44th AIAA Plasmadynamics and Lasers Conference June 24-27, 2013, San Diego, CA
AIAA 2013-2627
Modelling plasma flow on a self-field MPD thruster using a PISO based method Carlos M. Xisto∗, Jos´e C. P´ascoa† and Paulo J. Oliveira‡ Departamento de Eng. Electromecˆ anica, Universidade da Beira Interior, Covilh˜ a, 6201-001, Portugal
Downloaded by Carlos Xisto on July 6, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-2627
E-mail:
[email protected],
[email protected],
[email protected] A self-field MPD thruster, in its most basic form, comprises a central cathode surrounded by a concentric anode. With this coaxial geometric shape it is very difficult to use experimental techniques to analyse the plasma flow parameters. In the following paper we present some preliminary results for a 2D self-field MPD thruster. These were obtained with a new algorithm developed for solving the resistive magnetohydrodynamics (MHD) equations. The numerical method is based on the well known PISO algorithm and makes use of the AUSM-MHD scheme for flux calculation. Regarding validation three test cases will be addressed. We start with the Hunt problem, were we intend to validate our method for low speed resistive MHD flow. The second and third test cases are standard 2D ideal MHD problems. The last case is a 2D self-field MPD thruster were the effects of cathode geometry are analysed.
Nomenclature B E et F Ha Ispec m ˙ n p q T U V ρ η σ µ0 ϑef f γ
Magnetic field vector, T Electric field vector,T Total energy, J Momentum Thrust, N Hartmann number Specific impulse, s Mass flow rate, s Particle density Thermodynamic pressure, Pa Electron charge, C Static temperature, K Velocity vector, m/s Voltage drop, V Density, kg/m3 Electric resistivity, Ω · m Electric conductivity, S/m Permeability of free space, H/m Thruster efficiency Adiabatic index
I.
Introduction
Nowadays numerical codes are very important tools for the research and project phase of electric propulsion systems. The application of such codes for modelling magnetoplasmadynamic (MPD) thrusters is very ∗ PhD
student, Student Member AIAA Auxiliar ‡ Professor Catedr´ atico † Professor
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useful because it allows us to understand relevant flow-field parameters that are very difficult to visualize and analyse in experiments. This kind of devices in their most basic form consists in a central cathode surrounded by a concentric anode.1 However, with this coaxial geometry shape is quite hard to use experimental visualization techniques to analyse the plasma fluid flow. The use of measuring probes can also be problematic since they tend to disturb the plasma flow. To overcome such problems Toki et al.2 developed a multichannel two-dimensional device. With their approach an almost uniform current distribution can be obtained along the line of sight, which ensures easier optical accessibility to the discharge region. It is obvious that a 2D configuration does not fully reflects the features of coaxial devices. Nevertheless, the easier optical accessibility makes it possible to use advanced visualization techniques in order to relate the flow data with the performance data.3 The huge amount of data that can be collected within a quasi-bi-dimensional system, is also very useful for the validation of numerical tools that can be applied for the computation of 3D or 2D axisymmetric MPD nozzles. Computing the complex physics associated with an MPD device is not a trivial task. Some of the earlier models have assumed that the plasma flow in MPD nozzles behaves like a single conducting fluid.4, 5 However, this assumption leads to some ambiguity in the definition of the plasma transport coefficients, since these are usually defined in a different way for electrons and heavy species.6 Nevertheless, a single fluid approach can still be useful for a better understanding of the main acceleration mechanisms that are generated by this kind of systems.7 Another approach is to consider the plasma as a single fluid but with different temperatures, for the heavy species and electrons.8–11 With this last approach it is possible to compatibilize the simplicity of the single fluid approach and, at the same time, compute the two distinct transport coefficients in a more realistic way. In the following paper we present some preliminary results for a self-field MPD thruster. These were obtained with a new algorithm developed for the single fluid resistive magnetohydrodynamics (MHD) equations. This new algorithm is based on an all-Mach version of PISO,12 that was extended and validated for ideal MHD flows by Xisto et al.13, 14 In the next section we will present the governing equations and in section III the numerical method is briefly summarized. In section IV some validation cases are addressed. We start with the Hunt flow problem, were we intend to validate our method for low speed resistive MHD. Regarding ideal MHD validation, two standard bi-dimensional problems are addressed, namely the high density cloud shock wave interaction and the MHD rotor. Finally, in section V the effects of cathode geometry in the 2D self-field MPD thruster performance were analysed.
II.
