants can be used to evaluate what is missing. In particular, we have: III.A.5.

Boundary conditions in T-E mode: Bx , By , Ez

Bt is known, Ez is unknown: Ezbc

= Ezin −


c˜ Btin − Btbc K

(25)

Btbc

= Btin −


K Ezin − Ezbc c˜

(26)

Ez is known, Bt is unknown:

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III.A.6.

Boundary conditions in T-M mode: Ex , Ey , Bz

Et is known, Bz is unknown: Bzbc

= Bzin +


K Etin − Etbc c˜

(27)

Etbc

= Etin +


c˜ Btin − Btbc K

(28)

Bz is known, Et is unknown:

(29) III.A.7.

Implicit treatment of the electric current density

The electric current density, j, that appears in the Amp`ere equation (Eq.(2) is treated implicitly with respect to the electromagnetic field. The jacobian matrix of j with respect to B and E is computed and used in the time integration of the Maxwell equations. III.A.8.

Enforcement of the ∇ · B = 0 or ε0 ∇ · E = ρc constraint

The nature of the Maxwell equations is such that an ordinary finite volumes method such as the one just described may be disturbed by some particular grid configurations. For example, using a grid that is aligned with the Cartesian axes, some components of n × E in TE mode or of n × B in TM mode will not contribute to the integrals just because of the grid orientation. This fact is linked to the incapability of the scheme to numerically reproduce the condition ∇ · B in TE mode and the condition ε0 ∇ · E = ρc in TM mode, which are analytically satisfied by the Maxwell equations provided that they are satisfied by the initial conditions also. To solve such a numerical problem, the two conditions are enforced using a projection method.20 In general, the vector field F that has to numerically satisfy the condition ∇·F=0

(30)

can be written as the sum of two contributions. One is the (possibly inaccurate) numerical solution Fnum and the second is the correction that it may need to exactly satisfy Eq.(30): F = Fnum + Fcorr

(31)

Now, we can write Fnum as the sum of the gradient of a potential function and the curl of a vector potential (which is the physically meaningful part): Fnum = ∇φ + ∇ × A (32) Applying Eq.(30) to Eqn.(32), we obtain: ∇ · Fnum = ∇ · ∇φ

(33)

∇2 φ = ∇ · Fnum

(34)

that we can solve for φ The necessary correction Fcorr can finally be obtained as Fcorr = −∇φ, so that F = Fnum − ∇φ

(35)

Therefore, the numerical technique that we use to solve the Maxwell equations is composed of two steps. In the first step, we integrate in time the Maxwell equations as described above, and in the second step we correct B or E using the projection scheme. It is not really necessary to use the projection scheme after each integration step: its application after a certain number of time steps is sufficient.

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III.B.

Numerical solution of the Navier-Stokes equations

The Navier-Stokes equations are discretized according to a finite volumes method based on the same grid where the Maxwell equations are computed. Convective fluxes are evaluated using an upwind flux-difference splitting technique19 and diffusive fluxes through a centered scheme. Second order accuracy in space and in time is achieved using an Essentially Non-Oscillatory scheme. Chemical source terms are treated implicitly to avoid instabilities due to fast chemical reactions. The volumetric force j × B and heating j · E due to the electromagnetic field are also treated implicitly, at least as far as fluid dynamics variables are concerned. To this purpose, the Jacobian of the electromagnetic source terms is computed and then summed to the Jacobian of chemical source terms, so that a tight coupling between fluid dynamics and chemical and electromagnetic source terms is enforced. III.C.

Coupling Maxwell and Navier-Stokes solvers

The coupling between the different solvers is carried out as follows. After each Navier-Stokes integration step, the Maxwell equations are relaxed to a time-local quasi-steady-state, where the divergence constraints have been enforced using the projection method. Overall, the numerical technique is not time-accurate, but provides accurate results at steady state. The Maxwell equations are solved inside the body also and therefore do not need any wall boundary condition. The only boundary conditions that are needed by the Maxwell solver are at far field, where typically a zero magnetic field is imposed, and on the surface of the applied magnetic field source, that might be a solenoid or a permanent magnet. In the latter case, the magnetic field distribution on such surface should be provided.

IV.

A numerical experiment

To test the capability of the numerical method in a realistic case, we consider a geometrical setup that is similar to the one that was recently adopted at DLR-Koeln to carry out MFD experiments in an Argon flow. A cylinder containing a solenoid is immersed in pre-ionized hypersonic Argon flow. The cylinder axis is in the same direction of flow, so that the incoming stream faces a flat disk. The solenoid windings rotate around the symmetry axis and generate an axially symmetric magnetic field with axial and radial components only. The used computational grid is shown in Fig. 1. The typical temperature distribution without MFD interaction is shown in Fig.2. IV.A.