Governing Equations
Magnetohydrodynamics is related to the interaction of a conducting moving fluid with one or more magnetic fields. This interaction can be described by the MHD equations, which couple the magnetic field given by Maxwell equations, with the flow of a conducting fluid ruled by the Navier-Stokes equations. The resistive form of the MHD system of equations is given by: ∂ρ + ∇ · (ρU) ∂t BB B2 ∂ρU I− − τvisc + ∇ · ρUU + p + ∂t 2µ0 µ0 2 ∂ρet BB B U−U· + ∇ · ρet + p + ∂t 2µ0 µ0 B × ηj −∇ · (k∇T ) − ∇ · µ0 ∂B + ∇ · (UB − BU) + ∇ × (ηj) ∂t
=
0,
(1)
=
0,
(2)
=
0,
(3)
=
0,
(4)
where k is the plasma thermal conductivity, η = 1/σ represents the plasma electric resistivity and µ0 is the permeability of free space. This system expresses the conservation of mass (1), momentum (2), total energy (3), and propagation of the magnetic field (4). The viscous stress tensor for compressible flow is given by: h i 2 T µ∇ · U I + µ ∇U + (∇U) , τvisc = − (5) 3 2 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
where µ is the dynamic viscosity of the plasma. Temperature is a derived quantity and can be obtained from an equation of state, B2 1 1 2 U + et − . (6) T = cv 2 ρµ0 Using Amp`ere’s law, and some vector identities, the last term of Eq. (4) can be rewritten in terms of B: B ∇×B = −∇ · η∇ . (7) ∇ × (ηj) = ∇ × η µ0 µ0 The electrical conductivity is calculated as a scalar field using the Spitzer-Harm formulation, σ = 1, 53 × 10−2 where,
! √ 3/2 12 2π (kB ǫ0 T ) , q 3 n1/2
ln Λ = ln Downloaded by Carlos Xisto on July 6, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-2627
T 3/2 , ln Λ
(8)
(9)
is the Coulomb logarithm. In Eq. (9) kB is the Boltzmann constant, q is related to electron particle charge, ǫ0 is the permittivity of free space and n is the particle density expressed in particles per cubic meter of plasma.
III.
Numerical Model
The numerical method that was used for the MPD thruster analysis was developed and validated for the ideal MHD equations by Xisto et al.13, 14 This algorithm is based on a PISO method previously developed for the Euler equations and validated for all Mach number flows.12 In order to improve accuracy and stability in the computation of all kinds of MHD discontinuities we have adapted the AUSM-MHD scheme that was proposed by Han et al.15 in the framework of density based solvers, for variable interpolation the CUBISTA limiter of Alves et al.16 was used. In multidimensional MHD cases, we must ensure ∇ · B = 0 , here we have used the hyperbolic divergence cleaning technique proposed by Dedner et al.17 A.
Thruster parameters
Several parameters can be used to assess the performance of an MPD thruster. The overall thrust force can be computed from the momentum flux at thruster exit, Z ρU 2 + p dS. (10) F = out
The volume integral of the axial component of the Lorentz force gives us the electromagnetic (ELM) component of the overall thrust, Z (j × B)x dV. (11) FELM = V
The thermal contribution can be easily obtained as the difference between total and ELM forces, FT erm = F − FELM .
(12)
The thruster specific impulse can then be computed in the following way: Ispec =
F , mg ˙ 0
(13)
where g0 = 9, 8 m/s2 represents acceleration due to gravity. From Eq. (13) we can conclude that a system that is able to produce an higher specific impulse is more efficient, since it can generate more thrust for the same mass of gas propellant. The efficiency of a self-field 2D MPD thruster is thus given by.18 ϑef f =
F2 , 2mI ˙ (V + Velet ) 3 of 11
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(14)
where m ˙ is the mass flow rate and I is the discharge current. The electrical voltage across the plasma, V , can be calculated as the line integral of Ohm’s law from anode to cathode, Z Z V = E · dl = (ηj − U × B) · dl. (15) The numerical model is not able to predict the potential drop due to electric sheath formation near the electrode walls. For that reason a constant value of Velet = 20 V was specified.18, 19
IV.
Validation
In the following subsections we intend to validate the proposed method for resistive and ideal MHD flow. For that purpose three test cases will be addressed. We start with the well known Hunt flow problem, were we intend to validate our model for low speed resistive MHD flow. The two remaining cases are standard 2D ideal MHD problems, that can be use to validate our model for complex MHD shock interaction.
Downloaded by Carlos Xisto on July 6, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-2627
A.
Resistive MHD validation
The Hunt flow problem is a well know test case for incompressible resistive MHD viscous flow. This case allows us to validate the effect of the Lorentz force (j × B) in the flow of a conducting fluid in a square duct. The channel is composed by two electrical insulated walls, that are parallel to the applied magnetic field (in this case Bz ), and two conducting walls that are perpendicular to the applied field, see Fig. 1-a). We consider a fully developed flow with a Reynolds number equal to Re = 10 and a Hartmann number equal to Ha = 300. The flow is unidirectional (x direction) and is driven by a constant pressure gradient equal to −300 [Pa/m]. For the conducting walls an electrical conductance ratio equal to 0.1 [S] is specified. An analytical solution can be obtained with the hyperbolic functions that were proposed by Hunt,20 or by the exponential version of those equations.21 The solution was computed in a 2D non-uniform mesh with 46 × 46 nodes, see Fig. 1-a,b). Although this is a 3D problem, it can be easily converted in a two-dimensional case if we only consider one cross-section of the channel with periodic boundary conditions in the fluid flow direction. In Fig. 2 we present a comparison between analytical and numerical results. We can see that, for both boundary layers, the numerical results tend to agree very well with the analytical solution. The side-wall jets (Fig. 2-a) are related to the balance between the pressure gradient and the Lorentz force. In these insulated walls the current lines are parallel to the applied magnetic field, thus the electromagnetic braking force is significantly reduced. As a consequence, in these region, the pressure gradient is mainly balanced by the viscous force and the flow protrudes through the magnetic field in thin jets along the side-walls.22 B.
Ideal MHD validation
In this subsection we will address standard ideal MHD problems. The first test case is related to a well know problem in astrophysics, see.23 It comprises the interaction of an high density cloud with a strong shock wave. The computational domain is a square box with x, y ∈ [0; 1] and a total of 400 × 400 nodes. The discontinuity lies at x = 0.6 with the following left and right states for the dependent variables: 1 ρ[kg/m3 ] 3, 86859 −11.2536 U [m/s] 0 x Uy [m/s] 0 0 Uz [m/s] 0 0 = = , (16) p[Pa] 1 167.345 √ Bx [T/ µ0 ] 0 0 0.56418958 By [T/√µ0 ] 2.1826182 √ −2.1826182 L Bz [T/ µ0 ] 0.56418958 R the adiabatic index is equal to γ = 5/3. The high density cloud (ρ = 10[kg/m3]) assumes a circular shape with centre in (x, y) ∈ (0.8; 0.5) and radius equal to 0.15[m]. The value of pressure inside and outside the circular cloud is p = 1[Pa], so the cloud is in hydrostatic equilibrium with the environment. Regarding 4 of 11 American Institute of Aeronautics and Astronautics
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(a)
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1
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(b)
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Figure 1. a) Hunt flow problem channel. Global (b) and detailed view (c) of the numerical grid.
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Numerical results Analytical results
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Figure 2. Comparison between numerical and analytical results: a) side-wall boundary layer; b) Hartmann boundary layer.
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boundary conditions, in the right there is a fixed value boundary condition due to supersonic flow, and in the remaining boundaries the variables are extrapolated from the inside of the solution domain. The results were obtained for a final time of t = 0.06 [s], before the cloud leaves the numerical domain. In Fig. 3-a) we present the contour plots for density in a logarithmic colour scale and in Fig. 3-b) the magnetic field lines are plotted. We can see that the discontinuities are calculated with a good resolution and no unrealistic flow patters are observed. Notice that this case does not have any analytical solution and it can only be compared with the results obtained from different authors, see.23, 24
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y
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y
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Figure 3. Numerical results obtained for the cloud-shock interaction.
The second test case comprises the so called MHD rotor. This case can be found in25 and it allows us to test our model for the calculation of torsional Alfv´en waves. The rotation speed is an initial condition of the problem an no other external forces are applied to the rotor. This problem is again solved in a square domain x, y ∈ [0; 1] with a total of 400 × 400 nodes. The initial conditions for the magnetic field and pressure are uniform, √ √ √ (17) By = 0 [T/ µ0 ]; Bx = 5/ 4π [T/ µ0 ]; p = 1 [Pa]; γ = 1.4. The initial conditions for the rotating cylinder of radius r0 = 0.1 [m] are the following: (y − 0, 5) (x − 0, 5) ; Uy = U0 . (18) r0 r0 i1/2 h 2 2 Notice that Ux and Uy are functions of r < r0 , where r = (x − 0, 5) − (y − 0, 5) and U0 = 2 [m/s]. The environment is stationary (Ux = Uy = 0 [m/s] ) with a density ten times lower than the cylinder (ρ = 1 [kg/m3 ]) and is defined by the region r > r1 (r1 = 0.115 [m]). In order to obtain a soft transition between the rotating cylinder and the environmental fluid another region was created, with linear profiles for density and velocity, ρ = 10 [kg/m3]; Ux = −U0
ρ = 1 + 9f ; Ux = −f U0
(x − 0, 5) (y − 0, 5) ; Uy = f U0 ; r r
(19)
where, f = (r1 − r) / (r1 − r0 ) .