Applied field computation

The applied magnetic field generated by the solenoid was computed first through the numerical integration of the generalized Biot-Savart-Laplace law ZZZ j (r0 ) × (r − r0 ) 0 µ0 dr (36) B (r) = 3 4π |r − r0 | V The obtained result, which is based on the knowledge of the cross section of the solenoid windings (which was assumed axially symmetric), was then used to define the internal boundary conditions for the Maxwell solver. In particular, the hollow annular region that contains the solenoid was chosen as the surface on which the known value of the magnetic field had to be set. After having enforced the magnetic field distribution calculated using the generalized Biot-Savart-Laplace law on the walls of the solenoid seat, the Maxwell solver was run until a steady state solution was obtained. The two solutions are compared in Fig. 3. This was an interesting test for the Maxwell solver, which proved to be capable of calculating the applied magnetic field starting from the magnetic field distribution along a surface that contains the source of the applied field. This capability is important because it allows us to compute the applied field consistently. IV.B.

Magneto-fluid dynamics interaction

We carried out numerical experiments on the above described configuration using a constant value for the electrical conductivity, which was considered as scalar. This choice destroys the physical validity of the obtained results, but it is a useful simplification to check the capabilities of the numerical method. In particular, we used two values for λe , 100 and 1000 S/m respectively. In the free stream, this corresponds to a magnetic Reynolds number based on the cylinder diameter of 0.02 and 0.2, respectively, and to a magnetic interaction parameter based on the maximum absolute value of the magnetic field of 0.312 and 3.12. Two other numerical experiments were carried out using λe = 100 S/m and 6 of 8 American Institute of Aeronautics and Astronautics

Figure 1. Computational grid for the coupled Maxwell/Navier-Stokes simulation.

Figure 2. Temperature distribution without applied magnetic field.

λe = 1000 S/m and a magnetic field 10 times larger than in the previous two cases, as if the current in the solenoid was increased by a factor ten. In these cases, the magnetic interaction parameter is 31.2 and 312, respectively. 7 of 8 American Institute of Aeronautics and Astronautics

Figure 3. Computed applied magnetic field.

The temperature fields in the four cases are shown in Fig. 4, where they are compared with the temperature distribution without MFD interaction. In the case of strong magnetic field, the MFD interaction becomes strong also before the shock wave. In all cases, the shock stand-off distance increases with growing electrical conductivity, very slightly for small values of Q, but noticeably for large values of Q. The effect on the wall temperature can be appreciated in Fig. 5. With growing electrical conductivity, the wall temperature at stagnation point tends to increase, but it decreases around the corner of the cylinder, where the magnetofluid dynamics interaction is the strongest. Again, we wish to stress on the concept that all these results do not represent the complete physics of the problem, because the magnitude of the electrical conductivity is arbitrarily assigned and it is not affected by the changes that occur in the flowfield due to the presence of an electromagnetic field.

V.

Conclusions and future work

We demonstrated that the Maxwell and the Navier-Stokes equations can be solved in a coupled fashion by relaxing the Maxwell equations to quasi-steady state at each flow field integration step. Fast convergence of the Maxwell equations to a steady solution can be obtained by altering the speed of light. We showed some numerical experiments carried out on an axial symmetric configuration, which proved the capa-

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bilities of the numerical method but provided results which are not in complete qualitative agreement with available experimental data. We think that this is due to the fact that a fundamental ingredient of the magneto-fluid dynamics interaction, that is the correct evaluation of transport phenomena for computing the electric current density, was missing in our tests. Here, we did not exploit an important characteristic of the FMFD model, that is the possibility of using the electric current density j as it is computed by the routines that evaluate diffusive fluxes without invoking the generalized Ohm law. In fact,despite the flow solver accounts for the non-equilibrium ionization of the gas, we did not even compute the electrical conductivity, but just used a constant value. This was done to test the numerical methodology, but in this way the physical meaning of the obtained results was lost. In the very next future, subroutines for computing transport phenomena in the presence of an electromagnetic field will be included in our computer codes for solving the FMFD equations. This will affect not only the electric current density, but also thermal conductivity, which becomes anisotrope in the presence of a magnetic field. A further improvement that will be made in the Maxwell solver will be the addition of the possibility of computing the electromagnetic field within polarized and magnetized media. This is important because the applied magnetic field is frequently obtained by using soft magnetic materials for the yoke of the solenoid.

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Figure 4. Temperature fields with different electrical conductivities and magnetic field intensities.

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Figure 5. Wall temperature distribution in front of the cylinder

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