(20)
In Fig. 4 we present the results obtained after t = 0.15 [s], before the torsional waves reach the computational domain boundary. We can see from the Mach number distribution (Fig. 4-a), that the rotor is still rotating to a certain radial distance with an uniform angular velocity. Beyond that distance the the rotational speed decreases because the rotor exchange angular momentum with the environment. We can see that the proposed method can calculate torsional Alfv´en waves, those waves are particularly visible in the plot of magnetic pressure (B · B/2), Fig. 4-b). 6 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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(a)
x
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1
(b)
Figure 4. Results obtained for the MHD rotor after t = 0.15 [s]: a) Mach number distribution; b) Magnetic pressure distribution (B · B/2).
V.
Numerical Modelling of a 2D MPD Thruster
The previously validated MHD model will now be used to compute a 2D self-field MPD thruster. In the following section the effects of cathode size in the performance of a MPD thruster are analysed. This kind of analysis has been done at an experimental level26, 27 and several numerical studies were also performed.5, 7, 8, 19 We will demonstrate that the geometric characteristics of the cathode can influence the discharge current pattern, which can result in an increase of the electro-thermal component of thrust. Regarding the numerical MHD model several assumptions were made. The propellent gas is argon and we assume that it is injected in the discharge chamber in a state of total ionization. We can then accept that the plasma can be treated as a single, quasi-neutral fluid, in a state of thermal equilibrium (T = Te = Ti ). Several phenomena are also neglected, namely: viscosity; thermal conductivity; electrical sheath; Hall effect; and radiation processes. Electrical resistivity is given by Spitzer-Harm’s formulation, see Eq. (8). Also, the plasma flow is purely bi-dimensional in the xy-plane, this means that the only component of the magnetic field will act in the z direction. For validation purposes we have used the same boundary conditions as Funaki et al.5 At inlet, because of its subsonic regime, a mass flow rate of 2.5 g/s is imposed and a temperature of T = 5000 K is fixed. Since the flow is supersonic at outlet, all variables are extrapolated from inside the solution domain. In the electrode walls a slip condition is imposed for velocity and a perfectly electrical conducting wall is specified for the magnetic field. For both the insulated wall and symmetric boundary the magnetic field is fixed to zero. At inlet the magnetic field can be calculated as a function of the discharge current, B0 =
µ0 I , 2W
(21)
where I is the discharge current and W represents the width of the thruster, in this case W = 84 [mm]. In Fig. 5-b) are presented the two analysed configurations. Both geometries are composed by flared anodes (FA), where the only difference between the two devices is related to their cathode length. The first configuration, that we refer as FASC, is composed by a short cathode with a length equal to 13 [mm], the second geometry (FALC) is composed by a long cathode with 40 [mm] of length. The proposed method was validated against experimental3 and numerical results from different authors.5, 7 In Fig. 6-a) the axial velocity distribution results, along the section y = 0.009 [m] for a discharge current of I = 8000 [A], are compared with the numerical results obtained for the FASC configuration by two different authors. We can see that the calculated velocity is slightly lower but the plot is showing the same pattern. A more realistic validation requires a comparison with experiments. For that purpose, in Fig. 6 b), the numerical results 7 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
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0
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(a)
0.04
x
0.06
0.08
(b)
Figure 5. 2D self-field MPD thruster geometry and boundary conditions.
obtained for the momentum thrust in the FASC configuration are compared with the experimental ones. We can see that the present MHD model over-predicts the propulsion force. Nevertheless, the trend is quite similar and when the solution is compared with the results obtained by others authors,7 while using a similar numerical model, we can observe that our results agree very well. 6000
25
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Numerical (Funaki et al., 1997) Numerical (Mahendhran and Kumar, 2011)
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Numerical, current results - FASC Experimental (Funaki et al., 1997) Numerical (Mahendhran and Kumar, 2011)
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I (kA)
(a)
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Figure 6. Comparison between experimental and numerical results.
Let us now use several thruster parameters to analyse the effects of cathode length in the plasma flow. In Fig. 7-a) we compared the overall thrust values obtained with both configurations for several values of discharge current. We can see that with the short cathode configuration we can reach higher values of thrust. In Fig. 7-b) we can see that the ELM component of thrust is almost the same for both devices, and the biggest discrepancy appears in the thermal component of the thrust. This is a related to the geometric shape of the cathode, that allow’s a bigger concentration of discharge current lines in is tip, see Fig. 8-a). The results for thruster efficiency and specific impulse are presented in Fig. 9. The FASC configuration can achieve the biggest efficiency for the calculated range of discharge current. This was expected since, for the same mass flow rate and discharge current, this is the configuration that results in higher value of thrust.
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(a)
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Figure 7. Numerical results obtained for thrust in the FASC and FALC configurations.
(a)
(b)
Figure 8. Magnetic field distribution calculated for both configurations.
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Figure 9. Efficiency and specific impulse results.
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15
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VI.
Conclusions
A previously proposed method for the ideal MHD equations was here extended to compute viscous and resistive MHD flow. The numerical model was validated for resistive MHD using the analytical Hunt flow problem, and was also validated for ideal MHD flow using two standard bi-dimensional test cases. Regarding the modelling of the MPD thruster, a validation against experimental results for a 2D self-field MPD nozzle was also performed. The proposed numerical model over-predicts the values of thrust. However, assuming a state of total ionization and thermal equilibrium, while using an ideal gas law, can result in a huge increase of temperature, that will result in a higher thermal component of thrust. The MPD electrode geometric effects in the thruster performance were also analysed. We have demonstrated that an enhancement in the electro-thermal component of thrust will result in higher overall thrust values. In a self-field MPD thruster this can be achieved by reducing the cathode length.
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Acknowledgements This work was supported by FCT grant SFRH/BD/60285/2009 and by FCT project PTDC/ CTESPA/ 114163/2009 “SpaceProp - MHD Numerical Modelling in nozzles of MPD Thrusters for Space Propulsion”. Additional financial support was provided by CAST.
References 1 LaPointe, M. R. and Mikellides, P. G., “High Power MPD Thruster Development at the NASA Glenn Research Center,” 37th Joint Propulsion Conference and Exhibit, No. AIAA-2001-3499, 2001. 2 Toki, K., Sumida, M., and Kuriki, K., “Multichannel two-dimensional magnetoplasmadynamic arcjet,” Journal of Propulsion and Power , Vol. 8, 1992, pp. 93–97. 3 Funaki, I., Toki, K., and Kuriki, K., “Electrode Configuration Effect on the Performance of a Two-Dimensional Magnetoplasmadynamic Arcjet,” Journal of Propulsion and Power , Vol. 14, No. 6, 1998, pp. 1043–1048. 4 Sleziona, P., Auweter-Kurtz, M., and Schrade, H., “Numerical codes for cylindrical MPD trhusters,” Proc. 20th International Electric Propulsion Physics Conference, No. IEPS-88-043, Garmisch-Partenkirchen, 1988. 5 Funaki, I., Toki, K., and Kuriki, K., “Numerical analysis of a two-dimensional magnetoplasmadynamic arcjet,” Journal of propulsion and power , Vol. 13, No. 6, 1997, pp. 789–795. 6 Myers, R. M., Mantenieks, M. A., and Lapointe, M. R., “MPD thruster technology,” Tech. Rep. NASA-TM-105242, NASA Glenn Research Center, 1991. 7 Mahendhran, M. and Kumar, A., “Numerical Study on the effect of electrode geometry in MPD thrusters,” 32nd International Electric Propulsion Conference, No. IEPC-2011-228, Wiesbaden, Germany, 2011. 8 LaPointe, M. R., “Numerical Simulation of Geometric Scale Effects in Cylindrical Self-Field MPD Thrusters,” Tech. Rep. CR-189224, NASA Lewis Research Centre, 1992. 9 Sleziona, P. C., Auweter-Kurtz, M., and Schrade, H. O., “Numerical calculation of a Cylindrical MPD thruster,” Procedings 23 International Electric Propulsion Conference, No. IEPC-93-066, Seattle, 1993. 10 Miyasaka, T. and Fujiwara, T., “Numerical prediction of onset phenomenon in a 2-dimensional axisymmetric MPD thruster,” Joint Propulsion Conferences, American Institute of Aeronautics and Astronautics, 1999. 11 Sankaran, K., Martinelli, L., Jardin, S. C., and Choueiri, E. Y., “A flux-limited numerical method for solving the MHD equations to simulate propulsive plasma flows,” International Journal for Numerical Methods in Engineering, Vol. 53, No. 6, 2002, pp. 1415–1432. 12 Xisto, C. M., P´ ascoa, J. C., Oliveira, P. J., and Nicolini, D. A., “A hybrid pressure density-based algorithm for the Euler equations at all Mach number regimes,” International Journal for Numerical Methods in Fluids, Vol. 70:8, 2012, pp. 961–976. 13 Xisto, C. M., P´ ascoa, J. C., and Oliveira, P. J., “Preliminary assessment of a new algorithm for the MHD equations at all Mach number regimes,” European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012 , No. 3212, 2012. 14 Xisto, C., P´ ascoa, J., and Oliveira, P., “A pressure-based method with AUSM-type fluxes for MHD flows at arbitrary Mach numbers,” International Journal for Numerical Methods in Fluids, 2013, pp. in press. 15 Han, S., Lee, J., and Kim, K., “Accurate and Robust Pressure Weight Advection Upstream Splitting Method for Magnetohydrodynamics Equations,” AIAA Journal , Vol. 47, April 2009, pp. 970–981. 16 Alves, M. A., Oliveira, P. J., and Pinho, F. T., “A convergent and universally bounded interpolation scheme for the treatment of advection,” International Journal for Numerical Methods in Fluids, Vol. 41, No. 1, 2003, pp. 47–75. 17 Dedner, A., Kemm, F., Kr¨ oner, D., Munz, C.-D., Schnitzer, T., and Wesenberg, M., “Hyperbolic Divergence Cleaning for the MHD Equations,” Journal of Computational Physics, Vol. 175, 2002, pp. 645–673. 18 Kubota, K., Funaki, I., and Okuno, Y., “Comparison of Simulated Plasma Flow Field in a Two-Dimensional Magnetoplasmadynamic Thruster With Experimental Data,” IEEE Transactions onPlasma Science, Vol. 37, No. 12, 2009, pp. 2390–2398. 19 Kubota, K., Funaki, I., and Okuno, Y., “Numerical Study of Electrode Geometry effects on flowfield in two-dimensional MPD Thrusters,” 30th International Electric Propulsion Conference, No. IEPC-07-87, Florence, Italy, 2007.
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20 Hunt, J. C. R., “Magnetohydrodynamic flow in rectangular ducts,” Journal of Fluid Mechanics, Vol. 21, No. 04, 1965, pp. 577–590. 21 Ni, M.-J., Munipalli, R., Huang, P., Morley, N. B., and Abdou, M. A., “A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: On an arbitrary collocated mesh,” Journal of Computational Physics, Vol. 227, No. 1, 2007, pp. 205 – 228. 22 Priede, J., Aleksandrova, S., and Molokov, S., “Linear stability of Hunt’s flow,” Journal of Fluid Mechanics, Vol. 649, No. 3, 2010, pp. 115–134. 23 Dai, W. and Woodward, P. R., “A Simple Finite Difference Scheme for Multidimensional Magnetohydrodynamical Equations,” Journal of Computational Physics, Vol. 142, No. 2, 1998, pp. 331 – 369. 24 T´ oth, G., “The ∇ · B = 0 Constraint in Shock-Capturing Magnetohydrodynamics Codes,” Journal of Computational Physics, Vol. 161, 2000, pp. 605–652. 25 Balsara, D. S. and Spicer, D. S., “A Staggered Mesh Algorithm Using High Order Godunov Fluxes to Ensure Solenoidal Magnetic Fields in Magnetohydrodynamic Simulations,” Journal of Computational Physics, Vol. 149, No. 2, 1999, pp. 270–292. 26 Funaki, I., Kuriki, K., and Toki, K., “Electrode Configuration Effect on the Performance of a Two-Dimensional Magnetoplasmadynamic Arcjet,” Journal of Propulsion and Power , Vol. 14, No. 6, 1998, pp. 1043–1048. 27 Nakata, D., Toki, K., Funaki, I., Shimizu, Y., Kuninaka, H., and Arakawa, Y., “Experimental verification for the nozzle shape optimization of the self-field MPD thruster,” 29th International Electric Propulsion Conference, No. IEPC-05-163, 2005.
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